2. Multimodal Independent Component Analysis (MICA) and Group Independent Component Analysis (GroupICA)

Koki Tsuyuzaki

Department of Artificial Intelligence Medicine, Graduate School of Medicine, Chiba University
k.t.the-answer@hotmail.co.jp

2023-04-28

Introduction

In this vignette we consider approximating multiple data matrices as a product of multiple low-rank matrices (a.k.a., factor matrices).

Test data is available from toyModel.

library("iTensor")
data1 <- iTensor::toyModel("MICA")
data2 <- iTensor::toyModel("GroupICA")
str(data1, 2)
## List of 2
##  $ X: num [1:10001, 1:3] -0.0613 -0.0593 0.5027 0.3647 0.357 ...
##  $ Y: num [1:10001, 1:3] 0.342 0.385 0.473 0.898 1.001 ...
str(data2, 2)
## List of 2
##  $ X: num [1:5000, 1:6] -2.43 4.07 8.83 -1.56 6.1 ...
##  $ Y: num [1:5000, 1:6] -9.644 -2.587 -0.261 -2.316 -15.774 ...

Both data1 and data2 contain two time-series data X and Y as follows.

plot.ts(data1$X[7700:8000,], main="data1 (X)")

plot.ts(data1$Y[7700:8000,], main="data1 (Y)")

plot.ts(data2$X[4700:5000,], main="data2 (X)")

plot.ts(data2$Y[4700:5000,], main="data2 (Y)")

Multimodal Independent Component Analysis (MICA)

As a formulation that extends ICA (independent component analysis) to the multiple matrices case, Multimodal Independent Component Analysis (MICA) was proposed ((Akaho 1999)). MICA extracts statistically dependent pairs of features from the sources, where the components of feature vector extracted from each source are independent.

MICA can be performed as follows.

t_series <- seq(from = 0.00, to = 1.000, by = 1e-4)
out.MICA <- MICA(data1$X, data1$Y, J=3, gamma_ts = 1 - 1 / (1 + exp(-100 * (t_series - 0.3))))

J is the rank parameter for ICA and gamma_ts is the weighting factor for dependence on independence. You will see that MICA could extract some time-series signals.

plot.ts(out.MICA$U[7700:8000, ], main="Source Signal (X)")

plot.ts(out.MICA$V[7700:8000, ], main="Source Signal (Y)")

Group Independent Component Analysis (GroupICA)

Another formulation of the decomposition is Group Independent Component Analysis (GroupICA (Calhourn 2009; Pfister 2018)). GroupICA can be performed as follows.

out_groupica <- GroupICA(data2, J1=6, algorithm="pooled")

The rank for each factor matrix can be set as J1 and the decomposition algorithm can be easily switched by ica.algorithm (algorithm of ICA and ICA2 can be specified). To pool the results of ICA against each data matrix, we implemented three algorithms such as pooled, Calhoun2009, and Pfister2018. For the details, see the references (Calhourn 2009; Pfister 2018).

You will see that GroupICA could extract some time-series signals.

plot.ts(out_groupica$Ss[[1]], main="Source Signal (X)")

plot.ts(out_groupica$Ss[[2]], main="Source Signal (Y)")

Unlike MICA, GroupICA can also be applied to more than three matrices.

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
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## 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] mixOmics_6.24.0 ggplot2_3.4.2   lattice_0.21-8  MASS_7.3-59    
## [5] iTensor_1.0.2  
## 
## loaded via a namespace (and not attached):
##  [1] tidyr_1.3.0         sass_0.4.5          utf8_1.2.3         
##  [4] generics_0.1.3      stringi_1.7.12      digest_0.6.31      
##  [7] magrittr_2.0.3      evaluate_0.20       grid_4.3.0         
## [10] RColorBrewer_1.1-3  fastmap_1.1.1       plyr_1.8.8         
## [13] jsonlite_1.8.4      Matrix_1.5-4        ggrepel_0.9.3      
## [16] RSpectra_0.16-1     gridExtra_2.3       mgcv_1.8-42        
## [19] purrr_1.0.1         fansi_1.0.4         scales_1.2.1       
## [22] codetools_0.2-19    jquerylib_0.1.4     cli_3.6.1          
## [25] rlang_1.1.0         splines_4.3.0       munsell_0.5.0      
## [28] withr_2.5.0         cachem_1.0.7        yaml_2.3.7         
## [31] ellipse_0.4.5       tools_4.3.0         parallel_4.3.0     
## [34] reshape2_1.4.4      BiocParallel_1.34.0 einsum_0.1.0       
## [37] dplyr_1.1.2         colorspace_2.1-0    corpcor_1.6.10     
## [40] groupICA_0.1.1      vctrs_0.6.2         R6_2.5.1           
## [43] matrixStats_0.63.0  lifecycle_1.0.3     stringr_1.5.0      
## [46] pkgconfig_2.0.3     rTensor_1.4.8       geigen_2.3         
## [49] pillar_1.9.0        bslib_0.4.2         gtable_0.3.3       
## [52] jointDiag_0.4       glue_1.6.2          rARPACK_0.11-0     
## [55] Rcpp_1.0.10         highr_0.10          xfun_0.39          
## [58] tibble_3.2.1        tidyselect_1.2.0    knitr_1.42         
## [61] nlme_3.1-162        htmltools_0.5.5     igraph_1.4.2       
## [64] rmarkdown_2.21      compiler_4.3.0

References

Akaho, S. et al. 1999. “MICA: Multimodal Independent Component Analysis.” IJCNN’99 2: 927–32.
Calhourn, V. D. et al. 2009. “A Review of Group ICA for fMRI Data and ICA for Joint Inference of Imaging, Genetic, and ERP Data.” Neuroimage 45(1 Suppl): S163–72.
Pfister, N. et al. 2018. “groupICA: Independent Component Analysis for Grouped Data.” arXiv.