4700:
3714:
4695:{\displaystyle \operatorname {cov} (\mathbf {X} )^{-1}={\begin{bmatrix}{\frac {1}{\sigma _{x_{1}|x_{2}...}}}&&&0\\&{\frac {1}{\sigma _{x_{2}|x_{1},x_{3}...}}}\\&&\ddots \\0&&&{\frac {1}{\sigma _{x_{n}|x_{1}...x_{n-1}}}}\end{bmatrix}}{\begin{bmatrix}1&-\rho _{x_{1},x_{2}\mid x_{3}...}&\cdots &-\rho _{x_{1},x_{n}\mid x_{2}...x_{n-1}}\\-\rho _{x_{2},x_{1}\mid x_{3}...}&1&\cdots &-\rho _{x_{2},x_{n}\mid x_{1},x_{3}...x_{n-1}}\\\vdots &\vdots &\ddots &\vdots \\-\rho _{x_{n},x_{1}\mid x_{2}...x_{n-1}}&-\rho _{x_{n},x_{2}\mid x_{1},x_{3}...x_{n-1}}&\cdots &1\\\end{bmatrix}}{\begin{bmatrix}{\frac {1}{\sigma _{x_{1}|x_{2}...}}}&&&0\\&{\frac {1}{\sigma _{x_{2}|x_{1},x_{3}...}}}\\&&\ddots \\0&&&{\frac {1}{\sigma _{x_{n}|x_{1}...x_{n-1}}}}\end{bmatrix}}}
3703:
3137:
3211:
2315:
3698:{\displaystyle \operatorname {cov} (\mathbf {X} )={\begin{bmatrix}\sigma _{x_{1}}&&&0\\&\sigma _{x_{2}}\\&&\ddots \\0&&&\sigma _{x_{n}}\end{bmatrix}}{\begin{bmatrix}1&\rho _{x_{1},x_{2}}&\cdots &\rho _{x_{1},x_{n}}\\\rho _{x_{2},x_{1}}&1&\cdots &\rho _{x_{2},x_{n}}\\\vdots &\vdots &\ddots &\vdots \\\rho _{x_{n},x_{1}}&\rho _{x_{n},x_{2}}&\cdots &1\\\end{bmatrix}}{\begin{bmatrix}\sigma _{x_{1}}&&&0\\&\sigma _{x_{2}}\\&&\ddots \\0&&&\sigma _{x_{n}}\end{bmatrix}}}
15835:
14864:
3132:{\displaystyle \operatorname {corr} (\mathbf {X} )={\begin{bmatrix}1&{\frac {\operatorname {E} }{\sigma (X_{1})\sigma (X_{2})}}&\cdots &{\frac {\operatorname {E} }{\sigma (X_{1})\sigma (X_{n})}}\\\\{\frac {\operatorname {E} }{\sigma (X_{2})\sigma (X_{1})}}&1&\cdots &{\frac {\operatorname {E} }{\sigma (X_{2})\sigma (X_{n})}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\operatorname {E} }{\sigma (X_{n})\sigma (X_{1})}}&{\frac {\operatorname {E} }{\sigma (X_{n})\sigma (X_{2})}}&\cdots &1\end{bmatrix}}.}
11524:
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7443:
255:
2066:
11510:
41:
8752:
11063:
14888:
5899:
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7255:
10198:
1892:
10352:
11343:
6763:
191:
5704:
8421:
1779:
10671:
6886:
5109:
10036:
4860:
10203:
1510:
11330:
6054:{\displaystyle {\boldsymbol {\mu }}={\begin{bmatrix}{\boldsymbol {\mu }}_{X}\\{\boldsymbol {\mu }}_{Y}\end{bmatrix}},\qquad {\boldsymbol {\Sigma }}={\begin{bmatrix}\operatorname {K} _{\mathbf {XX} }&\operatorname {K} _{\mathbf {XY} }\\\operatorname {K} _{\mathbf {YX} }&\operatorname {K} _{\mathbf {YY} }\end{bmatrix}}}
7840:
6631:
1305:
8139:
7438:{\displaystyle \operatorname {K} _{\mathbf {XY\mid I} }=\operatorname {pcov} (\mathbf {X} ,\mathbf {Y} \mid \mathbf {I} )=\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )-\operatorname {cov} (\mathbf {X} ,\mathbf {I} )\operatorname {cov} (\mathbf {I} ,\mathbf {I} )^{-1}\operatorname {cov} (\mathbf {I} ,\mathbf {Y} ).}
1598:
2061:{\displaystyle \operatorname {corr} (\mathbf {X} )={\big (}\operatorname {diag} (\operatorname {K} _{\mathbf {X} \mathbf {X} }){\big )}^{-{\frac {1}{2}}}\,\operatorname {K} _{\mathbf {X} \mathbf {X} }\,{\big (}\operatorname {diag} (\operatorname {K} _{\mathbf {X} \mathbf {X} }){\big )}^{-{\frac {1}{2}}},}
12471:
O Kornilov, M Eckstein, M Rosenblatt, C P Schulz, K Motomura, A RouzĂ©e, J Klei, L Foucar, M Siano, A LĂŒbcke, F. Schapper, P Johnsson, D M P Holland, T Schlatholter, T Marchenko, S DĂŒsterer, K Ueda, M J J Vrakking and L J Frasinski "Coulomb explosion of diatomic molecules in intense XUV fields mapped
9094:
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a scalar factor and small random fluctuations (proven for a single-parent strategy and a static model, as the population size increases, relying on the quadratic approximation). Intuitively, this result is supported by the rationale that the optimal covariance distribution can offer mutation steps
5571:
12356:
W J Krzanowski "Principles of
Multivariate Analysis" (Oxford University Press, New York, 1988), Chap. 14.4; K V Mardia, J T Kent and J M Bibby "Multivariate Analysis (Academic Press, London, 1997), Chap. 6.5.3; T W Anderson "An Introduction to Multivariate Statistical Analysis" (Wiley, New York,
11505:{\displaystyle \operatorname {pcov} (\mathbf {X} ,\mathbf {Y} \mid \mathbf {I} )=\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )-\operatorname {cov} (\mathbf {X} ,\mathbf {I} )\left(\operatorname {cov} (\mathbf {I} ,\mathbf {I} )\backslash \operatorname {cov} (\mathbf {I} ,\mathbf {Y} )\right),}
8405:
5235:
6248:
10428:, a particular family of Randomized Search Heuristics, fundamentally relies on a covariance matrix in its mechanism. The characteristic mutation operator draws the update step from a multivariate normal distribution using an evolving covariance matrix. There is a formal proof that the
8747:{\displaystyle {\begin{aligned}&w^{\mathsf {T}}\operatorname {E} \left)(\mathbf {X} -\operatorname {E} )^{\mathsf {T}}\right]w=\operatorname {E} \left)(\mathbf {X} -\operatorname {E} )^{\mathsf {T}}w\right]\\&=\operatorname {E} {\big ){\big )}^{2}{\big ]}\geq 0,\end{aligned}}}
6623:
5520:
11058:{\displaystyle \left={\begin{bmatrix}X_{1}(t_{1})&X_{2}(t_{1})&\cdots &X_{n}(t_{1})\\\\X_{1}(t_{2})&X_{2}(t_{2})&\cdots &X_{n}(t_{2})\\\\\vdots &\vdots &\ddots &\vdots \\\\X_{1}(t_{m})&X_{2}(t_{m})&\cdots &X_{n}(t_{m})\end{bmatrix}},}
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11558:
shows that 10% overcorrection improves the map and makes ion-ion correlations clearly visible. Owing to momentum conservation these correlations appear as lines approximately perpendicular to the autocorrelation line (and to the periodic modulations which are caused by detector
1360:
11230:
7657:
10593:, the map shows statistical relations between different regions of the random functions. Statistically independent regions of the functions show up on the map as zero-level flatland, while positive or negative correlations show up, respectively, as hills or valleys.
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5010:
1155:
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391:
Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the
1028:
7177:
A covariance matrix with all non-zero elements tells us that all the individual random variables are interrelated. This means that the variables are not only directly correlated, but also correlated via other variables indirectly. Often such indirect,
10193:{\displaystyle \mathbf {Q} _{\mathbf {XX} }={\frac {1}{n-1}}\mathbf {M} _{\mathbf {X} }\mathbf {M} _{\mathbf {X} }^{\mathsf {T}},\qquad \mathbf {Q} _{\mathbf {XY} }={\frac {1}{n-1}}\mathbf {M} _{\mathbf {X} }\mathbf {M} _{\mathbf {Y} }^{\mathsf {T}}}
9559:
9813:
10347:{\displaystyle \mathbf {Q} _{\mathbf {XX} }={\frac {1}{n}}\mathbf {M} _{\mathbf {X} }\mathbf {M} _{\mathbf {X} }^{\mathsf {T}},\qquad \mathbf {Q} _{\mathbf {XY} }={\frac {1}{n}}\mathbf {M} _{\mathbf {X} }\mathbf {M} _{\mathbf {Y} }^{\mathsf {T}}.}
8047:
10356:
These empirical sample covariance matrices are the most straightforward and most often used estimators for the covariance matrices, but other estimators also exist, including regularised or shrinkage estimators, which may have better properties.
8972:
8319:
6758:{\displaystyle {\boldsymbol {\mu }}_{\mathbf {Y} |\mathbf {X} }={\boldsymbol {\mu }}_{\mathbf {Y} }+\operatorname {K} _{\mathbf {YX} }\operatorname {K} _{\mathbf {XX} }^{-1}\left(\mathbf {X} -{\boldsymbol {\mu }}_{\mathbf {X} }\right)}
11221:
7106:
6946:
12113:
show. The suppression of the uninteresting correlations is, however, imperfect because there are other sources of common-mode fluctuations than the laser intensity and in principle all these sources should be monitored in vector
11958:
5699:{\displaystyle \operatorname {var} (\mathbf {X} +\mathbf {Y} )=\operatorname {var} (\mathbf {X} )+\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )+\operatorname {cov} (\mathbf {Y} ,\mathbf {X} )+\operatorname {var} (\mathbf {Y} )}
6531:
6163:
6111:
12006:(note a change in the colour scale). Unfortunately, this map is overwhelmed by uninteresting, common-mode correlations induced by laser intensity fluctuating from shot to shot. To suppress such correlations the laser intensity
2112:
1774:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {E} )(\mathbf {X} -\operatorname {E} )^{\mathsf {T}}]=\operatorname {R} _{\mathbf {X} \mathbf {X} }-\operatorname {E} \operatorname {E} ^{\mathsf {T}}}
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1847:
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of nitrogen molecules multiply ionised by a laser pulse. Since only a few hundreds of molecules are ionised at each laser pulse, the single-shot spectra are highly fluctuating. However, collecting typically
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correlations are trivial and uninteresting. They can be suppressed by calculating the partial covariance matrix, that is the part of covariance matrix that shows only the interesting part of correlations.
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1593:
1556:
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814:
534:
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reveals several nitrogen ions in a form of peaks broadened by their kinetic energy, but to find the correlations between the ionisation stages and the ion momenta requires calculating a covariance map.
854:
8244:
7481:
7882:
6881:{\displaystyle \operatorname {K} _{\mathbf {Y|X} }=\operatorname {K} _{\mathbf {YY} }-\operatorname {K} _{\mathbf {YX} }\operatorname {K} _{\mathbf {XX} }^{-1}\operatorname {K} _{\mathbf {XY} }.}
8426:
6987:
5104:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {E} (\mathbf {XX^{\mathsf {T}}} )-{\boldsymbol {\mu }}_{\mathbf {X} }{\boldsymbol {\mu }}_{\mathbf {X} }^{\mathsf {T}}}
9415:
7967:
7513:
7023:
6314:
6282:
9665:
4855:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {var} (\mathbf {X} )=\operatorname {E} \left\right)\left(\mathbf {X} -\operatorname {E} \right)^{\mathsf {T}}\right]}
9584:
7630:
5280:
11734:
1505:{\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )=\operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} \left)(\mathbf {Y} -\operatorname {E} )^{\mathsf {T}}\right].}
11762:
10031:
9997:
11325:{\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )\approx \langle \mathbf {XY^{\mathsf {T}}} \rangle -\langle \mathbf {X} \rangle \langle \mathbf {Y} ^{\mathsf {T}}\rangle ,}
9895:
9864:
9402:
7137:
10624:
7914:
7835:{\displaystyle \operatorname {f} (\mathbf {X} )=(2\pi )^{-n/2}|{\boldsymbol {\Sigma }}|^{-1/2}\exp \left(-{\tfrac {1}{2}}\mathbf {(X-\mu )^{\mathsf {T}}\Sigma ^{-1}(X-\mu )} \right),}
7940:
7652:
7045:
5850:
9817:
In contrast to the covariance matrix defined above, Hermitian transposition gets replaced by transposition in the definition. Its diagonal elements may be complex valued; it is a
5825:
2272:
11842:
11804:
11677:
1300:{\displaystyle \operatorname {var} (\mathbf {X} )=\operatorname {cov} (\mathbf {X} ,\mathbf {X} )=\operatorname {E} \left)(\mathbf {X} -\operatorname {E} )^{\mathsf {T}}\right].}
8134:{\displaystyle \mathbf {d} ^{\mathsf {T}}{\boldsymbol {\Sigma }}\mathbf {c} =\operatorname {cov} (\mathbf {d} ^{\mathsf {T}}\mathbf {X} ,\mathbf {c} ^{\mathsf {T}}\mathbf {X} )}
1048:
11152:
10404:. The matrix of covariances among various assets' returns is used to determine, under certain assumptions, the relative amounts of different assets that investors should (in a
9296:
12134:
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5434:
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2310:
2215:
1887:
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1124:
626:
604:
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142:
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456:
334:
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554:
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2177:
846:
680:
653:
12157:
11597:
11143:
9138:
10373:, that allows one to completely decorrelate the data or, from a different point of view, to find an optimal basis for representing the data in a compact way (see
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9269:
8885:
8835:
7585:
5002:
574:
430:
410:
308:
288:
7053:
6893:
9089:{\displaystyle \operatorname {var} (\mathbf {M} ^{1/2}\mathbf {X} )=\mathbf {M} ^{1/2}\,\operatorname {var} (\mathbf {X} )\,\mathbf {M} ^{1/2}=\mathbf {M} .}
11915:
6116:
6064:
691:
170:
2071:
8400:{\displaystyle \operatorname {var} (\mathbf {b} ^{\mathsf {T}}\mathbf {X} )=\mathbf {b} ^{\mathsf {T}}\operatorname {var} (\mathbf {X} )\mathbf {b} ,\,}
12058:
10498:
9147:
5230:{\displaystyle \mathbf {a} ^{T}\operatorname {K} _{\mathbf {X} \mathbf {X} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {R} ^{n}}
1784:
15493:
9408:
transposing and conjugating. In the following expression, the product of a vector with its conjugate transpose results in a square matrix called the
8144:
6243:{\displaystyle \operatorname {K} _{\mathbf {XY} }=\operatorname {K} _{\mathbf {YX} }^{\mathsf {T}}=\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )}
9965:
rows of variables, from which the row means have been subtracted, then, if the row means were estimated from the data, sample covariance matrices
12140:
shows, where interesting correlations of ion momenta are now clearly visible as straight lines centred on ionisation stages of atomic nitrogen.
3141:
Each element on the principal diagonal of a correlation matrix is the correlation of a random variable with itself, which always equals 1. Each
13985:
1864:
6618:{\displaystyle \mathbf {Y} \mid \mathbf {X} \sim \ {\mathcal {N}}({\boldsymbol {\mu }}_{\mathbf {Y|X} },\operatorname {K} _{\mathbf {Y|X} }),}
14490:
5515:{\displaystyle \operatorname {var} (\mathbf {AX} +\mathbf {a} )=\mathbf {A} \,\operatorname {var} (\mathbf {X} )\,\mathbf {A} ^{\mathsf {T}}}
9104:
4865:
12164:. Mathematically, the former is expressed in terms of the sample covariance matrix and the technique is equivalent to covariance mapping.
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201:
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9304:
8757:
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1527:
1312:
1069:
785:
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262:
with a standard deviation of 3 in roughly the lower leftâupper right direction and of 1 in the orthogonal direction. Because the
132:
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97:
7448:
12376:
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12237:
9898:
7845:
6473:{\displaystyle \mathbf {X} ,\mathbf {Y} \sim \ {\mathcal {N}}({\boldsymbol {\mu }},\operatorname {\boldsymbol {\Sigma }} ),}
107:
15717:
15483:
12900:
12600:
12149:
8033:{\displaystyle \mathbf {c} ^{\mathsf {T}}\Sigma =\operatorname {cov} (\mathbf {c} ^{\mathsf {T}}\mathbf {X} ,\mathbf {X} )}
127:
92:
15871:
13504:
12652:
9635:{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {Z} }^{\mathsf {H}}=\operatorname {K} _{\mathbf {Z} \mathbf {Z} }}
6955:
5331:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }^{\mathsf {T}}=\operatorname {K} _{\mathbf {X} \mathbf {X} }}
4702:
This duality motivates a number of other dualities between marginalizing and conditioning for
Gaussian random variables.
156:
7486:
6996:
6287:
6255:
12491:
I Noda "Generalized two-dimensional correlation method applicable to infrared, Raman, and other types of spectroscopy"
11513:
102:
10441:
whose equidensity probability contours match the level sets of the landscape, and so they maximize the progress rate.
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14287:
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12338:
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11702:
9830:
11516:
operator, which bypasses the requirement to invert a matrix and is available in some computational packages such as
14892:
14465:
14339:
11600:
7588:
6412:
1023:{\displaystyle \operatorname {K} _{X_{i}X_{j}}=\operatorname {cov} =\operatorname {E} )(X_{j}-\operatorname {E} )]}
11739:
10002:
9968:
3208:
Just as the covariance matrix can be written as the rescaling of a correlation matrix by the marginal variances:
1126:, because it is the natural generalization to higher dimensions of the 1-dimensional variance. Others call it the
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Fig. 1 illustrates how a partial covariance map is constructed on an example of an experiment performed at the
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9565:
9554:{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {Z} }=\operatorname {cov} =\operatorname {E} \left,}
8412:
7596:
5148:
9808:{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {Z} }=\operatorname {cov} =\operatorname {E} \left}
5808:
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8817:
Conversely, every symmetric positive semi-definite matrix is a covariance matrix. To see this, suppose
32:
12136:. Yet in practice it is often sufficient to overcompensate the partial covariance correction as panel
3195:, if it exists, is the inverse covariance matrix (or inverse concentration matrix), also known as the
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maps the partial covariance matrix that is corrected for the intensity fluctuations. Panel
11337:
8247:
7161:. In this form they correspond to the coefficients obtained by inverting the matrix of the
6766:
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819:
658:
631:
377:
11573:
11216:{\displaystyle \langle \mathbf {X} \rangle ={\frac {1}{n}}\sum _{j=1}^{n}\mathbf {X} _{j}}
11119:
9123:
8:
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7101:{\displaystyle \operatorname {K} _{\mathbf {XX} }^{-1}\operatorname {K} _{\mathbf {XY} }}
6949:
6941:{\displaystyle \operatorname {K} _{\mathbf {YX} }\operatorname {K} _{\mathbf {XX} }^{-1}}
3708:
12230:
Fundamentals of
Probability and Stochastic Processes with Applications to Communications
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12541:
11953:{\displaystyle \langle \mathbf {X} \rangle \langle \mathbf {Y} ^{\mathsf {T}}\rangle }
1130:, because it is the matrix of covariances between the scalar components of the vector
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12294:
12260:
12233:
11604:
11224:
10377:
for a formal proof and additional properties of covariance matrices). This is called
10374:
6158:{\displaystyle \operatorname {K} _{\mathbf {YY} }=\operatorname {var} (\mathbf {Y} )}
6106:{\displaystyle \operatorname {K} _{\mathbf {XX} }=\operatorname {var} (\mathbf {X} )}
3711:, and partial variance, the inverse covariance matrix can be expressed analogously:
2107:{\displaystyle \operatorname {diag} (\operatorname {K} _{\mathbf {X} \mathbf {X} })}
15845:
15813:
15742:
15681:
15676:
15656:
15592:
15498:
15468:
15453:
15433:
15372:
15325:
15300:
15290:
15261:
15180:
15175:
15150:
15080:
15060:
14970:
14950:
14799:
14754:
14518:
14505:
14398:
14373:
14307:
14239:
14117:
13725:
13618:
13551:
13464:
13411:
13230:
13101:
12895:
12779:
12694:
12661:
12430:
9578:
9562:
9110:
8864:
7162:
6990:
6626:
5274:
3197:
466:
15438:
1309:
Both forms are quite standard, and there is no ambiguity between them. The matrix
15543:
15478:
15458:
15443:
15423:
15407:
15305:
15236:
15226:
15185:
15070:
15040:
14716:
14460:
14322:
14249:
13924:
13798:
13771:
13748:
13717:
13344:
13339:
13293:
13023:
12674:
12254:
12098:{\displaystyle \operatorname {pcov} (\mathbf {X} ,\mathbf {Y} \mid \mathbf {I} )}
10590:
10538:{\displaystyle \operatorname {pcov} (\mathbf {X} ,\mathbf {Y} \mid \mathbf {I} )}
2220:
Equivalently, the correlation matrix can be seen as the covariance matrix of the
2149:
771:
14206:
8863:
symmetric positive-semidefinite matrix. From the finite-dimensional case of the
1842:{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {X} }=\operatorname {E} }
15803:
15747:
15727:
15712:
15671:
15548:
15508:
15473:
15397:
15336:
15315:
15256:
15246:
15231:
15165:
15110:
15100:
15095:
15005:
14665:
14660:
13123:
13053:
12699:
12183:
11145:. The expected values needed in the covariance formula are estimated using the
10433:
9114:
1059:
779:
458:
matrix would be necessary to fully characterize the two-dimensional variation.
