def solve_for_params(params, x_min, x_max):
lower_mass = 0.01
upper_mass = 0.99
# Trial parameters
alpha, beta = params
# Equation for the roots defining params which satisfy the constraint
return (
invgamma.cdf(x_min, alpha, scale=beta) - lower_mass,
invgamma.cdf(x_max, alpha, scale=beta) - upper_mass,
)
def test(self, time, cutoff, probability, less_than=True):
""" Test posterior belief that median time-to-event parameter is less than or greater than some boundary value.
:param time: test at this time
:type time: float
:param cutoff: test median time against this critical value
:type cutoff: float
:param probability: require at least this degree of posterior certainty to declare significance
:type probability: float
:param less_than: True, to test parameter is less than cut-off, a-posteriori. False to test greater than
:type less_than: bool
:return: JSON-able dict object reporting test output
:rtype: dict
"""
event_time = np.array(self._times_to_event)
recruit_time = np.array(self._recruitment_times)
# Filter to just patients who are registered by time
registered_patients = recruit_time <= time
has_failed = time - recruit_time[registered_patients] > event_time[registered_patients]
survival_time = np.array([min(x, y) for (x, y) in
zip(time - recruit_time[registered_patients], event_time[registered_patients])
])
# Update posterior beliefs for mu_E
alpha_post = self.alpha_prior + sum(has_failed)
beta_post = self.beta_prior + np.log(2) * sum(survival_time)
mu_post = beta_post / (alpha_post-1)
# Run test:
test_probability = invgamma.cdf(cutoff, a=alpha_post, scale=beta_post) if less_than \
else 1 - invgamma.cdf(cutoff, a=alpha_post, scale=beta_post)
stop_trial = test_probability > probability if less_than else test_probability < probability
test_report = OrderedDict()
test_report['Time'] = time
test_report['Patients'] = sum(registered_patients)
test_report['Events'] = sum(has_failed)
test_report['TotalEventTime'] = sum(survival_time)
test_report['AlphaPosterior'] = alpha_post
test_report['BetaPosterior'] = beta_post
test_report['MeanEventTimePosterior'] = mu_post
test_report['MedianEventTimePosterior'] = mu_post * np.log(2)
test_report['Cutoff'] = cutoff
test_report['Certainty'] = probability
test_report['Probability'] = test_probability
test_report['LessThan'] = atomic_to_json(less_than)
test_report['Stop'] = atomic_to_json(stop_trial)
return test_report
def testInvGamma1(seed):
# Check that Gamma is parameterized correctly
ripl = get_ripl(seed=seed)
# samples
ripl.assume("a", "(inv_gamma 10.0 10.0)", label="pid")
observed = collectSamples(ripl, "pid")
# true CDF
inv_gamma_cdf = lambda x: invgamma.cdf(x, a=10, scale=10)
return reportKnownContinuous(inv_gamma_cdf, observed)
def test_cdf(self):
shapes = np.arange(0.01, 20, 0.5)
scales = np.arange(0.01, 10, 0.5)
xs = np.arange(0.01, 10, 0.5)
test_params = cartesian([xs, shapes, scales]).astype(np.float64)
python_results = np.zeros(test_params.shape[0])
for ind in range(test_params.shape[0]):
x = test_params[ind, 0]
shape = test_params[ind, 1]
scale = test_params[ind, 2]
python_results[ind] = invgamma.cdf(x, shape, scale=scale)
opencl_results = invgamma_cdf().evaluate(
{
'x': test_params[:, 0],
'shape': test_params[:, 1],
'scale': test_params[:, 2]
}, test_params.shape[0])
assert_allclose(opencl_results, python_results, atol=1e-7, rtol=1e-7)
def inverseIG(u, alpha, q):
res = norm.ppf(invgamma.cdf(u.local_data, alpha, scale=q))
return Field.from_local_data(u.domain, res)
def cdf(self, x: float) -> float:
# TODO: where is self.beta
return float(invgamma.cdf(x, self.alpha))
# Display the probability density function (``pdf``): x = np.linspace(invgamma.ppf(0.01, a), invgamma.ppf(0.99, a), 100) ax.plot(x, invgamma.pdf(x, a), 'r-', lw=5, alpha=0.6, label='invgamma pdf') # Alternatively, the distribution object can be called (as a function) # to fix the shape, location and scale parameters. This returns a "frozen" # RV object holding the given parameters fixed. # Freeze the distribution and display the frozen ``pdf``: rv = invgamma(a) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') # Check accuracy of ``cdf`` and ``ppf``: vals = invgamma.ppf([0.001, 0.5, 0.999], a) np.allclose([0.001, 0.5, 0.999], invgamma.cdf(vals, a)) # True # Generate random numbers: r = invgamma.rvs(a, size=1000) # And compare the histogram: ax.hist(r, density=True, histtype='stepfilled', alpha=0.2) ax.legend(loc='best', frameon=False) plt.show()
def fit_and_plot(args):
"""
Read in the TGAS parallax error data (or an already calculated KDE of the parallax error
distribution) and the "fit" the KDE by with a Gamma distribution. Fit quality is simply judged by
eye!
