Principal component analysis

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will be almost the same as that variable, with a small contribution from the other variable, whereas the second component will be almost aligned with the second original variable. This means that whenever the different variables have different units (like temperature and mass), PCA is a somewhat arbitrary method of analysis. (Different results would be obtained if one used Fahrenheit rather than Celsius for example.) Pearson's original paper was entitled "On Lines and Planes of Closest Fit to Systems of Points in Space" – "in space" implies physical Euclidean space where such concerns do not arise. One way of making the PCA less arbitrary is to use variables scaled so as to have unit variance, by standardizing the data and hence use the autocorrelation matrix instead of the autocovariance matrix as a basis for PCA. However, this compresses (or expands) the fluctuations in all dimensions of the signal space to unit variance.

12826:(DCA) is a method used in the atmospheric sciences for analysing multivariate datasets. Like PCA, it allows for dimension reduction, improved visualization and improved interpretability of large data-sets. Also like PCA, it is based on a covariance matrix derived from the input dataset. The difference between PCA and DCA is that DCA additionally requires the input of a vector direction, referred to as the impact. Whereas PCA maximises explained variance, DCA maximises probability density given impact. The motivation for DCA is to find components of a multivariate dataset that are both likely (measured using probability density) and important (measured using the impact). DCA has been used to find the most likely and most serious heat-wave patterns in weather prediction ensembles , and the most likely and most impactful changes in rainfall due to climate change . 11750:, which dominated studies of residential differentiation from the 1950s to the 1970s. Neighbourhoods in a city were recognizable or could be distinguished from one another by various characteristics which could be reduced to three by factor analysis. These were known as 'social rank' (an index of occupational status), 'familism' or family size, and 'ethnicity'; Cluster analysis could then be applied to divide the city into clusters or precincts according to values of the three key factor variables. An extensive literature developed around factorial ecology in urban geography, but the approach went out of fashion after 1980 as being methodologically primitive and having little place in postmodern geographical paradigms. 12131:

variables, excluding unique variance". In terms of the correlation matrix, this corresponds with focusing on explaining the off-diagonal terms (that is, shared co-variance), while PCA focuses on explaining the terms that sit on the diagonal. However, as a side result, when trying to reproduce the on-diagonal terms, PCA also tends to fit relatively well the off-diagonal correlations. Results given by PCA and factor analysis are very similar in most situations, but this is not always the case, and there are some problems where the results are significantly different. Factor analysis is generally used when the research purpose is detecting data structure (that is, latent constructs or factors) or

12172: 5507:. Each eigenvalue is proportional to the portion of the "variance" (more correctly of the sum of the squared distances of the points from their multidimensional mean) that is associated with each eigenvector. The sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean. PCA essentially rotates the set of points around their mean in order to align with the principal components. This moves as much of the variance as possible (using an orthogonal transformation) into the first few dimensions. The values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information (see 3888: 19085: 4606: 5033: 12337: 12808:

are linear combinations of alleles which best separate the clusters. Alleles that most contribute to this discrimination are therefore those that are the most markedly different across groups. The contributions of alleles to the groupings identified by DAPC can allow identifying regions of the genome driving the genetic divergence among groups In DAPC, data is first transformed using a principal components analysis (PCA) and subsequently clusters are identified using discriminant analysis (DA).

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principal components were actually dual variables or shadow prices of 'forces' pushing people together or apart in cities. The first component was 'accessibility', the classic trade-off between demand for travel and demand for space, around which classical urban economics is based. The next two components were 'disadvantage', which keeps people of similar status in separate neighbourhoods (mediated by planning), and ethnicity, where people of similar ethnic backgrounds try to co-locate.

3530: 3284: 19071: 1084: 4836: 12411: 3883:{\displaystyle {\begin{aligned}Q(\mathrm {PC} _{(j)},\mathrm {PC} _{(k)})&\propto (\mathbf {X} \mathbf {w} _{(j)})^{\mathsf {T}}(\mathbf {X} \mathbf {w} _{(k)})\\&=\mathbf {w} _{(j)}^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {X} \mathbf {w} _{(k)}\\&=\mathbf {w} _{(j)}^{\mathsf {T}}\lambda _{(k)}\mathbf {w} _{(k)}\\&=\lambda _{(k)}\mathbf {w} _{(j)}^{\mathsf {T}}\mathbf {w} _{(k)}\end{aligned}}} 19109: 19097: 3042: 1376: 2648: 11778:

indicators but was a good predictor of many more variables. Its comparative value agreed very well with a subjective assessment of the condition of each city. The coefficients on items of infrastructure were roughly proportional to the average costs of providing the underlying services, suggesting the Index was actually a measure of effective physical and social investment in the city.

2460: 5028:{\displaystyle {\begin{aligned}\mathbf {X} ^{T}\mathbf {X} &=\mathbf {W} \mathbf {\Sigma } ^{\mathsf {T}}\mathbf {U} ^{\mathsf {T}}\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\\&=\mathbf {W} \mathbf {\Sigma } ^{\mathsf {T}}\mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\\&=\mathbf {W} \mathbf {\hat {\Sigma }} ^{2}\mathbf {W} ^{\mathsf {T}}\end{aligned}}} 4671:. PCA thus can have the effect of concentrating much of the signal into the first few principal components, which can usefully be captured by dimensionality reduction; while the later principal components may be dominated by noise, and so disposed of without great loss. If the dataset is not too large, the significance of the principal components can be tested using 8861:

results. "If the number of subjects or blocks is smaller than 30, and/or the researcher is interested in PC's beyond the first, it may be better to first correct for the serial correlation, before PCA is conducted". The researchers at Kansas State also found that PCA could be "seriously biased if the autocorrelation structure of the data is not correctly handled".

12183:(NMF) is a dimension reduction method where only non-negative elements in the matrices are used, which is therefore a promising method in astronomy, in the sense that astrophysical signals are non-negative. The PCA components are orthogonal to each other, while the NMF components are all non-negative and therefore constructs a non-orthogonal basis. 5324: 3279:{\displaystyle \mathbf {w} _{(k)}=\mathop {\operatorname {arg\,max} } _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {\hat {X}} _{k}\mathbf {w} \right\|^{2}\right\}=\arg \max \left\{{\tfrac {\mathbf {w} ^{\mathsf {T}}\mathbf {\hat {X}} _{k}^{\mathsf {T}}\mathbf {\hat {X}} _{k}\mathbf {w} }{\mathbf {w} ^{T}\mathbf {w} }}\right\}} 4108: 11148: 2779: 12124:

seeking to reproduce the total variable variance, in which components reflect both common and unique variance of the variable. PCA is generally preferred for purposes of data reduction (that is, translating variable space into optimal factor space) but not when the goal is to detect the latent construct or factors.

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is similar to principal component analysis, in that factor analysis also involves linear combinations of variables. Different from PCA, factor analysis is a correlation-focused approach seeking to reproduce the inter-correlations among variables, in which the factors "represent the common variance of

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of stimuli along which the variance of the spike-triggered ensemble differed the most from that of the prior stimulus ensemble. Specifically, the eigenvectors with the largest positive eigenvalues correspond to the directions along which the variance of the spike-triggered ensemble showed the largest

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in 2008 extracted an attitudinal index toward housing from 28 attitude questions in a national survey of 2697 households in Australia. The first principal component represented a general attitude toward property and home ownership. The index, or the attitude questions it embodied, could be fed into a

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PCA relies on a linear model. If a dataset has a pattern hidden inside it that is nonlinear, then PCA can actually steer the analysis in the complete opposite direction of progress. Researchers at Kansas State University discovered that the sampling error in their experiments impacted the bias of PCA

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PCA is at a disadvantage if the data has not been standardized before applying the algorithm to it. PCA transforms original data into data that is relevant to the principal components of that data, which means that the new data variables cannot be interpreted in the same ways that the originals were.

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Discriminant analysis of principal components (DAPC) is a multivariate method used to identify and describe clusters of genetically related individuals. Genetic variation is partitioned into two components: variation between groups and within groups, and it maximizes the former. Linear discriminants

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A strong correlation is not "remarkable" if it is not direct, but caused by the effect of a third variable. Conversely, weak correlations can be "remarkable". For example, if a variable Y depends on several independent variables, the correlations of Y with each of them are weak and yet "remarkable".

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The above picture is an example of the difference between PCA and Factor Analysis. In the top diagram the "factor" (e.g., career path) represents the three observed variables (e.g., doctor, lawyer, teacher) whereas in the bottom diagram the observed variables (e.g., pre-school teacher, middle school

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PCA rapidly transforms large amounts of data into smaller, easier-to-digest variables that can be more rapidly and readily analyzed. In any consumer questionnaire, there are series of questions designed to elicit consumer attitudes, and principal components seek out latent variables underlying these

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PCA in genetics has been technically controversial, in that the technique has been performed on discrete non-normal variables and often on binary allele markers. The lack of any measures of standard error in PCA are also an impediment to more consistent usage. In August 2022, the molecular biologist

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Since then, PCA has been ubiquitous in population genetics, with thousands of papers using PCA as a display mechanism. Genetics varies largely according to proximity, so the first two principal components actually show spatial distribution and may be used to map the relative geographical location of

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Subsequent principal components can be computed one-by-one via deflation or simultaneously as a block. In the former approach, imprecisions in already computed approximate principal components additively affect the accuracy of the subsequently computed principal components, thus increasing the error

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Another limitation is the mean-removal process before constructing the covariance matrix for PCA. In fields such as astronomy, all the signals are non-negative, and the mean-removal process will force the mean of some astrophysical exposures to be zero, which consequently creates unphysical negative

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The applicability of PCA as described above is limited by certain (tacit) assumptions made in its derivation. In particular, PCA can capture linear correlations between the features but fails when this assumption is violated (see Figure 6a in the reference). In some cases, coordinate transformations

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each of the orthogonal eigenvectors to turn them into unit vectors. Once this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform the covariance matrix into a diagonalized form, in which the

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data: a) Configuration of nodes and 2D Principal Surface in the 3D PCA linear manifold. The dataset is curved and cannot be mapped adequately on a 2D principal plane; b) The distribution in the internal 2D non-linear principal surface coordinates (ELMap2D) together with an estimation of the density

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It is often difficult to interpret the principal components when the data include many variables of various origins, or when some variables are qualitative. This leads the PCA user to a delicate elimination of several variables. If observations or variables have an excessive impact on the direction

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Market research has been an extensive user of PCA. It is used to develop customer satisfaction or customer loyalty scores for products, and with clustering, to develop market segments that may be targeted with advertising campaigns, in much the same way as factorial ecology will locate geographical

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One of the problems with factor analysis has always been finding convincing names for the various artificial factors. In 2000, Flood revived the factorial ecology approach to show that principal components analysis actually gave meaningful answers directly, without resorting to factor rotation. The

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Mean subtraction (a.k.a. "mean centering") is necessary for performing classical PCA to ensure that the first principal component describes the direction of maximum variance. If mean subtraction is not performed, the first principal component might instead correspond more or less to the mean of the

4667:, which can be thought of as a high-dimensional rotation of the co-ordinate axes). However, with more of the total variance concentrated in the first few principal components compared to the same noise variance, the proportionate effect of the noise is less—the first few components achieve a higher 12186:

In PCA, the contribution of each component is ranked based on the magnitude of its corresponding eigenvalue, which is equivalent to the fractional residual variance (FRV) in analyzing empirical data. For NMF, its components are ranked based only on the empirical FRV curves. The residual fractional

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The earliest application of factor analysis was in locating and measuring components of human intelligence. It was believed that intelligence had various uncorrelated components such as spatial intelligence, verbal intelligence, induction, deduction etc and that scores on these could be adduced by

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In PCA, it is common that we want to introduce qualitative variables as supplementary elements. For example, many quantitative variables have been measured on plants. For these plants, some qualitative variables are available as, for example, the species to which the plant belongs. These data were

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and are completely correlated, then the PCA will entail a rotation by 45° and the "weights" (they are the cosines of rotation) for the two variables with respect to the principal component will be equal. But if we multiply all values of the first variable by 100, then the first principal component

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To find the axes of the ellipsoid, we must first center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we

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these too may be most spread out, and therefore most visible to be plotted out in a two-dimensional diagram; whereas if two directions through the data (or two of the original variables) are chosen at random, the clusters may be much less spread apart from each other, and may in fact be much more

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iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. The first principal component can equivalently be defined as a direction that maximizes the

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overcomes this disadvantage by finding linear combinations that contain just a few input variables. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by adding sparsity constraint on the input variables. Several approaches have been

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Principal component analysis creates variables that are linear combinations of the original variables. The new variables have the property that the variables are all orthogonal. The PCA transformation can be helpful as a pre-processing step before clustering. PCA is a variance-focused approach

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The statistical implication of this property is that the last few PCs are not simply unstructured left-overs after removing the important PCs. Because these last PCs have variances as small as possible they are useful in their own right. They can help to detect unsuspected near-constant linear

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and others pioneered the use of principal components analysis (PCA) to summarise data on variation in human gene frequencies across regions. The components showed distinctive patterns, including gradients and sinusoidal waves. They interpreted these patterns as resulting from specific ancient

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was developed by PCA from about 200 indicators of city outcomes in a 1996 survey of 254 global cities. The first principal component was subject to iterative regression, adding the original variables singly until about 90% of its variation was accounted for. The index ultimately used about 15

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About the same time, the Australian Bureau of Statistics defined distinct indexes of advantage and disadvantage taking the first principal component of sets of key variables that were thought to be important. These SEIFA indexes are regularly published for various jurisdictions, and are used

5582:. It is not, however, optimized for class separability. However, it has been used to quantify the distance between two or more classes by calculating center of mass for each class in principal component space and reporting Euclidean distance between center of mass of two or more classes. The 12175:

Fractional residual variance (FRV) plots for PCA and NMF; for PCA, the theoretical values are the contribution from the residual eigenvalues. In comparison, the FRV curves for PCA reaches a flat plateau where no signal are captured effectively; while the NMF FRV curves decline continuously,

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the model, producing conclusions that fail to generalise to other datasets. One approach, especially when there are strong correlations between different possible explanatory variables, is to reduce them to a few principal components and then run the regression against them, a method called

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corresponding to eigenvalues of a symmetric matrix are orthogonal (if the eigenvalues are different), or can be orthogonalised (if the vectors happen to share an equal repeated value). The product in the final line is therefore zero; there is no sample covariance between different principal

12480:(MPCA) that extracts features directly from tensor representations. MPCA is solved by performing PCA in each mode of the tensor iteratively. MPCA has been applied to face recognition, gait recognition, etc. MPCA is further extended to uncorrelated MPCA, non-negative MPCA and robust MPCA. 5542:

Mean-centering is unnecessary if performing a principal components analysis on a correlation matrix, as the data are already centered after calculating correlations. Correlations are derived from the cross-product of two standard scores (Z-scores) or statistical moments (hence the name:

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analyzing 12 PCA applications. He concluded that it was easy to manipulate the method, which, in his view, generated results that were 'erroneous, contradictory, and absurd.' Specifically, he argued, the results achieved in population genetics were characterized by cherry-picking and

5515:. PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest "variance" (as defined above). This advantage, however, comes at the price of greater computational requirements if compared, for example, and when applicable, to the 1379:

The above picture is of a scree plot that is meant to help interpret the PCA and decide how many components to retain. The start of the bend in the line (point of inflexion or "knee") should indicate how many components are retained, hence in this example, three factors should be

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attitudes. For example, the Oxford Internet Survey in 2013 asked 2000 people about their attitudes and beliefs, and from these analysts extracted four principal component dimensions, which they identified as 'escape', 'social networking', 'efficiency', and 'problem creating'.

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of points; c) The same as b), but for the linear 2D PCA manifold (PCA2D). The "basal" breast cancer subtype is visualized more adequately with ELMap2D and some features of the distribution become better resolved in comparison to PCA2D. Principal manifolds are produced by the

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variables is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through

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injected directly into the neuron) and records a train of action potentials, or spikes, produced by the neuron as a result. Presumably, certain features of the stimulus make the neuron more likely to spike. In order to extract these features, the experimenter calculates the

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In an "online" or "streaming" situation with data arriving piece by piece rather than being stored in a single batch, it is useful to make an estimate of the PCA projection that can be updated sequentially. This can be done efficiently, but requires different algorithms.

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components, for PCA have a flat plateau, where no data is captured to remove the quasi-static noise, then the curves drop quickly as an indication of over-fitting (random noise). The FRV curves for NMF is decreasing continuously when the NMF components are constructed

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Wang, Y.; Klijn, J. G.; Zhang, Y.; Sieuwerts, A. M.; Look, M. P.; Yang, F.; Talantov, D.; Timmermans, M.; Meijer-van Gelder, M. E.; Yu, J.; et al. (2005). "Gene expression profiles to predict distant metastasis of lymph-node-negative primary breast cancer".

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in the data that produce large errors, something that the method tries to avoid in the first place. It is therefore common practice to remove outliers before computing PCA. However, in some contexts, outliers can be difficult to identify. For example, in

12515:, the assignment of points to clusters and outliers is not known beforehand. A recently proposed generalization of PCA based on a weighted PCA increases robustness by assigning different weights to data objects based on their estimated relevancy. 2643:{\displaystyle \mathbf {w} _{(1)}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {Xw} \right\|^{2}\right\}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} \right\}} 2666: 5456: 11633:

leaving the deflated residual matrix used to calculate the subsequent leading PCs. For large data matrices, or matrices that have a high degree of column collinearity, NIPALS suffers from loss of orthogonality of PCs due to machine precision

5132: 4545: 2455:{\displaystyle \mathbf {w} _{(1)}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}(t_{1})_{(i)}^{2}\right\}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}} 10635: 12773:). The justification for this criterion is that if a node is removed from the regulatory layer along with all the output nodes connected to it, the result must still be characterized by a connectivity matrix with full column rank. 8524: 7749: 1358:

diagonal elements represent the variance of each axis. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues.

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re-orthogonalization algorithm is applied to both the scores and the loadings at each iteration step to eliminate this loss of orthogonality. NIPALS reliance on single-vector multiplications cannot take advantage of high-level

4412: 12157:, specified by the cluster indicators, is given by the principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace. However, that PCA is a useful relaxation of 11647:

and results in slow convergence for clustered leading singular values—both these deficiencies are resolved in more sophisticated matrix-free block solvers, such as the Locally Optimal Block Preconditioned Conjugate Gradient

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associated to this table into orthogonal factors. Because CA is a descriptive technique, it can be applied to tables for which the chi-squared statistic is appropriate or not. Several variants of CA are available including

11887:, comprising numerous highly correlated instruments, and PCA is used to define a set of components or factors that explain rate movements, thereby facilitating the modelling. One common risk management application is to 16305:

T. Bouwmans; A. Sobral; S. Javed; S. Jung; E. Zahzah (2015). "Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset".

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positive change compared to the variance of the prior. Since these were the directions in which varying the stimulus led to a spike, they are often good approximations of the sought after relevant stimulus features.

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As noted above, the results of PCA depend on the scaling of the variables. This can be cured by scaling each feature by its standard deviation, so that one ends up with dimensionless features with unital variance.

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such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on.

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of the axes, they should be removed and then projected as supplementary elements. In addition, it is necessary to avoid interpreting the proximities between the points close to the center of the factorial plane.

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Mean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data. Hence we proceed by centering the data as follows:

10247: 11958:. A second approach is to enhance portfolio return, using the principal components to select companies' stocks with upside potential. PCA has also been used to understand relationships between international 9889: 11254:), the naive covariance method is rarely used because it is not efficient due to high computational and memory costs of explicitly determining the covariance matrix. The covariance-free approach avoids the 11789:, which has been published since 1990 and is very extensively used in development studies, has very similar coefficients on similar indicators, strongly suggesting it was originally constructed using PCA. 9141: 4306: 1744: 1651: 1565: 10309: 4658:

Dimensionality reduction may also be appropriate when the variables in a dataset are noisy. If each column of the dataset contains independent identically distributed Gaussian noise, then the columns of

10794: 9431: 8822: 8179: 7847: 5319:{\displaystyle {\begin{aligned}\mathbf {T} &=\mathbf {X} \mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{\mathsf {T}}\mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \end{aligned}}} 3939: 2914: 1752: 9705: 4600: 8705: 8635: 8585: 12050:

recording techniques often pick up signals from more than one neuron. In spike sorting, one first uses PCA to reduce the dimensionality of the space of action potential waveforms, and then performs

9583: 9502: 8914: 4630: = 2 and keeping only the first two principal components finds the two-dimensional plane through the high-dimensional dataset in which the data is most spread out, so if the data contains 3455: 2033: 10928: 5228: 4841: 3535: 8224: 4103:{\displaystyle \mathbf {W} ^{\mathsf {T}}\mathbf {Q} \mathbf {W} \propto \mathbf {W} ^{\mathsf {T}}\mathbf {W} \,\mathbf {\Lambda } \,\mathbf {W} ^{\mathsf {T}}\mathbf {W} =\mathbf {\Lambda } } 11143:{\displaystyle {\begin{aligned}\operatorname {cov} (PX)&=\operatorname {E} \\&=\operatorname {E} \\&=P\operatorname {E} P^{*}\\&=P\operatorname {cov} (X)P^{-1}\\\end{aligned}}} 2222: 2160: 2098: 10747: 13244: 7884: 12057:

PCA as a dimension reduction technique is particularly suited to detect coordinated activities of large neuronal ensembles. It has been used in determining collective variables, that is,

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Dimensionality reduction results in a loss of information, in general. PCA-based dimensionality reduction tends to minimize that information loss, under certain signal and noise models.

8109: 12163:-means clustering was not a new result, and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions. 13012:– Integrates PCA in its visual programming environment. PCA displays a scree plot (degree of explained variance) where user can interactively select the number of principal components. 8326: 5066: 1230:. Factor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to 11209: 12438:

find their theoretical and algorithmic roots in PCA or K-means. Pearson's original idea was to take a straight line (or plane) which will be "the best fit" to a set of data points.

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can be a very useful step for visualising and processing high-dimensional datasets, while still retaining as much of the variance in the dataset as possible. For example, selecting

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The principle of the diagram is to underline the "remarkable" correlations of the correlation matrix, by a solid line (positive correlation) or dotted line (negative correlation).

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PCA has successfully found linear combinations of the markers that separate out different clusters corresponding to different lines of individuals' Y-chromosomal genetic descent.

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Another way to characterise the principal components transformation is therefore as the transformation to coordinates which diagonalise the empirical sample covariance matrix.

9747: 9531: 7913: 2774:{\displaystyle \mathbf {w} _{(1)}=\arg \max \left\{{\frac {\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} }{\mathbf {w} ^{\mathsf {T}}\mathbf {w} }}\right\}} 14905:

Flood, J (2000). Sydney divided: factorial ecology revisited. Paper to the APA Conference 2000, Melbourne, November and to the 24th ANZRSAI Conference, Hobart, December 2000.

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into a more manageable data set, which can then for analysis." Here, the resulting factors are linked to e.g. interest rates – based on the largest elements of the factor's

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and is conceptually similar to PCA, but scales the data (which should be non-negative) so that rows and columns are treated equivalently. It is traditionally applied to

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General Linear Model of tenure choice. The strongest determinant of private renting by far was the attitude index, rather than income, marital status or household type.

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to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small.

16108:. Lecture Notes in Computer Science 2350; (Presented at Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark). Springer, Berlin, Heidelberg. 7486: 7332: 7174: 6985: 6825: 6671: 6522: 6373: 6256: 6149: 6034: 5947: 5858: 5800: 5682: 5077: 1432: 15819:

Peter Richtarik; Martin Takac; S. Damla Ahipasaoglu (2012). "Alternating Maximization: Unifying Framework for 8 Sparse PCA Formulations and Efficient Parallel Codes".

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is termed the regulatory layer. While in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied :

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Markopoulos, Panos P.; Kundu, Sandipan; Chamadia, Shubham; Pados, Dimitris A. (15 August 2017). "Efficient L1-Norm Principal-Component Analysis via Bit Flipping".

12599: 4466: 1080:. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. 904: 12711: 5359:, so computing the SVD is now the standard way to calculate a principal components analysis from a data matrix, unless only a handful of components are required. 1209: 1063: 16059: 10425: 16190:

Kriegel, H. P.; Kröger, P.; Schubert, E.; Zimek, A. (2008). "A General Framework for Increasing the Robustness of PCA-Based Correlation Clustering Algorithms".

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indicating a better ability to capture signal. The FRV curves for NMF also converges to higher levels than PCA, indicating the less-overfitting property of NMF.

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Soummer, Rémi; Pueyo, Laurent; Larkin, James (2012). "Detection and Characterization of Exoplanets and Disks Using Projections on Karhunen-Loève Eigenimages".

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Chapin, John; Nicolelis, Miguel (1999). "Principal component analysis of neuronal ensemble activity reveals multidimensional somatosensory representations".

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on the left and on the right, that is, calculation of the covariance matrix is avoided, just as in the matrix-free implementation of the power iterations to

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centered at (1,3) with a standard deviation of 3 in roughly the (0.866, 0.5) direction and of 1 in the orthogonal direction. The vectors shown are the

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The methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies were recently reviewed in a survey paper.

11692:. This procedure is detailed in and Husson, Lê & Pagès 2009 and Pagès 2013. Few software offer this option in an "automatic" way. This is the case of 5481: 4329: 851: 12544:(ICA) is directed to similar problems as principal component analysis, but finds additively separable components rather than successive approximations. 12324:, indicating the continuous capturing of quasi-static noise; then converge to higher levels than PCA, indicating the less over-fitting property of NMF. 10194: 7702: 4463:

columns, this score matrix maximises the variance in the original data that has been preserved, while minimising the total squared reconstruction error

11670:

subjected to PCA for quantitative variables. When analyzing the results, it is natural to connect the principal components to the qualitative variable

11530:

approximates one of the leading principal components, while all columns are iterated simultaneously. The main calculation is evaluation of the product

10820:

is the projection of the data points onto the first principal component, the second column is the projection onto the second principal component, etc.

9832: 12190: 12026:, the set of all stimuli (defined and discretized over a finite time window, typically on the order of 100 ms) that immediately preceded a spike. The 15674: 13082:– Python library for machine learning which contains PCA, Probabilistic PCA, Kernel PCA, Sparse PCA and other techniques in the decomposition module. 16176: 13956: 13354:"Origins and levels of monthly and seasonal forecast skill for United States surface air temperatures determined by canonical correlation analysis" 8857:

They are linear interpretations of the original variables. Also, if PCA is not performed properly, there is a high likelihood of information loss.

1321:(EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al., 1988), 692: 16475:"An Alternative to PCA for Estimating Dominant Patterns of Climate Variability and Extremes, with Application to U.S. and China Seasonal Rainfall" 12528:(RPCA) via decomposition in low-rank and sparse matrices is a modification of PCA that works well with respect to grossly corrupted observations. 10260: 16032: 13641:

Kanade, T.; Ke, Qifa (June 2005). "Robust L₁ Norm Factorization in the Presence of Outliers and Missing Data by Alternative Convex Programming".

10761: 9389: 4703: 14731:

Roweis, Sam. "EM Algorithms for PCA and SPCA." Advances in Neural Information Processing Systems. Ed. Michael I. Jordan, Michael J. Kearns, and

9204: 18206: 14676: 14628: 9636: 899: 18711: 8529:

Then, perhaps the main statistical implication of the result is that not only can we decompose the combined variances of all the elements of

3296:, with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the weight vectors are eigenvectors of 11896: 9902:). This step affects the calculated principal components, but makes them independent of the units used to measure the different variables. 856: 707: 16278:

T. Bouwmans; E. Zahzah (2014). "Robust PCA via Principal Component Pursuit: A Review for a Comparative Evaluation in Video Surveillance".

12348:, on the contrary, which is not a projection on a system of axes, does not have these drawbacks. We can therefore keep all the variables. 18861: 16100: 13531:

Markopoulos, Panos P.; Karystinos, George N.; Pados, Dimitris A. (October 2014). "Optimal Algorithms for L1-subspace Signal Processing".

438: 16565:"Developing Representative Impact Scenarios From Climate Projection Ensembles, With Application to UKCP18 and EURO-CORDEX Precipitation" 16146:. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’05). Vol. 1. San Diego, CA. pp. 547–553. 18485: 17126: 12884:– A java based nodal arranging software for Analysis, in this the nodes called PCA, PCA compute, PCA Apply, PCA inverse make it easily. 11684:

For each center of gravity and each axis, p-value to judge the significance of the difference between the center of gravity and origin.

11572: 939: 742: 9146:

If the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (that is, the components of the vector

5508: 10406:

The eigenvalues represent the distribution of the source data's energy among each of the eigenvectors, where the eigenvectors form a

4663:

will also contain similarly identically distributed Gaussian noise (such a distribution is invariant under the effects of the matrix

4248:

variables which are uncorrelated over the dataset. However, not all the principal components need to be kept. Keeping only the first

14231:

Blanton, Michael R.; Roweis, Sam (2007). "K-corrections and filter transformations in the ultraviolet, optical, and near infrared".

12321: 10705: 4000:{\displaystyle \mathbf {Q} \propto \mathbf {X} ^{\mathsf {T}}\mathbf {X} =\mathbf {W} \mathbf {\Lambda } \mathbf {W} ^{\mathsf {T}}} 3026:{\displaystyle \mathbf {\hat {X}} _{k}=\mathbf {X} -\sum _{s=1}^{k-1}\mathbf {X} \mathbf {w} _{(s)}\mathbf {w} _{(s)}^{\mathsf {T}}} 18259: 12477: 9099: 4262: 1882:{\displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,l} 1663: 1570: 1484: 12135:. If the factor model is incorrectly formulated or the assumptions are not met, then factor analysis will give erroneous results. 11738:

looked for 56 factors of intelligence, developing the notion of Mental Age. Standard IQ tests today are based on this early work.

18698: 13188: 8143: 7811: 818: 15380:

Jirsa, Victor; Friedrich, R; Haken, Herman; Kelso, Scott (1994). "A theoretical model of phase transitions in the human brain".

11544:

matrix-matrix product functions, and typically leads to faster convergence, compared to the single-vector one-by-one technique.

10018:

consists entirely of real numbers, which is the case in many applications, the "conjugate transpose" is the same as the regular

14702: 13172: 4550: 367: 5469:

using a truncated singular value decomposition in this way produces a truncated matrix that is the nearest possible matrix of

16722: 16704: 16675: 16217: 16121: 16067: 15447: 15195: 14289:

Zhu, Guangtun B. (2016-12-19). "Nonnegative Matrix Factorization (NMF) with Heteroscedastic Uncertainties and Missing data".

14032: 13989: 13854: 13668: 12896:– Implements principal component analysis with the PrincipalComponents command using both covariance and correlation methods. 9544: 9463: 8878: 8849:

focusing only on the non-negative elements in the matrices, which is well-suited for astrophysical observations. See more at

11912: 8757: 4672: 17121: 16821: 15782: 15048:"Principal Component Analyses (PCA)‑based findings in population genetic studies are highly biased and must be reevaluated" 13208: 12519: 11786: 3425: 2003: 876: 639: 174: 16729: 14942:

Schamberger, Tamara; Schuberth, Florian; Henseler, Jörg. "Confirmatory composite analysis in human development research".

12374:

A particular disadvantage of PCA is that the principal components are usually linear combinations of all input variables.

17725: 16873: 13050:

which generally gives better numerical accuracy. Some packages that implement PCA in R, include, but are not limited to:

12525: 11238:) is guaranteed to be a non-negative definite matrix and thus is guaranteed to be diagonalisable by some unitary matrix. 8845:

fluxes, and forward modeling has to be performed to recover the true magnitude of the signals. As an alternative method,

8838: 8660: 8590: 8540: 3491:

multiplied by the square root of corresponding eigenvalues, that is, eigenvectors scaled up by the variances, are called

894: 13102:– Proprietary software most commonly used by social scientists for PCA, factor analysis and associated cluster analysis. 9319:. In general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise 8184: 5552: 15246: 13228: 12469:, which corresponds to PCA performed in a reproducing kernel Hilbert space associated with a positive definite kernel. 12435: 5520: 2165: 2103: 2041: 1088: 727: 702: 651: 11291:

One way to compute the first principal component efficiently is shown in the following pseudo-code, for a data matrix

1226:

of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to

981:

such that the directions (principal components) capturing the largest variation in the data can be easily identified.

