def calculate_a(self, benchmark_weights):
"""
Calculates the first-order coefficients a_i(x) in
`dsigma(x | theta, nu) / dsigma(x | theta, 0) = exp[ sum_i (a_i(x) nu_i + b_i(x) nu_i^2 )]`.
Parameters
----------
benchmark_weights : ndarray
Event weights `dsigma(x | theta_i, nu_i)` with shape `(n_events, n_benchmarks)`. The benchmarks are expected
to be sorted in the same order as the keyword benchmark_names used during initialization, and the
nuisance benchmarks are expected to be rescaled to have the same physics parameters theta as the
reference_benchmark given during initialization.
Returns
-------
a : ndarray
Coefficients a_i(x) with shape `(n_nuisance_parameters, n_events)`.
"""
a = []
with np.errstate(divide="ignore", invalid="ignore"):
for i_pos, i_neg, degree in zip(self.i_benchmarks_pos, self.i_benchmarks_neg, self.degrees):
if degree == 1:
a.append(np.log(benchmark_weights[:, i_pos] / benchmark_weights[:, self.i_benchmark_ref]))
elif degree == 2:
a.append(0.5 * np.log(benchmark_weights[:, i_pos] / benchmark_weights[:, i_neg]))
a = np.array(a) # Shape (n_nuisance_parameters, n_events)
a = sanitize_array(a, min_value=-10.0, max_value=10.0)
return a
def calculate_b(self, benchmark_weights):
"""
Calculates the second-order coefficients b_i(x) in
`dsigma(x | theta, nu) / dsigma(x | theta, 0) = exp[ sum_i (a_i(x) nu_i + b_i(x) nu_i^2 )]`.
Parameters
----------
benchmark_weights : ndarray
Event weights `dsigma(x | theta_i, nu_i)` with shape `(n_events, n_benchmarks)`. The benchmarks are expected
to be sorted in the same order as the keyword benchmark_names used during initialization, and the
nuisance benchmarks are expected to be rescaled to have the same physics parameters theta as the
reference_benchmark given during initialization.
Returns
-------
b : ndarray
Coefficients b_i(x) with shape `(n_nuisance_parameters, n_events)`.
"""
b = []
for i_pos, i_neg, degree in zip(self.i_benchmarks_pos,
self.i_benchmarks_neg, self.degrees):
if degree == 1:
b.append(np.zeros(benchmark_weights.shape[0]))
elif degree == 2:
b.append(0.5 * np.log(
benchmark_weights[:, i_pos] * benchmark_weights[:, i_neg] /
benchmark_weights[:, self.i_benchmark_ref]**2))
b = np.array(b) # Shape (n_nuisance_parameters, n_events)
b = sanitize_array(b, min_value=-10.0, max_value=10.0)
return b
def plot_distributions(
filename,
observables=None,
parameter_points=None,
uncertainties="nuisance",
nuisance_parameters=None,
draw_nuisance_toys=None,
normalize=False,
log=False,
observable_labels=None,
n_bins=50,
line_labels=None,
colors=None,
linestyles=None,
linewidths=1.5,
toy_linewidths=0.5,
alpha=0.15,
toy_alpha=0.75,
n_events=None,
n_toys=100,
n_cols=3,
):
"""
Plots one-dimensional histograms of observables in a MadMiner file for a given set of benchmarks.
Parameters
----------
filename : str
Filename of a MadMiner HDF5 file.
observables : list of str or None, optional
Which observables to plot, given by a list of their names. If None, all observables in the file
are plotted. Default value: None.
parameter_points : list of (str or ndarray) or None, optional
Which parameter points to use for histogramming the data. Given by a list, each element can either be the name
of a benchmark in the MadMiner file, or an ndarray specifying any parameter point in a morphing setup. If None,
all physics (non-nuisance) benchmarks defined in the MadMiner file are plotted. Default value: None.
