scholarly journals Significance tests for the wavelet power and the wavelet power spectrum

2007 ◽  
Vol 25 (11) ◽  
pp. 2259-2269 ◽  
Author(s):  
Z. Ge
Keyword(s):  
Wavelet Analysis ◽  
Power Spectrum ◽  
Lake Michigan ◽  
Random Noise ◽  
Gaussian White Noise ◽  
Covariance Structure ◽  
Sampling Period ◽  
Wavelet Power Spectrum ◽  
Significance Tests ◽  
Significance Levels

Abstract. Significance tests usually address the issue how to distinguish statistically significant results from those due to pure randomness when only one sample of the population is studied. This issue is also important when the results obtained using the wavelet analysis are to be interpreted. Torrence and Compo (1998) is one of the earliest works that has systematically discussed this problem. Their results, however, were based on Monte Carlo simulations, and hence, failed to unveil many interesting and important properties of the wavelet analysis. In the present work, the sampling distributions of the wavelet power and power spectrum of a Gaussian White Noise (GWN) were derived in a rigorous statistical framework, through which the significance tests for these two fundamental quantities in the wavelet analysis were established. It was found that the results given by Torrence and Compo (1998) are numerically accurate when adjusted by a factor of the sampling period, while some of their statements require reassessment. More importantly, the sampling distribution of the wavelet power spectrum of a GWN was found to be highly dependent on the local covariance structure of the wavelets, a fact that makes the significance levels intimately related to the specific wavelet family. In addition to simulated signals, the significance tests developed in this work were demonstrated on an actual wave elevation time series observed from a buoy on Lake Michigan. In this simple application in geophysics, we showed how proper significance tests helped to sort out physically meaningful peaks from those created by random noise. The derivations in the present work can be readily extended to other wavelet-based quantities or analyses using other wavelet families.

scholarly journals Comment on "Significance tests for the wavelet power and the wavelet power spectrum" by Ge (2007)

2012 ◽  
Vol 30 (12) ◽  
pp. 1743-1750 ◽  
Author(s):  
Z. Zhang ◽  
J. C. Moore
Keyword(s):  
Power Spectrum ◽  
White Noise ◽  
Continuous Time ◽  
Gaussian White Noise ◽  
Morlet Wavelet ◽  
Wavelet Power Spectrum ◽  
Significance Tests ◽  
Statistical Framework ◽  
Correct Formula

Abstract. As the main result in Ge's paper, Ge announced that he proved a formula on the distribution of Morlet wavelet power spectrum of continuous-time Gaussian white noise in a rigorous statistical framework. In this paper, we will show that Ge's formula is wrong and each step of Ge's proof is wrong. Moreover, we give and prove a correct formula in a rigorous statistical framework.

scholarly journals Significance tests for the wavelet cross spectrum and wavelet linear coherence

2008 ◽  
Vol 26 (12) ◽  
pp. 3819-3829 ◽  
Author(s):  
Z. Ge
Keyword(s):  
Phase Angle ◽  
Lake Michigan ◽  
Gaussian White Noise ◽  
Cauchy Distribution ◽  
Significance Tests ◽  
Data Set ◽  
Cross Spectrum ◽  
Sampling Distributions ◽  
Significance Levels ◽  
The Waves

Abstract. This work attempts to develop significance tests for the wavelet cross spectrum and the wavelet linear coherence as a follow-up study on Ge (2007). Conventional approaches that are used by Torrence and Compo (1998) based on stationary background noise time series were used here in estimating the sampling distributions of the wavelet cross spectrum and the wavelet linear coherence. The sampling distributions are then used for establishing significance levels for these two wavelet-based quantities. In addition to these two wavelet quantities, properties of the phase angle of the wavelet cross spectrum of, or the phase difference between, two Gaussian white noise series are discussed. It is found that the tangent of the principal part of the phase angle approximately has a standard Cauchy distribution and the phase angle is uniformly distributed, which makes it impossible to establish significance levels for the phase angle. The simulated signals clearly show that, when there is no linear relation between the two analysed signals, the phase angle disperses into the entire range of [−π,π] with fairly high probabilities for values close to ±π to occur. Conversely, when linear relations are present, the phase angle of the wavelet cross spectrum settles around an associated value with considerably reduced fluctuations. When two signals are linearly coupled, their wavelet linear coherence will attain values close to one. The significance test of the wavelet linear coherence can therefore be used to complement the inspection of the phase angle of the wavelet cross spectrum. The developed significance tests are also applied to actual data sets, simultaneously recorded wind speed and wave elevation series measured from a NOAA buoy on Lake Michigan. Significance levels of the wavelet cross spectrum and the wavelet linear coherence between the winds and the waves reasonably separated meaningful peaks from those generated by randomness in the data set. As with simulated signals, nearly constant phase angles of the wavelet cross spectrum are found to coincide with large values in the wavelet linear coherence between the winds and the waves. Not limited to geophysics, the significance tests developed in the present work can also be applied to many other quantitative studies using the continuous wavelet transform.

