nonPoisson

NonPoisson is a term used in statistics and stochastic modeling to describe models, processes, or data that do not follow the standard Poisson assumptions. In its typical form, a Poisson model assumes that counts in a fixed interval are Poisson-distributed with a mean equal to the variance, and that events occur independently with a constant rate. NonPoisson models depart from these assumptions in various ways.

Common departures include overdispersion or underdispersion, where the observed variance differs from the mean; autocorrelation or clustering of events over time; and non-exponential interarrival times, which violate the memoryless property of a Poisson process. NonPoisson descriptions are also used when the event rate is not constant, or when the process exhibits bursty or self-exciting behavior.

Several widely used non-Poisson models address these departures. The negative binomial distribution (often viewed as a Poisson-Gamma mixture) handles overdispersion. Zero-inflated and hurdle models address excess zeros. Renewal processes allow non-exponential interarrival times. Hawkes processes model clustering and self-excitation. Non-homogeneous or time-varying Poisson processes allow a rate function that changes over time, though some authors reserve “non-Poisson” for departures beyond rate variation.

Applications span ecology, epidemiology, genetics, finance, and network traffic, wherever count data or event timing deviate from Poisson assumptions. Model selection typically involves diagnostic checks for dispersion, autocorrelation, and goodness-of-fit, with likelihood-based or Bayesian methods guiding comparison among Poisson and non-Poisson alternatives. See also Poisson distribution, Poisson process, negative binomial distribution, Hawkes process, and renewal processes.