uncertainties : {"nuisance", "none"}, optional
Defines how uncertainty bands are drawn. With "nuisance", the variation in cross section from all nuisance
parameters is added in quadrature. With "none", no error bands are drawn.
nuisance_parameters : None or list of int, optional
If uncertainties is "nuisance", this can restrict which nuisance parameters are used to draw the uncertainty
bands. Each entry of this list is the index of one nuisance parameter (same order as in the MadMiner file).
draw_nuisance_toys : None or int, optional
If not None and uncertainties is "nuisance", sets the number of nuisance toy distributions that are drawn
(in addition to the error bands).
normalize : bool, optional
Whether the distribution is normalized to the total cross section. Default value: False.
log : bool, optional
Whether to draw the y axes on a logarithmic scale. Defaul value: False.
observable_labels : None or list of (str or None), optional
x-axis labels naming the observables. If None, the observable names from the MadMiner file are used. Default
value: None.
n_bins : int, optional
Number of histogram bins. Default value: 50.
line_labels : None or list of (str or None), optional
Labels for the different parameter points. If None and if parameter_points is None, the benchmark names from
the MadMiner file are used. Default value: None.
colors : None or str or list of str, optional
Matplotlib line (and error band) colors for the distributions. If None, uses default colors. Default value:
None.
linestyles : None or str or list of str, optional
Matplotlib line styles for the distributions. If None, uses default linestyles. Default value: None.
linewidths : float or list of float, optional
Line widths for the contours. Default value: 1.5.
toy_linewidths : float or list of float or None, optional
Line widths for the toy replicas, if uncertainties is "nuisance" and draw_nuisance_toys is not None. If None,
linewidths is used. Default value: 1.
alpha : float, optional
alpha value for the uncertainty bands. Default value: 0.25.
toy_alpha : float, optional
alpha value for the toy replicas, if uncertainties is "nuisance" and draw_nuisance_toys is not None. Default
value: 0.75.
n_events : None or int, optional
If not None, sets the number of events from the MadMiner file that will be analyzed and plotted. Default value:
None.
n_toys : int, optional
Number of toy nuisance parameter vectors used to estimate the systematic uncertainties. Default value: 100.
n_cols : int, optional
Number of columns of subfigures in the plot. Default value: 3.
Returns
-------
figure : Figure
Plot as Matplotlib Figure instance.
"""
# Load data
sa = SampleAugmenter(filename, include_nuisance_parameters=True)
if uncertainties == "nuisance":
nuisance_morpher = NuisanceMorpher(
sa.nuisance_parameters,
list(sa.benchmarks.keys()),
reference_benchmark=sa.reference_benchmark)
# Default settings
if parameter_points is None:
parameter_points = []
for key, is_nuisance in zip(sa.benchmarks, sa.benchmark_is_nuisance):
if not is_nuisance:
parameter_points.append(key)
if line_labels is None:
line_labels = parameter_points
n_parameter_points = len(parameter_points)
if colors is None:
colors = ["C" + str(i)
for i in range(10)] * (n_parameter_points // 10 + 1)
elif not isinstance(colors, list):
colors = [colors for _ in range(n_parameter_points)]
if linestyles is None:
linestyles = ["solid", "dashed", "dotted", "dashdot"
