CANONICAL VARIATE ANALYSIS WITH SPATIALLY-CORRELATED DATA

A general canonical variate model is derived when the observations are spatially correlated. For spatial covariance structures resulting from dependence of a pixel on its nearest neighbours, the solution reduces to an analysis of neighbour-corrected values. The usual analysis, in which spatial correlation is ignored, gives similar canonical vectors but over-estimates the canonical roots. A formula for approximating the reduction in the canonical roots to adjust for the spatial correlation is given.     

Some Practical Aspects of Canonical Variate Analysis

Canonical variate analysis can be viewed as a two-stage principal component analysis. Explicit consideration of the principal components from the first stage, formalized in the content of shrunken estimators, leads to a number of practical advantages. In morphometric studies, the first eigenvector is often a size vector, with the remaining vectors contrast or shape-type vectors, so that a decomposition of the canonical variates into size and shape components can be achieved. In applied studies, often a small number of the principal components effect most of the separation between groups; plots of group means and associated concentration ellipses (ideally these should be circular) for important principal components facilitate graphical inspection. Of considerable practical importance is the potential for improved stability of the estimated canonical vectors. When the between-groups sum of squares for a particular principal component is small, and the corresponding eigenvalue of the within-groups correlation matrix is also small, marked instability of the canonical vectors can be expected. The introduction of shrunken estimators, by adding shrinkage constrants to the eigenvalues, leads to more stable coefficients.     

CANONICAL VARIATE ANALYSIS—A GENERAL MODEL FORMULATION

A general model, specifying the population means as a function of the population canonical vectors, provides a natural basis for considering many aspects of canonical variate analysis. These aspects include: estimation for elliptical densities, and robust estimation; estimation of missing values; adequacy of hypothetical variables; regression in dummy variable space; and comparison of solutions.     

Variable selection and interpretation in canonical correlation analysis

The canonical variates in canonical correlation analysis are often interpreted by looking at the weights or loadings of the variables in each canonical variate and effectively ignoring those variables whose weights or loadings are small. It is shown that such a procedure can be misleading. The related problem of selecting a subset of the original variables which preserves the information in the most important canonical variates is also examined. Because of different possible definitions of ‘the information in canonical variates’, any such subset selection needs very careful consideration.     

On certain unbiased product estimators

When data on an auxiliary variate is available on all the units of the population, negatively correlated with the study variate, Robson (1957) and Murthy (1964) proposed product method of estimation for the estimation of the population total (mean) of the study variate. In this paper, we discuss a method given in Rao (1983) and obtain a simpler dervation of the class of unbiased product estimators for the case of Simple Random Sampling WithOut Replacement design as well as for the case of interpenetrating subsamples design which follows as a limiting case. Finally, we shall illustrate the results by means of two simple numerical example from live data.     

A generalization of random vectors and a correlation theory for them - a trial (with discussions) 2

There are defined generalized generalized random vectors. This notion contains usual random vectors, some classical and generalized stochastic processes and certain generalizations of them. There is developped a theory of correlation on basis of a theory of linear prediction (approximation) for such rendom vectors which includes especially canonical correlations.     

Antedependence models in the analysis of multi-group high-dimensional data

Antedependence modelling has previously been shown to be useful for twogroup discriminant analysis of high-dimensional data. In this paper, the theory of such models is extended to multi-group discriminant analysis and to canonical variate analysis for data display. The application of antedependence models of orders 1, 2 and 3 to spectroscopic analyses of rice samples is described, and the results are compared with those from standard methods based on principal component scores calculated from the data.     

Non-linear canonical correlation analysis with a simulated annealing solution

An exploratory tool is introduced to examine potential non-linear relation-ships between two sets of variables, X andY, in a sample of multivariate data. Simulated annealing is applied to find canonical coefficient vectors a and b such that a squared non-linear correlation between a'Xand b'Y is maximiSed. A measure of non-linear correlation is developed for this optimization which utilies a nearest-neighbor regression estimate for the unknown functional relationship. In addition to examining potential relations between the canonical variables, this method can identify the important variables in each set.     