12435:
12408:
8408:
15865:
15808:
15666:
15607:
15538:
15528:
15523:
15448:
15377:
15251:
15241:
15170:
15090:
15075:
15010:
14822:
14789:
14652:
14613:
14424:
14393:
13857:
13811:
13416:
13118:
12945:
12709:
12704:
11550:
maps common-mode correlations via intensity fluctuations of the laser. Panel
11332:
where the angular brackets denote sample averaging as before except that the
8175:{\displaystyle \mathbf {c} ^{\mathsf {T}}{\boldsymbol {\Sigma }}\mathbf {c} }
8041:
686:
385:
8185:
Similarly, the (pseudo-)inverse covariance matrix provides an inner product
7050:
The matrix of regression coefficients may often be given in transpose form,
336:
covariance matrix is needed; the directions of the arrows correspond to the
15691:
15648:
15553:
15266:
15205:
15115:
14995:
14764:
14697:
14674:
14589:
13919:
13215:
13113:
13048:
12990:
12975:
12912:
12867:
11340:. Using this estimation the partial covariance matrix can be calculated as
9568:, with real numbers in the main diagonal and complex numbers off-diagonal.
9105:
Complex random vector § Covariance matrix and pseudo-covariance matrix
10365:
The covariance matrix is a useful tool in many different areas. From it a
15533:
15503:
15271:
15105:
14975:
14807:
14769:
14452:
14353:
14215:
14028:
13995:
13487:
13404:
13399:
13043:
13000:
12980:
12960:
12950:
12719:
11146:
7917:
341:
337:
11532:
molecules undergoing
Coulomb explosion induced by a free-electron laser.
15584:
15045:
13653:
13133:
12833:
12764:
12714:
12689:
12609:
9654:
For complex random vectors, another kind of second central moment, the
7956:
Applied to one vector, the covariance matrix maps a linear combination
849:
462:
381:
353:
340:
of this covariance matrix and their lengths to the square roots of the
25:
12480:
12315:
11736:, which is shown in red at the bottom of Fig. 1. The average spectrum
7108:, suitable for post-multiplying a row vector of explanatory variables
4908:{\displaystyle {\boldsymbol {\mu }}_{\mathbf {X} }=\operatorname {E} }
15818:
15392:
13806:
13658:
13278:
13073:
12985:
12970:
12965:
12930:
12546:
8411:
of a real-valued random variable, so a covariance matrix is always a
7540:
763:{\displaystyle \mathbf {X} =(X_{1},X_{2},\dots ,X_{n})^{\mathsf {T}}}
254:
11542:
map the two terms of the covariance matrix, which is shown in panel
8943:
column vector-valued random variable whose covariance matrix is the
5004:-dimensional random variable, the following basic properties apply:
1863:
An entity closely related to the covariance matrix is the matrix of
1058:
Nomenclatures differ. Some statisticians, following the probabilist
15752:
13322:
12940:
12817:
12812:
12807:
12425:
9298:; thus the variance of a complex random variable is a real number.
775:
474:
3188:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }^{-1}}
244:{\displaystyle {\begin{bmatrix}1&0.5\\0.5&1\end{bmatrix}}}
40:
14904:
14827:
14528:
11564:
9370:
is a column vector of complex-valued random variables, then the
9363:{\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{n})^{\mathsf {T}}}
9244:{\displaystyle \operatorname {var} (Z)=\operatorname {E} \left,}
8807:{\displaystyle w^{\mathsf {T}}(\mathbf {X} -\operatorname {E} )}
8044:, it yields the covariance between the two linear combinations:
4977:{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})^{\mathsf {T}}}
14749:
13730:
13704:
13684:
12935:
12726:
11999:{\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )}
11517:
10488:{\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )}
5741:{\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )}
5266:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }\,}
5140:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }\,}
190:
12578:
10437:
2141:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
1588:{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {X} }}
1551:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
1336:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
1093:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
809:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
529:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
12409:"On the covariance-Hessian relation in evolution strategies"
12316:"Lectures on probability theory and mathematical statistics"
12290:
All of
Statistics: A Concise Course in Statistical Inference
8754:
where the last inequality follows from the observation that
12669:
11901:{\displaystyle \langle \mathbf {XY^{\mathsf {T}}} \rangle }
628:
are used to refer to random vectors, and Roman subscripted
12536:
12256:
An introduction to probability theory and its applications
11844:
are the same, except that the range of the time-of-flight
8257:
1867:
between each of the random variables in the random vector
1064:
1053:
8239:{\displaystyle \langle c-\mu |\Sigma ^{+}|c-\mu \rangle }
7945:
1519:
12143:
10545:
matrix are plotted as a 2-dimensional map. When vectors
9645:
The diagonal elements of the covariance matrix are real.
7476:{\displaystyle \operatorname {K} _{\mathbf {XY\mid I} }}
11528:
Figure 1: Construction of a partial covariance map of N
7877:{\displaystyle {\boldsymbol {\mu }}=\operatorname {E} }
432:
directions contain all of the necessary information; a
11466:
10741:
7759:
5972:
5916:
4476:
3974:
3753:
3601:
3339:
3240:
2344:
210:
18:
Measure of covariance of components of a random vector
12120:
12061:
12039:
12012:
11970:
11918:
11874:
11850:
11812:
11774:
11742:
11705:
11685:
11647:
11614:
11576:
11346:
11233:
11155:
11122:
11071:
10674:
10654:
10632:
10602:
10573:
10551:
10501:
10459:
10206:
10039:
10005:
9971:
9933:
9907:
9872:
9841:
9668:
9587:
9418:
9379:
9307:
9277:
9257:
9150:
9126:
8975:
8949:
8923:
8901:
8873:
8843:
8823:
8760:
8424:
8322:
8290:
8268:
8191:
8147:
8050:
7970:
7926:
7890:
7848:
7660:
7638:
7604:
7573:
7551:
7521:
7489:
7451:
7258:
7252:, the latter correlations are suppressed in a matrix
7236:
7214:
7192:
7145:
7114:
7056:
7031:
6999:
6958:
6896:
6774:
6634:
6534:
6512:
6490:
6421:
6395:
6373:
6348:
6326:
6290:
6258:
6171:
6119:
6067:
5902:
5880:
5858:
5836:
5811:
5780:
5758:
5712:
5574:
5552:
5530:
5442:
5420:
5394:
5372:
5346:
5283:
5244:
5157:
5118:
5013:
4990:
4921:
4868:
4716:
3717:
3214:
3159:
2318:
2280:
2229:
2185:
2158:
2120:
2074:
1895:
1873:
1852:
1787:
1601:
1567:
1530:
1363:
1315:
1158:
1136:
1110:
1072:
1036:
857:
822:
788:
694:
661:
634:
612:
590:
562:
542:
508:
486:
438:
418:
398:
316:
296:
276:
204:
14491:
Autoregressive conditional heteroskedasticity (ARCH)
5546:
is another random vector with the same dimension as
477:(i.e., the covariance of each element with itself).
198:
centered at (0, 0), with covariance matrix given by
7964:onto a vector of covariances with those variables:
7632:can be expressed in terms of the covariance matrix
6982:{\displaystyle \operatorname {K} _{\mathbf {Y|X} }}
3148:
1050:denotes the expected value (mean) of its argument.
13953:
12557:
12128:
12097:
12047:
12025:
11998:
11952:
11900:
11856:
11836:
11798:
11756:
11728:
11691:
11671:
11633:
11591:
11504:
11324:
11215:
11137:
11100:
11057:
10660:
10640:
10618:
10581:
10559:
10537:
10487:
10432:'s covariance matrix adapts to the inverse of the
10346:
10192:
10025:
9991:
9945:
9919:
9889:
9858:
9807:
9634:
9553:
9396:
9362:
9290:
9263:
9243:
9132:
9088:
8961:
8935:
8909:
8879:
8855:
8829:
8806:
8746:
8407:which must always be nonnegative, since it is the
8399:
8308:
8276:
8238:
8174:
8133:
8032:
7934:
7908:
7876:
7834:
7646:
7624:
7579:
7559:
7541:Covariance matrix as a parameter of a distribution
7529:
7508:{\displaystyle \operatorname {K} _{\mathbf {XY} }}
7507:
7475:
7437:
7244:
7222:
7200:
7153:
7131:
7100:
7039:
7018:{\displaystyle \operatorname {K} _{\mathbf {XX} }}
7017:
6981:
6940:
6880:
6757:
6617:
6520:
6498:
6472:
6403:
6381:
6356:
6334:
6316:can be identified as the variance matrices of the
6309:{\displaystyle \operatorname {K} _{\mathbf {YY} }}
6308:
6277:{\displaystyle \operatorname {K} _{\mathbf {XX} }}
6276:
6242:
6157:
6105:
6053:
5888:
5866:
5844:
5819:
5788:
5766:
5740:
5698:
5560:
5538:
5514:
5428:
5406:
5380:
5358:
5330:
5265:
5229:
5139:
5103:
4996:
4976:
4907:
4854:
4694:
3697:
3187:
3131:
2304:
2266:
2209:
2171:
2140:
2106:
2060:
1881:
1841:
1773:
1587:
1550:
1504:
1347:, since the diagonal terms are in fact variances.
1335:
1299:
1144:
1118:
1092:
1042:
1022:
840:
808:
762:
674:
647:
620:
598:
568:
548:
528:
494:
450:
424:
404:
328:
302:
282:
243:
12406:
7820:
7808:
7784:
7771:
6969:
6785:
6599:
6576:
584:Throughout this article, boldfaced unsubscripted
15863:
12533:", an easy way to visualize covariance matrices!
12331:Multivariate Statistics: a Vector Space Approach
9251:where the complex conjugate of a complex number
7625:{\displaystyle \operatorname {f} (\mathbf {X} )}
14039:Multivariate adaptive regression splines (MARS)
11729:{\displaystyle \langle \mathbf {X} (t)\rangle }
8141:. The variance of a linear combination is then
1865:Pearson product-moment correlation coefficients
1781:where the autocorrelation matrix is defined as
196:bivariate Gaussian probability density function
12449:L J Frasinski "Covariance mapping techniques"
12252:
12148:Two-dimensional infrared spectroscopy employs
11223:and the covariance matrix is estimated by the
8418:The above argument can be expanded as follows:
682:are used to refer to scalar random variables.
14920:
12594:
12560:Stochastic processes in physics and chemistry
8726:
8713:
8662:
8655:
2034:
1997:
1955:
1918:
164:
11947:
11930:
11927:
11919:
11895:
11875:
11757:{\displaystyle \langle \mathbf {X} \rangle }
11751:
11743:
11723:
11706:
11316:
11299:
11296:
11288:
11282:
11262:
11164:
11156:
10026:{\displaystyle \mathbf {Q} _{\mathbf {XY} }}
9992:{\displaystyle \mathbf {Q} _{\mathbf {XX} }}
8233:
8192:
7483:is effectively the simple covariance matrix
7172:
7139:rather than pre-multiplying a column vector
12352:
12350:
12223:
12221:
12219:
12156:. There are two versions of this analysis:
9890:{\displaystyle \mathbf {M} _{\mathbf {Y} }}
9859:{\displaystyle \mathbf {M} _{\mathbf {X} }}
9649:
15494:Fundamental (linear differential equation)
14927:
14913:
12639:
12601:
12587:
12555:
10388:The covariance matrix plays a key role in
10200:or, if the row means were known a priori,
9397:{\displaystyle \mathbf {Z} ^{\mathsf {H}}}
7132:{\displaystyle \mathbf {X} ^{\mathsf {T}}}
2114:is the matrix of the diagonal elements of
310:do not fully describe the distribution. A
171:
157:
13252:
12531:Covariance Matrix Explained With Pictures
12434:
12424:
12333:. John Wiley and Sons. pp. 116â117.
12286:
12246:
10619:{\displaystyle \mathbf {X} ,\mathbf {Y} }
9098:
9054:
9036:
8396:
7909:{\displaystyle |{\boldsymbol {\Sigma }}|}
7515:as if the uninteresting random variables
5497:
5479:
5262:
5217:
5136:
1994:
1976:
480:The covariance matrix of a random vector
384:between each pair of elements of a given
12366:
12347:
12216:
11522:
7587:possibly correlated random variables is
253:
189:
15799:Matrix representation of conic sections
12467:
12465:
12360:
12357:2003), 3rd ed., Chaps. 2.5.1 and 4.3.1.
10648:are acquired experimentally as rows of
9776:
9748:
9519:
9491:
8258:Which matrices are covariance matrices?
7935:{\displaystyle {\boldsymbol {\Sigma }}}
7850:
7647:{\displaystyle {\boldsymbol {\Sigma }}}
7040:{\displaystyle {\boldsymbol {\Sigma }}}
6738:
6664:
6637:
6566:
6452:
5937:
5921:
5904:
5845:{\displaystyle {\boldsymbol {\Sigma }}}
5813:
5082:
5068:
4871:
1054:Conflicting nomenclatures and notations
15864:
14565:KaplanâMeier estimator (product limit)
12313:
12189:Lewandowski-Kurowicka-Joe distribution
11941:
11888:
11310:
11275:
10419:
10335:
10266:
10184:
10107:
9794:
9606:
9537:
9388:
9354:
8767:
8673:
8623:
8552:
8518:
8436:
8365:
8340:
8156:
8117:
8095:
8059:
8008:
7979:
7946:Covariance matrix as a linear operator
7789:
7123:
6952:coefficients, while in linear algebra
6206:
5506:
5302:
5095:
5053:
4968:
4841:
1830:
1765:
1696:
1520:Relation to the autocorrelation matrix
1488:
1283:
754:
14908:
14638:
14205:
13952:
13251:
13021:
12638:
12582:
12537:
12443:
12369:A Foundation in Digital Communication
12328:
12307:
12144:Two-dimensional infrared spectroscopy
10444:
8250:, a measure of the "unlikelihood" of
270:components co-vary, the variances of
14875:
14575:Accelerated failure time (AFT) model
12462:
12322:
12227:
12033:is recorded at every shot, put into
9120:random variable with expected value
5820:{\displaystyle {\boldsymbol {\mu }}}
2267:{\displaystyle X_{i}/\sigma (X_{i})}
1350:By comparison, the notation for the
14887:
14170:Analysis of variance (ANOVA, anova)
13022:
12391:
11837:{\displaystyle \mathbf {Y} _{j}(t)}
11799:{\displaystyle \mathbf {X} _{j}(t)}
11672:{\displaystyle \mathbf {X} _{j}(t)}
7186:If two vectors of random variables
5340:For any constant (i.e. non-random)
4897:
4705:
13:
14934:
14265:CochranâMantelâHaenszel statistics
12891:Pearson product-moment correlation
12505:
12407:Shir, O.M.; A. Yehudayoff (2020).
9724:
9670:
9616:
9589:
9467:
9420:
9169:
8784:
8690:
8647:
8600:
8569:
8535:
8495:
8464:
8442:
8262:From the identity just above, let
8210:
7985:
7857:
7661:
7605:
7491:
7453:
7260:
7230:are correlated via another vector
7084:
7058:
7001:
6960:
6916:
6898:
6861:
6835:
6817:
6799:
6776:
6698:
6680:
6590:
6556:
6443:
6292:
6260:
6191:
6173:
6121:
6069:
6029:
6012:
5993:
5976:
5312:
5285:
5246:
5171:
5120:
5034:
5015:
4886:
4817:
4781:
4757:
4718:
3161:
2995:
2877:
2729:
2601:
2478:
2355:
2122:
2085:
2012:
1978:
1933:
1853:Relation to the correlation matrix
1808:
1789:
1745:
1728:
1709:
1673:
1642:
1622:
1603:
1569:
1532:
1465:
1434:
1412:
1393:
1317:
1260:
1229:
1207:
1074:
1043:{\displaystyle \operatorname {E} }
1037:
992:
951:
926:
859:
790:
543:
510:
14:
15893:
11964:shows their difference, which is
11768:In the example of Fig. 1 spectra
10412:) choose to hold in a context of
9831:Estimation of covariance matrices
5800:
15833:
14886:
14874:
14862:
14849:
14848:
14639:
12122:
12088:
12080:
12072:
12041:
11989:
11981:
11935:
11923:
11883:
11879:
11815:
11777:
11747:
11710:
11650:
11570:in Hamburg. The random function
11512:where the backslash denotes the
11487:
11479:
11459:
11451:
11429:
11421:
11401:
11393:
11373:
11365:
11357:
11304:
11292:
11270:
11266:
11252:
11244:
11203:
11160:
10718:
10697:
10682:
10634:
10612:
10604:
10575:
10553:
10528:
10520:
10512:
10478:
10470:
10328:
10322:
10314:
10308:
10287:
10284:
10278:
10259:
10253:
10245:
10239:
10218:
10215:
10209:
10177:
10171:
10163:
10157:
10128:
10125:
10119:
10100:
10094:
10086:
10080:
10051:
10048:
10042:
10017:
10014:
10008:
9983:
9980:
9974:
9881:
9875:
9850:
9844:
9782:
9767:
9754:
9739:
9709:
9699:
9680:
9675:
9626:
9621:
9599:
9594:
9525:
9510:
9497:
9482:
9457:
9449:
9430:
9425:
9382:
9309:
9140:is conventionally defined using
9079:
9057:
9047:
9018:
9006:
8987:
8903:
8794:
8777:
8700:
8683:
8610:
8593:
8579:
8562:
8505:
8488:
8474:
8457:
8389:
8381:
8359:
8347:
8334:
8270:
8168:
8163:
8150:
8124:
8111:
8102:
8089:
8071:
8066:
8053:
8023:
8015:
8002:
7973:
7928:
7897:
7867:
7814:
7811:
7803:
7800:
7796:
7777:
7774:
7717:
7671:
7640:
7615:
7553:
7523:
7499:
7496:
7467:
7461:
7458:
7425:
7417:
7390:
7382:
7365:
7357:
7337:
7329:
7309:
7301:
7293:
7274:
7268:
7265:
7238:
7216:
7194:
7147:
7117:
7092:
7089:
7066:
7063:
7033:
7009:
7006:
6973:
6965:
6924:
6921:
6906:
6903:
6869:
6866:
6843:
6840:
6825:
6822:
6807:
6804:
6789:
6781:
6744:
6729:
6706:
6703:
6688:
6685:
6670:
6653:
6643:
6603:
6595:
6580:
6572:
6544:
6536:
6514:
6492:
6460:
6431:
6423:
6397:
6375:
6350:
6328:
6300:
6297:
6268:
6265:
6233:
6225:
6199:
6196:
6181:
6178:
6148:
6129:
6126:
6096:
6077:
6074:
6037:
6034:
6020:
6017:
6001:
5998:
5984:
5981:
5960:
5882:
5860:
5838:
5782:
5760:
5731:
5723:
5689:
5669:
5661:
5641:
5633:
5613:
5593:
5585:
5554:
5532:
5500:
5490:
5475:
5464:
5456:
5453:
5422:
5374:
5322:
5317:
5295:
5290:
5256:
5251:
5208:
5191:
5181:
5176:
5160:
5130:
5125:
5088:
5074:
5048:
5044:
5025:
5020:
4923:
4877:
4827:
4810:
4791:
4774:
4747:
4728:
4723:
3728:
3225:
3171:
3166:
3149:Inverse of the covariance matrix
3145:is between â1 and +1 inclusive.
2329:
2132:
2127:
2095:
2090:
2022:
2017:
1988:
1983:
1943:
1938:
1906:
1875:
1824:
1818:
1799:
1794:
1755:
1738:
1719:
1714:
1683:
1666:
1652:
1635:
1613:
1608:
1579:
1574:
1542:
1537:
1475:
1458:
1444:
1427:
1403:
1398:
1382:
1374:
1327:
1322:
1270:
1253:
1239:
1222:
1197:
1189:
1169:
1138:
1112:
1084:
1079:
800:
795:
696:
614:
592:
520:
515:
488:
39:
15701:Used in science and engineering
14524:Least-squares spectral analysis
12485:
12474:J. Phys. B: At. Mol. Opt. Phys.
12451:J. Phys. B: At. Mol. Opt. Phys.
10596:In practice the column vectors
10360:
10275:
10116:
9561:The matrix so obtained will be
9291:{\displaystyle {\overline {z}}}
5958:
5201:
473:and its main diagonal contains
260:bivariate Gaussian distribution
14944:Explicitly constrained entries
13505:Mean-unbiased minimum-variance
12608:
12400:
12385:
12371:. Cambridge University Press.
12280:
12092:
12068:
11993:
11977:
11831:
11825:
11793:
11787:
11720:
11714:
11666:
11660:
11586:
11580:
11491:
11475:
11463:
11447:
11433:
11417:
11405:
11389:
11377:
11353:
11256:
11240:
11132:
11126:
11095:
11082:
11041:
11028:
11008:
10995:
10980:
10967:
10922:
10909:
10889:
10876:
10861:
10848:
10828:
10815:
10795:
10782:
10767:
10754:
10532:
10508:
10482:
10466:
10398:mutual fund separation theorem
9789:
9763:
9760:
9735:
9718:
9695:
9532:
9506:
9503:
9478:
9461:
9445:
9349:
9316:
9224:
9205:
9199:
9180:
9163:
9157:
9051:
9043:
9010:
8982:
8801:
8798:
8790:
8773:
8707:
8704:
8696:
8679:
8618:
8614:
8606:
8589:
8586:
8583:
8575:
8558:
8513:
8509:
8501:
8484:
8481:
8478:
8470:
8453:
8385:
8377:
8351:
8329:
8303:
8291:
8220:
8205:
8182:, its covariance with itself.