] * (n_parameter_points // 4 + 1)
elif not isinstance(linestyles, list):
linestyles = [linestyles for _ in range(n_parameter_points)]
if not isinstance(linewidths, list):
linewidths = [linewidths for _ in range(n_parameter_points)]
if toy_linewidths is None:
toy_linewidths = linewidths
if not isinstance(toy_linewidths, list):
toy_linewidths = [toy_linewidths for _ in range(n_parameter_points)]
# Observables
observable_indices = []
if observables is None:
observable_indices = list(range(len(sa.observables)))
else:
all_observables = list(sa.observables.keys())
for obs in observables:
try:
observable_indices.append(all_observables.index(str(obs)))
except ValueError:
logging.warning("Ignoring unknown observable %s", obs)
logger.debug("Observable indices: %s", observable_indices)
n_observables = len(observable_indices)
if observable_labels is None:
all_observables = list(sa.observables.keys())
observable_labels = [
all_observables[obs] for obs in observable_indices
]
# Get event data (observations and weights)
x, weights_benchmarks = sa.extract_raw_data()
logger.debug("Loaded raw data with shapes %s, %s", x.shape,
weights_benchmarks.shape)
# Remove negative weights
sane_event_filter = np.all(weights_benchmarks >= 0.0, axis=1)
n_events_before = weights_benchmarks.shape[0]
x = x[sane_event_filter]
weights_benchmarks = weights_benchmarks[sane_event_filter]
n_events_removed = n_events_before - weights_benchmarks.shape[0]
if int(np.sum(sane_event_filter, dtype=np.int)) < len(sane_event_filter):
logger.warning("Removed %s / %s events with negative weights",
n_events_removed, n_events_before)
# Shuffle events
x, weights_benchmarks = shuffle(x, weights_benchmarks)
# Only analyze n_events
if n_events is not None and n_events < x.shape[0]:
logger.debug("Only analyzing first %s / %s events", n_events,
x.shape[0])
x = x[:n_events]
weights_benchmarks = weights_benchmarks[:n_events]
if uncertainties != "nuisance":
n_toys = 0
n_nuisance_toys_drawn = 0
if draw_nuisance_toys is not None:
n_nuisance_toys_drawn = draw_nuisance_toys
theta_matrices = []
for theta in parameter_points:
if isinstance(theta, six.string_types):
matrix = get_theta_benchmark_matrix("benchmark", theta,
sa.benchmarks)
else:
matrix = get_theta_benchmark_matrix("morphing", theta,
sa.benchmarks, sa.morpher)
theta_matrices.append(matrix)
logger.debug("Calculated %s theta matrices", len(theta_matrices))
# Nuisance parameters
nuisance_toy_factors = []
if uncertainties == "nuisance":
n_nuisance_params = sa.n_nuisance_parameters
if not n_nuisance_params > 0:
raise RuntimeError(
"Cannot draw systematic uncertainties -- no nuisance parameters found!"
)
logger.debug("Drawing nuisance toys")
nuisance_toys = np.random.normal(loc=0.0,
scale=1.0,
size=n_nuisance_params * n_toys)
nuisance_toys = nuisance_toys.reshape(n_toys, n_nuisance_params)
# Restrict nuisance parameters
if nuisance_parameters is not None:
for i in range(n_nuisance_params):
if i not in nuisance_parameters:
nuisance_toys[:, i] = 0.0
logger.debug("Drew %s toy values for nuisance parameters",
n_toys * n_nuisance_params)
nuisance_toy_factors = np.array([
nuisance_morpher.calculate_nuisance_factors(
nuisance_toy, weights_benchmarks)
for nuisance_toy in nuisance_toys
]) # Shape (n_toys, n_events)
nuisance_toy_factors = sanitize_array(nuisance_toy_factors,
min_value=1.0e-2,
max_value=100.0)
# Shape (n_toys, n_events)
# Preparing plot
n_rows = (n_observables + n_cols - 1) // n_cols
