Showing papers on "Multiresolution analysis published in 2017"
TL;DR: A wavelet-based detection algorithm to deal with the nonstationary signatures of phasor measurement units (PMU) signals and successful results of detection and classification in real-world cases are presented.
Abstract: In order to deal with the nonstationary signatures of phasor measurement units (PMU) signals, this paper presents a wavelet-based detection algorithm. Moreover, for an application to PMU for event detection purpose, it is necessary for us to classify detected events into unexpected real power related accidents, such as generator trip or automated control, such as reactive power injection. The proposed normalized wavelet energy function calculates the root mean square (RMS) of detail coefficients from time-synchronized voltage and frequency that reflect nonstationary occurrence of significant changes in signals. For a robust detection, wavelet-based detection parameter is designed with consideration of nonstationary characteristics of events. Also, there are distinct transients in voltage and frequency caused by different event types, and distinct results are key-idea of event classification. Besides the determination of event occurrences, one can obtain the information of event characteristics that include event types and zonal information of event from the proposed method. Moreover, successful results of detection and classification in real-world cases are presented in this paper.
114 citations
TL;DR: A new approach based on semi-supervised machine learning and wavelet design applied to non-intrusive load monitoring is presented, using co-training of two machine learning classifiers to automate the process of learning the load pattern after designing new wavelets.
Abstract: This paper presents a new approach based on semi-supervised machine learning and wavelet design applied to non-intrusive load monitoring. Co-training of two machine learning classifiers is used to automate the process of learning the load pattern after designing new wavelets. The numerical results demonstrating the effectiveness of the proposed approach are discussed and conclusions are drawn.
79 citations
TL;DR: A novel wavelet multiresolution complex network (WMCN) is proposed for analyzing multivariate nonlinear time series and it is proposed that each time series is analyzed as a node and the connections are determined in terms of the distance among the feature vectors extracted from wavelet coefficients series.
Abstract: Characterizing complicated behavior from time series constitutes a fundamental problem of continuing interest and it has attracted a great deal of attention from a wide variety of fields on account of its significant importance. We in this paper propose a novel wavelet multiresolution complex network (WMCN) for analyzing multivariate nonlinear time series. In particular, we first employ wavelet multiresolution decomposition to obtain the wavelet coefficients series at different resolutions for each time series. We then infer the complex network by regarding each time series as a node and determining the connections in terms of the distance among the feature vectors extracted from wavelet coefficients series. We apply our method to analyze the multivariate nonlinear time series from our oil–water two-phase flow experiment. We construct various wavelet multiresolution complex networks and use the weighted average clustering coefficient and the weighted average shortest path length to characterize the nonlin...
65 citations
TL;DR: This paper proposes a novel pansharpening algorithm, which combines the conceptions of CS and MRA, and can be regarded as a generalized version of the existing band-dependent spatial-detail (BDSD) algorithm.
Abstract: Modern optical satellites can acquire bundles of panchSromatic (PAN) and multispectral (MS) images of the scene simultaneously. Because of the complexity of the sensors and amount of data involved, an MS image always has lower spatial resolution than the corresponding PAN image. Pansharpening aims at fusing MS images and PAN images, characterized by the spectral content of the former and the spatial details of the latter. There are two main large families of pansharpening algorithms, i.e., component substitution (CS) and multiresolution analysis (MRA). Generally speaking, the CS algorithms have better performance on spatial detail injection, while the MRA shows better spectral content preservation. In this paper, we propose a novel pansharpening algorithm, which combines the conceptions of CS and MRA. This proposed algorithm can be regarded as a generalized version of the existing band-dependent spatial-detail (BDSD) algorithm. A semisimulated dataset and three real datasets are adopted to compare the performance among the generalized-BDSD algorithm and six existing popular pansharpening algorithms. It shows that the proposed method has much lower spectral distortion and good visual appearance. In other words, the proposed method aggregates the advantages of CS and MRA, which shows effectiveness in practice.
