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 most common representation is

where ${\displaystyle {\widehat {x}}(n)}$ is the predicted signal value, ${\displaystyle x(n-i)}$ the previous observed values, with ${\displaystyle p\leq n}$, and ${\displaystyle a_{i}}$ the predictor coefficients. The error generated by this estimate is

where ${\displaystyle x(n)}$ 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 ${\displaystyle a_{i}}$ are chosen.

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

where ${\displaystyle \|\cdot \|}$ is a suitable chosen vector norm. Predictions such as ${\displaystyle {\widehat {x}}(n)}$ are routinely used within Kalman filters and smoothers to estimate current and past signal values, respectively, from noisy measurements.