randomly modulated periodic signals
DESCRIPTION
Randomly Modulated Periodic Signals. Melvin J. Hinich. Applied Research Laboratories University of Texas at Austin. [email protected]. http://web.austin.utexas.edu/hinich/. Rotating Cylinder Data. Cat Brain EEG Seizure. Daily Sunspots. Hourly Alberta Electricity Demand. - PowerPoint PPT PresentationTRANSCRIPT
1
Randomly Modulated Periodic Randomly Modulated Periodic SignalsSignals
Melvin J. HinichMelvin J. Hinich
[email protected]@austin.utexas.edu
Applied Research Applied Research LaboratoriesLaboratoriesUniversity of Texas at AustinUniversity of Texas at Austin
http://web.austin.utexas.edu/hinich/
2
Rotating Cylinder DataRotating Cylinder DataFluid Nonlinearity
-2000
-1500
-1000
-500
0
500
1000
1500
Time
Am
pli
tud
e
3
Cat Brain EEG SeizureCat Brain EEG Seizure
-4000
-2000
0
2000
4000
6000
8000
Time
mV
4
Daily SunspotsDaily Sunspots
Daily Sunspot Measurements
0
50
100
150
200
250
300
1850
1852
1855
1857
1860
1862
1865
1867
1870
1872
1874
1877
1879
1882
1884
1887
1889
1892
1894
1897
1899
1901
1904
1906
1909
1911
1914
1916
1919
1921
1923
1926
1928
1931
1933
1936
Days
Su
nsp
ots
5
Hourly Alberta Electricity DemandHourly Alberta Electricity DemandElectricity Demand ( 5/4/1996 - 6/15/1996)
0
1000
2000
3000
4000
5000
6000
7000
1 169 337 505 673 841
Time
kW H
ou
rs
6
Hourly Alberta Electricity PricesHourly Alberta Electricity PricesA Section of the Alberta Electric Spot Prices Time Series
-200
0
200
400
600
800
1000
1200
1 1681 3361 5041 6721 8401 10081 11761 13441 15121 16801 18481 20161 21841 23521 25201 26881 28561 30241 31921
Time
Pri
ce
7
One Week of Yen/US $ ReturnsOne Week of Yen/US $ Returns
Second Week of Yen/$ Log Returns
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
1 20 39 58 77 96 115 134 153 172 191 210 229
Time
log
x(t
) -
log
x(t
-1)
8
Canadian $/US $ Rates of ReturnCanadian $/US $ Rates of Return
Canadian$/US$ 2/11/2002 - 2/12/2003
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
1/11/2002 2/15/2002 3/22/2002 4/26/2002 5/31/2002 7/5/2002 8/9/2002 9/13/2002 10/18/2002 11/22/2002 12/27/2002 1/31/2003
Days
Rat
es o
f R
etu
rn
9
Bassoon NoteBassoon Note
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time
Am
pli
tud
e
10
Flute NoteFlute Note
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Time
Am
pli
tud
e
11
Definition of a RMPDefinition of a RMP
A signal is called a randomly modulated periodicity with period T=N if it is of the form
1 110
1 2 2
( ) cos 2 ( )
( ) sin 2
Kk k n k n
nk k k n k n
s u t f tx t N
s u t f t
1 2( ) ( ) 0k n k nEu t Eu t
for each 1, , where 2T
k K K
L
k
kf
T
nt n
12
Random ModulationsRandom Modulations
are jointly dependent random processes that satisfy two conditions:
The vector of the modulationsK
Periodic block stationarity
1 1( ), , ( ) = ( ), , ( ) m mf t t f t T t T u u u u
1 for all 0 mt t T
1 2( ), : 1, ,n k n k nt u t u t k K u
13
Finite DependenceFinite Dependence
' '1 1( ), , ( ) & ( ), , ( ) m rt t t tu u u u
'1 for some & and allmt D t D
' '1 1 & m nt t t t
Condition needed to ensure that averaging over frames yields asymptotically gaussian estimates
are independently distributed if
14
Fourier Series for ComponentsFourier Series for Components
Thus ( ) ( ) ( ) en n nx t s t u t wher
10 1 2
1
( ) cos 2 sin 2K
n k k n k k nk
s t s N s f t s f t
11 2
1
( ) cos 2 sin 2K
n k k n k k nk
u t N u f t u f t
15
