Caltech 40 metre Interferometer
Detector Characterisation
SPECTRAL PROPERTIES OF THE DATA

Department of Physics
The Australian National University (ANU)
Sub-group of ACIGA

Susan ScottDavid McClelland
Bernard Whiting

Philip Charlton
Benjamin EvansJohn Sandeman

Data analysis sub-group came into existence six months agoPreliminaries• Installed and ran GRASP on several different SGI operating systems• Read all available 40 metre data into ANU structured Mass Data Store• Provision of interfaces between Matlab, FRAME and GRASP codes

Problem of Out of Lock Data

Our analysis requires the use of "good data"
i.e. data taken when the instrument is in lockUse lock channel IFO_Lock?
Yes, but it's too coarse for our purposes

PROBLEM : the interferometer is not actually in lockfor all sections of the lock channel showing "in lock data"

Manual fix : eliminate bad data by inspection
How could this process be carried out automatically?

Automatic Procedure
Step 1 :Only consider data which the lock channel indicates is"in lock"

Step 2 :Segment this "in lock" data into blocks of equal lengthe.g. 1,000 data points

Step 3 :Compute the mean signal µ i for each block and theassociated standard deviation σi for each block

Step 4 :Histogram the standard deviations σi for all the blocks

Step 5 :Using the histogram, select suitable (non-zero) minimumand maximum standard deviations

Step 6 :Blocks with standard deviation lying between theminimum and maximum are marked as good

Step 7 :The remaining blocks are marked as bad and discarded

Objective: to deliver code for this procedure to LIGO by June 2000

Frequency Histograms• Use good blocks of 1,000 data points• Pad the blocks out to 8,192 = 213 points by adding zeros• Consider the first 550 blocks of data
• Perform a Fast Fourier Transform (FFT) on each block• Compile the FFT data into a 550 × 4097 array• For each frequency:(a) histogram the real part of the FFT (50 bins)
calculate the mean and standard deviation of the sample(b) histogram the imaginary part of the FFT (50 bins)
calculate the mean and standard deviation of the sample• Execute the same procedure for the next 550 blocks of data
• Combine with the first 550 block to compile histograms:
µ→
σ2 →N
N +M

σ2 +NN +M
M
N +M

µ+M
N+M

σ 2new +µ newNM(N + M )2 (µ − µ new ) 2

Likelihood Ratio as a Measure of Gaussianity• Our samples of the real and imaginary components of the FFT fora given frequency are binned• We can therefore assume that they are distributed multinomially,
with probability pi of a point falling into bin i

The likelihood function L for a multinomial distribution is given by
L=N!

n1

n1 !n 2 !K n k !

n2nk

p1 p 2 K p k

k is the number of bins
ni is the number of points in bin ikN = ∑ n i is the total number of points
i =1

A likelihood ratio l is obtained by taking the ratio of L to themaximum value attainable by L as the ni’s vary
kN.B. subject to the constraint N = ∑ n i
This gives a value 0 ≤ l ≤ 1i =1Example

For 2M tosses of an unbiased coin, the likelihood of n heads isn 1  1
n!( 2 M − n )!  2   2 (2 M )!

2 M −nThe maximum likelihood is attained when n = M. Hence1l=n!(2 M − n)!  2 M!M!

=n− M

 1
 2

( 2 M −n )− M( M !)2n!(2 M − n)!

The likelihood ratio is used to measure the Gaussianity of binned data
• Take the probabilities pi from a normal distribution with mean µand standard deviation σ :x +∆ x / 2
 1  x − µ21exp  −pi = dx2
σx − ∆x / 2 σ 2 π



∫

ii

xi is the centre of the i th bin∆x is the bin widthµ and σ are obtained from the frequency data
L attains its maximum at nˆ i = Np iThus l is given byl=nˆ 1 ! nˆ 2 !K nˆ k !
n1 !n 2 !K n k !

n1 −nˆ1p1

n2 −nˆ 2p2n k − nˆ kK pk

In practice it is simpler to calculate log l• Values of l close to 1 indicate a good fit to Gaussian• For large N, l is extremely sharply peaked• A more useful measure of Gaussianity is given by l1/N
the N th root of lWork in Progress• Speed tests on the SGI PowerChallenge• Further measures of normality
e.g. skew, kurtosis, χ2-statistic• Apply line removal techniques to 40 metre data

• Use the statistical analysis techniques outlined to measure theeffects of different line removal techniques(40 metre, Glasgow)