Clustering with selected Principal Components

December 29, 2012
By

(This article was originally published at Systematic Investor » R, and syndicated at StatsBlogs.)

In the Visualizing Principal Components post, I looked at the Principal Components of the companies in the Dow Jones Industrial Average index over 2012. Today, I want to show how we can use Principal Components to create Clusters (i.e. form groups of similar companies based on their distance from each other)

Let’s start by loading the historical prices for the the companies in the Dow Jones Industrial Average index that we saved in the Visualizing Principal Components post.

###############################################################################
# Load Systematic Investor Toolbox (SIT)
# http://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
setInternet2(TRUE)
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
    source(con)
close(con)

	#*****************************************************************
	# Load historical data
	#****************************************************************** 
	load.packages('quantmod')	
	
	# load data saved in the bt.pca.test() function
	load(file='bt.pca.test.Rdata')

	#*****************************************************************
	# Principal component analysis (PCA), for interesting discussion
	# http://machine-master.blogspot.ca/2012/08/pca-or-polluting-your-clever-analysis.html
	#****************************************************************** 
	prices = data$prices	
	ret = prices / mlag(prices) - 1
	
	p = princomp(na.omit(ret))
	
	loadings = p$loadings[]
	
	x = loadings[,1]
	y = loadings[,2]
	z = loadings[,3]	

To create Clusters, I will use the hierarchical cluster analysis, hclust function, in stats package. The first argument in the hclust function is the distance (dissimilarity) matrix. To compute distance matrix, let’s take the first 2 principal components and compute the Euclidean distance between each company:

	#*****************************************************************
	# Create clusters
	#****************************************************************** 		
	# create and plot clusters based on the first and second principal components
	hc = hclust(dist(cbind(x,y)), method = 'ward')
	plot(hc, axes=F,xlab='', ylab='',sub ='', main='Comp 1/2')
	rect.hclust(hc, k=3, border='red')

plot1.png.small

Similarly we can use the first three principal components:

	# create and plot clusters based on the first, second, and third principal components
	hc = hclust(dist(cbind(x,y,z)), method = 'ward')
	plot(hc, axes=F,xlab='', ylab='',sub ='', main='Comp 1/2/3')
	rect.hclust(hc, k=3, border='red')

plot2.png.small

Another option is to use the Correlation matrix as a proxy for a distance matrix:

	# create and plot clusters based on the correlation among companies
	hc = hclust(as.dist(1-cor(na.omit(ret))), method = 'ward')
	plot(hc, axes=F,xlab='', ylab='',sub ='', main='Correlation')
	rect.hclust(hc, k=3, border='red')

plot3.png.small

Please note that Clusters will be quite different, depending on the distance matrix you use.

To view the complete source code for this example, please have a look at the bt.clustering.test() function in bt.test.r at github.




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