def myloglike(cube, ndim, nparams):
efac = cube[0]
equad = 10**cube[1]
gam = cube[2]
A = 10**cube[3]
fs = np.zeros(args.ss)
for ii in range(args.ss):
fs[ii] = 10**cube[4 + ii]
rho2 = np.zeros(args.ss)
for ii in range(args.ss):
rho2[ii] = cube[ii + 4 + args.ss]
# check to make sure frequencies are ordered
ordered = np.all([fs[ii] < fs[ii + 1] for ii in range(args.ss - 1)])
if ordered:
F1 = list(
PALutils.createfourierdesignmatrix(psr.toas, args.nmodes).T)
tmp, f = PALutils.createfourierdesignmatrix(psr.toas,
args.nmodes,
freq=True)
for ii in range(args.ss):
F1.append(np.cos(2 * np.pi * fs[ii] * psr.toas))
F1.append(np.sin(2 * np.pi * fs[ii] * psr.toas))
F = np.array(F1).T
F = np.dot(proj, F)
# compute rho from A and gam# compute total time span of data
Tspan = psr.toas.max() - psr.toas.min()
# get power spectrum coefficients
f1yr = 1 / 3.16e7
rho = list(
np.log10(A**2 / 12 / np.pi**2 * f1yr**(gam - 3) * f**(-gam) /
Tspan))
# compute total rho
for ii in range(args.ss):
rho.append(rho2[ii])
loglike = PALLikelihoods.lentatiMarginalizedLike(
psr, F, s, np.array(rho), efac**2, equad)
else:
loglike = -np.inf
#print efac, rho, loglike
return loglike
def optStat(psr, ORF, gam=4.33333):
"""
Computes the Optimal statistic as defined in Chamberlin, Creighton, Demorest et al (2013)
@param psr: List of pulsar object instances
@param ORF: Vector of pairwise overlap reduction values
@param gam: Power Spectral index of GBW (default = 13/3, ie SMBMBs)
@return: Opt: Optimal statistic value (A_gw^2)
@return: sigma: 1-sigma uncertanty on Optimal statistic
@return: snr: signal-to-noise ratio of cross correlations
"""
#TODO: maybe compute ORF in code instead of reading it in. Would be less
# of a risk but a bit slower...
k = 0
npsr = len(psr)
top = 0
bot = 0
for ll in xrange(0, npsr):
for kk in xrange(ll + 1, npsr):
# form matrix of toa residuals and compute SigmaIJ
tm = PALutils.createTimeLags(psr[ll].toas, psr[kk].toas)
# create cross covariance matrix without overall amplitude A^2
SIJ = ORF[k] / 2 * PALutils.createRedNoiseCovarianceMatrix(
tm, 1, gam)
# construct numerator and denominator of optimal statistic
bot += np.trace(
np.dot(psr[ll].invCov,
np.dot(SIJ, np.dot(psr[kk].invCov, SIJ.T))))
top += np.dot(psr[ll].res, np.dot(psr[ll].invCov, np.dot(SIJ, \
np.dot(psr[kk].invCov, psr[kk].res))))
k += 1
# compute optimal statistic
Opt = top / bot
# compute uncertainty
sigma = 1 / np.sqrt(bot)
# compute SNR
snr = top / np.sqrt(bot)
# return optimal statistic and snr
return Opt, sigma, snr
def myloglike(cube, ndim, nparams):
efac = cube[0]
equad = 10**cube[1]
gam = cube[2]
A = 10**cube[3]
fs = np.zeros(args.ss)
for ii in range(args.ss):
fs[ii] = 10**cube[4+ii]
rho2 = np.zeros(args.ss)
for ii in range(args.ss):
rho2[ii] = cube[ii+4+args.ss]
# check to make sure frequencies are ordered
ordered = np.all([fs[ii] < fs[ii+1] for ii in range(args.ss-1)])
if ordered:
F1 = list(PALutils.createfourierdesignmatrix(psr.toas, args.nmodes).T)
tmp, f = PALutils.createfourierdesignmatrix(psr.toas, args.nmodes, freq=True)
for ii in range(args.ss):
F1.append(np.cos(2*np.pi*fs[ii]*psr.toas))
F1.append(np.sin(2*np.pi*fs[ii]*psr.toas))
F = np.array(F1).T
F = np.dot(proj, F)
# compute rho from A and gam# compute total time span of data
Tspan = psr.toas.max() - psr.toas.min()
# get power spectrum coefficients
f1yr = 1/3.16e7
rho = list(np.log10(A**2/12/np.pi**2 * f1yr**(gam-3) * f**(-gam)/Tspan))
# compute total rho
for ii in range(args.ss):
rho.append(rho2[ii])
loglike = PALLikelihoods.lentatiMarginalizedLike(psr, F, s, np.array(rho), efac**2, equad)
else:
loglike = -np.inf
#print efac, rho, loglike
return loglike
def loglike(x):
start = time.time()
# parameter values
theta = x[0]
phi = x[1]
freq = 10 ** x[2]
mc = 10 ** x[3]
dist = 10 ** x[4]
psi = x[5]
inc = x[6]
phase = x[7]
pdist = x[8:]
# generate list of waveforms for all puslars
s = PALutils.createResidualsFast(
psr, theta, phi, mc, dist, freq, phase, psi, inc, pdist=pdist, evolve=False, phase_approx=True
)
loglike = 0
for ct, p in enumerate(psr):
diff = p.res - s[ct]
loglike += -0.5 * (logdetTerm[ct] + np.dot(diff, np.dot(p.invCov, diff)))
if np.isnan(loglike):
print "NaN log-likelihood. Not good..."
return -np.inf
else:
return loglike
def myloglike(cube, ndim, nparams):
efac = cube[0]
equad = 10**cube[1]
fs = np.zeros(args.ss)
for ii in range(args.ss):
fs[ii] = 10**cube[2+ii]
rho = np.zeros(args.nmodes+args.ss)
for ii in range(args.nmodes+args.ss):
rho[ii] = cube[ii+2+args.ss]
F1 = list(PALutils.createfourierdesignmatrix(psr.toas, args.nmodes).T)
for ii in range(args.ss):
F1.append(np.cos(2*np.pi*fs[ii]*psr.toas))
F1.append(np.sin(2*np.pi*fs[ii]*psr.toas))
F = np.array(F1).T
F = np.dot(proj, F)
loglike = PALLikelihoods.lentatiMarginalizedLike(psr, F, s, rho, efac, equad)
#print efac, rho, loglike
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 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):
start = time.time()
# parameter values
theta = x[0]
phi = x[1]
freq = 10**x[2]
mc = 10**x[3]
dist = 10**x[4]
psi = x[5]
inc = x[6]
phase = x[7]
pdist = x[8:]
# generate list of waveforms for all puslars
s = PALutils.createResidualsFast(psr, theta, phi, mc, dist, freq, phase, psi, inc,\
pdist=pdist, evolve=False, phase_approx=True)
loglike = 0
for ct, p in enumerate(psr):
diff = p.res - s[ct]
loglike += -0.5 * (logdetTerm[ct] + np.dot(diff, np.dot(p.invCov, diff)))
if np.isnan(loglike):
print 'NaN log-likelihood. Not good...'
return -np.inf
else:
return loglike
def optStat(psr, ORF, gam=4.33333):
"""
Computes the Optimal statistic as defined in Chamberlin, Creighton, Demorest et al (2013)
@param psr: List of pulsar object instances
@param ORF: Vector of pairwise overlap reduction values
@param gam: Power Spectral index of GBW (default = 13/3, ie SMBMBs)
@return: Opt: Optimal statistic value (A_gw^2)
@return: sigma: 1-sigma uncertanty on Optimal statistic
@return: snr: signal-to-noise ratio of cross correlations
"""
#TODO: maybe compute ORF in code instead of reading it in. Would be less
# of a risk but a bit slower...
k = 0
npsr = len(psr)
top = 0
bot = 0
for ll in xrange(0, npsr):
for kk in xrange(ll+1, npsr):
# form matrix of toa residuals and compute SigmaIJ
tm = PALutils.createTimeLags(psr[ll].toas, psr[kk].toas)
# create cross covariance matrix without overall amplitude A^2
SIJ = ORF[k]/2 * PALutils.createRedNoiseCovarianceMatrix(tm, 1, gam)
# construct numerator and denominator of optimal statistic
bot += np.trace(np.dot(psr[ll].invCov, np.dot(SIJ, np.dot(psr[kk].invCov, SIJ.T))))
top += np.dot(psr[ll].res, np.dot(psr[ll].invCov, np.dot(SIJ, \
np.dot(psr[kk].invCov, psr[kk].res))))
k+=1
# compute optimal statistic
Opt = top/bot
# compute uncertainty
sigma = 1/np.sqrt(bot)
# compute SNR
snr = top/np.sqrt(bot)
# return optimal statistic and snr
return Opt, sigma, snr
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 marginalizedPulsarPhaseLike(psr, theta, phi, phase, inc, psi, freq, h, maximize=False):
"""
Compute the log-likelihood marginalized over pulsar phases
@param psr: List of pulsar object instances
@param theta: GW polar angle [radian]
@param phi: GW azimuthal angle [radian]
@param phase: Initial GW phase [radian]
@param inc: GW inclination angle [radian]
@param psi: GW polarization angle [radian]
@param freq: GW initial frequency [Hz]
@param h: GW strain
@param maximize: Option to maximize over pulsar phases instead of marginalize
"""
# get number of pulsars
npsr = len(psr)
# get c and d
c = np.cos(phase)
d = np.sin(phase)
# construct xi = M**5/3/D and omega
xi = 0.25 * np.sqrt(5/2) * (np.pi*freq)**(-2/3) * h
omega = np.pi*freq
lnlike = 0
for ct, pp in enumerate(psr):
# compute relevant inner products
cip = np.dot(np.cos(2*omega*pp.toas), np.dot(pp.invCov, pp.res))
sip = np.dot(np.sin(2*omega*pp.toas), np.dot(pp.invCov, pp.res))
N = np.dot(np.cos(2*omega*pp.toas), np.dot(pp.invCov, np.cos(2*omega*pp.toas)))
# compute fplus and fcross
fplus, fcross, cosMu = PALutils.createAntennaPatternFuncs(pp, theta, phi)
# mind you p's and q's
p = (1+np.cos(inc)**2) * (fplus*np.cos(2*psi) + fcross*np.sin(2*psi))
q = 2*np.cos(inc) * (fplus*np.sin(2*psi) - fcross*np.cos(2*psi))
# construct X Y and Z
X = -xi/omega**(1/3) * (p*sip + q*cip - 0.5*xi/omega**(1/3)*N*c*(p**2+q**2))
Y = -xi/omega**(1/3) * (q*sip - p*cip - 0.5*xi/omega**(1/3)*N*d*(p**2+q**2))
Z = xi/omega**(1/3) * ((p*c+q*d)*sip - (p*d-q*c)*cip \
-0.5*xi/omega**(1/3)*N*(p**2+q**2))
# add to log-likelihood
#print X, Y
if maximize:
lnlike += Z + np.sqrt(X**2 + Y**2)
else:
lnlike += Z + np.log(ss.iv(0, np.sqrt(X**2 + Y**2)))
return lnlike
def crossPower(psr, gam=13 / 3):
"""
Compute the cross power as defined in Eq 9 and uncertainty of Eq 10 in
Demorest et al (2012).
