Projection estimators of Pickands dependence functions

The authors consider the construction of intrinsic estimators for the Pickands dependence function of an extreme‐value copula. They show how an arbitrary initial estimator can be modified to satisfy the required shape constraints. Their solution consists in projecting this estimator in the space of Pickands functions, which forms a closed and convex subset of a Hilbert space. As the solution is not explicit, they replace this functional parameter space by a sieve of finite‐dimensional subsets. They establish the asymptotic distribution of the projection estimator and its finite‐dimensional approximations, from which they conclude that the projected estimator is at least as efficient as the initial one.     

A continuous repeated measurements model:the experimental design problem with fixed time constraints

A repeated measurements model where the within-subject response is modeled as a continuous time regression is analyzed using reproducing kernel Hilbert space methods, Parzen (1961). The basic goal is to study the statistical design problem when experimental units and measurements are "free", but the total measurement time is fixed. This kind of situation arises when the experiment uses scarce resources such as space or oceanographic platforms or a magnetic resonance imaging (MRI) device. In this situation the only constraint is the total time allocated to the experiment.     

Optimal estimation in functional linear regression for sparse noise‐contaminated data

In this article, we propose a novel approach to fit a functional linear regression in which both the response and the predictor are functions. We consider the case where the response and the predictor processes are both sparsely sampled at random time points and are contaminated with random errors. In addition, the random times are allowed to be different for the measurements of the predictor and the response functions. The aforementioned situation often occurs in longitudinal data settings. To estimate the covariance and the cross‐covariance functions, we use a regularization method over a reproducing kernel Hilbert space. The estimate of the cross‐covariance function is used to obtain estimates of the regression coefficient function and of the functional singular components. We derive the convergence rates of the proposed cross‐covariance, the regression coefficient, and the singular component function estimators. Furthermore, we show that, under some regularity conditions, the estimator of the coefficient function has a minimax optimal rate. We conduct a simulation study and demonstrate merits of the proposed method by comparing it to some other existing methods in the literature. We illustrate the method by an example of an application to a real‐world air quality dataset. The Canadian Journal of Statistics 47: 524–559; 2019 © 2019 Statistical Society of Canada     

基于fused惩罚的稀疏主成分分析

本文旨在研究基于fused惩罚的稀疏主成分分析方法，以适用于相邻变量之间高度相关甚至完全相等的数据情形。首先，从回归分析角度出发，提出一种求解稀疏主成分的简便思路，给出一种广义的稀疏主成分模型—— GSPCA模型及其求解算法，并证明在惩罚函数取1-范数时，该模型与现有的稀疏主成分模型——SPC模型的求解结果一致。其次，本文提出将fused惩罚与主成分分析相结合，得到一种fused稀疏主成分分析方法，并从惩罚性矩阵分解和回归分析两个角度，给出两种模型形式。在理论上证明了两种模型的求解结果是一致的，故将其统称为FSPCA模型。模拟实验显示，FSPCA模型在处理相邻变量之间高度相关甚至完全相等的数据集上的表现良好。最后，将FSPCA模型应用于手写数字识别，发现与SPC模型相比，FSPCA模型所提取的主成分具备更好的解释性，这使得该模型更具实用价值。     

Nonlinear functional models for functional responses in reproducing kernel hilbert spaces

The author proposes an extension of reproducing kernel Hilbert space theory which provides a new framework for analyzing functional responses with regression models. The approach only presumes a general nonlinear regression structure, as opposed to existing linear regression models. The author proposes generalized cross‐validation for automatic smoothing parameter estimation. He illustrates the use of the new estimator both on real and simulated data.     

Some solutions to the multivariate Behrens–Fisher problem for dissimilarity‐based analyses

The essence of the generalised multivariate Behrens–Fisher problem (BFP) is how to test the null hypothesis of equality of mean vectors for two or more populations when their dispersion matrices differ. Solutions to the BFP usually assume variables are multivariate normal and do not handle high‐dimensional data. In ecology, species' count data are often high‐dimensional, non‐normal and heterogeneous. Also, interest lies in analysing compositional dissimilarities among whole communities in non‐Euclidean (semi‐metric or non‐metric) multivariate space. Hence, dissimilarity‐based tests by permutation (e.g., PERMANOVA, ANOSIM) are used to detect differences among groups of multivariate samples. Such tests are not robust, however, to heterogeneity of dispersions in the space of the chosen dissimilarity measure, most conspicuously for unbalanced designs. Here, we propose a modification to the PERMANOVA test statistic, coupled with either permutation or bootstrap resampling methods, as a solution to the BFP for dissimilarity‐based tests. Empirical simulations demonstrate that the type I error remains close to nominal significance levels under classical scenarios known to cause problems for the un‐modified test. Furthermore, the permutation approach is found to be more powerful than the (more conservative) bootstrap for detecting changes in community structure for real ecological datasets. The utility of the approach is shown through analysis of 809 species of benthic soft‐sediment invertebrates from 101 sites in five areas spanning 1960 km along the Norwegian continental shelf, based on the Jaccard dissimilarity measure.     

