def spline_MC(xin, yin, xout, yin_err, n_trial):
# Change 'yin' and 'yin_err' inputs to 2D arrays and augment for number of trials
yin = np.atleast_2d(yin).T
yin_err = np.atleast_2d(yin_err)
y = np.tile(yin, (n_trial, 1))
yin_err = np.tile(yin_err, (n_trial, 1))
# Randomly assign y values to the input data within the error bounds
y = y + yin_err * np.random.standard_normal(y.shape)
# Interpolate onto locations specified in 'xout'
f = scipy.interpolate.interp1d(xin, y, kind='cubic')
y_fit = f(xout)
# Calculate mean and standard deviation at each xout
m = np.mean(y_fit, axis=0)
sigma = np.std(y_fit, axis=0)
# Plot mean and error in graph similar to the GPR curves using the 'errorfill' function defined here
#plt.errorbar(xout,m,sigma)
#errorfill(xout,m,sigma)
#plt.show()
gptools.univariate_envelope_plot(xout, m, sigma)
#plt.plot(xin,yin,'.')
#plt.xlabel('time [s]')
#plt.ylabel('Signal Amplitude [A.U.]')
return m, sigma
def profile_fitting(x,
y,
err_y=None,
optimize=True,
method='GPR',
kernel='SE',
num_dim=1,
debug_plots=True,
noiseLevel=2.,
**kwargs): #sigma_max=10.0, l_min = 0.005,
"""Interpolate profiles and uncertainties over a dense grid. Also return the maximum
value of the smoothed data.
This function can use Gaussian process regression or splines. When the former is adopted, both the
mean and the standard deviation of the updated profile are returned. Use the spline interpolation
only as a cross-check.
We allow the use of several GPR kernels so as to assess their performance in confidently
predicting profiles and assessing their uncertainty consistency.
Parameters
----------
x : array of float
Abscissa of data to be interpolated.
y : array of float
Data to be interpolated.
err_y : array of float, optional
Uncertainty in `y`. If absent, the data are interpolated.
optimize : bool, optional
Specify whether optimization over hyperparameters should occur or not. Default is True.
method : {'GPR', 'spline'}, optional
Method to use when interpolating. Default is 'GPR' (Gaussian process
regression). Can also use a cubic spline.
kernel : str, optional
Type of kernel to be used. At this stage, we create the kernel internally, but in the future
it would be better to do it externally and just give a gptools kernel object as an argument
More kernels should be added over time.
num_dim : int, optional
Number of dimensions of the input/output data. Default is 1
debug_plots : bool, optional
Set to True to plot the data, the smoothed curve (with uncertainty) and
the location of the peak value.
noiseLevel : float, optional
Initial guess for a noise multiplier. Default: 2
kwargs : dictionary
arguments to be passed on to set the hyper-prior bounds for the kernel of choice.
"""
# grid = scipy.linspace(max(0, x.min()), min(0.08, x.max()), 1000)
#grid = scipy.linspace(x.min(), x.max(), 1000)
grid = x
# Create empty object for results:
res = type('', (), {})()
if method == 'GPR':
# hp is the hyperprior. A product of kernels is a kernel, so the joint hyperprior is
# just the product of hyperpriors for each of the hyperparameters of the individual
# priors. gptools offers convenient functionalities to create joint hyperpriors.
# Define the kernel type amongst the implemented options.
if kernel == 'SE':
assert len(kwargs) == 4
hparams = hyperparams(**kwargs)
#hparams.set_kwargs(**kwargs)
# Defaults:
if not hasattr(hparams, 'sigma_mean'): hparams.sigma_mean = 2.0
if not hasattr(hparams, 'l_mean'): hparams.l_mean = 0.005
if not hasattr(hparams, 'sigma_sd'): hparams.sigma_sd = 10.0
if not hasattr(hparams, 'l_sd'): hparams.l_sd = 0.1
hprior = (gptools.GammaJointPriorAlt(
[hparams.sigma_mean, hparams.l_mean],
[hparams.sigma_sd, hparams.l_sd]))
k = gptools.SquaredExponentialKernel(
#= ====== =======================================================================
#0 sigma Amplitude of the covariance function
#1 l1 Small-X saturation value of the length scale.
#2 l2 Large-X saturation value of the length scale.
