def predict(self, xpred):
""" Realize the GP using the current values of the hyperparameters at values x=xpred.
Used for making GP plots. Wrapper around `celerite.GP.predict()`.
Args:
xpred (np.array): numpy array of x values for realizing the GP
Returns:
tuple: tuple containing:
np.array: numpy array of predictive means \n
np.array: numpy array of predictive standard deviations
"""
self.update_kernel_params()
# build celerite kernel with current values of hparams
kernel = celerite.terms.JitterTerm(
log_sigma=np.log(self.params[self.jit_param].value))
for i in np.arange(self.kernel.num_terms):
kernel = kernel + celerite.terms.ComplexTerm(
log_a=self.kernel.hparams[i, 0],
log_b=self.kernel.hparams[i, 1],
log_c=self.kernel.hparams[i, 2],
log_d=self.kernel.hparams[i, 3])
gp = celerite.GP(kernel)
gp.compute(self.x, self.yerr)
mu, var = gp.predict(self.y - self.params[self.gamma_param].value,
xpred,
return_var=True)
stdev = np.sqrt(var)
return mu, stdev
def GPfit(times, fluxes, errors, nonmask_idxs):
#t, y, yerr = times[np.isfinite(times)], fluxes[np.isfinite(fluxes)], errors[np.isfinite(errors)]
t, y, yerr = times[nonmask_idxs], fluxes[nonmask_idxs], errors[nonmask_idxs]
t, y, yerr = t[np.isfinite(t)], y[np.isfinite(y)], yerr[np.isfinite(yerr)]
Q = 1.0
w0 = 3.0
S0 = np.var(y) / (w0 * Q)
bounds = dict(log_S0=(-15, 15), log_Q=(-15, 15), log_omega0=(-15, 15))
kernel = terms.SHOTerm(log_S0=np.log(S0), log_Q=np.log(Q), log_omega0=np.log(w0), bounds=bounds)
#kernel += terms.SHOTerm(log_S0=np.log(S0), log_Q=np.log(Q), log_omega0=np.log(w0) bounds=bounds)
gp = celerite.GP(kernel, mean=np.mean(y))
gp.compute(t, yerr) # You always need to call compute once.
print("Initial log likelihood: {0}".format(gp.log_likelihood(y)))
initial_params = gp.get_parameter_vector()
bounds = gp.get_parameter_bounds()
r = minimize(neg_log_like, initial_params, method="L-BFGS-B", bounds=bounds, args=(y, gp))
gp.set_parameter_vector(r.x)
print(r)
#interpolated_times = np.linspace(np.nanmin(t), np.nanmax(t), 1000)
#interpolated_epochs = np.linspace(np.nanmin(t), np.nanmax(t), 1000)
#pred_mean, pred_var = gp.predict(y, interpolated_times, return_var=True)
pred_mean, pred_var = gp.predict(y, times, return_var=True)
pred_std = np.sqrt(pred_var)
return pred_mean, pred_std ### should be the same dimension as times.
def _lnlikelihood(self, theta, resample=False):
"""Log-likelihood function for the MCMC."""
# Unpack the free parameters.
vrot, noise, lnsigma, lnrho = theta
# Deproject the data and resample if requested.
if resample:
x, y = self.deprojected_spectrum(vrot)
else:
x, y = self.deprojected_spectra(vrot)
# Remove pesky points. This is not necessary but speeds it up
# and is useful to remove regions where there is a low
# sampling of points.
mask = np.logical_and(x > self.velax_mask[0], x < self.velax_mask[1])
x, y = x[mask], y[mask]
# Build the GP model.
k_noise = celerite.terms.JitterTerm(log_sigma=np.log(noise))
k_line = celerite.terms.Matern32Term(log_sigma=lnsigma, log_rho=lnrho)
gp = celerite.GP(k_noise + k_line, mean=np.nanmean(y), fit_mean=True)
# Calculate and the log-likelihood.
try:
gp.compute(x)
except Exception:
return -np.inf
ll = gp.log_likelihood(y, quiet=True)
return ll if np.isfinite(ll) else -np.inf
def baseline_hybrid_GP(*args):
x, y, yerr_w, xx, params, inst, key = args
kernel = terms.Matern32Term(log_sigma=1., log_rho=1.)
