Multiple Linear Regression

Description

Another useful case of regression, specially for experimental data, is when the dependent variable yy depends on 2 or more independent variables (x1x_1, x2x_2, …, xnx_n) as:

y=a0+a1x1+a2x2+⋯+anxnBidimensional:y=a0+a1x1+a2x2y = a_0 + a_1x_1 + a_2x_2 + \cdots + a_nx_n \\ \text{Bidimensional:} \\ y = a_0 + a_1x_1 + a_2x_2

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

y=a0+a1x1+a2x2E=yi−a0−a1x1−a2x2y = a_0 + a_1x_1 + a_2x_2 \\ E = y_i - a_0 - a_1x_1 - a_2x_2

To get the coefficients of the equation, we can represent the sums as a matrix and solve it:

[n∑x1∑x2∑yi∑x1∑x12∑x1x2∑x1yi∑x2∑x1x2∑x22∑x2yi]\left[\begin{array}{rrr:r} n & \sum x_1 & \sum x_2 & \sum y_i \\ \sum x_1 & \sum x_1^2 & \sum x_1x_2 & \sum x_1y_i \\ \sum x_2 & \sum x_1x_2 & \sum x_2^2 & \sum x_2y_i \end{array}\right]

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−a1x1−a2x2)2S_r = \sum_{i=1}^{n} (y_i - a_0 - a_1x_1 - a_2x_2)^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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