Probability distributions for Poisson processes with pile-up

Abstract: In this paper, two parametric probability distributions capable to describe the statistics of X-ray photon detection by a CCD are presented. They are formulated from simple models that account for the pile-up phenomenon, in which two or more photons are counted as one. These models are based on the Poisson process, but they have an extra parameter which includes all the detailed mechanisms of the pile-up process that must be fitted to the data statistics simultaneously with the rate parameter. The new probability distributions, one for number of counts per time bins (Poisson-like), and the other for waiting times (exponential-like) are tested fitting them to statistics of real data, and between them through numerical simulations, and their results are analyzed and compared. The probability distributions presented here can be used as background statistical models to derive likelihood functions for statistical methods in signal analysis.

• The paper introduces two parametric probability distributions that account for photon pile-up in CCD detectors.
• It derives a Poisson-like model and a discrete exponential model to accurately characterize photon counts and waiting times.
• Empirical validation with RX J0720.4-3125 data and simulations demonstrates the models’ robustness and practical utility.

Probability Distributions for Poisson Processes with Pile-Up

Introduction

This paper presents a rigorous statistical treatment of photon count data affected by the pile-up phenomenon in CCD-based X-ray detectors. Pile-up occurs when multiple photons are registered as a single event within a detector frame, resulting in systematic deviations from the Poissonian statistics typically assumed in high-energy astrophysics data analysis. The author introduces two parametric probability distributions to capture the influence of pile-up: a Poisson-like distribution governing the number of counts per frame and a pile-up-adjusted exponential-like distribution for photon waiting times. The formulations explicitly model pile-up through an additional parameter, allowing for simultaneous estimation of event rate and pile-up characteristics from data using standard statistical frameworks.

Model Formulation

The starting point is the recognition that photon arrivals to an X-ray telescope's focal plane are inherently a Poisson process characterized by a rate $r$. Under ideal detection, the count statistics per frame and waiting time distributions would strictly follow Poisson and exponential laws, respectively. However, due to pile-up, these distributions are distorted as the events corresponding to simultaneous or nearly simultaneous arrivals merge into a single count.

Poisson Distribution with Pile-Up

A combinatorial derivation accounts for the probability that, of $N$ photons collected within a given frame, only $n$ will be counted due to pile-up. The pile-up mechanism is encoded using parameters $\alpha_k$, representing the probability a new photon is grouped with an existing count when $k$ counts are already present in the frame. The resulting PMF is given by a sum over all pathways leading from $N$ collected photons to $n$ detected counts, each weighted by the piling-up probabilities. Under reasonable simplifying assumptions ($\alpha_k = k\alpha$ for small $r$), the final PMF depends on the event rate and a single pile-up parameter $\alpha$, involving Stirling numbers of the second kind—critical for capturing the merged-event combinatorics. Analytical and normalization issues for high rates and pile-up are mitigated by adjusting the PMF to ensure valid probability values.

Discrete Exponential Distribution with Pile-Up

For waiting time analysis, pile-up manifests as missing zero-waiting-time events (successive photons in the same frame), leading to a reduced fraction of zeros and a redistribution of longer waits. The author derives a general form for the pile-up-affected waiting time distribution, parametrized by the event rate $r$ and the fraction $X$ of lost events due to pile-up. This form does not rely on the restrictive assumptions made in the Poisson pile-up model, making it more general and robust for all observed pile-up regimes.

Statistical Estimation Framework

Both models are amenable to parameter estimation via least-squares matching of model probabilities to empirical frequencies, using chi-square minimization. Parameter uncertainties are determined from the Hessian of the chi-square surface. For each subset of data (divided according to stellar rotational phase for RX J0720.4-3125), counts per frame and waiting time distributions are statistically characterized and fitted independently with both models. The effective collection rate and fraction of lost events are checked for consistency and further improved using weighted averages based on the derived relations between observed and true rates.

Empirical Application

The models are validated using archival X-ray data from RX J0720.4-3125, observed with XMM-Newton's EPIC pn detector in full frame mode. The photon arrival statistics exhibit clear deviations from Poisson and exponential expectations in the presence of pile-up, notably a deficit of multiple-count frames and zero waiting-time events. Fitting the pile-up-augmented models yields excellent agreement with empirical distributions for both number of counts and waiting times, over the dynamic range of observed luminosities. The recovered parameters (event rate, pile-up fraction, and pile-up probability per photon) are shown to track variations in source luminosity as expected, with the pile-up parameter remaining instrument- and setup-dependent, and relatively constant across data subsets.

Model Comparison and Simulation

Numerical simulations are used to explore domains of applicability and generality of the two models. Simulated data generated according to the Poisson pile-up model is well-fitted by both models for a broad range of parameter values, with the discrete exponential pile-up model consistently recovering the simulation parameters. Conversely, the Poisson pile-up model fails to fit data generated under the more general (and assumption-free) discrete exponential pile-up law for a significant portion of parameter space, especially at high event rates or large pile-up. The lossless exponential-pile-up model provides a more universal description, confirming that the Poisson-based pile-up PMF is a special-case or approximation valid only under restrictive physical conditions (low pile-up, small frame count rates, or accurate $\alpha$).

Implications and Future Directions

The existence of simple, closed-form probability laws for pile-up-affected data critically enables robust likelihood functions for use in statistical inference tasks (e.g., period searches, burst detection, and time-domain modeling) that require background models consistent with instrumental systematics. Incorporating such distributions into standard X-ray data analysis pipelines would result in more accurate parameter estimation, especially for high-brightness sources or instruments operating near saturation. The generality of the discrete exponential with pile-up suggests applicability to a wider range of time-resolved photon counting problems across astrophysical instrumentation, wherever pile-up or detector nonlinearity is present.

Future work may focus on relaxing simplifying assumptions in the Poisson pile-up model, explicitly incorporating detector energy response or spatial dependencies, or extending the discrete pile-up exponential model to handle compound pile-up effects, grade migration, or time-variable pile-up. Additionally, further theoretical investigation into the connection or divergence of these models with existing mixture or compound Poisson frameworks could lead to broader unification in the statistical treatment of non-ideal detector response.

Conclusion

The paper provides an authoritative derivation and validation of parametric probability distributions for modeling photon counting data affected by pile-up in CCD detectors. The discrete exponential distribution with pile-up is demonstrated to be a general and robust statistical model, subsuming the Poisson pile-up framework. Empirical fits to real astronomical data and extensive simulations confirm the accuracy and utility of these models in practical observation scenarios. Integrating these statistical descriptions into signal analysis and inference methods is a promising avenue for improved analysis in high-count-rate time-domain astrophysics.