gp = celerite.GP(kernel, mean=np.nanmean(y))
gp.compute(x, yerr=yerr_w) #constrain on x/y/yerr
def neg_log_like(gp_params, y, gp):
gp.set_parameter_vector(gp_params)
return -gp.log_likelihood(y)
def grad_neg_log_like(gp_params, y, gp):
gp.set_parameter_vector(gp_params)
return -gp.grad_log_likelihood(y)[1]
initial_params = gp.get_parameter_vector()
bounds = gp.get_parameter_bounds()
soln = minimize(neg_log_like,
initial_params,
jac=grad_neg_log_like,
method="L-BFGS-B",
bounds=bounds,
args=(y, gp))
gp.set_parameter_vector(soln.x)
baseline = gp.predict(y, xx)[0] #constrain on x/y/yerr, evaluate on xx (!)
return baseline
def gp_sho(idx, data, resid, params):
logQ, logw, logS, log_jit = params
err = data.err / data.flux
#logerr = np.log(np.median(err))
logerr = log_jit
#print "logQ, logw, logS", logQ, logw, logS
mean_resid = np.mean(resid)
#resid -= mean_resid
t = data.t_vis[idx]
kernel = (terms.SHOTerm(log_S0=logS, log_Q=logQ, log_omega0=logw) +
terms.JitterTerm(log_sigma=logerr))
gp = celerite.GP(kernel, fit_mean=True)
gp.compute(t, err, check_sorted=False) #t must be ascending!
mu = gp.predict(resid, t, return_cov=False)
gp_lnlike = gp.log_likelihood(resid)
"""plt.errorbar(t, resid, err, fmt = '.k')
plt.plot(t, mu)
x = np.linspace(np.min(t), np.max(t), 1000)
pred_mean, pred_var = gp.predict(resid, x, return_var = True)
pred_std = np.sqrt(pred_var)
plt.fill_between(x, pred_mean+pred_std, pred_mean-pred_std, color='blue', alpha=0.3)
plt.show()"""
#np.save("resid", [t, resid, err])
#return [1.0 + np.array(mu) + mean_resid, gp_lnlike]
return [1.0 + np.array(mu), gp_lnlike]
def compute_gp(self, params, white=0, return_kernel=True):
kernel = self._compute_modes_kernel(params, white=white)
gp = celerite.GP(kernel)
if return_kernel:
return kernel, gp
return gp
def sim_lc(t, ppm, per_aper_ratio, periodic_freq, aperiodic_freq, A, planet=False, mean=1.0, planet_params=[0, 0, 0, 0, 0, 0, 0, 0, 0]):
if not planet:
mean=mean
if planet:
mean=transit_model(*planet_params, t)
aperiodic_sigma = 1/per_aper_ratio
periodic_sigma = 1
white_noise_sigma = ppm/1e6
# set up a gaussian process with two components and white noise
# non-periodic component
Q = 1. / np.sqrt(2.0) # related to the frequency of the variability
w0 = 2*np.pi*aperiodic_freq
S0 = (aperiodic_sigma**2.) / (w0 * Q)
bounds = dict(log_S0=(-15, 15), log_Q=(-15, 15), log_omega0=(-15, 15))
kernel = terms.SHOTerm(log_S0=np.log(S0), log_Q=np.log(Q), log_omega0=np.log(w0), bounds=bounds)
# periodic component
Q = 1.0
w0 = 2*np.pi*periodic_freq
S0 = (periodic_sigma**2.) / (w0 * Q)
kernel += terms.SHOTerm(log_S0=np.log(S0), log_Q=np.log(Q), log_omega0=np.log(w0), bounds=bounds)
# white noise
# kernel += terms.JitterTerm(log_sigma=np.log(white_noise_sigma), bounds=dict(log_sigma=(-15,15)))
gp = celerite.GP(kernel, mean=0, fit_mean=True, fit_white_noise=True)
gp.compute(t, white_noise_sigma)
return A * gp.sample() + mean + white_noise_sigma * np.random.randn(len(t))
def test_gp(celerite_kernel, seed=1234):
import celerite
import celerite.terms as cterms # NOQA
celerite_kernel = eval(celerite_kernel)
np.random.seed(seed)
x = np.sort(np.random.rand(100))
yerr = np.random.uniform(0.1, 0.5, len(x))
y = np.sin(x)
diag = yerr**2
celerite_gp = celerite.GP(celerite_kernel)
celerite_gp.compute(x, yerr)
celerite_loglike = celerite_gp.log_likelihood(y)
celerite_mu, celerite_cov = celerite_gp.predict(y)
_, celerite_var = celerite_gp.predict(y, return_cov=False, return_var=True)
kernel = _get_theano_kernel(celerite_kernel)
gp = GP(kernel, x, diag)
loglike = gp.log_likelihood(y).eval()
assert np.allclose(loglike, celerite_loglike)
mu = gp.predict()
_, var = gp.predict(return_var=True)
_, cov = gp.predict(return_cov=True)
assert np.allclose(mu.eval(), celerite_mu)
assert np.allclose(var.eval(), celerite_var)
assert np.allclose(cov.eval(), celerite_cov)
def generate_solar_fluxes(duration, cadence=60 * u.s, seed=None):
"""
Generate an array of fluxes with zero mean which mimic the power spectrum of
the SOHO/VIRGO SPM observations.