8128:
8084:
8027:
7997:
7902:
7892:
7871:
7863:
7723:
7712:
7691:
7681:
7675:
7667:
7619:
7611:
7445:The partial covariance matrix
7429:
7413:
7395:
7378:
7369:
7353:
7341:
7325:
7313:
7289:
6648:
6609:
6561:
6464:
6448:
6237:
6221:
6152:
6144:
6100:
6092:
5735:
5719:
5693:
5685:
5673:
5657:
5645:
5629:
5617:
5609:
5597:
5581:
5494:
5486:
5468:
5449:
5060:
5040:
4963:
4930:
4902:
4892:
4831:
4823:
4795:
4787:
4751:
4743:
4641:
4560:
4500:
3918:
3837:
3777:
3733:
3724:
3229:
3221:
3102:
3089:
3083:
3070:
3062:
3059:
3033:
3030:
3004:
3001:
2984:
2971:
2965:
2952:
2944:
2941:
2915:
2912:
2886:
2883:
2836:
2823:
2817:
2804:
2796:
2793:
2767:
2764:
2738:
2735:
2708:
2695:
2689:
2676:
2668:
2665:
2639:
2636:
2610:
2607:
2585:
2572:
2566:
2553:
2545:
2542:
2516:
2513:
2487:
2484:
2462:
2449:
2443:
2430:
2422:
2419:
2393:
2390:
2364:
2361:
2333:
2325:
2261:
2248:
2101:
2081:
2028:
2008:
1949:
1929:
1910:
1902:
1836:
1814:
1760:
1751:
1742:
1734:
1702:
1691:
1687:
1679:
1662:
1659:
1656:
1648:
1631:
1628:
1483:
1479:
1471:
1454:
1451:
1448:
1440:
1423:
1386:
1370:
1278:
1274:
1266:
1249:
1246:
1243:
1235:
1218:
1201:
1185:
1173:
1165:
1017:
1014:
1011:
998:
976:
973:
970:
957:
935:
932:
920:
894:
835:
823:
749:
703:
1:
15718:Fundamental (computer vision)
14818:Geographic information system
14034:Simultaneous equations models
12394:"The Matrix Reference Manual"
12209:
11112:-th discrete value in sample
9824:
5896:can be written in block form
2222:standardized random variables
1514:
782:, then the covariance matrix
579:
14001:Coefficient of determination
13612:Uniformly most powerful test
12413:Theoretical Computer Science
12152:to obtain 2D spectra of the
12129:{\displaystyle \mathbf {I} }
12048:{\displaystyle \mathbf {I} }
11101:{\displaystyle X_{j}(t_{i})}
10641:{\displaystyle \mathbf {I} }
10582:{\displaystyle \mathbf {Y} }
10560:{\displaystyle \mathbf {X} }
10408:) or are predicted to (in a
10379:principal component analysis
9713:
9283:
9228:
8910:{\displaystyle \mathbf {X} }
8887:has a nonnegative symmetric
8413:positive-semidefinite matrix
8277:{\displaystyle \mathbf {b} }
7597:probability density function
7589:jointly normally distributed
7560:{\displaystyle \mathbf {X} }
7530:{\displaystyle \mathbf {I} }
7245:{\displaystyle \mathbf {I} }
7223:{\displaystyle \mathbf {Y} }
7201:{\displaystyle \mathbf {X} }
7154:{\displaystyle \mathbf {X} }
6521:{\displaystyle \mathbf {X} }
6499:{\displaystyle \mathbf {Y} }
6413:jointly normally distributed
6404:{\displaystyle \mathbf {Y} }
6382:{\displaystyle \mathbf {X} }
6357:{\displaystyle \mathbf {Y} }
6335:{\displaystyle \mathbf {X} }
5889:{\displaystyle \mathbf {Y} }
5867:{\displaystyle \mathbf {X} }
5789:{\displaystyle \mathbf {Y} }
5767:{\displaystyle \mathbf {X} }
5561:{\displaystyle \mathbf {X} }
5539:{\displaystyle \mathbf {Y} }
5429:{\displaystyle \mathbf {a} }
5381:{\displaystyle \mathbf {A} }
3153:The inverse of this matrix,
2305:{\displaystyle i=1,\dots ,n}
2210:{\displaystyle i=1,\dots ,n}
1882:{\displaystyle \mathbf {X} }
1145:{\displaystyle \mathbf {X} }
1119:{\displaystyle \mathbf {X} }
621:{\displaystyle \mathbf {Y} }
599:{\displaystyle \mathbf {X} }
495:{\displaystyle \mathbf {X} }
7:
15484:Duplication and elimination
15283:eigenvalues or eigenvectors
14570:Proportional hazards models
14514:Spectral density estimation
14496:Vector autoregression (VAR)
13930:Maximum posterior estimator
13162:Randomized controlled trial
12564:. New York: North-Holland.
12518:Encyclopedia of Mathematics
12204:Quadratic form (statistics)
12167:
11699:produces a smooth spectrum
10402:capital asset pricing model
9957:columns of observations of
9577:The covariance matrix is a
8309:{\displaystyle (p\times 1)}
1524:The auto-covariance matrix
10:
15898:
15872:Covariance and correlation
15417:With specific applications
15046:Discrete Fourier Transform
14330:Multivariate distributions
12750:Average absolute deviation
12556:van Kampen, N. G. (1981).
11679:, and averaging them over
9828:
9102:
8891:, which can be denoted by
7949:
6948:is known as the matrix of
1889:, which can be written as
1856:
1345:variance-covariance matrix
374:varianceâcovariance matrix
182:
133:Cross-correlation function
98:Cross-correlation function
33:Correlation and covariance
15827:
15776:
15708:CabibboâKobayashiâMaskawa
15700:
15646:
15582:
15416:
15335:Satisfying conditions on
15334:
15280:
15219:
14943:
14844:
14798:
14735:
14688:
14651:
14647:
14634:
14606:
14588:
14555:
14546:
14504:
14451:
14412:
14361:
14352:
14318:Structural equation model
14273:
14230:
14226:
14201:
14160:
14126:
14080:
14047:
14009:
13976:
13972:
13948:
13888:
13797:
13716:
13680:
13671:
13654:Score/Lagrange multiplier
13639:
13592:
13537:
13463:
13454:
13264:
13260:
13247:
13206:
13180:
13132:
13087:
13069:Sample size determination
13034:
13030:
13017:
12921:
12876:
12850:
12832:
12788:
12740:
12660:
12651:
12647:
12634:
12616:
12436:10.1016/j.tcs.2019.09.002
12329:Eaton, Morris L. (1983).
12287:Wasserman, Larry (2004).
10436:of the search landscape,
10369:can be derived, called a
9946:{\displaystyle q\times n}
9920:{\displaystyle p\times n}
9662:) is defined as follows:
8962:{\displaystyle p\times p}
8936:{\displaystyle p\times 1}
8856:{\displaystyle p\times p}
8316:real-valued vector, then
7173:Partial covariance matrix
5407:{\displaystyle m\times 1}
5359:{\displaystyle m\times n}
1343:is also often called the
451:{\displaystyle 2\times 2}
329:{\displaystyle 2\times 2}
143:Cross-covariance function
121:For deterministic signals
108:Cross-covariance function
14813:Environmental statistics
14335:Elliptical distributions
14128:Generalized linear model
14057:Simple linear regression
13827:HodgesâLehmann estimator
13284:Probability distribution
13193:Stochastic approximation
12755:Coefficient of variation
12179:Eigenvalue decomposition
12105:is calculated as panels
11634:{\displaystyle m=10^{4}}
11603:spectrum of ions from a
11336:should be made to avoid
10383:KarhunenâLoĂšve transform
10371:whitening transformation
9953:respectively, i.e. with
9819:complex symmetric matrix
9656:pseudo-covariance matrix
9650:Pseudo-covariance matrix
7960:of the random variables
7593:elliptically distributed
6482:conditional distribution
502:is typically denoted by
183:Not to be confused with
128:Autocorrelation function
93:Autocorrelation function
86:For stochastic processes
63:Cross-correlation matrix
15066:Generalized permutation
14473:Cross-correlation (XCF)
14081:Non-standard predictors
13515:LehmannâScheffĂ© theorem
13188:Adaptive clinical trial
12472:by partial covariance"
12367:Lapidoth, Amos (2009).
12253:William Feller (1971).
12194:Multivariate statistics
11116:of the random function
5829:joint covariance matrix
5750:cross-covariance matrix
1352:cross-covariance matrix
1062:in his two-volume book
549:{\displaystyle \Sigma }
185:Cross-covariance matrix
138:Autocovariance function
103:Autocovariance function
73:Cross-covariance matrix
15840:Mathematics portal
14869:Mathematics portal
14690:Engineering statistics
14598:NelsonâAalen estimator
14175:Analysis of covariance
14062:Ordinary least squares
13986:Pearson product-moment
13390:Statistical functional
13301:Empirical distribution
13134:Controlled experiments
12863:Frequency distribution
12641:Descriptive statistics
12314:Taboga, Marco (2010).
12130:
12099:
12049:
12027:
12000:
11954:
11902:
11858:
11838:
11800:
11758:
11730:
11693:
11673:
11635:
11593:
11560:
11506:
11326:
11217:
11200:
11139:
11102:
11059:
10662:
10642:
10620:
10583:
10561:
10539:
10489:
10348:
10194:
10027:
9993:
9947:
9921:
9891:
9860:
9809:
9636:
9555:
9412:, as its expectation:
9398:
9364:
9292:
9265:
9245:
9134:
9099:Complex random vectors
9090:
8969:identity matrix. Then
8963:
8937:
8911:
8881:
8857:
8831:
8808:
8748:
8401:
8310:
8278:
8240:
8176:
8135:
8034:
7936:
7910:
7878:
7836:
7648:
7626:
7581:
7561:
7531:
7509:
7477:
7439:
7246:
7224:
7202:
7167:ordinary least squares
7155:
7133:
7102:
7041:
7019:
6983:
6942:
6882:
6759:
6619:
6522:
6500:
6474:
6405:
6383:
6358:
6336:
6318:marginal distributions
6310:
6278:
6244:
6159:
6107:
6055:
5890:
5868:
5846:
5821:
5790:
5768:
5742:
5700:
5562:
5540:
5516:
5430:
5408:
5382:
5360:
5332:
5267:
5231:
5141:
5105:
4998:
4978:
4909:
4856:
4696:
3707:So, using the idea of
3699:
3189:
3133:
2306:
2268:
2211:
2173:
2142:
2108:
2062:
1883:
1843:
1775:
1589:
1560:autocorrelation matrix
1552:
1506:
1337:
1301:
1146:
1120:
1094:
1044:
1024:
842:
810:
764:
685:If the entries in the
676:
649:
622:
600:
570:
550:
530:
496:
471:positive semi-definite
452:
426:
406:
362:auto-covariance matrix
345:
330:
304:
284:
251:
245:
68:Auto-covariance matrix
58:Autocorrelation matrix
14785:Population statistics
14727:System identification
14461:Autocorrelation (ACF)
14389:Exponential smoothing
14303:Discriminant analysis
14298:Canonical correlation
14162:Partition of variance
14024:Regression validation
13868:(JonckheereâTerpstra)
13767:Likelihood-ratio test
13456:Frequentist inference
13368:Locationâscale family
13289:Sampling distribution
13254:Statistical inference
13221:Cross-sectional study
13208:Observational studies
13167:Randomized experiment
12996:Stem-and-leaf display
12798:Central limit theorem
12419:. Elsevier: 157â174.
12228:Park, Kun Il (2018).
12131:
12100:
12050:
12028:
12026:{\displaystyle I_{j}}
12001:
11955:
11903:
11859:
11839:
11801:
11759:
11731:
11694:
11674:
11636:
11594:
11526:
11507:
11327:
11218:
11180:
11140:
11103:
11060:
10663:
10643:
10621:
10584:
10562:
10540:
10490:
10367:transformation matrix
10349:
10195:
10033:can be defined to be
10028:
9994:
9948:
9922:
9892:
9861:
9810:
9637:
9566:positive-semidefinite
9556:
9399:
9365:
9293:
9266:
9246:
9135:
9103:Further information:
9091:
8964:
8938:
8912:
8882:
8858:
8832:
8809:
8749:
8402:
8311:
8279:
8241:
8177:
8136:
8035:
7937:
7911:
7879:
7837:
7649:
7627:
7582:
7562:
7532:
7510:
7478:
7440:
7247:
7225:
7203:
7156:
7134:
7103:
7042:
7020:
6984:
6943:
6883:
6760:
6620:
6523:
6501:
6475:
6406:
6384:
6359:
6337:
6311:
6279:
6245:
6160:
6108:
6056:
5891:
5869:
5847:
5822:
5791:
5769:
5743:
5701:
5563:
5541:
5517:
5431:
5409:
5383:
5361:
5333:
5268:
5232:
5149:positive-semidefinite
5142:
5106:
4999:
4979:
4910:
4857:
4697:
3700:
3190:
3134:
2307:
2269:
2212:
2174:
2172:{\displaystyle X_{i}}
2143:
2109:
2063:
1884:
1857:Further information:
1844:
1776:
1590:
1553:
1507:
1338:
1302:
1147:
1121:
1104:of the random vector
1095:
1045:
1025:
843:
841:{\displaystyle (i,j)}
811:
765:
677:
675:{\displaystyle Y_{i}}
650:
648:{\displaystyle X_{i}}
623:
601:
571:
551:
531:
497:
453:
427:
407:
331:
305:
285:
258:Sample points from a
257:
246:
193:
14708:Probabilistic design
14293:Principal components
14136:Exponential families
14088:Nonlinear regression
14067:General linear model
14029:Mixed effects models
14019:Errors and residuals
13996:Confounding variable
13898:Bayesian probability
13876:Van der Waerden test
13866:Ordered alternative
13631:Multiple comparisons
13510:RaoâBlackwellization
13473:Estimating equations
13429:Statistical distance
13147:Factorial experiment
12680:Arithmetic-Geometric
12199:Principal components
12150:correlation analysis
12118:
12059:
12037:
12010:
11968:
11916:
11872:
11848:
11810:
11772:
11740:
11703:
11683:
11645:
11612:
11592:{\displaystyle X(t)}
11574:
11514:left matrix division
11344:
11231:
11153:
11138:{\displaystyle X(t)}
11120:
11069:
10672:
10652:
10630:
10600:
10571:
10549:
10499:
10457:
10204:
10037:
10003:
9969:
9931:
9905:
9870:
9839:
9666:
9585:
9416:
9377:
9305:
9275:
9255:
9148:
9133:{\displaystyle \mu }
9124:
8973:
8947:
8921:
8899:
8871:
8841:
8821:
8758:
8422:
8320:
8288:
8266:
8248:Mahalanobis distance
8246:, which induces the
8189:
8145:
8048:
7968:
7924:
7888:
7846:
7658:
7636:
7602:
7591:, or more generally
7571:
7549:
7537:were held constant.
7519:
7487:
7449:
7256:
7234:
7212:
7190:
7143:
7112:
7054:
7029:
6997:
6956:
6894:
6772:
6767:conditional variance
6632:
6532:
6510:
6488:
6419:
6393:
6371:
6346:
6324:
6288:
6256:
6169:
6117:
6065:
5900:
5878:
5856:
5834:
5809:
5778:
5756:
5710:
5572:
5550:
5528:
5440:
5418:
5392:
5370:
5344:
5281:
5242:
5155:
5116:
5011:
4988:
4919:
4866:
4714:
3715:
3212:
3203:concentration matrix
3157:
3143:off-diagonal element
2316:
2278:
2227:
2183:
2156:
2152:of the variances of
2118:
2072:
1893:
1871:
1785:
1599:
1565:
1528:
1361:
1313:
1156:
1134:
1108:
1070:
1034:
855:
820:
816:is the matrix whose
786:
692:
659:
632:
610:
588:
560:
540:
506:
484:
436:
416:
396:
314:
294:
274:
202:
24:Part of a series on
15789:Linear independence
15036:Diagonally dominant
14780:Official statistics
14703:Methods engineering
14384:Seasonal adjustment
14152:Poisson regressions
14072:Bayesian regression
14011:Regression analysis
13991:Partial correlation
13963:Regression analysis
13562:Prediction interval
13557:Likelihood interval
13547:Confidence interval
13539:Interval estimation
13500:Unbiased estimators
13318:Model specification
13198:Up-and-down designs
12886:Partial correlation
12842:Index of dispersion
12760:Interquartile range
12542:"Covariance Matrix"
12513:"Covariance matrix"
12174:Covariance function
11568:free-electron laser
11334:Bessel's correction
10420:Use in optimization
10390:financial economics
10340:
10271:
10189:
10112:
9611:
9372:conjugate transpose
9142:complex conjugation
7952:Covariance operator
7545:If a column vector
7079:
6937:
6856:
6719:
6211:
5307:
5100:
3709:partial correlation
3184:
1066:, call the matrix
1030:where the operator
774:, each with finite
15882:Summary statistics
15794:Matrix exponential
15784:Jordan normal form
15618:Fisher information
15489:Euclidean distance
15403:Totally unimodular
14800:Spatial statistics
14680:Medical statistics
14580:First hitting time
14534:Whittle likelihood
14185:Degrees of freedom
14180:Multivariate ANOVA
14113:Heteroscedasticity
13925:Bayesian estimator
13890:Bayesian inference
13739:KolmogorovâSmirnov
13624:Randomization test
13594:Testing hypotheses
13567:Tolerance interval
13478:Maximum likelihood
13373:Exponential family
13306:Density estimation
13266:Statistical theory
13226:Natural experiment
13172:Scientific control
13089:Survey methodology
12775:Standard deviation
12539:Weisstein, Eric W.
12126:
12095:
12045:
12023:
11996:
11950:
11898:
11854:
11834:
11796:
11754:
11726:
11689:
11669:
11631:
11589:
11561:
11502:
11322:
11213:
11135:
11098:
11055:
11046:
10658:
10638:
10616:
10579:
10557:
10535:
10485:
10453:the values of the
10451:covariance mapping
10445:Covariance mapping
10430:evolution strategy
10426:evolution strategy
10406:normative analysis
10344:
10320:
10251:
10190:
10169:
10092:
10023:
9989:
9943:
9917:
9887:
9856:
9805:
9632:
9588:
9551:
9394:
9360:
9288:
9261:
9241:
9130:
9086:
8959:
8933:
8907:
8877:
8867:, it follows that
8853:
8827:
8804:
8744:
8742:
8397:
8306:
8274:
8236:
8172:
8131:
8030:
7932:
7906:
7874:
7832:
7768:
7644:
7622:
7577:
7557:
7527:
7505:
7473:
7435:
7242:
7220:
7198:
7151:
7129:
7098:
7057:
7037:
7015:
6979:
6938:
6915:
6878:
6834:
6755:
6697:
6615:
6518:
6496:
6470:
6401:
6379:
6354:
6332:
6306:
6274:
6240:
6190:
6155:
6103:
6051:
6045:
5949:
5886:
5864:
5842:
5817:
5786:
5764:
5738:
5696:
5558:
5536:
5512:
5426:
5404:
5378:
5356:
5328:
5284:
5263:
5227:
5137:
5101:
5080:
4994:
4974:
4905:
4852:
4692:
4686:
4465:
3963:
3695:
3689:
3590:
3328:
3185:
3160:
3129:
3120:
2302:
2264:
2207:
2169:
2138:
2104:
2058:
1879:
1859:Correlation matrix
1839:
1771:
1585:
1558:is related to the
1548:
1502:
1333:
1297:
1142:
1116:
1090:
1040:
1020:
838:
806:
760:
672:
645:
618:
596:
566:
546:
526:
492:
448:
422:
402:
350:probability theory
346:
326:
300:
280:
252:
241:
235:
51:For random vectors
15859:
15858:
15851:Category:Matrices
15723:Fuzzy associative
15613:Doubly stochastic
15321:Positive-definite
15001:Block tridiagonal
14902:
14901:
14840:
14839:
14836:
14835:
14775:National accounts
14745:Actuarial science
14737:Social statistics
14630:
14629:
14626:
14625:
14622:
14621:
14557:Survival function
14542:
14541:
14404:Granger causality
14245:Contingency table
14220:Survival analysis
14197:
14196:
14193:
14192:
14049:Linear regression
13944:
13943:
13940:
13939:
13915:Credible interval
13884:
13883:
13667:
13666:
13483:Method of moments
13352:Parametric family
13313:Statistical model
13243:
13242:
13239:
13238:
13157:Random assignment
13079:Statistical power
13013:
13012:
13009:
13008:
12858:Contingency table
12828:
12827:
12695:Generalized/power
12378:978-0-521-19395-5
12266:978-0-471-25709-7
12239:978-3-319-68074-3
11857:{\displaystyle t}
11692:{\displaystyle j}
11605:Coulomb explosion
11225:sample covariance
11178:
10661:{\displaystyle n}
10410:positive analysis
10375:Rayleigh quotient
10304:
10235:
10153:
10076:
9716:
9410:covariance matrix
9286:
9264:{\displaystyle z}
9231:
8880:{\displaystyle M}
8830:{\displaystyle M}
7767:
7580:{\displaystyle n}
6553:
6440:
5205:
4997:{\displaystyle n}
4899:
4682:
4598:
4525:
3959:
3875:
3802:
3106:
2988:
2840:
2712:
2589:
2466:
2051:
1972:
1128:covariance matrix
569:{\displaystyle S}
425:{\displaystyle y}
405:{\displaystyle x}
366:dispersion matrix
358:covariance matrix
303:{\displaystyle y}
283:{\displaystyle x}
181:
180:
15889:
15846:List of matrices
15838:
15837:
15814:Row echelon form
15758:State transition
15687:Seidel adjacency
15569:Totally positive
15429:Alternating sign
15026:Complex Hadamard
14929:
14922:
14915:
14906:
14905:
14890:
14889:
14878:
14877:
14867:
14866:
14852:
14851:
14755:Crime statistics
14649:
14648:
14636:
14635:
14553:
14552:
14519:Fourier analysis
14506:Frequency domain
14486:
14433:
14399:Structural break
14359:
14358:
14308:Cluster analysis
14255:Log-linear model
14228:
14227:
14203:
14202:
14144:
14118:Homoscedasticity
13974:
13973:
13950:
13949:
13869:
13861:
13853:
13852:(KruskalâWallis)
13837:
13822:
13777:Cross validation
13762:
13744:AndersonâDarling
13691:
13678:
13677:
13649:Likelihood-ratio
13641:Parametric tests
13619:Permutation test
13602:1- & 2-tails
13493:Minimum distance
13465:Point estimation
13461:
13460:
13412:Optimal decision
13363:
13262:
13261:
13249:
13248:
13231:Quasi-experiment
13181:Adaptive designs
13032:
13031:
13019:
13018:
12896:Rank correlation
12658:
12657:
12649:
12648:
12636:
12635:
12603:
12596:
12589:
12580:
12579:
12575:
12563:
12552:
12551:
12526:
12499:
12493:Appl. Spectrosc.
