I have a set of data with two independent variables (x, y) and one dependent variable (z). I want to find a regression line that minimizes the perpendicular distances to the regression line, rather than the vertical distances as is customary for simple linear regression. From what I have read, this is a special case of Deming regression called orthogonal regression, which is a special case of total least squares.
I really only need the slope of the regression line for the x and y components. So, I was thinking I might be able to transform the problem into one with one independent variable (x) and one dependent variable (y), weighting each data point by the old z component. From what I have read, orthogonal regression of "2-dimensional" data such as this, has a convenient analytic formula in the complex plane. It also seems to be more efficient, computationally, than performing the regression with all three components. Admittedly, I don't know how to derive the regression formula for all three components.
Would the transformed problem yield the same results as the original? Since orthogonal regression computes errors using both dependent and independent variables, my intuition says they'd be the same. However, I lack the mathematical skills to confirm this.
I realize now that my usage of "projection" is ambiguous. If you did principal components analysis on the data, you'd get three eigenvectors. I want to project the eigenvector with the
smallest biggest z component, onto the xy plane, then rotate it 90 degrees.