Читать книгу Artificial Intelligence and Quantum Computing for Advanced Wireless Networks - Savo G. Glisic - Страница 44

In general, here we deal with the problem of predicting future samples of a time series using a number of samples from the past [6, 7]. Given M samples of the series, autoregression (AR) is fit to the data as

(3.41)

The model assumes that y(k) is obtained by summing up the past values of the sequence plus a modeling error e(k). This error represents the difference between the actual series y(k) and the single‐step prediction

(3.42)

In nonlinear prediction, the model is based on the following nonlinear AR:

(3.43)

where g is a nonlinear function. The model can be used for both scalar and vector sequences. The one time step prediction can be represented as

(3.44)

Given the focus of this chapter, we discuss how the network models studied so far can be used when the inputs to the network correspond to the time window y(k − 1) through (k − M). Using neural networks for predictions in this case has become increasingly popular. For our purposes, g will be realized with an FIR network. Thus, the prediction corresponding to the output of an FIR network with input y(k − 1) can be represented as

(3.45)

where N_M is an FIR network with total memory length M.