Neutral data fitting in two and three dimensions

We consider fitting a line (or plane) to data on two (or three) variables for the purpose of representing scientific or law-like relationships. If these variables are related in a manner in which no variable plays a special role (i.e. no distinction between dependent and independent variables), then it is appropriate that the fitting procedure should treat all variables in the same way. Starting from obviously desirable properties for the error measure we deduce the form of this error measure and show it to be unique. We derive fitting procedures that aggregate the proposed measure over all points using the sum of squares. For the case of three variables our procedure is new--and we analyse completely the solution to the resulting minimisation problem.