18508: 18400: 15474: 14783: 14517: 14415: 13735: 13223: 13152: 12180: 12099: 12095: 11911:

and other sensitivities. Under both, the first three, typically, principal components of the system are of interest (

11727: 8846: 775: 770: 423: 16424:"Discriminant analysis of principal components: a new method for the analysis of genetically structured populations" 11963: 11559:

with matrix deflation by subtraction implemented for computing the first few components in a principal component or

10640: 19113: 18686: 18560: 14549:

Geiger, Bernhard; Kubin, Gernot (January 2013). "Signal Enhancement as Minimization of Relevant Information Loss".

13125: 12103: 7852: 433: 71: 12449:

as the natural extension for the geometric interpretation of PCA, which explicitly constructs a manifold for data

9038: 8416:{\displaystyle \mathbf {\Sigma } =\lambda _{1}\alpha _{1}\alpha _{1}'+\cdots +\lambda _{p}\alpha _{p}\alpha _{p}'} 18744: 18405: 18150: 17521: 17111: 15273: 13015: 12502:

While PCA finds the mathematically optimal method (as in minimizing the squared error), it is still sensitive to

11767: 9349:

The following is a detailed description of PCA using the covariance method as opposed to the correlation method.

9294: 8078: 1353:

of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must

1283: 1069: 14171:"Detection and Characterization of Exoplanets using Projections on Karhunen Loeve Eigenimages: Forward Modeling" 13378: 13353: 13177: 11985:– and it is then observed how a "shock" to each of the factors affects the implied assets of each of the banks. 8985:

one can show that PCA can be optimal for dimensionality reduction, from an information-theoretic point-of-view.

19135: 18795: 18007: 17814: 17703: 17661: 15718: 15222: 14582: 13250: 13198: 13157: 12823: 12541: 11892: 11484:

convergence can be accelerated without noticeably sacrificing the small cost per iteration using more advanced

10133: 5042: 4652: 1318: 932: 828: 592: 413: 17735: 14906: 12042:

In neuroscience, PCA is also used to discern the identity of a neuron from the shape of its action potential.

11915:"shift", "twist", and "curvature"). These principal components are derived from an eigen-decomposition of the 19038: 17997: 16900: 15503:

Meglen, R.R. (1991). "Examining Large Databases: A Chemometric Approach Using Principal Component Analysis".

13131: 12931: 12473: 12047: 11182: 803: 505: 281: 15312: 14689: 11861: 9377: 8114: 7782: 1271: 19140: 18589: 18538: 18523: 18382: 18254: 18221: 18047: 18002: 17832: 15928: 13260: 13047: 12027: 11681:

Representation, on the factorial planes, of the centers of gravity of plants belonging to the same species.

11563:

analysis. For very-high-dimensional datasets, such as those generated in the *omics sciences (for example,

5583: 4690: 4684: 2789: 1287: 1239: 1223: 1222:. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or 1092: 760: 697: 607: 585: 418: 11946:

is maximized for a given level of risk, or alternatively, where risk is minimized for a given return; see

11806:

different population groups, thereby showing individuals who have wandered from their original locations.

8229: 19101: 18933: 18734: 18658: 17959: 17713: 17382: 16846: 15316: 15257: 12462: 12345: 7129: 911: 823: 808: 269: 91: 16604: 12430:

algorithm. Data are available for public competition. Software is available for free non-commercial use.

12171: 3357:

in the transformed coordinates, or as the corresponding vector in the space of the original variables, {

1895: 18818: 18790: 18785: 18533: 18292: 18198: 18178: 18086: 17797: 17615: 17098: 16970: 16788: 16776: 16760: 16749: 16088:. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’03). Madison, WI. 13255: 13162: 12034:(the set of all stimuli, defined over the same length time window) then indicate the directions in the 12006: 11840: 11678:

Identification, on the factorial planes, of the different species, for example, using different colors.

10161: 8031: 5526:

PCA is sensitive to the scaling of the variables. If we have just two variables and they have the same

1354: 871: 798: 548: 443: 231: 164: 124: 15867: 15774: 9723: 9507: 9032:

is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the

8038: 7889: 7603: 7295:

matrix of basis vectors, one vector per column, where each basis vector is one of the eigenvectors of

4689:

The principal components transformation can also be associated with another matrix factorization, the

1099:

scaled by the square root of the corresponding eigenvalue, and shifted so their tails are at the mean.

18550: 18318: 18039: 17964: 17893: 17822: 17742: 17730: 17600: 17588: 17581: 17289: 17010: 14380: 11974: 11857: 11798: 10165: 5536: 5516: 1390: 967: 925: 531: 299: 169: 16658: 16233:

Emmanuel J. Candes; Xiaodong Li; Yi Ma; John Wright (2011). "Robust Principal Component Analysis?".

16200: 15937:

Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods and Techniques

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to the original matrix, in the sense of the difference between the two having the smallest possible

5451:{\displaystyle \mathbf {T} _{L}=\mathbf {U} _{L}\mathbf {\Sigma } _{L}=\mathbf {X} \mathbf {W} _{L}} 5379:

can be obtained by considering only the first L largest singular values and their singular vectors:

19033: 18800: 18663: 18348: 18313: 18277: 18062: 17504: 17413: 17372: 17284: 16975: 16814: 16170: 15279: 15261: 15210: 13651: 13238: 13146: 13027: 12454: 12415: 11899:, are then reconstructed; with VaR calculated, finally, over the entire run. PCA is also used in 11880: 10145: 7973: 7667: 7405: 7252: 7063: 6903: 6749: 6595: 6446: 6297: 6190: 6075: 5960: 5893: 5613: 5512: 4623: 3484: 1314: 963: 553: 473: 396: 314: 144: 106: 101: 61: 56: 15939:, Olivas E.S. et al Eds. Information Science Reference, IGI Global: Hershey, PA, USA, 2009. 28–59. 15688: 14594: 13301:

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

11693: 11500:

with every new computation. The latter approach in the block power method replaces single-vectors

10801: 9322: 9300: 9274: 9175: 9149: 9077: 9013: 8991: 8966: 8944: 8922: 8267: 7998: 7922: 7212: 7023: 6863: 6709: 5172: 5127:{\displaystyle \mathbf {{\hat {\Sigma }}^{2}} =\mathbf {\Sigma } ^{\mathsf {T}}\mathbf {\Sigma } } 1309:(for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's 18942: 18555: 18495: 18432: 18070: 18054: 17792: 17654: 17644: 17494: 17408: 16004: 15900: 13193: 13119: 13105: 12078: 12030:

of the difference between the spike-triggered covariance matrix and the covariance matrix of the

11970: 11888: 11782: 8730: 7524: 7492: 7370: 7338: 7180: 6991: 6831: 6677: 6560: 6528: 6411: 6379: 6262: 6155: 6040: 5720: 5688: 3036:

and then finding the weight vector which extracts the maximum variance from this new data matrix

500: 349: 249: 76: 16649: 16642: 14719: 13733:

Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components.

18980: 18910: 18703: 18640: 18395: 18282: 17279: 17176: 17083: 16962: 16861: 16653: 16195: 16156: 16024: 15720: 15683: 14853: 13646: 13275: 13213: 12512: 12233: 11865: 11774: 10407: 10037: 9444:

variables, and you want to reduce the data so that each observation can be described with only

7465: 7311: 7153: 6964: 6804: 6650: 6501: 6352: 6235: 6128: 6013: 5926: 5837: 5779: 5661: 5564: 1411: 1263: 1066: 680: 656: 558: 319: 294: 254: 66: 14970: 14917: 13108:– Java library for machine learning which contains modules for computing principal components. 7452:

row vectors, where each vector is the projection of the corresponding data vector from matrix

19005: 18947: 18890: 18716: 18609: 18518: 18244: 18128: 17987: 17979: 17869: 17861: 17676: 17572: 17550: 17509: 17474: 17441: 17387: 17362: 17317: 17256: 17216: 17018: 16841: 15841: 15772: 14087: 13950: 13141: 12961: 12106:, which may be seen as the counterpart of principal component analysis for categorical data. 12090: 11719: 11560: 11247: 11156: 10847: 10418:

th eigenvector is the sum of the energy content across all of the eigenvalues from 1 through

10059: 7596: 5470: 4668: 4540:{\displaystyle \|\mathbf {T} \mathbf {W} ^{T}-\mathbf {T} _{L}\mathbf {W} _{L}^{T}\|_{2}^{2}} 1393: 1231: 989: 634: 456: 408: 264: 179: 51: 13485:

Chachlakis, Dimitris G.; Prater-Bennette, Ashley; Markopoulos, Panos P. (22 November 2019).

12082: 2863:

in the transformed co-ordinates, or as the corresponding vector in the original variables, {

18928: 18503: 18452: 18428: 18390: 18308: 18287: 18239: 18118: 18096: 18065: 17974: 17851: 17802: 17720: 17693: 17649: 17605: 17367: 17143: 17023: 16527: 16486: 16376: 16325: 15801: 15666: 15647: 15061: 14854:"Randomized online PCA algorithms with regret bounds that are logarithmic in the dimension" 14564: 14344: 14250: 14192: 14135: 13607: 13550: 13448: 13408: 13365: 13308: 13218: 12841: 12487: 11853: 11735: 9194:

is non-Gaussian (which is a common scenario), PCA at least minimizes an upper bound on the

5556: 4643: 4427: 1435: 563: 513: 15719:

Alexandre d'Aspremont; Laurent El Ghaoui; Michael I. Jordan; Gert R. G. Lanckriet (2007).

14615: 14005: 13870:

Bengio, Y.; et al. (2013). "Representation Learning: A Review and New Perspectives".

13643:

2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05)

12575: 12518:

Outlier-resistant variants of PCA have also been proposed, based on L1-norm formulations (

10837:-dimensional random vector expressed as column vector. Without loss of generality, assume 10698:

is above a certain threshold, like 90 percent. In this case, choose the smallest value of

4616:

calculated from repeat-count values for 37 Y-chromosomal STR markers from 354 individuals.

3933:

In matrix form, the empirical covariance matrix for the original variables can be written

8: 19075: 19000: 18923: 18604: 18368: 18361: 18323: 18231: 18211: 18183: 17916: 17782: 17777: 17767: 17759: 17577: 17538: 17428: 17418: 17327: 17106: 17062: 16980: 16905: 16807: 16564: 16080: 15603: 12912:– The SVD function is part of the basic system. In the Statistics Toolbox, the functions 12690: 12119:

teacher, high school teacher) are reduced into the component of interest (e.g., teacher).

11978: 11485: 11267: 10011: 8296: 5579: 5338: 4639: 4459:) that are decorrelated. By construction, of all the transformed data matrices with only 1397: 1188: 1112: 1104: 1042: 666: 602: 573: 478: 304: 237: 223: 209: 184: 134: 86: 46: 16531: 16490: 16380: 16365:"Network component analysis: Reconstruction of regulatory signals in biological systems" 16329: 15805: 15651: 15065: 14568: 14348: 14254: 14196: 14139: 13611: 13554: 13452: 13412: 13369: 13312: 11540:, efficient blocking eliminates the accumulation of the errors, allows using high-level 11454:. If the largest singular value is well separated from the next largest one, the vector 10630:{\displaystyle W_{kl}=V_{k\ell }\qquad {\text{for }}k=1,\dots ,p\qquad \ell =1,\dots ,L} 1454:

columns gives a particular kind of feature (say, the results from a particular sensor).

1065:

vectors. Here, a best-fitting line is defined as one that minimizes the average squared

19089: 18900: 18754: 18650: 18599: 18475: 18372: 18356: 18333: 18110: 17844: 17827: 17787: 17698: 17593: 17555: 17526: 17486: 17446: 17392: 17309: 16995: 16990: 16586: 16545: 16450: 16423: 16341: 16315: 16260: 16242: 15977: 15820: 15791: 15755: 15737: 15701: 15637: 15634:

Dimensionality reduction for k-means clustering and low rank approximation (Appendix B)

15608: 15561: 15539: 15520: 15405: 15353: 15097: 15084: 15052: 15047: 15020: 14995: 14834: 14808: 14658: 14640: 14554: 14488: 14362: 14334: 14290: 14266: 14240: 14210: 14182: 14151: 14125: 14018: 13905: 13879: 13765: 13716: 13674: 13623: 13597: 13566: 13540: 13498: 13464: 13438: 13398: 13329: 13296: 12778: 12756: 12736: 12716: 12670: 12647: 12624: 12604: 12555: 12302: 12282: 12149: 12086: 11904: 11820: 11815: 11575:(NIPALS) algorithm updates iterative approximations to the leading scores and loadings 11214: 10900: 9993: 9033: 8710: 8640: 8519:{\displaystyle \operatorname {Var} (x_{j})=\sum _{k=1}^{P}\lambda _{k}\alpha _{kj}^{2}} 7762: 7643: 7123: 6797: 6340: 6116: 5871: 5813: 5755: 1977: 1957: 1460: 1369: 1295: 1168: 1147: 1126: 1077: 1018: 995: 971: 644: 568: 354: 149: 16399: 16364: 15965: 15341: 14205: 14170: 14147: 13024:– Commercial software for analyzing multivariate data with instant response using PCA. 12984:– Free software computational environment mostly compatible with MATLAB, the function 12850:– includes PCA for projection, including robust variants of PCA, as well as PCA-based 12601:. A key difference from techniques such as PCA and ICA is that some of the entries of 19084: 18995: 18965: 18957: 18777: 18768: 18693: 18624: 18480: 18465: 18440: 18328: 18269: 18135: 18123: 17749: 17666: 17610: 17533: 17377: 17299: 17078: 16952: 16754: 16718: 16700: 16671: 16590: 16563:

Jewson, S.; Messori, G.; Barbato, G.; Mercogliano, P.; Mysiak, J.; Sassi, M. (2022).

16549: 16455: 16404: 16213: 16117: 16063: 15969: 15583: 15524: 15470: 15443: 15397: 15345: 15311:

See Ch. 25 § "Scenario testing using principal component analysis" in Li Ong (2014).

15242: 15218: 15191: 15116: 15101: 15089: 15025: 14826: 14779: 14758: 14745:

Geladi, Paul; Kowalski, Bruce (1986). "Partial Least Squares Regression:A Tutorial".

14662: 14513: 14453: 14411: 14214: 14069: 14028: 13985: 13897: 13850: 13720: 13664: 13334: 13088:– Free and open-source, cross-platform numerical computational package, the function 13009: 12951: 12890:– The PCA command is used to perform a principal component analysis on a set of data. 12062: 12019: 11916: 11908: 11489: 11439: 10880: 9920: 8534: 6643: 3510: 2785: 1942:

considered over the data set successively inherit the maximum possible variance from

1401: 1350: 1275: 1243: 1219: 1096: 1073: 978: 737: 580: 493: 289: 259: 204: 199: 154: 96: 16345: 15981: 15491:

Confirmatory Factor Analysis for Applied Research Methodology in the social sciences

15357: 14270: 14155: 13678: 12013:

process as a stimulus (usually either as a sensory input to a test subject, or as a

4605: 19020: 18975: 18739: 18726: 18619: 18594: 18528: 18460: 18338: 17946: 17839: 17772: 17685: 17632: 17451: 17322: 17116: 17000: 16915: 16882: 16692: 16663: 16576: 16535: 16494: 16445: 16435: 16394: 16384: 16333: 16287: 16264: 16252: 16205: 16158: 16157:

Kirill Simonov, Fedor V. Fomin, Petr A. Golovach, Fahad Panolan (June 9–15, 2019).

16109: 16045: 16041: 15999: 15961: 15909: 15759: 15747: 15705: 15693: 15612: 15598: 15512: 15409: 15389: 15337: 15079: 15069: 15015: 15007: 14996:"Interpreting principal component analyses of spatial population genetic variation" 14951: 14838: 14818: 14754: 14650: 14492: 14480: 14443: 14366: 14352: 14258: 14200: 14143: 14061: 13977: 13936: 13925:"Hypothesis tests for principal component analysis when variables are standardized" 13889: 13842: 13806: 13798: 13757: 13708: 13656: 13627: 13615: 13570: 13558: 13508: 13468: 13456: 13373: 13324: 13316: 13270: 12887: 12851: 12395:

forward-backward greedy search and exact methods using branch-and-bound techniques,

12058: 12051: 12014: 12002: 11951: 11939: 11723: 11718:

factor analysis from results on various tests, to give a single index known as the

11639: 11635: 11616:

The matrix deflation by subtraction is performed by subtracting the outer product,

4631: 4407:{\displaystyle t=W_{L}^{\mathsf {T}}x,x\in \mathbb {R} ^{p},t\in \mathbb {R} ^{L},} 4146:

is equal to the sum of the squares over the dataset associated with each component

1322: 1279: 1270:

in the 1930s. Depending on the field of application, it is also named the discrete

1267: 1235: 765: 518: 468: 378: 362: 332: 194: 189: 139: 129: 27: 14050:"Measuring systematic changes in invasive cancer cell shape using Zernike moments" 13909: 8533:

into decreasing contributions due to each PC, but we can also decompose the whole

7744:{\displaystyle \mathbf {\Sigma } _{y}=\mathbf {B'} \mathbf {\Sigma } \mathbf {B} } 18937: 18681: 18543: 18470: 18145: 18019: 17992: 17969: 17938: 17565: 17560: 17514: 17244: 16895: 16337: 16209: 16138: 15490: 15466:

Geometric Data Analysis, From Correspondence Analysis to Structured Data Analysis

15464: 15292: 14773: 14507: 12996: 12132: 12127: 11947: 11943: 11900: 11590: 11556: 11481: 11417: 10121: 7111: 5527: 4762: 1306: 1227: 793: 597: 463: 403: 18427: 16163:

Proceedings of the 36th International Conference on Machine Learning (ICML 2019)

15986: 14799:

Andrecut, M. (2009). "Parallel GPU Implementation of Iterative PCA Algorithms".

13513: 13486: 12272:{\displaystyle 1-\sum _{i=1}^{k}\lambda _{i}{\Big /}\sum _{j=1}^{n}\lambda _{j}} 11895:. Here, for each simulation-sample, the components are stressed, and rates, and 3524:

between two of the different principal components over the dataset is given by:

18886: 18881: 17344: 17274: 16920: 16360: 16291: 15914: 15895: 15818: 15074: 14357: 14322: 13941: 13924: 13183: 13073: 12975: 12871: 12336: 11697: 10141: 10137: 5477: 1443: 813: 344: 81: 16782: 16770: 16743: 14955: 13971: 13712: 12924:

gives the residuals and reconstructed matrix for a low-rank PCA approximation.

10396:

Make sure to maintain the correct pairings between the columns in each matrix.

19129: 19043: 19010: 18873: 18834: 18645: 18614: 18078: 18032: 17637: 17339: 17166: 16930: 16925: 16232: 16113: 16020: 15697: 14732: 14457: 14323:"Non-negative Matrix Factorization: Robust Extraction of Extended Structures" 14048:

Alizadeh, Elaheh; Lyons, Samanthe M; Castle, Jordan M; Prasad, Ashok (2016).

13981: 13619: 13562: 13460: 13233: 13031: 12450: 12439: 12419: 12114: 12043: 11959: 11955: 11731: 10132:. This step will typically involve the use of a computer-based algorithm for 1446:(the sample mean of each column has been shifted to zero), where each of the 732: 661: 543: 274: 159: 16540: 16515: 16440: 16389: 16256: 15773:

Michel Journee; Yurii Nesterov; Peter Richtarik; Rodolphe Sepulchre (2010).

15275:

An Application of Principal Component Analysis to Stock Portfolio Management

14022: 10894:

is a random vector with all its distinct components pairwise uncorrelated).

1185:-th principal component can be taken as a direction orthogonal to the first 18985: 18918: 18895: 18810: 18140: 17436: 17334: 17269: 17211: 17196: 17133: 17088: 16459: 16408: 15973: 15516: 15349: 15093: 15029: 14830: 14448: 14073: 13901: 13692: 13585: 13338: 13320: 13079: 12899: 12035: 11994: 11872: 11568: 10506:{\displaystyle g_{j}=\sum _{k=1}^{j}D_{kk}\qquad {\text{for }}j=1,\dots ,p} 8754:

tend to stay about the same size because of the normalization constraints:

6952: 5656:

data matrix, consisting of the set of all data vectors, one vector per row

4740:{\displaystyle \mathbf {X} =\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{T}} 4635:

likely to substantially overlay each other, making them indistinguishable.

1259: 1215: 16686: 16499: 16474: 15401: 14822: 14471:

Linsker, Ralph (March 1988). "Self-organization in a perceptual network".

13893: 13836: 13660: 12812: 12410: 9261:{\displaystyle I(\mathbf {x} ;\mathbf {s} )-I(\mathbf {y} ;\mathbf {s} ).} 1450:

rows represents a different repetition of the experiment, and each of the

1035:-th vector is the direction of a line that best fits the data while being 19028: 18990: 18673: 18574: 18436: 18249: 18216: 17708: 17625: 17620: 17264: 17221: 17201: 17181: 17171: 16940: 16581: 15863: 15728: 15043: 14245: 13789: 12955: 12893: 12508: 12458: 12427: 12010: 11982: 11920: 11884: 11811: 11766:

PCA is a formal method for the development of indexes. As an alternative

10153: 10055: 7115: 4647: 4010:

The empirical covariance matrix between the principal components becomes

2810: 1951: 1012: 538: 32: 15876:

Journal of Machine Learning Research Workshop and Conference Proceedings

9983:{\displaystyle \mathbf {C} ={1 \over {n-1}}\mathbf {B} ^{*}\mathbf {B} } 1214:

For either objective, it can be shown that the principal components are

1211:

principal components that maximizes the variance of the projected data.

17874: 17354: 17054: 16985: 16935: 16910: 16830: 16194:. Lecture Notes in Computer Science. Vol. 5069. pp. 418–435. 15952: 15393: 14618:, Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer 14065: 14008:, Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer 13769: 13748: 13265: 13203: 12981: 12927: 12494:, multiple factor analysis, co-inertia analysis, STATIS, and DISTATIS. 12486:-way principal component analysis may be performed with models such as 12466: 12422: 12375: 12369: 11932: 11928: 11260:

operations of explicitly calculating and storing the covariance matrix

10169: 10125: 7756: 5497: 2800: 1365: 1326: 1108: 1036: 960: 687: 383: 309: 16784:

StatQuest: StatQuest: Principal Component Analysis (PCA), Step-by-Step

16165:. Vol. 97. Long Beach, California, USA: PMLR. pp. 5818–5826. 15751: 14436:"Bias in Principal Components Analysis Due to Correlated Observations" 12947: 11950:

for discussion. Thus, one approach is to reduce portfolio risk, where

10109:{\displaystyle \mathbf {V} ^{-1}\mathbf {C} \mathbf {V} =\mathbf {D} } 9811:{\displaystyle \mathbf {B} =\mathbf {X} -\mathbf {h} \mathbf {u} ^{T}} 8837:

can restore the linearity assumption and PCA can then be applied (see

5535:

data. A mean of zero is needed for finding a basis that minimizes the

5519:, and in particular to the DCT-II which is simply known as the "DCT". 5337:

multiplied by the corresponding singular value. This form is also the

18027: 17879: 17499: 17294: 17206: 17191: 17186: 17151: 15839: 15664: 15370:

Brenner, N., Bialek, W., & de Ruyter van Steveninck, R.R. (2000).

15164:

Paper to the European Network for Housing Research Conference, Dublin

14654: 14321:

Ren, Bin; Pueyo, Laurent; Zhu, Guangtun B.; Duchêne, Gaspard (2018).

13167: 11876: 11638:

accumulated in each iteration and matrix deflation by subtraction. A

10019: 4613: 2660:

has been defined to be a unit vector, it equivalently also satisfies

1342: 1103:

Principal component analysis has applications in many fields such as

1083: 846: 627: 15721:"A Direct Formulation for Sparse PCA Using Semidefinite Programming" 15159: 14088:

Estimating Invariant Principal Components Using Diagonal Regression.

13922: 13811: 13802: 13761: 13696: 13484: 12811:

A DAPC can be realized on R using the package Adegenet. (more info:

11954:

are applied to the "principal portfolios" instead of the underlying

4675:, as an aid in determining how many principal components to retain. 17543: 17161: 17038: 17033: 17028: 16696: 16667: 16320: 16304: 15742: 15581: 15233:§III.A.3.7.2 in Carol Alexander and Elizabeth Sheedy, eds. (2004). 15011: 14616:

New Routes from Minimal Approximation Error to Principal Components

14339: 14295: 14262: 14187: 14006:

New Routes from Minimal Approximation Error to Principal Components

13846: 13503: 13443: 12799:

then the decomposition is unique up to multiplication by a scalar.

11997:

to identify the specific properties of a stimulus that increases a

11924: 11564: 10136:. These algorithms are readily available as sub-components of most 8429: 7752: 4792:

matrix, the columns of which are orthogonal unit vectors of length

1247: 16765: 16247: 16060:

Principal Manifolds for Data Visualisation and Dimension Reduction

15932: 15825: 15796: 15775:"Generalized Power Method for Sparse Principal Component Analysis" 15642: 14813: 14645: 14559: 14130: 14099: 13884: 13697:"On Lines and Planes of Closest Fit to Systems of Points in Space" 13602: 13545: 13403: 13096:

computes principal component analysis with standardized variables.

12844:– The built-in EigenDecomp function computes principal components. 11962:, and within markets between groups of companies in industries or 10242:{\displaystyle D_{k\ell }=\lambda _{k}\qquad {\text{for }}k=\ell } 5547:). Also see the article by Kromrey & Foster-Johnson (1998) on 3904:

has been used to move from line 2 to line 3. However eigenvectors

19048: 18749: 15462: 15117:"Restricted principal components analysis for marketing research" 14484: 13929:

Journal of Agricultural, Biological, and Environmental Statistics

13021: 12503: 12491: 12054:

to associate specific action potentials with individual neurons.

11571:) it is usually only necessary to compute the first few PCs. The 10367:

eigenvectors). In general, the matrix of right eigenvectors need

9899: 9884:{\displaystyle h_{i}=1\,\qquad \qquad {\text{for }}i=1,\ldots ,n} 6492:, computed using the mean and standard deviation for each column 6489: 3509:

itself can be recognized as proportional to the empirical sample

622: 15582:

Drineas, P.; A. Frieze; R. Kannan; S. Vempala; V. Vinay (2004).

13297:"Principal component analysis: a review and recent developments" 12968:

routine (available in both the Fortran versions of the Library).

12940:– Provides an implementation of principal component analysis in 12802: 11935:, no correlation need be incorporated in subsequent modelling). 11492:

or the Locally Optimal Block Preconditioned Conjugate Gradient (

10686:

as small as possible while achieving a reasonably high value of

10371:

be the (conjugate) transpose of the matrix of left eigenvectors.

8587:

from each PC. Although not strictly decreasing, the elements of

5890:

the number of dimensions in the dimensionally reduced subspace,

5549:"Mean-centering in Moderated Regression: Much Ado About Nothing" 1266:

in mechanics; it was later independently developed and named by

18970: 17951: 17925: 17905: 17156: 16947: 16562: 15425:

L'Analyse des Données. Volume II. L'Analyse des Correspondances

13746:

Hotelling, H (1936). "Relations between two sets of variates".

13351: 13085: 12991: 12937: 12909: 12860:– principal component analysis can be performed either via the 12835: 11998: 11649: 11537: 11493: 10173: 10149: 9625:

Place the calculated mean values into an empirical mean vector

5523:

techniques tend to be more computationally demanding than PCA.

1457:

Mathematically, the transformation is defined by a set of size

1361: 1123:

When performing PCA, the first principal component of a set of

373: 15584:"Clustering large graphs via the singular value decomposition" 13872:

IEEE Transactions on Pattern Analysis and Machine Intelligence

11977:. Its utility is in "distilling the information contained in 11297:

with zero mean, without ever computing its covariance matrix.

10046:

Find the eigenvectors and eigenvalues of the covariance matrix

1072:. These directions (i.e., principal components) constitute an 16799: 15896:"A Selective Overview of Sparse Principal Component Analysis" 15842:"Spectral Bounds for Sparse PCA: Exact and Greedy Algorithms" 15631: 15238: 14631:& Williams, L.J. (2010). "Principal component analysis". 13428: 12971: 12941: 12881: 12857: 12445: 12340:

Iconography of correlations – Geochemistry of marine aerosols

11730:

of intelligence, adding a formal technique to the science of

11701: 10344:

The eigenvalues and eigenvectors are ordered and paired. The

10157: 9136:{\displaystyle \mathbf {y} =\mathbf {W} _{L}^{T}\mathbf {x} } 5586:

is an alternative which is optimized for class separability.

4610: 4301:{\displaystyle \mathbf {T} _{L}=\mathbf {X} \mathbf {W} _{L}} 2799:

is that the quotient's maximum possible value is the largest

1739:{\displaystyle \mathbf {t} _{(i)}=(t_{1},\dots ,t_{l})_{(i)}} 1646:{\displaystyle \mathbf {x} _{(i)}=(x_{1},\dots ,x_{p})_{(i)}} 1560:{\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}} 1375: 1234:. CCA defines coordinate systems that optimally describe the 617: 612: 339: 16691:. Springer Series in Statistics. New York: Springer-Verlag. 16062:, LNCSE 58, Springer, Berlin – Heidelberg – New York, 2007. 14879:

Psychological Testing: Principles, Applications, and Issues.

13841:. Springer Series in Statistics. New York: Springer-Verlag. 12838:– a C++ and C# library that implements PCA and truncated PCA 11270:, for example, based on the function evaluating the product 10304:{\displaystyle D_{k\ell }=0\qquad {\text{for }}k\neq \ell .} 4326:

columns. In other words, PCA learns a linear transformation

16890: 16648:. Springer Series in Statistics. Springer-Verlag. pp. 16421: 15293:

Principal Component Analysis for Stock Portfolio Management

13784: 13379:

10.1175/1520-0493(1987)115<1825:oaloma>2.0.co;2

13099: 12920:(R2012b) give the principal components, while the function 12847: 11690:

introducing a qualitative variable as supplementary element

11644: 11541: 10789:{\displaystyle \mathbf {T} =\mathbf {B} \cdot \mathbf {W} } 10690:

on a percentage basis. For example, you may want to choose

9426:{\displaystyle \mathbf {Y} =\mathbb {KLT} \{\mathbf {X} \}} 8174:{\displaystyle \operatorname {tr} (\mathbf {\Sigma } _{y})} 7842:{\displaystyle \operatorname {tr} (\mathbf {\Sigma } _{y})} 6001: 4812:

matrix whose columns are orthogonal unit vectors of length

3289:

It turns out that this gives the remaining eigenvectors of

16189: 15632:

Cohen, M.; S. Elder; C. Musco; C. Musco; M. Persu (2014).

15115:

DeSarbo, Wayne; Hausmann, Robert; Kukitz, Jeffrey (2007).

13785:"On the early history of the singular value decomposition" 13530: 11826: 11700:, was the first to propose this option, and the R package 10402:

Compute the cumulative energy content for each eigenvector

9456:. Suppose further, that the data are arranged as a set of 9440:

Suppose you have data comprising a set of observations of

9344: 8851:

Relation between PCA and Non-negative Matrix Factorization

7456:

onto the basis vectors contained in the columns of matrix

5069:

is the square diagonal matrix with the singular values of

16713:

Husson François, Lê Sébastien & Pagès Jérôme (2009).

