Time Series Analysis using Hilbert-Huang Transform

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1 Time Series Analysis using Hilbert-Huang Transform [HHT] is one of the most important discoveries in the field of applied mathematics in NASA history. Presented by Nathan Taylor & Brent Davis

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3 Terminology Hilbert-Huang Transform (HHT) Empirical Mode Decomposition (EMD) Ensemble Empirical Mode Decomposition (EEMD) Intrinsic Mode Function (IMF) Empirical Relying on derived from observation or experiment Mode A particular form, variety, or manner Decomposition The separation of a whole into basic parts Intrinsic Belonging naturally

4 Computing the high frequency IMF Find all local maxima of TS Find all local minima of TS Fit a curve through maxima Fit a curve through minima Find the mean of these two curves

5 Sifting: Subtract mean from Original Time Series Subtract the mean from original TS The goal is after a number of siftings, to minimize the mean

6 Repeat this process of sifting Three Notes: The mean will approach zero. The two envelopes are becoming more symmetric.

7 After multiple siftings Q: What property does it have? The mean is now minimized, so we would like to develop a stopping criterion a) b) The number of extrema and zero crossings are alternating.

8 EPICA C Ice Core IMF 1 The first IMF should have the highest frequency component.

9 Decomposing TS after multiple siftings Residual + IMF1 = Original TS, Residual = Original TS IMF1... Trend = Original TS (IMF1 + IMF2 + + IMFN)

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11 About the trend An application of EMD is to find a empirical trend line. The trend shouldn't have periodic behavior Here's the last IMF and original TS

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20 What is mode mixing? Cubic spline are not perfect if envelopes are close together and at a height away from zero Each IMF should have a dominant signal. Rather than get clean IMF signals there are combinations of frequencies

21 Explanation of EEMD Ensemble - A unit or group of complementary parts that contribute to a single effect. Outline: Add white noise to original TS Create a set of IMFs using EMD Do this N times Take the average of each individual IMF Result: New set of IMFs with less mode mixing

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23 EEMD with 500 iterations

24 FFT Frequency Power Spectrum: Mathematical artifacts occur when you assume linear, stationary, global, and/or nonadaptive Nonlinear Nonstationary (mean, variance are functions of time Local (frequency is a function t) Adaptive

25 Analysis of each IMF

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27 Mathematical Artifacts of Time Series Analysis using HHT Spacing: -Even vs. Uneven -Length of spacing Curve Fitting: -Cubic Spline vs. Linear Interpolation Level of white noise added We'll use the Vostok & Epica C Ice Core Data for examples

28 Uneven versus Even Spacing IMF 8 Notice the EEMD on the evenly spaced data pulls out a more uniform signal

29 EMD v. EEMD Cubic Spline IMF 8 Notice the EEMD pulls out a more uniform signal

30 Linear Interpolation v. Cubic Spline IMF 6 The cubic spline and the linear interpolation return almost the exact same signal

31 Choices in Length of Spacing IMF 6 The difference in spacing has little effect on the signal pulled out.

32 Varying Levels of Noise Notice the decompositions with higher levels of noise pull out a more uniform signal.

33 An Application of EEMD to Paleoclimatology Paleoclimatology is the study of climate change on a long time scale typically using a proxy measurement. We looked at two: EPICA C Ice Core Data Measures Deuterium Content

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36 Noise of 0.5 Ensemble of 500 IMF 3 IMF 4 IMF 5

37 Welch Power Spectrum and Mean Periods of IMFs Notice there are three noticeable peaks in each 100,000 year cycle? 40,000 year cycle? 20,000 year cycle?

38 Milankovitch cycles: The earth axis completes one full cycle of precession every 26,000 years. The angle between Earth's rotational axis and the normal to the plane of its orbit, its obliquity, completes a full cycle every 41,000 years. The eccentricity is a measure of the departure of this ellipse from circularity Which combines to an approximate 100,000 year full cycle.

39 The angle between Earth's rotational axis and the normal to the plane of its orbit, its obliquity, completes a full cycle every 41,000 years.

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41 Lag results: Epica C Correlation Results for Obliquity & IMF 4 6,820 years ahead yields Choice: If the obliquity is a forcing function with which the earth responds, then the lag is 6,820 years ahead.

42 Correlation between IMF 4 and Obliquity without lag: coefficient: significance: with lag: coefficient: significance:

43 The eccentricity is a measure of the departure of this ellipse from circularity which combines to an approximate 100,000 year full cycle.

44 Correlation between IMF 5 and Eccentricity coefficient: significance:

45 The earth axis completes one full cycle of precession every 26,000 years.

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48 Epica C Correlation Results for Climatic Precession & IMF 3 Lag results: 16,368 years ahead yields Choice: If the precession is a forcing function with which the earth responds, then the lag is 16,368 years ahead.

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50 without lag: coefficient: significance: 0.35 Correlation between IMF 3 and Climatic Precession with lag: coefficient: significance: