"""
Statistics helper functions.
"""
__all__ = ["VarStats", "var_stats", "format_vars", "parse_var",
"stats", "credible_interval", "shortest_credible_interval"]
import re
import json
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()
integer = draw.integers is not None and draw.integers[var]
if integer:
values = np.floor(values)
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_interval(x=values, weights=weights,
ci=[0.95, ONE_SIGMA, 0.0])
## reporting uncertainty on credible intervals?
## might be nice to pair sd on credible intervals
## with the actual CIs, rather than use a separate param
#from .digits import credible_inderval_sd
#p95sd = credible_interval_sd(values, 0.95)
#p68sd = credible_interval_sd(values, ONE_SIGMA)
#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, p95_range=(p95[0], p95[1]+integer*0.9999999999),
p68=p68, p68_range=(p68[0], p68[1]+integer*0.9999999999),
# p95sd=p95sd, p68sd=p68sd,
median=p0[0], mean=mean, std=std, best=best,
integer=integer)
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)
def save_vars(all_vstats, filename):
with open(filename, 'w') as fid:
json.dump(
dict((v.label, v.__dict__) for v in all_vstats),
fid,
default=numpy_json,
sort_keys=True,
indent=2,
)
def numpy_json(o):
"""
JSON encoder for numpy data.
To automatically convert numpy data to lists when writing a datastream
use json.dumps(object, default=numpy_json).
"""
try:
return o.tolist()
except AttributeError:
raise TypeError
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)
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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_interval(x, ci, weights=None):
r"""
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 for 2 digits of precision. At least 1000
points are needed for an unbiased result, otherwise the resulting interval
will be shorter than expected (tested on a variety of distributions
including exponential, cauchy, gaussian, beta and gamma).
*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.
"""
n = x.size
ci = np.asarray(ci, 'd')
target = (1 + np.vstack((-ci, +ci))).T/2
if weights is None:
cdf = np.linspace(0.5/n, 1-0.5/n, n)
#cdf = np.linspace(1, n, n)/(n+1)
result = np.interp(target, cdf, np.sort(x))
else:
index = np.argsort(x)
x, weights = x[index], weights[index]
# convert weights to cdf
cdf = np.cumsum(weights)
cdf /= cdf[-1]
cdf -= 0.5*cdf[0]
#cdf *= n/(cdf[-1]*(n+1))
result = np.interp(target, cdf, x)
return result if ci.shape else result[0]
[docs]
def shortest_credible_interval(x, ci=0.95, weights=None):
"""
Find the credible interval covering the portion *ci* of the data.
*x* are samples from the posterior distribution.
*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.
Returns the minimum and maximum values of the interval.
If *ci* is a vector, return a vector of intervals.
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.
To remove bias towards toward smaller intervals, the midpoints between
the surrounding intervals are used as the end points.
"""
if weights is None:
x = np.sort(x)
# 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 _unweighted_hpd(x, ci)
else:
return [_unweighted_hpd(x, ci_k) for ci_k in ci]
else:
index = np.argsort(x)
x, weights = x[index], weights[index]
# Work from the empirical cdf, finding the corresponding right
# interval for each possible left interval and choosing that with
# the shortest distance.
cdf = np.cumsum(weights)
cdf /= cdf[-1]
#jcdf -= 0.5*cdf[0]
if np.isscalar(ci):
return _weighted_hpd(x, cdf, ci)
else:
return [_weighted_hpd(x, cdf, ci_k) for ci_k in ci]
def _unweighted_hpd(x, ci):
"""
Find shortest credible interval ci in sorted, unweighted x
"""
n = len(x)
size = int(ci*n)
if size >= n:
return x[0], x[-1]
else:
width = x[size:] - x[:-size]
index = np.argmin(width)
#left, right = x[idx], x[idx+size]
left = x[0] if index == 0 else (x[index-1] + x[index])/2
right = x[-1] if index+size == n-1 else (x[index+size] + x[index+size+1])/2
return left, right
def _weighted_hpd(z, cdf, ci): # extra one-half interval
"""
Find shortest credible interval ci in sorted, weighted x
"""
size = np.searchsorted(cdf, 1 - ci)
if size == 0:
return z[0], z[-1]
p_left = cdf[:size]
z_left = z[:size]
# avoid spurious floating point bugs, e.g., where .1+0.9 > 1.0
i_right = np.searchsorted(cdf[:-1], p_left + ci)
z_right = z[i_right]
index = np.argmin(z_right - z_left)
left = z_left[0] if index == 0 else (z_left[index-1] + z_left[index])/2
right = z_right[-1] if index+1 == len(z_right) else (z_right[index] + z_right[index+1])/2
return left, right