cw-sensitivity (OctApps Reference Manual)

2.9 `cw-sensitivity`

Function File: [ rhob, tms ] = AnalyticSensitivityConstSNRChiSqr ( paNt, pd, Ns, nu )

Calculate sensitivty in terms of the root-mean-square SNR for a population of constant-SNR signals, and for a chi^2 detection statistic

Arguments

rhob: detectable r.m.s. SNR (per segment)
tms: terms of the second factor of the expression
paNt: false alarm probability (per template)
pd: false dismissal probability
Ns: number of segments
nu: degrees of freedom per segment

Function File: [ rhoh, iter ] = AnalyticSensitivitySNRChiSqr ( paNt, pd, Ns, nu )

Calculate sensitivty in terms of the root-mean-square SNR for a population of isotropically distributed and oriented signals, and for a chi^2 detection statistic

Arguments

rhoh: detectable r.m.s. SNR (per segment)
iter: number of iterations needed to solve for rhoh
paNt: false alarm probability (per template)
pd: false dismissal probability
Ns: number of segments
nu: degrees of freedom per segment

Function File: [ rho, tms ] = AnalyticSensitivitySNRExpr ( za, pd, Ns, nu )

Implements an expression used in analytic sensitivity estimation for a chi^2 detection statistic

Arguments

rho: detectable r.m.s. SNR (per segment)
tms: terms of the second factor of the expression
za: normalised false alarm threshold
pd: false dismissal probability
Ns: number of segments
nu: degrees of freedom per segment

Examples

assert(AnalyticSensitivitySNRExpr(0.01, 0.1, 100, 4), 0.6312, 1e-3)

Function File: F = AntennaPattern ( a, b, x, y, zeta )

Calculate the antenna pattern of an interferometer

Arguments

F: antenna pattern
a
b: detector null vectors in equatorial coordinates
x
y: polarisation null vectors in equatorial coordinates
zeta: angle between interferometer arms in radians

Examples

assert(AntennaPattern([1,0,0], [0,1,0], [0.5,0.5,0], [0.5,-0.5,0], pi/2), [0,0,0], 1e-3)

Function File: rng = CreateRandParam ( p, p, … )

Parses random parameters specs, which may be either

‘constant’: denoting a single value, or
[‘min’, ‘max’]: denoting a range of values

Arguments

rng: random parameter generator
p: random parameter spec

Examples

assert(isstruct(CreateRandParam([0, 5.5], [2.2, 7])))

Function File: pDET = DetectionProbabilityStackSlide ( opt, val, … )

Estimate detection probability for given fixed sensitivity-depth signal Depth = sqrt(S)/h0

Options

Nseg: number of StackSlide segments
Tdata: total amount of data used, in seconds (Note: Tdata = Nsft * Tsft, where ’Nsft’ is the total number of SFTs of length ’Tsft’ used in the search, from all detectors)
misHist: mismatch histogram, produced using Hist()
pFA: false-alarm probability (-ies) *per template* (can be a vector)
avg2Fth: ALTERNATIVE to pFA: specify average-2F threshold directly (can be a vector)
detectors: CSV list of detectors to use ("H1"=Hanford, "L1"=Livingston, "V1"=Virgo, ...)
alpha: source right ascension in radians (default: all-sky = [0, 2pi])
delta: source declination (default: all-sky = [-pi/2, pi/2])
Depth: fixed sensitivity-depth of signal population (can be a vector)
detweights: detector weights on S_h to use (default: uniform weights)

Examples

See the tutorial on SensitivityDepthStackSlide().

Function File: [ L, slambda, gamma, zeta ] = DetectorLocations ( detID )

Return parameters of various gravitational-wave interferometers

Arguments

L

detector’s longitude in radians

slambda

sine of the detector’s latitude

gamma

detector orientation in radians

zeta

angle between interferometer arms in radians

detID

identifier of a gravitational-wave interferometer:

‘H’: LIGO Hanford
‘L’: LIGO Livingston
‘V’: VIRGO
‘G’: GEO
‘K’: KAGRA

Examples

assert(DetectorLocations("H"))

assert(DetectorLocations("L"))

assert(DetectorLocations("V"))

Function File: [ a, b ] = DetectorNullVectors ( Phis, slambda, gamma )

Calculate the vectors along which an interferometric detector is insensitive to gravitational waves

Arguments

a
b: detector null vectors in equatorial coordinates
Phis: local sidereal time at the detector
slambda: sine of the detector’s latitude
gamma: detector orientation in radians

