def calcNoiseBias(zcluster=0.5, concen=4.):
m200s = np.arange(1, 50)
r_mpc = np.arange(0.1, 3.0, 0.1)
duffy = basicMassCon.Duffy(None)
background = 1.0
beta_s = nfwutils.global_cosmology.beta_s([background], zcluster)
beta_s2 = beta_s**2
beta_cor = beta_s2 / beta_s
rho_c_over_sigma_c = 1.5 * nfwutils.global_cosmology.angulardist(
zcluster) * nfwutils.global_cosmology.beta(
[1e6], zcluster)[0] * nfwutils.global_cosmology.hubble2(
zcluster) / nfwutils.global_cosmology.v_c**2
def reducedShear(scaledm200):
m200 = 1e14 * scaledm200
c = duffy(m200, zcluster)
rs = nfwutils.rscaleConstM(m200, c, zcluster, 200)
gamma = tools.NFWShear(r_mpc, c, rs, rho_c_over_sigma_c)
kappa = tools.NFWKappa(r_mpc, c, rs, rho_c_over_sigma_c)
g = beta_s * gamma / (1 - beta_cor * kappa)
return g
shearprime = np.zeros((len(m200s), len(r_mpc)))
shearprimeprime = np.zeros((len(m200s), len(r_mpc)))
for i, m200 in enumerate(m200s):
shearprime[i, :] = scipy.misc.derivative(reducedShear,
m200,
dx=.01,
order=5)
shearprimeprime[i, :] = scipy.misc.derivative(reducedShear,
m200,
dx=.01,
n=2,
order=7)
return shearprime, shearprimeprime, -0.5 * shearprimeprime / (shearprime**
3)
def likelihoodPlots():
'''Show the lensing likelihoods, to make the point that yes, they are almost symetric in linear space, but Gaussian approximations are poor.'''
linfig = pylab.figure()
linax = linfig.add_subplot(1, 1, 1)
logfig = pylab.figure()
logax = logfig.add_subplot(1, 1, 1)
figs = [linfig, logfig]
true_masses = [2e14, 1e15]
linestyles = ['-', '--']
for true_mass, linestyle in zip(true_masses, linestyles):
m200s = np.arange(-5.05e14, 3e15, 1e13)
r_mpc_edges = np.linspace(0.75, 3., 13)
r_mpc = (r_mpc_edges[1:] + r_mpc_edges[:-1]) / 2.
zcluster = 0.415 #median of megacam
duffy = basicMassCon.Duffy()
c_duff = duffy(true_mass * nfwutils.global_cosmology.h, zcluster, 200.)
beta_s = 0.49 #median of megacam
g_truth = nfwnoise.createPerfectProfile(true_mass, c_duff, zcluster,
r_mpc, beta_s)
ngalsperarcmin = 9.7 #median of megacam set
r_arcmin_edges = (r_mpc_edges /
nfwutils.global_cosmology.angulardist(zcluster)) * (
180. / np.pi) * 60
bin_areas = np.pi * (r_arcmin_edges[1:]**2 - r_arcmin_edges[:-1]**2)
ngals_per_bin = ngalsperarcmin * bin_areas
g_err = 0.25 / np.sqrt(ngals_per_bin)
#calcualte pdf for duffy
config = {}
config['model'] = nfwfit.NFW_MC_Model()
config['massconRelation'] = basicMassCon.Duffy()
simutils.runConfigure(config)
likelihood = nfwnoise.Likelihood(config)
likelihood.bindData(r_mpc, g_truth, g_err, beta_s, zcluster)
duffy_logpdf = np.array([likelihood(m200) for m200 in m200s])
duffy_pdf = np.exp(duffy_logpdf - np.max(duffy_logpdf))
duffy_pdf = duffy_pdf / np.trapz(duffy_pdf, m200s)
#plot
linax.plot(m200s,
duffy_pdf,
marker='None',
linestyle=linestyle,
color='k',
linewidth=2.,
label='$M_{200} = $%1.1f x $10^{14} M_{\odot}$' %
(true_mass / 1e14))
linax.axvline(true_mass,
c=(0.6, 0.6, 0.6),
linewidth=2,
linestyle=linestyle)
posmass = m200s > 0
log10m200s = np.log10(m200s[posmass])
log10pdf = duffy_pdf[posmass]
log10pdf = log10pdf / np.trapz(log10pdf, log10m200s)
logax.plot(log10m200s,
log10pdf,
marker='None',
linestyle=linestyle,
color='k',
linewidth=2.,
label='$M_{200} = $%1.1f x $10^{14} M_{\odot}$' %
(true_mass / 1e14))
logax.axvline(np.log10(true_mass),
c=(0.6, 0.6, 0.6),
linewidth=2,
linestyle=linestyle)
linax.legend()
linax.set_xlabel(r'$M_{200} [M_{\odot}]$', fontsize=18)
linax.set_ylabel(r'Likelihood($M_{200}$)', fontsize=18)
linax.set_xlim(-0.5e15, 2.5e15)
linfig.tight_layout()
linfig.savefig('docs/figures/likelihood_shape.png')
linfig.savefig('docs/figures/likelihood_shape.eps')
linfig.savefig('docs/figures/likelihood_shape.pdf')
logax.legend(loc='upper left')
logax.set_xlabel(r'$\mathrm{log}_{10}M_{200}$', fontsize=18)
logax.set_ylabel(r'Likelihood($\mathrm{log}_{10}M_{200}$)', fontsize=18)
logax.set_xlim(np.log10(np.min(m200s[posmass])), np.log10(2.5e15))
logfig.tight_layout()
logfig.savefig('docs/figures/likelihood_logshape.png')
logfig.savefig('docs/figures/likelihood_logshape.eps')
logfig.savefig('docs/figures/likelihood_logshape.pdf')
return figs
def calcMaxlikeDistro(outputfile='maxlikedistro_paperplot_lowsn.pkl',
ngalsperarcmin=5):
'''show that using maximum likelihood estimators does not recover truth'''
figs = []
#megacamstyle
ntrials = 2000
samplesize = 25
r_mpc_edges = np.linspace(0.75, 3., 13)
r_mpc = (r_mpc_edges[1:] + r_mpc_edges[:-1]) / 2.
