Wavelet Variance Research Articles

A multiscale wavelet-based test for isotropy of random fields on a regular lattice.

A test for isotropy of images modeled as stationary or intrinsically stationary random fields on a lattice is developed. The test is based on the wavelet theory, and can operate on the horizontal and vertical scale of choice, or on any combination of scales. Scale is introduced through the wavelet variances (sometimes called as the wavelet power spectrum), which decompose the variance over different horizontal and vertical spatial scales. The method is more general than existing tests for isotropy, since it handles intrinsically stationary random fields as well as second-order stationary fields. The performance of the method is demonstrated on samples from different random fields, and compared with three existing methods. It is competitive with or outperforms existing methods since it consistently rejects close to the nominal level for isotropic fields while having a rejection rate for anisotropic fields comparable with the existing methods in the stationary case, and superior in the intrinsic case. As practical examples, paper density images of handsheets and mammogram images are analyzed.

Assessing Characteristic Scales Using Wavelets

Summary Characteristic scale is a notion that pervades the geophysical sciences, but it has no widely accepted precise definition. Motivated by the facts that the wavelet transform decomposes a time series into coefficients that are associated with different scales and that the variance of these coefficients (the so-called wavelet variance) decomposes the variance of the time series across scales, the paper proposes a definition for characteristic scale in terms of peaks in plots of the wavelet variance versus scale. After presenting basic theory for wavelet-based characteristic scales, a natural estimator for these scales is considered. Large sample theory for this estimator permits the construction of confidence intervals for a true unknown characteristic scale. Computer experiments are presented that demonstrate the efficacy of the large sample theory for finite sample sizes. Characteristic scale estimates are calculated for medium multiyear Arctic sea ice, global temperature records, coherent structures in river flows and the Madden–Julian oscillation in an atmospheric time series.

Long-range memory in internal and forced dynamics of millennium-long climate model simulations

Abstract. Northern Hemisphere (NH) temperature records from a palaeoclimate reconstruction and a number of millennium-long climate model experiments are investigated for long-range memory (LRM). The models are two Earth system models and two atmosphere–ocean general circulation models. The periodogram, detrended fluctuation analysis and wavelet variance analysis are applied to examine scaling properties and to estimate a scaling exponent of the temperature records. A simple linear model for the climate response to external forcing is also applied to the reconstruction and the forced climate model runs, and then compared to unforced control runs to extract the LRM generated by internal dynamics of the climate system. The climate models show strong persistent scaling with power spectral densities of the form S(f) ~ f −β with 0.8 < β < 1 on timescales from years to several centuries. This is somewhat stronger persistence than found in the reconstruction (β ≈ 0.7). We find no indication that LRM found in these model runs is induced by external forcing, which suggests that LRM on sub-decadal to century time scales in NH mean temperatures is a property of the internal dynamics of the climate system. Reconstructed and instrumental sea surface temperature records for a local site, Reykjanes Ridge, are also studied, showing that strong persistence is found also for local ocean temperature.

A new approach for crop identification with wavelet variance and JM distance.

This paper develops a new crop mapping method through combined utilization of both time and frequency information based on wavelet variance and Jeffries-Matusita (JM) distance (CIWJ for short). A two-dimensional wavelet spectrum was obtained from datasets of daily continuous vegetation indices through a continuous wavelet transform using the Mexican hat and the Morlet mother wavelets. The time-average wavelet variance (TAWV) and the scale-average wavelet variance (SAWV) were then calculated based on the wavelet spectrum of the Mexican hat and the Morlet wavelet, respectively. The class separability based on the JM distance was evaluated to discriminate the proper period or scale range applied. Finally, a procedure for criteria quantification was developed using the TAWV and SAWV as the major metrics, and the similarity between unclassified pixels and established land use/cover types was calculated. The proposed CIWJ method was applied to the middle Hexi Corridor in northwest China using 250-m 8-day composite moderate-resolution imaging spectroradiometer (MODIS) enhanced vegetation index (EVI) time series datasets in 2012. The CIWJ method was shown to be efficient in crop field mapping, with an overall accuracy of 83.6% and kappa coefficient of 0.7009, assessed with 30m Chinese Environmental Disaster Reduction Satellite (HJ-1)-derived data. Compared with methods utilizing information on either frequency or time, the CIWJ method demonstrates tremendous potential for efficient crop mapping and for further applications. This method could be applied to either coarse or high spatial resolution images for agricultural crop identification, as well as other more general or specific land use classifications.

