7463:. In order to bypass the "curse" and the metric selection problem, we are motivated to consider nonlinear functional regression models, which are subject to some structural constraints but do not overly infringe flexibility. One desires models that retain polynomial rates of convergence, while being more flexible than, say, functional linear models. Such models are particularly useful when diagnostics for the functional linear model indicate lack of fit, which is often encountered in real life situations. In particular, functional polynomial models, functional
9743:
24:
often time, but may also be spatial location, wavelength, probability, etc. Intrinsically, functional data are infinite dimensional. The high intrinsic dimensionality of these data brings challenges for theory as well as computation, where these challenges vary with how the functional data were sampled. However, the high or infinite dimensional structure of the data is a rich source of information and there are many interesting challenges for research and data analysis.
9755:, where the amplitude variation is the growth rate and the time variation explains the difference in children's biological age at which the pubertal and the pre-pubertal growth spurt occurred. In the presence of time variation, the cross-sectional mean function may not be an efficient estimate as peaks and troughs are located randomly and thus meaningful signals may be distorted or hidden.
9724:
covariates as predictors. For regression based functional classification models, functional generalized linear models or more specifically, functional binary regression, such as functional logistic regression for binary responses, are commonly used classification approaches. More generally, the generalized functional linear regression model based on the
8038:, respectively. In addition to the parameter function β that the above functional quadratic regression model shares with the FLM, it also features a parameter surface γ. By analogy to FLMs with scalar responses, estimation of functional polynomial models can be obtained through expanding both the centered covariate
10184:
Landmark registration (or feature alignment) assumes well-expressed features are present in all sample curves and uses the location of such features as a gold-standard. Special features such as peak or trough locations in functions or derivatives are aligned to their average locations on the template
44:
was done in the 1970s by Kleffe, Dauxois and Pousse including results about the asymptotic distribution of the eigenvalues. More recently in the 1990s and 2000s the field has focused more on applications and understanding the effects of dense and sparse observations schemes. The term "Functional Data
9714:
are two main approaches. These classical clustering concepts for vector-valued multivariate data have been extended to functional data. For clustering of functional data, k-means clustering methods are more popular than hierarchical clustering methods. For k-means clustering on functional data, mean
23:
that analyses data providing information about curves, surfaces or anything else varying over a continuum. In its most general form, under an FDA framework, each sample element of functional data is considered to be a random function. The physical continuum over which these functions are defined is
9723:
Functional classification assigns a group membership to a new data object either based on functional regression or functional discriminant analysis. Functional data classification methods based on functional regression models use class levels as responses and the observed functional data and other
9728:
is used. Functional Linear
Discriminant Analysis (FLDA) has also been considered as a classification method for functional data. Functional data classification involving density ratios has also been proposed. A study of the asymptotic behavior of the proposed classifiers in the large sample limit
10175:
The template function is determined through an iteration process, starting from cross-sectional mean, performing registration and recalculating the cross-sectional mean for the warped curves, expecting convergence after a few iterations. DTW minimizes a cost function through dynamic programming.
2073:
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function. Then the warping function is introduced through a smooth transformation from the average location to the subject-specific locations. A problem of landmark registration is that the features may be missing or hard to identify due to the noise in the data.
5547:
7102:
9719:
is also widely used in clustering vector-valued multivariate data and has been extended to functional data clustering. Furthermore, Bayesian hierarchical clustering also plays an important role in the development of model-based functional clustering.
6923:
1305:
9758:
Time warping, also known as curve registration, curve alignment or time synchronization, aims to identify and separate amplitude variation and time variation. If both time and amplitude variation are present, then the observed functional data
1810:
10115:
of the domain to itself, that is, loosely speaking, a class of invertible functions that maps the compact domain to itself such that both the function and its inverse are smooth. The set of linear transformation is contained in the set of
9750:
In addition to amplitude variation, time variation may also be assumed to present in functional data. Time variation occurs when the subject-specific timing of certain events of interest varies among subjects. One classical example is the
300:
3731:
10403:
There are Python packages to work with functional data, and its representation, perform exploratory analysis, or preprocessing, and among other tasks such as inference, classification, regression or clustering of functional data.
8737:
863:
420:
8937:
10136:. Another traditional method for time warping is landmark registration, which aligns special features such as peak locations to an average location. Other relevant warping methods include pairwise warping, registration using
7594:
1605:
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3905:
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5105:
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4018:
65:. The former is mathematically convenient, whereas the latter is somewhat more suitable from an applied perspective. These two approaches coincide if the random functions are continuous and a condition called
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Direct nonlinear extensions of the classical functional linear regression models (FLMs) still involve a linear predictor, but combine it with a nonlinear link function, analogous to the idea of
3022:
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5794:
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Functional polynomial regression models may be viewed as a natural extension of the
Functional Linear Models (FLMs) with scalar responses, analogous to extending linear regression model to
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functions are usually regarded as the cluster centers. Covariance structures have also been taken into consideration. Besides k-means type clustering, functional clustering based on
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Structures in cross-sectional mean destroyed if time variation is ignored. On the contrary, structures in cross-sectional mean is well-captured after restoring time variation.
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1130:
32:
Functional data analysis has roots going back to work by
Grenander and Karhunen in the 1940s and 1950s. They considered the decomposition of square-integrable continuous time
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7447:
or future value. Hence, it is a "concurrent regression model", which is also referred as "varying-coefficient" model. Further, various estimation methods have been proposed.
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Pigoli, D; Hadjipantelis, PZ; Coleman, JS; Aston, JAD (2017). "The statistical analysis of acoustic phonetic data: exploring differences between spoken
Romance languages".
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Coffey, N; Hinde, J; Holian, E. (2014). "Clustering longitudinal profiles using P-splines and mixed effects models applied to time-course gene expression data".
4361:
10363:
10006:
9620:
9115:
7501:
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7591:, the simplest and the most prominent member in the family of functional polynomial regression models is the quadratic functional regression given as follows,
783:
354:
10120:. One challenge in time warping is identifiability of amplitude and phase variation. Specific assumptions are required to break this non-identifiability.
6086:. Two major models have been considered in this setup. One of these two models, generally referred to as functional linear model (FLM), can be written as:
9729:
shows that under certain conditions the misclassification rate converges to zero, a phenomenon that has been referred to as "perfect classification".
10830:
Shi, M; Weiss, RE; Taylor, JMG. (1996). "An analysis of paediatric CD4 counts for acquired immune deficiency syndrome using flexible random curves".
11547:
11496:
10111:, which warps the time of an underlying template function by subjected-specific shift and scale. More general class of warping functions includes
3803:
10176:
Problems of non-smooth differentiable warps or greedy computation in DTW can be resolved by adding a regularization term to the cost function.
9081:
ensures identifiability in the sense that the estimates of these additive component functions do not interfere with that of the intercept term
11256:
Huang, JZ; Wu, CO; Zhou, L. (2002). "Varying-coefficient models and basis function approximations for the analysis of repeated measurements".
7459:
from the conventional linear model. Developments towards fully nonparametric regression models for functional data encounter problems such as
12725:
Carroll, C; Müller, HG; Kneip, A (2021). "Cross-component registration for multivariate functional data, with application to growth curves".
3910:
2068:{\displaystyle \sup _{s,t\in }\left|\Sigma (s,t)-\sum _{j=1}^{K}\lambda _{j}\varphi _{j}(s)\varphi _{j}(t)\right|\to 0,\qquad K\to \infty .}
7788:{\displaystyle \mathbb {E} (Y|X)=\alpha +\int _{0}^{1}\beta (t)X^{c}(t)\,dt+\int _{0}^{1}\int _{0}^{1}\gamma (s,t)X^{c}(s)X^{c}(t)\,ds\,dt}
12579:
Anirudh, R; Turaga, P; Su, J; Srivastava, A (2015). "Elastic functional coding of human actions: From vector-fields to latent variables".
11811:
Angelini, C; Canditiis, DD; Pensky, M. (2012). "Clustering time-course microarray data using functional
Bayesian infinite mixture model".
9403:
12661:
Happ, C; Greven, S (2018). "Multivariate
Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains".
11283:
Huang, JZ; Wu, CO; Zhou, L. (2004). "Polynomial spline estimation and inference for varying coefficient models with longitudinal data".
9222:
6280:{\displaystyle Y(s)=\alpha _{0}(s)+\sum _{j=1}^{p}\int _{0}^{1}\alpha _{j}(s,t)X_{j}^{c}(t)\,dt+\varepsilon (s),\ {\text{for}}\ s\in }
2808:
11337:
Eggermont, PPB; Eubank, RL; LaRiccia, VN. (2010). "Convergence rates for smoothing spline estimators in varying coefficient models".
8509:
8290:{\displaystyle \mathbb {E} (Y|X)=g\left(\int _{0}^{1}X^{c}(t)\beta _{1}(t)\,dt,\ldots ,\int _{0}^{1}X^{c}(t)\beta _{p}(t)\,dt\right)}
7798:
12815:
Chen, K; Delicado, P; Müller, HG (2017). "Modelling function-valued stochastic processes, with applications to fertility dynamics".
11981:
41:
5237:{\displaystyle Y=\beta _{0}+\langle X^{c},\beta \rangle +\varepsilon =\beta _{0}+\int _{0}^{1}X^{c}(t)\beta (t)\,dt+\varepsilon .}
4711:{\displaystyle Y=\beta _{0}+\langle X,\beta \rangle +\varepsilon =\beta _{0}+X_{1}\beta _{1}+\dots +X_{p}\beta _{p}+\varepsilon ,}
10473:
9725:
3403:
952:
582:
10760:"Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference"
4027:
2551:
963:
The
Hilbertian point of view is mathematically convenient, but abstract; the above considerations do not necessarily even view
7201:
10564:
10550:
2352:
10656:
Rice, JA; Silverman, BW. (1991). "Estimating the mean and covariance structure nonparametrically when the data are curves".
3260:
3040:
3473:
4020:
are the functional principal components (FPCs), sometimes referred to as scores. The
Karhunen–Loève expansion facilitates
12417:
Marron, JS; Ramsay, JO; Sangalli, LM; Srivastava, A (2015). "Functional data analysis of amplitude and phase variation".
10493:
9387:
3410:
of the inherently infinite-dimensional functional data to finite-dimensional random vector of scores. More specifically,
2972:
12706:
Chiou, JM; Yang, YF; Chen, YT (2014). "Multivariate functional principal component analysis: a normalization approach".
12352:
Gasser, T; Müller, HG; Kohler, W; Molinari, L; Prader, A. (1984). "Nonparametric regression analysis of growth curves".
11673:
11434:
Chen, D; Hall, P; Müller HG. (2011). "Single and multiple index functional regression models with nonparametric link".
6380:
5714:
10536:
9988:
is a latent time warping function that corresponds to a cumulative distribution function. The time warping functions
3557:
10196:
8113:
A functional multiple index model is given as below, with symbols having their usual meanings as formerly described,
7378:
2761:
12770:
Dai, X; Müller, HG (2018). "Principal component analysis for functional data on
Riemannian manifolds and spheres".
9943:
9895:
8416:
7286:
is usually assumed to be a random process with mean zero and finite variance. This model assumes that the value of
5542:{\displaystyle Y=\langle Z,\theta \rangle +\sum _{j=1}^{p}\int _{0}^{1}X_{j}^{c}(t)\beta _{j}(t)\,dt+\varepsilon ,}
4737:
466:
8969:
7464:
12866:
11070:
He, G; Müller, HG; Wang, JL. (2003). "Functional canonical analysis for square integrable stochastic processes".
5568:
1654:
682:
10067:
The simplest case of a family of warping functions to specify phase variation is linear transformation, that is
4812:
4777:
2297:
775:
10011:
9497:
4279:
3150:
7097:{\displaystyle Y(s)=\beta _{0}(s)+\sum _{j=1}^{p}\beta _{j}(s)X_{j}(s)+\varepsilon (s),\ {\text{for}}\ s\in ,}
3760:
are real-valued nonnegative eigenvalues in descending order with the corresponding orthonormal eigenfunctions
3209:
12871:
10396:
6918:{\displaystyle Y(s)=\alpha _{0}(s)+\sum _{j=1}^{p}X_{j}\alpha _{j}(s)+\varepsilon (s),\ {\text{for}}\ s\in ,}
5320:
2165:
1313:
10478:
9551:
7924:
7318:
6655:
3797:
3594:
2164:
and the
Hilbert space machinery can be subsequently applied. Continuity of sample paths can be shown using
37:
12065:
Hall, P; Poskitt, DS; Presnell, B. (2001). "A Functional Data—Analytic Approach to Signal Discrimination".
10588:
10070:
4929:
4550:
4519:
that associates vector responses with vector covariates. The traditional linear model with scalar response
2486:
1467:
543:
11221:
Wu, CO; Yu, KF. (2002). "Nonparametric varying-coefficient models for the analysis of longitudinal data".
9789:
4186:
2639:
10894:
Kong, D; Xue, K; Yao, F; Zhang, HH. (2016). "Partially functional linear regression in high dimensions".
8830:, analogous to the extension of multiple linear regression models to additive models and is expressed as,
7123:
5269:
12225:
Dai, X; Müller, HG; Yao, F. (2017). "Optimal Bayes classifiers for functional data and density ratios".
10731:
Kleffe, J. (1973). "Principal components of random variables with values in a seperable hilbert space".
10139:
2101:
435:
9029:
4522:
138:
12462:
Sakoe, H; Chiba, S. (1978). "Dynamic programming algorithm optimization for spoken word recognition".
11135:
He, G; Müller, HG; Wang, JL; Yang, WJ. (2010). "Functional linear regression via canonical analysis".
6497:
4995:
1300:{\displaystyle \mu (t)=\mathbb {E} X(t),\qquad \Sigma (s,t)={\textrm {Cov}}(X(s),X(t)),\qquad s,t\in }
1075:
11903:
Petrone, S; Guindani, M; Gelfand, AE. (2009). "Hybrid Dirichlet mixture models for functional data".
11469:
Müller HG; Wu Y; Yao, F. (2013). "Continuously additive models for nonlinear functional regression".
9398:
9084:
8769:
7260:
6539:
4872:
random error (noise). Functional linear models can be divided into two types based on the responses.
3763:
2920:
1818:
10446:
9276:
8297:
Here g represents an (unknown) general smooth function defined on a p-dimensional domain. The case
7895:
6306:
3570:
2731:
1704:
1630:
1138:
930:
902:
10849:
Hilgert, N; Mas, A; Verzelen, N. (2013). "Minimax adaptive tests for the functional linear model".
10415:
9665:
9391:
7456:
6729:
6342:
6051:
5858:
4021:
3411:
3407:
2430:
11940:"Clustering in linear mixed models with approximate Dirichlet process mixtures using EM algorithm"
11768:
Heinzl, F; Tutz, G. (2014). "Clustering in linear-mixed models with a group fused lasso penalty".
4847:
3736:
2182:
1871:
871:
9711:
9594:
8376:
7460:
1805:{\displaystyle {\mathcal {C}}=\sum _{j=1}^{\infty }\lambda _{j}\varphi _{j}\otimes \varphi _{j}.}
6565:
is usually a random process with mean zero and finite variance. In this case, at any given time
5799:
295:{\displaystyle \mathbb {E} \|X\|_{L^{2}}^{2}=\mathbb {E} (\int _{0}^{1}|X(t)|^{2}dt)<\infty }
12178:"Robust Classification of Functional and Quantitative Image Data Using Functional Mixed Models"
11302:Şentürk, D; Müller, HG. (2010). "Functional varying coefficient models for longitudinal data".
11100:
Yao, F; Müller, HG; Wang, JL. (2005). "Functional data analysis for sparse longitudinal data".
10284:
10252:
8405:
and relatively small sample sizes, the estimator given by this model often has large variance.
7506:
5826:
5682:
325:
11674:"Funclust: A curves clustering method using functional random variables density approximation"
5977:
3417:
12329:
11541:
11490:
10969:
10817:
Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators
10557:
Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators
10488:
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10436:
8088:
7991:
7875:
7480:
5378:
4512:
1447:
1427:
10431:
10248:
8382:
8352:
8326:
8068:
4903:
3726:{\displaystyle \Sigma (s,t)=\sum _{k=1}^{\infty }\lambda _{k}\varphi _{k}(s)\varphi _{k}(t)}
3338:
3252:
3118:
2701:
1845:
12316:
12100:
Ferraty, F; Vieu, P. (2003). "Curves discrimination: a nonparametric functional approach".
11820:
10956:
10620:
10582:
10573:
10498:
10368:
10313:
10129:
9762:
9645:
9337:
9057:
8942:
8806:
8742:
8467:
8323:
yields a functional single index model while multiple index models correspond to the case
8041:
7567:
6568:
6013:
5623:
5072:
5030:
4493:
4232:
3368:
2238:
1366:
1006:
143:
12597:
Dubey, P; Müller, HG (2021). "Modeling Time-Varying Random Objects and Dynamic Networks".
9625:
7289:
6626:
5916:
1407:
951:, in a non-increasing order. Truncating this infinite series to a finite order underpins
8:
11714:
Jacques, J; Preda, C. (2014). "Model-based clustering for multivariate functional data".
10456:
10426:
8300:
4767:
3564:
1896:
1049:
consist of equivalence classes, not functions. The stochastic process perspective views
514:
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12128:
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10960:
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12202:
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12047:
11962:
11939:
11920:
11880:
11855:
11836:
11793:
11696:
11654:
11619:
11602:
Banfield, JD; Raftery, AE. (1993). "Model-based Gaussian and non-Gaussian clustering".
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11519:
11417:
11397:
11319:
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11234:
11203:
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11144:
11117:
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11044:
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11005:
10974:
10876:
10858:
10669:
10638:
10483:
10348:
10133:
9991:
9707:
9605:
8732:{\displaystyle \mathbb {E} (Y|X)=\mathbb {E} (Y)+\sum _{k=1}^{\infty }\beta _{k}x_{k}.}
7486:
6606:
5603:
4883:
4501:
4259:
4212:
3453:
2892:
2872:
2466:
2177:
2147:
2081:
1052:
986:
966:
858:{\displaystyle X=\mu +\sum _{i=1}^{\infty }\langle X,\varphi _{i}\rangle \varphi _{i},}
757:
520:
305:
120:
100:
80:
62:
33:
12113:
11083:
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10759:
10642:
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10546:
10532:
10524:
10508:
9752:
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4516:
3450:
in a functional basis consisting of the eigenfunctions of the covariance operator on
3243:
415:{\displaystyle \mathbb {E} \langle X,h\rangle =\langle \mu ,h\rangle ,\qquad h\in H.}
12846:
12801:
12692:
12483:
12448:
12398:
12338:
12086:
12051:
12001:
11966:
11924:
11797:
11700:
11451:
Jiang, CR; Wang JL. (2011). "Functional single index models for longitudinal data".
