Fitting a line to data:

Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a greater degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data.

Fitting a line

Lets start with a first degree polynomial equation:

y= ax+b

This is a line with slope a. We know that a line will connect any two points. So, a first degree polynomial equation is an exact fit through any two points.

If we increase the order of the equation to a second degree polynomial, we get:

y= ax²+bx+c

This will exactly fit a simple curve to three points.

If we increase the order of the equation to a third degree polynomial, we get:

y= ax³+bx²+cx+d

This will exactly fit four points.

Different techniques of fitting lines

There are three different techniques for fitting straight lines to experimental data and discusses the corresponding evaluation of uncertainty. The techniques are (i) traditional fitting by least-squares, (ii) a Bayesian linear-regression analysis and (iii) an analysis according to the propagation of probability density functions attributed to the points measured. The material is presented to clarify assumptions underlying the techniques, to highlight differences between the techniques and to point to difficulties associated with applying the techniques under current views of 'uncertainty analysis'. Considerable attention is given to the estimation of values of the function and not just to the estimation of parameters of the function. The paper gives a summary of many results of least-squares fitting, including some unfamiliar results for the simultaneous estimation of the unknown function at all points. On many occasions the unknown function will only be approximately linear, in which case we must define a unique unknown gradient to give proper meaning to our 'estimate' of slope. This can be achieved by defining an interval of interest and then applying a least-squares-type result.

The data fitting procedure aims to obtain a reasonably good representation of the data in terms of a function of one or more variables. Actually, the required representation has to have some theoretical basis ( or prejudice ) behind it. Otherwise, there are a large number ( strictly infinite ) of functions which may be used to represent the data. The additional problem is, since the experiment is done at finite number of values of the independent variable, one does not have a unique function to describe the data. The problem is further enhanced by the errors in the measurements. One therefore has to choose a function having certain number of parameters to describe the data.

The method of determining the set of parameters of the fitting function which gives a best fit to the data involves three things. These are:

• The values of the parameters giving the best fit.
• An estimate of an error associated with these parameters.
• Finally we also need to have some statistical measure to decide on the goodness' of the fit.

One of the standard procedure used in data fitting is the method of least squares. This method basically minimizes the square of the difference between the data and the fitting function.