Generalized method of wavelet moments for inertial navigation filter design

The integration of observations issued from a satellite-based system (GNSS) with an inertial navigation system (INS) is usually performed through a Bayesian filter such as the extended Kalman filter (EKF). The task of designing the navigation EKF is strongly related to the inertial sensor error modeling problem. Accelerometers and gyroscopes may be corrupted by random errors of complex spectral structure. Consequently, identifying correct error-state parameters in the INS/GNSS EKF becomes difficult when several stochastic processes are superposed. In such situations, classical approaches like the Allan variance (AV) or power spectral density (PSD) analysis fail due to the difficulty of separating the error processes in the spectral domain. For this purpose, we propose applying a recently developed estimator based on the generalized method of wavelet moments (GMWM), which was proven to be consistent and asymptotically normally distributed. The GMWM estimator matches theoretical and sample-based wavelet variances (WVs), and can be computed using the method of indirect inference. This article mainly focuses on the implementation aspects related to the GMWM, and its integration within a general navigation filter alibration procedure. Regarding this, we apply the GMWM on error signals issued from MEMS-based inertial sensors by building and estimating composite stochastic processes for which classical methods cannot be used. In a first stage, we validate the resulting models using AV and PSD analyses and then, in a second stage, we study the impact of the resulting stochastic models design in terms of positioning accuracy using an emulated scenario with statically observed error signatures. We demonstrate that the GMWM-based calibration framework enables to estimate complex stochastic models in terms of the resulting navigation accuracy that are relevant for the observed structure of errors.

Estimation of Time Series Models via Robust Wavelet Variance

A robust approach to the estimation of time series models is proposed. Taking froma new estimation method called the Generalized Method of Wavelet Moments (GMWM)which is an indirect method based on the Wavelet Variance (WV), we replace the classicalestimator of the WV with a recently proposed robust M-estimator to obtain a robustversion of the GMWM. The simulation results show that the proposed approach can beconsidered as a valid robust approach to the estimation of time series and state-spacemodels.

Detecting land use and land management influences on catchment hydrology by modelling and wavelets

Summary There exists a widespread recognition that land use and management change should affect catchment response to precipitation (and therefore stream flow characteristics). Previous studies have shown that this affect exists at a local scale, but there is a paucity of evidence that local scale effects aggregate to detectable impacts within downstream catchments. This paper describes a novel wavelet-based analysis of hydrological model residuals to examine the effect of land-use change on the catchment hydrology across four UK catchments (the Kird, Lod, Coalburn and Wye), of which all but the Wye experienced significant changes in their land use. The HySim rainfall–runoff model was calibrated against observed long term flow series assuming a static land hydrology to allow for the effects of climatic variability. Deviations of model fit were assessed in relation to changes to catchment land hydrology. The wavelet transform was used to decompose both simulated and observed flows into different scale components and to assess changes in catchment hydrology across different temporal scales; model residuals and model performance were tested for significant changes in wavelet variance and wavelet correlation respectively. Significant changes in wavelet variance and wavelet correlation corresponded with changes in catchment land use for two of the three catchments that experience land use change, in the third there were significant changes in wavelet variation that may have resulted from land-use change, but no changes in wavelet correlation. The control catchment showed no significant features of variances and no significant change in wavelet correlation across the scales. This new approach holds great potential for separating the influences of land use and management change on catchment stream flow based on frequency scale.

Fault detection in rolling element bearings using wavelet-based variance analysis and novelty detection

Fractal signal processing and novelty detection are used for fault detection in rolling element bearings. The former applies the concept of self-similarity based on wavelet variance, and the latter is based on machine learning and utilises artificial neural networks. The method is demonstrated using simulated and experimental vibration data. The work presented involves validation both on laboratory test rig data and industrial wind turbine data. The results show that the method can be used successfully for automated fault detection in ball bearings under real operational conditions.

