On the Trend, Detrend and the Variability of Nonlinear and Nonstationary Time Series

A new application of HHT

Satellite Altimeter Data : Greenland

Two Sets of Data

The State-of-the-Arts

“One economist’s trend is another economist’s cycle” Engle, R. F. and Granger, C. W. J. 1991 Long-run Economic Relationships.

Cambridge University Press.

• Simple trend – straight line

• Stochastic trend – straight line for each quarter

Philosophical Problem

名不正則言不順

言不順則事不成 

                      ——孔夫子

On Definition

Without a proper definition, logic discourse would be impossible.

Without logic discourse, nothing can be accomplished.

Confucius

Definition of the Trend

Within the given data span, the trend is an intrinsically fitted monotonic function, or a function in which there can be at most one extremum.

The trend should be determined by the same mechanisms that generate the data; it should be an intrinsic and local property.

Being intrinsic, the method for defining the trend has to be adaptive. The results should be intrinsic (objective); all traditional trend determination methods give extrinsic (subjective) results.

Being local, it has to associate with a local length scale, and be valid only within that length span as a part of a full wave cycle.

Definition of Detrend and Variability

Within the given data span, detrend is an operation to remove the trend.

Within the given data span, the Variability is the residue of the data after the removal of the trend.

As the trend should be intrinsic and local properties of the data; Detrend and Variability are also local properties.

All traditional trend determination methods are extrinsic and/or subjective.

The Need for HHT

HHT is an adaptive (local, intrinsic, and objective) method to find the intrinsic local properties of the given data set, therefore, it is ideal for defining the trend and variability.

History of HHT1998: 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 EMD.

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 of IMFs.

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.

Two Sets of Data

Global Temperature Anomaly

Annual Data from 1856 to 2003

Global Temperature Anomaly 1856 to 2003

IMF Mean of 10 Sifts : CC(1000, I)

Mean IMF

STD IMF

Statistical Significance Test

Data and Trend C6

Data and Overall Trends : EMD and Linear

Rate of Change Overall Trends : EMD and Linear

Variability with Respect to Overall trend

Data and Trend C5:6

Data and Trends: C5:6

Rate of Change Trend C5:6

Trend Period C5

Variability with Respect to 65-Year trend

Data and Trend C4:6

Data and Trend C4:6

Rate of Change Trend C4:6

Trend Period C4

Variability with Respect to 20-Year trend

Data and Trend C3:6

Trend Period C3

Histogram of Trend Period C3

Variability with Respect to 10-Year trend

Hilbert Spectrum Global Temperature Anomaly

Marginal Hilbert Spectrum

Morlet Wavelet Spectrum

Hilbert and Morlet Wavelet Spectra

Financial Data : NasDaqSC

October 11, 1984 – December 29, 2000

October 12, 2004

NasDaq Data

NasDaq IMF

NasDaq IMF Reconstruction : A

NasDaq IMF Reconstruction : B

NasDaq Various Overall Trends

NasDaq various Overall Detrends

Mean : L = 0 Exp = 73.1187 EMD = 0.3588

STD : L = 559.09 Exp = 426.66 EMD = 238.10

NasDaq Trend IMF (C8-C9)

NasDaq Local Period for Trend IMF (C8-C9)mean = 796.6

NasDaq Trend IMF (C7-C9)

NasDaq Local Period for Trend IMF (C7-C9)Mean = 425.7

NasDaq Trend IMF (C6-C9)

NasDaq Local Period for Trend IMF (C6-C9)Mean = 196.5

NasDaq Traditional Moving Mean Trends: Details

NasDaq Trends: Moving Mean and EMD : Details

NasDaq Period of EMD Trend (C4)Mean = 35.56

NasDaq Distribution of Period for EMD Trend (C4)

NasDaq Detrended Data (C4-C9)

NasDaq Detrended Data (C4-C9) : Details

NasDaq Histogram Detrended Data (C1-C3)

Various Definitions of Variability

• Variability defined by percentage Gain is the absolute value of the Gain.

• Variability defined by daily high-low is the percentage of absolute value of High-Low.

• Variability defined by Empirical Mode Decomposition is the percentage of the absolute value of the sum from selected IMFs.

• Financial data do not look like ARIMA.

NasDaq Variability defined by EMD : C1

NasDaq Variability defined by Gain

NasDaq Variability defined by Daily High-Low

NasDaq Period of Variability defined by EMD : C1Mean = 8.38

NasDaq Histogram Period of EMD Variability : C1

NASDAQ Price gradient vs. Gain Variability

NASDAQ Price gradient vs. High-Low Variability

NASDAQ Price gradient vs. EMD Variability

Relationship between Variability: Gain vs. EMD

Relationship between Variability: Gain vs. High-Low

Relationship between Variability: EMD vs. High-Low

Statistical Significance

Test for IMF

Methodology

• Based on observations from Monte Carlo numerical experiments on 1 million white noise data points.