14633:

Wiley Interdisciplinary Reviews: Computational Statistics

14047: 12903: 10824: 9700:{\displaystyle u_{j}={\frac {1}{n}}\sum _{i=1}^{n}X_{ij}} 9169: 4595:{\displaystyle \|\mathbf {X} -\mathbf {X} _{L}\|_{2}^{2}} 1997:

The above may equivalently be written in matrix form as

1250:-based variants of standard PCA have also been proposed. 1076:

in which different individual dimensions of the data are

16092: 15547:

Neural Information Processing Systems Vol.14 (NIPS 2001)

15379: 15188:

Mathematics and Statistics for Financial Risk Management

14941: 11973:, essentially an analysis of a bank's ability to endure 9898:) may also be scaled to have a variance equal to 1 (see 5213:

Using the singular value decomposition the score matrix

3472:

matrix of weights whose columns are the eigenvectors of

905:

List of datasets in computer vision and image processing

16772:

A layman's introduction to principal component analysis

16058:

A.N. Gorban, B. Kegl, D.C. Wunsch, A. Zinovyev (Eds.),

16000:"ViDaExpert – Multidimensional Data Visualization Tool" 15948: 15537: 12974:– Proprietary numerical library containing PCA for the 12964:– Principal components analysis is implemented via the 12572:, it tries to decompose it into two matrices such that 12386:

a convex relaxation/semidefinite programming framework,

9578:{\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}} 9497:{\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}} 8909:{\displaystyle \mathbf {x} =\mathbf {s} +\mathbf {n} ,} 4252:

principal components, produced by using only the first

2232:

In order to maximize variance, the first weight vector

1242:

that optimally describes variance in a single dataset.

16422:

Liao, T.; Jombart, S.; Devillard, F.; Balloux (2010).

15538:

H. Zha; C. Ding; M. Gu; X. He; H.D. Simon (Dec 2001).

14893:

The Social Areas of Los Angeles: Analysis and Typology

14627: 13395:

A spectral algorithm for learning hidden markov models

12009:. In a typical application an experimenter presents a 11993:

A variant of principal components analysis is used in

11746:

In 1949, Shevky and Williams introduced the theory of

8850: 8817:{\displaystyle \alpha _{k}'\alpha _{k}=1,k=1,\dots ,p} 5832:

the number of elements in each row vector (dimension)

3182: 2823:

found, the first principal component of a data vector

16717:. Chapman & Hall/CRC The R Series, London. 224p. 16516:"Robust Worst-Case Scenarios from Ensemble Forecasts" 16161:. In Kamalika Chaudhuri, Ruslan Salakhutdinov (ed.). 15562:"K-means Clustering via Principal Component Analysis" 15559: 15297:

International Journal of Pure and Applied Mathematics

15114: 14968: 14509:

An Information-Theoretic Approach to Neural Computing

12781: 12759: 12739: 12719: 12693: 12673: 12650: 12627: 12607: 12578: 12558: 12305: 12285: 12193: 11217: 11185: 11159: 10926: 10903: 10850: 10804: 10764: 10758:

The projected data points are the rows of the matrix

10708: 10643: 10548: 10428: 10263: 10197: 10072: 9996: 9936: 9835: 9774: 9726: 9639: 9547: 9510: 9466: 9392: 9325: 9303: 9277: 9271:

The optimality of PCA is also preserved if the noise

9207: 9178: 9152: 9102: 9080: 9041: 9016: 8994: 8969: 8947: 8941:

is the sum of the desired information-bearing signal

8925: 8881: 8760: 8733: 8713: 8663: 8643: 8593: 8543: 8441: 8329: 8270: 8232: 8187: 8146: 8117: 8081: 8041: 8001: 7976: 7946: 7925: 7892: 7855: 7814: 7785: 7765: 7705: 7670: 7646: 7606: 7527: 7495: 7468: 7408: 7373: 7341: 7314: 7255: 7215: 7183: 7156: 7132: 7066: 7026: 6994: 6967: 6906: 6866: 6834: 6807: 6752: 6712: 6680: 6653: 6598: 6563: 6531: 6504: 6449: 6414: 6382: 6355: 6300: 6265: 6238: 6193: 6158: 6131: 6078: 6043: 6016: 5963: 5929: 5896: 5874: 5840: 5816: 5782: 5758: 5723: 5691: 5664: 5616: 5589: 5388: 5226: 5175: 5080: 5045: 4839: 4706: 4553: 4469: 4430: 4332: 4265: 4019: 3942: 3533: 3450:{\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} } 3428: 3045: 2917: 2669: 2474: 2248: 2168: 2106: 2044: 2028:{\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} } 2006: 1980: 1960: 1898: 1755: 1666: 1573: 1487: 1463: 1414: 1191: 1171: 1150: 1129: 1045: 1021: 998: 18712:

Autoregressive conditional heteroskedasticity (ARCH)

16715:

Exploratory Multivariate Analysis by Example Using R

16359:

Liao, J. C.; Boscolo, R.; Yang, Y.-L.; Tran, L. M.;

16298: 16277: 16136: 16102:

Multilinear Analysis of Image Ensembles: TensorFaces

16098: 16078: 14881:(8th ed.). Belmont, CA: Wadsworth, Cengage Learning. 14551:

Proc. ITG Conf. On Systems, Communication and Coding

14505: 13092:

computes principal component analysis, the function

12414:

Linear PCA versus nonlinear Principal Manifolds for

12166: 10518:

Select a subset of the eigenvectors as basis vectors

5553:

covariances are correlations of normalized variables

2900:-th component can be found by subtracting the first 2784:

The quantity to be maximised can be recognised as a

1294:(invented in the last quarter of the 19th century), 16358: 15893: 15868:"Sparse Probabilistic Principal Component Analysis" 15861: 14548: 14535:

Information theory and unsupervised neural networks

11726:actually developed factor analysis in 1904 for his 11553:

Non-linear iterative partial least squares (NIPALS)

