Assignment 4

Author

Yi Wang

Workshop: Principal Components Analysis & Clustering Methods

Principal Components Analysis

library(datasets)
library(ISLR)
Warning: package 'ISLR' was built under R version 4.3.3
arrest = USArrests
states = row.names(USArrests)
names(USArrests)
[1] "Murder"   "Assault"  "UrbanPop" "Rape"    
apply(USArrests, 2, mean)
  Murder  Assault UrbanPop     Rape 
   7.788  170.760   65.540   21.232 
apply(USArrests, 2, var)
    Murder    Assault   UrbanPop       Rape 
  18.97047 6945.16571  209.51878   87.72916 
pr.out=prcomp(USArrests, scale=TRUE)
names(pr.out)
[1] "sdev"     "rotation" "center"   "scale"    "x"       
pr.out$center
  Murder  Assault UrbanPop     Rape 
   7.788  170.760   65.540   21.232 
pr.out$scale
   Murder   Assault  UrbanPop      Rape 
 4.355510 83.337661 14.474763  9.366385 
pr.out$rotation
                PC1        PC2        PC3         PC4
Murder   -0.5358995 -0.4181809  0.3412327  0.64922780
Assault  -0.5831836 -0.1879856  0.2681484 -0.74340748
UrbanPop -0.2781909  0.8728062  0.3780158  0.13387773
Rape     -0.5434321  0.1673186 -0.8177779  0.08902432
dim(pr.out$x)
[1] 50  4
pr.out$rotation=-pr.out$rotation
pr.out$x=-pr.out$x
biplot(pr.out, scale=0)

pr.out$sdev
[1] 1.5748783 0.9948694 0.5971291 0.4164494
pr.var=pr.out$sdev^2
pr.var
[1] 2.4802416 0.9897652 0.3565632 0.1734301
pve=pr.var/sum(pr.var)
pve
[1] 0.62006039 0.24744129 0.08914080 0.04335752
plot(pve, xlab="Principal Component", ylab="Proportion of Variance Explained", ylim=c(0,1),type='b')

plot(cumsum(pve), xlab="Principal Component", ylab="Cumulative Proportion of Variance Explained", ylim=c(0,1),type='b')

library(factoextra)
Warning: package 'factoextra' was built under R version 4.3.3
Loading required package: ggplot2
Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa

fviz(pr.out, "ind", geom = "auto", mean.point = TRUE, font.family = "Georgia")
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fviz_pca_biplot(pr.out, font.family = "Georgia", col.var="pink")
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K Means Clustering

library(dplyr)

Attaching package: 'dplyr'
The following objects are masked from 'package:stats':

    filter, lag
The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union
library(ggplot2)
library(RColorBrewer)

computers = read.csv("https://raw.githubusercontent.com/guru99-edu/R-Programming/master/computers.csv") 

# Only retain two variables for illustration
rescaled_comp <- computers[4:5] %>%
  mutate(hd_scal = scale(hd),
         ram_scal = scale(ram)) %>%
  select(c(hd_scal, ram_scal))
        
ggplot(data = rescaled_comp, aes(x = hd_scal, y = ram_scal)) +
  geom_point(pch=20, col = "cyan") + theme_bw() +
  labs(x = "Hard drive size (Scaled)", y ="RAM size (Scaled)" ) +
  theme(text = element_text(family="Georgia")) 
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library(animation)
Warning: package 'animation' was built under R version 4.3.3
set.seed(2345)
library(animation)

# Animate the K-mean clustering process, cluster no. = 4
kmeans.ani(rescaled_comp[1:2], centers = 4, pch = 15:18, col = 1:4) 

ggplot(iris, aes(Petal.Length, Petal.Width)) + geom_point() + 
  theme_bw() +
  scale_color_manual(values=c("pink","lightgreen","cyan"))

ggplot(iris, aes(Petal.Length, Petal.Width, color = Species)) + geom_point() + 
  theme_bw() +
  scale_color_manual(values=c("orange","forestgreen","lightblue"))

set.seed(20)
irisCluster <- kmeans(iris[, 3:4], 3, nstart = 20)
irisCluster
K-means clustering with 3 clusters of sizes 52, 48, 50

