def loglike(x):
efac = np.ones(npsr)
equad = np.zeros(npsr)
theta = x[0]
phi = x[1]
f = 10**x[2]
mc = 10**x[3]
dist = 10**x[4]
psi = x[5]
inc = x[6]
phase = x[7]
pdist = x[8:(8+npsr)]
loglike = 0
for ct, p in enumerate(psr):
# make waveform with pulsar term
s = PALutils.createResiduals(p, theta, phi, mc, dist, f, phase, psi, inc, pdist=pdist[ct], \
psrTerm=True)
# project onto white noise basis
s = np.dot(projList[ct], s)
loglike += np.sum(p.res*s/(efac[ct]*Diag[ct] + equad[ct]**2))
loglike -= 0.5 * np.sum(s**2/(efac[ct]*Diag[ct] + equad[ct]**2))
if np.isnan(loglike):
print 'NaN log-likelihood. Not good...'
return -np.inf
else:
return loglike
def loglike(x):
efac = np.ones(npsr)
equad = np.zeros(npsr)
theta = x[0]
phi = x[1]
f = 10**x[2]
mc = 10**x[3]
dist = 10**x[4]
psi = x[5]
inc = x[6]
phase = x[7]
pdist = x[8:(8 + npsr)]
loglike = 0
for ct, p in enumerate(psr):
# make waveform with pulsar term
s = PALutils.createResiduals(p, theta, phi, mc, dist, f, phase, psi, inc, pdist=pdist[ct], \
psrTerm=True)
# project onto white noise basis
s = np.dot(projList[ct], s)
loglike += np.sum(p.res * s / (efac[ct] * Diag[ct] + equad[ct]**2))
loglike -= 0.5 * np.sum(s**2 / (efac[ct] * Diag[ct] + equad[ct]**2))
if np.isnan(loglike):
print 'NaN log-likelihood. Not good...'
return -np.inf
else:
return loglike
def upperLimitFunc(h, fstat_ref, freq, nreal):
"""
Compute the value of the fstat for a range of parameters, with fixed
amplitude over many realizations.
@param h: value of the strain amplitude to keep constant
@param fstat_ref: value of fstat for real data set
@param freq: GW frequency
@param nreal: number of realizations
"""
Tmaxyr = np.array([(p.toas.max() - p.toas.min())/3.16e7 for p in psr]).max()
count = 0
for ii in range(nreal):
# draw parameter values
gwtheta = np.arccos(np.random.uniform(-1, 1))
gwphi = np.random.uniform(0, 2*np.pi)
gwphase = np.random.uniform(0, 2*np.pi)
gwinc = np.arccos(np.random.uniform(0, 1))
gwpsi = np.random.uniform(-np.pi/4, np.pi/4)
# check to make sure source has not coalesced during observation time
coal = True
while coal:
gwmc = 10**np.random.uniform(7, 10)
tcoal = 2e6 * (gwmc/1e8)**(-5/3) * (freq/1e-8)**(-8/3)
if tcoal > Tmaxyr:
coal = False
# determine distance in order to keep strain fixed
gwdist = 4 * np.sqrt(2/5) * (gwmc*4.9e-6)**(5/3) * (np.pi*freq)**(2/3) / h
# convert back to Mpc
gwdist /= 1.0267e14
# create residuals
for ct,p in enumerate(psr):
inducedRes = PALutils.createResiduals(p, gwtheta, gwphi, gwmc, gwdist, \
freq, gwphase, gwpsi, gwinc, evolve=True)
# replace residuals in pulsar object
p.res = res[ct] + np.dot(R[ct], inducedRes)
# compute f-statistic
fpstat = PALLikelihoods.fpStat(psr, freq)
# check to see if larger than in real data
if fpstat > fstat_ref:
count += 1
# now get detection probability
detProb = count/nreal
print freq, h, detProb
return detProb - 0.95
def loglike(x):
tstart = time.time()
theta = np.arccos(x[0])
phi = x[1]
f = 10**x[2]
h = 10**x[3]
psi = x[4]
inc = np.arccos(x[5])
phase = x[6]
# get pulsar phase
pphase = np.zeros(npsr)
for ii in range(npsr):
pphase[ii] = x[(ndim-npsr) + ii]
# pick a distance and mass from the strain. Doesnt matter since non-evolving
mc = 5e8
dist = 4 * np.sqrt(2/5) * (mc*4.9e-6)**(5/3) * (np.pi*f)**(2/3) / h
dist /= 1.0267e14
loglike = 0
for ct, p in enumerate(psr):
# make waveform with no frequency evolution
s = PALutils.createResiduals(p, theta, phi, mc, dist, f, phase, psi, inc,\
pphase=pphase[ct], evolve=False)
diff = p.res - s
loglike += -0.5 * logdetTerm[ct]
loglike += -0.5 * np.dot(diff, np.dot(p.invCov, diff))
#print 'Evaluation time = {0} s'.format(time.time() - tstart)
if np.isnan(loglike):
print 'NaN log-likelihood. Not good...'