@param psr: List of pulsar object instances
@param gam: Power spectral index of GWB
@return: vector of cross power for each pulsar pair
@return: vector of cross power uncertainties for each pulsar pair
"""
# initialization
npsr = len(psr)
# now compute cross power
rho = []
sig = []
xi = []
for ll in range(npsr):
for kk in range(ll + 1, npsr):
# matrix of time lags
tm = PALutils.createTimeLags(psr[ll].toas, psr[kk].toas)
# create cross covariance matrix without overall amplitude A^2
SIJ = PALutils.createRedNoiseCovarianceMatrix(tm, 1, gam)
# construct numerator and denominator of optimal statistic
bot = np.trace(
np.dot(psr[ll].invCov,
np.dot(SIJ, np.dot(psr[kk].invCov, SIJ.T))))
top = np.dot(psr[ll].res, np.dot(psr[ll].invCov, np.dot(SIJ, \
np.dot(psr[kk].invCov, psr[kk].res))))
# cross correlation and uncertainty
rho.append(top / bot)
sig.append(1 / np.sqrt(bot))
return np.array(rho), np.array(sig)
def crossPower(psr, gam=13/3):
"""
Compute the cross power as defined in Eq 9 and uncertainty of Eq 10 in
Demorest et al (2012).
@param psr: List of pulsar object instances
@param gam: Power spectral index of GWB
@return: vector of cross power for each pulsar pair
@return: vector of cross power uncertainties for each pulsar pair
"""
# initialization
npsr = len(psr)
# now compute cross power
rho = []
sig = []
xi = []
for ll in range(npsr):
for kk in range(ll+1, npsr):
# matrix of time lags
tm = PALutils.createTimeLags(psr[ll].toas, psr[kk].toas)
# create cross covariance matrix without overall amplitude A^2
SIJ = PALutils.createRedNoiseCovarianceMatrix(tm, 1, gam)
# construct numerator and denominator of optimal statistic
bot = np.trace(np.dot(psr[ll].invCov, np.dot(SIJ, np.dot(psr[kk].invCov, SIJ.T))))
top = np.dot(psr[ll].res, np.dot(psr[ll].invCov, np.dot(SIJ, \
np.dot(psr[kk].invCov, psr[kk].res))))
# cross correlation and uncertainty
rho.append(top/bot)
sig.append(1/np.sqrt(bot))
return np.array(rho), np.array(sig)
def bigPulsarPhaseJump(x, iter, beta):
# get old parameters
q = x.copy()
# pick pulsar index at random
ind = np.random.randint(0, npsr, npsr)
ind = np.unique(ind)
# get relevant parameters
freq = 10**x[2]
pdist = x[8 + ind]
pdistErr = np.array([psr[ii].distErr for ii in list(ind)])
phi = x[1]
theta = x[0]
# put pulsar distance in correct units
pdist *= 1.0267e11
pdistErr *= 1.0267e11
# get cosMu
cosMu = np.zeros(len(ind))
for ii in range(len(ind)):
tmp1, temp2, cosMu[ii] = PALutils.createAntennaPatternFuncs(
psr[ind[ii]], theta, phi)
# construct pulsar phase
phase_old = 2 * np.pi * freq * pdist * (1 - cosMu)
# gaussian jump
phase_jump = np.random.randn(
np.size(pdist)) * pdistErr * freq * (1 - cosMu)
# make jump multiple of 2 pi
phase_jump = np.array([int(phase_jump[ii]) \
for ii in range(np.size(pdist))])
# new phase
phase_new = phase_old + 2 * np.pi * phase_jump
# solve for new pulsar distances from phase_new
L_new = phase_new / (2 * np.pi * freq * (1 - cosMu))
# convert back to Kpc
L_new /= 1.0267e11
q[8 + ind] = L_new
return q
def bigPulsarPhaseJump(x, iter, beta):
# get old parameters
q = x.copy()
# pick pulsar index at random
ind = np.random.randint(0, npsr, npsr)
ind = np.unique(ind)
# get relevant parameters
freq = 10**x[2]
pdist = x[8+ind]
pdistErr = np.array([psr[ii].distErr for ii in list(ind)])
phi = x[1]
theta = x[0]
# put pulsar distance in correct units
pdist *= 1.0267e11
pdistErr *= 1.0267e11
# get cosMu
cosMu = np.zeros(len(ind))
for ii in range(len(ind)):
tmp1, temp2, cosMu[ii] = PALutils.createAntennaPatternFuncs(psr[ind[ii]], theta, phi)
# construct pulsar phase
phase_old = 2*np.pi*freq*pdist*(1-cosMu)
# gaussian jump
phase_jump = np.random.randn(np.size(pdist))*pdistErr*freq*(1-cosMu)
# make jump multiple of 2 pi
phase_jump = np.array([int(phase_jump[ii]) \
for ii in range(np.size(pdist))])
# new phase
phase_new = phase_old + 2*np.pi*phase_jump
# solve for new pulsar distances from phase_new
L_new = phase_new/(2*np.pi*freq*(1-cosMu))
# convert back to Kpc
L_new /= 1.0267e11
q[8+ind] = L_new
return q
def smallPulsarPhaseJump(x, iter, beta):
# get old parameters
q = x.copy()
# jump size
sigma = np.sqrt(beta) * 0.5
# pick pulsar index at random
ind = np.random.randint(0, npsr, npsr)
ind = np.unique(ind)
# get relevant parameters
freq = 10**x[2]
pdist = x[8 + ind]
phi = x[1]
theta = x[0]
# put pulsar distance in correct units
pdist *= 1.0267e11
# get cosMu
cosMu = np.zeros(len(ind))
for ii in range(len(ind)):
tmp1, temp2, cosMu[ii] = PALutils.createAntennaPatternFuncs(
psr[ind[ii]], theta, phi)
# construct pulsar phase
phase_old = 2 * np.pi * freq * pdist * (1 - cosMu)
# gaussian jump
phase_new = phase_old + np.random.randn(np.size(pdist)) * sigma
# solve for new pulsar distances from phase_new
L_new = phase_new / (2 * np.pi * freq * (1 - cosMu))
# convert back to Kpc
L_new /= 1.0267e11
q[8 + ind] = L_new
return q
def smallPulsarPhaseJump(x, iter, beta):
# get old parameters
q = x.copy()
# jump size
sigma = np.sqrt(beta) * 0.5
# pick pulsar index at random
ind = np.random.randint(0, npsr, npsr)
ind = np.unique(ind)
# get relevant parameters
freq = 10**x[2]
pdist = x[8+ind]
phi = x[1]
theta = x[0]
# put pulsar distance in correct units
pdist *= 1.0267e11
# get cosMu
cosMu = np.zeros(len(ind))
for ii in range(len(ind)):
tmp1, temp2, cosMu[ii] = PALutils.createAntennaPatternFuncs(psr[ind[ii]], theta, phi)
# construct pulsar phase
phase_old = 2*np.pi*freq*pdist*(1-cosMu)
# gaussian jump
phase_new = phase_old + np.random.randn(np.size(pdist))*sigma
# solve for new pulsar distances from phase_new
L_new = phase_new/(2*np.pi*freq*(1-cosMu))
# convert back to Kpc
L_new /= 1.0267e11
q[8+ind] = L_new
return q
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
tt=[]
for p in psr:
tt.append(np.min(p.toas))
# find reference time
tref = np.min(tt)
# now scale pulsar time
for p in psr:
p.toas -= tref
# create list of reference residuals
res = [p.res for p in psr]
# get list of R matrices
R = [PALutils.createRmatrix(p.dmatrix, p.err) for p in psr]
L = []
for ct, p in enumerate(psr):
Amp = p.Amp
gam = p.gam
efac = p.efac
equad = p.equad
try:
cequad = p.cequad
except AttributeError:
cequad = 0
avetoas, U = PALutils.exploderMatrix(p.toas)
Tspan = p.toas.max()-p.toas.min()
def addInverseCovFromNoiseFile(self, parfile, timfile, noisefile, DMOFF=None, DMXOFF=None, dailyAverage=False):
"""
Add noise covariance matrix after timing model subtraction.
"""
# Check whether the two files exist
if not os.path.isfile(parfile) or not os.path.isfile(timfile):
raise IOError, "Cannot find parfile (%s) or timfile (%s)!" % (parfile, timfile)
assert(self.filename != None), "ERROR: HDF5 file not set!"
# 'a' means: read/write if exists, create otherwise
self.h5file = h5.File(self.filename, 'a')
# Create the data subgroup if it does not exist
if "Data" in self.h5file:
datagroup = self.h5file["Data"]
else:
raise IOError, "Cannot add noise parameters if Data group does not exist!"
# Load pulsar data from the JPL Cython tempo2 library
t2pulsar = t2.tempopulsar(parfile, timfile)
# turn off DMMODEL fitting
if DMOFF is not None:
t2pulsar['DMMODEL'].fit = False
# turn off DMX fitting
if DMXOFF is not None:
DMXFlag = False
print 'Turning off DMX fitting and turning DM fitting on'
for par in t2pulsar.pars:
if 'DMX' in par:
t2pulsar[par].fit = False
t2pulsar['DM'].fit = True
DMXFlag = True
if DMXFlag== False:
print 'NO DMX for pulsar {0}'.format(t2pulsar.name)
# refit 5 times to make sure we are converged
t2pulsar.fit(iters=5)
# Create the pulsar subgroup if it does not exist
if "Pulsars" in datagroup:
pulsarsgroup = datagroup["Pulsars"]
else:
raise IOError, "Cannot add noise parameters if pulsar group does not exist!"
