Investigating Higher Order
Statistics and Gaussianity
Steve Penn, Syracuse University
LIGO-G010132-00-Z
Synopsis

Introduction to Higher Order Statistics
» 1D: Correlation, Coherence, Power Spectra
» 2D: Bicorrelation, Bicoherence, Bispectrum
» 3D…

Bispectrum diagnostic

Gaussianity Test

Linearity Test
LSC • March 2001
Syracuse University Experimental Relativity Group
2
What are Higher Order
Statistics?

1D Statistics:

» Correlation: Cxy t    x   yt    d  X f  Y  f   Sxy  f 
*
» Power Spectral Density: C2 x t 
» Coherence:
C xy  f  

X f  X  f   S2x  f 
*
Sxy  f 
S2 x  f  S2 y  f 
– Tells us power and phase coherence at a given frequency
LSC • March 2001
Syracuse University Experimental Relativity Group
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Second Order Statistics

2D Statistics:
» Cumulant:

Cxyz t, t    x  yt    z t    d  X  f1  Y  f2  Z  f1  f2   Sxyz  f1 , f2 
*

» Bispectral Density:
C3x t   X  f1  X  f2  X  f1  f2   S3x  f1 , f2 
*
» Bicoherence:
C xyz  f  
Sxyz  f1 , f2 
S2 x  f1  S2y  f2  S2z  f1 , f2 
– Tells us power and phase coherence at a coupled frequency
LSC • March 2001
Syracuse University Experimental Relativity Group
4
Zero-lag Cumulants
Mean
Cx 0
Variance
C2 x 0 
Skewness
C3x 0 
0 if Symmetric
Kurtosis
C4 x 0 
0 if Gaussian
Useful statictical values, but…
Skewness = 0 does not prove symmetry
Kurtosis = 0 does not prove Gaussianity
Variations in skew and kurtosis not well quantified.
LSC • March 2001
Syracuse University Experimental Relativity Group
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Why Higher Order Statistics?

For a Gaussian process:Cnx t   0, for n  2

For independent processes:
z t   x t   yt ,

Cnz t   Cnx t   Cny t  Cny t 
n2
Allows for separation of Gaussian process for n>2
» Visual check of frequency coupling and phase noise
» Statistical test for the probability of gaussianity and linearity
» Iterative process to reconstruct nongaussian signal from the higher
order cumulants
LSC • March 2001
Syracuse University Experimental Relativity Group
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Gaussianity Monitor

Diagnostic Plot:
» Time series, Power spectrum, Two perspectives of the histogram of
time series with gaussian fit

Bispectrum and Bicoherence Plot

Gaussianity test:
 C 3x  f1 , f2  dU   2D

Summary file:
2
» Frame, channel, mean, variance, skew, kurtosis, gaussianity
probability
LSC • March 2001
Syracuse University Experimental Relativity Group
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Bispectrum Unique Area
fN
f2=2fN-f1
0
LSC • March 2001
f1=f2
fN
Syracuse University Experimental Relativity Group
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LSC • March 2001
Syracuse University Experimental Relativity Group
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LSC • March 2001
Syracuse University Experimental Relativity Group
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LSC • March 2001
Syracuse University Experimental Relativity Group
11
Status and Conclusions

Using the HOSA package in Matlab was not practical.
We needed a monitor running in DMT.

Old Bispectral algorithm extremely slow
(100*realtime)

New algorithm using FFTW appears to be near
realtime

Now we can analyze some data!!

Upcoming additions:
» Better windowing
» User selectable time interval
Record
user
selectedRelativity
s limit Group
LSC •»March
2001 # events
Syracuseabove
University
Experimental
» Plot/Set reference distribution for each channel.
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LSC • March 2001
Syracuse University Experimental Relativity Group
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