TABLE OF CONTENTS In time series, a powerful model is called the prediction method. It is assumed that the present value can be predicted from past values. For example, prediction can be used in many areas (i.e., in environmental trend, weather forecasting, stock market movement), although the reliability is questionable.
If it is used in spectrum estimation, the present value can be written as a linear combination of input and output
| (14.1) |  |
where x( n) is the digitized data, a i, and b l are constants, G is the gain of the system, and u( n) represents white noise. In statistics, this equation is called the autoregressive moving average (ARMA) model.
If one takes the z transform of this equation, the result is
| (14.2) |  |
In this equation, the white noise is usually considered as input and the data are the output. Thus, the transfer function H( z) of this equation is defined as the output divided by the input. The result is
| (14.3) |  |
This equation is called the general pole-zero form because the transfer function has both zeros and poles. The zeros are the z values that cause the numerator to be zero and the poles are the z values that cause the denominator to be zero.
If all the a i values are zero in (14.1), the equation becomes
| (14.4) |  |
This equation is called the moving average (MA) model. Its corresponding transfer function is
| (14.5) |