# Look up the name of the pulsar, and see if it exist
if t2pulsar.name in pulsarsgroup:
pass
else:
raise IOError, "%s must already exists in %s to add noise parameters!"\
% (t2pulsar.name, self.filename)
pulsarsgroup = pulsarsgroup[t2pulsar.name]
# first create G matrix from design matrix and toas
designmatrix = np.double(t2pulsar.designmatrix())
toas = np.double(t2pulsar.toas()*86400)
errs = np.double(t2pulsar.toaerrs*1e-6)
# if doing daily averaging
if dailyAverage:
# get average quantities
toas, qmatrix, errs, dmatrix, freqs, bands = PALutils.dailyAverage(t2pulsar)
# construct new daily averaged residuals and designmatrix
toas *= 86400
designmatrix = np.dot(qmatrix, dmatrix)
G = PALutils.createGmatrix(designmatrix)
# create matrix of time lags
tm = PALutils.createTimeLags(toas, toas, round=True)
# now read noise file to get model and parameters
file = open(noisefile,'r')
fH = None
tau = None
DMAmp = None
DMgam = None
for line in file.readlines():
# default parameters for different models other than pure PL
key = line.split()[0]
# get amplitude
if "Amp" == key:
Amp = float(line.split()[-1])
# get spectral index
elif "gam" == key:
gam = float(line.split()[-1])
# get efac
elif "efac" == key:
efac = float(line.split()[-1])
# get quad
elif "equad" == key:
equad = float(line.split()[-1])
# get high frequency cutoff if available
elif "fH" == key:
fH = float(line.split()[-1])
# get correlation time scale if available
elif "tau" == key:
tau = float(line.split()[-1])
# get DM Amplitude if available
elif "DMAmp" == key:
DMAmp = float(line.split()[-1])
# get DM Spectral Index if available
elif "DMgam" == key:
DMgam = float(line.split()[-1])
# cosstruct red and white noise covariance matrices
red = PALutils.createRedNoiseCovarianceMatrix(tm, Amp, gam, fH=fH)
white = PALutils.createWhiteNoiseCovarianceMatrix(errs, efac, equad, tau=tau)
# construct post timing model marginalization covariance matrix
cov = red + white
pcov = np.dot(G.T, np.dot(cov, G))
# finally construct "inverse"
invCov = np.dot(G, np.dot(np.linalg.inv(pcov), G.T))
# create dataset for inverse covariance matrix
pulsarsgroup.create_dataset('invCov', data = invCov)
# create dataset for G matrix
pulsarsgroup.create_dataset('Gmatrix', data = G)
# record noise parameter values
pulsarsgroup.create_dataset('Amp', data = Amp)
pulsarsgroup.create_dataset('gam', data = gam)
pulsarsgroup.create_dataset('efac', data = efac)
pulsarsgroup.create_dataset('equad', data = equad)
if fH is not None:
pulsarsgroup.create_dataset('fH', data = fH)
if tau is not None:
pulsarsgroup.create_dataset('tau', data = tau)
if DMAmp is not None:
pulsarsgroup.create_dataset('DMAmp', data = DMAmp)
if DMgam is not None:
pulsarsgroup.create_dataset('DMgam', data = DMgam)
# Close the hdf5 file
self.h5file.close()
def addInverseCovFromNoiseFile(self,
parfile,
timfile,
noisefile,
DMOFF=None,
DMXOFF=None,
dailyAverage=False):
"""
Add noise covariance matrix after timing model subtraction.
"""
# Check whether the two files exist
if not os.path.isfile(parfile) or not os.path.isfile(timfile):
raise IOError, "Cannot find parfile (%s) or timfile (%s)!" % (
parfile, timfile)
assert (self.filename != None), "ERROR: HDF5 file not set!"
# 'a' means: read/write if exists, create otherwise
self.h5file = h5.File(self.filename, 'a')
# Create the data subgroup if it does not exist
if "Data" in self.h5file:
datagroup = self.h5file["Data"]
else:
raise IOError, "Cannot add noise parameters if Data group does not exist!"
# Load pulsar data from the JPL Cython tempo2 library
t2pulsar = t2.tempopulsar(parfile, timfile)
# turn off DMMODEL fitting
if DMOFF is not None:
t2pulsar['DMMODEL'].fit = False
# turn off DMX fitting
if DMXOFF is not None:
DMXFlag = False
print 'Turning off DMX fitting and turning DM fitting on'
for par in t2pulsar.pars:
if 'DMX' in par:
t2pulsar[par].fit = False
t2pulsar['DM'].fit = True
DMXFlag = True
if DMXFlag == False:
print 'NO DMX for pulsar {0}'.format(t2pulsar.name)
# refit 5 times to make sure we are converged
t2pulsar.fit(iters=5)
# Create the pulsar subgroup if it does not exist
if "Pulsars" in datagroup:
pulsarsgroup = datagroup["Pulsars"]
else:
raise IOError, "Cannot add noise parameters if pulsar group does not exist!"
# Look up the name of the pulsar, and see if it exist
if t2pulsar.name in pulsarsgroup:
pass
else:
raise IOError, "%s must already exists in %s to add noise parameters!"\
% (t2pulsar.name, self.filename)
pulsarsgroup = pulsarsgroup[t2pulsar.name]
# first create G matrix from design matrix and toas
designmatrix = np.double(t2pulsar.designmatrix())
toas = np.double(t2pulsar.toas() * 86400)
errs = np.double(t2pulsar.toaerrs * 1e-6)
# if doing daily averaging
if dailyAverage:
# get average quantities
toas, qmatrix, errs, dmatrix, freqs, bands = PALutils.dailyAverage(
t2pulsar)
# construct new daily averaged residuals and designmatrix
toas *= 86400
designmatrix = np.dot(qmatrix, dmatrix)
G = PALutils.createGmatrix(designmatrix)
# create matrix of time lags
tm = PALutils.createTimeLags(toas, toas, round=True)
# now read noise file to get model and parameters
file = open(noisefile, 'r')
fH = None
tau = None
DMAmp = None
DMgam = None
for line in file.readlines():
# default parameters for different models other than pure PL
key = line.split()[0]
# get amplitude
if "Amp" == key:
Amp = float(line.split()[-1])
# get spectral index
elif "gam" == key:
gam = float(line.split()[-1])
# get efac
elif "efac" == key:
efac = float(line.split()[-1])
# get quad
elif "equad" == key:
equad = float(line.split()[-1])
# get high frequency cutoff if available
elif "fH" == key:
fH = float(line.split()[-1])
# get correlation time scale if available
elif "tau" == key:
tau = float(line.split()[-1])
# get DM Amplitude if available
elif "DMAmp" == key:
DMAmp = float(line.split()[-1])
# get DM Spectral Index if available
elif "DMgam" == key:
DMgam = float(line.split()[-1])
# cosstruct red and white noise covariance matrices
red = PALutils.createRedNoiseCovarianceMatrix(tm, Amp, gam, fH=fH)
white = PALutils.createWhiteNoiseCovarianceMatrix(errs,
efac,
equad,
tau=tau)
# construct post timing model marginalization covariance matrix
cov = red + white
pcov = np.dot(G.T, np.dot(cov, G))
# finally construct "inverse"
invCov = np.dot(G, np.dot(np.linalg.inv(pcov), G.T))
# create dataset for inverse covariance matrix
pulsarsgroup.create_dataset('invCov', data=invCov)
# create dataset for G matrix
pulsarsgroup.create_dataset('Gmatrix', data=G)
# record noise parameter values
pulsarsgroup.create_dataset('Amp', data=Amp)
pulsarsgroup.create_dataset('gam', data=gam)
pulsarsgroup.create_dataset('efac', data=efac)
pulsarsgroup.create_dataset('equad', data=equad)
if fH is not None:
pulsarsgroup.create_dataset('fH', data=fH)
if tau is not None:
pulsarsgroup.create_dataset('tau', data=tau)
if DMAmp is not None:
pulsarsgroup.create_dataset('DMAmp', data=DMAmp)
if DMgam is not None:
pulsarsgroup.create_dataset('DMgam', data=DMgam)
# Close the hdf5 file
self.h5file.close()
def __init__(self, pulsargroup, addNoise=False, addGmatrix=True):
# loop though keys in pulsargroup and fill in psr attributes that are needed for GW analysis
self.dist = None
self.distErr = None
self.fH = None
for key in pulsargroup:
# look for TOAs
if key == "TOAs":
self.toas = pulsargroup[key].value
# residuals
elif key == "residuals":
self.res = pulsargroup[key].value
# toa error bars
elif key == "toaErr":
self.err = pulsargroup[key].value
# frequencies in Hz
elif key == "freqs":
self.freqs = pulsargroup[key].value
# design matrix
elif key == "designmatrix":
self.dmatrix = pulsargroup[key].value
self.ntoa, self.nfit = self.dmatrix.shape
if addGmatrix:
self.G = PALutils.createGmatrix(self.dmatrix)
# design matrix
elif key == "pname":
self.name = pulsargroup[key].value
# pulsar distance in kpc
elif key == "dist":
self.dist = pulsargroup[key].value
# pulsar distance uncertainty in kpc
elif key == "distErr":
self.distErr = pulsargroup[key].value
# right ascension and declination
elif key == 'tmp_name':
par_names = list(pulsargroup[key].value)
for ct, name in enumerate(par_names):
# right ascension and phi
if name == "RAJ":
self.ra = pulsargroup["tmp_valpost"].value[ct]
self.phi = self.ra
# right ascension
if name == "DECJ":
self.dec = pulsargroup["tmp_valpost"].value[ct]
self.theta = np.pi / 2 - self.dec
# inverse covariance matrix
elif key == "invCov":
if addNoise:
self.invCov = pulsargroup[key].value
## noise parameters ##
elif key == "Amp":
self.Amp = pulsargroup[key].value
# red noise spectral
elif key == "gam":
self.gam = pulsargroup[key].value
# efac
elif key == "efac":
self.efac = pulsargroup[key].value
# equad
elif key == "equad":
self.equad = pulsargroup[key].value
elif key == "cequad":
self.cequad = pulsargroup[key].value
# fH
elif key == "fH":
self.fH = pulsargroup[key].value
# pulsar distance uncertainty in kpc
elif key == "distErr":
self.distErr = pulsargroup[key].value
if self.dist is None:
print 'WARNING: No distance info, using d = 1 kpc'
self.dist = 1.0
if self.distErr is None:
print 'WARNING: No distance error info, using sigma_d = 0.1 kpc'
self.distErr = 0.1
# make sure all pulsar have same reference time
tt=[]
for p in psr:
tt.append(np.min(p.toas))
# find reference time
tref = np.min(tt)
# now scale pulsar time
for p in psr:
p.toas -= tref
# get G matrices
for p in psr:
p.G = PALutils.createGmatrix(p.dmatrix)
# run Fp statistic to determine starting frequency
if args.null == False:
print 'Running initial Fpstat search'
fsearch = np.logspace(-9, -7, 1000)
fpstat = np.zeros(len(fsearch))
for ii in range(len(fsearch)):
fpstat[ii] = PALLikelihoods.fpStat(psr, fsearch[ii])
# determine maximum likelihood frequency
fmaxlike = fsearch[np.argmax(fpstat)]
print 'Maximum likelihood from f-stat search = {0}\n'.format(fmaxlike)
# get Tmax
Tmax = np.max([p.toas.max() - p.toas.min() for p in psr])
def firstOrderLikelihood(psr, ORF, Agw, gamgw, Ared, gred, efac, equad, \
interpolate=False):
"""
Compute the value of the first-order likelihood as defined in
Ellis, Siemens, van Haasteren (2013).
@param psr: List of pulsar object instances
@param ORF: Vector of pairwise overlap reduction values
@param Agw: Amplitude of GWB in standard strain amplitude units
@param gamgw: Power spectral index of GWB
@param Ared: Vector of amplitudes of intrinsic red noise in GWB strain units
@param gamgw: Vector of power spectral index of red noise
@param efac: Vector of efacs
@param equad: Vector of equads
@param interpolate: Boolean to perform interpolation only with compressed
data. (default = False)
@return: Log-likelihood value
"""
npsr = len(psr)
loglike = 0
tmp = []
# start loop to evaluate auto-terms
for ll in range(npsr):
r1 = np.dot(psr[ll].G.T, psr[ll].res)
# create time lags
tm = PALutils.createTimeLags(psr[ll].toas, psr[ll].toas)
#TODO: finish option to do interpolation when using compression
# calculate auto GW covariance matrix
SC = PALutils.createRedNoiseCovarianceMatrix(tm, Agw, gamgw)
# calculate auto red noise covariance matrix
SA = PALutils.createRedNoiseCovarianceMatrix(tm, Ared[ll], gred[ll])
# create white noise covariance matrix
#TODO: add ability to use multiple efacs for different backends
white = PALutils.createWhiteNoiseCovarianceMatrix(
psr[ll].err, efac[ll], equad[ll])
# total auto-covariance matrix
P = SC + SA + white
# sandwich with G matrices
Ppost = np.dot(psr[ll].G.T, np.dot(P, psr[ll].G))
# do cholesky solve
cf = sl.cho_factor(Ppost)
# solution vector P^_1 r
rr = sl.cho_solve(cf, r1)
# temporarily store P^-1 r
tmp.append(np.dot(psr[ll].G, rr))
# add to log-likelihood
loglike += -0.5 * (np.sum(np.log(2 * np.pi * np.diag(cf[0])**2)) +
np.dot(r1, rr))
# now compute cross terms
k = 0
for ll in range(npsr):
for kk in range(ll + 1, npsr):
# create time lags
tm = PALutils.createTimeLags(psr[ll].toas, psr[kk].toas)
# create cross covariance matrix
SIJ = PALutils.createRedNoiseCovarianceMatrix(tm, 1, gamgw)
# carry out matrix-vetor operations
tmp1 = np.dot(SIJ, tmp[kk])
# add to likelihood
loglike += ORF[k] / 2 * Agw**2 * np.dot(tmp[ll], tmp1)
# increment ORF counter
k += 1
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
# make sure all pulsar have same reference time
tt = []
for p in psr:
tt.append(np.min(p.toas))
# find reference time
tref = np.min(tt)
# now scale pulsar time
for p in psr:
p.toas -= tref
# get G matrices
for p in psr:
p.G = PALutils.createGmatrix(p.dmatrix)
# run Fp statistic to determine starting frequency
if args.freq is None:
print "Running initial Fpstat search"
fsearch = np.logspace(-9, -7, 1000)
fpstat = np.zeros(len(fsearch))
for ii in range(len(fsearch)):
fpstat[ii] = PALLikelihoods.fpStat(psr, fsearch[ii])
# determine maximum likelihood frequency
fmaxlike = fsearch[np.argmax(fpstat)]
print "Maximum likelihood from f-stat search = {0}\n".format(fmaxlike)
# get determinant of covariance matrix for use in likelihood
logdetTerm = []
# define the pulsargroup
pulsargroup = pfile['Data']['Pulsars']
# fill in pulsar class
psr = [PALpulsarInit.pulsar(pulsargroup[key],addNoise=True, addGmatrix=True) \
for key in pulsargroup]
# number of pulsars
npsr = len(psr)
# create list of reference residuals
res = [p.res for p in psr]
# get list of R matrices
R = [PALutils.createRmatrix(p.dmatrix, p.err) for p in psr]
# pre-compute noise covariance matrices
ct = 0
D = []
Dmatrix = []
print 'Computing diagonalized auto-covariance matrices'
for key in pulsargroup:
# get noise values from file TODO: change this to read directly from pulsar class
Amp = pulsargroup[key]['Amp'].value
gam = pulsargroup[key]['gam'].value
efac = pulsargroup[key]['efac'].value
equad = pulsargroup[key]['equad'].value
try:
def upperLimitFunc(A, optstat_ref, nreal):
"""
Compute the value of the Optimal Statistic for different signal realizations
@param A: value of GWB amplitude
@param optstat_ref: value of optimal statistic with no injection
@param nreal: number of realizations
"""
count = 0
for ii in range(nreal):
# create residuals
inducedRes = PALutils.createGWB(psr, A, 4.3333)
Pinvr = []
Pinv = []
for ct, p in enumerate(psr):
# replace residuals in pulsar object
p.res = res[ct] + np.dot(R[ct], inducedRes[ct])
# determine injected amplitude by minimizing Likelihood function
c = np.dot(Dmatrix[ct], p.res)
f = lambda x: -PALutils.twoComponentNoiseLike(x, D[ct], c)
fbounded = minimize_scalar(f, bounds=(0, 1e-14, 3.0e-13), method='Brent')
Amp = np.abs(fbounded.x)
#print Amp
#Amp = A
# construct P^-1 r
Pinvr.append(c/(Amp**2 * D[ct] + 1))
Pinv.append(1/(Amp**2 * D[ct] + 1))
# construct optimal statstic
k = 0
top = 0
bot = 0
for ll in range(npsr):
for kk in range(ll+1, npsr):
# compute numerator of optimal statisic
top += ORF[k]/2 * np.dot(Pinvr[ll], np.dot(SIJ[k], Pinvr[kk]))
# compute trace term
bot += (ORF[k]/2)**2 * np.trace(np.dot((Pinv[ll]*SIJ[k].T).T, (Pinv[kk]*SIJ[k]).T))
# iterate counter
k += 1
# get optimal statistic and SNR
optStat = top/bot
snr = top/np.sqrt(bot)
# check to see if larger than in real data
if optStat > optstat_ref:
count += 1
# now get detection probability
detProb = count/nreal
print A, detProb
injAmp.append(A)
injDetProb.append(detProb)
return detProb - 0.95
tt = []
for p in psr:
tt.append(np.min(p.toas))
# find reference time
tref = np.min(tt)
# now scale pulsar time
for p in psr:
p.toas -= tref
# create list of reference residuals
res = [p.res for p in psr]
# get list of R matrices
R = [PALutils.createRmatrix(p.dmatrix, p.err) for p in psr]
L = []
for ct, p in enumerate(psr):
Amp = p.Amp
gam = p.gam
efac = p.efac
equad = p.equad
try:
cequad = p.cequad
except AttributeError:
cequad = 0
avetoas, U = PALutils.exploderMatrix(p.toas)
Tspan = p.toas.max() - p.toas.min()
tt=[]
for p in psr:
tt.append(np.min(p.toas))
# find reference time
tref = np.min(tt)
# now scale pulsar time
for p in psr:
p.toas -= tref
# create list of reference residuals
res = [p.res for p in psr]
# get list of R matrices
R = [PALutils.createRmatrix(p.dmatrix, p.err) for p in psr]
#############################################################################################
#### DEFINE UPPER LIMIT FUNCTION #####
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
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 __init__(self,pulsargroup, addNoise=False, addGmatrix=True):
# loop though keys in pulsargroup and fill in psr attributes that are needed for GW analysis
self.dist = None
self.distErr = None
self.fH = None
for key in pulsargroup:
# look for TOAs
if key == "TOAs":
self.toas = pulsargroup[key].value
# residuals
elif key == "residuals":
self.res = pulsargroup[key].value
# toa error bars
elif key == "toaErr":
self.err = pulsargroup[key].value
# frequencies in Hz
elif key == "freqs":
self.freqs = pulsargroup[key].value
# design matrix
elif key == "designmatrix":
self.dmatrix = pulsargroup[key].value
self.ntoa, self.nfit = self.dmatrix.shape
if addGmatrix:
self.G = PALutils.createGmatrix(self.dmatrix)
# design matrix
elif key == "pname":
self.name = pulsargroup[key].value
# pulsar distance in kpc
elif key == "dist":
self.dist = pulsargroup[key].value
# pulsar distance uncertainty in kpc
elif key == "distErr":
self.distErr = pulsargroup[key].value
# right ascension and declination
elif key == 'tmp_name':
par_names = list(pulsargroup[key].value)
for ct,name in enumerate(par_names):
# right ascension and phi
if name == "RAJ":
self.ra = pulsargroup["tmp_valpost"].value[ct]
self.phi = self.ra
# right ascension
if name == "DECJ":
self.dec = pulsargroup["tmp_valpost"].value[ct]
self.theta = np.pi/2 - self.dec
# inverse covariance matrix
elif key == "invCov":
if addNoise:
self.invCov = pulsargroup[key].value
## noise parameters ##
elif key == "Amp":
self.Amp = pulsargroup[key].value
# red noise spectral
elif key == "gam":
self.gam = pulsargroup[key].value
# efac
elif key == "efac":
self.efac = pulsargroup[key].value
# equad
elif key == "equad":
self.equad = pulsargroup[key].value
# fH
elif key == "fH":
self.fH = pulsargroup[key].value
# pulsar distance uncertainty in kpc
elif key == "distErr":
self.distErr = pulsargroup[key].value
if self.dist is None:
print 'WARNING: No distance info, using d = 1 kpc'
self.dist = 1.0
if self.distErr is None:
print 'WARNING: No distance error info, using sigma_d = 0.1 kpc'
self.distErr = 0.1
def covarianceJumpProposal(x, iter, beta):
# get scale
scale = 1
# medium size jump every 1000 steps
if np.random.rand() < 1 / 1000:
scale = 5
# large size jump every 10000 steps
if np.random.rand() < 1 / 10000:
scale = 50
# get parmeters in new diagonalized basis
y = np.dot(U.T, x)
# make correlated componentwise adaptive jump
if iter < 100000:
ind = np.unique(np.random.randint(0, 8, 1))
else:
ind = np.unique(np.random.randint(0, 8, ndim))
neff = len(ind)
cd = 2.4 * np.sqrt(1 / beta) / np.sqrt(neff) * scale
#y[ind] = y[ind] + np.random.randn(neff) * cd * np.sqrt(S[ind])
q = x.copy()
q[ind] = q[ind] + np.random.randn(neff) * cd * np.sqrt(
np.diag(cov[ind, ind]))
#q = np.dot(U, y)
# need to make sure that we keep the pulsar phase constant, plus small offset
# get parameters before jump
omega0 = 2 * np.pi * 10**x[2]
L0 = x[8:] * 1.0267e11
phi0 = x[1]
theta0 = x[0]
omega1 = 2 * np.pi * 10**q[2]
L1 = q[8:] * 1.0267e11
phi1 = q[1]
theta1 = q[0]
# get cosMu
cosMu0 = np.zeros(npsr)
cosMu1 = np.zeros(npsr)
for ii in range(npsr):
tmp1, temp2, cosMu0[ii] = PALutils.createAntennaPatternFuncs(
psr[ii], theta0, phi0)
tmp1, temp2, cosMu1[ii] = PALutils.createAntennaPatternFuncs(
psr[ii], theta1, phi1)
# construct new pulsar distances to keep the pulsar phases constant
sigma = np.sqrt(1 / beta) * 0.0 * np.random.randn(npsr)
phase0 = omega0 * L0 * (1 - cosMu0)
phase1 = omega1 * L1 * (1 - cosMu1)
L_new = L1 - (np.mod(phase1, 2 * np.pi) - np.mod(phase0, 2 * np.pi) +
sigma) / (omega1 * (1 - cosMu1))
#phasenew = omega1*L_new*(1-cosMu1)
#print 'New phase = ', np.mod(phasenew ,2*np.pi)
#print 'Old phase = ', np.mod(phase0,2*np.pi)
#print 'New L = ', L_new
#print 'Old L = ', L0, '\n'
# convert back to Kpc
L_new /= 1.0267e11
q[8:] = L_new
#print x[0]
#print q[0], '\n'