12489:
12483:
12469:
12460:
12447:
12441:
12440:
12438:
12428:
12404:
12398:
12397:
12389:
12383:
12382:
12364:
12358:
12354:
12345:
12344:
12326:
12320:
12319:
12311:
12305:
12304:
12284:
12278:
12277:
12275:
12273:
12250:
12244:
12243:
12225:
12135:
12133:
12132:
12127:
12125:
12104:
12102:
12101:
12096:
12091:
12083:
12075:
12054:
12052:
12051:
12046:
12044:
12032:
12030:
12029:
12024:
12022:
12021:
12005:
12003:
12002:
11997:
11992:
11984:
11959:
11957:
11956:
11951:
11946:
11945:
11944:
11938:
11926:
11907:
11905:
11904:
11899:
11894:
11893:
11892:
11891:
11863:
11861:
11860:
11855:
11843:
11841:
11840:
11835:
11824:
11823:
11818:
11805:
11803:
11802:
11797:
11786:
11785:
11780:
11763:
11761:
11760:
11755:
11750:
11735:
11733:
11732:
11727:
11713:
11698:
11696:
11695:
11690:
11678:
11676:
11675:
11670:
11659:
11658:
11653:
11640:
11638:
11637:
11632:
11630:
11629:
11598:
11596:
11595:
11590:
11511:
11509:
11508:
11503:
11498:
11494:
11490:
11482:
11462:
11454:
11432:
11424:
11404:
11396:
11376:
11368:
11360:
11331:
11329:
11328:
11323:
11315:
11314:
11313:
11307:
11295:
11281:
11280:
11279:
11278:
11255:
11247:
11222:
11220:
11219:
11214:
11212:
11211:
11206:
11199:
11194:
11179:
11171:
11163:
11144:
11142:
11141:
11136:
11107:
11105:
11104:
11099:
11094:
11093:
11081:
11080:
11064:
11062:
11061:
11056:
11051:
11050:
11040:
11039:
11027:
11026:
11007:
11006:
10994:
10993:
10979:
10978:
10966:
10965:
10953:
10928:
10921:
10920:
10908:
10907:
10888:
10887:
10875:
10874:
10860:
10859:
10847:
10846:
10834:
10827:
10826:
10814:
10813:
10794:
10793:
10781:
10780:
10766:
10765:
10753:
10752:
10732:
10728:
10727:
10726:
10721:
10706:
10705:
10700:
10691:
10690:
10685:
10667:
10665:
10664:
10659:
10647:
10645:
10644:
10639:
10637:
10625:
10623:
10622:
10617:
10615:
10607:
10591:random functions
10588:
10586:
10585:
10580:
10578:
10566:
10564:
10563:
10558:
10556:
10544:
10542:
10541:
10536:
10531:
10523:
10515:
10494:
10492:
10491:
10486:
10481:
10473:
10394:portfolio theory
10392:, especially in
10385:(KL-transform).
10353:
10351:
10350:
10345:
10339:
10338:
10332:
10331:
10325:
10319:
10318:
10317:
10311:
10305:
10297:
10292:
10291:
10290:
10281:
10270:
10269:
10263:
10262:
10256:
10250:
10249:
10248:
10242:
10236:
10228:
10223:
10222:
10221:
10212:
10199:
10197:
10196:
10191:
10188:
10187:
10181:
10180:
10174:
10168:
10167:
10166:
10160:
10154:
10152:
10138:
10133:
10132:
10131:
10122:
10111:
10110:
10104:
10103:
10097:
10091:
10090:
10089:
10083:
10077:
10075:
10061:
10056:
10055:
10054:
10045:
10032:
10030:
10029:
10024:
10022:
10021:
10020:
10011:
9998:
9996:
9995:
9990:
9988:
9987:
9986:
9977:
9952:
9950:
9949:
9944:
9926:
9924:
9923:
9918:
9896:
9894:
9893:
9888:
9886:
9885:
9884:
9878:
9865:
9863:
9862:
9857:
9855:
9854:
9853:
9847:
9814:
9812:
9811:
9806:
9804:
9800:
9799:
9798:
9797:
9787:
9786:
9785:
9779:
9770:
9759:
9758:
9757:
9751:
9742:
9717:
9712:
9707:
9702:
9685:
9684:
9683:
9678:
9641:
9639:
9638:
9633:
9631:
9630:
9629:
9624:
9610:
9609:
9603:
9602:
9597:
9579:Hermitian matrix
9560:
9558:
9557:
9552:
9547:
9543:
9542:
9541:
9540:
9530:
9529:
9528:
9522:
9513:
9502:
9501:
9500:
9494:
9485:
9460:
9452:
9435:
9434:
9433:
9428:
9403:
9401:
9400:
9395:
9393:
9392:
9391:
9385:
9369:
9367:
9366:
9361:
9359:
9358:
9357:
9347:
9346:
9328:
9327:
9312:
9297:
9295:
9294:
9289:
9287:
9279:
9270:
9268:
9267:
9262:
9250:
9248:
9247:
9242:
9237:
9233:
9232:
9227:
9223:
9222:
9203:
9198:
9197:
9139:
9137:
9136:
9131:
9095:
9093:
9092:
9087:
9082:
9074:
9073:
9069:
9060:
9050:
9035:
9034:
9030:
9021:
9009:
9004:
9003:
8999:
8990:
8968:
8966:
8965:
8960:
8942:
8940:
8939:
8934:
8916:
8914:
8913:
8908:
8906:
8886:
8884:
8883:
8878:
8865:spectral theorem
8862:
8860:
8859:
8854:
8836:
8834:
8833:
8828:
8813:
8811:
8810:
8805:
8797:
8780:
8772:
8771:
8770:
8753:
8751:
8750:
8745:
8743:
8730:
8729:
8723:
8722:
8717:
8716:
8703:
8686:
8678:
8677:
8676:
8666:
8665:
8659:
8658:
8640:
8636:
8632:
8628:
8627:
8626:
8613:
8596:
8582:
8565:
8557:
8556:
8555:
8528:
8524:
8523:
8522:
8521:
8508:
8491:
8477:
8460:
8441:
8440:
8439:
8428:
8406:
8404:
8403:
8398:
8392:
8384:
8370:
8369:
8368:
8362:
8350:
8345:
8344:
8343:
8337:
8315:
8313:
8312:
8307:
8283:
8281:
8280:
8275:
8273:
8245:
8243:
8242:
8237:
8223:
8218:
8217:
8208:
8181:
8179:
8178:
8173:
8171:
8166:
8161:
8160:
8159:
8153:
8140:
8138:
8137:
8132:
8127:
8122:
8121:
8120:
8114:
8105:
8100:
8099:
8098:
8092:
8074:
8069:
8064:
8063:
8062:
8056:
8039:
8037:
8036:
8031:
8026:
8018:
8013:
8012:
8011:
8005:
7984:
7983:
7982:
7976:
7941:
7939:
7938:
7933:
7931:
7915:
7913:
7912:
7907:
7905:
7900:
7895:
7883:
7881:
7880:
7875:
7870:
7853:
7841:
7839:
7838:
7833:
7828:
7824:
7823:
7807:
7806:
7794:
7793:
7792:
7769:
7760:
7743:
7742:
7738:
7726:
7720:
7715:
7710:
7709:
7705:
7674:
7653:
7651:
7650:
7645:
7643:
7631:
7629:
7628:
7623:
7618:
7586:
7584:
7583:
7578:
7566:
7564:
7563:
7558:
7556:
7536:
7534:
7533:
7528:
7526:
7514:
7512:
7511:
7506:
7504:
7503:
7502:
7482:
7480:
7479:
7474:
7472:
7471:
7470:
7444:
7442:
7441:
7436:
7428:
7420:
7406:
7405:
7393:
7385:
7368:
7360:
7340:
7332:
7312:
7304:
7296:
7279:
7278:
7277:
7251:
7249:
7248:
7243:
7241:
7229:
7227:
7226:
7221:
7219:
7207:
7205:
7204:
7199:
7197:
7163:normal equations
7160:
7158:
7157:
7152:
7150:
7138:
7136:
7135:
7130:
7128:
7127:
7126:
7120:
7107:
7105:
7104:
7099:
7097:
7096:
7095:
7078:
7070:
7069:
7046:
7044:
7043:
7038:
7036:
7024:
7022:
7021:
7016:
7014:
7013:
7012:
6991:Schur complement
6988:
6986:
6985:
6980:
6978:
6977:
6976:
6972:
6947:
6945:
6944:
6939:
6936:
6928:
6927:
6911:
6910:
6909:
6887:
6885:
6884:
6879:
6874:
6873:
6872:
6855:
6847:
6846:
6830:
6829:
6828:
6812:
6811:
6810:
6794:
6793:
6792:
6788:
6764:
6762:
6761:
6756:
6754:
6750:
6749:
6748:
6747:
6741:
6732:
6718:
6710:
6709:
6693:
6692:
6691:
6675:
6674:
6673:
6667:
6658:
6657:
6656:
6651:
6646:
6640:
6627:conditional mean
6624:
6622:
6621:
6616:
6608:
6607:
6606:
6602:
6585:
6584:
6583:
6579:
6569:
6560:
6559:
6551:
6547:
6539:
6527:
6525:
6524:
6519:
6517:
6505:
6503:
6502:
6497:
6495:
6479:
6477:
6476:
6471:
6463:
6455:
6447:
6446:
6438:
6434:
6426:
6410:
6408:
6407:
6402:
6400:
6388:
6386:
6385:
6380:
6378:
6363:
6361:
6360:
6355:
6353:
6341:
6339:
6338:
6333:
6331:
6315:
6313:
6312:
6307:
6305:
6304:
6303:
6283:
6281:
6280:
6275:
6273:
6272:
6271:
6249:
6247:
6246:
6241:
6236:
6228:
6210:
6209:
6203:
6202:
6186:
6185:
6184:
6164:
6162:
6161:
6156:
6151:
6134:
6133:
6132:
6112:
6110:
6109:
6104:
6099:
6082:
6081:
6080:
6060:
6058:
6057:
6052:
6050:
6049:
6042:
6041:
6040:
6025:
6024:
6023:
6006:
6005:
6004:
5989:
5988:
5987:
5963:
5954:
5953:
5946:
5945:
5940:
5930:
5929:
5924:
5907:
5895:
5893:
5892:
5887:
5885:
5873:
5871:
5870:
5865:
5863:
5851:
5849:
5848:
5843:
5841:
5826:
5824:
5823:
5818:
5816:
5795:
5793:
5792:
5787:
5785:
5773:
5771:
5770:
5765:
5763:
5747:
5745:
5744:
5739:
5734:
5726:
5705:
5703:
5702:
5697:
5692:
5672:
5664:
5644:
5636:
5616:
5596:
5588:
5567:
5565:
5564:
5559:
5557:
5545:
5543:
5542:
5537:
5535:
5521:
5519:
5518:
5513:
5511:
5510:
5509:
5503:
5493:
5478:
5467:
5459:
5435:
5433:
5432:
5427:
5425:
5413:
5411:
5410:
5405:
5387:
5385:
5384:
5379:
5377:
5365:
5363:
5362:
5357:
5337:
5335:
5334:
5329:
5327:
5326:
5325:
5320:
5306:
5305:
5299:
5298:
5293:
5272:
5270:
5269:
5264:
5261:
5260:
5259:
5254:
5236:
5234:
5233:
5228:
5226:
5225:
5220:
5211:
5206:
5203:
5194:
5186:
5185:
5184:
5179:
5169:
5168:
5163:
5146:
5144:
5143:
5138:
5135:
5134:
5133:
5128:
5110:
5108:
5107:
5102:
5099:
5098:
5092:
5091:
5085:
5079:
5078:
5077:
5071:
5059:
5058:
5057:
5056:
5030:
5029:
5028:
5023:
5003:
5001:
5000:
4995:
4983:
4981:
4980:
4975:
4973:
4972:
4971:
4961:
4960:
4942:
4941:
4926:
4914:
4912:
4911:
4906:
4901:
4900:
4882:
4881:
4880:
4874:
4861:
4859:
4858:
4853:
4851:
4847:
4846:
4845:
4844:
4838:
4834:
4830:
4813:
4802:
4798:
4794:
4777:
4750:
4733:
4732:
4731:
4726:
4706:Basic properties
4701:
4699:
4698:
4693:
4691:
4690:
4683:
4681:
4680:
4679:
4678:
4654:
4653:
4644:
4639:
4638:
4621:
4618:
4617:
4604:
4603:
4599:
4597:
4596:
4586:
4585:
4573:
4572:
4563:
4558:
4557:
4540:
4537:
4529:
4528:
4526:
4524:
4523:
4513:
4512:
4503:
4498:
4497:
4480:
4470:
4469:
4452:
4451:
4450:
4449:
4425:
4424:
4412:
4411:
4399:
4398:
4386:
4385:
4366:
4365:
4364:
4363:
4339:
4338:
4326:
4325:
4313:
4312:
4269:
4268:
4267:
4266:
4242:
4241:
4229:
4228:
4216:
4215:
4203:
4202:
4173:
4172:
4162:
4161:
4149:
4148:
4136:
4135:
4114:
4113:
4112:
4111:
4087:
4086:
4074:
4073:
4061:
4060:
4036:
4035:
4025:
4024:
4012:
4011:
3999:
3998:
3968:
3967:
3960:
3958:
3957:
3956:
3955:
3931:
3930:
3921:
3916:
3915:
3898:
3895:
3894:
3881:
3880:
3876:
3874:
3873:
3863:
3862:
3850:
3849:
3840:
3835:
3834:
3817:
3814:
3806:
3805:
3803:
3801:
3800:
3790:
3789:
3780:
3775:
3774:
3757:
3744:
3743:
3731:
3704:
3702:
3701:
3696:
3694:
3693:
3686:
3685:
3684:
3683:
3668:
3667:
3654:
3653:
3649:
3648:
3647:
3646:
3631:
3623:
3622:
3620:
3619:
3618:
3617:
3595:
3594:
3577:
3576:
3575:
3574:
3562:
3561:
3545:
3544:
3543:
3542:
3530:
3529:
3489:
3488:
3487:
3486:
3474:
3473:
3447:
3446:
3445:
3444:
3432:
3431:
3413:
3412:
3411:
3410:
3398:
3397:
3376:
3375:
3374:
3373:
3361:
3360:
3333:
3332:
3325:
3324:
3323:
3322:
3307:
3306:
3293:
3292:
3288:
3287:
3286:
3285:
3270:
3262:
3261:
3259:
3258:
3257:
3256:
3228:
3198:precision matrix
3194:
3192:
3191:
3186:
3183:
3175:
3174:
3169:
3138:
3136:
3135:
3130:
3125:
3124:
3107:
3105:
3101:
3100:
3082:
3081:
3065:
3058:
3057:
3045:
3044:
3029:
3028:
3016:
3015:
2993:
2989:
2987:
2983:
2982:
2964:
2963:
2947:
2940:
2939:
2927:
2926:
2911:
2910:
2898:
2897:
2875:
2870:
2845:
2841:
2839:
2835:
2834:
2816:
2815:
2799:
2792:
2791:
2779:
2778:
2763:
2762:
2750:
2749:
2727:
2713:
2711:
2707:
2706:
2688:
2687:
2671:
2664:
2663:
2651:
2650:
2635:
2634:
2622:
2621:
2599:
2594:
2590:
2588:
2584:
2583:
2565:
2564:
2548:
2541:
2540:
2528:
2527:
2512:
2511:
2499:
2498:
2476:
2467:
2465:
2461:
2460:
2442:
2441:
2425:
2418:
2417:
2405:
2404:
2389:
2388:
2376:
2375:
2353:
2332:
2311:
2309:
2308:
2303:
2273:
2271:
2270:
2265:
2260:
2259:
2244:
2239:
2238:
2216:
2214:
2213:
2208:
2178:
2176:
2175:
2170:
2168:
2167:
2147:
2145:
2144:
2139:
2137:
2136:
2135:
2130:
2113:
2111:
2110:
2105:
2100:
2099:
2098:
2093:
2067:
2065:
2064:
2059:
2054:
2053:
2052:
2044:
2038:
2037:
2027:
2026:
2025:
2020:
2001:
2000:
1993:
1992:
1991:
1986:
1975:
1974:
1973:
1965:
1959:
1958:
1948:
1947:
1946:
1941:
1922:
1921:
1909:
1888:
1886:
1885:
1880:
1878:
1848:
1846:
1845:
1840:
1835:
1834:
1833:
1827:
1821:
1804:
1803:
1802:
1797:
1780:
1778:
1777:
1772:
1770:
1769:
1768:
1758:
1741:
1724:
1723:
1722:
1717:
1701:
1700:
1699:
1686:
1669:
1655:
1638:
1618:
1617:
1616:
1611:
1594:
1592:
1591:
1586:
1584:
1583:
1582:
1577:
1557:
1555:
1554:
1549:
1547:
1546:
1545:
1540:
1511:
1509:
1508:
1503:
1498:
1494:
1493:
1492:
1491:
1478:
1461:
1447:
1430:
1408:
1407:
1406:
1401:
1385:
1377:
1357:two vectors is
1342:
1340:
1339:
1334:
1332:
1331:
1330:
1325:
1306:
1304:
1303:
1298:
1293:
1289:
1288:
1287:
1286:
1273:
1256:
1242:
1225:
1200:
1192:
1172:
1151:
1149:
1148:
1143:
1141:
1125:
1123:
1122:
1117:
1115:
1099:
1097:
1096:
1091:
1089:
1088:
1087:
1082:
1049:
1047:
1046:
1041:
1029:
1027:
1026:
1021:
1010:
1009:
988:
987:
969:
968:
947:
946:
919:
918:
906:
905:
884:
883:
882:
881:
872:
871:
847:
845:
844:
839:
815:
813:
812:
807:
805:
804:
803:
798:
772:random variables
769:
767:
766:
761:
759:
758:
757:
747:
746:
728:
727:
715:
714:
699:
681:
679:
678:
673:
671:
670:
654:
652:
651:
646:
644:
643:
627:
625:
624:
619:
617:
605:
603:
602:
597:
595:
575:
573:
572:
567:
555:
553:
552:
547:
535:
533:
532:
527:
525:
524:
523:
518:
501:
499:
498:
493:
491:
457:
455:
454:
449:
431:
429:
428:
423:
411:
409:
408:
403:
335:
333:
332:
327:
309:
307:
306:
301:
289:
287:
286:
281:
250:
248:
247:
242:
240:
239:
173:
166:
159:
43:
21:
20:
15897:
15896:
15892:
15891:
15890:
15888:
15887:
15886:
15862:
15861:
15860:
15855:
15832:
15823:
15772:
15696:
15642:
15578:
15412:
15330:
15276:
15215:
15016:Centrosymmetric
14939:
14933:
14903:
14898:
14861:
14832:
14794:
14731:
14717:quality control
14684:
14666:Clinical trials
14643:
14618:
14602:
14590:Hazard function
14584:
14538:
14500:
14484:
14447:
14443:BreuschâGodfrey
14431:
14408:
14348:
14323:Factor analysis
14269:
14250:Graphical model
14222:
14189:
14156:
14142:
14122:
14076:
14043:
14005:
13968:
13967:
13936:
13880:
13867:
13859:
13851:
13835:
13820:
13799:Rank statistics
13793:
13772:Model selection
13760:
13718:Goodness of fit
13712:
13689:
13663:
13635:
13588:
13533:
13522:Median unbiased
13450:
13361:
13294:Order statistic
13256:
13235:
13202:
13176:
13128:
13083:
13026:
13024:Data collection
13005:
12917:
12872:
12846:
12824:
12784:
12736:
12653:Continuous data
12643:
12630:
12612:
12607:
12572:
12511:
12508:
12506:Further reading
12503:
12502:
12490:
12486:
12479:164028 (2013),
12470:
12463:
12456:152004 (2016),
12448:
12444:
12405:
12401:
12392:Brookes, Mike.
