Holder Exponent Research Articles

The Monitoring of Milling Tool Tipping by Estimating Holder Exponents of Vibration

The gradual tool wear is unavoidable in the machining process, it directly influences the surface integrity and dimensional tolerances of the components. The tool condition monitoring (TCM) systems have the capacity to make full use of the cutting potential of the cutting tools, which is of great significance for improving production efficiency and ensuring product quality. In milling titanium alloys, tool tipping is one of the main failure modes. Therefore, this study focuses on establishing a tool tipping monitoring approach for the milling process. The singularity analysis method based on wavelet transform was employed to characterize the variation of vibration waveforms quantitatively with the Holder Exponent (HE) index. The probability density distribution and statistical analysis were adopted to extract effective HE features from the HE indexes to correlate with the different tool conditions. The mutual information method was adopted to rank the discriminability of the HE statistical parameters. Then, several machine learning (ML) models were established with the screened HE features. Finally, the results of the experiments indicate that the Support Vector Machine (SVM) model has the highest classification accuracy and can provide practical guides on the tool changes.

Differentiation of early mild cognitive impairment in brainstem MR images using multifractal detrended moving average singularity spectral features

Brainstem texture analysis can provide valuable information in the diagnosis of Early Mild Cognitive Impairment (EMCI) condition. In this work, 3D brainstem structure is segmented and analysed for texture alterations using multifractal features to differentiate EMCI from other Alzheimer’s disease stages. The images obtained from public domain database are preprocessed for spatial registration, skull stripping and contrast enhancement. White matter volume is segmented from the preprocessed images using fuzzy ‘C’ means clustering algorithm. Midsagittal white matter tissue is used as the initial seed to segment the brainstem volume using sparse field level set method. Multifractal detrended moving average algorithm is used to compute the fluctuation function, generalized Hurst exponent and mass exponent to study the multifractal characteristics of brainstem structure. Features extracted from the multifractal spectrum are analysed to differentiate the images pertaining to EMCI subject group. Results indicate that the proposed technique is able to segment the brainstem structure from all the considered images. The fluctuation function is observed to have linear relationship with scale. The generalized Hurst exponent decreases with order and mass exponent follows a non-linear trend demonstrating the multifractal nature of brainstem. Singularity spectral features namely strength of multifractality, Holder exponent at f(2.8), tangent slope and maximum Holder exponent are found to be most significant in differentiating EMCI from subject groups. As this complex EMCI distinction is clinically important, the proposed approach is useful for early diagnosis of Alzheimer’s condition.

Weak formulation and scaling properties of energy fluxes in three-dimensional numerical turbulent Rayleigh–Bénard convection

We apply the weak formalism on the Boussinesq equations, to characterize scaling properties of the mean and the standard deviation of the potential, kinetic and viscous energy flux in very high resolution numerical simulations. The local Bolgiano-Oboukhov length $L_{BO}$ is investigated and it is found that its value may change of an order of magnitude through the domain, in agreement with previous results. We investigate the scale-by-scale averaged terms of the weak equations, which are a generalization of the Karman-Howarth-Monin and Yaglom equations. We have not found the classical Bolgiano-Oboukhov picture, but evidence of a mixture of Bolgiano-Oboukhov and Kolmogorov scalings. In particular, all the terms are compatible with a Bolgiano-Oboukhov local H\"older exponent for the temperature and a Kolmogorov 41 for the velocity. This behavior may be related to anisotropy and to the strong heterogeneity of the convective flow, reflected in the wide distribution of Bolgiano-Oboukhov local scales. The scale-by-scale analysis allows us also to compare the theoretical Bolgiano-Oboukhov length $L_{BO}$ computed from its definition with that empirically extracted through scalings obtained from weak analysis. The key result of the work is to show that the analysis of local weak formulation of the problem is powerful to characterize the fluctuation properties.

SAR Target Detection in Complex Scene Based on 2-D Singularity Power Spectrum Analysis

The synthetic aperture radar (SAR) target detection method in the background of a complex scene and extremely low signal-to-noise ratio (EL-SNR) is studied in this paper. With theoretical derivation and analysis, the singularity power spectrum (SPS) is developed into two-dimensional SPS (2D-SPS). Developed from the 2-D Holder exponent and the SPS, the 2D-SPS can be applied for the singularity power analysis of images. Furthermore, a novel SAR target detection method based on 2D-SPS is proposed. The proposed methodology is tested on sea clutters, both with and without SAR ship target, from the OpenSAR data sets. The experimental results indicate that the SAR target detection based on 2D-SPS performs better than conventional multifractal spectrum (MFS) and constant false alarm rate (CFAR) methods, and can achieve more than 95% and 98% detection probability, respectively, under EL-SNR with SNR = −30 dB, SNR = −10 dB, and $P_{f} = 10^{-2}$ , and almost 100% detection probability for weak and multiple targets within sea clutters. The proposed method can be applied to target detection under the background of fractal noise and provides a reference for target detection in related fields.