Signal Plus NoiseSignal Plus Noise
( ) ( )ns t is the mean of x t
int nu t has a periodic jo distribution
The modulation is part of the signalThe modulation is part of the signal
It is not measurement noiseIt is not measurement noise
16
Artificial Data ExamplesArtificial Data Examples
110
1 2
1 cos 2 ( )
1 sin 2
Kk n k n
nk k n k n
u t f tx t s N
u t f t
1 1 1k n k n nu t u t T e t
2 2 2k n k n nu t u t T e t
17
Five Standard DeviationsFive Standard Deviations 10 Harmonics Modulation = 5 = 0.9 Frame=100
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
18
Three Standard DeviationsThree Standard Deviations10 Harmonics Modulation = 3 = 0.9 Frame=100
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time
Am
plit
ud
e
19
Two Standard DeviationsTwo Standard Deviations10 Harmonics Modulation = 2 = 0.9 Frame =100
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time
Am
pliu
tde
20
One Standard DeviationOne Standard DeviationRandomly Modulated Pulses
10 Harmonics Modulation = 1 = 0.9 Frame =100
-1
-0.5
0
0.5
1
1.5
Time
Am
plit
ud
e
21
No Correlation in the ModulationNo Correlation in the Modulation
Four Randomly Modulated Pulses Frame = 100 =5
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time
Am
pli
tud
e
22
Block Data into FramesBlock Data into Frames
The data block is divided into M frames of length T
The t-th observation in the mth frame is
( 1) n 0, , 1x m T n N L
T is chosen by the user to be the period of the periodic component
23
Signal Coherence FunctionSignal Coherence Function1
0
( ) (( 1) )exp( 2 )N
m n k nn
X k x m T t i f t
( )m k mX k s U k 1
0
( ) ( )exp( 2 )N
m m n k nn
U k u t i f t
1 2
1( )
2k k ks s is
22 ( )u mk E U k
24
Frame Rate SynchronizationFrame 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 SpectrumSignal Coherence Spectrum
The signal-to-noise ratio is The signal-to-noise ratio is
2 2( ) ( )x k uk s k
2
21x
x
x
kk
k
2
2 2( )
( )k
x
k u
sk
s k
26
Estimating Signal CoherenceEstimating Signal Coherence
1
0
ˆ ( ) ( )exp( 2 )ˆN
n k nn
X k x t i f t
2
2 2
ˆ ( )( )ˆ
ˆ ( ) ( )ˆx
u
X kk
X k k
22 1
1
ˆ( ) ( )ˆM
u mm
k M X k X k
: 0, , 1 ˆ
nx t n N is the mean frame
averaged over the M frames
L
27
StatisticalStatistical MeasureMeasure of Modulation SNRof Modulation SNR
2
2
2
ˆ
ˆ
ˆ
x
x
X kMZ k
N k
Mk
N
22 2ˆ( ) ( ) ( )ˆ ˆx uk X k k
28
Chi Squared StatisticChi Squared Statistic
2ˆ x
MZ k k
N
2
k x
Mk
N
22
mod
&
buted.k
If the ulation is stationary the distribution of
each Z k is approximately
they are independently distri
29
Coherent Part of the Mean FrameCoherent Part of the Mean Frame
1
0
1 ˆ ( ) ( )exp( 2 )ˆK
coherent kk
x t c k X k i f tN
1 if
0 xc k k threshold
c k otherwise
30
Spectrum of the VarianceSpectrum of the Variance
01 1
2 cos 2 2 sin 2K K
n k k n k k nk k
s t a a f t b f t
Fourier Series ExpansionFourier Series Expansion
- s t a stationary random process
- k k kVar a Var b S f Spectrum
2
1
K
s kk
S f
31
Sunspot Spectra
Power Spectrum of Detrended Daily Sunspot Data Prewhitened by an AR(10) Fit
10
15
20
25
30
35
40
3966 152.5 77.76 52.18 39.27 31.48 26.26 22.53 19.73 17.55 15.8 14.37 13.18 12.17 11.3 10.55 9.89 9.31 8.794 8.332 7.916 7.54 7.198
Period
dB
32
Signal Coherence Spectrum of Detrended & Prewhitened Daily Sunspot Data
0.75
0.77
0.79
0.81
0.83
0.85
0.87
0.89
0.91
0.93
3966
.00
39.27
19.73
13.18 9.
897.
926.
605.
664.
954.
403.
963.
603.
303.
052.
832.
642.
482.
332.
202.
