An Introduction to Hilbert-Huang Transform A Plea for Adaptive ...

An Introduction to Hilbert-Huang Transform: A Plea for Adaptive Data Analysis Norden E. Huang Research Center for Adaptive Data Analysis National Central University Data Processing and Data Analysis • Processing [proces < L. Processus < pp of Procedere = Proceed: pro- forward + cedere, to go] : A particular method of doing something. • 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 • Why we do it? • How did we do it? • What should we do? Why? Why do we have to analyze data? Data are the only connects we have with the reality; data analysis is the only means we can find the truth and deepen our understanding of the problems. Ever since the advance of computer and sensor technology, there is an explosion of very complicate data. The situation has changed from a thirsty for data to that of drinking from a fire hydrant. Henri Poincaré Science is built up of facts*, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house. * Here facts are indeed our data. Data and Data Analysis Data Analysis is the key step in converting the ‘facts’ into the edifice of science. It infuses meanings to the cold numbers, and lets data telling their own stories and singing their own songs. Science vs. Philosophy Data and Data Analysis are what separate science from philosophy: With data we are talking about sciences; Without data we can only discuss philosophy. Scientific Activities Collecting, analyzing, synthesizing, and theorizing are the core of scientific activities. Theory without data to prove is just hypothesis. Therefore, data analysis is a key link in this continuous loop. Data Analysis Data analysis is too important to be left to the mathematicians. Why?! Different Paradigms I Mathematics vs. Science/Engineering • Mathematicians • • • • Scientists/Engineers • • • Absolute proofs Logic consistency Mathematical rigor Agreement with observations Physical meaning Working Approximations Different Paradigms II Mathematics vs. Science/Engineering • Mathematicians • • • Scientists/Engineers • • Idealized Spaces Perfect world in which everything is known Real Space Real world in which knowledge is incomplete and limited • Inconsistency in the different spaces and the real world • Constancy in the real world within allowable approximation Rigor vs. Reality As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein How? Data Processing vs. Analysis All traditional ‘data analysis’ methods are really for ‘data processing’. They are either developed by or established according to mathematician’s rigorous rules. Most of the methods consist of standard algorithms, which produce a set of simple parameters. They can only be qualified as ‘data processing’, not really ‘data analysis’. Data processing produces mathematical meaningful parameters; data analysis reveals physical characteristics of the underlying processes. Data Processing vs. Analysis In pursue of mathematic rigor and certainty, however, we lost sight of physics and are forced to idealize, but also deviate from, the reality. As a result, we are forced to live in a pseudo-real world, in which all processes are Linear and Stationary 削足適履 Trimming the foot to fit the shoe. Available Data Analysis Methods for Nonstationary (but Linear) time series • • • • Spectrogram Wavelet Analysis Wigner-Ville Distributions Empirical Orthogonal Functions aka Singular Spectral Analysis • Moving means • Successive differentiations Available Data Analysis Methods for Nonlinear (but Stationary and Deterministic) time series • Phase space method • Delay reconstruction and embedding • Poincaré surface of section • Self-similarity, attractor geometry & fractals • Nonlinear Prediction • Lyapunov Exponents for stability Typical Apologia • Assuming the process is stationary …. • Assuming the process is locally stationary …. • As the nonlinearity is weak, we can use perturbation approach …. Though we can assume all we want, but the reality cannot be bent by the assumptions. The Real World Mathematics are well and good but nature keeps dragging us around by the nose. Albert Einstein 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. What? The Job of a Scientist The job of a scientist is to listen carefully to nature, not to tell nature how to behave. Richard Feynman To listen is to use adaptive methods