2.1.2 Time Series and Dynamic Systems

A special type of function approximation problem is one where the input data are time dependent. This means that the function at hand has "memory", is thus dynamic, and is referred to as a dynamic system. For such systems, past information can be used to predict its future behavior. Two examples of dynamic system problems are: (1) predicting the price of a state bond, or that of some other financial instrument; and (2) describing the speed of an engine as a function of the applied voltage and load.

In both of these examples the output signal at some time instant depends on what has happened earlier. The first example is a time-series problem modeled as a system involving no inputs. In the second example there are two inputs, the applied voltage and the load. Examples of these kinds can be found in Section 8.2.1, Identifying the Dynamic of a DC Motor and in Section 12.2, Prediction of Currency Exchange Rate.

The process of finding a model of a system from observed inputs and outputs is generally known as system identification. The special case involving time series is more commonly known as time-series analysis. This is an applied science field that employs many different models and methods. The Neural Network package supports both linear and nonlinear models and methods in the form of neural network structures and associated learning algorithms.

A neural network models a dynamic system by employing memory in its inputs; specifically, storing a number of past input and output data. Such neural network structures are often referred to as tapped-delay-line neural networks, or NFIR, NARX, and NAR models.

Dynamic neural networks can be either feedforward in structure or employ radial basis functions, and they must accommodate memory for past information. This is further described in Section 2.6.

The Neural Networks package contains many useful Mathematica functions for working with dynamic neural networks. These built-in functions facilitate the training and use of the dynamic neural networks for prediction and simulation.