"""
Statistics helper functions.
"""
__all__ = ["VarStats", "var_stats", "format_vars", "parse_var",
"stats", "credible_intervals"]
import re
import numpy as np
from .formatnum import format_uncertainty
[docs]class VarStats(object):
def __init__(self, **kw):
self.__dict__ = kw
[docs]def var_stats(draw, vars=None):
if vars is None:
vars = range(draw.points.shape[1])
return [_var_stats_one(draw, v) for v in vars]
ONE_SIGMA = 1 - 2*0.15865525393145705
def _var_stats_one(draw, var):
weights, values = draw.weights, draw.points[:, var].flatten()
best_idx = np.argmax(draw.logp)
best = values[best_idx]
# Choose the interval for the histogram
#credible_interval = shortest_credible_interval
p95, p68, p0 = credible_intervals(x=values, weights=weights,
ci=[0.95, ONE_SIGMA, 0.0])
#open('/tmp/out','a').write(
# "in vstats: p68=%s, p95=%s, p0=%s, value range=%s\n"
# % (p68,p95,p0,(min(values),max(values))))
#if p0[0] != p0[1]: raise RuntimeError("wrong median %s"%(str(p0),))
mean, std = stats(x=values, weights=weights)
vstats = VarStats(label=draw.labels[var], index=var+1,
p95=p95, p68=p68,
median=p0[0], mean=mean, std=std, best=best)
return vstats
def format_num(x, place):
precision = 10**place
digits_after_decimal = abs(place) if place < 0 else 0
return "%.*f" % (digits_after_decimal, np.round(x/precision)*precision)
VAR_PATTERN = re.compile(r"""
^\ *
(?P<parnum>[0-9]+)\ +
(?P<parname>.+?)\ +
(?P<mean>[0-9.-]+?)
\((?P<err>[0-9]+)\)
(e(?P<exp>[+-]?[0-9]+))?\ +
(?P<median>[0-9.eE+-]+?)\ +
(?P<best>[0-9.eE+-]+?)\ +
\[\ *(?P<lo68>[0-9.eE+-]+?)\ +
(?P<hi68>[0-9.eE+-]+?)\]\ +
\[\ *(?P<lo95>[0-9.eE+-]+?)\ +
(?P<hi95>[0-9.eE+-]+?)\]
\ *$
""", re.VERBOSE)
[docs]def parse_var(line):
"""
Parse a line returned by format_vars back into the statistics for the
variable on that line.
"""
m = VAR_PATTERN.match(line)
if m:
exp = int(m.group('exp')) if m.group('exp') else 0
return VarStats(index=int(m.group('parnum')),
name=m.group('parname'),
mean=float(m.group('mean')) * 10**exp,
median=float(m.group('median')),
best=float(m.group('best')),
p68=(float(m.group('lo68')), float(m.group('hi68'))),
p95=(float(m.group('lo95')), float(m.group('hi95'))),
)
else:
return None
[docs]def stats(x, weights=None):
"""
Find mean and standard deviation of a set of weighted samples.
Note that the median is not strictly correct (we choose an endpoint
of the sample for the case where the median falls between two values
in the sample), but this is good enough when the sample size is large.
"""
if weights is None:
x = np.sort(x)
mean, std = np.mean(x), np.std(x, ddof=1)
else:
mean = np.mean(x*weights)/np.sum(weights)
# TODO: this is biased by selection of mean; need an unbiased formula
var = np.sum((weights*(x-mean))**2)/np.sum(weights)
std = np.sqrt(var)
return mean, std
[docs]def credible_intervals(x, ci, weights=None):
"""
Find the credible interval covering the portion *ci* of the data.
*x* are samples from the posterior distribution.
*ci* is a set of intervals in [0,1]. For a $1-\sigma$ interval use
*ci=erf(1/sqrt(2))*, or 0.68. About 1e5 samples are needed for 2 digits
of precision on a $1-\sigma$ credible interval. For a 95% interval,
about 1e6 samples are needed.
*weights* is a vector of weights for each x, or None for unweighted.
One could weight points according to temperature in a parallel tempering
dataset.
Returns an array *[[x1_low, x1_high], [l2_low, x2_high], ...]* where
*[xi_low, xi_high]* are the starting and ending values for credible
interval *i*.
This function is faster if the inputs are already sorted.
"""
from numpy import asarray, vstack, sort, cumsum, searchsorted, round, clip
ci = asarray(ci, 'd')
target = (1 + vstack((-ci, +ci))).T/2
if weights is None:
idx = clip(round(target*(x.size-1)), 0, x.size-1).astype('i')
return sort(x)[idx]
else:
idx = np.argsort(x)
x, weights = x[idx], weights[idx]
# convert weights to cdf
w = cumsum(weights/sum(weights))
return x[searchsorted(w, target)]
def shortest_credible_interval(x, ci=0.95, weights=None):
"""
Find the credible interval covering the portion *ci* of the data.
Returns the minimum and maximum values of the interval.
If *ci* is a vector, return a vector of intervals.
*x* are samples from the posterior distribution.
This function is faster if the inputs are already sorted.
About 1e6 samples are needed for 2 digits of precision on a 95%
credible interval, or 1e5 for 2 digits on a 1-sigma credible interval.
*ci* is the interval size in (0,1], and defaults to 0.95. For a
1-sigma interval use *ci=erf(1/sqrt(2))*.
*weights* is a vector of weights for each x, or None for unweighted.
For log likelihood data, setting weights to exp(max(logp)-logp) should
give reasonable results.
"""
sorted = np.all(x[1:]>=x[:-1])
if not sorted:
idx = np.argsort(x)
x = x[idx]
if weights is not None:
weights = weights[idx]
# w = exp(max(logp)-logp)
if weights is not None:
# convert weights to cdf
w = np.cumsum(weights/sum(weights))
# sample the cdf at every 0.001
idx = np.searchsorted(w, np.arange(0,1,0.001))
x = x[idx]
# Simple solution: ci*N is the number of points in the interval, so
# find the width of every interval of that size and return the smallest.
if np.isscalar(ci):
return _find_interval(x, ci)
else:
return [_find_interval(x, i) for i in ci]
def _find_interval(x, ci):
"""
Find credible interval ci in sorted, unweighted x
"""
n = len(x)
size = int( ci*n + np.sqrt(1-ci)*np.log(n) )
if size >= n:
return x[0],x[-1]
else:
width = x[size:] - x[:-size]
idx = np.argmin(width)
return x[idx],x[idx+size]
