A Plea for Adaptive Data Analysis An Introduction

A Plea for Adaptive Data Analysis: An Introduction to HHT for Nonlinear and Nonstationary Data Norden E. Huang Research Center for Adaptive Data Analysis National Central University Nanjing October 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 • Scientists/Engineers • Absolute proofs • Agreement with observations • Logic consistency • Physical meaning • Mathematical rigor • 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.

Hilbert Transform : Definition

The Traditional View of the Hilbert Transform for Data Analysis

Traditional View a la Hahn (1995) : Data LOD

Traditional View a 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 m 1

Empirical Mode Decomposition Sifting : to get one IMF component

The Stoppage Criteria The Cauchy type criterion: when SD is small than a preset value, where Or, simply pre-determine the number of iterations.

Empirical Mode Decomposition: Methodology : IMF c 1

Empirical Mode Decomposition Sifting : to get all the IMF components

Definition of Instantaneous Frequency

The Idea and the need of Instantaneous Frequency 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

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 % • Overall % • • • 0. 0003 0. 0001 0. 0215 0. 0117 0. 0022 0. 0031 0. 0026 0. 0083 0. 0042 0. 0369 0. 0400 • 0. 0452

LOD : Data & c 12

LOD : Data & Sum c 11 -12

LOD : Data & sum c 10 -12

LOD : Data & c 9 - 12

LOD : Data & c 8 - 12

LOD : Detailed Data and Sum c 8 -c 12

LOD : Data & c 7 - 12

LOD : Detail Data and Sum IMF c 7 -c 12

LOD : Difference Data – sum all IMFs

Traditional View a 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 sin 2 wt)

Duffing Type Wave Perturbation Expansion

Duffing Type Wavelet Spectrum

Duffing Type Wave Hilbert Spectrum

Duffing Type Wave Marginal Spectra

Ensemble EMD Noise 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 1 D 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 Adaptive Frequency Integral transform: Global Integral transform: Regional Differentiation: Local Presentation Energy-frequency Energy-timefrequency Nonlinear no no yes Non-stationary no yes Uncertainty yes no Harmonics 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 problems And 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, A 454, 903 -995. 1999: A New View of Nonlinear Water Waves – The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31, 417457. 2003: A confidence Limit for the Empirical mode decomposition and the Hilbert spectral analysis, Proc. of Roy. Soc. London, A 459, 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 I TECHNICAL PROPOSAL AND MANAGEMENT APPROACH Mathematical Analysis of the Empirical Mode Decomposition Ingrid Daubechies 1 and Norden Huang 2 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 timefrequency 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. …….