2. 1 Randomly Modulated Periodic Signals Melvin J. Hinich hinich@austin.utexas.edu Applied Research Laboratories University of Texas at Austin http://web.austin.utexas.edu/hinich/

3. 2 Rotating Cylinder Data

4. 3 Cat Brain EEG Seizure

5. 4 Daily Sunspots

6. 5 Hourly Alberta Electricity Demand

7. 6 Hourly Alberta Electricity Prices

8. 7 One Week of Yen/US $ Returns

9. 8 Canadian $/US $ Rates of Return

10. 9 Bassoon Note

11. 10 Flute Note

12. 11 Definition of a RMP A signal is called a randomly modulated periodicity with period T=N  if it is of the form

13. 12 Random Modulations are jointly dependent random processes that satisfy two conditions: Periodic block stationarity

14. 13 Finite Dependence Condition needed to ensure that averaging over frames yields asymptotically gaussian estimates are independently distributed if

15. 14 Fourier Series for Components

16. 15 Signal Plus Noise The modulation is part of the signal It is not measurement noise

17. 16 Artificial Data Examples

18. 17 Five Standard Deviations

19. 18 Three Standard Deviations

20. 19 Two Standard Deviations

21. 20 One Standard Deviation

22. 21 No Correlation in the Modulation

23. 22 Block Data into Frames The data block is divided into M frames of length T The t-th observation in the mth frame is T is chosen by the user to be the period of the periodic component

24. 23 Signal Coherence Function

25. 24 Frame Rate Synchronization The frame length T is chosen by the user to be the hypothetical period of the randomly modulated periodic signal. If T is not an integer multiple of the true period then coherence is lost.

26. 25 Signal Coherence Spectrum The signal-to-noise ratio is

27. 26 Estimating Signal Coherence

28. 27 Statistical Measure of Modulation SNR

29. 28 Chi Squared Statistic

30. 29 Coherent Part of the Mean Frame

31. 30 Spectrum of the Variance Fourier Series Expansion

32. 31 Sunspot Spectra

33. 32

34. 33

35. 34 Power & Signal Coherence Spectra - Demand

36. 35 Power & Signal Coherence Spectra - Prices

37. 36 Yen/$ Spectrum & Coherencies

38. 37 Exchange Rates Data 25 exchange rate series sampled half-hourly for the whole of 1996 with weekends removed. The weekend period is from 23:00 GMT on Friday when North American financial centres close until 23:00 GMT on Sunday when Australasian markets open. The incorporation of such prices would lead to spurious zero returns and would potentially render trading strategies which recommended a buy or sell at this time to be nonsensical. Removal of these weekend observations leaves 12,575 observations for subsequent analysis.

39. 38 Summary Statistics

40. 39 Coherence Results I

41. 40 Coherence Results II

42. 41 Coherent Part of Yen/$ Mean Frame

43. 42 Descriptive Statistics of Canadian$/US$ Daily Data 20.7 Sample size = 7569 20.7 Years Mean = 0.557E-04 Std Dev = 0.270E-02 Skew = 0.933E-01 Kurtosis = 3.62 Max value = 0.191E-01 Min value = -0.184E-01 40186.8 Frames = 40 Frame Length = 186.8 Frequencies in band = 93 Maximum coherence probability = 0.999999 Coherence threshold = 0.35

44. 43 Canadian$/US$ Daily Data Spectra

45. 44 Waterfall Spectrograms of RMP+Noise SNR = - 38 dB

46. 45 Sigcoh Probability Waterfall of RMP+Noise SNR = - 38 dB

47. 46 Waterfall Spectrograms of RMP+Noise SNR = - 44 dB

48. 47 Sigcoh Probability Waterfall of RMP+Noise SNR = - 44 dB

49. 48 Winglet Vibration Data

50. 49 Winglet Spectrum & Coherencies

51. 50 Prewhitened Winglet Spectra

52. 51 Tuned Prewhitened Winglet Spectra

53. 52 Noise Like Signal