11421:
11323:
11207:
11166:
11056:
10978:
10880:
10422:
Some packages can handle functional data under both dense and longitudinal designs.
10392:
and further to nonlinear manifolds, Hilbert spaces and eventually to metric spaces.
8932:{\displaystyle \mathbb {E} (Y|X)=\mathbb {E} (Y)+\sum _{k=1}^{\infty }f_{k}(x_{k}),}
6925:
which is a functional linear model with functional responses and scalar covariates.
12832:
12824:
12789:
12744:
12680:
12616:
12558:
12549:
Tang, R; Müller, HG. (2008). "Pairwise curve synchronization for functional data".
12510:
12471:
12436:
12384:
12324:
12267:
12197:
12189:
12148:
12140:
12109:
12074:
12037:
11996:
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11912:
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2098:
has continuous sample paths, namely that with probability one, the random function
1624:
751:
430:
12684:
12620:
1600:{\displaystyle ({\mathcal {C}}f)(t)=\int _{0}^{1}\Sigma (s,t)f(s)\,\mathrm {d} s.}
11982:"Classification using functional data analysis for temporal gene expression data"
11832:
11692:
11637:
James, GM; Sugar, CA. (2003). "Clustering for sparsely sampled functional data".
10520:
4771:
2483:
independent subjects. The sampling schedule may vary across subjects, denoted as
538:
426:
46:
12475:
11754:
11727:
11533:
11350:
11113:
10117:
10112:
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1698:
676:
12144:
12078:
11315:
11269:
10744:
12860:
12581:
Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
12515:
12498:
12281:
11958:
11198:
11181:
9742:
9716:
9600:
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4229:
yields a good approximation to the infinite sum. Thereby, the information in
1046:
58:
12817:
Journal of the Royal Statistical Society. Series B (Statistical Methodology)
12562:
12389:
12372:
12260:
Journal of the Royal Statistical Society. Series B (Statistical Methodology)
12042:
12025:
11871:
11856:"Bayesian nonparametric functional data analysis through density estimation"
11650:
11567:
Journal of the Royal Statistical Society, Series B (Statistical Methodology)
11482:
10992:
Ramsay, JO; Dalzell, CJ. (1991). "Some tools for functional data analysis".
10907:
12211:
12010:
11889:
11789:
11781:
6536:
is the corresponding functional slopes with same domain, respectively, and
4469:{\displaystyle X_{i}^{(K)}(t)=\mu (t)+\sum _{k=1}^{K}A_{ik}\varphi _{k}(t)}
3900:{\displaystyle X_{i}(t)=\mu (t)+\sum _{k=1}^{\infty }A_{ik}\varphi _{k}(t)}
1159:). The mean and covariance functions are defined in a pointwise manner as
896:
11563:"Functional clustering and identifying substructures of longitudinal data"
11510:
Müller HG; Stadmüller, U. (2005). "Generalized Functional Linear Models".
11027:
Malfait, N; Ramsay, JO. (2003). "The historical functional linear model".
10559:, Wiley series in probability and statistics, John Wiley & Sons, Ltd,
9212:{\displaystyle \mathbb {E} (Y|X)=\mathbb {E} (Y)+\int _{0}^{1}g(t,X(t))dt}
3406:(FPCA) is the most prevalent tool in FDA, partly because FPCA facilitates
3253:
3. Sparsely sampled functions with noisy measurements (longitudinal data)
924:
12026:"Functional linear discriminant analysis for irregularly sampled curves"
10408:
5908:
12837:
12828:
12793:
12748:
12644:
Journal of the Royal Statistical Society. Series C (Applied Statistics)
12530:
Gasser, T; Kneip, A (1995). "Searching for structure in curve sample".
11623:
11048:
10872:
10832:
Journal of the Royal Statistical Society. Series C (Applied Statistics)
10701:
10684:
10633:
10608:
9112:. Another form of FAM is the continuously additive model, expressed as,
4013:{\displaystyle A_{ik}=\int _{0}^{1}(X_{i}(t)-\mu (t))\varphi _{k}(t)dt}
2548:
for the i-th subject. The corresponding i-th observation is denoted as
20:
12440:
11412:
11385:
11158:
11524:
3414:
is achieved by expanding the underlying observed random trajectories
57:
Random functions can be viewed as random elements taking values in a
11615:
11040:
9380:
An obvious and direct extension of FLMs with scalar responses (see (
4875:
12784:
12739:
12675:
12611:
12499:"Statistical tools to analyze data representing a sample of curves"
12431:
12304:
12239:
10994:
Journal of the Royal Statistical Society, Series B (Methodological)
10944:
10571:
Annual Review of Statistics and Its Application, Vol. 2, 321 - 359,
10503:
9701:
9482:{\displaystyle \eta =\beta _{0}+\int _{0}^{1}X^{c}(t)\beta (t)\,dt}
7471:
are three special cases of functional nonlinear regression models.
5069:
4869:
3033:
2. Densely sampled functions with noisy measurements (dense design)
2913:
1. Fully observed functions without noise at arbitrarily dense grid
11402:
11149:
10863:
10580:
Annual Review of Statistics and Its Application, Vol. 3, 257-295,
9266:{\displaystyle g:\times \mathbb {R} \longrightarrow \mathbb {R} }
8739:
One form of FAMs is obtained by replacing the linear function of
2862:{\displaystyle {\textrm {Var}}(\epsilon _{ij})=\sigma _{ij}^{2}}
12641:
11366:
Yao, F; Müller, HG. (2010). "Functional quadratic regression".
8588:{\displaystyle X^{c}(t)=\sum _{k=1}^{\infty }x_{k}\phi _{k}(t)}
7865:{\displaystyle X^{c}(\cdot )=X(\cdot )-\mathbb {E} (X(\cdot ))}
7257:
are the coefficient functions defined on the same interval and
12416:
8108:
5065:
and replacing the inner product in Euclidean space by that in
4500:
and wavelet bases. Important applications of FPCA include the
12464:
IEEE Transactions on Acoustics, Speech, and Signal Processing
10345:
The range set of the stochastic process may be extended from
10337:, for example the data could be a sample of random surfaces.
4024:
in the sense that the partial sum converges uniformly, i.e.,
3733:, where the series convergence is absolute and uniform, and
2144:
is continuous, the Karhunen-Loève expansion above holds for
12256:"Achieving near perfect classification for functional data"
11386:"A test of significance in functional quadratic regression"
3800:, the FPCA expansion of an underlying random trajectory is
665:{\displaystyle {\mathcal {C}}h=\mathbb {E} ,\qquad h\in H,}
12351:
7474:
4176:{\displaystyle \sup _{t\in }\mathbb {E} ^{2}\rightarrow 0}
3398:
2427:. The realizations of the process for the i-th subject is
1444:
are continuous functions and then the covariance function
12578:
10685:"Peter Hall, functional data analysis and random objects"
7450:
2629:{\displaystyle {\textbf {X}}_{i}=(X_{i1},...,X_{iN_{i}})}
1135:
indexed by the unit interval (or more generally interval
10583:
https://doi.org/10.1146/annurev-statistics-041715-033624
10574:
https://doi.org/10.1146/annurev-statistics-010814-020413
7250:{\displaystyle \beta _{0},\beta _{1},\ldots ,\beta _{p}}
11810:
11336:
10193:
So far we considered scalar valued stochastic process,
6762:
as a constant function yields a special case of model (
2420:{\displaystyle \Sigma (s,t)={\textrm {Cov}}(X(s),X(t))}
11902:
11853:
11182:"Statistical estimation in varying coefficient models"
9375:
8630:
A functional linear model with scalar responses (see (
4507:
3328:{\displaystyle Y_{ij}=X_{i}(T_{ij})+\varepsilon _{ij}}
3108:{\displaystyle Y_{ij}=X_{i}(T_{ij})+\varepsilon _{ij}}
11509:
10371:
10351:
10316:
10287:
10255:
10199:
10142:
10073:
10014:
9994:
9946:
9898:
9792:
9765:
9668:
9648:
9628:
9608:
9554:
9500:
9406:
9340:
9279:
9225:
9118:
9087:
9060:
9032:
8972:
8945:
8836:
8809:
8772:
8745:
8642:
8601:
8512:
8470:
8419:
8385:
8355:
8329:
8303:
8119:
8091:
8071:
8044:
7994:
7962:
7927:
7898:
7878:
7801:
7597:
7570:
7538:
7509:
7489:
7381:
7321:
7292:
7263:
7204:
7172:
7126:
6944:
6774:
6732:
6658:
6629:
6609:
6571:
6542:
6500:
6468:
6383:
6345:
6309:
6097:
6054:
6016:
5980:
5948:
5919:
5909:
Functional regression models with functional response
5861:
5829:
5802:
5717:
5685:
5653:
5626:
5606:
5571:
5419:
5381:
5323:
5272:
5266:) can be extended to multiple functional covariates,
5108:
5075:
5033:
4998:
4932:
4906:
4886:
4850:
4815:
4780:
4740:
4593:
4553:
4525:
4364:
4282:
4262:
4235:
4215:
4189:
4030:
3913:
3806:
3766:
3739:
3632:
3597:
3573:
3543:{\displaystyle {\mathcal {C}}:L^{2}\rightarrow L^{2}}
3476:
3456:
3420:
3395:
Real life example: CD4 count data for AIDS patients.
3371:
3341:
3263:
3212:
3153:
3121:
3043:
2975:
2923:
2895:
2875:
2811:
2764:
2734:
2704:
2642:
2554:
2489:
2469:
2433:
2355:
2300:
2268:
2241:
2185:
2150:
2104:
2084:
1908:
1874:
1848:
1821:
1734:
1707:
1657:
1633:
1511:
1470:
1450:
1430:
1410:
1369:
1316:
1168:
1141:
1078:
1055:
1009:
989:
969:
933:
905:
874:
786:
760:
685:
585:
546:
523:
469:
438:
357:
328:
308:
188:
146:
123:
103:
83:
12129:"Functional data classification: a wavelet approach"
11468:
8408:
2176:
Functional data are considered as realizations of a
12064:
10757:
7564:and the corresponding centered predictor processes
3017:{\displaystyle t\in {\mathcal {I}},\,i=1,\ldots ,n}
12814:
12724:
12377:Journal of the Royal Statistical Society, Series B
10848:
10384:
10357:
10340:
10329:
10302:
10270:
10237:
10159:
10103:
10056:
10000:
9980:
9932:
9884:
9778:
9689:
9654:
9634:
9614:
9585:
9540:
9481:
9364:
9326:
9265:
9211:
9104:
9073:
9054:. This constraint on the general smooth functions
9046:
9018:
8958:
8931:
8822:
8795:
8758:
8731:
8619:
8587:
8498:
8456:
8397:
8367:
8341:
8315:
8289:
8097:
8077:
8057:
8030:
7980:
7948:
7913:
7884:
7864:
7787:
7583:
7556:
7524:
7495:
7439:
7367:
7307:
7278:
7249:
7190:
7158:
7096:
6917:
6754:
6704:
6644:
6615:
6595:
6557:
6528:
6486:
6454:
6369:
6331:
6279:
6078:
6040:
6002:
5966:
5934:
5885:
5847:
5815:
5788:
5703:
5671:
5639:
5612:
5592:
5541:
5400:
5367:
5309:
5236:
5088:
5057:
5019:
4984:
4912:
4892:
4856:
4836:
4801:
4758:
4710:
4574:
4539:
4468:
4345:
4268:
4248:
4221:
4201:
4175:
4012:
3899:
3788:
3752:
3725:
3618:
3583:
3542:
3462:
3442:
3384:
3357:
3327:
3231:
3198:
3137:
3107:
3016:
2961:
2901:
2881:
2861:
2797:
2750:
2720:
2690:
2628:
2540:
2475:
2455:
2419:
2341:
2286:
2254:
2227:
2156:
2136:
2090:
2067:
1887:
1860:
1834:
1804:
1717:
1689:
1643:
1599:
1492:
1456:
1436:
1416:
1393:
1355:
1299:
1151:
1124:
1061:
1037:
995:
975:
943:
915:
887:
857:
766:
742:
664:
568:
529:
505:
455:
414:
340:
314:
294:
174:
137:is a separable Hilbert space such as the space of
129:
109:
89:
11740:
10925:. Springer Series in Statistics. Springer-Verlag.
6455:{\displaystyle X_{j}^{c}(t)=X_{j}(t)-\mu _{j}(t)}
5789:{\displaystyle X_{j}^{c}(t)=X_{j}(t)-\mu _{j}(t)}
4876:Functional regression models with scalar response
4515:can be viewed as an extension of the traditional
3026:Often unrealistic but mathematically convenient.
77:In the Hilbert space viewpoint, one considers an
12858:
11134:
10609:"Stochastic processes and statistical inference"
9702:Clustering and classification of functional data
6721:
4032:
1910:
12663:Journal of the American Statistical Association
12599:Journal of the American Statistical Association
12532:Journal of the American Statistical Association
12309:Annual Review of Statistics and Its Application
12302:
11854:Rodríguez, A; Dunson, DB; Gelfand, AE. (2009).
11639:Journal of the American Statistical Association
11601:
11304:Journal of the American Statistical Association
11102:Journal of the American Statistical Association
10949:Annual Review of Statistics and Its Application
10942:
10923:Inference for functional data with applications
10920:
10893:
10799:
10733:Mathematische Operationsforschung und Statistik
10543:Inference for Functional Data with Applications
6928:
4504:and functional principal component regression.
12175:
12126:
10829:
10238:{\displaystyle \{X(t)\}_{t\in {\mathcal {T}}}}
7440:{\displaystyle \{X_{j}(t):t\leq s\}_{j=1}^{p}}
5711:is the centered functional covariate given by
5317:, also including additional vector covariates
2798:{\displaystyle \mathbb {E} (\epsilon _{ij})=0}
983:as a function at all, since common choices of
72:
12705:
11383:
11339:Journal of Statistical Planning and Inference
11301:
11099:
11026:
10991:
10819:. Wiley Series in Probability and Statistics.
10655:
10578:Wang et al. (2016) Functional Data Analysis,
9981:{\displaystyle h_{i}{\overset {iid}{\sim }}h}
9933:{\displaystyle X_{i}{\overset {iid}{\sim }}X}
9394:(GLM). The three components of the GFLM are:
8457:{\displaystyle \{\phi _{k}\}_{k=1}^{\infty }}
4759:{\displaystyle \langle \cdot ,\cdot \rangle }
4209:and thus the partial sum with a large enough
506:{\displaystyle \mathbb {E} \|X\|_{L^{2}}^{2}}
12253:
12224:
12102:Computational Statistics & Data Analysis
11743:Computational Statistics & Data Analysis
11716:Computational Statistics & Data Analysis
11713:
11671:
11546:: CS1 maint: multiple names: authors list (
11495:: CS1 maint: multiple names: authors list (
11069:
10216:
10200:
10008:are assumed to be invertible and to satisfy
9019:{\displaystyle \mathbb {E} (f_{k}(x_{k}))=0}
8434:
8420:
7417:
7382:
7345:
7322:
6682:
6659:
5438:
5426:
5287:
5273:
5147:
5128:
5097:, one arrives at the functional linear model
4753:
4741:
4625:
4613:
1095:
1079:
839:
820:
625:
607:
482:
475:
393:
381:
375:
363:
201:
194:
12596:
12529:
12496:
12099:
12023:
11282:
11255:
10938:
10936:
10934:
10932:
10814:
10718:Zur Spektraltheorie stochastischer Prozesse
8109:Functional single and multiple index models
5593:{\displaystyle \theta \in \mathbb {R^{q}} }
1690:{\displaystyle (\lambda _{j},\varphi _{j})}
743:{\displaystyle {\mathcal {C}}=\mathbb {E} }
12660:
12548:
12461:
11979:
11937:
11767:
11636:
10758:Dauxois, J; Pousse, A; Romain, Y. (1982).
10245:, defined on one dimensional time domain.
9386:)) is to add a link function leading to a
4837:{\displaystyle \beta \in \mathbb {R} ^{p}}
4802:{\displaystyle \beta _{0}\in \mathbb {R} }
4256:is reduced from infinite dimensional to a
3029:Real life example: Tecator spectral data.
2463:, and the sample is assumed to consist of
2342:{\displaystyle \mu (t)=\mathbb {E} (X(t))}
2171:
182:. Under the integrability condition that
12836:
12783:
12769:
12738:
12674:
12610:
12514:
12430:
12388:
12328:
12303:Wang, JL; Chiou, JM; Müller, HG. (2016).
12271:
12238:
12201:
12152:
12041:
12000:
11879:
11578:
11523:
11411:
11401:
11362:
11360:
11197:
11148:
10968:
10943:Wang, JL; Chiou, JM; Müller, HG. (2016).
10862:
10775:
10720:. Annales Academiae scientiarum Fennicae.
10700:
10632:
10606:
10569:Morris, J. (2015) Functional Regression,
10057:{\displaystyle \mathbb {E} (h^{-1}(t))=t}
10016:
9562:
9541:{\displaystyle {\text{Var}}(Y|X)=V(\mu )}
9472:
9281:
9259:
9251:
9145:
9120:
9089:
9040:
8974:
8863:
8838:
8669:
8644:
8275:
8206:
8121:
7840:
7778:
7771:
7675:
7599:
6220:
5584:
5580:
5523:
5218:
4824:
4795:
4562:
4533:
4346:{\displaystyle A_{i}=(A_{i1},...,A_{iK})}
4060:
3199:{\displaystyle T_{i1},\ldots ,T_{iN_{i}}}
2992:
2766:
2317:
2262:process on a bounded and closed interval
2130:
2078:Finally, under the extra assumption that
1585:
1318:
1185:
697:
600:
576:that is uniquely defined by the relation
471:
463:. Under the integrability condition that
446:
359:
226:
190:
52:
12370:
12330:10.1146/annurev-statistics-041715-033624
12030:Journal of the Royal Statistical Society
11905:Journal of the Royal Statistical Society
11179:
10970:10.1146/annurev-statistics-041715-033624
10929:
10715:
10658:Journal of the Royal Statistical Society
10179:
9741:
9219:for a bivariate smooth additive surface
6652:, depends on the entire trajectories of
4844:denote the regression coefficients, and
3232:{\displaystyle N_{i}\rightarrow \infty }
42:functional principal components analysis
12296:
12176:Zhu, H; Brown, PJ; Morris, JS. (2012).
