Orthogonal Regression to a Plane

The considerations made for the line can also be extended to the plane. It should be emphasized that the orthogonal regressions of a line, a plane, or a hyperplane are to be regarded as an eigenvalue problem and can be solved through Singular Value Decomposition (SVD), which is precisely the main application of Principal Component Analysis (PCA).

Let $\mathbf{p_0}=\E[\mathbf{p}]$ be the centroid of the points involved in the regression. Given the equation of the plane (1.49) and using the sum of distances as the error function (1.52), we immediately obtain the constraint:

$$k = - \mathbf{p_0} \cdot \hat{n}$$ (3.78)

that is, as already noted in the linear case, the centroid of the distribution lies on the plane. Starting from this initial constraint, it is possible to describe the plane as

$$(\mathbf{p} - \mathbf{p_0}) \cdot \hat{n} = 0$$ (3.79)

an overdetermined homogeneous system, whose solution can be obtained using the pseudoinverse (for example, through QR factorization or SVD). The value of $\hat{n}$ thus derived will be known up to a multiplicative factor, and for this reason, it can always be normalized, forcing it to unit length (the solutions obtained through factorizations are usually already normalized).