Wavelet Band Research Articles

Revisiting the Fractional Cointegrating Dynamics of Implied-Realized Volatility Relation with Wavelet Band Spectrum Regression

This paper revisits the fractional cointegrating relationship between ex-ante implied volatility and ex-post realized volatility. We argue that the concept of corridor implied volatility (CIV) should be used instead of the popular model-free option-implied volatility (MFIV) when assessing the fractional cointegrating relation as the latter may introduce bias to the estimation. For the realized volatility, we use recently proposed methods which are robust to noise as well as jumps and interestingly we find that it does not affect the implied-realized volatility relation. In addition, we develop a new tool for the estimation of fractional cointegrating relation between implied and realized volatility based on wavelets, a wavelet band least squares (WBLS). The main advantage of WBLS in comparison to other frequency domain methods is that it allows us to work conveniently with potentially non-stationary volatility due to the properties of wavelets. We study the dynamics of the relationship in the time-frequency domain with the wavelet coherence confirming that the dependence comes solely from the lower frequencies of the spectra. Motivated by this result we estimate the relationship only on this part of the spectra using WBLS and compare our results to the fully modified narrow-band least squares (FMNBLS) based on the Fourier frequencies. In the estimation, we use the S&P 500 and DAX monthly and bi-weekly option prices covering the recent financial crisis and we conclude that in the long-run, volatility inferred from the option prices using the corridor implied volatility (CIV) provides an unbiased forecast of the realized volatility.

Static Detection of Thin Layers into Concrete with Ground Penetrating Radar

Ground Penetrating Radar (GPR) is a nondestructive technique particularly well adapted to the inspection of concrete structures and can help to determine the structure inner geometry or to detect damaged areas. When the GPR is used on structures containing thin layers, for example, the sealing layer of a concrete bridge deck or the void into a masonry wall, it is important for the radar user to know the minimum thickness required to detect and estimate the thickness of those layers. The theory of thin layer detection is based on a sine wave but, in reality, most commercial GPR systems emit a large frequency band wavelet, which undergoes attenuation into the layer. To analyze the influence of those realistic conditions on the reflection coefficient of a thin layer, the authors combined experimental measurements and numercial FDTD simulations. The experimental results matched the numerical predictions well, presenting a fast attenuation compared to the theoretical predictions. Nevertheless, for thicknesses inferior to lambda 11, the reflection coefficient could still be considered as linearly dependent of the thickness to wavelength ratio. (A)

Bearing system health condition monitoring using a wavelet cross-spectrum analysis technique

Rolling-element bearings are widely used in rotary machinery systems. Accordingly, a reliable bearing fault detection technique is critically needed in industries to prevent the machinery system's performance degradation, malfunction, or even catastrophic failures. Bearing fault detection, however, still remains a very challenging task because most of the bearing fault related signatures are non-stationary. In this paper, a wavelet cross-spectrum (WCS) technique is proposed to tackle the challenge of feature extraction from these non-stationary signatures for bearing fault detection. The vibration signals are first analyzed by a wavelet transform to demodulate primary representative features; the periodic features are then enhanced by cross-correlating the resulting wavelet coefficient functions over several contributive neighboring wavelet bands. A Jarque-Bera statistic index is suggested for the bandwidth selection. The effectiveness ofthe proposed technique is examined by a series of experimental tests corresponding to different bearing conditions. Test results show that the developed WCS technique is an effective signal processing approach for not only stationary but also non-stationary feature extraction and analysis, and it can be applied effectively for bearing fault detection.

A Wavelet Spectrum Technique for Machinery Fault Diagnosis

Rotary machines are widely used in various applications. A reliable machinery fault detection technique is critically needed in industries to prevent the machinery system’s performance degradation, malfunction, or even catastrophic failures. The challenge for reliable fault diagnosis is related to the analysis of non-stationary features. In this paper, a wavelet spectrum (WS) technique is proposed to tackle the challenge of feature extraction from these non-stationary signatures; this work will focus on fault detection in rolling element bearings. The vibration signatures are first analyzed by a wavelet transform to demodulate representative features; the periodic features are then enhanced by cross-correlating the resulting wavelet coefficient functions over several contributive neighboring wavelet bands. The effectiveness of the proposed technique is examined by experimental tests corresponding to different bearing conditions. Test results show that the developed WS technique is an effective signal processing approach for non-stationary feature extraction and analysis, and it can be applied effectively for bearing fault detection.