9293:is iid and at least more Gaussian (in terms of the 8700:{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'} 8630:{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'} 8580:{\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'} 5348:Efficient algorithms exist to calculate the SVD of 5135:. Comparison with the eigenvector factorization of 18174: 16733:. Chapman & Hall/CRC The R Series London 272 p 16641: 16513: 15840:Baback Moghaddam; Yair Weiss; Shai Avidan (2005). 15665:Hui Zou; Trevor Hastie; Robert Tibshirani (2006). 15440:Theory and Applications of Correspondence Analysis 15437: 15313:"A Guide to IMF Stress Testing Methods and Models" 14320: 14226: 14224: 14115: 13970:Boyd, Stephen; Vandenberghe, Lieven (2004-03-08). 13393:Hsu, Daniel; Kakade, Sham M.; Zhang, Tong (2008). 12787: 12765: 12745: 12725: 12705: 12679: 12656: 12633: 12613: 12593: 12564: 12311: 12291: 12271: 12148:It has been asserted that the relaxed solution of 11770:has been proposed to develop and assess indexes. 11223: 11203: 11171: 11142: 10909: 10862: 10812: 10788: 10741: 10664: 10629: 10505: 10303: 10241: 10108: 10002: 9982: 9883: 9810: 9741: 9699: 9577: 9525: 9496: 9425: 9368:. Equivalently, we are seeking to find the matrix 9333: 9311: 9285: 9260: 9186: 9160: 9135: 9088: 9066: 9024: 9002: 8977: 8955: 8933: 8908: 8816: 8746: 8719: 8699: 8649: 8629: 8579: 8518: 8415: 8278: 8252: 8219:{\displaystyle \mathbf {B} =\mathbf {A} _{q}^{*},} 8218: 8173: 8132: 8103: 8063: 8009: 7987: 7962: 7933: 7907: 7878: 7841: 7800: 7771: 7743: 7683: 7652: 7628: 7545: 7513: 7480: 7438: 7391: 7359: 7326: 7285: 7238: 7201: 7168: 7140: 7101: 7049: 7012: 6979: 6941: 6889: 6852: 6819: 6787: 6735: 6698: 6665: 6633: 6581: 6549: 6516: 6479: 6432: 6400: 6367: 6330: 6283: 6250: 6220: 6176: 6143: 6105: 6061: 6028: 5990: 5941: 5914: 5880: 5852: 5822: 5794: 5764: 5741: 5709: 5676: 5646: 5450: 5318: 5183: 5126: 5060: 5027: 4739: 4594: 4539: 4443: 4406: 4300: 4102: 3999: 3882: 3449: 3278: 3025: 2773: 2642: 2454: 2216: 2154: 2092: 2027: 1986: 1966: 1930: 1881: 1738: 1645: 1559: 1469: 1426: 1203: 1177: 1156: 1135: 1057: 1027: 1004: 15849:Advances in Neural Information Processing Systems 15675:Journal of Computational and Graphical Statistics 15569:Proc. Of Int'l Conf. Machine Learning (ICML 2004) 15160:"Multinomial Analysis for Housing Careers Survey" 15141:Cultures of the Internet: The Internet in Britain 14871: 14595:"Engineering Statistics Handbook Section 6.5.5.2" 14004:A. A. Miranda, Y. A. Le Borgne, and G. Bontempi. 12818: 12536: 9533:representing a single grouped observation of the 5089: 5052: 4995: 4256:eigenvectors, gives the truncated transformation 3232: 3206: 3133: 2925: 2217:{\displaystyle {\mathbf {W} }_{jk}={w_{j}}_{(k)}} 2155:{\displaystyle {\mathbf {X} }_{ij}={x_{j}}_{(i)}} 2093:{\displaystyle {\mathbf {T} }_{ik}={t_{k}}_{(i)}} 19127: 16137:Vasilescu, M.A.O.; Terzopoulos, D. (June 2005). 16082:Multilinear Subspace Analysis of Image Ensembles 14993: 14775:Chemometric Techniques for Quantitative Analysis 14614:A.A. Miranda, Y.-A. Le Borgne, and G. Bontempi. 13969: 13241:(PCA applied to morphometry and computer vision) 11674:. For this, the following results are produced. 10678:as a guide in choosing an appropriate value for 5333:is given by one of the left singular vectors of 5192:are equal to the square-root of the eigenvalues 3174: 2697: 2569: 2503: 2465:Equivalently, writing this in matrix form gives 2365: 2277: 1481:-dimensional vectors of weights or coefficients 18260:Multivariate adaptive regression splines (MARS) 16369:Proceedings of the National Academy of Sciences 16033:Journal of the American Statistical Association 15422: 15327: 14944:International Journal of Behavioral Development 14440:Conference on Applied Statistics in Agriculture 14221: 14024:Introduction to Statistical Pattern Recognition 13830: 13828: 13826: 13824: 13822: 13352:Barnett, T. P. & R. Preisendorfer. (1987). 12934:library have a PCA package in the .mlab module. 11655: 11603:, based on the function evaluating the product 11460:gets close to the first principal component of 9894:In some applications, each variable (column of 5073:and the excess zeros chopped off that satisfies 4678: 4609:A principal components analysis scatterplot of 3415:The full principal components decomposition of 16192:Scientific and Statistical Database Management 16019: 14890: 14744: 13392: 13295:Jolliffe, Ian T.; Cadima, Jorge (2016-04-13). 13294: 13042:can be used for principal component analysis; 12327: 12068: 11741: 11722:(IQ). The pioneering statistical psychologist 11246:In practical implementations, especially with 11241: 10742:{\displaystyle {\frac {g_{L}}{g_{p}}}\geq 0.9} 8428:Before we look at its usage, we first look at 1372:) are used to interpret findings of the PCA. 900:List of datasets for machine-learning research 16815: 16569:Journal of Advances in Modeling Earth Systems 16271: 16130: 16072: 14851: 14433: 14230: 13923:Forkman J., Josse, J., Piepho, H. P. (2019). 12803:Discriminant analysis of principal components 12547: 8864: 7879:{\displaystyle \mathbf {B} =\mathbf {A} _{q}} 7561: 5362:As with the eigen-decomposition, a truncated 1974:is usually selected to be strictly less than 1238:between two datasets while PCA defines a new 933: 16745:University of Copenhagen video by Rasmus Bro 16609:Institute for Digital Research and Education 16175:: CS1 maint: multiple names: authors list ( 15540:"Spectral Relaxation for K-means Clustering" 15463:Le Roux; Brigitte and Henry Rouanet (2004). 14111: 14109: 14107: 13955:: CS1 maint: multiple names: authors list ( 13819: 13586:"Robust PCA With Partial Subspace Knowledge" 11426:, normalizes, and places the result back in 10410:for the data. The cumulative energy content 9420: 9412: 9067:{\displaystyle I(\mathbf {y} ;\mathbf {s} )} 5142:establishes that the right singular vectors 4578: 4554: 4523: 4470: 2377: 2369: 2289: 2281: 1892:in such a way that the individual variables 977:The data is linearly transformed onto a new 16730:Multiple Factor Analysis by Example Using R 16556: 16514:Scher, S.; Jewson, S.; Messori, G. (2021). 16099:Vasilescu, M.A.O.; Terzopoulos, D. (2002). 16079:Vasilescu, M.A.O.; Terzopoulos, D. (2003). 15215:Risk Management and Financial Institutions, 15137: 14877:Kaplan, R.M., & Saccuzzo, D.P. (2010). 14100:A Tutorial on Principal Component Analysis. 13691: 13583: 13149:(can replace of low-rank SVD approximation) 12958:can perform PCA; including robust variants. 11871:PCA is commonly used in problems involving 10890:has a diagonal covariance matrix (that is, 10381:Sort the columns of the eigenvector matrix 9592:Place the row vectors into a single matrix 8104:{\displaystyle x,\mathbf {B} ,\mathbf {A} } 5567:to a PCA based on the covariance matrix of 5559:) a PCA based on the correlation matrix of 4823:In terms of this factorization, the matrix 4455:features (the components of representation 4213: 1282:transform in multivariate quality control, 16860: 16822: 16808: 16766:A Tutorial on Principal Component Analysis 16140:Multilinear Independent Component Analysis 14994:Novembre, John; Stephens, Matthew (2008). 14891:Shevky, Eshref; Williams, Marilyn (1949). 13869: 13526: 13524: 13480: 13478: 13424: 13422: 12073: 11573:non-linear iterative partial least squares 10377:Rearrange the eigenvectors and eigenvalues 9614:Find the empirical mean along each column 9352:The goal is to transform a given data set 8299:, in selecting a subset of variables from 5774:the number of row vectors in the data set 940: 926: 17473: 16794: 16657: 16580: 16539: 16507: 16498: 16449: 16439: 16398: 16388: 16319: 16246: 16199: 16159:"Refined Complexity of PCA with Outliers" 15913: 15824: 15795: 15741: 15687: 15641: 15602: 15364: 15182: 15180: 15083: 15073: 15019: 14812: 14644: 14558: 14447: 14356: 14338: 14294: 14244: 14204: 14186: 14129: 14104: 13940: 13883: 13810: 13745: 13650: 13601: 13544: 13512: 13502: 13442: 13402: 13377: 13328: 12906:mathematics library with support for PCA. 12878:function in the MultivariateStats package 11923:at predefined maturities; and where the 11696:that historically, following the work of 9852: 9408: 9405: 9402: 6119:, one standard deviation for each column 5487: 5061:{\displaystyle \mathbf {\hat {\Sigma }} } 4816:and called the right singular vectors of 4391: 4370: 4072: 4066: 3310:-th principal component of a data vector 3079: 2904: − 1 principal components from 2388: 2300: 16684: 16639: 15235:The Professional Risk Managers’ Handbook 15138:Dutton, William H; Blank, Grant (2013). 14798: 14532: 14429: 14427: 14381:"What are the Pros and cons of the PCA?" 14316: 14314: 14312: 14310: 14308: 14306: 14284: 14282: 14280: 14017: 13834: 13640: 13076:– Proprietary software; for example, see 12753:(or alternatively the number of rows of 12409: 12335: 12170: 12113: 11975:a hypothetical adverse economic scenario 11942:, an optimal portfolio is one where the 11883:. Valuations here depend on the entire 11664: 11286: 5511:). PCA is often used in this manner for 4604: 1374: 1082: 16280:Computer Vision and Image Understanding 15278:. Department of Economics and Finance, 15205: 15203: 15186:See Ch. 9 in Michael B. Miller (2013). 15147:. Oxford Internet Institute. p. 6. 14470: 14434:Jiang, Hong; Eskridge, Kent M. (2000). 14162: 13782: 13645:. Vol. 1. IEEE. pp. 739–746. 13521: 13475: 13419: 13189:Functional principal component analysis 12829: 11847: 11827:Market research and indexes of attitude 11420:algorithm simply calculates the vector 11204:{\displaystyle \operatorname {cov} (X)} 10253:th eigenvalue of the covariance matrix 9345:Computation using the covariance method 1657:to a new vector of principal component 19128: 18786:Kaplan–Meier estimator (product limit) 16756:Stanford University video by Andrew Ng 16634:A User's Guide to Principal Components 16472: 16466: 15502: 15307: 15305: 15177: 15042: 14771: 14725: 14405: 14399: 14080: 13590:IEEE Transactions on Signal Processing 13533:IEEE Transactions on Signal Processing 13431:IEEE Transactions on Signal Processing 12950:– A high performance math library for 12442:expanded on this concept by proposing 11792: 11761: 10825:Derivation using the covariance method 10337:eigenvectors of the covariance matrix 10134:computing eigenvectors and eigenvalues 9712:Calculate the deviations from the mean 9297:) than the information-bearing signal 9172:), but the information-bearing signal 9096:and the dimensionality-reduced output 8988:In particular, Linsker showed that if 8291:relationships between the elements of 8133:{\displaystyle \mathbf {\Sigma } _{y}} 7801:{\displaystyle \mathbf {\Sigma } _{y}} 7126:, and 0 for all other elements ( note 5578:PCA is a popular primary technique in 5281: 5150:are equivalent to the eigenvectors of 5113: 5015: 4967: 4948: 4919: 4895: 4881: 4646:allowed, the greater is the chance of 4350: 4117:is the diagonal matrix of eigenvalues 4081: 4055: 4028: 3991: 3959: 3852: 3767: 3709: 3695: 3631: 3219: 3193: 3017: 2753: 2729: 2715: 2621: 2607: 18859: 18426: 18173: 17472: 17242: 16859: 16803: 15667:"Sparse principal component analysis" 15560:Chris Ding; Xiaofeng He (July 2004). 15157: 14424: 14303: 14277: 14168: 13487:"L1-norm Tucker Tensor Decomposition" 12531: 12392:an alternating maximization framework 12138: 2891: 19096: 18796:Accelerated failure time (AFT) model 16226: 15997: 15783:Journal of Machine Learning Research 15200: 14975:United Nations Development Programme 14861:Journal of Machine Learning Research 13209:L1-norm principal component analysis 12465:. Another popular generalization is 12389:a generalized power method framework 11832:areas with similar characteristics. 11547: 11432:. The eigenvalue is approximated by 8253:{\displaystyle \mathbf {A} _{q}^{*}} 6951:matrix consisting of the set of all 4796:called the left singular vectors of 3485:whitening or sphering transformation 1232:canonical correlation analysis (CCA) 1165:variance of the projected data. The 1070:distance from the points to the line 19108: 18391:Analysis of variance (ANOVA, anova) 17243: 16779:(a video of less than 100 seconds.) 15712: 15302: 15127:: 305–328 – via Researchgate. 14852:Warmuth, M. K.; Kuzmin, D. (2008). 14703:"Face Recognition System-PCA based" 14288: 13034:statistical package, the functions 12526:Robust principal component analysis 12007:spike-triggered covariance analysis 10754:Project the data onto the new basis 10682:. The goal is to choose a value of 9720:Subtract the empirical mean vector 7141:{\displaystyle \mathbf {\Lambda } } 1337:PCA can be thought of as fitting a 895:Glossary of artificial intelligence 13: 18486:Cochran–Mantel–Haenszel statistics 17112:Pearson product-moment correlation 16626: 15604:10.1023/b:mach.0000033113.59016.96 15121:Journal of Marketing in Management 14918:"Socio-Economic Indexes for Areas" 13229:Nonlinear dimensionality reduction 13173:Expectation–maximization algorithm 13018:– Contains PCA in its Pro version. 13001:DBMS_DATA_MINING.SVDS_SCORING_MODE 12436:nonlinear dimensionality reduction 12358: 12279:as a function of component number 12109: 12046:is an important procedure because 11593:multiplying on every iteration by 11234:This is very constructive, as cov( 11060: 11006: 10956: 9756:Store mean-subtracted data in the 5590:Table of symbols and abbreviations 5545:Pearson Product-Moment Correlation 5539:of the approximation of the data. 5521:Nonlinear dimensionality reduction 5352:without having to form the matrix 3573: 3570: 3549: 3546: 3321:can therefore be given as a score 3086: 3083: 3080: 3076: 3073: 3070: 2227: 1931:{\displaystyle t_{1},\dots ,t_{l}} 1831: 1828: 1825: 1400:that transforms the data to a new 1089:multivariate Gaussian distribution 14: 19152: 16737: 14895:. University of California Press. 14118:The Astrophysical Journal Letters 13736:Journal of Educational Psychology 13224:Non-negative matrix factorization 13153:Detrended correspondence analysis 12457:the points onto it. See also the 12181:Non-negative matrix factorization 12167:Non-negative matrix factorization 12100:canonical correspondence analysis 12096:detrended correspondence analysis 11814:published a theoretical paper in 11688:These results are what is called 10881:orthonormal transformation matrix 10348:th eigenvalue corresponds to the 9749:from each row of the data matrix 8847:non-negative matrix factorization 8295:, and they may also be useful in 4451:form an orthogonal basis for the 3893:where the eigenvalue property of 2803:of the matrix, which occurs when 1286:(POD) in mechanical engineering, 19107: 19095: 19083: 19070: 19069: 18860: 16597: 15933:"Principal Graphs and Manifolds" 15808:. CORE Discussion Paper 2008/70. 15299:. Volume 115 No. 1 2017, 153–167 14801:Journal of Computational Biology 13126:Multiple correspondence analysis 12405: 12104:multiple correspondence analysis 12001:'s probability of generating an 11758:frequently in spatial analysis. 11466:within the number of iterations 10806: 10782: 10774: 10766: 10102: 10094: 10089: 10075: 9976: 9965: 9938: 9798: 9792: 9784: 9776: 9742:{\displaystyle \mathbf {u} ^{T}} 9729: 9565: 9550: 9526:{\displaystyle \mathbf {x} _{i}} 9513: 9484: 9469: 9416: 9394: 9327: 9305: 9279: 9248: 9240: 9223: 9215: 9180: 9154: 9129: 9113: 9104: 9082: 9074:between the desired information 9057: 9049: 9018: 8996: 8971: 8949: 8927: 8899: 8891: 8883: 8707:is nonincreasing for increasing 8331: 8272: 8235: 8198: 8189: 8158: 8120: 8097: 8089: 8064:{\displaystyle y=\mathbf {B'} x} 8050: 8003: 7978: 7952: 7927: 7908:{\displaystyle \mathbf {A} _{q}} 7895: 7866: 7857: 7826: 7788: 7737: 7732: 7723: 7708: 7673: 7629:{\displaystyle y=\mathbf {B'} x} 7615: 7571:Some properties of PCA include: 7410: 7257: 7134: 7068: 6908: 6754: 6600: 6451: 6302: 6195: 6080: 5965: 5618: 5438: 5432: 5418: 5406: 5391: 5308: 5303: 5288: 5275: 5269: 5264: 5249: 5244: 5232: 5177: 5120: 5107: 5096: 5086: 5049: 5009: 4992: 4984: 4961: 4955: 4942: 4936: 4913: 4907: 4902: 4889: 4875: 4869: 4857: 4846: 4776:, called the singular values of 4727: 4721: 4716: 4708: 4567: 4558: 4507: 4495: 4480: 4474: 4288: 4282: 4268: 4096: 4088: 4075: 4068: 4062: 4049: 4040: 4035: 4022: 3985: 3979: 3974: 3966: 3953: 3944: 3860: 3835: 3791: 3750: 3722: 3716: 3703: 3678: 3647: 3641: 3609: 3603: 3443: 3438: 3430: 3264: 3253: 3245: 3229: 3203: 3187: 3146: 3130: 3097: 3048: 3000: 2982: 2976: 2941: 2922: 2760: 2747: 2739: 2736: 2723: 2709: 2672: 2631: 2628: 2615: 2601: 2578: 2541: 2538: 2512: 2477: 2432: 2412: 2373: 2285: 2251: 2172: 2110: 2048: 2021: 2016: 2008: 1807: 1786: 1669: 1576: 1490: 18745:Least-squares spectral analysis 16605:"Principal Components Analysis" 16415: 16352: 16183: 16150: 16052: 16013: 15991: 15942: 15922: 15887: 15855: 15833: 15812: 15766: 15658: 15625: 15575: 15553: 15531: 15496: 15483: 15456: 15431: 15416: 15373: 15330:Journal of Neuroscience Methods 15321: 15285: 15266: 15251: 15227: 15151: 15131: 15108: 15036: 14987: 14962: 14935: 14922:Australian Bureau of Statistics 14910: 14899: 14884: 14845: 14792: 14765: 14738: 14713: 14695: 14683: 14669: 14621: 14608: 14587: 14575: 14542: 14526: 14499: 14464: 14373: 14092: 14041: 14011: 13998: 13963: 13916: 13863: 13776: 13727: 12434:Most of the modern methods for 12398:Bayesian formulation framework. 11988: 11768:confirmatory composite analysis 11712: 11707: 11536:. Implemented, for example, in 10605: 10578: 10476: 10329:column vectors, each of length 10283: 10224: 10036:to calculate the covariance is 9854: 9853: 8637:will tend to become smaller as 1946:, with each coefficient vector 1857: 1835: 1823: 1284:proper orthogonal decomposition 988:of a collection of points in a 966:technique with applications in 17726:Mean-unbiased minimum-variance 16829: 16363:; Roychowdhury, V. P. (2003). 16046:10.1080/01621459.1989.10478797 15894:Hui Zou; Lingzhou Xue (2018). 14677:"SAS/STAT(R) 9.3 User's Guide" 13976:. Cambridge University Press. 13685: 13634: 13584:Zhan, J.; Vaswani, N. (2015). 13577: 13386: 13345: 13288: 13251:Principal component regression 13199:Independent component analysis 13158:Directional component analysis 12988:gives the principal component. 12824:Directional component analysis 12819:Directional component analysis 12621:are constrained to be 0. Here 12542:Independent component analysis 12537:Independent component analysis 11198: 11192: 11166: 11160: 11120: 11114: 11082: 11066: 11044: 11012: 10993: 10984: 10974: 10962: 10946: 10937: 10857: 10851: 10694:so that the cumulative energy 10665:{\displaystyle 1\leq L\leq p.} 9829:column vector of all 1s: 9252: 9236: 9227: 9211: 9061: 9045: 8919:that is, that the data vector 8827: 8461: 8448: 8168: 8153: 7982: 7963:{\displaystyle (\mathbf {B'} } 7947: 7836: 7821: 7433: 7417: 7280: 7264: 7096: 7075: 6936: 6915: 6782: 6761: 6628: 6607: 6474: 6458: 6325: 6309: 6215: 6202: 6100: 6087: 5985: 5972: 5641: 5625: 5571:, the standardized version of 5496:) are the square roots of the 4653:principal component regression 3871: 3865: 3846: 3840: 3828: 3822: 3802: 3796: 3784: 3778: 3761: 3755: 3733: 3727: 3689: 3683: 3663: 3658: 3652: 3637: 3626: 3620: 3614: 3599: 3589: 3584: 3578: 3560: 3554: 3541: 3498: 3495:in PCA or in Factor analysis. 3151: 3122: 3101: 3093: 3059: 3053: 3011: 3005: 2993: 2987: 2683: 2677: 2582: 2574: 2545: 2534: 2516: 2508: 2488: 2482: 2423: 2417: 2340: 2334: 2330: 2316: 2262: 2256: 2209: 2203: 2147: 2141: 2085: 2079: 1818: 1812: 1797: 1791: 1776: 1770: 1731: 1725: 1721: 1688: 1680: 1674: 1638: 1632: 1628: 1595: 1587: 1581: 1552: 1546: 1542: 1509: 1501: 1495: 1319:empirical orthogonal functions 315:Relevance vector machine (RVM) 1: 19039:Geographic information system 18255:Simultaneous equations models 16010:(free for non-commercial use) 15966:10.1016/S0140-6736(05)17947-1 15342:10.1016/S0165-0270(99)00130-2 14506:Deco & Obradovic (1996). 14086:Leznik, M; Tofallis, C. 2005 13282: 13132:Factor analysis of mixed data 12497: 12474:multilinear subspace learning 12363: 12005:. This technique is known as 11512:with block-vectors, matrices 11470:, which is small relative to 10897:A quick computation assuming 10798:That is, the first column of 7988:{\displaystyle \mathbf {B} )} 7684:{\displaystyle \mathbf {B'} } 7566: 7439:{\displaystyle \mathbf {T} =} 7286:{\displaystyle \mathbf {W} =} 7114:consisting of the set of all 7102:{\displaystyle \mathbf {D} =} 6959:, one eigenvector per column 6942:{\displaystyle \mathbf {V} =} 6788:{\displaystyle \mathbf {R} =} 6634:{\displaystyle \mathbf {C} =} 6480:{\displaystyle \mathbf {Z} =} 6343:from the mean of each column 6331:{\displaystyle \mathbf {B} =} 6221:{\displaystyle \mathbf {h} =} 6106:{\displaystyle \mathbf {s} =} 5991:{\displaystyle \mathbf {u} =} 5915:{\displaystyle 1\leq L\leq p} 5647:{\displaystyle \mathbf {X} =} 3927:components over the dataset. 2834:can then be given as a score 804:Computational learning theory 368:Expectation–maximization (EM) 18222:Coefficient of determination 17833:Uniformly most powerful test 16688:Principal Component Analysis 16644:Principal Component Analysis 16338:10.1016/j.cosrev.2016.11.001 16210:10.1007/978-3-540-69497-7_27 16023:; Stuetzle, W. (June 1989). 14759:10.1016/0003-2670(86)80028-9 14408:Applied Predictive Analytics 13838:Principal Component Analysis 13261:Singular value decomposition 13245:Principal component analysis 13048:singular value decomposition 13003:by specifying setting value 12733:is the number of columns of 11656:Online/sequential estimation 11405:exit if error < tolerance 11305:= a random vector of length 10813:{\displaystyle \mathbf {T} } 10363:eigenvectors (as opposed to 9610:Calculate the empirical mean 9334:{\displaystyle \mathbf {n} } 9312:{\displaystyle \mathbf {s} } 9286:{\displaystyle \mathbf {n} } 9187:{\displaystyle \mathbf {s} } 9161:{\displaystyle \mathbf {n} } 9089:{\displaystyle \mathbf {s} } 9025:{\displaystyle \mathbf {n} } 9003:{\displaystyle \mathbf {s} } 8978:{\displaystyle \mathbf {n} } 8956:{\displaystyle \mathbf {s} } 8934:{\displaystyle \mathbf {x} } 8303:, and in outlier detection. 8279:{\displaystyle \mathbf {A} } 8010:{\displaystyle \mathbf {A} } 7934:{\displaystyle \mathbf {A} } 7239:{\displaystyle j'=1\ldots p} 7050:{\displaystyle j'=1\ldots p} 6890:{\displaystyle j'=1\ldots p} 6736:{\displaystyle j'=1\ldots p} 5584:linear discriminant analysis 5184:{\displaystyle \mathbf {X} } 5157:, while the singular values 4691:singular value decomposition 4685:Singular value decomposition 4679:Singular value decomposition 4244:variables to a new space of 2790:positive semidefinite matrix 1332: 1325:in noise and vibration, and 1311:Principal Component Analysis 1288:singular value decomposition 1258:PCA was invented in 1901 by 1240:orthogonal coordinate system 1224:singular value decomposition 953:Principal component analysis 761:Coefficient of determination 608:Convolutional neural network 320:Support vector machine (SVM) 7: 18791:Proportional hazards models 18735:Spectral density estimation 18717:Vector autoregression (VAR) 18151:Maximum posterior estimator 17383:Randomized controlled trial 15438:Greenacre, Michael (1983). 15317:International Monetary Fund 14969:Human Development Reports. 14206:10.3847/0004-637X/824/2/117 14148:10.1088/2041-8205/755/2/L28 13514:10.1109/ACCESS.2019.2955134 13178:Exploratory factor analysis 13128:(for qualitative variables) 13112: 12463:principal geodesic analysis 12346:iconography of correlations 12328:Iconography of correlations 12187:eigenvalue plots, that is, 12102:. One special extension is 12069:Relation with other methods 11969:PCA may also be applied to 11931:(and as the components are 11891:, VaR, applying PCA to the 11742:Residential differentiation 11555:is a variant the classical 11266:, instead utilizing one of 11242:Covariance-free computation 10025:The reasoning behind using 9360:to an alternative data set 9295:Kullback–Leibler divergence 8747:{\displaystyle \alpha _{k}} 8315:(Spectral decomposition of 7546:{\displaystyle l=1\ldots L} 7514:{\displaystyle i=1\ldots n} 7392:{\displaystyle l=1\ldots L} 7360:{\displaystyle j=1\ldots p} 7299:, and where the vectors in 7202:{\displaystyle j=1\ldots p} 7013:{\displaystyle j=1\ldots p} 6853:{\displaystyle j=1\ldots p} 6699:{\displaystyle j=1\ldots p} 6582:{\displaystyle j=1\ldots p} 6550:{\displaystyle i=1\ldots n} 6433:{\displaystyle j=1\ldots p} 6401:{\displaystyle i=1\ldots n} 6284:{\displaystyle i=1\ldots n} 6177:{\displaystyle j=1\ldots p} 6062:{\displaystyle j=1\ldots p} 6004:, one mean for each column 5742:{\displaystyle j=1\ldots p} 5710:{\displaystyle i=1\ldots n} 5461:The truncation of a matrix 4763:rectangular diagonal matrix 4642:, the larger the number of 1994:to reduce dimensionality). 1118: 912:Outline of machine learning 809:Empirical risk minimization 10: 19157: 18551:Multivariate distributions 16971:Average absolute deviation 16292:10.1016/j.cviu.2013.11.009 15915:10.1109/JPROC.2018.2846588 15851:. Vol. 18. MIT Press. 15442:. London: Academic Press. 15075:10.1038/s41598-022-14395-4 14512:. New York, NY: Springer. 13942:10.1007/s13253-019-00355-5 13256:Singular spectrum analysis 13163:Dynamic mode decomposition 12548:Network component analysis 12367: 11860:, and has been applied to 11448:for the covariance matrix 9907:Find the covariance matrix 9585:as row vectors, each with 9380:transform (KLT) of matrix 8872:Under the assumption that 8865:PCA and information theory 8727:, whereas the elements of 8032:orthonormal transformation 7595:, consider the orthogonal 7562:Properties and limitations 7303:are a sub-set of those in 4682: 4240:from an original space of 3419:can therefore be given as 2788:. A standard result for a 1384: 1253: 549:Feedforward neural network 300:Artificial neural networks 19065: 19019: 18956: 18909: 18872: 18868: 18855: 18827: 18809: 18776: 18767: 18725: 18672: 18633: 18582: 18573: 18539:Structural equation model 18494: 18451: 18447: 18422: 18381: 18347: 18301: 18268: 18230: 18197: 18193: 18169: 18109: 18018: 17937: 17901: 17892: 17875:Score/Lagrange multiplier 17860: 17813: 17758: 17684: 17675: 17485: 17481: 17468: 17427: 17401: 17353: 17308: 17290:Sample size determination 17255: 17251: 17238: 17142: 17097: 17071: 17053: 17009: 16961: 16881: 16872: 16868: 16855: 16837: 14971:"Human Development Index" 14956:10.1177/01650254221117506 14722:Mathematica documentation 14406:Abbott, Dean (May 2014). 14327:The Astrophysical Journal 14175:The Astrophysical Journal 13713:10.1080/14786440109462720 11927:of each component is its 11889:calculating value at risk 11881:interest rate derivatives 11858:financial risk management 10183:will take the form of an 10166:Interactive Data Language 7849:, is maximized by taking 7481:{\displaystyle n\times L} 7327:{\displaystyle p\times L} 7169:{\displaystyle p\times p} 6980:{\displaystyle p\times p} 6820:{\displaystyle p\times p} 6666:{\displaystyle p\times p} 6517:{\displaystyle n\times p} 6368:{\displaystyle n\times p} 6251:{\displaystyle 1\times n} 6144:{\displaystyle p\times 1} 6029:{\displaystyle p\times 1} 5942:{\displaystyle 1\times 1} 5853:{\displaystyle 1\times 1} 5795:{\displaystyle 1\times 1} 5677:{\displaystyle n\times p} 5517:discrete cosine transform 1567:that map each row vector 1427:{\displaystyle n\times p} 968:exploratory data analysis 532:Artificial neural network 19034:Environmental statistics 18556:Elliptical distributions 18349:Generalized linear model 18278:Simple linear regression 18048:Hodges–Lehmann estimator 17505:Probability distribution 17414:Stochastic approximation 16976:Coefficient of variation 16795:Software implementations 16685:Jolliffe, I. T. (2002). 