Wavelet Transform Modulus Maxima Research Articles

DETECTION OF ATMOSPHERIC TURBULENCE BY MULTIFRACTAL ANALYSIS USING WAVELETS

We study the singularities of a temperature profile obtained by means of balloon measurements in the troposphere and lower stratosphere. The data give the evolution of the temperature as the altitude of the probe increases. We compare the scaling exponents deduced from the Wavelet Transform Modulus Maxima (WTMM) method and the structure function method. In the lower stratosphere, the variations in the multifractal properties with the altitude deduced from wavelets allow us to detect thin layers of about 200 m depth exhibiting atmospheric turbulence.

Chaotic pairwise competition

We investigate a kind of competition possible in a system of at least three populations competing for the same limited resource. As a model we use generalised Volterra equations in which the growth rates and competition coefficients of populations depend on the number of members of all populations. Because of the nonconstant values of the last quantities the system could be repelled from the state of cyclic pairwise competition described by May and Leonard (SIAM J. Appl. Math. 29 (1975) 243.). We investigate the competition in a chaotic regime of evolution of the number of members of populations. We show that the nonconstant competition coefficients can lead to a regularisation of the time intervals of domination of each population and the non-constant growth rates can lead to decreasing length of the time intervals of domination as well as to chaotisation of the occurrence of these intervals. A quantity characterising the time intervals between the successive maxima of the number of the populations individuals is discussed. By means of the wavelet transform modulus maxima method we calculate the τ( q)-spectrum and the Hölder exponent for the time series of this quantity. The results of the theory are illustrated by an example of competition among the three main political parties in Bulgaria and we discuss qualitative aspects of the dynamics of change of preferences of voters.

Decreased Fractal Correlation in Diurnal Physical Activity in Chronic Fatigue Syndrome

Our objectives were to study the temporal correlation of physical activity time series in patients with chronic fatigue syndrome (CFS) during normal daily life and to examine if it could identify the altered physical activity in these patients. Fractal scaling exponents of diurnal and nocturnal physical activity time series in 10 CFS patients and 6 healthy control subjects (CON) were calculated by the detrended fluctuation analysis (DFA) and the wavelet transform modulus maxima (WTMM) method. We hypothesized that, due to their illness- and/or fatigue-induced resting episodes, altered physical activity patterns in CFS patients might be observed at the interruption of activity bursts. Thus, we further developed a new method, the wavelet transform negative modulus maxima (WTNMM) method, which could evaluate the temporal correlation at the interruption of activities. We compared the fractal scaling exponents for CFS and CON by each method. Both for CFS and CON, we found the fractal time structures in their diurnal physical activity records for at least up to 35 minutes. No group difference was found in nocturnal activities. The WTNMM method revealed that, in diurnal activities, CFS patients had significantly (p < 0.01) smaller fractal scaling exponent (0.87 +/- 0.03) compared to controls (1.01 +/- 0.03). Such a difference was identified neither by the DFA nor WTMM method. CFS patients had more abrupt interruptions of voluntary physical activity during diurnal periods in normal daily life, probed by the decreased correlation in the negative modulus maxima of the wavelet-transformed activity data, possibly due to their exaggerated fatigue.

Joint application of continuous and discrete wavelet transform on gravity data to identify shallow and deep sources

The discrete wavelet transform (dwt), using the good property of localization of wavelet bases has been used as a powerful tool in filtering and denoising problems. The continuous wavelet transform (cwt) exploits the upward continuation properties of the field horizontal derivative and allows the location of potential field singularities in a simple geometrical manner. Within the cwt space-scale framework, the lines formed by joining, at different scales, the modulus maxima of the wavelet coefficients (multiscale edge detection method) intersect each other at the position of the point source or along the edges of the causative body. As long as the multiscale edge detection method is applied to experimental data the procedure may, however, fail, since the observed anomalies are the superposition of effects of sources having different density contrast, geometrical size and depths. We show that wavelet transform modulus maxima lines attributed to deep sources do not converge toward the true depths, but yield completely erroneous solutions. On the other hand, use of nth-order derivatives of the potential field allows the enhancement of the shallowest source effects, preventing us from obtaining information on the deeper ones. In this paper we therefore try to overcome this problem by a joint application of cwt and dwt. A localized dwt filter coupled to compactness criterion allows the separation of the effects due to the deeper sources from those of the shallower ones. Hence, the multiscale edge detection method, applied separately to the original and the filtered signals enabled the estimation of the depth of shallower and deeper sources, respectively. This analysis, performed on the gravity anomalies of Sardinia (Italy), has given estimations of the depths to both the Campidano graben and the Moho discontinuity, in good agreement with previous interpretations of gravity and seismic data.