Vibration response of spalled rolling element bearings: Observations, simulations and signal processing techniques to track the spall size

Fatigue in rolling element bearings, resulting in spalling of the races and/or rolling elements, is the most common cause of bearing failure. The useful life of the bearing may extend considerably beyond the appearance of the first spall and a premature removal of the bearing from service can be very expensive, but on the other hand chances cannot be taken with safety of machines or personnel. Previous studies indicated that there might be two parts to the defect vibration signal of a spalled bearing, the first part being originating from the entry of the rolling element into the fault (de-stress) and the second part being due to the departure of the rolling element from the fault (re-stress). This is investigated in this paper using vibration signatures of seeded faults at different speeds. The acceleration signals resulting from the entry of the rolling element into the spall and exit from it were found to be of different natures. The entry into the fault can be described as a step response, with mainly low frequency content, while the impact excites a much broader frequency impulse response. The latter is the most noticeable and prominent event, especially when examining the high pass filtered response or the enveloped signal. In order to enable a clear separation of the two events, and produce an averaged estimate of the size of the fault, two approaches are proposed to enhance the entry event while keeping the impulse response. The first approach (joint treatment) utilizes pre-whitening to balance the low and high frequency energy, then octave band wavelet analysis to allow selection of the best band (or scale) to balance the two pulses with similar frequency content. In the second approach, a separate treatment is applied to the step and the impulse responses, so that they can be equally represented in the signal. Cepstrum analysis can be used to give an average estimate of the spacing between the entry and impact events, but the latter can also be assessed by an arithmetic estimation of the mean and standard deviation of the event separation for a number of realizations, in particular for the second approach. In order to determine the effects of various simulations and signal processing parameters on the estimated delay times, the entry and exit events were simulated as modified step and impulse responses with precisely known starting times. The simulation was also found useful in pointing to artefacts associated with the cepstrum calculation, which affect even the simulated signals, and have thus prompted modifications of the processing of real signals. The results presented for the two approaches give a reasonable approximation of the measured fault widths (double the spacing between the entry and impact events) under different speed conditions, but the method of separate treatment is somewhat better and is thus recommended.

ECG signal denoising using higher order statistics in Wavelet subbands

In this work, we propose a novel denoising method based on evaluation of higher-order statistics at different Wavelet bands for an electrocardiogram (ECG) signal. Higher-order statistics at different Wavelet bands provides significant information about the statistical nature of the data in time and frequency. The fourth order cumulant, Kurtosis, and the Energy Contribution Efficiency (ECE) of signal in a Wavelet subband are combined to assess the noise content in the signal. Accordingly, four denoising factors are proposed. Performance of the denoising factors is evaluated and compared with the soft thresholding method. The filtered signal quality is assessed using Percentage Root Mean Square Difference (PRD), Wavelet Weighted Percentage Root Mean Square Difference (WWPRD), and Wavelet Energy-based Diagnostic Distortion (WEDD) measures. It is observed that the proposed denoising scheme not only filters the signal effectively but also helps retain the diagnostic information.

Harmonic wavelet-based data filtering for enhanced machine defect identification

A filter construction technique is presented for enhanced defect identification in rotary machine systems. Based on the generalized harmonic wavelet transform, a series of sub-frequency band wavelet coefficients are constructed by choosing different harmonic wavelet parameter pairs. The energy and entropy associated with each sub-frequency band are then calculated. The filtered signal is obtained by choosing the wavelet coefficients whose corresponding sub-frequency band has the maximum energy-to-entropy ratio. Experimental studies using rolling bearings that contain different types of structural defects have confirmed that the developed new technique enables high signal-to-noise ratio for effective machine defect identification.

