Last updated: 2022-12-08

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This is another simple illustration of the perform_cv cross-validation interface in which we use cross-validation to select the \(k\) (the number of clusters) in \(k\)-means. This example introduces two slight complications that didn’t arise in the Elastic Net demo:

  1. The \(k\)-means output depends on initialization. We address this by providing a common initialization for all the \(k\)-means runs. This ensures that perform_cv always produces the same result.

  2. The \(k\)-means problem is an unsupervised learning problem, so \(Y\) (which we define to be the unknown cluster centers) is not used in the “fit” function, and is only used for evaluating the quality of the fit.

Load the perform_cv code.

library(glmnet)
library(parallel)
source("../code/cv.R")

Initialize the sequence of pseudorandom numbers.

set.seed(1)

Simulate a clustering data set.

n <- 400
k <- 5
centers <- matrix(rnorm(2*k),k,2)
membership <- sample(k,n,replace = TRUE)
X <- matrix(0,n,2)
for (i in 1:n) {
  j     <- membership[i]
  X[i,] <- centers[j,] + rnorm(2)/4
}
par(mar = c(4,4,0,0))
plot(X[,1],X[,2],col = "royalblue",pch = 1,cex = 0.75,xlab = "x1",ylab = "x2")
points(centers[,1],centers[,2],col = "red",pch = 20,cex = 1)

Version Author Date
b6a0140 Peter Carbonetto 2022-12-08

The solid red points are the cluster centers.

Now run k-means once with \(k = 10\) clusters. We will use this to initialize the other runs of k-means.

fit_k10 <- kmeans(X,centers = 10,iter.max = 100)

This function runs k-means, initializing the cluster centers using the k-means clustering result with \(k = 10\) clusters.

run_kmeans <- function (x, y, cvpar)
  kmeans(x,fit_k10$centers[1:cvpar,],iter.max = 100)

This function assigns the “best-fit” cluster centers to the data points.

predict_kmeans <- function (x, model) {
  k <- nrow(model$centers)
  D <- as.matrix(dist(rbind(model$centers,x)))
  D <- D[1:k,-(1:k)]
  i <- apply(D,2,which.min)
  return(model$centers[i,])
}

This function computes the mean squared error (MSE) between the estimated cluster centers and the true cluster centers.

compute_mse <- function (pred, true)
  mean((pred - true)^2)

Having defined these three functions, and determined a common initialization for all the k-means runs, we are now ready to use perform_cv.

k <- 2:10
cv <- perform_cv(run_kmeans,predict_kmeans,compute_mse,X,
                 centers[membership,],k)

Now let’s see how the prediction error evolves as we change \(k\).

par(mar = c(4,4,0,0))
plot(k,rowMeans(cv),type = "l",lwd = 2,xlab = "k",ylab = "mse")
points(k,rowMeans(cv),pch = 20)

Version Author Date
676df0a Peter Carbonetto 2022-12-08

Reassuringly, the lowest error is achieved at the correct number of clusters (\(k = 5\)).


sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.7
# 
# Matrix products: default
# BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
# LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
# 
# locale:
# [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
# 
# attached base packages:
# [1] parallel  stats     graphics  grDevices utils     datasets  methods  
# [8] base     
# 
# other attached packages:
# [1] glmnet_4.0-2  Matrix_1.2-18
# 
# loaded via a namespace (and not attached):
#  [1] Rcpp_1.0.8        highr_0.8         pillar_1.6.2      compiler_3.6.2   
#  [5] bslib_0.3.1       later_1.0.0       jquerylib_0.1.4   git2r_0.29.0     
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# [13] lattice_0.20-38   jsonlite_1.7.2    evaluate_0.14     lifecycle_1.0.0  
# [17] tibble_3.1.3      pkgconfig_2.0.3   rlang_0.4.11      foreach_1.4.7    
# [21] yaml_2.2.0        xfun_0.29         fastmap_1.1.0     stringr_1.4.0    
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# [33] fansi_0.4.0       survival_3.1-8    rmarkdown_2.11    magrittr_2.0.1   
# [37] whisker_0.4       splines_3.6.2     codetools_0.2-16  backports_1.1.5  
# [41] promises_1.1.0    ellipsis_0.3.2    htmltools_0.5.2   shape_1.4.4      
# [45] httpuv_1.5.2      utf8_1.1.4        stringi_1.4.3     crayon_1.4.1