A frequently encountered engineering problem is that of having a set of data which one wishes to describe by a mathematical model and determine a set of parameters that characterize the model. This is similar to the topic addressed in Chapter 7 of curve fitting and data plotting. However, in that chapter the emphasis was only on obtaining a smooth curve representation of a set of data with noise. The intent of the curve fitting there was to obtain a function that could be used in subsequent manipulations of the data or in visually presenting the data in graphical form. The particular form of the fitting function was secondary to the goal of obtaining a good smooth approximation to the data. In the present chapter, the emphasis will still be on obtaining a “good” approximation to the data set; however, the major emphasis will shift to the fitting parameters of a data model which will be assumed to have some particular physical or mathematical significance. For example the fitting parameters could be some time constant of a physical process or some saturated value of a physical process or some time period for a physical process. The emphasis will thus be on estimating the value of the parameters used in describing a data set. This will naturally lead to question about how accurately the values of the parameters can be obtained and questions associated with confidence limits on the estimated parameters. Thus this chapter continues the discussion begun in Chapter 7 on curve fitting and brings in many of the statistical concepts developed in Chapter 8 to address the general area of modeling of data and the estimation of parameter values from the data for a given model of the data. In some engineering fields, this process is also called parameter extraction.
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