• All IMF generated by 10 siftings.• Fourier spectra based on 200 realizations of

4,000 data points sections.• Probability density based on 50,000 data

points data sections.

IMF Period Statistics 

IMF1 2 3 4 5 6 7 8 9

number of peaks

347042 168176 83456 41632 20877 10471 5290 2658 1348

Mean period 2.881 5.946 11.98 24.02 47.90 95.50 189.0 376.2 741.8

period in year 0.240 0.496 0.998 2.000 3.992 7.958 15.75 31.35 61.75

 

Fourier Spectra of IMFs

0 1 2 3 4 5 6 7 8 90

0.5

1

1.5

spectr

um

(10**

-3)

Fourier Spectra of IMFs

1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

ln T

spectr

um

(10**

-3)

Shifted Fourier Spectra of IMFs

Empirical Observations : INormalized spectral area is constant

constTdS nT ln,ln

Empirical Observations : IIComputation of mean period

n

nT

nTnTnn T

TdS

T

TdS

T

dTSdSNE

lnln ,ln

,ln2,,

T

TdS

TdST

nT

nT

n ln

ln

,ln

,ln

Empirical Observations : IIIThe product of the mean energy and period is

constant

constTE nn

constTE nn lnln

Monte Carlo Result : IMF Energy vs. Period

Empirical Observation: Histograms IMFs By Central Limit theory IMF should be normally distributed.

-1 0 10

5000

-1 -0.5 0 0.5 10

5000

-0.5 0 0.50

5000

-0.5 0 0.50

5000

-0.4 -0.2 0 0.2 0.40

5000

-0.2 0 0.20

5000

-0.2 -0.1 0 0.1 0.20

5000

-0.1 0 0.10

5000

mode 2 mode 3

mode 4 mode 5

mode 6 mode 7

mode 8 mode 9

Histograms : IMF Energy Density

0.15 0.2 0.250

100

200

0.05 0.1 0.150

100

200

0.02 0.04 0.06 0.080

100

200

0.01 0.02 0.03 0.04 0.050

100

200

0 0.01 0.02 0.030

100

200

0 0.01 0.020

100

200

0 0.005 0.010

100

200

0 0.005 0.010

100

200

300

mode 2 mode 3

mode 4 mode 5

mode 6 mode 7

mode 8 mode 9

By Central Limit theory, IMF should be normally distributed; therefore, its energy should be Chi-squared distributed.

Chi-Squared Energy Density Distributions

212)( nn NEENnn eNENE

By Central Limit theory, IMF should be normally distributed; therefore, its energy should be Chi-squared distributed.

Formula of Confidence Limit for IMF Distributions

Ey ln yeE

Introduce new variable y:

Then,

!3!21

2exp

32 yyyyy

ENCy

Confidence Limit for IMF Distributions

Data and IMFs SOI

1930 1940 1950 1960 1970 1980 1990 2000

-0.4-0.2

00.2

R

-0.5

0

0.5

C9

-0.5

0

0.5

C8

-10

1

C7

-10

1

C6

-10

1

C5

-2

0

2

C4

-2

0

2

C3

-20

2

C2

-20

2

C1

-50

5

Raw

SO

I

Statistical Significance for SOI IMFs

1 mon 1 yr 10 yr 100 yr

IMF 4, 5, 6 and 7 are 99% statistical significance signals.

Summary

• Not all IMF have the same statistical significance.

• Based on the white noise study, we have established a method to determine the statistical significant components.

• References:• Wu, Zhaohua and N. E. Huang, 2003: A Study of the

Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proceedings of the Royal Society of London (in press)

• Flandrin, P., G. Rilling, and P. Gonçalvès, 2003: Empirical Mode Decomposition as a Filterbank, IEEE Signal Processing, (in press).

Statistical Significance Test

Only the statistical Significant IMF components are signal above noise; therefore, they might be predictable.

Statistical Significance Test : Gain

Statistical Significance Test : High-Low

Statistical Significance Test : EMD

Statistical Significance Test : All Variability Definitions

The Sum of all the Statistical Significance IMFs

Relationship among Trends: Gain vs. EMD

Relationship among Trends: Gain vs. High-Low

Relationship among Trends: EMD vs. High-Low

Summary

• A working definition for the trend is established; it is a function of the local time scale.

• Need adaptive method to analysis nonstationary and nonlinear data for trend and variability

• Various definitions for variability should be compared in details to determine their significance.