and let the data sing, and not to force the data to fit preconceived modes. How to define nonlinearity? Based on Linear Algebra: nonlinearity is defined based on input vs. output. But in reality, such an approach is not practical. The alternative is to define nonlinearity based on data characteristics. Characteristics of Data from Nonlinear Processes d 2 x 2  x   x 3   co s  t dt  d 2 x 2  x dt  1   x 2    co s  t S p rin g w ith p o sitio n d ep en d en t co n s ta n t , in t ra  w a ve freq u en cy m o d u la tio n ; th erefo re , w e n eed in s ta n ta n eo u s freq u en c y . Duffing Pendulum x d x dt 2 2  x (1   x 2 )   co s  t . Hilbert Transform : Definition F or any x ( t )  L p , y( t )  1     x( ) t  d , th en , x ( t ) a n d y ( t ) fo rm th e a n a lytic p a irs: i ( t ) z( t )  x( t )  i y( t )  a( t ) e , w h ere a( t )   x 2  y 2  1/ 2 a n d  ( t )  ta n 1 y( t ) x( t ) . Hilbert Transform Fit Conformation to reality rather then to Mathematics We do not have to apologize, we should use methods that can analyze data generated by nonlinear and nonstationary processes. That means we have to deal with the intrawave frequency modulations, intermittencies, and finite rate of irregular drifts. Any method satisfies this call will have to be adaptive. The Traditional Approach of Hilbert Transform for Data Analysis Traditional Approach a la Hahn (1995) : Data LOD Traditional Approach a la Hahn (1995) : Hilbert Traditional Approach a la Hahn (1995) : Phase Angle Traditional Approach a la Hahn (1995) : Phase Angle Details Traditional Approach a la Hahn (1995) : Frequency Why the traditional approach does not work? Hilbert Transform a cos  + b : Data Hilbert Transform a cos  + b : Phase Diagram Hilbert Transform a cos  + b : Phase Angle Details Hilbert Transform a cos  + b : Frequency 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 Decomposition: Methodology : data & h1 Empirical Mode Decomposition: Methodology : h1 & m2 Empirical Mode Decomposition: Methodology : h3 & m4 Empirical Mode Decomposition: Methodology : h4 & m5 Empirical Mode Decomposition Sifting : to get one IMF component x ( t )  m 1  h1 , h1  ..... ..... hk  1  m k  hk .  hk  c 1 m2  h2 , . Two Stoppage Criteria : S and SD A. The S number : S is defined as the consecutive number of siftings, in which the numbers of zerocrossing and extrema are the same for these S siftings. B. SD is small than a pre-set value, where  SD  T hk  1 ( t )  hk ( t ) 2 t0  T hk  1 ( t ) 2 t0 Empirical Mode Decomposition: Methodology : IMF c1 Definition of the Intrinsic Mode Function (IMF) A n y fu n ctio n h a vin g th e sa m e n u m b ers o f zero  cro s sin g s a n d ex trem a , a n d a lso h a vin g sym m etric en velo p es d efin ed b y lo ca l m a x im a a n d m in im a resp ectively is d efin ed a s a n In trin sic M o d e F u n ctio n ( IM F ). A ll IM F en jo ys g o o d H ilb ert T ra n sfo rm : i ( t )  c( t )  a( t ) e Empirical Mode Decomposition Sifting : to get all the IMF components x ( t )  c 1  r1 , r1  c 2  r2 , . . . rn  1  c n  rn .  x( t )   n c j  rn . j1 Empirical Mode Decomposition: Methodology : data & r1 Empirical Mode Decomposition: Methodology : data and m1 Empirical Mode Decomposition: Methodology : data, r1 and m1 Empirical Mode Decomposition: Methodology : IMFs Definition of Instantaneous Frequency T h e F o u rier T ra n sfo rm o f th e In strin sic M o d e F u n n ctio n , c ( t ), g ives i (  t ) W ( )   t a( t ) e dt B y S ta tio n a ry p h a se a p p ro x im a tio n w e h a ve d ( t ) dt T h is is d efin ed a s th e In s ta n ta n eo u s F req u en cy .   , Definition of Frequency Given the period of a wave as T ; the frequency is defined as   1 T . Equivalence : The definition of frequency is equivalent to defining velocity as Velocity = Distance / Time Instantaneous Frequency V elocity  distan ce tim e dx dt 1 period d dt S o th at both v an d  can appear in differen tial equ ation s . ; m ean velocity N ew ton  v  F requ en cy  ; m ean frequ en cy H H T defin es th e p h ase fu n ction    The combination of Hilbert Spectral Analysis and Empirical Mode Decomposition is designated as HHT (HHT vs. FFT) Jean-Baptiste-Joseph Fourier 1807 “On the Propagation of Heat in Solid Bodies” 1812 Grand Prize of Paris Institute “Théorie analytique de la chaleur” ‘... the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigor.’ 1817 1822 Elected to Académie des Sciences Appointed as Secretary of Math Section paper published Fourier’s work is a great mathematical poem. Lord Kelvin Comparison between FFT and HHT 1. F F T : x( t )    j a j e i j t . 