Estimates Of Exponents Research Articles

Estimating Lyapunov exponents on a noisy environment by global and local Jacobian indirect algorithms

Most of the existing methods and techniques for the detection of chaotic behaviour from empirical time series try to quantify the well-known sensitivity to initial conditions through the estimation of the so-called Lyapunov exponents corresponding to the data generating system, even if this system is unknown. Some of these methods are designed to operate in noise-free environments, such as those methods that directly quantify the separation rate of two initially close trajectories. As an alternative, this paper provides two nonlinear indirect regression methods for estimating the Lyapunov exponents on a noisy environment. We extend the global Jacobian method, by using local polynomial kernel regressions and local neural net kernel models. We apply such methods to several noise-contaminated time series coming from different data generating processes. The results show that in general, the Jacobian indirect methods provide better results than the traditional direct methods for both clean and noisy time series. Moreover, the local Jacobian indirect methods provide more robust and accurate fit than the global ones, with the methods using local networks obtaining more accurate results than those using local polynomials.

Estimation of hurst exponent for sequential monitoring of clinical trials with covariate adaptive randomization

BackgroundClassical Brownian motion (BM) has been commonly used in monitoring clinical trials including those with covariate adaptive randomization (CAR). Independent increment property is commonly assumed in the sequential monitoring process of the clinical trials with CAR designs. However, in reality, correlation may exist in the error terms of the underlying model, resulting in dependent increment in the sequential monitoring process. MethodsWe conducted simulations for estimating the Hurst exponent to evaluate the stochastic property in the covariate adaptive randomized clinical trials under two scenarios: 1. CAR designs with independent and identically distributed error terms. 2. CAR designs with correlated error terms. The theoretical properties of covariate adaptive randomized clinical trials with correlated error structure were investigated. A test statistic including the covariance pattern of the error terms was proposed. ConclusionIn our study, the sequential test statistics under CAR procedure is shown to be asymptotically Brownian motion when the error structure is correctly specified. Further, Brownian motion is a special case of fractional Brownian motion when Hurst exponent equals to 0.5. Our simulations are consistent with the theoretical asymptotic results.

Multifractional Brownian motion characterization based on Hurst exponent estimation and statistical learning.

This paper proposes an approach for the estimation of a time-varying Hurst exponent to allow accurate identification of multifractional Brownian motion (MFBM). The contribution provides a prescription for how to deal with the MFBM measurement data to solve regression and classification problems. Theoretical studies are supplemented with computer simulations and real-world examples. Those prove that the procedure proposed in this paper outperforms the best-in-class algorithm.

Oscillatory and Aperiodic Neural Activity Jointly Predict Language Learning.

Memory formation involves the synchronous firing of neurons in task-relevant networks, with recent models postulating that a decrease in low-frequency oscillatory activity underlies successful memory encoding and retrieval. However, to date, this relationship has been investigated primarily with face and image stimuli; considerably less is known about the oscillatory correlates of complex rule learning, as in language. Furthermore, recent work has shown that nonoscillatory (1/ƒ) activity is functionally relevant to cognition, yet its interaction with oscillatory activity during complex rule learning remains unknown. Using spectral decomposition and power-law exponent estimation of human EEG data (17 women, 18 men), we show for the first time that 1/ƒ and oscillatory activity jointly influence the learning of word order rules of a miniature artificial language system. Flexible word-order rules were associated with a steeper 1/ƒ slope, whereas fixed word-order rules were associated with a shallower slope. We also show that increased theta and alpha power predicts fixed relative to flexible word-order rule learning and behavioral performance. Together, these results suggest that 1/ƒ activity plays an important role in higher-order cognition, including language processing, and that grammar learning is modulated by different word-order permutations, which manifest in distinct oscillatory profiles.

Finite-size scaling of critical avalanches.

