This repository contains functions to implement DMMD method in Dongbang Yuan and Irina Gaynanova. "Double-matched matrix decomposition for multi-view data", as well as scripts to reproduce the results in the paper.
Given two data matrices X1 and X2 with matched n samples, and matched p features, the method extracts additive decomposition Xk = Jkc + Ikc + Ek = Jkr + Ikr + Ek, where Jkc is joint structure with respect to matched samples (joint column-space), Ikc is individual structure with respect to matched samples (individual column-space), Jkr is joint structure with respect to matched features (joint row-space), Ikr is individual structure with respect to matched features (individual row-space), and Ek is residual noise.
To reproduce the results, please clone this project locally and open DMMD_Code.Rproj in Rstudio (this makes all relative paths in scripts work as expected).
DMMDFunctions folder contains function scripts needed for implementation of DMMD.
DoubleMatchedMatrixDecomposition.R - DMMD_v2 is the main function that performs DMMD, which includes rank selection and estimation of signal matrices given the ranks
Double_iterative.R - modification of DMMD to allow iterative updates of joint column and row spaces, DMMD_i is the main function
FindOptMatrix.R - FindOpt_DM_Iterative estimates the signal matrices given the ranks, this corresponds to Algorithm 1 in the paper
Profile_Likelihood_Rank_selection.R - contains functions for total rank estimation using profile likelihood
Angle_Calculation.R - contains functions for joint rank estimation using profile likelihood.
Preliminary_Functions.R - small supplementary functions used by the main functions above
To illustrate application of DMMD, we will simulate data based on DMMD model using scripts from DoubleMatchedDataGen.R (located in OtherFunctions folder)
source("OtherFunctions/DoubleMatchedDataGen.R")
source("DMMDFunctions/Preliminary_Functions.R")
set.seed(37)
# Generate data
data = DoubleDataGen3(n = 20, p = 16, rank = c(4, 3), joint_rank_col = 2, joint_rank_row = 1, nrep = 1, std1 = 0.01, std2 = 0.01)
X1 = data$X1_list[[1]]
X2 = data$X2_list[[1]]
We then illustrate application of DMMD.
# Source all necessary functions to apply DMMD
source("DMMDFunctions/Angle_Calculation.R")
source("DMMDFunctions/Profile_Likelihood_Rank_Selection.R")
source("DMMDFunctions/DoubleMatchedMatrixDecomposition.R")
source("DMMDFunctions/DMMD_iterative.R")
source("DMMDFunctions/FindOptMatrix.R")
# Apply DMMD
DMMD_result = DMMD_v2(X1, X2)
# Extract estimated ranks
DMMD_result$"Rank"
# Extract common and individual structures for 1st signal
J1c = DMMD_result$"Column Decomposition"$"Joint Column 1"
I1c = DMMD_result$"Column Decomposition"$"Individual Column 1"
J1r = DMMD_result$"Row Decomposition"$"Joint Row 1"
I1r = DMMD_result$"Row Decomposition"$"Individual Row 1"
# Verify that both decompositions give the same estimated signal
sum((J1c + I1c - J1r - I1r)^2)
# Get estimated 1st signal matrix of DMMD
A1_est = X1 - DMMD_result$Error$Error1
# Get the true signal
A1 = data$Signal1_list[[1]]
# Check the difference between estimated signal of DMMD with true signal
sum((A1_est - A1)^2)
Alternatively, DMMD-i can be applied (empirically this gives slight improvement over DMMD, but has considerably higher computation cost, not recommended with large datasets). The output is a little different from DMMD as it directly returns estimated signal matrices and joint row/column spaces for convenience of later analyses.
# Apply DMMD-i
DMMD_i_result = DMMD_i(X1, X2)
# Get estimated signal matrices of DMMD-i
A1_est_i = DMMD_i_result$A1
A2_est_i = DMMD_i_result$A2
# Get estimated basis vectors for joint subspaces
M_est = DMMD_i_result$M # joint column space (matched samples)
N_est = DMMD_i_result$N # joint row space (matched features)
# Get the joint column structure for view 1.
J1c = M_est %*% t(M_est) %*% X1
# Get individual column structure
I1c = A1_est_i - J1c
# Get the joint row structure for view 1
J1r = X1 %*% N_est %*% t(N_est)
# Get individual crow structure
I1r = A1_est_i - J1r
# Check the difference between estimated signal of DMMD-i with true signal
sum((A1_est_i - A1)^2)
DMMD algorithm does not rely on external libraries, however, for comparing different methods, corresponding libraries need to be installed. Please refer to the head of each script to make sure the packages are installed locally on your computer.
Model_Demo_Picture folder - codes to generate example of DMMD decomposition on toy data from Figure 1
SimulationData_Setting1, SimulationData_Setting2, SimulationData_Setting3, Data_Setting4_FixRank, Data_Setting5_FixRank_SNR0.5, SimulationData_Setting6 folders within Simulations:
- Main.R within each respective folder generates simulation data for corresponding setting. See Section 3 of the paper for additional details on the settings.
RankEstimation_Setting1, RankEstimation_Setting2, RankEstimation_Setting3_Special, Signal_Identification_Setting4, Signal_Identification_Setting4, Signal_Identification_Setting5, Signal_Identification_Setting6_Special folders within Simulations:
- AJIVE.R has implementation of AJIVE, SLIDE.R has implementation of SLIDE; Main.R runs DMMD and JIVE, PlotSimulations_Draft.R combines all the results from AJIVE, SLIDE, DMMD and JIVE and plot the figures. They are saved in Figures/Draft.
Rank estimation comparisons also include edge distribution (ED) method in Shu, Hai, Xiao Wang, and Hongtu Zhu. "D-CCA: A decomposition-based canonical correlation analysis for high-dimensional datasets." Journal of the American Statistical Association 115.529 (2020): 292-306. Also refer to Onatski, Alexei. “Determining the number of factors from empirical distribution of eigenvalues.” The Review of Economics and Statistics, vol. 92, no. 4, 2010, pp. 1004–1016. JSTOR for more information on the edge distribution method.
New_Data_TCGA_Dim folder - High-dimensional simulation setting - supplement S3.4
RankMisspecification folder - Signal estimation when ranks are misspecified - supplement S3.5
Speed_test folder - Computational comparisons - supplement S3.6.
Application/Soccer folder - soccer data sets and corresponding analyses
Application/TCGA folder - code for analyses of TCGA miRNA dataset (the data are in Data folder )