13
0

Histogram Transform Ensembles for Density Estimation

Abstract

We investigate an algorithm named histogram transform ensembles (HTE) density estimator whose effectiveness is supported by both solid theoretical analysis and significant experimental performance. On the theoretical side, by decomposing the error term into approximation error and estimation error, we are able to conduct the following analysis: First of all, we establish the universal consistency under L1(μ)L_1(\mu)-norm. Secondly, under the assumption that the underlying density function resides in the H\"{o}lder space C0,αC^{0,\alpha}, we prove almost optimal convergence rates for both single and ensemble density estimators under L1(μ)L_1(\mu)-norm and L(μ)L_{\infty}(\mu)-norm for different tail distributions, whereas in contrast, for its subspace C1,αC^{1,\alpha} consisting of smoother functions, almost optimal convergence rates can only be established for the ensembles and the lower bound of the single estimators illustrates the benefits of ensembles over single density estimators. In the experiments, we first carry out simulations to illustrate that histogram transform ensembles surpass single histogram transforms, which offers powerful evidence to support the theoretical results in the space C1,αC^{1,\alpha}. Moreover, to further exert the experimental performances, we propose an adaptive version of HTE and study the parameters by generating several synthetic datasets with diversities in dimensions and distributions. Last but not least, real data experiments with other state-of-the-art density estimators demonstrate the accuracy of the adaptive HTE algorithm.

View on arXiv
Comments on this paper