12390:
12386:
12379:
12365:
12361:
12355:
12348:
12341:
12327:
12323:
12312:
12308:
12301:
12285:
12281:
12271:
12269:
12267:
12251:
12247:
12240:
12226:
12217:
12212:
12170:
12154:condensed phase
12146:
12121:
12119:
12116:
12115:
12087:
12079:
12071:
12060:
12057:
12056:
12040:
12038:
12035:
12034:
12017:
12013:
12011:
12008:
12007:
11988:
11980:
11969:
11966:
11965:
11940:
11939:
11934:
11933:
11922:
11917:
11914:
11913:
11887:
11886:
11882:
11878:
11873:
11870:
11869:
11864:differs. Panel
11849:
11846:
11845:
11819:
11814:
11813:
11811:
11808:
11807:
11781:
11776:
11775:
11773:
11770:
11769:
11746:
11741:
11738:
11737:
11709:
11704:
11701:
11700:
11684:
11681:
11680:
11654:
11649:
11648:
11646:
11643:
11642:
11625:
11621:
11613:
11610:
11609:
11575:
11572:
11571:
11531:
11486:
11478:
11458:
11450:
11440:
11436:
11428:
11420:
11400:
11392:
11372:
11364:
11356:
11345:
11342:
11341:
11309:
11308:
11303:
11302:
11291:
11274:
11273:
11269:
11265:
11251:
11243:
11232:
11229:
11228:
11207:
11202:
11201:
11195:
11184:
11170:
11159:
11154:
11151:
11150:
11121:
11118:
11117:
11089:
11085:
11076:
11072:
11070:
11067:
11066:
11045:
11044:
11035:
11031:
11022:
11018:
11016:
11011:
11002:
10998:
10989:
10985:
10983:
10974:
10970:
10961:
10957:
10954:
10951:
10950:
10945:
10940:
10935:
10929:
10926:
10925:
10916:
10912:
10903:
10899:
10897:
10892:
10883:
10879:
10870:
10866:
10864:
10855:
10851:
10842:
10838:
10835:
10832:
10831:
10822:
10818:
10809:
10805:
10803:
10798:
10789:
10785:
10776:
10772:
10770:
10761:
10757:
10748:
10744:
10737:
10736:
10722:
10717:
10716:
10701:
10696:
10695:
10686:
10681:
10680:
10679:
10675:
10673:
10670:
10669:
10653:
10650:
10649:
10633:
10631:
10628:
10627:
10611:
10603:
10601:
10598:
10597:
10574:
10572:
10569:
10568:
10552:
10550:
10547:
10546:
10527:
10519:
10511:
10500:
10497:
10496:
10477:
10469:
10458:
10455:
10454:
10447:
10422:
10414:diversification
10363:
10334:
10333:
10327:
10326:
10321:
10313:
10312:
10307:
10306:
10296:
10283:
10282:
10277:
10276:
10265:
10264:
10258:
10257:
10252:
10244:
10243:
10238:
10237:
10227:
10214:
10213:
10208:
10207:
10205:
10202:
10201:
10183:
10182:
10176:
10175:
10170:
10162:
10161:
10156:
10155:
10142:
10137:
10124:
10123:
10118:
10117:
10106:
10105:
10099:
10098:
10093:
10085:
10084:
10079:
10078:
10065:
10060:
10047:
10046:
10041:
10040:
10038:
10035:
10034:
10013:
10012:
10007:
10006:
10004:
10001:
10000:
9979:
9978:
9973:
9972:
9970:
9967:
9966:
9932:
9929:
9928:
9906:
9903:
9902:
9880:
9879:
9874:
9873:
9871:
9868:
9867:
9849:
9848:
9843:
9842:
9840:
9837:
9836:
9833:
9827:
9793:
9792:
9788:
9781:
9780:
9775:
9774:
9766:
9753:
9752:
9747:
9746:
9738:
9734:
9730:
9708:
9706:
9698:
9679:
9674:
9673:
9669:
9667:
9664:
9663:
9660:relation matrix
9652:
9625:
9620:
9619:
9615:
9605:
9604:
9598:
9593:
9592:
9586:
9583:
9582:
9536:
9535:
9531:
9524:
9523:
9518:
9517:
9509:
9496:
9495:
9490:
9489:
9481:
9477:
9473:
9456:
9448:
9429:
9424:
9423:
9419:
9417:
9414:
9413:
9387:
9386:
9381:
9380:
9378:
9375:
9374:
9353:
9352:
9348:
9342:
9338:
9323:
9319:
9308:
9306:
9303:
9302:
9278:
9276:
9273:
9272:
9256:
9253:
9252:
9218:
9214:
9204:
9202:
9193:
9189:
9179:
9175:
9149:
9146:
9145:
9125:
9122:
9121:
9107:
9101:
9078:
9065:
9061:
9056:
9055:
9046:
9026:
9022:
9017:
9016:
9005:
8995:
8991:
8986:
8985:
8974:
8971:
8970:
8948:
8945:
8944:
8922:
8919:
8918:
8902:
8900:
8897:
8896:
8872:
8869:
8868:
8842:
8839:
8838:
8822:
8819:
8818:
8793:
8776:
8766:
8765:
8761:
8759:
8756:
8755:
8741:
8740:
8725:
8724:
8718:
8712:
8711:
8710:
8699:
8682:
8672:
8671:
8667:
8661:
8660:
8654:
8653:
8638:
8637:
8622:
8621:
8617:
8609:
8592:
8578:
8561:
8551:
8550:
8546:
8545:
8541:
8517:
8516:
8512:
8504:
8487:
8473:
8456:
8452:
8448:
8435:
8434:
8430:
8425:
8423:
8420:
8419:
8388:
8380:
8364:
8363:
8358:
8357:
8346:
8339:
8338:
8333:
8332:
8321:
8318:
8317:
8289:
8286:
8285:
8269:
8267:
8264:
8263:
8260:
8219:
8213:
8209:
8204:
8190:
8187:
8186:
8167:
8162:
8155:
8154:
8149:
8148:
8146:
8143:
8142:
8123:
8116:
8115:
8110:
8109:
8101:
8094:
8093:
8088:
8087:
8070:
8065:
8058:
8057:
8052:
8051:
8049:
8046:
8045:
8040:. Treated as a
8022:
8014:
8007:
8006:
8001:
8000:
7978:
7977:
7972:
7971:
7969:
7966:
7965:
7954:
7948:
7927:
7925:
7922:
7921:
7901:
7896:
7891:
7889:
7886:
7885:
7866:
7849:
7847:
7844:
7843:
7799:
7795:
7788:
7787:
7783:
7770:
7758:
7754:
7750:
7734:
7727:
7722:
7721:
7716:
7711:
7701:
7694:
7690:
7670:
7659:
7656:
7655:
7639:
7637:
7634:
7633:
7614:
7603:
7600:
7599:
7572:
7569:
7568:
7552:
7550:
7547:
7546:
7543:
7522:
7520:
7517:
7516:
7495:
7494:
7490:
7488:
7485:
7484:
7457:
7456:
7452:
7450:
7447:
7446:
7424:
7416:
7398:
7394:
7389:
7381:
7364:
7356:
7336:
7328:
7308:
7300:
7292:
7264:
7263:
7259:
7257:
7254:
7253:
7237:
7235:
7232:
7231:
7215:
7213:
7210:
7209:
7193:
7191:
7188:
7187:
7175:
7146:
7144:
7141:
7140:
7122:
7121:
7116:
7115:
7113:
7110:
7109:
7088:
7087:
7083:
7071:
7062:
7061:
7055:
7052:
7051:
7032:
7030:
7027:
7026:
7005:
7004:
7000:
6998:
6995:
6994:
6968:
6964:
6963:
6959:
6957:
6954:
6953:
6929:
6920:
6919:
6902:
6901:
6897:
6895:
6892:
6891:
6865:
6864:
6860:
6848:
6839:
6838:
6821:
6820:
6816:
6803:
6802:
6798:
6784:
6780:
6779:
6775:
6773:
6770:
6769:
6743:
6742:
6737:
6736:
6728:
6727:
6723:
6711:
6702:
6701:
6684:
6683:
6679:
6669:
6668:
6663:
6662:
6652:
6647:
6642:
6641:
6636:
6635:
6633:
6630:
6629:
6598:
6594:
6593:
6589:
6575:
6571:
6570:
6565:
6564:
6555:
6554:
6543:
6535:
6533:
6530:
6529:
6513:
6511:
6508:
6507:
6491:
6489:
6486:
6485:
6459:
6451:
6442:
6441:
6430:
6422:
6420:
6417:
6416:
6396:
6394:
6391:
6390:
6374:
6372:
6369:
6368:
6349:
6347:
6344:
6343:
6327:
6325:
6322:
6321:
6296:
6295:
6291:
6289:
6286:
6285:
6264:
6263:
6259:
6257:
6254:
6253:
6232:
6224:
6205:
6204:
6195:
6194:
6177:
6176:
6172:
6170:
6167:
6166:
6147:
6125:
6124:
6120:
6118:
6115:
6114:
6095:
6073:
6072:
6068:
6066:
6063:
6062:
6044:
6043:
6033:
6032:
6028:
6026:
6016:
6015:
6011:
6008:
6007:
5997:
5996:
5992:
5990:
5980:
5979:
5975:
5968:
5967:
5959:
5948:
5947:
5941:
5936:
5935:
5932:
5931:
5925:
5920:
5919:
5912:
5911:
5903:
5901:
5898:
5897:
5881:
5879:
5876:
5875:
5859:
5857:
5854:
5853:
5837:
5835:
5832:
5831:
5812:
5810:
5807:
5806:
5805:The joint mean
5803:
5781:
5779:
5776:
5775:
5759:
5757:
5754:
5753:
5730:
5722:
5711:
5708:
5707:
5688:
5668:
5660:
5640:
5632:
5612:
5592:
5584:
5573:
5570:
5569:
5553:
5551:
5548:
5547:
5531:
5529:
5526:
5525:
5505:
5504:
5499:
5498:
5489:
5474:
5463:
5452:
5441:
5438:
5437:
5421:
5419:
5416:
5415:
5393:
5390:
5389:
5373:
5371:
5368:
5367:
5345:
5342:
5341:
5321:
5316:
5315:
5311:
5301:
5300:
5294:
5289:
5288:
5282:
5279:
5278:
5255:
5250:
5249:
5245:
5243:
5240:
5239:
5221:
5216:
5215:
5207:
5202:
5190:
5180:
5175:
5174:
5170:
5164:
5159:
5158:
5156:
5153:
5152:
5129:
5124:
5123:
5119:
5117:
5114:
5113:
5094:
5093:
5087:
5086:
5081:
5073:
5072:
5067:
5066:
5052:
5051:
5047:
5043:
5024:
5019:
5018:
5014:
5012:
5009:
5008:
4989:
4986:
4985:
4967:
4966:
4962:
4956:
4952:
4937:
4933:
4922:
4920:
4917:
4916:
4896:
4895:
4876:
4875:
4870:
4869:
4867:
4864:
4863:
4840:
4839:
4826:
4809:
4808:
4804:
4803:
4790:
4773:
4772:
4768:
4767:
4763:
4746:
4727:
4722:
4721:
4717:
4715:
4712:
4711:
4708:
4685:
4684:
4668:
4664:
4649:
4645:
4640:
4634:
4630:
4629:
4625:
4620:
4616:
4610:
4609:
4601:
4600:
4581:
4577:
4568:
4564:
4559:
4553:
4549:
4548:
4544:
4539:
4535:
4534:
4527:
4508:
4504:
4499:
4493:
4489:
4488:
4484:
4479:
4472:
4471:
4464:
4463:
4458:
4453:
4439:
4435:
4420:
4416:
4407:
4403:
4394:
4390:
4381:
4377:
4376:
4372:
4367:
4353:
4349:
4334:
4330:
4321:
4317:
4308:
4304:
4303:
4299:
4293:
4292:
4287:
4282:
4277:
4271:
4270:
4256:
4252:
4237:
4233:
4224:
4220:
4211:
4207:
4198:
4194:
4193:
4189:
4184:
4179:
4174:
4157:
4153:
4144:
4140:
4131:
4127:
4126:
4122:
4116:
4115:
4101:
4097:
4082:
4078:
4069:
4065:
4056:
4052:
4051:
4047:
4042:
4037:
4020:
4016:
4007:
4003:
3994:
3990:
3989:
3985:
3980:
3970:
3969:
3962:
3961:
3945:
3941:
3926:
3922:
3917:
3911:
3907:
3906:
3902:
3897:
3893:
3887:
3886:
3878:
3877:
3858:
3854:
3845:
3841:
3836:
3830:
3826:
3825:
3821:
3816:
3812:
3811:
3804:
3785:
3781:
3776:
3770:
3766:
3765:
3761:
3756:
3749:
3748:
3736:
3732:
3727:
3716:
3713:
3712:
3688:
3687:
3679:
3675:
3674:
3670:
3666:
3660:
3659:
3651:
3650:
3642:
3638:
3637:
3633:
3629:
3628:
3621:
3613:
3609:
3608:
3604:
3597:
3596:
3589:
3588:
3583:
3578:
3570:
3566:
3557:
3553:
3552:
3548:
3546:
3538:
3534:
3525:
3521:
3520:
3516:
3513:
3512:
3507:
3502:
3497:
3491:
3490:
3482:
3478:
3469:
3465:
3464:
3460:
3458:
3453:
3448:
3440:
3436:
3427:
3423:
3422:
3418:
3415:
3414:
3406:
3402:
3393:
3389:
3388:
3384:
3382:
3377:
3369:
3365:
3356:
3352:
3351:
3347:
3345:
3335:
3334:
3327:
3326:
3318:
3314:
3313:
3309:
3305:
3299:
3298:
3290:
3289:
3281:
3277:
3276:
3272:
3268:
3267:
3260:
3252:
3248:
3247:
3243:
3236:
3235:
3224:
3213:
3210:
3209:
3176:
3170:
3165:
3164:
3158:
3155:
3154:
3151:
3119:
3118:
3113:
3108:
3096:
3092:
3077:
3073:
3066:
3053:
3049:
3040:
3036:
3024:
3020:
3011:
3007:
2994:
2992:
2990:
2978:
2974:
2959:
2955:
2948:
2935:
2931:
2922:
2918:
2906:
2902:
2893:
2889:
2876:
2874:
2871:
2868:
2867:
2862:
2857:
2852:
2846:
2843:
2842:
2830:
2826:
2811:
2807:
2800:
2787:
2783:
2774:
2770:
2758:
2754:
2745:
2741:
2728:
2726:
2724:
2719:
2714:
2702:
2698:
2683:
2679:
2672:
2659:
2655:
2646:
2642:
2630:
2626:
2617:
2613:
2600:
2598:
2595:
2592:
2591:
2579:
2575:
2560:
2556:
2549:
2536:
2532:
2523:
2519:
2507:
2503:
2494:
2490:
2477:
2475:
2473:
2468:
2456:
2452:
2437:
2433:
2426:
2413:
2409:
2400:
2396:
2384:
2380:
2371:
2367:
2354:
2352:
2350:
2340:
2339:
2328:
2317:
2314:
2313:
2279:
2276:
2275:
2255:
2251:
2240:
2234:
2230:
2228:
2225:
2224:
2184:
2181:
2180:
2163:
2159:
2157:
2154:
2153:
2150:diagonal matrix
2131:
2126:
2125:
2121:
2119:
2116:
2115:
2094:
2089:
2088:
2084:
2073:
2070:
2069:
2043:
2039:
2033:
2032:
2031:
2021:
2016:
2015:
2011:
1996:
1995:
1987:
1982:
1981:
1977:
1964:
1960:
1954:
1953:
1952:
1942:
1937:
1936:
1932:
1917:
1916:
1905:
1894:
1891:
1890:
1874:
1872:
1869:
1868:
1861:
1855:
1829:
1828:
1823:
1822:
1817:
1798:
1793:
1792:
1788:
1786:
1783:
1782:
1764:
1763:
1759:
1754:
1737:
1718:
1713:
1712:
1708:
1695:
1694:
1690:
1682:
1665:
1651:
1634:
1612:
1607:
1606:
1602:
1600:
1597:
1596:
1578:
1573:
1572:
1568:
1566:
1563:
1562:
1541:
1536:
1535:
1531:
1529:
1526:
1525:
1522:
1517:
1487:
1486:
1482:
1474:
1457:
1443:
1426:
1422:
1418:
1402:
1397:
1396:
1392:
1381:
1373:
1362:
1359:
1358:
1326:
1321:
1320:
1316:
1314:
1311:
1310:
1282:
1281:
1277:
1269:
1252:
1238:
1221:
1217:
1213:
1196:
1188:
1168:
1157:
1154:
1153:
1137:
1135:
1132:
1131:
1111:
1109:
1106:
1105:
1083:
1078:
1077:
1073:
1071:
1068:
1067:
1056:
1035:
1032:
1031:
1005:
1001:
983:
979:
964:
960:
942:
938:
914:
910:
901:
897:
877:
873:
867:
863:
862:
858:
856:
853:
852:
821:
818:
817:
799:
794:
793:
789:
787:
784:
783:
753:
752:
748:
742:
738:
723:
719:
710:
706:
695:
693:
690:
689:
666:
662:
660:
657:
656:
639:
635:
633:
630:
629:
613:
611:
608:
607:
591:
589:
586:
585:
582:
561:
558:
557:
541:
538:
537:
519:
514:
513:
509:
507:
504:
503:
487:
485:
482:
481:
437:
434:
433:
417:
414:
413:
397:
394:
393:
370:variance matrix
360:(also known as
315:
312:
311:
295:
292:
291:
275:
272:
271:
234:
233:
228:
222:
221:
216:
206:
205:
203:
200:
199:
188:
177:
148:
147:
123:
113:
112:
88:
78:
77:
53:
19:
12:
11:
5:
15895:
15885:
15884:
15879:
15874:
15857:
15856:
15854:
15853:
15848:
15843:
15828:
15825:
15824:
15822:
15821:
15816:
15811:
15806:
15804:Perfect matrix
15801:
15796:
15791:
15786:
15780:
15778:
15774:
15773:
15771:
15770:
15765:
15760:
15755:
15750:
15745:
15740:
15735:
15730:
15725:
15720:
15715:
15710:
15704:
15702:
15698:
15697:
15695:
15694:
15689:
15684:
15679:
15674:
15669:
15664:
15659:
15653:
15651:
15644:
15643:
15641:
15640:
15635:
15630:
15625:
15620:
15615:
15610:
15605:
15600:
15595:
15589:
15587:
15580:
15579:
15577:
15576:
15574:Transformation
15571:
15566:
15561:
15556:
15551:
15546:
15541:
15536:
15531:
15526:
15521:
15516:
15511:
15506:
15501:
15496:
15491:
15486:
15481:
15476:
15471:
15466:
15461:
15456:
15451:
15446:
15441:
15436:
15431:
15426:
15420:
15418:
15414:
15413:
15411:
15410:
15405:
15400:
15395:
15390:
15385:
15380:
15375:
15370:
15365:
15360:
15351:
15345:
15343:
15332:
15331:
15329:
15328:
15323:
15318:
15313:
15311:Diagonalizable
15308:
15303:
15298:
15293:
15287:
15285:
15281:Conditions on
15278:
15277:
15275:
15274:
15269:
15264:
15259:
15254:
15249:
15244:
15239:
15234:
15229:
15223:
15221:
15217:
15216:
15214:
15213:
15208:
15203:
15198:
15193:
15188:
15183:
15178:
15173:
15168:
15163:
15161:Skew-symmetric
15158:
15156:Skew-Hermitian
15153:
15148:
15143:
15138:
15133:
15128:
15123:
15118:
15113:
15108:
15103:
15098:
15093:
15088:
15083:
15078:
15073:
15068:
15063:
15058:
15053:
15048:
15043:
15038:
15033:
15028:
15023:
15018:
15013:
15008:
15003:
14998:
14993:
14991:Block-diagonal
14988:
14983:
14978:
14973:
14968:
14966:Anti-symmetric
14963:
14961:Anti-Hermitian
14958:
14953:
14947:
14945:
14941:
14940:
14932:
14931:
14924:
14917:
14909:
14900:
14899:
14897:
14896:
14884:
14872:
14858:
14845:
14842:
14841:
14838:
14837:
14834:
14833:
14831:
14830:
14825:
14820:
14815:
14810:
14804:
14802:
14796:
14795:
14793:
14792:
14787:
14782:
14777:
14772:
14767:
14762:
14757:
14752:
14747:
14741:
14739:
14733:
14732:
14730:
14729:
14724:
14719:
14710:
14705:
14700:
14694:
14692:
14686:
14685:
14683:
14682:
14677:
14672:
14663:
14661:Bioinformatics
14657:
14655:
14645:
14644:
14632:
14631:
14628:
14627:
14624:
14623:
14620:
14619:
14617:
14616:
14610:
14608:
14604:
14603:
14601:
14600:
14594:
14592:
14586:
14585:
14583:
14582:
14577:
14572:
14567:
14561:
14559:
14550:
14544:
14543:
14540:
14539:
14537:
14536:
14531:
14526:
14521:
14516:
14510:
14508:
14502:
14501:
14499:
14498:
14493:
14488:
14480:
14475:
14470:
14469:
14468:
14466:partial (PACF)
14457:
14455:
14449:
14448:
14446:
14445:
14440:
14435:
14427:
14422:
14416:
14414:
14413:Specific tests
14410:
14409:
14407:
14406:
14401:
14396:
14391:
14386:
14381:
14376:
14371:
14365:
14363:
14356:
14350:
14349:
14347:
14346:
14345:
14344:
14343:
14342:
14327:
14326:
14325:
14315:
14313:Classification
14310:
14305:
14300:
14295:
14290:
14285:
14279:
14277:
14271:
14270:
14268:
14267:
14262:
14260:McNemar's test
14257:
14252:
14247:
14242:
14236:
14234:
14224:
14223:
14199:
14198:
14195:
14194:
14191:
14190:
14188:
14187:
14182:
14177:
14172:
14166:
14164:
14158:
14157:
14155:
14154:
14138:
14132:
14130:
14124:
14123:
14121:
14120:
14115:
14110:
14105:
14100:
14098:Semiparametric
14095:
14090:
14084:
14082:
14078:
14077:
14075:
14074:
14069:
14064:
14059:
14053:
14051:
14045:
14044:
14042:
14041:
14036:
14031:
14026:
14021:
14015:
14013:
14007:
14006:
14004:
14003:
13998:
13993:
13988:
13982:
13980:
13970:
13969:
13966:
13965:
13960:
13954:
13946:
13945:
13942:
13941:
13938:
13937:
13935:
13934:
13933:
13932:
13922:
13917:
13912:
13911:
13910:
13905:
13894:
13892:
13886:
13885:
13882:
13881:
13879:
13878:
13873:
13872:
13871:
13863:
13855:
13839:
13836:(MannâWhitney)
13831:
13830:
13829:
13816:
13815:
13814:
13803:
13801:
13795:
13794:
13792:
13791:
13790:
13789:
13784:
13779:
13769:
13764:
13761:(ShapiroâWilk)
13756:
13751:
13746:
13741:
13736:
13728:
13722:
13720:
13714:
13713:
13711:
13710:
13702:
13693:
13681:
13675:
13673:Specific tests
13669:
13668:
13665:
13664:
13662:
13661:
13656:
13651:
13645:
13643:
13637:
13636:
13634:
13633:
13628:
13627:
13626:
13616:
13615:
13614:
13604:
13598:
13596:
13590:
13589:
13587:
13586:
13585:
13584:
13579:
13569:
13564:
13559:
13554:
13549:
13543:
13541:
13535:
13534:
13532:
13531:
13526:
13525:
13524:
13519:
13518:
13517:
13512:
13497:
13496:
13495:
13490:
13485:
13480:
13469:
13467:
13458:
13452:
13451:
13449:
13448:
13443:
13438:
13437:
13436:
13426:
13421:
13420:
13419:
13409:
13408:
13407:
13402:
13397:
13387:
13382:
13377:
13376:
13375:
13370:
13365:
13349:
13348:
13347:
13342:
13337:
13327:
13326:
13325:
13320:
13310:
13309:
13308:
13298:
13297:
13296:
13286:
13281:
13276:
13270:
13268:
13258:
13257:
13245:
13244:
13241:
13240:
13237:
13236:
13234:
13233:
13228:
13223:
13218:
13212:
13210:
13204:
13203:
13201:
13200:
13195:
13190:
13184:
13182:
13178:
13177:
13175:
13174:
13169:
13164:
13159:
13154:
13149:
13144:
13138:
13136:
13130:
13129:
13127:
13126:
13124:Standard error
13121:
13116:
13111:
13110:
13109:
13104:
13093:
13091:
13085:
13084:
13082:
13081:
13076:
13071:
13066:
13061:
13056:
13054:Optimal design
13051:
13046:
13040:
13038:
13028:
13027:
13015:
13014:
13011:
13010:
13007:
13006:
13004:
13003:
12998:
12993:
12988:
12983:
12978:
12973:
12968:
12963:
12958:
12953:
12948:
12943:
12938:
12933:
12927:
12925:
12919:
12918:
12916:
12915:
12910:
12909:
12908:
12903:
12893:
12888:
12882:
12880:
12874:
12873:
12871:
12870:
12865:
12860:
12854:
12852:
12851:Summary tables
12848:
12847:
12845:
12844:
12838:
12836:
12830:
12829:
12826:
12825:
12823:
12822:
12821:
12820:
12815:
12810:
12800:
12794:
12792:
12786:
12785:
12783:
12782:
12777:
12772:
12767:
12762:
12757:
12752:
12746:
12744:
12738:
12737:
12735:
12734:
12729:
12724:
12723:
12722:
12717:
12712:
12707:
12702:
12697:
12692:
12687:
12685:Contraharmonic
12682:
12677:
12666:
12664:
12655:
12645:
12644:
12632:
12631:
12629:
12628:
12623:
12617:
12614:
12613:
12606:
12605:
12598:
12591:
12583:
12577:
12576:
12570:
12553:
12534:
12527:
12507:
12504:
12501:
12500:
12498:1329â36 (1993)
12484:
12461:
12442:
12399:
12384:
12377:
12359:
12346:
12339:
12321:
12306:
12299:
12279:
12265:
12245:
12238:
12214:
12213:
12211:
12208:
12207:
12206:
12201:
12196:
12191:
12186:
12184:Gramian matrix
12181:
12176:
12169:
12166:
12145:
12142:
12124:
12094:
12090:
12086:
12082:
12078:
12074:
12070:
12067:
12064:
12043:
12020:
12016:
11995:
11991:
11987:
11983:
11979:
11976:
11973:
11949:
11943:
11937:
11932:
11929:
11925:
11921:
11897:
11890:
11885:
11881:
11877:
11853:
11833:
11830:
11827:
11822:
11817:
11795:
11792:
11789:
11784:
11779:
11753:
11749:
11745:
11725:
11722:
11719:
11716:
11712:
11708:
11688:
11668:
11665:
11662:
11657:
11652:
11641:such spectra,
11628:
11624:
11620:
11617:
11601:time-of-flight
11588:
11585:
11582:
11579:
11529:
11501:
11497:
11493:
11489:
11485:
11481:
11477:
11474:
11471:
11468:
11465:
11461:
11457:
11453:
11449:
11446:
11443:
11439:
11435:
11431:
11427:
11423:
11419:
11416:
11413:
11410:
11407:
11403:
11399:
11395:
11391:
11388:
11385:
11382:
11379:
11375:
11371:
11367:
11363:
11359:
11355:
11352:
11349:
11321:
11318:
11312:
11306:
11301:
11298:
11294:
11290:
11287:
11284:
11277:
11272:
11268:
11264:
11261:
11258:
11254:
11250:
11246:
11242:
11239:
11236:
11210:
11205:
11198:
11193:
11190:
11187:
11183:
11177:
11174:
11169:
11166:
11162:
11158:
11134:
11131:
11128:
11125:
11097:
11092:
11088:
11084:
11079:
11075:
11054:
11049:
11043:
11038:
11034:
11030:
11025:
11021:
11017:
11015:
11012:
11010:
11005:
11001:
10997:
10992:
10988:
10984:
10982:
10977:
10973:
10969:
10964:
10960:
10956:
10955:
10952:
10949:
10946:
10944:
10941:
10939:
10936:
10934:
10931:
10930:
10927:
10924:
10919:
10915:
10911:
10906:
10902:
10898:
10896:
10893:
10891:
10886:
10882:
10878:
10873:
10869:
10865:
10863:
10858:
10854:
10850:
10845:
10841:
10837:
10836:
10833:
10830:
10825:
10821:
10817:
10812:
10808:
10804:
10802:
10799:
10797:
10792:
10788:
10784:
10779:
10775:
10771:
10769:
10764:
10760:
10756:
10751:
10747:
10743:
10742:
10740:
10735:
10731:
10725:
10720:
10715:
10712:
10709:
10704:
10699:
10694:
10689:
10684:
10678:
10668:samples, e.g.