Efficient generation of random vectors by using the ratio-of-uniforms method with ellipsoidal envelopes

Stochastic simulation is widely used to validate procedures and provide guidance for both theoretical and practical problems. Random variate generation is the basis of stochastic simulation. Applying the ratio-of-uniforms method to generate random vectors requires the ability to generate points uniformly in a suitable region of the space. Starting from the observation that, for many multivariate distributions, the multidimensional objective region can be covered by a hyper-ellipsoid more tightly than by a hyper-rectangle, a new algorithm to generate from multivariate distributions is proposed. Due to the computational saving it can produce, this method becomes an appealing statistical tool to generate random vectors from families of standard and nonstandard multivariate distributions. It is particularly interesting to generate from densities known up to a multiplicative constant, for example, from those arising in Bayesian computation. The proposed method is applied and its efficiency is shown for some classes of distributions.     

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.     

Selecting the best predictor variate

The Bechhofer indifference-zone approach is used to determine the sample size for selecting the best predictor variate from a set of k variates. A multivariate normal model is assumed and the best predictor variate is defined to be that variate for which the predictand has the smallest population conditional variance. Asymptotic distribution theory and probability bounds are employed to obtain sample-size approximations, which are compared with (numerical) exact results.     

Canonical correlation analysis for the vector AR(1) model with ARCH innovations

This paper extends the results of canonical correlation analysis of Anderson [2002. Canonical correlation analysis and reduced-rank regression in autoregressive models. Ann. Statist. 30, 1134–1154] to a vector AR(1) process with a vector ARCH(1) innovations. We obtain the limiting distributions of the sample matrices, the canonical correlations and the canonical vectors of the process. The extension is important because many time series in economics and finance exhibit conditional heteroscedasticity. We also use simulation to demonstrate the effects of ARCH innovations on the canonical correlation analysis in finite sample. Both the limiting distributions and simulation results show that overlooking the ARCH effects in canonical correlation analysis can easily lead to erroneous inference.     

An RKHS framework for functional data analysis

Linear combinations of random variables play a crucial role in multivariate analysis. Two extension of this concept are considered for functional data and shown to coincide using the Loève–Parzen reproducing kernel Hilbert space representation of a stochastic process. This theory is then used to provide an extension of the multivariate concept of canonical correlation. A solution to the regression problem of best linear unbiased prediction is obtained from this abstract canonical correlation formulation. The classical identities of Lawley and Rao that lead to canonical factor analysis are also generalized to the functional data setting. Finally, the relationship between Fisher's linear discriminant analysis and canonical correlation analysis for random vectors is extended to include situations with function-valued random elements. This allows for classification using the canonical Y scores and related distance measures.     

Linear generalizations of various matric-t distributions

The concept of a matric-t variate is extended to cases where the positive (definite) part of the variate, which is usually Wishart distributed independently of the normal part, is a linear sum of positive (definite) variates with positive coefficients. These distributions and their quadratic forms are of importance i.a, for the exact solution to the multi¬variate Behrens-Fisher problem. A few useful identities con¬cerning the invariant polynomials with matrix arguments are derived     

A comparative study for robust canonical correlation methods

The aim of this study is to obtain robust canonical vectors and correlation coefficients based on the percentage bend correlation and winsorized correlation in the correlation matrix and fast consistent high breakdown (FCH), reweighted fast consistent high breakdown (RFCH), and reweighted multivariate normal (RMVN) estimators to estimate the covariance matrix and then compare these estimators with the existing estimators. In the correlation matrix of canonical correlation analysis (CCA), we present an approach that substitutes the percentage bend correlation and the winsorized correlation in place of the widely employed the Pearson correlation. Moreover, we employ the FCH, RFCH, and RMVN estimators to estimate the covariance matrix in the CCA. We conduct a simulation study and employ real data with the objective of comparing the performance of the different estimators for canonical vectors and correlation with that of our proposed approaches. The breakdown plots and independent tests are employed as differentiating criteria of the robustness and performance of the estimators. Based on our computational and real data studies, we propose suggestions and guidelines on the practical implications of our findings.     