Parameters
----------
duration : `~astropy.units.Quantity`
Duration of simulated observations to generate.
cadence : `~astropy.units.Quantity`
Length of time between fluxes
seed : float, optional
Random seed.
Returns
-------
times : `~astropy.units.Quantity`
Array of times at cadence ``cadence`` of length ``duration/cadence``
fluxes : `~numpy.ndarray`
Array of fluxes at cadence ``cadence`` of length ``duration/cadence``
kernel : `~celerite.terms.TermSum`
Celerite kernel used to approximate the solar power spectrum.
"""
if seed is not None:
np.random.seed(seed)
_process_inputs(duration, cadence, 5777 * u.K)
##########################
# Assemble celerite kernel
##########################
parameter_vector = np.copy(PARAM_VECTOR)
nterms = len(parameter_vector) // 3
kernel = terms.SHOTerm(log_S0=0, log_omega0=0, log_Q=0)
for term in range(nterms - 1):
kernel += terms.SHOTerm(log_S0=0, log_omega0=0, log_Q=0)
kernel.set_parameter_vector(parameter_vector)
gp = celerite.GP(kernel)
times = np.arange(0, duration.to(u.s).value, cadence.to(u.s).value) * u.s
x = times.value
gp.compute(x, check_sorted=False)
###################################
# Get samples with the kernel's PSD
###################################
y = gp.sample()
# Remove a linear trend:
y -= np.polyval(np.polyfit(x - x.mean(), y, 1), x - x.mean())
return times, y, kernel
def test_consistency(oterm, mean, data):
x, diag, y, t = data
# Setup the original GP
original_gp = original_celerite.GP(oterm, mean=mean)
original_gp.compute(x, np.sqrt(diag))
# Setup the new GP
term = terms.OriginalCeleriteTerm(oterm)
gp = celerite2.GaussianProcess(term, mean=mean)
gp.compute(x, diag=diag)
# "log_likelihood" method
assert np.allclose(original_gp.log_likelihood(y), gp.log_likelihood(y))
# "predict" method
for args in [
dict(return_cov=False, return_var=False),
dict(return_cov=False, return_var=True),
dict(return_cov=True, return_var=False),
]:
assert all(
np.allclose(a, b) for a, b in zip(
original_gp.predict(y, **args),
gp.predict(y, **args),
))
assert all(
np.allclose(a, b) for a, b in zip(
original_gp.predict(y, t=t, **args),
gp.predict(y, t=t, **args),
))
# "sample" method
seed = 5938
np.random.seed(seed)
a = original_gp.sample()
np.random.seed(seed)
b = gp.sample()
assert np.allclose(a, b)
np.random.seed(seed)
a = original_gp.sample(size=10)
np.random.seed(seed)
b = gp.sample(size=10)
assert np.allclose(a, b)
# "sample_conditional" method, numerics make this one a little unstable;
# just check the shape
a = original_gp.sample_conditional(y, t=t)
b = gp.sample_conditional(y, t=t)
assert a.shape == b.shape
a = original_gp.sample_conditional(y, size=10)
b = gp.sample_conditional(y, size=10)
assert a.shape == b.shape
def test_consistency(oterm, mean, data):
x, diag, y, t = data
# Setup the original GP
original_gp = original_celerite.GP(oterm, mean=mean)
original_gp.compute(x, np.sqrt(diag))
# Setup the new GP
term = terms.OriginalCeleriteTerm(oterm)
gp = celerite2.GaussianProcess(term, mean=mean)
gp.compute(x, diag=diag)
# "log_likelihood" method
assert np.allclose(original_gp.log_likelihood(y), gp.log_likelihood(y))
# Apply inverse
assert np.allclose(
np.squeeze(original_gp.apply_inverse(y)), gp.apply_inverse(y)
)
conditional_t = gp.condition(y, t=t)
mu, cov = original_gp.predict(y, t=t, return_cov=True)
assert np.allclose(conditional_t.mean, mu)
assert np.allclose(conditional_t.variance, np.diag(cov))
assert np.allclose(conditional_t.covariance, cov)
