Sara Taskinen

So Many Variables: Joint Modeling in Community Ecology.

Technological advances have enabled a new class of multivariate models for ecology, with the potential now to specify a statistical model for abundances jointly across many taxa, to simultaneously explore interactions across taxa and the response of abundance to environmental variables. Joint models can be used for several purposes of interest to ecologists, including estimating patterns of residual correlation across taxa, ordination, multivariate inference about environmental effects and environment-by-trait interactions, accounting for missing predictors, and improving predictions in situations where one can leverage knowledge of some species to predict others. We demonstrate this by example and discuss recent computation tools and future directions.

Fall Incidence in Frail Older Women after Individualized Visual Feedback-Based Balance Training

Background: The knowledge concerning balance training actually lowering fall rates among frail older persons is limited. Objective: The aim of this study was to examine the effects of a 4-week individualized visual feedback-based balance training on the fall incidence during 1-year follow-up among frail older women living in residential care. Methods: Twenty-seven older women from 2 residential care homes were randomized into exercise (n = 20) and control (n = 7) groups. Balance measurements were carried out before and after a 4-week training period and falls were monitored by monthly diaries for 1 year. An interview about fear of falling and physical activity was completed before and after the intervention and after the 1-year follow-up. Results: A positive effect of balance training on fall incidence was found. A dynamic Poisson regression model showed that during the follow-up the monthly risk of falling was decreased in the exercise group compared to controls (risk ratio 0.398, 95% CI 0.174–0.911, p = 0.029). In addition, the exercise group reported a reduced fear of falling and increased physical activity after a training period but these changes declined during the follow-up period. Conclusion: Individualized visual feedback-based balance training was shown to be a promising method for fall prevention among frail older women. High compliance (97.5%) with the training program showed that carefully targeted training programs can be carried out among older people with health limitations.

Model‐based approaches to unconstrained ordination

Summary Unconstrained ordination is commonly used in ecology to visualize multivariate data, in particular, to visualize the main trends between different sites in terms of their species composition or relative abundance. Methods of unconstrained ordination currently used, such as non-metric multidimensional scaling, are algorithm-based techniques developed and implemented without directly accommodating the statistical properties of the data at hand. Failure to account for these key data properties can lead to misleading results. A model-based approach to unconstrained ordination can address this issue, and in this study, two types of models for ordination are proposed based on finite mixture models and latent variable models. Each method is capable of handling different data types and different forms of species response to latent gradients. Further strengths of the models are demonstrated via example and simulation. Advantages of model-based approaches to ordination include the following: residual analysis tools for checking assumptions to ensure the fitted model is appropriate for the data; model selection tools to choose the most appropriate model for ordination; methods for formal statistical inference to draw conclusions from the ordination; and improved efficiency, that is model-based ordination better recovers true relationships between sites, when used appropriately.

Multivariate Nonparametric Tests of Independence

New test statistics are proposed for testing whether two random vectors are independent. Gieser and Randles, as well as Taskinen, Kankainen, and Oja have introduced and discussed multivariate extensions of the quadrant test of Blomqvist. This article serves as a sequel to this work and presents new multivariate extensions of Kendalls tau and Spearmans rho statistics. Two different approaches are discussed. First, interdirection proportions are used to estimate the cosines of angles between centered observation vectors and between differences of observation vectors. Second, covariances between affine-equivariant multivariate signs and ranks are used. The test statistics arising from these two approaches appear to be asymptotically equivalent if each vector is elliptically symmetric. The spatial sign versions are easy to compute for data in common dimensions, and they provide practical, robust alternatives to normal-theory methods. Asymptotic theory is developed to approximate the finite-sample null distributions as well, as to calculate limiting Pitman efficiencies. Small-sample null permutation distributions are also described. A simple simulation study is used to compare the proposed tests with the classical Wilks test. Finally, the theory is illustrated by an example.

Tests of multinormality based on location vectors and scatter matrices

Classical univariate measures of asymmetry such as Pearson’s (mean-median)/σ or (mean-mode)/σ often measure the standardized distance between two separate location parameters and have been widely used in assessing univariate normality. Similarly, measures of univariate kurtosis are often just ratios of two scale measures. The classical standardized fourth moment and the ratio of the mean deviation to the standard deviation serve as examples. In this paper we consider tests of multinormality which are based on the Mahalanobis distance between two multivariate location vector estimates or on the (matrix) distance between two scatter matrix estimates, respectively. Asymptotic theory is developed to provide approximate null distributions as well as to consider asymptotic efficiencies. Limiting Pitman efficiencies for contiguous sequences of contaminated normal distributions are calculated and the efficiencies are compared to those of the classical tests by Mardia. Simulations are used to compare finite sample efficiencies. The theory is also illustrated by an example.