16640:Jolliffe, I. T. (1986). 16114:10.1007/3-540-47969-4_30 15698:10.1198/106186006x113430 15423:Benzécri, J.-P. (1973). 15280:University of Canterbury 14358:10.3847/1538-4357/aaa1f2 14233:The Astronomical Journal 13982:10.1017/cbo9780511804441 13835:Jolliffe, I. T. (2002). 13620:10.1109/tsp.2015.2421485 13563:10.1109/TSP.2014.2338077 13461:10.1109/TSP.2017.2708023 13239:Point distribution model 13147:CUR matrix approximation 13122:(for contingency tables) 12874:– Supports PCA with the 12795:must have full row rank. 12476:, PCA is generalized to 12024:spike-triggered ensemble 8140:defined as before. Then 5513:dimensionality reduction 5492:The singular values (in 5480:, a result known as the 4624:dimensionality reduction 4214:Dimensionality reduction 3483:is sometimes called the 1442:, with column-wise zero 1329:in structural dynamics. 1327:empirical modal analysis 1296:eigenvalue decomposition 1262:, as an analogue of the 964:dimensionality reduction 841:Journals and conferences 788:Mathematical foundations 698:Temporal difference (TD) 554:Recurrent neural network 474:Conditional random field 397:Dimensionality reduction 145:Dimensionality reduction 107:Quantum machine learning 102:Neuromorphic engineering 62:Self-supervised learning 57:Semi-supervised learning 18694:Cross-correlation (XCF) 18302:Non-standard predictors 17736:Lehmann–Scheffé theorem 17409:Adaptive clinical trial 16541:10.1175/WAF-D-20-0219.1 16520:Weather and Forecasting 16441:10.1186/1471-2156-11-94 16390:10.1073/pnas.2136632100 16308:Computer Science Review 16257:10.1145/1970392.1970395 15901:Proceedings of the IEEE 15505:Journal of Chemometrics 15427:. Paris, France: Dunod. 15291:Giorgia Pasini (2017); 14778:. New York: CRC Press. 14533:Plumbley, Mark (1991). 14169:Pueyo, Laurent (2016). 13783:Stewart, G. W. (1993). 13743:, 417–441, and 498–520. 13194:Geometric data analysis 13120:Correspondence analysis 12383:a regression framework, 12079:Correspondence analysis 12074:Correspondence analysis 12032:prior stimulus ensemble 11979:macroeconomic variables 11783:Human Development Index 11211:were diagonalisable by 11172:{\displaystyle (\ast )} 10863:{\displaystyle (\ast )} 10191:diagonal matrix, where 8181:is minimized by taking 250:Apprenticeship learning 16:Method of data analysis 19090:Mathematics portal 18911:Engineering statistics 18819:Nelson–Aalen estimator 18396:Analysis of covariance 18283:Ordinary least squares 18207:Pearson product-moment 17611:Statistical functional 17522:Empirical distribution 17355:Controlled experiments 17084:Frequency distribution 16862:Descriptive statistics 16632:Jackson, J.E. (1991). 15517:10.1002/cem.1180050305 15493:. Guilford Press, 2006 15382:Biological Cybernetics 14747:Analytica Chimica Acta 14581:See also the tutorial 14449:10.4148/2475-7772.1247 13701:Philosophical Magazine 13358:Monthly Weather Review 13321:10.1098/rsta.2015.0202 13276:Weighted least squares 13214:Low-rank approximation 13138:qualitative variables) 12999:12c – Implemented via 12789: 12767: 12747: 12727: 12707: 12681: 12658: 12635: 12615: 12595: 12566: 12513:correlation clustering 12431: 12341: 12313: 12293: 12273: 12258: 12220: 12177: 12120: 12081:(CA) was developed by 11893:Monte Carlo simulation 11866:portfolio optimization 11775:City Development Index 11377:// λ is the eigenvalue 11225: 11205: 11173: 11144: 10911: 10864: 10814: 10790: 10743: 10666: 10631: 10507: 10462: 10385:and eigenvalue matrix 10359:denotes the matrix of 10333:, which represent the 10305: 10243: 10110: 10062:the covariance matrix 10004: 9984: 9885: 9812: 9743: 9701: 9683: 9579: 9527: 9498: 9427: 9335: 9313: 9287: 9262: 9198:, which is defined as 9188: 9162: 9137: 9090: 9068: 9026: 9004: 8979: 8957: 8935: 8910: 8818: 8748: 8721: 8701: 8651: 8631: 8581: 8520: 8487: 8417: 8280: 8254: 8220: 8175: 8134: 8105: 8065: 8011: 7989: 7964: 7935: 7915:consists of the first 7909: 7880: 7843: 7802: 7773: 7745: 7685: 7654: 7630: 7547: 7515: 7482: 7440: 7393: 7361: 7328: 7287: 7240: 7203: 7170: 7142: 7103: 7051: 7014: 6981: 6943: 6891: 6854: 6821: 6789: 6737: 6700: 6667: 6635: 6583: 6551: 6518: 6481: 6434: 6402: 6369: 6332: 6285: 6252: 6222: 6178: 6145: 6107: 6063: 6030: 5992: 5943: 5916: 5882: 5854: 5824: 5796: 5766: 5743: 5711: 5678: 5648: 5488:Further considerations 5452: 5320: 5185: 5128: 5062: 5029: 4741: 4619: 4596: 4541: 4445: 4408: 4302: 4104: 4001: 3884: 3520:The sample covariance 3451: 3280: 3027: 2974: 2775: 2644: 2456: 2218: 2156: 2094: 2029: 1988: 1968: 1932: 1883: 1740: 1647: 1561: 1471: 1428: 1381: 1323:spectral decomposition 1264:principal axis theorem 1205: 1179: 1158: 1137: 1100: 1059: 1029: 1006: 799:Bias–variance tradeoff 681:Reinforcement learning 657:Spiking neural network 67:Reinforcement learning 19136:Matrix decompositions 19006:Population statistics 18948:System identification 18682:Autocorrelation (ACF) 18610:Exponential smoothing 18524:Discriminant analysis 18519:Canonical correlation 18383:Partition of variance 18245:Regression validation 18089:(Jonckheere–Terpstra) 17988:Likelihood-ratio test 17677:Frequentist inference 17589:Location–scale family 17510:Sampling distribution 17475:Statistical inference 17442:Cross-sectional study 17429:Observational studies 17388:Randomized experiment 17217:Stem-and-leaf display 17019:Central limit theorem 16793:See also the list of 16727:Pagès Jérôme (2014). 16500:10.3390/atmos11040354 15469:. Dordrecht: Kluwer. 15258:example decomposition 15190:, 2nd Edition. Wiley 14823:10.1089/cmb.2008.0221 13894:10.1109/TPAMI.2013.50 13661:10.1109/CVPR.2005.309 13142:Canonical correlation 12852:clustering algorithms 12790: 12768: 12748: 12728: 12708: 12682: 12659: 12636: 12616: 12596: 12567: 12413: 12339: 12314: 12294: 12274: 12238: 12200: 12174: 12117: 12091:chi-squared statistic 11952:allocation strategies 11897:in turn option values 11839:Another example from 11720:Intelligence Quotient 11665:Qualitative variables 11561:partial least squares 11287:Iterative computation 11248:high dimensional data 11226: 11206: 11179:holds if and only if 11174: 11145: 10917:were unitary yields: 10912: 10865: 10815: 10791: 10744: 10667: 10632: 10508: 10442: 10306: 10244: 10111: 10005: 9985: 9886: 9813: 9744: 9702: 9663: 9580: 9528: 9499: 9437:Organize the data set 9428: 9364:of smaller dimension 9336: 9314: 9288: 9263: 9189: 9163: 9138: 9091: 9069: 9027: 9005: 8980: 8958: 8936: 8911: 8819: 8749: 8722: 8702: 8652: 8632: 8582: 8521: 8467: 8418: 8281: 8260:consists of the last 8255: 8221: 8176: 8135: 8106: 8066: 8012: 7990: 7965: 7936: 7910: 7881: 7844: 7803: 7774: 7746: 7686: 7655: 7631: 7597:linear transformation 7548: 7516: 7483: 7448:matrix consisting of 7441: 7394: 7362: 7329: 7288: 7241: 7204: 7171: 7143: 7104: 7052: 7015: 6982: 6944: 6892: 6855: 6822: 6790: 6738: 6701: 6668: 6636: 6584: 6552: 6519: 6482: 6435: 6403: 6370: 6333: 6286: 6253: 6223: 6179: 6146: 6108: 6064: 6031: 5993: 5944: 5917: 5883: 5855: 5825: 5797: 5767: 5744: 5712: 5679: 5649: 5557:Z- or standard-scores 5453: 5321: 5186: 5129: 5063: 5030: 4742: 4669:signal-to-noise ratio 4644:explanatory variables 4608: 4597: 4542: 4446: 4444:{\displaystyle W_{L}} 4414:where the columns of 4409: 4303: 4105: 4002: 3885: 3452: 3281: 3028: 2948: 2809:is the corresponding 2776: 2645: 2457: 2219: 2157: 2095: 2030: 1989: 1969: 1933: 1884: 1741: 1648: 1562: 1472: 1429: 1394:linear transformation 1389:PCA is defined as an 1378: 1206: 1180: 1159: 1138: 1086: 1078:linearly uncorrelated 1060: 1030: 1007: 990:real coordinate space 635:Neural radiance field 457:Structured prediction 180:Structured prediction 52:Unsupervised learning 18929:Probabilistic design 18514:Principal components 18357:Exponential families 18309:Nonlinear regression 18288:General linear model 18250:Mixed effects models 18240:Errors and residuals 18217:Confounding variable 18119:Bayesian probability 18097:Van der Waerden test 18087:Ordered alternative 17852:Multiple comparisons 17731:Rao–Blackwellization 17694:Estimating equations 17650:Statistical distance 17368:Factorial experiment 16901:Arithmetic-Geometric 16582:10.1029/2022MS003038 15217:5th Edition. Wiley. 14735:The MIT Press, 1998. 14720:Eigenvalues function 14692:Matlab documentation 13219:Matrix decomposition 12830:Software/source code 12779: 12757: 12737: 12717: 12691: 12671: 12664:has full column rank 12648: 12625: 12605: 12594:{\displaystyle E=AP} 12576: 12556: 12488:Tucker decomposition 12379:proposed, including 12303: 12283: 12191: 12089:. CA decomposes the 11854:quantitative finance 11848:Quantitative finance 11474:, at the total cost 11331:(a vector of length 11215: 11183: 11157: 10924: 10901: 10848: 10802: 10762: 10706: 10641: 10546: 10426: 10317:, also of dimension 10261: 10195: 10070: 9994: 9934: 9833: 9772: 9724: 9637: 9618:= 1, ..., 9545: 9508: 9464: 9390: 9323: 9301: 9275: 9205: 9176: 9150: 9100: 9078: 9039: 9014: 8992: 8967: 8945: 8923: 8879: 8758: 8731: 8711: 8661: 8641: 8591: 8541: 8439: 8327: 8268: 8230: 8185: 8144: 8115: 8079: 8039: 8017:is not defined here) 7999: 7974: 7970:is the transpose of 7944: 7923: 7890: 7853: 7812: 7783: 7779:. Then the trace of 7763: 7703: 7668: 7644: 7604: 7525: 7493: 7466: 7406: 7371: 7339: 7312: 7253: 7213: 7181: 7154: 7130: 7064: 7024: 6992: 6965: 6904: 6864: 6832: 6805: 6750: 6710: 6678: 6651: 6596: 6561: 6529: 6502: 6447: 6412: 6380: 6353: 6298: 6263: 6236: 6191: 6156: 6129: 6115:vector of empirical 6076: 6041: 6014: 6000:vector of empirical 5961: 5927: 5894: 5872: 5838: 5814: 5780: 5756: 5721: 5689: 5662: 5614: 5482:Eckart–Young theorem 5386: 5224: 5173: 5078: 5043: 4837: 4765:of positive numbers 4704: 4673:parametric bootstrap 4551: 4467: 4428: 4330: 4263: 4017: 3940: 3531: 3426: 3043: 2915: 2667: 2472: 2246: 2239:thus has to satisfy 2166: 2104: 2042: 2004: 1978: 1958: 1950:constrained to be a 1896: 1753: 1664: 1571: 1485: 1461: 1412: 1315:Eckart–Young theorem 1189: 1169: 1148: 1127: 1043: 1019: 996: 986:principal components 970:, visualization and 824:Statistical learning 722:Learning with humans 514:Local outlier factor 19141:Dimension reduction 19001:Official statistics 18924:Methods engineering 18605:Seasonal adjustment 18373:Poisson regressions 18293:Bayesian regression 18232:Regression analysis 18212:Partial correlation 18184:Regression analysis 17783:Prediction interval 17778:Likelihood interval 17768:Confidence interval 17760:Interval estimation 17721:Unbiased estimators 17539:Model specification 17419:Up-and-down designs 17107:Partial correlation 17063:Index of dispersion 16981:Interquartile range 16532:2021WtFor..36.1357S 16491:2020Atmos..11..354J 16473:Jewson, S. (2020). 16381:2003PNAS..10015522L 16375:(26): 15522–15527. 16330:2015arXiv151101245B 15806:2008arXiv0811.4724J 15652:2014arXiv1410.6801C 15158:Flood, Joe (2008). 15066:2022NatSR..1214683E 14772:Kramer, R. (1998). 14569:2012arXiv1205.6935G 14387:. September 1, 2019 14349:2018ApJ...852..104R 14255:2007AJ....133..734B 14197:2016ApJ...824..117P 14140:2012ApJ...755L..28S 14054:Integrative Biology 14019:Fukunaga, Keinosuke 13973:Convex Optimization 13612:2015ITSP...63.3332Z 13555:2014ITSP...62.5046M 13453:2017ITSP...65.4252M 13413:2008arXiv0811.4413H 13370:1987MWRv..115.1825B 13313:2016RSPTA.37450202J 12864:command or via the 12813:adegenet on the web 12706:{\displaystyle L-1} 12687:must have at least 12052:clustering analysis 11793:Population genetics 11762:Development indexes 11486:matrix-free methods 11442:on the unit vector 11268:matrix-free methods 10050:Compute the matrix 10038:Bessel's correction 10012:conjugate transpose 9341:becomes dependent. 9127: 8963:and a noise signal 8773: 8696: 8626: 8576: 8537:into contributions 8515: 8412: 8370: 8249: 8212: 8030:Consider again the 6496:of the data matrix 6347:of the data matrix 6123:of the data matrix 6117:standard deviations 6008:of the data matrix 5580:pattern recognition 5339:polar decomposition 4640:regression analysis 4591: 4536: 4521: 4355: 4229:maps a data vector 4218:The transformation 3857: 3772: 3700: 3479:. The transpose of 3224: 3022: 2349: 1398:inner product space 1317:(Harman, 1960), or 1305:in linear algebra, 1274:transform (KLT) in 1204:{\displaystyle i-1} 1113:atmospheric science 1105:population genetics 1058:{\displaystyle i-1} 667:Electrochemical RAM 574:reservoir computing 305:Logistic regression 224:Supervised learning 210:Multimodal learning 185:Feature engineering 130:Generative modeling 92:Rule-based learning 87:Curriculum learning 47:Supervised learning 22:Part of a series on 19021:Spatial statistics 18901:Medical statistics 18801:First hitting time 18755:Whittle likelihood 18406:Degrees of freedom 18401:Multivariate ANOVA 18334:Heteroscedasticity 18146:Bayesian estimator 18111:Bayesian inference 17960:Kolmogorov–Smirnov 17845:Randomization test 17815:Testing hypotheses 17788:Tolerance interval 17699:Maximum likelihood 17594:Exponential family 17527:Density estimation 17487:Statistical theory 17447:Natural experiment 17393:Scientific control 17310:Survey methodology 16996:Standard deviation 16235:Journal of the ACM 16025:"Principal Curves" 15931:, A. Y. Zinovyev, 15489:Timothy A. Brown. 15394:10.1007/bf00198909 15053:Scientific Reports 14066:10.1039/C6IB00100A 13307:(2065): 20150202. 13134:(for quantitative 12785: 12763: 12743: 12723: 12703: 12677: 12654: 12631: 12611: 12591: 12562: 12532:Similar techniques 12432: 12342: 12309: 12289: 12269: 12178: 12121: 12087:contingency tables 12083:Jean-Paul Benzécri 11905:interest rate risk 11821:circular reasoning 11816:Scientific Reports 11802:migration events. 11781:The country-level 11524:. Every column of 11221: 11201: 11169: 11140: 11138: 10907: 10860: 10810: 10786: 10739: 10662: 10627: 10503: 10301: 10239: 10106: 10000: 9980: 9881: 9808: 9739: 9697: 9575: 9523: 9494: 9423: 9331: 9309: 9283: 9258: 9184: 9158: 9133: 9111: 9086: 9064: 9034:mutual information 9022: 9000: 8975: 8953: 8931: 8906: 8814: 8761: 8744: 8717: 8697: 8684: 8647: 8627: 8614: 8577: 8564: 8516: 8498: 8413: 8400: 8358: 8276: 8250: 8233: 8216: 8196: 8171: 8130: 8101: 8061: 8007: 7985: 7960: 7931: 7905: 7876: 7839: 7798: 7769: 7741: 7699:) matrix, and let 7681: 7650: 7626: 7543: 7511: 7478: 7436: 7389: 7357: 7324: 7283: 7236: 7199: 7166: 7138: 7124:principal diagonal 7099: 7047: 7010: 6977: 6939: 6887: 6850: 6817: 6798:correlation matrix 6785: 6733: 6696: 6663: 6631: 6579: 6547: 6514: 6477: 6430: 6398: 6365: 6328: 6281: 6248: 6230:vector of all 1's 6218: 6174: 6141: 6103: 6059: 6026: 5988: 5939: 5912: 5878: 5850: 5820: 5792: 5762: 5739: 5707: 5674: 5644: 5448: 5329:so each column of 5316: 5314: 5181: 5124: 5058: 5025: 5023: 4737: 4620: 4592: 4577: 4537: 4522: 4505: 4441: 4404: 4339: 4298: 4100: 3997: 3880: 3878: 3833: 3748: 3676: 3447: 3405:th eigenvector of 3276: 3270: 3199: 3112: 3023: 2998: 2892:Further components 2771: 2640: 2593: 2527: 2452: 2403: 2387: 2329: 2315: 2299: 2214: 2152: 2090: 2025: 1984: 1964: 1928: 1879: 1736: 1643: 1557: 1467: 1424: 1382: 1370:explained variance 1201: 1175: 1154: 1133: 1101: 1055: 1025: 1002: 992:are a sequence of 972:data preprocessing 235: • 150:Density estimation 19123: 19122: 19061: 19060: 19057: 19056: 18996:National accounts 18966:Actuarial science 18958:Social statistics 18851: 18850: 18847: 18846: 18843: 18842: 18778:Survival function 18763: 18762: 18625:Granger causality 18466:Contingency table 18441:Survival analysis 18418: 18417: 18414: 18413: 18270:Linear regression 18165: 18164: 18161: 18160: 18136:Credible interval 18105: 18104: 17888: 17887: 17704:Method of moments 17573:Parametric family 17534:Statistical model 17464: 17463: 17460: 17459: 17378:Random assignment 17300:Statistical power 17234: 17233: 17230: 17229: 17079:Contingency table 17049: 17048: 16916:Generalized/power 16723:978-2-7535-0938-2 16706:978-0-387-95442-4 16677:978-0-387-95442-4 16219:978-3-540-69476-2 16123:978-3-540-43745-1 16068:978-3-540-73749-0 15960:(9460): 671–679. 15752:10.1137/050645506 15449:978-0-12-299050-2 15196:978-1-118-75029-2 14807:(11): 1593–1599. 14707:www.mathworks.com 14098:Jonathon Shlens, 14060:(11): 1183–1193. 14034:978-0-12-269851-4 13991:978-0-521-83378-3 13856:978-0-387-95442-4 13670:978-0-7695-2372-9 13596:(13): 3332–3347. 13539:(19): 5046–5058. 13497:: 178454–178465. 13437:(16): 4252–4264. 13010:Orange (software) 12788:{\displaystyle P} 12766:{\displaystyle P} 12746:{\displaystyle A} 12726:{\displaystyle L} 12680:{\displaystyle A} 12657:{\displaystyle A} 12634:{\displaystyle P} 12614:{\displaystyle A} 12565:{\displaystyle E} 12312:{\displaystyle n} 12299:given a total of 12292:{\displaystyle k} 12155:-means clustering 12144:-means clustering 12063:phase transitions 12020:covariance matrix 11917:covariance matrix 11909:partial durations 11856:, PCA is used in 11748:factorial ecology 11728:two-factor theory 11548:The NIPALS method 11490:Lanczos algorithm 11440:Rayleigh quotient 11224:{\displaystyle P} 11023: 10973: 10910:{\displaystyle P} 10731: 10582: 10480: 10287: 10228: 10140:systems, such as 10003:{\displaystyle *} 9961: 9921:covariance matrix 9858: 9661: 8720:{\displaystyle k} 8650:{\displaystyle k} 8535:covariance matrix 7772:{\displaystyle y} 7653:{\displaystyle y} 7556: 7555: 6644:covariance matrix 5881:{\displaystyle L} 5823:{\displaystyle p} 5765:{\displaystyle n} 5537:mean square error 5092: 5055: 4998: 4311:where the matrix 3511:covariance matrix 3269: 3235: 3209: 3136: 3067: 2928: 2786:Rayleigh quotient 2765: 2568: 2502: 2394: 2364: 2306: 2276: 1987:{\displaystyle p} 1967:{\displaystyle l} 1470:{\displaystyle l} 1402:coordinate system 1351:covariance matrix 1276:signal processing 1220:covariance matrix 1178:{\displaystyle i} 1157:{\displaystyle p} 1136:{\displaystyle p} 1097:covariance matrix 1074:orthonormal basis 1028:{\displaystyle i} 1005:{\displaystyle p} 979:coordinate system 950: 949: 755:Model diagnostics 738:Human-in-the-loop 581:Boltzmann machine 494:Anomaly detection 290:Linear regression 205:Ontology learning 200:Grammar induction 175:Semantic analysis 170:Association rules 155:Anomaly detection 97:Neuro-symbolic AI 19148: 19111: 19110: 19099: 19098: 19088: 19087: 19073: 19072: 18976:Crime statistics 18870: 18869: 18857: 18856: 18774: 18773: 18740:Fourier analysis 18727:Frequency domain 18707: 18654: 18620:Structural break 18580: 18579: 18529:Cluster analysis 18476:Log-linear model 18449: 18448: 18424: 18423: 18365: 18339:Homoscedasticity 18195: 18194: 18171: 18170: 18090: 18082: 18074: 18073:(Kruskal–Wallis) 18058: 18043: 17998:Cross validation 17983: 17965:Anderson–Darling 17912: 17899: 17898: 17870:Likelihood-ratio 17862:Parametric tests 17840:Permutation test 17823:1- & 2-tails 17714:Minimum distance 17686:Point estimation 17682: 17681: 17633:Optimal decision 17584: 17483: 17482: 17470: 17469: 17452:Quasi-experiment 17402:Adaptive designs 17253: 17252: 17240: 17239: 17117:Rank correlation 16879: 16878: 16870: 16869: 16857: 16856: 16824: 16817: 16810: 16801: 16800: 16785: 16773: 16757: 16746: 16710: 16681: 16661: 16647: 16621: 16620: 16618: 16616: 16601: 16595: 16594: 16584: 16560: 16554: 16553: 16543: 16526:(4): 1357–1373. 16511: 16505: 16504: 16502: 16470: 16464: 16463: 16453: 16443: 16419: 16413: 16412: 16402: 16392: 16356: 16350: 16349: 16323: 16302: 16296: 16295: 16275: 16269: 16268: 16250: 16230: 16224: 16223: 16203: 16187: 16181: 16180: 16174: 16166: 16154: 16148: 16147: 16145: 16134: 16128: 16127: 16107: 16096: 16090: 16089: 16087: 16076: 16070: 16056: 16050: 16049: 16040:(406): 502–506. 16029: 16017: 16011: 16009: 15995: 15989: 15985: 15946: 15940: 15926: 15920: 15919: 15917: 15908:(8): 1311–1320. 15891: 15885: 15884: 15872: 15859: 15853: 15852: 15846: 15837: 15831: 15830: 15828: 15816: 15810: 15809: 15799: 15779: 15770: 15764: 15763: 15745: 15725: 15716: 15710: 15709: 15691: 15671: 15662: 15656: 15655: 15645: 15629: 15623: 15622: 15620: 15619: 15606: 15591:Machine Learning 15588: 15579: 15573: 15572: 15566: 15557: 15551: 15550: 15544: 15535: 15529: 15528: 15500: 15494: 15487: 15481: 15480: 15460: 15454: 15453: 15435: 15429: 15428: 15420: 15414: 15413: 15377: 15371: 15368: 15362: 15361: 15325: 15319: 15309: 15300: 15289: 15283: 15270: 15264: 15255: 15249: 15231: 15225: 15207: 15198: 15184: 15175: 15174: 15172: 15170: 15155: 15149: 15148: 15146: 15135: 15129: 15128: 15112: 15106: 15105: 15087: 15077: 15040: 15034: 15033: 15023: 14991: 14985: 14984: 14982: 14981: 14966: 14960: 14959: 14939: 14933: 14932: 14930: 14929: 14914: 14908: 14903: 14897: 14896: 14888: 14882: 14875: 14869: 14868: 14858: 14849: 14843: 14842: 14816: 14796: 14790: 14789: 14769: 14763: 14762: 14742: 14736: 14729: 14723: 14717: 14711: 14710: 14699: 14693: 14687: 14681: 14680: 14673: 14667: 14666: 14655:10.1002/wics.101 14648: 14625: 14619: 14612: 14606: 14605: 14603: 14601: 14591: 14585: 14579: 14573: 14572: 14562: 14546: 14540: 14538: 14530: 14524: 14523: 14503: 14497: 14496: 14468: 14462: 14461: 14451: 14431: 14422: 14421: 14403: 14397: 14396: 14394: 14392: 14377: 14371: 14370: 14360: 14342: 14318: 14301: 14300: 14298: 14286: 14275: 14274: 14248: 14246:astro-ph/0606170 14228: 14219: 14218: 14208: 14190: 14166: 14160: 14159: 14133: 14113: 14102: 14096: 14090: 14084: 14078: 14077: 14045: 14039: 14038: 14015: 14009: 14002: 13996: 13995: 13967: 13961: 13960: 13954: 13946: 13944: 13920: 13914: 13913: 13887: 13878:(8): 1798–1828. 13867: 13861: 13860: 13832: 13817: 13816: 13814: 13780: 13774: 13773: 13756:(3/4): 321–377. 13731: 13725: 13724: 13689: 13683: 13682: 13654: 13638: 13632: 13631: 13605: 13581: 13575: 13574: 13548: 13528: 13519: 13518: 13516: 13506: 13482: 13473: 13472: 13446: 13426: 13417: 13416: 13406: 13390: 13384: 13383: 13381: 13349: 13343: 13342: 13332: 13292: 13271:Transform coding 13095: 13091: 13069: 13065: 13061: 13057: 13053: 13045: 13041: 13037: 13006: 13005:SVDS_SCORING_PCA 13002: 12987: 12967: 12923: 12919: 12915: 12888:Maple (software) 12877: 12867: 12863: 12794: 12792: 12791: 12786: 12772: 12770: 12769: 12764: 12752: 12750: 12749: 12744: 12732: 12730: 12729: 12724: 12712: 12710: 12709: 12704: 12686: 12684: 12683: 12678: 12663: 12661: 12660: 12655: 12640: 12638: 12637: 12632: 12620: 12618: 12617: 12612: 12600: 12598: 12597: 12592: 12571: 12569: 12568: 12563: 12511:algorithms like 12318: 12316: 12315: 12310: 12298: 12296: 12295: 12290: 12278: 12276: 12275: 12270: 12268: 12267: 12257: 12252: 12237: 12236: 12230: 12229: 12219: 12214: 12162: 12154: 12143: 12059:order parameters 12003:action potential 11636:round-off errors 11612: 11602: 11535: 11529: 11523: 11517: 11511: 11505: 11479: 11473: 11469: 11465: 11459: 11453: 11447: 11437: 11431: 11425: 11413: 11406: 11403: 11389: 11378: 11375: 11366: 11345: 11334: 11330: 11323: 11308: 11304: 11296: 11282: 11275: 11265: 11259: 11253: 11230: 11228: 11227: 11222: 11210: 11208: 11207: 11202: 11178: 11176: 11175: 11170: 11149: 11147: 11146: 11141: 11139: 11135: 11134: 11098: 11094: 11093: 11081: 11080: 11050: 11043: 11042: 11033: 11032: 11021: 10999: 10992: 10991: 10971: 10916: 10914: 10913: 10908: 10879: 10869: 10867: 10866: 10861: 10844:We want to find 10819: 10817: 10816: 10811: 10809: 10795: 10793: 10792: 10787: 10785: 10777: 10769: 10748: 10746: 10745: 10740: 10732: 10730: 10729: 10720: 10719: 10710: 10671: 10669: 10668: 10663: 10636: 10634: 10633: 10628: 10583: 10580: 10577: 10576: 10561: 10560: 10512: 10510: 10509: 10504: 10481: 10478: 10475: 10474: 10461: 10456: 10438: 10437: 10310: 10308: 10307: 10302: 10288: 10285: 10276: 10275: 10248: 10246: 10245: 10240: 10229: 10226: 10223: 10222: 10210: 10209: 10115: 10113: 10112: 10107: 10105: 10097: 10092: 10087: 10086: 10078: 10031: 10009: 10007: 10006: 10001: 9989: 9987: 9986: 9981: 9979: 9974: 9973: 9968: 9962: 9960: 9946: 9941: 9890: 9888: 9887: 9882: 9859: 9856: 9845: 9844: 9828: 9817: 9815: 9814: 9809: 9807: 9806: 9801: 9795: 9787: 9779: 9748: 9746: 9745: 9740: 9738: 9737: 9732: 9706: 9704: 9703: 9698: 9696: 9695: 9682: 9677: 9662: 9654: 9649: 9648: 9584: 9582: 9581: 9576: 9574: 9573: 9568: 9559: 9558: 9553: 9532: 9530: 9529: 9524: 9522: 9521: 9516: 9503: 9501: 9500: 9495: 9493: 9492: 9487: 9478: 9477: 9472: 9432: 9430: 9429: 9424: 9419: 9411: 9397: 9340: 9338: 9337: 9332: 9330: 9318: 9316: 9315: 9310: 9308: 9292: 9290: 9289: 9284: 9282: 9267: 9265: 9264: 9259: 9251: 9243: 9226: 9218: 9196:information loss 9193: 9191: 9190: 9185: 9183: 9167: 9165: 9164: 9159: 9157: 9142: 9140: 9139: 9134: 9132: 9126: 9121: 9116: 9107: 9095: 9093: 9092: 9087: 9085: 9073: 9071: 9070: 9065: 9060: 9052: 9031: 9029: 9028: 9023: 9021: 9010:is Gaussian and 9009: 9007: 9006: 9001: 8999: 8984: 8982: 8981: 8976: 8974: 8962: 8960: 8959: 8954: 8952: 8940: 8938: 8937: 8932: 8930: 8915: 8913: 8912: 8907: 8902: 8894: 8886: 8823: 8821: 8820: 8815: 8783: 8782: 8769: 8753: 8751: 8750: 8745: 8743: 8742: 8726: 8724: 8723: 8718: 8706: 8704: 8703: 8698: 8692: 8683: 8682: 8673: 8672: 8656: 8654: 8653: 8648: 8636: 8634: 8633: 8628: 8622: 8613: 8612: 8603: 8602: 8586: 8584: 8583: 8578: 8572: 8563: 8562: 8553: 8552: 8532: 8525: 8523: 8522: 8517: 8514: 8509: 8497: 8496: 8486: 8481: 8460: 8459: 8422: 8420: 8419: 8414: 8408: 8399: 8398: 8389: 8388: 8366: 8357: 8356: 8347: 8346: 8334: 8320: 8302: 8294: 8285: 8283: 8282: 8277: 8275: 8259: 8257: 8256: 8251: 8248: 8243: 8238: 8225: 8223: 8222: 8217: 8211: 8206: 8201: 8192: 8180: 8178: 8177: 8172: 8167: 8166: 8161: 8139: 8137: 8136: 8131: 8129: 8128: 8123: 8110: 8108: 8107: 8102: 8100: 8092: 8070: 8068: 8067: 8062: 8057: 8056: 8016: 8014: 8013: 8008: 8006: 7994: 7992: 7991: 7986: 7981: 7969: 7967: 7966: 7961: 7959: 7958: 7940: 7938: 7937: 7932: 7930: 7914: 7912: 7911: 7906: 7904: 7903: 7898: 7885: 7883: 7882: 7877: 7875: 7874: 7869: 7860: 7848: 7846: 7845: 7840: 7835: 7834: 7829: 7807: 7805: 7804: 7799: 7797: 7796: 7791: 7778: 7776: 7775: 7770: 7750: 7748: 7747: 7742: 7740: 7735: 7730: 7729: 7717: 7716: 7711: 7690: 7688: 7687: 7682: 7680: 7679: 7659: 7657: 7656: 7651: 7635: 7633: 7632: 7627: 7622: 7621: 7583:For any integer 7552: 7550: 7549: 7544: 7520: 7518: 7517: 7512: 7487: 7485: 7484: 7479: 7445: 7443: 7442: 7437: 7432: 7431: 7413: 7398: 7396: 7395: 7390: 7366: 7364: 7363: 7358: 7333: 7331: 7330: 7325: 7292: 7290: 7289: 7284: 7279: 7278: 7260: 7245: 7243: 7242: 7237: 7223: 7208: 7206: 7205: 7200: 7175: 7173: 7172: 7167: 7147: 7145: 7144: 7139: 7137: 7108: 7106: 7105: 7100: 7095: 7094: 7093: 7071: 7056: 7054: 7053: 7048: 7034: 7019: 7017: 7016: 7011: 6986: 6984: 6983: 6978: 