Modal Inference: Halfway between clustering
and mixture analyses

        Given a set of n data points and an appropriate kernel K, one has the natural nonparametric density estimator formed by averaging the values of the kernel over the data. The smoothness of the estimator depends on a bandwidth parameter h. We consider the modes of the resulting density as indicators of important substructures within the data. There is a natural extension of the EM algorithm that can be used to find the modes. In addition, the method of steepest ascent can be used to assign the individual data points to modes, providing a clustering of data points through their modal association. If in addition we let the bandwidth parameter go from 0 to infinity, we can construct a hierarchical clustering of the data points. In addition to providing satisfying clustering results that lie somewhere between clustering algorithms and a formal mixture analysis, the estimation method raises interesting inferential questions that lie somewhere between the two points of view. One question: is it a mistake to use mean squared error as a bandwidth criterion in density estimation?