Linear prediction

Linear prediction ( engl. linear prediction ) is a mathematical method of time series analysis, which estimates future values ​​of a signal or a discrete time series as a linear function of the past values ​​of the same time series. A variant is the econometric method, which also takes into account values ​​of another time series, their values ​​depend on the considered time series.

For centered real and stationary time series, the coefficients of the estimation functions are given by the Yule- Walker equations, this corresponds to the model by means of an AR (p) process. Next find the orthogonal projection methods ( Gram-Schmidt procedure) application.

The term linear prediction is also an abbreviation used for the application of this theory in digital signal processing, see linear predictive coding.

Mathematical representation

A common (one-dimensional ) representation is

Present with and, with the predicted value, the previously observed values ​​and the estimated coefficients. The estimation error has the representation

Where the true value indicates the time.

The forecasting methods differ in the way how the parameters are determined.

For multivariate time series is an error metric of the form

Defined, being chosen for a suitable vector norm.