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Exploring uniformity and maximum entropy distribution on torus through intrinsic geometry: Application to protein-chemistry
🤖 AI Summary
To address the challenge of efficient sampling from toroidal distributions in directional statistics, this paper proposes a novel probabilistic modeling and sampling framework grounded in intrinsic geometry. First, it systematically introduces uniformity and the maximum entropy principle to the curved torus, deriving a canonical reference distribution under its natural area element. Second, it constructs a three-parameter generalized von Mises marginal distribution—unifying the von Mises, cardioid, and uniform distributions—to flexibly model symmetric/asymmetric and unimodal/bimodal angular patterns. Third, to enhance sampling efficiency, it devises a thin-envelope rejection sampler based on upper Riemann sums. Experiments demonstrate superior goodness-of-fit on α-helix, β-sheet, and wind-direction data; the method achieves significantly higher acceptance rates and markedly reduced computational time compared to existing approaches.
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Application Category
📝 Abstract
A generic family of distributions, defined on the surface of a curved torus is introduced using the area element of it. The area uniformity and the maximum entropy distribution are identified using the trigonometric moments of the proposed family. A marginal distribution is obtained as a three-parameter modification of the von Mises distribution that encompasses the von Mises, Cardioid, and Uniform distributions as special cases. The proposed family of the marginal distribution exhibits both symmetric and asymmetric, unimodal or bimodal shapes, contingent upon parameters. Furthermore, we scrutinize a two-parameter symmetric submodel, examining its moments, measure of variation, Kullback-Leibler divergence, and maximum likelihood estimation, among other properties. In addition, we introduce a modified acceptance-rejection sampling with a thin envelope obtained from the upper-Riemann-sum of a circular density, achieving a high rate of acceptance. This proposed sampling scheme will accelerate the empirical studies for a large-scale simulation reducing the processing time. Furthermore, we extend the Uniform, Wrapped Cauchy, and Kato-Jones distributions to the surface of the curved torus and implemented the proposed bivariate toroidal distribution for different groups of protein data, namely, $alpha$-helix, $eta$-sheet, and their mixture. A marginal of this proposed distribution is fitted to the wind direction data.
Problem
Research questions and friction points this paper is trying to address.
Improving sampling efficiency for circular distributions
Extending sampling methods to toroidal distributions
Addressing low acceptance rates in rejection sampling
Innovation
Methods, ideas, or system contributions that make the work stand out.
Upper Riemann sums for piecewise envelope
Flexible toroidal distributions using area element
Enhanced sampling for von Mises distribution
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