Fractals

This paper introduces a multiscale analysis based on optimal piecewise linear approximations of time series. An optimality criterion is formulated, and on its base, a computationally effective algorithm is constructed for decomposition of a time series into a hierarchy of trends (local linear approximations) at different scales. The top of the hierarchy is the global linear approximation over the whole observational interval, the bottom is the original time series. Each internal level of the hierarchy corresponds to a piecewise linear approximation of analyzed series. Possible applications of the introduced Multiscale Trend Analysis (MTA) go far beyond the linear interpolation problem: This paper develops and illustrates methods of self-affine, hierarchical, and correlation analyses of time series.

We develop an algorithm to construct new self-similar space-filling packings of spheres. Each topologically different configuration is characterized by its own fractal dimension. We also find the first bichromatic packing known up to now.

A sandpile model on the square lattice with periodic boundary conditions is studied as a problem of variable density of grains per site. We define an order parameter and we show the existence of a critical density. This model is the closed version of the deterministic BTW model and belongs to the "Fixed Energy Sandpiles" class. An algorithm identifying the periodic states of the system permits us to have a good estimation of the order parameter, which is free from transient effects. The universality of the model with respect to two different forms of oriented flows is studied and compared with that of the isotropic flow. We find two universality classes characterized by different critical exponents. The Manhattan flow appears to belong to the same universality class as the isotropic flow. However, the directed flow gives a completely different critical behavior. The complexity of the model for densities larger than the critical is also studied. It is observed that, both the Manhattan model and the isotropic model have an interesting structure characterized by several plateaus. The right edges of these plateaus are characterized by a sudden increase of the order parameter, giving rise to new critical points. The directed flow model has a simple structure without plateaus.

Infantile hypertrophic pyloric stenosis (IHPS) is a common surgical condition of unknown etiology. The oral mucosal vascular networks of IHPS patients (n=25) and their unaffected parents showed lower blood vessel-free areas, as well as higher box-counting dimensions at two box size scales [D(1–46), D(1–15)], and relative Lempel-Ziv values (P<0.000001), as compared to those of gender- and age-matched controls. These findings may provide a useful phenotypical marker for identifying couples potentially at risk for the birth of an affected infant, while supporting the importance of a genetic component in this condition.

We propose in this work a dynamic speckle descriptor based on a Hurst wavelet estimator. The dynamic speckle or biospeckle is a phenomenon produced by laser illumination of active materials, such as biological tissue or the drying process of paint. Dynamic speckle interferometry is a useful technique for assessing the time evolution of surfaces as also to segment the loci of different activity in living tissues. Considering previous biospeckle characterization based on the autocorrelation function and its relation with the Hurst coefficient, a wavelet-based estimator is proposed as a feature extraction of the dynamic speckle characteristic. Encouraging results of the descriptor performance are obtained via three different experiments: a time history of speckle pattern applied to the drying of painting, segmenting regions in whole field image applied to the viability test of a corn seed and also to the bruising in fruits.

In studying the scale invariance of an empirical time series a twofold problem arises: it is necessary to test the series for self-similarity and, once passed such a test, the goal becomes to estimate the parameter H₀ of self-similarity. The estimation is therefore correct only if the sequence is truly self-similar but in general this is just assumed and not tested in advance. In this paper we suggest a solution for this problem. Given the process {X(t),t∈T}, we propose a new test based on the diameter δ of the space of the rescaled probability distribution functions of X(t). Two necessary conditions are deduced which contribute to discriminate self-similar processes and a closed formula is provided for the diameter of the fractional Brownian motion (fBm). Furthermore, by properly choosing the distance function, we reduce the measure of self-similarity to the Smirnov statistics when the one-dimensional distributions of X(t) are considered. This permits the application of the well-known two-sided test due to Kolmogorov and Smirnov in order to evaluate the statistical significance of the diameter δ, even in the case of strongly dependent sequences. As a consequence, our approach both tests the series for self-similarity and provides an estimate of the self-similarity parameter.

An analysis of multifractal characteristics of the potential time series of electrochemical noise generated during surface corrosion is performed. The generalized Hurst exponent, called H_q, is computed for potential fluctuations induced for steel corroding by a sulphate reducing bacterial consortia. The results indicate evidence of multifractal behavior in the sense that the qth-order (roughness) Hurst exponent H_q varies with changes in q. This multifractal structure is corroborated by means of microscopy inspection of the corroded surface.