n_events_for_range = 10000 if n_events is None else min(10000, n_events)
fig = plt.figure(figsize=(4.0 * n_cols, 4.0 * n_rows))
for i_panel, (i_obs, xlabel) in enumerate(
zip(observable_indices, observable_labels)):
logger.debug("Plotting panel %s: observable %s, label %s", i_panel,
i_obs, xlabel)
# Figure out x range
xmins, xmaxs = [], []
for theta_matrix in theta_matrices:
x_small = x[:n_events_for_range]
weights_small = mdot(theta_matrix,
weights_benchmarks[:n_events_for_range])
xmin = weighted_quantile(x_small[:, i_obs], 0.05, weights_small)
xmax = weighted_quantile(x_small[:, i_obs], 0.95, weights_small)
xwidth = xmax - xmin
xmin -= xwidth * 0.1
xmax += xwidth * 0.1
xmin = max(xmin, np.min(x[:, i_obs]))
xmax = min(xmax, np.max(x[:, i_obs]))
xmins.append(xmin)
xmaxs.append(xmax)
xmin = min(xmins)
xmax = max(xmaxs)
x_range = (xmin, xmax)
logger.debug("Ranges for observable %s: min = %s, max = %s", xlabel,
xmins, xmaxs)
# Subfigure
ax = plt.subplot(n_rows, n_cols, i_panel + 1)
# Calculate histograms
bin_edges = None
histos = []
histos_up = []
histos_down = []
histos_toys = []
for i_theta, theta_matrix in enumerate(theta_matrices):
theta_weights = mdot(theta_matrix,
weights_benchmarks) # Shape (n_events,)
histo, bin_edges = np.histogram(x[:, i_obs],
bins=n_bins,
range=x_range,
weights=theta_weights,
density=normalize)
histos.append(histo)
if uncertainties == "nuisance":
histos_toys_this_theta = []
for i_toy, nuisance_toy_factors_this_toy in enumerate(
nuisance_toy_factors):
toy_histo, _ = np.histogram(
x[:, i_obs],
bins=n_bins,
range=x_range,
weights=theta_weights * nuisance_toy_factors_this_toy,
density=normalize,
)
histos_toys_this_theta.append(toy_histo)
histos_up.append(
np.percentile(histos_toys_this_theta, 84.0, axis=0))
histos_down.append(
np.percentile(histos_toys_this_theta, 16.0, axis=0))
histos_toys.append(
histos_toys_this_theta[:n_nuisance_toys_drawn])
# Draw error bands
if uncertainties == "nuisance":
for histo_up, histo_down, lw, color, label, ls in zip(
histos_up, histos_down, linewidths, colors, line_labels,
linestyles):
bin_edges_ = np.repeat(bin_edges, 2)[1:-1]
histo_down_ = np.repeat(histo_down, 2)
histo_up_ = np.repeat(histo_up, 2)
plt.fill_between(bin_edges_,
histo_down_,
histo_up_,
facecolor=color,
edgecolor="none",
alpha=alpha)
# Draw some toys
for histo_toys, lw, color, ls in zip(histos_toys, toy_linewidths,
colors, linestyles):
for k in range(n_nuisance_toys_drawn):
bin_edges_ = np.repeat(bin_edges, 2)[1:-1]
histo_ = np.repeat(histo_toys[k], 2)
plt.plot(bin_edges_,
histo_,
color=color,
alpha=toy_alpha,
lw=lw,
ls=ls)
# Draw central lines
for histo, lw, color, label, ls in zip(histos, linewidths, colors,
line_labels, linestyles):
bin_edges_ = np.repeat(bin_edges, 2)[1:-1]
histo_ = np.repeat(histo, 2)
plt.plot(bin_edges_,
histo_,
color=color,
lw=lw,
ls=ls,
label=label,
alpha=1.0)
plt.legend()
plt.xlabel(xlabel)
if normalize:
plt.ylabel("Normalized distribution")
else:
plt.ylabel(r"$\frac{d\sigma}{dx}$ [pb / bin]")
plt.xlim(x_range[0], x_range[1])
if log:
ax.set_yscale("log", nonposy="clip")
else:
plt.ylim(0.0, None)
plt.tight_layout()
return fig
def plot_uncertainty(
filename,
theta,
observable,
obs_label,
obs_range,
n_bins=50,
nuisance_parameters=None,
n_events=None,
n_toys=100,
linecolor="black",
bandcolor1="#CC002E",
bandcolor2="orange",
ratio_range=(0.8, 1.2),
):
"""
Plots absolute and relative uncertainty bands in a histogram of one observable in a MadMiner file.