6948: 6946: 6945: 6940: 6935: 6934: 6933: 6911: 6896: 6894: 6893: 6888: 6874: 6859: 6857: 6856: 6851: 6826: 6824: 6823: 6818: 6794: 6792: 6791: 6786: 6781: 6780: 6779: 6757: 6742: 6740: 6739: 6734: 6720: 6705: 6703: 6702: 6697: 6672: 6670: 6669: 6664: 6640: 6638: 6637: 6632: 6627: 6626: 6625: 6603: 6588: 6586: 6585: 6580: 6556: 6554: 6553: 6548: 6523: 6521: 6520: 6515: 6486: 6484: 6483: 6478: 6473: 6472: 6454: 6439: 6437: 6436: 6431: 6407: 6405: 6404: 6399: 6374: 6372: 6371: 6366: 6337: 6335: 6334: 6329: 6324: 6323: 6305: 6290: 6288: 6287: 6282: 6257: 6255: 6254: 6249: 6227: 6225: 6224: 6219: 6214: 6213: 6198: 6183: 6181: 6180: 6175: 6150: 6148: 6147: 6142: 6112: 6110: 6109: 6104: 6099: 6098: 6083: 6068: 6066: 6065: 6060: 6035: 6033: 6032: 6027: 5997: 5995: 5994: 5989: 5984: 5983: 5968: 5948: 5946: 5945: 5940: 5921: 5919: 5918: 5913: 5887: 5885: 5884: 5879: 5859: 5857: 5856: 5851: 5829: 5827: 5826: 5821: 5801: 5799: 5798: 5793: 5771: 5769: 5768: 5763: 5748: 5746: 5745: 5740: 5716: 5714: 5713: 5708: 5683: 5681: 5680: 5675: 5653: 5651: 5650: 5645: 5640: 5639: 5621: 5594: 5593: 5457: 5455: 5454: 5449: 5447: 5446: 5441: 5435: 5427: 5426: 5421: 5415: 5414: 5409: 5400: 5399: 5394: 5371: 5325: 5323: 5322: 5317: 5315: 5311: 5306: 5295: 5291: 5286: 5285: 5284: 5278: 5272: 5267: 5256: 5252: 5247: 5235: 5190: 5188: 5187: 5182: 5180: 5133: 5131: 5130: 5125: 5123: 5118: 5117: 5116: 5110: 5101: 5100: 5099: 5094: 5093: 5085: 5067: 5065: 5064: 5059: 5057: 5056: 5048: 5034: 5032: 5031: 5026: 5024: 5020: 5019: 5018: 5012: 5006: 5005: 5000: 4999: 4991: 4987: 4976: 4972: 4971: 4970: 4964: 4958: 4953: 4952: 4951: 4945: 4939: 4928: 4924: 4923: 4922: 4916: 4910: 4905: 4900: 4899: 4898: 4892: 4886: 4885: 4884: 4878: 4872: 4860: 4855: 4854: 4849: 4746: 4744: 4743: 4738: 4736: 4735: 4730: 4724: 4719: 4711: 4601: 4599: 4598: 4593: 4590: 4585: 4576: 4575: 4570: 4561: 4546: 4544: 4543: 4538: 4535: 4530: 4520: 4515: 4510: 4504: 4503: 4498: 4489: 4488: 4483: 4477: 4450: 4448: 4447: 4442: 4440: 4439: 4423: 4413: 4411: 4410: 4405: 4400: 4399: 4394: 4379: 4378: 4373: 4354: 4353: 4347: 4307: 4305: 4304: 4299: 4297: 4296: 4291: 4285: 4277: 4276: 4271: 4109: 4107: 4106: 4101: 4099: 4091: 4086: 4085: 4084: 4078: 4071: 4065: 4060: 4059: 4058: 4052: 4043: 4038: 4033: 4032: 4031: 4025: 4006: 4004: 4003: 3998: 3996: 3995: 3994: 3988: 3982: 3977: 3969: 3964: 3963: 3962: 3956: 3947: 3889: 3887: 3886: 3881: 3879: 3875: 3874: 3863: 3856: 3855: 3849: 3838: 3832: 3831: 3810: 3806: 3805: 3794: 3788: 3787: 3771: 3770: 3764: 3753: 3741: 3737: 3736: 3725: 3719: 3714: 3713: 3712: 3706: 3699: 3698: 3692: 3681: 3669: 3662: 3661: 3650: 3644: 3636: 3635: 3634: 3624: 3623: 3612: 3606: 3588: 3587: 3576: 3564: 3563: 3552: 3456: 3454: 3453: 3448: 3446: 3441: 3433: 3285: 3283: 3282: 3277: 3275: 3271: 3268: 3267: 3262: 3261: 3256: 3249: 3248: 3243: 3242: 3237: 3236: 3228: 3223: 3222: 3216: 3211: 3210: 3202: 3198: 3197: 3196: 3190: 3183: 3164: 3160: 3159: 3154: 3150: 3149: 3144: 3143: 3138: 3137: 3129: 3111: 3104: 3100: 3090: 3089: 3063: 3062: 3051: 3032: 3030: 3029: 3024: 3021: 3020: 3014: 3003: 2997: 2996: 2985: 2979: 2973: 2962: 2944: 2936: 2935: 2930: 2929: 2921: 2780: 2778: 2777: 2772: 2770: 2766: 2764: 2763: 2758: 2757: 2756: 2750: 2743: 2742: 2734: 2733: 2732: 2726: 2720: 2719: 2718: 2712: 2705: 2687: 2686: 2675: 2649: 2647: 2646: 2641: 2639: 2635: 2634: 2626: 2625: 2624: 2618: 2612: 2611: 2610: 2604: 2592: 2585: 2581: 2558: 2554: 2553: 2548: 2544: 2526: 2519: 2515: 2492: 2491: 2480: 2461: 2459: 2458: 2453: 2451: 2447: 2446: 2445: 2440: 2436: 2435: 2427: 2426: 2415: 2402: 2386: 2376: 2354: 2350: 2348: 2343: 2328: 2327: 2314: 2298: 2288: 2266: 2265: 2254: 2223: 2221: 2220: 2215: 2213: 2212: 2201: 2200: 2199: 2185: 2184: 2176: 2175: 2161: 2159: 2158: 2153: 2151: 2150: 2139: 2138: 2137: 2123: 2122: 2114: 2113: 2099: 2097: 2096: 2091: 2089: 2088: 2077: 2076: 2075: 2061: 2060: 2052: 2051: 2034: 2032: 2031: 2026: 2024: 2019: 2011: 1993: 1991: 1990: 1985: 1973: 1971: 1970: 1965: 1937: 1935: 1934: 1929: 1927: 1926: 1908: 1907: 1888: 1886: 1885: 1880: 1834: 1822: 1821: 1810: 1801: 1800: 1789: 1780: 1779: 1768: 1767: 1766: 1745: 1743: 1742: 1737: 1735: 1734: 1719: 1718: 1700: 1699: 1684: 1683: 1672: 1652: 1650: 1649: 1644: 1642: 1641: 1626: 1625: 1607: 1606: 1591: 1590: 1579: 1566: 1564: 1563: 1558: 1556: 1555: 1540: 1539: 1521: 1520: 1505: 1504: 1493: 1476: 1474: 1473: 1468: 1433: 1431: 1430: 1425: 1268:Harold Hotelling 1236:cross-covariance 1210: 1208: 1207: 1202: 1184: 1182: 1181: 1176: 1163: 1161: 1160: 1155: 1142: 1140: 1139: 1134: 1064: 1062: 1061: 1056: 1034: 1032: 1031: 1026: 1011: 1009: 1008: 1003: 942: 935: 928: 889:Related articles 766:Confusion matrix 519:Isolation forest 464:Graphical models 243: 242: 195:Learning to rank 190:Feature learning 28:Machine learning 19: 18: 19156: 19155: 19151: 19150: 19149: 19147: 19146: 19145: 19126: 19125: 19124: 19119: 19082: 19053: 19015: 18952: 18938:quality control 18905: 18887:Clinical trials 18864: 18839: 18823: 18811:Hazard function 18805: 18759: 18721: 18705: 18668: 18664:Breusch–Godfrey 18652: 18629: 18569: 18544:Factor analysis 18490: 18471:Graphical model 18443: 18410: 18377: 18363: 18343: 18297: 18264: 18226: 18189: 18188: 18157: 18101: 18088: 18080: 18072: 18056: 18041: 18020:Rank statistics 18014: 17993:Model selection 17981: 17939:Goodness of fit 17933: 17910: 17884: 17856: 17809: 17754: 17743:Median unbiased 17671: 17582: 17515:Order statistic 17477: 17456: 17423: 17397: 17349: 17304: 17247: 17245:Data collection 17226: 17138: 17093: 17067: 17045: 17005: 16957: 16874:Continuous data 16864: 16851: 16833: 16828: 16783: 16771: 16755: 16744: 16740: 16707: 16678: 16659:10.1.1.149.8828 16629: 16627:Further reading 16624: 16614: 16612: 16603: 16602: 16598: 16561: 16557: 16512: 16508: 16471: 16467: 16420: 16416: 16357: 16353: 16303: 16299: 16276: 16272: 16231: 16227: 16220: 16201:10.1.1.144.4864 16188: 16184: 16171:cite conference 16168: 16167: 16155: 16151: 16143: 16135: 16131: 16124: 16105: 16097: 16093: 16085: 16077: 16073: 16057: 16053: 16027: 16018: 16014: 15996: 15992: 15947: 15943: 15927: 15923: 15892: 15888: 15870: 15860: 15856: 15844: 15838: 15834: 15817: 15813: 15777: 15771: 15767: 15723: 15717: 15713: 15669: 15663: 15659: 15630: 15626: 15617: 15615: 15586: 15580: 15576: 15564: 15558: 15554: 15542: 15536: 15532: 15501: 15497: 15488: 15484: 15477: 15461: 15457: 15450: 15436: 15432: 15421: 15417: 15378: 15374: 15369: 15365: 15326: 15322: 15310: 15303: 15290: 15286: 15282:, January 2015. 15271: 15267: 15256: 15252: 15232: 15228: 15208: 15201: 15185: 15178: 15168: 15166: 15156: 15152: 15144: 15136: 15132: 15113: 15109: 15041: 15037: 14992: 14988: 14979: 14977: 14967: 14963: 14940: 14936: 14927: 14925: 14916: 14915: 14911: 14904: 14900: 14889: 14885: 14876: 14872: 14856: 14850: 14846: 14797: 14793: 14786: 14770: 14766: 14743: 14739: 14730: 14726: 14718: 14714: 14709:. 19 June 2023. 14701: 14700: 14696: 14688: 14684: 14675: 14674: 14670: 14626: 14622: 14613: 14609: 14599: 14597: 14593: 14592: 14588: 14580: 14576: 14547: 14543: 14531: 14527: 14520: 14504: 14500: 14469: 14465: 14432: 14425: 14418: 14404: 14400: 14390: 14388: 14379: 14378: 14374: 14319: 14304: 14287: 14278: 14229: 14222: 14167: 14163: 14114: 14105: 14097: 14093: 14085: 14081: 14046: 14042: 14035: 14016: 14012: 14003: 13999: 13992: 13968: 13964: 13948: 13947: 13921: 13917: 13868: 13864: 13857: 13833: 13820: 13803:10.1137/1035134 13781: 13777: 13762:10.2307/2333955 13744: 13732: 13728: 13707:(11): 559–572. 13690: 13686: 13671: 13639: 13635: 13582: 13578: 13529: 13522: 13483: 13476: 13427: 13420: 13391: 13387: 13350: 13346: 13293: 13289: 13285: 13280: 13115: 13093: 13089: 13067: 13063: 13059: 13055: 13051: 13043: 13039: 13035: 13004: 13000: 12997:Oracle Database 12985: 12965: 12921: 12917: 12913: 12875: 12865: 12861: 12832: 12821: 12805: 12780: 12777: 12776: 12758: 12755: 12754: 12738: 12735: 12734: 12718: 12715: 12714: 12692: 12689: 12688: 12672: 12669: 12668: 12667:Each column of 12649: 12646: 12645: 12626: 12623: 12622: 12606: 12603: 12602: 12577: 12574: 12573: 12557: 12554: 12553: 12552:Given a matrix 12550: 12539: 12534: 12500: 12478:multilinear PCA 12408: 12372: 12366: 12361: 12359:Generalizations 12330: 12304: 12301: 12300: 12284: 12281: 12280: 12263: 12259: 12253: 12242: 12232: 12231: 12225: 12221: 12215: 12204: 12192: 12189: 12188: 12169: 12161: 12158: 12153: 12150: 12146: 12142: 12139: 12133:causal modeling 12128:Factor analysis 12112: 12110:Factor analysis 12076: 12071: 11991: 11948:Markowitz model 11944:expected return 11875:securities and 11850: 11829: 11795: 11764: 11744: 11715: 11710: 11667: 11658: 11628: 11622: 11604: 11598: 11591:power iteration 11588: 11581: 11557:power iteration 11550: 11531: 11525: 11519: 11513: 11507: 11501: 11482:power iteration 11475: 11471: 11467: 11461: 11455: 11449: 11443: 11438:, which is the 11433: 11427: 11421: 11418:power iteration 11414: 11408: 11404: 11390: 11379: 11376: 11367: 11346: 11336: 11332: 11325: 11321: 11306: 11300: 11292: 11289: 11277: 11276:at the cost of 11271: 11261: 11255: 11251: 11244: 11216: 11213: 11212: 11184: 11181: 11180: 11158: 11155: 11154: 11137: 11136: 11127: 11123: 11096: 11095: 11089: 11085: 11076: 11072: 11048: 11047: 11038: 11034: 11028: 11024: 10997: 10996: 10987: 10983: 10949: 10927: 10925: 10922: 10921: 10902: 10899: 10898: 10871: 10849: 10846: 10845: 10841:has zero mean. 10827: 10805: 10803: 10800: 10799: 10781: 10773: 10765: 10763: 10760: 10759: 10725: 10721: 10715: 10711: 10709: 10707: 10704: 10703: 10674:Use the vector 10642: 10639: 10638: 10579: 10569: 10565: 10553: 10549: 10547: 10544: 10543: 10522:Save the first 10477: 10467: 10463: 10457: 10446: 10433: 10429: 10427: 10424: 10423: 10352:th eigenvector. 10284: 10268: 10264: 10262: 10259: 10258: 10225: 10218: 10214: 10202: 10198: 10196: 10193: 10192: 10122:diagonal matrix 10101: 10093: 10088: 10079: 10074: 10073: 10071: 10068: 10067: 10026: 9995: 9992: 9991: 9975: 9969: 9964: 9963: 9950: 9945: 9937: 9935: 9932: 9931: 9855: 9840: 9836: 9834: 9831: 9830: 9823: 9802: 9797: 9796: 9791: 9783: 9775: 9773: 9770: 9769: 9733: 9728: 9727: 9725: 9722: 9721: 9688: 9684: 9678: 9667: 9653: 9644: 9640: 9638: 9635: 9634: 9569: 9564: 9563: 9554: 9549: 9548: 9546: 9543: 9542: 9517: 9512: 9511: 9509: 9506: 9505: 9488: 9483: 9482: 9473: 9468: 9467: 9465: 9462: 9461: 9415: 9401: 9393: 9391: 9388: 9387: 9347: 9326: 9324: 9321: 9320: 9304: 9302: 9299: 9298: 9278: 9276: 9273: 9272: 9247: 9239: 9222: 9214: 9206: 9203: 9202: 9179: 9177: 9174: 9173: 9153: 9151: 9148: 9147: 9128: 9122: 9117: 9112: 9103: 9101: 9098: 9097: 9081: 9079: 9076: 9075: 9056: 9048: 9040: 9037: 9036: 9017: 9015: 9012: 9011: 8995: 8993: 8990: 8989: 8970: 8968: 8965: 8964: 8948: 8946: 8943: 8942: 8926: 8924: 8921: 8920: 8898: 8890: 8882: 8880: 8877: 8876: 8867: 8830: 8778: 8774: 8765: 8759: 8756: 8755: 8738: 8734: 8732: 8729: 8728: 8712: 8709: 8708: 8688: 8678: 8674: 8668: 8664: 8662: 8659: 8658: 8642: 8639: 8638: 8618: 8608: 8604: 8598: 8594: 8592: 8589: 8588: 8568: 8558: 8554: 8548: 8544: 8542: 8539: 8538: 8530: 8510: 8502: 8492: 8488: 8482: 8471: 8455: 8451: 8440: 8437: 8436: 8404: 8394: 8390: 8384: 8380: 8362: 8352: 8348: 8342: 8338: 8330: 8328: 8325: 8324: 8316: 8314: 8300: 8292: 8271: 8269: 8266: 8265: 8244: 8239: 8234: 8231: 8228: 8227: 8207: 8202: 8197: 8188: 8186: 8183: 8182: 8162: 8157: 8156: 8145: 8142: 8141: 8124: 8119: 8118: 8116: 8113: 8112: 8096: 8088: 8080: 8077: 8076: 8049: 8048: 8040: 8037: 8036: 8029: 8002: 8000: 7997: 7996: 7977: 7975: 7972: 7971: 7951: 7950: 7945: 7942: 7941: 7926: 7924: 7921: 7920: 7899: 7894: 7893: 7891: 7888: 7887: 7870: 7865: 7864: 7856: 7854: 7851: 7850: 7830: 7825: 7824: 7813: 7810: 7809: 7792: 7787: 7786: 7784: 7781: 7780: 7764: 7761: 7760: 7736: 7731: 7722: 7721: 7712: 7707: 7706: 7704: 7701: 7700: 7672: 7671: 7669: 7666: 7665: 7645: 7642: 7641: 7614: 7613: 7605: 7602: 7601: 7582: 7569: 7564: 7558: 7526: 7523: 7522: 7521: 7494: 7491: 7490: 7467: 7464: 7463: 7424: 7420: 7409: 7407: 7404: 7403: 7372: 7369: 7368: 7367: 7340: 7337: 7336: 7313: 7310: 7309: 7271: 7267: 7256: 7254: 7251: 7250: 7216: 7214: 7211: 7210: 7209: 7182: 7179: 7178: 7155: 7152: 7151: 7133: 7131: 7128: 7127: 7112:diagonal matrix 7086: 7082: 7078: 7067: 7065: 7062: 7061: 7027: 7025: 7022: 7021: 7020: 6993: 6990: 6989: 6966: 6963: 6962: 6926: 6922: 6918: 6907: 6905: 6902: 6901: 6867: 6865: 6862: 6861: 6860: 6833: 6830: 6829: 6806: 6803: 6802: 6772: 6768: 6764: 6753: 6751: 6748: 6747: 6713: 6711: 6708: 6707: 6706: 6679: 6676: 6675: 6652: 6649: 6648: 6618: 6614: 6610: 6599: 6597: 6594: 6593: 6562: 6559: 6558: 6557: 6530: 6527: 6526: 6503: 6500: 6499: 6465: 6461: 6450: 6448: 6445: 6444: 6413: 6410: 6409: 6408: 6381: 6378: 6377: 6354: 6351: 6350: 6316: 6312: 6301: 6299: 6296: 6295: 6264: 6261: 6260: 6237: 6234: 6233: 6209: 6205: 6194: 6192: 6189: 6188: 6157: 6154: 6153: 6130: 6127: 6126: 6094: 6090: 6079: 6077: 6074: 6073: 6042: 6039: 6038: 6015: 6012: 6011: 5979: 5975: 5964: 5962: 5959: 5958: 5928: 5925: 5924: 5895: 5892: 5891: 5873: 5870: 5869: 5839: 5836: 5835: 5815: 5812: 5811: 5781: 5778: 5777: 5757: 5754: 5753: 5722: 5719: 5718: 5717: 5690: 5687: 5686: 5663: 5660: 5659: 5632: 5628: 5617: 5615: 5612: 5611: 5592: 5528:sample variance 5490: 5442: 5437: 5436: 5431: 5422: 5417: 5416: 5410: 5405: 5404: 5395: 5390: 5389: 5387: 5384: 5383: 5378: 5363: 5313: 5312: 5307: 5302: 5293: 5292: 5287: 5280: 5279: 5274: 5273: 5268: 5263: 5254: 5253: 5248: 5243: 5236: 5231: 5227: 5225: 5222: 5221: 5217:can be written 5202: 5176: 5174: 5171: 5170: 5167: 5119: 5112: 5111: 5106: 5105: 5095: 5084: 5083: 5082: 5081: 5079: 5076: 5075: 5047: 5046: 5044: 5041: 5040: 5022: 5021: 5014: 5013: 5008: 5007: 5001: 4990: 4989: 4988: 4983: 4974: 4973: 4966: 4965: 4960: 4959: 4954: 4947: 4946: 4941: 4940: 4935: 4926: 4925: 4918: 4917: 4912: 4911: 4906: 4901: 4894: 4893: 4888: 4887: 4880: 4879: 4874: 4873: 4868: 4861: 4856: 4850: 4845: 4844: 4840: 4838: 4835: 4834: 4830:can be written 4775: 4731: 4726: 4725: 4720: 4715: 4707: 4705: 4702: 4701: 4687: 4681: 4617: 4586: 4581: 4571: 4566: 4565: 4557: 4552: 4549: 4548: 4531: 4526: 4516: 4511: 4506: 4499: 4494: 4493: 4484: 4479: 4478: 4473: 4468: 4465: 4464: 4435: 4431: 4429: 4426: 4425: 4415: 4395: 4390: 4389: 4374: 4369: 4368: 4349: 4348: 4343: 4331: 4328: 4327: 4317: 4292: 4287: 4286: 4281: 4272: 4267: 4266: 4264: 4261: 4260: 4239: 4216: 4209: 4198: 4187: 4181: 4174: 4166: 4160: 4145: 4127: 4095: 4087: 4080: 4079: 4074: 4073: 4067: 4061: 4054: 4053: 4048: 4047: 4039: 4034: 4027: 4026: 4021: 4020: 4018: 4015: 4014: 3990: 3989: 3984: 3983: 3978: 3973: 3965: 3958: 3957: 3952: 3951: 3943: 3941: 3938: 3937: 3925: 3914: 3903: 3877: 3876: 3864: 3859: 3858: 3851: 3850: 3839: 3834: 3821: 3817: 3808: 3807: 3795: 3790: 3789: 3777: 3773: 3766: 3765: 3754: 3749: 3739: 3738: 3726: 3721: 3720: 3715: 3708: 3707: 3702: 3701: 3694: 3693: 3682: 3677: 3667: 3666: 3651: 3646: 3645: 3640: 3630: 3629: 3625: 3613: 3608: 3607: 3602: 3592: 3577: 3569: 3568: 3553: 3545: 3544: 3534: 3532: 3529: 3528: 3513:of the dataset 3501: 3442: 3437: 3429: 3427: 3424: 3423: 3400: 3389: 3378: 3367: 3356: 3345: 3334: 3320: 3263: 3257: 3252: 3251: 3250: 3244: 3238: 3227: 3226: 3225: 3218: 3217: 3212: 3201: 3200: 3192: 3191: 3186: 3185: 3184: 3181: 3177: 3155: 3145: 3139: 3128: 3127: 3126: 3125: 3121: 3120: 3116: 3096: 3092: 3091: 3069: 3068: 3052: 3047: 3046: 3044: 3041: 3040: 3016: 3015: 3004: 2999: 2986: 2981: 2980: 2975: 2963: 2952: 2940: 2931: 2920: 2919: 2918: 2916: 2913: 2912: 2894: 2887: 2880: 2873: 2862: 2855: 2844: 2833: 2822: 2759: 2752: 2751: 2746: 2745: 2744: 2735: 2728: 2727: 2722: 2721: 2714: 2713: 2708: 2707: 2706: 2704: 2700: 2676: 2671: 2670: 2668: 2665: 2664: 2659: 2627: 2620: 2619: 2614: 2613: 2606: 2605: 2600: 2599: 2598: 2594: 2577: 2573: 2572: 2549: 2537: 2533: 2532: 2528: 2511: 2507: 2506: 2481: 2476: 2475: 2473: 2470: 2469: 2441: 2431: 2416: 2411: 2410: 2409: 2405: 2404: 2398: 2393: 2389: 2372: 2368: 2344: 2333: 2323: 2319: 2310: 2305: 2301: 2284: 2280: 2255: 2250: 2249: 2247: 2244: 2243: 2238: 2230: 2228:First component 2202: 2195: 2191: 2190: 2189: 2177: 2171: 2170: 2169: 2167: 2164: 2163: 2140: 2133: 2129: 2128: 2127: 2115: 2109: 2108: 2107: 2105: 2102: 2101: 2078: 2071: 2067: 2066: 2065: 2053: 2047: 2046: 2045: 2043: 2040: 2039: 2020: 2015: 2007: 2005: 2002: 2001: 1979: 1976: 1975: 1959: 1956: 1955: 1922: 1918: 1903: 1899: 1897: 1894: 1893: 1824: 1811: 1806: 1805: 1790: 1785: 1784: 1769: 1762: 1758: 1757: 1756: 1754: 1751: 1750: 1724: 1720: 1714: 1710: 1695: 1691: 1673: 1668: 1667: 1665: 1662: 1661: 1631: 1627: 1621: 1617: 1602: 1598: 1580: 1575: 1574: 1572: 1569: 1568: 1545: 1541: 1535: 1531: 1516: 1512: 1494: 1489: 1488: 1486: 1483: 1482: 1462: 1459: 1458: 1413: 1410: 1409: 1387: 1335: 1307:factor analysis 1256: 1228:factor analysis 1190: 1187: 1186: 1170: 1167: 1166: 1149: 1146: 1145: 1128: 1125: 1124: 1121: 1044: 1041: 1040: 1020: 1017: 1016: 997: 994: 993: 946: 917: 916: 890: 882: 881: 842: 834: 833: 794:Kernel machines 789: 781: 780: 756: 748: 747: 728:Active learning 723: 715: 714: 683: 673: 672: 598:Diffusion model 534: 524: 523: 496: 486: 485: 459: 449: 448: 404:Factor analysis 399: 389: 388: 372: 335: 325: 324: 245: 244: 228: 227: 226: 215: 214: 120: 112: 111: 77:Online learning 42: 30: 17: 12: 11: 5: 19154: 19144: 19143: 19138: 19121: 19120: 19118: 19117: 19105: 19093: 19079: 19066: 19063: 19062: 19059: 19058: 19055: 19054: 19052: 19051: 19046: 19041: 19036: 19031: 19025: 19023: 19017: 19016: 19014: 19013: 19008: 19003: 18998: 18993: 18988: 18983: 18978: 18973: 18968: 18962: 18960: 18954: 18953: 18951: 18950: 18945: 18940: 18931: 18926: 18921: 18915: 18913: 18907: 18906: 18904: 18903: 18898: 18893: 18884: 18882:Bioinformatics 18878: 18876: 18866: 18865: 18853: 18852: 18849: 18848: 18845: 18844: 18841: 18840: 18838: 18837: 18831: 18829: 18825: 18824: 18822: 18821: 18815: 18813: 18807: 18806: 18804: 18803: 18798: 18793: 18788: 18782: 18780: 18771: 18765: 18764: 18761: 18760: 18758: 18757: 18752: 18747: 18742: 18737: 18731: 18729: 18723: 18722: 18720: 18719: 18714: 18709: 18701: 18696: 18691: 18690: 18689: 18687:partial (PACF) 18678: 18676: 18670: 18669: 18667: 18666: 18661: 18656: 18648: 18643: 18637: 18635: 18634:Specific tests 18631: 18630: 18628: 18627: 18622: 18617: 18612: 18607: 18602: 18597: 18592: 18586: 18584: 18577: 18571: 18570: 18568: 18567: 18566: 18565: 18564: 18563: 18548: 18547: 18546: 18536: 18534:Classification 18531: 18526: 18521: 18516: 18511: 18506: 18500: 18498: 18492: 18491: 18489: 18488: 18483: 18481:McNemar's test 18478: 18473: 18468: 18463: 18457: 18455: 18445: 18444: 18420: 18419: 18416: 18415: 18412: 18411: 18409: 18408: 18403: 18398: 18393: 18387: 18385: 18379: 18378: 18376: 18375: 18359: 18353: 18351: 18345: 18344: 18342: 18341: 18336: 18331: 18326: 18321: 18319:Semiparametric 18316: 18311: 18305: 18303: 18299: 18298: 18296: 18295: 18290: 18285: 18280: 18274: 18272: 18266: 18265: 18263: 18262: 18257: 18252: 18247: 18242: 18236: 18234: 18228: 18227: 18225: 18224: 18219: 18214: 18209: 18203: 18201: 18191: 18190: 18187: 18186: 18181: 18175: 18167: 18166: 18163: 18162: 18159: 18158: 18156: 18155: 18154: 18153: 18143: 18138: 18133: 18132: 18131: 18126: 18115: 18113: 18107: 18106: 18103: 18102: 18100: 18099: 18094: 18093: 18092: 18084: 18076: 18060: 18057:(Mann–Whitney) 18052: 18051: 18050: 18037: 18036: 18035: 18024: 18022: 18016: 18015: 18013: 18012: 18011: 18010: 18005: 18000: 17990: 17985: 17982:(Shapiro–Wilk) 17977: 17972: 17967: 17962: 17957: 17949: 17943: 17941: 17935: 17934: 17932: 17931: 17923: 17914: 17902: 17896: 17894:Specific tests 17890: 17889: 17886: 17885: 17883: 17882: 17877: 17872: 17866: 17864: 17858: 17857: 17855: 17854: 17849: 17848: 17847: 17837: 17836: 17835: 17825: 17819: 17817: 17811: 17810: 17808: 17807: 17806: 17805: 17800: 17790: 17785: 17780: 17775: 17770: 17764: 17762: 17756: 17755: 17753: 17752: 17747: 17746: 17745: 17740: 17739: 17738: 17733: 17718: 17717: 17716: 17711: 17706: 17701: 17690: 17688: 17679: 17673: 17672: 17670: 17669: 17664: 17659: 17658: 17657: 17647: 17642: 17641: 17640: 17630: 17629: 17628: 17623: 17618: 17608: 17603: 17598: 17597: 17596: 17591: 17586: 17570: 17569: 17568: 17563: 17558: 17548: 17547: 17546: 17541: 17531: 17530: 17529: 17519: 17518: 17517: 17507: 17502: 17497: 17491: 17489: 17479: 17478: 17466: 17465: 17462: 17461: 17458: 17457: 17455: 17454: 17449: 17444: 17439: 17433: 17431: 17425: 17424: 17422: 17421: 17416: 17411: 17405: 17403: 17399: 17398: 17396: 17395: 17390: 17385: 17380: 17375: 17370: 17365: 17359: 17357: 17351: 17350: 17348: 17347: 17345:Standard error 17342: 17337: 17332: 17331: 17330: 17325: 17314: 17312: 17306: 17305: 17303: 17302: 17297: 17292: 17287: 17282: 17277: 17275:Optimal design 17272: 17267: 17261: 17259: 17249: 17248: 17236: 17235: 17232: 17231: 17228: 17227: 17225: 17224: 17219: 17214: 17209: 17204: 17199: 17194: 17189: 17184: 17179: 17174: 17169: 17164: 17159: 17154: 17148: 17146: 17140: 17139: 17137: 17136: 17131: 17130: 17129: 17124: 17114: 17109: 17103: 17101: 17095: 17094: 17092: 17091: 17086: 17081: 17075: 17073: 17072:Summary tables 17069: 17068: 17066: 17065: 17059: 17057: 17051: 17050: 17047: 17046: 17044: 17043: 17042: 17041: 17036: 17031: 17021: 17015: 17013: 17007: 17006: 17004: 17003: 16998: 16993: 16988: 16983: 16978: 16973: 16967: 16965: 16959: 16958: 16956: 16955: 16950: 16945: 16944: 16943: 16938: 16933: 16928: 16923: 16918: 16913: 16908: 16906:Contraharmonic 16903: 16898: 16887: 16885: 16876: 16866: 16865: 16853: 16852: 16850: 16849: 16844: 16838: 16835: 16834: 16827: 16826: 16819: 16812: 16804: 16798: 16797: 16791: 16780: 16768: 16763: 16752: 16739: 16738:External links 16736: 16735: 16734: 16725: 16711: 16705: 16697:10.1007/b98835 16682: 16676: 16668:10.1007/b98835 16637: 16628: 16625: 16623: 16622: 16596: 16555: 16506: 16465: 16414: 16351: 16297: 16270: 16225: 16218: 16182: 16149: 16129: 16122: 16091: 16071: 16051: 16012: 16005:Institut Curie 15990: 15941: 15921: 15886: 15854: 15832: 15811: 15765: 15736:(3): 434–448. 15711: 15682:(2): 262–286. 15657: 15624: 15574: 15552: 15530: 15511:(3): 163–179. 15495: 15482: 15475: 15455: 15448: 15430: 15415: 15372: 15363: 15336:(1): 121–140. 15320: 15301: 15284: 15265: 15250: 15247:978-0976609704 15226: 15199: 15176: 15150: 15130: 15107: 15035: 15012:10.1038/ng.139 14986: 14961: 14934: 14909: 14898: 14883: 14870: 14844: 14791: 14784: 14764: 14737: 14724: 14712: 14694: 14682: 14668: 14639:(4): 433–459. 14620: 14607: 14586: 14574: 14541: 14525: 14518: 14498: 14479:(3): 105–117. 14463: 14423: 14416: 14398: 14372: 14302: 14276: 14263:10.1086/510127 14239:(2): 734–754. 14220: 14161: 14103: 14091: 14079: 14040: 14033: 14010: 13997: 13990: 13962: 13935:(2): 289–308. 13915: 13862: 13855: 13847:10.1007/b98835 13818: 13797:(4): 551–566. 13775: 13726: 13684: 13669: 13652:10.1.1.63.4605 13633: 13576: 13520: 13474: 13418: 13385: 13344: 13286: 13284: 13281: 13279: 13278: 13273: 13268: 13263: 13258: 13253: 13248: 13242: 13236: 13231: 13226: 13221: 13216: 13211: 13206: 13201: 13196: 13191: 13186: 13184:Factorial code 13181: 13175: 13170: 13165: 13160: 13155: 13150: 13144: 13139: 13129: 13123: 13116: 13114: 13111: 13110: 13109: 13103: 13097: 13083: 13077: 13071: 13025: 13019: 13013: 13007: 12994: 12989: 12979: 12976:.NET Framework 12969: 12959: 12945: 12935: 12925: 12907: 12897: 12891: 12885: 12879: 12869: 12855: 12845: 12839: 12831: 12828: 12820: 12817: 12804: 12801: 12797: 12796: 12784: 12774: 12762: 12742: 12722: 12702: 12699: 12696: 12676: 12665: 12653: 12630: 12610: 12590: 12587: 12584: 12581: 12561: 12549: 12546: 12538: 12535: 12533: 12530: 12499: 12496: 12461:algorithm and 12407: 12404: 12400: 12399: 12396: 12393: 12390: 12387: 12384: 12368:Main article: 12365: 12362: 12360: 12357: 12329: 12326: 12308: 12288: 12266: 12262: 12256: 12251: 12248: 12245: 12241: 12235: 12228: 12224: 12218: 12213: 12210: 12207: 12203: 12199: 12196: 12168: 12165: 12159: 12151: 12145: 12140: 12137: 12111: 12108: 12075: 12072: 12070: 12067: 12065:in the brain. 