Local Regularity Analysis Research Articles

Bump Energy for Durability Prediction of Coil Spring Based on Local Regularity Analysis

Bump Energy for Durability Prediction of Coil Spring Based on Local Regularity Analysis

Diagnosing simultaneous faults using the local regularity of vibration signals

The regularity of the vibration signals measured from a rotating machine is often affected by the condition of the machine. The fractional order of regularity can be measured using the definition of Hölder continuity. In this paper, we review the connection between the pointwise Hölder regularity of a signal and its wavelet transform. We calculate the wavelet transform modulus of acceleration measurements from a test rig. The effects of different faults were recorded, such as unbalance, the coupling misalignment of a claw clutch, the absence of lubrication in a ball bearing, the absence of the bearing’s cage, and their combinations. An analysis of the estimated isolated pointwise regularities from the wavelet transform modulus maxima ridges shows that the faults often cause irregularities in the signals and that their locations and frequencies can be used in diagnosing the faults. Coupling misalignment and the absence of lubrication in a ball bearing both cause impact-like vibrations, but these impacts have positive and negative regularities in the case of a coupling misalignment and mainly negative in the case of a dry bearing. Unbalance is best diagnosed from the integrals of the acceleration signals using traditional methods. In diagnosing the misalignment, bearing problems and simultaneous faults, the local regularity analysis outperforms the use of high order norms of differentiated acceleration measurements (i.e. jerk and snap signals). Using just three features (the number of local irregularities in an acceleration signal, their mean Hölder regularity and the arithmetic mean of the absolute values of a velocity signal), a quadratic classifier can be constructed whose estimated classification error is only 0.3%.

The Singular Set for a Semilinear Unstable Problem

We study the regularity of solutions of the following semilinear problem $${\Delta}u = -\lambda_{+}(x) (u^{+})^{q}+\lambda_{-} (x) (u^{-})^{q} \qquad \text{in} \;\; B_{1}, $$ where B 1 is the unit ball in ℝ n , 0 < q < 1 and λ ± satisfy a Holder continuity condition. Our main results concern local regularity analysis of solutions and their nodal set {u = 0}. The desired regularity is C [κ],κ−[κ] for κ = 2/(1 − q) and we divide the singular points in two classes. The first class contains the points where at least one of the derivatives of order less than κ is nonzero, the second class which is named $\mathcal {S}_{\kappa }$ , is the set of points where all the derivatives of order less than κ exist and vanish. We prove that $\mathcal {S}_{\kappa }=\varnothing $ when κ is not an integer. Moreover, with an example we show that $\mathcal {S}_{\kappa }$ can be nonempty if κ ∈ ℕ. Finally, a regularity investigation in the plane shows that the singular points in $\mathcal {S}_{\kappa }$ are isolated.

Local Regularity Analysis with Wavelet Transform in Gear Tooth Failure Detection

Abstract Diagnosing gear tooth and bearing failures in industrial power transition situations has been studied a lot but challenges still remain. This study aims to look at the problem from a more theoretical perspective. Our goal is to find out if the local regularity i.e. smoothness of the measured signal can be estimated from the vibrations of epicyclic gearboxes and if the regularity can be linked to the meshing events of the gear teeth. Previously it has been shown that the decreasing local regularity of the measured acceleration signals can reveal the inner race faults in slowly rotating bearings. The local regularity is estimated from the modulus maxima ridges of the signal’s wavelet transform. In this study, the measurements come from the epicyclic gearboxes of the Kelukoski water power station (WPS). The very stable rotational speed of the WPS makes it possible to deduce that the gear mesh frequencies of the WPS and a frequency related to the rotation of the turbine blades are the most significant components in the spectra of the estimated local regularity signals.

Automatic Detection of Defects in Tire Radiographic Images

This paper is about the detection of tire defects in multi-textural radiographic images. We consider the tire defects characterization problem in ways of local regularity analysis and scale characteristic. Optimal scale and threshold parameters are selected using a defect edge measurement model to frame defect edge detection. This framework distinguishes the defects from the background textures. Finally, a novel method for detection of tire defects is proposed based on wavelet multiscale analysis. We provide examples with a consistent dataset of 400 images selected over 3700 industrial images in order to illustrate and validate the obtained results which demonstrate substantial improvement over the state of the art.

A semilinear PDE with free boundary

We study the semilinear problem Δu=λ+(x)(u+)q−1−λ−(x)(u−)q−1inB1, from a regularity point of view for solutions and the free boundary ∂{±u>0}. Here B1 is the unit ball, 1<q<2 and λ± are Lipschitz.Our main results concern local regularity analysis of solutions and their free boundaries. One of the main difficulties encountered in studying this equation is classification of global solutions. In dimension two we are able to present a fairly good analysis of global homogeneous solutions, and hence a better understanding of the behavior of the free boundary. In higher dimensions the problem becomes quite complicated, but we are still able to state partial results; e.g. we prove that if a solution is close to one-dimensional solution in a small ball, then in an even smaller ball the free boundary can be represented locally as two C1-regular graphs Γ+=∂{u>0} and Γ−=∂{u<0}, tangential to each other.It is noteworthy that the above problem (in contrast to the case q=1) introduces interesting and quite challenging features, that are not encountered in the case q=1. E.g. one obtains homogeneous global solutions that are not one-dimensional. This complicates the analysis of the free boundary substantially.