Asymmetrical singularities in real-world signals

We generalize the wavelet transform modulus maxima approach in order to analyze positive and negative changes separately and show different singularity spectra depending on the direction of changes in (i) human heartbeat interval data during sympathetic blockade, (ii) time series of daytime human physical activity of healthy individuals (but not of patients with debilitating fatigue), and (iii) daily stock price records of the Nikkei 225 in the period 1990-2002--but not of the S&P 500. We conclude that the analysis of asymmetrical singularities provides deeper insights into the underlying complexity of real-world signals that can greatly enhance our understanding of the mechanisms determining the systems' dynamics.

Three-dimensional wavelet-based multifractal method: the need for revisiting the multifractal description of turbulence dissipation data.

We generalize the wavelet transform modulus maxima (WTMM) method to multifractal analysis of 3D random fields. This method is calibrated on synthetic 3D monofractal fractional Brownian fields and on 3D multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration. Then we apply the 3D WTMM method to the dissipation field issued from 3D isotropic turbulence simulations. We comment on the need to revisit previous box-counting analyses which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master nonconservative singular cascade measures.

Multifractality of river runoff and precipitation: comparison of fluctuation analysis and wavelet methods

We study the multifractal temporal scaling properties of river discharge and precipitation records. We compare the results for the multifractal detrended fluctuation analysis method with the results for the wavelet-transform modulus maxima technique and obtain agreement within the error margins. In contrast to previous studies, we find non-universal behaviour: on long time scales, above a crossover time scale of several weeks, the runoff records are described by fluctuation exponents varying from river to river in a wide range. Similar variations are observed for the precipitation records which exhibit weaker, but still significant multifractality. For all runoff records the type of multifractality is consistent with a modified version of the binomial multifractal model, while several precipitation records seem to require different models.

A NEW METHOD FOR AUTOMATIC DETECTION OF THE ELECTROCARDIOGRAM (ECG) CHARACTERISTIC POINTS

In this article we propose a new method based on wavelet transform and on neural networks to detect the maximum, onset and offset of electrocardiogram (ECG) characteristic waves. First, the wavelet transform modulus maxima is used to extract main information of the ECG. We use Neural Networks to discern important modulus maxima and with them, the maximum point of a wave is detected. Then, a forward and backward search is made to detect the onset and offset of ECG waves. The detection degree of the maximum of the waves was proved using the QT Database and compared with another method. As a result, the proposed method can work with several morphologies of highly noisy ECGs.

A new iterative algorithm for reconstructing a signal from its dyadic wavelet transform modulus maxima

A new algorithm for reconstructing a signal from its wavelet transform modulus maxima is presented based on an iterative method for solutions to monotone operator equations in Hilbert spaces. The algorithm’s convergence is proved. Numerical simulations for different types of signals are given. The results indicate that compared with Mallat’s alternate projection method, the proposed algorithm is simpler, faster and more effective.

Wavelet-watershed automatic infrared image segmentation method

A novel approach for infrared image segmentation with applications to ship surveillance and classification is presented. It employs a watershed transformation using a marker that combines wavelet transform modulus maxima with morphological techniques. This approach is totally automatic and has many advantages compared with the traditional methods, such as its ability to extract objects beyond or below the environment temperature, robustness to noise inherent to infrared images, as well as a solution of the horizon problem.

Wavelet-based estimators of scaling behavior

Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are studied extensively. These estimators mainly include the (bi)orthogonal wavelet estimators and the wavelet transform modulus maxima (WTMM) estimator. This study focuses both on short and long time-series. In the framework of fractional autoregressive integrated moving average (FARIMA) processes, we advocate the use of approximately adapted wavelet estimators. For these ideal processes, the scaling behavior actually extends down to the smallest scale, i.e., the sampling period of the time series, if an adapted decomposition is used. But in practical situations, there generally exists a cutoff scale below which the scaling behavior no longer holds. We test the robustness of the set of wavelet-based estimators with respect to that cutoff scale as well as to the specific density of the underlying law of the process. In all situations, the WTMM estimator is shown to be the best or among the best estimators in terms of the mean-squared error (MSE). We also compare the wavelet estimators with the detrended fluctuation analysis (DFA) estimator which was previously proved to be among the best estimators which are not wavelet-based estimators. The WTMM estimator turns out to be a very competitive estimator which can be further generalized to characterize multiscaling behavior.