54 citations
TL;DR: There is no consensus about what quaternion wavelet transforms should look like and what their properties should be, so this article reviews what has been written and concludes with an analysis of what it is that should define a QWT as being truly quaternionic.
48 citations
TL;DR: The AWLP method has been revisited and its performance significantly improved by simply performing the histogram matching of Pan to the individual MS bands, rather than to the intensity component, thereby losing the original proportionality feature.
Abstract: In this paper, the authors investigate the statistical matching of the panchromatic (Pan) image to the multispectral (MS) bands, also known as the histogram matching , for the two main classes of pansharpening methods, ie, those based on component substitution (CS) or spectral methods and those based on multiresolution analysis (MRA) or spatial methods Also, hybrid methods combining CS with MRA, like the widespread additive wavelet luminance proportional (AWLP), are investigated It is shown that all spectral, spatial, and hybrid methods must perform a dynamics matching of the enhancing Pan to the individual MS bands for MRA or a combination of them (the component that shall be substituted) for CS For hybrid methods, the problem is more complex and both types of histogram matching may be suitable Such an intersensor balance may be either explicit or implicitly performed by the detail-injection model, eg, the popular projective and multiplicative injection models An experimental setup exploiting IKONOS and WorldView-2 data sets demonstrates that a correct histogram matching is the key to attain extra performance from established methods As a first result of this paper, the AWLP method has been revisited and its performance significantly improved by simply performing the histogram matching of Pan to the individual MS bands, rather than to the intensity component, thereby losing the original proportionality feature
47 citations
TL;DR: An adaptive saliency detection method based on clustering and spectral dissimilarity is presented and nonlinear intensity–hue–saturation transform with multiresolution analysis based on dual-tree complex wavelet transform is combined in order to complement each other’s advantages.
Abstract: In remote sensing images, demands for spectral and spatial resolution vary from region to region. Regions with abundant texture and well-defined boundaries (like residential areas and roads) need more spatial details to provide better descriptions of various ground objects while regions such as farmland and mountains are mainly discriminated by spectral characteristic. However, most existing fusion algorithms for remote sensing images execute a unified processing in the whole image, leaving those important needs out of consideration. The employment of diverse fusion strategy for regions with different needs can provide an effective solution to this problem. In this letter, we propose a new saliency-driven fusion method based on complex wavelet transform. First, an adaptive saliency detection method based on clustering and spectral dissimilarity is presented to generate saliency factor for indicating diverse needs of the two kinds of resolutions in regions. Then, we combine nonlinear intensity–hue–saturation transform with multiresolution analysis based on dual-tree complex wavelet transform in order to complement each other’s advantages. Finally, saliency factor is employed to control the detail injection in the fusion, helping to satisfy different needs of different regions. Experiments reveal the validity and advantages of our proposal.
40 citations
01 Jan 2017
TL;DR: The wavelet transform is a family of functions constructed by using translation and dilation of a single function, called the mother wavelet, for the analysis of nonstationary signals.
Abstract: Historically, the concept of wavelets started to appear more frequently only in the early 1980s. This new concept can be viewed as a synthesis of various ideas originating from different disciplines including mathematics, physics, and engineering. One of the main reasons for the discovery of wavelets and wavelet transforms is that the Fourier transform analysis does not contain the local information of signals. So the Fourier transform cannot be used for analyzing signals in a joint time and frequency domain. In 1982, Jean Morlet, in collaboration with a group of French engineers, first introduced the idea of wavelets as a family of functions constructed by using translation and dilation of a single function, called the mother wavelet, for the analysis of nonstationary signals. Wavelet transforms are relatively recent developments that have fascinated the scientific, engineering, and mathematics community with their versatile applicability. The application areas for wavelets have been growing for the last 20 years at a very rapid rate. They have been applied in a number of fields including signal and image processing, sampling theory, turbulence, differential equations, statistics, quality control, computer graphics, economics and finance, medicine, neural networks, geophysics, astrophysics, quantum mechanics, neuroscience, and chemistry. For more information about the history and applications of wavelet transforms, the reader is referred to Daubechies (1992), Chui (l992), Meyer (1993a,b), Kaiser (1994), Cohen (1995), Hubbard (1996), Strang and Nguyen (1996), Burrus et al. (1997), Wojtaszczyk (1997), Debnath (1998a,b,c), Mallat (1998), Pinsky (2001), Ali et al. (2015), Gomes and Velho (2015), and Debnath and Shah (2015).
