def test_calibrate_lrt_works_with_mvn(self):
m = 1
nfreq = 10000
freq = np.linspace(1, 10, nfreq)
rng = np.random.RandomState(100)
noise = rng.exponential(size=nfreq)
model = models.Const1D()
model.amplitude = 2.0
p = model(freq)
power = noise * p
ps = Powerspectrum()
ps.freq = freq
ps.power = power
ps.m = m
ps.df = freq[1] - freq[0]
ps.norm = "leahy"
loglike = PSDLogLikelihood(ps.freq, ps.power, model, m=1)
model2 = models.PowerLaw1D() + models.Const1D()
model2.x_0_0.fixed = True
loglike2 = PSDLogLikelihood(ps.freq, ps.power, model2, 1)
pe = PSDParEst(ps)
pval = pe.calibrate_lrt(loglike, [2.0], loglike2,
[2.0, 1.0, 2.0], sample=None,
max_post=False, nsim=10,
seed=100)
assert pval > 0.001
def test_simulate_highest_outlier_works(self):
m = 1
nfreq = 100
seed = 100
freq = np.linspace(1, 10, nfreq)
rng = np.random.RandomState(seed)
noise = rng.exponential(size=nfreq)
model = models.Const1D()
model.amplitude = 2.0
p = model(freq)
power = noise * p
ps = Powerspectrum()
ps.freq = freq
ps.power = power
ps.m = m
ps.df = freq[1] - freq[0]
ps.norm = "leahy"
nsim = 5
loglike = PSDLogLikelihood(ps.freq, ps.power, model, m=1)
s_all = np.atleast_2d(np.ones(nsim) * 2.0).T
pe = PSDParEst(ps)
maxpow_sim = pe.simulate_highest_outlier(s_all, loglike, [2.0],
max_post=False, seed=seed)
assert maxpow_sim.shape[0] == nsim
assert np.all(maxpow_sim > 9.00) and np.all(maxpow_sim < 31.0)
def test_generate_data_produces_correct_distribution(self):
model = models.Const1D()
model.amplitude = 2.0
p = model(self.ps.freq)
seed = 100
rng = np.random.RandomState(seed)
noise = rng.exponential(size=len(p))
power = noise*p
ps = Powerspectrum()
ps.freq = self.ps.freq
ps.power = power
ps.m = 1
ps.df = self.ps.freq[1]-self.ps.freq[0]
ps.norm = "leahy"
lpost = PSDLogLikelihood(ps.freq, ps.power, model, m=1)
pe = PSDParEst(ps)
rng2 = np.random.RandomState(seed)
sim_data = pe._generate_data(lpost, [2.0], rng2)
assert np.allclose(ps.power, sim_data.power)
def dask_fit_fourier_pl_c(power_spectrum):
"""
Fits the power law + constant observation model
Parameters
----------
power_spectrum :
Return
------
"""
# Make the random data into a Powerspectrum object
ps = Powerspectrum()
ps.freq = power_spectrum[0]
ps.power = power_spectrum[1]
ps.df = ps.freq[1] - ps.freq[0]
ps.m = 1
# Define the log-likelihood of the data given the model
loglike = PSDLogLikelihood(ps.freq, ps.power, observation_model, m=ps.m)
# Parameter estimation object
parameter_estimate = PSDParEst(ps, fitmethod="L-BFGS-B", max_post=False)
# Estimate the starting parameters
ipe = InitialParameterEstimatePlC(ps.freq, ps.power)
return parameter_estimate.fit(loglike, [ipe.amplitude, ipe.index, ipe.background],
scipy_optimize_options=scipy_optimize_options)
def test_object_works_with_loglikelihood_object(self):
llike = PSDLogLikelihood(self.ps.freq, self.ps.power,
self.model, m=self.ps.m)
pe = ParameterEstimation()
res = pe.fit(llike, [2.0])
pass
def test_calibrate_highest_outlier_works_with_mvn(self):
m = 1
nfreq = 10000
