1 Randomly Modulated Periodic Signals Melvin J. Hinich Applied Research Laboratories University of Texas at Austin

2 Rotating Cylinder Data

3 Cat Brain EEG Seizure

4 Daily Sunspots

5 Hourly Alberta Electricity Demand

6 Hourly Alberta Electricity Prices

7 One Week of Yen/US $ Returns

8 Canadian $/US $ Rates of Return

9 Bassoon Note

10 Flute Note

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

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

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

14 Fourier Series for Components

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

16 Artificial Data Examples

17 Five Standard Deviations

18 Three Standard Deviations

19 Two Standard Deviations

20 One Standard Deviation

21 No Correlation in the Modulation

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

23 Signal Coherence Function

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.

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

26 Estimating Signal Coherence

27 Statistical Measure of Modulation SNR

28 Chi Squared Statistic

29 Coherent Part of the Mean Frame

30 Spectrum of the Variance Fourier Series Expansion

31 Sunspot Spectra

32

33

34 Power & Signal Coherence Spectra - Demand

35 Power & Signal Coherence Spectra - Prices

36 Yen/$ Spectrum & Coherencies

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.

38 Summary Statistics

39 Coherence Results I

40 Coherence Results II

41 Coherent Part of Yen/$ Mean Frame

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

43 Canadian$/US$ Daily Data Spectra

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

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

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

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

48 Winglet Vibration Data

49 Winglet Spectrum & Coherencies

50 Prewhitened Winglet Spectra

51 Tuned Prewhitened Winglet Spectra

52 Noise Like Signal