conditional = gp.condition(y)
mu, cov = original_gp.predict(y, return_cov=True)
assert np.allclose(conditional.mean, mu)
assert np.allclose(conditional.variance, np.diag(cov))
assert np.allclose(conditional.covariance, cov)
# "sample" method
seed = 5938
np.random.seed(seed)
a = original_gp.sample()
np.random.seed(seed)
b = gp.sample()
assert np.allclose(a, b)
np.random.seed(seed)
a = original_gp.sample(size=10)
np.random.seed(seed)
b = gp.sample(size=10)
assert np.allclose(a, b)
# "sample_conditional" method, numerics make this one a little unstable;
# just check the shape
a = original_gp.sample_conditional(y, t=t)
b = conditional_t.sample()
assert a.shape == b.shape
a = original_gp.sample_conditional(y, size=10)
b = conditional.sample(size=10)
assert a.shape == b.shape
def generate_solar_fluxes(size,
cadence=60 * u.s,
parameter_vector=parameter_vector):
"""
Generate an array of fluxes with zero mean which mimic the power spectrum of
the SOHO/VIRGO SPM observations.
Parameters
----------
size : int
Number of fluxes to generate. Note: Assumes ``size``>>500.
cadence : `~astropy.units.Quantity`
Length of time between fluxes
Returns
-------
y : `~numpy.ndarray`
Array of fluxes at cadence ``cadence`` of length ``size``.
"""
nterms = len(parameter_vector) // 3
kernel = terms.SHOTerm(log_S0=0, log_omega0=0, log_Q=0)
for term in range(nterms - 1):
kernel += terms.SHOTerm(log_S0=0, log_omega0=0, log_Q=0)
kernel.set_parameter_vector(parameter_vector)
gp = celerite.GP(kernel)
x = np.arange(0, size // 500, cadence.to(u.s).value)
gp.compute(x, check_sorted=False)
y = gp.sample(500)
y_concatenated = []
for i, yi in enumerate(y):
xi = np.arange(len(yi))
fit = np.polyval(np.polyfit(xi - xi.mean(), yi, 1), xi - xi.mean())
yi -= fit
if i == 0:
y_concatenated.append(yi)
else:
offset = yi[0] - y_concatenated[i - 1][-1]
y_concatenated.append(yi - offset)
y_concatenated = np.hstack(y_concatenated)
x_c = np.arange(len(y_concatenated))
y_concatenated -= np.polyval(
np.polyfit(x_c - x_c.mean(), y_concatenated, 1), x_c - x_c.mean())
return y_concatenated, kernel
def find_celerite_MAP(t,
y,
yerr,
sigma0=0.1,
tau0=100,
prior='None',
set_bounds=True,
sig_lims=[0.02, 0.7],
tau_lims=[1, 550],
verbose=False):
kernel = terms.RealTerm(log_a=2 * np.log(sigma0), log_c=np.log(1.0 / tau0))
gp = celerite.GP(kernel, mean=np.mean(y))
gp.compute(t, yerr)
# set initial params
initial_params = gp.get_parameter_vector()
if verbose:
print(initial_params)
# set boundaries
if set_bounds:
if verbose:
print('sig_lims:', sig_lims, 'tau_lims:', tau_lims)
tau_bounds, sigma_bounds = tau_lims, sig_lims
loga_bounds = (2 * np.log(min(sigma_bounds)),
2 * np.log(max(sigma_bounds)))
logc_bounds = (np.log(1 / max(tau_bounds)),
np.log(1 / min(tau_bounds)))
bounds = [loga_bounds, logc_bounds]
else: # - inf to + inf
bounds = gp.get_parameter_bounds()
if verbose:
print(bounds)
# wrap the neg_log_posterior for a chosen prior
def neg_log_like(params, y, gp):
return neg_log_posterior(params, y, gp, prior, 'celerite')
# find MAP solution
r = minimize(neg_log_like,
initial_params,
method="L-BFGS-B",
bounds=bounds,
args=(y, gp))
gp.set_parameter_vector(r.x)
res = gp.get_parameter_dict()
tau_fit = np.exp(-res['kernel:log_c'])
sigma_fit = np.exp(res['kernel:log_a'] / 2)
if verbose:
print('sigma_fit', sigma_fit, 'tau_fit', tau_fit)
return sigma_fit, tau_fit, gp
def _build_kernel(x, y, hyperparams):
"""Build the GP kernel. Returns None if gp.compute(x) fails."""
