Pointwise Holder Exponent Research Articles

On fair entropy of the tent family

<p style='text-indent:20px;'>The notions of fair measure and fair entropy were introduced by Misiurewicz and Rodrigues [<xref ref-type="bibr" rid="b13">13</xref>] recently, and discussed in detail for piecewise monotone interval maps. In particular, they showed that the fair entropy <inline-formula><tex-math id="M1">\begin{document}$ h(a) $\end{document}</tex-math></inline-formula> of the tent map <inline-formula><tex-math id="M2">\begin{document}$ f_a $\end{document}</tex-math></inline-formula>, as a function of the parameter <inline-formula><tex-math id="M3">\begin{document}$ a = \exp(h_{top}(f_a)) $\end{document}</tex-math></inline-formula>, is continuous and strictly increasing on <inline-formula><tex-math id="M4">\begin{document}$ [\sqrt{2},2] $\end{document}</tex-math></inline-formula>. In this short note, we extend the last result and characterize regularity of the function <inline-formula><tex-math id="M5">\begin{document}$ h $\end{document}</tex-math></inline-formula> precisely. We prove that <inline-formula><tex-math id="M6">\begin{document}$ h $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M7">\begin{document}$ \frac{1}{2} $\end{document}</tex-math></inline-formula>-Hölder continuous on <inline-formula><tex-math id="M8">\begin{document}$ [\sqrt{2},2] $\end{document}</tex-math></inline-formula> and identify its best Hölder exponent on each subinterval of <inline-formula><tex-math id="M9">\begin{document}$ [\sqrt{2},2] $\end{document}</tex-math></inline-formula>. On the other hand, parallel to a recent result on topological entropy of the quadratic family due to Dobbs and Mihalache [<xref ref-type="bibr" rid="b7">7</xref>], we give a formula of pointwise Hölder exponents of <inline-formula><tex-math id="M10">\begin{document}$ h $\end{document}</tex-math></inline-formula> at parameters chosen in an explicitly constructed set of full measure. This formula particularly implies that the derivative of <inline-formula><tex-math id="M11">\begin{document}$ h $\end{document}</tex-math></inline-formula> vanishes almost everywhere.

Do Sentiment and Exchange Rate Share Memories? An Application of Multifractional Process Modeling

We use the notion of local Holder regularity to investigate the roughness of the pattern of EUR/USD exchange rate process as well as the corresponding investor sentiment dynamic process. Specifically, we use the pointwise Holder exponent to measure the local Holder regularity of a multifractional process. This application extends the increment ratio method to estimate the pointwise Holder exponent of multifractional Brownian motion function. Our testing and comparison results show that both the exchange rate and the sentiment proxy's local Holder regularities are time-varying and are more homogeneous during crises periods. More importantly, in contrast to the optimistic sentiment, the roughness of the pessimistic sentiment dynamic process displays only slightly higher similarity to that of EUR/USD exchange rate. These results suggest that the EUR/USD exchange rate process and the pessimistic sentiment dynamic process share a common pattern of smoothness, which implies that they share a similar memory pattern.

Multifractal Nature of Diesel Engine Rattle Noise in Vehicle

AbstractThe investigation results of the emission of noises accompanying the diesel engine powertrain running in unloaded condition at low or idle speed, are presented in the hereby paper. The multifractal nature of the tested signals was confirmed by means of two groups of parameters. The proposed local parameters are based on the distribution of the probability density of measures, segregated into subsets according to pointwise Holder exponents. The obtained spectra constitute a set of multifractal measures related to singularities representing the local scaling of measures in various points of the time series. Global parameters were defined on the basis of multifractal measures of the parametrised Renyi’s entropy. These are properties treating summarily the whole spectrum of the emitted noise and reflecting changes of the vibroacoustic energy levels generated by noise sources. The tests encompassing simultaneous application of both groups of parameters confirmed their efficiency in the comparative analysis as well as in the subjective assessment of the noise level generated in the drive system of vehicles with diesel engines. This opens new possibilities within the simulation range of vehicle noises, for the needs of constructing the passive and active reduction systems of the effects of the vibroacoustic energy propagation.

Inference of strata separation and gas emission paths in longwall overburden using continuous wavelet transform of well logs and geostatistical simulation

