Wavelet Transform Modulus Maxima Research Articles

Outliers Detection of Dam Displacement Monitoring Data Based on Wavelet Transform

Dam safety monitoring data can be viewed as a digital signal sequence which consists of different frequency components. Identifying outliers to ensure the reliability of observational data becomes a foundation work of dam monitoring data analysis. Outliers of time signal series can be detected by wavelet transform. Lipschitz index can be used to measure the local singularity of a function, and the original abnormal signal can be found in the position of wavelet transform modulus maxima. Take horizontal displacement values observed by Lijiaxia concrete dam as example, an assumed error are added to the time series signal deliberately. A 4-level decomposition of the observation data was done by using wavelet db1, the results show that the modulus maxima occur at the given time. Therefore, outliers can be detected and located accurately by wavelet transform, which is important to analyze the safety monitoring data of dam.

Multiscale Singularity Analysis of Cutting Forces for Micromilling Tool-Wear Monitoring

This paper introduces a novel approach for tool condition monitoring (TCM) in micromilling, which is based on the analysis of singularity of cutting force waveforms and the statistical distribution of the singularity measure. It is a multiscale approach with both localized and global cutting force analyses. The local singularity characteristics of the cutting force are measured by the Holder exponent at the point and determined by wavelet transform modulus maxima. The global characteristics are measured by the singularity spectrum, which represents the overall behavior of the singularity among different tool-wear states. Experimental studies from both micromilling of copper and steel show that the axial component of the cutting force is robust and most discriminant for TCM using its singularity measurements.

NONLINEAR FRACTURE SIGNAL ANALYSIS USING MULTIFRACTAL APPROACH COMBINED WITH WAVELETS

Based on the continuous Wavelet Transform Modulus Maxima method (WTMM), a multifractal analysis was introduced to discriminate the irregular fracture signals of materials. This method provides an efficient numerical technique to characterize statistically the local regularity of fractures. The results obtained by this nonlinear analysis suggest that multifractal parameters such as the capacity dimension D0, the average singularity strength α0, the aperture of the left side (α0 - α min ) and the total width (α max - α min ) of the D(α) spectra allow a better fit for the characterization of the different fracture stages. Discriminating the three principal stages of the fracture namely the fracture initiation, the fracture propagation and the final rupture, provides a powerful diagnostic tool to identify the crack initiation site, and thus delineates the causes of the cracking of the material.

Fractal analysis revisited by the continuous wavelet transform of AVO seismic data

The main goal of this paper is to establish reservoirs media heterogeneities by the wavelet transform modulus maxima lines. First, we gathered amplitude versus offsets (AVO) amplitudes at the top of the reservoirs, then we calculated the 2D wavelet transform after we calculated its maxima, and we estimated the Holder exponent at each maxima. Variation of the Holder exponent can give more information about lithology and fluid nature at any point.

Multifractal properties of large bubble paths in a single bubble column

Multifractal properties of large bubble paths in a single bubble columnIn the paper the paths of bubbles emitted from the brass nozzle with inner diameter equal to 1.6 mm have been analyzed. The mean frequency of bubble departure was in the range from 2 to 65.1 Hz. Bubble paths have been recorded using a high speed camera. The image analysis technique has been used to obtain the bubble paths for different mean frequencies of bubble departures. The multifractal analysis (WTMM - wavelet transform modulus maxima methodology) has been used to investigate the properties of bubble paths. It has been shown that bubble paths are the multifractals and the influence of previously departing bubbles on bubble trajectory is significant for bubble departure frequencyfb> 30 Hz.

Multifractal Analysis Revisited by the Continuous Wavelet Transform Applied in Lithofacies Segmentation from Well-Logs Data

The main goal of this paper is to use the wavelet transform modulus maxima lines (WTMM) and the detrended fluctuations analysis (DFA) methods to establish a new technique of lithofacies segmentation from well logs data. The WTMM is used to delimitate lithoafacies boundaries and the DFA is used to provide an exact estimation of the roughness coefficient of lithofacies. Application of the proposed idea at the synthetic and real data of a borehole located in Berkine basin shows that the proposed technique can enhance reservoirs characterization.