noise, lnsigma, lnrho = hyperparams
k_noise = celerite.terms.JitterTerm(log_sigma=np.log(noise))
k_line = celerite.terms.Matern32Term(log_sigma=lnsigma, log_rho=lnrho)
gp = celerite.GP(k_noise + k_line, mean=np.nanmean(y), fit_mean=True)
try:
gp.compute(x)
except Exception:
return None
return gp
def log_probability(self, params, band):
"""
Calculate log of posterior probability
Parameters
-----------
params : iterable
list of parameter values
band : string
SDSS/HiPERCAM band
"""
self.set_parameter_vector(params)
t, _, y, ye, _, _ = np.loadtxt(self.lightcurves[band]).T
# check model params are valid - checks against bounds
lp = self.log_prior()
if not np.isfinite(lp):
return -np.inf
# make a GP for this band, if it doesnt already exist
if not hasattr(self, 'gpdict'):
self.gpdict = dict()
# Oscillation params
pulsation_amp = self.pulse_amp * scale_pulsation(band, self.pulse_temp)
if band not in self.gpdict:
kernel = terms.SHOTerm(np.log(pulsation_amp), np.log(self.pulse_q),
np.log(self.pulse_omega))
gp = celerite.GP(kernel)
gp.compute(t, ye)
self.gpdict[band] = gp
else:
gp = self.gpdict[band]
gp.set_parameter_vector(
(np.log(pulsation_amp), np.log(self.pulse_q),
np.log(self.pulse_omega)))
gp.compute(t, ye)
# now add prior of Gaussian process
lp += gp.log_prior()
if not np.isfinite(lp):
return -np.inf
try:
ym = self.get_value(band)
except ValueError as err:
# invalid lcurve params
print('warning: model failed ', err)
return -np.inf
else:
return gp.log_likelihood(y - ym) + lp
def _lnlike(self, theta):
"""Log-likelihood with chi-squared likelihood function."""
models = self._calculatemodels(theta)
if not self.GP:
return self._chisquared(models)
lnx2 = 0.0
_, _, _, _, _, sigs, corrs = self._parse(theta)
for i in range(self.ntrans):
rho = M32(log_sigma=sigs[i], log_rho=corrs[i])
gp = celerite.GP(rho)
gp.compute(self.velaxs[i], self.rms[i])
lnx2 += gp.log_likelihood(models[i] - self.spectra[i])
return np.nansum(lnx2)
def _compute_gp(self):
# Compute the GP prior on the spectrum
if self._lnlam is None:
pass
if self.s_rho > 0.0:
kernel = celerite.terms.Matern32Term(np.log(self.s_sig),
np.log(self.s_rho))
gp = celerite.GP(kernel)
s_C = gp.get_matrix(self.lnlam_padded)
else:
s_C = np.eye(self.Kp) * self.s_sig**2
self._s_cho_C = cho_factor(s_C)
self._s_CInv = cho_solve(self._s_cho_C, np.eye(self.Kp))
def call_gp(params):
log_sigma, log_rho, log_error_scale = params
if GP_CODE=='celerite':
kernel = terms.Matern32Term(log_sigma=log_sigma, log_rho=log_rho)
gp = celerite.GP(kernel, mean=MEAN, fit_mean=False) #log_white_noise=np.log(yerr),
gp.compute(xx, yerr=yyerr/err_norm*np.exp(log_error_scale))
return gp
elif GP_CODE=='george':
kernel = np.exp(log_sigma) * kernels.Matern32Kernel(log_rho)
gp = george.GP(kernel, mean=MEAN, fit_mean=False) #log_white_noise=np.log(yerr),
gp.compute(xx, yerr=yyerr/err_norm*np.exp(log_error_scale))
return gp
else:
raise ValueError('A bad thing happened.')
