Orthogonal regression

Orthogonal regression

For the case of equal error variances, i.e., when ${\displaystyle \delta =1}$, Deming regression becomes orthogonal regression: it minimizes the sum of squared perpendicular distances from the data points to the regression line. In this case, denote each observation as a point ${\displaystyle z_{j}=x_{j}+iy_{j}}$ in the complex plane (i.e., the point ${\displaystyle (x_{j},y_{j})}$ where ${\displaystyle i}$ is the imaginary unit). Denote as ${\displaystyle S=\sum {(z_{j}-{\overline {z}})^{2}}}$ the sum of the squared differences of the data points from the centroid${\displaystyle {\overline {z}}={\tfrac {1}{n}}\sum z_{j}}$ (also denoted in complex coordinates), which is the point whose horizontal and vertical locations are the averages of those of the data points. Then: