Covariance operators of random functions are crucial tools to study the way random elements concentrate over their support. The principal component analysis of a random function X is well-known from a theoretical viewpoint and extensively used in practical situations. In this work we focus on local covariance operators. They provide some pieces of information about the distribution of X around a fixed point of the space x₀. A description of the asymptotic behaviour of the theoretical and empirical counterparts is carried out. Asymptotic developments are given under assumptions on the location of x₀ and on the distributions of projections of the data on the eigenspaces of the (non-local) covariance operator.
Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered for example as movement trajectories on the surface of the earth, are an important special case. We consider an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the time-varying Fréchet mean function, and then perfor…
There are often two important types of variation in functional data: the horizontal (or phase) variation and the vertical (or amplitude) variation. These two types of variation have been appropriately separated and modeled through a domain warping method (or curve registration) based on the Fisher Rao metric. This paper focuses on the analysis of the horizontal variation, captured by the domain warping functions. The square-root velocity function representation transforms the manifold of the warping functions to a Hilbert sphere. Motivated by recent results on manifold analogs of principal com…
Functional principal component analysis (FPCA) based on the Karhunen--Loève decomposition has been successfully applied in many applications, mainly for one sample problems. In this paper we consider common functional principal components for two sample problems. Our research is motivated not only by the theoretical challenge of this data situation, but also by the actual question of dynamics of implied volatility (IV) functions. For different maturities the log-returns of IVs are samples of (smooth) random functions and the methods proposed here study the similarities of their stochastic beha…
We analyse the prediction error of principal component regression (PCR) and prove non-asymptotic upper bounds for the corresponding squared risk. Under mild assumptions, we show that PCR performs as well as the oracle method obtained by replacing empirical principal components by their population counterparts. Our approach relies on upper bounds for the excess risk of principal component analysis.
We consider the sparse principal component analysis for high-dimensional stationary processes. The standard principal component analysis performs poorly when the dimension of the process is large. We establish the oracle inequalities for penalized principal component estimators for the processes including heavy-tailed time series. The rate of convergence of the estimators is established. We also elucidate the theoretical rate for choosing the tuning parameter in penalized estimators. The performance of the sparse principal component analysis is demonstrated by numerical simulations. The utilit…
We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations $X_1,\dots, X_n$ in a separable Hilbert space $\mathbb{H}$ with unknown covariance operator $Σ.$ The complexity of the problem is characterized by its effective rank ${\bf r}(Σ):= \frac{{\rm tr}(Σ)}{\|Σ\|},$ where ${\rm tr}(Σ)$ denotes the trace of $Σ$ and $\|Σ\|$ denotes its operator norm. We develop a method of bias reduction in the problem of estimation of linear functionals of eigenvectors of $Σ.$ Under the assumption that ${\bf r}(Σ)=o(n),$ we establish the asymptotic normality and asymptotic properties…
We propose a nonparametric method to explicitly model and represent the derivatives of smooth underlying trajectories for longitudinal data. This representation is based on a direct Karhunen--Loève expansion of the unobserved derivatives and leads to the notion of derivative principal component analysis, which complements functional principal component analysis, one of the most popular tools of functional data analysis. The proposed derivative principal component scores can be obtained for irregularly spaced and sparsely observed longitudinal data, as typically encountered in biomedical studie…
In this paper, we study the problem of sparse Principal Component Analysis (PCA) in the high-dimensional setting with missing observations. Our goal is to estimate the first principal component when we only have access to partial observations. Existing estimation techniques are usually derived for fully observed data sets and require a prior knowledge of the sparsity of the first principal component in order to achieve good statistical guarantees. Our contributions is threefold. First, we establish the first information-theoretic lower bound for the sparse PCA problem with missing observations…
Classical multivariate principal component analysis has been extended to functional data and termed functional principal component analysis (FPCA). Most existing FPCA approaches do not accommodate covariate information, and it is the goal of this paper to develop two methods that do. In the first approach, both the mean and covariance functions depend on the covariate $Z$ and time scale $t$ while in the second approach only the mean function depends on the covariate $Z$. Both new approaches accommodate additional measurement errors and functional data sampled at regular time grids as well as s…
In this paper we analyze different ways of performing principal component analysis throughout three different approaches: robust covariance and correlation matrix estimation, projection pursuit approach and non-parametric maximum entropy algorithm. The objective of these approaches is the correction of the well known sensitivity to outliers of the classical method for principal component analysis. Due to their robustness, they perform very well in contaminated data, while the classical approach fails to preserve the characteristics of the core information.
We consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are obtained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve. For sparse functional data, these rates are equivalent to the optimal rates in nonparametric reg…
Principal component analysis continues to be a powerful tool in dimension reduction of high dimensional data. We assume a variance-diverging model and use the high-dimension, low-sample-size asymptotics to show that even though the principal component directions are not consistent, the sample and prediction principal component scores can be useful in revealing the population structure. We further show that these scores are biased, and the bias is asymptotically decomposed into rotation and scaling parts. We propose methods of bias-adjustment that are shown to be consistent and work well in the…
In this paper, we address the problem of dimension reduction for time series of functional data $(X_t\colon t\in\mathbb{Z})$. Such {\it functional time series} frequently arise, e.g., when a continuous-time process is segmented into some smaller natural units, such as days. Then each~$X_t$ represents one intraday curve. We argue that functional principal component analysis (FPCA), though a key technique in the field and a benchmark for any competitor, does not provide an adequate dimension reduction in a time-series setting. FPCA indeed is a {\it static} procedure which ignores the essential i…
In the course of the last century, Principal Component Analysis (PCA) have become one of the pillars of modern scientific methods. Although PCA is normally addressed as a statistical tool aiming at finding orthogonal directions on which the variance is maximized, its first introduction by Pearson at 1901 was done through defining a non-linear least-squares minimization problem of fitting a plane to scattered data points. Thus, it seems natural that PCA and linear least-squares regression are somewhat related, as they both aim at fitting planes to data points. In this paper, we present a connec…
Level, slope, and curvature are three commonly-believed principal components in interest rate term structure and are thus widely used in modeling. This paper characterizes the heterogeneity of how misspecified such models are through time. Presenting the orthonormal basis in the Nelson-Siegel model interpretable as the three factors, we design two nonparametric tests for whether the basis is equivalent to the data-driven functional principal component basis underlying the yield curve dynamics, considering the ordering of eigenfunctions or not, respectively. Eventually, we discover high dispers…
This work aims to give non-asymptotic results for estimating the first principal component of a multivariate random process. We first define the covariance function and the covariance operator in the multivariate case. We then define a projection operator. This operator can be seen as a reconstruction step from the raw data in the functional data analysis context. Next, we show that the eigenelements can be expressed as the solution to an optimization problem, and we introduce the LASSO variant of this optimization problem and the associated plugin estimator. Finally, we assess the estimator's…
We consider spatially dependent functional data collected under a geostatistics setting, where locations are sampled from a spatial point process. The functional response is the sum of a spatially dependent functional effect and a spatially independent functional nugget effect. Observations on each function are made on discrete time points and contaminated with measurement errors. Under the assumption of spatial stationarity and isotropy, we propose a tensor product spline estimator for the spatio-temporal covariance function. When a coregionalization covariance structure is further assumed, w…
Functional data analysis (FDA) methods have computational and theoretical appeals for some high dimensional data, but lack the scalability to modern large sample datasets. To tackle the challenge, we develop randomized algorithms for two important FDA methods: functional principal component analysis (FPCA) and functional linear regression (FLR) with scalar response. The two methods are connected as they both rely on the accurate estimation of functional principal subspace. The proposed algorithms draw subsamples from the large dataset at hand and apply FPCA or FLR over the subsamples to reduce…
In this paper, we introduce a computational framework for recovering a high-resolution approximation of an unknown function from its low-resolution indirect measurements as well as high-resolution training observations by merging the frameworks of generalized sampling and functional principal component analysis. In particular, we increase the signal resolution via a data driven approach, which models the function of interest as a realization of a random field and leverages a training set of observations generated via the same underlying random process. We study the performance of the resulting…
In this paper, we revisit the problem of clustering 1318 new variable stars found in the Milky way. Our recent work distinguishes these stars based on their light curves which are univariate series of brightness from the stars observed at discrete time points. This work proposes a new approach to look at these discrete series as continuous curves over time by transforming them into functional data. Then, functional principal component analysis is performed using these functional light curves. Clustering based on the significant functional principal components reveals two distinct groups of ecl…
Functional principal component analysis (FPCA) has been widely used to capture major modes of variation and reduce dimensions in functional data analysis. However, standard FPCA based on the sample covariance estimator does not work well in the presence of outliers. To address this challenge, a new robust functional principal component analysis approach based on the functional pairwise spatial sign (PASS) operator, termed PASS FPCA, is introduced where we propose estimation procedures for both eigenfunctions and eigenvalues with and without measurement error. Compared to existing robust FPCA m…
Online Learning of Functional Principal Component Analysis for Multidimensional Functional Data
School of Data Science, Fudan University（复旦大学数据科学学院）；Department of Statistics and Actuarial Science, Simon Fraser University（西蒙菲莎大学统计与精算科学系）；Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University（复旦大学脑启发智能科学与技术研究院）