We examine probability distribution for avalanche sizes observed in self-organized critical systems. While a power-law distribution with a cutoff because of finite system size is typical behavior, a systematic investigation reveals that it may also decrease with increasing the system size at a fixed avalanche size. We implement the scaling method and identify scaling functions. The data collapse ensures a correct estimation of the critical exponents and distinguishes two exponents related to avalanche size and system size. Our simple analysis provides striking implications. While the exact value for avalanches size exponent remains elusive for the prototype sandpile on a square lattice, we suggest the exponent should be 1. The simulation results represent that the distribution shows a logarithmic system size dependence, consistent with the normalization condition. We also argue that for the train or Oslo sandpile model with bulk drive, the avalanche size exponent is slightly less than 1, which differs significantly from the previous estimate of 1.11.

Prediction of the geometric characteristics of the laser cladding of Inconel 718 on the Inconel 738 substrate via genetic algorithm and linear regression

Assessing the geometric characteristics of a track is the first step in indicating the results of the laser cladding process. In this research, the effect of three essential process parameters, namely scanning speed, laser power, and powder feed rate, on the geometric characteristics of the Inconel 718 track on the Inconel 738 substrate have been studied. The goal of this research was to obtain the optimal process parameters for Inconel 718 laser cladding. To investigate the impact of process parameters on the geometric characteristics, five levels of laser power, three levels of scanning velocity, and three levels of powder feed rate have been considered, and a full factorial design of experiment has been performed. Consequently, 45 tracks have been cladded, and their geometric characteristics, including penetration depth, wetting angle, track height, track width, and compound characteristics such as dilution, has been investigated. After analyzing the results, the behavior of the geometric characteristics for the same process parameters in other brackets have been predicted via linear regression and genetic optimization via MATLAB software. PαVβFγ is participating in Y=aX+b as a compound variable. The regression process is feasible via accurate estimation of exponents α, β and γ. Therefore, the genetic algorithm was used to reduce the errors of linear regression. It is concluded that all of the process parameters have quantitative and qualitative effects on the geometry of the track. Finally, based on the modeling results, a process map was drawn to predict the impact of process parameters on the track geometry.

Assessment of relative dispersion in the Gulf of Tonkin using numerical modeling and HF radar observations of surface currents

Particle pair statistics from synthetic drifter trajectories reconstructed from realistic, high-resolution numerical simulations (SYMPHONIE model) and HF radar velocity measurements are used to investigate the dispersion properties in the Gulf of Tonkin (GoT). This study takes an approach based on two-particles statistics providing the relative dispersion, relative diffusivity and Finite Size Lyapunov Exponent (FSLE) estimates. In the GoT, the relative dispersion follows the predictions from the theory of two-dimensional turbulence with two inertial subranges identified in the kinetic energy spectrum with the spectral slopes −5/3 and −3. The time evolution of dispersion shows an exponential growth during 5–8 days, followed by a power law regime during the next 6–20 days. Fixed-length indicators from the relative diffusivity and the FSLE reveal a local dispersion at large and intermediate scales (above Rossby radius of deformation) and non-local dispersion in sub-mesoscale range (below Rossby radius of deformation). The effect of river runoff on the local hydrodynamics and dispersion processes is assessed using the numerical model simulations without river discharge. The results show that in the model, the river plume, when present, highly impacts the Lagrangian statistics. High gradients of buoyancy reinforce the sub-mesoscale circulation in a large region along the Vietnamese coast and modify the scales and intensity of turbulent dispersion. However, a clear change of dispersion regime in the sub-mesoscale range is not identified, suggesting that the mesoscale circulation in the GoT largely governs particle spreading even at small scale.

Dynamic scaling in rotating turbulence: A shell model study.

We investigate the scaling form of appropriate timescales extracted from time-dependent correlation functions in rotating turbulent flows. In particular, we obtain precise estimates of the dynamic exponents z_{p}, associated with the timescales, and their relation with the more commonly measured equal-time exponents ζ_{p}. These theoretical predictions, obtained by using the multifractal formalism, are validated through extensive numerical simulations of a shell model for such rotating flows.

