arXiv subjects

David E. Tyler

Publications and source records attributed to David E. Tyler.

At least 19 recordsLinked to original sources

arXiv2023

Robust and Resistant Regularized Covariance Matrices

We introduce a class of regularized M-estimators of multivariate scatter and show, analogous to the popular spatial sign covariance matrix (SSCM), that they possess high breakdown points. We also show that the SSCM can be viewed as an extreme member of this class. Unlike the SSCM, this class of estimators takes into account the shape of the contours of the data cloud when down-weighing observations. We also propose a median based cross validation criterion for selecting the tuning parameter for this class of regularized M-estimators. This cross validation criterion helps assure the resulting tuned scatter estimator is a good fit to the data as well as having a high breakdown point. A motivation for this new median based criterion is that when it is optimized over all possible scatter parameters, rather than only over the tuned candidates, it results in a new high breakdown point affine equivariant multivariate scatter statistic.

stat.ME↗

arXiv2022

Linear pooling of sample covariance matrices

We consider the problem of estimating high-dimensional covariance matrices of $K$-populations or classes in the setting where the sample sizes are comparable to the data dimension. We propose estimating each class covariance matrix as a distinct linear combination of all class sample covariance matrices. This approach is shown to reduce the estimation error when the sample sizes are limited, and the true class covariance matrices share a somewhat similar structure. We develop an effective method for estimating the coefficients in the linear combination that minimize the mean squared error under the general assumption that the samples are drawn from (unspecified) elliptically symmetric distributions possessing finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be an asymptotically unbiased estimator of the normalized covariance matrix as the dimension grows to infinity. We also show how the proposed method can be used in choosing the regularization parameters for multiple target matrices in a single class covariance matrix estimation problem. We assess the proposed method via numerical simulation studies including an application in global minimum variance portfolio optimization using real stock data.

stat.ME↗

arXiv2021

On the variability of the sample covariance matrix under complex elliptical distributions

We derive the form of the variance-covariance matrix for any affine equivariant matrix-valued statistics when sampling from complex elliptical distributions. We then use this result to derive the variance-covariance matrix of the sample covariance matrix (SCM) as well as its theoretical mean squared error (MSE) when finite fourth-order moments exist. Finally, illustrative examples of the formulas are presented.

math.ST↗

arXiv2020

Breakdown points of penalized and hybrid M-estimators of covariance

We introduce a class of hybrid M-estimators of multivariate scatter which, analogous to the popular spatial sign covariance matrix (SSCM), possess high breakdown points. We also show that the SSCM can be viewed as an extreme member of this class. Unlike the SSCM, but like the regular M-estimators of scatter, this new class of estimators takes into account the shape of the contours of the data cloud for downweighting observations.

stat.ME↗

arXiv2019

Shrinking the Sample Covariance Matrix using Convex Penalties on the Matrix-Log Transformation

For $q$-dimensional data, penalized versions of the sample covariance matrix are important when the sample size is small or modest relative to $q$. Since the negative log-likelihood under multivariate normal sampling is convex in $Σ^{-1}$, the inverse of its covariance matrix, it is common to add to it a penalty which is also convex in $Σ^{-1}$. More recently, Deng-Tsui (2013) and Yu et al.(2017) have proposed penalties which are functions of the eigenvalues of $Σ$, and are convex in $\log Σ$, but not in $Σ^{-1}$. The resulting penalized optimization problem is not convex in either $\log Σ$ or $Σ^{-1}$. In this paper, we note that this optimization problem is geodesically convex in $Σ$, which allows us to establish the existence and uniqueness of the corresponding penalized covariance matrices. More generally, we show the equivalence of convexity in $\log Σ$ and geodesic convexity for penalties on $Σ$ which are strictly functions of their eigenvalues. In addition, when using such penalties, we show that the resulting optimization problem reduces to to a $q$-dimensional convex optimization problem on the eigenvalues of $Σ$, which can then be readily solved via Newton-Raphson. Finally, we argue that it is better to apply these penalties to the shape matrix $Σ/(\det Σ)^{1/q}$ rather than to $Σ$ itself. A simulation study and an example illustrate the advantages of applying the penalty to the shape matrix.

math.ST↗

arXiv2018

Lassoing Eigenvalues

The properties of penalized sample covariance matrices depend on the choice of the penalty function. In this paper, we introduce a class of non-smooth penalty functions for the sample covariance matrix, and demonstrate how this method results in a grouping of the estimated eigenvalues. We refer to this method as "lassoing eigenvalues" or as the "elasso".

stat.ME↗

arXiv2017

Asymptotic and bootstrap tests for subspace dimension

Most linear dimension reduction methods proposed in the literature can be formulated using an appropriate pair of scatter matrices, see e.g. Ye and Weiss (2003), Tyler et al. (2009), Bura and Yang (2011), Liski et al. (2014) and Luo and Li (2016). The eigen-decomposition of one scatter matrix with respect to another is then often used to determine the dimension of the signal subspace and to separate signal and noise parts of the data. Three popular dimension reduction methods, namely principal component analysis (PCA), fourth order blind identification (FOBI) and sliced inverse regression (SIR) are considered in detail and the first two moments of subsets of the eigenvalues are used to test for the dimension of the signal space. The limiting null distributions of the test statistics are discussed and novel bootstrap strategies are suggested for the small sample cases. In all three cases, consistent test-based estimates of the signal subspace dimension are introduced as well. The asymptotic and bootstrap tests are compared in simulations and illustrated in real data examples.

stat.ME↗

arXiv2017

Asymptotic and bootstrap tests for the dimension of the non-Gaussian subspace

Dimension reduction is often a preliminary step in the analysis of large data sets. The so-called non-Gaussian component analysis searches for a projection onto the non-Gaussian part of the data, and it is then important to know the correct dimension of the non-Gaussian signal subspace. In this paper we develop asymptotic as well as bootstrap tests for the dimension based on the popular fourth order blind identification (FOBI) method.

math.ST↗

arXiv2016

Simultaneous penalized M-estimation of covariance matrices using geodesically convex optimization

A common assumption when sampling $p$-dimensional observations from $K$ distinct group is the equality of the covariance matrices. In this paper, we propose two penalized $M$-estimation approaches for the estimation of the covariance or scatter matrices under the broader assumption that they may simply be close to each other, and hence roughly deviate from some positive definite "center". The first approach begins by generating a pooled $M$-estimator of scatter based on all the data, followed by a penalised $M$-estimator of scatter for each group, with the penalty term chosen so that the individual scatter matrices are shrunk towards the pooled scatter matrix. In the second approach, we minimize the sum of the individual group $M$-estimation cost functions together with an additive joint penalty term which enforces some similarity between the individual scatter estimators, i.e. shrinkage towards a mutual center. In both approaches, we utilize the concept of geodesic convexity to prove the existence and uniqueness of the penalized solution under general conditions. We consider three specific penalty functions based on the Euclidean, the Riemannian, and the Kullback-Leibler distances. In the second approach, the distance based penalties are shown to lead to estimators of the mutual center that are related to the arithmetic, the Riemannian and the harmonic means of positive definite matrices, respectively. A penalty based on an ellipticity measure is also considered which is particularly useful for shape matrix estimators. Fixed point equations are derived for each penalty function and the benefits of the estimators are illustrated in regularized discriminant analysis problem.

stat.ME↗

arXiv2016

Geodesic Convexity and Regularized Scatter Estimators

As observed by Auderset et al. (2005) and Wiesel (2012), viewing covariance matrices as elements of a Riemannian manifold and using the concept of geodesic convexity provide useful tools for studying M-estimators of multivariate scatter. In this paper, we begin with a mathematically rigorous self-contained overview of Riemannian geometry on the space of symmetric positive definite matrices and of the notion of geodesic convexity. The overview contains both a review as well as new results. In particular, we introduce and utilize first and second order Taylor expansions with respect to geodesic parametrizations. This enables us to give sufficient conditions for a function to be geodesically convex. In addition, we introduce the concept of geodesic coercivity, which is important in establishing the existence of a minimum to a geodesic convex function. We also develop a general partial Newton algorithm for minimizing smooth and strictly geodesically convex functions. We then use these results to generate a fairly complete picture of the existence, uniqueness and computation of regularized M-estimators of scatter defined using additive geodescially convex penalty terms. Various such penalties are demonstrated which shrink an estimator towards the identity matrix or multiples of the identity matrix. Finally, we propose a cross-validation method for choosing the scaling parameter for the penalty function, and illustrate our results using a numerical example.

stat.ME↗

arXiv2016

On the eigenvalues of the spatial sign covariance matrix in more than two dimensions

We gather several results on the eigenvalues of the spatial sign covariance matrix of an elliptical distribution. It is shown that the eigenvalues are a one-to-one function of the eigenvalues of the shape matrix and that they are closer together than the latter. We further provide a one-dimensional integral representation of the eigenvalues, which facilitates their numerical computation.