11990: 11987: 11971:stress testing 11960:equity markets 11862:other problems 11849: 11846: 11828: 11825: 11799:Cavalli-Sforza 11794: 11791: 11763: 11760: 11743: 11740: 11714: 11711: 11709: 11706: 11698:Ludovic Lebart 11686: 11685: 11682: 11679: 11666: 11663: 11657: 11654: 11626: 11620: 11586: 11579: 11549: 11546: 11488:, such as the 11299: 11288: 11285: 11243: 11240: 11220: 11200: 11197: 11194: 11191: 11188: 11168: 11165: 11162: 11151: 11150: 11133: 11130: 11126: 11122: 11119: 11116: 11113: 11110: 11107: 11104: 11101: 11099: 11097: 11092: 11088: 11084: 11079: 11075: 11071: 11068: 11065: 11062: 11059: 11056: 11053: 11051: 11049: 11046: 11041: 11037: 11031: 11027: 11020: 11017: 11014: 11011: 11008: 11005: 11002: 11000: 10998: 10995: 10990: 10986: 10982: 10979: 10976: 10970: 10967: 10964: 10961: 10958: 10955: 10952: 10950: 10948: 10945: 10942: 10939: 10936: 10933: 10930: 10929: 10906: 10859: 10856: 10853: 10826: 10823: 10822: 10821: 10808: 10797: 10796: 10784: 10780: 10776: 10772: 10768: 10751: 10750: 10749: 10738: 10735: 10728: 10724: 10718: 10714: 10672: 10661: 10658: 10655: 10652: 10649: 10646: 10626: 10623: 10620: 10617: 10614: 10611: 10608: 10604: 10601: 10598: 10595: 10592: 10589: 10586: 10575: 10572: 10568: 10564: 10559: 10556: 10552: 10515: 10514: 10513: 10502: 10499: 10496: 10493: 10490: 10487: 10484: 10473: 10470: 10466: 10460: 10455: 10452: 10449: 10445: 10441: 10436: 10432: 10399: 10398: 10397: 10394: 10374: 10373: 10372: 10353: 10342: 10311: 10300: 10297: 10294: 10291: 10282: 10279: 10274: 10271: 10267: 10238: 10235: 10232: 10221: 10217: 10213: 10208: 10205: 10201: 10177: 10138:matrix algebra 10104: 10100: 10096: 10091: 10085: 10082: 10077: 10043: 10042: 10041: 10023: 9999: 9978: 9972: 9967: 9959: 9956: 9953: 9949: 9944: 9940: 9904: 9892: 9891: 9880: 9877: 9874: 9871: 9868: 9865: 9862: 9851: 9848: 9843: 9839: 9805: 9800: 9794: 9790: 9786: 9782: 9778: 9754: 9736: 9731: 9709: 9708: 9707: 9694: 9691: 9687: 9681: 9676: 9673: 9670: 9666: 9660: 9657: 9652: 9647: 9643: 9629:of dimensions 9623: 9607: 9606: 9605: 9596:of dimensions 9590: 9572: 9567: 9562: 9557: 9552: 9520: 9515: 9491: 9486: 9481: 9476: 9471: 9422: 9418: 9414: 9410: 9407: 9404: 9400: 9396: 9378:Karhunen–Loève 9346: 9343: 9329: 9307: 9281: 9269: 9268: 9257: 9254: 9250: 9246: 9242: 9238: 9235: 9232: 9229: 9225: 9221: 9217: 9213: 9210: 9182: 9156: 9131: 9125: 9120: 9115: 9110: 9106: 9084: 9063: 9059: 9055: 9051: 9047: 9044: 9020: 8998: 8973: 8951: 8929: 8917: 8916: 8905: 8901: 8897: 8893: 8889: 8885: 8866: 8863: 8829: 8826: 8813: 8810: 8807: 8804: 8801: 8798: 8795: 8792: 8789: 8786: 8781: 8777: 8772: 8768: 8764: 8741: 8737: 8716: 8695: 8691: 8687: 8681: 8677: 8671: 8667: 8657:increases, as 8646: 8625: 8621: 8617: 8611: 8607: 8601: 8597: 8575: 8571: 8567: 8561: 8557: 8551: 8547: 8527: 8526: 8513: 8508: 8505: 8501: 8495: 8491: 8485: 8480: 8477: 8474: 8470: 8466: 8463: 8458: 8454: 8450: 8447: 8444: 8426: 8425: 8424: 8423: 8411: 8407: 8403: 8397: 8393: 8387: 8383: 8379: 8376: 8373: 8369: 8365: 8361: 8355: 8351: 8345: 8341: 8337: 8333: 8307: 8288: 8287: 8274: 8247: 8242: 8237: 8215: 8210: 8205: 8200: 8195: 8191: 8170: 8165: 8160: 8155: 8152: 8149: 8127: 8122: 8099: 8095: 8091: 8087: 8084: 8073: 8072: 8071: 8060: 8055: 8052: 8047: 8044: 8022: 8019: 8018: 8005: 7984: 7980: 7957: 7954: 7949: 7929: 7902: 7897: 7873: 7868: 7863: 7859: 7838: 7833: 7828: 7823: 7820: 7817: 7795: 7790: 7768: 7739: 7734: 7728: 7725: 7720: 7715: 7710: 7678: 7675: 7649: 7638: 7637: 7636: 7625: 7620: 7617: 7612: 7609: 7575: 7568: 7565: 7563: 7560: 7554: 7553: 7542: 7539: 7536: 7533: 7530: 7510: 7507: 7504: 7501: 7498: 7488: 7477: 7474: 7471: 7461: 7446: 7435: 7430: 7427: 7423: 7419: 7416: 7412: 7400: 7399: 7388: 7385: 7382: 7379: 7376: 7356: 7353: 7350: 7347: 7344: 7334: 7323: 7320: 7317: 7307: 7293: 7282: 7277: 7274: 7270: 7266: 7263: 7259: 7247: 7246: 7235: 7232: 7229: 7226: 7222: 7219: 7198: 7195: 7192: 7189: 7186: 7176: 7165: 7162: 7159: 7149: 7136: 7109: 7098: 7092: 7089: 7085: 7081: 7077: 7074: 7070: 7058: 7057: 7046: 7043: 7040: 7037: 7033: 7030: 7009: 7006: 7003: 7000: 6997: 6987: 6976: 6973: 6970: 6960: 6949: 6938: 6932: 6929: 6925: 6921: 6917: 6914: 6910: 6898: 6897: 6886: 6883: 6880: 6877: 6873: 6870: 6849: 6846: 6843: 6840: 6837: 6827: 6816: 6813: 6810: 6800: 6795: 6784: 6778: 6775: 6771: 6767: 6763: 6760: 6756: 6744: 6743: 6732: 6729: 6726: 6723: 6719: 6716: 6695: 6692: 6689: 6686: 6683: 6673: 6662: 6659: 6656: 6646: 6641: 6630: 6624: 6621: 6617: 6613: 6609: 6606: 6602: 6590: 6589: 6578: 6575: 6572: 6569: 6566: 6546: 6543: 6540: 6537: 6534: 6524: 6513: 6510: 6507: 6497: 6487: 6476: 6471: 6468: 6464: 6460: 6457: 6453: 6441: 6440: 6429: 6426: 6423: 6420: 6417: 6397: 6394: 6391: 6388: 6385: 6375: 6364: 6361: 6358: 6348: 6338: 6327: 6322: 6319: 6315: 6311: 6308: 6304: 6292: 6291: 6280: 6277: 6274: 6271: 6268: 6258: 6247: 6244: 6241: 6231: 6228: 6217: 6212: 6208: 6204: 6201: 6197: 6185: 6184: 6173: 6170: 6167: 6164: 6161: 6151: 6140: 6137: 6134: 6124: 6113: 6102: 6097: 6093: 6089: 6086: 6082: 6070: 6069: 6058: 6055: 6052: 6049: 6046: 6036: 6025: 6022: 6019: 6009: 5998: 5987: 5982: 5978: 5974: 5971: 5967: 5955: 5954: 5949: 5938: 5935: 5932: 5922: 5911: 5908: 5905: 5902: 5899: 5888: 5877: 5866: 5865: 5860: 5849: 5846: 5843: 5833: 5830: 5819: 5808: 5807: 5802: 5791: 5788: 5785: 5775: 5772: 5761: 5750: 5749: 5738: 5735: 5732: 5729: 5726: 5706: 5703: 5700: 5697: 5694: 5684: 5673: 5670: 5667: 5657: 5654: 5643: 5638: 5635: 5631: 5627: 5624: 5620: 5608: 5607: 5604: 5601: 5598: 5591: 5588: 5500:of the matrix 5489: 5486: 5478:Frobenius norm 5459: 5458: 5445: 5440: 5434: 5430: 5425: 5420: 5413: 5408: 5403: 5398: 5393: 5376: 5327: 5326: 5310: 5305: 5301: 5298: 5296: 5294: 5290: 5283: 5277: 5271: 5266: 5262: 5259: 5257: 5255: 5251: 5246: 5242: 5239: 5237: 5234: 5230: 5229: 5196: 5179: 5161: 5122: 5115: 5109: 5104: 5098: 5091: 5088: 5054: 5051: 5036: 5035: 5017: 5011: 5004: 4997: 4994: 4986: 4982: 4979: 4977: 4975: 4969: 4963: 4957: 4950: 4944: 4938: 4934: 4931: 4929: 4927: 4921: 4915: 4909: 4904: 4897: 4891: 4883: 4877: 4871: 4867: 4864: 4862: 4859: 4853: 4848: 4843: 4842: 4769: 4748: 4747: 4734: 4729: 4723: 4718: 4714: 4710: 4683:Main article: 4680: 4677: 4638:Similarly, in 4589: 4584: 4580: 4574: 4569: 4564: 4560: 4556: 4534: 4529: 4525: 4519: 4514: 4509: 4502: 4497: 4492: 4487: 4482: 4476: 4472: 4438: 4434: 4403: 4398: 4393: 4388: 4385: 4382: 4377: 4372: 4367: 4364: 4361: 4358: 4352: 4346: 4342: 4338: 4335: 4322:rows but only 4315: 4309: 4308: 4295: 4290: 4284: 4280: 4275: 4270: 4233: 4215: 4212: 4203: 4192: 4183: 4175: 4170: 4162: 4154: 4139: 4121: 4111: 4110: 4098: 4094: 4090: 4083: 4077: 4070: 4064: 4057: 4051: 4046: 4042: 4037: 4030: 4024: 4008: 4007: 3993: 3987: 3981: 3976: 3972: 3968: 3961: 3955: 3950: 3946: 3919: 3908: 3897: 3891: 3890: 3873: 3870: 3867: 3862: 3854: 3848: 3845: 3842: 3837: 3830: 3827: 3824: 3820: 3816: 3813: 3811: 3809: 3804: 3801: 3798: 3793: 3786: 3783: 3780: 3776: 3769: 3763: 3760: 3757: 3752: 3747: 3744: 3742: 3740: 3735: 3732: 3729: 3724: 3718: 3711: 3705: 3697: 3691: 3688: 3685: 3680: 3675: 3672: 3670: 3668: 3665: 3660: 3657: 3654: 3649: 3643: 3639: 3633: 3628: 3622: 3619: 3616: 3611: 3605: 3601: 3598: 3595: 3593: 3591: 3586: 3583: 3580: 3575: 3572: 3567: 3562: 3559: 3556: 3551: 3548: 3543: 3540: 3537: 3536: 3500: 3497: 3458: 3457: 3445: 3440: 3436: 3432: 3394: 3383: 3372: 3361: 3350: 3339: 3325: 3314: 3287: 3286: 3274: 3266: 3260: 3255: 3247: 3241: 3234: 3231: 3221: 3215: 3208: 3205: 3195: 3189: 3180: 3176: 3173: 3170: 3167: 3163: 3158: 3153: 3148: 3142: 3135: 3132: 3124: 3119: 3115: 3110: 3107: 3103: 3099: 3095: 3088: 3085: 3082: 3078: 3075: 3072: 3066: 3061: 3058: 3055: 3050: 3034: 3033: 3019: 3013: 3010: 3007: 3002: 2995: 2992: 2989: 2984: 2978: 2972: 2969: 2966: 2961: 2958: 2955: 2951: 2947: 2943: 2939: 2934: 2927: 2924: 2893: 2890: 2885: 2878: 2867: 2860: 2849: 2838: 2827: 2820: 2782: 2781: 2769: 2762: 2755: 2749: 2741: 2738: 2731: 2725: 2717: 2711: 2703: 2699: 2696: 2693: 2690: 2685: 2682: 2679: 2674: 2657: 2651: 2650: 2638: 2633: 2630: 2623: 2617: 2609: 2603: 2597: 2591: 2588: 2584: 2580: 2576: 2571: 2567: 2564: 2561: 2557: 2552: 2547: 2543: 2540: 2536: 2531: 2525: 2522: 2518: 2514: 2510: 2505: 2501: 2498: 2495: 2490: 2487: 2484: 2479: 2463: 2462: 2450: 2444: 2439: 2434: 2430: 2425: 2422: 2419: 2414: 2408: 2401: 2397: 2392: 2385: 2382: 2379: 2375: 2371: 2367: 2363: 2360: 2357: 2353: 2347: 2342: 2339: 2336: 2332: 2326: 2322: 2318: 2313: 2309: 2304: 2297: 2294: 2291: 2287: 2283: 2279: 2275: 2272: 2269: 2264: 2261: 2258: 2253: 2236: 2229: 2226: 2211: 2208: 2205: 2198: 2194: 2188: 2183: 2180: 2174: 2149: 2146: 2143: 2136: 2132: 2126: 2121: 2118: 2112: 2087: 2084: 2081: 2074: 2070: 2064: 2059: 2056: 2050: 2036: 2035: 2023: 2018: 2014: 2010: 1983: 1963: 1925: 1921: 1917: 1914: 1911: 1906: 1902: 1890: 1889: 1878: 1875: 1872: 1869: 1866: 1863: 1860: 1856: 1853: 1850: 1847: 1844: 1841: 1838: 1833: 1830: 1827: 1820: 1817: 1814: 1809: 1804: 1799: 1796: 1793: 1788: 1783: 1778: 1775: 1772: 1765: 1761: 1733: 1730: 1727: 1723: 1717: 1713: 1709: 1706: 1703: 1698: 1694: 1690: 1687: 1682: 1679: 1676: 1671: 1640: 1637: 1634: 1630: 1624: 1620: 1616: 1613: 1610: 1605: 1601: 1597: 1594: 1589: 1586: 1583: 1578: 1554: 1551: 1548: 1544: 1538: 1534: 1530: 1527: 1524: 1519: 1515: 1511: 1508: 1503: 1500: 1497: 1492: 1466: 1444:empirical mean 1423: 1420: 1417: 1386: 1383: 1334: 1331: 1272:Karhunen–Loève 1255: 1252: 1218:of the data's 1200: 1197: 1194: 1174: 1153: 1132: 1120: 1117: 1054: 1051: 1048: 1024: 1001: 948: 947: 945: 944: 937: 930: 922: 919: 918: 915: 914: 909: 908: 907: 897: 891: 888: 887: 884: 883: 880: 879: 874: 869: 864: 859: 854: 849: 843: 840: 839: 836: 835: 832: 831: 826: 821: 816: 814:Occam learning 811: 806: 801: 796: 790: 787: 786: 783: 782: 779: 778: 773: 771:Learning curve 768: 763: 757: 754: 753: 750: 749: 746: 745: 740: 735: 730: 724: 721: 720: 717: 716: 713: 712: 711: 710: 700: 695: 690: 684: 679: 678: 675: 674: 671: 670: 664: 659: 654: 649: 648: 647: 637: 632: 631: 630: 625: 620: 615: 605: 600: 595: 590: 589: 588: 578: 577: 576: 571: 566: 561: 551: 546: 541: 535: 530: 529: 526: 525: 522: 521: 516: 511: 503: 497: 492: 491: 488: 487: 484: 483: 482: 481: 476: 471: 460: 455: 454: 451: 450: 447: 446: 441: 436: 431: 426: 421: 416: 411: 406: 400: 395: 394: 391: 390: 387: 386: 381: 376: 370: 365: 360: 352: 347: 342: 336: 331: 330: 327: 326: 323: 322: 317: 312: 307: 302: 297: 292: 287: 279: 278: 277: 272: 267: 257: 255:Decision trees 252: 246: 232:classification 222: 221: 220: 217: 216: 213: 212: 207: 202: 197: 192: 187: 182: 177: 172: 167: 162: 157: 152: 147: 142: 137: 132: 127: 125:Classification 121: 118: 117: 114: 113: 110: 109: 104: 99: 94: 89: 84: 82:Batch learning 79: 74: 69: 64: 59: 54: 49: 43: 40: 39: 36: 35: 24: 23: 15: 9: 6: 4: 3: 2: 19153: 19142: 19139: 19137: 19134: 19133: 19131: 19116: 19115: 19106: 19104: 19103: 19094: 19092: 19091: 19086: 19080: 19078: 19077: 19068: 19067: 19064: 19050: 19047: 19045: 19044:Geostatistics 19042: 19040: 19037: 19035: 19032: 19030: 19027: 19026: 19024: 19022: 19018: 19012: 19011:Psychometrics 19009: 19007: 19004: 19002: 18999: 18997: 18994: 18992: 18989: 18987: 18984: 18982: 18979: 18977: 18974: 18972: 18969: 18967: 18964: 18963: 18961: 18959: 18955: 18949: 18946: 18944: 18941: 18939: 18935: 18932: 18930: 18927: 18925: 18922: 18920: 18917: 18916: 18914: 18912: 18908: 18902: 18899: 18897: 18894: 18892: 18888: 18885: 18883: 18880: 18879: 18877: 18875: 18874:Biostatistics 18871: 18867: 18863: 18858: 18854: 18836: 18835:Log-rank test 18833: 18832: 18830: 18826: 18820: 18817: 18816: 18814: 18812: 18808: 18802: 18799: 18797: 18794: 18792: 18789: 18787: 18784: 18783: 18781: 18779: 18775: 18772: 18770: 18766: 18756: 18753: 18751: 18748: 18746: 18743: 18741: 18738: 18736: 18733: 18732: 18730: 18728: 18724: 18718: 18715: 18713: 18710: 18708: 18706:(Box–Jenkins) 18702: 18700: 18697: 18695: 18692: 18688: 18685: 18684: 18683: 18680: 18679: 18677: 18675: 18671: 18665: 18662: 18660: 18659:Durbin–Watson 18657: 18655: 18649: 18647: 18644: 18642: 18641:Dickey–Fuller 18639: 18638: 18636: 18632: 18626: 18623: 18621: 18618: 18616: 18615:Cointegration 18613: 18611: 18608: 18606: 18603: 18601: 18598: 18596: 18593: 18591: 18590:Decomposition 18588: 18587: 18585: 18581: 18578: 18576: 18572: 18562: 18559: 18558: 18557: 18554: 18553: 18552: 18549: 18545: 18542: 18541: 18540: 18537: 18535: 18532: 18530: 18527: 18525: 18522: 18520: 18517: 18515: 18512: 18510: 18507: 18505: 18502: 18501: 18499: 18497: 18493: 18487: 18484: 18482: 18479: 18477: 18474: 18472: 18469: 18467: 18464: 18462: 18461:Cohen's kappa 18459: 18458: 18456: 18454: 18450: 18446: 18442: 18438: 18434: 18430: 18425: 18421: 18407: 18404: 18402: 18399: 18397: 18394: 18392: 18389: 18388: 18386: 18384: 18380: 18374: 18370: 18366: 18360: 18358: 18355: 18354: 18352: 18350: 18346: 18340: 18337: 18335: 18332: 18330: 18327: 18325: 18322: 18320: 18317: 18315: 18314:Nonparametric 18312: 18310: 18307: 18306: 18304: 18300: 18294: 18291: 18289: 18286: 18284: 18281: 18279: 18276: 18275: 18273: 18271: 18267: 18261: 18258: 18256: 18253: 18251: 18248: 18246: 18243: 18241: 18238: 18237: 18235: 18233: 18229: 18223: 18220: 18218: 18215: 18213: 18210: 18208: 18205: 18204: 18202: 18200: 18196: 18192: 18185: 18182: 18180: 18177: 18176: 18172: 18168: 18152: 18149: 18148: 18147: 18144: 18142: 18139: 18137: 18134: 18130: 18127: 18125: 18122: 18121: 18120: 18117: 18116: 18114: 18112: 18108: 18098: 18095: 18091: 18085: 18083: 18077: 18075: 18069: 18068: 18067: 18064: 18063:Nonparametric 18061: 18059: 18053: 18049: 18046: 18045: 18044: 18038: 18034: 18033:Sample median 18031: 18030: 18029: 18026: 18025: 18023: 18021: 18017: 18009: 18006: 18004: 18001: 17999: 17996: 17995: 17994: 17991: 17989: 17986: 17984: 17978: 17976: 17973: 17971: 17968: 17966: 17963: 17961: 17958: 17956: 17954: 17950: 17948: 17945: 17944: 17942: 17940: 17936: 17930: 17928: 17924: 17922: 17920: 17915: 17913: 17908: 17904: 17903: 17900: 17897: 17895: 17891: 17881: 17878: 17876: 17873: 17871: 17868: 17867: 17865: 17863: 17859: 17853: 17850: 17846: 17843: 17842: 17841: 17838: 17834: 17831: 17830: 17829: 17826: 17824: 17821: 17820: 17818: 17816: 17812: 17804: 17801: 17799: 17796: 17795: 17794: 17791: 17789: 17786: 17784: 17781: 17779: 17776: 17774: 17771: 17769: 17766: 17765: 17763: 17761: 17757: 17751: 17748: 17744: 17741: 17737: 17734: 17732: 17729: 17728: 17727: 17724: 17723: 17722: 17719: 17715: 17712: 17710: 17707: 17705: 17702: 17700: 17697: 17696: 17695: 17692: 17691: 17689: 17687: 17683: 17680: 17678: 17674: 17668: 17665: 17663: 17660: 17656: 17653: 17652: 17651: 17648: 17646: 17643: 17639: 17638:loss function 17636: 17635: 17634: 17631: 17627: 17624: 17622: 17619: 17617: 17614: 17613: 17612: 17609: 17607: 17604: 17602: 17599: 17595: 17592: 17590: 17587: 17585: 17579: 17576: 17575: 17574: 17571: 17567: 17564: 17562: 17559: 17557: 17554: 17553: 17552: 17549: 17545: 17542: 17540: 17537: 17536: 17535: 17532: 17528: 17525: 17524: 17523: 17520: 17516: 17513: 17512: 17511: 17508: 17506: 17503: 17501: 17498: 17496: 17493: 17492: 17490: 17488: 17484: 17480: 17476: 17471: 17467: 17453: 17450: 17448: 17445: 17443: 17440: 17438: 17435: 17434: 17432: 17430: 17426: 17420: 17417: 17415: 17412: 17410: 17407: 17406: 17404: 17400: 17394: 17391: 17389: 17386: 17384: 17381: 17379: 17376: 17374: 17371: 17369: 17366: 17364: 17361: 17360: 17358: 17356: 17352: 17346: 17343: 17341: 17340:Questionnaire 17338: 17336: 17333: 17329: 17326: 17324: 17321: 17320: 17319: 17316: 17315: 17313: 17311: 17307: 17301: 17298: 17296: 17293: 17291: 17288: 17286: 17283: 17281: 17278: 17276: 17273: 17271: 17268: 17266: 17263: 17262: 17260: 17258: 17254: 17250: 17246: 17241: 17237: 17223: 17220: 17218: 17215: 17213: 17210: 17208: 17205: 17203: 17200: 17198: 17195: 17193: 17190: 17188: 17185: 17183: 17180: 17178: 17175: 17173: 17170: 17168: 17167:Control chart 17165: 17163: 17160: 17158: 17155: 17153: 17150: 17149: 17147: 17145: 17141: 17135: 17132: 17128: 17125: 17123: 17120: 17119: 17118: 17115: 17113: 17110: 17108: 17105: 17104: 17102: 17100: 17096: 17090: 17087: 17085: 17082: 17080: 17077: 17076: 17074: 17070: 17064: 17061: 17060: 17058: 17056: 17052: 17040: 17037: 17035: 17032: 17030: 17027: 17026: 17025: 17022: 17020: 17017: 17016: 17014: 17012: 17008: 17002: 16999: 16997: 16994: 16992: 16989: 16987: 16984: 16982: 16979: 16977: 16974: 16972: 16969: 16968: 16966: 16964: 16960: 16954: 16951: 16949: 16946: 16942: 16939: 16937: 16934: 16932: 16929: 16927: 16924: 16922: 16919: 16917: 16914: 16912: 16909: 16907: 16904: 16902: 16899: 16897: 16894: 16893: 16892: 16889: 16888: 16886: 16884: 16880: 16877: 16875: 16871: 16867: 16863: 16858: 16854: 16848: 16845: 16843: 16840: 16839: 16836: 16832: 16825: 16820: 16818: 16813: 16811: 16806: 16805: 16802: 16796: 16792: 16790: 16786: 16781: 16778: 16774: 16769: 16767: 16764: 16762: 16758: 16753: 16751: 16747: 16742: 16741: 16732: 16731: 16726: 16724: 16720: 16716: 16712: 16708: 16702: 16698: 16694: 16690: 16689: 16683: 16679: 16673: 16669: 16665: 16660: 16655: 16651: 16646: 16645: 16638: 16635: 16631: 16630: 16610: 16606: 16600: 16592: 16588: 16583: 16578: 16574: 16570: 16566: 16559: 16551: 16547: 16542: 16537: 16533: 16529: 16525: 16521: 16517: 16510: 16501: 16496: 16492: 16488: 16484: 16480: 16476: 16469: 16461: 16457: 16452: 16447: 16442: 16437: 16433: 16429: 16425: 16418: 16410: 16406: 16401: 16396: 16391: 16386: 16382: 16378: 16374: 16370: 16366: 16362: 16355: 16347: 16343: 16339: 16335: 16331: 16327: 16322: 16317: 16313: 16309: 16301: 16293: 16289: 16285: 16281: 16274: 16266: 16262: 16258: 16254: 16249: 16244: 16240: 16236: 16229: 16221: 16215: 16211: 16207: 16202: 16197: 16193: 16186: 16178: 16172: 16164: 16160: 16153: 16142: 16141: 16133: 16125: 16119: 16115: 16111: 16104: 16103: 16095: 16084: 16083: 16075: 16069: 16065: 16061: 16055: 16047: 16043: 16039: 16035: 16034: 16026: 16022: 16016: 16007: 16006: 16001: 15998:Zinovyev, A. 15994: 15988: 15983: 15979: 15975: 15971: 15967: 15963: 15959: 15955: 15954: 15945: 15938: 15934: 15930: 15925: 15916: 15911: 15907: 15903: 15902: 15897: 15890: 15882: 15878: 15877: 15869: 15865: 15858: 15850: 15843: 15836: 15827: 15822: 15815: 15807: 15803: 15798: 15793: 15789: 15785: 15784: 15776: 15769: 15761: 15757: 15753: 15749: 15744: 15739: 15735: 15731: 15730: 15722: 15715: 15707: 15703: 15699: 15695: 15690: 15689:10.1.1.62.580 15685: 15681: 15677: 15676: 15668: 15661: 15653: 15649: 15644: 15639: 15635: 15628: 15614: 15610: 15605: 15600: 15597:(1–3): 9–33. 15596: 15592: 15585: 15578: 15570: 15563: 15556: 15548: 15541: 15534: 15526: 15522: 15518: 15514: 15510: 15506: 15499: 15492: 15486: 15478: 15476:9781402022357 15472: 15468: 15467: 15459: 15451: 15445: 15441: 15434: 15426: 15419: 15411: 15407: 15403: 15399: 15395: 15391: 15387: 15383: 15376: 15367: 15359: 15355: 15351: 15347: 15343: 15339: 15335: 15331: 15324: 15318: 15314: 15308: 15306: 15298: 15294: 15288: 15281: 15277: 15276: 15269: 15263: 15259: 15254: 15248: 15244: 15240: 15236: 15230: 15224: 15220: 15216: 15212: 15206: 15204: 15197: 15193: 15189: 15183: 15181: 15165: 15161: 15154: 15143: 15142: 15134: 15126: 15122: 15118: 15111: 15103: 15099: 15095: 15091: 15086: 15081: 15076: 15071: 15067: 15063: 15059: 15055: 15054: 15049: 15045: 15039: 15031: 15027: 15022: 15017: 15013: 15009: 15006:(5): 646–49. 15005: 15001: 14997: 14990: 14976: 14972: 14965: 14957: 14953: 14950:(1): 88–100. 14949: 14945: 14938: 14923: 14919: 14913: 14907: 14902: 14894: 14887: 14880: 14874: 14866: 14862: 14855: 14848: 14840: 14836: 14832: 14828: 14824: 14820: 14815: 14810: 14806: 14802: 14795: 14787: 14785:9780203909805 14781: 14777: 14776: 14768: 14760: 14756: 14752: 14748: 14741: 14734: 14733:Sara A. Solla 14728: 14721: 14716: 14708: 14704: 14698: 14691: 14686: 14678: 14672: 14664: 14660: 14656: 14652: 14647: 14642: 14638: 14634: 14630: 14624: 14617: 14611: 14596: 14590: 14584: 14578: 14570: 14566: 14561: 14556: 14552: 14545: 14536: 14529: 14521: 14519:9781461240167 14515: 14511: 14510: 14502: 14494: 14490: 14486: 14482: 14478: 14474: 14473:IEEE Computer 14467: 14459: 14455: 14450: 14445: 14441: 14437: 14430: 14428: 14419: 14417:9781118727966 14413: 14409: 14402: 14386: 14382: 14376: 14368: 14364: 14359: 14354: 14350: 14346: 14341: 14336: 14332: 14328: 14324: 14317: 14315: 14313: 14311: 14309: 14307: 14297: 14292: 14285: 14283: 14281: 14272: 14268: 14264: 14260: 14256: 14252: 14247: 14242: 14238: 14234: 14227: 14225: 14216: 14212: 14207: 14202: 14198: 14194: 14189: 14184: 14180: 14176: 14172: 14165: 14157: 14153: 14149: 14145: 14141: 14137: 14132: 14127: 14123: 14119: 14112: 14110: 14108: 14101: 14095: 14089: 14083: 14075: 14071: 14067: 14063: 14059: 14055: 14051: 14044: 14036: 14030: 14026: 14025: 14020: 14014: 14007: 14001: 13993: 13987: 13983: 13979: 13975: 13974: 13966: 13958: 13952: 13943: 13938: 13934: 13930: 13926: 13919: 13911: 13907: 13903: 13899: 13895: 13891: 13886: 13881: 13877: 13873: 13866: 13858: 13852: 13848: 13844: 13840: 13839: 13831: 13829: 13827: 13825: 13823: 13813: 13808: 13804: 13800: 13796: 13792: 13791: 13786: 13779: 13771: 13767: 13763: 13759: 13755: 13751: 13750: 13742: 13738: 13737: 13730: 13722: 13718: 13714: 13710: 13706: 13702: 13698: 13694: 13688: 13680: 13676: 13672: 13666: 13662: 13658: 13653: 13648: 13644: 13637: 13629: 13625: 13621: 13617: 13613: 13609: 13604: 13599: 13595: 13591: 13587: 13580: 13572: 13568: 13564: 13560: 13556: 13552: 13547: 13542: 13538: 13534: 13527: 13525: 13515: 13510: 13505: 13500: 13496: 13492: 13488: 13481: 13479: 13470: 13466: 13462: 13458: 13454: 13450: 13445: 13440: 13436: 13432: 13425: 13423: 13414: 13410: 13405: 13400: 13396: 13389: 13380: 13375: 13371: 13367: 13363: 13359: 13355: 13348: 13340: 13336: 13331: 13326: 13322: 13318: 13314: 13310: 13306: 13302: 13298: 13291: 13287: 13277: 13274: 13272: 13269: 13267: 13264: 13262: 13259: 13257: 13254: 13252: 13249: 13246: 13243: 13240: 13237: 13235: 13232: 13230: 13227: 13225: 13222: 13220: 13217: 13215: 13212: 13210: 13207: 13205: 13202: 13200: 13197: 13195: 13192: 13190: 13187: 13185: 13182: 13180:(Wikiversity) 13179: 13176: 13174: 13171: 13169: 13166: 13164: 13161: 13159: 13156: 13154: 13151: 13148: 13145: 13143: 13140: 13137: 13133: 13130: 13127: 13124: 13121: 13118: 13117: 13107: 13104: 13101: 13098: 13087: 13084: 13081: 13078: 13075: 13072: 13049: 13033: 13029: 13026: 13023: 13020: 13017: 13014: 13011: 13008: 12998: 12995: 12993: 12990: 12983: 12980: 12977: 12973: 12970: 12963: 12960: 12957: 12953: 12949: 12946: 12943: 12939: 12936: 12933: 12929: 12926: 12911: 12908: 12905: 12901: 12898: 12895: 12892: 12889: 12886: 12883: 12880: 12873: 12870: 12859: 12856: 12853: 12849: 12846: 12843: 12840: 12837: 12834: 12833: 12827: 12825: 12816: 12814: 12809: 12800: 12782: 12775: 12760: 12740: 12720: 12713:zeroes where 12700: 12697: 12694: 12674: 12666: 12651: 12644: 12643: 12642: 12628: 12608: 12588: 12585: 12582: 12579: 12559: 12545: 12543: 12529: 12527: 12523: 12521: 12516: 12514: 12510: 12505: 12495: 12493: 12489: 12485: 12481: 12479: 12475: 12470: 12468: 12464: 12460: 12456: 12452: 12451:approximation 12448: 12447: 12441: 12440:Trevor Hastie 12437: 12429: 12424: 12421: 12420:breast cancer 12417: 12416:visualization 12412: 12406:Nonlinear PCA 12403: 12397: 12394: 12391: 12388: 12385: 12382: 12381: 12380: 12377: 12371: 12356: 12352: 12349: 12347: 12338: 12334: 12325: 12323: 12306: 12286: 12264: 12260: 12254: 12249: 12246: 12243: 12239: 12226: 12222: 12216: 12211: 12208: 12205: 12201: 12197: 12194: 12184: 12182: 12173: 12164: 12156: 12136: 12134: 12129: 12125: 12116: 12107: 12105: 12101: 12097: 12092: 12088: 12084: 12080: 12066: 12064: 12060: 12055: 12053: 12049: 12048:extracellular 12045: 12044:Spike sorting 12040: 12037: 12033: 12029: 12025: 12021: 12016: 12012: 12008: 12004: 12000: 11996: 11986: 11984: 11980: 11976: 11972: 11967: 11965: 11961: 11957: 11953: 11949: 11945: 11941: 11936: 11934: 11930: 11926: 11922: 11918: 11914: 11910: 11906: 11902: 11898: 11894: 11890: 11886: 11882: 11878: 11874: 11869: 11867: 11863: 11859: 11855: 11845: 11842: 11837: 11833: 11824: 11822: 11817: 11813: 11807: 11803: 11800: 11790: 11788: 11784: 11779: 11776: 11771: 11769: 11759: 11755: 11751: 11749: 11739: 11737: 11733: 11732:psychometrics 11729: 11725: 11721: 11705: 11703: 11699: 11695: 11691: 11683: 11680: 11677: 11676: 11675: 11673: 11662: 11653: 11651: 11646: 11641: 11637: 11632: 11625: 11619: 11614: 11611: 11607: 11601: 11596: 11592: 11585: 11578: 11574: 11570: 11566: 11562: 11558: 11554: 11545: 11543: 11539: 11534: 11528: 11522: 11516: 11510: 11504: 11497: 11495: 11491: 11487: 11483: 11478: 11464: 11458: 11452: 11446: 11441: 11436: 11430: 11424: 11419: 11412: 11401: 11397: 11393: 11387: 11383: 11380:error = |λ ⋅ 11374: 11371: 11365: 11361: 11357: 11353: 11349: 11344: 11340: 11337:for each row 11328: 11324:times: 11319: 11315: 11311: 11303: 11298: 11295: 11284: 11281: 11274: 11269: 11264: 11258: 11249: 11239: 11237: 11232: 11218: 11195: 11189: 11186: 11163: 11131: 11128: 11124: 11117: 11111: 11108: 11105: 11102: 11100: 11090: 11086: 11077: 11073: 11069: 11063: 11057: 11054: 11052: 11039: 11035: 11029: 11025: 11018: 11015: 11009: 11003: 11001: 10988: 10980: 10977: 10968: 10965: 10959: 10953: 10951: 10943: 10940: 10934: 10931: 10920: 10919: 10918: 10904: 10895: 10893: 10889: 10885: 10882: 10878: 10874: 10854: 10842: 10840: 10836: 10832: 10778: 10770: 10757: 10756: 10755: 10752: 10736: 10733: 10726: 10722: 10716: 10712: 10701: 10697: 10693: 10689: 10685: 10681: 10677: 10673: 10659: 10656: 10653: 10650: 10647: 10644: 10624: 10621: 10618: 10615: 10612: 10609: 10606: 10602: 10599: 10596: 10593: 10590: 10587: 10584: 10573: 10570: 10566: 10562: 10557: 10554: 10550: 10541: 10537: 10533: 10529: 10525: 10521: 10520: 10519: 10516: 10500: 10497: 10494: 10491: 10488: 10485: 10482: 10471: 10468: 10464: 10458: 10453: 10450: 10447: 10443: 10439: 10434: 10430: 10421: 10417: 10413: 10409: 10405: 10404: 10403: 10400: 10395: 10392: 10388: 10384: 10380: 10379: 10378: 10375: 10370: 10366: 10362: 10358: 10354: 10351: 10347: 10343: 10340: 10336: 10332: 10328: 10324: 10320: 10316: 10312: 10298: 10295: 10292: 10289: 10280: 10277: 10272: 10269: 10265: 10256: 10252: 10236: 10233: 10230: 10219: 10215: 10211: 10206: 10203: 10199: 10190: 10186: 10182: 10178: 10175: 10171: 10167: 10163: 10159: 10155: 10151: 10147: 10143: 10139: 10135: 10131: 10127: 10123: 10119: 10098: 10083: 10080: 10065: 10061: 10057: 10053: 10049: 10048: 10047: 10044: 10039: 10035: 10029: 10024: 10021: 10017: 10014:operator. If 10013: 9997: 9970: 9957: 9954: 9951: 9947: 9942: 9929: 9925: 9922: 9918: 9914: 9910: 9909: 9908: 9905: 9903: 9901: 9897: 9878: 9875: 9872: 9869: 9866: 9863: 9860: 9849: 9846: 9841: 9837: 9826: 9821: 9803: 9788: 9780: 9767: 9763: 9759: 9755: 9752: 9734: 9719: 9718: 9717: 9713: 9710: 9692: 9689: 9685: 9679: 9674: 9671: 9668: 9664: 9658: 9655: 9650: 9645: 9641: 9632: 9628: 9624: 9621: 9617: 9613: 9612: 9611: 9608: 9603: 9599: 9595: 9591: 9588: 9570: 9560: 9555: 9540: 9539: 9538: 9536: 9518: 9489: 9479: 9474: 9460:data vectors 9459: 9455: 9451: 9447: 9443: 9438: 9435: 9434: 9433: 9398: 9385: 9383: 9379: 9375: 9371: 9367: 9363: 9359: 9356:of dimension 9355: 9350: 9342: 9296: 9255: 9244: 9233: 9230: 9219: 9208: 9201: 9200: 9199: 9197: 9171: 9144: 9123: 9118: 9108: 9053: 9042: 9035: 8986: 8903: 8895: 8887: 8875: 8874: 8873: 8870: 8862: 8858: 8854: 8852: 8848: 8842: 8840: 8834: 8825: 8811: 8808: 8805: 8802: 8799: 8796: 8793: 8790: 8787: 8784: 8779: 8775: 8770: 8766: 8762: 8739: 8735: 8714: 8693: 8689: 8685: 8679: 8675: 8669: 8665: 8644: 8623: 8619: 8615: 8609: 8605: 8599: 8595: 8573: 8569: 8565: 8559: 8555: 8549: 8545: 8536: 8511: 8506: 8503: 8499: 8493: 8489: 8483: 8478: 8475: 8472: 8468: 8464: 8456: 8452: 8445: 8442: 8435: 8434: 8433: 8431: 8409: 8405: 8401: 8395: 8391: 8385: 8381: 8377: 8374: 8371: 8367: 8363: 8359: 8353: 8349: 8343: 8339: 8335: 8323: 8322: 8319: 8313: 8311: 8306: 8305: 8304: 8298: 8263: 8245: 8240: 8213: 8208: 8203: 8193: 8163: 8150: 8147: 8125: 8093: 8085: 8082: 8074: 8058: 8053: 8045: 8042: 8035: 8034: 8033: 8028: 8026: 8021: 8020: 7955: 7918: 7900: 7871: 7861: 7831: 7818: 7815: 7793: 7766: 7758: 7754: 7726: 7718: 7713: 7698: 7694: 7676: 7663: 7647: 7639: 7623: 7618: 7610: 7607: 7600: 7599: 7598: 7594: 7590: 7586: 7581: 7579: 7574: 7573: 7572: 7559: 7540: 7537: 7534: 7531: 7528: 7508: 7505: 7502: 7499: 7496: 7489: 7475: 7472: 7469: 7462: 7459: 7455: 7451: 7447: 7428: 7425: 7421: 7414: 7402: 7401: 7386: 7383: 7380: 7377: 7374: 7354: 7351: 7348: 7345: 7342: 7335: 7321: 7318: 7315: 7308: 7306: 7302: 7298: 7294: 7275: 7272: 7268: 7261: 7249: 7248: 7233: 7230: 7227: 7224: 7220: 7217: 7196: 7193: 7190: 7187: 7184: 7177: 7163: 7160: 7157: 7150: 7148:used above ) 7125: 7121: 7117: 7113: 7110: 7090: 7087: 7083: 7079: 7072: 7060: 7059: 7044: 7041: 7038: 7035: 7031: 7028: 7007: 7004: 7001: 6998: 6995: 6988: 6974: 6971: 6968: 6961: 6958: 6954: 6950: 6930: 6927: 6923: 6919: 6912: 6900: 6899: 6884: 6881: 6878: 6875: 6871: 6868: 6847: 6844: 6841: 6838: 6835: 6828: 6814: 6811: 6808: 6801: 6799: 6796: 6776: 6773: 6769: 6765: 6758: 6746: 6745: 6730: 6727: 6724: 6721: 6717: 6714: 6693: 6690: 6687: 6684: 6681: 6674: 6660: 6657: 6654: 6647: 6645: 6642: 6622: 6619: 6615: 6611: 6604: 6592: 6591: 6576: 6573: 6570: 6567: 6564: 6544: 6541: 6538: 6535: 6532: 6525: 6511: 6508: 6505: 6498: 6495: 6491: 6488: 6469: 6466: 6462: 6455: 6443: 6442: 6427: 6424: 6421: 6418: 6415: 6395: 6392: 6389: 6386: 6383: 6376: 6362: 6359: 6356: 6349: 6346: 6342: 6339: 6320: 6317: 6313: 6306: 6294: 6293: 6278: 6275: 6272: 6269: 6266: 6259: 6245: 6242: 6239: 6232: 6229: 6210: 6206: 6199: 6187: 6186: 6171: 6168: 6165: 6162: 6159: 6152: 6138: 6135: 6132: 6125: 6122: 6118: 6114: 6095: 6091: 6084: 6072: 6071: 6056: 6053: 6050: 6047: 6044: 6037: 6023: 6020: 6017: 6010: 