Cluster means:
  Petal.Length Petal.Width
1     4.269231    1.342308
2     5.595833    2.037500
3     1.462000    0.246000

Clustering vector:
  [1] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
 [38] 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [75] 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 1 2 2 2 2
[112] 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2
[149] 2 2

Within cluster sum of squares by cluster:
[1] 13.05769 16.29167  2.02200
 (between_SS / total_SS =  94.3 %)

Available components:

[1] "cluster"      "centers"      "totss"        "withinss"     "tot.withinss"
[6] "betweenss"    "size"         "iter"         "ifault"      
class(irisCluster$cluster)
[1] "integer"
table(irisCluster$cluster, iris$Species)
   
    setosa versicolor virginica
  1      0         48         4
  2      0          2        46
  3     50          0         0
irisCluster$cluster <- as.factor(irisCluster$cluster)
ggplot(iris, aes(Petal.Length, Petal.Width, color = irisCluster$cluster)) + geom_point() +
  scale_color_manual(values=c("magenta","forestgreen","steelblue")) +
  theme_bw()

actual = ggplot(iris, aes(Petal.Length, Petal.Width, color = Species)) + geom_point() + 
  theme_bw() +
  scale_color_manual(values=c("firebrick1","lightgreen","lightblue")) +
  theme(legend.position="bottom") +
  theme(text = element_text(family="Georgia")) 
kmc = ggplot(iris, aes(Petal.Length, Petal.Width, color = irisCluster$cluster)) + geom_point() +
  theme_bw() +
  scale_color_manual(values=c("orange", "blue", "green")) +
  theme(legend.position="bottom") +
  theme(text = element_text(family="Georgia")) 
library(grid)
library(gridExtra)

Attaching package: 'gridExtra'
The following object is masked from 'package:dplyr':

    combine
grid.arrange(arrangeGrob(actual, kmc, ncol=2, widths=c(1,1)), nrow=1)
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library(readr)
wine <- read_csv("https://raw.githubusercontent.com/datageneration/gentlemachinelearning/master/data/wine.csv")
Rows: 178 Columns: 14
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (14): class, Alcohol, Malic, Ash, Ash_alcalinity, Magnesium, Total_pheno...

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
## Choose and scale variables
wine_subset <- scale(wine[ , c(2:4)])

## Create cluster using k-means, k = 3, with 25 initial configurations
wine_cluster <- kmeans(wine_subset, centers = 3,
                       iter.max = 10,
                       nstart = 25)
wine_cluster
K-means clustering with 3 clusters of sizes 48, 60, 70

Cluster means:
     Alcohol      Malic        Ash
1  0.1470536  1.3907328  0.2534220
2  0.8914655 -0.4522073  0.5406223
3 -0.8649501 -0.5660390 -0.6371656

Clustering vector:
  [1] 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2
 [38] 2 3 1 2 1 2 1 3 1 1 2 2 2 3 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 2 3 3 2 2 2
 [75] 3 3 3 3 3 1 3 3 3 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
[112] 3 1 3 3 3 3 3 1 3 3 2 1 1 1 3 3 3 3 1 3 1 3 1 3 3 1 1 1 1 1 2 1 1 1 1 1 1
[149] 1 1 1 1 2 1 3 1 1 1 2 2 1 1 1 1 2 1 1 1 2 1 3 3 2 1 1 1 2 1

Within cluster sum of squares by cluster:
[1]  73.71460  67.98619 111.63512
 (between_SS / total_SS =  52.3 %)

Available components:

[1] "cluster"      "centers"      "totss"        "withinss"     "tot.withinss"
[6] "betweenss"    "size"         "iter"         "ifault"      
wssplot <- function(data, nc=15, seed=1234){
  wss <- (nrow(data)-1)*sum(apply(data,2,var))
  for (i in 2:nc){
    set.seed(seed)
    wss[i] <- sum(kmeans(data, centers=i)$withinss)}
  plot(1:nc, wss, type="b", xlab="Number of Clusters",
       ylab="Within groups sum of squares")
}

# plotting values for each cluster starting from 1 to 9
wssplot(wine_subset, nc = 9)

wine_cluster$cluster = as.factor(wine_cluster$cluster)
pairs(wine[2:4],
      col = c("orange", "blue", "green")[wine_cluster$cluster],
      pch = c(15:17)[wine_cluster$cluster],
      main = "K-Means Clusters: Wine data")

table(wine_cluster$cluster)

 1  2  3 
48 60 70 
library(factoextra)
fviz_nbclust(wine_subset, kmeans, method = "wss")

wine.km <- eclust(wine_subset, "kmeans", nboot = 2)

wine.km
K-means clustering with 3 clusters of sizes 60, 70, 48

Cluster means:
     Alcohol      Malic        Ash
1  0.8914655 -0.4522073  0.5406223
2 -0.8649501 -0.5660390 -0.6371656
3  0.1470536  1.3907328  0.2534220

Clustering vector:
  [1] 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 3 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1
 [38] 1 2 3 1 3 1 3 2 3 3 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 1 2 2 1 1 1
 [75] 2 2 2 2 2 3 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
[112] 2 3 2 2 2 2 2 3 2 2 1 3 3 3 2 2 2 2 3 2 3 2 3 2 2 3 3 3 3 3 1 3 3 3 3 3 3
[149] 3 3 3 3 1 3 2 3 3 3 1 1 3 3 3 3 1 3 3 3 1 3 2 2 1 3 3 3 1 3

Within cluster sum of squares by cluster:
[1]  67.98619 111.63512  73.71460
 (between_SS / total_SS =  52.3 %)

Available components:

 [1] "cluster"      "centers"      "totss"        "withinss"     "tot.withinss"
 [6] "betweenss"    "size"         "iter"         "ifault"       "clust_plot"  
[11] "silinfo"      "nbclust"      "data"         "gap_stat"    
wine.km$nbclust
[1] 3
fviz_nbclust(wine_subset, kmeans, method = "gap_stat")

fviz_silhouette(wine.km)
  cluster size ave.sil.width
1       1   60          0.44
2       2   70          0.33
3       3   48          0.30

fviz_cluster(wine_cluster, data = wine_subset) + 
  theme_bw() +
  theme(text = element_text(family="Georgia")) 
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fviz_cluster(wine_cluster, data = wine_subset, ellipse.type = "norm") + 
  theme_bw() +
  theme(text = element_text(family="Georgia")) 
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Hierarchical Clustering

arrest.hc <- USArrests %>%
  scale() %>%                    # Scale all variables
  dist(method = "euclidean") %>% # Euclidean distance for dissimilarity 
  hclust(method = "ward.D2")     # Compute hierarchical clustering

# Generate dendrogram using factoextra package
fviz_dend(arrest.hc, k = 4, # Four groups
          cex = 0.5, 
          k_colors = c("orange","green","blue", "purple"),
          color_labels_by_k = TRUE, # color labels by groups
          rect = TRUE, # Add rectangle (cluster) around groups,
          main = "Cluster Dendrogram: USA Arrest data"
) + theme(text = element_text(family="Georgia")) 
Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
of ggplot2 3.3.4.
ℹ The deprecated feature was likely used in the factoextra package.
  Please report the issue at <https://github.com/kassambara/factoextra/issues>.
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family not found in Windows font database

Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
family not found in Windows font database

Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
family not found in Windows font database
Warning in grid.Call.graphics(C_text, as.graphicsAnnot(x$label), x$x, x$y, :
font family not found in Windows font database
Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
family not found in Windows font database

References

James, Gareth, Daniela Witten, Trevor Hastie, and Robert Tibshirani. 2013 An introduction to statistical learning. Vol. 112. New York: Springer.