Central Limit Theorems for Wavelet Packet Decompositions of Stationary Random Processes

This paper provides central limit theorems for the wavelet packet decomposition of stationary band-limited random processes. The asymptotic analysis is performed for the sequences of the wavelet packet coefficients returned at the nodes of any given path of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -band wavelet packet decomposition tree. It is shown that if the input process is strictly stationary, these sequences converge in distribution to white Gaussian processes when the resolution level increases, provided that the decomposition filters satisfy a suitable property of regularity. For any given path, the variance of the limit white Gaussian process directly relates to the value of the input process power spectral density at a specific frequency.

A wavelet‐based analysis of surface mechanomyographic signals from the quadriceps femoris

The purpose of this study was to use a wavelet analysis designed specifically for surface mechanomyographic (MMG) signals to examine the MMG responses of the vastus lateralis (VL), rectus femoris (RF), and vastus medialis (VM) muscles. Fifteen healthy men [age (mean +/- SD): 26.4 +/- 6.1 years] volunteered to perform isometric muscle actions of the dominant leg extensors at 20%, 40%, 60%, 80%, and 100% of the maximum voluntary contraction (MVC). During each muscle action, surface MMG signals were detected from the VL, RF, and VM and processed with the MMG wavelet analysis. The results show that, for the VL and VM muscles, there was compression of the total MMG intensity spectra toward low frequencies for most force levels above 20% MVC. For the RF, however, the peak of the total MMG intensity spectrum occurred at approximately 30-40 HZ for all force levels. Because the VL, RF, and VM are all innervated by the femoral nerve, the discrepancies among the three muscles for total MMG intensity in each wavelet band may have been due to differences in architecture, muscle stiffness, and/or intramuscular pressure.

Timing and localization of ionospheric signatures associated with substorm expansion phase onset

In this paper, we present case studies of the optical and magnetic signatures of the characteristics of the first minute of substorm expansion phase onset observed in the ionosphere. We find that for two isolated substorms, the onset of magnetic pulsations in the 24–96 s period wavelet band are colocated in time and space with the formation and development of small‐scale optical undulations along the most equatorward preexisting auroral arc prior to auroral breakup. These undulations undergo an inverse spatial cascade into vortices prior to the release of the westward traveling surge. We also present a case study of a multiple activation substorm, whereby discrete onsets of ULF wave power above a predetermined quiet time threshold are shown to be associated with specific optical intensifications and brightenings. Moreover, in the multiple activation substorm event, we show that neither the formation of the small‐scale undulations nor the formation of similar structures along a north–south aligned arc is sufficient to produce auroral breakup associated with expansion phase onset. It is only ∼10 min after these two disparate activation regions initiate that auroral breakup and the subsequent formation of a westward traveling surge occur. We discuss the implications of these results in terms of the triggering mechanisms likely to be occurring during these specific events.

A Novel Fractal Wavelet Image Compression Approach

By investigating the limitation of existing wavelet tree based image compression methods, we propose a novel wavelet fractal image compression method in this paper. Briefly, the initial errors are appointed given the different levels of importance accorded the frequency sublevel band wavelet coefficients. Higher frequency sublevel bands would lead to larger initial errors. As a result, the sizes of sublevel blocks and super blocks would be changed according to the initial errors. The matching sizes between sublevel blocks and super blocks would be changed according to the permitted errors and compression rates. Systematic analyses are performed and the experimental results demonstrate that the proposed method provides a satisfactory performance with a clearly increasing rate of compression and speed of encoding without reducing SNR and the quality of decoded images. Simulation results show that our method is superior to the traditional wavelet tree based methods of fractal image compression.

Automatic Visual Inspection for Leather Manufacture

The visual inspection system was developed for defects detection on leather surfaces, which is an important component of automatic CAD/CAM cutting systems. The main functions of the system are quality control and raw material cutting. An efficient algorithm, which combines multiresolution approach, energy and entropy matrices, is presented for detection of defects embedded in leather surface images. A wavelet band selection procedure was developed to automatically determine the number of resolution levels and decompose subimages for the best discrimination of defects and removals of repetitive texture patterns in the image. An adaptive binary thresholding is then used to separate the defective regions from the uniform gray-level background in the restored image. The proposed methodology is able to efficiently detect several types of defects that current approaches cannot detect, and is fast enough to be used for real-time leather inspection.

