Linear Prediction

Time series >

Linear prediction is a technique for anlayzing time series; It allows us to predict future values from historical data. It is often used in digital signal processing, because it allows the future values of a signal to be estimated in terms of a linear function of past samples.

Types of Linear Prediction

There are three main types of linear prediction. They are differentiated by the form of the transfer function; a function H(Z) which can generally be defined according to its characteristics:

  • The numerator of H(z) is constant: We call this an autoregressive (AR) or all-pole model.
  • The denominator of H(z) is constant: This we call a moving average or all-zero model.
  • No assumptions can be made about the characteristics of H(z): A model in which we can make no assumptions is called a autoregressive moving average (ARMA), or mixed pole/zero model.

Calculating Predicted Signal Values

The autoregressive model is the model most extensively and used and studied today. This is because of a couple of reasons:

  1. It produces equations that are relatively easy to solve,
  2. It accurately models many practical, real world applications, such as speech production.

In the autoregressive model, a predicted signal value x̂(n) can be calculated by:

linear prediction


This is an estimate; not an exact value, and the error term is referred to as e(n). By definition, where x(n) is the true signal value,

e(n)= x(n) – x̂(n)


Cinneide, Alan. Linear Prediction. The Technique, Its Solution and Application to Speech. Retrieved from on May 16, 2018.
Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics, Cambridge University Press.
Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.
Mahkonen, Katariina. Linear Prediction. SGN-14006 Course Notes. Retrieved from on May 16, 2018.
Vaidyanathan, P. P. The Theory of Linear Prediction. Retrieved from on May 14, 2018.

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