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Tests for separability in nonparametric covariance operators of random surfaces
Abstract
We consider the problem of testing for separability in nonparametric covariance operators of multidimensional functional data is considered. We cast the problem in a tensor product of Hilbert space framework, where the role of the partial trace operator is emphasized, and the tests proposed are computationally tractable. An applications to acoustic phonetic data is also presented.
... Mateu (2011) investigated the spatially correlated functional data based on geostatistical and point process contexts and provided a framework for extending multivariate geostatistical approaches in the functional context. Recently, Aston et al. (2015) used a nonparametric approach for estimating the spatial correlation along with providing a test for the separability of the spatial and temporal correlation. Paul and Peng (2011) discussed a nonparametric method similar to PACE to estimate fPCs and proved that the L 2 risk of their estimator achieves optimal nonparametric rate under mild correlation regime when the number of observations per curve is bounded. ...
This paper focuses on the analysis of spatially correlated functional data.
The between-curve correlation is modeled by correlating functional principal
component scores of the functional data. We propose a Spatial Principal
Analysis by Conditional Expectation framework to explicitly estimate spatial
correlations and reconstruct individual curves. This approach works even when
the observed data per curve are sparse. Assuming spatial stationarity,
empirical spatial correlations are calculated as the ratio of eigenvalues of
the smoothed covariance surface and cross-covariance
surface at locations indexed by i and j. Then a
anisotropy Mat\'ern spatial correlation model is fit to empirical correlations.
Finally, principal component scores are estimated to reconstruct the sparsely
observed curves. This framework can naturally accommodate arbitrary covariance
structures, but there is an enormous reduction in computation if one can assume
the separability of temporal and spatial components. We propose hypothesis
tests to examine the separability as well as the isotropy effect of spatial
correlation. Simulation studies and applications of empirical data show
improvements in the curve reconstruction using our framework over the method
where curves are assumed to be independent. In addition, we show that the
asymptotic properties of estimates in uncorrelated case still hold in our case
if 'mild' spatial correlation is assumed.
In this study, a random subspace-based function tree (RSFT) was developed for landslide susceptibility modeling, and by comparing with a bagging-based function tree (BFT), classification regression tree (CART), and Naïve-Bayes tree (NBTree) Classifier, to judge the performance difference between the hybrid model and the single models. In the first step, according to the characteristics of the geological environment and previous literature, 12 landslide conditioning factors were selected, including aspect, slope, profile curvature, plan curvature, elevation, topographic wetness index (TWI), lithology, and normalized difference vegetation index (NDVI), land use, soil, distance to river and distance to the road. Secondly, 328 historical landslides were randomly divided into a training group and a validation group in a ratio of 70/30, and the important analysis of landslide points and conditional factors was carried out using the functional tree (FT) model. In the third step, all data are loaded into FT, RSFT, BFT, CART, and NBTree models for the generation of landslide susceptibility maps (LSM). Comparisons were made by the area under the receiver operating characteristic curve (AUC) to determine efficiency and effectiveness. According to the verification results, the five models selected this time all perform reasonably, but the RSFT model has the highest prediction rate (AUC = 0.838), which is better than the other three single machine learning models. The results of this study also demonstrated that the hybrid model generally improves the predictive power of the benchmark landslide susceptibility models.
We present a new methodology and accompanying theory to test for separability
of spatio-temporal functional data. In spatio-temporal statistics, separability
is a common simplifying assumption concerning the covariance structure which,
if true, can greatly increase estimation accuracy and inferential power. While
our focus is on testing for the separation of space and time in spatio-temporal
data, our methods can be applied to any area where separability is useful,
including biomedical imaging. We present three tests, one being a functional
extension of the Monte Carlo likelihood method of Mitchell et. al. (2005),
while the other two are based on quadratic forms. Our tests are based on
asymptotic distributions of maximum likelihood estimators, and do not require
Monte Carlo or bootstrap replications. The specification of the joint
asymptotic distribution of these estimators is the main theoretical
contribution of this paper. It can be used to derive many other tests. The main
methodological finding is that one of the quadratic form methods, which we call
a norm approach, emerges as a clear winner in terms of finite sample
performance in nearly every setting we considered. The norm approach focuses
directly on the Frobenius distance between the spatio-temporal covariance
function and its separable approximation. We demonstrate the efficacy of our
methods via simulations and an application to Irish wind data.
