Introduction
This vignette presents the \(\textbf{MetabolSSMF}\) package, which
implements a framework for the simplex-structured matrix factorisation
(SSMF) algorithm in \(\mathrm{R}\).
This package provides a standard algorithm, which allows users to apply
the SSMF to perform soft clustering of the data set that they are
interested in. The usage of this package will be illustrated through an
application to a simulated metabolomic data set. This vignette also
includes the usage of:
- gap statistic to select the number of
clusters;
- bootstrap resampling for estimating confidence
intervals for the cluster prototypes;
- soft adjusted Rand index for comparing known and
estimated partitions;
- Shannon diversity index for evaluating the average
uncertainty in a soft clustering partition.
This document gives a quick tour of \(\textbf{MetabolSSMF}\) functionalities. It
was written in R Markdown, using the knitr package for production. See
help(package=“MetabolSSMF”) for further details.
\(\textbf{Installation:}\) The
latest version of the package can be installed from \(\textbf{Github}\):
# Install the package
devtools::install_github("WenxuanLiu1996/MetabolSSMF")
# Load the package
library(MetabolSSMF)
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
Data
The simulated data set is created by the product of a soft membership
matrix \(H\) and a prototype matrix
\(W\). A user can use a Dirichlet
distribution to generate the values for the soft membership matrix \(H\) with simplex structure. \(W\) can be defined using a designed
pattern. The package provides a simulated data set \(X\), simulated prototype matrix \(W\) and simulated membership matrix \(H\). These can be loaded through the code
data(SimulatedDataset),
data(SimulatedPrototypes) and
data(SimulatedMemberships), respectively.
# Simulated data set X
X <- as.matrix(SimulatedDataset)
# Simulated W matrix (prototype matrix)
W <- as.matrix(SimulatedPrototypes)
# Simulated H matrix (membership matrix)
H <- as.matrix(SimulatedMemberships)
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
Visualise the simulated data
Visualise the values of simulated prototype matrix
\(W\)
# Define 4 different colors
color <- c(ggsci::pal_jco("default", alpha=0.8)(7)[2], ggsci::pal_jama("default", alpha=0.8)(7)[c(3:5)])
op1 <- par(mar=c(5.1, 4.1, 4.1, 4.1), xpd=TRUE)
matplot(t(W), type='l', lty = 1, ylab='Prototypes', col = color, lwd=2, ylim = c(0,1))
legend('topright', inset=c(-0.15,0), legend = 1:4, fill=color[1:4], cex = 0.8)
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
Visualise the values of simulated soft membership
matrix \(H\) in each
observation
The scatter pies plot of the 2-dimensional projected clusters. It
represents the simulated data with 4 soft clusters, the area of colour
in the pies depicts the values of the soft membership of the
observations.
pca_X <- prcomp(X, center = F, scale. = F)
op1 <- par(mar=c(5.1, 4.1, 4.1, 4.1), xpd=TRUE)
caroline::pies(lapply(apply(H, 1, list), unlist), color.table = caroline::nv(color[1:4], paste0('Pty', 1:4)), x0=pca_X$x[,1], y0=pca_X$x[,2], radii = 1.5, ylab='PC2', xlab='PC1')
legend('topright', inset=c(-0.15,0), legend = 1:4, fill=color[1:4], cex = 0.8)
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
Choose the number of clusters
The SSMF algorithm can be run with \(k=1,
2, \ldots, K\) using a for loop and the results can
be saved to a list. The ‘elbow criterion’ or the gap statistic can be
done as a subsequent step.
# Set the maximum k that is used in loop.
K <- 10
# Run the SSMF algorithm with various k and save the results as a list
fit_SSMF <- list()
for(k in 1:K){
fit <- ssmf(X, k = k, lr = 0.001, meth='kmeans')
fit_SSMF[[k]] <- fit
}
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
Plot the residual sum of squares (“elbow
criterion”)
# Extract the RSS and save as a vector
rss_SSMF <- unlist(lapply(fit_SSMF, function(x) x$SSE))
# Plot RSS
plot(1:K, rss_SSMF, type="b", xlab = "K", ylab = "RSS", main='Elbow Criterion')
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
If the ‘elbow’ is not clear, the gap statistic is suggested.