Statistical representation of high-dimensional deformation fields with application to statistically constrained 3D warping

This paper proposes a 3D statistical model aiming at effectively capturing statistics of high-dimensional deformation fields and then uses this prior knowledge to constrain 3D image warping. The conventional statistical shape model methods, such as the active shape model (ASM), have been very successful in modeling shape variability. However, their accuracy and effectiveness typically drop dramatically in high-dimensionality problems involving relatively small training datasets, which is customary in 3D and 4D medical imaging applications. The proposed statistical model of deformation (SMD) uses wavelet-based decompositions coupled with PCA in each wavelet band, in order to more accurately estimate the pdf of high-dimensional deformation fields, when a relatively small number of training samples are available. SMD is further used as statistical prior to regularize the deformation field in an SMD-constrained deformable registration framework. As a result, more robust registration results are obtained relative to using generic smoothness constraints on deformation fields, such as Laplacian-based regularization. In experiments, we first illustrate the performance of SMD in representing the variability of deformation fields and then evaluate the performance of the SMD-constrained registration, via comparing a hierarchical volumetric image registration algorithm, HAMMER, with its SMD-constrained version, referred to as SMD+HAMMER. This SMD-constrained deformable registration framework can potentially incorporate various registration algorithms to improve robustness and stability via statistical shape constraints.

Wavelet-Domain Video Denoising Based on Reliability Measures

This paper proposes a novel video denoising method based on nondecimated wavelet band filtering. In the proposed method, motion estimation and adaptive recursive temporal filtering are performed in a closed loop, followed by an intra-frame spatially adaptive filter. All processing occurs in the wavelet domain. The paper introduces new wavelet-based motion reliability measures. We make a difference between motion reliability per orientation and reliability per wavelet band. These two reliability measures are employed in different stages of the proposed denoising scheme. The reliability per orientation (horizontal and vertical) measure is used in the proposed motion estimation scheme while the reliability of the estimated motion vectors (MVs) per wavelet band is utilized for subsequent adaptive temporal and spatial filtering. We propose a novel cost function for motion estimation which takes into account the spatial orientation of image structures and their motion matching values. Our motion estimation approach is a novel wavelet-domain three-step scheme, where the refinement of MVs in each step is determined based on the proposed motion reliabilities per orientation. The temporal filtering is performed separately in each wavelet band along the estimated motion trajectory and the parameters of the temporal filter depend on the motion reliabilities per wavelet band. The final spatial filtering step employs an adaptive smoothing of wavelet coefficients that yields a stronger filtering at the positions where the temporal filter was less effective. The results on various grayscale sequences demonstrate that the proposed filter outperforms several state-of-the-art filters visually (as judged by a small test panel) as well as in terms of peak signal-to-noise ratio

Image analysis using a dual-tree M-band wavelet transform

We propose a two-dimensional generalization to the M-band case of the dual-tree decomposition structure (initially proposed by Kingsbury and further investigated by Selesnick) based on a Hilbert pair of wavelets. We particularly address: 1) the construction of the dual basis and 2) the resulting directional analysis. We also revisit the necessary pre-processing stage in the M-band case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposed M-band decomposition is demonstrated via denoising comparisons on several image types (natural, texture, seismics), with various M-band wavelets and thresholding strategies. Significant improvements in terms of both overall noise reduction and direction preservation are observed.

Matrix factorizations for reversible integer implementation of orthonormal [formula omitted]-band wavelet transforms

This paper presents a matrix factorization method for implementing orthonormal M -band wavelet reversible integer transforms. Based on an algebraic construction approach, the polyphase matrix of orthonormal M -band wavelet transforms can be factorized into a finite sequence of elementary reversible matrices that map integers to integers reversibly. These elementary reversible matrices can be further factorized into lifting matrices, thus we extend the classical lifting scheme to a more flexible framework.