#= ====== =======================================================================
# param_bounds=[(0, sigma_max), (0, 2.0)],
hyperprior=hprior,
initial_params=[
10000.0, 400000.0
], # random, doesn't matter because we do random starts anyway
fixed_params=[False] * 2)
elif kernel == 'gibbs':
#if num_dim == 1: assert len(kwargs) == 10
hparams = hyperparams(**kwargs)
# Defaults:
if not hasattr(hparams, 'sigma_min'): hparams.sigma_min = 0.0
if not hasattr(hparams, 'sigma_max'): hparams.sigma_max = 10.0
if not hasattr(hparams, 'l1_mean'): hparams.l1_mean = 0.3
if not hasattr(hparams, 'l1_sd'): hparams.l1_sd = 0.3
if not hasattr(hparams, 'l2_mean'): hparams.l2_mean = 0.5
if not hasattr(hparams, 'l2_sd'): hparams.l2_sd = 0.25
if not hasattr(hparams, 'lw_mean'): hparams.lw_mean = 0.0
if not hasattr(hparams, 'lw_sd'): hparams.lw_sd = 0.3
if not hasattr(hparams, 'x0_mean'): hparams.x0_mean = 0.0
if not hasattr(hparams, 'x0_sd'): hparams.x0_sd = 0.3
hprior = (gptools.UniformJointPrior([
(hparams.sigma_min, hparams.sigma_max),
]) * gptools.GammaJointPriorAlt([
hparams.l1_mean, hparams.l2_mean, hparams.lw_mean,
hparams.x0_mean
], [hparams.l1_sd, hparams.l2_sd, hparams.lw_sd, hparams.x0_sd]))
k = gptools.GibbsKernel1dTanh(
#= ====== =======================================================================
#0 sigma Amplitude of the covariance function
#1 l1 Small-X saturation value of the length scale.
#2 l2 Large-X saturation value of the length scale.
#3 lw Length scale of the transition between the two length scales.
#4 x0 Location of the center of the transition between the two length scales.
#= ====== =======================================================================
initial_params=[
2.0, 0.5, 0.05, 0.1, 0.5
], # for random_starts!= 0, the initial state of the hyperparameters is not actually used.,
fixed_params=[False] * 5,
hyperprior=hprior,
)
elif kernel == 'matern52':
if num_dim == 1: assert len(kwargs) == 4
hparams = hyperparams(**kwargs)
# Defaults:
if not hasattr(hparams, 'sigma_mean'): hparams.sigma_mean = 2.0
if not hasattr(hparams, 'l_mean'): hparams.l_mean = 0.005
if not hasattr(hparams, 'sigma_sd'): hparams.sigma_sd = 10.0
if not hasattr(hparams, 'l_sd'): hparams.l_sd = 0.1
hprior = (gptools.GammaJointPriorAlt(
[hparams.sigma_mean, hparams.l_mean],
[hparams.sigma_sd, hparams.l_sd]))
k = gptools.Matern52Kernel( # this has 2 hyperparameters in 1D
#= ===== ===========================================
#0 sigma Prefactor to the kernel
#2 l1 Length scale for first dimension
#3 ... More length scales for more dimensions
#= ===== =======================================
hyperprior=hprior,
initial_params=[0.5, 0.5],
fixed_params=[False] * 2)
elif kernel == 'RQ': # rational quadratic
if num_dim == 1: assert len(kwargs) == 6
hparams = hyperparams(**kwargs)
# Defaults:
if not hasattr(hparams, 'sigma_mean'): hparams.sigma_mean = 2.0
if not hasattr(hparams, 'alpha_mean'): hparams.alpha_mean = 0.005
if not hasattr(hparams, 'l1_mean'): hparams.l1_mean = 0.005
if not hasattr(hparams, 'sigma_sd'): hparams.sigma_sd = 10.0
if not hasattr(hparams, 'alpha_sd'): hparams.alpha_sd = 0.1
if not hasattr(hparams, 'l1_sd'): hparams.l1_sd = 0.1
hprior = (gptools.GammaJointPriorAlt(
[hparams.sigma_mean, hparams.alpha_mean, hparams.l1_mean],
[hparams.sigma_sd, hparams.alpha_sd, hparams.l1_sd]))
k = gptools.RationalQuadraticKernel(
#= ===== ===========================================
#0 sigma Prefactor to the kernel
#1 alpha Order of kernel
#2 l1 Length scale for first dimension
#3 l2 Length scale for second dimension
#4 ... More length scales for more dimensions
#= ===== =======================================
hyperprior=hprior,
initial_params=[1.0, 0.5, 1.0],
fixed_params=[False] * 3)
elif isinstance(kernel, gptools.Kernel):
k = kernel
else:
ValueError('Only the SE kernel is currently defined! Break here.')