seed = 100
freq = np.linspace(1, 10, nfreq)
rng = np.random.RandomState(seed)
noise = rng.exponential(size=nfreq)
model = models.Const1D()
model.amplitude = 2.0
p = model(freq)
power = noise * p
ps = Powerspectrum()
ps.freq = freq
ps.power = power
ps.m = m
ps.df = freq[1] - freq[0]
ps.norm = "leahy"
nsim = 10
loglike = PSDLogLikelihood(ps.freq, ps.power, model, m=1)
pe = PSDParEst(ps)
pval = pe.calibrate_highest_outlier(loglike, [2.0], sample=None,
max_post=False, seed=seed,
nsim=nsim)
assert pval > 0.001
def test_object_works_with_loglikelihood_object(self):
llike = PSDLogLikelihood(self.ps.freq, self.ps.power,
self.model, m=self.ps.m)
pe = ParameterEstimation()
res = pe.fit(llike, [2.0])
assert isinstance(res,
OptimizationResults), "res must be of " \
"type OptimizationResults"
def fit_data_with_gaussian(x_values, y_values, amplitude=1., mean=0, stddev=1.):
g_init = Gaussian1D(amplitude, mean, stddev)
lpost = PSDLogLikelihood(x_values, y_values, g_init)
parest = ParameterEstimation()
res = parest.fit(lpost, [amplitude, mean, stddev], neg=True)
opt_amplitude = res.p_opt[0]
opt_mean = res.p_opt[1]
opt_stddev = res.p_opt[2]
return opt_amplitude, opt_mean, opt_stddev
def test_compute_lrt_works(self):
m = 1
nfreq = 100000
freq = np.linspace(1, 10, nfreq)
rng = np.random.RandomState(100)
noise = rng.exponential(size=nfreq)
model = models.Const1D()
model.amplitude = 2.0
p = model(freq)
power = noise * p
ps = Powerspectrum()
ps.freq = freq
ps.power = power
ps.m = m
ps.df = freq[1] - freq[0]
ps.norm = "leahy"
loglike = PSDLogLikelihood(ps.freq, ps.power, model, m=1)
loglike = PSDLogLikelihood(ps.freq, ps.power, model, m=1)
s_all = np.atleast_2d(np.ones(10) * 2.0).T
model2 = models.PowerLaw1D() + models.Const1D()
model2.x_0_0.fixed = True
loglike2 = PSDLogLikelihood(ps.freq, ps.power, model2, 1)
pe = PSDParEst(ps)
lrt_obs, res1, res2 = pe.compute_lrt(loglike, [2.0],
loglike2, [2.0, 1.0, 2.0],
neg=True)
lrt_sim = pe.simulate_lrts(s_all,
loglike, [2.0],
loglike2, [2.0, 1.0, 2.0],
max_post=False,
seed=100)
assert (lrt_obs > 0.4) and (lrt_obs < 0.6)
assert np.all(lrt_sim < 10.0) and np.all(lrt_sim > 0.01)
def test_calibrate_lrt_works_as_expected(self):
m = 1
df = 0.01
freq = np.arange(df, 5 + df, df)
nfreq = freq.size
rng = np.random.RandomState(100)
noise = rng.exponential(size=nfreq)
model = models.Const1D()
model.amplitude = 2.0
p = model(freq)
power = noise * p
ps = Powerspectrum()
ps.freq = freq
ps.power = power
ps.m = m
ps.df = df
ps.norm = "leahy"
loglike = PSDLogLikelihood(ps.freq, ps.power, model, m=1)
s_all = np.atleast_2d(np.ones(10) * 2.0).T
model2 = models.PowerLaw1D() + models.Const1D()
model2.x_0_0.fixed = True
loglike2 = PSDLogLikelihood(ps.freq, ps.power, model2, m=1)
pe = PSDParEst(ps)
pval = pe.calibrate_lrt(loglike, [2.0], loglike2,
[2.0, 1.0, 2.0], sample=s_all,
max_post=False, nsim=5,
seed=100)
assert pval > 0.001
def fit_data_with_lorentz_and_const(x_values, y_values):
amplitude = 5.
x_0 = 1
fwhm = 0.5
const = 5.
