Time Series Analysis: Forecasting and Control, Fourth Edition

CHAPTER ELEVEN

Transfer Function Models

In this chapter we introduce a class of discrete linear transfer function models. These models can be used to represent commonly occurring dynamic situations and are parsimonious in their use of parameters.

11.1 LINEAR TRANSFER FUNCTION MODELS

We suppose that pairs of observations (X_t, Y_t) are available at equispaced intervals of time of an input X and an output Y from some dynamic system, as illustrated in Figure 11.1. In some situations, both X and Y are essentially continuous but are observed only at discrete times. It then makes sense to consider not only what the data has to tell us about the model representing transfer from one discrete series to another, but also what the discrete model might be able to tell us about the corresponding continuous model. In other examples, the discrete series are all that exist, and there is no background continuous process. Where we relate continuous and discrete systems, we shall use the basic sampling interval as the unit of time. That is, periods of time will be measured by the number of sampling intervals they occupy. Also, a discrete observation X_t will be deemed to have occurred “at time t.”

When we consider the value of a continuous variable, say Y at time t, we denote it by Y(t). If t happens to be a time at which a discrete variable Y is observed, its value is denoted by Y_t. When we wish to emphasize the dependence of a discrete output Y, not only on time but also on the level of the input ...