Calculate total sum of squares between clusters in R - r

Question

My objective is to compare which of the two clustering methods I've used cluster_method_1 and cluster_method_2 has the largest between cluster sum of squares in order to identify which one achieves better separation.
I'm basically looking for an efficient way to calculate the distance between each point of cluster 1 and all points of cluster 2,3,4, and so on.
example dataframe:
structure(list(x1 = c(0.01762376, -1.147739752, 1.073605848,
2.000420899, 0.01762376, 0.944438811, 2.000420899, 0.01762376,
-1.147739752, -1.147739752), x2 = c(0.536193126, 0.885609849,
-0.944699546, -2.242627057, -1.809984553, 1.834120637, 0.885609849,
0.96883563, 0.186776403, -0.678508604), x3 = c(0.64707104, -0.603759684,
-0.603759684, -0.603759684, -0.603759684, 0.64707104, -0.603759684,
-0.603759684, -0.603759684, 1.617857394), x4 = c(-0.72712328,
0.72730861, 0.72730861, -0.72712328, -0.72712328, 0.72730861,
0.72730861, -0.72712328, -0.72712328, -0.72712328), cluster_method_1 = structure(c(1L,
3L, 3L, 3L, 2L, 2L, 3L, 2L, 1L, 4L), .Label = c("1", "2", "4",
"6"), class = "factor"), cluster_method_2 = structure(c(5L, 3L,
1L, 3L, 4L, 2L, 1L, 1L, 1L, 6L), .Label = c("1", "2", "3", "4",
"5", "6"), class = "factor")), row.names = c(NA, -10L), class = c("tbl_df",
"tbl", "data.frame"))
x1 x2 x3 x4 cluster_method_1 cluster_method_2
<dbl> <dbl> <dbl> <dbl> <fct> <fct>
1 0.0176 0.536 0.647 -0.727 1 5
2 -1.15 0.886 -0.604 0.727 4 3
3 1.07 -0.945 -0.604 0.727 4 1
4 2.00 -2.24 -0.604 -0.727 4 3
5 0.0176 -1.81 -0.604 -0.727 2 4
6 0.944 1.83 0.647 0.727 2 2
7 2.00 0.886 -0.604 0.727 4 1
8 0.0176 0.969 -0.604 -0.727 2 1
9 -1.15 0.187 -0.604 -0.727 1 1
10 -1.15 -0.679 1.62 -0.727 6 6

Answer 1

The within sum-of-squares for cluster Si can be written as the sum of all pairwise (Euclidean) distances squared, divided by twice the number of points in that cluster (see e.g. the Wikipedia article on k-means clustering)
For convenience we define a function calc_SS that returns the within sum-of-squares for a (numeric) data.frame
calc_SS <- function(df) sum(as.matrix(dist(df)^2)) / (2 * nrow(df))
It's then straightforward to calculate the within (cluster) sum-of-squares for every cluster for every method
library(tidyverse)
df %>%
gather(method, cluster, cluster_method_1, cluster_method_2) %>%
group_by(method, cluster) %>%
nest() %>%
transmute(
method,
cluster,
within_SS = map_dbl(data, ~calc_SS(.x))) %>%
spread(method, within_SS)
## A tibble: 6 x 3
# cluster cluster_method_1 cluster_method_2
# <chr> <dbl> <dbl>
#1 1 1.52 9.99
#2 2 10.3 0
#3 3 NA 10.9
#4 4 15.2 0
#5 5 NA 0
#6 6 0 0
The total within sum-of-squares is then just the sum of the within sum-of-squares for every cluster
df %>%
gather(method, cluster, cluster_method_1, cluster_method_2) %>%
group_by(method, cluster) %>%
nest() %>%
transmute(
method,
cluster,
within_SS = map_dbl(data, ~calc_SS(.x))) %>%
group_by(method) %>%
summarise(total_within_SS = sum(within_SS)) %>%
spread(method, total_within_SS)
## A tibble: 1 x 2
# cluster_method_1 cluster_method_2
# <dbl> <dbl>
#1 27.0 20.9
By the way, we can confirm that calc_SS does indeed return the within sum-of-squares using the iris dataset:
set.seed(2018)
df2 <- iris[, 1:4]
kmeans <- kmeans(as.matrix(df2), 3)
df2$cluster <- kmeans$cluster
df2 %>%
group_by(cluster) %>%
nest() %>%
mutate(within_SS = map_dbl(data, ~calc_SS(.x))) %>%
arrange(cluster)
## A tibble: 3 x 3
# cluster data within_SS
# <int> <list> <dbl>
#1 1 <tibble [38 × 4]> 23.9
#2 2 <tibble [62 × 4]> 39.8
#3 3 <tibble [50 × 4]> 15.2
kmeans$within
#[1] 23.87947 39.82097 15.15100

Answer 2

The total sum of squares, sum_x sum_y ||x-y||² is constant.
The total sum of squares can be computed trivially from variance.
If you now subtract the within-cluster sum of squares where x and y belong to the same cluster, then the between cluster sum of squares remains.
If you do this approach, it takes O(n) time instead of O(n²).
Corollary: the solution with the smallest WCSS has the largest BCSS.

Answer 3

Consider the package clValid. It calculates a large number of indexes for validating clustering. The Dunn index is particularly appropriate for what you are trying to do. The documentation says that the Dunn index is the ratio between the smallest distance between observation not in the same cluster to the largest intra-cluster distance. The documentation for it can be found at https://cran.r-project.org/web/packages/clValid/clValid.pdf.