Perinuclear distribution of heterochromatin in developing C. elegans embryos

Specific nuclear domains are nonrandomly positioned within the nuclear space, and this preferential positioning has been shown to play an important role in genome activity and stability. Well-known examples include the organization of repetitive DNA in telomere clusters or in the chromocenter of Drosophila and mammalian cells, which may provide a means to control the availability of general repressors, such as the heterochromatin protein 1 (HP1). We have specifically characterized the intranuclear positioning of in vivo fluorescence of the Caenorhabditis elegans HP1 homologue HPL-2 as a marker for heterochromatin domains in developing embryos. For this purpose, the wavelet transform modulus maxima (WTMM) segmentation method was generalized and adapted to segment the small embryonic cell nuclei in three dimensions. The implementation of a radial distribution algorithm revealed a preferential perinuclear positioning of HPL-2 fluorescence in wild-type embryos compared with the diffuse and homogeneous nuclear fluorescence observed in the lin-13 mutants. For all other genotypes analyzed, the quantitative analysis highlighted various degrees of preferential HPL-2 positioning at the nuclear periphery, which directly correlates with the number of HPL-2 foci previously counted on 2D projections. Using a probabilistic 3D cell nuclear model, we found that any two nuclei having the same number of foci, but with a different 3D probabilistic positioning scheme, can have significantly different counts in the 2D maximum projection, thus showing the deceptive limitations of using techniques of 2D maximum projection foci counts. By this approach, a strong perinuclear positioning of HPL-2 foci was brought into light upon inactivation of conserved chromatin-associated proteins, including the HAT cofactor TRAPP.

QUANTIFYING THE EVOLVING MAGNETIC STRUCTURE OF ACTIVE REGIONS

The topical and controversial issue of parameterizing the magnetic structure of solar active regions has vital implications in the understanding of how these structures form, evolve, produce solar flares, and decay. This interdisciplinary and ill-constrained problem of quantifying complexity is addressed by using a two-dimensional wavelet transform modulus maxima (WTMM) method to study the multifractal properties of active region photospheric magnetic fields. The WTMM method provides an adaptive space-scale partition of a fractal distribution, from which one can extract the multifractal spectra. The use of a novel segmentation procedure allows us to remove the quiet Sun component and reliably study the evolution of active region multifractal parameters. It is shown that prior to the onset of solar flares, the magnetic field undergoes restructuring as Dirac-like features (with a Holder exponent, h = –1) coalesce to form step functions (where h = 0). The resulting configuration has a higher concentration of gradients along neutral line features. We propose that when sufficient flux is present in an active region for a period of time, it must be structured with a fractal dimension greater than 1.2, and a Holder exponent greater than –0.7, in order to produce M- and X-class flares. This result has immediate applications in the study of the underlying physics of active region evolution and space weather forecasting.

CHARACTERIZING COMPLEXITY IN SOLAR MAGNETOGRAM DATA USING A WAVELET-BASED SEGMENTATION METHOD

The multifractal nature of solar photospheric magnetic structures are studied using the 2D wavelet transform modulus maxima (WTMM) method. This relies on computing partition functions from the wavelet transform skeleton defined by the WTMM method. This skeleton provides an adaptive space-scale partition of the fractal distribution under study, from which one can extract the multifractal singularity spectrum. We describe the implementation of a multiscale image processing segmentation procedure based on the partitioning of the WT skeleton which allows the disentangling of the information concerning the multifractal properties of active regions from the surrounding quiet-Sun field. The quiet Sun exhibits a average H\"older exponent $\sim -0.75$, with observed multifractal properties due to the supergranular structure. On the other hand, active region multifractal spectra exhibit an average H\"older exponent $\sim 0.38$ similar to those found when studying experimental data from turbulent flows.

Investigating the Multifractal Properties of Significant Wave Height Time Series Using a Wavelet-Based Approach

Singularities play a significant role in the characterization of time series. The temporal characteristics of fluctuating significant wave height time series are investigated in this study. The hourly time series from 24 stations located off the west coast of the United States are used for the analysis. The multifractal nature of these time series is unveiled by employing a wavelet-based method. The wavelet transform modulus maxima method is applied to obtain the multifractal spectra (singularity spectrum) of the significant wave height series. The multifractal spectra and their parameters such as peak, min and max Holder exponents, skewness coefficients, and support lengths are calculated. The peak Holder exponent ranged from 0.30 to 0.46 throughout the study area. Holder exponents that are less than 0.5 indicate that the time series exhibits an antipersistent random walk. The spatial variation of parameters is depicted through kriging maps. Different spatial variation patterns can be seen from the maps. It is clear that deep offshore stations have relatively higher Holder exponents than coastal areas. This change can be related to the wave generation mechanism, by way of physical interpretations. Since the stations located in the deep offshore can receive more swell waves than coastal zones and are open to large-scale storms, they may tend to be more persistent and have greater Holder exponents. Also, the type of the singularities occurring in deep offshore and in the coastal zones is assessed by considering the wave generating mechanisms.