10657:
10636:
10614:
10610:
10606:
10577:
10555:
10534:
10530:
10526:
10522:
10518:
10514:
10510:
10507:
10504:
10484:
10480:
10476:
10472:
10468:
10465:
10462:
10446:
10443:
10434:Hessian matrix
10421:
10418:
10381:(PCA) and the
10362:
10359:
10343:
10337:
10330:
10324:
10316:
10310:
10303:
10300:
10295:
10289:
10286:
10280:
10274:
10268:
10261:
10255:
10247:
10241:
10234:
10231:
10226:
10220:
10217:
10211:
10186:
10179:
10173:
10165:
10159:
10151:
10148:
10145:
10141:
10136:
10130:
10127:
10121:
10115:
10109:
10102:
10096:
10088:
10082:
10074:
10071:
10068:
10064:
10059:
10053:
10050:
10044:
10019:
10016:
10010:
9985:
9982:
9976:
9942:
9939:
9936:
9916:
9913:
9910:
9883:
9877:
9852:
9846:
9829:Main article:
9826:
9823:
9803:
9796:
9791:
9784:
9778:
9773:
9769:
9765:
9762:
9756:
9750:
9745:
9741:
9737:
9733:
9729:
9726:
9723:
9720:
9715:
9711:
9705:
9701:
9697:
9694:
9691:
9688:
9682:
9677:
9672:
9651:
9648:
9647:
9646:
9643:
9628:
9623:
9618:
9614:
9608:
9601:
9596:
9591:
9574:
9573:
9550:
9546:
9539:
9534:
9527:
9521:
9516:
9512:
9508:
9505:
9499:
9493:
9488:
9484:
9480:
9476:
9472:
9469:
9466:
9463:
9459:
9455:
9451:
9447:
9444:
9441:
9438:
9432:
9427:
9422:
9390:
9384:
9356:
9351:
9345:
9341:
9337:
9334:
9331:
9326:
9322:
9318:
9315:
9311:
9285:
9282:
9260:
9240:
9236:
9230:
9226:
9221:
9217:
9213:
9210:
9207:
9201:
9196:
9192:
9188:
9185:
9182:
9178:
9174:
9171:
9168:
9165:
9162:
9159:
9156:
9153:
9129:
9100:
9097:
9085:
9081:
9077:
9072:
9068:
9064:
9059:
9053:
9049:
9045:
9042:
9039:
9033:
9029:
9025:
9020:
9015:
9012:
9008:
9002:
8998:
8994:
8989:
8984:
8981:
8978:
8958:
8955:
8952:
8932:
8929:
8926:
8905:
8876:
8852:
8849:
8846:
8826:
8803:
8800:
8796:
8792:
8789:
8786:
8783:
8779:
8775:
8769:
8764:
8739:
8736:
8733:
8728:
8721:
8715:
8709:
8706:
8702:
8698:
8695:
8692:
8689:
8685:
8681:
8675:
8670:
8664:
8657:
8652:
8649:
8646:
8643:
8641:
8639:
8635:
8631:
8625:
8620:
8616:
8612:
8608:
8605:
8602:
8599:
8595:
8591:
8588:
8585:
8581:
8577:
8574:
8571:
8568:
8564:
8560:
8554:
8549:
8544:
8540:
8537:
8534:
8531:
8527:
8520:
8515:
8511:
8507:
8503:
8500:
8497:
8494:
8490:
8486:
8483:
8480:
8476:
8472:
8469:
8466:
8463:
8459:
8455:
8451:
8447:
8444:
8438:
8433:
8429:
8427:
8395:
8391:
8387:
8383:
8379:
8376:
8373:
8367:
8361:
8356:
8353:
8349:
8342:
8336:
8331:
8328:
8325:
8305:
8302:
8299:
8296:
8293:
8272:
8259:
8256:
8235:
8232:
8229:
8226:
8222:
8216:
8212:
8207:
8203:
8200:
8197:
8194:
8170:
8165:
8158:
8152:
8130:
8126:
8119:
8113:
8108:
8104:
8097:
8091:
8086:
8083:
8080:
8077:
8073:
8068:
8061:
8055:
8029:
8025:
8021:
8017:
8010:
8004:
7999:
7996:
7993:
7990:
7987:
7981:
7975:
7950:Main article:
7947:
7944:
7930:
7904:
7899:
7894:
7873:
7869:
7865:
7862:
7859:
7856:
7852:
7831:
7827:
7822:
7819:
7816:
7813:
7810:
7805:
7802:
7798:
7791:
7786:
7782:
7779:
7776:
7773:
7766:
7763:
7757:
7753:
7749:
7746:
7741:
7737:
7733:
7730:
7725:
7719:
7714:
7708:
7704:
7700:
7697:
7693:
7689:
7686:
7683:
7680:
7677:
7673:
7669:
7666:
7663:
7642:
7621:
7617:
7613:
7610:
7607:
7576:
7555:
7542:
7539:
7525:
7501:
7498:
7493:
7469:
7466:
7463:
7460:
7455:
7434:
7431:
7427:
7423:
7419:
7415:
7412:
7409:
7404:
7401:
7397:
7392:
7388:
7384:
7380:
7377:
7374:
7371:
7367:
7363:
7359:
7355:
7352:
7349:
7346:
7343:
7339:
7335:
7331:
7327:
7324:
7321:
7318:
7315:
7311:
7307:
7303:
7299:
7295:
7291:
7288:
7285:
7282:
7276:
7273:
7270:
7267:
7262:
7240:
7218:
7196:
7174:
7171:
7149:
7125:
7119:
7094:
7091:
7086:
7082:
7077:
7074:
7068:
7065:
7060:
7035:
7011:
7008:
7003:
6975:
6971:
6967:
6962:
6935:
6932:
6926:
6923:
6918:
6914:
6908:
6905:
6900:
6877:
6871:
6868:
6863:
6859:
6854:
6851:
6845:
6842:
6837:
6833:
6827:
6824:
6819:
6815:
6809:
6806:
6801:
6797:
6791:
6787:
6783:
6778:
6753:
6746:
6740:
6735:
6731:
6726:
6722:
6717:
6714:
6708:
6705:
6700:
6696:
6690:
6687:
6682:
6678:
6672:
6666:
6661:
6655:
6650:
6645:
6639:
6614:
6611:
6605:
6601:
6597:
6592:
6588:
6582:
6578:
6574:
6568:
6563:
6558:
6550:
6546:
6542:
6538:
6516:
6494:
6469:
6466:
6462:
6458:
6454:
6450:
6445:
6437:
6433:
6429:
6425:
6399:
6377:
6364:respectively.
6352:
6330:
6302:
6299:
6294:
6270:
6267:
6262:
6239:
6235:
6231:
6227:
6223:
6220:
6217:
6214:
6208:
6201:
6198:
6193:
6189:
6183:
6180:
6175:
6154:
6150:
6146:
6143:
6140:
6137:
6131:
6128:
6123:
6102:
6098:
6094:
6091:
6088:
6085:
6079:
6076:
6071:
6048:
6039:
6036:
6031:
6027:
6022:
6019:
6014:
6010:
6009:
6003:
6000:
5995:
5991:
5986:
5983:
5978:
5974:
5973:
5971:
5966:
5962:
5957:
5952:
5944:
5939:
5934:
5933:
5928:
5923:
5918:
5917:
5915:
5910:
5906:
5884:
5862:
5840:
5815:
5802:
5801:Block matrices
5799:
5798:
5797:
5784:
5762:
5737:
5733:
5729:
5725:
5721:
5718:
5715:
5695:
5691:
5687:
5684:
5681:
5678:
5675:
5671:
5667:
5663:
5659:
5656:
5653:
5650:
5647:
5643:
5639:
5635:
5631:
5628:
5625:
5622:
5619:
5615:
5611:
5608:
5605:
5602:
5599:
5595:
5591:
5587:
5583:
5580:
5577:
5556:
5534:
5522:
5508:
5502:
5496:
5492:
5488:
5485:
5482:
5477:
5473:
5470:
5466:
5462:
5458:
5455:
5451:
5448:
5445:
5424:
5403:
5400:
5397:
5376:
5355:
5352:
5349:
5338:
5324:
5319:
5314:
5310:
5304:
5297:
5292:
5287:
5258:
5253:
5248:
5237:
5224:
5219:
5214:
5210:
5200:
5197:
5193:
5189:
5183:
5178:
5173:
5167:
5162:
5132:
5127:
5122:
5111:
5097:
5090:
5084:
5076:
5070:
5065:
5062:
5055:
5050:
5046:
5042:
5039:
5036:
5033:
5027:
5022:
5017:
4993:
4970:
4965:
4959:
4955:
4951:
4948:
4945:
4940:
4936:
4932:
4929:
4925:
4904:
4894:
4891:
4888:
4885:
4879:
4873:
4850:
4843:
4837:
4833:
4829:
4825:
4822:
4819:
4816:
4812:
4807:
4801:
4797:
4793:
4789:
4786:
4783:
4780:
4776:
4771:
4766:
4762:
4759:
4756:
4753:
4749:
4745:
4742:
4739:
4736:
4730:
4725:
4720:
4707:
4704:
4689:
4677:
4674:
4671:
4667:
4663:
4660:
4657:
4652:
4648:
4643:
4637:
4633:
4628:
4624:
4619:
4615:
4612:
4611:
4608:
4605:
4602:
4595:
4592:
4589:
4584:
4580:
4576:
4571:
4567:
4562:
4556:
4552:
4547:
4543:
4538:
4536:
4533:
4530:
4522:
4519:
4516:
4511:
4507:
4502:
4496:
4492:
4487:
4483:
4478:
4477:
4475:
4468:
4462:
4459:
4457:
4454:
4448:
4445:
4442:
4438:
4434:
4431:
4428:
4423:
4419:
4415:
4410:
4406:
4402:
4397:
4393:
4389:
4384:
4380:
4375:
4371:
4368:
4362:
4359:
4356:
4352:
4348:
4345:
4342:
4337:
4333:
4329:
4324:
4320:
4316:
4311:
4307:
4302:
4298:
4295:
4294:
4291:
4288:
4286:
4283:
4281:
4278:
4276:
4273:
4272:
4265:
4262:
4259:
4255:
4251:
4248:
4245:
4240:
4236:
4232:
4227:
4223:
4219:
4214:
4210:
4206:
4201:
4197:
4192:
4188:
4185:
4183:
4180:
4178:
4175:
4171:
4168:
4165:
4160:
4156:
4152:
4147:
4143:
4139:
4134:
4130:
4125:
4121:
4118:
4117:
4110:
4107:
4104:
4100:
4096:
4093:
4090:
4085:
4081:
4077:
4072:
4068:
4064:
4059:
4055:
4050:
4046:
4043:
4041:
4038:
4034:
4031:
4028:
4023:
4019:
4015:
4010:
4006:
4002:
3997:
3993:
3988:
3984:
3981:
3979:
3976:
3975:
3973:
3966:
3954:
3951:
3948:
3944:
3940:
3937:
3934:
3929:
3925:
3920:
3914:
3910:
3905:
3901:
3896:
3892:
3889:
3888:
3885:
3882:
3879:
3872:
3869:
3866:
3861:
3857:
3853:
3848:
3844:
3839:
3833:
3829:
3824:
3820:
3815:
3813:
3810:
3807:
3799:
3796:
3793:
3788:
3784:
3779:
3773:
3769:
3764:
3760:
3755:
3754:
3752:
3747:
3742:
3739:
3735:
3730:
3726:
3723:
3720:
3692:
3682:
3678:
3673:
3669:
3665:
3662:
3661:
3658:
3655:
3652:
3645:
3641:
3636:
3632:
3630:
3627:
3624:
3616:
3612:
3607:
3603:
3602:
3600:
3593:
3587:
3584:
3582:
3579:
3573:
3569:
3565:
3560:
3556:
3551:
3547:
3541:
3537:
3533:
3528:
3524:
3519:
3515:
3514:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3492:
3485:
3481:
3477:
3472:
3468:
3463:
3459:
3457:
3454:
3452:
3449:
3443:
3439:
3435:
3430:
3426:
3421:
3417:
3416:
3409:
3405:
3401:
3396:
3392:
3387:
3383:
3381:
3378:
3372:
3368:
3364:
3359:
3355:
3350:
3346:
3344:
3341:
3340:
3338:
3331:
3321:
3317:
3312:
3308:
3304:
3301:
3300:
3297:
3294:
3291:
3284:
3280:
3275:
3271:
3269:
3266:
3263:
3255:
3251:
3246:
3242:
3241:
3239:
3234:
3231:
3227:
3223:
3220:
3217:
3182:
3179:
3173:
3168:
3163:
3150:
3147:
3128:
3123:
3117:
3114:
3112:
3109:
3104:
3099:
3095:
3091:
3088:
3085:
3080:
3076:
3072:
3069:
3064:
3061:
3056:
3052:
3048:
3043:
3039:
3035:
3032:
3027:
3023:
3019:
3014:
3010:
3006:
3003:
3000:
2997:
2991:
2986:
2981:
2977:
2973:
2970:
2967:
2962:
2958:
2954:
2951:
2946:
2943:
2938:
2934:
2930:
2925:
2921:
2917:
2914:
2909:
2905:
2901:
2896:
2892:
2888:
2885:
2882:
2879:
2873:
2872:
2869:
2866:
2863:
2861:
2858:
2856:
2853:
2851:
2848:
2847:
2844:
2838:
2833:
2829:
2825:
2822:
2819:
2814:
2810:
2806:
2803:
2798:
2795:
2790:
2786:
2782:
2777:
2773:
2769:
2766:
2761:
2757:
2753:
2748:
2744:
2740:
2737:
2734:
2731:
2725:
2723:
2720:
2718:
2715:
2710:
2705:
2701:
2697:
2694:
2691:
2686:
2682:
2678:
2675:
2670:
2667:
2662:
2658:
2654:
2649:
2645:
2641:
2638:
2633:
2629:
2625:
2620:
2616:
2612:
2609:
2606:
2603:
2597:
2596:
2593:
2587:
2582:
2578:
2574:
2571:
2568:
2563:
2559:
2555:
2552:
2547:
2544:
2539:
2535:
2531:
2526:
2522:
2518:
2515:
2510:
2506:
2502:
2497:
2493:
2489:
2486:
2483:
2480:
2474:
2472:
2469:
2464:
2459:
2455:
2451:
2448:
2445:
2440:
2436:
2432:
2429:
2424:
2421:
2416:
2412:
2408:
2403:
2399:
2395:
2392:
2387:
2383:
2379:
2374:
2370:
2366:
2363:
2360:
2357:
2351:
2349:
2346:
2345:
2343:
2338:
2335:
2331:
2327:
2324:
2321:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2263:
2258:
2254:
2250:
2247:
2243:
2237:
2233:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2166:
2162:
2134:
2129:
2124:
2103:
2097:
2092:
2087:
2083:
2080:
2077:
2057:
2050:
2047:
2042:
2036:
2030:
2024:
2019:
2014:
2010:
2007:
2004:
1999:
1990:
1985:
1980:
1971:
1968:
1963:
1957:
1951:
1945:
1940:
1935:
1931:
1928:
1925:
1920:
1915:
1912:
1908:
1904:
1901:
1898:
1877:
1854:
1851:
1838:
1832:
1826:
1820:
1816:
1813:
1810:
1807:
1801:
1796:
1791:
1767:
1762:
1757:
1753:
1750:
1747:
1744:
1740:
1736:
1733:
1730:
1727:
1721:
1716:
1711:
1707:
1704:
1698:
1693:
1689:
1685:
1681:
1678:
1675:
1672:
1668:
1664:
1661:
1658:
1654:
1650:
1647:
1644:
1641:
1637:
1633:
1630:
1627:
1624:
1621:
1615:
1610:
1605:
1581:
1576:
1571:
1544:
1539:
1534:
1521:
1518:
1516:
1513:
1501:
1497:
1490:
1485:
1481:
1477:
1473:
1470:
1467:
1464:
1460:
1456:
1453:
1450:
1446:
1442:
1439:
1436:
1433:
1429:
1425:
1421:
1417:
1414:
1411:
1405:
1400:
1395:
1391:
1388:
1384:
1380:
1376:
1372:
1369:
1366:
1329:
1324:
1319:
1296:
1292:
1285:
1280:
1276:
1272:
1268:
1265:
1262:
1259:
1255:
1251:
1248:
1245:
1241:
1237:
1234:
1231:
1228:
1224:
1220:
1216:
1212:
1209:
1206:
1203:
1199:
1195:
1191:
1187:
1184:
1181:
1178:
1175:
1171:
1167:
1164:
1161:
1140:
1114:
1086:
1081:
1076:
1060:William Feller
1055:
1052:
1039:
1019:
1016:
1013:
1008:
1004:
1000:
997:
994:
991:
986:
982:
978:
975:
972:
967:
963:
959:
956:
953:
950:
945:
941:
937:
934:
931:
928:
925:
922:
917:
913:
909:
904:
900:
896:
893:
890:
887:
880:
876:
870:
866:
861:
837:
834:
831:
828:
825:
802:
797:
792:
780:expected value
756:
751:
745:
741:
737:
734:
731:
726:
722:
718:
713:
709:
705:
702:
698:
669:
665:
642:
638:
616:
594:
581:
578:
565:
545:
522:
517:
512:
490:
447:
444:
441:
421:
401:
376:) is a square
325:
322:
319:
299:
279:
238:
232:
229:
227:
224:
223:
220:
217:
215:
212:
211:
209:
179:
178:
176:
175:
168:
161:
153:
150:
149:
146:
145:
140:
135:
130:
124:
119:
118:
115:
114:
111:
110:
105:
100:
95:
89:
84:
83:
80:
79:
76:
75:
70:
65:
60:
54:
49:
48:
45:
44:
36:
35:
29:
28:
17:
9:
6:
4:
3:
2:
15894:
15883:
15880:
15878:
15875:
15873:
15870:
15869:
15867:
15852:
15849:
15847:
15844:
15842:
15841:
15836:
15830:
15829:
15826:
