This lecture will provide an outline of data-based mathematical modeling, with a particular focus on parameter estimation for discrete-time (sampled data) Transfer Function (TF) models. These are statistical models obtained by direct reference to data. The data are utilised to both identify the model structure and to estimate the parameters which characterise this structure. To illustrate the generic methods considered, the lecture will utilize worked examples based on both biological and non-biological data sets.
Linear systems theory assumes a cause-and-effect relationship between the input and output variables. In this context, the TF is a mathematical tool that describes the transfer of the input signal to an output signal. In this lecture, attention will be restricted to multiple–input, single–output TF models, represented in terms of the backward shift operator. One advantage of such a model is its simplicity and ability to characterise the dominant model behaviour of a dynamic system. Such models are useful for classification, forecasting, control system design and other purposes. For example, by interpreting the parameter estimates in mechanistic terms, they can also be used to help improve our understanding of the physical or biological system in question. This is called a data-based mechanistic approach (e.g. Young, 1998).
The lecture will compare the Response Error and Equation Error approaches to parameter estimation, before using a worked example to develop the Normal Equations for Least Squares analysis. The limitations of this solution will be discussed, followed by consideration of alternatives and extensions, such as instrumental variables (Young, 1984) and prediction error minimization. Other model structures, including continuous-time and nonlinear systems will be mentioned. However, the lecture will necessarily concentrate on a brief overview, with pointers to further reading and to readily available software tools.