g_init = Lorentz1D(amplitude, x_0, fwhm)
g_init += Const1D(const)
lpost = PSDLogLikelihood(x_values, y_values, g_init)
parest = ParameterEstimation()
res = parest.fit(lpost, [amplitude, x_0, fwhm, const], neg=True)
opt_amplitude = res.p_opt[0]
opt_x_0 = res.p_opt[1]
opt_fwhm = res.p_opt[2]
opt_const = res.p_opt[3]
return opt_amplitude, opt_x_0, opt_fwhm, opt_const
def test_fitting_with_ties_and_bounds(self, capsys):
double_f = lambda model: model.x_0_0 * 2
model = self.model.copy()
model = self.model + models.Lorentz1D(amplitude=model.amplitude_0,
x_0=model.x_0_0 * 2,
fwhm=model.fwhm_0)
model.x_0_0 = self.model.x_0_0
model.amplitude_0 = self.model.amplitude_0
model.amplitude_1 = self.model.amplitude_1
model.fwhm_0 = self.model.fwhm_0
model.x_0_2.tied = double_f
model.fwhm_0.bounds = [0, 10]
model.amplitude_0.fixed = True
p = model(self.ps.freq)
noise = np.random.exponential(size=len(p))
power = noise * p
ps = Powerspectrum()
ps.freq = self.ps.freq
ps.power = power
ps.m = self.ps.m
ps.df = self.ps.df
ps.norm = "leahy"
pe = PSDParEst(ps)
llike = PSDLogLikelihood(ps.freq, ps.power, model)
true_pars = [
self.amplitude_0, self.x_0_0, self.fwhm_0, self.amplitude_1,
model.amplitude_2.value, model.x_0_2.value, model.fwhm_2.value
]
res = pe.fit(llike, true_pars)
res.print_summary(llike)
out, err = capsys.readouterr()
assert "100.00000 (Fixed)" in out
pattern = \
re.compile(r"5\) Parameter x_0_2\s+: [0-9]\.[0-9]{5}\s+\(Tied\)")
assert pattern.search(out)
compare_pars = [
self.x_0_0, self.fwhm_0, self.amplitude_1, model.amplitude_2.value,
model.fwhm_2.value
]
assert np.all(np.isclose(compare_pars, res.p_opt, rtol=0.5))
def test_fitting_with_ties_and_bounds(self, capsys, rebin):
double_f = lambda model : model.x_0_0 * 2
model = self.model.copy()
model += models.Lorentz1D(amplitude=model.amplitude_0,
x_0 = model.x_0_0 * 2,
fwhm = model.fwhm_0)
model.x_0_0 = self.model.x_0_0
model.amplitude_0 = self.model.amplitude_0
model.amplitude_1 = self.model.amplitude_1
model.fwhm_0 = self.model.fwhm_0
model.x_0_2.tied = double_f
model.fwhm_0.bounds = [0, 10]
model.amplitude_0.fixed = True
p = model(self.ps.freq)
noise = np.random.exponential(size=len(p))
power = noise*p
ps = Powerspectrum()
ps.freq = self.ps.freq
ps.power = power
ps.m = self.ps.m
ps.df = self.ps.df
ps.norm = "leahy"
if rebin != 0:
ps = ps.rebin_log(rebin)
pe = PSDParEst(ps, fitmethod="TNC")
llike = PSDLogLikelihood(ps.freq, ps.power, model)
true_pars = [self.x_0_0, self.fwhm_0,
self.amplitude_1,
model.amplitude_2.value,
model.fwhm_2.value]
res = pe.fit(llike, true_pars, neg=True)
compare_pars = [self.x_0_0, self.fwhm_0,
self.amplitude_1,
model.amplitude_2.value,
model.fwhm_2.value]
assert np.allclose(compare_pars, res.p_opt, rtol=0.5)
def fit_powerspectrum(ps,
model,
starting_pars,
max_post=False,
priors=None,
fitmethod="L-BFGS-B"):
"""
Fit a number of Lorentzians to a power spectrum, possibly including white
noise. Each Lorentzian has three parameters (amplitude, centroid position,
full-width at half maximum), plus one extra parameter if the white noise
level should be fit as well. Priors for each parameter can be included in
case `max_post = True`, in which case the function will attempt a
Maximum-A-Posteriori fit. Priors must be specified as a dictionary with one
entry for each parameter.
The parameter names are `(amplitude_i, x_0_i, fwhm_i)` for each `i` out of
a total of `N` Lorentzians. The white noise level has a parameter
`amplitude_(N+1)`. For example, a model with two Lorentzians and a
white noise level would have parameters:
[amplitude_0, x_0_0, fwhm_0, amplitude_1, x_0_1, fwhm_1, amplitude_2].
Parameters
----------
ps : Powerspectrum
A Powerspectrum object with the data to be fit
model: astropy.modeling.models class instance
The parametric model supposed to represent the data. For details
see the astropy.modeling documentation
starting_pars : iterable
The list of starting guesses for the optimizer. See explanation above
for ordering of parameters in this list.
fit_whitenoise : bool, optional, default True
If True, the code will attempt to fit a white noise level along with
the Lorentzians. Be sure to include a starting parameter for the
optimizer in `starting_pars`!
max_post : bool, optional, default False
If True, perform a Maximum-A-Posteriori fit of the data rather than a
Maximum Likelihood fit. Note that this requires priors to be specified,
otherwise this will cause an exception!
priors : {dict | None}, optional, default None
Dictionary with priors for the MAP fit. This should be of the form
{"parameter name": probability distribution, ...}
fitmethod : string, optional, default "L-BFGS-B"
Specifies an optimization algorithm to use. Supply any valid option for
`scipy.optimize.minimize`.
