2802 lines
67 KiB
Python
2802 lines
67 KiB
Python
"""This file contains code for use with "Think Stats" and
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"Think Bayes", both by Allen B. Downey, available from greenteapress.com
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Copyright 2014 Allen B. Downey
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License: GNU GPLv3 http://www.gnu.org/licenses/gpl.html
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"""
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from __future__ import print_function, division
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"""This file contains class definitions for:
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Hist: represents a histogram (map from values to integer frequencies).
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Pmf: represents a probability mass function (map from values to probs).
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_DictWrapper: private parent class for Hist and Pmf.
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Cdf: represents a discrete cumulative distribution function
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Pdf: represents a continuous probability density function
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"""
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import bisect
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import copy
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import logging
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import math
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import random
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import re
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from collections import Counter
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from operator import itemgetter
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import thinkplot
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import numpy as np
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import pandas
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import scipy
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from scipy import stats
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from scipy import special
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from scipy import ndimage
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from io import open
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ROOT2 = math.sqrt(2)
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def RandomSeed(x):
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"""Initialize the random and np.random generators.
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x: int seed
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"""
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random.seed(x)
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np.random.seed(x)
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def Odds(p):
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"""Computes odds for a given probability.
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Example: p=0.75 means 75 for and 25 against, or 3:1 odds in favor.
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Note: when p=1, the formula for odds divides by zero, which is
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normally undefined. But I think it is reasonable to define Odds(1)
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to be infinity, so that's what this function does.
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p: float 0-1
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Returns: float odds
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"""
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if p == 1:
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return float('inf')
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return p / (1 - p)
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def Probability(o):
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"""Computes the probability corresponding to given odds.
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Example: o=2 means 2:1 odds in favor, or 2/3 probability
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o: float odds, strictly positive
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Returns: float probability
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"""
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return o / (o + 1)
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def Probability2(yes, no):
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"""Computes the probability corresponding to given odds.
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Example: yes=2, no=1 means 2:1 odds in favor, or 2/3 probability.
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yes, no: int or float odds in favor
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"""
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return yes / (yes + no)
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class Interpolator(object):
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"""Represents a mapping between sorted sequences; performs linear interp.
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Attributes:
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xs: sorted list
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ys: sorted list
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"""
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def __init__(self, xs, ys):
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self.xs = xs
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self.ys = ys
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def Lookup(self, x):
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"""Looks up x and returns the corresponding value of y."""
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return self._Bisect(x, self.xs, self.ys)
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def Reverse(self, y):
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"""Looks up y and returns the corresponding value of x."""
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return self._Bisect(y, self.ys, self.xs)
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def _Bisect(self, x, xs, ys):
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"""Helper function."""
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if x <= xs[0]:
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return ys[0]
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if x >= xs[-1]:
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return ys[-1]
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i = bisect.bisect(xs, x)
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frac = 1.0 * (x - xs[i - 1]) / (xs[i] - xs[i - 1])
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y = ys[i - 1] + frac * 1.0 * (ys[i] - ys[i - 1])
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return y
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class _DictWrapper(object):
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"""An object that contains a dictionary."""
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def __init__(self, obj=None, label=None):
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"""Initializes the distribution.
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obj: Hist, Pmf, Cdf, Pdf, dict, pandas Series, list of pairs
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label: string label
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"""
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self.label = label if label is not None else '_nolegend_'
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self.d = {}
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# flag whether the distribution is under a log transform
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self.log = False
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if obj is None:
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return
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if isinstance(obj, (_DictWrapper, Cdf, Pdf)):
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self.label = label if label is not None else obj.label
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if isinstance(obj, dict):
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self.d.update(obj.items())
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elif isinstance(obj, (_DictWrapper, Cdf, Pdf)):
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self.d.update(obj.Items())
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elif isinstance(obj, pandas.Series):
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self.d.update(obj.value_counts().iteritems())
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else:
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# finally, treat it like a list
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self.d.update(Counter(obj))
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if len(self) > 0 and isinstance(self, Pmf):
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self.Normalize()
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def __hash__(self):
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return id(self)
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def __str__(self):
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cls = self.__class__.__name__
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return '%s(%s)' % (cls, str(self.d))
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__repr__ = __str__
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def __eq__(self, other):
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return self.d == other.d
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def __len__(self):
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return len(self.d)
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def __iter__(self):
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return iter(self.d)
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def iterkeys(self):
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"""Returns an iterator over keys."""