return q
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
# make sure pulsar names are in correct order
# TODO: is this a very round about way to do this?
index = []
for ct, p in enumerate(pp):
if p.name == psr[ct].name:
index.append(ct)
else:
for ii in range(npsr):
if pp[ii].name == psr[ct].name:
index.append(ii)
pp = [pp[ii] for ii in index]
M = [PALutils.createQSDdesignmatrix(p.toas) for p in psr]
RQ = [PALutils.createRmatrix(M[ct], p.err) for ct, p in enumerate(psr)]
# construct noise matrix for new noise realizations
print 'Constructing noise cholesky decompositions'
L = []
for ct, p in enumerate(psr):
Amp = p.Amp
gam = p.gam
efac = p.efac
equad = p.equad
cequad = p.cequad
avetoas, U = PALutils.exploderMatrix(p.toas)
# make sure all pulsar have same reference time
tt=[]
for p in psr:
tt.append(np.min(p.toas))
# find reference time
tref = np.min(tt)
# now scale pulsar time
for p in psr:
p.toas -= tref
# compute pairwise overlap reduction function values
print 'Computing Overlap Reduction Function Values'
ORF = PALutils.computeORF(psr)
# since we have defined our ORF to be normalized to 1
hdcoeff = ORF/2
# compute optimal statistic
print 'Running Cross correlation Statistic on {0} Pulsars'.format(npsr)
crosspower, crosspowererr = PALLikelihoods.crossPower(psr, args.gam)
# angular separation
xi = []
for ll in range(npsr):
for kk in range(ll+1, npsr):
xi.append(PALutils.angularSeparation(psr[ll].theta, psr[ll].phi, \
psr[kk].theta, psr[kk].phi))
for p in psr:
p.toas -= tref
#import glob
#invmat = glob.glob('/Users/Justin/Work/nanograv/nanograv/data_products/joris/*invCov*')
#
## get list of R matrices
#R = [PALutils.createRmatrix(p.dmatrix, p.err) for p in psr]
#
#for ct,p in enumerate(psr):
# p.invCov = np.dot(R[ct].T, np.dot(p.invCov, R[ct]))
# compute pairwise overlap reduction function values
print 'Computing Overlap Reduction Function Values'
ORF = PALutils.computeORF(psr)
# compute optimal statistic
print 'Running Optimal Statistic on {0} Pulsars'.format(npsr)
Opt, sigma, snr = PALLikelihoods.optStat(psr, ORF, gam=args.gam)
print 'Results of Search\n'
print '------------------------------------\n'
print 'A_gw^2 = {0}'.format(Opt)
print 'std. dev. = {0}'.format(sigma)
print 'SNR = {0}'.format(snr)
if snr > 3.0:
print 'SNR of {0} is above threshold!'.format(snr)
def firstOrderLikelihood(psr, ORF, Agw, gamgw, Ared, gred, efac, equad, \
interpolate=False):
"""
Compute the value of the first-order likelihood as defined in
Ellis, Siemens, van Haasteren (2013).
@param psr: List of pulsar object instances
@param ORF: Vector of pairwise overlap reduction values
@param Agw: Amplitude of GWB in standard strain amplitude units
@param gamgw: Power spectral index of GWB
@param Ared: Vector of amplitudes of intrinsic red noise in GWB strain units
@param gamgw: Vector of power spectral index of red noise
@param efac: Vector of efacs
@param equad: Vector of equads
@param interpolate: Boolean to perform interpolation only with compressed
data. (default = False)
@return: Log-likelihood value
"""
npsr = len(psr)
loglike = 0
tmp = []
# start loop to evaluate auto-terms
for ll in range(npsr):
r1 = np.dot(psr[ll].G.T, psr[ll].res)
# create time lags
tm = PALutils.createTimeLags(psr[ll].toas, psr[ll].toas)
#TODO: finish option to do interpolation when using compression
# calculate auto GW covariance matrix
SC = PALutils.createRedNoiseCovarianceMatrix(tm, Agw, gamgw)
# calculate auto red noise covariance matrix
SA = PALutils.createRedNoiseCovarianceMatrix(tm, Ared[ll], gred[ll])
# create white noise covariance matrix
#TODO: add ability to use multiple efacs for different backends
white = PALutils.createWhiteNoiseCovarianceMatrix(psr[ll].err, efac[ll], equad[ll])
# total auto-covariance matrix
P = SC + SA + white
# sandwich with G matrices
Ppost = np.dot(psr[ll].G.T, np.dot(P, psr[ll].G))
# do cholesky solve
cf = sl.cho_factor(Ppost)
# solution vector P^_1 r
rr = sl.cho_solve(cf, r1)
# temporarily store P^-1 r
tmp.append(np.dot(psr[ll].G, rr))
# add to log-likelihood
loglike += -0.5 * (np.sum(np.log(2*np.pi*np.diag(cf[0])**2)) + np.dot(r1, rr))
# now compute cross terms
k = 0
for ll in range(npsr):
for kk in range(ll+1, npsr):
# create time lags
tm = PALutils.createTimeLags(psr[ll].toas, psr[kk].toas)
# create cross covariance matrix
SIJ = PALutils.createRedNoiseCovarianceMatrix(tm, 1, gamgw)
# carry out matrix-vetor operations
tmp1 = np.dot(SIJ, tmp[kk])
# add to likelihood
loglike += ORF[k]/2 * Agw**2 * np.dot(tmp[ll], tmp1)
# increment ORF counter
k += 1
return loglike
for p in psr:
print 'Pulsar {0} has {1} ns weighted rms'.format(
p.name,
p.rms() * 1e9)
npsr = len(psr)
pfile.close()
# get Tmax
Tmax = np.max([p.toas.max() - p.toas.min() for p in psr])
# initialize fourier design matrix
F = [
PALutils.createfourierdesignmatrix(p.toas, args.nmodes, Tspan=Tmax)
for p in psr
]
if args.powerlaw:
tmp, f = PALutils.createfourierdesignmatrix(p.toas,
args.nmodes,
Tspan=Tmax,
freq=True)
# get G matrices
for p in psr:
p.G = PALutils.createGmatrix(p.dmatrix)
# pre compute diagonalized efac + equad white noise model
Diag = []
def modelIndependentFullPTASinglSource(psr, proj, s, f, theta, phi, rho, kappa, efac, equad, ORF):
"""
Model Independent single source testing function
"""
tstart = time.time()
# get the number of modes, should be the same for all pulsars
nmode = len(rho)
npsr = len(psr)
# get F matrices for all pulsars at given frequency
F = [np.array([np.sin(2*np.pi*f*p.toas), np.cos(2*np.pi*f*p.toas)]).T for p in psr]
F = [np.dot(proj[ii], F[ii]) for ii in range(len(proj))]
loglike1 = 0
FtNF = []
for ct,p in enumerate(psr):
# compute d
if ct == 0:
d = np.dot(F[ct].T, p.res/(efac[ct]*s[ct] + equad[ct]**2))
else:
d = np.append(d, np.dot(F[ct].T, p.res/(efac[ct]*s[ct] + equad[ct]**2)))
# compute FT N F
N = 1/(efac[ct]*s[ct] + equad[ct]**2)
right = (N*F[ct].T).T
FtNF.append(np.dot(F[ct].T, right))
# log determinant of N
logdet_N = np.sum(np.log(efac[ct]*s[ct] + equad[ct]**2))
# triple produce in likelihood function
dtNdt = np.sum(p.res**2/(efac[ct]*s[ct] + equad[ct]**2))
loglike1 += -0.5 * (logdet_N + dtNdt)
# construct elements of sigma array
sigdiag = []
sigoffdiag = []
fplus = np.zeros(npsr)
fcross = np.zeros(npsr)
for ii in range(npsr):
fplus[ii], fcross[ii], cosMu = PALutils.createAntennaPatternFuncs(psr[ii], theta, phi)
tot = np.zeros(2*nmode)
offdiag = np.zeros(2*nmode)
# off diagonal terms
offdiag[0::2] = 10**rho
offdiag[1::2] = 10**rho
# diagonal terms
tot[0::2] = 10**rho
tot[1::2] = 10**rho
# add in individual red noise
if len(kappa[ii]) > 0:
tot[0::2][0:len(kappa[ii])] += 10**kappa[ii]
tot[1::2][0:len(kappa[ii])] += 10**kappa[ii]
# fill in lists of arrays
sigdiag.append(tot)
sigoffdiag.append(offdiag)
tstart2 = time.time()
# compute Phi inverse from Lindley's code
smallMatrix = np.zeros((2*nmode, npsr, npsr))
for ii in range(npsr):
for jj in range(ii,npsr):
if ii == jj:
smallMatrix[:,ii,jj] = ORF[ii,jj] * sigdiag[jj] * (fplus[ii]**2 + fcross[ii]**2)
else:
smallMatrix[:,ii,jj] = ORF[ii,jj] * sigoffdiag[jj] * (fplus[ii]*fplus[jj] + fcross[ii]*fcross[jj])
smallMatrix[:,jj,ii] = smallMatrix[:,ii,jj]
# invert them
logdet_Phi = 0
for ii in range(2*nmode):
L = sl.cho_factor(smallMatrix[ii,:,:])
smallMatrix[ii,:,:] = sl.cho_solve(L, np.eye(npsr))
logdet_Phi += np.sum(2*np.log(np.diag(L[0])))
# now fill in real covariance matrix
Phi = np.zeros((2*npsr*nmode, 2*npsr*nmode))
for ii in range(npsr):
for jj in range(ii,npsr):
for kk in range(0,2*nmode):
Phi[kk+ii*2*nmode,kk+jj*2*nmode] = smallMatrix[kk,ii,jj]
# symmeterize Phi
Phi = Phi + Phi.T - np.diag(np.diag(Phi))
# compute sigma
Sigma = sl.block_diag(*FtNF) + Phi
tmatrix = time.time() - tstart2
tstart3 = time.time()
# 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])))
tinverse = time.time() - tstart3
logLike = -0.5 * (logdet_Phi + logdet_Sigma) + 0.5 * (np.dot(d, expval2)) + loglike1
#print 'Total time: {0}'.format(time.time() - tstart)
#print 'Matrix construction time: {0}'.format(tmatrix)
#print 'Inversion time: {0}\n'.format(tinverse)
return logLike
# make sure all pulsar have same reference time
tt=[]
for p in psr:
tt.append(np.min(p.toas))
# find reference time
tref = np.min(tt)
# now scale pulsar time
for p in psr:
p.toas -= tref
# get G matrices
for p in psr:
p.G = PALutils.createGmatrix(p.dmatrix)
# run Fp statistic to determine starting frequency
if args.freq is None:
print 'Running initial Fpstat search'
fsearch = np.logspace(-9, -7, 1000)
fpstat = np.zeros(len(fsearch))
for ii in range(len(fsearch)):
fpstat[ii] = PALLikelihoods.fpStat(psr, fsearch[ii])
# determine maximum likelihood frequency
fmaxlike = fsearch[np.argmax(fpstat)]
print 'Maximum likelihood from f-stat search = {0}\n'.format(fmaxlike)
# get determinant of covariance matrix for use in likelihood
logdetTerm = []
def addpulsar(self,
parfile,
timfile,
DMOFF=None,
DMXOFF=None,
dailyAverage=False):
"""
Add another pulsar to the HDF5 file, given a tempo2 par and tim file.
@param parfile: tempo2 par file
@param timfile: tempo2 tim file
@param DMOFF: Option to turn off DMMODEL fitting
@param DMOFF: Option to turn off DMX fitting
@param dailyAverage: Option to perform daily averaging to reduce the number
of points by consructing daily averaged TOAs that have
one residual per day per frequency band. (This has only
been tested on NANOGrav data thus far.)
"""
# Check whether the two files exist
if not os.path.isfile(parfile) or not os.path.isfile(timfile):
raise IOError, "Cannot find parfile (%s) or timfile (%s)!" % (
parfile, timfile)
assert (self.filename != None), "ERROR: HDF5 file not set!"
# 'a' means: read/write if exists, create otherwise
self.h5file = h5.File(self.filename, 'a')
if "Model" in self.h5file:
self.h5file.close()
self.h5file = None
raise IOError, "model already available in '%s'. Refusing to add data" % (
self.filename)
# Create the data subgroup if it does not exist
if "Data" in self.h5file:
datagroup = self.h5file["Data"]
else:
datagroup = self.h5file.create_group("Data")
# Load pulsar data from the JPL Cython tempo2 library
t2pulsar = t2.tempopulsar(parfile, timfile)
# do multiple fits
#t2pulsar.fit(iters=10)
# turn off DMMODEL fitting
if DMOFF is not None:
t2pulsar['DMMODEL'].fit = False
# turn off DMX fitting
if DMXOFF is not None:
DMXFlag = False
print 'Turning off DMX fitting and turning DM fitting on'
for par in t2pulsar.pars:
if 'DMX' in par:
print par
t2pulsar[par].fit = False
t2pulsar['DM'].fit = True
DMXFlag = True
if DMXFlag == False:
print 'NO DMX for pulsar {0}'.format(t2pulsar.name)
# refit 5 times to make sure we are converged
t2pulsar.fit(iters=5)
# Create the pulsar subgroup if it does not exist
if "Pulsars" in datagroup:
pulsarsgroup = datagroup["Pulsars"]
else:
pulsarsgroup = datagroup.create_group("Pulsars")
# Look up the name of the pulsar, and see if it exist
if t2pulsar.name in pulsarsgroup:
self.h5file.close()
raise IOError, "%s already exists in %s!" % (t2pulsar.name,
self.filename)
pulsarsgroup = pulsarsgroup.create_group(t2pulsar.name)
# Read the data from the tempo2 structure.
designmatrix = np.double(t2pulsar.designmatrix())
residuals = np.double(t2pulsar.residuals())
toas = np.double(t2pulsar.toas())
errs = np.double(t2pulsar.toaerrs * 1e-6)
pname = t2pulsar.name
try: # if tim file has frequencies
freqs = np.double(t2pulsar.freqs)
except AttributeError:
freqs = 0
try: # if tim file has frequency band flags
bands = t2pulsar.flags['B']
except KeyError:
bands = 0
# if doing daily averaging
if dailyAverage:
# get average quantities
toas, qmatrix, errs, dmatrix, freqs, bands = PALutils.dailyAverage(
t2pulsar)
# construct new daily averaged residuals and designmatrix
residuals = np.dot(qmatrix, residuals)
designmatrix = np.dot(qmatrix, dmatrix)
# Write the TOAs, residuals, and uncertainties.
spd = 24.0 * 3600 # seconds per day
pulsarsgroup.create_dataset('TOAs', data=toas *
spd) # days (MJD) * sec per day
pulsarsgroup.create_dataset('residuals', data=residuals) # seconds
pulsarsgroup.create_dataset('toaErr', data=errs) # seconds
pulsarsgroup.create_dataset('freqs', data=freqs * 1e6) # Hz
pulsarsgroup.create_dataset('bands', data=bands) # Hz
# add tim and par file paths
pulsarsgroup.create_dataset('parFile', data=parfile) # string
pulsarsgroup.create_dataset('timFile', data=timfile) # string
# Write the full design matrix
pulsarsgroup.create_dataset('designmatrix', data=designmatrix)
# Obtain the timing model parameters
tmpname = np.array(t2pulsar.pars)
tmpvalpre = np.double(
[t2pulsar.prefit[parname].val for parname in t2pulsar.pars])
tmpvalpost = np.double(
[t2pulsar[parname].val for parname in t2pulsar.pars])
tmperrpre = np.double(
[t2pulsar.prefit[parname].err for parname in t2pulsar.pars])
tmperrpost = np.double(
[t2pulsar[parname].err for parname in t2pulsar.pars])
# Write the timing model parameter (TMP) descriptions
pulsarsgroup.create_dataset('pname', data=pname) # pulsar name
pulsarsgroup.create_dataset('tmp_name', data=tmpname) # TMP name
pulsarsgroup.create_dataset('tmp_valpre',
data=tmpvalpre) # TMP pre-fit value
pulsarsgroup.create_dataset('tmp_valpost',
data=tmpvalpost) # TMP post-fit value
pulsarsgroup.create_dataset('tmp_errpre',
data=tmperrpre) # TMP pre-fit error
pulsarsgroup.create_dataset('tmp_errpost',
data=tmperrpost) # TMP post-fit error
# Close the hdf5 file
self.h5file.close()
# make sure all pulsar have same reference time
tt = []
for p in psr:
tt.append(np.min(p.toas))
# find reference time
tref = np.min(tt)
# now scale pulsar time
for p in psr:
p.toas -= tref
# get G matrices
for p in psr:
p.G = PALutils.createGmatrix(p.dmatrix)
# run Fp statistic to determine starting frequency
print 'Running initial Fpstat search'
fsearch = np.logspace(-9, -7, 200)
fpstat = np.zeros(len(fsearch))
for ii in range(len(fsearch)):
fpstat[ii] = PALLikelihoods.fpStat(psr, fsearch[ii])
# determine maximum likelihood frequency
fmaxlike = fsearch[np.argmax(fpstat)]
print 'Maximum likelihood from f-stat search = {0}\n'.format(fmaxlike)
# prior ranges
thmin = phasemin = incmin = psimin = 0
thmax = incmax = np.pi
# make sure all pulsar have same reference time
tt = []
for p in psr:
tt.append(np.min(p.toas))
# find reference time
tref = np.min(tt)
# now scale pulsar time
for p in psr:
p.toas -= tref
# compute pairwise overlap reduction function values
print 'Computing Overlap Reduction Function Values'
ORF = PALutils.computeORF(psr)
# since we have defined our ORF to be normalized to 1
hdcoeff = ORF / 2
# compute optimal statistic
print 'Running Cross correlation Statistic on {0} Pulsars'.format(npsr)
crosspower, crosspowererr = PALLikelihoods.crossPower(psr, args.gam)
# angular separation
xi = []
for ll in range(npsr):
for kk in range(ll + 1, npsr):
xi.append(PALutils.angularSeparation(psr[ll].theta, psr[ll].phi, \
psr[kk].theta, psr[kk].phi))
# make sure pulsar names are in correct order
# TODO: is this a very round about way to do this?
index = []
for ct,p in enumerate(pp):
if p.name == psr[ct].name:
index.append(ct)
else:
for ii in range(npsr):
if pp[ii].name == psr[ct].name:
index.append(ii)
pp = [pp[ii] for ii in index]
M = [PALutils.createQSDdesignmatrix(p.toas) for p in psr]
RQ = [PALutils.createRmatrix(M[ct], p.err) for ct, p in enumerate(psr)]
# construct noise matrix for new noise realizations
print 'Constructing noise cholesky decompositions'
L = []
for ct, p in enumerate(psr):
Amp = p.Amp
gam = p.gam
efac = p.efac
equad = p.equad
cequad = p.cequad
avetoas, U = PALutils.exploderMatrix(p.toas)
# carry out Fp search
if args.fpFlag:
print 'Beginning Fp Search with {0} pulsars, with frequency range {1} -- {2}'.format(npsr, f[0], f[-1])
fpstat = np.zeros(args.nfreqs)
for ii in range(args.nfreqs):
fpstat[ii] = PALLikelihoods.fpStat(psr, f[ii])
print 'Done Search. Computing False Alarm Probability'
# single template FAP
pf = np.array([PALutils.ptSum(npsr, fpstat[ii]) for ii in range(np.alen(f))])
# get total false alarm probability with trials factor
pfT = 1 - (1-pf)**np.alen(f)
# write results to file
if not os.path.exists(args.outDir):
os.makedirs(args.outDir)
# get filename from hdf5 file
fname = args.outDir + '/' + args.h5file.split('/')[-1].split('.')[0] + '.txt'
fout = open(fname, 'w')
print 'Writing results to file {0}'.format(fname)
for ii in range(np.alen(f)):
fout.write('%g %g %g\n'%(f[ii], fpstat[ii], pfT[ii]))
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 addpulsar(self, parfile, timfile, DMOFF=None, DMXOFF=None, dailyAverage=False):
"""
Add another pulsar to the HDF5 file, given a tempo2 par and tim file.
