A moving-box algorithm to estimate generalized dimensions and the f (α) spectrum

Estimated generalized dimensions of river networks

“…In fact, problems arise when the cells contain few points, namely they are characterised by low statistics, emphasized by the negative exponents; this, in turn, makes the measure to diverge exponentially (Yamaguti and Prado, 1995;Pastor-Satorras and Riedi, 1996;Feeny, 2000). The numerical check is difficult to apply, unless one redistributes the cells in such a way that they always cover a sufficient part of the fractal set, thus increasing the statistics of the points falling inside (Yamaguti and Prado, 1995;Pastor-Satorras and Riedi, 1996;Yamaguti and Prado, 1997;Oiwa and Fiedler-Ferrara, 1998). Some errors, due to the use of Legendre transforms (Chhabra and Jensen, 1989;Veneziano et al, 1995), are to be associated to such problems.…”

Morphological covering based generalized dimension for gear fault diagnosis

“…The generalized dimension, which consists of exponents that characterize nonuniformity of the measure, was introduced earlier than the singular spectrum and can be computed easier than the latter. It has been widely used for characterizing multifractal phenomena over the past few decades [12][13][14][15][16][17][18][19][20].…”

“…Box-counting method, by far, is the one of the most involved techniques for estimating the generalized dimension in literature [13,14,16,18,20,29]. By using boxes of size ε, the minimum number of the boxes N(ε) covered the set is counted.…”

“…Box-counting (BC) method is the most popular tool for estimating the generalized dimensions due to its ease to understand and implement. It has been applied widely for characterizing various fractal phenomena, such as image and nonstationary time series [12][13][14][15][16][17][18][19][20]. However, some researches have pointed out that the box-counting based algorithm may lead to less precise multifractal estimations due to its intrinsically regular partition rule [21][22][23].…”

Comparison of gliding box and box-counting methods in river network analysis

“…The number of samples at a given scale, applying BC, is restricted by the size of the partitioning space and data resolution, which is usually another main factor influencing statistical estimation in MFA (Cheng and Agterberg, 1996). To avoid these problems several methods have been proposed (Liebovitch and Toth, 1989;Block et al, 1990;Veneziano et al, 1995;Yamaguti and Prado, 1997;Oiwa and Fiedler-Ferrara, 1998;Feeny et al, 2000) and a few applied to the river network, such as the sand box method (De Bartolo et al, 2004), generalization of the correlation method Gaudio et al, 2006), and the fixed mass method .…”