SINGULARITY ANALYSIS USING CONTINUOUS WAVELET TRANSFORM FOR BEARING FAULT DIAGNOSIS

In this paper, wavelet transform is applied to detect abrupt changes in the vibration signals obtained from operating bearings being monitored. In particular, singularity analysis across all scales of the continuous wavelet transform is performed to identify the location (in time) of defect-induced bursts in the vibration signals. Through modifying the intensity of the wavelet transform modulus maxima, defect-related vibration signature is highlighted and can be easily associated with the bearing defect characteristic frequencies for diagnosis. Due to the fact that vibration characteristics of faulty bearings are complex and defect-related vibration signature is normally buried in the wideband noise and high frequency structural resonance, simple signal processing cannot be used to detect bearing fault. We show, through experimental results, that the proposed method has the ability to discriminate noise from the signal significantly and is robust to bearing operating conditions, such as load and speed, and severity of the bearing damage. These properties are desirable for automatic detection of machine faults.

Multifractal detrended fluctuation analysis of nonstationary time series

We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series with those for shuffled series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima method, and show that the results are equivalent.

Multiscality in the dynamics of coupled chaotic systems

We investigate the scaling features of complex motions in systems of two coupled chaotic oscillators by means of the wavelet-transform modulus maxima method and the detrended fluctuation analysis. We show that the transition from asynchronous to synchronous dynamics typically reduces the degree of multiscality in the characteristic temporal scales. Correlation properties caused by adjustment of the involved time scales are discussed, and experimental results for coupled functional units of the kidney are analyzed.

Wavelet transform based multifractal formalism in outlier detection and localisation for financial time series

We present a method of detecting and localising outliers in financial time series and other stochastic processes. The method checks the internal consistency of the scaling behaviour of the process within the paradigm of the multifractal spectrum. Deviation from the expected spectrum is interpreted as the potential presence of outliers. The detection part of the method is then supplemented by the localisation analysis part, using the local scaling properties of the time series. Localised outliers can then be removed one by one, with the possibility of dynamic verification of spectral properties. Both the multifractal spectrum formalism and the local scaling properties of the time series are implemented on the wavelet transform modulus maxima tree.

Spectral-temporal filtering of NDT data using wavelet transform modulus maxima

In this paper, a method is described for the filtering of non-destructive testing (NDT) signals. By employing wavelet transform modulus maxima filtering in the time–frequency plane, it allows the high frequency components associated with features within the signal to be retained while high frequency components associated with both signal ring down and noise are removed. The method is applied to sonic echo signals associated with the NDT of piled foundations. The method allows for spectral-temporal filtering of the signal and offers a superior partitioning of the signal of interest from both noise and ringdown artefact. The method has applications in the elucidation of other signals.

Focused local learning with wavelet neural networks

A novel objective function is presented that incorporates both local and global errors as well as model parsimony in the construction of wavelet neural networks. Two methods are presented to assist in the minimization of this objective function, especially the local error term. First, during network initialization, a locally adaptive grid is utilized to include candidate wavelet basis functions whose local support addresses the local error of the local feature set. This set can be either user-defined or determined using information derived from the wavelet transform modulus maxima representation. Next, during the network construction, a new selection procedure based on a subspace projection operator is presented to help focus the selection of wavelet basis functions to reduce the local error. Simulation results demonstrate the effectiveness of these methodologies in minimizing local and global error while maintaining model parsimony and incurring a minimal increase on computational complexity.

Wavelet analysis of transients in fractal superlattices

Fractal superlattices are designed by alternating dielectric layers according to an iterative fractal process. The discrete self-similarity of such stratified structures can be remotely detected from interrogation by an incident pulse. In this paper, the impulse response of one-dimensional Cantor superlattices is computed and the wavelet transform is applied to the reflected signal in order to explore its temporal distribution of singularities. For a sufficiently narrow pulse, the skeleton of the wavelet-transform modulus-maxima exhibits a hierarchical structure that makes apparent the iterative process governing the construction rule of the interrogated fractal superlattice and its geometry. In the reflected signal, such hierarchy reveals the existence of singularities that are distributed on the governing Cantor set. Finally, the similarity dimension is extracted from reflection data and a strategy for estimating the stage of growth is developed.

Improving training of radial basis function network for classification of power quality disturbances

Features extracted from non-stationary and transitory power quality disturbances using wavelet transform modulus maxima can serve as powerful discriminating features for wavelet-based classification of these disturbances. Using these features, a comprehensive 'knowledge-based' algorithm is proposed for the training of the radial basis function network classifier, so that at its convergence the network gives both the optimal feature weight vector as well as the cluster centres and scaling widths.

Wavelet-based partition function for remote analysis of fractal superlattices

A wavelet-based partition function is introduced for remotely estimating the similarity dimension of discrete self-similar (fractal) superlattices. From the impulse-response wavelet-transform of such multi-scale structures the partition function is computed by summing only the contribution of the wavelet-transform modulus-maxima that belong to a hierarchical structure in the time-scale domain.