@param parfile: tempo2 par file
@param timfile: tempo2 tim file
@param DMOFF: Option to turn off DMMODEL fitting
@param DMOFF: Option to turn off DMX fitting
@param dailyAverage: Option to perform daily averaging to reduce the number
of points by consructing daily averaged TOAs that have
one residual per day per frequency band. (This has only
been tested on NANOGrav data thus far.)
"""
# Check whether the two files exist
if not os.path.isfile(parfile) or not os.path.isfile(timfile):
raise IOError, "Cannot find parfile (%s) or timfile (%s)!" % (parfile, timfile)
assert(self.filename != None), "ERROR: HDF5 file not set!"
# 'a' means: read/write if exists, create otherwise
self.h5file = h5.File(self.filename, 'a')
if "Model" in self.h5file:
self.h5file.close()
self.h5file = None
raise IOError, "model already available in '%s'. Refusing to add data" % (self.filename)
# Create the data subgroup if it does not exist
if "Data" in self.h5file:
datagroup = self.h5file["Data"]
else:
datagroup = self.h5file.create_group("Data")
# Load pulsar data from the JPL Cython tempo2 library
t2pulsar = t2.tempopulsar(parfile, timfile)
# do multiple fits
#t2pulsar.fit(iters=10)
# turn off DMMODEL fitting
if DMOFF is not None:
t2pulsar['DMMODEL'].fit = False
# turn off DMX fitting
if DMXOFF is not None:
DMXFlag = False
print 'Turning off DMX fitting and turning DM fitting on'
for par in t2pulsar.pars:
if 'DMX' in par:
print par
t2pulsar[par].fit = False
t2pulsar['DM'].fit = True
DMXFlag = True
if DMXFlag== False:
print 'NO DMX for pulsar {0}'.format(t2pulsar.name)
# refit 5 times to make sure we are converged
t2pulsar.fit(iters=5)
# Create the pulsar subgroup if it does not exist
if "Pulsars" in datagroup:
pulsarsgroup = datagroup["Pulsars"]
else:
pulsarsgroup = datagroup.create_group("Pulsars")
# Look up the name of the pulsar, and see if it exist
if t2pulsar.name in pulsarsgroup:
self.h5file.close()
raise IOError, "%s already exists in %s!" % (t2pulsar.name, self.filename)
pulsarsgroup = pulsarsgroup.create_group(t2pulsar.name)
# Read the data from the tempo2 structure.
designmatrix = np.double(t2pulsar.designmatrix())
residuals = np.double(t2pulsar.residuals())
toas = np.double(t2pulsar.toas())
errs = np.double(t2pulsar.toaerrs*1e-6)
pname = t2pulsar.name
try: # if tim file has frequencies
freqs = np.double(t2pulsar.freqs)
except AttributeError:
freqs = 0
try: # if tim file has frequency band flags
bands = t2pulsar.flags['B']
except KeyError:
bands = 0
# if doing daily averaging
if dailyAverage:
# get average quantities
toas, qmatrix, errs, dmatrix, freqs, bands = PALutils.dailyAverage(t2pulsar)
# construct new daily averaged residuals and designmatrix
residuals = np.dot(qmatrix, residuals)
designmatrix = np.dot(qmatrix, dmatrix)
# Write the TOAs, residuals, and uncertainties.
spd = 24.0*3600 # seconds per day
pulsarsgroup.create_dataset('TOAs', data = toas*spd) # days (MJD) * sec per day
pulsarsgroup.create_dataset('residuals', data = residuals) # seconds
pulsarsgroup.create_dataset('toaErr', data = errs) # seconds
pulsarsgroup.create_dataset('freqs', data = freqs*1e6) # Hz
pulsarsgroup.create_dataset('bands', data = bands) # Hz
# add tim and par file paths
pulsarsgroup.create_dataset('parFile', data = parfile) # string
pulsarsgroup.create_dataset('timFile', data = timfile) # string
# Write the full design matrix
pulsarsgroup.create_dataset('designmatrix', data = designmatrix)
# Obtain the timing model parameters
tmpname = np.array(t2pulsar.pars)
tmpvalpre = np.double([t2pulsar.prefit[parname].val for parname in t2pulsar.pars])
tmpvalpost = np.double([t2pulsar[parname].val for parname in t2pulsar.pars])
tmperrpre = np.double([t2pulsar.prefit[parname].err for parname in t2pulsar.pars])
tmperrpost = np.double([t2pulsar[parname].err for parname in t2pulsar.pars])
# Write the timing model parameter (TMP) descriptions
pulsarsgroup.create_dataset('pname', data=pname) # pulsar name
pulsarsgroup.create_dataset('tmp_name', data=tmpname) # TMP name
pulsarsgroup.create_dataset('tmp_valpre', data=tmpvalpre) # TMP pre-fit value
pulsarsgroup.create_dataset('tmp_valpost', data=tmpvalpost) # TMP post-fit value
pulsarsgroup.create_dataset('tmp_errpre', data=tmperrpre) # TMP pre-fit error
pulsarsgroup.create_dataset('tmp_errpost', data=tmperrpost) # TMP post-fit error
# Close the hdf5 file
self.h5file.close()
# carry out Fp search
if args.fpFlag:
print 'Beginning Fp Search with {0} pulsars, with frequency range {1} -- {2}'.format(
npsr, f[0], f[-1])
fpstat = np.zeros(args.nfreqs)
for ii in range(args.nfreqs):
fpstat[ii] = PALLikelihoods.fpStat(psr, f[ii])
print 'Done Search. Computing False Alarm Probability'
# single template FAP
pf = np.array(
[PALutils.ptSum(npsr, fpstat[ii]) for ii in range(np.alen(f))])
# get total false alarm probability with trials factor
pfT = 1 - (1 - pf)**np.alen(f)
# write results to file
if not os.path.exists(args.outDir):
os.makedirs(args.outDir)
# get filename from hdf5 file
fname = args.outDir + '/' + args.h5file.split('/')[-1].split(
'.')[0] + '.txt'
fout = open(fname, 'w')
print 'Writing results to file {0}'.format(fname)
for ii in range(np.alen(f)):
# make sure all pulsar have same reference time
tt = []
for p in psr:
tt.append(np.min(p.toas))
# find reference time
tref = np.min(tt)
# now scale pulsar time
for p in psr:
p.toas -= tref
# get G matrices
for p in psr:
p.G = PALutils.createGmatrix(p.dmatrix)
# run Fp statistic to determine starting frequency
if args.null == False:
print "Running initial Fpstat search"
fsearch = np.logspace(-9, -7, 1000)
fpstat = np.zeros(len(fsearch))
for ii in range(len(fsearch)):
fpstat[ii] = PALLikelihoods.fpStat(psr, fsearch[ii])
# determine maximum likelihood frequency
fmaxlike = fsearch[np.argmax(fpstat)]
print "Maximum likelihood from f-stat search = {0}\n".format(fmaxlike)
# get Tmax
Tmax = np.max([p.toas.max() - p.toas.min() for p in psr])
def marginalizedPulsarPhaseLikeNumerical(psr, theta, phi, phase, inc, psi, freq, h,\
maximize=False):
"""
Compute the log-likelihood marginalized over pulsar phases
@param psr: List of pulsar object instances
@param theta: GW polar angle [radian]
@param phi: GW azimuthal angle [radian]
@param phase: Initial GW phase [radian]
@param inc: GW inclination angle [radian]
@param psi: GW polarization angle [radian]
@param freq: GW initial frequency [Hz]
@param h: GW strain
@param maximize: Option to maximize over pulsar phases instead of marginalize
"""
tstart = time.time()
# get number of pulsars
npsr = len(psr)
# construct xi = M**5/3/D and omega
xi = 0.25 * np.sqrt(5/2) * (np.pi*freq)**(-2/3) * h
omega = np.pi*freq
# get a values from Ellis et al 2012
a1 = xi * ((1+np.cos(inc)**2)*np.cos(phase)*np.cos(2*psi) + \
2*np.cos(inc)*np.sin(phase)*np.sin(2*psi))
a2 = -xi * ((1+np.cos(inc)**2)*np.sin(phase)*np.cos(2*psi) - \
2*np.cos(inc)*np.cos(phase)*np.sin(2*psi))
a3 = xi * ((1+np.cos(inc)**2)*np.cos(phase)*np.sin(2*psi) - \
2*np.cos(inc)*np.sin(phase)*np.cos(2*psi))
a4 = -xi * ((1+np.cos(inc)**2)*np.sin(phase)*np.sin(2*psi) + \
2*np.cos(inc)*np.cos(phase)*np.cos(2*psi))
lnlike = 0
tip = 0
tint = 0
tmax = 0
for ct, pp in enumerate(psr):
tstartip = time.time()
# compute relevant inner products
N1 = np.dot(np.cos(2*omega*pp.toas), np.dot(pp.invCov, pp.res))
N2 = np.dot(np.sin(2*omega*pp.toas), np.dot(pp.invCov, pp.res))
M11 = np.dot(np.sin(2*omega*pp.toas), np.dot(pp.invCov, np.sin(2*omega*pp.toas)))
M22 = np.dot(np.cos(2*omega*pp.toas), np.dot(pp.invCov, np.cos(2*omega*pp.toas)))
M12 = np.dot(np.cos(2*omega*pp.toas), np.dot(pp.invCov, np.sin(2*omega*pp.toas)))
# compute fplus and fcross
fplus, fcross, cosMu = PALutils.createAntennaPatternFuncs(pp, theta, phi)
# mind your p's and q's
p = fplus*a1 + fcross*a3
q = fplus*a2 + fcross*a4
# constuct multipliers of pulsar phase terms
X = p*N1 + q*N2 + p**2*M11 + q**2*M22 + 2*p*q*M12
Y = p*N1 + q*N2 + 2*p**2*M11 + 2*q**2*M22 + 4*p*q*M12
Z = p*N2 - q*N1 + 2*(p**2-q**2)*M12 - 2*p*q*(M11-M22)
W = q**2*M11 + p**2*M22 -2*p*q*M12
V = p*q*(M11-M22) - (p**2-q**2)*M12
#print X, Y, Z, W, V
tip += (time.time() - tstartip)
tstartint = time.time()
# find the maximum of argument of exponential function
phip = np.linspace(0, 2*np.pi, 10000)