A Outlier Identification and Correction Method Based on Wavelet Transform

There are many outliers in air pollution time series data for various reasons. It has a serious impact on the data analysis and use. There are three main ways to identify anomalies but they each have definite limitations, especially when identifying and correcting the first category and the second category of outlier at the same time. In order to solve this problem, this paper presents a new way to identify anomalies based on wavelet transform and identify outlier by the use of the wavelet transform modulus maxima , then pass the amendment of the outlier through inverse transform the wavelet transform coefficient. Evidence shows that this method can be used to identify and correct the two types of outlier simultaneously and the results are obvious.

2D Continuous Wavelet Transform of potential fields due to extended source distributions

We analyse the real Continuous Wavelet Transform 2D (CWT2D) of potential fields for the investigation of potential field singularities. We focus our attention to extended geological sources, in order to verify the reliability of this method with realistic fields. 3D space-scale representation (3D Scalogram) related to synthetic models were generated, showing the Wavelet Transform Modulus Maxima (WTMM) at each scale. The WTMM are related to the shape of the source, so defining some sort of source boundary analysis through the CWT. Wavelets of different order may help to gain resolution and define source features. Selecting a range of scales where the sources behave as if they are approximately isolated, the depth to the source may be estimated basing on the property that the lines joining the modulus maxima of the wavelet coefficients at different scales (WTMML) intersect each other at the edges of the causative body. Therefore, it is possible to manage the information contained in the wavelet transform of fields related to extended sources. In the real case of the anomaly gravity map of the Vesuvius area (Italy), we estimated the depth of the Mesozoic carbonate basement in the Pompei Basin. We showed also how the WTMML information can be integrated to that of another multiscale method, the Depth from Extreme Points (DEXP) transformation, which is also related to the source density distribution of a given region.

Automated Detection of Coronal Loops Using a Wavelet Transform Modulus Maxima Method

We propose and test a wavelet transform modulus maxima method for the automated detection and extraction of coronal loops in extreme ultraviolet images of the solar corona. This method decomposes an image into a number of size scales and tracks enhanced power along each ridge corresponding to a coronal loop at each scale. We compare the results across scales and suggest the optimum set of parameters to maximize completeness, while minimizing detection of noise. For a test coronal image, we compare the global statistics (e.g. number of loops at each length) to previous automated coronal-loop detection algorithms.

WEAK LFM SIGNAL DECTECTION BASED ON WAVELET TRANSFORM MODULUS MAXIMA DENOISING AND OTHER TECHNIQUES

A new method for detecting weak linear frequency modulated (LFM) pulse signals buried in additive white Gaussian noise (AWGN) is presented in this paper. The method is based on the features of wavelet transform modulus maxima (WTMM) denoising and auto-correlation filtering theory. Firstly, the frequency-domain information is extracted by auto-correlation matched filtering, and is used to deduce the optimal wavelet decomposition scales. Secondly, let the signal modulus dominate on the biggest scale after the optimal scales decomposition, then keeping the signal modulus and removing the noise modulus at each scale are performed by utilizing the different propagation properties of signal and noise wavelet modulus maxima across the scales. Finally, a reconstructed signal is obtained from the reserved signal modulus with an improved signal-to-noise ratio (SNR), and is used for time-domain information extraction. At the same time, wavelet denoising depends on selecting an optimum wavelet that matches well the shape of the signal. The cross correlation coefficients between signal and db wavelets are calculated and the optimal wavelet to analysis the LFM signal is selected. Simulations show that the method can extract time-frequency information of LFM signal when SNR ≤ -6 dB .

Wavelet-based multifractal analysis of DNA sequences by using chaos-game representation

Chaos game representation (CGR) is proposed as a scale-independent representation for DNA sequences and provides information about the statistical distribution of oligonucleotides in a DNA sequence. CGR images of DNA sequences represent some kinds of fractal patterns, but the common multifractal analysis based on the box counting method cannot deal with CGR images perfectly. Here, the wavelet transform modulus maxima (WTMM) method is applied to the multifractal analysis of CGR images. The results show that the scale-invariance range of CGR edge images can be extended to three orders of magnitude, and complete singularity spectra can be calculated. Spectrum parameters such as the singularity spectrum span are extracted to describe the statistical character of DNA sequences. Compared with the singularity spectrum span, exon sequences with a minimal spectrum span have the most uniform fractal structure. Also, the singularity spectrum parameters are related to oligonucleotide length, sequence component and species, thereby providing a method of studying the length polymorphism of repeat oligonucleotides.

