Detection limit for rate fluctuations in inhomogeneous Poisson processes

Area of Science:

• Statistical modeling
• Time series analysis
• Signal processing

Background:

• Estimating underlying rates from event data is challenging due to irregular event occurrences.
• Sparse data can obscure rapid or small fluctuations in event densities.
• Existing methods aim to avoid overfitting and determine constant rates from irregular data.

Purpose of the Study:

• To analytically examine the detectability of rate fluctuations in optimized histogram and Bayesian rate estimators.
• To determine if these methods share an identical formula for the detectable-undetectable phase transition point.
• To numerically investigate the variational Bayes hidden Markov model's detectability of rate fluctuations and compare its transition point.

Main Methods:

• Analytical examination of optimized histogram and Bayesian rate estimators.
• Numerical examination of the variational Bayes hidden Markov model.
• Focus on the phase transition as a measure of undetectable rate fluctuation.

Main Results:

• Optimized histogram and Bayesian rate estimators were analyzed for their phase transition points.
• The variational Bayes hidden Markov model's transition point was numerically evaluated.
• Consistency was observed among the three methods regarding the detectable-undetectable phase transition.

Conclusions:

• The study suggests a theoretical limit exists for detecting rate fluctuations in event data.
• The consistency across analytical and numerical methods indicates a fundamental property of rate estimation.
• Further research may explore the implications of this theoretical limit in various applications.