# Rate Distortion Minimization in a Python Clustering Algorithm

I'm attempting to solve for $\hat{k}$ clusters, such that the rate distortion is minimized, as described here, however, the answers that I am getting from my algorithm are not following the "Jump" behavior (as is needed to appropriately choose $k$).

My current process:

Given a $N$ by $d$ matrix of asset returns:

1. Calculate the correlation of the assets, resulting in a $d \times d$ correlation matrix.
2. For $k \in [1, d]$ calculate the $1 \times d$ vectors of cluster labels, using the correlation matrix as the input
3. For $k \in [1, d]$ calculate $k, 1 \times d$ vectors of cluster centroids, using the correlation matrix as the input
4. Based on the cluster label (see code below), construct $\mathbf{A}$, a $d \times d$ matrix of cluster centroids based on the cluster label of each asset. So for instance, say $k=3$ and there are 4 assets total, and the results of the k-means cluster is:

• asset 1 is in cluster 1
• asset 2 is in cluster 1
• asset 3 is in cluster 2
• asset 4 is in cluster 3

I would create a $4 \times 4$ matrix where $\textrm{row }1 = \textrm{centroid } 1$ (a 1, 4 row vector), $\textrm{row }2 = \textrm{centroid } 1$ $\textrm{row }3 = \textrm{centroid } 2$ (a 1, 4 row vector), $\textrm{row }4 = \textrm{centroid} 3$

5. Then calculate the rate distortion function, $\epsilon$, for each $k$, where:

$\epsilon_{k} \triangleq \frac{1}{p}\mathbf{A^\intercal}\mathbf{\Sigma^{-1}}\mathbf{A}$ where, $\mathbf{\Sigma} \triangleq$ The Covariance Matrix of the $N \times d$ asset returns NOTE: I'm not certain whether I take the covariance of the asset returns or the covariance of the covariance matrix (which didn't seem right to me).

6. This leaves me with a $d \times d$ matrix that I then sum all the values, and follow the remainder of the procedure listed here. This procedure is derived from the paper, Finding the number of clusters in a data set: An information theoretic approach

My Python Code is as follows:

import Pycluster

def rate_distortion(X, cov_matrix, num_clusters):
clusterid, error, nfound = Pycluster.kcluster(X, nclusters = num_clusters)
cdata, cmask = Pycluster.clustercentroids(X, clusterid = clusterid)
c_mat = pandas.DataFrame(numpy.empty(X.shape), columns = X.columns)
for i, cluster in enumerate(clusterid):
c_mat.ix[i, :] = cdata[cluster, :]

diff_mat = numpy.subtract(X, c_mat)
p = X.shape
Y = p/2.
distortion = 1./p * numpy.sum(numpy.dot(numpy.dot(diff_mat.transpose(),
numpy.linalg.inv(cov_matrix)),
diff_mat))
return distortion**(-Y)


To execute a script that exactly illustrates to what I'm referring to, use data located here and the following script:

import pandas, numpy, Pycluster

prices = pandas.DataFrame.from_csv('asset_prices.csv')
returns = prices.apply(numpy.log).diff()
k = numpy.arange(1,12)
d = []
for i in k:
d.append(rate_distortion(returns.corr(), returns.cov(), i)
d = pandas.Series(d, index = k)

In : d
Out:
1     8.845139e-33
2     3.969062e-29
3     9.387323e-28
4     9.200729e-28
5     4.675902e-18
6     2.412458e-21
7     3.582043e-18
8     8.094695e-17
9     1.424341e-16
10    4.320064e-14
11             inf
dtype: float64

• any chance you could post your data? – quasi May 7 '13 at 12:09
• Just posted the data, as well as a small script to show you exactly what I'm seeing (as well as the output). Thanks for the suggestion @quasi – benjaminmgross May 7 '13 at 13:53
• So I'm a little confused. You're using $\Sigma$ as your underlying data, with each row a data point. So you have $d$ points in $\mathbb{R}^d$. But in your dispersion calculation, you're not using the covariance matrix of $\Sigma$, you're using $\Sigma$ itself. This seems to not be the algorithm. – quasi May 7 '13 at 20:39
• The naive application of the paper you cited seems to be: calculate $\Sigma$ as you did, then use the distortion measure to cluster the $d$-dimensional returns as a function of time. – quasi May 7 '13 at 20:40
• @quasi, thanks for the reply. A little clarification, $\mathbf{\Sigma}$ as I've defined it, is the covariance matrix as an input to the distortion calculation. The clustering algorithm is being run on the correlation matrix of asset returns. So you're saying, I should set $\mathbf{\Sigma}$ equal to the Covariance of the correlation matrix (which is the input to my clustering algorithm), instead of the covariance of asset returns (which I'm currently doing), correct? – benjaminmgross May 7 '13 at 20:57