11560:
11428:
10474:Functional principal component analysis
10170:
7956:are coefficient functions with domains
7475:Functional polynomial regression models
5823:is regression coefficient function for
5368:{\displaystyle Z=(Z_{1},\cdots ,Z_{q})}
3404:Functional principal component analysis
3399:Functional principal component analysis
1356:{\displaystyle \mathbb {E} <\infty }
958:
953:functional principal component analysis
36:into eigencomponents, now known as the
12859:
12127:Chang, C; Chen, Y; Ogden, RT. (2014).
11357:
10730:
10682:
9586:{\displaystyle \mu =\mathbb {E} (Y|X)}
9372:, in order to ensure identifiability.
7949:{\displaystyle \gamma (\cdot ,\cdot )}
7872:is the centered functional covariate,
7451:Functional nonlinear regression models
7368:{\displaystyle \{X_{j}(s)\}_{j=1}^{p}}
6705:{\displaystyle \{X_{j}(t)\}_{j=1}^{p}}
6462:is a centered functional covariate on
3619:{\displaystyle \Sigma (\cdot ,\cdot )}
12592:
12590:
12574:
12572:
12412:
12410:
12408:
11095:
11093:
10104:{\displaystyle h(t)=\delta +\gamma t}
9706:For vector-valued multivariate data,
4985:{\displaystyle X^{c}(t)=X(t)-\mu (t)}
4926:) by a centered functional covariate
4575:{\displaystyle X\in \mathbb {R} ^{p}}
2541:{\displaystyle T_{i1},...,T_{iN_{i}}}
1493:{\displaystyle {\mathcal {C}}:H\to H}
569:{\displaystyle {\mathcal {C}}:H\to H}
11220:
10795:
10793:
10791:
10789:
10787:
10541:Horvath, L. and Kokoszka, P. (2012)
10132:(DTW) used for applications such as
9885:{\displaystyle Y_{i}(t)=X_{i},t\in }
6935:
6088:
5410:
5260:The simple functional linear model (
5099:
4584:
4355:
4202:{\displaystyle K\rightarrow \infty }
3239:applies to typical functional data.
2691:{\displaystyle X_{ij}=X_{i}(T_{ij})}
1502:
1069:as a collection of random variables
429:but the mean can also be defined as
11734:
10494:Generalized functional linear model
9940:is a latent amplitude function and
9388:generalized functional linear model
9376:Generalized functional linear model
8375:, this model is problematic due to
7159:{\displaystyle X_{1},\ldots ,X_{p}}
5974:and multiple functional covariates
5310:{\displaystyle \{X_{j}\}_{j=1}^{p}}
4508:Functional linear regression models
3470:. Consider the covariance operator
2558:
923:, corresponding to the nonnegative
13:
12587:
12569:
12542:
12405:
12218:
11235:10.1111/j.1751-5823.2002.tb00176.x
11090:
11029:The Canadian Journal of Statistics
11006:10.1111/j.2517-6161.1991.tb01844.x
10800:Ramsay, J; Silverman, BW. (2005).
10670:10.1111/j.2517-6161.1991.tb01821.x
10514:
10395:
10228:
10160:{\displaystyle {\mathcal {L}}^{2}}
10146:
8895:
8701:
8636:)) can thus be written as follows,
8551:
8449:
4196:
3860:
3670:
3633:
3598:
3576:
3479:
3392:per subject is random and finite.
3365:are random times and their number
3226:
2984:
2698:. In addition, the measurement of
2356:
2137:{\displaystyle X:\to \mathbb {R} }
2059:
1948:
1824:
1761:
1737:
1710:
1636:
1587:
1555:
1517:
1473:
1451:
1431:
1404:Under the mean square continuity,
1350:
1205:
1144:
936:
908:
815:
688:
588:
549:
456:{\displaystyle \mu =\mathbb {E} X}
289:
14:
12883:
10921:Horváth, L; Kokoszka, P. (2012).
10784:
10649:
9047:{\displaystyle k\in \mathbb {N} }
8409:Functional additive models (FAMs)
6339:is the functional intercept, for
5905:) have been studied extensively.
4540:{\displaystyle Y\in \mathbb {R} }
3558:compact operator on Hilbert space
12273:10.1111/j.1467-9868.2011.01003.x
12194:10.1111/j.1541-0420.2012.01765.x
11917:10.1111/j.1467-9868.2009.00708.x
11580:10.1111/j.1467-9868.2007.00605.x
11223:International Statistical Review
11137:Journal of Multivariate Analysis
11072:Journal of Multivariate Analysis
10802:Functional Data Analysis, 2nd ed
10764:Journal of Multivariate Analysis
10555:Hsing, T. and Eubank, R. (2015)
9622:connecting the conditional mean
8766:in the above expression ( i.e.,
7465:single and multiple index models
7315:depends on the current value of
6718:) has been studied extensively.
6529:{\displaystyle \alpha _{j}(s,t)}
5020:{\displaystyle \beta =\beta (t)}
3591:, i.e., the covariance function
3145:are recorded on a regular grid,
2728:is assumed to have random noise
1125:{\displaystyle \{X(t)\}_{t\in }}
12808:
12763:
12718:
12699:
12654:
12635:
12523:
12490:
12455:
12364:
12345:
12247:
12169:
12120:
12093:
12058:
12024:James, GM; Hastie, TJ. (2001).
12017:
11973:
11931:
11896:
11847:
11804:
11761:
11707:
11665:
11630:
11595:
11554:
11503:
11462:
11445:
11384:Horváth, L; Reeder, R. (2013).
11377:
11330:
11295:
11276:
11249:
11214:
11173:
11128:
11063:
11020:
10985:
10914:
10887:
10842:
10531:, 2nd ed., New York: Springer,
10341:Multivariate stochastic process
9732:
9105:{\displaystyle \mathbb {E} (Y)}
8803:) by a general smooth function
8796:{\displaystyle \beta _{k}x_{k}}
7279:{\displaystyle \varepsilon (s)}
6558:{\displaystyle \varepsilon (s)}
5913:Consider a functional response
4880:Replacing the vector covariate
4353:with the approximated process：
3789:{\displaystyle \varphi _{k}(t)}
2962:{\displaystyle Y_{it}=X_{i}(t)}
2869:, which are independent across
2052:
1835:{\displaystyle {\mathcal {C}}f}
1269:
1204:
649:
399:
11672:Jacques, J; Preda, C. (2013).
10823:
10808:
10751:
10724:
10709:
10676:
10600:
10297:
10291:
10265:
10259:
10212:
10206:
10167:distance and elastic warping.
10083:
10077:
10045:
10042:
10036:
10020:
9879:
9867:
9855:
9852:
9846:
9825:
9809:
9803:
9737:
9684:
9678:
9580:
9573:
9566:
9535:
9529:
9520:
9513:
9506:
9469:
9463:
9457:
9451:
9359:
9347:
9327:{\displaystyle \mathbb {E} =0}
9315:
9312:
9309:
9303:
9291:
9285:
9255:
9244:
9232:
9200:
9197:
9191:
9179:
9155:
9149:
9138:
9131:
9124:
9099:
9093:
9007:
9004:
8991:
8978:
8923:
8910:
8873:
8867:
8856:
8849:
8842:
8679:
8673:
8662:
8655:
8648:
8614:
8602:
8582:
8576:
8529:
8523:
8493:
8481:
8413:For a given orthonormal basis
8272:
8266:
8253:
8247:
8203:
8197:
8184:
8178:
8139:
8132:
8125:
8065:and the coefficient functions
8025:
8013:
8007:
7995:
7975:
7963:
7943:
7931:
7914:{\displaystyle \beta (\cdot )}
7908:
7902:
7859:
7856:
7850:
7844:
7833:
7827:
7818:
7812:
7768:
7762:
7749:
7743:
7730:
7718:
7672:
7666:
7653:
7647:
7617:
7610:
7603:
7551:
7539:
7519:
7513:
7401:
7395:
7341:
7335:
7302:
7296:
7273:
7267:
7185:
7173:
7088:
7076:
7053:
7047:
7038:
7032:
7019:
7013:
6976:
6970:
6954:
6948:
6909:
6897:
6874:
6868:
6859:
6853:
6806:
6800:
6784:
6778:
6749:
6743:
6678:
6672:
6639:
6633:
6590:
6578:
6552:
6546:
6523:
6511:
6481:
6469:
6449:
6443:
6427:
6421:
6405:
6399:
6332:{\displaystyle \alpha _{0}(s)}
6326:
6320:
6274:
6262:
6239:
6233:
6217:
6211:
6193:
6181:
6129:
6123:
6107:
6101:
6035:
6023:
5997:
5991:
5961:
5949:
5929:
5923:
5783:
5777:
5761:
5755:
5739:
5733:
5666:
5654:
5600:is regression coefficient for
5520:
5514:
5501:
5495:
5362:
5330:
5215:
5209:
5203:
5197:
5052:
5040:
5014:
5008:
4979:
4973:
4964:
4958:
4949:
4943:
4463:
4457:
4407:
4401:
4392:
4386:
4381:
4375:
4340:
4296:
4193:
4167:
4158:
4154:
4148:
4098:
4092:
4083:
4077:
4064:
4054:
4042:
4001:
3995:
3982:
3979:
3973:
3964:
3958:
3945:
3894:
3888:
3838:
3832:
3823:
3817:
3783:
3777:
3720:
3714:
3701:
3695:
3648:
3636:
3613:
3601:
3584:{\displaystyle {\mathcal {C}}}
3537:
3525:
3512:
3509:
3497:
3437:
3431:
3306:
3290:
3223:
3086:
3070:
2956:
2950:
2835:
2819:
2786:
2770:
2751:{\displaystyle \epsilon _{ij}}
2685:
2669:
2623:
2572:
2450:
2444:
2414:
2411:
2405:
2396:
2390:
2384:
2371:
2359:
2336:
2333:
2327:
2321:
2310:
2304:
2281:
2269:
2222:
2210:
2195:
2189:
2126:
2123:
2111:
2056:
2043:
2035:
2029:
2016:
2010:
1963:
1951:
1938:
1926:
1718:{\displaystyle {\mathcal {C}}}
1684:
1658:
1644:{\displaystyle {\mathcal {C}}}
1582:
1576:
1570:
1558:
1534:
1528:
1525:
1512:
1484:
1464:defines a covariance operator
1388:
1376:
1344:
1335:
1328:
1322:
1294:
1282:
1263:
1260:
1254:
1245:
1239:
1233:
1220:
1208:
1198:
1192:
1178:
1172:
1152:{\displaystyle {\mathcal {T}}}
1117:
1105:
1091:
1085:
1032:
1020:
944:{\displaystyle {\mathcal {C}}}
916:{\displaystyle {\mathcal {C}}}
737:
734:
722:
716:
704:
701:
643:
640:
628:
604:
560:
283:
267:
262:
256:
249:
230:
169:
157:
1:
12685:10.1080/01621459.2016.1273115
12621:10.1080/01621459.2021.1917416
12254:Delaigle, A; Hall, P (2012).
12114:10.1016/S0167-9473(03)00032-X
12002:10.1093/bioinformatics/bti742
11980:Leng, X; Müller, HG. (2006).
11813:Journal of Applied Statistics
11084:10.1016/S0047-259X(02)00056-8
10593:
10414:
10188:
9690:{\displaystyle \mu =g(\eta )}
9273:which is required to satisfy
7483:model. For a scalar response
7166:are functional covariates on
6755:{\displaystyle X_{j}(\cdot )}
6722:Function-on-scalar regression
6370:{\displaystyle j=1,\ldots ,p}
6079:{\displaystyle j=1,\ldots ,p}
5886:{\displaystyle j=1,\ldots ,p}
3626:, has spectral decomposition
2456:{\displaystyle X_{i}(\cdot )}
2166:Kolmogorov continuity theorem
302:, one can define the mean of
12497:Kneip, A; Gasser, T (1992).
11938:Heinzl, F; Tutz, G. (2013).
11833:10.1080/02664763.2011.578620
11693:10.1016/j.neucom.2012.11.042
10815:Hsing, T; Eubank, R (2015).
10777:10.1016/0047-259X(82)90088-4
10589:Category:Regression analysis
9708:k-means partitioning methods
6929:Concurrent regression models
4857:{\displaystyle \varepsilon }
4492:Other popular bases include
3753:{\displaystyle \lambda _{k}}
2228:{\displaystyle X(t),\ t\in }
1888:{\displaystyle \varphi _{j}}
888:{\displaystyle \varphi _{i}}
776:Karhunen-Loève decomposition
38:Karhunen-Loève decomposition
7:
12371:Ramsay, JO; Li, X. (1998).
11561:Chiou, JM; Li, PL. (2007).
10467:
10249:Multidimensional domain of
10128:Earlier approaches include
9382:
8632:
7503:and a functional covariate
7110:
6764:
6714:
6293:
5901:
5895:
5555:
5262:
5250:
4922:
4900:and the coefficient vector
4724:
4482:
3552:
1613:
139:square-integrable functions
73:Hilbertian random variables
10:
12888:
12476:10.1109/TASSP.1978.1163055
12305:"Functional Data Analysis"
11755:10.1016/j.csda.2013.04.001
11728:10.1016/j.csda.2012.12.004
11534:10.1214/009053604000001156
11351:10.1016/j.jspi.2009.06.017
11180:Fan, J; Zhang, W. (1999).
11114:10.1198/016214504000001745
10945:"Functional data analysis"
10123:
9753:Berkeley Growth Study Data
5816:{\displaystyle \beta _{j}}
4517:multivariate linear models
3244:Berkeley Growth Study Data
27:
12145:10.1007/s00180-014-0503-4
12079:10.1198/00401700152404273
11316:10.1198/jasa.2010.tm09228
10745:10.1080/02331887308801137
10303:{\displaystyle X(\cdot )}
10271:{\displaystyle X(\cdot )}
9642:and the linear predictor
9390:(GFLM) in analogy to the
8105:in an orthonormal basis.
7892:is a scalar coefficient,
7525:{\displaystyle X(\cdot )}
7375:only and not the history
5848:{\displaystyle X_{j}^{c}}
5704:{\displaystyle X_{j}^{c}}
4992:and coefficient function
3567:, the kernel function of
341:{\displaystyle \mu \in H}
40:. A rigorous analysis of
12772:The Annals of Statistics
12354:The Annals of Statistics
12133:Computational Statistics
11959:10.1177/1471082X12471372
11512:The Annals of Statistics
11436:The Annals of Statistics
11186:The Annals of Statistics
10529:Functional data analysis
9392:generalized linear model
7457:generalized linear model
6003:{\displaystyle X_{j}(t)}
4513:Functional linear models
3443:{\displaystyle X_{i}(t)}
2349:and covariance function
425:This formulation is the
45:Analysis" was coined by
17:Functional data analysis
12390:10.1111/1467-9868.00129
12043:10.1111/1467-9868.00297
11651:10.1198/016214503000189
11453:he Annals of Statistics
11270:10.1093/biomet/89.1.111
10545:, New York: Springer,
9712:hierarchical clustering
8377:curse of dimensionality
8098:{\displaystyle \gamma }
8031:{\displaystyle \times }
7885:{\displaystyle \alpha }
7461:curse of dimensionality
6933:This model is given by,
5401:{\displaystyle Z_{1}=1}
2172:Functional data designs
1457:{\displaystyle \Sigma }
1437:{\displaystyle \Sigma }
97:-valued random element
67:mean-squared continuity
12867:Statistical data types
12516:10.1214/aos/1176348769
11782:10.1002/bimj.201200111
11199:10.1214/aos/1017939139
10607:Grenander, U. (1950).
