# GMD: Generalized Matrix Decomposition In taryue/GMDecomp: Visualization tools for the Generalized Matrix Decomposition (GMD).

## Description

Computes the generalized matrix decomposition of X.

## Usage

 `1` ```GMD(X, H, Q, K) ```

## Arguments

 `X` An n x p data matrix. `H` An n x n positive semi-definite similarity kernel. `Q` An p x p positive semi-definite similarity kernel. `K` a scalar specifying the dimension of GMD components (see Details).

## Details

The K-dimensional GMD of X with respect to H and Q is given by X = USV^T, where

(U, S, V) = argmin_{U,S,V} ||X - USV^T||^2_{H, Q},

subject to U^THU = I_K, V^TQV = I_K and diag(S) ≥ 0. Here, for any n x p matrix A, ||A||^2_H,Q = tr(A^THAQ).

## Value

A list of the GMD components U, S, V, H and Q (see Details).

## Author(s)

Parker Knight and Yue Wang ywang2310@fredhutch.org

## References

Allen, G. I., L. Grosenick, and J. Taylor (2014). A generalized least-square matrix decom- position. Journal of the American Statistical Association 109(505), 145–159.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```## Not run: X = matrix(rnorm(1000), 50, 20) autocorr.mat <- function(p, rho) { mat <- diag(p) return(rho^abs(row(mat)-col(mat))) } H = autocorr.mat(50, 0.6) Q = autocorr.mat(20, 0.7) GMD.fit = GMD(X, H, Q, 2) ## End(Not run) ```

taryue/GMDecomp documentation built on Nov. 5, 2019, 10 a.m.