38 citations
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TL;DR: It is shown how the discovery of robust scalable numerical solvers for arbitrary bounded linear operators can be automated as a Game Theory problem by reformulating the process of computing with partial information and limited resources as that of playing underlying hierarchies of adversarial information games.
Abstract: We show how the discovery of robust scalable numerical solvers for arbitrary bounded linear operators can be automated as a Game Theory problem by reformulating the process of computing with partial information and limited resources as that of playing underlying hierarchies of adversarial information games. When the solution space is a Banach space B endowed with a quadratic norm ∥⋅∥, the optimal measure (mixed strategy) for such games (e.g. the adversarial recovery of u ∈ B, given partial measurements [ϕ_i,u] with ϕ_i ∈ B^∗, using relative error in ∥⋅∥-norm as a loss) is a centered Gaussian field ξ solely determined by the norm ∥⋅∥, whose conditioning (on measurements) produces optimal bets. When measurements are hierarchical, the process of conditioning this Gaussian field produces a hierarchy of elementary bets (gamblets). These gamblets generalize the notion of Wavelets and Wannier functions in the sense that they are adapted to the norm ∥⋅∥ and induce a multi-resolution decomposition of B that is adapted to the eigensubspaces of the operator defining the norm ∥⋅∥. When the operator is localized, we show that the resulting gamblets are localized both in space and frequency and introduce the Fast Gamblet Transform (FGT) with rigorous accuracy and (near-linear) complexity estimates. As the FFT can be used to solve and diagonalize arbitrary PDEs with constant coefficients, the FGT can be used to decompose a wide range of continuous linear operators (including arbitrary continuous linear bijections from H^s_0 to H^(−s) or to L^2) into a sequence of independent linear systems with uniformly bounded condition numbers and leads to O(NpolylogN) solvers and eigenspace adapted Multiresolution Analysis (resulting in near linear complexity approximation of all eigensubspaces).
35 citations
01 Oct 2017
TL;DR: The results show clearly various forms of changes in amplitude and frequency of the signals, which shows that this method is fast, sensitive, and practical for detection and identification of power quality disturbance.
Abstract: With the growing use of sensitive and susceptive electronic and computing equipment, power quality is foreseen to cause a great concern to electric utilities. The best analysis on power quality is vital to provide better service to customers. Disturbances in power system usually produce continuity changes in the power signal. Wavelet transform is particularly useful in detecting discontinuities in signals, and this makes it appropriate for detection of disturbances in power quality. Wavelet transform is proposed to detect and identify the power quality disturbance at its instance of occurrence. Power quality disturbances are sag, swell, interruption, transient and harmonic. This study reviews various kinds of power quality disturbances with the goal of detecting them using wavelet transform. The results show clearly various forms of changes in amplitude and frequency of the signals. The application shows that this method is fast, sensitive, and practical for detection and identification of power quality disturbance.
33 citations
TL;DR: In this article, a new operational matrix method based on Haar wavelets is proposed to solve linear and non-linear differential equations of fractional order, which does not require the inverse of the Haar matrices.
Abstract: In this paper, a new operational matrix method based on Haar wavelets is proposed to solve linear and non-linear differential equations of fractional order. Contrary to wavelet operational methods available in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The main characteristics of our approach is that it converts fractional differential equations to system of algebraic equations and does not require the inverse of the Haar matrices. Illustrative examples are included to demonstrate the validity and applicability of the present method. Moreover, special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.
01 Oct 2017
TL;DR: A fault-diagnosis technique to identify shorted switches based on features generated through wavelet transform of the converter output and subsequent classification in support vector machines is proposed and analyzed.
