# Measure of goodness of fit of ellipsoid model to data

Consider a measurement $$m$$ of a physical property, the value of which depends on the direction investigated. If the property derives from the application of a second-rank symmetric tensor, the magnitude $$m_i$$ in a direction given by the unit vector $$u_i(x_i, y_i, z_i)$$ is such that:

$$\tag{1} Ax_i^2+By_i^2+Cz_i^2+Dx_iy_i+Ex_iz_i+Fy_iz_i=m_i$$

which is a quadratic equation that describes the surface of an ellipsoid centered at the origin of an Cartesian x,y,z coordinate system. The six parameters $$A$$-$$F$$ define the shape and orientation of the ellipsoid representing the spatial variation of the property in the measurement reference.

The Question: Given values for the six parameters $$A$$-$$F$$, and an arbitrary number $$N$$ of measurements $$m_i$$ (with their associated $$x_i, y_i, z_i$$ component values), how do we define a measure of goodness of fit, e.g., a coefficient of determination $$R^2$$?

• Could you explain the sense in which this is not a routine multiple regression problem, whose answer is "use the usual $R^2$"?
– whuber
Commented Oct 11, 2018 at 14:49