Linear prediction

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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.


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