10479:Karhunen–Loève theorem
10386:
10359:
10331:
10304:
10272:
10239:
10161:
10105:
10058:
10002:
9982:
9934:
9886:
9780:
9747:
9691:
9656:
9636:
9616:
9587:
9542:
9483:
9366:
9328:
9267:
9213:
9106:
9075:
9048:
9020:
8960:
8933:
8899:
8824:
8797:
8760:
8733:
8705:
8621:
8589:
8555:
8500:
8458:
8399:
8398:{\displaystyle p>1}
8369:
8368:{\displaystyle p>1}
8343:
8342:{\displaystyle p>1}
8317:
8291:
8099:
8079:
8078:{\displaystyle \beta }
8059:
8032:
7982:
7950:
7915:
7886:
7866:
7789:
7585:
7558:
7526:
7497:
7441:
7369:
7309:
7280:
7251:
7192:
7160:
7098:
7002:
6919:
6832:
6756:
6726:In particular, taking
6706:
6646:
6617:
6597:
6559:
6530:
6488:
6456:
6371:
6333:
6281:
6155:
6080:
6042:
6004:
5968:
5936:
5887:
5849:
5817:
5790:
5705:
5673:
5641:
5614:
5594:
5543:
5464:
5402:
5369:
5311:
5238:
5090:
5059:
5021:
4986:
4914:
4913:{\displaystyle \beta }
4894:
4858:
4838:
4803:
4760:
4712:
4576:
4541:
4470:
4433:
4347:
4270:
4250:
4223:
4203:
4177:
4124:
4014:
3901:
3864:
3798:Karhunen–Loève theorem
3790:
3754:
3727:
3674:
3620:
3585:
3544:
3464:
3444:
3386:
3359:
3358:{\displaystyle T_{ij}}
3329:
3233:
3200:
3139:
3138:{\displaystyle T_{ij}}
3109:
3018:
2963:
2903:
2883:
2863:
2799:
2752:
2722:
2721:{\displaystyle X_{ij}}
2692:
2630:
2542:
2477:
2457:
2421:
2343:
2288:
2256:
2229:
2158:
2138:
2092:
2069:
1989:
1889:
1862:
1861:{\displaystyle f\in H}
1842:is continuous for all
1836:
1806:
1765:
1719:
1691:
1651:, yielding eigenpairs
1645:
1601:
1494:
1458:
1438:
1418:
1395:
1357:
1301:
1153:
1126:
1063:
1039:
997:
977:
945:
917:
889:
859:
819:
768:
744:
666:
570:
531:
507:
457:
416:
342:
322:as the unique element
316:
296:
176:
131:
111:
91:
53:Mathematical formalism
12563:10.1093/biomet/asn047
11947:Statistical Modelling
11872:10.1093/biomet/asn054
11483:10.1093/biomet/ast004
10908:10.1093/biomet/asv062
10489:Functional regression
10387:
10385:{\displaystyle R^{p}}
10360:
10332:
10330:{\displaystyle R^{p}}
10305:
10273:
10240:
10180:Landmark registration
10162:
10106:
10059:
10003:
9983:
9935:
9887:
9781:
9779:{\displaystyle Y_{i}}
9745:
9692:
9657:
9655:{\displaystyle \eta }
9637:
9617:
9588:
9543:
9484:
9367:
9365:{\displaystyle t\in }
9329:
9268:
9214:
9107:
9076:
9074:{\displaystyle f_{k}}
9049:
9021:
8961:
8959:{\displaystyle f_{k}}
8934:
8879:
8825:
8823:{\displaystyle f_{k}}
8798:
8761:
8759:{\displaystyle x_{k}}
8734:
8685:
8622:
8590:
8535:
8501:
8499:{\displaystyle L^{2}}
8459:
8400:
8370:
8344:
8318:
8292:
8100:
8080:
8060:
8058:{\displaystyle X^{c}}
8033:
7983:
7951:
7916:
7887:
7867:
7790:
7586:
7584:{\displaystyle X^{c}}
7559:
7527:
7498:
7481:polynomial regression
7442:
7370:
7310:
7281:
7252:
7193:
7161:
7099:
6982:
6920:
6812:
6757:
6707:
6647:
6618:
6598:
6596:{\displaystyle s\in }
6560:
6531:
6489:
6457:
6372:
6334:
6282:
6135:
6081:
6043:
6041:{\displaystyle t\in }
6005:
5969:
5937:
5888:
5850:
5818:
5791:
5706:
5674:
5642:
5640:{\displaystyle X_{j}}
5615:
5595:
5544:
5444:
5403:
5370:
5312:
5239:
5091:
5089:{\displaystyle L^{2}}
5060:
5058:{\displaystyle t\in }
5022:
4987:
4915:
4895:
4859:
4839:
4804:
4761:
4713:
4577:
4547:and vector covariate
4542:
4471:
4413:
4348:
4271:
4251:
4249:{\displaystyle X_{i}}
4224:
4204:
4178:
4104:
4015:
3902:
3844:
3791:
3755:
3728:
3654:
3621:
3586:
3545:
3465:
3445:
3387:
3385:{\displaystyle N_{i}}
3360:
3330:
3234:
3201:
3140:
3110:
3019:
2964:
2904:
2884:
2864:
2800:
2753:
2723:
2693:
2631:
2543:
2478:
2458:
2422:
2344:
2289:
2257:
2255:{\displaystyle L^{2}}
2230:
2159:
2139:
2093:
2070:
1969:
1890:
1863:
1837:
1807:
1745:
1720:
1692:
1646:
1602:
1495:
1459:
1439:
1419:
1396:
1394:{\displaystyle t\in }
1358:
1302:
1154:
1127:
1064:
1040:
1038:{\displaystyle L^{2}}
998:
978:
946:
918:
890:
860:
799:
769:
745:
667:
571:
532:
508:
458:
417:
343:
317:
297:
177:
175:{\displaystyle L^{2}}
132:
112:
92:
19:(FDA) is a branch of
12872:Statistical analysis
12503:Annals of Statistics
12373:"Curve registration"
10851:Annals of Statistics
10716:Karhunen, K (1946).
10689:Annals of Statistics
10683:Müller, HG. (2016).
10499:Stochastic processes
10369:
10349:
10314:
10285:
10253:
10197:
10171:Dynamic time warping
10140:
10130:dynamic time warping
10071:
10012:
9992:
9944:
9896:
9790:
9763:
9666:
9646:
9635:{\displaystyle \mu }
9626:
9606:
9552:
9498:
9404:
9338:
9277:
9223:
9116:
9085:
9058:
9030:
8970:
8943:
8834:
8807:
8770:
8743:
8640:
8599:
8510:
8468:
8417:
8383:
8353:
8327:
8301:
8117:
8089:
8069:
8042:
7992:
7960:
7925:
7896:
7876:
7799:
7595:
7568:
7536:
7507:
7487:
7379:
7319:
7308:{\displaystyle Y(s)}
7290:
7261:
7202:
7170:
7124:
6942:
6772:
6730:
6656:
6645:{\displaystyle Y(s)}
6627:
6607:
6569:
6540:
6498:
6466:
6381:
6343:
6307:
6095:
6052:
6014:
5978:
5946:
5935:{\displaystyle Y(s)}
5917:
5859:
5827:
5800:
5715:
5683:
5651:
5624:
5604:
5569:
5417:
5379:
5321:
5270:
5106:
5073:
5031:
4996:
4930:
4904:
4884:
4848:
4813:
4778:
4738:
4591:
4551:
4523:
4362:
4280:
4276:-dimensional vector
4260:
4233:
4213:
4187:
4028:
3911:
3804:
3764:
3737:
3630:
3595:
3571:
3474:
3454:
3418:
3369:
3339:
3261:
3210:
3151:
3119:
3041:
2973:
2921:
2893:
2873:
2809:
2762:
2732:
2702:
2640:
2552:
2487:
2467:
2431:
2353:
2298:
2266:
2239:
2183:
2148:
2102:
2082:
1906:
1872:
1846:
1819:
1732:
1705:
1655:
1631:
1509:
1468:
1448:
1428:
1417:{\displaystyle \mu }
1408:
1367:
1314:
1166:
1139:
1076:
1053:
1007:
987:
967:
959:Stochastic processes
931:
903:
872:
784:
758:
754:allows to decompose
683:
583:
544:
521:
467:
436:
355:
326:
306:
186:
144:
121:
101:
81:
12419:Statistical Science
12321:2016AnRSA...3..257W
11825:2012JApSt..39..129A
11770:Biometrical Journal
10961:2016AnRSA...3..257W
10625:1950ArM.....1..195G
10613:Arkiv för Matematik
9845:
9440:
9175:
8453:
8316:{\displaystyle p=1}
8236:
8167:
7714:
7699:
7643:
7436:
7364:
6701:
6398:
6210:
6170:
5844:
5732:
5700:
5494:
5479:
5306:
5186:
4582:can be expressed as
4385:
4022:dimension reduction
3944:
3412:dimension reduction
3408:dimension reduction
3242:Real life example:
2858:
2294:with mean function
1554:
515:covariance operator
502:
247:
221:
12829:10.1111/rssb.12160
12794:10.1214/17-AOS1660
12749:10.1111/biom.13340
12605:(540): 2252–2267.
11310:(491): 1256–1264.
10873:10.1214/13-AOS1093
10702:10.1214/16-AOS1492
10634:10.1007/BF02590638
10484:Modes of variation
10382:
10355:
10327:
10300:
10268:
10235:
10157:
10134:speech recognition
10101:
10054:
9998:
9978:
9930:
9882:
9828:
9786:can be modeled as
9776:
9748:
9687:
9652:
9632:
9612:
9583:
9538:
9479:
9426:
9362:
9324:
9263:
9209:
9161:
9102:
9071:
9044:
9016:
8956:
8929:
8820:
8793:
8756:
8729:
8617:
8585:
8496:
8454:
8433:
8395:
8365:
8339:
8313:
8287:
8222:
8153:
8095:
8075:
8055:
8028:
7978:
7946:
7911:
7882:
7862:
7785:
7700:
7685:
7629:
7581:
7554:
7522:
7493:
7437:
7416:
7365:
7344:
7305:
7276:
7247:
7188:
7156:
7094:
6915:
6752:
6702:
6681:
6642:
6613:
6593:
6555:
6526:
6484:
6452:
6384:
6367:
6329:
6277:
6196:
6156:
6076:
6038:
6000:
5964:
5932:
5883:
5845:
5830:
5813:
5786:
5718:
5701:
5686:
5669:
5637:
5610:
5590:
5539:
5480:
5465:
5398:
5365:
5307:
5286:
5234:
5172:
5086:
5055:
5017:
4982:
4910:
4890:
4854:
4834:
4799:
4756:
4708:
4572:
4537:
4502:modes of variation
4466:
4365:
4343:
4266:
4246:
4219:
4199:
4173:
4058:
4010:
3930:
3897:
3786:
3750:
3723:
3616:
3581:
3540:
3460:
3440:
3382:
3355:
3325:
3229:
3196:
3135:
3105:
3014:
2969:available for all
2959:
2899:
2879:
2859:
2841:
2795:
2748:
2718:
2688:
2626:
2538:
2473:
2453:
2417:
2339:
2284:
2252:
2225:
2178:stochastic process
2154:
2134:
2088:
2065:
1942:
1885:
1858:
1832:
1802:
1715:
1687:
1641:
1597:
1540:
1490:
1454:
1434:
1414:
1391:
1353:
1297:
1149:
1122:
1059:
1035:
993:
973:
941:
913:
885:
855:
764:
740:
662:
566:
527:
503:
481:
453:
412:
338:
312:
292:
233:
200:
172:
127:
107:
87:
63:stochastic process
34:stochastic process
12778:(6B): 3334–3361.
12708:Statistica Sinica
12538:(432): 1179–1188.
12441:10.1214/15-STS524
11413:10.3150/12-BEJ446
11396:(5A): 2120–2151.
11285:Statistica Sinica
11159:10.3150/09-BEJ228
10565:978-0-470-01691-6
10551:978-1-4614-3654-6
10509:Variance function
10358:{\displaystyle R}
10001:{\displaystyle h}
9973:
9925:
9615:{\displaystyle g}
9504:
9493:Variance function
7496:{\displaystyle Y}
7118:
7117:
7069:
7065:
7061:
6890:
6886:
6882:
6616:{\displaystyle Y}
6301:
6300:
6255:
6251:
6247:
5613:{\displaystyle Z}
5563:
5562:
5258:
5257:
4893:{\displaystyle X}
4732:
4731:
4490:
4489:
4269:{\displaystyle K}
4222:{\displaystyle K}
4031:
3463:{\displaystyle X}
2902:{\displaystyle j}
2882:{\displaystyle i}
2816:
2560:
2476:{\displaystyle n}
2381:
2203:
2157:{\displaystyle X}
2091:{\displaystyle X}
1909:
1899:then states that
1895:are continuous.
1621:
1620:
1230:
1062:{\displaystyle X}
996:{\displaystyle H}
976:{\displaystyle X}
767:{\displaystyle X}
530:{\displaystyle X}
315:{\displaystyle X}
130:{\displaystyle H}
110:{\displaystyle X}
90:{\displaystyle H}
12879:
12851:
12850:
12840:
12812:
12806:
12805:
12787:
12767:
12761:
12760:
12742:
12722:
12716:
12715:
12703:
12697:
12696:
12678:
12669:(522): 649–659.
12658:
12652:
12651:
12639:
12633:
12632:
12614:
12594:
12585:
12584:
12576:
12567:
12566:
12546:
12540:
12539:
12527:
12521:
12520:
12518:
12509:(3): 1266–1305.
12494:
12488:
12487:
12459:
12453:
12452:
12434:
12414:
12403:
12402:
12392:
12368:
12362:
12361:
12349:
12343:
12342:
12332:
12300:
12294:
12293:
12275:
12251:
12245:
12244:
12242:
12222:
12216:
12215:
12205:
12188:(4): 1260–1268.
12173:
12167:
12166:
12156:
12139:(6): 1497–1513.
12124:
12118:
12117:
12108:(1–2): 161–173.
12097:
12091:
12090:
12062:
12056:
12055:
12045:
12021:
12015:
12014:
12004:
11986:
11977:
11971:
11970:
11944:
11935:
11929:
11928:
11900:
11894:
11893:
11883:
11851:
11845:
11844:
11808:
11802:
11801:
11765:
11759:
11758:
11738:
11732:
11731:
11711:
11705:
11704:
11678:
11669:
11663:
11662:
11645:(462): 397–408.
11634:
11628:
11627:
11599:
11593:
11592:
11582:
11558:
11552:
11551:
11545:
11537:
11527:
11507:
11501:
11500:
11494:
11486:
11466:
11460:
11449:
11443:
11432:
11426:
11425:
11415:
11405:
11381:
11375:
11364:
11355:
11354:
11334:
11328:
11327:
11299:
11293:
11292:
11280:
11274:
11273:
11253:
11247:
11246:
11218:
11212:
11211:
11201:
11192:(5): 1491–1518.
11177:
11171:
11170:
11152:
11132:
11126:
11125:
11108:(470): 577–590.
11097:
11088:
11087:
11067:
11061:
11060:
11024:
11018:
11017:
10989:
10983:
10982:
10972:
10940:
10927:
10926:
10918:
10912:
10911:
10891:
10885:
10884:
10866:
10846:
10840:
10839:
10827:
10821:
10820:
10812:
10806:
10805:
10797:
10782:
10781:
10779:
10755:
10749:
10748:
10728:
10722:
10721:
10713:
10707:
10706:
10704:
10695:(5): 1867–1887.
