Principal Canonical Correlation Analysis Research Articles

Principal Canonical Correlation Analysis with Missing Data in Small Samples

Missing data occur in various fields, such as clinical trials and social science. Canonical correlation analysis often used to analyze the correlation between two random vectors, cannot be performed on a dataset with missing data. Canonical correlation coefficients (CCCs) can also be calculated from a covariance matrix. When the covariance matrix can be estimated by excluding (complete-case and available-case analyses) or imputing (multivariate imputation by chained equations, k-nearest neighbor (kNN), and iterative robust model-based imputation) missing data, CCCs are estimated from this covariance matrix. CCCs have bias even with all observation data. Usually, estimated CCCs are even larger than population CCCs when a covariance matrix estimated from a dataset with missing data is used. The purpose is to bring the CCCs estimated from the dataset with missing data close to the population CCCs. The procedure involves three steps. First, principal component analysis is performed on the covariance matrix from the dataset with missing data to obtain the eigenvectors. Second, the covariance matrix is transformed using first to fourth eigenvectors. Finally, the CCCs are calculated from the transformed covariance matrix. CCCs derived using with this procedure are called the principal CCCs (PCCCs), and simulation studies and numerical examples confirmed the effectiveness of the PCCCs estimated from the dataset with missing data. There were many cases in the simulation results where the bias and root-mean-squared error of the PCCC estimated from the missing data based on kNN were the smallest. In the numerical example results, the first PCCC estimated from the missing data based on kNN is close to the first CCC estimated from the dataset comprising all observation data when the correlation between two vectors is low. Therefore, PCCCs based on kNN were recommended.

A Variable Selection Criterion for Two Sets of Principal Component Scores in Principal Canonical Correlation Analysis

Canonical correlation analysis (CCA) is often used to analyze the correlation between two random vectors. However, sometimes interpretation of CCA results may be hard. In an attempt to address these difficulties, principal canonical correlation analysis (PCCA) was proposed. PCCA is CCA between two sets of principal component (PC) scores. We consider the problem of selecting useful PC scores in CCA. A variable selection criterion for one set of PC scores has been proposed by Ogura (2010), here, we propose a variable selection criterion for two sets of PC scores in PCCA. Furthermore, we demonstrate the effectiveness of this criterion.

A variable selection method in principal canonical correlation analysis

We propose a variable selection procedure for the canonical correlation analysis (CCA) between two sets of principal components. We attempt to create predictive models for selecting such variables by combining principal component analysis (PCA) and CCA, and we refer to them collectively as principal canonical correlation analysis (PCCA). We derive a model selection criterion of one set of principal components, based on the selection of a covariance structure analysis within the framework of the PCCA. Compared to the variable selection procedure used in the CCA, the procedure used in the PCCA return a smaller number of variables. This is because the principal components derived from a PCA descend in order of the amount of information that they contain. The principal components with the smallest variance contributions are disregarded because their information contribution becomes negligible. Herein, we demonstrate the effectiveness of this criterion by using an example. Moreover, we investigate the properties of a variable selection criterion using the bootstrap resampling. The variable selection procedure used with the PCCA is compared to that used for the CCA.

Failure Detection in Large-Scale Internet Services by Principal Subspace Mapping

Fast and accurate failure detection is becoming essential in managing large-scale Internet services. This paper proposes a novel detection approach based on the subspace mapping between the system inputs and internal measurements. By exploring these contextual dependencies, our detector can initiate repair actions accurately, increasing the availability of the system. Although a classical statistical method, the canonical correlation analysis (CCA), is presented in the paper to achieve subspace mapping, we also propose a more advanced technique, the principal canonical correlation analysis (PCCA), to improve the performance of the CCA-based detector. PCCA extracts a principal subspace from internal measurements that is not only highly correlated with the inputs but also a significant representative of the original measurements. Experimental results on a Java 2 platform, enterprise edition (J2EE)- based Web application demonstrate that such property of PCCA is especially beneficial to failure detection tasks.

Distributions and the Bootstrap Method of Some Statistics in Principal Canonical Correlation Analysis

We consider two sets of variables with a joint distribution and analyze the canonical correlations between the variables in the two sets. One of the analyses used is the canonical correlation analysis, which finds linear combinations of variables in the sets that have the maximum correlation, and these linear combinations are the first coordinates in new systems. Then, a second linear combination in each set is obtained such that the linear combination is uncorrelated with the first linear combination. The procedure is continued until two new coordinate systems are specified completely. This theory was developed by Hotelling (1935, 1936). In this paper, we first determine the principal components of the two sets and then calculate the canonical correlation between the two principal components. Principal components analysis is a procedure used for analyzing multivariate data that transforms the original variables into new ones that are uncorrelated and account for decreasing proportions of the variance in the data. This analysis attempts to characterize or explain the variability in a vector variable by replacing it with a new variable with fewer components with large variance. We know that the interpretation of principal components is easier than the canonical variate. Therefore, comparing canonical correlation analysis with principal component analysis, we can say that the canonical correlations of two principal components are more useful for understanding the relationships of the given data sets. This paper derives the limiting distribution of the canonical correlation