Polynomial Method

Description

Some discrete data is poorly represented by a straight line, therefore is better to fit them to a curve making the polynomial regression a better option.

To create the polynomial, the least-square method can be easily extended with the polynomial function adding the error:

y=a0+a1xi+a2xi2+⋯+amxim+EE=yi−a0−a1xi−a2xi2−⋯−amximy = a_0 + a_1x_i + a_2x_i^2 + \cdots + a_mx_i^m + E \\ E = y_i - a_0 - a_1x_i - a_2x_i^2 - \cdots - a_mx_i^m \\

To get the coefficients of the equation:

∑yi=na0+a1∑xi+a2∑xi2+⋯+am∑xim∑xiyi=a0∑xi+a1∑xi2+a2∑xi3+⋯+am∑xim+1∑xi2yi=a0∑xi2+a1∑xi3+a2∑xi4+⋯+am∑xim+2⋮∑ximyi=a0∑xim+a1∑xim+1+a2∑xim+2+⋯+am∑xi2m\begin{aligned} \sum y_i & = & na_0 & + & a_1\sum x_i & + & a_2\sum x_i^2 & + & \cdots & + & a_m\sum x_i^m \\ \sum x_iy_i & = & a_0\sum x_i & + & a_1\sum x_i^2 & + & a_2\sum x_i^3 & + & \cdots & + & a_m\sum x_i^{m + 1} \\ \sum x_i^2y_i & = & a_0\sum x_i^2 & + & a_1\sum x_i^3 & + & a_2\sum x_i^4 & + & \cdots & + & a_m\sum x_i^{m + 2} \\ & & & & & \vdots \\ \sum x_i^my_i & = & a_0\sum x_i^m & + & a_1\sum x_i^{m+1} & + & a_2\sum x_i^{m+2} & + & \cdots & + & a_m\sum x_i^{2m} \\ \end{aligned}

Making these equations equal to zero, to get the coefficients of a mthm^{th} degree polynomial becomes a problem of solving a system of m+1m + 1 linear equations.

Error calculation

The best adjusted data is the one that minimizes the sum of the squared residues SrS_r, meaning the error between the model and the experimental data, this is the vertical distance between the points and the line:

Sr=∑i=1n(yi−a0−a1xi−a2xi2−⋯−amxim)2S_r = \sum_{i=1}^{n} (y_i - a_0 - a_1x_i - a_2x_i^2 - \cdots - a_mx_i^m)^2

To calculate the error, we need to sum the squared difference of each point with the mean on the y-axis:

St=∑i+1n(yi−y‾)2S_t = \sum_{i+1}^{n} (y_i - \overline{y})^2

The coefficient of corelation (how good is the regression) is calculated as:

r=St−SrStr = \sqrt{\dfrac{S_t - S_r}{S_t}}

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