Fractal dimension8/23/2023 Following the same construction, but randomly choosing the side of the old interval on which each new pair of intervals is placed, yields one of many “statistically self-similar” curves (F). Even with n = 10 (E), zooming in on the pinnacle of the curve by a factor of three yields an image visually indistinguishable from the largest magnification five times over, meaning the curve is approximately self-similar over a finite range of scales. The curve is thus self-similar with a similarity dimension of log(4)/log(3). The limit curve can be subdivided into four quarters, each an exact copy of the whole, scaled down by a factor of three. The Koch curve is the limit approached as n → ∞. The process is repeated for each newly created interval, yielding the second (C), third (D), and n th iterations. Each interval (A) is divided evenly into three and the middle section is replaced by the complementary two sides of an equilateral triangle (B). Construction of the Koch curve, following Falconer (2003). It can thus still be described by the same FD on average and we say that it exhibits “statistical self-similarity.” This example illustrates that “fractal” and “self-similar” are not synonyms: self-similarity, or statistical self-similarity, is just one mode of fractal regularity.įigure 1. The placement of its details at any scale is random, and can be highly variable from one implementation to the next, but the rate at which detail emerges as its image is magnified is the same as for the original Koch curve. An even less regular example is a random Koch curve (Figure 1F), which does not consist of scaled images of itself, but rather of stochastic variations on these. Although a Koch curve is too irregular to be well described by Euclidean measures, its self-similarity allows for a simple description of its scaling properties via this FD. For example, a Koch curve (see Figures 1A–E for construction) has infinite length, zero area, and a finite similarity (fractal) dimension of log(4)∕log(3). A FD will hold useful information on the geometry of such sets, while Euclidean measures may not. Our findings serve as a caution against applying FD under the assumption of statistical self-similarity without rigorously evaluating it first.įractal dimensions (FD) are metrics useful in characterizing the geometry of sets too irregular to be described in more classical ways that nevertheless exhibit sufficient fractal regularity ( Falconer, 2003). FD estimates did not characterize the scaling of our digitizations well: the scaling exponent was a function of scale. Our representations of coarse root system digitizations did not exhibit details over a sufficient range of scales to be considered statistically self-similar and informatively approximated as fractals, suggesting a lack of sufficient ramification of the coarse root systems for reiteration to be thought of as a dominant force in their development. Pattern search had higher initial computational cost but converged on lower error values more efficiently than the commonly employed brute force method. QE, due to both grid position and orientation, was a significant source of error in FD estimates, but pattern search provided an efficient means of minimizing it. The degree of statistical self-similarity was evaluated using linear regression residuals and local slope estimates. A pattern search algorithm was used to minimize QE by optimizing grid placement and its efficiency was compared to the brute force method. Coarse root systems of 36 shrubs were digitized in 3D and subjected to box-counts. The goals of this study were to characterize the effect of QE due to translation and rotation on FD estimates, to provide an efficient method of reducing QE, and to evaluate the assumption of statistical self-similarity of coarse root datasets typical of those used in recent trait studies. Previous studies either ignore QE or employ inefficient brute-force grid translations to reduce it. The box-counting procedure is also subject to error arising from arbitrary grid placement, known as quantization error (QE), which is strictly positive and varies as a function of scale, making it problematic for the procedure's slope estimation step. The vast majority of published studies fail to evaluate the assumption of statistical self-similarity, which underpins the validity of the procedure. 2Department of Ecology, Evolution and Natural Resources, Rutgers, The State University of New Jersey, New Brunswick, NJ, USAįractal dimension (FD), estimated by box-counting, is a metric used to characterize plant anatomical complexity or space-filling characteristic for a variety of purposes.1Saiers Lab, School of Forestry and Environmental Studies, Yale University, New Haven, CT, USA.
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