Derivation for Linear Fit in nD

In nD a line is an intersection of n-1 planes so we can let the total variance be the sum of variances for the deviations from each plane. The distance of closest approach to the origin for each plane is then a projection of the average of Δx and we get a number of terms for the variance similar to that in two dimensions.



For the changes of the e's to be independent they would have to be perpendicular to each other. Each e is then an eigenvector of the covariance matrix.