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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
Time Series Analysis using Hilbert-Huang Transform [HHT] is one of - - PowerPoint PPT Presentation
Time Series Analysis using Hilbert-Huang Transform [HHT] is one of - - PowerPoint PPT Presentation
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 Terminology Hilbert-Huang Transform (HHT)
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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
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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
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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
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Repeat this process of sifting
Three Notes: The mean will approach zero. The two envelopes are becoming more symmetric.
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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.
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EPICA C Ice Core IMF 1
The first IMF should have the highest frequency component.
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Decomposing TS after multiple siftings
Residual + IMF1 = Original TS, Residual = Original TS – IMF1 . . .
Trend = Original TS – (IMF1 + IMF2 + … + IMFN)
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About the trend
An application
- f 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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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
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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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EMD
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EEMD with 500 iterations
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Nonlinear Nonstationary (mean, variance
are functions of time
Local (frequency is
a function t)
Adaptive
FFT Frequency Power Spectrum: Mathematical artifacts
- ccur when you
assume linear, stationary, global, and/or nonadaptive
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Analysis of each IMF
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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
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Uneven versus Even Spacing – IMF 8
Notice the EEMD on the evenly spaced data pulls out a more uniform signal
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EMD v. EEMD – Cubic Spline IMF 8
Notice the EEMD pulls out a more uniform signal
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Linear Interpolation v. Cubic Spline – IMF 6
The cubic spline and the linear interpolation return almost the exact same signal
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Choices in Length of Spacing – IMF 6
The difference in spacing has little effect on the signal pulled out.
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Varying Levels of Noise
Notice the decompositions with higher levels of noise pull out a more uniform signal.
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An Application of EEMD to Paleoclimatology
Paleoclimatology is the study
- f 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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Noise of 0.5 Ensemble of 500 IMF 5 IMF 4 IMF 3
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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?
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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.
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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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Epica C Correlation Results for Obliquity & IMF 4
Lag results: 6,820 years ahead yields 0.6964 Choice: If the obliquity is a forcing function with which the earth responds, then the lag is 6,820 years ahead.
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Correlation between IMF 4 and Obliquity
without lag: coefficient: 0.3444 significance: 0.6220 with lag: coefficient: 0.6954 significance: 1.0000
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The eccentricity is a measure of the departure of this ellipse from circularity which combines to an approximate 100,000 year full cycle.
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Correlation between IMF 5 and Eccentricity
coefficient: 0.5992 significance: 0.9940
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The earth axis completes one full cycle of precession every 26,000 years.
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Epica C Correlation Results for Climatic Precession & IMF 3
Lag results: 16,368 years ahead yields 0.4225 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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