def dSHO_maxlikelihood(lc, npeaks=1):
# Now let's do some of the GP stuff on this
# A non-periodic component
Q = 1.0 / np.sqrt(2.0)
w0 = 3.0
S0 = np.var(lc['NormFlux']) / (w0 * Q)
bounds = dict(log_S0=(-16, 16), log_Q=(-15, 15), log_omega0=(-15, 15))
kernel = terms.SHOTerm(log_S0=np.log(S0),
log_Q=np.log(Q),
log_omega0=np.log(w0),
bounds=bounds)
kernel.freeze_parameter(
"log_Q") # We don't want to fit for "Q" in this term
# A periodic component
for i in range(npeaks):
Q = 1.0
w0 = 3.0
S0 = np.var(lc['NormFlux']) / (w0 * Q)
kernel += terms.SHOTerm(log_S0=np.log(S0),
log_Q=np.log(Q),
log_omega0=np.log(w0),
bounds=bounds)
sigma = np.median(lc['NormErr'])
kernel += terms.JitterTerm(log_sigma=np.log(sigma))
gp = celerite.GP(kernel, mean=np.mean(lc['NormFlux']))
gp.compute(lc['Time'],
lc['NormErr']) # You always need to call compute once.
print("Initial log likelihood: {0}".format(
gp.log_likelihood(lc['NormFlux'])))
initial_params = gp.get_parameter_vector()
bounds = gp.get_parameter_bounds()
r = minimize(neg_log_like,
initial_params,
method="L-BFGS-B",
bounds=bounds,
args=(lc['NormFlux'], gp))
gp.set_parameter_vector(r.x)
print("Final log likelihood: {0}".format(gp.log_likelihood(
lc['NormFlux'])))
print("Maximum Likelihood Soln: {}".format(gp.get_parameter_dict()))
return gp
def wavelet(df):
# Gaussian Regression
result = pd.DataFrame()
mjds = df['mjd'].unique()
# Two observations per unique mjd value
t = np.arange(np.min(mjds), np.max(mjds), 0.5)
if (len(t) % 2) == 0:
t = np.insert(t, len(t), t[len(t) - 1] + 0.5)
for obj, agg_df in df.groupby('object_id'):
agg_df = agg_df.sort_values(by=['mjd'])
X = agg_df['mjd']
Y = agg_df['flux']
Yerr = agg_df['flux_err']
# Start by setting hyperparamaters to unit:
log_sigma = 0
log_rho = 0
kernel = celerite.terms.Matern32Term(log_sigma, log_rho)
# According to the paper from Narayan et al, 2018, we will use the Matern 3/2 Kernel.
gp = celerite.GP(kernel, mean=0.0)
gp.compute(X, Yerr)
# extract our initial guess at parameters
# from the celerite kernel and put it in a
# vector:
p0 = gp.get_parameter_vector()
# run optimization:
results = minimize(nll,
p0,
method='L-BFGS-B',
jac=grad_nll,
args=(Y, gp))
# set your initial guess parameters
# as the output from the scipy optimiser
# remember celerite keeps these in ln() form!
gp.set_parameter_vector(np.abs(results.x))
# Predict posterior mean and variance
mu, var = gp.predict(Y, t, return_var=True)
if (sum(np.isnan(mu)) != 0):
print('NANs exist in mu vector')
return [obj, results.x, mu]
# Wavelet Transform
# calculate wavelet transform using even numbered array
(cA2, cD2), (cA1, cD1) = pywt.swt(mu[1:, ], 'sym2', level=2)
obj_df = pd.DataFrame(list(cA2) + list(cA1) + list(cD2) +
list(cD1)).transpose()
obj_df['object_id'] = obj
result = pd.concat([result, obj_df])
result.reset_index(inplace=True)
result.drop("index", axis=1, inplace=True)
return result
def call_gp(params):
log_sigma, log_rho, log_error_scale, contrast, rv, fwhm = params
if GP_CODE=='celerite':
mean_model = Model_celerite(**dict(**dict(Contrast=contrast, RV=rv, FWHM=fwhm)))
kernel = terms.Matern32Term(log_sigma=log_sigma, log_rho=log_rho)
gp = celerite.GP(kernel, mean=mean_model)
gp.compute(xx, yerr=yyerr/err_norm*np.exp(log_error_scale))
return gp
elif GP_CODE=='george':
mean_model = Model_george(**dict(**dict(Contrast=contrast, RV=rv, FWHM=fwhm)))
kernel = np.exp(log_sigma) * kernels.Matern32Kernel(log_rho)
gp = george.GP(kernel, mean=mean_model)
gp.compute(xx, yerr=yyerr/err_norm*np.exp(log_error_scale))
return gp
else:
raise ValueError('gp_code must be "celerite" or "george".')