return -np.inf
else:
return loglike
def loglike(x):
tstart = time.time()
theta = np.arccos(x[0])
phi = x[1]
f = 10**x[2]
h = 10**x[3]
psi = x[4]
inc = np.arccos(x[5])
phase = x[6]
# get pulsar phase
pphase = np.zeros(npsr)
for ii in range(npsr):
pphase[ii] = x[(ndim - npsr) + ii]
# pick a distance and mass from the strain. Doesnt matter since non-evolving
mc = 5e8
dist = 4 * np.sqrt(
2 / 5) * (mc * 4.9e-6)**(5 / 3) * (np.pi * f)**(2 / 3) / h
dist /= 1.0267e14
loglike = 0
for ct, p in enumerate(psr):
# make waveform with no frequency evolution
s = PALutils.createResiduals(p, theta, phi, mc, dist, f, phase, psi, inc,\
pphase=pphase[ct], evolve=False)
diff = p.res - s
loglike += -0.5 * logdetTerm[ct]
loglike += -0.5 * np.dot(diff, np.dot(p.invCov, diff))
#print 'Evaluation time = {0} s'.format(time.time() - tstart)
if np.isnan(loglike):
print 'NaN log-likelihood. Not good...'
return -np.inf
else:
return loglike
def HTFSingleInd(
psr,
F,
proj,
SS,
rho,
efac,
equad,
gwtheta,
gwphi,
mc,
dist,
fgw,
phase0,
psi,
inc,
pphase=None,
pdist=None,
evolve=True,
psrTerm=True,
phase_approx=False,
):
"""
Lentati marginalized likelihood function only including efac and equad
and power law coefficients
@param psr: Pulsar class
@param F: Fourier design matrix constructed in PALutils
@param proj: Projection operator from white noise
@param SS: Diagonalized white noise matrix
@param rho: Power spectrum coefficients
@param efac: constant multipier on error bar covaraince matrix term
@param equad: Additional white noise added in quadrature to efac
@param gwtheta: Polar angle of GW source in celestial coords [radians]
@param gwphi: Azimuthal angle of GW source in celestial coords [radians]
@param mc: Chirp mass of SMBMB [solar masses]
@param dist: Luminosity distance to SMBMB [Mpc]
@param fgw: Frequency of GW (twice the orbital frequency) [Hz]
@param phase0: Initial Phase of GW source [radians]
@param psi: Polarization of GW source [radians]
@param inc: Inclination of GW source [radians]
@param pdist: Pulsar distance to use other than those in psr [kpc]
@param pphase: Use pulsar phase to determine distance [radian]
@param psrTerm: Option to include pulsar term [boolean]
@param evolve: Option to exclude evolution [boolean]
@return: LogLike: loglikelihood
"""
# make waveform with no frequency evolution
s = PALutils.createResiduals(
psr,
gwtheta,
gwphi,
mc,
dist,
fgw,
phase0,
psi,
inc,
pphase=pphase,
evolve=evolve,
pdist=pdist,
psrTerm=psrTerm,
phase_approx=phase_approx,
)
diff = np.dot(proj, (psr.res - s))
# compute total time span of data
Tspan = psr.toas.max() - psr.toas.min()
# compute d
d = np.dot(F.T, diff / (efac * SS + equad ** 2))
# compute Sigma
N = 1 / (efac * SS + equad ** 2)
right = (N * F.T).T
FNF = np.dot(F.T, right)
arr = np.zeros(2 * len(rho))
ct = 0
for ii in range(0, 2 * len(rho), 2):
arr[ii] = rho[ct]
arr[ii + 1] = rho[ct]
ct += 1
Sigma = FNF + np.diag(1 / arr)
# cholesky decomp for second term in exponential
cf = sl.cho_factor(Sigma)
expval2 = sl.cho_solve(cf, d)
logdet_Sigma = np.sum(2 * np.log(np.diag(cf[0])))
dtNdt = np.sum(diff ** 2 / (efac * SS + equad ** 2))
logdet_Phi = np.sum(np.log(arr))
logdet_N = np.sum(np.log(efac * SS + equad ** 2))
logLike = -0.5 * (logdet_N + logdet_Phi + logdet_Sigma) - 0.5 * (dtNdt - np.dot(d, expval2))
return logLike
def upperLimitFunc(h):
"""
Compute the value of the fstat for a range of parameters, with fixed
amplitude over many realizations.