zcluster = 0.3
duffy = basicMassCon.Duffy(None)
c200 = 4.0
beta_s = nfwutils.global_cosmology.beta_s([1.0], zcluster)[0]
r_arcmin_edges = (r_mpc_edges /
nfwutils.global_cosmology.angulardist(zcluster)) * (
180. / np.pi) * 60
bin_areas = np.pi * (r_arcmin_edges[1:]**2 - r_arcmin_edges[:-1]**2)
ngals_per_bin = ngalsperarcmin * bin_areas
g_err = 0.25 / np.sqrt(ngals_per_bin)
m200s = np.arange(1e12, 5e15, 1e13)
config = varcontainer.VarContainer()
config.massconmodule = 'basicMassCon'
config.massconrelation = 'constant'
config.concentration = c200
model = nfwfit.buildModel(config)
likelihood = nfwnoise.Likelihood(config)
unweightedbootstraps_mean = np.zeros(ntrials)
weightedaverages = np.zeros(ntrials)
wlssaverage = np.zeros(ntrials)
linear_fit = np.zeros(ntrials)
smithaverages = np.zeros(ntrials)
pdfaverages = np.zeros(ntrials)
for curtrial in range(ntrials):
true_masses = 10**np.random.uniform(np.log10(5e14), np.log10(2e15),
samplesize)
lens_masses = true_masses * np.exp(
0.2 * np.random.standard_normal(size=samplesize))
#compute individual cluster likelihoods
maxlike_masses = np.zeros(samplesize)
maxlike_errs = np.zeros((2, samplesize))
pdfs = np.zeros((samplesize, len(m200s)))
logpdf = np.zeros(len(m200s))
for curcluster in range(samplesize):
g_obs = nfwnoise.createPerfectProfile(lens_masses[curcluster], c200,
zcluster, r_mpc, beta_s) + \
g_err*np.random.standard_normal(size=len(g_err))
fitter = nfwfit.NFWFitter(None, model, config)
maxlikemass, maxlikeerr = fitter.minChisqMethod(
r_mpc,
g_obs,
g_err,
beta_s,
beta_s**2,
zcluster,
guess=[true_masses[curcluster] / model.massScale])
maxlike_masses[curcluster] = maxlikemass['m200']
try:
maxlike_errs[:, curcluster] = maxlikeerr['m200']
except TypeError:
maxlike_errs[:, curcluster] = (maxlikeerr['m200'],
maxlikeerr['m200'])
likelihood.bindData(r_mpc, g_obs, g_err, beta_s, zcluster)
for i, m200 in enumerate(m200s):
logpdf[i] = likelihood(m200)
pdf = np.exp(logpdf - np.max(logpdf))
pdf /= np.trapz(pdf, m200s)
pdfs[curcluster, :] = pdf
maxlike_masses *= model.massScale
maxlike_errs = np.abs(maxlike_errs * model.massScale)
maxlike_logratio = np.log(maxlike_masses / true_masses)
#combine using different formulisms
bootstraps = np.zeros(1000)
for curboot in range(1000):
booted = np.random.randint(0, samplesize, samplesize)
bootstraps[curboot] = np.mean(maxlike_logratio[booted])
unweightedbootstraps_mean[curtrial] = np.mean(bootstraps)
maxlike_simpleerr = np.sum(maxlike_errs,
axis=0) / (2. * maxlike_masses)
weight = 1. / maxlike_simpleerr**2
weightedaverages[curtrial] = np.sum(
weight * maxlike_logratio) / np.sum(weight)
wlssaverage_weights = 1. / (maxlike_simpleerr**2 +
(0.2 * maxlike_masses)**2)
wlssaverage[curtrial] = np.sum(
wlssaverage_weights *
maxlike_logratio) / np.sum(wlssaverage_weights)
smithweight = 1. / (np.sum(maxlike_errs, axis=0) / 2.)**2
smithaverages[curtrial] = np.sum(
smithweight * maxlike_logratio) / np.sum(smithweight)
halos = [
dict(id=curcluster,
true_mass=true_masses[curcluster],
masses=m200s,
pdf=pdfs[curcluster, :]) for curcluster in range(samplesize)
]
pdfmodel = deconvolvedlognorm.buildPDFModel(halos)
pdfmap = pymc.MAP(pdfmodel)
pdfmap.fit()
pdfaverages[curtrial] = pdfmap.logmu.value
linefitter = fitmodel.FitModel(true_masses / 1e14,
maxlike_masses / 1e14,
np.sum(maxlike_errs / 1e14, axis=0) /
2.,
fitmodel.RatioModel,
guess=[1.])
linefitter.fit()
linear_fit[curtrial] = linefitter.par_vals['a0']
results = dict(unweightedbootstrapsmean=np.exp(unweightedbootstraps_mean),
weightedaverages=np.exp(weightedaverages),
wlssaverage=np.exp(wlssaverage),
smithaverages=np.exp(smithaverages),
pdfaverages=np.exp(pdfaverages),
linearfit=linear_fit)
with open(outputfile, 'wb') as output:
cPickle.dump(results, output, -1)