# Linear prediction

**Linear prediction** (engl. *Linear prediction* ) is a mathematical method of time series analysis , which future values of a signal or a discrete time series as a linear function of the past values of the same time series estimates .

One variant is the econometric method, which also takes into account the values of another time series on which the time series under consideration depends.

For centered , real and stationary time series, the coefficients of the estimation functions are given by the Yule-Walker equations ; this corresponds to the modeling by an AR (p) process . Orthogonal projection methods ( Gram-Schmidt method ) are also used.

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

## Mathematical representation

A common (one-dimensional) representation is

- ,

with and , where represent the predicted value, the values already observed and the estimation coefficients. The estimation error has the representation

- ,

where denotes the true value at the time .

The forecasting methods differ in the way in which the parameters are determined. The parameters are usually determined in such a way that the mean square error is minimized. Then one speaks of a **best linear ****expectation faithful ****prediction** , shortly **Blev** ( English *Best Linear Unbiased Prediction* shortly *BLUP* ). BLUP and BLUE were introduced by Charles Roy Henderson in the 1950s .
** **

For multi-dimensional time series, an error metric of the shape

defined, with a suitable vector norm being selected.

## literature

- Jens-Peter Kreiß and Georg Neuhaus:
*Introduction to time series analysis.*Springer-Verlag, Berlin 2006, ISBN 3-540-33571-4 . - Robinson, GK (1991). That BLUP is a Good Thing: The Estimation of Random Effects.
*Statistical Science*6 (1): 15-32. doi : 10.1214 / ss / 1177011926 JSTOR 2245695

## Web links

- Prediction for stationary time series (accessed on July 20, 2018)
- The Theory of Linear Prediction (accessed July 20, 2018)
- Linear Prediction (accessed July 20, 2018)
- The History of Linear Prediction (accessed July 20, 2018)
- Linear prediction. The Technique, Its Solution and Application to Speech (accessed July 20, 2018)