Poisson Regression
Overview
The poisson regression is used when the variable of interest is hypothesized to follow a
Poisson Distribution, where the
distribution is conditioned on a set of exogenous factors. Examples would include
modeling the
number bank deposit accounts opened in a month.
Modeling Count Rate
The Poisson distribution is defined by the following probability function.
{% P(x) = \mu^x e^{-\mu} / x! %}
The Poisson regression models the Poisson rate {% \mu %} as dependent on a set of external factors.
{% \mu_t = exp[\alpha + \beta_1 X_1 + \beta_2 X_2 + ... + \beta_n X_n] %}
{% log \mu_t = \vec{\beta}^T \vec{x_t} %}
{% \alpha %} can be renamed to be {% \beta_0 %}
{% \mu_t = exp[\beta_0 X_0 + \beta_1 X_1 + \beta_2 X_2 + ... + \beta_n X_n] %}
where we have set {% X_0 = 1 %} and {% \beta_0 = \alpha %}
Accumulated Lengths
The equations above work fine if the counts in each interval can be compared on an apples to apples basis. This is usually not the
case. For example, some counts occur over different time frames. Or, in the example of number of diseases detected in a population,
the population sizes are different. One would think that the rate, {% \mu %} would be different for different period lengths and
population sizes.
To correct for these effects, {% \mu %} is often scaled by an accumulation factor, in this case denoted {% l_t %}. It is
weight that is applied to the counts in question.
{% \mu_t' = l_t \mu_t %}
Topics
