2. Guided Partial Least Squares (guided-PLS)
Introduction
In this vignette, we consider a novel supervised dimensional reduction method guided partial least squares (guided-PLS).
Test data is available from toyModel.
## List of 8
## $ X1 : int [1:100, 1:300] 86 101 95 106 113 85 88 103 106 84 ...
## $ X2 : int [1:200, 1:150] 106 81 91 101 91 105 111 81 113 105 ...
## $ Y1 : int [1:100, 1:50] 101 77 77 87 101 89 111 113 101 112 ...
## $ Y1_dummy: num [1:100, 1:3] 1 1 1 1 1 1 1 1 1 1 ...
## $ Y2 : int [1:200, 1:50] 107 81 102 90 84 106 97 90 88 115 ...
## $ Y2_dummy: num [1:200, 1:3] 1 1 1 1 1 1 1 1 1 1 ...
## $ col1 : chr [1:100] "#66C2A5" "#66C2A5" "#66C2A5" "#66C2A5" ...
## $ col2 : chr [1:200] "#66C2A5" "#66C2A5" "#66C2A5" "#66C2A5" ...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 (Argelaguet 2021).
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:
\[ \max_{W_{1},W_{2}} \mathrm{tr} \left( W_{1}^{T} X_{1}^{T} Y_{1}^{T} Y_{2} X_{2} W_{2} \right)\ \mathrm{s.t.}\ W_{1}^{T}W_{1} = W_{2}^{T}W_{2} = I_{K} \]
Session Information
## R version 4.6.0 (2026-04-24)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 24.04.4 LTS
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so; LAPACK version 3.12.0
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## time zone: Etc/UTC
## tzcode source: system (glibc)
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] fields_17.3 RColorBrewer_1.1-3 viridisLite_0.4.3 spam_2.11-3
## [5] guidedPLS_1.2.0 rmarkdown_2.31
##
## loaded via a namespace (and not attached):
## [1] cli_3.6.6 knitr_1.51 rlang_1.2.0 xfun_0.57
## [5] dotCall64_1.2 jsonlite_2.0.0 buildtools_1.0.0 htmltools_0.5.9
## [9] maketools_1.3.2 sys_3.4.3 sass_0.4.10 grid_4.6.0
## [13] evaluate_1.0.5 jquerylib_0.1.4 fastmap_1.2.0 yaml_2.3.12
## [17] lifecycle_1.0.5 compiler_4.6.0 irlba_2.3.7 Rcpp_1.1.1-1.1
## [21] maps_3.4.3 lattice_0.22-9 digest_0.6.39 R6_2.6.1
## [25] bslib_0.11.0 Matrix_1.7-5 tools_4.6.0 cachem_1.1.0