The assumption of separability of covariance operators in the different
directions for a random image or hypersurface can be of substantial use in
applications, especially in situations where the accurate estimation of the
full covariance structure is unfeasible, either for computational reasons or
due to a small sample size. However, inferential tools to verify this
assumption are somewhat lacking in high-dimensional or functional settings
where this assumption is most relevant. We propose here to test separability by
focusing on K-dimensional projections of the difference between the covariance
operator and its separable approximation. The subspace we project onto is one
generated by the eigenfunctions estimated under the separability hypothesis,
negating the need to ever estimate the full nonseparable covariance. This
approach also allows for a completely non-parametric form for the covariance
structure. We show that the rescaled difference of the sample covariance
operator with its separable approximation is asymptotically Gaussian, without
Gaussian assumptions on the observations, with a variance that depends on the
second and fourth order moments of the observations. As a by-product of this
result, we derive asymptotically pivotal tests under Gaussian assumptions and
we propose bootstrap methods for approximating the distribution of the test
statistics when multiple eigendirections are taken into account. We probe the
finite sample performance through simulations studies and we present the
application of our methodology to the covariance structure of log-spectrogram
images from a phonetic linguistics dataset.
A framework is developed for inference concerning the covariance operator of a functional random process, where the covariance
operator itself is an object of interest for statistical analysis. Distances for comparing positive-definite covariance matrices
are either extended or shown to be inapplicable to functional data. In particular, an infinite-dimensional analogue of the
Procrustes size-and-shape distance is developed. Convergence of finite-dimensional approximations to the infinite-dimensional
distance metrics is also shown. For inference, a Fréchet estimator of both the covariance operator itself and the average
covariance operator is introduced. A permutation procedure to test the equality of the covariance operators between two groups
is also considered. Additionally, the use of such distances for extrapolation to make predictions is explored. As an example
of the proposed methodology, the use of covariance operators has been suggested in a philological study of cross-linguistic
dependence as a way to incorporate quantitative phonetic information. It is shown that distances between languages derived
from phonetic covariance functions can provide insight into the relationships between the Romance languages.
Geostatistical approaches to modeling spatio-temporal data rely on parametric covariance models and rather stringent assumptions, such as stationarity, separability and full symmetry. This paper reviews recent advances in the literature on space-time covariance functions in light of the aforementioned notions, which are illustrated using wind data from Ireland. Experiments with time-forward kriging predictors suggest that the use of more complex and more realistic covariance models results in improved predictive performance.
A fully automated smoothing procedure for uniformly sampled datasets is described. The algorithm, based on a penalized least squares method, allows fast smoothing of data in one and higher dimensions by means of the discrete cosine transform. Automatic choice of the amount of smoothing is carried out by minimizing the generalized cross-validation score. An iteratively weighted robust version of the algorithm is proposed to deal with occurrences of missing and outlying values. Simplified Matlab codes with typical examples in one to three dimensions are provided. A complete user-friendly Matlab program is also supplied. The proposed algorithm, which is very fast, automatic, robust and requiring low storage, provides an efficient smoother for numerous applications in the area of data analysis.
In recent years there has been explosive growth in the number of neuroimaging studies performed using functional Magnetic Resonance Imaging (fMRI). The field that has grown around the acquisition and analysis of fMRI data is intrinsically interdisciplinary in nature and involves contributions from researchers in neuroscience, psychology, physics and statistics, among others. A standard fMRI study gives rise to massive amounts of noisy data with a complicated spatio-temporal correlation structure. Statistics plays a crucial role in understanding the nature of the data and obtaining relevant results that can be used and interpreted by neuroscientists. In this paper we discuss the analysis of fMRI data, from the initial acquisition of the raw data to its use in locating brain activity, making inference about brain connectivity and predictions about psychological or disease states. Along the way, we illustrate interesting and important issues where statistics already plays a crucial role. We also seek to illustrate areas where statistics has perhaps been underutilized and will have an increased role in the future. Comment: Published in at http://dx.doi.org/10.1214/09-STS282 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)
Data collected by scientists are increasingly in the form of trajectories or curves. Often these can be viewed as realizations of a composite process driven by both amplitude and time variation. We consider the situation in which functional variation is dominated by time variation, and develop a curve-synchronization method that uses every trajectory in the sample as a reference to obtain pairwise warping functions in the first step. These initial pairwise warping functions are then used to create improved estimators of the underlying individual warping functions in the second step. A truncated averaging process is used to obtain robust estimation of individual warping functions. The method compares well with other available time-synchronization approaches and is illustrated with Berkeley growth data and gene expression data for multiple sclerosis. Copyright 2008, Oxford University Press.
