An Improved Multifractal Box-Counting Algorithm, Virtual Phase Transitions, and Negative Dimensions

It is often taken for granted that the analysis of multifractals with fixed-size algorithms leads to faithful results although this is not the case in practice. We present a novel and self-consistent way to obtain from a partition sum Sq(l) the spectrum of singularities f(α) and its confidence intervals. With this tool we gain new insights into the intricacies of fixed-size algorithms and propose consequent improvements. We give a numerical analysis of the Hénon attractor which displays theoretical predictions of a phase transitionlike plateau. 05.45.+b, 47.53.+n, 47.10.+g Typeset using REVTEX 1 Following the seminal paper of Halsey et al. [1] on multifractals, a continuing surge of interest in multifractals has arisen. In the course of this developement fixed box-size algorithms have acquired a widespread popularity despite their poor performance. Some work has been done to demonstrate that the performance of fixed-size or box-counting algorithms (BCAs) is far from perfect [2], several natural limitations have often been pointed out [3–6], and many improvements have been proposed [7–12]. In this Letter, we first show a way how to make generalised dimension estimates more reliable and also how to arrive at reasonable error estimates, starting from an intermediate standard BCA (c.f. [13]) result. We then add to this new algorithm a more advanced counting procedure and demonstrate the efficient interplay of these techniques. Let μ be a probability measure defined on a (possibly fractal) support E ⊂ R. We are interested in the spectrum of singularities f(α) of (local) Hölder exponent α : α(x) = lim l→0 logμ(B(l,x)) log l , (1) where B(l,x) is the ball centered at x ∈ E with radius l [1,14,15,13]. In many cases, f(α) can be interpreted as the Hausdorff dimension of the set of x ∈ E with local Hölder exponent α. To these ends we cover E with a grid of boxes (hypercubes) B(l) = ∏d j=1[njl, (nj+1)l) of size l, where the nj are integers. We then calculate the q-th moment (or partition function) Sq(li) of μ for several li ∈ L = {l1, l2, ...ln} and q ∈ Q = {q1, q2, ..., qn} by: Sq(li) = ∑ μ(B)6=0 (μ(B)) =< μ > . (2) Brackets <> indicate sample averages as usual. The generalized dimensions D(q) (for q 6= 1) and the scaling function τ(q) are defined by: τ(q) = D(q)(q − 1) = lim l→0 logSq(l) log l . (3) In practice one might hope that Sq scales with l : logSq(li) = τ(q) log(li) + c (4) 2 and so obtain τ(q) via a least squares line fit to the plot of logSq(li) against log li for all suitable li. Frequently, the Legendre transformation of τ(q) is considered [1] : α = ∂τ ∂q , f = qα− τ , (5) which shall not be the main subject here. Apart from obvious questions of how the resolution and finiteness of the sample and the presence of noise affect the algorithm, we find a number of flaws with the procedure itself. A fixed-size approach compensates the lack of knowledge about the fine structure of a fractal or about its generating process with taking the limit of infinitesimal length scales , equations (1),(3). Even if the thermodynamic formalism [1,16] is in theory capable of giving asymptotic averages of the moments of the probability distribution of local Hölder exponents, we have no measure for the accuracy of the scaling behaviour of a single average (eq. (4)), and thus of τ(q) for a given q. Furthermore, it has to remain open for which length scales li the assumed scaling behaviour is valid. The errors of the counting procedure are not quantifiable. In particular, errors that originate from boxes with spuriously small mass (clipping errors) for q < 0 have often been addressed [2,3,12]. An awkward but crucial question is, how well advised is the assumption of a Gaussian error distribution for the Sq(li) when there are systematic errors as well as large deviations present? Similarly, does the resulting standard deviation have any meaning at all? It has been pointed out [2,4] that the error obtained from the least-squares fit grossly underestimates the error present. We have to realize that we cannot in general make a priori geometric assumptions about inherent length scales or suitable covers for a set, and dubious assumptions about the convergence of the limit (3) do not overcome this. Although the fixed-size approach seems to be rather ad hoc at times, in the face of the complexity of the task, using BCAs is often unavoidable. The following algorithm is designed to add some trustworthiness to the highly volatile 3 results of standard BCAs. In practice, we find a much better description of eq. (4) to be: logSq(li) = τ(q) log li + C(li, q) . (6) The function C(li, q) was termed by Cutler [4,5] the wandering intercept. Our development depends on two assumptions about the wandering intercept: • for fixed l, C(l, q) is a slowly varying function of q. • for ‘most’ q we find : C(l, q) = F (l)×G(q) . (7) These assumptions amount to saying that each moment “knows” about the corresponding deviations from the ideal scaling of moments for nearby q. The idea is to use this information to compensate for the adverse effects of these deviatons on the fitting procedure (Symmetric Scaling-error Compensation). Of course, it remains to be shown in each case that these assumptions hold. As we will see, even if this is not the case, one still can learn much about the sample. Inspired by a related method introduced by Benzi et al. [17] in the statistical description of fully developed turbulence, by using (6) and (7) we express logSq as a function of logSq′ instead of log l: logSq(li) = τ(q) τ(q′) logSq′(li) + F (li)(G(q)− τ(q) τ(q′) G(q)) . (8) We can now proceed to fit a line to eq.(8) to obtain the quotient τ(q)/τ(q). Although we still cannot assume that the error distribution is normal, in the absence of at least some of the systematic errors this deserves more credence than the standard procedure. To make best use of the mutual information contained in all moments, it is desirable to plot logSq as a function of as many logSq′ as possible. We choose qi ∈ Q, q1 < q2 < . . . < qmax such