Pernicious effect of physical cutoffs in fractal analysis.
Fractal scaling appears ubiquitous, but the typical extension of the scaling range observed is just one to two decades. A recent study has shown that an apparent fractal scaling spanning a similar range can emerge from the randomness in dilute sets. We show that this occurs also in most kinds of nonfractal sets irrespective of defining the fractal dimension by box counting, minimal covering, the Minkowski sausage, Walker's ruler, or the correlation dimension. We trace this to the presence of physical cutoffs, which induce smooth changes in the scaling, and a bias over a couple of decades around some characteristic length. The latter affects also the practical measure of fractality of truly fractal objects. A defensive strategy against artifacts and bias consists in carefully identifying the cutoffs and a quick-and-dirty thumb rule requires to observe fractal scaling over at least three decades.
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Settore di Geofisica, Dipartimento di Fisica, Università di Bologna, Viale Berti Pichat, 8 Bologna, Italy. matteo@ibogeo.df.unibo.it
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