Fractals

A new class of scaling random processes is introduced. The processes are obtained as sums of non-negative pulses and are generated through a recursive scheme that proceeds from coarser to finer scales. The representation at any given scale has the form of a filtered doubly-stochastic Poisson point process whose intensity depends linearly on the process at the immediately coarser scale. The filter functions (the pulses) at different scales are related through contractive affine transformations. If the transformations are isotropic in physical space, then the spectral density of the infinitely iterated process, , is found to decay as along any radial direction, where the exponent c depends on the dimension of physical space and the parameters of the affine transformation. Various extensions of the basic scheme are considered, including pulses of random shape and anisotropic affine relations among the pulses at different scales. Processes of this type have several applications, including the representation of width functions in river basin hydrology.

We propose a new method to describe scaling behavior of time series. We introduce an extension of extreme values. Using these extreme values determined by a scale, we define some functions. Moreover, using these functions, we can measure a kind of fractal dimension — fold dimension. In financial high frequency data, observations can occur at varying time intervals. Using these functions, we can analyze non-equidistant data without interpolation or evenly sampling. Further, the problem of choosing the appropriate time scale is avoided. Lastly, these functions are related to a viewpoint of investor whose transaction costs coincide with the spread.

Chromatin appearance in breast epithelial cells has been shown to have fractal properties, and detrended fluctuation analysis (DFA) is an effective method for characterizing the scaling in non-stationary fractal signals in terms of a scaling exponent. This study examines the use of DFA for the characterization of chromatin appearance in breast epithelial cells. Images of nuclei representative of fine-needle aspiration biopsies of the breast are characterized in terms of the scaling exponent for 19 patients with benign lesions and 22 patients with invasive ductal carcinoma. Characterizing patients by the standard deviations of the values of the scaling exponent for their representative nuclei, a statistically significant difference is noted between benign and malignant cases. This reflects that malignancies exhibit less variability in chromatin roughness than do benign cases. Previous logistic regression models for the diagnosis of breast epithelial cell lesions are improved upon by incorporating the standard deviation of the scaling exponent. Using leave-one-out cross-validation, the best logistic regression classifiers demonstrate a sensitivity of 95% and a specificity of 100%. A combination of DFA and lacunarity analysis is seen to provide the best approach to characterizing chromatin in breast epithelial cell nuclei.

Empirical mass estimates for the enigmatic galactic dark matter objects (GDMOs), which account for most of the mass of the universe, but remain unidentified, are compared with the definitive predictions of a discrete fractal model of the cosmos. Trends in the reported mass estimates are identified and uncertainties are discussed. Intriguing similarities among various stellar and dark matter mass functions are noted, and a possible identification of the galactic dark matter objects is suggested.

In the Mandelbrot set, the bulbs attached directly to the main cardioid are called the p/q-bulbs. The reason for this is that the largest component of the interior of these bulbs consists of c-values for which the quadratic function Q_c(z) = z² + c admits an attracting cycle with rotation number p/q. In this paper, we give a geometric method to read off p/q from the geometry of the antenna attached to the bulb.

Following the Mandelbrot's theory of fractals, many shapes and phenomena in nature have been suggested to be fractal. Even animal behavior and human physiological responses can also be represented as fractal. Here, we show the evidence that it is possible to apply the concept of fractals even to the facial expression recognition, which is one of the most important parts of human recognition. Rating data derived from judging morphed facial images were represented in the two-dimensional psychological space by multidimensional scaling of four different scales. The resultant perimeter of the structure of the emotion circumplex was fluctuated and was judged to have a fractal dimension of 1.18. The smaller the unit of measurement, the longer the length of the perimeter of the circumplex. In this study, we provide interdisciplinarily important evidence of fractality through its application to facial expression recognition.

In this paper, a method of construction of fractal interpolation functions (FIF) on random grids in ℝ² is examined.

We extend a theorem of Falconer to give a complete result on the dimension and Lebesgue measure of typical self-similar attractors on the line. Further, we present some examples to point out that in higher dimensions, one cannot expect the same result in this generality.

Fractal analysis has been performed to investigate the aftershock activity that followed the 1994 M_W 6.7 Northridge earthquake (Southern California). The sequence analyzed is very long and the obtained fractal estimates seem very reliable. High degree of clustering is revealed by the calculation of the coefficient of variation C_V ≈ 3.2, and the autocorrelation function shows the presence of dependence among the inter-event times. The obtained value of the capacity dimension D ≈ 0.05 demonstrates that the temporal distribution of the aftershocks is strongly clusterized. Furthermore, we calculated the power-law exponent of the power spectral density of the seismic process, by means of the Allan Factor method and the count-based Periodogram method; the very high obtained values α_AF ≈ 1.53 and α_S ≈ 1.43 reveal the presence of high correlated time structures in the process.

We prove for a subclass of recurrent iterated function systems (also called graph-directed iterated function systems) that the boundary of their attractor is again the attractor of a recurrent IFS. Our method is constructive and permits computation of the Hausdorff dimension of the attractor and its boundary.

Successful usage of a large family of deterministically generated measures to model complex nonlinear phenomena (e.g. rainfall, turbulence and groundwater contaminant transport) has been reported recently.^1–7 As these measures, generated as derived distributions of multifractal measures via fractal interpolating functions (FIF), i.e. the fractal-multifractal (FM) approach, have been found to share the inherent character of natural sets, the present study further investigates their dynamical (chaotic or stochastic) properties. It is shown, through a variety of examples and via the use of power spectrum, mass exponents and false nearest neighbors in the state-space, that the FM approach indeed generates deterministic measures whose behavior (depending on their parameters) may be classified as low-dimensional and chaotic or as high-dimensional and stochastic. These results suggest the general suitability of the FM approach for understanding and modeling nonlinear natural phenomena.

Human posture is the result of a complex control system. The joint output of several physiological — most likely nonlinearly interacting — processes leads to constant correctional movements which enable humans to stand upright. These correctional body movements reflect features of the underlying control mechanisms and have recently been shown to be multifractal processes. We analyze the movements of healthy quiet standing persons by means of wavelet-based multifractal spectra. We show that the functional form of these spectra depends on the mode of balance control. We compare the findings from a group of healthy persons to a group of patients suffering from balance disorders. As the main result we find that if balance control inputs, such as visual cues or tactile information are reduced, the dominant fractal exponent becomes smaller and the range of fractal exponents strongly narrows. We suggest to use a set of multifractal spectrum parameters as a "measure of complexity."

We analyze the extremes of stationary multifractal measures using large deviation theory. We consider various cases involving discrete multiplicative cascades: scalar or vector cascades with dependent or independent generators, bare or dressed measures, and marginal (single-point) or joint (multi-point) extremes. In each case, we obtain the scaling behavior of the probability of large deviations as the resolution of the cascade diverges. Existing rough exponential limits for scalar cascades are confirmed, whereas for other cases our scaling relationships differ from previously published results. For scalar cascades, we refine the rough limits by obtaining the asymptotic pre-factor to the exponential term. Based on these refined asymptotics, we propose a variant to the Probability Distribution/Multiple Scaling (PDMS) technique to estimate the co-dimension function c(γ).