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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 $R^2$?

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

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