A Confidence Limit for Hilbert Spectrum Through stoppage criteria

Need a confidence limit As we have presented here, EMD could generate infinite many sets of IMFs. In this case, which one of the infinite many sets really represents the true physics? To answer this question, we need a confidence limit on our result. Traditional methods also had the similar problem of generating many answers. Take Fourier analysis for example; we have to assume trigonometric series is basis. How about other basis? Why not slightly distorted sinusoidal wave as basis? ….

Confidence Limit for Fourier Spectrum The Confidence limit for Fourier Spectral analysis is based on ergodic assumption. It is derived by dividing the data into M sections, and substituting temporal (or spatial) average as ensemble average. This approach is valid for linear and stationary processes, and the sub-sections have to be statistically independent. By dividing the data into subsections, the resolution will suffer.

Statistical Independence The probability function of x and y jointly, f(x, y), is equal to f(x) times f(y) if x and y are statistically independent. For any number of variables, x 1, x 2,..., x n, if the joint probability is the product of the several probability functions, then the variables are all statistically independent. Independent variables are noncorrelated, but not necessarily conversely. James & James : Mathematics Dictionary

LOD Data

Confidence Limit for Fourier Spectrum Confidence Limit from 7 sections, each 2048 points.

Are the sub-sections statistically independent? For narrow band signals, most likely they would not be independent.

Confidence Limit for Hilbert Spectrum Any data can be decomposed into infinitely many different constituting component sets. EMD is a method to generate infinitely many different IMF representations based on different sifting parameters. Some of the IMFs are better than others based on various properties: for example, Orthogonal Index. A Confidence Limit for Hilbert Spectral analysis can be based on an ensemble of ‘ valid IMF ’ resulting from different sifting parameters S covering the parameter space fairly. It is valid for nonlinear and nonstationary processes.

Different Kinds of Confidence Limit The basic idea is to generate various IMFs, treat the mean as the true answer, and obtain the confidence limit based on the STD from the various solutions. By stoppage criteria By Ensemble EMD By down sampling Different spline methods

Empirical Mode Decomposition Sifting : to get one IMF component

None of the above methods depends on ergodic assumption. We are truly achieving an ensemble mean.

The Stoppage Criteria : S and SD A. The S number : S is defined as the consecutive number of siftings, in which the numbers of zero- crossing and extrema are the same for these S siftings. B. If the mean is smaller than a pre-assigned value. C. Fixed sifting (iterating) time. D. SD is small than a pre-set value, where

Critical Parameters for EMD The maximum number of sifting allowed to extract an IMF, N. –Note: N is originally set to guarantee convergence of sifting, but later found to be superfluous. The criterion for accepting a sifting component as an IMF, the Stoppage criterion S. Therefore, the nomenclature for the IMF are CE(N, S) : for extrema sifting CC(N, S) : for curvature sifting

Sifting with Intermittence Test To avoid mode mixing, we have to institute a special criterion to separate oscillation of different time scales into different IMF components. The criteria is to select time scale so that oscillations with time scale shorter than this pre- selected criterion is not included in the IMF.

Intermittence Sifting : Data

Intermittence Sifting : IMF

Intermittence Sifting : Hilbert Spectra

Intermittence Sifting : Hilbert Spectra (Low)

Intermittence Sifting : Marginal Spectra

Intermittence Sifting : Marginal spectra (Low)

Intermittence Sifting : Marginal spectra (High)

Critical Parameters for Sifting Because of the inclusion of intermittence test there will be one set of intermittence criteria. Therefore, the Nomenclature for IMF here are CEI(N, S: n1, n2, … ) CCI(N, S: n1, n2, … ) with n1, n2 as the intermittence test criteria. Note: N is originally set to guarantee convergence of sifting, but later found to be superfluous.

Effects of EMD (Sifting) To separate data into components of similar scale. To eliminate ridding waves. To make the results symmetric with respect to the x-axis and the amplitude more even. –Note: The first two are necessary for valid IMF, the last effect actually cause the IMF to lost its intrinsic properties.

LOD Data

IMF CE(100, 2)

IMF CE(100, 10)

Orthogonal Index as function of N and S Contour

Orthogonality Index as function of N and S

Confidence Limit without Intermittence Criteria Number of IMF for different siftings may not be the same; therefore, average of IMF is, in general, not possible. However, we can take the mean of the Hilbert Spectra, for we can make all the spectra having the same frequency and time ranges.

Hilbert Spectrum CE(100, 2)

Mean Hilbert Spectrum : All CEs

STD Hilbert Spectrum : All CEs

Marginal Mean & STD Hilbert Spectra : All CEs

Mean and STD of Marginal Hilbert Spectra

Confidence Limit with Intermittence Criteria In general, the number of IMFs can be controlled to the same; therefore, averages of IMFs and Hilbert Spectra are all possible.

IMF CEI(100,2; 4,-1^3,45^2,-10)

IMF CEI(100,10; 4,-1^3,45^2,-10)

Envelopes of Selected Annual Cycle IMFs

Orthogonal Indices for CEI cases

IMF : Mean CEI 9 cases

IMF : STD CEI 9 cases

Mean Hilbert Spectrum : All CEIs

STD Hilbert Spectrum for All CEIs

Marginal mean & STD Hilbert Spectra : All CEIs

Mean Marginal Hilbert Spectrum & Confidence Limit : All CEIs

Mean Marginal Hilbert Spectrum & Confidence Limit : All CEs

Individual Annual Cycle IMFs : 9 CEI Cases

Details of Individual Annual Cycle : CEIs

Mean Annual Cycle & Envelope: 9 CEI Cases

Individual Envelopes for Annual Cycle IMFs

Mean Envelopes for Annual Cycle IMFs

Optimal Sifting Parameters The Maximum sifting number should be set very high to guarantee that the stoppage criterion is always satisfied. The Stoppage criterion should be selected by considering the difference between the individual case with the mean to see if there is an optimal range where the difference is minimum. The difference can be computed from the Hilbert spectra or IMF components. It turn out that the IMF is a more sensitive way to determine the optimal sifting parameters.

Computation of the Differences where V(t) can be IMF or Hilbert Spectrum.

IMF for CEI Cases : Annual Cycle

IMF for CEI Cases : Half-monthly tidal Cycle

Hilbert Spectrum : Deviation Individual form the mean CEI

Hilbert Spectrum : Deviation Individual form the mean CE

Another Example using Earthquake Data Earthquake data has no fixed time scale; therefore, it is not possible to sift with intermittence. The only way to compute the confidence limit is use an ensemble of Hilbert Spectra.

Earthquake Data

Mean Hilbert Spectrum

Another Example using Earthquake Data

Optimal Selection of Stoppage Criterion From the above tests, we can see that the Hilbert spectrum difference is less sensitive to the changes of stoppage criterion S than IMFs. From the IMF tests, we suggest that the S number should be set in the range of 3 to 10. This selection is in agreement with our past experiences; however, additional quantitative tests should be conduct for other data types.

Summary The Confidence limit presented here exists only with respect to the EMD method used. The Confidence limit presented here is only one of many possibilities. Instead of using OI as criterion, we can also use the STD of different trials to get a feeling of the stability of the analysis. Instead of Stoppage criteria, we can us different spline methods, down sampling and study their variations. Most interestingly, we could use Ensemble EMD, to be discussed next.

Envelope of IMF : c1