scholarly journals Artificial Detection of Lower-Frequency Periodicity in Climatic Studies by Wavelet Analysis Demonstrated on Synthetic Time Series

2019 ◽  
Vol 58 (9) ◽  
pp. 2077-2086 ◽  
Author(s):  
Assaf Hochman ◽  
Hadas Saaroni ◽  
Felix Abramovich ◽  
Pinhas Alpert
Keyword(s):  
Time Series ◽  
Wavelet Analysis ◽  
Power Spectrum ◽  
Low Frequency ◽  
Real Data ◽  
Careful Analysis ◽  
Wavelet Power Spectrum ◽  
Natural Oscillations ◽  
Local Singularity ◽  
Climatic Time Series

AbstractThe continuous wavelet transform (CWT) is a frequently used tool to study periodicity in climate and other time series. Periodicity plays a significant role in climate reconstruction and prediction. In numerous studies, the use of CWT revealed dominant periodicity (DP) in climatic time series. Several studies suggested that these “natural oscillations” would even reverse global warming. It is shown here that the results of wavelet analysis for detecting DPs can be misinterpreted in the presence of local singularities that are manifested in lower frequencies. This may lead to false DP detection. CWT analysis of synthetic and real-data climatic time series, with local singularities, indicates a low-frequency DP even if there is no true periodicity in the time series. Therefore, it is argued that this is an inherent general property of CWT. Hence, applying CWT to climatic time series should be reevaluated, and more careful analysis of the entire wavelet power spectrum is required, with a focus on high frequencies as well. A conelike shape in the wavelet power spectrum most likely indicates the presence of a local singularity in the time series rather than a DP, even if the local singularity has an observational or a physical basis. It is shown that analyzing the derivatives of the time series may be helpful in interpreting the wavelet power spectrum. Nevertheless, these tests are only a partial remedy that does not completely neutralize the effects caused by the presence of local singularities.

scholarly journals Wavelet Analysis of Red Noise and Its Application in Climate Diagnosis

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Zhihua Zhang
Keyword(s):  
Wavelet Analysis ◽  
Power Spectrum ◽  
General Formula ◽  
Spectral Peak ◽  
Wavelet Power Spectrum ◽  
Statistical Framework ◽  
Analytic Wavelet ◽  
Real Analytic

Signals are often destroyed by various kinds of noises. A common way to statistically assess the significance of a broad spectral peak in signals and the synchronization between signals is to compare with simple noise processes. At present, wavelet analysis of red noise is studied limitedly and there is no general formula on the distribution of the wavelet power spectrum of red noise. Moreover, the distribution of the wavelet phase of red noise is also unknown. In this paper, for any given real/analytic wavelet, we will use a rigorous statistical framework to obtain the distribution of the wavelet power spectrum and wavelet phase of red noise and apply these formulas in climate diagnosis.

scholarly journals Sufficiency of a Gaussian power spectrum likelihood for accurate cosmology from upcoming weak lensing surveys

2021 ◽  
Author(s):  
Robin E Upham ◽  
Michael L Brown ◽  
Lee Whittaker
Keyword(s):  
Power Spectrum ◽  
Random Noise ◽  
Power Spectra ◽  
Stage Iv ◽  
Wishart Distribution ◽  
Weak Lensing ◽  
Dependence Structure ◽  
Gaussian Fields ◽  
Non Gaussian ◽  
Parameter Constraints