A comparison of maximum likelihood and absolute moments for the estimation of Hurst exponents in a stationary framework

The absolute-moment method is widespread for estimating the Hurst exponent of a fractional Brownian motion X. But this method is biased when applied to a stationary version of X, in particular an inverse Lamperti transform of X, with a linear time contraction of parameter θ. We present an adaptation of the absolute-moment method to this framework and we compare it to the maximum likelihood method, with simulations and an application to a financial time series. While it appears that the maximum-likelihood method is more accurate than the adapted absolute-moment estimation, this last method is not uninteresting for two reasons: it makes it possible to confirm visually that the model is well specified and it is computationally more performing.

Evidence for pressure induced unconventional quantum criticality in the coupled spin ladder antiferromagnet C9H18N2CuBr4

Quantum phase transitions in quantum matter occur at zero temperature between distinct ground states by tuning a nonthermal control parameter. Often, they can be accurately described within the Landau theory of phase transitions, similarly to conventional thermal phase transitions. However, this picture can break down under certain circumstances. Here, we present a comprehensive study of the effect of hydrostatic pressure on the magnetic structure and spin dynamics of the spin-1/2 ladder compound C9H18N2CuBr4. Single-crystal heat capacity and neutron diffraction measurements reveal that the Néel-ordered phase breaks down beyond a critical pressure of Pc ∼ 1.0 GPa through a continuous quantum phase transition. Estimates of the critical exponents suggest that this transition may fall outside the traditional Landau paradigm. The inelastic neutron scattering spectra at 1.3 GPa are characterized by two well-separated gapped modes, including one continuum-like and another resolution-limited excitation in distinct scattering channels, which further indicates an exotic quantum-disordered phase above Pc.

Atlas of urban scaling laws

Accurate estimates of the urban fractal dimension D f are obtained by implementing the detrended moving average algorithm on high-resolution multi-spectral satellite images from the WorldView2 (WV2) database covering the largest European cities. Fractal dimension D f varies between 1.65 and 1.90 with high values for highly urbanised urban sectors and low ones for suburban and peripheral ones. Based on recently proposed models, the values of the fractal dimension D f are checked against the exponents β s and β i of the scaling law Y ∼ N β , respectively for socio-economic and infrastructural variables Y, with N the population size. The exponents β s and β i are traditionally derived as if cities were zero-dimensional objects, with the relevant feature Y related to a single homogeneous population value N, thus neglecting the microscopic heterogeneity of the urban structure. Our findings go beyond this limit. High sensitive and repeatable satellite records yield robust local estimates of the urban scaling exponents. Furthermore, the work discusses how to discriminate among different scaling theories, shedding light on the debated issue of scaling phenomena contradictory perspectives and pave paths to a more systematic adoption of the complex system science methods to urban landscape analysis.

Opinion dynamics involving contrarian and independence behaviors based on the Sznajd model with two-two and three-one agent interactions

We investigate the opinion evolution of outflow dynamics based on the Sznajd model on a complete graph involving contrarian and independence behaviors. We consider a group of four spins representing the social agents with the following scenarios: (1) scenario two-two with contrarian agents or independence agents and (2) scenario three-one with contrarian or independence agents. All of them undergo a second-order phase transition according to our simulation. The critical point decreases exponentially as λ and f increase, where λ and f are contrarian and flexibility factors, respectively. Furthermore, we find that the critical point of scenario three-one is smaller than that of scenario two-two. For the same level of λ and f, the critical point of the scenario involving independence is smaller than the scenario with contrarian agents. From a sociophysics point of view, we observe that scenario three-one can likely reach a stalemate situation rather than scenario two-two. Surprisingly, the scenarios involving contrarians have a higher probability of achieving a consensus than a scenario involving independence. Our estimates of the critical exponents indicate that the model is still in the same universality class as the mean-field Ising model.