Multifractal-based lacunarity analysis of trabecular bone in radiography

This study presents textural characterization techniques for effective osteoporosis diagnosis using bone radiograph images. The automatic classification of osteoporosis and healthy (control) cases using bone radiograph images in this work presents a major challenge as the images show no visual differences for both cases. The proposed work utilizes multifractals to characterize the trabecular bone texture in the radiographs. Initially, Holder exponents are computed, then Hausdorff dimensions are determined, which quantify the global regularity of the pixels. Finally, lacunarity is computed from the Hausdorff dimensions. Performance metrics show that estimating lacunarity from the Hausdorff dimensions, rather than the input image, directly helps in achieving better textural characterization of bone radiographs, leading to better performance in osteoporosis classification. The proposed lacunarity-based trabecular bone textural characterization method is compared with other multifractal-based methods for trabecular bone textural characterization, such as box-counting and regularization dimensions. The proposed method is also evaluated with the textural characterization of a bone radiograph challenge dataset to demonstrate its effectiveness compared to the other methods used in the challenge.

Vibration singularity analysis for milling tool condition monitoring

Tool condition monitoring (TCM) is a very effective way to enhance productivity and ensure work-piece quality. This paper introduces a cutting condition independent TCM approach for milling with vibration singularity analysis. Holder Exponents (HE) are chosen as index to estimate the singularity of vibration signals. Wavelet transform modulus maxima (WTMM) are employed to estimate HEs. A wavelet basis selection method is established to select the optimal wavelet bases for the estimation of HEs. Means of HE values and Number of singular points of feed direction vibration components are found to be the most correlated with tool conditions based on experimental study. The sensitivity between these HE features and the transition points of different tool wear states is discovered. Verified by a public database, this sensitivity is found independent of the cutting conditions. Then a TCM approach is proposed which utilized a Support Vector Machine (SVM) model and a transition point identification method (TPIM). Experiment results indicate that this approach is efficient and the TPIM helps to reach more precise classification results.

Ship target detection and segmentation method based on multi‐fractal analysis

A novel automatic ship target detection and segmentation approach based on modified multi-fractal method in synthetic aperture radar (SAR) image is proposed in this study. Firstly, singular power spectrum (SPS) is developed to two-dimensional SPS (2D-SPS), and applied to feature extraction and SAR target detection. Secondly, 2D Holder exponent and 2D multi-fractal spectrum are jointly used to ship target segmentation in SAR image. The algorithm steps and detection flow is here conceptually assessed, analytically derived, numerically verified, and also tested on actual SAR images. Results indicate that the proposed method has significant advantages in ship target detection and segmentation in SAR image.

Fractal behaviour of distances between consecutive aftershocks: the examples of Landers, Northridge and Hector Mine Southern California mainshocks

Distances between consecutive aftershocks are analysed by means of mono- and multifractal theory with the aim of quantifying the complexity of the physical mechanism governing them, as well as their predictability and predictive instability. Hausdorff, Ha, and Hurst, H, exponents are determined by semivariograms and rescaled analysis, respectively. The exponent β of the power law describing power spectral contents is also quantified. These three parameters permit a generation of fractional Gaussian noise, fGn, simulating distances. The complexity and predictive instability of physical mechanism generating the series of distances is quantified by means of the correlation dimension, μ*, the Kolmogorov entropy, κ, and the Lyapunov exponents, λi, which are based on the reconstruction theorem formulation. Additionally, the multifractal detrended fluctuation analysis, MF-DFA, contributes with a different point of view to quantify the complexity of the series, in terms of fractal spectral width, W, spectral asymmetry, B, and the critical Holder exponent, a0. By one hand, the MF-DFA is applied to the complete set of distances characterising the whole aftershock process. By the other hand, the MF-DFA is applied to segments of the series of distances with the aim of determining the evolution of the complexity since the mainshock up to the end of the stress relaxation process. Finally, an ARIMA multilinear regression process is applied to obtain some improvements, in comparison with fGn simulations, on the prediction of distances. The database for this analysis is obtained from the Southern California Seismic Network (SCSN) catalogue. Three series of aftershocks equalling to or exceeding magnitudes of 2.0, assuring seismic catalogue completeness, and associated with Landers (06/28/1992), Northridge (01/17/1994) and Hector Mine (10/16/1999) mainshocks are obtained. It is worth mentioning that common mono-multifractal behaviour for the three aftershocks series is not detected, whatever aftershock periods or segments of them are considered.