A novel variability-based method for quasar selection: evidence for a rest-frame ∼54 d characteristic time-scale★

We compare quasar selection techniques based on their optical variability using data from the Catalina Real-time Transient Survey (CRTS). We introduce a new technique based on Slepian wavelet variance (SWV) that shows comparable or better performance to structure functions and damped random walk models but with fewer assumptions. Combining these methods with WISE mid-IR colors produces a highly efficient quasar selection technique which we have validated spectroscopically. The SWV technique also identifies characteristic timescales in a time series and we find a characteristic rest frame timescale of ~54 days, confirmed in the light curves of ~18000 quasars from CRTS, SDSS and MACHO data, and anticorrelated with absolute magnitude. This indicates a transition between a damped random walk and $P(f) \propto f^{-1/3}$ behaviours and is the first strong indication that a damped random walk model may be too simplistic to describe optical quasar variability.

Stability Dependence and Diurnal Change of Large-Scale Turbulence Structures in the Near-Neutral Atmospheric Boundary Layer Observed from a Meteorological Tower

Based on the analysis of observations from a 213-m tall meteorological tower at Tsukuba, Japan, we have investigated the favourable conditions for the predominant existence of large-scale turbulence structures in the near-neutral atmospheric boundary layer (ABL). From the wavelet variance spectrum for the streamwise velocity component ( $$u$$ ) measured by a sonic anemometer-thermometer at the highest level (200 m), large-scale structures (time-scale range of 100–300 s) predominantly exist under slightly unstable and close to neutral conditions. The emergence of large-scale structures also can be related to the diurnal evolution of the ABL. The large-scale structures play an important role in the overall flow structure of the lower boundary layer. For example, $$u$$ velocity components at the 200-m and 50-m levels show relatively high correlation with the existence of large-scale structures. Under slightly unstable (near-neutral) conditions, a low-speed region in advance of the high-speed structure shows a positive deviation of temperature and appears as the plume structure that is forced by buoyancy in the heated lower layer. In spite of the difference in buoyancy effects between the near-neutral and unstable cases, large-scale structures are frequently observed in both cases and the same vertical correlation of $$u$$ components is indicated. However, the vertical wind shear is smaller in the unstable cases. On the other hand, in near-neutral cases, the transport efficiency of momentum at the higher level and the flux contribution of sweep motions are larger than those in the unstable cases.

Wavelet‐based self‐organizing maps for classifying multivariate time series

Following a nonparametric approach, we suggest a time‐series clustering method. Our clustering approach combines the benefits connected to the interpretative power of the nonparametric representation of the time series, and the clustering and vector quantization informational gain produced by the adopted unsupervised neural networks technique, enhanced with the self‐organizing maps ordering and topological preservation abilities. The proposed clustering method takes into account a composite wavelet‐based information of the multivariate time series by adding to the information connected to the wavelet variance, namely the influence of variability of individual univariate components of the multivariate time series across scales, the information associated to wavelet correlation, represented by the interaction between pairs of univariate components of the multivariate time series at each scale, and then suitably tuning the combination of these pieces of information. In order to assess the effectiveness of the proposed clustering approach, a simulation study and an empirical application are shown. Copyright © 2013 John Wiley & Sons, Ltd.

Discriminant analysis of multivariate time series: Application to diagnosis based on ECG signals

In analysing ECG data, the main aim is to differentiate between the signal patterns of healthy subjects and those of individuals with specific heart conditions. We propose an approach for classifying multivariate ECG signals based on discriminant and wavelet analyses. For this purpose we use multiple-scale wavelet variances and wavelet correlations to distinguish between the patterns of multivariate ECG signals based on the variability of the individual components of each ECG signal and on the relationships between every pair of these components. Using the results of other ECG classification studies in the literature as references, we demonstrate that our approach applied to 12-lead ECG signals from a particular database compares favourably. We also demonstrate with real and synthetic ECG data that our approach to classifying multivariate time series out-performs other well-known approaches for classifying multivariate time series.

