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 some constant, usually 1. Here θ is simply a concatenated to b.
Thus, when given a Poisson regression model θ and an input vector, the predicted mean of the associated Poisson distribution is given by
If Yi are independent observations with corresponding values xi of the predictor variable, then θ can be estimated by maximum likelihood. The maximum-likelihood estimates lack a closed-form expression and must be found by numerical methods. The probability surface for maximum-likelihood Poisson regression is always convex, making Newton–Raphson or other gradient-based methods appropriate estimation techniques.
Read more about this topic: Poisson Regression
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