09
Period
Sig
nal
Co
her
ence
33
3966
.00
360.5
5
188.8
6
127.9
4
96.73
77.76
65.02
55.86
48.96
43.58
39.27
35.73
32.78
30.27
28.13
26.26
24.63
23.19
21.91
20.76
S1
Period in Days
Time Segments
Coherence Probabilitygram of Four Segments of the Sunspots Data
0.96-0.97 0.97-0.98 0.98-0.99 0.99-1
34
Power & Signal Coherence Spectra - DemandPower & Signal Coherence Spectra - DemandPower & Signal Coherence Spectra of the Residuals from an AR(12) Fit of the Alberta
Electricity Hourly Spot Demand
0
0.25
0.5
0.75
1
168.
0
42.0
24.0
16.8
12.9
10.5 8.8
7.6
6.7
6.0
5.4
4.9
4.5
4.2
3.9
3.7
3.4
3.2
3.1
2.9
2.8
2.6
2.5
2.4
2.3
2.2
2.1
2.0
Period in Hours
Signal Coherence
0
10
20
30
40
50
60
dBCoherence Log Spectrum
35
Power & Signal Coherence Spectra - PricesPower & Signal Coherence Spectra - PricesPower & Signal Coherence Spectra of the Residuals from an AR(12) Fit of the
Alberta Electricity Hourly Spot Prices
10
20
30
40
50
60
168.0 24.0 12.9 8.8 6.7 5.4 4.5 3.9 3.4 3.1 2.8 2.5 2.3 2.1
Period in Hours
dB
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Signal CoherenceSpectrum Coherence
36
Yen/$ Spectrum & CoherenciesYen/$ Spectrum & CoherenciesYen/$ Spectrum & Peaks of the Coherence Probabilities
5 dB in Spectrum = 1.78 * Amplitude
-3
-2
-1
0
1
2
3
240.00 21.82 11.43 7.74 5.85 4.71 3.93 3.38 2.96 2.64 2.38 2.16
Period
dB
0.9975
0.998
0.9985
0.999
Coherence Probability
37
Exchange Rates DataExchange 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 StatisticsSummary Statistics
DEM_ JPY
GBP_ DEM
GBP_ USD
USD_ ITL
USD_ JPY
Mean 3.4E-4 9.7E-4 5.6E-4 -2.1E-4 6.5E-4
Var 6.5E-3 4.6E-3 4.8E-4 9.0E-3 6.2E-3
Skew -0.049 -0.004 -0.167 -0.011 -0.019
Kurt 5.642 83.51 13.01 15.719 9.723
Min -0.707 -1.966 -1.137 -0.924 -0.770
Max 0.659 1.992 1.203 0.966 0.758
Lag 1 -0.198 -0.306 -0.205 -0.315 -0.150
39
Coherence Results ICoherence Results I
Period 30 min
Spectrum (dB)
Coherence p-value
DEM_JPY
60 hours -2.164 0.314 0.003 2 hours 2 minutes -0.165 0.307 0.004
GBP_DEM
6 hours -1.357 0.312 0.003 2 hours 8 minutes -0.775 0.329 0.002 1 hour 38 minutes -1.534 0.317 0.003
GBP_USD
12 hours -3.688 0.302 0.005 8 hours -2.882 0.295 0.007
USD_CHF
15 hours -1.320 0.336 0.001 8 hours -2.110 0.345 0.001
3 hours 32 minutes -0.803 0.299 0.006 2 hours 0.096 0.283 0.010
40
Coherence Results IICoherence Results IIPanel E: USD_DEM
15 hours 0.390 0.352 0.001
8 hours -0.365 0.331 0.002
Panel F: USD_ITL
12 hours -4.166 0.308 0.004
5 hours 43 min -3.135 0.329 0.002
1 hour 49 min 0.139 0.353 0.001
Panel G: USD_JPY
60 hours -1.669 0.341 0.001
15 hours -0.956 0.362 0.000
8 hours -1.399 0.349 0.001
2 hours 37 min 0.046 0.298 0.006
1 hour 5 min 0.907 0.297 0.006
41
Coherent Part of Yen/$ Mean FrameCoherent Part of Yen/$ Mean FrameYen/$ Coherent Part of the Mean Frame
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
1 20 39 58 77 96 115 134 153 172 191 210 229
Time
42
Descriptive Statistics of Canadian$/US$ Daily DataDescriptive Statistics of Canadian$/US$ Daily Data
• Sample size = 7569 20.720.7 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
• Frames = 4040 Frame Length = 186.8186.8
• Frequencies in band = 93
• Maximum coherence probability = 0.999999
• Coherence threshold = 0.35
43
Canadian$/US$ Daily Data SpectraCanadian$/US$ Daily Data Spectra
-2
0
2
186.80 31.13 16.98 11.68 8.90 7.18 6.03 5.19 4.56 4.06 3.66 3.34 3.06 2.83 2.63 2.46 2.31 2.17 2.05
Period in Days
dB
0.99
0.995
1
Coherence ProbabilitySpectrum - US$/Canadian$ Coherence Spectrum
44
Waterfall SpectrogramsWaterfall Spectrograms of RMP+Noise of RMP+Noise SNR = - 38 dBSNR = - 38 dB
3.0 3.5 4.0 4.4 4.9 5.4 5.9 6.4 6.8 7.3 7.8 8.3 8.8 9.2 9.7 10.2 10.7 11.2 11.6 12.1 12.6 13.1S1
S16
S31
S46
S61
Frequency Hz
Time Slices
RMP Waterfall Plot of Normalized Spectrograms - u = 20 & e = 10 SNR = - 38 dB
0.95 < Peak Values < 1.0
0.99-1
0.98-0.99
0.97-0.98
0.96-0.97
0.95-0.96
0.94-0.95
45
Sigcoh Probability Waterfall of RMP+Noise Sigcoh Probability Waterfall of RMP+Noise SNR = - 38 dBSNR = - 38 dB
1.0
5.0
9.0
13.0
17.0
21.0
25.0
29.0
S1
S16
S31
S46
S61
S76
Frequency Hz
Time Slices
Waterfall Plot of Signal Coherence Probabilities - u = 20 & e = 10 - 38 dB
0.95 < Peaks < 1.0
0.99-1
0.98-0.99
0.97-0.98
0.96-0.97
0.95-0.96
0.94-0.95
46
Waterfall SpectrogramsWaterfall Spectrograms of RMP+Noise of RMP+Noise SNR = - 44 dBSNR = - 44 dB
3.0 3.5 4.0 4.4 4.9 5.4 5.9 6.4 6.8 7.3 7.8 8.3 8.8 9.2 9.7 10.2 10.7 11.2 11.6 12.1 12.6 13.11
16
31
46
61
Frequency Hz
Figure 3
Time Slices
RMP Waterfall Plot of Normalized Spectrograms - u = 20 & e = 1 SNR = - 44 dB
0.95 < Peak Values < 1.0
0.99-1
0.98-0.99
0.97-0.98
0.96-0.97
0.95-0.96
0.94-0.95
47
Sigcoh Probability Waterfall of RMP+Noise Sigcoh Probability Waterfall of RMP+Noise SNR = - 44 dBSNR = - 44 dB
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
1
16
31
46
61
Frequency Hz
Figure 4
Time Slices
Waterfall Plot of Signal Coherence Probabilities - u = 20 & e = 20 SNR = - 44 dB
0.95 < Peaks < 1.0
0.99-1
0.98-0.99
0.97-0.98
0.96-0.97
0.95-0.96
0.94-0.95
48
Winglet Vibration DataWinglet Vibration Data
Block of the Winglet Data
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Am
pli
tud
e
49
Winglet Spectrum & CoherenciesWinglet Spectrum & Coherencies
Winglet Spectrum & Peaks of the Coherence Probabilities 20 dB in Spectrum = 10 * Amplitude
-14
-9
-4
1
6
11
0.0010.0220.043 0.0640.0850.1060.127 0.1480.1690.1900.2110.232 0.2530.2740.295
Frequency in kHz
dB
0.995
0.996
0.997
0.998
0.999
1
Coherence Probability Spectrum Coherence Probability
50
Prewhitened Winglet SpectraPrewhitened Winglet SpectraAR Prewhitened Winglet Data
-24
-22
-20
-18
-16
-14
-12
-10
0.070 0.082 0.094 0.106 0.118 0.130 0.142 0.154 0.166 0.178 0.190 0.202 0.214 0.226 0.238 0.250 0.262 0.274 0.286
Frequency in kHz
dB
0.995
0.996
0.997
0.998
0.999
1
Coherence ProbabilitySpectrum Coherence Probability
51
Tuned Prewhitened Winglet SpectraTuned Prewhitened Winglet SpectraAR Prewhitened Winglet Spectra Tuned to 1.018 Hz
-24
-22
-20
-18
-16
-14
-12
-10
0.070 0.088 0.105 0.122 0.140 0.157 0.174 0.191 0.209 0.226 0.243 0.261 0.278 0.295
Frequency in kHz
dB
0.995
0.996
0.997
0.998
0.999
1
Coherence Probability
Spectrum Coherence Probability
52
Noise Like SignalNoise Like Signal
50 Harmonics Modulation = 40 = 0.9 Frame = 200
-6
-4
-2
0
2
4
6
Time
Am
pli
tud
e