Wavelet Formulation Research Articles

Formulation of Wavelet Based Multi‐Scale Multi‐Objective Performance Evaluation (WMMPE) Metric for Improved Calibration of Hydrological Models

AbstractEstimation of the hydrological model parameters, known as model calibration, is an important step in application of these models for various water resources problems. In the auto‐calibration approach, the match between the simulated and observed variables is assessed using the statistical performance measures, which are used as objective functions. Earlier studies show that the use of traditional single‐objective functions alone cannot capture all the features of the hydrographs, motivating the utilization of multi‐objective calibration. Further, wavelet‐based approaches can capture features that occur at different time scales. In this study, Wavelet Transforms are employed to formulate a Wavelet based Multi‐Scale Multi‐Objective Performance Evaluation (WMMPE) metric that combines multiple standard metrics at multiple time scales. The performance of WMMPE along with other objective functions such as Nash Sutcliffe Efficiency (NSE), Kling Gupta Efficiency (KGE)), Euclidean distance based Multi Objective Calibration Function and wavelet based Multi‐Scale NSE in calibrating two conceptual hydrologic models were tested on 53 watersheds. The performance of the five calibrated models were evaluated for their efficacy in simulating different characteristics of hydrograph such as, low and high flows, water balance components, frequency domain characteristics and average error magnitude. The results show that WMMPE calibrated models are performing well in matching all the flow segments of hydrograph in >22 basins for the two conceptual models. Further, results of the analysis suggested that the presence of outliers have minimum impact on the simulation of WMMPE calibrated models in the validation period.

Neural Network Renormalization Group.

We present a variational renormalization group (RG) approach based on a reversible generative model with hierarchical architecture. The model performs hierarchical change-of-variables transformations from the physical space to a latent space with reduced mutual information. Conversely, the neural network directly maps independent Gaussian noises to physical configurations following the inverse RG flow. The model has an exact and tractable likelihood, which allows unbiased training and direct access to the renormalized energy function of the latent variables. To train the model, we employ probability density distillation for the bare energy function of the physical problem, in which the training loss provides a variational upper bound of the physical free energy. We demonstrate practical usage of the approach by identifying mutually independent collective variables of the Ising model and performing accelerated hybrid MonteCarlo sampling in the latent space. Lastly, we comment on the connection of the present approach to the wavelet formulation of RG and the modern pursuit of information preserving RG.

Modal Analysis of Ambient PMU Measurements Using Orthogonal Wavelet Bases

This paper proposes a new method, called wavelet scale decomposition, for modal analysis of ambient synchrophasor data using orthogonal wavelet bases. Wavelet formulation of the problem enables reliable estimation results even with short data analysis windows that are a few minutes long. Also, in addition to finding the mode frequency and damping ratio of oscillatory modes, the proposed method is able to estimate their mode shapes as well. The method is tested on some archived real phasor measurement unit data sets and on simulations from Kundur two area system. The results show that the proposed method is able to track damping ratio variations in power systems effectively with short analysis windows.

Modelling of flow‐dependent ensemble‐based background‐error correlations using a wavelet formulation in 4D‐Var at Météo‐France

The use of a wavelet formulation and ensemble data assimilation is considered in this study, in order to specify flow‐dependent background‐error correlations for the Météo‐France global 4D‐Var system. This wavelet model is calibrated by using a temporal series of recent perturbed backgrounds, which is sampled in a sliding way. This approach is compared to a homogeneous and static spectral formulation of error correlations. Diagnostic studies are conducted in order to illustrate the representation of heterogeneous and flow‐dependent correlation features, corresponding for instance to smaller horizontal length‐scales in the vicinity of midlatitude lows and to larger vertical correlations in vertically unstable regions. The positive impact of the flow‐dependent wavelet formulation on the forecast quality is then shown during two contrasting three‐week periods. The impact of an increased ensemble size when using this approach is also illustrated, in terms of length‐scale dynamics and forecast quality.

Interplay between functional connectivity and scale-free dynamics in intrinsic fMRI networks

Studies employing functional connectivity-type analyses have established that spontaneous fluctuations in functional magnetic resonance imaging (fMRI) signals are organized within large-scale brain networks. Meanwhile, fMRI signals have been shown to exhibit 1/f-type power spectra — a hallmark of scale-free dynamics. We studied the interplay between functional connectivity and scale-free dynamics in fMRI signals, utilizing the fractal connectivity framework — a multivariate extension of the univariate fractional Gaussian noise model, which relies on a wavelet formulation for robust parameter estimation. We applied this framework to fMRI data acquired from healthy young adults at rest and while performing a visual detection task. First, we found that scale-invariance existed beyond univariate dynamics, being present also in bivariate cross-temporal dynamics. Second, we observed that frequencies within the scale-free range do not contribute evenly to inter-regional connectivity, with a systematically stronger contribution of the lowest frequencies, both at rest and during task. Third, in addition to a decrease of the Hurst exponent and inter-regional correlations, task performance modified cross-temporal dynamics, inducing a larger contribution of the highest frequencies within the scale-free range to global correlation. Lastly, we found that across individuals, a weaker task modulation of the frequency contribution to inter-regional connectivity was associated with better task performance manifesting as shorter and less variable reaction times. These findings bring together two related fields that have hitherto been studied separately — resting-state networks and scale-free dynamics, and show that scale-free dynamics of human brain activity manifest in cross-regional interactions as well.

Identification of DNA-Binding and Protein-Binding Proteins Using Enhanced Graph Wavelet Features

Interactions between biomolecules play an essential role in various biological processes. For predicting DNA-binding or protein-binding proteins, many machine-learning-based techniques have used various types of features to represent the interface of the complexes, but they only deal with the properties of a single atom in the interface and do not take into account the information of neighborhood atoms directly. This paper proposes a new feature representation method for biomolecular interfaces based on the theory of graph wavelet. The enhanced graph wavelet features (EGWF) provides an effective way to characterize interface feature through adding physicochemical features and exploiting a graph wavelet formulation. Particularly, graph wavelet condenses the information around the center atom, and thus enhances the discrimination of features of biomolecule binding proteins in the feature space. Experiment results show that EGWF performs effectively for predicting DNA-binding and protein-binding proteins in terms of Matthew's correlation coefficient (MCC) score and the area value under the receiver operating characteristic curve (AUC).

Progressive compression of manifold polygon meshes

This paper presents a new algorithm for the progressive compression of manifold polygon meshes. The input surface is decimated by several traversals that generate successive levels of detail through a specific patch decimation operator which combines vertex removal and local remeshing. The mesh connectivity is encoded by two lists of Boolean error predictions based on the mesh geometry: one for the inserted edges and the other for the faces with a removed center vertex. The mesh geometry is encoded with a barycentric error prediction of the removed vertex coordinates and a local curvature prediction. We also include two methods that improve the rate-distortion performance: a wavelet formulation with a lifting scheme and an adaptive quantization technique. Experimental results demonstrate the effectiveness of our approach in terms of compression rates and rate-distortion performance.

Modal identification based on continuous wavelet transform and ambient excitation tests

This paper deals with the use of the continuous wavelet transform for modal identification from ambient vibration tests. The wavelet analysis of the free responses of a linear mechanical system has been used to estimate its natural frequencies, viscous damping ratios, and mode shapes, using either the modulus or the phase of the wavelet transform with “ridge” and “skeleton” notions. This technique is extended in this paper, to the response from ambient excitation tests based on the equivalent formulation between the free response and the correlation function of a linear mechanical system under stationary stochastic excitation. The main novelties involve the new formulation of continuous wavelet transform and the use of the singular value decomposition algorithm for ridges and skeleton detection under the noisy conditions of ambient vibration tests. A complete procedure for modal identification of ambient response, including these new formulations, is presented and then applied to numerical examples and experimental test to illustrate its validity.

B-Spline Wavelet on the Interval Element for the Solution of Hamilton Canonical Equation

Based on the modified Hellinger-Reissner (H-R) variational principle for the elastic material, the formulation of B-spline wavelet on the interval (BSWI) element of the Hamilton canonical equation was derived through the use of the interpolation scaling function of BSWI. To demonstrate the excellent predictive capability of the formulation, numerical studies were conducted on a thick plate and a three-layer plate with various common boundary conditions. The basic steps from which the formulation of the BSWI element of the Hamilton canonical equation was derived can also be extended to deduce the expressions of other wavelet elements in the Hamiltonian system, such as Daubechies orthogonal wavelet, biorthogonal wavelet based on lifting scheme, and so on.

New density estimation methods for charged particle beams with applications to microbunching instability

In this paper we discuss representations of charge particle densities in particle-in-cell simulations, analyze the sources and profiles of the intrinsic numerical noise, and present efficient methods for their removal. We devise two alternative estimation methods for charged particle distribution which represent significant improvement over the Monte Carlo cosine expansion used in the 2D code of Bassi et al. [G. Bassi, J. A. Ellison, K. Heinemann, and R. Warnock, Phys. Rev. ST Accel. Beams 12, 080704 (2009); G. Bassi and B. Terzi\ifmmode \acute{c}\else \'{c}\fi{}, in Proceedings of the 23rd Particle Accelerator Conference, Vancouver, Canada, 2009 (IEEE, Piscataway, NJ, 2009), TH5PFP043], designed to simulate coherent synchrotron radiation (CSR) in charged particle beams. The improvement is achieved by employing an alternative beam density estimation to the Monte Carlo cosine expansion. The representation is first binned onto a finite grid, after which two grid-based methods are employed to approximate particle distributions: (i) truncated fast cosine transform; and (ii) thresholded wavelet transform (TWT). We demonstrate that these alternative methods represent a staggering upgrade over the original Monte Carlo cosine expansion in terms of efficiency, while the TWT approximation also provides an appreciable improvement in accuracy. The improvement in accuracy comes from a judicious removal of the numerical noise enabled by the wavelet formulation. The TWT method is then integrated into the CSR code [G. Bassi, J. A. Ellison, K. Heinemann, and R. Warnock, Phys. Rev. ST Accel. Beams 12, 080704 (2009)], and benchmarked against the original version. We show that the new density estimation method provides a superior performance in terms of efficiency and spatial resolution, thus enabling high-fidelity simulations of CSR effects, including microbunching instability.

Diagnostic and impact studies of a wavelet formulation of background‐error correlations in a global model

AbstractA wavelet formulation on the sphere is being considered for modelling heterogeneous background‐error correlations for the Météo‐France global numerical weather prediction (NWP) model. This approach is compared with the operational spectral formulation, which is horizontally homogeneous to a large extent. Diagnostic studies have been conducted to examine geographical variations of three‐dimensional correlations over the whole globe. Results indicate that the contrast between relatively broad horizontal correlations in the Tropics and sharp ones in midlatitudes is well represented by the wavelet formulation. Heterogeneities in vertical correlations are also better captured in the wavelet approach than in the spectral one, with visible changes as functions of e.g. latitude and land/sea contrasts. In addition, wavelet‐based correlation estimates are shown to be partly sensitive to the choice of the calibration period. The impact of the wavelet formulation on the forecast quality has been investigated during a three‐week calibration period, and also during the following three weeks. While the impact of the wavelet formulation is globally positive during the two periods, it tends to be more spectacular during the calibration time interval, as expected. These results indicate that an on‐line calibration should be considered in the future, in order to exploit fully the ability of wavelets to extract correlation heterogeneities from ensemble data. Copyright © 2011 Royal Meteorological Society

Heterogeneous Correlation Modeling Based on the Wavelet Diagonal Assumption and on the Diffusion Operator

Abstract This article discusses several models for background error correlation matrices using the wavelet diagonal assumption and the diffusion operator. The most general properties of filtering local correlation functions, with wavelet formulations, are recalled. Two spherical wavelet transforms based on Legendre spectrum and a gridpoint spherical wavelet transform are compared. The latter belongs to the class of second-generation wavelets. In addition, a nonseparable formulation that merges the wavelets and the diffusion operator model is formally proposed. This hybrid formulation is illustrated in a simple two-dimensional framework. These three formulations are tested in a toy experiment on the sphere: a large ensemble of perturbed forecasts is used to simulate a true background error ensemble, which gives a reference. This ensemble is then applied to compute the required parameters for each model. A randomization method is utilized in order to diagnose these different models. In particular, their ability to represent the geographical variations of the local correlation functions is studied by diagnosis of the local length scale. The results from these experiments show that the spectrally based wavelet formulation filters the geographical variations of the local correlation length scale but it is less able to represent the anisotropy. The gridpoint-based wavelet formulation is also able to represent some parts of the geographical variations but it appears that the correlation functions are dependent on the grid. Finally, the formulation based on the diffusion represents quite well the local length scale.

Deciphering plume–lithosphere interactions beneath Europe from topographic signatures

When a mantle plume rises and impinges on the base of the lithosphere, it expectably produces variations in surface topography. Taking into consideration a realistic mantle rheology, plume ascent rates can reach tens to hundreds of metres per year, whereupon the impingement of the plume head at the base of the lithosphere can be considered as an “impact". Recent numerical experiments based on tectonically realistic formulations for the lithosphere and a representative mantle rheology, have shown that plume-induced undulations exhibit temporal successions of uplift and subsidence at various wavelengths. From spectral (Fourier) analyses of the undulations would appear that two groups of wavelengths (200–400 km and 60–100 km) predominate. Interestingly, a spectral analysis of Europe's topography also reveals two dominant groups. In the present study, we have used a spectral analysis with a wavelet formulation in order to discriminate between tectonically-induced undulations (uni-directional deformation) and plume-induced undulations (omni-directional deformation). The European lithosphere is well-suited for this approach since it has been suggested that two mantle plumes (the Massif Central and the Eifel area) underlie Western Europe, where Alpine compression has folded the lithosphere over several hundreds of kilometres. The wavelet analysis of Europe's surface topography confirms that the energy distribution of the topographic undulations outside the two main volcanic provinces is homogeneous, thus contrasting with the coexistence of both large-scale and medium-scale high-energy features that are obtained for the Massif Central and Eifel areas. Similar signatures are also found beneath the northern Sudetes area. The wavelet approach dedicated to the analysis of plume-related topographic signatures needs, however, detailed theoretical grounds and would probably benefit from two-dimensional analyses.

Elastic thickness structure of South America estimated using wavelets and satellite-derived gravity data

We used a wavelet formulation of the classical spectral isostatic analysis to invert satellite-derived gravity and topography/bathymetry for elastic thickness ( T e) over South America and its surrounding plates. To provide a homogeneous representation of the gravity field for this vast region, we corrected free-air anomalies derived from a combination of terrestrial/marine gravity data with data from the GRACE and CHAMP satellite missions (model EIGEN-CG03C) by a simple Bouguer slab using a smoothed representation of surface relief (wavelengths > 125 km). The resulting Bouguer anomaly compares well with terrestrial data acquired in the Central Andes and allows T e to be confidently estimated for values greater than 10 km. The T e map resolves regional-scale features that are well-correlated with known surface structures and shows maximum values of 100 ± 15 km over the Archean–Neoproterozoic core of the continent, decreasing to less than 30 km around continental margins. Several regions of the oceanic plates and continental margins have an elastic thickness less than 10 km. We performed a quantitative analysis by comparing the elastic thickness with the thermal structure predicted from the age of oceanic crust and igneous–metamorphic rocks. This demonstrates that oceanic plates have been weakened by thermal interaction with hotspots and locally by fracturing and hydration near the trench. We observe that only the nucleus of the continent has resisted the thermomechanical weakening induced by the rifting of Africa and South America along the passive margin and the Andean orogeny along the active margin. This latter region shows along-strike variations in T e that correlate with the geotectonic segmentation of the margin and with the pattern of crustal seismicity. Our results reveal that the rigidity structure follows the segmentation of the seismogenic zone along the subduction fault, suggesting a causal relationship that should be investigated in order to improve the understanding and predictability of great earthquakes and tsunamis.

Wavelet‐Based Poisson Solver for Use in Particle‐in‐Cell Simulations

We report on a successful implementation of a wavelet-based Poisson solver for use in three-dimensional particle-in-cell simulations. Our method harnesses advantages afforded by the wavelet formulation, such as sparsity of operators and data sets, existence of effective preconditioners, and the ability simultaneously to remove numerical noise and additional compression of relevant data sets. We present and discuss preliminary results relating to the application of the new solver to test problems in accelerator physics and astrophysics.

Generalized hierarchical bases: a Wavelet‐Ritz‐Galerkin framework for Lagrangian FEM

PurposeThis paper presents a novel framework for solving elliptic partial differential equations (PDEs) over irregularly spaced meshes on bounded domains.Design/methodology/approachSecond‐generation wavelet construction gives rise to a powerful generalization of the traditional hierarchical basis (HB) finite element method (FEM). A framework based on piecewise polynomial Lagrangian multiwavelets is used to generate customized multiresolution bases that have not only HB properties but also additional qualities.FindingsFor the 1D Poisson problem, we propose – for any given order of approximation – a compact closed‐form wavelet basis that block‐diagonalizes the stiffness matrix. With this wavelet choice, all coupling between the coarse scale and detail scales in the matrix is eliminated. In contrast, traditional higher‐order (n>1) HB do not exhibit this property. We also achieve full scale‐decoupling for the 2D Poisson problem on an irregular mesh. No traditional HB has this quality in 2D.Research limitations/implicationsSimilar techniques may be applied to scale‐decouple the multiresolution finite element (FE) matrices associated with more general elliptic PDEs.Practical implicationsBy decoupling scales in the FE matrix, the wavelet formulation lends itself particularly well to adaptive refinement schemes.Originality/valueThe paper explains second‐generation wavelet construction in a Lagrangian FE context. For 1D higher‐order and 2D first‐order bases, we propose a particular choice of wavelet, customized to the Poisson problem. The approach generalizes to other elliptic PDE problems.

Wavelet formulation of path integral Monte Carlo

A wavelet formulation of path integral Monte Carlo (PIMC) is constructed. Comparison with Fourier path integral Monte Carlo is presented using simple one-dimensional examples. Wavelet path integral Monte Carlo exhibits a few advantages over previous methods for PIMC. The efficiency of the current method is at least comparable to other techniques.

Dynamics of the transition zone under Europe inferred from wavelet cross-spectra of seismic tomography

Tomographical results are commonly presented in the form of color images and not much statistical quantification has been carried out on the derived models. Correlation between different depths can shed important light concerning the dynamics. We have generalized the application of multidimensional wavelets to investigate the products of two field variables, such as the cross-spectrum, which is of paramount importance for quantifying the correlation between two depth levels of seismic tomography with a multiple-scale character. For two multidimensional fields A and B, we calculate the correlation C by projecting this as an Hermitian inner product in physical space with a two-dimensional (2D), fourth derivative of the Gaussian wavelet as the weighting function. The correlation function C becomes now a multi-scaled function, a map cast in terms of both the scale and location of the wavelet transform. Having calculated C, we can delineate the locations and length-scales of the prominent features in the landscape of the correlation function. This wavelet formulation is very general and can be extended to other types of statistical analysis, for example in a Kalman filter system. We have used a high-resolution (finer than 1°) seismic tomographical model for analyzing the extent of mantle layering under Europe by focussing on the different length-scales in the correlation function involving the 3D seismic anomalies lying between 400 and 600 km depth. Between the depths of 500 and 600 km under Europe, the wavelet correlation analysis shows that an ellipse-shaped object exists with an area of 2000 km×4000 km having a strong correlation for length-scales of around 400 km, and weaker correlation for shorter length scales of around 150 km. On the other hand, between depths of 400 and 600 km, the correlation deteriorates on the long length scales and becomes even worse at the short length scales. From the wavelet correlation spectra, we can extract an horizontal characteristic length scale of around 100 km, which may be related to the boundary interaction between the slab and the ambient mantle. The correlation results suggest that the thickness of the recumbent fast (cold) material in the transition zone is between 100 and 150 km. This large elliptical pattern of presumably cold material would act to inhibit the vigor of mantle convection locally beneath Europe today.

A wavelet formulation of the finite-difference method: full-vector analysis of optical waveguide junctions

We have developed an efficient, large-stencil finite-difference scheme of the time-dependent Maxwell's curl equations based on the wavelet-collocation formulation in the time-domain. The proposed scheme enables, for the first time within a limited computational resource, full-vector analysis of three-dimensional rib waveguides that are typically used in integrated planar optical devices. The formulation takes advantage of compactly-supported interpolating bases to expand and represent the electric and magnetic fields. Moreover, unlike the well-known beam propagation methods, the numerical scheme is based on the first-principle algorithm with no explicit approximation, and thus rigorous and versatile for various types of boundary conditions. We demonstrate the efficiency of the method by first analyzing a straight rib-waveguide and examining the convergence of the results. Then we investigate a Y-shaped junction structure that is electrically too large to analyze with the conventional finite-difference time-domain scheme.

Robust Hierarchical Indexing based on Texture Features

In this paper, we present a hierarchical indexing method based on texture characterization for image retrieval. The novelty of our contribution is the hierarchical structure of the index: it exploits the multiresolution formulation of Wavelet Transforms to define a new set of approximated versions of the images for each level of resolution. On this set, the algorithm extracts significant signatures by means of statistical correlations; the experimental results and the analysis of computational complexity have proved that the algorithm presents the best performance at the highest level of the indexing hierarchy, where the computational complexity is the lowest. Our method has been evaluated by the following methodologies: (a) the study of the computational complexity for signature generation; (b) the comparison with analogous methods based on texture analysis by reporting the performance obtained on the same database (Brodatz); and (c) the evaluation of the robustness of the hierarchical indexing in different color spaces, by querying the database with different versions of the original images obtained by noise addition (gaussian and scanner acquisition noise and lossy compression distortion), brightness and contrast enhancement, color and scale adjustment and rotation. Even if our method is designed for texture databases, experiments show satisfactory results also on a real heterogeneous photographic database. This confirms the possibility of exploiting our method as a low computational complexity indexing tool based on texture characterization in a broader system for hierarchical content-based retrieval.