10680:
10674:
10673:
10653:
10647:
10646:
10636:
10604:
10391:
10389:
10388:
10383:
10381:
10380:
10364:
10362:
10361:
10356:
10336:
10334:
10333:
10328:
10326:
10325:
10309:
10307:
10306:
10301:
10277:
10275:
10274:
10269:
10244:
10242:
10241:
10236:
10234:
10233:
10232:
10231:
10166:
10164:
10163:
10158:
10156:
10155:
10150:
10149:
10110:
10108:
10107:
10102:
10063:
10061:
10060:
10055:
10035:
10034:
10019:
10007:
10005:
10004:
9999:
9987:
9985:
9984:
9979:
9974:
9972:
9958:
9956:
9955:
9939:
9937:
9936:
9931:
9926:
9924:
9910:
9908:
9907:
9891:
9889:
9888:
9883:
9844:
9836:
9824:
9823:
9802:
9801:
9785:
9783:
9782:
9777:
9775:
9774:
9696:
9694:
9693:
9688:
9661:
9659:
9658:
9653:
9641:
9639:
9638:
9633:
9621:
9619:
9618:
9613:
9595:conditional mean
9592:
9590:
9589:
9584:
9576:
9565:
9547:
9545:
9544:
9539:
9516:
9505:
9502:
9488:
9486:
9485:
9480:
9450:
9449:
9439:
9434:
9422:
9421:
9399:Linear predictor
9371:
9369:
9368:
9363:
9333:
9331:
9330:
9325:
9284:
9272:
9270:
9269:
9264:
9262:
9254:
9218:
9216:
9215:
9210:
9174:
9169:
9148:
9134:
9123:
9111:
9109:
9108:
9103:
9092:
9080:
9078:
9077:
9072:
9070:
9069:
9053:
9051:
9050:
9045:
9043:
9025:
9023:
9022:
9017:
9003:
9002:
8990:
8989:
8977:
8965:
8963:
8962:
8957:
8955:
8954:
8938:
8936:
8935:
8930:
8922:
8921:
8909:
8908:
8898:
8893:
8866:
8852:
8841:
8829:
8827:
8826:
8821:
8819:
8818:
8802:
8800:
8799:
8794:
8792:
8791:
8782:
8781:
8765:
8763:
8762:
8757:
8755:
8754:
8738:
8736:
8735:
8730:
8725:
8724:
8715:
8714:
8704:
8699:
8672:
8658:
8647:
8626:
8624:
8623:
8620:{\displaystyle }
8618:
8594:
8592:
8591:
8586:
8575:
8574:
8565:
8564:
8554:
8549:
8522:
8521:
8506:, we can expand
8505:
8503:
8502:
8497:
8480:
8479:
8463:
8461:
8460:
8455:
8452:
8447:
8432:
8431:
8404:
8402:
8401:
8396:
8374:
8372:
8371:
8366:
8348:
8346:
8345:
8340:
8322:
8320:
8319:
8314:
8296:
8294:
8293:
8288:
8286:
8282:
8265:
8264:
8246:
8245:
8235:
8230:
8196:
8195:
8177:
8176:
8166:
8161:
8135:
8124:
8104:
8102:
8101:
8096:
8084:
8082:
8081:
8076:
8064:
8062:
8061:
8056:
8054:
8053:
8037:
8035:
8034:
8029:
7987:
7985:
7984:
7981:{\displaystyle }
7979:
7955:
7953:
7952:
7947:
7920:
7918:
7917:
7912:
7891:
7889:
7888:
7883:
7871:
7869:
7868:
7863:
7843:
7811:
7810:
7794:
7792:
7791:
7786:
7761:
7760:
7742:
7741:
7713:
7708:
7698:
7693:
7665:
7664:
7642:
7637:
7613:
7602:
7590:
7588:
7587:
7582:
7580:
7579:
7563:
7561:
7560:
7557:{\displaystyle }
7555:
7531:
7529:
7528:
7523:
7502:
7500:
7499:
7494:
7446:
7444:
7443:
7438:
7435:
7430:
7394:
7393:
7374:
7372:
7371:
7366:
7363:
7358:
7334:
7333:
7314:
7312:
7311:
7306:
7285:
7283:
7282:
7277:
7256:
7254:
7253:
7248:
7246:
7245:
7227:
7226:
7214:
7213:
7197:
7195:
7194:
7191:{\displaystyle }
7189:
7165:
7163:
7162:
7157:
7155:
7154:
7136:
7135:
7112:
7103:
7101:
7100:
7095:
7067:
7066:
7063:
7059:
7031:
7030:
7012:
7011:
7001:
6996:
6969:
6968:
6936:
6924:
6922:
6921:
6916:
6888:
6887:
6884:
6880:
6852:
6851:
6842:
6841:
6831:
6826:
6799:
6798:
6761:
6759:
6758:
6753:
6742:
6741:
6711:
6709:
6708:
6703:
6700:
6695:
6671:
6670:
6651:
6649:
6648:
6643:
6622:
6620:
6619:
6614:
6602:
6600:
6599:
6594:
6564:
6562:
6561:
6556:
6535:
6533:
6532:
6527:
6510:
6509:
6493:
6491:
6490:
6487:{\displaystyle }
6485:
6461:
6459:
6458:
6453:
6442:
6441:
6420:
6419:
6397:
6392:
6376:
6374:
6373:
6368:
6338:
6336:
6335:
6330:
6319:
6318:
6295:
6286:
6284:
6283:
6278:
6253:
6252:
6249:
6245:
6209:
6204:
6180:
6179:
6169:
6164:
6154:
6149:
6122:
6121:
6089:
6085:
6083:
6082:
6077:
6047:
6045:
6044:
6039:
6009:
6007:
6006:
6001:
5990:
5989:
5973:
5971:
5970:
5967:{\displaystyle }
5965:
5941:
5939:
5938:
5933:
5892:
5890:
5889:
5884:
5854:
5852:
5851:
5846:
5843:
5838:
5822:
5820:
5819:
5814:
5812:
5811:
5795:
5793:
5792:
5787:
5776:
5775:
5754:
5753:
5731:
5726:
5710:
5708:
5707:
5702:
5699:
5694:
5678:
5676:
5675:
5672:{\displaystyle }
5670:
5646:
5644:
5643:
5638:
5636:
5635:
5620:, the domain of
5619:
5617:
5616:
5611:
5599:
5597:
5596:
5591:
5589:
5588:
5587:
5557:
5548:
5546:
5545:
5540:
5513:
5512:
5493:
5488:
5478:
5473:
5463:
5458:
5411:
5407:
5405:
5404:
5399:
5391:
5390:
5374:
5372:
5371:
5366:
5361:
5360:
5342:
5341:
5316:
5314:
5313:
5308:
5305:
5300:
5285:
5284:
5252:
5243:
5241:
5240:
5235:
5196:
5195:
5185:
5180:
5168:
5167:
5140:
5139:
5124:
5123:
5100:
5095:
5093:
5092:
5087:
5085:
5084:
5064:
5062:
5061:
5056:
5026:
5024:
5023:
5018:
4991:
4989:
4988:
4983:
4942:
4941:
4919:
4917:
4916:
4911:
4899:
4897:
4896:
4891:
4863:
4861:
4860:
4855:
4843:
4841:
4840:
4835:
4833:
4832:
4827:
4808:
4806:
4805:
4800:
4798:
4790:
4789:
4765:
4763:
4762:
4757:
4726:
4717:
4715:
4714:
4709:
4698:
4697:
4688:
4687:
4669:
4668:
4659:
4658:
4646:
4645:
4609:
4608:
4585:
4581:
4579:
4578:
4573:
4571:
4570:
4565:
4546:
4544:
4543:
4538:
4536:
4484:
4475:
4473:
4472:
4467:
4456:
4455:
4446:
4445:
4432:
4427:
4384:
4373:
4356:
4352:
4350:
4349:
4344:
4339:
4338:
4311:
4310:
4292:
4291:
4275:
4273:
4272:
4267:
4255:
4253:
4252:
4247:
4245:
4244:
4228:
4226:
4225:
4220:
4208:
4206:
4205:
4200:
4182:
4180:
4179:
4174:
4166:
4165:
4147:
4146:
4137:
4136:
4123:
4118:
4076:
4075:
4063:
4057:
4019:
4017:
4016:
4011:
3994:
3993:
3957:
3956:
3943:
3938:
3926:
3925:
3906:
3904:
3903:
3898:
3887:
3886:
3877:
3876:
3863:
3858:
3816:
3815:
3795:
3793:
3792:
3787:
3776:
3775:
3759:
3757:
3756:
3751:
3749:
3748:
3732:
3730:
3729:
3724:
3713:
3712:
3694:
3693:
3684:
3683:
3673:
3668:
3625:
3623:
3622:
3617:
3590:
3588:
3587:
3582:
3580:
3579:
3565:Mercer's theorem
3549:
3547:
3546:
3541:
3524:
3523:
3496:
3495:
3483:
3482:
3469:
3467:
3466:
3461:
3449:
3447:
3446:
3441:
3430:
3429:
3391:
3389:
3388:
3383:
3381:
3380:
3364:
3362:
3361:
3356:
3354:
3353:
3334:
3332:
3331:
3326:
3324:
3323:
3305:
3304:
3289:
3288:
3276:
3275:
3238:
3236:
3235:
3230:
3222:
3221:
3205:
3203:
3202:
3197:
3195:
3194:
3193:
3192:
3166:
3165:
3144:
3142:
3141:
3136:
3134:
3133:
3114:
3112:
3111:
3106:
3104:
3103:
3085:
3084:
3069:
3068:
3056:
3055:
3023:
3021:
3020:
3015:
2988:
2987:
2968:
2966:
2965:
2960:
2949:
2948:
2936:
2935:
2908:
2906:
2905:
2900:
2888:
2886:
2885:
2880:
2868:
2866:
2865:
2860:
2857:
2852:
2834:
2833:
2818:
2817:
2814:
2804:
2802:
2801:
2796:
2785:
2784:
2769:
2757:
2755:
2754:
2749:
2747:
2746:
2727:
2725:
2724:
2719:
2717:
2716:
2697:
2695:
2694:
2689:
2684:
2683:
2668:
2667:
2655:
2654:
2635:
2633:
2632:
2627:
2622:
2621:
2620:
2619:
2587:
2586:
2568:
2567:
2562:
2561:
2547:
2545:
2544:
2539:
2537:
2536:
2535:
2534:
2502:
2501:
2482:
2480:
2479:
2474:
2462:
2460:
2459:
2454:
2443:
2442:
2426:
2424:
2423:
2418:
2383:
2382:
2379:
2348:
2346:
2345:
2340:
2320:
2293:
2291:
2290:
2287:{\displaystyle }
2285:
2261:
2259:
2258:
2253:
2251:
2250:
2234:
2232:
2231:
2226:
2201:
2163:
2161:
2160:
2155:
2143:
2141:
2140:
2135:
2133:
2097:
2095:
2094:
2089:
2074:
2072:
2071:
2066:
2042:
2038:
2028:
2027:
2009:
2008:
1999:
1998:
1988:
1983:
1941:
1897:Mercer's theorem
1894:
1892:
1891:
1886:
1884:
1883:
1867:
1865:
1864:
1859:
1841:
1839:
1838:
1833:
1828:
1827:
1815:Moreover, since
1811:
1809:
1808:
1803:
1798:
1797:
1785:
1784:
1775:
1774:
1764:
1759:
1741:
1740:
1724:
1722:
1721:
1716:
1714:
1713:
1696:
1694:
1693:
1688:
1683:
1682:
1670:
1669:
1650:
1648:
1647:
1642:
1640:
1639:
1625:spectral theorem
1615:
1606:
1604:
1603:
1598:
1590:
1553:
1548:
1521:
1520:
1503:
1499:
1497:
1496:
1491:
1477:
1476:
1463:
1461:
1460:
1455:
1443:
1441:
1440:
1435:
1423:
1421:
1420:
1415:
1400:
1398:
1397:
1392:
1362:
1360:
1359:
1354:
1343:
1342:
1321:
1306:
1304:
1303:
1298:
1232:
1231:
1228:
1188:
1158:
1156:
1155:
1150:
1148:
1147:
1131:
1129:
1128:
1123:
1121:
1120:
1068:
1066:
1065:
1060:
1044:
1042:
1041:
1036:
1019:
1018:
1002:
1000:
999:
994:
982:
980:
979:
974:
950:
948:
947:
942:
940:
939:
922:
920:
919:
914:
912:
911:
894:
892:
891:
886:
884:
883:
864:
862:
861:
856:
851:
850:
838:
837:
818:
813:
773:
771:
770:
765:
752:spectral theorem
749:
747:
746:
741:
700:
692:
691:
671:
669:
668:
663:
603:
592:
591:
575:
573:
572:
567:
553:
552:
536:
534:
533:
528:
512:
510:
509:
504:
501:
496:
495:
494:
474:
462:
460:
459:
454:
449:
431:Bochner integral
421:
419:
418:
413:
362:
347:
345:
344:
339:
321:
319:
318:
313:
301:
299:
298:
293:
276:
275:
270:
252:
246:
241:
229:
220:
215:
214:
213:
193:
181:
179:
178:
173:
156:
155:
136:
134:
133:
128:
116:
114:
113:
108:
96:
94:
93:
88:
12887:
12886:
12882:
12881:
12880:
12878:
12877:
12876:
12857:
12856:
12855:
12854:
12813:
12809:
12768:
12764:
12723:
12719:
12704:
12700:
12659:
12655:
12650:(5): 1130–1145.
12640:
12636:
12595:
12588:
12577:
12570:
12547:
12543:
12528:
12524:
12495:
12491:
12460:
12456:
12415:
12406:
12369:
12365:
12350:
12346:
12301:
12297:
12252:
12248:
12223:
12219:
12174:
12170:
12125:
12121:
12098:
12094:
12063:
12059:
12022:
12018:
11984:
11978:
11974:
11942:
11936:
11932:
11901:
11897:
11852:
11848:
11809:
11805:
11766:
11762:
11739:
11735:
11712:
11708:
11676:
11670:
11666:
11635:
11631:
11616:10.2307/2532201
11600:
11596:
11559:
11555:
11539:
11538:
11508:
11504:
11488:
11487:
11467:
11463:
11450:
11446:
11433:
11429:
11382:
11378:
11365:
11358:
11335:
11331:
11300:
11296:
11281:
11277:
11254:
11250:
11219:
11215:
11178:
11174:
11133:
11129:
11098:
11091:
11068:
11064:
11041:10.2307/3316063
11025:
11021:
10990:
10986:
10941:
10930:
10919:
10915:
10892:
10888:
10847:
10843:
10828:
10824:
10813:
10809:
10798:
10785:
10756:
10752:
10729:
10725:
10714:
10710:
10681:
10677:
10654:
10650:
10605:
10601:
10596:
10525:Silverman, B.W.
10517:
10515:Further reading
10470:
10420:
10401:
10376:
10372:
10370:
10367:
10366:
10350:
10347:
10346:
10343:
10321:
10317:
10315:
10312:
10311:
10286:
10283:
10282:
10279:
10254:
10251:
10250:
10227:
10226:
10219:
10215:
10198:
10195:
10194:
10191:
10182:
10173:
10151:
10145:
10144:
10143:
10141:
10138:
10137:
10126:
10118:diffeomorphisms
10113:diffeomorphisms
10072:
10069:
10068:
10027:
10023:
10015:
10013:
10010:
10009:
9993:
9990:
9989:
9962:
9957:
9951:
9947:
9945:
9942:
9941:
9914:
9909:
9903:
9899:
9897:
9894:
9893:
9837:
9832:
9819:
9815:
9797:
9793:
9791:
9788:
9787:
9770:
9766:
9764:
9761:
9760:
9740:
9735:
9704:
9667:
9664:
9663:
9647:
9644:
9643:
9627:
9624:
9623:
9607:
9604:
9603:
9572:
9561:
9553:
9550:
9549:
9512:
9501:
9499:
9496:
9495:
9445:
9441:
9435:
9430:
9417:
9413:
9405:
9402:
9401:
9378:
9339:
9336:
9335:
9280:
9278:
9275:
9274:
9258:
9250:
9224:
9221:
9220:
9170:
9165:
9144:
9130:
9119:
9117:
9114:
9113:
9088:
9086:
9083:
9082:
9065:
9061:
9059:
9056:
9055:
9039:
9031:
9028:
9027:
8998:
8994:
8985:
8981:
8973:
8971:
8968:
8967:
8950:
8946:
8944:
8941:
8940:
8917:
8913:
8904:
8900:
8894:
8883:
8862:
8848:
8837:
8835:
8832:
8831:
8814:
8810:
8808:
8805:
8804:
8787:
8783:
8777:
8773:
8771:
8768:
8767:
8750:
8746:
8744:
8741:
8740:
8720:
8716:
8710:
8706:
8700:
8689:
8668:
8654:
8643:
8641:
8638:
8637:
8600:
8597:
8596:
8570:
8566:
8560:
8556:
8550:
8539:
8517:
8513:
8511:
8508:
8507:
8475:
8471:
8469:
8466:
8465:
8448:
8437:
8427:
8423:
8418:
8415:
8414:
8411:
8384:
8381:
8380:
8354:
8351:
8350:
8349:. However, for
8328:
8325:
8324:
8302:
8299:
8298:
8260:
8256:
8241:
8237:
8231:
8226:
8191:
8187:
8172:
8168:
8162:
8157:
8152:
8148:
8131:
8120:
8118:
8115:
8114:
8111:
8090:
8087:
8086:
8070:
8067:
8066:
8049:
8045:
8043:
8040:
8039:
7993:
7990:
7989:
7961:
7958:
7957:
7926:
7923:
7922:
7897:
7894:
7893:
7877:
7874:
7873:
7839:
7806:
7802:
7800:
7797:
7796:
7756:
7752:
7737:
7733:
7709:
7704:
7694:
7689:
7660:
7656:
7638:
7633:
7609:
7598:
7596:
7593:
7592:
7575:
7571:
7569:
7566:
7565:
7537:
7534:
7533:
7508:
7505:
7504:
7488:
7485:
7484:
7477:
7469:additive models
7467:and functional
7453:
7431:
7420:
7389:
7385:
7380:
7377:
7376:
7359:
7348:
7329:
7325:
7320:
7317:
7316:
7291:
7288:
7287:
7262:
7259:
7258:
7241:
7237:
7222:
7218:
7209:
7205:
7203:
7200:
7199:
7171:
7168:
7167:
7150:
7146:
7131:
7127:
7125:
7122:
7121:
7062:
7026:
7022:
7007:
7003:
6997:
6986:
6964:
6960:
6943:
6940:
6939:
6931:
6883:
6847:
6843:
6837:
6833:
6827:
6816:
6794:
6790:
6773:
6770:
6769:
6737:
6733:
6731:
6728:
6727:
6724:
6696:
6685:
6666:
6662:
6657:
6654:
6653:
6628:
6625:
6624:
6608:
6605:
6604:
6603:, the value of
6570:
6567:
6566:
6541:
6538:
6537:
6505:
6501:
6499:
6496:
6495:
6467:
6464:
6463:
6437:
6433:
6415:
6411:
6393:
6388:
6382:
6379:
6378:
6344:
6341:
6340:
6314:
6310:
6308:
6305:
6304:
6248:
6205:
6200:
6175:
6171:
6165:
6160:
6150:
6139:
6117:
6113:
6096:
6093:
6092:
6053:
6050:
6049:
6015:
6012:
6011:
5985:
5981:
5979:
5976:
5975:
5947:
5944:
5943:
5918:
5915:
5914:
5911:
5860:
5857:
5856:
5839:
5834:
5828:
5825:
5824:
5807:
5803:
5801:
5798:
5797:
5771:
5767:
5749:
5745:
5727:
5722:
5716:
5713:
5712:
5695:
5690:
5684:
5681:
5680:
5652:
5649:
5648:
5631:
5627:
5625:
5622:
5621:
5605:
5602:
5601:
5583:
5579:
5578:
5570:
5567:
5566:
5508:
5504:
5489:
5484:
5474:
5469:
5459:
5448:
5418:
5415:
5414:
5386:
5382:
5380:
5377:
5376:
5356:
5352:
5337:
5333:
5322:
5319:
5318:
5301:
5290:
5280:
5276:
5271:
5268:
5267:
5191:
5187:
5181:
5176:
5163:
5159:
5135:
5131:
5119:
5115:
5107:
5104:
5103:
5080:
5076:
5074:
5071:
5070:
5032:
5029:
5028:
4997:
4994:
4993:
4937:
4933:
4931:
4928:
4927:
4905:
4902:
4901:
4885:
4882:
4881:
4878:
4849:
4846:
4845:
4828:
4823:
4822:
4814:
4811:
4810:
4794:
4785:
4781:
4779:
4776:
4775:
4772:Euclidean space
4739:
4736:
4735:
4693:
4689:
4683:
4679:
4664:
4660:
4654:
4650:
4641:
4637:
4604:
4600:
4592:
4589:
4588:
4566:
4561:
4560:
4552:
4549:
4548:
4532:
4524:
4521:
4520:
4510:
4451:
4447:
4438:
4434:
4428:
4417:
4374:
4369:
4363:
4360:
4359:
4331:
4327:
4303:
4299:
4287:
4283:
4281:
4278:
4277:
4261:
4258:
4257:
4240:
4236:
4234:
4231:
4230:
4214:
4211:
4210:
4188:
4185:
4184:
4161:
4157:
4142:
4138:
4129:
4125:
4119:
4108:
4071:
4067:
4059:
4035:
4029:
4026:
4025:
3989:
3985:
3952:
3948:
3939:
3934:
3918:
3914:
3912:
3909:
3908:
3882:
3878:
3869:
3865:
3859:
3848:
3811:
3807:
3805:
3802:
3801:
3771:
3767:
3765:
3762:
3761:
3744:
3740:
3738:
3735:
3734:
3708:
3704:
3689:
3685:
3679:
3675:
3669:
3658:
3631:
3628:
3627:
3596:
3593:
3592:
3575:
3574:
3572:
3569:
3568:
3519:
3515:
3491:
3487:
3478:
3477:
3475:
3472:
3471:
3455:
3452:
3451:
3425:
3421:
3419:
3416:
3415:
3401:
3376:
3372:
3370:
3367:
3366:
3346:
3342:
3340:
3337:
3336:
3316:
3312:
3297:
3293:
3284:
3280:
3268:
3264:
3262:
3259:
3258:
3255:
3217:
3213:
3211:
3208:
3207:
3188:
3184:
3180:
3176:
3158:
3154:
3152:
3149:
3148:
3126:
3122:
3120:
3117:
3116:
3096:
3092:
3077:
3073:
3064:
3060:
3048:
3044:
3042:
3039:
3038:
3035:
2983:
2982:
2974:
2971:
2970:
2944:
2940:
2928:
2924:
2922:
2919:
2918:
2915:
2894:
2891:
2890:
2874:
2871:
2870:
2853:
2845:
2826:
2822:
2813:
2812:
2810:
2807:
2806:
2777:
2773:
2765:
2763:
2760:
2759:
2739:
2735:
2733:
2730:
2729:
2709:
2705:
2703:
2700:
2699:
2676:
2672:
2663:
2659:
2647:
2643:
2641:
2638:
2637:
2615:
2611:
2607:
2603:
2579:
2575:
2563:
2557:
2556:
2555:
2553:
2550:
2549:
2530:
2526:
2522:
2518:
2494:
2490:
2488:
2485:
2484:
2468:
2465:
2464:
2438:
2434:
2432:
2429:
2428:
2378:
2377:
2354:
2351:
2350:
2316:
2299:
2296:
2295:
2267:
2264:
2263:
2246:
2242:
2240:
2237:
2236:
2184:
2181:
2180:
2174:
2149:
2146:
2145:
2129:
2103:
2100:
2099:
2083:
2080:
2079:
2023:
2019:
2004:
2000:
1994:
1990:
1984:
1973:
1947:
1943:
1913:
1907:
1904:
1903:
1879:
1875:
1873:
1870:
1869:
1847:
1844:
1843:
1823:
1822:
1820:
1817:
1816:
1793:
1789:
1780:
1776:
1770:
1766:
1760:
1749:
1736:
1735:
1733:
1730:
1729:
1709:
1708:
1706:
1703:
1702:
1678:
1674:
1665:
1661:
1656:
1653:
1652:
1635:
1634:
1632:
1629:
1628:
1586:
1549:
1544:
1516:
1515:
1510:
1507:
1506:
1472:
1471:
1469:
1466:
1465:
1449:
1446:
1445:
1429:
1426:
1425:
1409:
1406:
1405:
1368:
1365:
1364:
1338:
1334:
1317:
1315:
1312:
1311:
1227:
1226:
1184:
1167:
1164:
1163:
1143:
1142:
1140:
1137:
1136:
1098:
1094:
1077:
1074:
1073:
1054:
1051:
1050:
1014:
1010:
1008:
1005:
1004:
988:
985:
984:
968:
965:
964:
961:
935:
934:
932:
929:
928:
907:
906:
904:
901:
900:
879:
875:
873:
870:
869:
846:
842:
833:
829:
814:
803:
785:
782:
781:
759:
756:
755:
696:
687:
686:
684:
681:
680:
599:
587:
586:
584:
581:
580:
548:
547:
545:
542:
541:
539:linear operator
522:
519:
518:
513:is finite, the
497:
490:
486:
485:
470:
468:
465:
464:
445:
437:
434:
433:
427:Pettis integral
358:
356:
353:
352:
327:
324:
323:
307:
304:
303:
271:
266:
265:
248:
242:
237:
225:
216:
209:
205:
204:
189:
187:
184:
183:
151:
147:
145:
142:
141:
122:
119:
118:
102:
99:
98:
82:
79:
78:
75:
55:
47:James O. Ramsay
30:
12:
11:
5:
12885:
12875:
12874:
12869:
12853:
12852:
12823:(1): 177–196.
