Modality-Constrained Density Estimation via Deformable Templates

Estimation of a probability density function (pdf) from its samples, while satisfying certain shape constraints, is an important problem that lacks coverage in the literature. This article introduces a novel geometric, deformable template constrained density estimator (dtcode) for estimating pdfs constrained to have a given number of modes. Our approach explores the space of thus-constrained pdfs using the set of shape-preserving transformations: an arbitrary template from the given shape class is transformed via a shape-preserving transformation to obtain the final optimal estimate. The search for this optimal transformation, under the maximum-likelihood criterion, is performed by mapping transformations to the tangent space of a Hilbert sphere, where they are effectively linearized, and can be expressed using an orthogonal basis. This framework is first applied to (univariate) unconditional densities and then extended to conditional densities. We provide asymptotic convergence rates for dtcode, and an application of the framework to the speed distributions for different traffic flows on Californian highways.