AN ENTROPY BASED IN WAVELET LEADERS TO QUANTIFY THE LOCAL REGULARITY OF A SIGNAL AND ITS APPLICATION TO ANALIZE THE DOW JONES INDEX

Local regularity analysis is useful in many fields, such as financial analysis, fluid mechanics, PDE theory, signal and image processing. Different quantifiers have been proposed to measure the local regularity of a function. In this paper we present a new quantifier of local regularity of a signal: the pointwise wavelet leaders entropy. We define this new measure of regularity by combining the concept of entropy, coming from the information theory and statistical mechanics, with the wavelet leaders coefficients. Also we establish its inverse relation with one of the well-known regularity exponents, the pointwise Hölder exponent. Finally, we apply this methodology to the financial data series of the Dow Jones Industrial Average Index, registered in the period 1928–2011, in order to compare the temporal evolution of the pointwise Hölder exponent and the pointwise wavelet leaders entropy. The analysis reveals that temporal variation of these quantifiers reflects the evolution of the Dow Jones Industrial Average Index and identifies historical crisis events. We propose a new approach to analyze the local regularity variation of a signal and we apply this procedure to a financial data series, attempting to make a contribution to understand the dynamics of financial markets.

Local regularity analysis of market index for the 2008 economical crisis

There is evidence that signals from financial markets, such as stock indices, interest rates or commodities, have a multifractal nature. In recent years, many efforts have been made to relate the inefficiency of markets with the multifractal characteristics of this corresponding signals. These characteristics are summarized in the knowledge of the spectrum of singularities or multifractal spectrum that relates to the set of singular points of the signal with its corresponding Hausdorff dimension. The novel approach proposed in this paper, to study the dynamics of financial markets, is to analyze the evolution of the set of singular points or Hölder exponents of the series of exchanges, measured daily. We examined the “logarithmic returns” of stock indices from 9 countries in developed markets and 12 belonging to emerging markets from February 2006 to March 2009.The analysis reveals that the temporal variation of the local Hölder exponent point reflects the evolution of the crisis and identifies the historical events which have occurred during this phenomenon, from the minimum values of the Hölder exponent.

Local regularity analysis of strata heterogeneities from sonic logs

Abstract. Borehole logs provide geological information about the rocks crossed by the wells. Several properties of rocks can be interpreted in terms of lithology, type and quantity of the fluid filling the pores and fractures. Here, the logs are assumed to be nonhomogeneous Brownian motions (nhBms) which are generalized fractional Brownian motions (fBms) indexed by depth-dependent Hurst parameters H(z). Three techniques, the local wavelet approach (LWA), the average-local wavelet approach (ALWA), and Peltier Algorithm (PA), are suggested to estimate the Hurst functions (or the regularity profiles) from the logs. First, two synthetic sonic logs with different parameters, shaped by the successive random additions (SRA) algorithm, are used to demonstrate the potential of the proposed methods. The obtained Hurst functions are close to the theoretical Hurst functions. Besides, the transitions between the modeled layers are marked by Hurst values discontinuities. It is also shown that PA leads to the best Hurst value estimations. Second, we investigate the multifractional property of sonic logs data recorded at two scientific deep boreholes: the pilot hole VB and the ultra deep main hole HB, drilled for the German Continental Deep Drilling Program (KTB). All the regularity profiles independently obtained for the logs provide a clear correlation with lithology, and from each regularity profile, we derive a similar segmentation in terms of lithological units. The lithological discontinuities (strata' bounds and faults contacts) are located at the local extrema of the Hurst functions. Moreover, the regularity profiles are compared with the KTB estimated porosity logs, showing a significant relation between the local extrema of the Hurst functions and the fluid-filled fractures. The Hurst function may then constitute a tool to characterize underground heterogeneities.

Detection of glitches and signal reconstruction using Hölder and wavelet analysis

We present a method based on a local regularity analysis for detecting and removing artefact signatures in noisy interferometric signals. Using Hölder and wavelet transform modulus maxima lines analysis (WTMML) [S. Mallat, W. Hwang, Singularities detection and processing with wavelets, IEEE Transaction on Information Theory 38 (1992) 617–643] in suitably selected regions of the time-scale half-plane, we can estimate the regularity degree of the signal. Glitches that are considered as a discontinuity on the signal show Hölder component lower than a fixed threshold defined for a continuous signal. After detection and signature removal, the signal is then locally reconstructed using Mallat reconstruction formulae. The method has been tested with Herschel SPIRE FTS proto-flight model calibration observations and shows remarkable results. Optimization of the detection parameters has been performed on the correlation coefficient, the scale domain for Hölder exponent estimation and reconstruction for SPIRE FTS signals.

The genetic algorithm for a signal enhancement

The paper is devoted to the problem of time series enhancement, which is based on the analysis of local regularity. The model construction using this analysis does not require any a priori assumption on the structure of the noise and the functional relationship between original signal and noise. The signal itself may be nowhere differentiable with rapidly varying local regularity, what is overcome with the help of the new technique of increasing the local Hölder regularity of the signal under research. A new signal with prescribed regularity is constructed using the genetic algorithm. This approach is applied to enhancement of time series in the paleoclimatology, solar physics, dendrochronology, meteorology and hydrology.

More on the analysis of local regularity through wavelets

In this paper we want to extend the results of pointwise analysis through wavelet transforms to the class of functions where the local fluctuation is bounded by any submultiplicative function. This generalizez the results obtained before in the well-known case of Holder regularity.