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Register a new userProbabilistic Canonical Correlation Analysis for Sparse Count Data
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Added by
Lin Qiu
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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Canonical correlation analysis (CCA) is a classical and important multivariate technique for exploring the relationship between two sets of continuous variables. CCA has applications in many fields, such as genomics and neuroimaging. It can extract meaningful features as well as use these features for subsequent analysis. Although some sparse CCA methods have been developed to deal with high-dimensional problems, they are designed specifically for continuous data and do not consider the integer-valued data from next-generation sequencing platforms that exhibit very low counts for some important features. We propose a model-based probabilistic approach for correlation and canonical correlation estimation for two sparse count data sets (PSCCA). PSCCA demonstrates that correlations and canonical correlations estimated at the natural parameter level are more appropriate than traditional estimation methods applied to the raw data. We demonstrate through simulation studies that PSCCA outperforms other standard correlation approaches and sparse CCA approaches in estimating the true correlations and canonical correlations at the natural parameter level. We further apply the PSCCA method to study the association of miRNA and mRNA expression data sets from a squamous cell lung cancer study, finding that PSCCA can uncover a large number of strongly correlated pairs than standard correlation and other sparse CCA approaches.
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Canonical correlation analysis investigates linear relationships between two sets of variables, but often works poorly on modern data sets due to high-dimensionality and mixed data types such as continuous, binary and zero-inflated. To overcome these challenges, we propose a semiparametric approach for sparse canonical correlation analysis based on Gaussian copula. Our main contribution is a truncated latent Gaussian copula model for data with excess zeros, which allows us to derive a rank-based estimator of the latent correlation matrix for mixed variable types without the estimation of marginal transformation functions. The resulting canonical correlation analysis method works well in high-dimensional settings as demonstrated via numerical studies, as well as in application to the analysis of association between gene expression and micro RNA data of breast cancer patients.
Classical canonical correlation analysis (CCA) requires matrices to be low dimensional, i.e. the number of features cannot exceed the sample size. Recent developments in CCA have mainly focused on the high-dimensional setting, where the number of features in both matrices under analysis greatly exceeds the sample size. These approaches impose penalties in the optimization problems that are needed to be solve iteratively, and estimate multiple canonical vectors sequentially. In this work, we provide an explicit link between sparse multiple regression with sparse canonical correlation analysis, and an efficient algorithm that can estimate multiple canonical pairs simultaneously rather than sequentially. Furthermore, the algorithm naturally allows parallel computing. These properties make the algorithm much efficient. We provide theoretical results on the consistency of canonical pairs. The algorithm and theoretical development are based on solving an eigenvectors problem, which significantly differentiate our method with existing methods. Simulation results support the improved performance of the proposed approach. We apply eigenvector-based CCA to analysis of the GTEx thyroid histology images, analysis of SNPs and RNA-seq gene expression data, and a microbiome study. The real data analysis also shows improved performance compared to traditional sparse CCA.
We propose novel first-order stochastic approximation algorithms for canonical correlation analysis (CCA). Algorithms presented are instances of inexact matrix stochastic gradient (MSG) and inexact matrix exponentiated gradient (MEG), and achieve $epsilon$-suboptimality in the population objective in $operatorname{poly}(frac{1}{epsilon})$ iterations. We also consider practical variants of the proposed algorithms and compare them with other methods for CCA both theoretically and empirically.
Count data are collected in many scientific and engineering tasks including image processing, single-cell RNA sequencing and ecological studies. Such data sets often contain missing values, for example because some ecological sites cannot be reached in a certain year. In addition, in many instances, side information is also available, for example covariates about ecological sites or species. Low-rank methods are popular to denoise and impute count data, and benefit from a substantial theoretical background. Extensions accounting for covariates have been proposed, but to the best of our knowledge their theoretical and empirical properties have not been thoroughly studied, and few softwares are available for practitioners. We propose a complete methodology called LORI (Low-Rank Interaction), including a Poisson model, an algorithm, and automatic selection of the regularization parameter, to analyze count tables with covariates. We also derive an upper bound on the estimation error. We provide a simulation study with synthetic data, revealing empirically that LORI improves on state of the art methods in terms of estimation and imputation of the missing values. We illustrate how the method can be interpreted through visual displays with the analysis of a well-know plant abundance data set, and show that the LORI outputs are consistent with known results. Finally we demonstrate the relevance of the methodology by analyzing a water-birds abundance table from the French national agency for wildlife and hunting management (ONCFS). The method is available in the R package lori on the Comprehensive Archive Network (CRAN).
Research on Poisson regression analysis for dependent data has been developed rapidly in the last decade. One of difficult problems in a multivariate case is how to construct a cross-correlation structure and at the meantime make sure that the covariance matrix is positive definite. To address the issue, we propose to use convolved Gaussian process (CGP) in this paper. The approach provides a semi-parametric model and offers a natural framework for modeling common mean structure and covariance structure simultaneously. The CGP enables the model to define different covariance structure for each component of the response variables. This flexibility ensures the model to cope with data coming from different resources or having different data structures, and thus to provide accurate estimation and prediction. In addition, the model is able to accommodate large-dimensional covariates. The definition of the model, the inference and the implementation, as well as its asymptotic properties, are discussed. Comprehensive numerical examples with both simulation studies and real data are presented.
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