Guide to linear Least Squares curve fitting techniques
Abstract:The purpose of this Item is to promote 'good practice' when choosing and fitting a function with a linear Least Squares error profile to represent a set of data. There are a host of spreadsheets, mathematical calculation software and dedicated plotting packages that enable the automatic fitting of a chosen function to a set of data. Care and judgement however must always be exercised when choosing the form of equation that is best suited to follow the overall trends of the data. Curve fitting is a procedure for the fitting of algebraic functions to a set of known data points in order that the approximating function can be used in place of the data points for estimation of the dependent variable for all desired values of the independent variable. It is essentially a smoothing process in which the fitted curve follows the general trends of the data rather than passing through each individual point. It is not to be confused with the various types of interpolation procedure, such as splines, that produce curves constrained to pass through all the data points. Section 3 of this Item introduces the Least Squares method of curve fitting, the equations for which are derived in Appendix A. Four types of function, from the many available, have been chosen to illustrate the curve fitting process; the power series polynomial function of a single independent variable, the logarithmic function, the rational function and the peak function. Section 3 also details five criteria for the quality of fit of a function, introduces scaling of variables and presents notes relating to the preparation of data for curve fitting. Section 4 presents examples to illustrate what is meant by underfitting and overfitting of functions to data sets and the fitting of functions to sparse data sets. The curve fitting procedure is developed in further sections of this Item by illustration using a series of examples using data from aeronautical and other engineering applications.
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