Biplots of compositional data

Summary. The singular value decomposition and its interpretation as a linear biplot have proved to be a powerful tool for analysing many forms of multivariate data. Here we adapt biplot methodology to the specific case of compositional data consisting of positive vectors each of which is constrained to have unit sum. These relative variation biplots have properties relating to the special features of compositional data: the study of ratios, subcompositions and models of compositional relationships. The methodology is applied to a data set consisting of six-part colour compositions in 22 abstract paintings, showing how the singular value decomposition can achieve an accurate biplot of the colour ratios and how possible models interrelating the colours can be diagnosed.     

Separating trend and cyclical dynamics in state space models with exogenous inputs

In the present study we compare three state rotation methods in modelling the impact of the US economy on the Finnish economy, i.e. Schur decomposition, eigenvalue analysis and singular value decomposition. Singular value decomposition is seen to provide a robust approximation of the state rotation in most cases studied, irrespective of whether the characteristic roots of the state transition matrix are complex. Thus, singular value decomposition seems to be a viable computational device not only in estimating the system matrices of the state space model, but also in state rotation, as compared to the more involved techniques based on eigenvalue analysis or Schur decomposition.     

Enriched biplots for canonical correlation analysis

This paper discusses biplots of the between-set correlation matrix obtained by canonical correlation analysis. It is shown that these biplots can be enriched with the representation of the cases of the original data matrices. A representation of the cases that is optimal in the generalized least squares sense is obtained by the superposition of a scatterplot of the canonical variates on the biplot of the between-set correlation matrix. Goodness of fit statistics for all correlation and data matrices involved in canonical correlation analysis are discussed. It is shown that adequacy and redundancy coefficients are in fact statistics that express the goodness of fit of the original data matrices in the biplot. The within-set correlation matrix that is represented in standard coordinates always has a better goodness of fit than the within-set correlation matrix that is represented in principal coordinates. Given certain scalings, the scalar products between variable vectors approximate correlations better than the cosines of angles between variable vectors. Several data sets are used to illustrate the results.     

Some new aspects of taxicab correspondence analysis

Correspondence analysis (CA) and nonsymmetric correspondence analysis are based on generalized singular value decomposition, and, in general, they are not equivalent. Taxicab correspondence analysis (TCA) is a \(\hbox {L}_{1}\) variant of CA, and it is based on the generalized taxicab singular value decomposition (GTSVD). Our aim is to study the taxicab variant of nonsymmetric correspondence analysis. We find that for diagonal metric matrices GTSVDs of a given data set are equivalent; from which we deduce the equivalence of TCA and taxicab nonsymmetric correspondence analysis. We also attempt to show that TCA stays as close as possible to the original correspondence matrix without calculating a dissimilarity (or similarity) measure between rows or columns. Further, we discuss some new geometric and distance aspects of TCA.     

On equivalence of fractional factorial designs based on singular value decomposition

The singular value decomposition of a real matrix always exists and is essentially unique. Based on the singular value decomposition of the design matrices of two general 2-level fractional factorial designs, new necessary and sufficient conditions for the determination of combinatorial equivalence or non-equivalence of the corresponding designs are derived. Equivalent fractional factorial designs have identical statistical properties for estimation of factorial contrasts and for model fitting. Non-equivalent designs, however, may have the same statistical properties under one particular model but different properties under a different model. Results extend to designs with factors at larger number of levels.     

A note on the analysis of association in cross-classifications having ordered categories

The use of the singular value decomposition of a matrix in the analysis of cross-classifications having ordered categories la presented? Utilizing some matrix properties of a two-way contingency table, the singular value decomposition approach la applied on models such as the null association, uniform association and row-column effect models discussed recently in the literature. Some properties of estimates resulting from the singular value decomposition approach are discussed     

Factor analysis regression

The paper describes two regression models—principal components and maximum-likelihood factor analysis—which may be used when the stochastic predictor varibles are highly intereorrelated and/or contain measurement error. The two problems can occur jointly, for example in social-survey data where the true (but unobserved) covariance matrix can be singular. Departure from singularity of the sample dispersion matrix is then due to measurement error. We first consider the more elementary principal components regression model, where it is shown that it can be derived as a special case of (i) canonical correlation, and (ii) restricted least squares. The second part consists of the more general maximum-likelihood factor-analysis regression model, which is derived from the generalized inverse of the product of two singular matrices. Also, it is proved that factor-analysis regression can be considered as an instrumental variables estimator and therefore does not depend on whether factors have been “properly” identified in terms of substantive behaviour. Consequently the additional task of rotating factors to “simple structure” does not arise.     