def Gaussian_process(self, kernel, plot=False):
mean = np.mean(self.flux)
gp = celerite.GP(kernel, mean=mean)
gp.compute(self.time, self.flux_err)
initial_params = gp.get_parameter_vector()
bounds = gp.get_parameter_bounds()
r = minimize(self.neg_log_like,
initial_params,
method="L-BFGS-B",
bounds=bounds,
args=(self.flux, gp))
gp.set_parameter_vector(r.x)
#pred_mean, pred_var = gp.predict(self.flux, self.time, return_var=True)
if (plot):
x = np.linspace(self.time[0], self.time[-1], 1000)
pred_mean, pred_var = gp.predict(self.flux, x, return_var=True)
pred_std = np.sqrt(pred_var)
color = "#ff7f0e"
plt.plot(x, pred_mean, label='GP model')
plt.errorbar(self.time,
self.flux,
fmt='o',
yerr=self.flux_err,
label=self.band + ' band',
markersize=2.5)
plt.fill_between(x,
pred_mean + pred_std,
pred_mean - pred_std,
color=color,
alpha=0.3,
edgecolor="none")
plt.legend()
plt.title('')
plt.xlabel('time (days)')
plt.ylabel('relative flux')
plt.ylim(min(pred_mean), max(pred_mean))
plt.xlim(0 - self.time[2], np.max(self.time) + 10)
plt.title(self.name + ' ' + self.band + ' data')
plt.show()
return self.calc_Rchi2_GP(gp)
def trappist1_variability(times):
alpha = 0.973460343001
log_a = np.log(np.exp(-26.88111923) * alpha)
log_c = -1.0890621571818671
# log_sigma = -5.6551601053314622
# kernel = (terms.JitterTerm(log_sigma=log_sigma) +
# terms.RealTerm(log_a=log_a, log_c=log_c))
kernel = terms.RealTerm(log_a=log_a, log_c=log_c)
gp = celerite.GP(kernel, mean=0, fit_white_noise=True, fit_mean=True)
gp.compute(times)
sample = gp.sample()
sample -= np.median(sample)
return sample + 1
def k296_variability(times):
alpha = 0.854646217641
log_a = np.log(np.exp(-13.821195) * alpha)
log_c = -1.0890621571818671
# log_sigma = -7.3950524
# kernel = (terms.JitterTerm(log_sigma=log_sigma) +
# terms.RealTerm(log_a=log_a, log_c=log_c))
kernel = terms.RealTerm(log_a=log_a, log_c=log_c)
gp = celerite.GP(kernel, mean=0, fit_white_noise=True, fit_mean=True)
gp.compute(times)
sample = gp.sample()
sample -= np.median(sample)
return sample + 1
def compute_background(t, amp, freqs, white=0):
"""
Compute granulation background using Gaussian process
"""
if white == 0:
white = 1e-6
kernel = terms.JitterTerm(log_sigma=np.log(white))
S0 = calculateS0(amp, freqs)
print(f"S0: {S0}")
for i in range(len(amp)):
kernel += terms.SHOTerm(log_S0=np.log(S0[i]),
log_Q=np.log(1 / np.sqrt(2)),
log_omega0=np.log(2 * np.pi * freqs[i]))
gp = celerite.GP(kernel)
return kernel, gp, S0
def pred_lc(t, y, yerr, params, p, t_pred, return_var=True):
"""
Generate predicted values at particular time stamps given the initial
time series and a best-fit model.
Args:
t (array(float)): Time stamps of the initial time series.
y (array(float)): y values (i.e., flux) of the initial time series.
yerr (array(float)): Measurement errors of the initial time series.
params (array(float)): Best-fit CARMA parameters
p (int): The AR order (p) of the given best-fit model.
t_pred (array(float)): Time stamps to generate predicted time series.
return_var (bool, optional): Whether to return uncertainties in the mean
prediction. Defaults to True.
Returns:
(array(float), array(float), array(float)): t_pred, mean prediction at t_pred
and uncertainties (variance) of the mean prediction.
"""
assert p >= len(
params) - p, "The dimension of AR must be greater than that of MA"
# get ar, ma
ar = params[:p]
ma = params[p:]
# reposition lc
y_aln = y - np.median(y)
# init kernel, gp and compute matrix
kernel = CARMA_term(np.log(ar), np.log(ma))
gp = celerite.GP(kernel, mean=0)
gp.compute(t, yerr)
try:
mu, var = gp.predict(y_aln, t_pred, return_var=return_var)
except FloatingPointError as e:
print(e)
print("No (super small) variance will be returned")
return_var = False
mu, var = gp.predict(y_aln, t_pred, return_var=return_var)
return t_pred, mu + np.median(y), var
def autocorr_ml(y, thin=1, c=5.0):
"""Compute the autocorrelation using a GP model."""