@param h: value of the strain amplitude to keep constant
@param fstat_ref: value of fstat for real data set
@param freq: GW frequency
@param nreal: number of realizations
"""
Tmaxyr = np.array([(p.toas.max() - p.toas.min())/3.16e7 for p in psr]).max()
count = 0
for ii in range(nreal):
# draw parameter values
gwtheta = np.arccos(np.random.uniform(-1, 1))
gwphi = np.random.uniform(0, 2*np.pi)
gwphase = np.random.uniform(0, 2*np.pi)
gwinc = np.arccos(np.random.uniform(-1, 1))
gwpsi = np.random.uniform(-np.pi/4, np.pi/4)
# check to make sure source has not coalesced during observation time
gwmc = 10**np.random.uniform(7, 10)
tcoal = 2e6 * (gwmc/1e8)**(-5/3) * (freq/1e-8)**(-8/3)
if tcoal < Tmaxyr:
gwmc = 1e5
# determine distance in order to keep strain fixed
gwdist = 4 * np.sqrt(2/5) * (gwmc*4.9e-6)**(5/3) * (np.pi*freq)**(2/3) / h
# convert back to Mpc
gwdist /= 1.0267e14
# create residuals and refit for all pulsars
for ct,p in enumerate(psr):
inducedRes = PALutils.createResiduals(p, gwtheta, gwphi, gwmc, gwdist, \
freq, gwphase, gwpsi, gwinc)
# create simulated data set
noise = np.dot(L[ct], np.random.randn(L[ct].shape[0]))
pp[ct].stoas[:] -= pp[ct].residuals()/86400
pp[ct].stoas[:] += np.longdouble(np.dot(RQ[ct], noise)/86400)
pp[ct].stoas[:] += np.longdouble(np.dot(RQ[ct], inducedRes)/86400)
# refit
pp[ct].fit(iters=3)
# replace residuals in pulsar object
p.res = pp[ct].residuals()
print p.name, p.rms()*1e6
# compute f-statistic
fpstat = PALLikelihoods.fpStat(psr, freq)
# check to see if larger than in real data
if fpstat > fstat_ref:
count += 1
# now get detection probability
detProb = count/nreal
print h, detProb
return detProb - 0.95
def HTFSingleInd(psr, F, proj, SS, rho, efac, equad, gwtheta, gwphi, mc, \
dist, fgw, phase0, psi, inc, pphase=None, pdist=None, \
evolve=True, psrTerm=True, phase_approx=False):
"""
Lentati marginalized likelihood function only including efac and equad
and power law coefficients
@param psr: Pulsar class
@param F: Fourier design matrix constructed in PALutils
@param proj: Projection operator from white noise
@param SS: Diagonalized white noise matrix
@param rho: Power spectrum coefficients
@param efac: constant multipier on error bar covaraince matrix term
@param equad: Additional white noise added in quadrature to efac
@param gwtheta: Polar angle of GW source in celestial coords [radians]
@param gwphi: Azimuthal angle of GW source in celestial coords [radians]
@param mc: Chirp mass of SMBMB [solar masses]
@param dist: Luminosity distance to SMBMB [Mpc]
@param fgw: Frequency of GW (twice the orbital frequency) [Hz]
@param phase0: Initial Phase of GW source [radians]
@param psi: Polarization of GW source [radians]
@param inc: Inclination of GW source [radians]
@param pdist: Pulsar distance to use other than those in psr [kpc]
@param pphase: Use pulsar phase to determine distance [radian]
@param psrTerm: Option to include pulsar term [boolean]
@param evolve: Option to exclude evolution [boolean]
@return: LogLike: loglikelihood
"""
# make waveform with no frequency evolution
s = PALutils.createResiduals(psr, gwtheta, gwphi, mc, dist, fgw, phase0, psi, inc,\
