Parzen-Window Kernel Parameter

The Parzen-window parameter $\sigma$, effectively controls the smoothing of the data in the feature space $< {\bf z} >$ of neighborhood-intensity vectors. However, $\sigma$ must be commensurate with the number and density of observations in that space, and thus it should adapt to different sampling strategies and applications. We have found that the optimal (cross-validated ML) $\sigma$, estimated from limited data, does not properly ``connect'' all of the configurations of gray matter neighborhoods in a single class, thereby breaking the manifold into many distinct pieces prone to misclassification. Indeed, this method of regularization is known to under-smooth the PDF and be sensitive to outliers. In practice, to obtain desirable results with finite data, we impose additional smoothness on the Markov PDFs of each class, by multiplying the optimal $\sigma$ by a factor $\alpha$ larger than unity. This strategy is somewhat ad hoc and a different strategy based on plug-in bandwidth estimators [156,171] that produces over-smooth, but more robust, PDF estimates might work better. We have found that the choice of the precise value of this multiplicative factor $\alpha$ is not critical and Table 6.1 in the next section confirms that the algorithm is quite robust to small changes in $\alpha$, i.e., $\alpha$ varying between $5$ and $10$. All of the results in this chapter employ $\alpha = 10$.