Measures of fit in principal component and canonical variate analyses

Treating principal component analysis (PCA) and canonical variate analysis (CVA) as methods for approximating tables, we develop measures, collectively termed predictivity, that assess the quality of fit independently for each variable and for all dimensionalities. We illustrate their use with data from aircraft development, the African timber industry and copper froth measurements from the mining industry. Similar measures are described for assessing the predictivity associated with the individual samples (in the case of PCA and CVA) or group means (in the case of CVA). For these measures to be meaningful, certain essential orthogonality conditions must hold that are shown to be satisfied by predictivity.     

On the geometry of multivariate L 1 objective functions

Summary: Two multivariate L ₁ objective functions, namely the k–variate extensions of the classical mean deviation and mean difference, are considered. The duality between the original data vectors and the hyperplanes going through the origin and k – 1 data points is discussed and, consequently, different interesting representations and interpretations of the multivariate mean deviation are introduced. A similar duality is found between the lift data vectors and the hyperplanes going through k data points leading to different representations of the multivariate mean difference. The objective functions are also shown to have interpretations in terms of the centers of facets of the data based zonotopes and lift zonotopes. Moreover, interchanging the roles of the data vectors and the data hyperplanes yields nonparametric measures of (angular) distances between the data vectors as well as between the hyperplanes. Finally, multivariate sign and rank based tests and estimates in the one–sample and several–samples multivariate cases are discussed to illustrate the theory.*The authors wish to thank the referees for valuable comments and suggestions. The research was partially supported by the Academy of Finland.     

A statistical approach to multivariate evaluation of diversity with respect to quantitative characteristics in cereal germplasm collections

SUMMARY The aim of this paper is to undertake the problem of adapting some multivariate statistical methods (MANOVA, cluster analysis with simultaneous test procedures T 2 based on Roy's union-intersection rule and canonical variate analysis) max and describing their possible usage in the evaluation and interpretation of the phenotypic diversity with regard to quantitative traits in cereal collections. The presented procedures are used in a case where experimental data have been obtained from single-replicated trials conducted at the same location over a few years. In such cases, data can be nonorthogonal connected accessions x years cross-classification with none or one observation in a given subclass. The application of the suggested procedures is illustrated by a numerical example of a winter rye collection from the Plant Breeding and Acclimatization Institute in Radzikow near Warsaw (Poland).     

Estimating a distribution for each category of a sensitive variate

The use of a Randomized Response (RR) design makes it possible to estimate the distribution of a sensitive variate. In this paper, the estimation of the distribution of a non-sensitive variate for each category of a sensitive variate is considered for the case where data on the sensitive variate is obtained by use of an RR procedure. Simple estimators are developed without making any distributional assumptions about the non-sensitive variate. However, if distributional assumptions are made, it is shown that the EM algorithm may be used to compute Maximum Likelihood estimates. Computational comparisons of the estimators, using simulation, indicate that the simple estimators perform well, particularly for large sample sizes.     

Study of redundancy of vector variables in canonical correlations

Fujikoshi (1982) obtained the necessary and sufficient conditions for the increased number of variables in the two sets of vectors not affecting the original nonzero canonical correlations and used these to obtain the likelihood ratio test procedure. He assumed a nonsingular covariance matrix due to random variables. Here, we study the same problem when the covariance matrix is singular and establish some further results. In this study, we note that the unit canonical correlations have to be separated in some of the situations. These results are valid for complex random vector variables and in some situations, the test for redundancy is given for complex random variables.