Detection limit for rate fluctuations in inhomogeneous Poisson processes

Toshiaki Shintani and Shigeru Shinomoto

Phys. Rev. E 85, 041139 – Published 24 April 2012

Abstract

Estimations of an underlying rate from data points are inevitably disturbed by the irregular occurrence of events. Proper estimation methods are designed to avoid overfitting by discounting the irregular occurrence of data, and to determine a constant rate from irregular data derived from a constant probability distribution. However, it can occur that rapid or small fluctuations in the underlying density are undetectable when the data are sparse. For an estimation method, the maximum degree of undetectable rate fluctuations is uniquely determined as a phase transition, when considering an infinitely long series of events drawn from a fluctuating density. In this study, we analytically examine an optimized histogram and a Bayesian rate estimator with respect to their detectability of rate fluctuation, and determine whether their detectable-undetectable phase transition points are given by an identical formula defining a degree of fluctuation in an underlying rate. In addition, we numerically examine the variational Bayes hidden Markov model in its detectability of rate fluctuation, and determine whether the numerically obtained transition point is comparable to those of the other two methods. Such consistency among these three principled methods suggests the presence of a theoretical limit for detecting rate fluctuations.

Received 27 February 2012

DOI:https://doi.org/10.1103/PhysRevE.85.041139

Authors & Affiliations

Toshiaki Shintani^* and Shigeru Shinomoto^†

Department of Physics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan

^*shintani@ton.scphys.kyoto-u.ac.jp
^†shinomoto@scphys.kyoto-u.ac.jp

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Vol. 85, Iss. 4 — April 2012

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Images

Figure 1
Switching Poisson process. (a) Transition between up and down states. The state switches between the high rate $μ + σ$ and the low rate $μ - σ$ randomly in a time scale of $τ$ . (b) Raster diagram of events derived from the switching rate. (c) The rate estimated from a time series of events using a two-state hidden Markov model.Reuse & Permissions
Figure 2
OUP modulated Poisson process. (a) Rate modulated with the Ornstein-Uhlenbeck process. (b) Raster diagram of events derived from fluctuating rate. (c) The rate estimated from a time series of events using a two-state hidden Markov model.Reuse & Permissions
Figure 3
Estimating underlying rate from a time series of events derived from the switching Poisson process (left) and the OUP modulated Poisson process (right). (a) Rate processes. (b) Sample event sequences derived from a rate. (c) Histograms optimized according to the MISE minimizing principle. (d) Rates estimated by the empirical Bayes estimator. (e) Rates estimated by the variational Bayes HMM. Parameters: $μ = 25 Hz$ and $σ = 5 Hz$ . From left to right, the time scales of rate processes are $τ = 0.275$ , 2, $0.5$ , and $3 \sec$ .Reuse & Permissions
Figure 4
Detectable-undetectable phase transitions of an MISE-optimal histogram and a Bayesian rate estimator for the switching Poisson process and the OUP modulated Poisson process. (a) An MISE-optimal histogram. Left: MISE [Eq. (12)] for the cases of $τ < τ_{c}$ , $= τ_{c}$ , and $> τ_{c}$ . Right: Inverse of the optimal bin size $1 / Δ^{*}$ computed for the switching Poisson process and the OUP modulated Poisson process. In both cases, the critical point is given by $τ_{c} = μ / σ^{2}$ . (b) An empirical Bayes estimator. Left: Free energy defined by the log likelihood Eq. (23) for the cases of $τ < τ_{c}$ , $= τ_{c}$ , and $> τ_{c}$ . Right: An optimal hyperparameter $γ^{*}$ representing the degree of flatness. The conditions of phase transition for the switching Poisson process and the OUP modulated Poisson process are the same as those for the MISE-optimal histogram. The error bars of numerical data represent the average and standard deviation of the inverse bin size $1 / Δ^{*}$ or hyperparameter $γ^{*}$ determined by applying practical optimization algorithms to ten synthetic data numerically generated by simulating the inhomogeneous Poisson processes. Parameters of synthetic data used for numerical analysis are the same as those of Fig. 3.Reuse & Permissions
Figure 5
Model selection in the variational Bayes HMM. (a) Rate processes. Left: switching Poisson process. Right: OUP modulated Poisson process. (b) The rates estimated by one-state, two-state, and three-state HMMs. (c) The likelihood lower bounds of two-state HMM and three-state HMM measured from that of the one-state HMM. Numerical analysis is performed under the parameters of $μ = 25 Hz$ and $σ = 5 Hz$ .Reuse & Permissions
Figure 6
Assessments of rate estimation methods in their goodness of fit to the underlying rate, measuring the distance in $L^{2}$ and $L^{1}$ norms, the Kullback-Leibler distance, and likelihood. The results for the switching Poisson process or OUP modulated Poisson process are shown on the left and right, respectively.Reuse & Permissions

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