Let us take a curve, suppose we want to fit a smooth curve which goes through “p + I” given points when the value of ‘p’ is large, then an interpolating Polynomials which have high degree ‘p’ tends to high oscillatory, which gives an unsatisfactory fit. In some location interpolation is designed by linking the lower degree polynomials, where all the data points are known as nodes or knots. If we get matching Derivatives at the nodes then this interpolation is smooth and by matching higher order derivatives we enhanced the smoothness. Now we will see how to do Curve Fitting:
Let we have data points (p0), (f0), (p1), (f2),….(pn), (fn), be arranged according to their magnitude so that
P0 < P0 < …. Pn-1 < Pn,
We will include a function S which is polynomial of degree d on each interval [Pq-1, pq],
q = 1, 2,…, n where
S (pq) = fq;
In order to get maximum smoothness at the nodes, than we shall allow S to have up to (d – 1) continuous derivatives. Such Functions are known as splines.
To perform fitting, we have to define some function which depends on the parameter and parameter measures the closeness between the data and the model. Then this function is minimized up to the lowest value with respect to the parameter. Whatever value we get that value minimized the function, and we get best fitting parameters.
Curve fitting is used to find the best fit line for a series of data points. It is also known as regression analysis. It is used to smooth the data and improve the appearance. It is also used to construct a curve or mathematical function, which is used to find the best fit to a series of data points. It shows the relationship among the different variables.