Multi-fractal patterns occur widely in nature. In developing new algorithms
to determine multi-fractal spectra of experimental data I am lead to the
conclusion that generalised dimensions $D_q$ of order $q\leq0$, including the
Hausdorff dimension, are effectively \emph{irrelevant}. The reason is that
these dimensions are extraordinarily sensitive to regions of low density in the
multi-fractal data. Instead, one should concentrate attention on generalised
dimensions $D_q$ for $q\geq 1$, and of these the information dimension $D_1$
seems the most robustly estimated from a finite amount of data.Comment: 11 page

Roberts, A. J.

English

arXiv

Multi-fractal patterns occur widely in nature. In developing new algorithms to determine multi-fractal spectra of experimental data I am lead to the conclusion that generalised dimensions $D_q$ of order $q\leq0$, including the Hausdorff dimension, are effectively \emph{irrelevant}. The reason is that these dimensions are extraordinarily sensitive to regions of low density in the multi-fractal data. Instead, one should concentrate attention on generalised dimensions $D_q$ for $q\geq 1$, and of these the information dimension $D_1$ seems the most robustly estimated from a finite amount of data.A. J. Robert

Roberts, Anthony John

Adelaide Research &amp; Scholarship

arXiv:nlin/0512014v1  [nlin.PS]  7 Dec 2005Use the information dimension, not theHausdorffA. J. Roberts∗Original version 1997, revised February 8, 2008AbstractMulti-fractal patterns occur widely in nature. In developing newalgorithms to determine multi-fractal spectra of experimental data Iam lead to the conclusion that generalised dimensions Dq of orderq ≤ 0, including the Hausdorff dimension, are effectively irrelevant.The reason is that these dimensions are extraordinarily sensitive toregions of low density in the multi-fractal data. Instead, one shouldconcentrate attention on generalised dimensions Dq for q ≥ 1, and ofthese the information dimension D1 seems the most robustly estimatedfrom a finite amount of data.Contents1 Introduction 22 Poor conditioning of generalised dimensions of negative order 23 Fractal dimensions unbiased by finite size of data sets 6References 7∗Dept. Mathematics & Computing, University of Southern Queensland, Toowoomba,Queensland 4350, Australia. mailto:aroberts@usq.edu.au11 Introduction 21 IntroductionThe characterisation of spatial distributions in terms of fractal concepts [12,3] is becoming increasingly important. In particular, many distributionsin nature are found to have the characteristics of a multi-fractal [6, 5, 14]:among many examples are galaxy clustering [1, 13], strange attractors [15],fluid turbulence [17], percolation [7], the shapes of neurons [9, 8], and plantdistributions [2] and shapes [10].In application, methods for estimating fractal dimensions are often unre-liable. One source of error lies in largely unknown biases introduced by thefinite size of data sets, addressed by Grassberger [4], and in the associatedfinite range of length-scales inherent in gathered data. In situations wherethousands or tens of thousands of data points are known such biases maybe minor; however, in some interesting problems, for example in the spatialclustering of underwater plants [2], only of the order of 100 data points areknown and confidence in the fractal characterisation may be misplaced. Weneed to know more about factors that cause errors in dimension estimates.Section 2 discusses the sensitivity of the multiplicative multi-fractal pro-cess to regions of very low probability (measure). Since such regions onlyrarely contribute a data point, an experimental sample cannot discern thembut such regions do affect the generalised dimensions. Hence I argue thatthe determination from experimental data of generalised dimensions, Dq, fornon-positive q is meaningless; for 0 < q < 1 computations are very sensitiveto the sample; and thus the most robust fractal dimension is the informationdimension D1. The argument is supported in Section 3 by a maximum like-lihood method [16] of estimating the multi-fractal properties of a data set.The method shows the enormous sensitivity of Dq for negative q. In contrastthe information dimension is reliably estimated.2 Poor conditioning of generalised dimensionsof negative orderFor example, consider the Hausdorff dimension, D0, of multifractals gener-ated by two different ternary multiplicative process.