An integrated condition monitoring scheme for health state identification of a multi-stage gearbox through Hurst exponent estimates

The vibration and acoustic signals collected from rotating machinery are often non-stationary and aperiodic, and it is a challenge to post-process and extract the defect sensitive health indicators. In this paper, we demonstrate how the estimated Hurst exponent of such measured data can be advantageous for analyzing non-stationary and aperiodic data due to its self-similarity and scale-invariance properties. To illustrate this, the paper demonstrates detection of fault diagnostics of a multi-stage gearbox operating under fluctuating speeds through estimated Hurst exponent of the raw vibration and acoustic signals as a health indicator. Thirteen health states of the gearbox are considered, and the raw vibration and acoustic signals are collected. The Hurst exponents are calculated using three different approaches: generalized Hurst exponent (q = 1, 2, 3, and 4), rescale range statistical (R/S) analysis, and dispersion analysis from the vibration and acoustic signals. Three different health indicator datasets are formulated and subjected to feature learning through conventional machine-learning (decision tree and support vector machine) and advanced machine-learning (deep-learning) classifiers. The effectiveness of these datasets while discriminating between the health states of the gearbox is investigated, yielding classification accuracies of 96.4% when compared with the individual health indicator datasets. The ability of the fault diagnosis and defect severity analysis with reduced reliance on the signal post-processing algorithms is demonstrated.

Estimation of the Hurst Parameter in Spot Volatility

This paper contributes in three stages in a logic of the cognitive process: we firstly propose a new estimation of Hurst exponent by changing frequency method which is purely mathematical. Then we want to check if the new Hurst is efficient, so we prove the advantages of this new Hurst in asymptotic variance in the perspective compared with other two Hurst estimator. However, a purely mathematical game is not enough, a good estimation should be proven by reality, so we apply the new Hurst estimator into truncated and non-truncated spot volatility which fills the gap of previous literatures using 5-min price data (Source: Wind Financial Terminal) of 10 Chinese A-share industry indices from 1 January 2005 until 31 December 2020.

Recession curve power-law exponent estimation: is there a perfect approach?

The coefficient (k) of (discharge at time t) power-law relationship is typically computed after fixing the value of the exponent (α). However, recession analysis poses the challenge of finding a suitable α for a basin. Although many recent studies have suggested to consider the median of the α distribution as the representative value, a purely logical reasoning has not been yet provided. In this study, we argue that there is no perfect approach to estimate α and that it should depend on the final objective. Recession flow prediction is considered as the objective in this study. We employ a model to predict recession discharge for 408 USGS basins for a wide range of α values. Considering Nash-Sutcliffe efficiency as the indicator of model performance, we observed that the optimum value of α is substantially lower than the α median for most of the basins. Overall, our study establishes that there is no single value of preferable for all practical purposes.

An improved strain-life model based on the Walker equation to describe tensile and compressive mean stress effects

Strain-life models that include Walker-like products can reproduce the higher mean stress effects in fatigue crack initiation observed in high-strength materials. These models are able to improve Smith-Watson-Topper’s strain-life predictions, at the cost of additional data-fitting exponents. This work reviews and compares several such models using 34 metallic alloy data sets from the literature that include non-zero mean stresses, totaling 1,355 uniaxial fatigue specimens. This comprehensive evaluation suggests that two different calibrations of Walker’s exponent are required to properly fit data measured under tensile and compressive mean stresses. Hence, an equivalent strain amplitude is proposed based on these two exponents, as well as on the ratio between the stress peak and amplitude. The new strain-life model resulting from this equivalent strain amplitude is able to describe uniaxial fatigue data under all stress ratios R < 1. Estimates for both exponents are provided as a function of the ultimate tensile strength of the material, which could be used in the absence of fatigue data under non-zero mean stresses, an asset for practical applications. A modified version of the proposed strain-life model, which considers mean stress effects only in the elastic damage term, is evaluated as well. Explicit equations are presented for the optimal calibration of all parameters from both models.