A Note on the Factorization of a Matrix Inverse

The decomposition of a matrix as a product of a lower triangular with ones on the diagonal and an upper triangular matrix is useful for solving systems of linear equations. For a given non singular matrix, this type of decomposition is unique and algorithms exist to obtain the two factors. However, in certain problems the factorization of the inverse matrix may be of interest. This note presents an algorithm for factoring the inverse matrix using simple operations of elements from the original matrix. As examples, we give factorizations for several well-known and widely used correlation matrices. The usefulness and practicality of these factorizations are provided in an application of statistical modeling using unbiased estimating equations.     

A Tale of Two Matrix Factorizations

In statistical practice, rectangular tables of numeric data are commonplace, and are often analyzed using dimension-reduction methods like the singular value decomposition and its close cousin, principal component analysis (PCA). This analysis produces score and loading matrices representing the rows and the columns of the original table and these matrices may be used for both prediction purposes and to gain structural understanding of the data. In some tables, the data entries are necessarily nonnegative (apart, perhaps, from some small random noise), and so the matrix factors meant to represent them should arguably also contain only nonnegative elements. This thinking, and the desire for parsimony, underlies such techniques as rotating factors in a search for “simple structure.” These attempts to transform score or loading matrices of mixed sign into nonnegative, parsimonious forms are, however, indirect and at best imperfect. The recent development of nonnegative matrix factorization, or NMF, is an attractive alternative. Rather than attempt to transform a loading or score matrix of mixed signs into one with only nonnegative elements, it directly seeks matrix factors containing only nonnegative elements. The resulting factorization often leads to substantial improvements in interpretability of the factors. We illustrate this potential by synthetic examples and a real dataset. The question of exactly when NMF is effective is not fully resolved, but some indicators of its domain of success are given. It is pointed out that the NMF factors can be used in much the same way as those coming from PCA for such tasks as ordination, clustering, and prediction. Supplementary materials for this article are available online.     

Adaptive shrinkage of singular values

To recover a low-rank structure from a noisy matrix, truncated singular value decomposition has been extensively used and studied. Recent studies suggested that the signal can be better estimated by shrinking the singular values as well. We pursue this line of research and propose a new estimator offering a continuum of thresholding and shrinking functions. To avoid an unstable and costly cross-validation search, we propose new rules to select two thresholding and shrinking parameters from the data. In particular we propose a generalized Stein unbiased risk estimation criterion that does not require knowledge of the variance of the noise and that is computationally fast. A Monte Carlo simulation reveals that our estimator outperforms the tested methods in terms of mean squared error on both low-rank and general signal matrices across different signal-to-noise ratio regimes. In addition, it accurately estimates the rank of the signal when it is detectable.     

Q plots,a graphical aid for regression analysis

Plots are presented which are based on the singular value decomposition of the augmented data matrix in regression. In general, these plots assist in identifying discrepant observations, and in conjunction with associated diagnostics they are useful for identifying influential observations.     

Estimation of gaussian mixtures with rotationally invariant covariance matrices

Homoscedastic and heteroscedastic Gaussian mixtures differ in the constraints placed on the covariance matrices of the mixture components. A new mixture, called herein a strophoscedastic mixture, is defined by a new constraint, This constraint requires the matrices to be identical under orthogonal trans¬formations, where different transformations are allowed for different matrices. It is shown that the M-step of the EM method for estimating the parameters of strophoscedastic mixtures from sample data is explicitly solvable using singular value decompositions. Consequently, the EM-based maximum likelihood estimation algorithm is as easily implemented for strophoscedastic mixtures as it is for homoscedastic and heteroscedastic mixtures. An example of a “noisy” Archimedian spiral is presented.     

