Bayesian regression of piecewise homogeneous Poisson processes

 

Diego J. R. Sevilla¹*

* E-mail: dsevilla@fceia.unr.edu.ar

¹ Departamento de Física y Química, Escuela de Formación Básica. Facultad de Ciencias Exactas, Ingeniería y Agrimensura. Universidad Nacional de Rosario, Av. Pellegrini 250, S2000BTP Rosario, Argentina.

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In this paper, a Bayesian method for piecewise regression is adapted to handle counting processes data distributed as Poisson. A numerical code in Mathematica is developed and tested analyzing simulated data. The resulting method is valuable for detecting breaking points in the count rate of time series for Poisson processes.

Keywords Poisson processes; statistical methods; piecewise constant regression

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I. Introduction

Bayesian statistics have revolutionized data analysis 1. Techniques like the Generalized Lomb-Scargle Periodogram 2 allow us to obtain oscillation frequencies of time series with unprecedented accuracy. The Gregory and Loredo method 3 goes further allowing us to find and characterize periodic signals of any period and shape.

To detect non-periodical variations, the Exact Bayesian Regression of Piecewise Constant Functions by Marcus Hutter (hereafter Hutter's method) 4 is valuable. It permits to estimate the most probable partition of a data set in segments of constant signals, determining the number of segments and their borders, and in-segments means and variances. Hutter's method works with two continuous distributions: Normal, and Cauchy-Lorentz. The latter -the canonical example of a pathological distribution with undefined moments-, is also suitable to analyze data with other symmetric probability distributions, especially with heavy tails.

In the case of counting processes, especially for low rates, when data consist in non-negative small integers, methods specially designed to discrete probability distributions are necessary. Some regression methods, specially for non-homogeneous Poisson processes 5, were developed.

In this paper, Hutter's method is adapted for analyzing data distributed as Poisson. The results are summarized in a code in Mathematica 6. It can be used to analyze data of several physical processes which follow the Poisson distribution (e.g., detection of photons in X-ray Astronomy, particles in nuclear disintegration, etc.), if sudden changes in detection rates are suspected.

II. Method

Hutter's method is summarized in Table 1 of Ref. 4 in a pseudo C code which is divided in two blocks. The first one calculates moments with k = 0,1, 2 of the PDF of the statistical models for segments of data := |n_i+i,... ,nj}. The second one performs the regression from moments A -. The code developed in this work is divided in three blocks.

As the members of the Poisson distributions family are identified by one parameter -the mean rate r of the Poisson process-, the PDF of the models

 

Acknowledgements - This work was partially supported by the National University of Rosario.

1 P C Gregory, Bayesian logical data analysis for the physical sciences, Cambridge University Press, Cambridge, UK (2004).         [ Links ]

2 G L Bretthorst, Lecture notes in statistics, Springer, Berlin (1988).

3 P C Gregory, T J Loredo, A new method for the detection of a periodic signal of unknown shape and period, Astrophys. J. 398, 146 (1992).

4 M Hutter, Exact Bayesian regression of piece-wise constant functions, Bayesian Analysis 2, 635 (2007).

5 J F Lawless, Regression methods for Poisson process data, J. Am. Stat. Assoc. 82, 399 (1987).

6 Wolfram Research Inc., Mathematica version 9.0, Wolfram Research, Inc., Champaign, Illinois (2012).

7 A Gelman, J B Carlin, H S Stern, D B Rubin, Bayesian data analysis, Taylor & Francis, UK (2014).

8 Mathematica codes and examples by the author can be found at http://www.papersinphysics.org.