Multiple linear regression
Coefficients b are estimated by the method of least squares (Subsection 10.3.1), which minimizes the sum of squares of the differences between observed values y and values y calculated using the regression equation. In order to obtain a least-squares best fit, each member (left and right) of matrix equation y = Xb is multiplied by the transpose of matrix X, i.e. X'y = X'Xb. By doing so, the rectangular matrix X produces a square matrix X'X, which can be inverted. The values of coefficients b0 and b1 are computed directly after inverting the square matrix [X'X]:
Using the same approach, it is easy to compute coefficients bo, bj, ..., bp of a multiple linear regression (Subsection 10.3.3). In this type of regression, variable y is a linear function of several (p) variables x, so that one can write: y = b0 + b^ + ... + bpXp. Vectors y and b and matrix X are defined as follows:
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