12807:
12762:
12733:(3): 839–851.
12717:
12698:
12653:
12634:
12586:
12568:
12557:(4): 875–889.
12541:
12522:
12489:
12454:
12425:(4): 468–484.
12404:
12383:(2): 351–363.
12363:
12344:
12315:(1): 257–295.
12295:
12266:(2): 267–286.
12246:
12233:(3): 545–560.
12217:
12168:
12119:
12092:
12057:
12036:(3): 533–550.
12016:
11989:Bioinformatics
11972:
11930:
11911:(4): 755–782.
11895:
11866:(1): 149–162.
11846:
11819:(1): 129–149.
11803:
11760:
11733:
11706:
11681:Neurocomputing
11664:
11629:
11610:(3): 803–821.
11594:
11573:(4): 679–699.
11553:
11518:(2): 774–805.
11502:
11477:(3): 607–622.
11461:
11444:
11442:(3):1720–1747.
11427:
11376:
11356:
11345:(2): 369–381.
11329:
11294:
11275:
11264:(1): 111–128.
11248:
11229:(3): 373–393.
11213:
11172:
11143:(3): 705–729.
11127:
11089:
11062:
11035:(2): 115–128.
11019:
11000:(3): 539–561.
10984:
10955:(1): 257–295.
10928:
10913:
10902:(1): 147–159.
10886:
10857:(2): 838–869.
10841:
10822:
10807:
10783:
10770:(1): 136–154.
10750:
10739:(5): 391–406.
10723:
10708:
10675:
10664:(1): 233–243.
10648:
10619:(3): 195–277.
10598:
10597:
10595:
10592:
10586:
10585:
10576:
10567:
10553:
10539:
10516:
10513:
10512:
10511:
10506:
10501:
10496:
10491:
10486:
10481:
10476:
10469:
10466:
10465:
10464:
10459:
10454:
10449:
10444:
10439:
10434:
10429:
10419:
10413:
10412:
10411:
10400:
10394:
10379:
10375:
10354:
10342:
10339:
10324:
10320:
10299:
10296:
10293:
10290:
10281:The domain of
10278:
10267:
10264:
10261:
10258:
10247:
10230:
10225:
10222:
10218:
10214:
10211:
10208:
10205:
10202:
10190:
10187:
10181:
10178:
10172:
10169:
10154:
10148:
10125:
10122:
10100:
10097:
10094:
10091:
10088:
10085:
10082:
10079:
10076:
10053:
10050:
10047:
10044:
10041:
10038:
10033:
10030:
10026:
10022:
10018:
9997:
9977:
9971:
9968:
9965:
9961:
9954:
9950:
9929:
9923:
9920:
9917:
9913:
9906:
9902:
9881:
9878:
9875:
9872:
9869:
9866:
9863:
9860:
9857:
9854:
9851:
9848:
9843:
9840:
9835:
9831:
9827:
9822:
9818:
9814:
9811:
9808:
9805:
9800:
9796:
9773:
9769:
9739:
9736:
9734:
9731:
9717:mixture models
9703:
9700:
9699:
9698:
9686:
9683:
9680:
9677:
9674:
9671:
9651:
9631:
9611:
9598:
9582:
9579:
9575:
9571:
9568:
9564:
9560:
9557:
9537:
9534:
9531:
9528:
9525:
9522:
9519:
9515:
9511:
9508:
9490:
9478:
9475:
9471:
9468:
9465:
9462:
9459:
9456:
9453:
9448:
9444:
9438:
9433:
9429:
9425:
9420:
9416:
9412:
9409:
9377:
9374:
9361:
9358:
9355:
9352:
9349:
9346:
9343:
9323:
9320:
9317:
9314:
9311:
9308:
9305:
9302:
9299:
9296:
9293:
9290:
9287:
9283:
9261:
9257:
9253:
9249:
9246:
9243:
9240:
9237:
9234:
9231:
9228:
9208:
9205:
9202:
9199:
9196:
9193:
9190:
9187:
9184:
9181:
9178:
9173:
9168:
9164:
9160:
9157:
9154:
9151:
9147:
9143:
9140:
9137:
9133:
9129:
9126:
9122:
9101:
9098:
9095:
9091:
9068:
9064:
9042:
9038:
9035:
9015:
9012:
9009:
9006:
9001:
8997:
8993:
8988:
8984:
8980:
8976:
8953:
8949:
8928:
8925:
8920:
8916:
8912:
8907:
8903:
8897:
8892:
8889:
8886:
8882:
8878:
8875:
8872:
8869:
8865:
8861:
8858:
8855:
8851:
8847:
8844:
8840:
8817:
8813:
8790:
8786:
8780:
8776:
8753:
8749:
8728:
8723:
8719:
8713:
8709:
8703:
8698:
8695:
8692:
8688:
8684:
8681:
8678:
8675:
8671:
8667:
8664:
8661:
8657:
8653:
8650:
8646:
8616:
8613:
8610:
8607:
8604:
8595:on the domain
8584:
8581:
8578:
8573:
8569:
8563:
8559:
8553:
8548:
8545:
8542:
8538:
8534:
8531:
8528:
8525:
8520:
8516:
8495:
8492:
8489:
8486:
8483:
8478:
8474:
8451:
8446:
8443:
8440:
8436:
8430:
8426:
8422:
8410:
8407:
8394:
8391:
8388:
8364:
8361:
8358:
8338:
8335:
8332:
8312:
8309:
8306:
8285:
8281:
8278:
8274:
8271:
8268:
8263:
8259:
8255:
8252:
8249:
8244:
8240:
8234:
8229:
8225:
8221:
8218:
8215:
8212:
8209:
8205:
8202:
8199:
8194:
8190:
8186:
8183:
8180:
8175:
8171:
8165:
8160:
8156:
8151:
8147:
8144:
8141:
8138:
8134:
8130:
8127:
8123:
8110:
8107:
8094:
8074:
8052:
8048:
8027:
8024:
8021:
8018:
8015:
8012:
8009:
8006:
8003:
8000:
7997:
7977:
7974:
7971:
7968:
7965:
7945:
7942:
7939:
7936:
7933:
7930:
7910:
7907:
7904:
7901:
7881:
7861:
7858:
7855:
7852:
7849:
7846:
7842:
7838:
7835:
7832:
7829:
7826:
7823:
7820:
7817:
7814:
7809:
7805:
7784:
7781:
7777:
7774:
7770:
7767:
7764:
7759:
7755:
7751:
7748:
7745:
7740:
7736:
7732:
7729:
7726:
7723:
7720:
7717:
7712:
7707:
7703:
7697:
7692:
7688:
7684:
7681:
7678:
7674:
7671:
7668:
7663:
7659:
7655:
7652:
7649:
7646:
7641:
7636:
7632:
7628:
7625:
7622:
7619:
7616:
7612:
7608:
7605:
7601:
7578:
7574:
7553:
7550:
7547:
7544:
7541:
7521:
7518:
7515:
7512:
7492:
7476:
7473:
7452:
7449:
7434:
7429:
7426:
7423:
7419:
7415:
7412:
7409:
7406:
7403:
7400:
7397:
7392:
7388:
7384:
7362:
7357:
7354:
7351:
7347:
7343:
7340:
7337:
7332:
7328:
7324:
7304:
7301:
7298:
7295:
7275:
7272:
7269:
7266:
7244:
7240:
7236:
7233:
7230:
7225:
7221:
7217:
7212:
7208:
7187:
7184:
7181:
7178:
7175:
7153:
7149:
7145:
7142:
7139:
7134:
7130:
7116:
7115:
7106:
7104:
7093:
7090:
7087:
7084:
7081:
7078:
7075:
7072:
7058:
7055:
7052:
7049:
7046:
7043:
7040:
7037:
7034:
7029:
7025:
7021:
7018:
7015:
7010:
7006:
7000:
6995:
6992:
6989:
6985:
6981:
6978:
6975:
6972:
6967:
6963:
6959:
6956:
6953:
6950:
6947:
6930:
6927:
6914:
6911:
6908:
6905:
6902:
6899:
6896:
6893:
6879:
6876:
6873:
6870:
6867:
6864:
6861:
6858:
6855:
6850:
6846:
6840:
6836:
6830:
6825:
6822:
6819:
6815:
6811:
6808:
6805:
6802:
6797:
6793:
6789:
6786:
6783:
6780:
6777:
6751:
6748:
6745:
6740:
6736:
6723:
6720:
6699:
6694:
6691:
6688:
6684:
6680:
6677:
6674:
6669:
6665:
6661:
6641:
6638:
6635:
6632:
6612:
6592:
6589:
6586:
6583:
6580:
6577:
6574:
6554:
6551:
6548:
6545:
6525:
6522:
6519:
6516:
6513:
6508:
6504:
6483:
6480:
6477:
6474:
6471:
6451:
6448:
6445:
6440:
6436:
6432:
6429:
6426:
6423:
6418:
6414:
6410:
6407:
6404:
6401:
6396:
6391:
6387:
6366:
6363:
6360:
6357:
6354:
6351:
6348:
6328:
6325:
6322:
6317:
6313:
6299:
6298:
6289:
6287:
6276:
6273:
6270:
6267:
6264:
6261:
6258:
6244:
6241:
6238:
6235:
6232:
6229:
6226:
6223:
6219:
6216:
6213:
6208:
6203:
6199:
6195:
6192:
6189:
6186:
6183:
6178:
6174:
6168:
6163:
6159:
6153:
6148:
6145:
6142:
6138:
6134:
6131:
6128:
6125:
6120:
6116:
6112:
6109:
6106:
6103:
6100:
6075:
6072:
6069:
6066:
6063:
6060:
6057:
6037:
6034:
6031:
6028:
6025:
6022:
6019:
5999:
5996:
5993:
5988:
5984:
5963:
5960:
5957:
5954:
5951:
5931:
5928:
5925:
5922:
5910:
5907:
5882:
5879:
5876:
5873:
5870:
5867:
5864:
5842:
5837:
5833:
5810:
5806:
5785:
5782:
5779:
5774:
5770:
5766:
5763:
5760:
5757:
5752:
5748:
5744:
5741:
5738:
5735:
5730:
5725:
5721:
5698:
5693:
5689:
5668:
5665:
5662:
5659:
5656:
5634:
5630:
5609:
5586:
5582:
5577:
5574:
5561:
5560:
5551:
5549:
5538:
5535:
5532:
5529:
5526:
5522:
5519:
5516:
5511:
5507:
5503:
5500:
5497:
5492:
5487:
5483:
5477:
5472:
5468:
5462:
5457:
5454:
5451:
5447:
5443:
5440:
5437:
5434:
5431:
5428:
5425:
5422:
5397:
5394:
5389:
5385:
5364:
5359:
5355:
5351:
5348:
5345:
5340:
5336:
5332:
5329:
5326:
5304:
5299:
5296:
5293:
5289:
5283:
5279:
5275:
5256:
5255:
5246:
5244:
5233:
5230:
5227:
5224:
5221:
5217:
5214:
5211:
5208:
5205:
5202:
5199:
5194:
5190:
5184:
5179:
5175:
5171:
5166:
5162:
5158:
5155:
5152:
5149:
5146:
5143:
5138:
5134:
5130:
5127:
5122:
5118:
5114:
5111:
5083:
5079:
5054:
5051:
5048:
5045:
5042:
5039:
5036:
5016:
5013:
5010:
5007:
5004:
5001:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4960:
4957:
4954:
4951:
4948:
4945:
4940:
4936:
4909:
4889:
4877:
4874:
4853:
4831:
4826:
4821:
4818:
4797:
4793:
4788:
4784:
4755:
4752:
4749:
4746:
4743:
4730:
4729:
4720:
4718:
4707:
4704:
4701:
4696:
4692:
4686:
4682:
4678:
4675:
4672:
4667:
4663:
4657:
4653:
4649:
4644:
4640:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4607:
4603:
4599:
4596:
4569:
4564:
4559:
4556:
4535:
4531:
4528:
4509:
4506:
4498:Fourier series
4488:
4487:
4478:
4476:
4465:
4462:
4459:
4454:
4450:
4444:
4441:
4437:
4431:
4426:
4423:
4420:
4416:
4412:
4409:
4406:
4403:
4400:
4397:
4394:
4391:
4388:
4383:
4380:
4377:
4372:
4368:
4342:
4337:
4334:
4330:
4326:
4323:
4320:
4317:
4314:
4309:
4306:
4302:
4298:
4295:
4290:
4286:
4265:
4243:
4239:
4218:
4198:
4195:
4192:
4172:
4169:
4164:
4160:
4156:
4153:
4150:
4145:
4141:
4135:
4132:
4128:
4122:
4117:
4114:
4111:
4107:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4074:
4070:
4066:
4062:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4034:
4009:
4006:
4003:
4000:
3997:
3992:
3988:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3955:
3951:
3947:
3942:
3937:
3933:
3929:
3924:
3921:
3917:
3896:
3893:
3890:
3885:
3881:
3875:
3872:
3868:
3862:
3857:
3854:
3851:
3847:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3814:
3810:
3785:
3782:
3779:
3774:
3770:
3747:
3743:
3722:
3719:
3716:
3711:
3707:
3703:
3700:
3697:
3692:
3688:
3682:
3678:
3672:
3667:
3664:
3661:
3657:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3615:
3612:
3609:
3606:
3603:
3600:
3578:
3556:), which is a
3539:
3536:
3533:
3530:
3527:
3522:
3518:
3514:
3511:
3508:
3505:
3502:
3499:
3494:
3490:
3486:
3481:
3459:
3439:
3436:
3433:
3428:
3424:
3400:
3397:
3379:
3375:
3352:
3349:
3345:
3322:
3319:
3315:
3311:
3308:
3303:
3300:
3296:
3292:
3287:
3283:
3279:
3274:
3271:
3267:
3254:
3251:
3228:
3225:
3220:
3216:
3191:
3187:
3183:
3179:
3175:
3172:
3169:
3164:
3161:
3157:
3132:
3129:
3125:
3102:
3099:
3095:
3091:
3088:
3083:
3080:
3076:
3072:
3067:
3063:
3059:
3054:
3051:
3047:
3034:
3031:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2991:
2986:
2981:
2978:
2958:
2955:
2952:
2947:
2943:
2939:
2934:
2931:
2927:
2914:
2911:
2898:
2878:
2856:
2851:
2848:
2844:
2840:
2837:
2832:
2829:
2825:
2821:
2794:
2791:
2788:
2783:
2780:
2776:
2772:
2768:
2745:
2742:
2738:
2715:
2712:
2708:
2687:
2682:
2679:
2675:
2671:
2666:
2662:
2658:
2653:
2650:
2646:
2625:
2618:
2614:
2610:
2606:
2602:
2599:
2596:
2593:
2590:
2585:
2582:
2578:
2574:
2571:
2566:
2533:
2529:
2525:
2521:
2517:
2514:
2511:
2508:
2505:
2500:
2497:
2493:
2472:
2452:
2449:
2446:
2441:
2437:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2338:
2335:
2332:
2329:
2326:
2323:
2319:
2315:
2312:
2309:
2306:
2303:
2283:
2280:
2277:
2274:
2271:
2249:
2245:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2200:
2197:
2194:
2191:
2188:
2173:
2170:
2153:
2132:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2087:
2076:
2075:
2064:
2061:
2058:
2055:
2051:
2048:
2045:
2041:
2037:
2034:
2031:
2026:
2022:
2018:
2015:
2012:
2007:
2003:
1997:
1993:
1987:
1982:
1979:
1976:
1972:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1946:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1912:
1882:
1878:
1857:
1854:
1851:
1831:
1826:
1813:
1812:
1801:
1796:
1792:
1788:
1783:
1779:
1773:
1769:
1763:
1758:
1755:
1752:
1748:
1744:
1739:
1712:
1699:tensor product
1686:
1681:
1677:
1673:
1668:
1664:
1660:
1638:
1619:
1618:
1609:
1607:
1596:
1593:
1589:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1552:
1547:
1543:
1539:
1536:
1533:
1530:
1527:
1524:
1519:
1514:
1489:
1486:
1483:
1480:
1475:
1453:
1433:
1413:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1352:
1349:
1346:
1341:
1337:
1333:
1330:
1327:
1324:
1320:
1308:
1307:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1203:
1200:
1197:
1194:
1191:
1187:
1183:
1180:
1177:
1174:
1171:
1146:
1133:
1132:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1097:
1093:
1090:
1087:
1084:
1081:
1058:
1047:Sobolev spaces
1034:
1031:
1028:
1025:
1022:
1017:
1013:
992:
972:
960:
957:
938:
910:
882:
878:
866:
865:
854:
849:
845:
841:
836:
832:
828:
825:
822:
817:
812:
809:
806:
802:
798:
795:
792:
789:
763:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
699:
695:
690:
673:
672:
661:
658:
655:
652:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
602:
598:
595:
590:
565:
562:
559:
556:
551:
526:
500:
493:
489:
484:
480:
477:
473:
452:
448:
444:
441:
423:
422:
411:
408:
405:
402:
398:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
361:
337:
334:
331:
311:
291:
288:
285:
282:
279:
274:
269:
264:
261:
258:
255:
251:
245:
240:
236:
232:
228:
224:
219:
212:
208:
203:
199:
196:
192:
171:
168:
165:
162:
159:
154:
150:
126:
106:
86:
74:
71:
69:is satisfied.
