A quasi-likelihood method for fractal-dimension estimation (original) (raw)
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The inverse power law distributions are used as the model for fractal probability distributions that have fractional exponents (λ) and such that the transformation X(1-λ) is uniformly distributed. The paper examines aspects of point estimation and tests of hypotheses about statistical fractals. It is shown that the maximum likelihood estimator of the fractional dimension λ is uniformly minimum variance unbiased estimator (UMVUE) using the Lehmann-Scheffe's theorem and also that the likelihood ratio test for H0: λ= λ0 is uniformly most powerful (UMPT) by the Neymann-Pearson Lemma. The paper likewise explains that the test for equality of two medians of two fractal distributions is equivalent to a test for the equality of the fractal dimensions. In fact, the result is generalized to the test for the equality of two αth quantiles of two fractal distributions. The test for the equality of two fractal distributions is compared with the Mann-Whitney U test and with the Student's t...
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scale-free measure of irregularity or ‘roughness’ of patterns. If ( ) { } d t : t Z R ∈ is a stationary Gaussian stochastic process, whose realization is a d-dimensional geometric structure in 1 + R d (i.e. a curve or profile if d=1 and a surface if d=2), the fractal dimension is always 1 + ≤ < d D d . If Z(t) is very smooth we have a fractal dimension close to d, while D approaches d+1 if the structure is extremely rough. A variety of methods have been proposed to determine the fractal dimension. Some of the most well known methods are: box-counting method, walking-dividers method, spectral method and variogram method. Each of them requires the estimation of a slope coefficient in a log-linear regression based on m points near the origin (Taylor and Taylor, 1991; Kent and Wood, 1997). In any case, the problem is that an estimate of the fractal dimension is difficult to produce directly, so an alternative approach is to calculate an estimate of a quantity called fractal index, δ,...
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The multifractal concept was introduced in the 1980s by Mandelbrot. This theory arose from the analysis of complex and=or discontinuous objects. In this study, we analyzed the data scatter obtained by a modiÿed box-counting method. Considering the curved shape of the data scatter, it is noticeable that there is more than one slope corresponding to di erent fractal behavior of an object. In this work, to discriminate di erent fractal dimensions from data scatter obtained by box counting, we suggest a rigorous selection of data points. The results show that large "'s usually characterize the embedding surface of the whole object and that small "'s approximate the dimension of the substructure for discontinuous objects. They also show that a dimension can be associated with a density distribution of singularities.
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Pure and Applied Geophysics, 1998
Within the fractal approach to studying the distribution of seismic event locations, different fractal dimension definitions and estimation algorithms are in use. Although one expects that for the same data set, values of different dimensions will be different, it is usually anticipated that the direction of fractal dimension changes among different data sets will be the same for every fractal dimension. Mutual relations between the three most popular fractal dimensions, namely: the capacity, cluster and correlation dimensions, have been investigated in the present work. The studies were performed on the Monte Carlo generated data sets. The analysis has shown that dependence of the fractal dimensions on epicenter distribution, and relations among the fractal dimensions, are complex and variable. Neither values nor even inequalities among dimension estimates are preserved when different fractal dimensions are used. The correlation and the capacity dimensions seem to be good tools to trace collinear tendencies of eipicenters while the cluster dimension is more appropriate to studying uniform clustering of points.
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The natural world is recognized to be fractal: from the growth of a leaf to how the trees propagate to form a forest, the neural networks, the DNA and the galaxies in space. The patterns and geometry that nature creates seem familiar and predetermined; actually it is random and unpredictable. It is this characteristic that made fractals a convenient method for such studies. Insights into the unpredictability could be a key in understanding the natural random events, like earthquakes, typhoons and other more subtle, natural occurrence like growths and cell developments. Data in different studies have always been thought of as normally distributed; however, recent investigations found that most of these statistical data are non-normal, and in a lot of cases are found to be fractal. According to Padua et al 2013, statistical fractal observations are random observations that possess stochastically self-similar patterns at various scale. In this light, this paper aims to illustrate the pervasiveness of statistical fractal observations in real-life by examining old data sets that used to be modeled in the framework of normal distribution theory. Samples of such observations will be presented in this paper.
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Within beds of blue mussel (Mytilus edulis L.), individuals are aggregated into small patches, which in turn are incorporated into bigger patches, revealing a complex hierarchy of spatial structure. The present study was done to find the different scales of variation in the distribution of mussel biomass, and to describe the spatial heterogeneity on these scales. The three approaches compared for this purpose were fractal analysis, spatial autocorrelation and hierarchical (or nested) analysis of variances (ANOVA). The complexity (i.e. patchiness) of mussel aggregations was described with fractal dimension, calculated with the semivariogram method. Three intertidal mussel beds were studied on the west coast of Sweden. The distribution of wet biomass was studied along transects up to 128 m. The average biomasses of blue mussels on the three mussel beds were 1825±210, 179±21 and 576±66 g per0.1 m2, respectively, and the fractal dimensions of the mussel distribution were 1.726±0.010, 1.842±0.014 and 1.939±0.029 on transects 1–3, respectively. Distributions of mussels revealed multiscaling behaviour. The fractal dimension significantly changed twice on different scales on the first bed (thus showing three scaling regions), the second and third beds revealed two and three scaling regions, respectively. High fractal dimension was followed by significant spatial autocorrelation on smaller scales. The fractal analysis detects the multiple scaling regions of spatial variance even when the spatial structure may not be distinguished significantly by conventional statistical inference. The study shows that the fractal analysis, the spatial autocorrelation analysis and the hierarchical ANOVA give complementary information about the spatial variability in mussel populations.