A naive approach would be to select the optimal value of according to the objective function, namely the value of that minimizes RSS. Defining as the minimal RSS of all clusterings with clusters, we observe that is a monotonically decreasing function in (Exercise 16.7 ), which reaches its minimum 0 for where is the number of documents. We would end up with each document being in its own cluster. Clearly, this is not an optimal clustering.
A heuristic method that gets around this problem is to estimate as follows. We first perform (e.g., ) clusterings with clusters (each with a different initialization) and compute the RSS of each. Then we take the minimum of the RSS values. We denote this minimum by . Now we can inspect the values as increases and find the ``knee'' in the curve - the point where successive decreases in become noticeably smaller. There are two such points in Figure 16.8 , one at , where the gradient flattens slightly, and a clearer flattening at . This is typical: there is seldom a single best number of clusters. We still need to employ an external constraint to choose from a number of possible values of (4 and 9 in this case).
A second type of criterion for cluster cardinality imposes a
penalty for each new cluster - where conceptually we start
with a single cluster containing all documents and then
search for the optimal number of clusters by
successively incrementing by one. To determine the cluster
cardinality in this way, we create a generalized
objective function that combines two elements: distortion ,
a measure of how much documents deviate from the prototype
of their clusters
(e.g., RSS
for -means); and a
measure of model complexity . We interpret a
clustering here as a model of the data. Model complexity in
clustering is usually the number of clusters or a function
thereof. For -means, we then get this
selection criterion for :
The obvious difficulty with Equation 195 is that we need to determine . Unless this is easier than determining directly, then we are back to square one. In some cases, we can choose values of that have worked well for similar data sets in the past. For example, if we periodically cluster news stories from a newswire, there is likely to be a fixed value of that gives us the right in each successive clustering. In this application, we would not be able to determine based on past experience since changes.
A theoretical justification for Equation 195 is the
Akaike Information Criterion or AIC, an
information-theoretic measure that trades off distortion against model complexity.
The general form of AIC is:
For -means, the AIC can be stated as follows:
To derive Equation 197 from Equation 196 observe that in -means since each element of the centroids is a parameter that can be varied independently; and that (modulo a constant) if we view the model underlying -means as a Gaussian mixture with hard assignment, uniform cluster priors and identical spherical covariance matrices (see Exercise 16.7 ).
The derivation of AIC is based on a number of assumptions, e.g., that the data are . These assumptions are only approximately true for data sets in information retrieval. As a consequence, the AIC can rarely be applied without modification in text clustering. In Figure 16.8 , the dimensionality of the vector space is 50,000. Thus, dominates the smaller RSS-based term ( , not shown in the figure) and the minimum of the expression is reached for . But as we know, (corresponding to the four classes China, Germany, Russia and Sports) is a better choice than . In practice, Equation 195 is often more useful than Equation 197 - with the caveat that we need to come up with an estimate for .
Exercises.