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return iter(self.d)
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def __contains__(self, value):
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return value in self.d
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def __getitem__(self, value):
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return self.d.get(value, 0)
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def __setitem__(self, value, prob):
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self.d[value] = prob
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def __delitem__(self, value):
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del self.d[value]
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def Copy(self, label=None):
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"""Returns a copy.
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Make a shallow copy of d. If you want a deep copy of d,
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use copy.deepcopy on the whole object.
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label: string label for the new Hist
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returns: new _DictWrapper with the same type
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"""
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new = copy.copy(self)
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new.d = copy.copy(self.d)
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new.label = label if label is not None else self.label
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return new
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def Scale(self, factor):
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"""Multiplies the values by a factor.
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factor: what to multiply by
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Returns: new object
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"""
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new = self.Copy()
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new.d.clear()
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for val, prob in self.Items():
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new.Set(val * factor, prob)
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return new
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def Log(self, m=None):
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"""Log transforms the probabilities.
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Removes values with probability 0.
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Normalizes so that the largest logprob is 0.
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"""
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if self.log:
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raise ValueError("Pmf/Hist already under a log transform")
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self.log = True
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if m is None:
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m = self.MaxLike()
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for x, p in self.d.items():
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if p:
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self.Set(x, math.log(p / m))
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else:
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self.Remove(x)
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def Exp(self, m=None):
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"""Exponentiates the probabilities.
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m: how much to shift the ps before exponentiating
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If m is None, normalizes so that the largest prob is 1.
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"""
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if not self.log:
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raise ValueError("Pmf/Hist not under a log transform")
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self.log = False
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if m is None:
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m = self.MaxLike()
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for x, p in self.d.items():
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self.Set(x, math.exp(p - m))
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def GetDict(self):
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"""Gets the dictionary."""
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return self.d
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def SetDict(self, d):
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"""Sets the dictionary."""
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self.d = d
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def Values(self):
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"""Gets an unsorted sequence of values.
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Note: one source of confusion is that the keys of this
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dictionary are the values of the Hist/Pmf, and the
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values of the dictionary are frequencies/probabilities.
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"""
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return self.d.keys()
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def Items(self):
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"""Gets an unsorted sequence of (value, freq/prob) pairs."""
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return self.d.items()
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def Render(self, **options):
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"""Generates a sequence of points suitable for plotting.
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Note: options are ignored
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Returns:
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tuple of (sorted value sequence, freq/prob sequence)
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"""
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if min(self.d.keys()) is np.nan:
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logging.warning('Hist: contains NaN, may not render correctly.')
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return zip(*sorted(self.Items()))
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def MakeCdf(self, label=None):
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"""Makes a Cdf."""
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label = label if label is not None else self.label
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return Cdf(self, label=label)
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def Print(self):
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"""Prints the values and freqs/probs in ascending order."""
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for val, prob in sorted(self.d.items()):
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print(val, prob)
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def Set(self, x, y=0):
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"""Sets the freq/prob associated with the value x.
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Args:
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x: number value
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y: number freq or prob
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"""
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self.d[x] = y
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def Incr(self, x, term=1):
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"""Increments the freq/prob associated with the value x.
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Args:
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x: number value
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term: how much to increment by
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"""
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self.d[x] = self.d.get(x, 0) + term
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def Mult(self, x, factor):
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"""Scales the freq/prob associated with the value x.
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Args:
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x: number value
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factor: how much to multiply by
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"""
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self.d[x] = self.d.get(x, 0) * factor
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def Remove(self, x):
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"""Removes a value.
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Throws an exception if the value is not there.
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Args:
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x: value to remove
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"""
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del self.d[x]
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def Total(self):
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"""Returns the total of the frequencies/probabilities in the map."""
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total = sum(self.d.values())
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return total
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def MaxLike(self):
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"""Returns the largest frequency/probability in the map."""
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return max(self.d.values())
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def Largest(self, n=10):
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"""Returns the largest n values, with frequency/probability.
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n: number of items to return
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"""
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return sorted(self.d.items(), reverse=True)[:n]
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def Smallest(self, n=10):
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"""Returns the smallest n values, with frequency/probability.