# Create additional noise to optimize over (the first argument is n_dims)
nk = gptools.DiagonalNoiseKernel(
1,
n=0,
initial_noise=np.mean(err_y) * noiseLevel,
fixed_noise=True
) #, noise_bound=(np.mean(err_y)*noiseLevel*(4.0/5.0),np.mean(err_y)*noiseLevel*(6.0/5.0))) #(np.min(err_y), np.max(err_y)*noiseLevel))#, enforce_bounds=True)
#print "noise_bound= [", np.min(err_y), ",",np.max(err_y)*noiseLevel,"]"
gp = gptools.GaussianProcess(k, X=x, y=y, err_y=err_y, noise_k=nk)
for i in range(len(y)):
if y[i] == 0:
gp.add_data(x[i], 0, n=0, err_y=0.0)
gp.add_data(x[i], 0, n=0, err_y=0.0)
gp.add_data(x[i], 0, n=0, err_y=0.0)
gp.add_data(x[i], 0, n=1, err_y=0.0)
with warnings.catch_warnings():
warnings.filterwarnings(
"ignore", message="invalid value encountered in subtract")
if optimize:
res_min, ll_trials = gp.optimize_hyperparameters(
verbose=False, random_starts=100)
else:
print 'Optimization is turned off. Using initial guesses for hyperparameters!'
m_gp, s_gp = gp.predict(grid, noise=True)
res.free_params = gp.free_params[:]
res.free_param_names = gp.free_param_names[:]
res.free_param_bounds = gp.free_param_bounds[:]
# Check percentage of points within 3 sd:
points_in_1sd = 0.0
points_in_2sd = 0.0
points_in_3sd = 0.0
for i in range(len(y)):
# Find value of grid that is the closest to x[i]:
gidx = np.argmin(abs(grid - x[i]))
if abs(m_gp[gidx] - y[i]) < s_gp[gidx]:
points_in_1sd += 1.0
if abs(m_gp[gidx] - y[i]) > s_gp[gidx] and abs(
m_gp[gidx] - y[i]) < 2 * s_gp[gidx]:
points_in_2sd += 1.0
if abs(m_gp[gidx] - y[i]) > 2 * s_gp[gidx] and abs(
m_gp[gidx] - y[i]) < 3 * s_gp[gidx]:
points_in_3sd += 1.0
frac_within_1sd = float(points_in_1sd) / len(y)
frac_within_2sd = float(points_in_2sd) / len(y)
frac_within_3sd = float(points_in_3sd) / len(y)
###
print("Estimating AIC, BIC...")
sum2_diff = 0
for i in range(len(y)):
# Find value of grid that is the closest to x[i]:
gidx = np.argmin(abs(grid - x[i]))
sum2_diff = (m_gp[gidx] - y[i])**2
chi_squared = float(sum2_diff) / len(y)
num_params = len(hparams.__dict__) / 2
num_data = len(y)
AIC = chi_squared + 2.0 * num_params
BIC = chi_squared + num_params * scipy.log(num_data)
elif method == 'spline':
m_gp = scipy.interpolate.UnivariateSpline(
x,
y,
w=1.0 / err_y,
s=None #2*len(x)
)(grid)
if scipy.isnan(m_gp).any():
print(x)
print(y)
print(err_y)
#i = m_gp.argmax()
else:
raise ValueError("Undefined method: %s" % (method, ))
if debug_plots:
f = plt.figure()
a = f.add_subplot(1, 1, 1)
a.errorbar(x, y, yerr=err_y, fmt='.', color='b')
a.plot(grid, m_gp, color='g')
if method == 'GPR':
gptools.univariate_envelope_plot(grid,
m_gp,
s_gp,
ax=a,
label='Inferred')
#a.fill_between(grid, m_gp - s_gp, m_gp + s_gp, color='g', alpha=0.5)
#plt.plot(grid[m_gp.argmax()],m_gp.max(),'r*')
plt.xlabel('time (s)', fontsize=14)
plt.ylabel('Signal Amplitude (A.U.)', fontsize=14)
plt.tick_params(axis='both', which='major', labelsize=14)
if method == 'GPR':
res.m_gp = m_gp
res.s_gp = s_gp
res.frac_within_1sd = frac_within_1sd
res.frac_within_2sd = frac_within_2sd
res.frac_within_3sd = frac_within_3sd
if optimize:
res.ll = res_min.fun
res.ll_trials = ll_trials
res.BIC = BIC
res.AIC = AIC
else:
