skip to main content

Explore Research Products in the NSF-PAR

Title: Nonparametric covariance estimation with shrinkage toward stationary models
Abstract

Estimation of an unstructured covariance matrix is difficult because of the challenges posed by parameter space dimensionality and the positive‐definiteness constraint that estimates should satisfy. We consider a general nonparametric covariance estimation framework for longitudinal data using the Cholesky decomposition of a positive‐definite matrix. The covariance matrix of time‐ordered measurements is diagonalized by a lower triangular matrix with unconstrained entries that are statistically interpretable as parameters for a varying coefficient autoregressive model. Using this dual interpretation of the Cholesky decomposition and allowing for irregular sampling time points, we treat covariance estimation as bivariate smoothing and cast it in a regularization framework for desired forms of simplicity in covariance models. Viewing stationarity as a form of simplicity or parsimony in covariance, we model the varying coefficient function with components depending on time lag and its orthogonal direction separately and penalize the components that capture the nonstationarity in the fitted function. We demonstrate construction of a covariance estimator using the smoothing spline framework. Simulation studies establish the advantage of our approach over alternative estimators proposed in the longitudinal data setting. We analyze a longitudinal dataset to illustrate application of the methodology and compare our estimates to those resulting from alternative models.

This article is categorized under:

Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

Statistical and Graphical Methods of Data Analysis > Nonparametric Methods

Algorithms and Computational Methods > Maximum Likelihood Methods

  more » « less
NSF-PAR ID:
10448013
Author(s) / Creator(s):
 ;  
Publisher / Repository:
Wiley Blackwell (John Wiley & Sons)
Date Published:
Journal Name:
WIREs Computational Statistics
Volume:
12
Issue:
6
ISSN:
1939-5108
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ( , WIREs Computational Statistics)
    Abstract

    Gaussian process (GP) is a staple in the toolkit of a spatial statistician. Well‐documented computing roadblocks in the analysis of large geospatial datasets using GPs have now largely been mitigated via several recent statistical innovations. Nearest neighbor Gaussian process (NNGP) has emerged as one of the leading candidates for such massive‐scale geospatial analysis owing to their empirical success. This article reviews the connection of NNGP to sparse Cholesky factors of the spatial precision (inverse‐covariance) matrix. Focus of the review is on these sparse Cholesky matrices which are versatile and have recently found many diverse applications beyond the primary usage of NNGP for fast parameter estimation and prediction in the spatial (generalized) linear models. In particular, we discuss applications of sparse NNGP Cholesky matrices to address multifaceted computational issues in spatial bootstrapping, simulation of large‐scale realizations of Gaussian random fields, and extensions to nonparametric mean function estimation of a GP using random forests. We also review a sparse‐Cholesky‐based model for areal (geographically aggregated) data that addresses long‐established interpretability issues of existing areal models. Finally, we highlight some yet‐to‐be‐addressed issues of such sparse Cholesky approximations that warrant further research.

    This article is categorized under:

    Algorithms and Computational Methods > Algorithms

    Algorithms and Computational Methods > Numerical Methods

     
    more » « less
  2. ; ( , Journal of the Royal Statistical Society Series B: Statistical Methodology)
    Summary

    We introduce a flexible marginal modelling approach for statistical inference for clustered and longitudinal data under minimal assumptions. This estimated estimating equations approach is semiparametric and the proposed models are fitted by quasi-likelihood regression, where the unknown marginal means are a function of the fixed effects linear predictor with unknown smooth link, and variance–covariance is an unknown smooth function of the marginal means. We propose to estimate the nonparametric link and variance–covariance functions via smoothing methods, whereas the regression parameters are obtained via the estimated estimating equations. These are score equations that contain nonparametric function estimates. The proposed estimated estimating equations approach is motivated by its flexibility and easy implementation. Moreover, if data follow a generalized linear mixed model, with either a specified or an unspecified distribution of random effects and link function, the model proposed emerges as the corresponding marginal (population-average) version and can be used to obtain inference for the fixed effects in the underlying generalized linear mixed model, without the need to specify any other components of this generalized linear mixed model. Among marginal models, the estimated estimating equations approach provides a flexible alternative to modelling with generalized estimating equations. Applications of estimated estimating equations include diagnostics and link selection. The asymptotic distribution of the proposed estimators for the model parameters is derived, enabling statistical inference. Practical illustrations include Poisson modelling of repeated epileptic seizure counts and simulations for clustered binomial responses.

     
    more » « less
  3. ; ; ( , Methods in Ecology and Evolution)
    Abstract

    The quantification of Hutchinson's n‐dimensional hypervolume has enabled substantial progress in community ecology, species niche analysis and beyond. However, most existing methods do not support a partitioning of the different components of hypervolume. Such a partitioning is crucial to address the ‘curse of dimensionality’ in hypervolume measures and interpret the metrics on the original niche axes instead of principal components. Here, we propose the use of multivariate normal distributions for the comparison of niche hypervolumes and introduce this as the multivariate‐normal hypervolume (MVNH) framework (R package available onhttps://github.com/lvmuyang/MVNH).