2. H H T : x( t )    j i  t  j (  ) d a j( t ) e . Comparisons: Fourier, Hilbert & Wavelet 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 View a la Hahn (1995) : Hilbert Mean Annual Cycle & Envelope: 9 CEI Cases Mean Hilbert Spectrum : All CEs Tidal Machine 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 a posteriori: Complete Convergent Orthogonal Unique Hilbert’s View on Nonlinear Data Duffing Type Wave Data: x = cos(wt+0.3 sin2wt) Duffing Type Wave Perturbation Expansion F or   1 , w e can h ave x ( t )  cos   t   sin 2  t   cos  t cos   sin 2  t   sin  t sin   sin 2  t   cos  t   sin  t sin 2  t  ....       1   cos  t  cos 3  t  .... 2 2  T h is is very sim ilar to th e solu tion of D u ffin g e qu ation . Duffing Type Wave Wavelet Spectrum Duffing Type Wave Hilbert Spectrum Duffing Type Wave Marginal Spectra Duffing Equation d 2 x 2  x   x 3   cos  t . dt S o lved w ith o d e 2 3 tb fo r t  0 to 2 0 0 w ith   1   0 .1   0 .0 4 H z In itia l co n d itio n : [ x ( o ) , x '( 0 ) ]  [ 1 , 1 ] Duffing Equation : Data Duffing Equation : IMFs Duffing Equation : Hilbert Spectrum Duffing Equation : Detailed Hilbert Spectrum Duffing Equation : Wavelet Spectrum Duffing Equation : Hilbert & Wavelet Spectra Speech Analysis Nonlinear and nonstationary data Speech Analysis Hello : Data Four comparsions D Global Temperature Anomaly Annual Data from 1856 to 2003 Global Temperature Anomaly 1856 to 2003 IMF Mean of 10 Sifts : CC(1000, I) Statistical Significance Test Data and Trend C6 Rate of Change Overall Trends : EMD and Linear What This Means • Instantaneous Frequency offers a total different view for nonlinear data: instantaneous frequency with no need for harmonics and unlimited by uncertainty. • Adaptive basis is indispensable for nonstationary and nonlinear data analysis • HHT establishes a new paradigm of data analysis Comparisons Fourier Basis a priori Wavelet a priori Hilbert Adaptive Frequency Presentation Nonlinear Non-stationary Uncertainty Harmonics Convolution: Global Energy-frequency no no yes yes Convolution: Regional Energy-timefrequency no yes yes yes Differentiation: Local Energy-timefrequency yes yes no no Conclusion Adaptive method is the only scientifically meaningful way to analyze data. It is the only way to find out the underlying physical processes; therefore, it is indispensable in scientific research. It is physical, direct, and simple. 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. The invention of the basic method of EMD, and Hilbert transform for determining the Instantaneous Frequency and energy. 1999: A New View of Nonlinear Water Waves – The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31, 417-457. Introduction of the intermittence in decomposition. 2003: A confidence Limit for the Empirical mode decomposition and the Hilbert spectral analysis, Proc. of Roy. Soc. London, A459, 2317-2345. Establishment of a confidence limit without the ergodic assumption. 2004: A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proc. Roy. Soc. London, (in press) Defined statistical significance and predictability. 2004: On the Instantaneous Frequency, Proc. Roy. Soc. London, (Under review) Removal of the limitations posted by Bedrosian and Nuttall theorems for instantaneous Frequency computations. Current Applications • • • Non-destructive Evaluation for Structural Health Monitoring – (DOT, NSWC, and DFRC/NASA, KSC/NASA Shuttle) Vibration, speech, and acoustic signal analyses – (FBI, MIT, and DARPA) Earthquake Engineering – (DOT) Bio-medical applications – (Harvard, UCSD, Johns Hopkins) Global Primary Productivity Evolution map from LandSat data – (NASA Goddard, NOAA) Cosmological Gravity Wave – (NASA Goddard) Financial market data analysis – (NCU) • • • • Advances in Adaptive data Analysis: Theory and Applications A new journal to be published by the World Scientific Under the joint Co-Editor-in-Chief Norden E. Huang, RCADA NCU Thomas Yizhao Hou, CALTECH in the January 2008 Oliver Heaviside 1850 - 1925 Why should I refuse a good dinner simply because I don't understand the digestive processes involved.