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.
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.
- 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
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