Jyrki Möttönen

Multivariate spatial sign and rank methods

Rotation invariant multivariate spatial sign and rank tests and corresponding rotation equivariant estimates based on L1 type objective function with Euclidean distance are considered. Multivariate spatial analogues of sign and rank concepts, one- and two-sample sign tests, Wilcoxon rank sum and signed rank tests, median and Hodges-Lehmann estimates are reviewed and discussed. Connections with other generalization are considered. The testing theory is illustrated by two examples.

Affine-Invariant Multivariate One-Sample Signed-Rank Tests

Abstract Brown and Hettmansperger introduced affine-invariant bivariate analogs of the sign, rank, and signed-rank tests based on the Oja median. In this article affine-invariant k-variate extensions of the one-sample signed-rank test and the Hodges-Lehmann estimate are considered. The necessary distribution theory is developed, and asymptotic Pitman efficiencies with respect to Hotellings T 2 test under multivariate t distributions are tabulated. An application of the signed-rank tests to a repeated-measurement setting is presented.

Affine equivariant multivariate rank methods

The classical multivariate statistical methods (MANOVA, principal component analysis, multivariate multiple regression, canonical correlation, factor analysis, etc.) assume that the data come from a multivariate normal distribution and the derivations are based on the sample covariance matrix. The conventional sample covariance matrix and consequently the standard multivariate techniques based on it are, however, highly sensitive to outlying observations. In the paper a new, more robust and highly efficient, approach based on an affine equivariant rank covariance matrix is proposed and outlined. Affine equivariant multivariate rank concept is based on the multivariate Oja (Statist. Probab. Lett. 1 (1983) 327) median.

Multivariate nonparametric tests in a randomized complete block design

In this paper multivariate extensions of the Friedman and Page tests for the comparison of several treatments are introduced. Related unadjusted and adjusted treatment effect estimates for the multivariate response variable are also found and their properties discussed. The test statistics and estimates are analogous to the traditional univariate methods. In test constructions, the univariate ranks are replaced by multivariate spatial ranks (J. Nonparam. Statist. 5 (1995) 201). Asymptotic theory is developed to provide approximations for the limiting distributions of the test statistics and estimates. Limiting efficiencies of the tests and treatment effect estimates are found in the multivariate normal and t distribution cases. The tests are rotation invariant only, but affine invariant versions can be easily constructed. The theory is illustrated by an example.

The geometry of the affine invariant multivariate sign and rank methods

For k-variate data sets, hyperplanes determined by the k-subsets of observations and the normals of these hyperplanes are discussed. Their use in defining multi-variate invariant concepts of sign and rank and multivariate median (based on the Oja criterion) as well as in the computation of these methods is illustrated.

Prediction of Stem Measurements of Scots Pine

The aim of this study was to investigate prediction of stem measurements of Scots pine(Pinus sylvestris L.) for a modern computerized forest harvester. We are interested in the prediction of stem curve measurements when measurements of stems already processed and a short section of the stem under process are known. The techniques presented here are based on cubic smoothing splines and on multivariate regression models. One advantage of these methods is that they do not assume any special functional form of the stem curve. They can also be applied to the prediction of branch limits and stem height of pine stems.

Robust autocovariance estimation based on sign and rank correlation coefficients

This paper addresses the problem of estimating autocorrelation coefficients in the presence of outliers. Tools for characterizing the robustness are developed as well. Autocorrelation coefficients are obtained recursively by computing partial correlation (PARCOR) coefficients first. In order to achieve robustness, product-moment correlation coefficients are replaced by correlations computed using rank and sign correlation coefficients. Transformations relating rank and sign correlations and conventional correlations are exploited in the process. Finally, robust estimates of autocorrelation coefficients are obtained. They are used to construct an autocovariance matrix. Examples of the performance of the method are given by using a matrix constructed from autocorrelation coefficients and MUSIC subspace frequency estimator. The influence of outliers on conventional estimators and the robustness of the proposed method are illustrated in simulations as well.

A robust multiple-locus method for quantitative trait locus analysis of non-normally distributed multiple traits

Linear regression-based quantitative trait loci/association mapping methods such as least squares commonly assume normality of residuals. In genetics studies of plants or animals, some quantitative traits may not follow normal distribution because the data include outlying observations or data that are collected from multiple sources, and in such cases the normal regression methods may lose some statistical power to detect quantitative trait loci. In this work, we propose a robust multiple-locus regression approach for analyzing multiple quantitative traits without normality assumption. In our method, the objective function is least absolute deviation (LAD), which corresponds to the assumption of multivariate Laplace distributed residual errors. This distribution has heavier tails than the normal distribution. In addition, we adopt a group LASSO penalty to produce shrinkage estimation of the marker effects and to describe the genetic correlation among phenotypes. Our LAD-LASSO approach is less sensitive to the outliers and is more appropriate for the analysis of data with skewedly distributed phenotypes. Another application of our robust approach is on missing phenotype problem in multiple-trait analysis, where the missing phenotype items can simply be filled with some extreme values, and be treated as outliers. The efficiency of the LAD-LASSO approach is illustrated on both simulated and real data sets.

Robust Variable Selection and Coefficient Estimation in Multivariate Multiple Regression Using LAD-Lasso

Univariate and multivariate lasso estimation methods are highly sensitive to outlying observations because of the sum of squared norms term in the objective function. Using sum of norms (least absolute deviations, LAD) instead of sum of squared norms gives us a considerably more robust estimate for the regression coefficients. In this paper we combine LAD with the multivariate lasso method and illustrate its estimation using simulated data set that are similar to those typically seen in association genetics. We will shortly consider also how the significance testing is done for non-zero coefficients and how the tuning parameter value can be determined.

Testing of Multivariate Spline Growth Model

In this paper we present a new method for testing multivariate growth curves which is based on spline approximation and on F-test. We show how the basic spline regression model can easily be extended to the multiple response case. The method is illustrated using a real data set.