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2. Guided Partial Least Squares (guided-PLS)

Koki Tsuyuzaki

2023-05-12

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)

## 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:

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

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))

Session Information

## R version 4.3.0 (2023-04-21)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Debian GNU/Linux bookworm/sid
## 
## 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-r0.3.21.so;  LAPACK version 3.11.0
## 
## 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_14.1       viridis_0.6.2     viridisLite_0.4.1 spam_2.9-1       
## [5] guidedPLS_1.0.0  
## 
## loaded via a namespace (and not attached):
##  [1] Matrix_1.5-3     gtable_0.3.3     jsonlite_1.8.4   highr_0.10      
##  [5] compiler_4.3.0   maps_3.4.1       gridExtra_2.3    jquerylib_0.1.4 
##  [9] scales_1.2.1     yaml_2.3.7       fastmap_1.1.1    lattice_0.20-45 
## [13] ggplot2_3.4.2    R6_2.5.1         knitr_1.42       dotCall64_1.0-2 
## [17] tibble_3.2.1     munsell_0.5.0    bslib_0.4.2      pillar_1.9.0    
## [21] rlang_1.1.0      utf8_1.2.3       cachem_1.0.7     xfun_0.39       
## [25] sass_0.4.5       cli_3.6.1        magrittr_2.0.3   digest_0.6.31   
## [29] grid_4.3.0       irlba_2.3.5.1    lifecycle_1.0.3  vctrs_0.6.2     
## [33] evaluate_0.20    glue_1.6.2       fansi_1.0.4      colorspace_2.1-0
## [37] rmarkdown_2.21   tools_4.3.0      pkgconfig_2.0.3  htmltools_0.5.5

References

Argelaguet, et al., R. 2021. “Computational Principles and Challenges in Single-Cell Data Integration.” Nature Biotechnology 39: 1202–15.

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