Fourth moments and independent component analysis

In independent component analysis it is assumed that the components of the observed random vector are linear combinations of latent independent random variables, and the aim is then to find an estimate for a transformation matrix back to these independent components. In the engineering literature, there are several traditional estimation procedures based on the use of fourth moments, such as FOBI (fourth order blind identification), JADE (joint approximate diagonalization of eigenmatrices), and FastICA, but the statistical properties of these estimates are not well known. In this paper various independent component functionals based on the fourth moments are discussed in detail, starting with the corresponding optimization problems, deriving the estimating equations and estimation algorithms, and finding asymptotic statistical properties of the estimates. Comparisons of the asymptotic variances of the estimates in wide independent component models show that in most cases JADE and the symmetric version of FastICA perform better than their competitors.

Deflation-based FastICA with adaptive choices of nonlinearities

Deflation-based FastICA is a popular method for independent component analysis. In the standard deflation-based approach the row vectors of the unmixing matrix are extracted one after another always using the same nonlinearities. In practice the user has to choose the nonlinearities and the efficiency and robustness of the estimation procedure then strongly depends on this choice as well as on the order in which the components are extracted. In this paper we propose a novel adaptive two-stage deflation-based FastICA algorithm that (i) allows one to use different nonlinearities for different components and (ii) optimizes the order in which the components are extracted. Based on a consistent preliminary unmixing matrix estimate and our theoretical results, the algorithm selects in an optimal way the order and the nonlinearities for each component from a finite set of candidates specified by the user. It is also shown that, for each component, the best possible nonlinearity is obtained by using the log-density function. The resulting ICA estimate is affine equivariant with a known asymptotic distribution. The excellent performance of the new procedure is shown with asymptotic efficiency and finite-sample simulation studies.

Robust estimation and inference for bivariate line-fitting in allometry

In allometry, bivariate techniques related to principal component analysis are often used in place of linear regression, and primary interest is in making inferences about the slope. We demonstrate that the current inferential methods are not robust to bivariate contamination, and consider four robust alternatives to the current methods -- a novel sandwich estimator approach, using robust covariance matrices derived via an influence function approach, Hubers M-estimator and the fast-and-robust bootstrap. Simulations demonstrate that Hubers M-estimators are highly efficient and robust against bivariate contamination, and when combined with the fast-and-robust bootstrap, we can make accurate inferences even from small samples.

Tests and estimates of shape based on spatial signs and ranks

Nonparametric procedures for testing and estimation of the shape matrix in the case of multivariate elliptic distribution are considered. Testing for sphericity is an important special case. The tests and estimates are based on the spatial sign and rank covariance matrices. The estimates based on the spatial sign covariance matrix and symmetrized spatial sign covariance matrix are Tylers [A distribution-free M-estimator of multivariate scatter, Ann. Statist. 15 (1987), pp. 234–251] shape matrix and and Dümbgens [On Tylers M-functional of scatter in high dimension, Ann. Inst. Statist. Math. 50 (1998), pp. 471–491] shape matrix, respectively. The test based on the spatial sign covariance matrix is the sign test statistic in the class of nonparametric tests proposed by Hallin and Paindaveine [Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity, Ann. Statist. 34 (2006), pp. 2707–2756]. New tests and estimates based on the spatial rank covariance matrix are proposed. The shape estimates introduced in the paper play an important role in the inner standardisation of the spatial sign and rank tests for multivariate location. Limiting distributions of the tests and estimates are reviewed and derived, and asymptotic efficiencies as well as finite-sample efficiencies of the proposed tests are compared with those of the classical modified Johns [Some optimal multivariate tests, Biometrika 58 (1971), pp. 123–127; The distribution of a statistic used for testing sphericity of normal distributions, Biometrika 59 (1972), pp. 169–173] test and the van der Waerden test (Hallin and Paindaveine, [Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity, Ann. Statist. 34 (2006), pp. 2707–2756]). The symmetrised spatial sign- and rank-based estimates and tests seem to have a very high efficiency in the multivariate normal case, and they are much better than the classical estimate (shape matrix based on the regular covariance matrix) and test (Johns test) for distributions with heavy tails.

Separation of Uncorrelated Stationary Time Series Using Autocovariance Matrices

In blind source separation, one assumes that the observed p time series are linear combinations of p latent uncorrelated weakly stationary time series. To estimate the unmixing matrix, which transforms the observed time series back to uncorrelated latent time series, second‐order blind identification (SOBI) uses joint diagonalization of the covariance matrix and autocovariance matrices with several lags. In this article, we find the limiting distribution of the well‐known symmetric SOBI estimator under general conditions and compare its asymptotical efficiencies to those of the recently introduced deflation‐based SOBI estimator. The theory is illustrated by some finite‐sample simulation studies.