15820:
15817:
15815:
15812:
15810:
15809:Pseudoinverse
15807:
15805:
15802:
15800:
15797:
15795:
15792:
15790:
15787:
15785:
15782:
15781:
15779:
15777:Related terms
15775:
15769:
15768:Z (chemistry)
15766:
15764:
15761:
15759:
15756:
15754:
15751:
15749:
15746:
15744:
15741:
15739:
15736:
15734:
15731:
15729:
15726:
15724:
15721:
15719:
15716:
15714:
15711:
15709:
15706:
15705:
15703:
15699:
15693:
15690:
15688:
15685:
15683:
15680:
15678:
15675:
15673:
15670:
15668:
15665:
15663:
15660:
15658:
15655:
15654:
15652:
15650:
15645:
15639:
15636:
15634:
15631:
15629:
15626:
15624:
15621:
15619:
15616:
15614:
15611:
15609:
15606:
15604:
15601:
15599:
15596:
15594:
15591:
15590:
15588:
15586:
15581:
15575:
15572:
15570:
15567:
15565:
15562:
15560:
15557:
15555:
15552:
15550:
15547:
15545:
15542:
15540:
15537:
15535:
15532:
15530:
15527:
15525:
15522:
15520:
15517:
15515:
15512:
15510:
15507:
15505:
15502:
15500:
15497:
15495:
15492:
15490:
15487:
15485:
15482:
15480:
15477:
15475:
15472:
15470:
15467:
15465:
15462:
15460:
15457:
15455:
15452:
15450:
15447:
15445:
15442:
15440:
15437:
15435:
15432:
15430:
15427:
15425:
15422:
15421:
15419:
15415:
15409:
15406:
15404:
15401:
15399:
15396:
15394:
15391:
15389:
15386:
15384:
15381:
15379:
15376:
15374:
15371:
15369:
15366:
15364:
15361:
15359:
15355:
15352:
15350:
15347:
15346:
15344:
15342:
15338:
15333:
15327:
15324:
15322:
15319:
15317:
15314:
15312:
15309:
15307:
15304:
15302:
15299:
15297:
15294:
15292:
15289:
15288:
15286:
15284:
15279:
15273:
15270:
15268:
15265:
15263:
15260:
15258:
15255:
15253:
15250:
15248:
15245:
15243:
15240:
15238:
15235:
15233:
15230:
15228:
15225:
15224:
15222:
15218:
15212:
15209:
15207:
15204:
15202:
15199:
15197:
15194:
15192:
15189:
15187:
15184:
15182:
15179:
15177:
15174:
15172:
15169:
15167:
15164:
15162:
15159:
15157:
15154:
15152:
15149:
15147:
15144:
15142:
15139:
15137:
15134:
15132:
15129:
15127:
15126:Pentadiagonal
15124:
15122:
15119:
15117:
15114:
15112:
15109:
15107:
15104:
15102:
15099:
15097:
15094:
15092:
15089:
15087:
15084:
15082:
15079:
15077:
15074:
15072:
15069:
15067:
15064:
15062:
15059:
15057:
15054:
15052:
15049:
15047:
15044:
15042:
15039:
15037:
15034:
15032:
15029:
15027:
15024:
15022:
15019:
15017:
15014:
15012:
15009:
15007:
15004:
15002:
14999:
14997:
14994:
14992:
14989:
14987:
14984:
14982:
14979:
14977:
14974:
14972:
14969:
14967:
14964:
14962:
14959:
14957:
14956:Anti-diagonal
14954:
14952:
14949:
14948:
14946:
14942:
14937:
14930:
14925:
14923:
14918:
14916:
14911:
14910:
14907:
14895:
14894:
14885:
14883:
14882:
14873:
14871:
14870:
14865:
14859:
14857:
14856:
14847:
14846:
14843:
14829:
14826:
14824:
14823:Geostatistics
14821:
14819:
14816:
14814:
14811:
14809:
14806:
14805:
14803:
14801:
14797:
14791:
14790:Psychometrics
14788:
14786:
14783:
14781:
14778:
14776:
14773:
14771:
14768:
14766:
14763:
14761:
14758:
14756:
14753:
14751:
14748:
14746:
14743:
14742:
14740:
14738:
14734:
14728:
14725:
14723:
14720:
14718:
14714:
14711:
14709:
14706:
14704:
14701:
14699:
14696:
14695:
14693:
14691:
14687:
14681:
14678:
14676:
14673:
14671:
14667:
14664:
14662:
14659:
14658:
14656:
14654:
14653:Biostatistics
14650:
14646:
14642:
14637:
14633:
14615:
14614:Log-rank test
14612:
14611:
14609:
14605:
14599:
14596:
14595:
14593:
14591:
14587:
14581:
14578:
14576:
14573:
14571:
14568:
14566:
14563:
14562:
14560:
14558:
14554:
14551:
14549:
14545:
14535:
14532:
14530:
14527:
14525:
14522:
14520:
14517:
14515:
14512:
14511:
14509:
14507:
14503:
14497:
14494:
14492:
14489:
14487:
14485:(BoxâJenkins)
14481:
14479:
14476:
14474:
14471:
14467:
14464:
14463:
14462:
14459:
14458:
14456:
14454:
14450:
14444:
14441:
14439:
14438:DurbinâWatson
14436:
14434:
14428:
14426:
14423:
14421:
14420:DickeyâFuller
14418:
14417:
14415:
14411:
14405:
14402:
14400:
14397:
14395:
14394:Cointegration
14392:
14390:
14387:
14385:
14382:
14380:
14377:
14375:
14372:
14370:
14369:Decomposition
14367:
14366:
14364:
14360:
14357:
14355:
14351:
14341:
14338:
14337:
14336:
14333:
14332:
14331:
14328:
14324:
14321:
14320:
14319:
14316:
14314:
14311:
14309:
14306:
14304:
14301:
14299:
14296:
14294:
14291:
14289:
14286:
14284:
14281:
14280:
14278:
14276:
14272:
14266:
14263:
14261:
14258:
14256:
14253:
14251:
14248:
14246:
14243:
14241:
14240:Cohen's kappa
14238:
14237:
14235:
14233:
14229:
14225:
14221:
14217:
14213:
14209:
14204:
14200:
14186:
14183:
14181:
14178:
14176:
14173:
14171:
14168:
14167:
14165:
14163:
14159:
14153:
14149:
14145:
14139:
14137:
14134:
14133:
14131:
14129:
14125:
14119:
14116:
14114:
14111:
14109:
14106:
14104:
14101:
14099:
14096:
14094:
14093:Nonparametric
14091:
14089:
14086:
14085:
14083:
14079:
14073:
14070:
14068:
14065:
14063:
14060:
14058:
14055:
14054:
14052:
14050:
14046:
14040:
14037:
14035:
14032:
14030:
14027:
14025:
14022:
14020:
14017:
14016:
14014:
14012:
14008:
14002:
13999:
13997:
13994:
13992:
13989:
13987:
13984:
13983:
13981:
13979:
13975:
13971:
13964:
13961:
13959:
13956:
13955:
13951:
13947:
13931:
13928:
13927:
13926:
13923:
13921:
13918:
13916:
13913:
13909:
13906:
13904:
13901:
13900:
13899:
13896:
13895:
13893:
13891:
13887:
13877:
13874:
13870:
13864:
13862:
13856:
13854:
13848:
13847:
13846:
13843:
13842:Nonparametric
13840:
13838:
13832:
13828:
13825:
13824:
13823:
13817:
13813:
13812:Sample median
13810:
13809:
13808:
13805:
13804:
13802:
13800:
13796:
13788:
13785:
13783:
13780:
13778:
13775:
13774:
13773:
13770:
13768:
13765:
13763:
13757:
13755:
13752:
13750:
13747:
13745:
13742:
13740:
13737:
13735:
13733:
13729:
13727:
13724:
13723:
13721:
13719:
13715:
13709:
13707:
13703:
13701:
13699:
13694:
13692:
13687:
13683:
13682:
13679:
13676:
13674:
13670:
13660:
13657:
13655:
13652:
13650:
13647:
13646:
13644:
13642:
13638:
13632:
13629:
13625:
13622:
13621:
13620:
13617:
13613:
13610:
13609:
13608:
13605:
13603:
13600:
13599:
13597:
13595:
13591:
13583:
13580:
13578:
13575:
13574:
13573:
13570:
13568:
13565:
13563:
13560:
13558:
13555:
13553:
13550:
13548:
13545:
13544:
13542:
13540:
13536:
13530:
13527:
13523:
13520:
13516:
13513:
13511:
13508:
13507:
13506:
13503:
13502:
13501:
13498:
13494:
13491:
13489:
13486:
13484:
13481:
13479:
13476:
13475:
13474:
13471:
13470:
13468:
13466:
13462:
13459:
13457:
13453:
13447:
13444:
13442:
13439:
13435:
13432:
13431:
13430:
13427:
13425:
13422:
13418:
13417:loss function
13415:
13414:
13413:
13410:
13406:
13403:
13401:
13398:
13396:
13393:
13392:
13391:
13388:
13386:
13383:
13381:
13378:
13374:
13371:
13369:
13366:
13364:
13358:
13355:
13354:
13353:
13350:
13346:
13343:
13341:
13338:
13336:
13333:
13332:
13331:
13328:
13324:
13321:
13319:
13316:
13315:
13314:
13311:
13307:
13304:
13303:
13302:
13299:
13295:
13292:
13291:
13290:
13287:
13285:
13282:
13280:
13277:
13275:
13272:
13271:
13269:
13267:
13263:
13259:
13255:
13250:
13246:
13232:
13229:
13227:
13224:
13222:
13219:
13217:
13214:
13213:
13211:
13209:
13205:
13199:
13196:
13194:
13191:
13189:
13186:
13185:
13183:
13179:
13173:
13170:
13168:
13165:
13163:
13160:
13158:
13155:
13153:
13150:
13148:
13145:
13143:
13140:
13139:
13137:
13135:
13131:
13125:
13122:
13120:
13119:Questionnaire
13117:
13115:
13112:
13108:
13105:
13103:
13100:
13099:
13098:
13095:
13094:
13092:
13090:
13086:
13080:
13077:
13075:
13072:
13070:
13067:
13065:
13062:
13060:
13057:
13055:
13052:
13050:
13047:
13045:
13042:
13041:
13039:
13037:
13033:
13029:
13025:
13020:
13016:
13002:
12999:
12997:
12994:
12992:
12989:
12987:
12984:
12982:
12979:
12977:
12974:
12972:
12969:
12967:
12964:
12962:
12959:
12957:
12954:
12952:
12949:
12947:
12946:Control chart
12944:
12942:
12939:
12937:
12934:
12932:
12929:
12928:
12926:
12924:
12920:
12914:
12911:
12907:
12904:
12902:
12899:
12898:
12897:
12894:
12892:
12889:
12887:
12884:
12883:
12881:
12879:
12875:
12869:
12866:
12864:
12861:
12859:
12856:
12855:
12853:
12849:
12843:
12840:
12839:
12837:
12835:
12831:
12819:
12816:
12814:
12811:
12809:
12806:
12805:
12804:
12801:
12799:
12796:
12795:
12793:
12791:
12787:
12781:
12778:
12776:
12773:
12771:
12768:
12766:
12763:
12761:
12758:
12756:
12753:
12751:
12748:
12747:
12745:
12743:
12739:
12733:
12730:
12728:
12725:
12721:
12718:
12716:
12713:
12711:
12708:
12706:
12703:
12701:
12698:
12696:
12693:
12691:
12688:
12686:
12683:
12681:
12678:
12676:
12673:
12672:
12671:
12668:
12667:
12665:
12663:
12659:
12656:
12654:
12650:
12646:
12642:
12637:
12633:
12627:
12624:
12622:
12619:
12618:
12615:
12611:
12604:
12599:
12597:
12592:
12590:
12585:
12584:
12581:
12573:
12571:0-444-86200-5
12567:
12562:
12561:
12554:
12549:
12548:
12543:
12540:
12535:
12532:
12528:
12524:
12520:
12519:
12514:
12510:
12509:
12497:
12494:
12488:
12482:
12478:
12475:
12468:
12466:
12459:
12455:
12452:
12446:
12437:
12432:
12427:
12422:
12418:
12414:
12410:
12403:
12395:
12388:
12380:
12374:
12370:
12363:
12353:
12351:
12342:
12340:0-471-02776-6
12336:
12332:
12325:
12317:
12310:
12302:
12300:0-387-40272-1
12296:
12292:
12291:
12283:
12268:
12262:
12258:
12257:
12249:
12241:
12235:
12231:
12224:
12222:
12220:
12215:
12205:
12202:
12200:
12197:
12195:
12192:
12190:
12187:
12185:
12182:
12180:
12177:
12175:
12172:
12171:
12165:
12163:
12159:
12155:
12151:
12141:
12139:
12112:
12108:
12084:
12076:
12065:
12062:
12018:
12014:
11985:
11974:
11971:
11963:
11911:
11867:
11851:
11828:
11820:
11790:
11782:
11766:
11717:
11686:
11663:
11655:
11626:
11622:
11618:
11615:
11606:
11602:
11583:
11577:
11569:
11566:
11557:
11553:
11549:
11545:
11541:
11537:
11533:
11525:
11521:
11519:
11515:
11499:
11495:
11483:
11472:
11469:
11455:
11444:
11441:
11437:
11425:
11414:
11411:
11408:
11397:
11386:
11383:
11380:
11369:
11361:
11350:
11347:
11339:
11335:
11319:
11285:
11259:
11248:
11237:
11234:
11226:
11208:
11196:
11191:
11188:
11185:
11181:
11175:
11172:
11167:
11148:
11129:
11123:
11115:
11111:
11090:
11086:
11077:
11073:
11052:
11047:
11036:
11032:
11023:
11019:
11013:
11003:
10999:
10990:
10986:
10975:
10971:
10962:
10958:
10947:
10942:
10937:
10932:
10917:
10913:
10904:
10900:
10894:
10884:
10880:
10871:
10867:
10856:
10852:
10843:
10839:
10823:
10819:
10810:
10806:
10800:
10790:
10786:
10777:
10773:
10762:
10758:
10749:
10745:
10738:
10733:
10729:
10723:
10713:
10710:
10707:
10702:
10692:
10687:
10676:
10655:
10608:
10594:
10592:
10589:are discrete
10524:
10516:
10505:
10502:
10474:
10463:
10460:
10452:
10442:
10439:
10435:
10431:
10427:
10417:
10415:
10411:
10407:
10403:
10399:
10395:
10391:
10386:
10384:
10380:
10376:
10372:
10368:
10358:
10354:
10341:
10301:
10298:
10293:
10272:
10232:
10229:
10224:
10149:
10146:
10143:
10139:
10134:
10113:
10072:
10069:
10066:
10062:
10057:
9964:
9960:
9956:
9940:
9937:
9934:
9914:
9911:
9908:
9901:of dimension
9900:
9899:data matrices
9897:are centered
9832:
9822:
9820:
9815:
9801:
9771:
9743:
9731:
9727:
9721:
9703:
9692:
9689:
9686:
9661:
9658:(also called
9657:
9644:
9612:
9580:
9576:
9575:
9571:
9570:
9569:
9567:
9564:
9548:
9544:
9514:
9486:
9474:
9470:
9464:
9453:
9442:
9439:
9436:
9411:
9407:
9404:is formed by
9373:
9343:
9339:
9335:
9332:
9329:
9324:
9320:
9313:
9299:
9280:
9258:
9238:
9234:
9219:
9215:
9211:
9208:
9194:
9190:
9186:
9183:
9176:
9172:
9166:
9160:
9154:
9151:
9143:
9127:
9119:
9118:scalar-valued
9116:
9112:
9106:
9096:
9083:
9075:
9070:
9066:
9062:
9040:
9037:
9031:
9027:
9023:
9013:
9000:
8996:
8992:
8979:
8976:
8956:
8953:
8950:
8930:
8927:
8924:
8894:
8890:
8874:
8866:
8850:
8847:
8844:
8824:
8815:
8814:is a scalar.