Numerical treatment of restricted gauss-markov model 1

The singular value decomposition (SVD) has been widely used in the ordinary linear model and other statistical problems. In this paper, we shall introduce the generalized singular value decomposition (GSVD) of any two matrices X and H having the same number of columns to moti-vate the numerical treatment of large scale restricted Gauss-Markov model (y,Xβ\Hβ = r,σ²1), a situation to reveal the relationship (or restriction) existing among the parameters of the model. Many approaches to restricted linear model are already available. Those approaches apply the generalized inverse of matrices and emphasize the the-oretical solution of the problem rather than the development of efficient and numerical stable algorithm for the computation of estimators. The possible merit of the method present here might lie in the facts that they directly lead to an efficient, numerically stable and easily programmed algorithm for     

Performance evaluation of likelihood-ratio tests for assessing similarity of the covariance matrices of two multivariate normal populations

Abstract

In analyzing two multivariate normal data sets, the assumption about equality of covariance matrices is usually used as a default for doing subsequence inferences. If this equality doesn’t hold, later inferences will be more complex and usually approximate. If one detects some identical components between two decomposed non equal covariance matrices and uses this extra information, one expects that subsequence inferences can be more accurately performed. For this purpose, in this article we consider some statistical tests about the equality of components of decomposed covariance matrices of two multivariate normal populations. Our emphasis is on the spectral decomposition of these matrices. Hypotheses about the equalities of sizes, shapes, and set of directions as components of these two covariance matrices are tested by the likelihood ratio test (LRT). Some simulation studies are carried out to investigate the accuracy and power of the LRT. Finally, analyses of two real data sets are illustrated.     

An approach to non-linear principal components analysis using radially symmetric kernel functions

An approach to non-linear principal components using radially symmetric kernel basis functions is described. The procedure consists of two steps: a projection of the data set to a reduced dimension using a non-linear transformation whose parameters are determined by the solution of a generalized symmetric eigenvector equation. This is achieved by demanding a maximum variance transformation subject to a normalization condition (Hotelling's approach) and can be related to the homogeneity analysis approach of Gifi through the minimization of a loss function. The transformed variables are the principal components whose values define contours, or more generally hypersurfaces, in the data space. The second stage of the procedure defines the fitting surface, the principal surface, in the data space (again as a weighted sum of kernel basis functions) using the definition of self-consistency of Hastie and Stuetzle. The parameters of this principal surface are determined by a singular value decomposition and crossvalidation is used to obtain the kernel bandwidths. The approach is assessed on four data sets.     

Recent developments in high dimensional covariance estimation and its related issues,a review

In this paper we review some of recent developments in high dimensional data analysis, especially in the estimation of covariance and precision matrix, asymptotic results on the eigenstructure in the principal components analysis, and some relevant issues such as test on the equality of two covariance matrices, determination of the number of principal components, and detection of hubs in a complex network.     

Biplots in correspondence analysis

Conditions under which correspondence analysis maps are biplots are discussed, as well as the interpretation of such biplots. It is shown that the asymmetric map which jointly displays the profiles and the vertices which define the unit vectors in the profile space is a biplot. A number of different ways of interpreting this joint plot are discussed, some of these being dependent on the choice of the x2 metric in the profile space. Biplot axes can be defined and calibrated on the zero-to-one profile scale in the usual way to recover approximations to the individual profile elements. Finally, the biplot interpretation in the context of multiple correspondence analysis is discussed. It is pointed out that joint correspondence analysis leads to a joint display of several variables which can be calibrated in a similar fashion to recover profile elements of the subtables of the Burt matrix.     

Difference-based Methods for Truncating the Singular Value Decomposition

Given a noisy time series (or signal), one may wish to remove the noise from the observed series. Assuming that the noise-free series lies in some low-dimensional subspace of rank r, a common approach is to embed the noisy time series into a Hankel trajectory matrix. The singular value decomposition is then used to deconstruct the Hankel matrix into a sum of rank-one components. We wish to demonstrate that there may be some potential in using difference-based methods of the observed series in order to provide guidance regarding the separation of the noise from the signal, and to estimate the rank of the low-dimensional subspace in which the true signal is assumed to lie.     

A robust biplot

This paper introduces a robust biplot which is related to multivariate M-estimates. The n × p data matrix is first considered as a sample of size n from some p-variate population, and robust M-estimates of the population location vector and scatter matrix are calculated. In the construction of the biplot, each row of the data matrix is assigned a weight determined in the preliminary robust estimation. In a robust biplot, one can plot the variables in order to represent characteristics of the robust variance-covariance matrix: the length of the vector representing a variable is proportional to its robust standard deviation, while the cosine of the angle between two variables is approximately equal to their robust correlation. The proposed biplot also permits a meaningful representation of the variables in a robust principal-component analysis. The discrepancies between least-squares and robust biplots are illustrated in an example.