Orthogonal Regression to a Plane

The considerations made for a line can also be extended to a plane. It should be emphasized that orthogonal regression of a line, plane, or hyperplane is an eigenvalue problem and can be solved using SVD decomposition (this is precisely the main application of PCA).

Let $\mathbf{p_0}=\E[\mathbf{p}]$ be the centroid of the points involved in the regression. Given the plane equation (1.85) and using the sum of the distances (1.88) as the error function, one immediately obtains the constraint:

$$k = - \mathbf{p_0} \cdot \hat{n}$$ (4.95)

that is, as already observed in the linear case, the centroid of the distribution belongs to the plane. Starting from this first constraint, the plane can be described as

$$(\mathbf{p} - \mathbf{p_0}) \cdot \hat{n} = 0$$ (4.96)

an overdetermined homogeneous system, whose solution can be obtained using the pseudoinverse (for example, through QR or SVD factorization). The value of $\hat{n}$ obtained in this way is defined only up to a multiplicative factor and can therefore always be normalized by imposing unit length (solutions obtained through factorizations are usually already normalized).