The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure is unfeasible, either for computational reasons, or due to a small sample size. However, inferential tools to verify this assumption are somewhat lacking in high-dimensional or functional data analysis settings, where this assumption is most relevant. We propose here to test separability by focusing on K-dimensional projections of the difference between the covariance operator and a nonparametric separable approximation. The subspace we project onto is one generated by the eigenfunctions of the covariance operator estimated under the separability hypothesis, negating the need to ever estimate the full nonseparable covariance. We show that the rescaled difference of the sample covariance operator with its separable approximation is asymptotically Gaussian. As a by-product of this result, we derive asymptotically pivotal tests under Gaussian assumptions, and propose bootstrap methods for approximating the distribution of the test statistics. We probe the finite sample performance through simulations studies, and present an application to log-spectrogram images from a phonetic linguistics dataset.
Many modern neuroimaging studies acquire large spatial images of the brain observed sequentially over time. Such data are often stored in the forms of matrices. To model these matrix-variate data we introduce a class of separable processes using explicit latent process modeling. To account for the size and two-way structure of the data, we extend principal component analysis to achieve dimensionality reduction at the individual level. We introduce necessary identifiability conditions for each model and develop scalable estimation procedures. The method is motivated by and applied to a functional magnetic resonance imaging study designed to analyze the relationship between pain and brain activity.
With the advance of modern technology, more and more data are being recorded continuously during a time interval or intermittently at several discrete time points. These are both examples of functional data, which has become a commonly encountered type of data. Functional data analysis (FDA) encompasses the statistical methodology for such data. Broadly interpreted, FDA deals with the analysis and theory of data that are in the form of functions. This paper provides an overview of FDA, starting with simple statistical notions such as mean and covariance functions, then covering some core techniques, the most popular of which is functional principal component analysis (FPCA). FPCA is an important dimension reduction tool, and in sparse data situations it can be used to impute functional data that are sparsely observed. Other dimension reduction approaches are also discussed. In addition, we review another core technique, functional linear regression, as well as clustering and classification of functional data. Beyond linear and single- or multiple- index methods, we touch upon a few nonlinear approaches that are promising for certain applications. They include additive and other nonlinear functional regression models and models that feature time warping, manifold learning, and empirical differential equations. The paper concludes with a brief discussion of future directions.
We introduce a simple and interpretable model for functional data analysis for situations where the observations at each location are functional rather than scalar. This new approach is based on a tensor product representation of the function-valued process and utilizes eigenfunctions of marginal kernels. The resulting marginal principal components and product principal components are shown to have nice properties. Given a sample of independent realizations of the underlying function-valued stochastic process, we propose straightforward fitting methods to obtain the components of this model and to establish asymptotic consistency and rates of convergence for the estimates proposed. The methods are illustrated by modelling the dynamics of annual fertility profile functions for 17 countries. This analysis demonstrates that the approach proposed leads to insightful interpretations of the model components and interesting conclusions.
We present a unified statistical theory for assessing the significance of apparent signal observed in noisy difference images. The results are usable in a wide range of applications, including fMRI, but are discussed with particular reference to PET images which represent changes in cerebral blood flow elicited by a specific cognitive or sensorimotor task. Our main result is an estimate of the P-value for local maxima of Gaussian, t, χ2 and F fields over search regions of any shape or size in any number of dimensions. This unifies the P-values for large search areas in 2-D (Friston et al. [1991]: J Cereb Blood Flow Metab 11:690–699) large search regions in 3-D (Worsley et al. [1992]: J Cereb Blood Flow Metab 12:900–918) and the usual uncorrected P-value at a single pixel or voxel.