Functional principal component analysis is essential in functional data analysis, but the inferences will become unconvincing when some non-Gaussian characteristics occur, such as heavy tail and skewness. The focus of this paper is to develop a robust functional principal component analysis methodology in dealing with non-Gaussian longitudinal data, for which sparsity and irregularity along with non-negligible measurement errors must be considered. We introduce a Kendall's $τ$ function whose particular properties make it a nice proxy for the covariance function in the eigenequation when handli…
Which components of the singular value decomposition of a signal-plus-noise data matrix are most informative for the inferential task of detecting or estimating an embedded low-rank signal matrix? Principal component analysis ascribes greater importance to the components that capture the greatest variation, i.e., the singular vectors associated with the largest singular values. This choice is often justified by invoking the Eckart-Young theorem even though that work addresses the problem of how to best represent a signal-plus-noise matrix using a low-rank approximation and not how to best_infe…
In recent times, functional data analysis (FDA) has been successfully applied in the field of high dimensional data classification. In this paper, we present a novel classification framework using functional data and classwise Principal Component Analysis (PCA). Our proposed method can be used in high dimensional time series data which typically suffers from small sample size problem. Our method extracts a piece wise linear functional feature space and is particularly suitable for hard classification problems.The proposed framework converts time series data into functional data and uses classw…
Functional binary datasets occur frequently in real practice, whereas discrete characteristics of the data can bring challenges to model estimation. In this paper, we propose a sparse logistic functional principal component analysis (SLFPCA) method to handle the functional binary data. The SLFPCA looks for local sparsity of the eigenfunctions to obtain convenience in interpretation. We formulate the problem through a penalized Bernoulli likelihood with both roughness penalty and sparseness penalty terms. An efficient algorithm is developed for the optimization of the penalized likelihood using…
This paper presents a new statistical method for clustering step data, a popular form of health record data easily obtained from wearable devices. Since step data are high-dimensional and zero-inflated, classical methods such as K-means and partitioning around medoid (PAM) cannot be applied directly. The proposed method is a novel combination of newly constructed variables that reflect the inherent features of step data, such as quantity, strength, and pattern, and a multivariate functional principal component analysis that can integrate all the features of the step data for clustering. The pr…
Existing approaches for multivariate functional principal component analysis are restricted to data on the same one-dimensional interval. The presented approach focuses on multivariate functional data on different domains that may differ in dimension, e.g. functions and images. The theoretical basis for multivariate functional principal component analysis is given in terms of a Karhunen-Loève Theorem. For the practically relevant case of a finite Karhunen-Loève representation, a relationship between univariate and multivariate functional principal component analysis is established. This offers…
Functional principal component analysis (FPCA) could become invalid when data involve non-Gaussian features. Therefore, we aim to develop a general FPCA method to adapt to such non-Gaussian cases. A Kenall's $τ$ function, which possesses identical eigenfunctions as covariance function, is constructed. The particular formulation of Kendall's $τ$ function makes it less insensitive to data distribution. We further apply it to the estimation of FPCA and study the corresponding asymptotic consistency. Moreover, the effectiveness of the proposed method is demonstrated through a comprehensive simulat…
Within the framework of functional data analysis, we develop principal component analysis for periodically correlated time series of functions. We define the components of the above analysis including periodic, operator-valued filters, score processes and the inversion formulas. We show that these objects are defined via convergent series under a simple condition requiring summability of the Hilbert-Schmidt norms of the filter coefficients, and that they poses optimality properties. We explain how the Hilbert space theory reduces to an approximate finite-dimensional setting which is implemente…
Stanford University（斯坦福大学）；MIT（麻省理工学院）；Microsoft Research, New England（新英格兰微软研究院）
We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any black-box routine for ridge regression. By avoiding explicit principal component analysis (PCA), our algorithm is the first with no runtime dependence on the number of top principal components. We show that it can be used to give a fast iterative method for the popular principal component regression problem, giving the first major runtime improvemen…
Functional principal component analysis has become the most important dimension reduction technique in functional data analysis. Based on B-spline approximation, functional principal components (FPCs) can be efficiently estimated by the expectation-maximization (EM) and the geometric restricted maximum likelihood (REML) algorithms under the strong assumption of Gaussianity on the principal component scores and observational errors. When computing the solution, the EM algorithm does not exploit the underlying geometric manifold structure, while the performance of REML is known to be unstable. I…
Conventional principal component analysis (PCA) finds a principal vector that maximizes the sum of second powers of principal components. We consider a generalized PCA that aims at maximizing the sum of an arbitrary convex function of principal components. We present a gradient ascent algorithm to solve the problem. For the kernel version of generalized PCA, we show that the solutions can be obtained as fixed points of a simple single-layer recurrent neural network. We also evaluate our algorithms on different datasets.