Parameters
----------
filename : str
Filename of a MadMiner HDF5 file.
theta : ndarray, optional
Which parameter points to use for histogramming the data.
observable : str
Which observable to plot, given by its name in the MadMiner file.
obs_label : str
x-axis label naming the observable.
obs_range : tuple of two float
Range to be plotted for the observable.
n_bins : int
Number of bins. Default value: 50.
nuisance_parameters : None or list of int, optional
This can restrict which nuisance parameters are used to draw the uncertainty
bands. Each entry of this list is the index of one nuisance parameter (same order as in the MadMiner file).
n_events : None or int, optional
If not None, sets the number of events from the MadMiner file that will be analyzed and plotted. Default value:
None.
n_toys : int, optional
Number of toy nuisance parameter vectors used to estimate the systematic uncertainties. Default value: 100.
linecolor : str, optional
Line color for central prediction. Default value: "black".
bandcolor1 : str, optional
Error band color for 1 sigma uncertainty. Default value: "#CC002E".
bandcolor2 : str, optional
Error band color for 2 sigma uncertainty. Default value: "orange".
ratio_range : tuple of two floar
y-axis range for the plots of the ratio to the central prediction. Default value: (0.8, 1.2).
Returns
-------
figure : Figure
Plot as Matplotlib Figure instance.
"""
# Load data
sa = SampleAugmenter(filename, include_nuisance_parameters=True)
nuisance_morpher = NuisanceMorpher(
sa.nuisance_parameters, list(sa.benchmarks.keys()), reference_benchmark=sa.reference_benchmark
)
# Observable index
obs_idx = list(sa.observables.keys()).index(observable)
# Get event data (observations and weights)
x, weights_benchmarks = sa.weighted_events()
x = x[:, obs_idx]
# Theta matrix
theta_matrix = sa._get_theta_benchmark_matrix(theta)
weights = mdot(theta_matrix, weights_benchmarks)
# Remove negative weights
x = x[weights >= 0.0]
weights_benchmarks = weights_benchmarks[weights >= 0.0]
weights = weights[weights >= 0.0]
# Shuffle events
x, weights, weights_benchmarks = shuffle(x, weights, weights_benchmarks)
# Only analyze n_events
if n_events is not None and n_events < x.shape[0]:
x = x[:n_events]
weights_benchmarks = weights_benchmarks[:n_events]
weights = weights[:n_events]
# Nuisance parameters
n_nuisance_params = sa.n_nuisance_parameters
nuisance_toys = np.random.normal(loc=0.0, scale=1.0, size=n_nuisance_params * n_toys)
nuisance_toys = nuisance_toys.reshape(n_toys, n_nuisance_params)
# Restrict nuisance parameters
if nuisance_parameters is not None:
for i in range(n_nuisance_params):
if i not in nuisance_parameters:
nuisance_toys[:, i] = 0.0
nuisance_toy_factors = np.array(
[
nuisance_morpher.calculate_nuisance_factors(nuisance_toy, weights_benchmarks)
for nuisance_toy in nuisance_toys
]
) # Shape (n_toys, n_events)
nuisance_toy_factors = sanitize_array(nuisance_toy_factors, min_value=1.0e-2, max_value=100.0)
# Shape (n_toys, n_events)
# Calculate histogram for central prediction, not normalized
histo, bin_edges = np.histogram(x, bins=n_bins, range=obs_range, weights=weights, density=False)
# Calculate toy histograms, not normalized
histos_toys_this_theta = []
for i_toy, nuisance_toy_factors_this_toy in enumerate(nuisance_toy_factors):