Model‐independent Reconstruction of the Primordial Power Spectrum fromWilkinson Microwave Anistropy ProbeData

Reconstructing the shape of the primordial power spectrum in a model independent way from cosmological data is a useful consistency check on what is usually assumed regarding early universe physics. It is also our primary window to unknown physics during the inflationary era. Using a power-law form for the primordial power spectrum $P_{in}(k)$ and constraining the scalar spectral index and its running, \cite{Peiris03} found that the first year WMAP data seem to indicate a preferred scale in $P_{in}(k)$. We use two complementary methods: the wavelet band powers method of \cite{pia1}, and the top-hat binning method of \cite{Wang99} to reconstruct $P_{in}(k)$ as a free function from CMB data alone (WMAP, CBI, and ACBAR), or from CMB data together with large scale structure data (2dFGRS and PCSZ). The shape of the reconstructed $P_{in}(k)$ is consistent with scale-invariance, although it allows some indication of a preferred scale at $k \sim 0.01$Mpc$^{-1}$. While consistent with the possible evidence for a running of the scalar spectral index found by the WMAP team, our results highlight the need of more stringent and independent constraints on cosmological parameters (the Hubble constant in particular) in order to more definitively constrain deviations of $P_{in}(k)$ from scale-invariance without making assumptions about the inflationary model.

Direct Wavelet Expansion of the Primordial Power Spectrum

In order to constrain and possibly detect unusual physics during inflation, we allow the power spectrum of primordial matter density fluctuations, P_{in}(k), to be an arbitrary function in the estimation of cosmological parameters from data. The multi-resolution and good localization properties of orthogonal wavelets make them suitable for detecting features in P_{in}(k). We expand P_{in}(k) directly in wavelet basis functions. The likelihood of the data is thus a function of the wavelet coefficients of P_{in}(k), as well as the H_0, \Omega_b h^2, $\Omega_c h^2 and the \tau_{ri}, in a flat $\Lambda$CDM cosmology. We derive constraints on these parameters from CMB anisotropy data (WMAP, CBI, and ACBAR) and large scale structure (LSS) data (2dFGRS and PSCZ) using the Markov Chain Monte Carlo (MCMC) technique. The direct wavelet expansion method is different and complimentary to the wavelet band power method of Mukherjee & Wang (2003a,b), and results from the two methods are consistent. In addition, as we demonstrate, the direct wavelet expansion method has the advantage that once the wavelet coefficients have been constrained, the reconstruction of P_{in}(k) can be effectively denoised, i.e., P_{in}(k) can be reconstructed using only the coefficients that, say, deviate from zero at greater than 1\sigma. In doing so the essential properties of P_{in}(k) are retained. The reconstruction also suffers much less from the correlated errors of binning methods. The shape of the primordial power spectrum, as reconstructed in detail here, reveals an interesting new feature at 0.001 \la k/{Mpc}^{-1} \la 0.005. It will be interesting to see if this feature is confirmed by future data. The reconstructed and denoised P_{in}(k) is favored over the scale-invariant and power-law forms at \ga 1\sigma.

Wavelet Band Powers of the Primordial Power Spectrum from Cosmic Microwave Background Data

Measuring the primordial matter power spectrum is our primary means of probing unknown physics in the very early universe. We allow the primordial power spectrum to be an arbitrary function and parameterize it in terms of wavelet band powers. Current cosmological data correspond to 11 such wavelet bands. We derive constraints on these band powers as well as H0, Ωbh2, and Ωmh2 from current cosmic microwave background anisotropy data using the Markov chain Monte Carlo technique. Our results indicate a feature in the primordial power spectrum at 0.008 k/(h Mpc-1) 0.1. Wilkinson Microwave Anisotropy Probe and Planck data should allow us to put tighter constraints on the primordial power spectrum.

Automatic band selection for wavelet reconstruction in the application of defect detection

In this paper, we present a multiresolution approach for the inspection of local defects embedded in homogeneously textured surfaces. It is based on an efficient image restoration scheme using the wavelet transforms. By properly selecting the smooth subimage or the combination of detail subimages at different resolution levels for image reconstruction, the global repetitive texture pattern can be effectively removed and only local anomalies are preserved in the restored image. A wavelet band selection procedure is developed to automatically determine the best reconstruction parameters based on the energy distribution of wavelet coefficients. Experimental results show that the decomposed subimages and the number of resolution levels determined by the automatic band selection scheme are similar to the manual selection results, and the defects in a variety of real textures including machined surfaces, natural wood, sandpaper and textile fabrics are well detected.