Abstract: The modular multilevel converter with series and parallel connectivity was shown to provide advantages in several industrial applications. Its reliability largely depends on the absence of failures in the power semiconductors. We propose and analyze a fault-diagnosis technique to identify shorted switches based on features generated through wavelet transform of the converter output and subsequent classification in support vector machines. The multi-class support vector machine is trained with multiple recordings of the output of each fault condition as well as the converter under normal operation. Simulation results reveal that the proposed method has high classification latency and high robustness. Except for the monitoring of the output, which is required for the converter control in any case, this method does not require additional module sensors.
TL;DR: The selection of the optimal wavelet and thresholding for PD pulses in order to apply the wavelet technique to PD detection under HVDC revealed that processing PD pulses with the mother wavelet of bior2.6, automatic threshold, and intermediate thresholding function presented the best performance.
Abstract: With the rapid development of HVDC technology, the detection and analysis of partial discharge (PD) under HVDC are new challenges to ensure reliable operation of the related power apparatus. The wavelet technique has been proposed for analyzing PD pulses under HVAC and ultra-high frequency signal, but its application for PD under HVDC has not been discussed. This paper dealt with the selection of the optimal wavelet and thresholding for PD pulses in order to apply the wavelet technique to PD detection under HVDC. Four electrode systems, namely protrusion on conductor, protrusion on enclosure, free particle, and crack inside spacer were fabricated to simulate typical defects in a gas insulated switchgear. The detected PD pulses were decomposed by multiresolution analysis. The correlation coefficient and dynamic time warping methods were used to select the optimal wavelet. The optimal threshold and thresholding function were chosen from various combinations with the simulated pulses. The results revealed that processing PD pulses with the mother wavelet of bior2.6, automatic threshold, and intermediate thresholding function presented the best performance.
TL;DR: An optimized wavelet designed by fractionally delaying the coefficients of the unit delay filter is introduced, named as fraclet, and the proposed algorithm relying on fraclets has shown better results as compared with wavelet.
Abstract: Summary
This work introduces an optimized wavelet designed by fractionally delaying the coefficients of the unit delay filter. The wavelets produced by optimized fractionally delayed filter coefficients are named as fraclets. Fraclet has been utilized in the real-time power quality events' detection and classification which has been hitherto addressed with the help of unit delay filters. Normal signal, voltage sag, swell, harmonics, sag with harmonics, swell with harmonics, sag and swell with harmonics, and sag and swell with interruption are considered in this work to validate the performance of proposed algorithm based on fraclet. Along with real-time generation of various power quality events with TMS320C6748 DSP board, different events have also been simulated by using parametric equations. The upper hand of fraclets over wavelets has been highlighted by maximally flat frequency response of fraclets. Moreover, fraclets facilitate better multiresolution analysis by providing low energy compaction ratio (ECR). Further, 11 characteristic features have also been extracted from each decomposition level of PQ signal up to fifth level and fed to the modified probabilistic neural network (MPNN) for validating the proposed power quality event detection and classification algorithm. MPNN has outperformed the support vector machine (SVM), both with fraclet and wavelet. The proposed algorithm relying on fraclets has shown better results as compared with wavelet.
TL;DR: The memetic algorithm identifies the position for each sensor while the 2-D discrete Haar wavelet enhances the quality of coverage by moving the sensors in the optimal position.
Abstract: In recent days, wireless sensor network has been playing a significant role in various applications. Sensor placement imposes a real challenge when the number of sensors is limited. For a target coverage problem, the critical part is achieving maximum quality of coverage with limited sensors. The quality of coverage can be improved by placing the sensors in the optimal position such that it monitors entire targets. In this paper, a hybrid memetic algorithm with 2-D discrete Haar wavelet is proposed for identifying the best position for each sensor. The memetic algorithm identifies the position for each sensor while the 2-D discrete Haar wavelet enhances the quality of coverage by moving the sensors in the optimal position. Simulation results have been carried out to prove the efficiency of proposed hybrid algorithm, and is compared with the available algorithm in the terms of quality of coverage, the number of sensors, and the optimal positions.
