{r, echo = FALSE} knitr::opts_chunk$set( collapse = TRUE, comment = "#>", fig.path = "man/figures/README-" )
The goal of pivmet is to propose some pivotal methods in order to:
undo the label switching problem which naturally arises during the MCMC sampling in Bayesian mixture models \(\rightarrow\) pivotal relabelling (Egidi et al. 2018a)
fit sparse finite Gaussian mixtures
initialize the K-means algorithm aimed at obtaining a good clustering solution \(\rightarrow\) pivotal seeding (Egidi et al. 2018b)
You can install the CRAN version of pivmet with:
{r, eval = FALSE} install.packages("pivmet") library(pivmet)
You can install the development version of pivmet from Github with:
{r gh-installation, eval = FALSE} # install.packages("devtools") devtools::install_github("leoegidi/pivmet")
First of all, we load the package and we import the fish dataset belonging to the bayesmix package:
{r example} library(bayesmix) library(pivmet) data(fish) y <- fish[,1] N <- length(y) # sample size k <- 5 # fixed number of clusters nMC <- 12000 # MCMC iterations
Then we fit a Bayesian Gaussian mixture using the piv_MCMC function:
{r fit, message =FALSE, warning = FALSE} res <- piv_MCMC(y = y, k = k, nMC = nMC)
Finally, we can apply pivotal relabelling and inspect the new posterior estimates with the functions piv_rel and piv_plot, respectively:
{r plot, message =FALSE, warning = FALSE} rel <- piv_rel(mcmc=res) piv_plot(y = y, mcmc = res, rel_est = rel, type = "chains") piv_plot(y = y, mcmc = res, rel_est = rel, type = "hist")
To allow sparse finite mixture fit, we could select the argument sparsity = TRUE:
{r sparsity, message =FALSE, warning = FALSE} res2 <- piv_MCMC(y, k, nMC, sparsity = TRUE, priors = list(alpha = rep(0.001, k))) # sparse on eta barplot(table(res2$nclusters), xlab= expression(K["+"]), col = "blue", border = "red", main = expression(paste("p(",K["+"], "|y)")), cex.main=3, yaxt ="n", cex.axis=2.4, cex.names=2.4, cex.lab=2)
Sometimes K-means algorithm does not provide an optimal clustering solution. Suppose to generate some clustered data and to detect one pivotal unit for each group with the MUS (Maxima Units Search algorithm) function:
```{r mus, echo =TRUE, eval = TRUE, message = FALSE, warning = FALSE} library(mvtnorm)
#generate some data
set.seed(123) n <- 620 centers <- 3 n1 <- 20 n2 <- 100 n3 <- 500 x <- matrix(NA, n,2) truegroup <- c( rep(1,n1), rep(2, n2), rep(3, n3))
for (i in 1:n1){ x[i,]=rmvnorm(1, c(1,5), sigma=diag(2))} for (i in 1:n2){ x[n1+i,]=rmvnorm(1, c(4,0), sigma=diag(2))} for (i in 1:n3){ x[n1+n2+i,]=rmvnorm(1, c(6,6), sigma=diag(2))}
H <- 1000 a <- matrix(NA, H, n)
for (h in 1:H){ a[h,] <- kmeans(x,centers)$cluster }
#build the similarity matrix sim_matr <- matrix(NA, n,n) for (i in 1:(n-1)){ for (j in (i+1):n){ sim_matr[i,j] <- sum(a[,i]==a[,j])/H sim_matr[j,i] <- sim_matr[i,j] } }
cl <- kmeans(x, centers, nstart=10)$cluster mus_alg <- MUS(C = sim_matr, clusters = cl, prec_par = 5) ```
Quite often, classical K-means fails in recognizing the true groups:
```{r kmeans_plots, echo =TRUE, fig.show=‘hold’, eval = TRUE, message = FALSE, warning = FALSE} # launch classical kmeans kmeans_res <- kmeans(x, centers, nstart = 10) # plots par(mfrow=c(1,2)) colors_cluster <- c(“grey”, “darkolivegreen3”, “coral”) colors_centers <- c(“black”, “darkgreen”, “firebrick”)
graphics::plot(x, col = colors_cluster[truegroup] ,bg= colors_cluster[truegroup], pch=21, xlab=“y[,1]”, ylab=“y[,2]”, cex.lab=1.5, main=“True data”, cex.main=1.5)
graphics::plot(x, col = colors_cluster[kmeans_res$cluster], bg=colors_cluster[kmeans_res$cluster], pch=21, xlab=“y[,1]”, ylab=“y[,2]”, cex.lab=1.5,main=“K-means”, cex.main=1.5) points(kmeans_res$centers, col = colors_centers[1:centers], pch = 8, cex = 2) ```
In such situations, we may need a more robust version of the classical K-means. The pivots may be used as initial seeds for a classical K-means algorithm. The function piv_KMeans works as the classical kmeans function, with some optional arguments (in the figure below, the colored triangles represent the pivots).
```{r musk, fig.show=‘hold’} # launch piv_KMeans piv_res <- piv_KMeans(x, centers) # plots par(mfrow=c(1,2), pty=“s”) colors_cluster <- c(“grey”, “darkolivegreen3”, “coral”) colors_centers <- c(“black”, “darkgreen”, “firebrick”) graphics::plot(x, col = colors_cluster[truegroup], bg= colors_cluster[truegroup], pch=21, xlab=“x[,1]”, ylab=“x[,2]”, cex.lab=1.5, main=“True data”, cex.main=1.5)
graphics::plot(x, col = colors_cluster[piv_res$cluster], bg=colors_cluster[piv_res$cluster], pch=21, xlab=“x[,1]”, ylab=“x[,2]”, cex.lab=1.5, main=“piv_Kmeans”, cex.main=1.5) points(x[piv_res$pivots[1],1], x[piv_res$pivots[1],2], pch=24, col=colors_centers[1],bg=colors_centers[1], cex=1.5) points(x[piv_res$pivots[2],1], x[piv_res$pivots[2],2], pch=24, col=colors_centers[2], bg=colors_centers[2], cex=1.5) points(x[piv_res$pivots[3],1], x[piv_res$pivots[3],2], pch=24, col=colors_centers[3], bg=colors_centers[3], cex=1.5) points(piv_res$centers, col = colors_centers[1:centers], pch = 8, cex = 2)
```
Egidi, L., Pappadà, R., Pauli, F. and Torelli, N. (2018a). Relabelling in Bayesian Mixture Models by Pivotal Units. Statistics and Computing, 28(4), 957-969.
Egidi, L., Pappadà, R., Pauli, F., Torelli, N. (2018b). K-means seeding via MUS algorithm. Conference Paper, Book of Short Papers, SIS2018, ISBN: 9788891910233.