Multivariate locally adaptive kernel density estimation

Abstract

When the underlying density exhibits multiple modes with different scales and orientations, density estimators with locally adaptive smoothing parameters show substantial gains over those with fixed bandwidths. However, it is a concern that the local smoothing matrices may not be well parameterized, and the corresponding optimization problems will be difficult. In this paper, we build a more promising and practical algorithm. The local bandwidth factors are chosen through clustering, and the global smoothing parameter is achieved by optimizing the Asymptotic Mean Integrated Squared Error. Most importantly, our locally adaptive estimator involves optimizing a scalar rather than solving a costly multivariate optimization problem. Our method, which can also be applied to manifold density estimation, is an improvement and generalization of the binned version estimator of Sain.

Keywords:

• Variable bandwidth
• Clustering
• Local bandwidth factor
• Mean integrated squared error
• Manifold

Acknowledgements

We greatly appreciate the helpful comments and suggestions from the editors and the reviewers. We would like to thank Dr. Zhuangyan Fang from Peking University for valuable discussions. Our heartfelt thanks also go to Prof. Minghua Deng for offering computational resources.

Disclosure statement

All the authors have no conflict of interest.

Additional information

Funding

This work was supported by the [NSFC] under Grant [number 11871079].