Three-dimensional O(N) -invariant ϕ4 models at criticality for N≥4

We study the $O(N)$-invariant $\phi^4$ model on the simple cubic lattice by using Monte Carlo simulations. By using a finite size scaling analysis, we obtain accurate estimates for the critical exponents $\nu$ and $\eta$ for $N=4$, $5$, $6$, $8$, $10$, and $12$. We study the model for each $N$ for at least three different values of the parameter $\lambda$ to control leading corrections to scaling. We compare our results with those obtained by other theoretical methods.

Methods of Retrieving Large-Variable Exponents

Methods of determining, from small-variable asymptotic expansions, the characteristic exponents for variables tending to infinity are analyzed. The following methods are considered: diff-log Padé summation, self-similar factor approximation, self-similar diff-log summation, self-similar Borel summation, and self-similar Borel–Leroy summation. Several typical problems are treated. The comparison of the results shows that all these methods provide close estimates for the large-variable exponents. The reliable estimates are obtained when different methods of summation are compatible with each other.

Geo-Hydrological Events and Temporal Trends in CAPE and TCWV over the Main Cities Facing the Mediterranean Sea in the Period 1979–2018

The Mediterranean region is regarded as the meeting point between Europe, Africa and the Middle East. Due to favourable climatic conditions, many civilizations have flourished here. Approximately, about half a billion people live in the Mediterranean region, which provides a key passage for trading between Europe and Asia. Belonging to the middle latitude zone, this region experiences high meteorological variability that is mostly induced by contrasting hot and cold air masses that generally come from the west. Due to such phenomenon, this region is subject to frequent intensive precipitation events. Besides, in this complex physiographic and orographic region, human activities have contributed to enhance the geo-hydrologic risk. Further, in terms of climate change, the Mediterranean is a hot spot, probably exposing it to future damaging events. In this framework, this research focuses on the analysis of precipitation related events recorded in the EM–DAT disasters database for the period 1979–2018. An increasing trend emerges in both event records and related deaths. Then a possible linkage with two meteorological variables was investigated. Significant trends were studied for CAPE (Convective Available Potential Energy) and TCWV (Total Column Water Vapor) data, as monthly means in 100 km2 cells for 18 major cities facing the Mediterranean Sea. The Mann–Kendall trend test, Sen’s slope estimation and the Hurst exponent estimation for the investigation of persistency in time series were applied. The research provides new evidence and quantification for the increasing trend of climate related disasters at the Mediterranean scale: recorded events in 1999–2018 are about four times the ones in 1979–1998. Besides, it relates this rise with the trend of two meteorological variables associated with high intensity precipitation events, which shows a statistically significative increasing trend in many of the analysed cities facing the Mediterranean Sea.

Spreading, pinching, and coalescence: the Ohnesorge units.

Understanding the kinematics and dynamics of spreading, pinching, and coalescence of drops is critically important for a diverse range of applications involving spraying, printing, coating, dispensing, emulsification, and atomization. Hence experimental studies visualize and characterize the increase in size over time for drops spreading over substrates, or liquid bridges between coalescing drops, or the decrease in the radius of pinching necks during drop formation. Even for Newtonian fluids, the interplay of inertial, viscous, and capillary stresses can lead to a number of scaling laws, with three limiting self-similar cases: visco-inertial (VI), visco-capillary (VC) and inertio-capillary (IC). Though experiments are presented as examples of the methods of dimensional analysis, the lack of precise values or estimates for pre-factors, transitions, and scaling exponents presents difficulties for quantitative analysis and material characterization. In this tutorial review, we reanalyze and summarize an elaborate set of landmark published experimental studies on a wide range of Newtonian fluids. We show that moving beyond VI, VC, and IC units in favor of intrinsic timescale and lengthscale determined by all three material properties (viscosity, surface tension and density), creates a complementary system that we call the Ohnesorge units. We find that in spite of large differences in topological features, timescales, and material properties, the analysis of spreading, pinching and coalescing drops in the Ohnesorge units results in a remarkable collapse of the experimental datasets, highlighting the shared and universal features displayed in such flows.