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n: number of items to return
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"""
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return sorted(self.d.items(), reverse=False)[:n]
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class Hist(_DictWrapper):
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"""Represents a histogram, which is a map from values to frequencies.
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Values can be any hashable type; frequencies are integer counters.
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"""
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def Freq(self, x):
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"""Gets the frequency associated with the value x.
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Args:
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x: number value
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Returns:
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int frequency
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"""
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return self.d.get(x, 0)
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def Freqs(self, xs):
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"""Gets frequencies for a sequence of values."""
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return [self.Freq(x) for x in xs]
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def IsSubset(self, other):
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"""Checks whether the values in this histogram are a subset of
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the values in the given histogram."""
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for val, freq in self.Items():
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if freq > other.Freq(val):
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return False
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return True
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def Subtract(self, other):
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"""Subtracts the values in the given histogram from this histogram."""
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for val, freq in other.Items():
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self.Incr(val, -freq)
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class Pmf(_DictWrapper):
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"""Represents a probability mass function.
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Values can be any hashable type; probabilities are floating-point.
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Pmfs are not necessarily normalized.
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"""
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def Prob(self, x, default=0):
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"""Gets the probability associated with the value x.
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Args:
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x: number value
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default: value to return if the key is not there
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Returns:
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float probability
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"""
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return self.d.get(x, default)
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def Probs(self, xs):
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"""Gets probabilities for a sequence of values."""
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return [self.Prob(x) for x in xs]
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def Percentile(self, percentage):
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"""Computes a percentile of a given Pmf.
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Note: this is not super efficient. If you are planning
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to compute more than a few percentiles, compute the Cdf.
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percentage: float 0-100
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returns: value from the Pmf
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"""
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p = percentage / 100.0
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total = 0
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for val, prob in sorted(self.Items()):
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total += prob
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if total >= p:
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return val
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def ProbGreater(self, x):
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"""Probability that a sample from this Pmf exceeds x.
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x: number
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returns: float probability
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"""
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if isinstance(x, _DictWrapper):
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return PmfProbGreater(self, x)
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else:
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t = [prob for (val, prob) in self.d.items() if val > x]
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return sum(t)
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def ProbLess(self, x):
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"""Probability that a sample from this Pmf is less than x.
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x: number
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returns: float probability
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"""
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if isinstance(x, _DictWrapper):
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return PmfProbLess(self, x)
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else:
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t = [prob for (val, prob) in self.d.items() if val < x]
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return sum(t)
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def __lt__(self, obj):
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"""Less than.
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obj: number or _DictWrapper
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returns: float probability
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"""
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return self.ProbLess(obj)
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def __gt__(self, obj):
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"""Greater than.
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obj: number or _DictWrapper
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returns: float probability
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"""
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return self.ProbGreater(obj)
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def __ge__(self, obj):
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"""Greater than or equal.
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obj: number or _DictWrapper
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returns: float probability
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"""
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return 1 - (self < obj)
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def __le__(self, obj):
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"""Less than or equal.
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obj: number or _DictWrapper
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returns: float probability
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"""
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return 1 - (self > obj)
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def Normalize(self, fraction=1.0):
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"""Normalizes this PMF so the sum of all probs is fraction.
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Args:
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fraction: what the total should be after normalization
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Returns: the total probability before normalizing
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"""
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if self.log:
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raise ValueError("Normalize: Pmf is under a log transform")
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total = self.Total()
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if total == 0.0:
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raise ValueError('Normalize: total probability is zero.')
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#logging.warning('Normalize: total probability is zero.')
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#return total
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factor = fraction / total
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for x in self.d:
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self.d[x] *= factor
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return total
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def Random(self):
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"""Chooses a random element from this PMF.
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Note: this is not very efficient. If you plan to call
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this more than a few times, consider converting to a CDF.
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Returns:
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float value from the Pmf
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"""
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target = random.random()
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total = 0.0
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for x, p in self.d.items():
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total += p
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if total >= target:
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return x
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# we shouldn't get here
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raise ValueError('Random: Pmf might not be normalized.')
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def Mean(self):
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"""Computes the mean of a PMF.