Examples

[a,b] = DetectorNullVectors(0.0, 0.0, pi/2);
assert(a, [0;1;0], 1e-3)
assert(b, [0;0;1], 1e-3)

Function File: [ v, v, … ] = NextRandParam ( rng, N )

Generates values for random parameters, given a generator

Arguments

rng: random parameter generator
N: number of values to generate
v: values of random parameter

Examples

gsl;
[a, b] = NextRandParam(CreateRandParam([0, 5.5], [2.2, 7]), 100);
assert(0 <= min(a) && max(a) <= 5.5);
assert(2.2 <= min(b) && max(b) <= 7);

Function File: [ xp, yp, xx, yx ] = PolarisationNullVectors ( alpha, sdelta, psi )

Calculate the vectors along which a pure plus/cross gravitational wave create no space-time peturbation

Arguments

xp
yp: plus polarisation null vectors in equatorial coordinates
xx
yx: cross polarisation null vectors in equatorial coordinates
alpha: source right ascension in radians
sdelta: sine of source declination
psi: source polarisation angle in radians

Examples

[xp, yp, xx, yx] = PolarisationNullVectors(0, 0, pi/2);
assert(xp, [0; -1/sqrt(2); 1/sqrt(2)], 1e-3);
assert(yp, [0; 1/sqrt(2); 1/sqrt(2)], 1e-3);
assert(xx, [0; 0; 1], 1e-3);
assert(yx, [0; 1; 0], 1e-3);

Function File: [ Depth, pd_Depth ] = SensitivityDepth ( opt, val, … )

Calculate sensitivity in terms of the sensitivity depth.

Arguments

Depth: SensitivityDepth
pd_Depth: calculated false dismissal probability

Options

pd

false dismissal probability

Ns

number of segments

Tdata

total amount of data used in seconds

Rsqr

histogram of SNR "geometric factor" R^2, computed using SqrSNRGeometricFactorHist(), or scalar giving mean value of R^2

stat

detection statistic, one of:

{ChiSqr, opt, val, …}

chi^2 statistic, e.g. the F-statistic, with options:

paNt: false alarm probability per template
sa: false alarm threshold
dof: degrees of freedom per segment (default: 4)
norm: use normal approximation to chi^2 (default: false)

{HoughFstat, opt, val, …}

Hough on the F-statistic, with options:

paNt: false alarm probability per template
nth: number count false alarm threshold
Fth: F-statistic threshold per segment
zero: use zeroth-order approximation (default: false)

prog

show progress updates

misHist

mismatch histograms (default: no mismatch)

Examples

Rsqr = SqrSNRGeometricFactorHist;

Function File: sensDepth = SensitivityDepthHoughF ( opt, val, … )

Estimate Hough-on-Fstat sensitivity depth, defined as

sensDepth = sqrt(Sdata)/h0,

where

Sdata: an estimate of the noise PSD over all the data used (which should be computed as the harmonic mean over all the SFTs from all detectors)
h0: the smallest detectable GW amplitude at the given false-alarm (pFA) and false-dismissal probability (pFD)

Options

Nseg: number of Hough segments
Tdata: total amount of data used, in seconds (Note: Tdata = Nsft * Tsft, where Nsft is the total number of SFTs of length Tsft used in the search, from all detectors)
misHist: mismatch histogram, produced using Hist()
pFD: false-dismissal probability = 1 - pDet
pFA: false-alarm probability (-ies) *per template* (can be a vector)
Fth: F-stat threshold (on F, not 2F!) in each segment for "pixel" selection
detectors: CSV list of detectors to use ("H1"=Hanford, "L1"=Livingston, "V1"=Virgo, ...)
detweights: detector weights on S_h to use (default: uniform weights)
alpha: source right ascension in radians (default: all-sky)
delta: source declination (default: all-sky)

Examples

Nseg = 20;
Tdata = 60*3600*Nseg;
misHist = createDeltaHist(0.1);
pFD = 0.1;
pFA = [1e-10; 1e-8; 1e-6];
Fth = 2.5;
dets = "H1,L1";
sigma = SensitivityDepthHoughF("Nseg", Nseg, "Tdata", Tdata, "misHist", misHist, "pFD", pFD, "pFA", pFA, "Fth", Fth, "detectors", dets);
assert(max(abs(sigma - [27.442; 35.784; 42.490])) < 0.05);

Function File: sensDepth = SensitivityDepthStackSlide ( opt, val, … )

