2. Guided Partial Least Squares (guided-PLS)
In guidedPLS: Supervised Dimensional Reduction by Guided Partial Least Squares

Introduction

In this vignette, we consider a novel supervised dimensional reduction method guided partial least squares (guided-PLS).

Test data is available from toyModel.

library("guidedPLS")
data <- guidedPLS::toyModel("Easy")
str(data, 2)

You will see that there are three blocks in the data matrix as follows.

suppressMessages(library("fields"))
layout(c(1,2,3))
image.plot(data$Y1_dummy, main="Y1 (Dummy)", legend.mar=8)
image.plot(data$Y1, main="Y1", legend.mar=8)
image.plot(data$X1, main="X1", legend.mar=8)

Guided Partial Least Squares (guided-PLS)

Here, suppose that we have two data matrices $X_1$ ($N \times M$) and $X_2$ ($S \times T$), and the row vectors of them are assumed to be centered. Since these two matrices have no common row or column, integration of them is not trivial. Such a data structure is called "diagonal" and known as a barrier to omics data integration [@diagonal].

Here is a simpler way to set up the problem; suppose that we have another set of matrices $Y_1$ ($M \times I$) and $Y_2$ ($T \times I$), which are the label matrices for $X_1$ and $X_2$, respectively.

In guided-PLS, the data matrices $X_1$ and $X_2$ are projected into lower dimension via $Y_1$ and $Y_2$, and then PLS-SVD are performed against the $Y_{1} X_{1}$ and $Y_{2} X_{2}$ as follows:

$$ \max_{W_{1},W_{2}} \mathrm{tr} \left( W_{1}^{T} X_{1}^{T} Y_{1}^{T} Y_{2} X_{2} W_{2} \right)\ \mathrm{s.t.}\ W_{1}^{T}W_{1} = W_{2}^{T}W_{2} = I_{K} $$

Basic Usage

guidedPLS is performed as follows.

out <- guidedPLS(X1=data$X1, X2=data$X2, Y1=data$Y1, Y2=data$Y2, k=2)

plot(rbind(out$scoreX1, out$scoreX2), col=c(data$col1, data$col2),
pch=c(rep(2, length=nrow(out$scoreX1)), rep(3, length=nrow(out$scoreX2))))
legend("bottomleft", legend=c("XY1", "XY2"), pch=c(2,3))