Abstract We investigate whether a Gaussian likelihood is sufficient to obtain accurate parameter constraints from a Euclid-like combined tomographic power spectrum analysis of weak lensing, galaxy clustering and their cross-correlation. Testing its performance on the full sky against the Wishart distribution, which is the exact likelihood under the assumption of Gaussian fields, we find that the Gaussian likelihood returns accurate parameter constraints. This accuracy is robust to the choices made in the likelihood analysis, including the choice of fiducial cosmology, the range of scales included, and the random noise level. We extend our results to the cut sky by evaluating the additional non-Gaussianity of the joint cut-sky likelihood in both its marginal distributions and dependence structure. We find that the cut-sky likelihood is more non-Gaussian than the full-sky likelihood, but at a level insufficient to introduce significant inaccuracy into parameter constraints obtained using the Gaussian likelihood. Our results should not be affected by the assumption of Gaussian fields, as this approximation only becomes inaccurate on small scales, which in turn corresponds to the limit in which any non-Gaussianity of the likelihood becomes negligible. We nevertheless compare against N-body weak lensing simulations and find no evidence of significant additional non-Gaussianity in the likelihood. Our results indicate that a Gaussian likelihood will be sufficient for robust parameter constraints with power spectra from Stage IV weak lensing surveys.

scholarly journals Wavelet analysis applied on temporal data sets in order to reveal possible pre-seismic radio anomalies and comparison with the trend of the raw data 

2021 ◽  
Author(s):  
Giovanni Nico ◽  
Pier Francesco Biagi ◽  
Anita Ermini ◽  
Mohammed Yahia Boudjada ◽  
Hans Ulrich Eichelberger ◽  
...  
Keyword(s):  
Time Series ◽  
Wavelet Analysis ◽  
Power Spectrum ◽  
Power Spectra ◽  
Wavelet Spectrum ◽  
Time Signal ◽  
Time Behavior ◽  
Sudden Increase ◽  
Night Time ◽  
Raw Data

<p>Since 2009, several radio receivers have been installed throughout Europe in order to realize the INFREP European radio network for studying the VLF (10-50 kHz) and LF (150-300 kHz) radio precursors of earthquakes. Precursors can be related to “anomalies” in the night-time behavior of  VLF signals. A suitable method of analysis is the use of the Wavelet spectra.  Using the “Morlet function”, the Wavelet transform of a time signal is a complex series that can be usefully represented by its square amplitude, i.e. considering the so-called Wavelet power spectrum.</p><p>The power spectrum is a 2D diagram that, once properly normalized with respect to the power of the white noise, gives information on the strength and precise time of occurrence of the various Fourier components, which are present in the original time series. The main difference between the Wavelet power spectra and the Fourier power spectra for the time series is that the former identifies the frequency content along the operational time, which cannot be done with the latter. Anomalies are identified as regions of the Wavelet spectrogram characterized by a sudden increase in the power strength.</p><p>On January 30, 2020 an earthquake with Mw= 6.0 occurred in Dodecanese Islands. The results of the Wavelet analysis carried out on data collected some INFREP receivers is compared with the trends of the raw data. The time series from January 24, 2020 till January 31, 2000 was analyzed. The Wavelet spectrogram shows a peak corresponding to a period of 1 day on the days before January 30. This anomaly was found for signals transmitted at the frequencies 19,58 kHz, 20, 27 kHz, 23,40 kHz with an energy in the peak increasing from 19,58 kHz to 23,40 kHz. In particular, the signal at the frequency 19,58 kHz, shows a peak on January 29, while the frequencies 20,27 kHz and 23,40 kHz are characterized by a peak starting on January 28 and continuing to January 29. The results presented in this work shows the perspective use of the Wavelet spectrum analysis as an operational tool for the detection of anomalies in VLF and LF signal potentially related to EQ precursors.</p>

scholarly journals Storm Wave Characteristics

1967 ◽  
Vol 7 (01) ◽  
pp. 87-98 ◽  
Author(s):  
R.J. Robinson ◽  
H.R. Brannon ◽  
G.W. Kattawar
Keyword(s):  
Power Spectrum ◽  
Gulf Of Mexico ◽  
Random Noise ◽  
Offshore Structures ◽  
Wave Profile ◽  
Wave Crest ◽  
Wave Characteristics ◽  
Storm Wave ◽  
Wave Profiles ◽  
Wave Shapes