# Compute the initial estimate of tau using the standard method
init = autocorr_new(y, c=c)
z = y[:, ::thin]
N = z.shape[1]
# Build the GP model
tau = max(1.0, init / thin)
bounds = [(-5.0, 5.0), (-np.log(N), 0.0)]
kernel = terms.RealTerm(
np.log(0.9 * np.var(z)),
-np.log(tau),
bounds=bounds
)
kernel += terms.RealTerm(
np.log(0.1 * np.var(z)),
-np.log(0.5 * tau),
bounds=bounds,
)
gp = celerite.GP(kernel, mean=np.mean(z))
gp.compute(np.arange(z.shape[1]))
# Define the objective
def nll(p):
# Update the GP model
gp.set_parameter_vector(p)
# Loop over the chains and compute the likelihoods
v, g = zip(*(gp.grad_log_likelihood(z0, quiet=True) for zo in z))
# Combine the datasets
return -np.sum(v), -np.sum(g, axis=0)
# Optimize the model
p0 = gp.get_parameter_vector()
bounds = gp.get_parameter_bounds()
soln = minimize(nll, p0, jac=True, bounds=bounds)
gp.set_parameter_vector(soln.x)
# Compute the maximum likelihood tau
a, c = kernel.coefficients[:2]
tau = thin * 2 * np.sum(a / c) / np.sum(a)
return tau
def predict(self,xpred):
""" Realize the GP using the current values of the hyperparameters at values x=xpred.
Used for making GP plots. Wrapper for `celerite.GP.predict()`.
Args:
xpred (np.array): numpy array of x values for realizing the GP
Returns:
tuple: tuple containing:
np.array: numpy array of predictive means \n
np.array: numpy array of predictive standard deviations
"""
self.update_kernel_params()
B = self.kernel.hparams['gp_B'].value
C = self.kernel.hparams['gp_C'].value
L = self.kernel.hparams['gp_L'].value
Prot = self.kernel.hparams['gp_Prot'].value
# build celerite kernel with current values of hparams
kernel = celerite.terms.JitterTerm(
log_sigma = np.log(self.params[self.jit_param].value)
)
kernel += celerite.terms.RealTerm(
log_a=np.log(B*(1+C)/(2+C)),
log_c=np.log(1/L)
)
kernel += celerite.terms.ComplexTerm(
log_a=np.log(B/(2+C)),
log_b=-np.inf,
log_c=np.log(1/L),
log_d=np.log(2*np.pi/Prot)
)
gp = celerite.GP(kernel)
gp.compute(self.x, self.yerr)
# mu, var = gp.predict(self.y-self.params[self.gamma_param].value, xpred, return_var=True)
mu, var = gp.predict(self._resids(), xpred, return_var=True)
stdev = np.sqrt(var)
return mu, stdev
def get_rotation_gp(t, y, yerr, period, min_period, max_period):
kernel = get_basic_kernel(t, y, yerr)
kernel += MixtureOfSHOsTerm(log_a=np.log(np.var(y)),
log_Q1=np.log(15),
mix_par=-1.0,
log_Q2=np.log(15),
log_P=np.log(period),
bounds=dict(
log_a=(-20.0, 10.0),
log_Q1=(-0.5 * np.log(2.0), 11.0),
mix_par=(-5.0, 5.0),
log_Q2=(-0.5 * np.log(2.0), 11.0),
log_P=(np.log(min_period),
np.log(max_period)),
))
gp = celerite.GP(kernel=kernel, mean=0.)
gp.compute(t)
return gp
def test_gp(celerite_kernel, seed=1234):
import celerite
import celerite.terms as cterms # NOQA
celerite_kernel = eval(celerite_kernel)
np.random.seed(seed)
x = np.sort(np.random.rand(100))
yerr = np.random.uniform(0.1, 0.5, len(x))
y = np.sin(x)
diag = yerr**2
celerite_gp = celerite.GP(celerite_kernel)
celerite_gp.compute(x, yerr)
celerite_loglike = celerite_gp.log_likelihood(y)
kernel = _get_theano_kernel(celerite_kernel)
gp = GP(kernel, x, diag)
loglike = gp.log_likelihood(y).eval()
assert np.allclose(loglike, celerite_loglike)