Clustering a correlation matrix

Question

I have a correlation matrix which states how every item is correlated to the other item. Hence for a N items, I already have a N*N correlation matrix. Using this correlation matrix how do I cluster the N items in M bins so that I can say that the Nk Items in the kth bin behave the same. Kindly help me out. All item values are categorical.

Thanks. Do let me know if you need any more information. I need a solution in Python but any help in pushing me towards the requirements will be a big help.

Answer 1

Looks like a job for block modeling. Google for "block modeling" and the first few hits are helpful.

Say we have a covariance matrix where N=100 and there are actually 5 clusters:

What block modelling is trying to do is find an ordering of the rows, so that the clusters become apparent as 'blocks':

Below is a code example that performs a basic greedy search to accomplish this. It's probably too slow for your 250-300 variables, but it's a start. See if you can follow along with the comments:

import numpy as np
from matplotlib import pyplot as plt

# This generates 100 variables that could possibly be assigned to 5 clusters
n_variables = 100
n_clusters = 5
n_samples = 1000

# To keep this example simple, each cluster will have a fixed size
cluster_size = n_variables // n_clusters

# Assign each variable to a cluster
belongs_to_cluster = np.repeat(range(n_clusters), cluster_size)
np.random.shuffle(belongs_to_cluster)

# This latent data is used to make variables that belong
# to the same cluster correlated.
latent = np.random.randn(n_clusters, n_samples)

variables = []
for i in range(n_variables):
    variables.append(
        np.random.randn(n_samples) + latent[belongs_to_cluster[i], :]
    )

variables = np.array(variables)

C = np.cov(variables)

def score(C):
    '''
    Function to assign a score to an ordered covariance matrix.
    High correlations within a cluster improve the score.
    High correlations between clusters decease the score.
    '''
    score = 0
    for cluster in range(n_clusters):
        inside_cluster = np.arange(cluster_size) + cluster * cluster_size
        outside_cluster = np.setdiff1d(range(n_variables), inside_cluster)

        # Belonging to the same cluster
        score += np.sum(C[inside_cluster, :][:, inside_cluster])

        # Belonging to different clusters
        score -= np.sum(C[inside_cluster, :][:, outside_cluster])
        score -= np.sum(C[outside_cluster, :][:, inside_cluster])

    return score


initial_C = C
initial_score = score(C)
initial_ordering = np.arange(n_variables)

plt.figure()
plt.imshow(C, interpolation='nearest')
plt.title('Initial C')
print 'Initial ordering:', initial_ordering
print 'Initial covariance matrix score:', initial_score

# Pretty dumb greedy optimization algorithm that continuously
# swaps rows to improve the score
def swap_rows(C, var1, var2):
    '''
    Function to swap two rows in a covariance matrix,
    updating the appropriate columns as well.
    '''
    D = C.copy()
    D[var2, :] = C[var1, :]
    D[var1, :] = C[var2, :]

    E = D.copy()
    E[:, var2] = D[:, var1]
    E[:, var1] = D[:, var2]

    return E

current_C = C
current_ordering = initial_ordering
current_score = initial_score

max_iter = 1000
for i in range(max_iter):
    # Find the best row swap to make
    best_C = current_C
    best_ordering = current_ordering
    best_score = current_score
    for row1 in range(n_variables):
        for row2 in range(n_variables):
            if row1 == row2:
                continue
            option_ordering = best_ordering.copy()
            option_ordering[row1] = best_ordering[row2]
            option_ordering[row2] = best_ordering[row1]
            option_C = swap_rows(best_C, row1, row2)
            option_score = score(option_C)

            if option_score > best_score:
                best_C = option_C
                best_ordering = option_ordering
                best_score = option_score

    if best_score > current_score:
        # Perform the best row swap
        current_C = best_C
        current_ordering = best_ordering
        current_score = best_score
    else:
        # No row swap found that improves the solution, we're done
        break

# Output the result
plt.figure()
plt.imshow(current_C, interpolation='nearest')
plt.title('Best C')
print 'Best ordering:', current_ordering
print 'Best score:', current_score
print
print 'Cluster     [variables assigned to this cluster]'
print '------------------------------------------------'
for cluster in range(n_clusters):
    print 'Cluster %02d  %s' % (cluster + 1, current_ordering[cluster*cluster_size:(cluster+1)*cluster_size])

Answer 2

Have you looked at hierarchical clustering? It can work with similarities, not only distances. You can cut the dendrogram at a height where it splits into k clusters, but usually it is better to visually inspect the dendrogram and decide on a height to cut.

Hierarchical clustering is also often used to produce a clever reordering for a similarity matrix visualization as seen in the other answer: it places more similar entries next to each other. This can serve as a validation tool for the user, too!

Answer 3

Have you looked into correlation clustering? This clustering algorithm uses the pair-wise positive/negative correlation information to automatically propose the optimal number of clusters with a well defined functional and a rigorous generative probabilistic interpretation.

Answer 4

I would filter at some meaningful (statistical significance) threshold and then use the dulmage-mendelsohn decomposition to get the connected components. Maybe before you can try to remove some problem like transitive correlations (A highly correlated to B, B to C, C to D, so there is a component containing all of them, but in fact D to A is low). you can use some betweenness based algorithm. It's not a biclustering problem as someone suggested, as the correlation matrix is symmetrical and therefore there is no bi-something.