Predictive information and distance between past and future of a time series

A characterization for the nullity of the cosine angle between two subspaces of a Hilbert space is established. Given a time series x, we use this characterization in order to investigate the relationship between the notions of predictor space and distance between the information contained in the past and in the future of x. In particular, we prove that the predictor space of x coincides with the zero vector space {0} if and only if this distance achieves its maximum value.     

Truncated regular vines in high dimensions with application to financial data

Using only bivariate copulas as building blocks, regular vine copulas constitute a flexible class of high‐dimensional dependency models. However, the flexibility comes along with an exponentially increasing complexity in larger dimensions. In order to counteract this problem, we propose using statistical model selection techniques to either truncate or simplify a regular vine copula. As a special case, we consider the simplification of a canonical vine copula using a multivariate copula as previously treated by Heinen & Valdesogo ( 2009 ) and Valdesogo ( 2009 ). We validate the proposed approaches by extensive simulation studies and use them to investigate a 19‐dimensional financial data set of Norwegian and international market variables. The Canadian Journal of Statistics 40: 68–85; 2012 © 2012 Statistical Society of Canada     

Estimation for High‐Dimensional Linear Mixed‐Effects Models Using ℓ1‐Penalization

Abstract. We propose an ?₁‐penalized estimation procedure for high‐dimensional linear mixed‐effects models. The models are useful whenever there is a grouping structure among high‐dimensional observations, that is, for clustered data. We prove a consistency and an oracle optimality result and we develop an algorithm with provable numerical convergence. Furthermore, we demonstrate the performance of the method on simulated and a real high‐dimensional data set.     

再生核Hilbert空间展开的函数型回归模型变量选择

针对协变量是函数型、响应变量是标量的多元函数型回归模型,文章提出了函数系数基于再生核Hilbert空间展开的变量选择方法。首先,利用带积分余项的泰勒展开式和再生核Hilbert空间内积性质将模型转化为结构化形式,其次,通过自适应弹性网惩罚对结构化模型中的组间和组内系数同时进行压缩。结果证明了这种压缩估计具有Oracle性质,蒙特卡罗模拟结果也显示新方法在不同样本量、不同噪声和变量相关性干扰下均优于基于普通基函数展开的变量选择方法,且尤其适用于原始协变量高度相关的情形。最后,通过分析一个商品房平均销售价格影响因素数据演示了新方法的应用。     

Cross-validation methods in principal component analysis: A comparison

In principal component analysis (PCA), it is crucial to know how many principal components (PCs) should be retained in order to account for most of the data variability. A class of “objective” rules for finding this quantity is the class of cross-validation (CV) methods. In this work we compare three CV techniques showing how the performance of these methods depends on the covariance matrix structure. Finally we propose a rule for the choice of the “best” CV method and give an application to real data.     

A Note on Estimation in Hilbertian Linear Models

We study estimation and prediction in linear models where the response and the regressor variable both take values in some Hilbert space. Our main objective is to obtain consistency of a principal component‐based estimator for the regression operator under minimal assumptions. In particular, we avoid some inconvenient technical restrictions that have been used throughout the literature. We develop our theory in a time‐dependent setup that comprises as important special case the autoregressive Hilbertian model.     

Functional analysis of variance for Hilbert-valued multivariate fixed effect models

This paper presents new results on functional analysis of variance for fixed effect models with correlated Hilbert-valued Gaussian error components. The geometry of the reproducing kernel Hilbert space of the error term is considered in the computation of the total sum of squares, the residual sum of squares, and the sum of squares due to the regression. Under suitable linear transformation of the correlated functional data, the distributional characteristics of these statistics, their moment generating and characteristic functions, are derived. Fixed effect linear hypothesis testing is finally formulated in the Hilbert-valued multivariate Gaussian context considered.     

Kernel partial correlation: a novel approach to capturing conditional independence in graphical models for noisy data

Graphical models capture the conditional independence structure among random variables via existence of edges among vertices. One way of inferring a graph is to identify zero partial correlation coefficients, which is an effective way of finding conditional independence under a multivariate Gaussian setting. For more general settings, we propose kernel partial correlation which extends partial correlation with a combination of two kernel methods. First, a nonparametric function estimation is employed to remove effects from other variables, and then the dependence between remaining random components is assessed through a nonparametric association measure. The proposed approach is not only flexible but also robust under high levels of noise owing to the robustness of the nonparametric approaches.     

A self-reliant projected information criterion for the number of factors

Abstract

In this article, we propose a new projected PCA to determine the number of factors. We project variables of interest into the space spanned by cross sectional averages of variables. And then we construct the eigenvalue tests and the information criteria to estimate the number of factors. We derive the large sample consistency and conduct finite sample simulations to demonstrate the better performances of our estimators. In order to show the edge of our estimators in real data analysis, we revisit a large house price data set for which the number of factors is hard to select.     

