Linear prediction

Linear prediction is a mathematical operation where future values of a discrete-time signal are 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

x

^

(

n

)

=

∑

i

=

1

p

a

i

x

(

n

−

i

)

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

where

x

^

(

n

)

{\displaystyle {\widehat {x}}(n)}

is the predicted signal value,

x

(

n

−

i

)

{\displaystyle x(n-i)}

the previous observed values, with

p

≤

n

{\displaystyle p\leq n}

, and

a

i

{\displaystyle a_{i}}

the predictor coefficients. The error generated by this estimate is

e

(

n

)

=

x

(

n

)

−

x

^

(

n

)

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

where

x

(

n

)

{\displaystyle x(n)}

is the true signal value.

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

a

i

{\displaystyle a_{i}}

are chosen.

For multi-dimensional signals the error metric is often defined as

e

(

n

)

=

‖

x

(

n

)

−

x

^

(

n

)

‖

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

where

‖

⋅

‖

{\displaystyle \|\cdot \|}

is a suitable chosen vector norm. Predictions such as

x

^

(

n

)

{\displaystyle {\widehat {x}}(n)}

are routinely used within Kalman filters and smoothers to estimate current and past signal values, respectively, from noisy measurements.

Estimating the parameters

The most common choice in optimization of parameters

a

i

{\displaystyle a_{i}}

is the root mean square criterion which is also called the autocorrelation criterion.