**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:

- It produces equations that are relatively easy to solve,
- 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:

*x(n-i)*are the previous observed values,*a*are the predictor variable’s coefficients._{i}

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)**

## References

Cinneide, Alan. Linear Prediction. The Technique, Its Solution and Application to Speech. Retrieved from https://www.dit.ie/media/electricalengineering/documents/mikelgainza/92.pdf 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 http://www.cs.tut.fi/~sgn14006/PDF/S03-LP.pdf on May 16, 2018.

Vaidyanathan, P. P. The Theory of Linear Prediction. Retrieved from https://authors.library.caltech.edu/25063/1/S00086ED1V01Y200712SPR003.pdf on May 14, 2018.

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