Poisson regression
In statistics, Poisson regression is a form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. A Poisson regression model is sometimes known as a log-linear model, especially when used to model contingency tables.
Poisson regression models are generalized linear models with the logarithm as the (canonical) link function, and the Poisson distribution function as the assumed probability distribution of the response.
Regression models
If 
is a vector of independent variables, then the model takes the form
where 
and 
. Sometimes this is written more compactly as
where x is now an (n + 1)-dimensional vector consisting of n independent variables concatenated to a vector of ones. Here θ is simply α concatenated to β.
Thus, when given a Poisson regression model θ and an input vector x, the predicted mean of the associated Poisson distribution is given by
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