Estimate StackSlide sensitivity depth, defined as

sensDepth = sqrt(Sdata)/h0,

where

Sdata: an estimate of the noise PSD over all the data used (which should be computed as the harmonic mean over all the SFTs from all detectors)
h0: the smallest detectable GW amplitude at the given false-alarm (pFA) and false-dismissal probability (pFD)

Options

Nseg

number of StackSlide segments (every row is one trial, every column is for one stage )

Tdata

total amount of data used, in seconds can be a row vector for different amounts of data in each stage or a column vector for different trial setups or matrix for both combined (Note: Tdata = Nsft * Tsft, where Nsft is the total number of SFTs of length Tsft used in the search, from all detectors)

## two different setups with 5 stages (5 columns, 2 rows)
Nseg = [90,90,44,44,22;100,100,50,50,25]
## different for every stage but the same for every trial
Tdata = [ NSFT*1800, NSFT*900, NSFT*3600, NSFT*1800, NSFT*1800]
## as we have 5 stages there must be five thrqesholds
avg2Fth = [6.109,6.109,7.38,8.82,15]
## we also need one mismatch histogram per stage
misHist = {mismatch1, mismatch2, mismatch3, mismatch4, mismatch5}
## a column with two false dimissal probabilitites, one for each trial
pFD = [0.1,0.05]'

misHist

cell array of mismatch histograms, one for every stage, produced using Hist()

pFD

false-dismissal probability = 1 - pDet = 1 - ’confidence’

pFA

false-alarm probability (-ies) *per template* (every row is one trial, every column is for one stage )

avg2Fth

ALTERNATIVE to pFA: average-2F threshold (every row is one trial, every column is for one stage )

detectors

CSV list of detectors to use ("H1"=Hanford, "L1"=Livingston, "V1"=Virgo, ...)

detweights

detector weights on S_h to use (default: uniform weights)

alpha

source right ascension in radians (default: all-sky)

delta

source declination (default: all-sky)

cosi

orientation angle (default: isotropic average)

psi

polarization angle (default: isotropic average)

Examples

See the tutorial on SensitivityDepthStackSlide().

Function File: [ ap, ax ] = SignalAmplitudes ( nonax, cosi )

Function File: apxnorm = SignalAmplitudes ( nonax )

Calculate the amplitudes of each polarisation from a signal emitted by a particular emission mechanism: "nonax": nonaxisymmetric distortion at 2f

Arguments

ap
ax: signal polarisation amplitudes
apxnorm: normalisation constant for R^2
cosi: cosine of the inclination angle

Examples

assert(SignalAmplitudes("nonax"), 4/25)

Function File: Rsqr = SqrSNRGeometricFactorHist ( opt, val, … )

Calculate a histogram of the squared SNR "geometric factor", R^2

Arguments

Rsqr: histogram of R^2

Options

T: observation time in sidereal days (default: inf)
detectors: detectors to use; either e.g. "H1,L1" or "HL" (default: L1)
detweights: detector weights on S_h to use (default: uniform weights)
alpha: source right ascension in radians (default: all-sky)
sdelta: sine of source declination (default: all-sky)
psi: source orientation in radians (default: all)
cosi: cosine of inclination angle (default: all)
emission: emission mechanism (default: nonax)
zmstime: sidereal time of the zero meridian at observation mid-point
hist_dx: histogram bin size
hist_N: number of histogram points to calculate at a time
hist_err: histogram error target
use_cache: if true [default], use a cached version of the histogram if available for the given input parameters

Examples

Rsqr0  = SqrSNRGeometricFactorHist("use_cache", false);	## stores results in cache
Rsqr0c = SqrSNRGeometricFactorHist("use_cache", true );
assert ( isequal ( Rsqr0, Rsqr0c ) );

Function File: Fsqr_t = TimeAvgSqrAntennaPattern ( a0, b0, x, y, zeta, OmegaT, nmax )

Calculate the time-averaged squared antenna pattern of an interferometer

Arguments

Fsqr_t: time-averaged squared antenna pattern
a0
b0: detector null vectors at observation mid-point, in equatorial coordinates
x
y: polarisation null vectors in equatorial coordinates
zeta: angle between interferometer arms in radians
OmegaT: product of angular sidereal frequency and observation time
nmax: maximum sinc term to add up (0 to 4; default is 4)

Examples

assert(TimeAvgSqrAntennaPattern([1;0;0], [0;1;0], [0;0.5;0.5], [0;0.5;-0.5], pi/2, inf), 0.03125)

2.9 cw-sensitivity

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2.9 `cw-sensitivity`