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Returns:
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float mean
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"""
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mean = 0.0
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for x, p in self.d.items():
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mean += p * x
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return mean
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def Var(self, mu=None):
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"""Computes the variance of a PMF.
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mu: the point around which the variance is computed;
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if omitted, computes the mean
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returns: float variance
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"""
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if mu is None:
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mu = self.Mean()
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var = 0.0
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for x, p in self.d.items():
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var += p * (x - mu) ** 2
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return var
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def Std(self, mu=None):
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"""Computes the standard deviation of a PMF.
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mu: the point around which the variance is computed;
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if omitted, computes the mean
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returns: float standard deviation
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"""
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var = self.Var(mu)
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return math.sqrt(var)
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def MaximumLikelihood(self):
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"""Returns the value with the highest probability.
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Returns: float probability
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"""
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_, val = max((prob, val) for val, prob in self.Items())
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return val
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def CredibleInterval(self, percentage=90):
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"""Computes the central credible interval.
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If percentage=90, computes the 90% CI.
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Args:
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percentage: float between 0 and 100
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Returns:
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sequence of two floats, low and high
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"""
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cdf = self.MakeCdf()
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return cdf.CredibleInterval(percentage)
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def __add__(self, other):
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"""Computes the Pmf of the sum of values drawn from self and other.
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other: another Pmf or a scalar
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returns: new Pmf
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"""
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try:
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return self.AddPmf(other)
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except AttributeError:
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return self.AddConstant(other)
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def AddPmf(self, other):
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"""Computes the Pmf of the sum of values drawn from self and other.
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other: another Pmf
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returns: new Pmf
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"""
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pmf = Pmf()
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|
for v1, p1 in self.Items():
|
|
for v2, p2 in other.Items():
|
|
pmf.Incr(v1 + v2, p1 * p2)
|
|
return pmf
|
|
|
|
def AddConstant(self, other):
|
|
"""Computes the Pmf of the sum a constant and values from self.
|
|
|
|
other: a number
|
|
|
|
returns: new Pmf
|
|
"""
|
|
pmf = Pmf()
|
|
for v1, p1 in self.Items():
|
|
pmf.Set(v1 + other, p1)
|
|
return pmf
|
|
|
|
def __sub__(self, other):
|
|
"""Computes the Pmf of the diff of values drawn from self and other.
|
|
|
|
other: another Pmf
|
|
|
|
returns: new Pmf
|
|
"""
|
|
try:
|
|
return self.SubPmf(other)
|
|
except AttributeError:
|
|
return self.AddConstant(-other)
|
|
|
|
def SubPmf(self, other):
|
|
"""Computes the Pmf of the diff of values drawn from self and other.
|
|
|
|
other: another Pmf
|
|
|
|
returns: new Pmf
|
|
"""
|
|
pmf = Pmf()
|
|
for v1, p1 in self.Items():
|
|
for v2, p2 in other.Items():
|
|
pmf.Incr(v1 - v2, p1 * p2)
|
|
return pmf
|
|
|
|
def __mul__(self, other):
|
|
"""Computes the Pmf of the product of values drawn from self and other.
|
|
|
|
other: another Pmf
|
|
|
|
returns: new Pmf
|
|
"""
|
|
try:
|
|
return self.MulPmf(other)
|
|
except AttributeError:
|
|
return self.MulConstant(other)
|
|
|
|
def MulPmf(self, other):
|
|
"""Computes the Pmf of the diff of values drawn from self and other.
|
|
|
|
other: another Pmf
|
|
|
|
returns: new Pmf
|
|
"""
|
|
pmf = Pmf()
|
|
for v1, p1 in self.Items():
|
|
for v2, p2 in other.Items():
|
|
pmf.Incr(v1 * v2, p1 * p2)
|
|
return pmf
|
|
|
|
def MulConstant(self, other):
|
|
"""Computes the Pmf of the product of a constant and values from self.
|
|
|
|
other: a number
|
|
|
|
returns: new Pmf
|
|
"""
|
|
pmf = Pmf()
|
|
for v1, p1 in self.Items():
|
|
pmf.Set(v1 * other, p1)
|
|
return pmf
|
|
|
|
def __div__(self, other):
|
|
"""Computes the Pmf of the ratio of values drawn from self and other.