TL;DR: Adapt numerical schemes for the Vlasov equation are developed by combining discontinuous Galerkin discretisation, multiresolution analysis and semi-Lagrangian time integration by implementing a tree based structure to achieve adaptivity.
TL;DR: A framework to calculate CC2 approximated coupled-cluster ground state correlation energies in a multiresolution basis is derived and implemented into the MADNESS library, and singularities arising from the nuclear and electronic potentials are regularized by explicitly taking thenuclear and electronic cusps into account.
Abstract: A framework to calculate CC2 approximated coupled-cluster ground state correlation energies in a multiresolution basis is derived and implemented into the MADNESS library. The CC2 working equations are formulated in first quantization which makes them suitable for real-space methods. The first quantized equations can be interpreted diagrammatically using the usual diagrams from second quantization with adjusted interpretation rules. Singularities arising from the nuclear and electronic potentials are regularized by explicitly taking the nuclear and electronic cusps into account. The regularized three- and six-dimensional cluster functions are represented directly on an adaptive grid. The resulting equations are free of singularities and virtual orbitals, which results in a low intrinsic scaling. Correlation energies close to the basis set limit are computed for small molecules. This work is the first step toward CC2 excitation energies in a multiresolution basis.
TL;DR: The numerical results show that the adaptive meshfree spectral graph wavelet method can accurately capture the emergence of the localized patterns at all the scales and the node arrangement is accordingly adapted.
TL;DR: In this article, a new methodology is proposed based on assigning weights to each selection criteria based on analytic hierarchy process (AHP) depending on the nature of vibration data and then finding the overall ranking of each wavelet.
Abstract: Vibration in machine tools beyond safe limits is a serious concern which affects the performance of the machine tools. For predicting the degradation in performance, periodic measurement of vibration of machine tools is an important step. For dynamic signal analysis, wavelet transform has been increasingly applied for system health monitoring. One of the key steps in processing a signal with wavelet transform is a selection of appropriate mother wavelet. Different quantitative criteria being used in earlier studies for selecting mother wavelets are maximum energy, minimum Shannon entropy, and their ratio. But there is no unique agreement over mother wavelet with different wavelet selection criteria. This study endeavors to use benchmarks in signal denoising such as Peak signal to noise ratio (PSNR), Mean squared error (MSE), and Max error as wavelet selection criteria in addition to maximum energy and minimum Shannon entropy for selecting appropriate mother wavelet. The new methodology is based on assigning weights to each selection criteria based on analytic hierarchy process (AHP) depending on the nature of vibration data and then finding the overall ranking of each wavelet. Mother wavelets from Daubechies, Symlet, Coiflet, and Bior families are investigated in this study. The best mother wavelets proposed by the weighting schemes are further used for processing vibration signals using a multiresolution analysis. This methodology is successfully implemented for assessing health of bearings of critical subsystem of a lathe machine tool. The new wavelet selection strategy may be implemented with equal success for health assessment of a broad range of machine tools such as CNC lathes, milling machines, machining centers, and other delicate machine tools.
TL;DR: In this paper, the Fourier transform is used to construct wavelet packets for nonuniform multiresolution analysis on local field of positive characteristic (MRA) and investigate their properties.
Abstract: The concept of nonuniform multiresolution analysis on local field of positive characteristic was considered by Shah and Abdullah for which the translation set is a discrete set which is not a group. We construct the associated wavelet packets for such an MRA and investigate their properties by means of the Fourier transform.
TL;DR: In this article, wavelet packet decomposition is applied to perform multiresolution analysis of the nonstationary current signals for detection and extraction of the sideband frequency components superimposed on the fundamental.