Multivariate functional response low‐rank regression with an application to brain imaging data

We propose a multivariate functional response low‐rank regression model with possible high‐dimensional functional responses and scalar covariates. By expanding the slope functions on a set of sieve bases, we reconstruct the basis coefficients as a matrix. To estimate these coefficients, we propose an efficient procedure using nuclear norm regularization. We also derive error bounds for our estimates and evaluate our method using simulations. We further apply our method to the Human Connectome Project neuroimaging data to predict cortical surface motor task‐evoked functional magnetic resonance imaging signals using various clinical covariates to illustrate the usefulness of our results.     

A Potts‐mixture spatiotemporal joint model for combined magnetoencephalography and electroencephalography data

We develop a new methodology for determining the location and dynamics of brain activity from combined magnetoencephalography (MEG) and electroencephalography (EEG) data. The resulting inverse problem is ill‐posed and is one of the most difficult problems in neuroimaging data analysis. In our development we propose a solution that combines the data from three different modalities, magnetic resonance imaging (MRI), MEG and EEG, together. We propose a new Bayesian spatial finite mixture model that builds on the mesostate‐space model developed by Daunizeau & Friston [Daunizeau and Friston, NeuroImage 2007; 38, 67–81]. Our new model incorporates two major extensions: (i) We combine EEG and MEG data together and formulate a joint model for dealing with the two modalities simultaneously; (ii) we incorporate the Potts model to represent the spatial dependence in an allocation process that partitions the cortical surface into a small number of latent states termed mesostates. The cortical surface is obtained from MRI. We formulate the new spatiotemporal model and derive an efficient procedure for simultaneous point estimation and model selection based on the iterated conditional modes algorithm combined with local polynomial smoothing. The proposed method results in a novel estimator for the number of mixture components and is able to select active brain regions, which correspond to active variables in a high‐dimensional dynamic linear model. The methodology is investigated using synthetic data and simulation studies and then demonstrated on an application examining the neural response to the perception of scrambled faces. R software implementing the methodology along with several sample datasets are available at the following GitHub repository https://github.com/v2south/PottsMix . The Canadian Journal of Statistics 47: 688–711; 2019 © 2019 Statistical Society of Canada     

On a random series in a hilbert space

Let (S_n) be partial sums of a non-degenerate sequence of Identically and independently distributed random variables taking values in a separable Hilbert space. Then for 0 ≤ β ≤ 3/2, the series converges almost nowhere. For β > 3/2 this may not be true.     

Dihedral angles principal geodesic analysis using nonlinear statistics

Statistics, as one of the applied sciences, has great impacts in vast area of other sciences. Prediction of protein structures with great emphasize on their geometrical features using dihedral angles has invoked the new branch of statistics, known as directional statistics. One of the available biological techniques to predict is molecular dynamics simulations producing high-dimensional molecular structure data. Hence, it is expected that the principal component analysis (PCA) can response some related statistical problems particulary to reduce dimensions of the involved variables. Since the dihedral angles are variables on non-Euclidean space (their locus is the torus), it is expected that direct implementation of PCA does not provide great information in this case. The principal geodesic analysis is one of the recent methods to reduce the dimensions in the non-Euclidean case. A procedure to utilize this technique for reducing the dimension of a set of dihedral angles is highlighted in this paper. We further propose an extension of this tool, implemented in such way the torus is approximated by the product of two unit circle and evaluate its application in studying a real data set. A comparison of this technique with some previous methods is also undertaken.     

Principal component analysis of canine hip dysplasia phenotypes and their statistical power for genome-wide association mapping

The aims of this study were to undertake principal component analysis (PCA) of hip dysplasia (HD) and to examine the power of the principal components (PCs) in genome-wide association studies. A cohort of 278 dogs for PCA and that of 369 dogs for genotyping were used. The distraction index (DI), the dorsolateral subluxation (DLS) score, the Norberg angle (NA), and the extended-hip radiographic (EHR) score were used for the PCA. One thousand single-nucleotide polymorphisms (SNPs) (of 23,500) were used to simulate genetic locus sharing between the HD phenotypes and 1000 SNPs were used to calculate the genetic mapping power of the PCs. The DI and the DLS score (first group) reflected hip laxity and the NA and the EHR score (second group) reflected the congruency between the femoral head and acetabulum. The average hip measurements of the two groups reflected in the first PC captured 55% of total radiographic variation. The first four PCs captured 90% of the total variation. The PCs had higher statistical mapping power to detect pleiotropic quantitative trait loci (QTL) than the raw phenotypes. The PCA demonstrated for the first time that HD can be reduced mathematically into simpler components essential for its genetic dissection. Genes that contribute jointly to all four radiographic hip phenotypes can be detected by mapping their first four PCs, while those contributing to individual phenotypes can be mapped by association with the individual raw phenotype.