Imagine that an insurance company needs to create a risk analysis model of the number of car crashes that will occur in the country in the next 52 weeks. A reasonable assumption (which can be checked by analyzing the historic data) is that the number of car crashes n(t) over a period of time follows a Poisson process, i.e. each car crash is independent of any other. This is, of course, not exactly true since many of the car crashes involve at least two cars, and sometimes more than 10, but probably not from the same insurance company. Here we will neglect this small approximation, so:
n(t) = Poisson(λ(t))
The Poisson intensity parameter - λ(t) - is the mean, or expected, number of events per unit time. In this model it is not constant throughout the year because of two factors:
The number of crashes depends on the number of cars in the country. Let's assume that the number of cars in the country will grow within a period of one year by 15%. And since the correlation between the two parameters is probably not perfect, the number of car crashes is expected to increase by 10% over the same period.
The seasonality factor. The number of car crashes increases in the winter season due to several reasons like slippery roads and low visibility, and with certain yearly events like summer holidays, Christmas, etc. Seasonality is a repeated underlying pattern (perhaps disguised by overlying randomness) from one year to the next.
We can model seasonality as follows:
where f(t) - is a trend function and Si - is a seasonality factor for period i.
The model Poisson Series shows an example of the above technique.
The Poisson intensity parameter may also include other factors - in fact, as many factors as needed in order to give a fair estimate to the mean number of events over a period of time. For example, if the same insurance company was to model the number of old people deaths in transition-economy country X, λ(t) might consist of the following factors:
The trend factor, which is influenced by the changes in the population size and improvement of medical care;
The seasonality factor. The old people tend to die more often in hot and cold seasons, and less in other seasons; and
The economic factor. As Country X is going through economic hardships, many old people are affected by instability in the country and their death can be caused by factors like: stress, cold (as they are not able to pay for central heating), malnutrition.
The next model provides an example: Seasonal Poisson Random Walk.
The links to the Poisson Series & Seasonal Poisson Random Walk software specific models are provided here: