A Plea for Adaptive Data Analysis:An Introduction to HHT for Nonlinear and

Nonstationary Data

Norden E. HuangResearch Center for Adaptive Data Analysis

National Central University

NanjingOctober 2009

Data Processing and Data Analysis

• Processing [proces < L. Processus < pp of Procedere = Proceed: pro- forward + cedere, to go] : A particular method of doing something.

• Data Processing >>>> Mathematically meaningful parameters

• Analysis [Gr. ana, up, throughout + lysis, a loosing] : A separating of any whole into its parts, especially with an examination of the parts to find out their nature, proportion, function, interrelationship etc.

• Data Analysis >>>> Physical understandings

Scientific Activities

Collecting and analyzing data, synthesizing and theorizing the analyzed results are the core of scientific activities.

Therefore, data analysis is a key link in this continuous loop.

Data Analysis

There are, unfortunately, tensions between sciences and mathematics.

Data analysis is too important to be left to the mathematicians.

Why?!

Different Paradigms Mathematics vs. Science/Engineering

• Mathematicians

• Absolute proofs

• Logic consistency

• Mathematical rigor

• Scientists/Engineers

• Agreement with observations

• Physical meaning

• Working Approximations

Motivations for alternatives: Problems for Traditional Methods

• Physical processes are mostly nonstationary

• Physical Processes are mostly nonlinear

• Data from observations are invariably too short

• Physical processes are mostly non-repeatable.

Ensemble mean impossible, and temporal mean might not be meaningful for lack of stationarity and ergodicity. Traditional methods are inadequate.

p

2 2 1 / 2 1

i ( t )

For any x( t ) L ,

1 x( )y( t ) d ,

t

then, x( t )and y( t ) form the analytic pairs:

z( t ) x( t ) i y( t ) ,

where

y( t )a( t ) x y and ( t ) tan .

x( t )

a( t ) e

Hilbert Transform : Definition

The Traditional View of the Hilbert Transform for Data Analysis

Traditional Viewa la Hahn (1995) : Data LOD

Traditional Viewa la Hahn (1995) : Hilbert

The Empirical Mode Decomposition Method and Hilbert Spectral Analysis

Sifting

Empirical Mode Decomposition: Methodology : Test Data

Empirical Mode Decomposition: Methodology : data and m1

Empirical Mode DecompositionSifting : to get one IMF component

1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,

h m h ,

.....

.....

h m h

.h c

.

The Stoppage Criteria

The Cauchy type criterion: when SD is small than a pre-set value, where

T2

k 1 kt 0

T2

k 1t 0

h ( t ) h ( t )SD

h ( t )

Or, simply pre-determine the number of iterations.

Empirical Mode Decomposition: Methodology : IMF c1

Empirical Mode DecompositionSifting : to get all the IMF components

1 1

1

n

n

jj

2 2

n

1

n

n

1

x( t ) c r ,

r c r ,

r

r

. . .

r c .

c .x( t )

Definition of Instantaneous Frequency

i ( t )

t

The Fourier Transform of the Instrinsic Mode

Funnction, c( t ), gives

W ( ) a( t ) e dt

By Sta

d ( t ),

d

tionary phase approximation we have

This is defined as the Ins tan taneous Frequency .

t

The Idea and the need of Instantaneous Frequency

k , ;t

k0 .

t

According to the classic wave theory, the wave conservation law is based on a gradually changing φ(x,t) such that

Therefore, both wave number and frequency must have instantaneous values. But how to find φ(x, t)?

The combination of Hilbert Spectral Analysis and Empirical Mode Decomposition has been

designated by NASA as

HHT

(HHT vs. FFT)

Comparison between FFT and HHT

j

j

t

i t

jj

i ( )d

jj

1. FFT :

x( t ) a e .

2. HHT :

x( t ) a ( t ) e .

Comparisons: Fourier, Hilbert & Wavelet

Speech Analysis Hello : Data

Four comparsions D

An Example of Sifting

Length Of Day Data

LOD : IMF

Orthogonality Check

• Pair-wise % • 0.0003• 0.0001• 0.0215• 0.0117• 0.0022• 0.0031• 0.0026• 0.0083• 0.0042• 0.0369• 0.0400

• Overall %

• 0.0452

LOD : Data & c12

LOD : Data & Sum c11-12

LOD : Data & sum c10-12

LOD : Data & c9 - 12

LOD : Data & c8 - 12

LOD : Detailed Data and Sum c8-c12

LOD : Data & c7 - 12

LOD : Detail Data and Sum IMF c7-c12

LOD : Difference Data – sum all IMFs

Traditional Viewa la Hahn (1995) : Hilbert

Mean Annual Cycle & Envelope: 9 CEI Cases

Properties of EMD Basis

The Adaptive Basis based on and derived from the data by the empirical method satisfy nearly all the traditional requirements for basis empirically and a posteriori:

Complete

Convergent

Orthogonal

Unique

Hilbert’s View on Nonlinear Data

Duffing Type Wave

Data: x = cos(wt+0.3 sin2wt)

Duffing Type WavePerturbation Expansion

For 1 , we can have

cos t cos sin 2 t sin t sin sin 2 t

cos t sin t sin 2 t ....

This is very similar to the solutionof D

x( t ) cos t sin 2 t

1 cos t cos 3

uffing equ

t ....2

atio

2

n .

Duffing Type WaveWavelet Spectrum

Duffing Type WaveHilbert Spectrum

Duffing Type WaveMarginal Spectra

Ensemble EMDNoise Assisted Signal Analysis (nasa)

Utilizing the uniformly distributed reference frame based on the white noise to eliminate the mode mixing

Enable EMD to apply to function with spiky or flat portion

The true result of EMD is the ensemble of infinite trials.

Wu and Huang, Adv. Adapt. Data Ana., 2009

New Multi-dimensional EEMD

• Extrema defined easily• Computationally inexpensive, relatively• Ensemble approach removed the Mode

Mixing• Edge effects easier to fix in each 1D slice• Results are 2-directional

Wu, Huang and Chen, AADA, 2009

What This Means• EMD separates scales in physical space; it generates

an extremely sparse representation for any given data.

• Added noises help to make the decomposition more robust with uniform scale separations.

• Instantaneous Frequency offers a total different view for nonlinear data: instantaneous frequency needs no harmonics and is unlimited by uncertainty principle.

• Adaptive basis is indispensable for nonstationary and nonlinear data analysis

• EMD establishes a new paradigm of data analysis

Comparisons

Fourier Wavelet Hilbert

Basis a priori a priori Adaptive

Frequency Integral transform: Global

Integral transform: Regional

Differentiation:

Local

Presentation Energy-frequency Energy-time-frequency

Energy-time-frequency

Nonlinear no no yes

Non-stationary no yes yes

Uncertainty yes yes no

Harmonics yes yes no

Conclusion

Adaptive method is the only scientifically meaningful way to analyze nonlinear and nonstationary data.

It is the only way to find out the underlying physical processes; therefore, it is indispensable in scientific research.

EMD is adaptive; It is physical, direct, and simple.

But, we have a lot of problemsAnd need a lot of helps!

National Central University

Research Center for Adaptive Data Analysis

History of HHT

1998: The Empirical Mode Decomposition Method and the Hilbert Spectrum for Non-stationary Time Series Analysis, Proc. Roy. Soc. London, A454, 903-995.

1999: A New View of Nonlinear Water Waves – The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31, 417-457.

2003: A confidence Limit for the Empirical mode decomposition and the Hilbert spectral analysis, Proc. of Roy. Soc. London, A459, 2317-2345.

2004: A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proc. Roy. Soc. London, (in press)

Recent Developments in HHT

2007: On the trend, detrending, and variability of nonlinear and nonstationary time series. Proc. Natl. Acad. Sci., 104, 14,889-14,894.

2009: On Ensemble Empirical Mode Decomposition. Advances in Adaptive Data Analysis. Advances in Adaptive data Analysis, 1, 1-41

2009: On instantaneous Frequency. Advances in Adaptive Data Analysis 1, 177-229.

2009: Multi-Dimensional Ensemble Empirical Mode Decomposition. Advances in Adaptive Data Analysis, 1, 339-372.

VOLUME ITECHNICAL PROPOSAL AND MANAGEMENT

APPROACHMathematical Analysis of the Empirical Mode Decomposition

Ingrid Daubechies1 and Norden Huang2

1 Program in Applied and Computational Mathematics (Princeton)2 Research Center for Adaptive Data Analysis,

(National Central University)

Since its invention by PI Huang over ten years ago, the Empirical Mode Decomposition (EMD) has been applied to a wide range of applications. The EMD is a two-stage, adaptive method that provides a nonlinear time-frequency analysis that has been remarkably successful in the analysis of nonstationary signals. It has been used in a wide range of fields, including (among many others) biology, geophysics, ocean research, radar and medicine. …….