Lectures for Advanced Statistics

Advanced Statistics - Biology 6030

Bowling Green State University, Fall 2019

Cluster Analysis

Cluster Analysis is a set of methods for grouping objects into different categories based on similarities. As a descriptive technique it discovers structures in data without the need for an explaination as to why they exist.

Uses

identify groups of similar cases based on a set attributes
classify cases into relatively homogeneous groups

Example

cluster multiple individuals of one species from different localities to determine population membership
include all variables that are believed to be significant for characterizing the cases
compute measures of similarity among a number of cases. Note that the inverse of similarity is distance
plot clusters as a function of coefficient of similarity: dendrogram, icicle plot

How this is done

standardize variables that are measured in different units
based on this information group cases into clusters using one of several clustering algorithms
terminate the formation of clusters when all objects are included in one big cluster (agglomerative) or have been split into individual cases (divise)
A variety of different distance/similarity measures can be used. For the following pair of cases consider advantages and disadvantages of different methods
Distance Measures
- Euclidian Distance
  - Squared Euclidian Distance D² = (x1₁-x1₂)² + (x2₁-x2₂)² + ... + (xn₁-xn₂)²
  - standardized squared Euclidian Distance
- Manhattan Distance
- Hamming Distance
- Mahalanobis Distance
- Inner Product Measure using the angle between two vectors
Clustering algorithms
- hierarchical clustering: agglomerative or divise
- criteria for combining/dividing clusters:
  - single linkage, nearest neighbor
  - complete linkage, furthest neighbor
  - UPGMA method - average linkage between, unweighted pair-group method using arithmetic averages
  - Wards method
  - Centroid method

In R import datafile "BodyMeasures.txt" and decide how many distinct clusters seem to be present in your data. Standardize the data, then plot the sums of squares within for different numbers of groups. Similar to a screeplot, check at what number of clusters your SSwithin levels out. Based on this, five clusters looks like a useful number to extracts.

> bodyMeasures <- read.table("/BodyMeasures.txt", header=TRUE, sep=",", na.strings="NA", dec=".")
> bodyMeasures2 <- bodyMeasures[2:12]
> zbodyMeasures2 <- scale(bodyMeasures2)
> wss <- (nrow(zbodyMeasures2)-1)*sum(apply(zbodyMeasures2,2,var))
> for (i in 2:15) wss[i] <- sum(kmeans(zbodyMeasures2,centers=i)$withinss)
> plot(1:15, wss, type="b", xlab="Number of Clusters",ylab="Within groups sum of squares")

Use a kMeans clustering procedure, then get the cluster means and append them to the data set

> fit <- kmeans(zbodyMeasures2, 5) # 5 cluster solution
> aggregate(zbodyMeasures2,by=list(fit$cluster),FUN=mean)
> zbodyMeasures2 <- data.frame(zbodyMeasures2, fit$cluster)

Create a hierarchical conglommeration using Ward's method. First calculate a specific distance matrix, then obtain the clusters and list number of items in each cluster

> distMat <- dist(zbodyMeasures2, method = "euclidean")
> fit <- hclust(distMat, method="ward")
> plot(fit)
> groups <- cutree(fit, k=5)
> table(groups)

last modified: 03/27/14