• Consider first the process shown in Figure 1(a) where an interval isdivided into three thirds and the “mass” of the original interval isassigned as follows: a fraction f1 > 0 to the left third; a fractionTony Roberts, February 8, 20082 Poor conditioning of generalised dimensions of negative order 3ǫf2ǫf1 ǫ2 f2ǫf1ǫǫf 22f2f1f1f2f21f2f1f 22f2f1f1f2f21f2f1(b)(a)Figure 1: schematic diagram of the first few stages in the multiplicative multi-fractal process to illustrate the sensitivity of the Hausdorff dimension D0 withrespect to low density regions, (b), as a perturbation of the same process withzero density regions, (a).f2 = 1 − f1 > 0 to the right third; and none to the middle third.Repeat this subdivision recursively. This generates a multiplicativemultifractal whose Hausdorff dimension of D0 = log3 2 = 0.6309 isprecisely the same as the Cantor set because there is no “mass” in themiddle thirds.• Conversely, and perversely, consider the process shown in Figure 1(b)where for some small ǫ the “mass” is assigned as follows: a fractionf1 > 0 is assigned to the left third; a fraction f2 > 0 is assignedto the rightmost third; and a small fraction ǫ > 0 is assigned to themiddle third (such that f1 + f2 + ǫ = 1). Repeat recursively. Thisgenerates a multiplicative multi-fractal whose Hausdorff dimension isD0 = 1 because there is “mass” everywhere along the whole interval!Although the vast bulk of the “mass” can be covered by 2n intervals oflength 3−n, we definitely do need 3n intervals in order to ensure coverageof the thinly spread “mass” that fills most of the original interval.The importance of this for the analysis of an experimental data set of N sam-pled points is that one cannot tell the difference from the data between thesetwo multi-fractal generating processes for an ǫ = o(1/N). Thus one cannotestimate the Hausdorff dimension D0 with any accuracy since either answer,0.6309 or 1 could be correct.Tony Roberts, February 8, 20082 Poor conditioning of generalised dimensions of negative order 4Similar reasoning applies to generalised dimensions with negative q. El-ementary arguments give that the generalised dimensions [6] of the multi-fractal generated by the second process above areDq ={−1q−1log3 [fq1 + fq2 + ǫq] if q 6= 1 ,− [f1 log3 f1 + f2 log3 f2 + ǫ log3 ǫ] if q = 1 .(1)It is readily appreciated that for negative order q and small ǫ, the term ǫq in-side the logarithm dominates the evaluation of the generalised dimension Dq.Hence, all generalised dimensions for negative q are also extremely sensitiveto small ǫ. In a data set obtained from experiments, one cannot expect todistinguish between zero ǫ and small non-zero ǫ = o(1/N), and yet the gen-eralised exponents and multi-fractal spectrum are markedly different. SeeFigure 2 which plots the generalised dimensions for f1 ≈ 1/4, f2 ≈ 3/4 andvarious small ǫ.We can be more precise about the sensitivity to low density regions bycomputing the derivative of Dq with respect to ǫ. For definiteness, supposef1 = φ1(1− ǫ) and f2 = φ2(1− ǫ). Then∂Dq∂ǫ=−qq − 1ǫq−1 − (φq1 + φq2) (1− ǫ)q−1log 3 [ǫq + (φq1 + φq2) (1− ǫ)q]. (2)For small, but non-zero, ǫ→ 0 this asymptotes to∂Dq∂ǫ∼1log 3qq−1if 1 < q ,q(1−q)(φq1+φq2)ǫq−1 if 0 < q < 1 ,q1−qǫ−1 if q < 0 .(3)This derivative is unbounded as ǫ → 0 for q < 1, and so any computationof Dq is only robust if q ≥ 1.The reason for this aberrant behaviour is clear. With a finite numberof data points, it is impossible to tell the difference between truly emptyspace and space which is visited so rarely that no data point happens to fallwithin it. That is, one cannot tell the difference between empty space andspace that should be filled in with very low probability. These differencesdramatically affect the generalised dimensions Dq for q < 1. Thus for anyexperimental data set:• estimating Dq for q ≤ 0 is nonsense (including the Hausdorff dimen-sion);• estimates of Dq for small positive q are sensitive; andTony Roberts, February 8, 20082 Poor conditioning of generalised dimensions of negative order 5-4 -3 -2 -1 0 1 2 3 4 500.511.522.533.5 q DqFigure 2: multi-fractal generalised dimensions Dq for the ternary multi-fractal process with f1 = (1 − ǫ)/4, f2 = (1 − ǫ)3/4 and ǫ = 0 (solid),0.01 (dashed) and 0.05 (dotted). This figure shows that Dq for negative or-der q is extraordinarily sensitive to small influences: the curve of smaller ǫ isthe most changed.