Data Depth for Classical and Orthogonal Regression

We present a comparison of different depth notions which are appropriate for classical and orthogonal regression with and without intercept. We consider the global depth and tangential depth introduced by Mizera (2002) and the simplicial depth studied for regression in detail at first by Müller (2005). The global depth and the tangential depth are based on quality functions. These quality functions can be based on likelihood functions as was used in Mizera and Müller (2004) and Müller (2005) or on residuals. For orthogonal regression the approach based on residuals is more appropriate. Classical regression and orthogonal regression differ only by the definition of the residuals. In classical regression, residuals are defined by the distance of the data points to the regression line or plane in vertical direction, i.e. parallel to the y-axis. In orthogonal regression, the residual is the distance between the data point and the regression line or plane in perpendicular direction to the line or plane. This residual is invariant with respect to rotations of the axis since only the direction perpendicular to the regression line or plane is used. Orthogonal regression should be used in cases where no natural xand y-axes exists like in image analysis.