## Curve Fitting

**Curve Fitting:**

The curve fitting process fits the equation of approximating curves to the raw field data. Nevertheless, for a given set of data, the fitting curves of a given type are generally NOT unique. Thus, a curve with a minimal deviation from all data points is desired.

**The Method of Least Squares:**

Principle of Least Square: The curve of best fit is that for which the sum of squares of the residuals is minimum. The method of least squares assumes that the best-fit curve of a given type is the curve that has the minimal sum of the deviations squared (least square error) from a given set of data. Let the set of data points be (xi, yi), i = 1,2, ...,m, and let the curve given by y = f (x) be fitted to this data. At x = xi, the observed value of the ordinate is yi and the corresponding value on the fitting curve is f (xi). If εi is the error of approximation at x = xi, then we have

** εi = yi− f (xi)**

The sum of square of errors,

** ε = Σe _{i} ^{2}_{ }= Σ[yi− f (xi)]^{2}**

Now if ε = 0 then yi = f (xi), i.e. all the points lie on the curve. Otherwise, the minimum e results the best fitting of the curve to the data.

**Fitting of A Straight Line:**

Let (xi, yi) be n sets of observations of related data and y = a bx be the straight line to be fitted.Then equation

By the principle of least square, ε is minimum for some a and b, i.e.,

These above equations are called normal equations. Since x_{i} and y_{i} are known, these equations result in equations in a and b. We can solve these for a and b.

**Remark**: For the sake of simplicity leave suffix notation to obtain the following form of normal equations.