Abstract: A line-start interior permanent magnet (LSIPM) motor operates with inherent hunting phenomena. This paper presents a novel procedure for real-time diagnosis of hunting in LSIPM motors based on stator current signatures. Torsional vibration associated with hunting in a LSIPM motor influences the electrical supply, and introduces variable amplitude lower and upper sidebands in the stator current. In this paper, wavelet packet decomposition is applied to perform multiresolution analysis of the nonstationary current signals for detection and extraction of the sideband frequency components superimposed on the fundamental. Statistical techniques are applied on the extracted data signals and a signature for the diagnosis of hunting phenomenon is established. The proposed technique is validated by performing finite element analysis (FEA) of two three-phase four-pole LSIPM motors of different power ratings for various case scenarios. Experimental investigations have been carried out for a three-phase four-pole 208-V, 1-HP LSIPM motor drive in order to validate the performance of the proposed method under various operating conditions. Based on FEA and experimental results, the proposed technique can successfully diagnose the hunting phenomenon without a vibration sensor.
01 Aug 2017
TL;DR: A better way for finding fault location using discrete wavelet transform (DWT) and artificial neural network (ANN) and one algorithm is developed for finding the type of fault usingWavelet transform.
Abstract: Transmission lines are the essential link between power stations and consumers, which carries bulk amount of power to the required premises. So protection of transmission lines become more predominant and need exact fault location on transmission line. This paper presents a better way for finding fault location using discrete wavelet transform (DWT) and artificial neural network (ANN) and one algorithm is developed for finding the type of fault using wavelet transform. The simulation is carried out using MATLAB software.
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TL;DR: This work proposes a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph that relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of theSet of functions.
Abstract: We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of the set of functions. Our construction involves two parameters q and q'. The first one controls the mean number of kept vertices in the downsampling, while the second one is a tuning parameter between space localization and frequency localization. We provide an explicit reconstruction formula, bounds on the reconstruction operator norm and on the error in the intertwining relation, and a Jackson-like inequality. These bounds lead to recommend a way to choose the parameters q and q'. We illustrate the method by numerical experiments.
TL;DR: A multi resolution based noise removal in magnetic resonance images for abnormality detection and recognition within the brain has been proposed.
Abstract: Modern medical diagnosis equipments included with digital signal processing capabilities have been used for fast and accurate diagnosis of brain structure abnormalities. In this paper a multi resolution based noise removal in magnetic resonance images for abnormality detection and recognition within the brain has been proposed. Wavelet and curvelet based multi resolution approximation has been used to decompose the inter-object relationships into different levels of detail. Contourlet based multi resolution approximation is presented in this work for better abnormality detection. Comparison of extracted feature points between the reference image and the image under study has been made in detection of the abnormality.
TL;DR: A first quantized approach to calculate approximate coupled-cluster singles and doubles CC2 excitation energies in real space using multiresolution analysis.
Abstract: We report a first quantized approach to calculate approximate coupled-cluster singles and doubles CC2 excitation energies in real space. The cluster functions are directly represented on an adaptive grid using multiresolution analysis. Virtual orbitals are neither calculated nor needed in this approach. The nuclear and electronic cusps are taken into account explicitly regularizing the corresponding equations exactly. First calculations on small molecules are in excellent agreement with the best available LCAO results.
TL;DR: In this article, a wavelet base is used to decompose a local field theory by spatial resolution, which can be used to eliminate short-distance degrees of freedom in truncated theories.
Abstract: We investigate both the theoretical and computational aspects of using wavelet bases to perform an exact decomposition of a local field theory by spatial resolution. The decomposition admits natural volume and resolution truncations. We demonstrate that flow equation methods can be used to eliminate short-distance degrees of freedom in truncated theories. The method is tested on a free scalar field in one dimension, where the spatial derivatives couple the degrees of freedom on different scales, although the method is applicable to more complex field theories. The flow equation method is shown to decouple both distance and energy scales in this example. The response to changing the volume and resolution cutoffs and the mass is discussed.
03 Jul 2017
TL;DR: In this article, the wavelet packet transform is used for estimating the Holder continuity of a function f ∈ (CM ∩ L2(ℝ),M > 0.
Abstract: In the present paper, we show that, under some conditions wavelet packet basis of L2(ℝ) can be used as a tool for the uniform approximation of a function f ∈ (CM ∩ L2)(ℝ),M > 0. The necessary and sufficient condition on the wavelet packet transform for estimating the Holder continuity of a function f has been given. Wavelets associated with Riesz projectors have been proposed and various results related to vanishing moments have been presented. Further, we use such wavelets and prove that Holder continuity of a function aids in the decay of wavelet coefficients and thus helps in approximating it. Finally, we give some properties of wavelets associated with Riesz projectors.
TL;DR: In this paper, boundary adapted wavelets are introduced, which are orthogonal and have the same scale in the three spatial directions of the turbulent channel flow, and the role of coherent vorticity is investigated.
Abstract: We introduce boundary adapted wavelets, which are orthogonal and have the same scale in the three spatial directions. The construction thus yields a multiresolution analysis. We analyse direct numerical simulation data of turbulent channel flow computed at a friction Reynolds number of 395, and investigate the role of coherent vorticity. Thresholding of the vorticity wavelet coefficients allows us to split the flow into two parts, coherent and incoherent flows. The coherent vorticity is reconstructed from its few intense wavelet coefficients and the coherent velocity is reconstructed using Biot–Savart's law. The statistics of the coherent flow, i.e. energy and enstrophy spectra, are close to the statistics of the total flow, and moreover, the nonlinear energy budgets of the total flow are very well preserved. The remaining incoherent part, represented by the large majority of the weak wavelet coefficients, corresponds to a structureless, i.e. noise-like, background flow whose energy is equidistrib...
TL;DR: A new orthogonal projection operator wavelet kernel, aiming at developing an efficient computational learning approach for nonlinear dynamical system identification, has a closed-form expression, which greatly facilitates its application in kernel learning.
Abstract: A giant leap has been made in the past couple of decades with the introduction of kernel-based learning as a mainstay for designing effective nonlinear computational learning algorithms. In view of the geometric interpretation of conditional expectation and the ubiquity of multiscale characteristics in highly complex nonlinear dynamic systems [1] – [3] , this paper presents a new orthogonal projection operator wavelet kernel, aiming at developing an efficient computational learning approach for nonlinear dynamical system identification. In the framework of multiresolution analysis, the proposed projection operator wavelet kernel can fulfill the multiscale, multidimensional learning to estimate complex dependencies. The special advantage of the projection operator wavelet kernel developed in this paper lies in the fact that it has a closed-form expression, which greatly facilitates its application in kernel learning. To the best of our knowledge, it is the first closed-form orthogonal projection wavelet kernel reported in the literature. It provides a link between grid-based wavelets and mesh-free kernel-based methods. Simulation studies for identifying the parallel models of two benchmark nonlinear dynamical systems confirm its superiority in model accuracy and sparsity.
TL;DR: In this paper, the Dynamic Mode Decomposition (DMD) with sparse sampling is used for the diagnostic analysis of multiscale physics, which is an ideal spatiotemporal matrix decomposition that correlates spatial features of computational or experimental data to periodic temporal behavior.
Abstract: The characterization of intermittent, multiscale and transient dynamics using data-driven analysis remains an open challenge. We demonstrate an application of the Dynamic Mode Decomposition (DMD) with sparse sampling for the diagnostic analysis of multiscale physics. The DMD method is an ideal spatiotemporal matrix decomposition that correlates spatial features of computational or experimental data to periodic temporal behavior. DMD can be modified into a multiresolution analysis to separate complex dynamics into a hierarchy of multiresolution timescale components, where each level of the hierarchy divides dynamics into distinct background (slow) and foreground (fast) timescales. The multiresolution DMD is capable of characterizing nonlinear dynamical systems in an equation-free manner by recursively decomposing the state of the system into low-rank spatial modes and their temporal Fourier dynamics. Moreover, these multiresolution DMD modes can be used to determined sparse sampling locations which are nearly optimal for dynamic regime classification and full state reconstruction. Specifically, optimized sensors are efficiently chosen using QR column pivots of the DMD library, thus avoiding an NP-hard selection process. We demonstrate the efficacy of the method on several examples, including global sea-surface temperature data, and show that only a small number of sensors are needed for accurate global reconstructions and classification of El Ni\~no events.