# 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 labour, 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 loose 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 centre 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 modelled two different ways:

1. A set of seasonality indices {I1 to In} where you are modeling n individual forecasts within the seasonality period. This is the method implemented in VoseTimeSeasonalGBM.

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

Read on: Bounded random walk

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