UF
How to run
the sign correlation analysis
in LDAS?
S.Klimenko
University of Florida
Abbreviations:
z
XC – cross-check, which allows to confirm validity of the data and the
data analysis.
S.Klimenko, 05/06/02
Cross-Correlation in Wavelet Domain
z
z
UF
x-correlation is a sum over wavelet layers
¾ τ – time lag
S = ∑ N k wk (τ )rk (τ )
¾ n – wavelet layer number
n ,τ
¾ Nk – number of samples in layer k
¾ rk(τ) – correlation coefficients as a function of lag time t
wk(τ) – optimal weight
∞
wk (τ ) = ∫ df ψ k ( f ) f
−∞
2
−3
ΩGW ( f ) ⋅ γ ( f , Ω L , Ω H )
exp(− j 2πfτ )
L
H
RL ( f ) PL ( f ) / σ k ⋅ RH ( f ) PH ( f ) / σ k
9Ψk – Fourier image of mother wavelet for layer k
9 γ – overlap reduction function
L
H
9 σ n ,σ n - noise rms in wavelet domain for detector L (H)
9R(f) – detector responses
S.Klimenko, 05/06/02
Sign Cross-Correlation
z
UF
S s = ∑ N kω k (τ ) ρ k (τ )
Sign x-correlation
n ,τ
¾ ρk(τ) – sign correlation coefficients
¾ ωk(τ) – optimal filter
z
optimal filter
ω k (τ ) =
ε k ⋅ wk (τ )
vk (τ )
¾ εk – sign correlation efficiency
¾ vk(τ) – contribution from correlated noise
¾ ε k ⋅ wk (τ ) should not vary with time
z
Variance of ρ
S.Klimenko, 05/06/02
var( ρ k (τ )) =
1
Nk
vk (τ )
What we can calculate with LDAS?
Process data segments T sec long (T=1024sec)
T
T
T
T
xh
- data
hh(Ω) - simulation
- data
xl
hl(Ω) - simulation
ρκ(τ)
------ - sign correlation coefficient for layer k and lag τ
vκ(τ)
------ - variance of the sign correlation coefficients
σ Lk
------ - rms of wavelet coefficients for IFO L
σ Hk
------ - rms of wavelet coefficients for IFO H
Wκ(τ)
------ - optimal coefficients (output of DSO)
rκ(τ,Ω) ------ - linear correlation coefficients for (xh+hh)*(xl+hl)
ρκ(τ,Ω) ------ - sign correlation coefficients (xh+hh)*(xl+ hl)
S.Klimenko, 05/06/02
UF
What we calculate in datacondAPI?
1.
Do wavelet transform – WaveletForward()
2.
Extract wavelet layer – GetLayer(x,k)
3.
Calculate rms of wavelet coefficients σIk (I=H,L)
4.
Convert to sign time series – u=signum(x)
5.
Calculate sign x-statistics – s=mul(uH,uL)
6.
Calculate x-correlation coefficient - ρ k(τ)=mean(s)
7.
Calculate ρ variance
z psd = mul( fft(s), conj(fft(s)) ) – need to be fixed
z auto-correlation function of s - a(τ) = fft-1(psd)
z calculate and save v(k) – estimate of correlated noise
S.Klimenko, 05/06/02
UF
What we get from v?
v = 1+
Ts / ∆t
∑ (n − m)a (m∆t ) = var ( ρ
n
c
UF
n)
m =1
z
v takes in to account the second-order statistics P(si,sj)
¾ s – sign cross-correlation statistics
z
v depends on
¾ an – correlated noise autocorrelation function (estimated from s)
¾ Ts - correlation time scale
z
v is a measure of correlated noise, or quality of data.
¾ XC: look at v value and its variation with time.
¾ XC: check contribution of correlated noise at different time scales Ts
z
v is used for calculation of the variance of ρ
S.Klimenko, 05/06/02
How to calculate εk?
z
εk is the efficiency of the sign x-correlation
z
εk does not depend on
UF
¾ SGW model and strength of SGW signal
¾ detector response
¾ overlap reduction function
z
Therefore εk can be calculated by adding a white Gaussian
noise to the output of the interferometers, the same both
for L and H. Then by measuring the sign and linear
correlation coefficients we can find εk as
εk =
S.Klimenko, 05/06/02
ρ k (τ , Ω)
rk (τ , Ω )
How to calculate optimal filter?
z
Optimal filter
UF
wk (τ )
σ kH σ kLWk (τ )
ω k (τ ) = ε k ⋅
= εk ⋅
vk (τ )
vk (τ )
¾ where Wk(τ) is the output of stochastic DSO
Wk (τ ) =
∞
∫ df ψ k ( f ) f
−∞
2
−3
Ω GW ( f ) ⋅ γ ( f , Ω L , Ω H )
exp (− j 2π fτ )
RL ( f ) PL ( f ) ⋅ RH ( f ) PH ( f )
9 just input Ψk(t) and Ψk(t+τ) instead of the interferometer data
z
XC: optimal filter should not vary with time if the detector
calibration is correct
S.Klimenko, 05/06/02
Upper Limit
z
Cross-correlation & variance:
Vs = ∑ N kω k2 (τ )vk (τ )
S s = ∑ N kω k (τ ) ρ k (τ )
n ,τ
n ,τ
z
x-correlation expectation value:
z
signal to noise ratio:
z
z
UF
confidence level:
upper limit:
S.Klimenko, 05/06/02
SNR = Ω
µ = Ω∑ N kω k2 (τ )vk (τ ) = ΩVs
n ,τ
2
N
ω
∑ k k (τ )vk (τ ) = Ω Vs
n ,τ
 Ss 
1
 → S~s (95CL)
CL = erf 
 V 
2
 s
~
~ S s (95CL)
Ω=
Vs