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

Analyzing Nonlinear Time Series with

Hilbert-Huang Transform

Sai-Ping LiLunch Seminar

December 7, 2011

Introduction

Hilbert-Huang Transform

Empirical Mode Decomposition

Some Examples and Applications

Summary and Future Works

Some Examples of Wave Forms

Stationary and Non-Stationary Time Series

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

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.

Consider the non-dissipative Duffing equation:

γ is the amplitude of a periodic forcing function with a frequency ω.

If = 0, the system is linear and the solution can be found easily.𝜖If 𝜖 ≠ 0, the system becomes nonlinear. If 𝜖 is not small, perturbation method cannot be used. The system is highly nonlinear and new phenomena such as bifurcation and chaos can occur.

Rewrite the above equation as:

The quantity in the parenthesis above can be regarded as a varying spring constant, or varying pendulum length.

The frequency of the system can change from location to location, and also from time to time, even within one oscillation cycle.

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.

Financial Time Series

Stylized Facts :

Stylized Facts :

An Empirical data analysis method:

Hilbert-Huang Transform

Empirical Mode Decomposition (EMD)+

Hilbert Spectral Analysis (HSA)

The Hilbert-Huang Transform

Empirical Mode Decomposition:

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.

Example: A test Dataset

Identify all local extrema, then connect all the local maxima by a cubic spline line in the upper envelope. Repeat the procedure for the local minima to produce the lower envelope. The upper and lower envelopes should cover all the data between them. Their mean is designated as .

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.

The data (red) and (blue).

The difference between the data and the mean is the first component , i.e.,

Left: Repeated sifting steps with and . Right: Repeated sifting steps with and .

Repeat the Sifting Process:

The Sifting Process for the first IMF:

In the next step,

Repeat Sifting,●

●

●

And is designated as,

The first IMF component after 12 steps.

Separate from the original data, and call the residue ,

The original data (blue) and the residue .

The procedure is repeated to get all the subsequent ’s ,

●

●

●

The original dataset is decomposed into a sum of IMF’s and a residue,

A decomposition of the data into n-empirical modes is achieved, and a residue obtained which can either be the mean trend or a constant.

A test dataset shown in the above. On the right is the original dataset decomposed into 8 IMF’s and a trend ().

Empirical Mode Decomposition

Example: Gold Price Data Analysis

-100

100

IMF5

-100

100

IMF6

-100

100IMF7

-200

200

IMF8

0

1400

data

-50

50

IMF1

-50

50

IMF2

-50

50

IMF3

-100

100

IMF4

1968M1 1976M5 1984M9 1992M1 2000M5 2008M90

900

res.

Date

Statistics: LME gold prices

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≧

Trend

Low frequency term

Mean period < 12 months

High frequency term

12 months• Boundary:

Composition of LME Gold Prices

Trend: inflation

Pearson

correlation

Kendall

correlation

Trend of CPI 0.939 0.917

Trend of PPI 0.906 0.917

• 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”.

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.

500

10001500

data

-20020

C1

-20020

C2

-50050

C3

-50050

C4

-50050

C5

-50050

C6

-2000

200

C7

2-Jan-07 2-Jan-08 2-Jan-09 4-Jan-10 30-Dec-10500

10001500

Day

Residue

Example:

Electroencephalography (EEG) and

Heart Rate Variability (HRV)

Example: Electroencephalography (EEG)

Active Wake Stage

Example: Electroencephalography (EEG)

Stage I: Light Sleep

Stage II

Stage IIISlow Wave Sleep

Stage IVQuiet Sleep

Example: Electroencephalography (EEG)

Left: A dataset of active wake stage.Right: Its corresponding IMF’s after EMD.

Electrocardiography (ECG)

Example: Heart Rate Variability (HRV)

Left: Original datasetRight: HRV after EMD

Example: Heart Rate Variability (HRV)

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

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]

Comparison of Fourier, Wavelet and HHT

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…….

Application: Price Fluctuations in Financial Time Series

All parameters are time dependent. is the drift, is the volatility and is a standard Wiener process with zero mean and unit rate of variance.

Stoppage Criterion: The criterion to stop the sifting of IMF’s.

Historically, there are two methods:

(1) The normalized squared difference between two successive sifting operations defined as

If this squared difference is less than a preset value, the sifting process will stop.

(2) The sifting process will stop only after S consecutive times, when the numbers of zero-crossings and extrema stay the same and are equal or differ at most by one. The number S is preset.

Orthogonality:

The orthogonality of the EMD components should also be checked a posteriorinumerically as follows: let us first write equation

as

Form the square of the signal as

And define IO as

If the IMF’s are orthogonal, IO should be zero.