Curve Fitting Methods: A Survey

Sreehari Gopalakrishnan, Nikolaos Bourbakis
DOI: 10.4018/IJMSTR.2016100103
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Abstract

The main idea of this paper is to provide an insight to the reader and create awareness on some of the basic Curve Fitting techniques that have evolved and existed over the past few decades. A critical survey has been done on the various Curve Fitting methodologies proposed by various Mathematicians and Researchers who had been working in the fields of Computer Graphics, Image Processing and Pattern Recognition etc. to name a few over the past. Most of the methodologies that have evolved over time have their core structure derived from few significantly basic concepts and it is these basic concepts that have been tried to grasp through this survey and its tried to present the same as the essence of the survey. Note that the same physical variables may be named differently under different methodologies. The purpose was to maintain the originality of the work published by the original authors and this survey to only report their work as it is.
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1. Introduction And A Brief Review

Curve fitting is a significant research problem with no perfect solution. Many researchers (Niemann (1981), Walton (1989), de Boor (1978), Pavlidis (1982), Dierckx (1993), Cox & Hayes (1973), Hayes (1970), Lancaster & Salkauskas (1986), Forsyth et.al. (1999), Rogers & Adams (1990), Hearn & Baker (1998), Imai & Iri (1986), Taubin & Ronfard (1996), Pham (1989), Zucker & Parent (1982), Tsai & Chen (1994), Pei & Horng (1995), Mathews & Fink (2004), Nelson (1997), www.google.com/curve-fitting,) have offered notable solutions in time. Here we try to offer a brief overview for some of these methods and address their challenges.

To get a gist of what is being talked through the length of the paper, consider the mathematical representation of smooth continuous relationships between one physical variable (the Dependent variable, say x), and another (the Independent variable, say y). These relationships are defined initially either by a set of data points or by a graph and the process by which a mathematical function is determined which passes satisfactorily close to the points or graph is Curve Fitting. The degree of difficulty in Curve Fitting varies widely from one set of data to another. One good and fortunate thing about curve fitting is that there exists a significant safeguard available with it and it’s that the fitted curve can be tested for acceptability by comparing it graphically with the given set of data or curve. This procedure should always be followed. If necessary, graphical accuracy can usually be enhanced by subtracting a simple smooth curve, such as a straight line or quadratic, from both the data and the fitted curve and plotting the differences.

In plotting the fitted curve, it is important to recognize that it is not sufficient to plot only the values of the curve at the data abscissa. At the very least, its values at points about halfway between all adjacent abscissa values should also be plotted as unwanted fluctuations may otherwise go undetected. Its very important that data is available, spanning the whole range of the independent variable required in the subsequent use of the fitted function. Extrapolation of the function beyond the range of the available data is not always preferred. Generally, the fitting strategies approached first are with single low-degree polynomial, if these fail to produce acceptable fit, then piece-wise polynomials, like Cubic-Spline functions are tried.

Before actually beginning with the Curve-Fitting process, the variables used should be appropriately chosen; often its fine to use the variables as they exist, but sometimes the use of the logarithm, square-root or some other function of the variable may lead to better-behaved relationship. This is a criterion worth consideration while preparing graphs or tables etc. For practical realization reasons, it is usually better to avoid having to deal with a Curve whose behavior in one part of the range is radically different from that in another part. According to the specific case, either the dependent variable or the independent variable or both may be transformed to give a better-behaved relationship, in cases where there is a choice, the most convenient choice to transform the independent variable. Another point worth consideration is that checking whether the data points are of same or different accuracy and so whether they should be assigned different weights in the fitting process. Often, the data points will all be of equal or nearly equal accuracy. If the data values yr (r = 1, 2, ..., m) of the dependent variable are of substantially differing (absolute) accuracy's, then appropriate weights must be calculated from estimates of the absolute accuracy's of yr, expressed as standard deviations, probable errors or by some other measure which is of the same dimensions as y. specifically, the weight of yr which is denoted by wr, is the reciprocal of the accuracy estimate of yr .

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