Parameters
----------
args - command line arguments.
"""
offset, alpha, beta = args['distroParams']
tgasPlxErrPdfFile = 'TGAS-parallax-errors-pdf.csv'
if path.isfile(tgasPlxErrPdfFile):
errpdf = np.genfromtxt(tgasPlxErrPdfFile,
comments='#',
skip_header=1,
delimiter=',',
names=['err', 'logdens'],
dtype=None)
x = errpdf['err'][:, np.newaxis]
logdens = errpdf['logdens']
else:
tgasData = fits.open('TGAS-allPlxErrorsVsGmag.fits')[1].data
eplx = tgasData['parallax_error'][:, np.newaxis]
kde = KernelDensity(kernel='epanechnikov', bandwidth=0.008).fit(eplx)
x = np.linspace(0, 1, 1001)[:, np.newaxis]
logdens = kde.score_samples(x)
f = open('TGAS-errors-kde.csv', mode='w')
f.write('#err,logdens\n')
for xx, yy in zip(x[:, 0], logdens):
f.write("{0},{1}\n".format(xx, yy))
f.close()
xx = np.linspace(0.004, 1, 1000)
ig = invgamma.pdf(xx, alpha, loc=offset, scale=beta)
norm = invgamma.cdf(1.0, alpha, loc=offset, scale=beta)
ig = ig / norm
useagab(usetex=False, fontfam='sans', sroncolours=False)
fig = plt.figure(figsize=(12, 10))
ax = fig.add_subplot(111)
apply_tufte(ax)
ax.plot(x[:, 0],
np.exp(logdens),
label='KDE of $\\sigma_\\varpi$ distribution',
lw=3)
ax.plot(
xx,
ig,
label="Fit with InvGamma($x-{0:.3f}|\\alpha={1:.3f}$, $\\beta={2:.3f})$"
.format(offset, alpha, beta))
ax.set_xlabel('$\\varpi$ [mas]')
ax.set_ylabel('pdf')
ax.legend(loc='upper right', fontsize=12)
ax.set_title('Parallax error distribution for TGAS', fontsize=14)
basename = 'FitOfParallaxErrorDistribution'
if args['pdfOutput']:
plt.savefig(basename + '.pdf')
elif args['pngOutput']:
plt.savefig(basename + '.png')
else:
plt.show()
def cdf(self, X):
cdf = invgamma.cdf(X, self.alpha, scale=self.beta)
return cdf
'gamma(1.25, 1) pdf', 'gamma_pdf', (1.25, 1)),
TestParams(-1, 10, 1000, lambda x: gamma.pdf(x, 1.25, 0, 1 / 2),
'gamma(1.25, 2) pdf', 'gamma_pdf', (1.25, 2)),
TestParams(-1, 10, 1000, lambda x: gamma.cdf(x, 1), 'gamma(1, 1) cdf',
'gamma_cdf', (1, 1)),
TestParams(-1, 10, 1000, lambda x: gamma.cdf(x, 1.25, 0, 1),
'gamma(1.25, 1) cdf', 'gamma_cdf', (1.25, 1)),
TestParams(-1, 10, 1000, lambda x: gamma.cdf(x, 1.25, 0, 1 / 2),
'gamma(1.25, 2) cdf', 'gamma_cdf', (1.25, 2)),
TestParams(-1, 10, 1000, lambda x: invgamma.pdf(x, 1),
'inverse_gamma(1, 1) pdf', 'inverse_gamma_pdf', (1, 1)),
TestParams(-1, 10, 1000, lambda x: invgamma.pdf(x, 1.25, 0, 1),
'inverse_gamma(1.25, 1) pdf', 'inverse_gamma_pdf', (1.25, 1)),
TestParams(-1, 10, 1000, lambda x: invgamma.pdf(x, 1.25, 0, 2),
'inverse_gamma(1.25, 2) pdf', 'inverse_gamma_pdf', (1.25, 2)),
TestParams(-1, 10, 1000, lambda x: invgamma.cdf(x, 1),
'inverse_gamma(1, 1) cdf', 'inverse_gamma_cdf', (1, 1)),
TestParams(-1, 10, 1000, lambda x: invgamma.cdf(x, 1.25, 0, 1),
'inverse_gamma(1.25, 1) cdf', 'inverse_gamma_cdf', (1.25, 1)),
TestParams(-1, 10, 1000, lambda x: invgamma.cdf(x, 1.25, 0, 1 / 2),
'inverse_gamma(1.25, 2) cdf', 'inverse_gamma_cdf', (1.25, 2)),
]
head = r"""#include <stdio.h>
#include <stdlib.h>
#include "../src/distrs.h"
#define EPS 1e-5
int test_fn(DTYPE *x, DTYPE *y, size_t n, DTYPE (*f)(DTYPE x), char *name)
