Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Time Series - Trends

When measuring the temperature twice a day, the resulting time series may be as follows:

Temperature Time Series

Two typical phenomena can be observed: an upward trend and daily cycles. Both the trend and seasonal patterns have to be removed before any model can be established, because typical models require stationary time series. When the characteristics of a time series (for instance, the mean and the variance) differ at different points in time, the time series is non-stationary. After removing the trend of the time series in our example, the time series looks like this:

Trend Filtering

After averaging out the daily cycles, which are caused by measuring twice a day, the following time series is obtained:

Removing Cyclic Patterns

Finally, removing both the trend and the cycles results in the following series:

Removing Cyclic Patterns

Now, it is time to check the properties of the resulting time series. If it is actually stationary, it provides the basis for setting up a model. In order to forecast future values, the result obtained by using the model has to be transformed so that the trend and the daily cycle are again taken into account.