toy_histo, _ = np.histogram(
x, bins=n_bins, range=obs_range, weights=weights * nuisance_toy_factors_this_toy, density=False
)
histos_toys_this_theta.append(toy_histo)
histo_plus2sigma = np.percentile(histos_toys_this_theta, 97.5, axis=0)
histo_plus1sigma = np.percentile(histos_toys_this_theta, 84.0, axis=0)
histo_minus1sigma = np.percentile(histos_toys_this_theta, 16.0, axis=0)
histo_minus2sigma = np.percentile(histos_toys_this_theta, 2.5, axis=0)
# Calculate histogram for central prediction, normalized
histo_norm, bin_edges_norm = np.histogram(x, bins=n_bins, range=obs_range, weights=weights, density=True)
# Calculate toy histograms, normalized
histos_toys_this_theta = []
for i_toy, nuisance_toy_factors_this_toy in enumerate(nuisance_toy_factors):
toy_histo, _ = np.histogram(
x, bins=n_bins, range=obs_range, weights=weights * nuisance_toy_factors_this_toy, density=True
)
histos_toys_this_theta.append(toy_histo)
histo_plus2sigma_norm = np.percentile(histos_toys_this_theta, 97.5, axis=0)
histo_plus1sigma_norm = np.percentile(histos_toys_this_theta, 84.0, axis=0)
histo_minus1sigma_norm = np.percentile(histos_toys_this_theta, 16.0, axis=0)
histo_minus2sigma_norm = np.percentile(histos_toys_this_theta, 2.5, axis=0)
# Prepare plotting
def plot_mc(edges, histo_central, histo_m2, histo_m1, histo_p1, histo_p2, relative=False):
bin_edges_ = np.repeat(edges, 2)[1:-1]
histo_ = np.repeat(histo_central, 2)
histo_m2_ = np.repeat(histo_m2, 2)
histo_m1_ = np.repeat(histo_m1, 2)
histo_p1_ = np.repeat(histo_p1, 2)
histo_p2_ = np.repeat(histo_p2, 2)
if relative:
histo_m2_ /= histo_
histo_m1_ /= histo_
histo_p1_ /= histo_
histo_p2_ /= histo_
histo_ /= histo_
plt.fill_between(bin_edges_, histo_m2_, histo_p2_, facecolor=bandcolor2, edgecolor="none")
plt.fill_between(bin_edges_, histo_m1_, histo_p1_, facecolor=bandcolor1, edgecolor="none")
plt.plot(bin_edges_, histo_, color=linecolor, lw=1.5, ls="-")
# Make plot
fig = plt.figure(figsize=(10, 7))
gs = gridspec.GridSpec(2, 2, height_ratios=[2, 1])
# MC, absolute residuals
ax = plt.subplot(gs[2])
plot_mc(bin_edges, histo, histo_minus2sigma, histo_minus1sigma, histo_plus1sigma, histo_plus2sigma, relative=True)
plt.xlabel(obs_label)
plt.ylabel(r"Relative to central pred.")
plt.xlim(obs_range[0], obs_range[1])
plt.ylim(ratio_range[0], ratio_range[1])
# MC, absolute
ax = plt.subplot(gs[0], sharex=ax)
plot_mc(bin_edges, histo, histo_minus2sigma, histo_minus1sigma, histo_plus1sigma, histo_plus2sigma)
plt.ylabel(r"Differential cross section [pb/bin]")
plt.ylim(0.0, None)
plt.setp(ax.get_xticklabels(), visible=False)
# MC, relative residuals
ax = plt.subplot(gs[3])
plot_mc(
bin_edges_norm,
histo_norm,
histo_minus2sigma_norm,
histo_minus1sigma_norm,
histo_plus1sigma_norm,
histo_plus2sigma_norm,
relative=True,
)
plt.xlabel(r"$p_{T,\gamma}$ [GeV]")
plt.ylabel(r"Relative to central pred.")
plt.xlim(obs_range[0], obs_range[1])
plt.ylim(ratio_range[0], ratio_range[1])
# MC, relative
ax = plt.subplot(gs[1], sharex=ax)
plot_mc(
bin_edges_norm,
histo_norm,
histo_minus2sigma_norm,
histo_minus1sigma_norm,
histo_plus1sigma_norm,
histo_plus2sigma_norm,
)
plt.ylabel(r"Normalized distribution")
plt.ylim(0.0, None)
plt.setp(ax.get_xticklabels(), visible=False)
# Return
plt.tight_layout()
return fig
