def lnLikelihood(image, flux, x, y, psf, invvar):
star = MixtureOfGaussians(np.array([
flux,
]), np.array([x, y]), np.array([
0.,
])) # delta-function
cstar = star.convolve(psf)
modelimage = cstar.evaluate_grid(-5, 7, -5, 7)
return -0.5 * np.sum(invvar * (image - modelimage)**2)
def main():
# make a simple psf
amp = np.array([0.9, 0.1])
mean = np.random.uniform(size=(2, 2))
var = np.array([2.0, 20.])
psf = MixtureOfGaussians(amp, mean, var)
psf.normalize()
trueimage = psf.evaluate_grid(-5, 7, -5, 7)
plt.clf()
plt.imshow(trueimage, interpolation='nearest', cmap='gray')
plt.savefig('truth.png')
# do some simple test fitting to check likelihood code
invvar = 0.01 * np.ones_like(trueimage)
flux = 1.0
y = 0.5
xlist = np.arange(-1.0, 1.0, 0.01)
lnLlist = np.array(
[lnLikelihood(trueimage, flux, x, y, psf, invvar) for x in xlist])
plt.clf()
plt.plot(xlist, lnLlist, 'k-', alpha=0.5)
plt.xlabel('model star position')
plt.ylabel('$\ln$ likelihood')
plt.savefig('lnL.png')
# try an optimization
if False:
pars = np.array([0.9, -1.0, 1.0])
notpars = (trueimage, psf, invvar)
bestpars = op.fmin(cost, pars, args=(notpars, ))
print(bestpars)
# now loop over noisiness
minx = -3.5
maxx = -1.5
picklefn = 'fluxes.pickle'
if not os.path.exists(picklefn):
logvarlist = np.random.uniform(minx, maxx, size=(3000))
fluxlist = np.zeros_like(logvarlist)
for i, logvar in enumerate(logvarlist):
noise = np.random.normal(size=trueimage.shape)
var = 10.**logvar
image = trueimage + var * noise
invvar = np.ones_like(image) / var / var
pars = np.array([1.0, 0.0, 0.0])
notpars = (image, psf, invvar)
bestpars = op.fmin(cost, pars, args=(notpars, ))
fluxlist[i] = bestpars[0]
print(i, logvar, bestpars)
an.pickle_to_file((logvarlist, fluxlist), picklefn)
(logvarlist, fluxlist) = an.unpickle_from_file(picklefn)
# make plots
plt.clf()
plt.axhline(1.1, color='k', alpha=0.25)
plt.axhline(1.0, color='k', alpha=0.25)
plt.axhline(0.9, color='k', alpha=0.25)
plt.plot(logvarlist, fluxlist, 'k.', alpha=0.5)
plt.xlabel('$\log_{10}$ pixel noise variance')
plt.ylabel('inferred flux')
plt.xlim(minx, maxx)
plt.ylim(0.8, 1.2)
plt.savefig('flux.png')
medbins = np.arange(minx, maxx + 0.0001, 0.25)
for i in range(len(medbins) - 1):
a, b = medbins[[i, i + 1]]
inside = np.sort(fluxlist[(logvarlist > a) * (logvarlist < b)])
nin = len(inside)
plt.plot([a, b], [inside[0.025 * nin], inside[0.025 * nin]],
'k-',
alpha=0.5)
plt.plot([a, b], [inside[0.16 * nin], inside[0.16 * nin]],
'k-',
alpha=0.5)
plt.plot([a, b], [inside[0.50 * nin], inside[0.50 * nin]],
'k-',
alpha=0.5)
plt.plot([a, b], [inside[0.84 * nin], inside[0.84 * nin]],
'k-',
alpha=0.5)
plt.plot([a, b], [inside[0.975 * nin], inside[0.975 * nin]],
'k-',
alpha=0.5)
plt.savefig('fluxquant.png')
def main():
# make a simple psf
amp = np.array([0.9, 0.1])
mean = np.random.uniform(size=(2, 2))
var = np.array([2.0, 20.])
psf = MixtureOfGaussians(amp, mean, var)
psf.normalize()
trueimage = psf.evaluate_grid(-5, 7, -5, 7)
plt.clf()
plt.imshow(trueimage, interpolation='nearest', cmap='gray')
plt.savefig('truth.png')
# do some simple test fitting to check likelihood code
invvar = 0.01 * np.ones_like(trueimage)
flux = 1.0
y = 0.5
xlist = np.arange(-1.0, 1.0, 0.01)
lnLlist = np.array([lnLikelihood(trueimage, flux, x, y, psf, invvar) for x in xlist])
plt.clf()
plt.plot(xlist, lnLlist, 'k-', alpha=0.5)
plt.xlabel('model star position')
plt.ylabel('$\ln$ likelihood')
plt.savefig('lnL.png')
# try an optimization
if False:
pars = np.array([0.9, -1.0, 1.0])
notpars = (trueimage, psf, invvar)
bestpars = op.fmin(cost, pars, args=(notpars, ))
print bestpars
# now loop over noisiness
minx = -3.5
maxx = -1.5
picklefn = 'fluxes.pickle'
if not os.path.exists(picklefn):
logvarlist = np.random.uniform(minx, maxx, size=(3000))
fluxlist = np.zeros_like(logvarlist)
for i, logvar in enumerate(logvarlist):
noise = np.random.normal(size=trueimage.shape)
var = 10.**logvar
image = trueimage + var * noise
invvar = np.ones_like(image) / var / var
pars = np.array([1.0, 0.0, 0.0])
notpars = (image, psf, invvar)
bestpars = op.fmin(cost, pars, args=(notpars, ))
fluxlist[i] = bestpars[0]
print i, logvar, bestpars
an.pickle_to_file((logvarlist, fluxlist), picklefn)
(logvarlist, fluxlist) = an.unpickle_from_file(picklefn)
# make plots
plt.clf()
plt.axhline(1.1, color='k', alpha=0.25)
plt.axhline(1.0, color='k', alpha=0.25)
plt.axhline(0.9, color='k', alpha=0.25)
plt.plot(logvarlist, fluxlist, 'k.', alpha=0.5)
plt.xlabel('$\log_{10}$ pixel noise variance')
plt.ylabel('inferred flux')
plt.xlim(minx, maxx)
plt.ylim(0.8, 1.2)
plt.savefig('flux.png')
medbins = np.arange(minx, maxx+0.0001, 0.25)
for i in range(len(medbins)-1):
a, b = medbins[[i, i+1]]
inside = np.sort(fluxlist[(logvarlist > a) * (logvarlist < b)])
nin = len(inside)
plt.plot([a, b], [inside[0.025* nin], inside[0.025* nin]], 'k-', alpha=0.5)
plt.plot([a, b], [inside[0.16 * nin], inside[0.16 * nin]], 'k-', alpha=0.5)
plt.plot([a, b], [inside[0.50 * nin], inside[0.50 * nin]], 'k-', alpha=0.5)
plt.plot([a, b], [inside[0.84 * nin], inside[0.84 * nin]], 'k-', alpha=0.5)
plt.plot([a, b], [inside[0.975* nin], inside[0.975* nin]], 'k-', alpha=0.5)
plt.savefig('fluxquant.png')
def lnLikelihood(image, flux, x, y, psf, invvar):
star = MixtureOfGaussians(np.array([flux, ]), np.array([x, y]), np.array([0., ])) # delta-function
cstar = star.convolve(psf)
modelimage = cstar.evaluate_grid(-5, 7, -5, 7)
return -0.5 * np.sum(invvar * (image - modelimage)**2)
