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x-correlation in wavelet domain
for detection of stochastic gravitational waves
S.Klimenko, University of Florida
Outline
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Introduction
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Optimal cross-correlation
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Correlation tests
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Cross-correlation in wavelet domain
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Correlated noise
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Conclusion
S.Klimenko
Introduction
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Stochastic Gravitational Waves
¾ from early universe or/and large
number of unresolved sources
(GW energy density)/(closure density)
ΩGW < 10
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Detection of SGW
−5
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Ω
Ω
Ω
=1
=
=
10 -
7
0 -9
10 -
11
¾ x-correlation of detectors output sL(t) and sH(t)
T
T
0
0
S = ∫ dt ∫ dt ′sL (t ′) sH (t )Q (t − t ′, Ω L , Ω H )
¾ Q - optimal kernel, T – observation time
¾ ΩH (ΩL) is the orientation of H (L) interferometer
S.Klimenko
al
i
t
i
w in
w
ad
v
ed
c
an
Optimal Cross-Correlation
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x-correlation in Fourier domain
Sep 96)
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(B.Allen,J.Romano gr- qc/ 9604033 v3 30
∞
S = ∫ df ~
sH ( f )~
sL* ( f )Q ( f , Ω L , Ω H )
−∞
¾ Optimal kernel:
9 ΩGW − SGW strength
| f | −3 Ω GW ( f )γ ( f , Ω L , Ω H )
Q( f , Ω L , Ω H ) =
PL ( f ) PH ( f )
9 PL, PH - spectral densities of detector noise
9 γ – detector overlap function
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(E. Flanagan, Phys. Rev. D48, 2389 (1993))
Questions:
¾ What is distribution of S if noise is not Gaussian?
¾ What to do if noise is not stationary?
¾ What to do if S is affected by correlated (©) noise?
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Correlation Tests
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linear correlation test
r=
∑ (x
∑ (x
− x )( y i − y )
i
i
i
− x)
2
∑ (y
i
− y)
2
w- correlation coefficient
¾ parametric: no universal way to compute r distribution
¾ if data is not Gaussian, r is a poor statistics to decide
9 correlation is statistically significant
9 one observed correlation is stronger then another.
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rank correlation test
¾ non-parametric: exactly known r distribution
¾ effective but CPU inefficient for large data sets
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Sign Correlation Test
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ui = sign( xi − xˆ )
Sign transform:
¾
x̂
- median of x
si = sign( xi − xˆ ) ⋅ sign( yi − yˆ )
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Sign statistics:
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Correlation coefficient γ:
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Distribution of γ (n – number of samples):
very robust:
¾ error from
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x̂
and
γ = mean( si )
 nγ 2 
n