Abstract Accurate prediction of storm wave characteristics are needed for design of offshore structures. Statistical methods of random noise analysis provide techniques for predicting required wave properties. These techniques have been used to analyze and characterize storm wave profiles from Gulf of Mexico recording installations in 34, 65 and 98 ft of water. Correlations based on the results can be used to predict wave crest probabilities and wave shapes for a range of Gull water depths and storm conditions. Example predictions of wave crest probability as a function of water depth for a particular set of storm conditions are given. Introduction Accurate predictions of wave crests and wave shapes are needed for design of offshore structures. With predictions of wave crests, platform deck elevations giving adequate protection from exposure to storm waves can be selected. Use of hydrodynamic wave theories with the wave crest and wave shape predictions permits an estimate of the wave forces an offshore structure must withstand. Observation of ocean waves suggests use of statistical analysis for studying wave characteristics for water depths beyond the breaking wave zone; recent wave generation theories lend support to this approach. One of the most powerful methods of statistically analyzing wave profiles was introduced by Cartwright and Longuet-Higgins; most of the basic relations used were developed by Rice in studies of random noise theory. This method predicts wave crest elevation probabilities from two parameters characterizing a wave profile. The statistical approach also provides a means for predicting wave shapes. In the random noise model of a wave profile, surface elevation is represented as an infinite sum of sine waves with closely spaced frequencies and random phase angles. Power density is proportional to the sum of the squares of the amplitudes of sine waves having frequencies within a narrow band, and the distribution of power density as a function of frequency is called the power density spectrum, or power spectrum. The power spectrum characterizes an irregular sea, and hence finds use in motion studies of ships, barges and semisubmersible drilling platforms. From the power spectrum, a wave profile of any duration of time can, in principle, be calculated; this long wave profile depicts the many wave shapes to which a structure may be exposed. Thus, the statistical methods of wave analysis provide an approach to selecting wave shapes as well as wave crest elevations needed for design of offshore structures. For practical application of these techniques, power spectra and related quantities must be predicted from storm properties. Barber and Tucker have reviewed correlations of wave properties and storm conditions. Their review summarizes work of Darbyshire, among others, in which a correlation of power spectra with wind intensity of the North Atlantic was developed. These results, plus more recent work by Pierson and Moskowitz and Kitaigorodskii, establish feasibility of correlation, but there is no theoretical basis for modifications extending the relations to other area's as remote as the Gulf of Mexico. Hence, to predict characteristics of Gulf of Mexico waves, a study of local observations is needed. This paper presentsa practical approach to computing wave profiles that depict shapes of waves for use in force calculations,a summary of relations for predicting probabilities of wave crest elevations,correlations of parameters needed to apply these methods in the Gulf of Mexico andexamples of application of the techniques. THEORETICAL BACKGROUND The following sections summarize relations needed to calculate wave profiles and to estimate wave crest probabilities. SPEJ P. 87ˆ

scholarly journals Enhancing Image Denoising Performance of Bidimensional Empirical Mode Decomposition by Improving the Edge Effect

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Feng-Ping An ◽  
Da-Chao Lin ◽  
Xian-Wei Zhou ◽  
Zhihui Sun
Keyword(s):  
Empirical Mode Decomposition ◽  
Edge Effect ◽  
Signal To Noise Ratio ◽  
Random Noise ◽  
Gaussian White Noise ◽  
Support Vector ◽  
Mode Decomposition ◽  
Expansion Technique ◽  
Bidimensional Empirical Mode Decomposition ◽  
Adaptive Ability

Bidimensional empirical mode decomposition (BEMD) algorithm, with high adaptive ability, provides a suitable tool for the noisy image processing, and, however, the edge effect involved in its operation gives rise to a problem—how to obtain reliable decomposition results to effectively remove noises from the image. Accordingly, we propose an approach to deal with the edge effect caused by BEMD in the decomposition of an image signal and then to enhance its denoising performance. This approach includes two steps, in which the first one is an extrapolation operation through the regression model constructed by the support vector machine (SVM) method with high generalization ability, based on the information of the original signal, and the second is an expansion by the closed-end mirror expansion technique with respect to the extrema nearest to and beyond the edge of the data resulting from the first operation. Applications to remove the Gaussian white noise, salt and pepper noise, and random noise from the noisy images show that the edge effect of the BEMD can be improved effectively by the proposed approach to meet requirement of the reliable decomposition results. They also illustrate a good denoising effect of the BEMD by improving the edge effect on the basis of the proposed approach. Additionally, the denoised image preserves information details sufficiently and also enlarges the peak signal-to-noise ratio.

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