Stationarity and

Degree of Stationarity

Norden Huang

Research Center for Adaptive Data Analysis

National Central University

Need to define the Degree Stationarity

• Traditionally, stationarity is taken for granted; it is given; it is an article of faith.

• All the definitions of stationarity are too restrictive.

• All definitions of stationarity are qualitative.• Good definition need to be quantitative to

give a Degree of Stationarity

Definition : Strictly Stationary

2

1 2 n 1 2 n

For a random var iable x( t ), if

x( t ) , x( t ) m, and that

x( t ), x( t ), ... x( t ) and x( t ), x( t ), ... x( t )

have the same joi nt distribution for all .

Definition : Wide Sense Stationary

2

1 2 1 2

1 2 1 2

For any random var iable x( t ), if

x( t ) , x( t ) m, and that

x( t ), x( t ) and x( t ), x( t )

have the same joi nt distribution for all .

Therefore , x( t ) x( t ) C( t t ) .

Definition : Statistically Stationary

• If the stationarity definitions are satisfied with certain degree of averaging.

• All averaging involves a time scale. The definition of this time scale is problematic.

Degree of Stationarity

t

2T

0

For a time frequency distribution, H( ,t ),

1n( ) H( ,t ) dt ;

T

1 H( ,t )DS( ) 1 dt .

T n( )

Degree of Statistical Stationarity

t

2Tt

0

For a time frequency distribution, H( ,t ),

1n( ) H( ,t ) dt ;

T

H( ,t )1DS( , t ) 1 dt .

T n( )

An Example

Ocean Wind Wave Data

Ocean Waves

• Water waves are nonlinear.

• Crests of breaking waves need many harmonics to fit

• Waves are nonstationary

• Spectrum full of Harmonics; it is hard to separate free from bound wave energy

Ocean Waves : data

Ocean Waves : IMF

Ocean Waves : Hilbert Spectrum

Ocean Waves : Hilbert Spectrum x10

Ocean Waves : Hilbert Spectrum x100

Ocean Waves : Degree of Stationarity

An Example

Earthquake DataChi-Chi, Taiwan

September 21, 1999

Huang, N. E. , et al. 2001 : A new spectral representation of earthquake data: Hilbert Spectral analysis of station TCU129, Chi-Chi, Taiwan, 21 September 1999, Bulletin of the Seismological Society of America, Volume 91, pp 1310-1338.

Earthquake

• Earthquake is definitely transient; therefore, nonstationary.

• For near field locations, the earth motion is also highly nonlinear.

• Traditional treatment of earthquake data by response spectral analysis is not adequate.

Response Spectrum

The response spectrum of a earthquake signal is defined through the maximum displacement of a linear single degree of freedom system with predetermined damping driven by the given earthquake signal. The displacement is given by the Duhamel Integral:

n

t( t )

n dd 0

n

2 1 / 2d n

1( t , , ) a( )e sin ( t )d ,

where

a( ) is the earthquake acceleration signal ,

is the undamped natural system frequency,

is the damping factor , and

( 1 ) is the damped system feequency.

Response Spectrum

n n

t

n nn 0

ti t i

n 0

n nn

For 0 , we have

1( t ; ) a( ) sin ( t ) d

1Im e a( )e d

1F( , t ) sin( t ) .

Response Spectrum

n n max n e nn

e

2e n n max n

Therefore ,

1F( ) ( ) A ( ) ,

where A , the equivalent acceleration is defined as

A ( , ) ( , ) .

Response Spectrum

• As the Duhamel Integral gives a quantity with the dimension of velocity, the response spectrum is also known as the pseudo-velocity spectrum.

• The linear single degree of freedom system is a linear filter; therefore,

• There is a definitive relationship between the Fourier Spectrum and Response spectrum.

Chi-Chi Earthquake : Data

Chi-Chi Earthquake : F & RS ; E

Chi-Chi Earthquake : F & RS ; N

Chi-Chi Earthquake : F & RS ; Z

Chi-Chi Earthquake : Hilbert E

Chi-Chi Earthquake : Hilbert N

Chi-Chi Earthquake : Hilbert Z

Chi-Chi Earthquake : MH & F : E

Chi-Chi Earthquake : MH & F : N

Chi-Chi Earthquake : MH & F : Z

Chi-Chi Earthquake : Hilbert : E200

Chi-Chi Earthquake : Hilbert E1000

Chi-Chi Earthquake : DS E

Chi-Chi Earthquake : DS N

Chi-Chi Earthquake : DS Z

Chi-Chi Earthquake : DS All

Chi-Chi Earthquake : DSS200 All

Chi-Chi Earthquake : DSS1000 All

Chi-Chi Earthquake : DS

• Hilbert spectral analysis reveals a ‘damaging-causing’ low frequency band of energy not properly shown in the Fourier Analysis.

• The strongest component, EW, is also the most nonstationary one.

• The weakest component, Z, is also the most stationary one.

• The Hilbert and Fourier spectra agree well for the most stationary case.

Heart Rate Variability : HRV

Normal heart rate is chaotic

Quiz on physiologic dynamics

Heart Failure Heart Failure

Normal Atrial Fibrillation

• Loss of dynamical fluctuations is bad

• Not all dynamical fluctuations are good

Hea

rt R

ate

(bpm

)H

eart

Rat

e (b

pm)

Hea

rt R

ate

(bpm

)H

eart

Rat

e (b

pm)

Time (min) Time (min)

Heart Rate Variability : 8 hours

Degree of Stationarity

Data White Noise

Data White Noise

Degree of stationary for nonlinear data

Inter- and intra-wave modulations

Duffing Chip Data

0 200 400 600 800 1000-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Duffing Chip Data

Time: Sec

Am

pli

tud

e

Duffing Chip : Hilbert ZCHilbert Spectrum : Duffing Chip ZC

Time: Sec

Fre

qu

en

cy

: C

ycle

/Se

c

0 200 400 600 800 10000

0.01

0.02

0.03

0.04

0.05

Duffing Chip : Hilbert QuadHilbert Spectrum : Duffing Chip Quad

Time: Sec

Fre

qu

en

cy

: C

ycle

/Se

c

0 200 400 600 800 10000

0.01

0.02

0.03

0.04

0.05

Duffing Chip : Hilbert HilbertHilbert Spectrum : Duffing Chip Hilbert

Time: Sec

Fre

qu

en

cy

: C

ycle

/Se

c

0 200 400 600 800 10000

0.01

0.02

0.03

0.04

0.05

Duffing Chip : Degree of Stationarity

10-4

10-3

10-2

10-1

100

101

102

103

104

Frequency : Cycle/Sec

De

gre

e o

f S

tati

on

ari

ty

Degree of Stationary : Duffing Chip

QuadHilbertZC

Duffing Chip : Degree of Stationarity

0 200 400 600 800 1000 12000

0.01

0.02

0.03

0.04

0.05

0.06

Time: Sec

Fre

qu

en

cy

: C

yc

le/S

ec

Duffing Chip : Instantaneous Frequency

QuadHilbertZC

Duffing Chip : Normalized Intra-wave Modulation

0 200 400 600 800 1000 1200-1

-0.5

0

0.5

1

1.5

Time: Sec

No

rma

lize

d I

ntr

a-w

av

e M

od

ula

tio

n

Normalized Intra-wave Modulation

QuadHilbert

Conclusions

• The high frequency range of the spectrum is highly intermittent.

• Even the Statistical Degree of Stationarity cannot smooth the variations.

• Before invoke the stationarity assumption, we should check the Degree of Stationarity.