The self-similarity properties and multifractal analysis of DNA sequences

Abstract In this work is presented a pedagogical point of view of multifractal analysis deoxyribonucleic acid (DNA) sequences is presented. The DNA sequences are formed by 4 nucleotides (adenine, cytosine, guanine, and tymine). Following Jeffrey’s paper we associated a simple contractive function to each nucleotide, and constructed the Hutchinson’s operator W, which was used to build covers of different sizes of the unitary square Q, thus Wk (Q) is a cover of Q, conformed by 4 k squares Qk of size 2 −k , as each Qk corresponds to a unique subsequence of nucleotides with length k : b 1 b 2 ...bk . Besides, it is obtained the optimal cover Ck to the fractal F generated for each DNA sequence was obtained. We made a multifractal decomposition of Ck in terms of the sets Jα conformed by the Qk ’s with the same value of the Holder exponent α, and determined f (α), the Hausdorff dimension of Jα , using the curdling theorem.

Multifractal properties of sample paths of ground state-transformed jump processes

We consider a class of L\'evy-type processes with unbounded coefficients, arising as Doob $h$-transforms of Feynman-Kac type representations of non-local Schr\"odinger operators, where the function $h$ is chosen to be the ground state of such an operator. First, we show the existence of a c\`adl\`ag version of the so-obtained ground state-transformed processes. Next, we prove that they satisfy a related stochastic differential equation with jumps. Making use of this SDE, we then derive and prove the multifractal spectrum of local H\"older exponents of sample paths of ground state-transformed processes.

Singularity Analysis of Cutting Force and Vibration for Tool Condition Monitoring in Milling

Tool wear is inevitable in manufacturing and affects the surface quality and geometric tolerance significantly. A robust and efficient tool condition monitoring (TCM) system is needed to maximize tool life, ensure work-piece quality, and benefit the cost control of manufacturers. This paper presents a systematic singularity analysis approach of cutting force and vibrations for feature extraction of TCM in milling. The singularity of sensory signals is estimated by Holder Exponents (HE), which are determined by wavelet transform modulus maxima (WTMM). A comprehensive wavelet basis selection approach is proposed to choose the appropriate wavelet basis for different sensory signals. A de-noising algorithm based on WTMMs' estimation was used as a pre-processing technique to improve noise reduction and preserve the singularities. The mutual information method was employed to rank HE features. The effectiveness of the singularity analysis approach is validated through the Support Vector Machine (SVM) models trained by these ranked features. The estimating results of case studies confirm the efficacy and efficiency of the proposed approach.

Hölder Exponents of Self-Similar Functions

We study the Holder exponents of self-affine similar functions on the interval [0; 1]. Closed-form expressions for the Holder exponents via the self-similarity parameters are obtained.

Calibrating Rough Volatility Models: A Convolutional Neural Network Approach

In this paper we use convolutional neural networks to find the Holder exponent of simulated sample paths of the rBergomi model, a recently proposed stock price model used in mathematical finance. We contextualise this as a calibration problem, thereby providing a very practical and useful application.

Partial Regularity of Stationary Navier-Stokes Systems under Natural Growth Condition

In this article, we consider the partial regularity of stationary Navier-Stokes system under the natural growth condition. Applying the method of A-harmonic approximation, we obtain some results about the partial regularity and establish the optimal Holder exponent for the derivative of a weak solution on its regular set.

Application of wavelet analysis to the analysis of geomagnetic field variations

Wavelet analysis is becoming more popular in geophysics. It is used for numerous researches, including tropical convection, the El Nino-Southern Oscillation, atmospheric cold fronts, temperature variations, the dispersion of ocean waves, and coherent structures in turbulent flows, number of sunspots etc. In this paper we research how informative is the application of wavelet analysis to the analysis of geomagnetic field variations at the mid-latitude observatory “Mikhnevo” of Institute of Geosphere Dynamics of Russian Academy of Science. We review continuous wavelet transform by focusing attention on such aspects as choice of mother wavelet, choice of scales, cone of influence, visualization of results, reconstruction of time series from wavelet transform and its application to estimate the Holder exponents and singularity spectra. In our work we use Morlet wavelet with frequency parameter of 6. In so doing, the reconstruction of the time series from the wavelet transform has a mean square error of 3.4%. The application of wavelet analysis made it possible to distinguish pronounced periodicities of the geomagnetic field with periods of 27, 13.5, 9, 6 days. In solar quiet-day variations is dominated by the 24- 12-, 8-, and 6-hour period components. An analysis of the modulus of the wavelet transform coefficients qualitatively indicates a scaling (close to the monofractal) character of the variations of the geomagnetic field in the diurnal range. Moreover, the intensity of periodic variations of geomagnetic variation isn’t constant in time. The application of the method of wavelet transform modulus maxima confirmed the monofractal character of the diurnal variation for any solar activity. In contrast to the 1-day variation, the 27-day variation and its harmonics show a higher degree of multifractality during a maximum of solar activity in comparison with the minimum of solar activity.