Analysis of flow structures and energy spectra in chemical process equipment

Studies have been carried out to extract relevant information on flow structures, their break-up distribution and the associated local turbulence phenomena using the 1D and 2D energy spectra. The 1D analysis uses the hot film anemometry (HFA) and large eddy simulation (LES) data sets, while the 2D analysis uses the PIV results. The chemical process equipment considered for the study are jet loop reactor (JLR), channel flow (CHA) and bubble column reactor (BCR). The work involves relative comparison of spectral methods, namely, fast Fourier transform (FFT), discrete wavelet transform (DWT), continuous wavelet transform (CWT), eddy isolation methodology (EIM) and proper orthogonal decomposition (POD) in evaluating the 1D and 2D spectra for the three equipment. FFT studies are used to identify the dominant frequencies, but they do not reveal the time dependent changes in the flow structure. This limitation is overcome by using the EIM, DWT and CWT and methodologies for identification of flow structures are discussed. Using DWT, a novel intermittency-based quantifier, namely, the scale relative local intermittency measure (SRLIM) is found to be particularly useful for identification of intermittent flow structures. For example, it has been used to track the intermittent bursts in the CHA, the vortex tube related intermittencies in the JLR and the bubble induced eddies in the BCR. The advantages of CWT for obtaining the eddy time-scale and length-scale distribution from the velocity time series is shown. The evaluation of distributions has been carried out using wavelet transform modulus maxima (WTMM) along with a symbolic analysis of the loci of the maxima lines. The structure break-up distribution and the energy break-up distribution that are obtained show a generalization to the known L and P models for intermittencies. The turbulence parameters, i.e., local turbulent kinetic energy, energy dissipation rate and turbulent viscosity, have been calculated and compared using the results obtained from the FFT, DWT, CWT and EIM. For the different approaches these parameters lie in reasonable ranges.

Non-extensive properties, multifractality, and inefficiency degree of the Athens Stock Exchange General Index

We examine the dynamic properties of the daily returns of the Athens Stock Exchange General Index. The probability density and cumulative distribution functions are studied using the generalized non-extensive statistics framework. The multifractal properties are investigated by application of wavelet transform modulus maxima and multifractal detrended fluctuation analysis methods. The generalized Hurst exponent, saturating for large moment values, shows a significant multifractality range which is connected to the inefficiency of the market, compared to the matured markets.

Continuous wavelet transform based time-scale and multifractal analysis of the nonlinear oscillations in a hollow cathode glow discharge plasma

Continuous wavelet transform (CWT) based time-scale and multifractal analyses have been carried out on the anode glow related nonlinear floating potential fluctuations in a hollow cathode glow discharge plasma. CWT has been used to obtain the contour and ridge plots. Scale shift (or inversely frequency shift), which is a typical nonlinear behavior, has been detected from the undulating contours. From the ridge plots, we have identified the presence of nonlinearity and degree of chaoticity. Using the wavelet transform modulus maxima technique we have obtained the multifractal spectrum for the fluctuations at different discharge voltages and the spectrum was observed to become a monofractal for periodic signals. These multifractal spectra were also used to estimate different quantities such as the correlation and fractal dimension, degree of multifractality, and complexity parameters. These estimations have been found to be consistent with the nonlinear time series analysis.

EMPIRICAL TESTING OF MULTIFRACTALITY OF FINANCIAL TIME SERIES BASED ON WTMM

The multifractal spectrum calculated with wavelet transform modulus maxima (WTMM) provides information on the higher moments of market returns distribution and the multiplicative cascade of volatilities. This paper applies a wavelet based methodology for calculation of the multifractal spectrum of financial time series. WTMM methodology provides a better measure of risk changes compared to the structure function approach. It is well founded in applied mathematics and physics with little popularity among finance researchers.

Lacunarity of Rough Surfaces From the Wavelet Analysis of Scattering Data

Many natural and man-made rough surfaces exhibit multiscale features and may be advantageously described by fractal models. The surface roughness is then characterized by the fractal dimension while the distribution of gap or scale sizes (or gapiness) is measured by the lacunarity. Fractal dimension and lacunarity are two independent descriptors, that is, two surfaces having the same fractal dimension may look different and differ according to the distribution of gap or scale sizes. The remote analysis of the scaling properties of such multiscale surfaces may be achieved from interrogation by an electromagnetic pulse. Numerous approaches have been developed to describe the relationship between the fractal dimension of rough surfaces and the properties of the scattered waves. But, despite its potential importance in engineering, the remote extraction of the lacunarity from scattering data has not received the same attention. This paper is devoted to the extraction of the lacunarity of rough (fractal) surface from the wavelet analysis of the near-field impulse response using a heuristic method. It is shown that the skeleton of the wavelet-transform modulus-maxima of the scattered signal allows the straightforward estimation of the surface lacunarity for the examples studied here. Finally the wavelet analyze is applied for studying rough surfaces of different lacunarities and various fractal dimensions.