stat.CO↗

arXiv2014

Breakdown Properties of the M-Estimators of Multivariate Scatter

The M-estimators of multivariate scatter are known to have breakdown points no greater than 1/(p+1), where p is the dimension of the data. In high dimension, the breakdown points are usually considered to be disappointingly low. This paper studies the breakdown problem in more detail. The exact breakdown points for the M-estimators of scatter are obtained and it is shown that their low values are primarily due to contamination restricted to some plane. If such "coplanar" contamination is not present, then there exists M-estimators which have breakdown points close to 1/2. The effect of coplanar contamination is further examined and is shown to be related to the singularity of the scatter matrix. Finally, the implications of the results of this paper on whether the low breakdown point is necessarily a bad feature and on multivariate outlier detection are briefly discussed. This paper is a reprint of an unpublished 1986 Rutgers Technical Report.

math.ST↗

arXiv2014

Regularized $M$-estimators of scatter matrix

In this paper, a general class of regularized $M$-estimators of scatter matrix are proposed which are suitable also for low or insufficient sample support (small $n$ and large $p$) problems. The considered class constitutes a natural generalization of $M$-estimators of scatter matrix (Maronna, 1976) and are defined as a solution to a penalized $M$-estimation cost function that depend on a pair $(α,β)$ of regularization parameters. We derive general conditions for uniqueness of the solution using concept of geodesic convexity. Since these conditions do not include Tyler's $M$-estimator, necessary and sufficient conditions for uniqueness of the penalized Tyler's cost function are established separately. For the regularized Tyler's $M$-estimator, we also derive a simple, closed form and data dependent solution for choosing the regularization parameter based on shape matrix matching in the mean squared sense. An iterative algorithm that converges to the solution of the regularized $M$-estimating equation is also provided. Finally, some simulations studies illustrate the improved accuracy of the proposed regularized $M$-estimators of scatter compared to their non-regularized counterparts in low sample support problems. An example of radar detection using normalized matched filter (NMF) illustrate that an adaptive NMF detector based on regularized $M$-estimators are able to maintain accurately the preset CFAR level and at at the same time provide similar probability of detection as the (theoretical) NMF detector.

stat.AP↗

arXiv2014

The spatial sign covariance matrix with unknown location

The consistency and asymptotic normality of the spatial sign covariance matrix with unknown location are shown. Simulations illustrate the different asymptotic behavior when using the mean and the spatial median as location estimator.

math.ST↗

arXiv2014

A cautionary note on robust covariance plug-in methods

Many multivariate statistical methods rely heavily on the sample covariance matrix. It is well known though that the sample covariance matrix is highly non-robust. One popular alternative approach for "robustifying" the multivariate method is to simply replace the role of the covariance matrix with some robust scatter matrix. The aim of this paper is to point out that in some situations certain properties of the covariance matrix are needed for the corresponding robust "plug-in" method to be a valid approach, and that not all scatter matrices necessarily possess these important properties. In particular, the following three multivariate methods are discussed in this paper: independent components analysis, observational regression and graphical modeling. For each case, it is shown that using a symmetrized robust scatter matrix in place of the covariance matrix results in a proper robust multivariate method.

stat.ME↗

arXiv2013

The asymptotic inadmissibility of the spatial sign covariance matrix for elliptically symmetric distributions

The asymptotic efficiency of the spatial sign covariance matrix (SSCM) relative to affine equivariant estimates of scatter is studied in detail. In particular, the SSCM is shown to be asymptoticaly inadmissible, i.e. the asymptotic variance-covariance matrix of the consistency corrected SSCM is uniformly smaller than that of its affine equivariant counterpart, namely Tyler's scatter matrix. Although the SSCM has often been recommended when one is interested in principal components analysis, the degree of the inefficiency of the SSCM is shown to be most severe in situations where principal components are of most interest. A finite sample simulation shows the inefficiency of the SSCM also holds for small sample sizes, and that the asymptotic relative efficiency is a good approximation to the finite sample efficiency for relatively modest sample sizes.

stat.ME↗

arXiv2013

Robust estimators for non-decomposable elliptical graphical models

Asymptotic properties of scatter estimators for elliptical graphical models are studied. Such models impose a given pattern of zeros on the inverse of the shape matrix of an elliptically distributed random vector. In particular, we introduce the class of graphical M-estimators and compare them to plug-in M-estimators. It turns out that, under suitable conditions, both approaches yield the same asymptotic efficiency. Furthermore, the results of this paper apply to both decomposable and non-decomposable graphical models and so generalize the results for decomposable models given by Vogel & Fried (2011) for the plug-in M-estimators.

math.ST↗

arXiv2012

Robust functional principal components: A projection-pursuit approach

In many situations, data are recorded over a period of time and may be regarded as realizations of a stochastic process. In this paper, robust estimators for the principal components are considered by adapting the projection pursuit approach to the functional data setting. Our approach combines robust projection-pursuit with different smoothing methods. Consistency of the estimators are shown under mild assumptions. The performance of the classical and robust procedures are compared in a simulation study under different contamination schemes.

math.ST↗