A general low-order partial regularity theory for asymptotically convex functionals with asymptotic dependence on the minimizer

We consider a minimizer $$u\in W^{1,p}(\Omega )$$ , where $$\Omega \subseteq \mathbb {R}^{n}$$ is a bounded open set, of the integral functional $$\begin{aligned} u\mapsto \int _{\Omega }f(x,u,Du)\ dx \end{aligned}$$ in the case when $$f : \Omega \subseteq \mathbb {R}^{n}\times \mathbb {R}^{N}\times \mathbb {R}^{N\times n}\rightarrow \mathbb {R}$$ is asymptotically related to a more regular function; since we assume that $$N\ge 2$$ , we study here the vectorial case. The asymptotical relatedness condition is such that dependence on u is allowed even as $$|\xi |\rightarrow +\infty $$ . Unlike in previous work f is allowed to satisfy a more general growth condition, which permits a coupling between the x and u variables. Due to the generality of the asymptotical relatedness condition this coupling induces a subtle restriction on the regularity required of the various functions in the growth condition, and we study these restrictions in this paper. In addition, here we do not obtain the full spectrum of Holder continuity for the minimizer, but rather a restricted range of Holder exponents that depend on the initial data.

Mutifractals based multimodal 3D image registration

Multimodal registration is a method to register the volumes of different modalities, for e.g., computed tomography (CT) and magnetic resonance (MR). Mutual information (MI) based methods are widely used for multimodal registration. The MI characterizes the statistical dependence between the voxel intensities of volumes. Robustness of the MI based registration is affected, when there is a low correspondence between the voxel intensities of volumes. This can be improved by integrating the geometric characteristics of volumes like complexity, singularity and irregularity with registration. A novel approach for 3D multimodal image registration based on the multifractal characterization of volumes is being proposed in this paper. The proposed method uses multifractal formalism to incorporate geometric characteristics into registration. Multifractal formalism involves determination of Holder exponent followed by computation of Hausdorff dimension. Holder exponents quantify the local regularity of the volumes and Hausdorff dimensions quantify the global regularity (multifractality) of the volumes. The performance of the proposed algorithm is evaluated using synthetic phantom images for different noise levels and 41 clinical 3D brain images of 7 different patients from a public domain database. The above-mentioned test platforms highlight the efficiency of the proposed method towards improving the robustness and accuracy of registration.

The local Hölder exponent for the entropy of real unimodal maps

We consider the topological entropy h(θ) of real unimodal maps as a function of the kneading parameter θ (equivalently, as a function of the external angle in the Mandelbrot set). We prove that this function is locally Holder continuous where h(θ) > 0, and more precisely for any θ which does not lie in a plateau the local Holder exponent equals exactly, up to a factor log 2, the value of the function at that point. This confirms a conjecture of Isola and Politi (1990), and extends a similar result for the dimension of invariant subsets of the circle.

Option Pricing with Fractional Stochastic Volatility and Discontinuous Payoff Function of Polynomial Growth

We consider the pricing problem related to payoffs of polynomial growth that can have discontinuities of the 1st kind. The asset price dynamic is modeled within the Black-Scholes framework characterized by a stochastic volatility term driven by a fractional Ornstein-Uhlenbeck process. In order to solve the aforementioned problem, we consider three approaches. The first one consists in a suitable transformation of the initial value of the asset price, in order to eliminate possible discontinuities. Then we discretize both the Wiener process and the fractional Brownian motion and estimate the rate of convergence of the related discretized price to its real value whose closed-form analytical expression is usually difficult to obtain. The second approach consists in considering the conditional expectation with respect to the entire trajectory of the fractional Brownian motion (fBm). Here we derive a presentation for the option price which involves only an integral functional depending on the fBm trajectory, and then discretize the fBm and estimate the rate of convergence of the associated numerical scheme. In both cases the rate of convergence is the same and equals n−rH, where n is the partition size, H is the Hurst index of the fBm, and r is the Holder exponent of the volatility function. The third method consists in calculating the density of the integral functional depending on the trajectory of the fBm via Malliavin calculus and providing the option price in terms of the associated probability density.

On the weak approximation of a skew diffusion by an Euler-type scheme

We study the weak approximation error of a skew diffusion with bounded measurable drift and Holder diffusion coefficient by an Euler-type scheme, which consists of iteratively simulating skew Brownian motions with constant drift. We first establish two sided Gaussian bounds for the density of this approximation scheme. Then, a bound for the difference between the densities of the skew diffusion and its Euler approximation is obtained. Notably, the weak approximation error is shown to be of order h η/2 , where h is the time step of the scheme, η being the Holder exponent of the diffusion coefficient.