How to Calculate the Goodness-of-Fit of a Fractal Dimension

How to calculate the goodness-of-fit of a fractal dimension To the Editor, Karyometric data are helpful for differential diagnosis and prognosis [1–4]. A new morphometric variable, the coefficient of determination of the Minkowski frac-tal dimensions (the R 2 value between the y of the regression lines and y values of the real data) has recently been introduced [1,2]. Adam et al. [1] described a negative correlation between the fractal dimension (FD) and the R 2 values. Yet at least part of this effect may under certain assumptions be expected for theoretical reasons. If the residuals of the regression are the y-residuals (i.e., log(fractal area)) – as is common in Least Squares regression – R 2 will vary systematically as the slope of the regression line (i.e., FD) changes, even though the distribution of observed points around the regression line may be the same. However, if the residuals were computed orthogonally to the regression line, at least some of this effect would vanish. A simple simulation of a point distribution can be made: homogeneous random on an interval [−L, L] in the x-direction, normal random in the y-direction, N (0, σ). Then rotate the point distribution around the origin to a given mean slope, −φ, and alter the slope stepwise by an angle Δφ around the mean slope, and compute the R 2 value for each step. The numerical value of the correlation between R 2 and FD = 2 + tg(φ) depends on the parameters used, σ/L and the range of φ. For FD < 2.5 it is positive, for FD > 2.5 it is negative, since R 2 peaks at a slope of 45 degrees. This is an obvious result if all the rotated distributions are identical except for the rotation angle. But the correlation is still there – although the numerical value is lower – if the distributions at each separate angle are allowed to be independent. I do not deny that the coefficient of determination of the Minkowski fractal dimensions might be a statistically significant prognostic variable, or that there may be a real variation in the distribution around the regression line as FD varies. I am simply commenting that the variation of R 2 described in [1,2] may be partially due to a projection effect in the residuals, and that an improved measure could be obtained by a regression based on orthogonal residuals. It is worth …