Noise estimation and reduction for chaotic time series by wavelet variance decomposition

In wavelet noise processing,signal decomposition catches more attention while noise itself is ignored.To solve this problem,by using the function of wavelet variance decomposition to analyze white noise,a new method of noise estimation and reduction by thresholding was proposed for chaotic time series.The noise level was estimated with wavelet variances at first and second scale,while the soft threshold was chosen by calculating wavelet variance of noise at every scale.The method was tested in Lorenz and Chen's system.The result shows that the proposed method is better than other wavelet noise estimation and reduction methods.At last,it is proved to be effective in de-noising Shanghai Stock Exchange(SSE) index and Shanghai Futures Exchange(SHFE) rubber futures time series.

Variation analysis of precipitation during past 286 years in Beijing area, China, using non‐parametric test and wavelet analysis

AbstractNon‐parametric methods including Mann–Kendall (M–K) test, continuous wavelet transform (CWT) and discrete wavelet transform analysis are applied in this paper to detect the trend and periodic trait of precipitation data series in Beijing area where the data set spans nearly 300 years from 1724 to 2009. First, the trend of precipitation variables is elaborated by the M–K test (Sequential M–K test). The results show that there is an increasing trend (the value of this trend is 1.98) at the 5%‐significance level and there are not turning points in the whole data series. Then, CWT and wavelet variance are used to check for significant periodic characteristics of data series. In the plots of wavelet transform coefficients and figure of wavelet variance, some periodic events affect the trend of the annual total precipitation series in Beijing area. 85‐year, 35‐year and 21‐year periodic events are found to be the main periodic series of long‐term precipitation data, and they are all statistically significant. Moreover, the results of non‐parametric M–K test are exhibited on seven different combinations of discrete wavelet components. D5 (32‐year periodicity) periodic component is the effective and significant component on data. It is coincident with the result (35‐year periodic event as one part of main periodicity) by using CWT analysis. Moreover, approximation mode shows potential trend of the whole data set because it is the residuals as all periodicities are removed from data series. Thus, the mode A + D5 is responsible for producing a real basic structure of the trend founded on the data. Copyright © 2012 John Wiley & Sons, Ltd.

Wavelet-Variance-Based Estimation for Composite Stochastic Processes

This article presents a new estimation method for the parameters of a time series model. We consider here composite Gaussian processes that are the sum of independent Gaussian processes which, in turn, explain an important aspect of the time series, as is the case in engineering and natural sciences. The proposed estimation method offers an alternative to classical estimation based on the likelihood, that is straightforward to implement and often the only feasible estimation method with complex models. The estimator furnishes results as the optimization of a criterion based on a standardized distance between the sample wavelet variances (WV) estimates and the model-based WV. Indeed, the WV provides a decomposition of the variance process through different scales, so that they contain the information about different features of the stochastic model. We derive the asymptotic properties of the proposed estimator for inference and perform a simulation study to compare our estimator to the MLE and the LSE with different models. We also set sufficient conditions on composite models for our estimator to be consistent, that are easy to verify. We use the new estimator to estimate the stochastic error's parameters of the sum of three first order Gauss–Markov processes by means of a sample of over 800, 000 issued from gyroscopes that compose inertial navigation systems. Supplementary materials for this article are available online.

ECoG Classification Research Based on Wavelet Variance and Probabilistic Neural Network

For a typical ECoG-based brain-computer interface system that the subjects task is to imagine movements of either the left small finger or the tongue, a feature extraction algorithm using wavelet variance was proposed. Firstly the wavelet transform was discussed, and the definition and significance of wavelet variance were bring out and taken as feature, 6 channels with most distinctive features were selected from 64 channels for analysis; consequently the EEG data were decomposed using db4 wavelet, the wavelet coefficients variances containing Mu rhythm and Beta rhythm were taken out as features based on ERD/ERS phenomenon, and the features were classified by probabilistic neural network under a optimal spread with an algorithm of cross validation. The result of off-line showed high average classification accuracies of 89.21% and 88% for training and test data were achieved, the wavelet variance has characteristics of more simple and effective and it is suitable for feature extraction in BCI research.

