Poisson Observation Model

Using the Poisson distribution to define an observation. This is effective for discrete count data.

In the above figure, the green dots represent occurences of plants in space (x,y point coordinates). These are "points" (in theory) and have no area, just a coordinate.

When we are counting these for a survey, we are surveying in a given area (Area 1 and Area 2). We are trying to determine the density (number of individuals per area). In the Poisson distribution, we call this parameter $λ$ - the number of individuals divided by area.

λ_{1} = \frac{no. individuals in 1}{area 1} = \frac{10}{5 m^{2}}

λ_{2} = \frac{no. individuals in 2}{area 2} = \frac{2}{15 m^{2}}

The expected value for any site $i$ is $E [y_{i}] = {area}_{i} λ_{i}$ - area times expected density.

In general:

n general, you can describe the Poisson distribution model as:

y_{i} \sim Poisson (E_{i} \times λ_{i})

Where $E_{i}$ denotes sampling effort for this site (for example, area sampled) and the density $λ_{i}$ . We call this "offset" and this is how it is used in R. In the upcoming example let's consider $E_{i} = 1$ so that $E [y_{i}] = λ_{i}$ (expected value of $y_{i}$ is just $λ_{i}$ ).

Even though it is technically possible, considering the possibility of density being zero would cause some mathematical problems, so we will just say that $λ$ is limited to positive non-zero numbers.

Since a regression in a linear model ( $x_{i} β$ ) can include negative numbers and zero $(- \infty, \infty)$ , we need to transform these values so they also have this domain in order for the regression to make sense. So:

\log (E [y_{i}]) = \log (λ_{i}) = x_{i} β

We call this a Link function. We are using it to link the density into "real space" e.g. $(- \infty, \infty)$ . The link function is denoted by $g (λ_{i})$

Since $\log λ_{i} = x_{i} β$ , and therefore $λ_{i} = e^{x_{i} β}$ , we can rewrite our model from earlier like:

y_{i} \sim Poisson (e^{x_{i} β})

The Poisson distribution is a probability statement for data, e.g. it has a Likelihood function to estimate the parameter $β$ . This is how the generalized linear models works.

In R:

glm(Y ~ x1 + x2 + x3 + x4 + offset(log(TransectLength)), family=poissons, data=dat)