54:
51:
29:
26:
9:
6:
4:
3:
2:
12884:
12873:
12870:
12868:
12865:
12864:
12862:
12848:
12844:
12839:
12834:
12830:
12826:
12822:
12818:
12811:
12803:
12799:
12795:
12791:
12786:
12781:
12777:
12773:
12766:
12758:
12754:
12750:
12746:
12741:
12736:
12732:
12728:
12721:
12713:
12709:
12702:
12694:
12690:
12686:
12682:
12677:
12672:
12668:
12664:
12657:
12649:
12645:
12638:
12630:
12626:
12622:
12618:
12613:
12608:
12604:
12600:
12593:
12591:
12582:
12575:
12573:
12564:
12560:
12556:
12552:
12545:
12537:
12533:
12526:
12517:
12512:
12508:
12504:
12500:
12493:
12485:
12481:
12477:
12473:
12469:
12465:
12458:
12450:
12446:
12442:
12438:
12433:
12428:
12424:
12420:
12413:
12411:
12409:
12400:
12396:
12391:
12386:
12382:
12378:
12374:
12367:
12360:(1): 210–229.
12359:
12355:
12348:
12340:
12336:
12331:
12326:
12322:
12318:
12314:
12310:
12306:
12299:
12291:
12287:
12283:
12279:
12274:
12269:
12265:
12261:
12257:
12250:
12241:
12236:
12232:
12228:
12221:
12213:
12209:
12204:
12199:
12195:
12191:
12187:
12183:
12179:
12172:
12164:
12160:
12155:
12150:
12146:
12142:
12138:
12134:
12130:
12123:
12115:
12111:
12107:
12103:
12096:
12088:
12084:
12080:
12076:
12072:
12068:
12067:Technometrics
12061:
12053:
12049:
12044:
12039:
12035:
12031:
12027:
12020:
12012:
12008:
12003:
11998:
11994:
11990:
11983:
11976:
11968:
11964:
11960:
11956:
11952:
11948:
11941:
11934:
11926:
11922:
11918:
11914:
11910:
11906:
11899:
11891:
11887:
11882:
11877:
11873:
11869:
11865:
11861:
11857:
11850:
11842:
11838:
11834:
11830:
11826:
11822:
11818:
11814:
11807:
11799:
11795:
11791:
11787:
11783:
11779:
11775:
11771:
11764:
11756:
11752:
11748:
11744:
11737:
11729:
11725:
11722:(C): 92–106.
11721:
11717:
11710:
11702:
11698:
11694:
11690:
11686:
11682:
11675:
11668:
11660:
11656:
11652:
11648:
11644:
11640:
11633:
11625:
11621:
11617:
11613:
11609:
11605:
11598:
11590:
11586:
11581:
11576:
11572:
11568:
11564:
11557:
11549:
11543:
11535:
11531:
11526:
11521:
11517:
11513:
11506:
11498:
11492:
11484:
11480:
11476:
11472:
11465:
11458:
11454:
11448:
11441:
11437:
11431:
11423:
11419:
11414:
11409:
11404:
11399:
11395:
11391:
11387:
11380:
11373:
11369:
11363:
11361:
11352:
11348:
11344:
11340:
11333:
11325:
11321:
11317:
11313:
11309:
11305:
11298:
11291:(3): 763–788.
11290:
11286:
11279:
11271:
11267:
11263:
11259:
11252:
11244:
11240:
11236:
11232:
11228:
11224:
11217:
11209:
11205:
11200:
11195:
11191:
11187:
11183:
11176:
11168:
11164:
11160:
11156:
11151:
11146:
11142:
11138:
11131:
11123:
11119:
11115:
11111:
11107:
11103:
11096:
11094:
11085:
11081:
11077:
11073:
11066:
11058:
11054:
11050:
11046:
11042:
11038:
11034:
11030:
11023:
11015:
11011:
11007:
11003:
10999:
10995:
10988:
10980:
10976:
10971:
10966:
10962:
10958:
10954:
10950:
10946:
10939:
10937:
10935:
10933:
10924:
10917:
10909:
10905:
10901:
10897:
10890:
10882:
10878:
10874:
10870:
10865:
10860:
10856:
10852:
10845:
10838:(2): 151–163.
10837:
10833:
10826:
10818:
10811:
10803:
10796:
10794:
10792:
10790:
10788:
10778:
10773:
10769:
10765:
10761:
10754:
10746:
10742:
10738:
10734:
10727:
10719:
10712:
10703:
10698:
10694:
10690:
10686:
10679:
10671:
10667:
10663:
10659:
10652:
10644:
10640:
10635:
10630:
10626:
10622:
10618:
10614:
10610:
10603:
10599:
10591:
10590:
10584:
10581:
10577:
10575:
10572:
10568:
10566:
10562:
10558:
10554:
10552:
10548:
10544:
10540:
10538:
10537:0-387-40080-X
10534:
10530:
10526:
10522:
10521:Ramsay, J. O.
10519:
10518:
10510:
10507:
10505:
10502:
10500:
10497:
10495:
10492:
10490:
10487:
10485:
10482:
10480:
10477:
10475:
10472:
10471:
10463:
10460:
10458:
10455:
10453:
10450:
10448:
10445:
10443:
10440:
10438:
10435:
10433:
10430:
10428:
10425:
10424:
10423:
10417:
10410:
10407:
10406:
10405:
10398:
10393:
10377:
10373:
10352:
10338:
10322:
10318:
10294:
10288:
10262:
10256:
10246:
10223:
10220:
10209:
10203:
10186:
10177:
10168:
10152:
10135:
10131:
10121:
10119:
10114:
10098:
10095:
10092:
10089:
10086:
10080:
10074:
10065:
10051:
10048:
10039:
10031:
10028:
10024:
9995:
9975:
9969:
9966:
9963:
9959:
9952:
9948:
9927:
9921:
9918:
9915:
9911:
9904:
9900:
9876:
9873:
9870:
9864:
9861:
9858:
9849:
9841:
9838:
9833:
9829:
9820:
9816:
9812:
9806:
9798:
9794:
9771:
9767:
9756:
9754:
9744:
9730:
9727:
9726:FPCA approach
9721:
9718:
9713:
9709:
9681:
9675:
9672:
9669:
9649:
9629:
9609:
9602:
9601:Link function
9599:
9596:
9577:
9569:
9558:
9555:
9532:
9526:
9523:
9517:
9509:
9494:
9491:
9476:
9473:
9466:
9460:
9454:
9446:
9442:
9436:
9431:
9427:
9423:
9418:
9414:
9410:
9407:
9400:
9397:
9396:
9395:
9393:
9389:
9385:
9384:
9373:
9356:
9353:
9350:
9344:
9341:
9321:
9318:
9306:
9300:
9297:
9294:
9288:
9247:
9241:
9238:
9235:
9229:
9226:
9206:
9203:
9194:
9188:
9185:
9182:
9176:
9171:
9166:
9162:
9158:
9152:
9141:
9135:
9127:
9096:
9066:
9062:
9036:
9033:
9013:
9010:
8999:
8995:
8986:
8982:
8951:
8947:
8926:
8918:
8914:
8905:
8901:
8890:
8887:
8884:
8880:
8876:
8870:
8859:
8853:
8845:
8815:
8811:
8788:
8784:
8778:
8774:
8751:
8747:
8726:
8721:
8717:
8711:
8707:
8696:
8693:
8690:
8686:
8682:
8676:
8665:
8659:
8651:
8635:
8634:
8628:
8611:
8608:
8605:
8579:
8571:
8567:
8561:
8557:
8546:
8543:
8540:
8536:
8532:
8526:
8518:
8514:
8490:
8487:
8484:
8476:
8472:
8444:
8441:
8438:
8428:
8424:
8406:
8392:
8389:
8386:
8378:
8362:
8359:
8356:
8336:
8333:
8330:
8310:
8307:
8304:
8283:
8279:
8276:
8269:
8261:
8257:
8250:
8242:
8238:
8232:
8227:
8223:
8219:
8216:
8213:
8210:
8207:
8200:
8192:
8188:
8181:
8173:
8169:
8163:
8158:
8154:
8149:
8145:
8142:
8136:
8128:
8106:
8092:
8072:
8050:
8046:
8022:
8019:
8016:
8010:
8004:
8001:
7998:
7972:
7969:
7966:
7940:
7937:
7934:
7928:
7905:
7899:
7879:
7853:
7847:
7836:
7830:
7824:
7821:
7815:
7807:
7803:
7782:
7779:
7775:
7772:
7765:
7757:
7753:
7746:
7738:
7734:
7727:
7724:
7721:
7715:
7710:
7705:
7701:
7695:
7690:
7686:
7682:
7679:
7676:
7669:
7661:
7657:
7650:
7644:
7639:
7634:
7630:
7626:
7623:
7620:
7614:
7606:
7576:
7572:
7548:
7545:
7542:
7516:
7510:
7490:
7482:
7472:
7470:
7466:
7462:
7458:
7448:
7432:
7427:
7424:
7421:
7413:
7410:
7407:
7404:
7398:
7390:
7386:
7360:
7355:
7352:
7349:
7338:
7330:
7326:
7299:
7293:
7270:
7264:
7242:
7238:
7234:
7231:
7228:
7223:
7219:
7215:
7210:
7206:
7182:
7179:
7176:
7151:
7147:
7143:
7140:
7137:
7132:
7128:
7114:
7107:
7105:
7091:
7085:
7082:
7079:
7073:
7070:
7056:
7050:
7044:
7041:
7035:
7027:
7023:
7016:
7008:
7004:
6998:
6993:
6990:
6987:
6983:
6979:
6973:
6965:
6961:
6957:
6951:
6945:
6938:
6937:
6934:
6926:
6912:
6906:
6903:
6900:
6894:
6891:
6877:
6871:
6865:
6862:
6856:
6848:
6844:
6838:
6834:
6828:
6823:
6820:
6817:
6813:
6809:
6803:
6795:
6791:
6787:
6781:
6775:
6767:
6766:
6746:
6738:
6734:
6719:
6717:
6716:
6697:
6692:
6689:
6686:
6675:
6667:
6663:
6636:
6630:
6610:
6587:
6584:
6581:
6575:
6572:
6549:
6543:
6520:
6517:
6514:
6506:
6502:
6478:
6475:
6472:
6446:
6438:
6434:
6430:
6424:
6416:
6412:
6408:
6402:
6394:
6389:
6385:
6364:
6361:
6358:
6355:
6352:
6349:
6346:
6323:
6315:
6311:
6297:
6290:
6288:
6271:
6268:
6265:
6259:
6256:
6242:
6236:
6230:
6227:
6224:
6221:
6214:
6206:
6201:
6197:
6190:
6187:
6184:
6176:
6172:
6166:
6161:
6157:
6151:
6146:
6143:
6140:
6136:
6132:
6126:
6118:
6114:
6110:
6104:
6098:
6091:
6090:
6087:
6073:
6070:
6067:
6064:
6061:
6058:
6055:
6032:
6029:
6026:
6020:
6017:
5994:
5986:
5982:
5958:
5955:
5952:
5926:
5920:
5906:
5904:
5903:
5898:
5897:
5880:
5877:
5874:
5871:
5868:
5865:
5862:
5840:
5835:
5831:
5808:
5804:
5780:
5772:
5768:
5764:
5758:
5750:
5746:
5742:
5736:
5728:
5723:
5719:
5696:
5691:
5687:
5663:
5660:
5657:
5632:
5628:
5607:
5575:
5572:
5559:
5552:
5550:
5536:
5533:
5530:
5527:
5524:
5517:
5509:
5505:
5498:
5490:
5485:
5481:
5475:
5470:
5466:
5460:
5455:
5452:
5449:
5445:
5441:
5435:
5432:
5429:
5423:
5420:
5413:
5412:
5409:
5395:
5392:
5387:
5383:
5357:
5353:
5349:
5346:
5343:
5338:
5334:
5327:
5324:
5302:
5297:
5294:
5291:
5281:
5277:
5265:
5264:
5254:
5247:
5245:
5231:
5228:
5225:
5222:
5219:
5212:
5206:
5200:
5192:
5188:
5182:
5177:
5173:
5169:
5164:
5160:
5156:
5153:
5150:
5144:
5141:
5136:
5132:
5125:
5120:
5116:
5112:
5109:
5102:
5101:
5098:
5096:
5081:
5077:
5068:
5067:Hilbert space
5049:
5046:
5043:
5037:
5034:
5011:
5005:
5002:
4999:
4976:
4970:
4967:
4961:
4955:
4952:
4946:
4938:
4934:
4925:
4924:
4907:
4887:
4873:
4871:
4867:
4851:
4829:
4819:
4816:
4791:
4786:
4782:
4773:
4769:
4768:inner product
4750:
4747:
4744:
4728:
4721:
4719:
4705:
4702:
4699:
4694:
4690:
4684:
4680:
4676:
4673:
4670:
4665:
4661:
4655:
4651:
4647:
4642:
4638:
4634:
4631:
4628:
4622:
4619:
4616:
4610:
4605:
4601:
4597:
4594:
4587:
4586:
4583:
4567:
4557:
4554:
4529:
4526:
4518:
4514:
4505:
4503:
4499:
4495:
4486:
4479:
4477:
4460:
4452:
4448:
4442:
4439:
4435:
4429:
4424:
4421:
4418:
4414:
4410:
4404:
4398:
4395:
4389:
4378:
4370:
4366:
4358:
4357:
4354:
4335:
4332:
4328:
4324:
4321:
4318:
4315:
4312:
4307:
4304:
4300:
4293:
4288:
4284:
4263:
4241:
4237:
4216:
4190:
4170:
4162:
4151:
4143:
4139:
4133:
4130:
4126:
4120:
4115:
4112:
4109:
4105:
4101:
4095:
4089:
4086:
4080:
4072:
4068:
4051:
4048:
4045:
4039:
4036:
4023:
4007:
4004:
3998:
3990:
3986:
3976:
3970:
3967:
3961:
3953:
3949:
3940:
3935:
3931:
3927:
3922:
3919:
3915:
3891:
3883:
3879:
3873:
3870:
3866:
3855:
3852:
3849:
3845:
3841:
3835:
3829:
3826:
3820:
3812:
3808:
3799:
3780:
3772:
3768:
3745:
3741:
3717:
3709:
3705:
3698:
3690:
3686:
3680:
3676:
3665:
3662:
3659:
3655:
3651:
3645:
3642:
3639:
3610:
3607:
3604:
3566:
3561:
3559:
3555:
3554:
3534:
3531:
3528:
3520:
3516:
3506:
3503:
3500:
3492:
3488:
3484:
3457:
3434:
3426:
3422:
3413:
3409:
3405:
3396:
3393:
3377:
3373:
3350:
3347:
3343:
3320:
3317:
3313:
3309:
3301:
3298:
3294:
3285:
3281:
3277:
3272:
3269:
3265:
3257:Measurements
3250:
3249:
3245:
3240:
3218:
3214:
3189:
3185:
3181:
3177:
3173:
3170:
3167:
3162:
3159:
3155:
3146:
3130:
3127:
3123:
3100:
3097:
3093:
3089:
3081:
3078:
3074:
3065:
3061:
3057:
3052:
3049:
3045:
3037:Measurements
3030:
3027:
3024:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2989:
2979:
2976:
2953:
2945:
2941:
2937:
2932:
2929:
2925:
2917:Measurements
2910:
2896:
2876:
2854:
2849:
2846:
2842:
2838:
2830:
2827:
2823:
2792:
2789:
2781:
2778:
2774:
2743:
2740:
2736:
2713:
2710:
2706:
2680:
2677:
2673:
2664:
2660:
2656:
2651:
2648:
2644:
2616:
2612:
2608:
2604:
2600:
2597:
2594:
2591:
2588:
2583:
2580:
2576:
2569:
2564:
2531:
2527:
2523:
2519:
2515:
2512:
2509:
2506:
2503:
2498:
2495:
2491:
2470:
2447:
2439:
2435:
2408:
2402:
2399:
2393:
2387:
2374:
2368:
2365:
2362:
2330:
2324:
2313:
2307:
2301:
2278:
2275:
2272:
2247:
2243:
2219:
2216:
2213:
2207:
2204:
2198:
2192:
2186:
2179:
2169:
2167:
2151:
2120:
2117:
2114:
2108:
2105:
2085:
2062:
2053:
2049:
2046:
2039:
2032:
2024:
2020:
2013:
2005:
2001:
1995:
1991:
1985:
1980:
1977:
1974:
1970:
1966:
1960:
1957:
1954:
1944:
1935:
1932:
1929:
1923:
1920:
1917:
1914:
1902:
1901:
1900:
1898:
1880:
1876:
1855:
1852:
1849:
1829:
1799:
1794:
1790:
1786:
1781:
1777:
1771:
1767:
1756:
1753:
1750:
1746:
1742:
1728:
1727:
1726:
1700:
1697:, so that in
1679:
1675:
1671:
1666:
1662:
1626:
1617:
1610:
1608:
1594:
1591:
1579:
1573:
1567:
1564:
1561:
1550:
1545:
1541:
1537:
1531:
1522:
1505:
1504:
1501:
1487:
1481:
1478:
1411:
1402:
1385:
1382:
1379:
1373:
1370:
1347:
1339:
1331:
1325:
1291:
1288:
1285:
1279:
1276:
1273:
1270:
1266:
1257:
1251:
1248:
1242:
1236:
1223:
1217:
1214:
1211:
1201:
1195:
1189:
1181:
1175:
1169:
1162:
1161:
1160:
1114:
1111:
1108:
1102:
1099:
1088:
1082:
1072:
1071:
1070:
1056:
1048:
1029:
1026:
1023:
1015:
1011:
990:
970:
956:
954:
926:
898:
880:
876:
852:
847:
843:
834:
830:
826:
823:
810:
807:
804:
800:
796:
793:
790:
787:
780:
779:
778:
777:
761:
753:
731:
728:
725:
719:
713:
710:
707:
693:
678:
659:
656:
653:
650:
646:
637:
634:
631:
622:
619:
616:
613:
610:
596:
593:
579:
578:
577:
563:
557:
554:
540:
524:
516:
498:
491:
487:
478:
450:
442:
439:
432:
428:
409:
406:
403:
400:
396:
390:
387:
384:
378:
372:
369:
366:
351:
350:
349:
335:
332:
329:
309:
286:
280:
277:
272:
259:
253:
243:
238:
234:
222:
217:
210:
206:
197:
166:
163:
160:
152:
148:
140:
124:
104:
84:
70:
68:
64:
60:
59:Hilbert space
50:
48:
43:
39:
35:
25:
22:
18:
12820:
12816:
12810:
12775:
12771:
12765:
12730:
12726:
12720:
12714:: 1571–1596.