8787:
8781:
8762:
8737:
8734:
8731:
8719:
8693:
8687:
8668:
8650:
8644:
8642:
8633:
8629:
8603:
8597:
8572:
8566:
8547:
8542:
8538:
8532:
8529:
8525:
8498:
8492:
8467:
8461:
8449:
8445:
8431:
8416:
8414:
8410:
8393:
8374:
8371:
8354:
8326:
8323:
8300:
8297:
8294:
8255:
8253:
8249:
8230:
8227:
8224:
8214:
8201:
8198:
8195:
8183:
8106:
8081:
8078:
8075:
8043:
8042:bilinear form
8019:
7994:
7991:
7988:
7963:
7959:
7953:
7943:
7919:
7860:
7854:
7829:
7825:
7817:
7780:
7764:
7761:
7755:
7751:
7747:
7744:
7739:
7735:
7731:
7728:
7706:
7702:
7698:
7695:
7687:
7684:
7678:
7664:
7608:
7598:
7594:
7590:
7574:
7538:
7464:
7432:
7421:
7410:
7407:
7402:
7399:
7386:
7375:
7372:
7361:
7350:
7347:
7344:
7333:
7322:
7319:
7316:
7305:
7297:
7286:
7283:
7280:
7271:
7184:
7181:
7170:
7168:
7164:
7080:
7075:
7072:
7048:
6992:
6951:
6933:
6930:
6912:
6888:
6875:
6857:
6852:
6849:
6831:
6813:
6795:
6768:
6751:
6733:
6724:
6720:
6715:
6712:
6694:
6676:
6659:
6628:
6612:
6586:
6548:
6540:
6483:
6467:
6456:
6435:
6427:
6414:
6365:
6319:
6251:
6229:
6218:
6215:
6212:
6187:
6141:
6138:
6135:
6089:
6086:
6083:
6046:
5969:
5964:
5955:
5950:
5942:
5926:
5913:
5908:
5830:
5751:
5727:
5716:
5713:
5682:
5679:
5676:
5665:
5654:
5651:
5648:
5637:
5626:
5623:
5620:
5606:
5603:
5600:
5589:
5578:
5575:
5523:
5483:
5480:
5471:
5460:
5446:
5443:
5401:
5398:
5395:
5388:and constant
5353:
5350:
5347:
5339:
5308:
5276:
5238:
5222:
5212:
5204:for all
5198:
5195:
5187:
5165:
5150:
5112:
5063:
5037:
5031:
5007:
5006:
5005:
4991:
4957:
4953:
4949:
4946:
4943:
4938:
4934:
4927:
4889:
4883:
4848:
4835:
4820:
4814:
4805:
4799:
4784:
4778:
4769:
4764:
4760:
4754:
4740:
4737:
4734:
4703:
4687:
4675:
4672:
4669:
4665:
4661:
4658:
4655:
4650:
4646:
4635:
4631:
4626:
4622:
4613:
4606:
4593:
4590:
4587:
4582:
4578:
4574:
4569:
4565:
4554:
4550:
4545:
4541:
4531:
4520:
4517:
4514:
4509:
4505:
4494:
4490:
4485:
4481:
4473:
4466:
4460:
4455:
4446:
4443:
4440:
4436:
4432:
4429:
4426:
4421:
4417:
4413:
4408:
4404:
4400:
4395:
4391:
4387:
4382:
4378:
4373:
4369:
4360:
4357:
4354:
4350:
4346:
4343:
4340:
4335:
4331:
4327:
4322:
4318:
4314:
4309:
4305:
4300:
4296:
4289:
4284:
4279:
4274:
4263:
4260:
4257:
4253:
4249:
4246:
4243:
4238:
4234:
4230:
4225:
4221:
4217:
4212:
4208:
4204:
4199:
4195:
4190:
4186:
4181:
4176:
4169:
4166:
4163:
4158:
4154:
4150:
4145:
4141:
4137:
4132:
4128:
4123:
4119:
4108:
4105:
4102:
4098:
4094:
4091:
4088:
4083:
4079:
4075:
4070:
4066:
4062:
4057:
4053:
4048:
4044:
4039:
4032:
4029:
4026:
4021:
4017:
4013:
4008:
4004:
4000:
3995:
3991:
3986:
3982:
3977:
3971:
3964:
3952:
3949:
3946:
3942:
3938:
3935:
3932:
3927:
3923:
3912:
3908:
3903:
3899:
3890:
3883:
3870:
3867:
3864:
3859:
3855:
3851:
3846:
3842:
3831:
3827:
3822:
3818:
3808:
3797:
3794:
3791:
3786:
3782:
3771:
3767:
3762:
3758:
3750:
3745:
3740:
3737:
3721:
3718:
3710:
3705:
3690:
3680:
3676:
3671:
3663:
3656:
3643:
3639:
3634:
3625:
3614:
3610:
3605:
3598:
3591:
3585:
3580:
3571:
3567:
3563:
3558:
3554:
3549:
3539:
3535:
3531:
3526:
3522:
3517:
3509:
3504:
3499:
3494:
3483:
3479:
3475:
3470:
3466:
3461:
3455:
3450:
3441:
3437:
3433:
3428:
3424:
3419:
3407:
3403:
3399:
3394:
3390:
3385:
3379:
3370:
3366:
3362:
3357:
3353:
3348:
3342:
3336:
3329:
3319:
3315:
3310:
3302:
3295:
3282:
3278:
3273:
3264:
3253:
3249:
3244:
3237:
3232:
3218:
3215:
3206:
3204:
3200:
3199:
3180:
3177:
3146:
3144:
3139:
3126:
3121:
3115:
3110:
3097:
3093:
3086:
3078:
3074:
3067:
3054:
3050:
3046:
3041:
3037:
3025:
3021:
3017:
3012:
3008:
2998:
2979:
2975:
2968:
2960:
2956:
2949:
2936:
2932:
2928:
2923:
2919:
2907:
2903:
2899:
2894:
2890:
2880:
2864:
2859:
2854:
2849:
2831:
2827:
2820:
2812:
2808:
2801:
2788:
2784:
2780:
2775:
2771:
2759:
2755:
2751:
2746:
2742:
2732:
2721:
2716:
2703:
2699:
2692:
2684:
2680:
2673:
2660:
2656:
2652:
2647:
2643:
2631:
2627:
2623:
2618:
2614:
2604:
2580:
2576:
2569:
2561:
2557:
2550:
2537:
2533:
2529:
2524:
2520:
2508:
2504:
2500:
2495:
2491:
2481:
2470:
2457:
2453:
2446:
2438:
2434:
2427:
2414:
2410:
2406:
2401:
2397:
2385:
2381:
2377:
2372:
2368:
2358:
2347:
2341:
2336:
2322:
2319:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2256:
2252:
2245:
2241:
2235:
2231:
2223:
2218:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2164:
2160:
2151:
2078:
2075:
2055:
2048:
2045:
2040:
2005:
2002:
1969:
1966:
1961:
1926:
1923:
1913:
1899:
1896:
1866:
1860:
1850:
1811:
1805:
1748:
1731:
1725:
1705:
1676:
1670:
1645:
1639:
1625:
1619:
1561:
1512:
1499:
1495:
1468:
1462:
1437:
1431:
1419:
1415:
1409:
1389:
1378:
1367:
1364:
1356:
1353:
1348:
1346:
1307:
1294:
1290:
1263:
1257:
1232:
1226:
1214:
1210:
1204:
1193:
1182:
1179:
1176:
1162:
1159:
1129:
1103:
1065:
1061:
1051:
1006:
1002:
995:
989:
984:
980:
965:
961:
954:
948:
943:
939:
929:
923:
915:
911:
907:
902:
898:
891:
888:
885:
878:
874:
868:
864:
851:
848:entry is the
832:
829:
826:
781:
777:
773:
743:
739:
735:
732:
729:
724:
720:
716:
711:
707:
700:
688:
687:column vector
683:
667:
663:
640:
636:
577:
563:
478:
476:
472:
468:
464:
459:
445:
442:
439:
419:
399:
389:
387:
386:random vector
383:
379:
375:
371:
367:
363:
359:
355:
351:
343:
339:
323:
320:
317:
297:
277:
269:
265:
261:
256:
236:
230:
225:
218:
213:
207:
197:
192:
186:
174:
169:
167:
162:
160:
155:
154:
152:
151:
144:
141:
139:
136:
134:
131:
129:
126:
125:
122:
117:
116:
109:
106:
104:
101:
99:
96:
94:
91:
90:
87:
82:
81:
74:
71:
69:
66:
64:
61:
59:
56:
55:
52:
47:
46:
42:
38:
37:
34:
31:
30:
27:
23:
22:
16:
15831:
15763:Substitution
15649:graph theory
15602:
15146:Quaternionic
15136:Persymmetric
14891:
14879:
14860:
14853:
14765:Econometrics
14715: /
14698:Chemometrics
14675:Epidemiology
14668: /
14641:Applications
14483:ARIMA model
14430:Q-statistic
14379:Stationarity
14275:Multivariate
14218: /
14214: /
14212:Multivariate
14210: /
14150: /
14146: /
13920:Bayes factor
13819:Signed rank
13731:
13705:
13697:
13685:
13380:Completeness
13216:Cohort study
13114:Opinion poll
13049:Missing data
13036:Study design
12991:Scatter plot
12913:Scatter plot
12906:Spearman's Ï
12868:Grouped data
12559:
12545:
12516:
12495:
12492:
12487:
12476:
12473:
12453:
12450:
12445:
12416:
12412:
12402:
12387:
12368:
12362:
12330:
12324:
12309:
12293:. Springer.
12289:
12282:
12270:. Retrieved
12255:
12248:
12232:. Springer.
12229:
12162:asynchronous
12147:
12137:
12110:
12106:
11961:
11909:
11865:
11767:
11562:
11555:
11551:
11547:
11543:
11539:
11535:
11527:
11113:
11109:
10595:
10450:
10448:
10423:
10387:
10364:
10361:Applications
10355:
9962:
9958:
9954:
9834:
9816:
9659:
9655:
9653:
9409:
9405:
9300:
9117:
9108:
8892:
8816:
8417:
8261:
8251:
8184:
7961:
7957:
7955:
7544:
7185:
7176:
7049:
6889:
6528:is given by
6366:
6252:
5804:
4709:
3706:
3207:
3202:
3196:
3152:
3140:
2219:
1862:
1523:
1354:
1349:
1344:
1308:
1127:
1101:
1063:
1057:
684:
583:
479:
460:
390:
373:
369:
365:
361:
357:
347:
338:eigenvectors
267:
263:
120:
85:
67:
50:
15:
15738:Hamiltonian
15662:Biadjacency
15598:Correlation
15514:Householder
15464:Commutation
15201:Vandermonde
15196:Tridiagonal
15131:Permutation
15121:Nonnegative
15106:Matrix unit
14986:Bisymmetric
14893:WikiProject
14808:Cartography
14770:Jurimetrics
14722:Reliability
14453:Time domain
14432:(LjungâBox)
14354:Time-series
14232:Categorical
14216:Time-series
14208:Categorical
14143:(Bernoulli)
13978:Correlation
13958:Correlation
13754:JarqueâBera
13726:Chi-squared
13488:M-estimator
13441:Asymptotics
13385:Sufficiency
13152:Interaction
13064:Replication
13044:Effect size
13001:Violin plot
12981:Radar chart
12961:Forest plot
12951:Correlogram
12901:Kendall's Ï
12481:open access
12458:open access
12158:synchronous
11147:sample mean
10400:and in the
9271:is denoted
8889:square root
7918:determinant
7654:as follows
7595:, then its
7180:common-mode
6890:The matrix
6625:defined by
380:giving the
342:eigenvalues
15866:Categories
15638:Transition
15633:Stochastic
15603:Covariance
15585:statistics
15564:Symplectic
15559:Similarity
15388:Unimodular
15383:Orthogonal
15368:Involutory
15363:Invertible
15358:Projection
15354:Idempotent
15296:Convergent
15191:Triangular
15141:Polynomial
15086:Hessenberg
15056:Equivalent
15051:Elementary
15031:Copositive
15021:Conference
14981:Bidiagonal
14760:Demography
14478:ARMA model
14283:Regression
13860:(Friedman)
13821:(Wilcoxon)
13759:Normality
13749:Lilliefors
13696:Student's
13572:Resampling
13446:Robustness
13434:divergence
13424:Efficiency
13362:(monotone)
13357:Likelihood
13274:Population
13107:Stratified
13059:Population
12878:Dependence
12834:Count data
12765:Percentile
12742:Dispersion
12675:Arithmetic
12610:Statistics
12426:1806.03674
12210:References
11960:and panel
9825:Estimation
9572:Properties
6950:regression
5436:, one has
1515:Properties
850:covariance
580:Definition
465:matrix is
463:covariance
382:covariance
354:statistics
26:Statistics
15819:Wronskian
15743:Irregular
15733:Gell-Mann
15682:Laplacian
15677:Incidence
15657:Adjacency
15628:Precision
15593:Centering
15499:Generator
15469:Confusion
15454:Circulant
15434:Augmented
15393:Unipotent
15373:Nilpotent
15349:Congruent
15326:Stieltjes
15301:Defective
15291:Companion
15262:Redheffer
15181:Symmetric
15176:Sylvester
15151:Signature
15081:Hermitian
15061:Frobenius
14971:Arrowhead
14951:Alternant
14141:Logistic
13908:posterior
13834:Rank sum
13582:Jackknife
13577:Bootstrap
13395:Bootstrap
13330:Parameter
13279:Statistic
13074:Statistic
12986:Run chart
12971:Pie chart
12966:Histogram
12956:Fan chart
12931:Bar chart
12813:L-moments
12700:Geometric
12547:MathWorld
12523:EMS Press
12272:10 August
12259:. Wiley.
12085:∣
12066:
11975:
11948:⟩
11931:⟨
11928:⟩
11920:⟨
11896:⟩
11876:⟨
11752:⟩
11744:⟨
11724:⟩
11707:⟨
11559:ringing).
11473:
11467:∖
11445:
11415:
11409:−
11387:
11370:∣
11351:
11317:⟩
11300:⟨
11297:⟩
11289:⟨
11286:−
11283:⟩
11263:⟨
11260:≈
11238:
11182:∑
11165:⟩
11157:⟨
11014:⋯
10948:⋮
10943:⋱
10938:⋮
10933:⋮
10895:⋯
10801:⋯
10711:…
10525:∣
10506:
10464:
10147:−
10070:−
9938:×
9912:×
9777:μ
9772:−
9749:μ
9744:−
9728:
9714:¯
9693:
9563:Hermitian
9520:μ
9515:−
9492:μ
9487:−
9471:
9443:
9333:…
9284:¯
9229:¯
9216:μ
9212:−
9191:μ
9187:−
9173:
9155:
9128:μ
9041:
8980:
8954:×
8928:×
8848:×
8788:
8782:−
8732:≥
8694:
8688:−
8651:
8604:
8598:−
8573:
8567:−
8539:
8499:
8493:−
8468:
8462:−
8446:
8375:
8327:
8298:×
8234:⟩
8231:μ
8228:−
8211:Σ
8202:μ
8199:−
8193:⟨
8164:Σ
8082:
8067:Σ
7995:
7986:Σ
7929:Σ
7898:Σ
7861:
7851:μ
7818:μ
7815:−
7801:−
7797:Σ
7781:μ
7778:−
7756:−
7748:
7729:−
7718:Σ
7696:−
7688:π
7665:
7641:Σ
7609:
7465:∣
7411:
7400:−
7376:
7351:
7345:−
7323:
7306:∣
7287:
7272:∣
7081:
7073:−
7034:Σ
6931:−
6913:
6858:
6850:−
6832:
6814:−
6739:μ
6734:−
6721:
6713:−
6695:
6665:μ
6638:μ
6567:μ
6549:∼
6541:∣
6480:then the
6461:Σ
6453:μ
6436:∼
6219:
6142:
6090:
5961:Σ
5938:μ
5922:μ
5905:μ
5839:Σ
5814:μ
5717:
5683:
5655:
5627:
5607:
5579:
5484:
5447:
5399:×
5351:×
5275:symmetric
5213:∈
5196:≥
5188:
5083:μ
5069:μ
5064:−
5038:
4947:…
4890:
4872:μ
4821:
4815:−
4785:
4779:−
4761:
4741:
4673:−
4627:σ
4607:⋱
4546:σ
4486:σ
4456:⋯
4444:−
4401:∣
4374:ρ
4370:−
4358:−
4328:∣
4301:ρ
4297:−
4290:⋮
4285:⋱
4280:⋮
4275:⋮
4261:−
4218:∣
4191:ρ
4187:−
4182:⋯
4151:∣
4124:ρ
4120:−
4106:−
4076:∣
4049:ρ
4045:−
4040:⋯
4014:∣
3987:ρ
3983:−
3950:−
3904:σ
3884:⋱
3823:σ
3763:σ
3738:−
3722:
3672:σ
3657:⋱
3635:σ
3606:σ
3581:⋯
3550:ρ
3518:ρ
3510:⋮
3505:⋱
3500:⋮
3495:⋮
3462:ρ
3456:⋯
3420:ρ
3386:ρ
3380:⋯
3349:ρ
3311:σ
3296:⋱
3274:σ
3245:σ
3219:
3178:−
3111:⋯
3087:σ
3068:σ
3051:μ
3047:−
3022:μ
3018:−
2999:
2969:σ
2950:σ
2933:μ
2929:−
2904:μ
2900:−
2881:
2865:⋮
2860:⋱
2855:⋮
2850:⋮
2821:σ
2802:σ
2785:μ
2781:−
2756:μ
2752:−
2733:
2722:⋯
2693:σ
2674:σ
2657:μ
2653:−
2628:μ
2624:−
2605:
2570:σ
2551:σ
2534:μ
2530:−
2505:μ
2501:−
2482:
2471:⋯
2447:σ
2428:σ
2411:μ
2407:−
2382:μ
2378:−
2359:
2323:
2294:…
2246:σ
2199:…
2148:(i.e., a
2079:
2041:−
2006:
1962:−
1927:
1900:
1812:
1749:
1732:
1726:−
1677:
1671:−
1646:
1640:−
1626:
1469:
1463:−
1438:
1432:−
1416:
1368:
1264:
1258:−
1233:
1227:−
1211:
1183:
1163:
996:
990:−
955:
949:−
930:
892:
733:…
544:Σ
475:variances
467:symmetric
443:×
321:×
15877:Matrices
15647:Used in
15583:Used in
15544:Rotation
15519:Jacobian
15479:Distance
15459:Cofactor
15444:Carleman
15424:Adjugate
15408:Weighing
15341:inverses
15337:products
15306:Definite
15237:Identity
15227:Exchange
15220:Constant
15186:Toeplitz
15071:Hadamard
15041:Diagonal
14855:Category
14548:Survival
14425:Johansen
14148:Binomial
14103:Isotonic
13690:(normal)
13335:location
13142:Blocking
13097:Sampling
12976:QâQ plot
12941:Box plot
12923:Graphics
12818:Skewness
12808:Kurtosis
12780:Variance
12710:Heronian
12705:Harmonic
12168:See also
11908:, panel
11546:. Panel
10396:and its
9111:variance
8409:variance
4915:, where
1102:variance
776:variance
15748:Overlap
15713:Density
15672:Edmonds
15549:Seifert
15509:Hessian
15474:Coxeter
15398:Unitary
15316:Hurwitz
15247:Of ones
15232:Hilbert
15166:Skyline
15111:Metzler
15101:Logical
15096:Integer
15006:Boolean
14938:classes
14881:Commons
14828:Kriging
14713:Process
14670:studies
14529:Wavelet
14362:General
13529:Plug-in
13323:L space
13102:Cluster
12803:Moments
12621:Outline
12525:, 2001
11599:is the
11534:Panels
11227:matrix
11149:, e.g.
11108:is the
9581:, i.e.
9115:complex
8917:be any
7916:is the
7169:(OLS).
6989:is the
5748:is the
5568:, then
5414:vector
5366:matrix
5277:, i.e.
5151:, i.e.
2068:where
1355:between
15667:Degree
15608:Design
15539:Random
15529:Payoff
15524:Moment
15449:Cartan
15439:BĂ©zout
15378:Normal
15252:Pascal
15242:Lehmer
15171:Sparse
15091:Hollow
15076:Hankel
15011:Cauchy
14936:Matrix
14750:Census
14340:Normal
14288:Manova
14108:Robust
13858:2-way
13850:1-way
13688:-test
13359:
12936:Biplot
12727:Median
12720:Lehmer
12662:Center
12568:
12375:
12337:
12297:
12263:
12236:
11912:shows
11868:shows
11518:Matlab
11065:where
10626:, and
8895:. Let
7842:where
6552:
6506:given
6439:
6061:where
5706:where
378:matrix
15728:Gamma
15692:Tutte
15554:Shear
15267:Shift
15257:Pauli
15206:Walsh
15116:Moore
14996:Block
14374:Trend
13903:prior
13845:anova
13734:-test
13708:-test
13700:-test
13607:Power
13552:Pivot
13345:shape
13340:scale
12790:Shape
12770:Range
12715:Heinz
12690:Cubic
12626:Index
12421:arXiv
11565:FLASH
10438:up to
9113:of a
8837:is a
8284:be a
4984:is a
372:, or
15534:Pick
15504:Gram
15272:Zero
14976:Band
14607:Test
13807:Sign
13659:Wald
12732:Mode
12670:Mean
12566:ISBN
12373:ISBN
12335:ISBN
12295:ISBN
12274:2012
12261:ISBN
12234:ISBN
12160:and
12109:and
12063:pcov
12055:and
11806:and
11538:and
11348:pcov
11338:bias
10567:and
10503:pcov
10424:The
9999:and
9961:and
9927:and
9866:and
9406:both
9109:The
7884:and
7284:pcov
7208:and
6765:and
6484:for
6411:are
6389:and
6342:and
6320:for
6284:and
6165:and
5874:and
5827:and
5774:and
4862:and
4710:For
3201:(or
2320:corr
2274:for
2179:for
2076:diag
2003:diag
1924:diag
1897:corr
1100:the
778:and
770:are
655:and
606:and
469:and
461:Any
412:and
356:, a
352:and
290:and
266:and
15623:Hat
15356:or
15339:or
13787:BIC
13782:AIC
12431:doi
12417:801
11972:cov
11470:cov
11442:cov
11412:cov
11384:cov
11235:cov
10495:or
10461:cov
10449:In
9835:If
9690:cov
9440:cov
9301:If
9152:var
9038:var
8977:var
8372:var
8324:var
8079:cov
7992:cov
7920:of
7745:exp
7567:of
7408:cov
7373:cov
7348:cov
7320:cov
7165:of
7025:in
6993:of
6367:If
6216:cov
6139:var
6087:var
5852:of
5752:of
5714:cov
5680:var
5652:cov
5624:cov
5604:var
5576:var
5524:If
5481:var
5444:var
5273:is
5147:is
4738:var
3719:cov
3216:cov
3205:).
2217:).
1595:by
1365:cov
1180:cov
1160:var
889:cov
556:or
348:In
226:0.5
219:0.5
15868::
12544:.
12521:,
12515:,
12496:47
12477:46
12464:^
12454:49
12429:.
12415:.
12411:.
12349:^
12218:^
11623:10
11520:.
10416:.
9821:.
9144::
8415:.
8254:.
7942:.
7047:.
6415:,
6250:.
6113:,
2312:.
1849:.
1152:.
576:.
536:,
388:.
368:,
364:,
194:A
15753:S
15211:Z
14928:e
14921:t
14914:v
13732:G
13706:F
13698:t
13686:Z
13405:V
13400:U
12602:e
12595:t
12588:v
12574:.
12550:.
12529:"
12439:.
12433::
12423::
12396:.
12381:.
12343:.
12318:.
12303:.
12276:.
12242:.
12138:f
12123:I
12111:e
12107:d
12093:)
12089:I
12081:Y
12077:,
12073:X
12069:(
12042:I
12019:j
12015:I
11994:)
11990:Y
11986:,
11982:X
11978:(
11962:c
11942:T
11936:Y
11924:X
11910:b
11889:T
11884:Y
11880:X
11866:a
11852:t
11832:)
11829:t
11826:(
11821:j
11816:Y
11794:)
11791:t
11788:(
11783:j
11778:X
11748:X
11721:)
11718:t
11715:(
11711:X
11687:j
11667:)
11664:t
11661:(
11656:j
11651:X
11627:4
11619:=
11616:m
11587:)
11584:t
11581:(
11578:X
11556:f
11552:e
11548:d
11544:c
11540:b
11536:a
11530:2
11500:,
11496:)
11492:)
11488:Y
11484:,
11480:I
11476:(
11464:)
11460:I
11456:,
11452:I
11448:(
11438:(
11434:)
11430:I
11426:,
11422:X
11418:(
11406:)
11402:Y
11398:,
11394:X
11390:(
11381:=
11378:)
11374:I
11366:Y
11362:,
11358:X
11354:(
11320:,
11311:T
11305:Y
11293:X
11276:T
11271:Y
11267:X
11257:)
11253:Y
11249:,
11245:X
11241:(
11209:j
11204:X
11197:n
11192:1
11189:=
11186:j
11176:n
11173:1
11168:=
11161:X
11133:)
11130:t
11127:(
11124:X
11114:j
11110:i
11096:)
11091:i
11087:t
11083:(
11078:j
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11053:,
11048:]
11042:)
11037:m
11033:t
11029:(
11024:n
11020:X
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11004:m
11000:t
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10991:2
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10877:(
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10844:1
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