Classical Canonical Correlation Analysis Research Articles

Prediction of East Asian summer precipitation via independent component analysis

A new statistical postprocessing method is proposed for seasonal climate prediction. The proposed method is based on a combination of independent component analysis (ICA) and canonical correlation analysis (CCA). Since the classical CCA cannot handle high-dimensional data wherein the number of variables is larger than the number of observations, ICA is pre-performed to reduce the dimension of the data. It is well known that empirical orthogonal function (EOF) analysis is a popular method for dimension reduction in the climatology community; however, loss of information occurs when the data is not Gaussian distributed. To extend the scope of distribution assumption and improve the prediction ability simultaneously, we propose the ICA-based method. This study focuses on the prediction of future precipitation for the boreal summer (June–July–August; JJA) through 29 years (1979–2007) on East Asia region. Results of the proposed ICA-based method show an improvement in seasonal climate prediction in terms of correlation and root mean square error as compared with those of the GCM simulation and the EOF/CCA method.

Restricted kernel canonical correlation analysis

Kernel canonical correlation analysis (KCCA) is a procedure for assessing the relationship between two sets of random variables when the classical method, canonical correlation analysis (CCA), fails because of the nonlinearity of the data. The KCCA method is mostly used in machine learning, especially for information retrieval and text mining. Because the data is often represented with non-negative numbers, we propose to incorporate the non-negativity restriction directly into the KCCA method. Similar restrictions have been studied in relation to the classical CCA and called restricted canonical correlation analysis (RCCA), so that we call the proposed method restricted kernel canonical correlation analysis (RKCCA). We also provide some possible approaches for solving the optimization problem to which our method translates. The motivation for introducing RKCCA is given in Section 2.

Integrative analysis of gene expression and copy number alterations using canonical correlation analysis

BackgroundWith the rapid development of new genetic measurement methods, several types of genetic alterations can be quantified in a high-throughput manner. While the initial focus has been on investigating each data set separately, there is an increasing interest in studying the correlation structure between two or more data sets. Multivariate methods based on Canonical Correlation Analysis (CCA) have been proposed for integrating paired genetic data sets. The high dimensionality of microarray data imposes computational difficulties, which have been addressed for instance by studying the covariance structure of the data, or by reducing the number of variables prior to applying the CCA. In this work, we propose a new method for analyzing high-dimensional paired genetic data sets, which mainly emphasizes the correlation structure and still permits efficient application to very large data sets. The method is implemented by translating a regularized CCA to its dual form, where the computational complexity depends mainly on the number of samples instead of the number of variables. The optimal regularization parameters are chosen by cross-validation. We apply the regularized dual CCA, as well as a classical CCA preceded by a dimension-reducing Principal Components Analysis (PCA), to a paired data set of gene expression changes and copy number alterations in leukemia.ResultsUsing the correlation-maximizing methods, regularized dual CCA and PCA+CCA, we show that without pre-selection of known disease-relevant genes, and without using information about clinical class membership, an exploratory analysis singles out two patient groups, corresponding to well-known leukemia subtypes. Furthermore, the variables showing the highest relevance to the extracted features agree with previous biological knowledge concerning copy number alterations and gene expression changes in these subtypes. Finally, the correlation-maximizing methods are shown to yield results which are more biologically interpretable than those resulting from a covariance-maximizing method, and provide different insight compared to when each variable set is studied separately using PCA.ConclusionsWe conclude that regularized dual CCA as well as PCA+CCA are useful methods for exploratory analysis of paired genetic data sets, and can be efficiently implemented also when the number of variables is very large.

Climate Prediction by a Hybrid Method with Emphasizing Future Precipitation Change of East Asia

A canonical correlation analysis(CCA)-based method is proposed for prediction of future climate change which combines information from ensembles of atmosphere-ocean general circulation models(AOGCMs) and observed climate values. This paper focuses on predictions of future climate on a regional scale which are of potential economic values. The proposed method is obtained by coupling the classical CCA with empirical orthogonal functions(EOF) for dimension reduction. Furthermore, we generate a distribution of climate responses, so that extreme events as well as a general feature such as long tails and unimodality can be revealed through the distribution. Results from real data examples demonstrate the promising empirical properties of the proposed approaches.

Representation-Constrained Canonical Correlation Analysis: A Hybridization of Canonical Correlation and Principal Component Analyses

The classical canonical correlation analysis is extremely greedy to maximize the squared correlation between two sets of variables. As a result, if one of the variables in the dataset-1 is very highly correlated with another variable in the dataset-2, the canonical correlation will be very high irrespective of the correlation among the rest of the variables in the two datasets. We intend here to propose an alternative measure of association between two sets of variables that will not permit the greed of a select few variables in the datasets to prevail upon the fellow variables so much as to deprive the latter of contributing to their representative variables or canonical variates. Our proposed Representation-Constrained Canonical correlation (RCCCA) Analysis has the Classical Canonical Correlation Analysis (CCCA) at its one end (λ=0) and the Classical Principal Component Analysis (CPCA) at the other (as λ tends to be very large). In between it gives us a compromise solution. By a proper choice of λ, one can avoid hijacking of the representation issue of two datasets by a lone couple of highly correlated variables across those datasets. This advantage of the RCCCA over the CCCA deserves a serious attention by the researchers using statistical tools for data analysis.

Nonlinear measures of association with kernel canonical correlation analysis and applications

Measures of association between two sets of random variables have long been of interest to statisticians. The classical canonical correlation analysis (LCCA) can characterize, but also is limited to, linear association. This article introduces a nonlinear and nonparametric kernel method for association study and proposes a new independence test for two sets of variables. This nonlinear kernel canonical correlation analysis (KCCA) can also be applied to the nonlinear discriminant analysis. Implementation issues are discussed. We place the implementation of KCCA in the framework of classical LCCA via a sequence of independent systems in the kernel associated Hilbert spaces. Such a placement provides an easy way to carry out the KCCA. Numerical experiments and comparison with other nonparametric methods are presented.