Motivated by modern observational studies, we introduce a class of functional models that expands nested and crossed designs. These models account for the natural inheritance of correlation structure from sampling design in studies where the fundamental sampling unit is a function or image. Inference is based on functional quadratics and their relationship with the underlying covariance structure of the latent processes. A computationally fast and scalable estimation procedure is developed for ultra-high dimensional data. Methods are illustrated in three examples: high-frequency accelerometer …
This tutorial reviews the main steps of the principal component analysis of a multivariate data set and its subsequent dimensional reduction on the grounds of identified dominant principal components. The underlying computations are demonstrated and performed by means of a script written in the statistical software package R.
Regularized variants of Principal Components Analysis, especially Sparse PCA and Functional PCA, are among the most useful tools for the analysis of complex high-dimensional data. Many examples of massive data, have both sparse and functional (smooth) aspects and may benefit from a regularization scheme that can capture both forms of structure. For example, in neuro-imaging data, the brain's response to a stimulus may be restricted to a discrete region of activation (spatial sparsity), while exhibiting a smooth response within that region. We propose a unified approach to regularized PCA which…
Understanding and predicting the electric consumption patterns in the short-, mid- and long-term, at the distribution and transmission level, is a fundamental asset for smart grids infrastructure planning, dynamic network reconfiguration, dynamic energy pricing and savings, and thus energy efficiency. This work introduces the Functional Principal Component Analysis (FPCA) as a versatile method to both investigate and predict, at different level of spatial aggregation, the consumption patterns. The method was applied to a unique and sensitive dataset that includes electric consumption and contr…
In this paper the exact linear relation between the leading eigenvectors of the modularity matrix and the singular vectors of an uncentered data matrix is developed. Based on this analysis the concept of a modularity component is defined, and its properties are developed. It is shown that modularity component analysis can be used to cluster data similar to how traditional principal component analysis is used except that modularity component analysis does not require data centering.
Principal component analysis (PCA) is a standard dimensionality reduction technique used in various research and applied fields. From an algorithmic point of view, classical PCA can be formulated in terms of operations on a multivariate Gaussian likelihood. As a consequence of the implied Gaussian formulation, the principal components are not robust to outliers. In this paper, we propose a modified formulation, based on the use of a multivariate Cauchy likelihood instead of the Gaussian likelihood, which has the effect of robustifying the principal components. We present an algorithm to comput…
Generalized principal component analysis (GLM-PCA) facilitates dimension reduction of non-normally distributed data. We provide a detailed derivation of GLM-PCA with a focus on optimization. We also demonstrate how to incorporate covariates, and suggest post-processing transformations to improve interpretability of latent factors.
We consider the problem of how many components to retain in the application of principal component analysis when the dimension is much higher than the number of observations. To estimate the number of components, we propose to sequentially test skewness of the squared lengths of residual scores that are obtained by removing leading principal components. The residual lengths are asymptotically left-skewed if all principal components with diverging variances are removed, and right-skewed if not. The proposed estimator is shown to be consistent, performs well in high-dimensional simulation studie…
