stationarity and degree of stationarity norden huang research center for adaptive data analysis...
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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.