Factor analysis provides linear factors that describe relationships between individual variables of a data set. We extend this classical formulation into linear factors that describe relationships between groups of variables, where each group represents either a set of related variables or a data set. The model also naturally extends canonical correlation analysis to more than two sets, in a way that is more flexible than previous extensions. Our solution is formulated as variational inference of a latent variable model with structural sparsity, and it consists of two hierarchical levels: The higher level models the relationships between the groups, whereas the lower models the observed variables given the higher level. We show that the resulting solution solves the group factor analysis problem accurately, outperforming alternative factor analysis based solutions as well as more straightforward implementations of group factor analysis. The method is demonstrated on two life science data sets, one on brain activation and the other on systems biology, illustrating its applicability to the analysis of different types of high-dimensional data sources.
Most multi-set extensions of CCA, however, are not equivalent to our model. For example, @cite_12 and @cite_18 extend CCA for @math , but instead of GFA they solve the more limited problem of multiple-battery factor analysis @cite_29 @cite_2 . The MBFA models provide one set of factors that describe the relationships between groups, and then model the variation specific to each group either with a free covariance matrix or a separate set of factors for that group. Besides the multi-set extensions of CCA, also the probabilistic interpretation of sparse matrix factorization @cite_11 , and the JIVE model for integrated analysis of multiple data types @cite_10 @cite_20 belong to the family of MBFA models. These models differ in their priors, parameterization and inference, but are all conceptually equivalent.
@cite_18 @cite_29 @cite_2 @cite_20 @cite_10 @cite_12 @cite_11
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Abstract In this paper we describe a method for functional connectivity analysis of fMRI data between given brain regions-of-interest (ROIs). The method relies on nonnegativity constrained- and spatially regularized multiset canonical correlation analysis (CCA), and assigns weights to the fMRI signals of the ROIs so that their representative signals become simultaneously maximally correlated. The different pairwise correlations between the representative signals of the ROIs are combined using the maxvar approach for multiset CCA, which has been shown to be equivalent to the generalized eigenvector formulation of CCA. The eigenvector in the maxvar approach gives an indication of the relative importance of each ROI in obtaining a maximal overall correlation, and hence, can be interpreted as a functional connectivity pattern of the ROIs. The successive canonical correlations define subsequent functional connectivity patterns, in decreasing order of importance. We apply our method on synthetic data and real fMRI data and show its advantages compared to unconstrained CCA and to PCA. Furthermore, since the representative signals for the ROIs are optimized for maximal correlation they are also ideally suited for further effective connectivity analyses, to assess the information flows between the ROIs in the brain. McDonald's (1970) generalization of Tucker's (1958) inter-battery factor analysis model to multiple batteries is considered. The identification of parameters is examined and an iterative algorithm for obtaining maximum-likelihood estimates is provided. Consideration of the relationship between inter-battery factor analysis and canonical correlation analysis in the case of two batteries suggests a generalization of canonical coefficients to the situation where there are several batteries. Examples of the application of the procedure are given. Research in several fields now requires the analysis of datasets in which multiple high-dimensional types of data are available for a common set of objects. In particular, The Cancer Genome Atlas (TCGA) includes data from several diverse genomic technologies on the same cancerous tumor samples. In this paper we introduce Joint and Individual Variation Explained (JIVE), a general decomposition of variation for the integrated analysis of such datasets. The decomposition consists of three terms: a low-rank approximation capturing joint variation across data types, low-rank approximations for structured variation individual to each data type, and residual noise. JIVE quantifies the amount of joint variation between data types, reduces the dimensionality of the data, and provides new directions for the visual exploration of joint and individual structure. The proposed method represents an extension of Principal Component Analysis and has clear advantages over popular two-block methods such as Canonical Correlation Analysis and Partial Least Squares. A JIVE analysis of gene expression and miRNA data on Glioblastoma Multiforme tumor samples reveals gene-miRNA associations and provides better characterization of tumor types. We present a generative model for performing sparse probabilistic projections, which includes sparse principal component analysis and sparse canonical correlation analysis as special cases. Sparsity is enforced by means of automatic relevance determination or by imposing appropriate prior distributions, such as generalised hyperbolic distributions. We derive a variational Expectation-Maximisation algorithm for the estimation of the hyperparameters and show that our novel probabilistic approach compares favourably to existing techniques. We illustrate how the proposed method can be applied in the context of cryptoanalysis as a preprocessing tool for the construction of template attacks. Building a common representation for several related data sets is an important problem in multi-view learning. CCA and its extensions have shown that they are effective in finding the shared variation among all data sets. However, these models generally fail to exploit the common structure of the data when the views are with private information. Recently, methods explicitly modeling the information into shared part and private parts have been proposed, but they presume to know the prior knowledge about the latent space, which is usually impossible to obtain. In this paper, we propose a probabilistic model, which could simultaneously learn the structure of the latent space whilst factorize the information correctly, therefore the prior knowledge of the latent space is unnecessary. Furthermore, as a probabilistic model, our method is able to deal with missing data problem in a natural way. We show that our approach attains the performance of state-of-art methods on the task of human pose estimation when the motion capture view is completely missing, and significantly improves the inference accuracy with only a few observed data.