ENSEMBLE EMPIRICAL MODE DECOMPOSITION Noise Assisted Signal Analysis (nasa) Part I Preliminary Zhaohua Wu and N. E. Huang: Ensemble Empirical Mode Decomposition: A Noise Assisted Data Analysis Method. Advances in Adaptive Data Analysis, 1, 1-41, 2009 The title of my talk is “The impact of ENSO on NAO variability”. This work was done with Drs. Kirtman and Schneider at COLA.

Theoretical Foundations Intermittency test, though ameliorates the mode mixing, destroys the adaptive nature of EMD. The EMD study of white noise guarantees a uniformed frame of scales. The cancellation of white noise with sufficient number of ensemble.

Theoretical Background I Intermittency

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 longer than this pre-selected criterion is not included in the IMF.

Observations Intermittency test ameliorates the mode mixing considerably. Intermittency test requires a set of subjective criteria. EMD with intermittency is no longer totally adaptive. For complicated data, the subjective criteria are hard, or impossible, to determine.

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.

Theoretical Background II A Study of White Noise

Wu, Zhaohua and N. E. Huang, 2004: A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proceedings of the Royal Society of London , A 460, 1597-1611.

Methodology Based on observations from Monte Carlo numerical experiments on 1 million white noise data points. All IMF generated by 10 siftings. Fourier spectra based on 200 realizations of 4,000 data points sections. Probability density based on 50,000 data points data sections.

IMF Period Statistics IMF 1 2 3 4 5 6 7 8 9 number of peaks   IMF 1 2 3 4 5 6 7 8 9 number of peaks 347042 168176 83456 41632 20877 10471 5290 2658 1348 Mean period 2.881 5.946 11.98 24.02 47.90 95.50 189.0 376.2 741.8 period in year 0.240 0.496 0.998 2.000 3.992 7.958 15.75 31.35 61.75  

Fourier Spectra of IMFs

Empirical Observations : I Mean Energy

Empirical Observations : II Normalized spectral area is constant

Empirical Observations : III Normalized spectral area is constant

Empirical Observations : IV Computation of mean period

Empirical Observations : III The product of the mean energy and period is constant

Monte Carlo Result : IMF Energy vs. Period

Empirical Observation: Histograms IMFs By Central Limit theory IMF should be normally distributed.

Fundamental Theorem of Probability If we know the density function of a random variable, x, then we can express the density function of any random variable, y, for a given y=g(x). The procedure is as follows:

Fundamental Theorem of Probability If we know the density function of a random variable, x, is normal, then x-square should be

Chi and Chi-Square Statistics

DEGREE OF FREEDOM Random samples of length N contains N degree of freedom Each Fourier component contains one degree of freedom For EMD, the shares of DOF is proportional to its share of energy; therefore, the degree of freedom for each IMF is given as

Chi-Squared Energy Density Distributions

Histograms : IMF Energy Density By Central Limit theory, IMF should be normally distributed; therefore, its energy should be Chi-squared distributed.

Chi-Squared Energy Density Distributions By Central Limit theory, IMF should be normally distributed; therefore, its energy should be Chi-squared distributed.

Formula of Confidence Limit for IMF Distributions I

Formula of Confidence Limit for IMF Distributions II

Formula of Confidence Limit for IMF Distributions III

Formula of Confidence Limit for IMF Distributions IV For a Gaussian distribution, it is often to relate α to the standard deviation, σ , i.e., α confidence level corresponds to k σ, where k varies with α. For example, having values -2.326, -0.675, -0.0, 0.675, and 2.326 for the first, 25th, 50th, 75th and 99th percentiles (with α being 0.01, 0.25, 0.5, 0.75, 0.99), respectively.

Formula of Confidence Limit for IMF Distributions V

Formula of Confidence Limit for IMF Distributions VI

Confidence Limit for IMF Distributions

Data and IMFs SOI

Statistical Significance for SOI IMFs IMF 4, 5, 6 and 7 are 99% statistical significance signals. 1 mon 1 yr 10 yr 100 yr

Summary Not all IMF have the same statistical significance. Based on the white noise study, we have established a method to determine the statistical significant components. References: Wu, Zhaohua and N. E. Huang, 2003: A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proceedings of the Royal Society of London A460, 1597-1611. Flandrin, P., G. Rilling, and P. Gonçalvès, 2003: Empirical Mode Decomposition as a Filterbank, IEEE Signal Proc Lett. 11 (2): 112-114.

Observations The white noise signal consists of signal of all scales. EMD separates the scale dyadically. The white noise provide a uniformly distributed frame of scales through EMD.

Different Approaches but reach the same end. Flandrin, P., G. Rilling and P. Goncalves, 2004: Empirical Mode Decomposition as a filter bank. IEEE Signal Process. Lett., 11, 112-114. Flandrin, P., P. Goncalves and G. Rilling, 2005: EMD equivalent filter banks, from interpretation to applications. Introduction to Hilbert-Huang Transform and its Applications, Ed. N. E. Huang and S. S. P. Shen, p. 57-74. World Scientific, New Jersey,

Fractional Gaussian Noise aka Fractional Brownian Motion

Examples

Flandrin’s results

Flandrin’s results

Flandrin’s results

Flandrin’s results

Flandrin’s results

Flandrin’s results : Delta Function

Flandrin’s results : Delta Function

Theoretical Background III Effects of adding White Noise

Some Preliminary Robert John Gledhill, 2003: Methods for Investigating Conformational Change in Biomolecular Simulations, University of Southampton, Department of Chemistry, Ph D Thesis. He investigated the effect of added noise as a tool for checking the stability of EMD.

Some Preliminary His basic assumption is that the correct result is the one without noise:

Test results Top Whole data perturbed; bottom only 10% perturbed.

Test results

Observations They made the critical assumption that the unperturbed signal gives the correct results. When the amplitude of the added perturbing noise is small, the discrepancy is small. When the amplitude of the added perturbing noise is large, the discrepancy becomes bi-modal.