6007: 6003: 5999: 5980: 5976: 5969: 5957: 5956: 5953: 5950: 5936: 5933: 5930: 5923: 5909: 5906: 5903: 5900: 5897: 5889: 5875: 5868: 5867: 5864: 5861: 5847: 5844: 5841: 5834: 5831: 5817: 5810: 5809: 5806: 5803: 5789: 5786: 5783: 5776: 5773: 5759: 5752: 5751: 5736: 5733: 5730: 5727: 5724: 5704: 5701: 5698: 5695: 5692: 5685: 5671: 5668: 5665: 5658: 5655: 5636: 5633: 5629: 5622: 5610: 5609: 5605: 5602: 5599: 5596: 5595: 5587: 5585: 5581: 5576: 5574: 5570: 5566: 5562: 5558: 5554: 5550: 5546: 5540: 5538: 5532: 5529: 5524: 5522: 5518: 5514: 5510: 5506: 5503: 5499: 5495: 5485: 5483: 5479: 5475: 5472: 5468: 5464: 5443: 5428: 5423: 5411: 5401: 5396: 5382: 5381: 5380: 5375: 5372:score matrix 5370: 5366: 5360: 5358: 5355: 5351: 5346: 5344: 5340: 5336: 5332: 5299: 5297: 5260: 5258: 5240: 5238: 5220: 5219: 5218: 5216: 5211: 5209: 5206: 5200: 5195: 5191: 5165: 5160: 5156: 5153: 5149: 5145: 5141: 5138: 5134: 5102: 5072: 5068: 5002: 4980: 4978: 4932: 4930: 4865: 4863: 4851: 4833: 4832: 4831: 4829: 4826: 4821: 4819: 4815: 4811: 4807: 4803: 4799: 4795: 4791: 4787: 4783: 4779: 4773: 4768: 4764: 4761: 4757: 4753: 4732: 4712: 4700: 4699: 4698: 4696: 4692: 4686: 4676: 4674: 4670: 4666: 4662: 4656: 4654: 4649: 4645: 4641: 4636: 4633: 4629: 4625: 4615: 4612: 4607: 4603: 4587: 4582: 4572: 4562: 4532: 4527: 4517: 4512: 4500: 4490: 4485: 4462: 4458: 4454: 4436: 4432: 4422: 4418: 4401: 4396: 4386: 4383: 4380: 4375: 4365: 4362: 4359: 4356: 4344: 4340: 4336: 4333: 4325: 4321: 4314: 4293: 4278: 4273: 4259: 4258: 4257: 4255: 4251: 4247: 4243: 4237: 4232: 4228: 4225: 4221: 4211: 4207: 4202: 4196: 4191: 4186: 4179: 4173: 4169: 4165: 4158: 4153: 4149: 4143: 4138: 4134: 4131: 4125: 4120: 4116: 4092: 4044: 4013: 4012: 4011: 3970: 3948: 3936: 3935: 3934: 3931: 3928: 3923: 3918: 3912: 3907: 3901: 3896: 3868: 3843: 3825: 3818: 3814: 3812: 3799: 3781: 3774: 3758: 3745: 3743: 3730: 3686: 3673: 3671: 3655: 3617: 3596: 3594: 3581: 3565: 3557: 3538: 3527: 3526: 3525: 3523: 3518: 3516: 3512: 3508: 3505: 3496: 3494: 3490: 3487:. Columns of 3486: 3482: 3478: 3475: 3471: 3467: 3463: 3434: 3422: 3421: 3420: 3418: 3413: 3411: 3408: 3404: 3398: 3393: 3387: 3382: 3376: 3371: 3365: 3360: 3354: 3349: 3343: 3338: 3332: 3328: 3324: 3318: 3313: 3309: 3304: 3302: 3299: 3295: 3292: 3272: 3258: 3239: 3213: 3178: 3171: 3168: 3165: 3161: 3156: 3140: 3117: 3113: 3108: 3105: 3064: 3056: 3039: 3038: 3037: 3008: 2990: 2970: 2967: 2964: 2959: 2956: 2953: 2949: 2945: 2937: 2932: 2911: 2910: 2909: 2907: 2903: 2899: 2889: 2884: 2877: 2871: 2866: 2859: 2853: 2848: 2842: 2837: 2831: 2826: 2819: 2814: 2812: 2808: 2807: 2802: 2798: 2795: 2791: 2787: 2767: 2701: 2694: 2691: 2688: 2680: 2663: 2662: 2661: 2656: 2636: 2595: 2589: 2586: 2565: 2562: 2559: 2555: 2550: 2529: 2523: 2520: 2499: 2496: 2493: 2485: 2468: 2467: 2466: 2448: 2442: 2437: 2428: 2420: 2406: 2399: 2395: 2390: 2383: 2380: 2361: 2358: 2355: 2351: 2345: 2337: 2324: 2320: 2311: 2307: 2302: 2295: 2292: 2273: 2270: 2267: 2259: 2242: 2241: 2240: 2235: 2225: 2206: 2196: 2192: 2186: 2181: 2178: 2144: 2134: 2130: 2124: 2119: 2116: 2082: 2072: 2068: 2062: 2057: 2054: 2012: 2000: 1999: 1998: 1995: 1981: 1961: 1953: 1949: 1945: 1941: 1923: 1919: 1915: 1912: 1909: 1904: 1900: 1876: 1873: 1870: 1867: 1864: 1861: 1858: 1854: 1851: 1848: 1845: 1842: 1839: 1836: 1815: 1802: 1794: 1781: 1773: 1763: 1759: 1749: 1748: 1747: 1728: 1715: 1711: 1707: 1704: 1701: 1696: 1692: 1685: 1677: 1660: 1656: 1635: 1622: 1618: 1614: 1611: 1608: 1603: 1599: 1592: 1584: 1549: 1536: 1532: 1528: 1525: 1522: 1517: 1513: 1506: 1498: 1480: 1464: 1455: 1453: 1449: 1445: 1441: 1437: 1421: 1418: 1415: 1406: 1403: 1399: 1395: 1392: 1377: 1373: 1371: 1367: 1363: 1359: 1356: 1352: 1346: 1344: 1341:-dimensional 1340: 1330: 1328: 1324: 1320: 1316: 1312: 1308: 1304: 1301: 1297: 1293: 1289: 1285: 1281: 1277: 1273: 1269: 1265: 1261: 1251: 1249: 1245: 1241: 1237: 1233: 1229: 1225: 1221: 1217: 1212: 1198: 1195: 1192: 1172: 1151: 1130: 1116: 1114: 1111:studies, and 1110: 1106: 1098: 1094: 1090: 1085: 1081: 1079: 1075: 1071: 1068: 1067:perpendicular 1052: 1049: 1046: 1039:to the first 1038: 1022: 1014: 999: 991: 987: 982: 980: 975: 973: 969: 965: 962: 958: 954: 943: 938: 936: 931: 929: 924: 923: 921: 920: 913: 910: 906: 903: 902: 901: 898: 896: 893: 892: 886: 885: 878: 875: 873: 870: 868: 865: 863: 860: 858: 855: 853: 850: 848: 845: 844: 838: 837: 830: 827: 825: 822: 820: 817: 815: 812: 810: 807: 805: 802: 800: 797: 795: 792: 791: 785: 784: 777: 774: 772: 769: 767: 764: 762: 759: 758: 752: 751: 744: 741: 739: 736: 734: 733:Crowdsourcing 731: 729: 726: 725: 719: 718: 709: 706: 705: 704: 701: 699: 696: 694: 691: 689: 686: 685: 682: 677: 676: 668: 665: 663: 662:Memtransistor 660: 658: 655: 653: 650: 646: 643: 642: 641: 638: 636: 633: 629: 626: 624: 621: 619: 616: 614: 611: 610: 609: 606: 604: 601: 599: 596: 594: 591: 587: 584: 583: 582: 579: 575: 572: 570: 567: 565: 562: 560: 557: 556: 555: 552: 550: 547: 545: 544:Deep learning 542: 540: 537: 536: 533: 528: 527: 520: 517: 515: 512: 510: 508: 504: 502: 499: 498: 495: 490: 489: 480: 479:Hidden Markov 477: 475: 472: 470: 467: 466: 465: 462: 461: 458: 453: 452: 445: 442: 440: 437: 435: 432: 430: 427: 425: 422: 420: 417: 415: 412: 410: 407: 405: 402: 401: 398: 393: 392: 385: 382: 380: 377: 375: 371: 369: 366: 364: 361: 359: 357: 353: 351: 348: 346: 343: 341: 338: 337: 334: 329: 328: 321: 318: 316: 313: 311: 308: 306: 303: 301: 298: 296: 293: 291: 288: 286: 284: 280: 276: 275:Random forest 273: 271: 268: 266: 263: 262: 261: 258: 256: 253: 251: 248: 247: 240: 239: 234: 233: 225: 219: 218: 211: 208: 206: 203: 201: 198: 196: 193: 191: 188: 186: 183: 181: 178: 176: 173: 171: 168: 166: 163: 161: 160:Data cleaning 158: 156: 153: 151: 148: 146: 143: 141: 138: 136: 133: 131: 128: 126: 123: 122: 116: 115: 108: 105: 103: 100: 98: 95: 93: 90: 88: 85: 83: 80: 78: 75: 73: 72:Meta-learning 70: 68: 65: 63: 60: 58: 55: 53: 50: 48: 45: 44: 38: 37: 34: 29: 26: 25: 21: 20: 19112: 19100: 19081: 19074: 18986:Econometrics 18936: / 18919:Chemometrics 18896:Epidemiology 18889: / 18862:Applications 18704:ARIMA model 18651:Q-statistic 18600:Stationarity 18513: 18496:Multivariate 18439: / 18435: / 18433:Multivariate 18431: / 18371: / 18367: / 18141:Bayes factor 18040:Signed rank 17952: 17926: 17918: 17906: 17601:Completeness 17437:Cohort study 17335:Opinion poll 17270:Missing data 17257:Study design 17212:Scatter plot 17134:Scatter plot 17127:Spearman's ρ 17089:Grouped data 16728: 16714: 16687: 16643: 16633: 16613:. Retrieved 16608: 16599: 16572: 16568: 16558: 16523: 16519: 16509: 16482: 16478: 16468: 16431: 16428:BMC Genetics 16427: 16417: 16372: 16368: 16354: 16311: 16307: 16300: 16283: 16279: 16273: 16238: 16234: 16228: 16191: 16185: 16162: 16152: 16139: 16132: 16101: 16094: 16081: 16074: 16054: 16037: 16031: 16015: 16003: 15993: 15957: 15951: 15944: 15936: 15929:A. N. Gorban 15924: 15905: 15899: 15889: 15880: 15874: 15857: 15848: 15835: 15814: 15787: 15781: 15768: 15733: 15727: 15714: 15679: 15673: 15660: 15633: 15627: 15616:. Retrieved 15594: 15590: 15577: 15568: 15555: 15549:: 1057–1064. 15546: 15533: 15508: 15504: 15498: 15485: 15465: 15458: 15439: 15433: 15424: 15418: 15388:(1): 27–35. 15385: 15381: 15375: 15366: 15333: 15329: 15323: 15296: 15287: 15274: 15272:Libin Yang. 15268: 15253: 15234: 15229: 15214: 15187: 15167:. Retrieved 15163: 15153: 15140: 15133: 15124: 15120: 15110: 15060:(1). 14683. 15057: 15051: 15044:Elhaik, Eran 15038: 15003: 14999: 14989: 14978:. Retrieved 14974: 14964: 14947: 14943: 14937: 14926:. Retrieved 14921: 14912: 14901: 14892: 14886: 14878: 14873: 14867:: 2287–2320. 14864: 14860: 14847: 14804: 14800: 14794: 14774: 14767: 14750: 14746: 14740: 14727: 14715: 14706: 14697: 14690:eig function 14685: 14671: 14636: 14632: 14623: 14610: 14598:. Retrieved 14589: 14577: 14550: 14544: 14534: 14528: 14508: 14501: 14485:10.1109/2.36 14476: 14472: 14466: 14439: 14407: 14401: 14389:. Retrieved 14384: 14375: 14330: 14326: 14236: 14232: 14178: 14174: 14164: 14121: 14117: 14094: 14082: 14057: 14053: 14043: 14027:. Elsevier. 14023: 14013: 14000: 13972: 13965: 13951:cite journal 13932: 13928: 13918: 13875: 13871: 13865: 13837: 13794: 13788: 13778: 13753: 13747: 13740: 13734: 13729: 13704: 13700: 13687: 13642: 13636: 13593: 13589: 13579: 13536: 13532: 13494: 13490: 13434: 13430: 13394: 13388: 13361: 13357: 13347: 13304: 13300: 13290: 13135: 13080:scikit-learn 12822: 12810: 12806: 12798: 12551: 12540: 12524: 12517: 12501: 12483: 12482: 12471: 12453:followed by 12443: 12433: 12428:elastic maps 12401: 12373: 12353: 12350: 12343: 12331: 12322:sequentially 12185: 12179: 12147: 12126: 12122: 12077: 12056: 12041: 12031: 12028:eigenvectors 12023: 11995:neuroscience 11992: 11989:Neuroscience 11968: 11937: 11913:representing 11903:exposure to 11873:fixed income 11870: 11851: 11838: 11834: 11830: 11808: 11804: 11796: 11780: 11772: 11765: 11756: 11752: 11747: 11745: 11716: 11713:Intelligence 11708:Applications 11689: 11687: 11671: 11668: 11659: 11640:Gram–Schmidt 11630: 11623: 11617: 11615: 11609: 11605: 11599: 11594: 11583: 11576: 11569:metabolomics 11552: 11551: 11532: 11526: 11520: 11514: 11508: 11502: 11498: 11476: 11462: 11456: 11450: 11444: 11434: 11428: 11422: 11415: 11410: 11399: 11395: 11391: 11385: 11381: 11372: 11369: 11363: 11359: 11355: 11351: 11347: 11342: 11338: 11326: 11317: 11313: 11309: 11301: 11293: 11290: 11283:operations. 11279: 11272: 11262: 11256: 11245: 11235: 11233: 11152: 10896: 10891: 10887: 10883: 10876: 10872: 10843: 10838: 10834: 10830: 10828: 10753: 10699: 10695: 10691: 10687: 10683: 10679: 10675: 10539: 10535: 10531: 10527: 10523: 10517: 10419: 10415: 10411: 10401: 10390: 10389:in order of 10386: 10382: 10376: 10368: 10364: 10360: 10356: 10349: 10345: 10338: 10334: 10330: 10326: 10322: 10318: 10314: 10254: 10250: 10188: 10184: 10180: 10129: 10117: 10063: 10060:diagonalizes 10056:eigenvectors 10051: 10045: 10033: 10027: 10015: 9927: 9926:from matrix 9923: 9916: 9912: 9906: 9895: 9893: 9824: 9819: 9765: 9761: 9757: 9750: 9714: 9711: 9630: 9626: 9619: 9615: 9609: 9601: 9597: 9593: 9586: 9534: 9457: 9453: 9449: 9445: 9441: 9439: 9436: 9386: 9381: 9373: 9369: 9365: 9361: 9357: 9353: 9351: 9348: 9270: 9195: 9145: 8987: 8918: 8871: 8868: 8859: 8855: 8843: 8835: 8831: 8528: 8427: 8317: 8309: 8308: 8289: 8261: 8024: 8023: 7916: 7696: 7692: 7661: 7592: 7588: 7584: 7577: 7576: 7570: 7557: 7457: 7453: 7449: 7304: 7300: 7296: 7119: 6956: 6953:eigenvectors 6493: 6344: 6120: 6005: 5951: 5862: 5804: 5577: 5572: 5568: 5560: 5548: 5544: 5541: 5533: 5525: 5504: 5501: 5493: 5491: 5473: 5466: 5462: 5460: 5373: 5368: 5364: 5361: 5356: 5353: 5349: 5347: 5342: 5334: 5330: 5328: 5214: 5212: 5207: 5204: 5198: 5193: 5169: 5163: 5158: 5154: 5151: 5147: 5143: 5139: 5136: 5074: 5070: 5039: 5037: 4827: 4824: 4822: 4817: 4813: 4809: 4805: 4801: 4797: 4793: 4789: 4785: 4781: 4777: 4771: 4766: 4759: 4755: 4751: 4749: 4694: 4688: 4664: 4660: 4657: 4637: 4627: 4621: 4460: 4456: 4452: 4420: 4416: 4323: 4319: 4312: 4310: 4253: 4249: 4245: 4241: 4235: 4230: 4226: 4223: 4219: 4217: 4205: 4200: 4194: 4189: 4184: 4177: 4171: 4167: 4163: 4156: 4151: 4147: 4141: 4136: 4132: 4129: 4123: 4118: 4114: 4112: 4009: 3932: 3929: 3921: 3916: 3910: 3905: 3899: 3894: 3892: 3521: 3519: 3514: 3506: 3503: 3502: 3492: 3488: 3480: 3476: 3473: 3469: 3465: 3461: 3459: 3416: 3414: 3409: 3406: 3402: 3396: 3391: 3385: 3380: 3374: 3369: 3363: 3358: 3352: 3347: 3341: 3336: 3330: 3326: 3322: 3316: 3311: 3307: 3305: 3300: 3297: 3293: 3290: 3288: 3035: 2905: 2901: 2897: 2895: 2882: 2875: 2869: 2864: 2857: 2851: 2846: 2840: 2835: 2829: 2824: 2817: 2815: 2805: 2804: 2796: 2793: 2783: 2654: 2652: 2464: 2233: 2231: 2037: 1996: 1947: 1943: 1939: 1891: 1658: 1654: 1478: 1456: 1451: 1447: 1439: 1408:Consider an 1407: 1388: 1360: 1349:compute the 1347: 1338: 1336: 1310: 1302: 1299: 1291: 1260:Karl Pearson 1257: 1216:eigenvectors 1213: 1122: 1102: 1093:eigenvectors 1015:, where the 1013:unit vectors 985: 983: 976: 956: 952: 951: 819:PAC learning 506: 428: 355: 350:Hierarchical 282: 236: 230: 19114:WikiProject 19029:Cartography 18991:Jurimetrics 18943:Reliability 18674:Time domain 18653:(Ljung–Box) 18575:Time-series 18453:Categorical 18437:Time-series 18429:Categorical 18364:(Bernoulli) 18199:Correlation 18179:Correlation 17975:Jarque–Bera 17947:Chi-squared 17709:M-estimator 17662:Asymptotics 17606:Sufficiency 17373:Interaction 17285:Replication 17265:Effect size 17222:Violin plot 17202:Radar chart 17182:Forest plot 17172:Correlogram 17122:Kendall's τ 16361:Sabatti, C. 15987:Data online 15864:Jennifer Dy 15790:: 517–553. 15729:SIAM Review 14385:i2tutorials 13790:SIAM Review 13693:Pearson, K. 13491:IEEE Access 13364:(9): 1825. 13247:(Wikibooks) 12962:NAG Library 12894:Mathematica 12509:data mining 12459:elastic map 12011:white noise 11983:eigenvector 11885:yield curve 11812:Eran Elhaik 11785:(HDI) from 10526:columns of 10393:eigenvalue. 10325:, contains 10172:as well as 10154:Mathematica 10126:eigenvalues 10032:instead of 9537:variables. 9448:variables, 8828:Limitations 8264:columns of 7919:columns of 7759:matrix for 7664:vector and 7116:eigenvalues 5603:Dimensions 5498:eigenvalues 4648:overfitting 4150:, that is, 3499:Covariances 2811:eigenvector 1952:unit vector 1746:, given by 1368:(degree of 1366:scree plots 703:Multi-agent 640:Transformer 539:Autoencoder 295:Naive Bayes 33:data mining 19130:Categories 18981:Demography 18699:ARMA model 18504:Regression 18081:(Friedman) 18042:(Wilcoxon) 17980:Normality 17970:Lilliefors 17917:Student's 17793:Resampling 17667:Robustness 17655:divergence 17645:Efficiency 17583:(monotone) 17578:Likelihood 17495:Population 17328:Stratified 17280:Population 17099:Dependence 17055:Count data 16986:Percentile 16963:Dispersion 16896:Arithmetic 16831:Statistics 16485:(4): 354. 16479:Atmosphere 16321:1511.01245 16021:Hastie, T. 15953:The Lancet 15862:Yue Guan; 15743:cs/0406021 15618:2012-08-02 15571:: 225–232. 15223:1119448115 14980:2022-05-06 14928:2022-05-05 14600:19 January 14340:1712.10317 14333:(2): 104. 14296:1612.06037 14188:1604.06097 14181:(2): 117. 14124:(2): L28. 13749:Biometrika 13504:1904.06455 13444:1610.01959 13283:References 13266:Sparse PCA 13234:Oja's rule 13204:Kernel PCA 13068:FactoMineR 13060:ExPosition 12982:GNU Octave 12956:FreePascal 12928:Matplotlib 12866:princomp() 12498:Robust PCA 12467:kernel PCA 12455:projecting 12444:Principal 12423:microarray 12376:Sparse PCA 12370:Sparse PCA 12364:Sparse PCA 11933:orthogonal 11929:eigenvalue 11877:portfolios 11734:. In 1924 11702:FactoMineR 11652:) method. 11496:) method. 10702:such that 10391:decreasing 10170:GNU Octave 9919:empirical 9504:with each 8839:kernel PCA 8432:elements, 8310:Property 3 8297:regression 8025:Property 2 7808:, denoted 7757:covariance 7578:Property 1 7567:Properties 7122:along its 6341:deviations 4614:haplotypes 2801:eigenvalue 1396:on a real 1391:orthogonal 1109:microbiome 1037:orthogonal 688:Q-learning 586:Restricted 384:Mean shift 333:Clustering 310:Perceptron 238:regression 140:Clustering 135:Regression 18362:Logistic 18129:posterior 18055:Rank sum 17803:Jackknife 17798:Bootstrap 17616:Bootstrap 17551:Parameter 17500:Statistic 17295:Statistic 17207:Run chart 17192:Pie chart 17187:Histogram 17177:Fan chart 17152:Bar chart 17034:L-moments 16921:Geometric 16654:CiteSeerX 16591:254965361 16550:236300040 16434:: 11:94. 16286:: 22–34. 16248:0912.3599 16241:(3): 11. 16196:CiteSeerX 15826:1212.4137 15797:0811.4724 15684:CiteSeerX 15643:1410.6801 15525:120886184 15262:John Hull 15211:John Hull 15102:251932226 15000:Nat Genet 14814:0811.1081 14663:122379222 14646:1108.4372 14560:1205.6935 14539:Tech Note 14458:2475-7772 14410:. Wiley. 14215:118349503 14131:1207.4197 13885:1206.5538 13721:125037489 13647:CiteSeerX 13603:1403.1591 13546:1405.6785 13404:0811.4413 13168:Eigenface 12868:function. 12842:Analytica 12698:− 12261:λ 12240:∑ 12223:λ 12202:∑ 12198:− 12061:, during 11841:Joe Flood 11736:Thurstone 11190:⁡ 11164:∗ 11129:− 11112:⁡ 11091:∗ 11078:∗ 11064:⁡ 11040:∗ 11030:∗ 11010:⁡ 10989:∗ 10960:⁡ 10935:⁡ 10855:∗ 10779:⋅ 10734:≥ 10654:≤ 10648:≤ 10619:… 10607:ℓ 10597:… 10581:for 10574:ℓ 10495:… 10479:for 10444:∑ 10296:ℓ 10293:≠ 10286:for 10273:ℓ 10237:ℓ 10227:for 10216:λ 10207:ℓ 10081:− 10020:transpose 9998:∗ 9971:∗ 9955:− 9911:Find the 9873:… 9857:for 9789:− 9665:∑ 9589:elements. 9561:… 9480:… 9231:− 8806:… 8776:α 8763:α 8736:α 8686:α 8676:α 8666:λ 8616:α 8606:α 8596:λ 8566:α 8556:α 8546:λ 8500:α 8490:λ 8469:∑ 8446:⁡ 8402:α 8392:α 8382:λ 8375:⋯ 8360:α 8350:α 8340:λ 8332:Σ 8246:∗ 8209:∗ 8159:Σ 8151:⁡ 8121:Σ 7827:Σ 7819:⁡ 7789:Σ 7733:Σ 7709:Σ 7662:q-element 7538:… 7506:… 7473:× 7384:… 7352:… 7319:× 7231:… 7194:… 7161:× 7135:Λ 7042:… 7005:… 6972:× 6882:… 6845:… 6812:× 6728:… 6691:… 6658:× 6574:… 6542:… 6509:× 6425:… 6393:… 6360:× 6276:… 6243:× 6169:… 6136:× 6054:… 6021:× 5934:× 5907:≤ 5901:≤ 5845:× 5787:× 5734:… 5702:… 5669:× 5419:Σ 5309:Σ 5270:Σ 5121:Σ 5108:Σ 5090:^ 5087:Σ 5053:^ 5050:Σ 4996:^ 4993:Σ 4956:Σ 4943:Σ 4908:Σ 4876:Σ 4722:Σ 4693:(SVD) of 4579:‖ 4563:− 4555:‖ 4524:‖ 4491:− 4471:‖ 4387:∈ 4366:∈ 4097:Λ 4069:Λ 4045:∝ 3980:Λ 3949:∝ 3819:λ 3775:λ 3597:∝ 3233:^ 3207:^ 3172:⁡ 3134:^ 3114:⁡ 2968:− 2950:∑ 2946:− 2926:^ 2695:⁡ 2566:⁡ 2500:⁡ 2429:⋅ 2396:∑ 2378:‖ 2370:‖ 2362:⁡ 2308:∑ 2290:‖ 2282:‖ 2274:⁡ 1913:… 1871:… 1849:… 1803:⋅ 1705:… 1612:… 1526:… 1419:× 1380:retained. 1355:normalize 1343:ellipsoid 1333:Intuition 1298:(EVD) of 1290:(SVD) of 1280:Hotelling 1196:− 1087:PCA of a 1050:− 847:ECML PKDD 829:VC theory 776:ROC curve 708:Self-play 628:DeepDream 469:Bayes net 260:Ensembles 41:Paradigms 19076:Category 18769:Survival 18646:Johansen 18369:Binomial 18324:Isotonic 17911:(normal) 17556:location 17363:Blocking 17318:Sampling 17197:Q–Q plot 17162:Box plot 17144:Graphics 17039:Skewness 17029:Kurtosis 17001:Variance 16931:Heronian 16926:Harmonic 16636:(Wiley). 16460:20950446 16409:14673099 16346:10420698 16314:: 1–71. 16008:. Paris. 15982:16358549 15974:15721472 15866:(2009). 15358:17786731 15350:10638820 15213:(2018). 15209:§9.7 in 15094:36038559 15046:(2022). 15030:18425127 14831:19772385 14753:: 1–17. 14629:Abdi. H. 14271:18561804 14156:51088743 14074:27735002 14021:(1990). 13902:23787338 13812:1903/566 13695:(1901). 13679:17144854 13339:26953178 13113:See also 13090:princomp 13036:princomp 12986:princomp 12914:princomp 12504:outliers 11925:variance 11907:, given 11864:such as 11797:In 1978 11724:Spearman 11610:((X r)X) 11565:genomics 11435:r (XX) r 11335:) 10886:so that 10414:for the 9372:, where 8771:′ 8694:′ 8624:′ 8574:′ 8430:diagonal 8410:′ 8368:′ 8054:′ 7956:′ 7886:, where 7753:variance 7727:′ 7677:′ 7619:′ 7221:′ 7091:′ 7032:′ 6931:′ 6872:′ 6777:′ 6718:′ 6623:′ 6490:z-scores 5606:Indices 5600:Meaning 5551:. 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10327:p 10323:p 10319:p 10315:V 10299:. 10290:k 10281:0 10278:= 10270:k 10266:D 10255:C 10251:j 10234:= 10231:k 10220:k 10212:= 10204:k 10200:D 10189:p 10185:p 10181:D 10176:. 10164:( 10146:R 10130:C 10118:D 10103:D 10099:= 10095:V 10090:C 10084:1 10076:V 10064:C 10052:V 10040:. 10034:n 10028:n 10022:. 10016:B 9977:B 9966:B 9958:1 9952:n 9948:1 9943:= 9939:C 9928:B 9924:C 9917:p 9913:p 9896:B 9879:n 9876:, 9870:, 9867:1 9864:= 9861:i 9850:1 9847:= 9842:i 9838:h 9825:n 9820:h 9804:T 9799:u 9793:h 9785:X 9781:= 9777:B 9766:B 9762:p 9758:n 9753:. 9751:X 9735:T 9730:u 9693:j 9690:i 9686:X 9680:n 9675:1 9672:= 9669:i 9659:n 9656:1 9651:= 9646:j 9642:u 9631:p 9627:u 9622:. 9620:p 9616:j 9604:. 9602:p 9598:n 9594:X 9587:p 9571:n 9566:x 9556:1 9551:x 9535:p 9519:i 9514:x 9490:n 9485:x 9475:1 9470:x 9458:n 9454:p 9450:L 9446:L 9442:p 9421:} 9417:X 9413:{ 9409:T 9406:L 9403:K 9399:= 9395:Y 9382:X 9374:Y 9370:Y 9366:L 9362:Y 9358:p 9354:X 9328:n 9306:s 9280:n 9256:. 9253:) 9249:s 9245:; 9241:y 9237:( 9234:I 9228:) 9224:s 9220:; 9216:x 9212:( 9209:I 9181:s 9155:n 9130:x 9124:T 9119:L 9114:W 9109:= 9105:y 9083:s 9062:) 9058:s 9054:; 9050:y 9046:( 9043:I 9019:n 8997:s 8972:n 8950:s 8928:x 8904:, 8900:n 8896:+ 8892:s 8888:= 8884:x 8812:p 8809:, 8803:, 8800:1 8797:= 8794:k 8791:, 8788:1 8785:= 8780:k 8767:k 8740:k 8715:k 8690:k 8680:k 8670:k 8645:k 8620:k 8610:k 8600:k 8570:k 8560:k 8550:k 8531:x 8512:2 8507:j 8504:k 8494:k 8484:P 8479:1 8476:= 8473:k 8465:= 8462:) 8457:j 8453:x 8449:( 8406:p 8396:p 8386:p 8378:+ 8372:+ 8364:1 8354:1 8344:1 8336:= 8318:Σ 8312:: 8301:x 8293:x 8286:. 8273:A 8262:q 8241:q 8236:A 8214:, 8204:q 8199:A 8194:= 8190:B 8169:) 8164:y 8154:( 8126:y 8098:A 8094:, 8090:B 8086:, 8083:x 8059:x 8051:B 8046:= 8043:y 8027:: 8004:A 7983:) 7979:B 7953:B 7948:( 7928:A 7917:q 7901:q 7896:A 7872:q 7867:A 7862:= 7858:B 7837:) 7832:y 7822:( 7794:y 7767:y 7755:- 7738:B 7724:B 7719:= 7714:y 7697:p 7693:q 7674:B 7648:y 7624:x 7616:B 7611:= 7608:y 7593:p 7589:q 7585:q 7580:: 7541:L 7535:1 7532:= 7529:l 7509:n 7503:1 7500:= 7497:i 7476:L 7470:n 7458:W 7454:X 7450:n 7434:] 7429:l 7426:i 7422:T 7418:[ 7415:= 7411:T 7387:L 7381:1 7378:= 7375:l 7355:p 7349:1 7346:= 7343:j 7322:L 7316:p 7305:V 7301:W 7297:C 7281:] 7276:l 7273:j 7269:W 7265:[ 7262:= 7258:W 7234:p 7228:1 7225:= 7218:j 7197:p 7191:1 7188:= 7185:j 7164:p 7158:p 7120:C 7097:] 7088:j 7084:j 7080:D 7076:[ 7073:= 7069:D 7045:p 7039:1 7036:= 7029:j 7008:p 7002:1 6999:= 6996:j 6975:p 6969:p 6957:C 6937:] 6928:j 6924:j 6920:V 6916:[ 6913:= 6909:V 6885:p 6879:1 6876:= 6869:j 6848:p 6842:1 6839:= 6836:j 6815:p 6809:p 6783:] 6774:j 6770:j 6766:R 6762:[ 6759:= 6755:R 6731:p 6725:1 6722:= 6715:j 6694:p 6688:1 6685:= 6682:j 6661:p 6655:p 6629:] 6620:j 6616:j 6612:C 6608:[ 6605:= 6601:C 6577:p 6571:1 6568:= 6565:j 6545:n 6539:1 6536:= 6533:i 6512:p 6506:n 6494:j 6475:] 6470:j 6467:i 6463:Z 6459:[ 6456:= 6452:Z 6428:p 6422:1 6419:= 6416:j 6396:n 6390:1 6387:= 6384:i 6363:p 6357:n 6345:j 6326:] 6321:j 6318:i 6314:B 6310:[ 6307:= 6303:B 6279:n 6273:1 6270:= 6267:i 6246:n 6240:1 6216:] 6211:i 6207:h 6203:[ 6200:= 6196:h 6172:p 6166:1 6163:= 6160:j 6139:1 6133:p 6121:j 6101:] 6096:j 6092:s 6088:[ 6085:= 6081:s 6057:p 6051:1 6048:= 6045:j 6024:1 6018:p 6006:j 5986:] 5981:j 5977:u 5973:[ 5970:= 5966:u 5937:1 5931:1 5910:p 5904:L 5898:1 5876:L 5848:1 5842:1 5818:p 5790:1 5784:1 5760:n 5737:p 5731:1 5728:= 5725:j 5705:n 5699:1 5696:= 5693:i 5672:p 5666:n 5642:] 5637:j 5634:i 5630:X 5626:[ 5623:= 5619:X 5573:X 5569:Z 5561:X 5555:( 5505:X 5502:X 5494:Σ 5474:L 5467:T 5463:M 5444:L 5439:W 5433:X 5429:= 5424:L 5412:L 5407:U 5402:= 5397:L 5392:T 5377:L 5374:T 5369:L 5365:n 5357:X 5354:X 5350:X 5343:T 5335:X 5331:T 5304:U 5300:= 5289:W 5282:T 5276:W 5265:U 5261:= 5250:W 5245:X 5241:= 5233:T 5215:T 5208:X 5205:X 5201:) 5199:k 5197:( 5194:λ 5178:X 5166:) 5164:k 5162:( 5159:σ 5155:X 5152:X 5148:X 5144:W 5140:X 5137:X 5114:T 5103:= 5097:2 5016:T 5010:W 5003:2 4985:W 4981:= 4968:T 4962:W 4949:T 4937:W 4933:= 4920:T 4914:W 4903:U 4896:T 4890:U 4882:T 4870:W 4866:= 4858:X 4852:T 4847:X 4828:X 4825:X 4818:X 4814:p 4810:p 4806:p 4802:W 4798:X 4794:n 4790:n 4786:n 4782:U 4778:X 4774:) 4772:k 4770:( 4767:σ 4760:p 4756:n 4752:Σ 4733:T 4728:W 4717:U 4713:= 4709:X 4695:X 4665:W 4661:T 4628:L 4588:2 4583:2 4573:L 4568:X 4559:X 4533:2 4528:2 4518:T 4513:L 4508:W 4501:L 4496:T 4486:T 4481:W 4475:T 4461:L 4457:t 4453:L 4437:L 4433:W 4421:L 4417:p 4402:, 4397:L 4392:R 4384:t 4381:, 4376:p 4371:R 4363:x 4360:, 4357:x 4351:T 4345:L 4341:W 4337:= 4334:t 4324:L 4320:n 4316:L 4313:T 4294:L 4289:W 4283:X 4279:= 4274:L 4269:T 4254:L 4250:L 4246:p 4242:p 4238:) 4236:i 4234:( 4231:x 4227:W 4224:X 4220:T 4208:) 4206:k 4204:( 4201:w 4197:) 4195:i 4193:( 4190:x 4188:( 4185:i 4180:) 4178:i 4176:( 4172:k 4168:t 4164:i 4159:) 4157:k 4155:( 4152:λ 4148:k 4144:) 4142:k 4140:( 4137:λ 4133:X 4130:X 4126:) 4124:k 4122:( 4119:λ 4115:Λ 4093:= 4089:W 4082:T 4076:W 4063:W 4056:T 4050:W 4041:W 4036:Q 4029:T 4023:W 3992:T 3986:W 3975:W 3971:= 3967:X 3960:T 3954:X 3945:Q 3924:) 3922:k 3920:( 3917:w 3913:) 3911:j 3909:( 3906:w 3902:) 3900:k 3898:( 3895:w 3872:) 3869:k 3866:( 3861:w 3853:T 3847:) 3844:j 3841:( 3836:w 3829:) 3826:k 3823:( 3815:= 3803:) 3800:k 3797:( 3792:w 3785:) 3782:k 3779:( 3768:T 3762:) 3759:j 3756:( 3751:w 3746:= 3734:) 3731:k 3728:( 3723:w 3717:X 3710:T 3704:X 3696:T 3690:) 3687:j 3684:( 3679:w 3674:= 3664:) 3659:) 3656:k 3653:( 3648:w 3642:X 3638:( 3632:T 3627:) 3621:) 3618:j 3615:( 3610:w 3604:X 3600:( 3590:) 3585:) 3582:k 3579:( 3574:C 3571:P 3566:, 3561:) 3558:j 3555:( 3550:C 3547:P 3542:( 3539:Q 3522:Q 3515:X 3507:X 3504:X 3489:W 3481:W 3477:X 3474:X 3470:p 3466:p 3462:W 3444:W 3439:X 3435:= 3431:T 3417:X 3410:X 3407:X 3403:k 3399:) 3397:k 3395:( 3392:w 3388:) 3386:k 3384:( 3381:w 3377:) 3375:k 3373:( 3370:w 3366:) 3364:i 3362:( 3359:x 3355:) 3353:k 3351:( 3348:w 3344:) 3342:i 3340:( 3337:x 3333:) 3331:i 3329:( 3327:k 3323:t 3319:) 3317:i 3315:( 3312:x 3308:k 3301:X 3298:X 3294:X 3291:X 3273:} 3265:w 3259:T 3254:w 3246:w 3240:k 3230:X 3220:T 3214:k 3204:X 3194:T 3188:w 3179:{ 3166:= 3162:} 3157:2 3147:w 3141:k 3131:X 3118:{ 3109:1 3106:= 3098:w 3087:x 3084:a 3081:m 3077:g 3074:r 3071:a 3065:= 3060:) 3057:k 3054:( 3049:w 3018:T 3012:) 3009:s 3006:( 3001:w 2994:) 2991:s 2988:( 2983:w 2977:X 2971:1 2965:k 2960:1 2957:= 2954:s 2942:X 2938:= 2933:k 2923:X 2906:X 2902:k 2898:k 2883:w 2876:w 2872:) 2870:i 2868:( 2865:x 2858:w 2854:) 2852:i 2850:( 2847:x 2843:) 2841:i 2836:t 2832:) 2830:i 2828:( 2825:x 2818:w 2806:w 2797:X 2794:X 2768:} 2761:w 2754:T 2748:w 2740:w 2737:X 2730:T 2724:X 2716:T 2710:w 2702:{ 2689:= 2684:) 2681:1 2678:( 2673:w 2655:w 2637:} 2632:w 2629:X 2622:T 2616:X 2608:T 2602:w 2596:{ 2590:1 2587:= 2579:w 2560:= 2556:} 2551:2 2542:w 2539:X 2530:{ 2524:1 2521:= 2513:w 2494:= 2489:) 2486:1 2483:( 2478:w 2449:} 2443:2 2438:) 2433:w 2424:) 2421:i 2418:( 2413:x 2407:( 2400:i 2391:{ 2384:1 2381:= 2374:w 2356:= 2352:} 2346:2 2341:) 2338:i 2335:( 2331:) 2325:1 2321:t 2317:( 2312:i 2303:{ 2296:1 2293:= 2286:w 2268:= 2263:) 2260:1 2257:( 2252:w 2234:w 2210:) 2207:k 2204:( 2197:j 2193:w 2187:= 2182:k 2179:j 2173:W 2148:) 2145:i 2142:( 2135:j 2131:x 2125:= 2120:j 2117:i 2111:X 2086:) 2083:i 2080:( 2073:k 2069:t 2063:= 2058:k 2055:i 2049:T 2022:W 2017:X 2013:= 2009:T 1982:p 1962:l 1948:w 1944:X 1940:t 1924:l 1920:t 1916:, 1910:, 1905:1 1901:t 1877:l 1874:, 1868:, 1865:1 1862:= 1859:k 1855:n 1852:, 1846:, 1843:1 1840:= 1837:i 1832:r 1829:o 1826:f 1819:) 1816:k 1813:( 1808:w 1798:) 1795:i 1792:( 1787:x 1782:= 1777:) 1774:i 1771:( 1764:k 1760:t 1732:) 1729:i 1726:( 1722:) 1716:l 1712:t 1708:, 1702:, 1697:1 1693:t 1689:( 1686:= 1681:) 1678:i 1675:( 1670:t 1655:X 1639:) 1636:i 1633:( 1629:) 1623:p 1619:x 1615:, 1609:, 1604:1 1600:x 1596:( 1593:= 1588:) 1585:i 1582:( 1577:x 1553:) 1550:k 1547:( 1543:) 1537:p 1533:w 1529:, 1523:, 1518:1 1514:w 1510:( 1507:= 1502:) 1499:k 1496:( 1491:w 1479:p 1465:l 1452:p 1448:n 1440:X 1422:p 1416:n 1339:p 1303:X 1300:X 1292:X 1199:1 1193:i 1173:i 1152:p 1131:p 1053:1 1047:i 1023:i 1000:p 955:( 941:e 934:t 927:v 507:k 356:k 283:k 241:) 229:(

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