Gap statistic
# Apply the gap statistic to the simulated data
fit_gap <- gap(X, rss = rss_SSMF)
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
# Visualise the results
op2 <- par(mar=c(5.1, 4.1, 4.1, 8.1), xpd=TRUE)
plot(x = c(1:K), y = fit_gap$gap, type='b', ylim=range((fit_gap$gap), (fit_gap$gap - fit_gap$standard.error)[-1]), xlab='K', ylab='GAP', main='Gap statistic')
lines(x = c(1:(K-1)), y = (fit_gap$gap - fit_gap$standard.error)[-1], col='blue', lty=2, type='b')
lines(x = rep(fit_gap$optimal.k, 2), y = range(fit_gap$gap), lty = 2, col='red')
legend('topright', inset=c(-0.45,0), col=c('black', 'blue', 'red'), lty=c(1, 2, 2), legend = c('Gap', '-1 SE', 'Optimal K'), cex=0.8)
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
The estimated number of prototypes is 4 from both the ‘elbow
criterion’ and gap statistic.
Bootstrap
# Set the number of times to bootstrap
M <- 1000
# Bootstrap
# Initial H (membership) matrix to start the algorithm
# Based on the results above, k=4 here
initialisation <- init(data = X, k = 4, method = 'kmeans')
fit_boot <- bootstrap(data = X, k = 4, H = initialisation$H, mtimes=M)
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
Note: Usually the \(H\) matrix is
the output of the function ssmf( ). If \(H\) is not supplied, the bootstrapped \(W\) matrix might have different prototype
orders from the outputs of the function ssmf( ).
Visualise the values of the bootstrap estimation and
confidence intervals for the prototypes
op1 <- par(mar=c(5.1, 4.1, 4.1, 4.1), xpd=TRUE)
matplot(t(fit_boot$W.est[c(2,4,1,3),]), type='n', xaxt='n', ylab='Features', ylim = range(fit_boot$lower,fit_boot$upper))
for(i in 1:4){
alpha <- 0.2
plat <- c(ggsci::pal_jco("default",alpha=alpha)(7)[2], ggsci::pal_jama("default", alpha=alpha)(7)[c(3:5)])
col <- switch(i, plat[1], plat[2], plat[3], plat[4])
polygon(c(1:138, 138:1), c(fit_boot$upper[c(2,4,1,3),][i,], rev(fit_boot$lower[c(2,4,1,3),][i,])), col = col, border= col, lty = 1)
}
matplot(t(fit_boot$W.est[c(2,4,1,3),]), type='l', lty = 1, ylab='Features', add=T, col=color, lwd=2)
legend('topright', inset=c(-0.2,0), legend = 1:4, fill=color[1:4])
axis(1, at=c(1, seq(10, 130, 10), 138), labels = c(1, seq(10, 130, 10), 138), cex=0.8)
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
Estimated prototypes are represented by the solid lines and the 4
colours represent 4 clusters. Shadows around the lines are confidence
intervals for the prototypes.
Visualise the values of the estimated soft membership
matrix
op1 <- par(mar=c(5.1, 4.1, 4.1, 4.1), xpd=TRUE)
caroline::pies(lapply(apply(fit_SSMF[[4]]$H, 1, list), unlist), color.table = caroline::nv(color[1:4], c(4,3,2,1)), x0=pca_X$x[,1], y0=pca_X$x[,2], radii = 1.5, ylab='PC2', xlab='PC1')
legend('topright', inset=c(-0.15,0), legend = 1:4, fill=color[1:4], cex=0.8)
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
Shannon diversity index
E <- rep(NA, nrow(fit_SSMF[[4]]$H))
for(i in 1:nrow(fit_SSMF[[4]]$H)){
E[i] <- diversity(fit_SSMF[[4]]$H[i,], two.power=T)
}
round(mean(E), 1)
#> [1] 2.7
/private/var/folders/l3/8md3czn13hz4w14zx7_lrr900000gn/T/Rtmp3ggjhG/Rbuild645639a314ca/MetabolSSMF/vignettes/MetabolSSMF.R
The estimated soft memberships of the simulated observations have an
average values of \(2^{\mathrm{E}(h_{i
\cdot})}=\) 2.7, indicating that most observations softly
clustered by SSMF are fuzzy between 2 and 3clusters.