Activity 4.2 - Kmeans, PAM, and DBSCAN clustering

SUBMISSION INSTRUCTIONS

1. Render to html
2. Publish your html to RPubs
3. Submit a link to your published solutions

Loading required packages:

library(cluster)
library(dbscan)
library(factoextra)
library(tidyverse)
library(patchwork)
library(ggrepel)

Question 1

Reconsider the three data sets below. We will now compare kmeans, PAM, and DBSCAN to cluster these data sets.

three_spheres <- read.csv('Data/cluster_data1.csv')
ring_moon_sphere <- read.csv('Data/cluster_data2.csv')
two_spirals_sphere <- read.csv('Data/cluster_data3.csv')

A)

With kmeans and PAM, we can specify that we want 3 clusters. But recall with DBSCAN we select minPts and eps, and the number of clusters is determined accordingly. Use k-nearest-neighbor distance plots to determine candidate epsilon values for each data set if minPts = 4. Add horizontal line(s) to each plot indicating your selected value(s) of \(\epsilon.\)

ts_scaled <- scale(three_spheres)
rms_scaled <- scale(ring_moon_sphere)
tss_scaled <- scale(two_spirals_sphere)

kNNdistplot(ts_scaled, minPts = 4)
abline(h = 0.3)

kNNdistplot(rms_scaled, minPts = 4)
abline(h = 0.18)

kNNdistplot(tss_scaled, minPts = 4)
abline(h = 0.03)
abline(h = 0.22)

There Spheres seems to have a clear elbow right around 0.3, Ring, Moon, Sphere seems to also have a clear elbow around 0.18, however Two Spiral Sphere seems to be two places that could be the correct elbow, one at 0.03 and the other at 0.22.

B)

Write a function called plot_dbscan_results(df, eps, minPts). This function takes a data frame, epsilon value, and minPts as arguments and does the following:

• Runs DBSCAN on the inputted data frame df, given the eps and minPts values;
• Creates a scatterplot of the data frame with points color-coded by assigned cluster membership. Make sure the title of the plot includes the value of eps and minPts used to create the clusters!!

Using this function, and your candidate eps values from A) as a starting point, implement DBSCAN to correctly identify the 3 cluster shapes in each of the three data sets. You will likely need to revise the eps values until you settle on a “correct” solution.

plot_dbscan_results <- function(df, eps, minPts){
  
  db <- dbscan::dbscan(df, eps = eps, minPts = minPts)
  
  plot_df <- as_tibble(df) %>%
    mutate(cluster = factor(db$cluster)) %>%
    rename(x = 1, y = 2)
  
  g <- ggplot(plot_df, aes(x = x, y = y, color = cluster)) +
    geom_point(size = 2, alpha = 0.8)  +
    labs(
      title = paste("DBSCAN"),
      color = "Cluster"
    ) +
    theme_minimal()
  
  return(g)
}

plot_dbscan_results(ts_scaled, eps = 0.42, minPts = 4)

plot_dbscan_results(rms_scaled, eps = 0.267, minPts = 4)

plot_dbscan_results(tss_scaled, eps = 0.27, minPts = 4)

C)

Compare your DBSCAN solutions to the 3-cluster solutions from k-means and PAM. Use the patchwork package and your function from B) to produce a 3x3 grid of plots: one plot per method/data set combo. Comment on your findings.

plot_kmeans <- function(df, k){
  km <- kmeans(df, centers = k, nstart = 10)
  
  as_tibble(df) %>%
    mutate(cluster = factor(km$cluster)) %>%
    rename(x = 1, y = 2) %>%
    ggplot(aes(x, y, color = cluster)) +
    geom_point(size = 2, alpha = 0.8) +
    labs(title = paste("K-means" ),
         color = "Cluster") +
    theme_minimal()
}

plot_pam <- function(df, k, nstart = 10){
  pam_res <- cluster::pam(df, k)
  
  as_tibble(df) %>%
    mutate(cluster = factor(pam_res$clustering)) %>%
    rename(x = 1, y = 2) %>%
    ggplot(aes(x, y, color = cluster)) +
    geom_point(size = 2, alpha = 0.8) +
    labs(title = paste("PAM"),
         color = "Cluster") +
    theme_minimal()
}

ts_db  <- plot_dbscan_results(ts_scaled,  eps = 0.42,  minPts = 4)
rms_db <- plot_dbscan_results(rms_scaled, eps = 0.267, minPts = 4)
tss_db <- plot_dbscan_results(tss_scaled, eps = 0.27,  minPts = 4)

ts_km  <- plot_kmeans(ts_scaled, 3)
ts_pam <- plot_pam(ts_scaled, 3)

rms_km  <- plot_kmeans(rms_scaled, 3)
rms_pam <- plot_pam(rms_scaled, 3)

tss_km  <- plot_kmeans(tss_scaled, 3)
tss_pam <- plot_pam(tss_scaled, 3)

(ts_db| rms_db | tss_db )  /
(ts_km| rms_km |tss_km ) /
(ts_pam| rms_pam |tss_pam )

Three Spheres:

• All three methods found the known clusters within the data

Ring, Moon, Sphere:

• DBSCAN did much better at identifying the 3 main clusters, both k-means and PAM reported similar results

Two Spiral Sphere:

• DBSCAN did the best at identifying the groupings, k-means made groupings in pizza shaped groups where PAM made stripes within the points

