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Learning Generalization and Regularization of Nonhomogeneous Temporal Poisson Processes

Son Nguyen Van, Hoai Nguyen Xuan

TL;DR

This paper proposes a framework for regularized learning of NHPPs with two new adaptive and data-driven binning methods that help to remove the ad-hoc tuning of binning parameters.

Abstract

The Poisson process, especially the nonhomogeneous Poisson process (NHPP), is an essentially important counting process with numerous real-world applications. Up to date, almost all works in the literature have been on the estimation of NHPPs with infinite data using non-data driven binning methods. In this paper, we formulate the problem of estimation of NHPPs from finite and limited data as a learning generalization problem. We mathematically show that while binning methods are essential for the estimation of NHPPs, they pose a threat of overfitting when the amount of data is limited. We propose a framework for regularized learning of NHPPs with two new adaptive and data-driven binning methods that help to remove the ad-hoc tuning of binning parameters. Our methods are experimentally tested on synthetic and real-world datasets and the results show their effectiveness.

Learning Generalization and Regularization of Nonhomogeneous Temporal Poisson Processes

TL;DR

This paper proposes a framework for regularized learning of NHPPs with two new adaptive and data-driven binning methods that help to remove the ad-hoc tuning of binning parameters.

Abstract

The Poisson process, especially the nonhomogeneous Poisson process (NHPP), is an essentially important counting process with numerous real-world applications. Up to date, almost all works in the literature have been on the estimation of NHPPs with infinite data using non-data driven binning methods. In this paper, we formulate the problem of estimation of NHPPs from finite and limited data as a learning generalization problem. We mathematically show that while binning methods are essential for the estimation of NHPPs, they pose a threat of overfitting when the amount of data is limited. We propose a framework for regularized learning of NHPPs with two new adaptive and data-driven binning methods that help to remove the ad-hoc tuning of binning parameters. Our methods are experimentally tested on synthetic and real-world datasets and the results show their effectiveness.
Paper Structure (40 sections, 2 theorems, 30 equations, 15 figures, 5 tables, 3 algorithms)

This paper contains 40 sections, 2 theorems, 30 equations, 15 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Background
  4. Poisson Distribution
  5. Temporal Poisson Process
  6. Temporal Homogeneous Poisson Process
  7. Temporal Nonhomogeneous Poisson Processes
  8. Distribution Tests
  9. Standard Kolmogorov-Smirnov Test
  10. Log Test
  11. Statistical Learning Theory
  12. Risk Functional Minimization
  13. The Problem of Regression Estimation/ Learning
  14. Empirical Risk Minimization (ERM) Principle
  15. Generalization of learning machines
  16. ...and 25 more sections

Key Result

Proposition 1

For any $n \geq 2$, the inequality holds true.

Figures (15)

  • Figure 1: Structural risk minimization principle of Vapnik Vapnik1998
  • Figure 2: Common estimation process
  • Figure 3: Overfitting in piecewise-polynomial regression on the San Francisco taxi request data (see Section \ref{['ex']})
  • Figure 4: The proposed learning framework
  • Figure 5: An example of the training and testing processes
  • ...and 10 more figures

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Proposition 2
  • proof