Website snapshot

content/research/clusteranalysis/notes/lec4-3.md

@@ -0,0 +1,40 @@
+# Cluster Tendency
+
+This is the assessment of the suitability of clustering. Cluster Tendency determines whether the data has any inherent grouping structure.
+
+This is a hard task since there are so many different definitions of clusters (portioning, hierarchical, density, graph, etc.) Even after fixing a cluster type, this is still hard in defining an appropriate null model for a data set.
+
+One way we can go about measuring cluster tendency is to compare the data against random data. On average, random data should not contain clusters.
+
+There are some clusterability assessment methods such as Spatial histogram, distance distribution and Hopkins statistic.
+
+## Hopkins Statistic
+
+Let $X$ be the set of $n$ data points in $d$ dimensional space. Consider a random sample (without replacement) of $m << n$ data points. Also generate a set $Y$ of $m$ uniformly randomly distributed data points.
+
+Now define two distance measures $u_i$ to be the distance of $y_i \in Y$ from its nearest neighbor in X and $w_i$ to be the distance of $x_i \in X$ from its nearest neighbor in X
+
+We can then define Hopkins statistic as
+$$
+H = \frac{\sum_{i = 1}^m{u_i^d}}{\sum_{i = 1}^m{u_i^d} + \sum_{i =1}^m{w_i^d}}
+$$
+
+### Properties
+
+With this definition, uniform random data should tend to have values near 0.5, and clustered data should tend to have values nearer to 1.
+
+### Drawbacks
+
+However, data containing a single Gaussian will also score close to one. As this statistic measures deviation from a uniform distribution. Making this statistic less useful in application as real data is usually not remotely uniform.
+
+
+
+## Spatial Histogram Approach
+
+For this method, I'm not too sure how this works, but here are some key points I found.
+
+Divide each dimension in equal width bins, and count how many points lie in each of the bins and obtain the empirical joint probability mass function.
+
+Do the same for the randomly sampled data
+
+Finally compute how much they differ using the Kullback-Leibler (KL) divergence value. If it differs greatly than we can say that the data is clusterable.