|
|
|
|
other: another Pmf
|
|
|
|
returns: new Pmf
|
|
"""
|
|
try:
|
|
return self.DivPmf(other)
|
|
except AttributeError:
|
|
return self.MulConstant(1/other)
|
|
|
|
__truediv__ = __div__
|
|
|
|
def DivPmf(self, other):
|
|
"""Computes the Pmf of the ratio of values drawn from self and other.
|
|
|
|
other: another Pmf
|
|
|
|
returns: new Pmf
|
|
"""
|
|
pmf = Pmf()
|
|
for v1, p1 in self.Items():
|
|
for v2, p2 in other.Items():
|
|
pmf.Incr(v1 / v2, p1 * p2)
|
|
return pmf
|
|
|
|
def Max(self, k):
|
|
"""Computes the CDF of the maximum of k selections from this dist.
|
|
|
|
k: int
|
|
|
|
returns: new Cdf
|
|
"""
|
|
cdf = self.MakeCdf()
|
|
return cdf.Max(k)
|
|
|
|
|
|
class Joint(Pmf):
|
|
"""Represents a joint distribution.
|
|
|
|
The values are sequences (usually tuples)
|
|
"""
|
|
|
|
def Marginal(self, i, label=None):
|
|
"""Gets the marginal distribution of the indicated variable.
|
|
|
|
i: index of the variable we want
|
|
|
|
Returns: Pmf
|
|
"""
|
|
pmf = Pmf(label=label)
|
|
for vs, prob in self.Items():
|
|
pmf.Incr(vs[i], prob)
|
|
return pmf
|
|
|
|
def Conditional(self, i, j, val, label=None):
|
|
"""Gets the conditional distribution of the indicated variable.
|
|
|
|
Distribution of vs[i], conditioned on vs[j] = val.
|
|
|
|
i: index of the variable we want
|
|
j: which variable is conditioned on
|
|
val: the value the jth variable has to have
|
|
|
|
Returns: Pmf
|
|
"""
|
|
pmf = Pmf(label=label)
|
|
for vs, prob in self.Items():
|
|
if vs[j] != val:
|
|
continue
|
|
pmf.Incr(vs[i], prob)
|
|
|
|
pmf.Normalize()
|
|
return pmf
|
|
|
|
def MaxLikeInterval(self, percentage=90):
|
|
"""Returns the maximum-likelihood credible interval.
|
|
|
|
If percentage=90, computes a 90% CI containing the values
|
|
with the highest likelihoods.
|
|
|
|
percentage: float between 0 and 100
|
|
|
|
Returns: list of values from the suite
|
|
"""
|
|
interval = []
|
|
total = 0
|
|
|
|
t = [(prob, val) for val, prob in self.Items()]
|
|
t.sort(reverse=True)
|
|
|
|
for prob, val in t:
|
|
interval.append(val)
|
|
total += prob
|
|
if total >= percentage / 100.0:
|
|
break
|
|
|
|
return interval
|
|
|
|
|
|
def MakeJoint(pmf1, pmf2):
|
|
"""Joint distribution of values from pmf1 and pmf2.
|
|
|
|
Assumes that the PMFs represent independent random variables.
|
|
|
|
Args:
|
|
pmf1: Pmf object
|
|
pmf2: Pmf object
|
|
|
|
Returns:
|
|
Joint pmf of value pairs
|
|
"""
|
|
joint = Joint()
|
|
for v1, p1 in pmf1.Items():
|
|
for v2, p2 in pmf2.Items():
|
|
joint.Set((v1, v2), p1 * p2)
|
|
return joint
|
|
|
|
|
|
def MakeHistFromList(t, label=None):
|
|
"""Makes a histogram from an unsorted sequence of values.
|
|
|
|
Args:
|
|
t: sequence of numbers
|
|
label: string label for this histogram
|
|
|
|
Returns:
|
|
Hist object
|
|
"""
|
|
return Hist(t, label=label)
|
|
|
|
|
|
def MakeHistFromDict(d, label=None):
|
|
"""Makes a histogram from a map from values to frequencies.
|
|
|
|
Args:
|
|
d: dictionary that maps values to frequencies
|
|
label: string label for this histogram
|
|
|
|
Returns:
|
|
Hist object
|
|
"""
|
|
return Hist(d, label)
|
|
|
|
|
|
def MakePmfFromList(t, label=None):
|
|
"""Makes a PMF from an unsorted sequence of values.
