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Analyzing Nonlinear Time Series with Hilbert-Huang Transform Sai-Ping Li Lunch Seminar December 7, 2011

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Introduction Introduction Hilbert-Huang Transform Hilbert-Huang Transform Empirical Mode Decomposition Empirical Mode Decomposition Some Examples and Applications Some Examples and Applications Summary and Future Works Summary and Future Works

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Some Examples of Wave Forms

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Stationary and Non-Stationary Time Series

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Time series: random data plus trend, with best-fit line and different smoothings

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Earthquake data of El Centro in 1985. Tidal data of Kahului Harbor, Maui, October 4-9, 1994. Examples of Non-Stationary Time Series Blood pressure of a rat. Difference of daily Non-stationary annual cycle and 30- year mean annual cycle of surface temperature at Victoria Station, Canada.

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Numerical Solution for the Duffing Equation Left: The numerical solution of x and dx/dt as a function of time. Right: The phase diagram with continuous winding indicates no fixed period of oscillations.

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Financial Time Series

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Stylized Facts :

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An Empirical data analysis method: Hilbert-Huang Transform Empirical Mode Decomposition (EMD) + Hilbert Spectral Analysis (HSA)

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The Hilbert-Huang Transform

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Empirical Mode Decomposition: IMF Based on the assumption that any dataset consists of different simple intrinsic modes of oscillations. Each of these intrinsic oscillatory modes is represented by an intrinsic mode function (IMF) with the following definition: (1)in the whole dataset, the number of extrema and the number of zero-crossings must either equal or differ at most by one, and (2) at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero.

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Example: A test Dataset

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The data (blue) upper and lower envelopes (green) defined by the local maxima and minima, respectively, and the mean value of the upper and lower envelopes given in red.

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Repeat the Sifting Process:

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Empirical Mode Decomposition

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Example: Gold Price Data Analysis

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Statistics: LME gold prices

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Composition: LME monthly gold prices Dividing the components into high, low and trend by reasonable boundary (12 months), and analyze the factors or economic meanings in different time scales. Mean period 12 months TrendLow frequency termMean period < 12 monthsHigh frequency term 12 months Boundary:

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Composition of LME Gold Prices

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Trend: inflation Pearson correlation Kendall correlation Trend of CPI0.9390.917 Trend of PPI0.9060.917 We assume the gold price trend relates to inflation at first. US monthly CPI and PPI are used to quantify the ordering of inflation. Since the gold price trend hold high correlation with trend of CPI and PPI, the economic meanings of trend can be described by inflation.

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Low frequency term: significant events 1973/10: 4th Middle East War 1979/11 1980/01: Iranian hostage crisis 1980/09: Iran/Iraq War 1982/08: Mexico External Debt Crisis 1987/10: New York Stock Market Crash 2007/02: USA Subprime Mortgage Crisis 2008/09: Lehman Brothers bankruptcy 1996 to 2006: Booming economic in USA The six obvious variations correspond to six significant events. The six significant events include wars, panic international situation, and financial crisis.

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Example: Electroencephalography (EEG) and Heart Rate Variability (HRV)

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Example: Electroencephalography (EEG) Active Wake Stage

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Example: Electroencephalography (EEG) Stage I: Light Sleep Stage II Stage III Slow Wave Sleep Stage IV Quiet Sleep

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Example: Electroencephalography (EEG) Left: A dataset of active wake stage. Right: Its corresponding IMFs after EMD.

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Electrocardiography (ECG)

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Example: Heart Rate Variability (HRV)

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Left: Original dataset Right: HRV after EMD Example: Heart Rate Variability (HRV)

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Relation of EEG and HRV of an Obstructive Sleep Apnea (OSA) patient using EMD analysis

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Partial List of Application of HHT Biomedical applications: Huang et al. [1999] Chemistry and chemical engineering: Phillips et al. [2003] Financial applications: Huang et al. [2003b] Image processing: Hariharan et al. [2006] Meteorological and atmospheric applications: Salisbury and Wimbush [2002] Ocean engineering:Schlurmann [2002] Seismic studies: Huang et al. [2001] Solar Physics: Barnhart and Eichinger [2010] Structural applications: Quek et al. [2003] Health monitoring: Pines and Salvino [2002] System identification: Chen and Xu [2002]

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Comparison of Fourier, Wavelet and HHT

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Problems related to Hilbert-Huang Transform (1)Adaptive data analysis methodology in general (2) Nonlinear system identification methods (3) Prediction problem for non-stationary processes (end effects) (4) Spline problems (best spline implementation for the HHT, convergence and 2-D) (5) Optimization problems (the best IMF selection and uniqueness) (6) Approximation problems (Hilbert transform and quadrature) (7) Other miscellaneous questions concerning the HHT…….

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TIME SERIES by H.V.S. DE SILVA DEPARTMENT OF MATHEMATICS

TIME SERIES by H.V.S. DE SILVA DEPARTMENT OF MATHEMATICS

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