res.m_gp = m_gp
return res
# hyperparameters -- this is because the noise is fixed to zero there. It is
# very reassuring that both converged to similar values for the prefactor and
# the covariance length scale, and that the fitted (homoscedastic) noise is
# comparable to the mean noise in `core_data['err_y']`.
# We'll come back to do the MCMC estimates in a bit -- first let's make some
# predictions with the trained GP.
# First, define a grid to evaluate on:
X_star = numpy.linspace(0, 1.1, 400)
# Now, we can make a prediction of the mean and standard deviation of the fit:
y_star, err_y_star = gp.predict(X_star)
# And we can plot this with our data:
gptools.univariate_envelope_plot(
X_star,
y_star,
err_y_star,
label='core, heteroscedastic noise, MAP',
ax=a_val
)
# And we can do the same for our case where we inferred the noise:
y_star_noise, err_y_star_noise = gp_noise.predict(X_star)
gptools.univariate_envelope_plot(
X_star,
y_star_noise,
err_y_star_noise,
label='core, inferred noise, MAP',
ax=a_val
)
# We can make predictions of the first derivative simply by using the `n`
# keyword, just like when adding data:
# hyperparameters -- this is because the noise is fixed to zero there. It is
# very reassuring that both converged to similar values for the prefactor and
# the covariance length scale, and that the fitted (homoscedastic) noise is
# comparable to the mean noise in `core_data['err_y']`.
### We'll come back to do the MCMC estimates in a bit -- first let's make some
### predictions with the trained GP.
# First, define a grid to evaluate on:
X_star = scipy.linspace(0, 1.1, 400)
# Now, we can make a prediction of the mean and standard deviation of the fit:
y_star, err_y_star = gp.predict(X_star)
# And we can plot this with our data:
gptools.univariate_envelope_plot(
X_star,
y_star,
err_y_star,
label='core, heteroscedastic noise, MAP',
ax=a_val
)
# And we can do the same for our case where we inferred the noise:
y_star_noise, err_y_star_noise = gp_noise.predict(X_star)
gptools.univariate_envelope_plot(
X_star,
y_star_noise,
err_y_star_noise,
label='core, inferred noise, MAP',
ax=a_val
)
# We can make predictions of the first derivative simply by using the `n`
# keyword, just like when adding data:
# find statistics for optimized result:
res_test_Te_final = profile_fitting(x_test_Te,
y_test_Te,
err_y=y_unc_test_Te,
optimize=True,
method='GPR',
kernel='gibbs',
noiseLevel=np.mean(psi_val_Te),
debug_plots=True,
**gibbs_params)
plt.xlabel(r'$\rho$', fontsize=16)
plt.ylabel(r'$T_e$ [keV]', fontsize=16)
print 'gibbs kernel free params for Te: ', res_test_Te_final.free_params
print 'gibbs kernel ll for Te: ', res_test_Te_final.ll
print 'gibbs kernel BIC for Te: ', res_test_Te_final.BIC
print 'Fraction of points within 1 sd for Te: {}'.format(
res_test_Te_final.frac_within_1sd)
print 'Fraction of points within 2 sd for Te: {}'.format(
res_test_Te_final.frac_within_2sd)
print 'Fraction of points within 3 sd for Te: {}'.format(
res_test_Te_final.frac_within_3sd)
left, bottom, width, height = [0.55, 0.55, 0.3, 0.3]
ax2 = plt.gcf().add_axes([left, bottom, width, height])
gptools.univariate_envelope_plot(x_test_Te[x_test_Te > 0.9],
res_test_Te_final.m_gp[x_test_Te > 0.9],
res_test_Te_final.s_gp[x_test_Te > 0.9],
ax=ax2,
color='green')
ax2.set_xlabel(r'$\rho$', fontsize=14)
ax2.set_ylabel(r'$T_e$ [keV]', fontsize=14)