    The framework provides parametric measures of the size and dissimilarity of niche hypervolumes, each of which can be partitioned into biologically interpretable components. Specifically, the determinant of the covariance matrix (i.e. the generalized variance) of a MVNH is a measure of total niche size, which can be partitioned into univariate niche variance components and a correlation component (a measure of dimensionality, i.e. the effective number of independent niche axes standardized by the number of dimensions). The Bhattacharyya distance (BD; a function of the geometric mean of two probability distributions) between two MVNHs is a measure of niche dissimilarity. The BD partitions total dissimilarity into the components of Mahalanobis distance (standardized Euclidean distance with correlated variables) between hypervolume centroids and the determinant ratio which measures hypervolume size difference. The Mahalanobis distance and determinant ratio can be further partitioned into univariate divergences and a correlation component.

    We use empirical examples of community‐ and species‐level analysis to demonstrate the new insights provided by these metrics. We show that the newly proposed framework enables us to quantify the relative contributions of different hypervolume components and to connect these analyses to the ecological drivers of functional diversity and environmental niche variation.

    Our approach overcomes several operational and computational limitations of popular nonparametric methods and provides a partitioning framework that has wide implications for understanding functional diversity, niche evolution, niche shifts and expansion during biotic invasions, etc.

     
    more » « less
  4. ; ; ; ( , Methods in Ecology and Evolution)
    Abstract

    Understanding animal movement often relies upon telemetry and biologging devices. These data are frequently used to estimate latent behavioural states to help understand why animals move across the landscape. While there are a variety of methods that make behavioural inferences from biotelemetry data, some features of these methods (e.g. analysis of a single data stream, use of parametric distributions) may limit their generality to reliably discriminate among behavioural states.

    To address some of the limitations of existing behavioural state estimation models, we introduce a nonparametric Bayesian framework called the mixed‐membership method for movement (M4), which is available within the open‐sourcebayesmoveR package. This framework can analyse multiple data streams (e.g. step length, turning angle, acceleration) without relying on parametric distributions, which may capture complex behaviours more successfully than current methods. We tested our Bayesian framework using simulated trajectories and compared model performance against two segmentation methods (behavioural change point analysis (BCPA) and segclust2d), one machine learning method [expectation‐maximization binary clustering (EMbC)] and one type of state‐space model [hidden Markov model (HMM)]. We also illustrated this Bayesian framework using movements of juvenile snail kitesRostrhamus sociabilisin Florida, USA.

    The Bayesian framework estimated breakpoints more accurately than the other segmentation methods for tracks of different lengths. Likewise, the Bayesian framework provided more accurate estimates of behaviour than the other state estimation methods when simulations were generated from less frequently considered distributions (e.g. truncated normal, beta, uniform). Three behavioural states were estimated from snail kite movements, which were labelled as ‘encamped’, ‘area‐restricted search’ and ‘transit’. Changes in these behaviours over time were associated with known dispersal events from the nest site, as well as movements to and from possible breeding locations.

    Our nonparametric Bayesian framework estimated behavioural states with comparable or superior accuracy compared to the other methods when step lengths and turning angles of simulations were generated from less frequently considered distributions. Since the most appropriate parametric distributions may not be obvious a priori, methods (such as M4) that are agnostic to the underlying distributions can provide powerful alternatives to address questions in movement ecology.

     
    more » « less
  5. ; ; ; ( , Canadian Journal of Statistics)
    Abstract

    The generalized semiparametric mixed varying‐coefficient effects model for longitudinal data can accommodate a variety of link functions and flexibly model different types of covariate effects, including time‐constant, time‐varying and covariate‐varying effects. The time‐varying effects are unspecified functions of time and the covariate‐varying effects are nonparametric functions of a possibly time‐dependent exposure variable. A semiparametric estimation procedure is developed that uses local linear smoothing and profile weighted least squares, which requires smoothing in the two different and yet connected domains of time and the time‐dependent exposure variable. The asymptotic properties of the estimators of both nonparametric and parametric effects are investigated. In addition, hypothesis testing procedures are developed to examine the covariate effects. The finite‐sample properties of the proposed estimators and testing procedures are examined through simulations, indicating satisfactory performances. The proposed methods are applied to analyze the AIDS Clinical Trial Group 244 clinical trial to investigate the effects of antiretroviral treatment switching in HIV‐infected patients before and after developing the T215Y antiretroviral drug resistance mutation.The Canadian Journal of Statistics47: 352–373; 2019 © 2019 Statistical Society of Canada

     
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email