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Asymptotic Error Rates for Point Process Classification

Xinhui Rong, Victor Solo

TL;DR

The paper tackles the multi-class classification problem for point processes and derives asymptotic upper and lower bounds on the misclassification probability $P_e(T)$ using pairwise affinities. It develops an asymptotic inverse Fano theorem and applies affinity-based entropy bounds to relate $P_e(T)$ to affinities $R_{kj}(T)$ and $K_{kj}(T)$. It specializes the results to renewal processes, deriving Laplace-transform expressions and showing exponential decay $R_{kj}(T) \sim \alpha_{kj} e^{-\gamma_{kj} T}$ and $K_{kj}(T) \sim c_{kj} e^{-{\rho_{kj}} T}$ with explicit rates, yielding asymptotic bounds $P_e(T) \lesssim \tfrac{\alpha^*}{2\ln 2} e^{-\gamma^* T}$ and $P_e(T) \sim \tfrac{c^*}{\rho^* T} e^{-\rho^* T}$. It is validated by simulations on a Gamma/moE example and suggests future work on heavy-tailed renewals and Hawkes processes.

Abstract

Point processes are finding growing applications in numerous fields, such as neuroscience, high frequency finance and social media. So classic problems of classification and clustering are of increasing interest. However, analytic study of misclassification error probability in multi-class classification has barely begun. In this paper, we tackle the multi-class likelihood classification problem for point processes and develop, for the first time, both asymptotic upper and lower bounds on the error rate in terms of computable pair-wise affinities. We apply these general results to classifying renewal processes. Under some technical conditions, we show that the bounds have exponential decay and give explicit associated constants. The results are illustrated with a non-trivial simulation.

Asymptotic Error Rates for Point Process Classification

TL;DR

The paper tackles the multi-class classification problem for point processes and derives asymptotic upper and lower bounds on the misclassification probability using pairwise affinities. It develops an asymptotic inverse Fano theorem and applies affinity-based entropy bounds to relate to affinities and . It specializes the results to renewal processes, deriving Laplace-transform expressions and showing exponential decay and with explicit rates, yielding asymptotic bounds and . It is validated by simulations on a Gamma/moE example and suggests future work on heavy-tailed renewals and Hawkes processes.

Abstract

Point processes are finding growing applications in numerous fields, such as neuroscience, high frequency finance and social media. So classic problems of classification and clustering are of increasing interest. However, analytic study of misclassification error probability in multi-class classification has barely begun. In this paper, we tackle the multi-class likelihood classification problem for point processes and develop, for the first time, both asymptotic upper and lower bounds on the error rate in terms of computable pair-wise affinities. We apply these general results to classifying renewal processes. Under some technical conditions, we show that the bounds have exponential decay and give explicit associated constants. The results are illustrated with a non-trivial simulation.
Paper Structure (17 sections, 11 theorems, 85 equations, 2 figures)

This paper contains 17 sections, 11 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Point Process Properties
  4. Likelihood Classification
  5. Likelihood Rule/Bayes Rule
  6. Misclassification Error Rate
  7. Error Rate Bounds: Affinities and Asymptotics
  8. Point Process Error Rate Entropy Bounds
  9. Point Process Error Rate Affinity Bounds
  10. Asymptotic Inverse Fano Theorem
  11. Asymptotic Affinities for Renewal Processes
  12. Laplace Transform of Affinities and Divergences
  13. $R_{kj}$ Asymptotics
  14. $K_{kj}$ Asymptotics
  15. Error Rate Asymptotics for Renewal Processes
  16. ...and 2 more sections

Key Result

Lemma 1

Multiclass Error Rate Entropy Bounds. The misclassification error rate $P_e$ is bounded as follows. where $\phi_T=\phi(\mathcal{H}(T))$ solves and $\mathcal{H}_b(\phi) = -\phi\log \phi - (1-\phi) \log(1-\phi)$ is the binary entropy.

Figures (2)

  • Figure 1: MoE IET densities.
  • Figure 2: MC error probabilities and error bounds: (a) For $T\geq5000$, the error probabilities are bounded by the asymptotic bounds. (b) The transformed plots show that the error probability $P_e(T)$ has asymptotically exponential decay.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Lemma 2
  • Theorem 1
  • Lemma 3
  • Lemma 4
  • Definition 4
  • Theorem 2
  • ...and 6 more