1. The Analysis of Non-Stationary Time Series with Time Varying Frequencies using Time Deformation The Analysis of Non-Stationary Time Series with Time Varying Frequencies using Time Deformation Southern Methodist University Research Group headed by Henry Gray and Wayne Woodward

2. Weakly Stationary Process

3. Weakly Stationary Process Models for Stationary Processes Models for Stationary Processes - AR(p), ARMA(p,q), etc - AR(p), ARMA(p,q), etc

4. Power Spectrum Power Spectrum F

5. Power Spectrum Power Spectrum Most important tool for analyzing frequency content of stationary time series F

6. Power Spectrum Power Spectrum Most important tool for analyzing frequency content of stationary time series - Fourier transform of autocovariance F

19. Note: Spectrum not really appropriate for series with time- varying frequencies

20. Data with Time-Varying Frequency Behavior Data with Time-Varying Frequency Behavior

21. Cyclical behavior is continuously changing across time, and the autocovariance will depend on both lag and time.

22. What Happens if We Ignore the Fact that Frequencies are Changing in Time? What Happens if We Ignore the Fact that Frequencies are Changing in Time?

23. traditional spectral analysis is not really a suitable tool to detect these cycles What Happens if We Ignore the Fact that Frequencies are Changing in Time? What Happens if We Ignore the Fact that Frequencies are Changing in Time?

24. traditional spectral analysis is not really a suitable tool to detect these cycles ARMA(p,q) models not appropriate What Happens if We Ignore the Fact that Frequencies are Changing in Time? What Happens if We Ignore the Fact that Frequencies are Changing in Time?

25. traditional spectral analysis is not really a suitable tool to detect these cycles ARMA(p,q) models not appropriate - forecasts from such models are poor What Happens if We Ignore the Fact that Frequencies are Changing in Time? What Happens if We Ignore the Fact that Frequencies are Changing in Time?

26. traditional spectral analysis is not really a suitable tool to detect these cycles ARMA(p,q) models not appropriate - forecasts from such models are poor whitening filters may not whiten What Happens if We Ignore the Fact that Frequencies are Changing in Time? What Happens if We Ignore the Fact that Frequencies are Changing in Time?

27. traditional spectral analysis is not really a suitable tool to detect these cycles ARMA(p,q) models not appropriate - forecasts from such models are poor whitening filters may not whiten standard filtering methods may be ineffective What Happens if We Ignore the Fact that Frequencies are Changing in Time? What Happens if We Ignore the Fact that Frequencies are Changing in Time?

28. Current Methods for Analyzing Data with Time-Varying Frequencies Current Methods for Analyzing Data with Time-Varying Frequencies windowed Fourier transforms (Gabor) smoothing spline ANOVA (Guo, et al., JASA, Sept. 2003) wavelet analysis autoregression with time-varying parameters

29. An Alternative Approach: TIME DEFORMATION (Gray and Zhang, 1988)

30. An Alternative Approach: TIME DEFORMATION (Gray and Zhang, 1988) -transform time ( instead of X ( t ))

31. An Alternative Approach: TIME DEFORMATION (Gray and Zhang, 1988) -transform time ( instead of X ( t )) -continuous case

32. M-Stationary Process (Gray and Zhang, 1988) M-Stationary Process (Gray and Zhang, 1988)

33. M-Stationary Process (Gray and Zhang, 1988) M-Stationary Process (Gray and Zhang, 1988) Note: is called the M-autocorrelation

34. Note M-stationary processes can be viewed as stationary processes by time deformation

35. Note M-stationary processes can be viewed as stationary processes by time deformation

36. Note M-stationary processes can be viewed as stationary processes by time deformation

37. Continuous Euler Process (EAR) (Gray and Zhang, 1988) The p th order continuous Euler process is given by

38. Continuous Euler Process (EAR) (Gray and Zhang, 1988) The p th order continuous Euler process is given by DUAL PROCESS:

39. M-Spectrum M-Spectrum Let X(t) be an M-stationary process. Then the M- spectrum is given by

40. Discrete M-Stationary Process

41. Discrete Euler Process – EAR(p) (Gray, Vijverberg, Woodward, 2003) Discrete Euler Process – EAR(p) (Gray, Vijverberg, Woodward, 2003)

43. Dual Process -- Y k Dual Process -- Y k Notes: this says we can induce stationarity by sampling properly i.e. we should sample at h k, then index on k the resulting process, Y k, is AR( p ) estimation of coefficients,  i, M-autocorrelation, and M-spectrum accomplished using Y k