12711:
12707:
12701:
12666:
12662:
12656:
12647:
12643:
12637:
12602:
12598:
12583:: 3147–3155.
12580:
12554:
12550:
12544:
12535:
12531:
12525:
12506:
12502:
12492:
12467:
12463:
12457:
12422:
12418:
12380:
12376:
12366:
12357:
12353:
12347:
12312:
12308:
12298:
12263:
12259:
12249:
12230:
12226:
12220:
12185:
12181:
12171:
12136:
12132:
12122:
12105:
12101:
12095:
12070:
12066:
12060:
12033:
12029:
12019:
11995:(1): 68–76.
11992:
11988:
11975:
11953:(1): 41–67.
11950:
11946:
11933:
11908:
11904:
11898:
11863:
11859:
11849:
11816:
11812:
11806:
11776:(1): 44–68.
11773:
11769:
11763:
11749:(C): 14–29.
11746:
11742:
11736:
11719:
11715:
11709:
11684:
11680:
11667:
11642:
11638:
11632:
11607:
11603:
11597:
11570:
11566:
11556:
11542:cite journal
11525:math/0505638
11515:
11511:
11505:
11491:cite journal
11474:
11470:
11464:
11459:(1):362–388.
11456:
11452:
11447:
11439:
11435:
11430:
11393:
11389:
11379:
11371:
11367:
11342:
11338:
11332:
11307:
11303:
11297:
11288:
11284:
11278:
11261:
11257:
11251:
11226:
11222:
11216:
11189:
11185:
11175:
11140:
11136:
11130:
11105:
11101:
11078:(1): 54–77.
11075:
11071:
11065:
11032:
11028:
11022:
10997:
10993:
10987:
10952:
10948:
10922:
10916:
10899:
10895:
10889:
10854:
10850:
10844:
10835:
10831:
10825:
10816:
10810:
10801:
10767:
10763:
10753:
10736:
10732:
10726:
10717:
10711:
10692:
10688:
10678:
10661:
10657:
10651:
10616:
10612:
10602:
10587:
10579:
10570:
10556:
10542:
10528:
10421:
10402:
10344:
10280:
10192:
10183:
10174:
10127:
10066:
9757:
9749:
9733:Time warping
9722:
9705:
9381:
9379:
8631:
8629:
8412:
8112:
7532:with domain
7478:
7454:
7119:
7108:
6932:
6763:
6725:
6713:
6302:
6291:
5912:
5900:
5894:
5564:
5553:
5261:
5259:
5248:
4921:
4879:
4766:denotes the
4733:
4722:
4511:
4491:
4480:
3562:
3551:
3402:
3394:
3256:
3241:
3147:
3036:
3028:
3025:
2916:
2175:
2077:
1814:
1622:
1611:
1403:
1309:
1134:
962:
897:eigenvectors
867:
674:
424:
76:
66:
56:
31:
16:
15:
12838:2117/126653
11687:: 164–171.
10804:. Springer.
9738:Motivations
2235:that is an
1627:applies to
925:eigenvalues
348:satisfying
12861:Categories
12785:1705.06226
12740:1811.01429
12727:Biometrics
12676:1509.02029
12612:2104.04628
12551:Biometrika
12432:1512.03216
12240:1605.03707
12227:Biometrika
12182:Biometrics
12073:(1): 1–9.
11860:Biometrika
11604:Biometrics
11471:Biometrika
11374:(1):49–64.
11368:Biometrika
11258:Biometrika
10896:Biometrika
10594:References
10447:classiFunc
10409:scikit-fda
10310:can be in
10189:Extensions
8966:satisfies
5893:. Models (
4920:in model (
4864:is a zero
3248:Stock data
1868:, all the
61:, or as a
21:statistics
12757:220687157
12629:233210300
12470:: 43–49.
12290:124261587
12282:1369-7412
12163:120454400
11589:120883171
11403:1105.0014
11390:Bernoulli
11243:122007787
11150:1102.5212
11014:118960346
10864:1206.1194
10643:120451372
10295:⋅
10263:⋅
10224:∈
10096:γ
10090:δ
10029:−
9960:∼
9912:∼
9865:∈
9839:−
9682:η
9670:μ
9650:η
9630:μ
9556:μ
9533:μ
9461:β
9428:∫
9415:β
9408:η
9345:∈
9256:⟶
9248:×
9163:∫
9037:∈
8896:∞
8881:∑
8775:β
8708:β
8702:∞
8687:∑
8568:ϕ
8552:∞
8537:∑
8450:∞
8425:ϕ
8258:β
8224:∫
8217:…
8189:β
8155:∫
8093:γ
8073:β
8011:×
7941:⋅
7935:⋅
7929:γ
7906:⋅
7900:β
7880:α
7854:⋅
7837:−
7831:⋅
7816:⋅
7716:γ
7702:∫
7687:∫
7645:β
7631:∫
7624:α
7517:⋅
7411:≤
7265:ε
7239:β
7232:…
7220:β
7207:β
7141:…
7074:∈
7045:ε
7005:β
6984:∑
6962:β
6895:∈
6866:ε
6845:α
6814:∑
6792:α
6747:⋅
6712:. Model (
6576:∈
6544:ε
6503:α
6435:μ
6431:−
6359:…
6312:α
6260:∈
6231:ε
6173:α
6158:∫
6137:∑
6115:α
6068:…
6021:∈
5875:…
5805:β
5769:μ
5765:−
5576:∈
5573:θ
5534:ε
5506:β
5467:∫
5446:∑
5439:⟩
5436:θ
5427:⟨
5347:⋯
5229:ε
5207:β
5174:∫
5161:β
5154:ε
5148:⟩
5145:β
5129:⟨
5117:β
5038:∈
5006:β
5000:β
4971:μ
4968:−
4908:β
4852:ε
4820:∈
4817:β
4792:∈
4783:β
4754:⟩
4751:⋅
4745:⋅
4742:⟨
4703:ε
4691:β
4674:⋯
4662:β
4639:β
4632:ε
4626:⟩
4623:β
4614:⟨
4602:β
4558:∈
4530:∈
4449:φ
4415:∑
4399:μ
4197:∞
4194:→
4168:→
4140:φ
4106:∑
4102:−
4090:μ
4087:−
4040:∈
3987:φ
3971:μ
3968:−
3932:∫
3880:φ
3861:∞
3846:∑
3830:μ
3796:. By the
3769:φ
3742:λ
3706:φ
3687:φ
3677:λ
3671:∞
3656:∑
3634:Σ
3611:⋅
3605:⋅
3599:Σ
3513:→
3314:ε
3227:∞
3224:→
3171:…
3094:ε
3006:…
2980:∈
2843:σ
2824:ϵ
2775:ϵ
2737:ϵ
2448:⋅
2357:Σ
2302:μ
2208:∈
2127:→
2060:∞
2057:→
2044:→
2021:φ
2002:φ
1992:λ
1971:∑
1967:−
1949:Σ
1924:∈
1877:φ
1853:∈
1791:φ
1787:⊗
1778:φ
1768:λ
1762:∞
1747:∑
1701:notation
1676:φ
1663:λ
1556:Σ
1542:∫
1500:given by
1485:→
1452:Σ
1432:Σ
1412:μ
1374:∈
1351:∞
1280:∈
1206:Σ
1170:μ
1103:∈
877:φ
844:φ
840:⟩
831:φ
821:⟨
816:∞
801:∑
794:μ
732:μ
729:−
720:⊗
714:μ
711:−
654:∈
638:μ
635:−
626:⟩
623:μ
620:−
608:⟨
561:→
483:‖
476:‖
440:μ
404:∈
394:⟩
385:μ
382:⟨
376:⟩
364:⟨
333:∈
330:μ
290:∞
235:∫
202:‖
195:‖
12847:13719492
12802:13671221
12693:88521295
12484:17900407
12449:55849758
12399:17175587
12339:13709250
12212:22670567
12154:11192549
12087:21662019
12052:16050693
12011:16257986
11967:11448616
11925:18638091
11890:19262739
11798:10969266
11790:24249100
11701:33591208
11422:88512527
11324:14296231
11208:16758288
11167:17843044
11057:55092204
10979:13709250
10881:13119710
10504:Lp space
10468:See also
10418:packages
10399:packages
9892:, where
9662:through
9548:, where
9334:for all
6623:, i.e.,
5375:, where
4870:variance
3907:, where
3335:, where
3115:, where
2636:, where
1363:for all
117:, where
12317:Bibcode
12203:3443537
11881:2650433
11841:8902492
11821:Bibcode
11659:9487422
11624:2532201
11122:1243975
11049:3316063
10957:Bibcode
10621:Bibcode
10527:(2005)
10462:fdasrvf
10452:fda.usc
10442:FDboost
10437:fdapace
10124:Methods
9593:is the
8379:. With
5899:) and (
4868:finite
3550:as in (
1725:writes
774:as the
750:. The
675:or, in
28:History
12845:
12800:
12755:
12691:
12627:
12482:
12447:
12397:
12337:
12288:
12280:
12210:
12200:
12161:
12151:
12085:
12050:
12009:
11965:
11923:
11888:
11878:
11839:
11796:
11788:
11699:
11657:
11622:
11587:
11420:
11322:
11241:
11206:
11165:
11120:
11055:
11047:
11012:
10977:
10879:
10641:
10563:
10549:
10535:
10432:refund
10397:Python
8939:where
7795:where
7120:where
7068:
7060:
6889:
6881:
6303:where
6254:
6246:
5855:, for
5796:, and
5565:where
4734:where
4494:spline
3206:, and
2202:
868:where
679:form,
677:tensor
12843:S2CID
12798:S2CID
12780:arXiv
12753:S2CID
12735:arXiv
12689:S2CID
12671:arXiv
12625:S2CID
12607:arXiv
12480:S2CID
12445:S2CID
12427:arXiv
12395:S2CID
12335:S2CID
12286:S2CID
12235:arXiv
12159:S2CID
12083:S2CID
12048:S2CID
11985:(PDF)
11963:S2CID
11943:(PDF)
11921:S2CID
11837:S2CID
11794:S2CID
11697:S2CID
11677:(PDF)
11655:S2CID
11620:JSTOR
11585:S2CID
11520:arXiv
11418:S2CID
11398:arXiv
11320:S2CID
11239:S2CID
11204:S2CID
11163:S2CID
11145:arXiv
11118:S2CID
11053:S2CID
11045:JSTOR
11010:S2CID
10975:S2CID
10877:S2CID
10859:arXiv
10639:S2CID
2758:with
1003:like
537:is a
12278:ISSN
12208:PMID
12007:PMID
11886:PMID
11786:PMID
11548:link
11497:link
10561:ISBN
10547:ISBN
10533:ISBN
10523:and
9710:and
9026:for
8390:>
8360:>
8334:>
8085:and
7988:and
7921:and
5408:, by
5027:for
4866:mean
4809:and
3560:.
3246:and
2889:and
2805:and
1623:The
1424:and
1401:).
1348:<
1310:(if
1045:and
895:are
287:<
12833:hdl
12825:doi
12790:doi
12745:doi
12681:doi
12667:113
12617:doi
12603:117
12559:doi
12511:doi
12472:doi
12437:doi
12385:doi
12325:doi
12268:doi
12231:104
12198:PMC
12190:doi
12149:PMC
12141:doi
12110:doi
12075:doi
12038:doi
11997:doi
11955:doi
11913:doi
11876:PMC
11868:doi
11829:doi
11778:doi
11751:doi
11724:doi
11689:doi
11685:112
11647:doi
11612:doi
11575:doi
11530:doi
11479:doi
11475:100
11408:doi
11347:doi
11343:140
11312:doi
11308:105
11266:doi
11231:doi
11194:doi
11155:doi
11110:doi
11106:100
11080:doi
11037:doi
11002:doi
10965:doi
10904:doi
10900:103
10869:doi
10772:doi
10741:doi
10697:doi
10666:doi
10629:doi
10457:dtw
10427:fda
10365:to
9697:.
9597:;
9503:Var
9489:;
8464:on
7064:for
6885:for
6250:for
5942:on
5647:is
4770:in
4183:as
4033:sup
3563:By
2815:Var
2380:Cov
1911:sup
1229:Cov
927:of
899:of
517:of
12863::
12841:.
12831:.
12821:79
12819:.
12796:.
12788:.
12776:46
12774:.
12751:.
12743:.
12731:77
12729:.
12712:24
12710:.
12687:.
12679:.
12665:.
12648:67
12646:.
12623:.
12615:.
12601:.
12589:^
12571:^
12555:95
12553:.
12536:90
12534:.
12507:20
12505:.
12501:.
12478:.
12468:26
12466:.
12443:.
12435:.
12423:30
12421:.
12407:^
12393:.
12381:60
12379:.
12375:.
12358:12
12356:.
12333:.
12323:.
12311:.
12307:.
12284:.
12276:.
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2187:X
2152:X
2131:R
2124:]
2121:1
2118:,
2115:0
2112:[
2109::
2106:X
2086:X
2063:.
2054:K
2050:,
2047:0
2040:|
2036:)
2033:t
2030:(
2025:j
2017:)
2014:s
2011:(
2006:j
1996:j
1986:K
1981:1
1978:=
1975:j
1964:)
1961:t
1958:,
1955:s
1952:(
1945:|
1939:]
1936:1
1933:,
1930:0
1927:[
1921:t
1918:,
1915:s
1881:j
1856:H
1850:f
1830:f
1825:C
1800:.
1795:j
1782:j
1772:j
1757:1
1754:=
1751:j
1743:=
1738:C
1711:C
1685:)
1680:j
1672:,
1667:j
1659:(
1637:C
1616:)
1614:1
1612:(
1595:.
1592:s
1588:d
1583:)
1580:s
1577:(
1574:f
1571:)
1568:t
1565:,
1562:s
1559:(
1551:1
1546:0
1538:=
1535:)
1532:t
1529:(
1526:)
1523:f
1518:C
1513:(
1488:H
1482:H
1479::
1474:C
1389:]
1386:1
1383:,
1380:0
1377:[
1371:t
1345:]
1340:2
1336:)
1332:t
1329:(
1326:X
1323:[
1319:E
1295:]
1292:1
1289:,
1286:0
1283:[
1277:t
1274:,
1271:s
1267:,
1264:)
1261:)
1258:t
1255:(
1252:X
1249:,
1246:)
1243:s
1240:(
1237:X
1234:(
1224:=
1221:)
1218:t
1215:,
1212:s
1209:(
1202:,
1199:)
1196:t
1193:(
1190:X
1186:E
1182:=
1179:)
1176:t
1173:(
1145:T
1118:]
1115:1
1112:,
1109:0
1106:[
1100:t
1096:}
1092:)
1089:t
1086:(
1083:X
1080:{
1057:X
1033:]
1030:1
1027:,
1024:0
1021:[
1016:2
1012:L
991:H
971:X
937:C
909:C
881:i
853:,
848:i
835:i
827:,
824:X
811:1
808:=
805:i
797:+
791:=
788:X
762:X
738:]
735:)
726:X
723:(
717:)
708:X
705:(
702:[
698:E
694:=
689:C
660:,
657:H
651:h
647:,
644:]
641:)
632:X
629:(
617:X
614:,
611:h
605:[
601:E
597:=
594:h
589:C
564:H
558:H
555::
550:C
525:X
499:2
492:2
488:L
479:X
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451:X
447:E
443:=
410:.
407:H
401:h
397:,
391:h
388:,
379:=
373:h
370:,
367:X
360:E
336:H
310:X
284:)
281:t
278:d
273:2
268:|
263:)
260:t
257:(
254:X
250:|
244:1
239:0
231:(
227:E
223:=
218:2
211:2
207:L
198:X
191:E
170:]
167:1
164:,
161:0
158:[
153:2
149:L
125:H
105:X
85:H
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