Stationarity and Degree of Stationarity

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Presentation transcript:

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

Definition : Wide Sense Stationary

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

Degree of Statistical Stationarity

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

Earthquake Data Chi-Chi, Taiwan September 21, 1999 An Example Earthquake Data Chi-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:

Response Spectrum

Response Spectrum

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 Heart Rate (bpm) Heart Rate (bpm) Normal Atrial Fibrillation Heart Rate (bpm) Heart Rate (bpm) Time (min) Time (min) Loss of dynamical fluctuations is bad Not all dynamical fluctuations are good

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

Duffing Chip : Hilbert ZC

Duffing Chip : Hilbert Quad

Duffing Chip : Hilbert Hilbert

Duffing Chip : Degree of Stationarity

Duffing Chip : Degree of Stationarity

Duffing Chip : Normalized Intra-wave Modulation

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.