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Linear Prediction

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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.

Stephanie Glen. "Linear Prediction" From Elementary Statistics for the rest of us!

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