Linear Regression.md

Linear Regression

Permanent Note Created: 01-20-2022 13:40

Linear regression is modeling a variable as a linear function of other variables.

$Y = \beta_0 + \beta_1 \cdot x_1 + \beta_2 \cdot x_2 + ...+ \epsilon$

This reads as Y is a weight linear combination of the x variables. The equation may also be written in matrix format. $$ X=[x_1 \space x_2 \space x_3] $$ $$ b=[\beta_1 \space \beta_2 \space \beta_3] $$ $$ Y=Xb $$ This equation is solved for b $$ X^T Y = X^T X b $$ $X^TX$ is now a square matrix.

$$ (X^TX)^{-1}X^T Y = (X^TX)^{-1}X^T X b $$ $$ (X^TX)^{-1}X^T Y = b $$

Precalculation

If the independent varivale $Y$ is brain imaging data, then the following is allowed:

	Nvoxels
	transX = transpose(X)
	pX = inverse(X*transX)*transX
	beta = size(Nvoxels, NcolumnsInX)
	for i in Y:
		beta = pX*i

The pX variable is also called the pseudoinverse of X

References