# Seasonal time series

Many random variables exhibit some degree of seasonality over time: that is, some quality of the probability distribution of their values (usually the mean and spread, but in principle the minimum, maximum, etc) has a repeated pattern with a defined period.

For example:

• A nation's unemployment rate has a yearly period because of seasonal labor, school and university leavers, etc. That's why seasonally-adjusted figures are presented on the news;

• Delays on a railway system have a yearly period because a sudden leaf fall causes the trains to lose grip, very high temperatures make electrical connections expand and short out, very cold temperatures cause freezing of points, etc;

• Some strikes have a yearly period, because pilots walk out just before the holidays, refuse collectors walk out when it's high summer (the smell), etc;

• Electricity demand is higher in some countries in summer (air conditioning) and winter (heating);

• Most of our lives follow a weekly work and school cycle, and along with that go shop revenue, traffic, TV viewing, etc;

• Electricity demand is higher in a city center from Monday to Friday (offices);

• Over a day, ... okay, you get the point.

## Handling seasonality

Seasonality is probably only relevant to us if:

• The decision option is to have a seasonal impact;

• Seasonal peaks or troughs represent a constraint on your system;

• The seasonal variation has an impact on other variables you are trying to estimate;

• The data we have covers a fraction of some period; or

• Breaking down a time series into components helps us better estimate the series as a whole;

if you can, aggregate estimates over complete seasonal periods which will allow you to use a simpler model.

## Seasonality index method

The effect of seasonality is modeled two different ways:

1. A set of seasonality indices {I1 to In} where you are modeling n individual forecasts within the seasonality period.

2. A set of periodic functions (like a sin function) with different amplitudes and frequencies (not recommended).

## Applying seasonality indices

The models below show examples each of an additive and a multiplicative seasonal forecast.

### Multiplicative

Seasonality indices are most naturally applied in a multiplicative fashion, i.e. if the base forecast is S'(t), the seasonal forecast would be S(t) = S'(t)*Ii in part i of the seasonality period. This means that the seasonality adjustment is proportional to the base value, and the indices are centered around 1. It also means, however, that if the there is a trend in the base estimate, the seasonality indices will introduce a shift in the seasonal estimate compared to the base.