Compositional Canonical Correlation Analysis

2009

Geological Society, London, Special Publications, 2006

Compositional data are those which contain only relative information. They are parts of some whole. In most cases they are recorded as closed data, i.e. data summing to a constant, such as 100% -whole-rock geochemical data being classic examples. Compositional data have important and particular properties that preclude the application of standard statistical techniques on such data in raw form. Standard techniques are designed to be used with data that are free to range from -oo to +oo. Compositional data are always positive and range only from 0 to 100, or any other constant, when given in closed form. If one component increases, others must, perforce, decrease, whether or not there is a genetic link between these components. This means that the results of standard statistical analysis of the relationships between raw components or parts in a compositional dataset are clouded by spurious effects. Although such analyses may give apparently interpretable results, they are, at best, approximations and need to be treated with considerable circumspection. The methods outlined in this volume are based on the premise that it is the relative variation of components which is of interest, rather than absolute variation. Log-ratios of components provide the natural means of studying compositional data. In this contribution the basic terms and operations are introduced using simple numerical examples to illustrate their computation and to familiarize the reader with their use.

2009

So far, similarity/diversity of objects has been widely studied in different research fields and a number of distance measures to estimate diversity between objects have been proposed. However, not much interest has been addressed to analysis of how diverse are configurations of objects in two different multivariate spaces. Since computerisation and automation nowadays lead to a large availability of information, it is apparent that a system could be described in different ways and, consequently, methods for comparison of the different viewpoints are required. These methods, for instance, may be usefully applied to Quantitative Structure–Activity Relationship (QSAR) studies. In this field, several thousands of molecular descriptors have been proposed in the literature and different selections of descriptors define different chemical spaces that need to be compared.Moreover, variable selection techniques such as Genetic Algorithms, Simulated Annealing, and Tabu Search are widely used to process available information in order to select optimal QSAR models. When more than one optimal model results, the problem arising is how to compare these models to find out whether they are really diverse or based on descriptors explaining almost the same information.In this paper, novel indices are proposed to measure similarity/diversity between pairs of data sets by the aid of the variable cross-correlation matrix.

The Burt matrix collects all bivariate cross tables, and/or covariance matrices, of m variables in a single matrix. Various forms of canonical analysis based on the Burt matrix are discussed.

ACM Computing Surveys

Canonical correlation analysis is a family of multivariate statistical methods for the analysis of paired sets of variables. Since its proposition, canonical correlation analysis has, for instance, been extended to extract relations between two sets of variables when the sample size is insufficient in relation to the data dimensionality, when the relations have been considered to be non-linear, and when the dimensionality is too large for human interpretation. This tutorial explains the theory of canonical correlation analysis, including its regularised, kernel, and sparse variants. Additionally, the deep and Bayesian CCA extensions are briefly reviewed. Together with the numerical examples, this overview provides a coherent compendium on the applicability of the variants of canonical correlation analysis. By bringing together techniques for solving the optimisation problems, evaluating the statistical significance and generalisability of the canonical correlation model, and interpr...

Recent statistical work on approaches to analysing compositional data-where variables sum to a constant for each row of a data matrix-may encounter dijiculties when applied to data of the kind typically arising in scientijic archaeology. The reason is that results obtained may be unsatisfactory from a substantive viewpoint for identijiable technical reasons. This paper explores and illustrates some possible resolutions of the problem. A feature of the approach used is to analyse subsets of the variables on separate scales. A synthesis of the results obtained from separate analyses is essential and the use of multiple correspondence analysis for this purpose is illustrated.

The Science of the total environment, 2010

Environmental sciences usually deal with compositional (closed) data. Whenever the concentration of chemical elements is measured, the data will be closed, i.e. the relevant information is contained in the ratios between the variables rather than in the data values reported for the variables. Data closure has severe consequences for statistical data analysis. Most classical statistical methods are based on the usual Euclidean geometry - compositional data, however, do not plot into Euclidean space because they have their own geometry which is not linear but curved in the Euclidean sense. This has severe consequences for bivariate statistical analysis: correlation coefficients computed in the traditional way are likely to be misleading, and the information contained in scatterplots must be used and interpreted differently from sets of non-compositional data. As a solution, the ilr transformation applied to a variable pair can be used to display the relationship and to compute a measu...

2011

Within the special geometry of the simplex, the sample space of compositional data, compositional orthonormal coordinates allow the application of any multivariate statistical approach. The search for meaningful coordinates has suggested balances (between two groups of parts)-based on a sequential binary partition of a D-part composition-and a representation in form of a CoDa-dendrogram. Projected samples are represented in a dendrogram-like graph showing: (a) the way of grouping parts; (b) the explanatory role of subcompositions generated in the partition process; (c) the decomposition of the variance; (d) the center and quantiles of each balance. The representation is useful for the interpretation of balances and to describe the sample in a single diagram independently of the number of parts. Also, samples of two or more populations, as well as several samples from the same population, can be represented in the same graph, as long as they have the same parts registered. The approach is illustrated with an example of food consumption in Europe.

Psychometrika, 2011

Regularized generalized canonical correlation analysis (RGCCA) is a generalization of regularized canonical correlation analysis to three or more sets of variables. It constitutes a general framework for many multi-block data analysis methods. It combines the power of multi-block data analysis methods (maximization of well identified criteria) and the flexibility of PLS path modeling (the researcher decides which blocks are connected and which are not). Searching for a fixed point of the stationary equations related to RGCCA, a new monotonically convergent algorithm, very similar to the PLS algorithm proposed by Herman Wold, is obtained. Finally, a practical example is discussed.

2008

Multiple-set canonical correlation analysis (Generalized CANO or GCANO for short) is an important technique because it subsumes a number of interesting multivariate data analysis techniques as special cases. More recently, it has also been recognized as an important technique for integrating information from multiple sources. In this paper, we present a simple regularization technique for GCANO and demonstrate its usefulness. Regularization is deemed important as a way of supplementing insufficient data by prior knowledge, and/or of incorporating certain desirable properties in the estimates of parameters in the model. Implications of regularized GCANO for multiple correspondence analysis are also discussed. Examples are given to illustrate the use of the proposed technique.