PoissonModelle

Poisson models are a class of statistical models designed for count data, where the observed counts Y_i are assumed to follow a Poisson distribution with rate lambda_i > 0. In many implementations the rate depends on covariates X_i through a generalized linear model with a log link: log(lambda_i) = X_i beta. This yields the standard Poisson regression framework, where the expected count is E[Y_i|X_i] = lambda_i = exp(X_i beta).

Estimation is typically by maximum likelihood, providing coefficients that can be interpreted on the multiplicative scale. A one-unit change in a covariate X_j changes the expected rate by a factor of exp(beta_j). When the observation units correspond to different exposure times or areas, an offset term log(exposure_i) is often included so that lambda_i = exposure_i * exp(X_i beta), modeling incident rates rather than raw counts.

Applications of Poisson models span many fields, including epidemiology, ecology, criminology, and public health. They are used to model disease case counts, accident frequencies, wildlife sightings, or crime incidents, particularly when data are counts governed by underlying rates over time or space. Variants include Poisson regression with random effects for hierarchical data (Poisson mixed models) and log-linear models for contingency tables. The Poisson process is a related concept describing the timing of independent events at a constant rate.

Assumptions include equidispersion (mean equals variance) and independence of counts. Real-world data often exhibit overdispersion or zero inflation, in which case quasi-Poisson, negative binomial, or zero-inflated Poisson models may be preferable.