44. Discrete EARMA(p,q) Processes -- Dual is ARMA(p,q)

45. EARMA M- Spectral Estimator

46. Practical Issues most data sets are observed at equally-spaced time points methods are used to obtain data at “Euler time points”, i.e. at t = h k -scientific sampling -interpolation -Kalman filter “realization offset” of the observed realization needs to be estimated

47. Realizations with Different Realization Offsets ( h j Realizations with Different Realization Offsets ( h j )

48. M-Periodic Function

49. A function g(t) is M-periodic with M-period  if  is the minimum value such that g(t) = g(t  ) for all t

50. M-Periodic Function A function g(t) is M-periodic with M-period  if  is the minimum value such that g(t) = g(t  ) for all tExample:

51. M-Periodic Function A function g(t) is M-periodic with M-period  if  is the minimum value such that g(t) = g(t  ) for all tExample:

52. M-Periodic Function A function g(t) is M-periodic with M-period  if  is the minimum value such that g(t) = g(t  ) for all t

53. M-Periodic Function A function g(t) is M-periodic with M-period  if  is the minimum value such that g(t) = g(t  ) for all t Example:

54. M-Periodic Function A function g(t) is M-periodic with M-period  if  is the minimum value such that g(t) = g(t  ) for all t

55. M-Periodic Function A function g(t) is M-periodic with M-period  if  is the minimum value such that g(t) = g(t  ) for all t

56. Instantaneous Period: Instantaneous Period:

57. Instantaneous Period Instantaneous Period Terminology

58. Instantaneous Frequency Instantaneous Frequency Terminology

59. Instantaneous Period Instantaneous Period Instantaneous Frequency Instantaneous Frequency Discrete Case Discrete Case Terminology

60. Note: Note: ip(t;f*) is in “calendar” or “regular” time and therefore f(t;f*) is in cycles per sampling unit based on the units of the original equally-spaced data set.

61. A Discrete Euler Model A Discrete Euler Model

63. Dual Process:

64. Plots for Original Data Original Data Sample ACF Spectral Estimators

66. Plots for Dual Data Plots for Dual Data M-AutocorrelationDual Data M-Spectrum

67. Forecasts Forecasts

68. Instantaneous Spectrum

71. Instantaneous Autocorrelation:  X (l, t)

72. - depends on l and t

73. Instantaneous Autocorrelation:  X (l, t) M-Autocorrelation

74. Instantaneous Autocorrelation:  X (l, t) M-Autocorrelation

75. Instantaneous Autocorrelation:  X (l, t) M-Autocorrelation

78. Large Brown Bat Echolocation Signal

79. First 100 and Last 60 Points First 100 and Last 60 Points

80. Sample Autocorrelation and Spectrum

81. Plots for Dual Data M-frequency

82. Forecasts of Points 40–80 Forecasts of Points 40–80

83. Forecasts of Last 24 Points Forecasts of Last 24 Points

84. Instantaneous Spectrum

85. First 100 and Last 60 Points First 100 and Last 60 Points

86. Animation

87. “Snapshots” “Snapshots” t = 1 t = 14 t = 115 t = 381

88. Gabor Transform Gabor Transform

89. Continuous Wavelet Transform

90. Filtering Example Filtering Example Original Data Butterworth Filtered Data

91. Filtering Example Filtering Example Original Data Butterworth Filtered Data

93. Final Filtering Results Final Filtering Results

94. Extension: G( )-Stationary Process

98. Effect of Varying Effect of Varying = - 1 = 0 = 1 = 1.5

99. Applications Applications Seismic dispersion curves -known to be a compacting signal Arterial blood flow data -Doppler signal -lesions create a second (higher frequency) Doppler signal Bat identification by echolocation signal Musical transients …

103. Instantaneous Autocorrelation Instantaneous Autocorrelation

105. Filtering Example Filtering Example

112. Instantaneous Autocorrelation Instantaneous Autocorrelation

116. G( )-Stationary Process G( )-Stationary Process Let X ( t ) be a stochastic process defined for - We call { X(t); t  (0,  )} a G( )-stationary process Dual Process : Y ( u) = X(t), where Y ( u ) is a stationary process.

117. G( )-Stationary Process G( )-Stationary Process Let X ( t ) be a stochastic process defined for - We call { X(t); t  (0,  )} a G( )-stationary process Dual Process : Y ( u) = X(t), where Y ( u ) is a stationary process.

118. Filtering Example Filtering Example

119. Simulated Euler(12) Data

120. Simulated Data - Dual

121. Simulated Data – Instantaneous Spectrum

122. Simulated Data – Gabor Transform

123. M-Spectrum (i.e. Spectrum of Dual)

124. Discrete M-Spectrum