pphase=pphase, evolve=evolve, pdist=pdist, \
psrTerm=psrTerm, phase_approx=phase_approx)
diff = np.dot(proj, (psr.res - s))
# compute total time span of data
Tspan = psr.toas.max() - psr.toas.min()
# compute d
d = np.dot(F.T, diff/(efac*SS + equad**2))
# compute Sigma
N = 1/(efac*SS + equad**2)
right = (N*F.T).T
FNF = np.dot(F.T, right)
arr = np.zeros(2*len(rho))
ct = 0
for ii in range(0, 2*len(rho), 2):
arr[ii] = rho[ct]
arr[ii+1] = rho[ct]
ct += 1
Sigma = FNF + np.diag(1/arr)
# cholesky decomp for second term in exponential
cf = sl.cho_factor(Sigma)
expval2 = sl.cho_solve(cf, d)
logdet_Sigma = np.sum(2*np.log(np.diag(cf[0])))
dtNdt = np.sum(diff**2/(efac*SS + equad**2))
logdet_Phi = np.sum(np.log(arr))
logdet_N = np.sum(np.log(efac*SS + equad**2))
logLike = -0.5 * (logdet_N + logdet_Phi + logdet_Sigma)\
- 0.5 * (dtNdt - np.dot(d, expval2))
return logLike
def upperLimitFunc(h, fstat_ref, freq, nreal, theta=None, phi=None, detect=False, \
dist=None):
"""
Compute the value of the fstat for a range of parameters, with fixed
amplitude over many realizations.
@param h: value of the strain amplitude to keep constant
@param fstat_ref: value of fstat for real data set
@param freq: GW frequency
@param nreal: number of realizations
"""
Tmaxyr = np.array([(p.toas.max() - p.toas.min()) / 3.16e7
for p in psr]).max()
count = 0
for ii in range(nreal):
# draw parameter values
gwtheta = np.arccos(np.random.uniform(-1, 1))
gwphi = np.random.uniform(0, 2 * np.pi)
gwphase = np.random.uniform(0, 2 * np.pi)
gwinc = np.arccos(np.random.uniform(-1, 1))
#gwpsi = np.random.uniform(-np.pi/4, np.pi/4)
gwpsi = np.random.uniform(0, np.pi)
# check to make sure source has not coalesced during observation time
coal = True
while coal:
gwmc = 10**np.random.uniform(7, 10)
tcoal = 2e6 * (gwmc / 1e8)**(-5 / 3) * (freq / 1e-8)**(-8 / 3)
if tcoal > Tmaxyr:
coal = False
# determine distance in order to keep strain fixed
gwdist = 4 * np.sqrt(
2 / 5) * (gwmc * 4.9e-6)**(5 / 3) * (np.pi * freq)**(2 / 3) / h
# convert back to Mpc
gwdist /= 1.0267e14
# check for fixed sky location
if theta is not None:
gwtheta = theta
if phi is not None:
gwphi = phi
if dist is not None:
gwdist = dist
gwmc = ((gwdist * 1.0267e14) / 4 / np.sqrt(2 / 5) /
(np.pi * freq)**(2 / 3) * h)**(3 / 5) / 4.9e-6
# create residuals
for ct, p in enumerate(psr):
inducedRes = PALutils.createResiduals(p, gwtheta, gwphi, gwmc, gwdist, \
freq, gwphase, gwpsi, gwinc, evolve=True)
# replace residuals in pulsar object
noise = np.dot(L[ct], np.random.randn(L[ct].shape[0]))
p.res = np.dot(R[ct], noise + inducedRes)
# compute f-statistic
fpstat = PALLikelihoods.fpStat(psr, freq)
# check to see if larger than in real data
if detect:
if PALutils.ptSum(npsr, fpstat) < 1e-4:
count += 1
else:
if fpstat > fstat_ref:
count += 1
# now get detection probability
detProb = count / nreal
if args.dist:
print '%e %e %f\n' % (freq, gwmc, detProb)
else:
print freq, h, detProb
return detProb - 0.95
def upperLimitFunc(h):
"""
Compute the value of the fstat for a range of parameters, with fixed
amplitude over many realizations.