Synoptic volumetric variations and flushing of the Tampa Bay estuary

Two types of analyses are used to investigate the synoptic wind-driven flushing of Tampa Bay in response to the El Nino-Southern Oscillation (ENSO) cycle from 1950 to 2007. Hourly sea level elevations from the St. Petersburg tide gauge, and wind speed and direction from three different sites around Tampa Bay are used for the study. The zonal (u) and meridional (v) wind components are rotated clockwise by 40° to obtain axial and co-axial components according to the layout of the bay. First, we use the subtidal observed water level as a proxy for mean tidal height to estimate the rate of volumetric bay outflow. Second, we use wavelet analysis to bandpass sea level and wind data in the time–frequency domain to isolate the synoptic sea level and surface wind variance. For both analyses the long-term monthly climatology is removed and we focus on the volumetric and wavelet variance anomalies. The overall correlation between the Oceanic Nino Index and volumetric analysis is small due to the seasonal dependence of the ENSO response. The mean monthly climatology between the synoptic wavelet variance of elevation and axial winds are in close agreement. During the winter, El Nino (La Nina) increases (decreases) the synoptic variability, but decreases (increases) it during the summer. The difference in winter El Nino/La Nina wavelet variances is about 20 % of the climatological value, meaning that ENSO can swing the synoptic flushing of the bay by 0.22 bay volumes per month. These changes in circulation associated with synoptic variability have the potential to impact mixing and transport within the bay.

Texture Classification of PolSAR Data Based on Sparse Coding of Wavelet Polarization Textons

This paper presents a frame for classifying polarimetric synthetic aperture radar (PolSAR) data. The frame is based on the combination of wavelet polarization information, textons, and sparse coding. Polarimetric synthesis unites with the discrete wavelet frame to obtain wavelet polarization variance through the calculation of the wavelet variance in the space of polarization states. The K-means cluster algorithm is implemented to cluster the wavelet polarization variance vectors of the training samples for the purpose of constructing a texton dictionary. A patch, in which all the wavelet polarization variance vectors match those in the texton dictionary, is used to obtain a statistical histogram. Sparse coding is applied to describe the histogram feature and generate a new texture feature called sparse coding of a wavelet polarization texton. Finally, support vector machine is used for the classification. All experiments are carried out on five sets of PolSAR data. The experimental results confirm that the proposed method effectively classifies PolSAR data.

Identifying scale-location specific control on vegetation distribution in mountain-hill region

Abstract The scale-location specific control on vegetation distribution was investigated through continuous wavelet transforms approaches in subtropical mountain-hill region, Fujian, China. The Normalized Difference Vegetation Index (NDVI) was calculated as an indicator of vegetation greenness using Chinese Environmental Disaster Reduction Satellite images along latitudinal and longitudinal transects. Four scales of variations were identified from the local wavelet spectrum of NDVI, with much stronger wavelet variances observed at larger scales. The characteristic scale of vegetation distribution within mountainous and hilly regions in Southeast China was around 20 km. Significantly strong wavelet coherency was generally examined in regions with very diverse topography, typically characterized as small mountains and hills fractured by rivers and residents. The continuous wavelet based approaches provided valuable insight on the hierarchical structure and its corresponding characteristic scales of ecosystems, which might be applied in defining proper levels in multilevel models and optimal bandwidths in Geographically Weighted Regression.

Continuous wavelet transform analysis for self-similarity properties of turbulence in magnetized DC glow discharge plasma

AbstractCharacterization of self-similarity properties of turbulence in magnetized plasma is being carried out in DC glow discharge plasma. The time series floating potential fluctuation experimental data are acquired from the plasma by Langmuir probe. Continuous wavelet transform (CWT) analysis considering db4 mother wavelet has been applied to the experimental data and self-similarity properties are detected by evaluating the Hurst exponent from the wavelet variance plotting. From the CWT spectrum, effort is made to extract a highly correlated frequency by locating the brightest spot. Accordingly, those signals are treated for finding out correlation dimension and the Liapunov exponent so that the exact frequency responsible for the chaotic behavior could be found out.