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

Weakly Stationary Process

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

Power Spectrum Power Spectrum F

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

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

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

Data with Time-Varying Frequency Behavior Data with Time-Varying Frequency Behavior

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

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?

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?

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?

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?

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?

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?

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

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

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

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

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

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

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

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

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

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

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

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

Discrete M-Stationary Process

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

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

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

EARMA M- Spectral Estimator

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

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

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

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:

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:

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

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:

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

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

Instantaneous Period: Instantaneous Period:

Instantaneous Period Instantaneous Period Terminology

Instantaneous Frequency Instantaneous Frequency Terminology

Instantaneous Period Instantaneous Period Instantaneous Frequency Instantaneous Frequency Discrete Case Discrete Case Terminology

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.

A Discrete Euler Model A Discrete Euler Model

Dual Process:

Plots for Original Data Original Data Sample ACF Spectral Estimators

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

Forecasts Forecasts

Instantaneous Spectrum

Instantaneous Autocorrelation:  X (l, t)

- depends on l and t

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

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

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

Large Brown Bat Echolocation Signal

First 100 and Last 60 Points First 100 and Last 60 Points

Sample Autocorrelation and Spectrum

Plots for Dual Data M-frequency

Forecasts of Points 40–80 Forecasts of Points 40–80

Forecasts of Last 24 Points Forecasts of Last 24 Points

Instantaneous Spectrum

First 100 and Last 60 Points First 100 and Last 60 Points

Animation

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

Gabor Transform Gabor Transform

Continuous Wavelet Transform

Filtering Example Filtering Example Original Data Butterworth Filtered Data

Filtering Example Filtering Example Original Data Butterworth Filtered Data

Final Filtering Results Final Filtering Results

Extension: G( )-Stationary Process

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

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 …

Instantaneous Autocorrelation Instantaneous Autocorrelation

Filtering Example Filtering Example

Instantaneous Autocorrelation Instantaneous Autocorrelation

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.

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.

Filtering Example Filtering Example

Simulated Euler(12) Data

Simulated Data - Dual

Simulated Data – Instantaneous Spectrum

Simulated Data – Gabor Transform

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

Discrete M-Spectrum