1. Stationarity and Degree of Stationarity
Norden Huang
Research Center for Adaptive Data Analysis
National Central University

2. 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

3. Definition : Strictly Stationary

4. Definition : Wide Sense Stationary

5. 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.

6. Degree of Stationarity

7. Degree of Statistical Stationarity

8. An Example
Ocean Wind Wave Data

9. 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

10. Ocean Waves : data

11. Ocean Waves : IMF

12. Ocean Waves : Hilbert Spectrum

13. Ocean Waves : Hilbert Spectrum x10

14. Ocean Waves : Hilbert Spectrum x100

15. Ocean Waves : Degree of Stationarity

16. Earthquake Data Chi-Chi, Taiwan September 21, 1999
An Example
Earthquake Data
Chi-Chi, Taiwan
September 21, 1999
Huang, N. E. , et al : 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

17. 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.

18. 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:

19. Response Spectrum

20. Response Spectrum

21. 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.

22. Chi-Chi Earthquake : Data

23. Chi-Chi Earthquake : F & RS ; E

24. Chi-Chi Earthquake : F & RS ; N

25. Chi-Chi Earthquake : F & RS ; Z

26. Chi-Chi Earthquake : Hilbert E

27. Chi-Chi Earthquake : Hilbert N

28. Chi-Chi Earthquake : Hilbert Z

29. Chi-Chi Earthquake : MH & F : E

30. Chi-Chi Earthquake : MH & F : N

31. Chi-Chi Earthquake : MH & F : Z

32. Chi-Chi Earthquake : Hilbert : E200

33. Chi-Chi Earthquake : Hilbert E1000

34. Chi-Chi Earthquake : DS E

35. Chi-Chi Earthquake : DS N

36. Chi-Chi Earthquake : DS Z

37. Chi-Chi Earthquake : DS All

38. Chi-Chi Earthquake : DSS200 All

39. Chi-Chi Earthquake : DSS1000 All

40. 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.

41. Heart Rate Variability : HRV
Normal heart rate is chaotic

42. 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

43. Heart Rate Variability : 8 hours

44. Degree of Stationarity

45. Data White Noise

46. Data White Noise

47. Degree of stationary for nonlinear data
Inter- and intra-wave modulations

48. Duffing Chip Data

49. Duffing Chip : Hilbert ZC

50. Duffing Chip : Hilbert Quad

51. Duffing Chip : Hilbert Hilbert

52. Duffing Chip : Degree of Stationarity

53. Duffing Chip : Degree of Stationarity

54. Duffing Chip : Normalized Intra-wave Modulation

55. 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.