• I only recommend the reporting of dimensions Dq for q ≥ 1 as beingrobust.Out of all the generalised dimensions for order q ≥ 1, D1 is most represen-tative of the fractal as a whole. For large order q, the computation of Dq isdetermined only by the very “densest” regions of the multi-fractal and so isnot representative of the whole fractal. In the above multiplicative process,Dq ∼ − log3 max(f1, f2, ǫ) as q →∞ ,showing that the large q behaviour is dictated by the one parameter of theprocess that determines the character of the very densest clusters in the frac-tal. The very dense clusters occur rarely in the fractal; they have low fractaldimension as seen in the low f value typically associated with low values of αTony Roberts, February 8, 20083 Fractal dimensions unbiased by finite size of data sets 6in the multi-fractal spectrum. Because of this rareness, the computation fromexperimental data of Dq for large positive order q is unreliable. Then, con-versely, the information dimension weights the data most uniformly, and so“knows” most about the fractal, without being overly sensitive to the possi-ble occurrence of regions of very low probability. The information dimensionseems most informative.3 Fractal dimensions unbiased by finite sizeof data setsCronin & Roberts [16] proposed a novel method to eliminate biases, causedby finite sized data sets, in determining the multi-fractal properties of a givendata set. Jelenik et al. [9, 8] used this method to explore the shape of neuroncells. The method compares characteristics of the inter-point distances in thedata set with those of artificially generated multi-fractals. By maximisingthe likelihood that the characteristics are the same we model the multi-fractal nature of the data by the parameters of the artificial multi-fractal.By searching among artificial multi-fractals with precisely the same numberof sample points as in the data, we anticipate that biases due to the finitesample size will be statistically the same in the data and in the artificial multi-fractals; hence predictions based upon the fitted multi-fractal parametersshould be unbiased by the finite sample size.The method also appears to give a reliable indication of the error in theestimates—a very desirable feature as also noted by Judd & Mees [11]. Mostimportantly for this paper, I generate finite size data sets with specific pa-rameters for the following specific multiplicative multi-fractal process. Givenparameters ρ ∈ [0, 0.5] and φ ∈ [0, 0.5] a binary multiplicative multi-fractalis generated by the recursive procedure of dividing each interval into twohalves, then assigning a fraction φ of the points in the interval to a ran-dom sub-interval of length ρ in the left half, and the complementary fractionφ′ = 1 − φ to a random sub-interval of length ρ in the right half. Such aprocess has generalised dimensionDq =log (φq + φ′q)(q − 1) log ρ, (4)and a multi-fractal spectrum f(α) [5, §4] given parameterically in terms of0 < ξ < 1 and ξ′ = 1− ξ asf =ξ log ξ + ξ′ log ξ′log ρ, α =ξ logφ+ ξ′ logφ′log ρ. (5)Tony Roberts, February 8, 2008References 7Here I chose ρ = 1/3 and φ = 1/4 and sample the process with N = 100; sucha multi-fractal forms a finite data set whose parameters we need to estimatefrom the sample.As explained in [16], we analyse such a sample by probing it with exactlythe same multiplicative multi-fractal process, and seek the best fit parame-ters. Here the resulting estimate of the original parameters is then in erroronly due to the finite size of the sample of the original multi-fractal process.Because we fit the data with a process which we know includes the one thatgenerated the data (a luxury rare in practise), there is no other error. Thusthe spread in errors that we see is characteristic of only the errors induced bya finite sized sample, nothing else. In particular, observe that the deductionsof the preceding section are indeed appropriate.I repeat the sampling of the multi-fractal followed by a maximum likeli-hood estimate of the parameters 16 times. Figure 3 plots the estimates ofthe parameters. Observe that the whole sampling and estimation processappears unbiased in that the mean of the predictions is reasonably close tothe correct values of the parameters.Ultimately, experimenters want to examine multi-fractal properties of thedata. Here these will be determined from the parameters (ρ, φ) of the best fitmulti-fractal substituted into analytic expressions such as (4) and (5). Foreach of the 16 realisations and their best-fit estimates, I plot