|
|
|
|
Args:
|
|
t: sequence of numbers
|
|
label: string label for this PMF
|
|
|
|
Returns:
|
|
Pmf object
|
|
"""
|
|
return Pmf(t, label=label)
|
|
|
|
|
|
def MakePmfFromDict(d, label=None):
|
|
"""Makes a PMF from a map from values to probabilities.
|
|
|
|
Args:
|
|
d: dictionary that maps values to probabilities
|
|
label: string label for this PMF
|
|
|
|
Returns:
|
|
Pmf object
|
|
"""
|
|
return Pmf(d, label=label)
|
|
|
|
|
|
def MakePmfFromItems(t, label=None):
|
|
"""Makes a PMF from a sequence of value-probability pairs
|
|
|
|
Args:
|
|
t: sequence of value-probability pairs
|
|
label: string label for this PMF
|
|
|
|
Returns:
|
|
Pmf object
|
|
"""
|
|
return Pmf(dict(t), label=label)
|
|
|
|
|
|
def MakePmfFromHist(hist, label=None):
|
|
"""Makes a normalized PMF from a Hist object.
|
|
|
|
Args:
|
|
hist: Hist object
|
|
label: string label
|
|
|
|
Returns:
|
|
Pmf object
|
|
"""
|
|
if label is None:
|
|
label = hist.label
|
|
|
|
return Pmf(hist, label=label)
|
|
|
|
|
|
def MakeMixture(metapmf, label='mix'):
|
|
"""Make a mixture distribution.
|
|
|
|
Args:
|
|
metapmf: Pmf that maps from Pmfs to probs.
|
|
label: string label for the new Pmf.
|
|
|
|
Returns: Pmf object.
|
|
"""
|
|
mix = Pmf(label=label)
|
|
for pmf, p1 in metapmf.Items():
|
|
for x, p2 in pmf.Items():
|
|
mix.Incr(x, p1 * p2)
|
|
return mix
|
|
|
|
|
|
def MakeUniformPmf(low, high, n):
|
|
"""Make a uniform Pmf.
|
|
|
|
low: lowest value (inclusive)
|
|
high: highest value (inclusize)
|
|
n: number of values
|
|
"""
|
|
pmf = Pmf()
|
|
for x in np.linspace(low, high, n):
|
|
pmf.Set(x, 1)
|
|
pmf.Normalize()
|
|
return pmf
|
|
|
|
|
|
class Cdf(object):
|
|
"""Represents a cumulative distribution function.
|
|
|
|
Attributes:
|
|
xs: sequence of values
|
|
ps: sequence of probabilities
|
|
label: string used as a graph label.
|
|
"""
|
|
def __init__(self, obj=None, ps=None, label=None):
|
|
"""Initializes.
|
|
|
|
If ps is provided, obj must be the corresponding list of values.
|
|
|
|
obj: Hist, Pmf, Cdf, Pdf, dict, pandas Series, list of pairs
|
|
ps: list of cumulative probabilities
|
|
label: string label
|
|
"""
|
|
self.label = label if label is not None else '_nolegend_'
|
|
|
|
if isinstance(obj, (_DictWrapper, Cdf, Pdf)):
|
|
if not label:
|
|
self.label = label if label is not None else obj.label
|
|
|
|
if obj is None:
|
|
# caller does not provide obj, make an empty Cdf
|
|
self.xs = np.asarray([])
|
|
self.ps = np.asarray([])
|
|
if ps is not None:
|
|
logging.warning("Cdf: can't pass ps without also passing xs.")
