Many things we are concerned about occur randomly in time: people arriving at a queue (customers, emergency patients, telephone calls into a centre, etc); accidents, natural disasters, shocks to a market, terrorist attacks, etc. Naturally we may want to model these over time, perhaps to figure out whether we will have enough stock vaccine, storage space, etc.
This model can be used, for example, to describe vehicle accident claims made to an insurance company, or cases of a disease for a health authority: as the number of cars increases, the number of car crashes increases correspondingly according to some function; as the pollution level in a city increases, the number of people with respiratory disease increases.
The fractional variation of the series is much bigger in the top panel than that in the bottom panel. This is because the standard deviation of Poisson(λ) counts equals √λ. Thus, the coefficient of variance (std.dev./mean) is 1/√λ. which gets smaller as λ gets bigger, meaning that the larger the expected number of events, the smaller the fractional variation one would observe. This property of a Poisson process is very useful to insurance companies: the more people they cover, the more stable their liabilities become, and the less margin they need to cover themselves at a certain risk level... an example of when big is actually better.
A variation of this model is to take account of seasonality by multiplying the expected number of events by seasonal indices (which should average to 1).
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.
where f(t) is a trend function and Si is a seasonality factor for period i.
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.
Notice the much greater peaks in the plot for this model compared to that of the previous model. Mixing a Poisson with a Gamma distribution to create the Pólya is a helpful tool because we can get the likelihood function directly from the pmf of the Pólya and therefore fit to historical data. If the MLE value for h is very small then the Poisson model will be as good a fit and has one less parameter to estimate, so the Pólya model is a useful first test.