Download
1 / 113
1.31k likes | 1.87k Views
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.
E N D
An Introduction to Hilbert-Huang Transform:A Plea for Adaptive Data AnalysisNorden E. HuangResearch Center for Adaptive Data AnalysisNational 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 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?!
Mathematicians Absolute proofs Logic consistency Mathematical rigor Scientists/Engineers Agreement with observations Physical meaning Working Approximations Different Paradigms IMathematics vs. Science/Engineering
Mathematicians Idealized Spaces Perfect world in which everything is known Inconsistency in the different spaces and the real world Scientists/Engineers Real Space Real world in which knowledge is incomplete and limited Constancy in the real world within allowable approximation Different Paradigms IIMathematics vs. Science/Engineering
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
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 Methodsfor 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 Methodsfor 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.
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.
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
The Empirical Mode Decomposition Method and Hilbert Spectral AnalysisSifting
Empirical Mode DecompositionSifting : to get one IMF component