Abstract Prediction of potential methane emission pathways from various sources into active mine workings or sealed gobs from longwall overburden is important for controlling methane and for improving mining safety. The aim of this paper is to infer strata separation intervals and thus gas emission pathways from standard well log data. The proposed technique was applied to well logs acquired through the Mary Lee/Blue Creek coal seam of the Upper Pottsville Formation in the Black Warrior Basin, Alabama, using well logs from a series of boreholes aligned along a nearly linear profile. For this purpose, continuous wavelet transform (CWT) of digitized gamma well logs was performed by using Mexican hat and Morlet, as the mother wavelets, to identify potential discontinuities in the signal. Pointwise Holder exponents (PHE) of gamma logs were also computed using the generalized quadratic variations (GQV) method to identify the location and strength of singularities of well log signals as a complementary analysis. PHEs and wavelet coefficients were analyzed to find the locations of singularities along the logs. Using the well logs in this study, locations of predicted singularities were used as indicators in single normal equation simulation (SNESIM) to generate equi-probable realizations of potential strata separation intervals. Horizontal and vertical variograms of realizations were then analyzed and compared with those of indicator data and training image (TI) data using the Kruskal–Wallis test. A sum of squared differences was employed to select the most probable realization representing the locations of potential strata separations and methane flow paths. Results indicated that singularities located in well log signals reliably correlated with strata transitions or discontinuities within the strata. Geostatistical simulation of these discontinuities provided information about the location and extents of the continuous channels that may form during mining. If there is a gas source within their zone of influence, paths may develop and allow methane movement towards sealed or active gobs under pressure differentials. Knowledge gained from this research will better prepare mine operations for potential methane inflows, thus improving mine safety.

Fraktalna analiza finansowych szeregów czasowych z wykorzystaniem wymiaru fraktalnego oraz punktowych wykładników Höldera

This paper presents a fractal analysis application to the verification of assumptions of Fractal Market Hypothesis and the presence of fractal properties in financial time series. In this research, the box-counting dimension and pointwise Holder exponents are used. Achieved results lead to interesting observations related to nonrandomness of price series and occurrence of relationships binding fractal properties and variability measures with the presence of trends and influence of the economic situation on financial instruments’ prices.

The influence of migrating bed forms on the velocity-intermittency structure of turbulent flow over a gravel bed

[1] Modeling turbulent flows at high Reynolds number requires solving simplified variants of the Navier-Stokes equations. The methods used to close the resulting Reynolds-averaged, or eddy simulation equations usually follow classical theory and, at small enough scales, postulate universal scaling for turbulence that is independent of the velocity itself. This may not be the best way to conceptualize geophysical turbulence. Turbulent intermittency may be defined in terms of the local “roughness” of the velocity signal as measured by pointwise Holder exponents. This study investigates the joint velocity-intermittency structure of flow over a gravel-bed surface with migrating bed forms. We report clear velocity-intermittency dependence and quantify its nature above the moving bed form profile. Our results imply differences in energy transfer close to bed forms at shorter wavelengths than those forced directly. Hence, progress in modeling flows of geophysical relevance may require a reconsideration of the principles on which turbulence closures are based.

Incremental moments and Hölder exponents of multifractional multistable processes

Multistable processes, that is, processes which are, at each “time”, tangent to a stable process, but where the index of stability varies along the path, have been recently introduced as models for phenomena where the intensity of jumps is non constant. In this work, we give further results on (multifractional) multistable processes related to their local structure. We show that, under certain conditions, the incremental moments display a scaling behaviour, and that the pointwise Holder exponent is, as expected, related to the local stability index. We compute the precise value of the almost sure Holder exponent in the case of the multistable Levy motion, which turns out to reveal an interesting phenomenon.

Self-regulating processes

We construct functions and stochastic processes for which a functional relation holds between amplitude and local regularity, as measured by the pointwise or local Holder exponent. We consider in particular functions and processes built by extending Weierstrass function, multifractional Brownian motion and the Levy construction of Brownian motion. Such processes have recently proved to be relevant models in various applications. The aim of this work is to provide a theoretical background to these studies and to provide a first step in the development of a theory for such self-regulating processes.

Estimations of Hölder Regularities and Direction of Singularity by Hart Smith and Curvelet Transforms

Using Hart Smith’s and curvelet transforms, new necessary and new sufficient conditions for an L2(ℝ2) function to possess Holder regularity, uniform and pointwise, with exponent α>0 are given. Similar to the characterization of Holder regularity by the continuous wavelet transform, the conditions here are in terms of bounds of the transforms across fine scales. However, due to the parabolic scaling, the sufficient and necessary conditions differ in both the uniform and pointwise cases. We also investigate square-integrable functions with sufficiently smooth background. Specifically, sufficient and necessary conditions, which include the special case with 1-dimensional singularity line, are derived for pointwise Holder exponent. Inside their “cones” of influence, these conditions are practically the same, giving near-characterization of direction of singularity.

An algorithm for the characterization of time-series based on local regularity

The pointwise Holder exponent describes the local regularity of a function. This description is particularly applicable to complex time-series representing natural phenomena that exhibit statistical self-similarity, where the scaling exponent fluctuates from point to point. In this work, an algorithm is proposed for the estimation of both global and local scaling exponents of time-series. The technique is based on scaled window variance method and its practical implementation relies on the convolution operation. The proposed algorithm is an attractive alternative to existing techniques because of its simplicity and computational efficiency. The accuracy of the algorithm is verified by application on synthetic signals with a prescribed time evolution of Holder exponents. Case studies are presented in the fields of fluid dynamics, room acoustics and machinery diagnostics.