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

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

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

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

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. ${\displaystyle \|\cdot \|}$

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)