# 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

${\ displaystyle {\ widehat {x}} (n): = \ sum _ {i = 1} ^ {p} a_ {i} x (ni) \,}$,

with and , where represent the predicted value, the values ​​already observed and the estimation coefficients. The estimation error has the representation ${\ displaystyle p, n \ in \ mathbb {N}}$${\ displaystyle x (i) \ in \ mathbb {R}}$${\ displaystyle {\ widehat {x}} (n)}$${\ displaystyle x (ni)}$${\ displaystyle a_ {i} \ in \ mathbb {R}}$

${\ displaystyle e (n) = x (n) - {\ widehat {x}} (n) \,}$,

where denotes the true value at the time . ${\ displaystyle x (n)}$${\ displaystyle n}$

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 . ${\ displaystyle a_ {i}}$

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

${\ displaystyle e (n): = \ | x (n) - {\ widehat {x}} (n) \ | \,}$

defined, with a suitable vector norm being selected. ${\ displaystyle \ | \ cdot \ |}$