Bico Mon Steve Penn Hobart and William Smith

Bico. Mon Steve Penn, Hobart and William Smith Colleges LSC Meeting — August 2002 LIGO Hanford Observatory LIGO-G 020390 -00 -Z LSC • August 2002 HWS LIGO Group

Synopsis l Introduction to Higher Order Statistics » 1 D: Correlation, Coherence, Power Spectra » 2 D: Bicorrelation, Bicoherence, Bispectrum l Monitor Update l Future work LSC • August 2002 HWS LIGO Group 2

What are Higher Order Statistics? l 1 D Statistics: » Correlation: » Power Spectral Density: » Coherence: – Tells us power and phase coherence at a given frequency LSC • August 2002 HWS LIGO Group 3

Second Order Statistics l 2 D Statistics: » Bicumulant: » Bispectral Density: » Bicoherence: – Tells us power and phase coherence at a coupled frequency LSC • August 2002 HWS LIGO Group 4

Zero-lag Cumulants Mean Variance Skewness Kurtosis 0 if Symmetric 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 • August 2002 HWS LIGO Group 5

Why Higher Order Statistics? l For a Gaussian process: l For independent processes: l 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 • August 2002 HWS LIGO Group 6

Auto-Bispectrum Unique Area f. N f 2=2 f. N-f 1 0 LSC • August 2002 f 1=f 2 f. N HWS LIGO Group 7

Monitor Features l Plots (cross-)bicoherence, (cross-)bispectrum, & PSD’s » Operates on 1– 3 channels » User-specified Bispectrum method (Direct / In. Direct) » Automatically decimates to the lowest channel rate. » User selected fmax and Df Limited to factor 2 n » User specifies the accuracy » Monitor Outputs the Time span (gives user the update time) » User-specified Bispectrum method (Direct / In. Direct) » Windowing: Optimized Rao-Gabr windowing » Outputs GIF files of the plots » Help facility LSC • August 2002 HWS LIGO Group 8

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LSC • August 2002 HWS LIGO Group 10

Improvements / Future Directions l GUI, frequency-selection method, freq. resolution l Vijay Chickarmane and I are working together to monitor the time variation of regions of high bicoherence. --> Background monitoring - we are devising a method to fit the bilinear features for tracking of coefficients and/or subtraction LSC • August 2002 HWS LIGO Group 11