P ( n, γ ) ≈
⋅ exp −

2π
2


¾ Gaussian (large n):
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ŷ
~2/n2, much less then var(γ )=1/n for large n
Comparison of Correlation Tests
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Data: simulated uncorrelated noise (n) + Gaussian signal (g)
x = nx + g ,
y = ny + g
Test efficiency: εs = rs/rL
wcorrelation coefficient
¾ for Gaussian noise
9 rank test efficiency – 95%
9 sign test efficiency - 64%
(2.5 times more data)
¾ independent on SNR
wnoise distributions
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wsign test efficiency
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Wavelet Transforms
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time-frequency representation of data in wavelet domain
¾ Pmn : n – scale (frequency) index, m – time index
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due to of locality of wavelet basis, wavelet layers are
decimated time series (similar to windowed FT).
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X-correlation
S=∑
nm
T
T
0
0
∑p
q I
kl mn kl , mn
k ,l
I kl ,mn = ∫ dt ∫ dt ′ψ kl (t ′)ψ mn (t )Q (t − t ′, Ω L , Ω H )
¾ Ψnm – basis of wavelet functions
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Cross-Correlation in Wavelet Domain
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x-correlation is a sum over wavelet layers
S = ∑ wn (τ )rn (τ )
n ,τ
∞
wn (τ ) = N n ∫ df ψ n ( f )
−∞
z
2
ΩGW ( f )γ ( f , Ω L , Ω H ) σ nL σ nH
.
exp(− j 2πfτ )
3
PL ( f ) PH ( f )
f
¾ τ – time lag
¾ n – wavelet layer number
¾ Nn – number of samples in layer n
¾ rn(τ) – correlation coefficients as a function of leg time τ
¾ wn(τ) – optimal weight
9Ψn – Fourier image of mother wavelet for layer n
9 σ nL ,σ nH - noise rms in wavelet domain for detector L (H)
equivalent to S calculated in Time & Fourier domains
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Sign X-Correlation in Wavelet Domain
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replace rn(τ) with γn(τ) - sign correlation coefficients
To keep the weights optimal, take into account the sign
correlation efficiency εn
~ (τ )γ (τ ), w
~ (τ ) = w (τ )ε
S = ∑w
s
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n ,τ
n
n
n
n
γn(τ) are normally distributed with variance 1/Nn
¾ then the x-correlation variance is:
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n
var( S s ) = ∑
n ,τ
1 ~2
wn (τ )
Nn
What we gain/loose
¾ sign test is less efficient (65%) when data is Gaussian:
¾ if data is not Gaussian
9 the sign test can be more efficient
9 gain confidence in calculation of S distribution
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Robust Spectral Amplitude
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Optimal weight
∞
2
~
wn (τ ) = N n ∫ df ψ n ( f ) f
−∞
z
−3
ΩGW ( f ) ⋅ γ ( f , Ω L , Ω H )
exp(− j 2πfτ )
AL ( f ) ⋅ AH ( f )
“noise amplitude”
AI ( f ) =
PI ( f )
σ nI ε n
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A(f) is more robust then P(f) if noise is non-stationary
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test with simulated noise
¾ Gaussian noise (σg) + tail :
σ
n
/σ
1.0
1.45
2.31
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g
P
0.45
0.94
2.40
total rms σn
A
0.0266
0.0274
0.0273
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© Noise
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C = hL + nL , hH + nH = hL , hH + nL , nH
wSGW (infinite CTS)
w“good” (uncorrelated) noise
w“bad” © noise
w“ugly” © noise
~T
~T
~ T
~ T
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remove “bad” © noise (likely to be data processing artifacts)
How to deal with “ugly” © noise?
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Autocorrelation Function
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sign statistics s(t)={uxuy}
a(t) - autocorrelation function of s(t)
¾ a measure of correlated noise.
wX-correlation of
wL1:AS_Q & H2:AS_Q
win wavelet domain:
w32-64 Hz band
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wE7 data
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Noise Model in SCT
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uncorrelated noise
a(0) = 1, a(τ ≥ ∆t ) = 0
¾ autocorrelation function:
¾ null hypothesis: data sets are not correlated
¾ variance:
z
1
var 0 ( γ ) =
n
correlated noise with time scale <Ts
¾ autocorrelation function:
a(τ < Ts ) = an (τ ), a(τ > Ts ) = 0
¾ null hypothesis: data sets are not correlated at time scale >Ts
1
R = 1 + ∑ ( n − m)an ( m∆t )
varTs (γ ) = R,
n
m =1
SCT allows calculate var(γ), depending on the noise model.
¾ variance:
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Ts / ∆t
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R
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variance ratio
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varTs (γ )
R=
var0 (γ )
¾ R is a measure of © noise, or quality of data.
¾ R times more data needed to reach same CL as for uncorrelated noise.
¾ If R is too large, the noise should be removed, if possible
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residual correlated noise is handled by
¾ reduced correlation coefficient:
γ ′ = γR −1
9 normally distributed with variance 1/n
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x-correlation in wavelet domain:
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~ (τ )γ ′ (τ ),
Ss = ∑ w
n
n
n ,τ
Variance Ratio (Ts)
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L1xH2: 11 data segments 4096 sec each (total 12.5 h of E7 data)
16-32Hz
32-64Hz
Ts
Ts
128-256Hz
64-128Hz
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Ts
Ts
Data with Lines Removed
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QMLR method was used
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32-64Hz
σ – rms for
uncorrelated noise
γ
σ
Ts
± 3σ
32-64Hz
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dots - before
solid - after
Setting Upper Limit
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1.
correlation coefficients γ – measured
2.
variance of γ – calculated for given model of © noise
¾ © noise can be estimated from data if Ts<<T
3.
optimal coefficients w – calculated for given SGW model.
¾ sign correlation efficiency ε – estimated from simulation.
~ (τ )γ ′ (τ ),
Ss = ∑ w
n
n
4.
as a result of 1,2,3 calculate x-correlation
5.
find from simulation the dependence S(Ωsim) (~ aΩsim)
6.
Set upper limit by calculating confidence belts.
S.Klimenko
n ,τ
Summary
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A robust correlation test with treatment of © noise is
described. It allows:
¾ calculate x-correlation distribution if noise is not Gaussian
¾ work with non-stationary noise
¾ use a simple model of correlated noise
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suggested method offers a good tool to estimate
contribution from © noise.
¾ On E7 data it is shown how © noise affects the x-correlation.
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we suggest to use sign x-correlation as a complementary
method for setting SGW upper limit
¾ very simple and CPU efficient
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R
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