Abstract
In this work, we incorporate matrix projections into the reduced rank regression method, and then develop reduced rank regression estimators based on random projection and orthogonal projection in high-dimensional multivariate linear regression model. We propose a consistent estimator of the rank of the coefficient matrix and achieve prediction performance bounds for the proposed estimators based on mean squared errors. Finally, some simulation studies and a real data analysis are carried out to demonstrate that the proposed methods possess good stability, prediction performance and rank consistency compared to some other existing methods.
Highlights
Multivariate linear regression methods are widely used statistical tools in regression analysis
The reduced rank regression would allow the rank of B to be less than min(p, r), and so the model parametrization can be expressed as B = B1B2, where B1 ∈ Rr×d, B2 ∈ Rd×p, and rank(B1)=rank(B2)=d
We propose three reduced rank estimators with a nuclear norm penalty in multivariate linear regression model in terms of single random projection, averaged random projection and principal components analysis, respectively
Summary
Multivariate linear regression methods are widely used statistical tools in regression analysis. A multivariate linear regression has n observations with r responses and p predictors, and can be expressed as. Where Y ∈ Rn×r denotes a multivariate response matrix, X ∈ Rn×p represents a matrix of predictors, ε ∈ Rn×r is an error matrix with its entry εij being independent of each other with mean zero and variance σi2j, and B ∈ Rp×r is the regression coefficient matrix. The model in (1) is the foundation of multivariate regression analysis with its aim being to study the relationship between X and. Y through the regression coefficient matrix B. where X+ denotes the Moore–Penrose inverse of X
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