Question 2

In this question we will apply cluster analysis to analyze economic development indicators (WDIs) from the World Bank. The data are all 2020 indicators and include:

• life_expectancy: average life expectancy at birth
• gdp: GDP per capita, in 2015 USD
• co2: CO2 emissions, in metric tons per capita
• fert_rate: annual births per 1000 women
• health: percentage of GDP spent on health care
• imports and exports: imports and exports as a percentage of GDP
• internet and electricity: percentage of population with access to internet and electricity, respectively
• infant_mort: infant mortality rate, infant deaths per 1000 live births
• inflation: consumer price inflation, as annual percentage
• income: annual per-capita income, in 2020 USD

wdi <- read.csv('Data/wdi_extract_clean.csv') 
head(wdi)

      country life_expectancy        gdp      co2 fert_rate    health internet
1 Afghanistan        61.45400   527.8346 0.180555     5.145 15.533614  17.0485
2     Albania        77.82400  4437.6535 1.607133     1.371  7.503894  72.2377
3     Algeria        73.25700  4363.6853 3.902928     2.940  5.638317  63.4727
4      Angola        63.11600  2433.3764 0.619139     5.371  3.274885  36.6347
5   Argentina        75.87800 11393.0506 3.764393     1.601 10.450306  85.5144
6     Armenia        73.37561  4032.0904 2.334560     1.700 12.240562  76.5077
  infant_mort electricity  imports inflation  exports    income
1        55.3        97.7 36.28908  5.601888 10.42082  475.7181
2         8.1       100.0 36.97995  1.620887 22.54076 4322.5497
3        20.4        99.7 24.85456  2.415131 15.53520 2689.8725
4        42.3        47.0 27.62749 22.271539 38.31454 1100.2175
5         8.7       100.0 13.59828 42.015095 16.60541 7241.0303
6        10.2       100.0 39.72382  1.211436 29.76499 3617.0320

Focus on using kmeans for this problem.

A)

My claim: 3-5 clusters appear optimal for this data set. Support or refute my claim using appropriate visualizations.

wdi_numeric <- wdi %>% select(-country)
wdi_scaled <- scale(wdi_numeric)

fviz_nbclust(wdi_scaled, 
             FUNcluster = kmeans,
             method='wss',
             ) + 
  labs(title = 'Plot of WSS vs k using kmeans') + 
fviz_nbclust(wdi_scaled, 
             FUNcluster = kmeans,
             method='silhouette',
             )+ 
  labs(title = 'Plot of avg silhouette vs k using kmeans')

I support your claim because the silhouette plot shows that the highest average silhouette width is at k=4, the WSS shows an elbow but it is hard to tell if it starts at 3,4, or 5 all which support your claim.

B)

Use k-means to identify 4 clusters. Characterize the 4 clusters using a dimension reduction technique. Provide examples of countries that are representative of each cluster. Be thorough.

wdi_numeric1 <- wdi %>% 
  column_to_rownames("country")

wdi_km <- kmeans(wdi_scaled, centers = 4, nstart = 10)
wdi$cluster <- factor(wdi_km$cluster)
  
wdi_pca <- prcomp(wdi_numeric1, center=TRUE, scale. = TRUE)


kmeans_biplot <- fviz_pca(wdi_pca, 
         habillage = factor(wdi_km$cluster),
         repel = TRUE) + 
      ggtitle('K-means 4-cluster solution') + 
      guides(color='none',shape='none')

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.
ℹ The deprecated feature was likely used in the ggpubr package.
  Please report the issue at <https://github.com/kassambara/ggpubr/issues>.

kmeans_biplot

Orange Cluster:

• Countries within the orange cluster have high fertility rate, infant mortality and inflation. Some countries that follow these characteristics are many countries in Africa like Angola, Niger, Zambia and many others.

Green Cluster:

• The green cluster tends to be countries that have either high or low amounts of imports and export and not large amounts of the other factors. Some countries that follow these characteristics are countries like China, Philippines, Cabo, Thailand and much more.

Purple Cluster:

• The purple cluster is the smallest cluster of them all only have 3 countries within it, Ireland, Singapore, and Luxembourg. These countries all have very high amounts of imports and exports.

Blue Cluster:

• The blue cluster are countries that have high life expectancy, income, GDP, electricity and other factors. Some countries that follow these characteristics are countries like Denmark, Iceland, Netherlands, and Norway.

C)

Remove Ireland, Singapore, and Luxembourg from the data set. Use k-means to find 4 clusters again, with these three countries removed. How do the cluster definitions change?

wdi_less <- wdi %>% 
  select(-cluster) %>% 
  filter(!country %in% c("Ireland", "Singapore", "Luxembourg")) %>% 
  column_to_rownames("country")

less_scaled <- scale(wdi_less)

less_km <- kmeans(less_scaled, centers = 4, nstart = 10)

wdi_less$cluster <- factor(less_km$cluster)

less_pca <- prcomp(less_scaled, center = TRUE, scale. = TRUE)

kmeans_biplot <- fviz_pca(less_pca, 
         habillage = factor(less_km$cluster),
         #label = "none", 
         repel = TRUE) + 
      ggtitle('K-means 4-cluster solution') + 
      guides(color='none',shape='none')

kmeans_biplot

The cluster definitions changed a lot because it got rid of the small 3 country cluster and remade the 4 clusters. It kept the two left most groups very similar but the right grouping or blue cluster above it split it into two. It split them based on if they had high imports and exports or if their health, income, or GDP were high.