@param h: value of the strain amplitude to keep constant
@param fstat_ref: value of fstat for real data set
@param freq: GW frequency
@param nreal: number of realizations
"""
Tmaxyr = np.array([(p.toas.max() - p.toas.min()) / 3.16e7
for p in psr]).max()
count = 0
for ii in range(nreal):
# draw parameter values
gwtheta = np.arccos(np.random.uniform(-1, 1))
gwphi = np.random.uniform(0, 2 * np.pi)
gwphase = np.random.uniform(0, 2 * np.pi)
gwinc = np.arccos(np.random.uniform(-1, 1))
gwpsi = np.random.uniform(-np.pi / 4, np.pi / 4)
# check to make sure source has not coalesced during observation time
gwmc = 10**np.random.uniform(7, 10)
tcoal = 2e6 * (gwmc / 1e8)**(-5 / 3) * (freq / 1e-8)**(-8 / 3)
if tcoal < Tmaxyr:
gwmc = 1e5
# determine distance in order to keep strain fixed
gwdist = 4 * np.sqrt(
2 / 5) * (gwmc * 4.9e-6)**(5 / 3) * (np.pi * freq)**(2 / 3) / h
# convert back to Mpc
gwdist /= 1.0267e14
# create residuals and refit for all pulsars
for ct, p in enumerate(psr):
inducedRes = PALutils.createResiduals(p, gwtheta, gwphi, gwmc, gwdist, \
freq, gwphase, gwpsi, gwinc)
# create simulated data set
noise = np.dot(L[ct], np.random.randn(L[ct].shape[0]))
pp[ct].stoas[:] -= pp[ct].residuals() / 86400
pp[ct].stoas[:] += np.longdouble(np.dot(RQ[ct], noise) / 86400)
pp[ct].stoas[:] += np.longdouble(
np.dot(RQ[ct], inducedRes) / 86400)
# refit
pp[ct].fit(iters=3)
# replace residuals in pulsar object
p.res = pp[ct].residuals()
print p.name, p.rms() * 1e6
# compute f-statistic
fpstat = PALLikelihoods.fpStat(psr, freq)
# check to see if larger than in real data
if fpstat > fstat_ref:
count += 1
# now get detection probability
detProb = count / nreal
print h, detProb
return detProb - 0.95
if args.nopterm:
print 'Not including pulsar term in single source waveform!'
pterm = False
else:
pterm = True
if args.snr is not None:
print 'Scaling distance to give SNR = {0}'.format(args.snr)
snr2 = 0
args.gwdist = 1
for ct, p in enumerate(pp):
inducedRes = (PALutils.createResiduals(psr[ct], np.pi/2-args.gwdec, args.gwra, \
args.gwchirpmass, args.gwdist, args.gwfreq, args.gwphase, \
args.gwpolarization, args.gwinc, psrTerm=pterm))
# compute snr
snr2 += PALutils.calculateMatchedFilterSNR(psr[ct], inducedRes, inducedRes)**2
# get total snr
snr = np.sqrt(snr2)
# scale distance appropiately
args.gwdist = snr/args.snr
print 'Scaled GW distance = {0} for SNR = {1}'.format(args.gwdist, args.snr)
# make residuals
for ct, p in enumerate(pp):