Returns
-------
parest : PSDParEst object
A PSDParEst object for further analysis
res : OptimizationResults object
The OptimizationResults object storing useful results and quantities
relating to the fit
Example
-------
We start by making an example power spectrum with three Lorentzians
>>> m = 1
>>> nfreq = 100000
>>> freq = np.linspace(1, 1000, nfreq)
>>> np.random.seed(100) # set the seed for the random number generator
>>> noise = np.random.exponential(size=nfreq)
>>> model = models.PowerLaw1D() + models.Const1D()
>>> model.x_0_0.fixed = True
>>> alpha_0 = 2.0
>>> amplitude_0 = 100.0
>>> amplitude_1 = 2.0
>>> model.alpha_0 = alpha_0
>>> model.amplitude_0 = amplitude_0
>>> model.amplitude_1 = amplitude_1
>>> p = model(freq)
>>> power = noise * p
>>> ps = Powerspectrum()
>>> ps.freq = freq
>>> ps.power = power
>>> ps.m = m
>>> ps.df = freq[1] - freq[0]
>>> ps.norm = "leahy"
Now we have to guess starting parameters. For each Lorentzian, we have
amplitude, centroid position and fwhm, and this pattern repeats for each
Lorentzian in the fit. The white noise level is the last parameter.
>>> t0 = [80, 1.5, 2.5]
Let's also make a model to test:
>>> model_to_test = models.PowerLaw1D() + models.Const1D()
>>> model_to_test.x_0_0.fixed = True
We're ready for doing the fit:
>>> parest, res = fit_powerspectrum(ps, model_to_test, t0)
`res` contains a whole array of useful information about the fit, for
example the parameters at the optimum:
>>> p_opt = res.p_opt
"""
if priors:
lpost = PSDPosterior(ps, model, priors=priors)
else:
lpost = PSDLogLikelihood(ps.freq, ps.power, model, m=ps.m)
parest = PSDParEst(ps, fitmethod=fitmethod, max_post=max_post)
res = parest.fit(lpost, starting_pars, neg=True)
return parest, res
# parameters for fake data.
alpha = 2.0
amplitude = 5.0
white_noise = 2.0
freq = np.linspace(0.01, 10.0, int(10.0 / 0.01))
from astropy.modeling.fitting import _fitter_to_model_params
_fitter_to_model_params(plc, [amplitude, alpha, white_noise])
psd_shape = plc(freq)
powers = psd_shape * np.random.chisquare(2, size=psd_shape.shape[0]) / 2.0
ps = Powerspectrum()
ps.freq = freq
ps.power = powers
ps.df = ps.freq[1] - ps.freq[0]
ps.m = 1
loglike = PSDLogLikelihood(ps.freq, ps.power, plc, m=ps.m)
test_pars = [1, 5, 100]
parest = PSDParEst(ps, fitmethod="L-BFGS-B", max_post=True)
# flat prior for the power law index
p_alpha = lambda alpha: ((-1. <= alpha) & (alpha <= 5.))
# flat prior for the power law amplitude
p_amplitude = lambda amplitude: ((0.01 <= amplitude) & (amplitude <= 10.0))
# normal prior for the white noise parameter
p_whitenoise = lambda white_noise: scipy.stats.norm(2.0, 0.1).pdf(white_noise)
priors = {}
priors["alpha_0"] = p_alpha
priors["amplitude_0"] = p_amplitude
psd_shape = observation_model(freq)
# Now randomize the true data
powers = psd_shape*np.random.chisquare(2, size=psd_shape.shape[0])/2.0
# Make the random data into a Powerspectrum object
from stingray import Powerspectrum
ps = Powerspectrum()
ps.freq = freq
ps.power = powers
ps.df = ps.freq[1] - ps.freq[0]
ps.m = 1
# Define the log-likelihood of the data given the model
from stingray.modeling import PSDLogLikelihood
loglike = PSDLogLikelihood(ps.freq, ps.power, observation_model, m=ps.m)
loglike(true_parameters)
# Parameter estimation
from stingray.modeling import PSDParEst
parest = PSDParEst(ps, fitmethod="L-BFGS-B", max_post=False)
starting_pars = true_parameters
res = parest.fit(loglike, starting_pars)
print(true_parameters)
print(res.p_opt)
print(res.err)
print("AIC: " + str(res.aic))
print("BIC: " + str(res.bic))
res.print_summary(loglike)
