**Linear prediction**is a mathematical operation where future values of a digital signal is estimated as a linear function of previous samples.

In digital signal processing linear prediction is often called linear predictive coding (LPC) and can thus be viewed as a subset of filter theory. In system analysis (a subfield of mathematics), linear prediction can be viewed as a part of mathematical modelling or optimization.

## The prediction model

The most common representation is

where is the estimated signal value, the previous values, and the predictor coefficients. The error generated by this estimate is

These equations are valid for all types of (one-dimensional) linear prediction. The differences are found in the way the parameters are chosen.

For multi-dimensional signals the error is often defined as

## Estimating the parameters

where is the autocorrelation of signal defined as In the multi-dimensional case this corresponds to minimising the L_{2}norm.

The above equations are called the normal equations or Yule-Walker equations. In matrix form the equations can be equivalently written as

- ,

Another more general approach is to minimise

Optimisation of the parameters is a wide topic and a large number of other approaches have been proposed.

Still, the autocorrelation method is the most common and it is used, for example, for speech coding in the GSM standard.

Solution of the matrix equation is computationally a relatively expensive process. The Gauss algorithm for matrix inversion is probably the oldest solution but this approach does not efficiently use the symmetry of and . A faster algorithm is the Levinson recursion proposed by N. Levinson in 1947, which recursively calculates the solution. Later, Delsarte et al proposed an improvement to this algorithm called the split Levinson recursion which requires about half the number of multiplications and divisions. It uses a special symmetrical property of parameter vectors on subsequent recursion levels.