Fractal analysis: Annual rainfall in Chennai

The annual rainfall data of Chennai is analyzed using the Fractal Construction Technique. According to Mandelbrot the dimension of any line including nautical lines may not be Euclidean but Fractional, Mandelbrot, 1982. This fractional dimension leads to a repetitive appearance of any pattern. Climate which is usually periodic by nature can be analyzed through this technique. Efforts are on to search the fractal geometry of climate and to predict its periodicity on different temporal scales. This paper estimates the various parameters like Lyapunov exponent, Maximum Lyapunov characteristic exponent, Lyapunov time, Kaplan-Yorke dimension for the annual rainfall of Chennai.

Estimates of heat kernels of non-symmetric Lévy processes

We investigate densities of vaguely continuous convolution semigroups of probability measures on ℝ d {{\mathbb{R}^{d}}} . First, we provide results that give upper estimates in a situation when the corresponding jump measure is allowed to be highly non-symmetric. Further, we prove upper estimates of the density and its derivatives if the jump measure compares with an isotropic unimodal measure and the characteristic exponent satisfies a certain scaling condition. Lower estimates are discussed in view of a recent development in that direction, and in such a way to complement upper estimates. We apply all those results to establish precise estimates of densities of non-symmetric Lévy processes.

Inference On Streamflow Predictability Using Horizontal Visibility Graph Based Networks

<p>Streamflow is a dynamical process that integrates water movement in space and time within basin boundaries. The authors characterize the dynamics associated with streamflow time series data from about seventy-one U.S. Geological Survey (USGS) stream-gauge stations in the state of Iowa. They employ a novel approach called visibility graph (VG). It uses the concept of mapping time series into complex networks to investigate the time evolutionary behavior of dynamical system. The authors focus on a simple variant of VG algorithm called horizontal visibility graph (HVG). The tracking of dynamics and hence, the predictability of streamflow processes, are carried out by extracting two key pieces of information called characteristic exponent, λ of degree distribution and global clustering coefficient, GC pertaining to HVG derived network. The authors use these two measures to identify whether streamflow process has its origin in random or chaotic processes. They show that the characterization of streamflow dynamics is sensitive to data attributes. Through a systematic and comprehensive analysis, the authors illustrate that streamflow dynamics characterization is sensitive to the normalization, and the time-scale of streamflow time-series. At daily scale, streamflow at all stations used in the analysis, reveals randomness with strong spatial scale (basin size) dependence. This has implications for predictability of streamflow and floods. The authors demonstrate that dynamics transition through potentially chaotic to randomly correlated process as the averaging time-scale increases. Finally, the temporal trends of λ and GC are statistically significant at about 40% of the total number of stations analyzed. Attributing this trend to factors such as changing climate or land use requires further research.</p>

Information Gain in Event Space Reflects Chance and Necessity Components of an Event

Information flow for occurrences in phase space can be assessed through the application of the Lyapunov characteristic exponent (multiplicative ergodic theorem), which is positive for non-linear systems that act as information sources and is negative for events that constitute information sinks. Attempts to unify the reversible descriptions of dynamics with the irreversible descriptions of thermodynamics have replaced phase space models with event space models. The introduction of operators for time and entropy in lieu of traditional trajectories has consequently limited—to eigenvectors and eigenvalues—the extent of knowable details about systems governed by such depictions. In this setting, a modified Lyapunov characteristic exponent for vector spaces can be used as a descriptor for the evolution of information, which is reflective of the associated extent of undetermined features. This novel application of the multiplicative ergodic theorem leads directly to the formulation of a dimension that is a measure for the information gain attributable to the occurrence. Thus, it provides a readout for the magnitudes of chance and necessity that contribute to an event. Related algorithms express a unification of information content, degree of randomness, and complexity (fractal dimension) in event space.

The Entrance Law of the Excursion Measure of the Reflected Process for Some Classes of Lévy Processes

We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.

A New Feature Extraction Method for Bearing Faults in Impulsive Noise Using Fractional Lower-Order Statistics

According to the performance degradation problem of feature extraction from higher-order statistics in the context of alpha-stable noise, a new feature extraction method is proposed. Firstly, the nonstationary vibration signal of rolling bearings is decomposed into several product functions by LMD to realize signal stability. Then, the distribution properties of product functions in the time domain are discussed by the comparison of heavy tails and characteristic exponent estimation. Fractional lower-order p-function optimization is obtained by the calculation of the distance ratio based on K-means algorithms. Finally, a fault feature dataset is established by the optimal FLOS and lower-dimensional mapping matrix of covariation to accurately and intuitively describe various bearing faults. Since the alpha-stable noise is effectively suppressed and state described precisely, the presented method has shown better performance than the traditional methods in bearing experiments via fractional lower-order feature extraction.

SINH-ACCELERATION: EFFICIENT EVALUATION OF PROBABILITY DISTRIBUTIONS, OPTION PRICING, AND MONTE CARLO SIMULATIONS

Characteristic functions of several popular classes of distributions and processes admit analytic continuation into unions of strips and open coni around [Formula: see text]. The Fourier transform techniques reduce calculation of probability distributions and option prices in the evaluation of integrals whose integrands are analytic in domains enjoying these properties. In the paper, we suggest to use changes of variables of the form [Formula: see text] and the simplified trapezoid rule to evaluate the integrals accurately and fast. We formulate the general scheme, and apply the scheme for calculation probability distributions and pricing European options in Lévy models, the Heston model, the CIR model, and a Lévy model with the CIR-subordinator. We outline applications to fast and accurate calibration procedures and Monte Carlo simulations in Lévy models, regime switching Lévy models that can account for stochastic drift, volatility and skewness, the Heston model, other affine models and quadratic term structure models. For calculation of quantiles in the tails using the Newton or bisection method, it suffices to precalculate several hundreds of values of the characteristic exponent at points on an appropriate grid (conformal principal components) and use these values in formulas for cpdf and pdf.