With the advance of modern technology, more and more data are being recorded
continuously during a time interval or intermittently at several discrete time
points. They are both examples of "functional data", which have become a
prevailing type of data. Functional Data Analysis (FDA) encompasses the
statistical methodology for such data. Broadly interpreted, FDA deals with the
analysis and theory of data that are in the form of functions. This paper
provides an overview of FDA, starting with simple statistical notions such as
mean and covariance functions, then covering some core techniques, the most
popular of which is Functional Principal Component Analysis (FPCA). FPCA is an
important dimension reduction tool and in sparse data situations can be used to
impute functional data that are sparsely observed. Other dimension reduction
approaches are also discussed. In addition, we review another core technique,
functional linear regression, as well as clustering and classification of
functional data. Beyond linear and single or multiple index methods we touch
upon a few nonlinear approaches that are promising for certain applications.
They include additive and other nonlinear functional regression models, such as
time warping, manifold learning, and dynamic modeling with empirical
differential equations. The paper concludes with a brief discussion of future
directions.
We analyze geo-referenced high-dimensional data describing the use over time of the mobile-phone network in the urban area of Milan, Italy. Aim of the analysis is to identify subregions of the metropolitan area of Milan sharing a similar pattern along time, and possibly related to activities taking place in specific locations and/or times within the city. To tackle this problem, we develop a non-parametric method for the analysis of spatially dependent functional data, named Bagging Voronoi Treelet analysis. This novel approach integrates the treelet decomposition with a proper treatment of spatial dependence, obtained through a Bagging Voronoi strategy. The latter relies on the aggregation of different replicates of the analysis, each involving a set of functional local representatives associated to random Voronoi-based neighborhoods covering the investigated area. Results clearly point out some interesting temporal patterns interpretable in terms of population density mobility (e.g., daily work activities in the tertiary district, leisure activities in residential areas in the evenings and in the weekend, commuters movements along the highways during rush hours, and localized mob concentrations related to occasional events). Moreover we perform simulation studies, aimed at investigating the properties and performances of the method, and whose description is available online as Supplementary material.
Longitudinal imaging studies have moved to the forefront of medical research due to their ability to characterize spatio-temporal features of biological structures across the lifespan. Valid inference in longitudinal imaging requires enough flexibility of the covariance model to allow reasonable fidelity to the true pattern. On the other hand, the existence of computable estimates demands a parsimonious parameterization of the covariance structure. Separable (Kronecker product) covariance models provide one such parameterization in which the spatial and temporal covariances are modeled separately. However, evaluating the validity of this parameterization in high-dimensions remains a challenge. Here we provide a scientifically informed approach to assessing the adequacy of separable (Kronecker product) covariance models when the number of observations is large relative to the number of independent sampling units (sample size). We address both the general case, in which unstructured matrices are considered for each covariance model, and the structured case, which assumes a particular structure for each model. For the structured case, we focus on the situation where the within subject correlation is believed to decrease exponentially in time and space as is common in longitudinal imaging studies. However, the provided framework equally applies to all covariance patterns used within the more general multivariate repeated measures context. Our approach provides useful guidance for high dimension, low sample size data that preclude using standard likelihood based tests. Longitudinal medical imaging data of caudate morphology in schizophrenia illustrates the approaches appeal.
Most statistical analyses involve one or more observations taken on each of a number of individuals in a sample, with the aim of making inferences about the general population from which the sample is drawn. In an increasing number of fields, these observations are curves or images. Curves and images are examples of functions, since an observed intensity is available at each point on a line segment, a portion of a plane, or a volume. For this reason, we call observed curves and images ‘functional data,’ and statistical methods for analyzing such data are described by the term ‘functional data analysis.’ It is the smoothness of the processes generating functional data that differentiates this type of data from more classical multivariate observations. This smoothness means that we can work with the information in the derivatives of functions or images. This article includes several illustrative examples.
We present a unified statistical theory for assessing the significance of apparent signal observed in noisy difference images. The results are usable in a wide range of applications, including fMRI, but are discussed with particular reference to PET images which represent changes in cerebral blood flow elicited by a specific cognitive or sensorimotor task. Our main result is an estimate of the P-value for local maxima of Gaussian, t, chi(2) and F fields over search regions of any shape or size in any number of dimensions. This unifies the P-values for large search areas in 2-D (Friston et al. [1991]: J Cereb Blood Flow Metab 11:690-699) large search regions in 3-D (Worsley et al. [1992]: J Cereb Blood Flow Metab 12:900-918) and the usual uncorrected P-value at a single pixel or voxel.
Fundamentals of the Theory of Operator Algebras: Elementary theory. Number v. 1 in Fundamentals of the Theory of Operator Algebras
- R V Kadison
- J R Ringrose