Modality-Constrained Density Estimation via Deformable Templates

Abstract

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.

Keywords:

• Conditional densities
• Deformation group
• Density estimation
• Modality constraints
• Shape constraints
• Sieve estimation

Supplementary Materials

Supplementary materials by sectionIn Section 1 of the supplementary materials, we present a proof of Theorem 1. In Section 2, we discuss the asymptotic properties of our estimator, and present a theorem which provides an upper bound on the convergence rate. We prove this theorem in Section 3. In Section 4, we include tables illustrating the average practical performance for our approach (dtcode), umd, and scdensity, for the examples considered in the simulation study in the main article. In Section 4.1, we discuss the effect of the number of basis elements on the final estimate. In Section 5, we include some examples of general shape-constrained density estimation beyond M-modality, like monotonicity, an upper bound on the number of modes, and so on. In Section 6, we include a simulation study for conditional density estimation. In Section 7, we discuss an application of shape-constrained density estimation to DNA methylation profiles.

Additional information

Funding

This research was supported in part by grants NSF DMS 1621787 and NSF CCF 1617397 to AS and NSF DMS 1613156 to DP.