the correspond-ing predicted generalised dimensions Dq in Figure 4. (The correspondinggraphs of the multi-fractal spectra f(α) are plotted in Figure 7 of [16] alongwith the true f(α) curve.) Observe that the predicted dimensions for posi-tive q (low α) are quite good for all realisations, especially near the informa-tion dimension, D1. However, predicted dimensions for negative q (high α)are very poor; this is also the case for the Hausdorff dimension D0 (the max-imum of the f(α) curve). The negative q predictions are poor despite thefitting process “knowing” that there are no very low probability regions inthis artificial process. In general applications one cannot know this and Iexpect the negative q (large α) predictions to be significantly worse. Thesenumerical results convincingly support the arguments of the preceding sec-tion that we should use the information dimension, not the Hausdorff.References[1] S. Borgani, G. Murante, A. Provenzale, and R. Valdarnini. Multifractalanalysis of the galaxy distribution: reliability of results from finite datasets. Phys Rev E, 47:3879–3888, 1993.Tony Roberts, February 8, 2008References 8[2] L. M. Emmerson and A. J. Roberts. Fractal and multi-fractal patternsof seaweed settlement. preprint, 1995.[3] J. Feder. Fractals. Plenum Press, 1988.[4] P. Grassberger. Finite sample corrections to entropy and dimensionestimates. Phys Lett A, 128:369–373, 1988.[5] T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I.Shraiman. Fractal measures and their singularities: the characterizationof strange sets. Phys Rev A, 33:1141–1151, 1986.[6] H. G. E. Hentschel and I. Procaccia. The infinite number of generalizeddimensions of fractals and strange attractors. Physica D, 8:435, 1983.[7] M. B. Isichenko. Percolation, statistical topography, and transport inrandom media. Rev. Mod. Phys, 64(4):961–1043, October 1992.[8] H. F. Jelinek, D. J. Cornforth, A. J. Roberts, G. Landini, P. Bourke, andA. Iorio. Image processing of finite size rat retinal ganglion cells usingmultifractal and local connected fractal analysis. In 17th AustralianJoint Conference on Artificial Intelligence, volume 3339 of Lecture Notesin Computer Science, pages 961–966. Springer–Verlag Heidelberg, 2004.[9] H. F. Jelinek, A. J. Roberts, and L. Peichl. Multifractal analysis: a pilotstudy using rat ganglion cells. Proc. Austr. Neurosci. Soc., 13:152, 2002.[10] C. L. Jones. 2-d wavelet packet analysis of structural self-organisationand morphogenic regulation in filamentous fungal colonies. Technical re-port, [http://www.swin.edu.au/chem/bio/cs96/camjones.htm], May1996.[11] K. Judd and A. I. Mees. Estimating dimensions with confidence. Tech-nical report, University of Western Australia, 1991.[12] B. B. Mandelbrot. Fractals: Form chance and dimension. J. FluidMech., 92:206–208, 1979.[13] V. J. Martinez. Fractal aspects of galaxy clustering. In Heck and Per-dang, editors, Applying fractals in astronomy, Lect Notes in Phys m3.Springer–Verlag, 1991.[14] G. Paladin and A. Vulpiani. Anomalous scaling laws in multifractalobjects. Phys Rep, 156:148–225, 1987.Tony Roberts, February 8, 2008References 9[15] I. Procaccia. Universal properties of dynamically complex systems: theorganisation of chaos. Nature, 333:618–623, 1988. 16th June.[16] A. J. Roberts and A. Cronin. Unbiased estimation of multi-fractal di-mensions of finite data sets. Physica A, 233:867–878, 1996.[17] K. R. Sreenivasan. Fractals and multifractals in fluid turbulence. Annu.Rev. Fluid Mech., 23:539–600, 1991.Tony Roberts, February 8, 2008References 100 0.1 0.2 0.3 0.4 0.500.050.10.150.20.250.30.350.40.450.5ρφFigure 3: predicted multi-fractal parameters (ρ, φ), indicated by ◦’s, fromthe maximum likelihood match to an ensemble of 16 different realisations,each of N = 100 data points, of a binary multiplicative multi-fractal withparameters ρ = 1/3 and φ = 1/4, indicated by +. The mean location of thepredictions is indicated by a ×.Tony Roberts, February 8, 2008References 11-4 -3 -2 -1 0 1 2 3 4 50.20.40.60.811.21.4qDqFigure 4: ensemble of multi-fractal generalised dimensions Dq, dotted, foreach of the predictions plotted in Figure 3 made from samples of N = 100data points. For comparison the generalised dimensions for the actual fractalis plotted as the solid line. Observe the good estimation near the informationdimension, but the large errors for negative order q.Tony Roberts, February 8, 2008

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