arg = X - Y*np.cos(phip) + Z*np.sin(phip) + W*np.sin(phip)**2 + 2*V*np.cos(phip)*np.sin(phip)
maxarg = np.max(arg)
if maximize:
tmax += maxarg
else:
# define integrand for numerical integration
f = lambda phi: np.exp(X - Y*np.cos(phi) + Z*np.sin(phi) + \
W*np.sin(phi)**2 + 2*V*np.cos(phi)*np.sin(phi) - maxarg)
# do numerical integration
integral = integrate.quad(f, 0, 2*np.pi)[0]
lnlike += maxarg + np.log(integral)
tint += (time.time() - tstartint)
print 'Loglike = {0}'.format(lnlike)
print 'Total Evaluation Time = {0} s'.format(time.time() - tstart)
print 'Total inner product evaluation Time = {0} s'.format(tip)
print 'Total Integration Time = {0} s\n'.format(tint)
if maximize:
lnlike = tmax
return lnlike
# now scale pulsar time
for p in psr:
p.toas -= tref
#import glob
#invmat = glob.glob('/Users/Justin/Work/nanograv/nanograv/data_products/joris/*invCov*')
#
## get list of R matrices
#R = [PALutils.createRmatrix(p.dmatrix, p.err) for p in psr]
#
#for ct,p in enumerate(psr):
# p.invCov = np.dot(R[ct].T, np.dot(p.invCov, R[ct]))
# compute pairwise overlap reduction function values
print 'Computing Overlap Reduction Function Values'
ORF = PALutils.computeORF(psr)
# compute optimal statistic
print 'Running Optimal Statistic on {0} Pulsars'.format(npsr)
Opt, sigma, snr = PALLikelihoods.optStat(psr, ORF, gam=args.gam)
print 'Results of Search\n'
print '------------------------------------\n'
print 'A_gw^2 = {0}'.format(Opt)
print 'std. dev. = {0}'.format(sigma)
print 'SNR = {0}'.format(snr)
if snr > 3.0:
print 'SNR of {0} is above threshold!'.format(snr)
pass
print 'Reading in HDF5 file'
# import hdf5 file
pfile = h5.File(args.h5file, 'r')
# define the pulsargroup
pulsargroup = pfile['Data']['Pulsars'][args.pname]
# fill in pulsar class
psr = PALpulsarInit.pulsar(pulsargroup, addGmatrix=True)
# initialize fourier design matrix
if args.nmodes != 0:
F, f = PALutils.createfourierdesignmatrix(psr.toas, args.nmodes, freq=True)
Tspan = psr.toas.max() - psr.toas.min()
##fsred = np.array([1.599558028614668e-07, 5.116818355403073e-08]) # 1855
#fsred = np.array([9.549925860214369e-08]) # 1909
#fsred = np.array([1/Tspan, 9.772372209558111e-08]) # 1909
#
#F = np.zeros((psr.ntoa, 2*len(fsred)))
#F[:,0::2] = np.cos(2*np.pi*np.outer(psr.toas, fsred))
#F[:,1::2] = np.sin(2*np.pi*np.outer(psr.toas, fsred))
# get G matrices
psr.G = PALutils.createGmatrix(psr.dmatrix)
# pre compute diagonalized efac + equad white noise model
efac = np.dot(psr.G.T, np.dot(np.diag(psr.err**2), psr.G))
def marginalizedPulsarPhaseLikeNumerical(psr, theta, phi, phase, inc, psi, freq, h,\
maximize=False):
"""
Compute the log-likelihood marginalized over pulsar phases
@param psr: List of pulsar object instances
@param theta: GW polar angle [radian]
@param phi: GW azimuthal angle [radian]
@param phase: Initial GW phase [radian]
@param inc: GW inclination angle [radian]
@param psi: GW polarization angle [radian]
@param freq: GW initial frequency [Hz]
@param h: GW strain
@param maximize: Option to maximize over pulsar phases instead of marginalize
"""
tstart = time.time()
# get number of pulsars
npsr = len(psr)
# construct xi = M**5/3/D and omega
xi = 0.25 * np.sqrt(5 / 2) * (np.pi * freq)**(-2 / 3) * h
omega = np.pi * freq
# get a values from Ellis et al 2012
a1 = xi * ((1+np.cos(inc)**2)*np.cos(phase)*np.cos(2*psi) + \
2*np.cos(inc)*np.sin(phase)*np.sin(2*psi))
a2 = -xi * ((1+np.cos(inc)**2)*np.sin(phase)*np.cos(2*psi) - \
2*np.cos(inc)*np.cos(phase)*np.sin(2*psi))
a3 = xi * ((1+np.cos(inc)**2)*np.cos(phase)*np.sin(2*psi) - \
2*np.cos(inc)*np.sin(phase)*np.cos(2*psi))
a4 = -xi * ((1+np.cos(inc)**2)*np.sin(phase)*np.sin(2*psi) + \
2*np.cos(inc)*np.cos(phase)*np.cos(2*psi))
lnlike = 0
tip = 0
tint = 0
tmax = 0
for ct, pp in enumerate(psr):
tstartip = time.time()
# compute relevant inner products
N1 = np.dot(np.cos(2 * omega * pp.toas), np.dot(pp.invCov, pp.res))
N2 = np.dot(np.sin(2 * omega * pp.toas), np.dot(pp.invCov, pp.res))
M11 = np.dot(np.sin(2 * omega * pp.toas),
np.dot(pp.invCov, np.sin(2 * omega * pp.toas)))
M22 = np.dot(np.cos(2 * omega * pp.toas),
np.dot(pp.invCov, np.cos(2 * omega * pp.toas)))
M12 = np.dot(np.cos(2 * omega * pp.toas),
np.dot(pp.invCov, np.sin(2 * omega * pp.toas)))
# compute fplus and fcross
fplus, fcross, cosMu = PALutils.createAntennaPatternFuncs(
pp, theta, phi)
# mind your p's and q's
p = fplus * a1 + fcross * a3
q = fplus * a2 + fcross * a4
# constuct multipliers of pulsar phase terms
X = p * N1 + q * N2 + p**2 * M11 + q**2 * M22 + 2 * p * q * M12
Y = p * N1 + q * N2 + 2 * p**2 * M11 + 2 * q**2 * M22 + 4 * p * q * M12
Z = p * N2 - q * N1 + 2 * (p**2 - q**2) * M12 - 2 * p * q * (M11 - M22)
W = q**2 * M11 + p**2 * M22 - 2 * p * q * M12
V = p * q * (M11 - M22) - (p**2 - q**2) * M12
#print X, Y, Z, W, V
tip += (time.time() - tstartip)
tstartint = time.time()
# find the maximum of argument of exponential function
phip = np.linspace(0, 2 * np.pi, 10000)
arg = X - Y * np.cos(phip) + Z * np.sin(phip) + W * np.sin(
phip)**2 + 2 * V * np.cos(phip) * np.sin(phip)
maxarg = np.max(arg)
if maximize:
tmax += maxarg
else:
# define integrand for numerical integration
f = lambda phi: np.exp(X - Y*np.cos(phi) + Z*np.sin(phi) + \
W*np.sin(phi)**2 + 2*V*np.cos(phi)*np.sin(phi) - maxarg)
# do numerical integration
integral = integrate.quad(f, 0, 2 * np.pi)[0]
lnlike += maxarg + np.log(integral)
tint += (time.time() - tstartint)
print 'Loglike = {0}'.format(lnlike)
print 'Total Evaluation Time = {0} s'.format(time.time() - tstart)
print 'Total inner product evaluation Time = {0} s'.format(tip)
print 'Total Integration Time = {0} s\n'.format(tint)
if maximize:
lnlike = tmax
return lnlike
def marginalizedPulsarPhaseLike(psr,
theta,
phi,
phase,
inc,
psi,
freq,
h,
maximize=False):
"""
Compute the log-likelihood marginalized over pulsar phases
@param psr: List of pulsar object instances
@param theta: GW polar angle [radian]
@param phi: GW azimuthal angle [radian]
@param phase: Initial GW phase [radian]
@param inc: GW inclination angle [radian]
@param psi: GW polarization angle [radian]
@param freq: GW initial frequency [Hz]
@param h: GW strain
@param maximize: Option to maximize over pulsar phases instead of marginalize
"""
# get number of pulsars
npsr = len(psr)
# get c and d
c = np.cos(phase)
d = np.sin(phase)
# construct xi = M**5/3/D and omega
xi = 0.25 * np.sqrt(5 / 2) * (np.pi * freq)**(-2 / 3) * h
omega = np.pi * freq
lnlike = 0
for ct, pp in enumerate(psr):
# compute relevant inner products
cip = np.dot(np.cos(2 * omega * pp.toas), np.dot(pp.invCov, pp.res))
sip = np.dot(np.sin(2 * omega * pp.toas), np.dot(pp.invCov, pp.res))
N = np.dot(np.cos(2 * omega * pp.toas),
np.dot(pp.invCov, np.cos(2 * omega * pp.toas)))
# compute fplus and fcross
fplus, fcross, cosMu = PALutils.createAntennaPatternFuncs(
pp, theta, phi)
# mind your p's and q's
p = (1 + np.cos(inc)**2) * (fplus * np.cos(2 * psi) +
fcross * np.sin(2 * psi))
q = 2 * np.cos(inc) * (fplus * np.sin(2 * psi) -
fcross * np.cos(2 * psi))
# construct X Y and Z
X = -xi / omega**(1 / 3) * (p * sip + q * cip -
0.5 * xi / omega**(1 / 3) * N * c *
(p**2 + q**2))
Y = -xi / omega**(1 / 3) * (q * sip - p * cip -
0.5 * xi / omega**(1 / 3) * N * d *
(p**2 + q**2))
Z = xi/omega**(1/3) * ((p*c+q*d)*sip - (p*d-q*c)*cip \
-0.5*xi/omega**(1/3)*N*(p**2+q**2))
# add to log-likelihood
#print X, Y
if maximize:
lnlike += Z + np.sqrt(X**2 + Y**2)
else:
lnlike += Z + np.log(ss.iv(0, np.sqrt(X**2 + Y**2)))
return lnlike
# make sure all pulsar have same reference time
tt=[]
for p in psr:
tt.append(np.min(p.toas))
# find reference time
tref = np.min(tt)
# now scale pulsar time
for p in psr:
p.toas -= tref
# get G matrices
for p in psr:
p.G = PALutils.createGmatrix(p.dmatrix)
# run Fp statistic to determine starting frequency
print 'Running initial Fpstat search'
fsearch = np.logspace(-9, -7, 200)
fpstat = np.zeros(len(fsearch))
for ii in range(len(fsearch)):
fpstat[ii] = PALLikelihoods.fpStat(psr, fsearch[ii])
# determine maximum likelihood frequency
fmaxlike = fsearch[np.argmax(fpstat)]
print 'Maximum likelihood from f-stat search = {0}\n'.format(fmaxlike)
# prior ranges
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