In statistics, time series analysis deals with the inferential analysis of time series and the prediction of trends for their future development. It is a special form of regression analysis. A time series is a chronologically ordered sequence (but usually not a series in the mathematical sense) of numbers or observations, in which the arrangement of the characteristic characteristics necessarily results from the passage of time (e.g. share prices, stock market prices in general, population development, price index, voting intention surveys, weather data, interest rate index). Objectives of time series analysis can be:
- the shortest possible description of a historical time series
- the prediction of future time series values (forecast) on the basis of knowledge of their previous values (weather forecast))
- the detection of changes in time series (EEG or ECG monitoring in medicine during surgical procedures, changes in global vegetation phenology due to anthropogenic climate change)
- the elimination of serial or seasonal dependencies or trends in time series (seasonal adjustment) in order to reliably estimate simple parameters such as averages
The procedure within the framework of time series analysis can be divided into the following work phases:
- Identification phase: Identification of a suitable model for modelling the time series
- Estimation phase: estimation of suitable parameters for the selected model
- Diagnostic phase: diagnosis and evaluation of the estimated model
- Deployment phase: Use of the estimated and found suitable model (especially for forecasting purposes)
There are differences in the individual phases, depending on whether linear models for time series analysis (Box-Jenkins method, component model) or nonlinear models are used. In the following, the Box-Jenkins method is discussed as an example.
First and foremost, the graphical representation of the empirical time series values should be performed. This is the simplest and most intuitive method. Within the framework of the graphical analysis, initial conclusions can be drawn about the presence of trends, seasonality, outliers, variance stationarity and other anomalies.
Before further work can be done, the fundamental question of whether the time series should be mapped in a deterministic model (trend model) or a stochastic model must be clarified. These two alternatives imply different methods of trend adjustment, see Trend Adjustment Fluctuation Analysis. In the trend model, the adjustment is carried out by means of a regression estimate, in the stochastic model by means of the formation of differences.
In the estimation phase, the model parameters and coefficients are estimated using different techniques. For the trend model, the least-squares estimation is suitable, for the models in the Box-Jenkins approach, the moment method, the nonlinear least-squares estimation, and the maximum-likelihood method are suitable for estimation.
In the diagnostic phase, the quality of the model or, if necessary, several selected models are assessed. The following procedure is recommended:
- Step 1: Check whether the estimated coefficients are significantly different from zero. In the case of individual coefficients, this is done with the help of a t-test, several coefficients together are examined with an F-test.
- Step 2: If the Box-Jenkins method is used, it is necessary to examine the extent to which the empirical autocorrelation coefficients correspond to those that should theoretically be obtained on the basis of the previously estimated coefficients. In addition, the partial autocorrelation coefficients as well as the spectrum can be analyzed.
- Step 3: Finally, a careful analysis of the residuals is carried out. The residuals should no longer have any structure. The centeredness of the residuals can be checked with a t-test. To clarify the latter, the so-called portmanteau tests can be used. For example, information criteria can be used for this purpose.
In the deployment phase, it is necessary to formulate a prediction equation from the model equation established in the identification phase and found to be useful. An optimality criterion must be defined in advance. For this purpose, the minimum mean squared error (MMSE) can be used.