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,
- ai are the predictor variable’s coefficients.
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)
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