|
|
return
|
|
else:
|
|
# if the caller provides xs and ps, just store them
|
|
if ps is not None:
|
|
if isinstance(ps, str):
|
|
logging.warning("Cdf: ps can't be a string")
|
|
|
|
self.xs = np.asarray(obj)
|
|
self.ps = np.asarray(ps)
|
|
return
|
|
|
|
# caller has provided just obj, not ps
|
|
if isinstance(obj, Cdf):
|
|
self.xs = copy.copy(obj.xs)
|
|
self.ps = copy.copy(obj.ps)
|
|
return
|
|
|
|
if isinstance(obj, _DictWrapper):
|
|
dw = obj
|
|
else:
|
|
dw = Hist(obj)
|
|
|
|
if len(dw) == 0:
|
|
self.xs = np.asarray([])
|
|
self.ps = np.asarray([])
|
|
return
|
|
|
|
xs, freqs = zip(*sorted(dw.Items()))
|
|
self.xs = np.asarray(xs)
|
|
self.ps = np.cumsum(freqs, dtype=np.float)
|
|
self.ps /= self.ps[-1]
|
|
|
|
def __str__(self):
|
|
return 'Cdf(%s, %s)' % (str(self.xs), str(self.ps))
|
|
|
|
__repr__ = __str__
|
|
|
|
def __len__(self):
|
|
return len(self.xs)
|
|
|
|
def __getitem__(self, x):
|
|
return self.Prob(x)
|
|
|
|
def __setitem__(self):
|
|
raise UnimplementedMethodException()
|
|
|
|
def __delitem__(self):
|
|
raise UnimplementedMethodException()
|
|
|
|
def __eq__(self, other):
|
|
return np.all(self.xs == other.xs) and np.all(self.ps == other.ps)
|
|
|
|
def Copy(self, label=None):
|
|
"""Returns a copy of this Cdf.
|
|
|
|
label: string label for the new Cdf
|
|
"""
|
|
if label is None:
|
|
label = self.label
|
|
return Cdf(list(self.xs), list(self.ps), label=label)
|
|
|
|
def MakePmf(self, label=None):
|
|
"""Makes a Pmf."""
|
|
if label is None:
|
|
label = self.label
|
|
return Pmf(self, label=label)
|
|
|
|
def Values(self):
|
|
"""Returns a sorted list of values.
|
|
"""
|
|
return self.xs
|
|
|
|
def Items(self):
|
|
"""Returns a sorted sequence of (value, probability) pairs.
|
|
|
|
Note: in Python3, returns an iterator.
|
|
"""
|
|
a = self.ps
|
|
b = np.roll(a, 1)
|
|
b[0] = 0
|
|
return zip(self.xs, a-b)
|
|
|
|
def Shift(self, term):
|
|
"""Adds a term to the xs.
|
|
|
|
term: how much to add
|
|
"""
|
|
new = self.Copy()
|
|
# don't use +=, or else an int array + float yields int array
|
|
new.xs = new.xs + term
|
|
return new
|
|
|
|
def Scale(self, factor):
|
|
"""Multiplies the xs by a factor.
|
|
|
|
factor: what to multiply by
|
|
"""
|
|
new = self.Copy()
|
|
# don't use *=, or else an int array * float yields int array
|
|
new.xs = new.xs * factor
|
|
return new
|
|
|
|
def Prob(self, x):
|
|
"""Returns CDF(x), the probability that corresponds to value x.
|
|
|
|
Args:
|
|
x: number
|
|
|
|
Returns:
|
|
float probability
|
|
"""
|
|
if x < self.xs[0]:
|
|
return 0.0
|
|
index = bisect.bisect(self.xs, x)
|
|
p = self.ps[index-1]
|
|
return p
|
|
|
|
def Probs(self, xs):
|
|
"""Gets probabilities for a sequence of values.
|
|
|
|
xs: any sequence that can be converted to NumPy array
|
|
|
|
returns: NumPy array of cumulative probabilities
|
|
"""
|
|
xs = np.asarray(xs)
|
|
index = np.searchsorted(self.xs, xs, side='right')
|
|
ps = self.ps[index-1]
|
|
ps[xs < self.xs[0]] = 0.0
|
|
return ps
|
|
|
|
ProbArray = Probs
|
|
|
|
def Value(self, p):
|
|
"""Returns InverseCDF(p), the value that corresponds to probability p.
|
|
|
|
Args:
|
|
p: number in the range [0, 1]
|
|
|
|
Returns:
|
|
number value
|
|
"""
|
|
if p < 0 or p > 1:
|
|
raise ValueError('Probability p must be in range [0, 1]')
|
|
|
|
index = bisect.bisect_left(self.ps, p)
|
|
return self.xs[index]
|
|
|
|
def ValueArray(self, ps):
|
|
"""Returns InverseCDF(p), the value that corresponds to probability p.
|
|
|
|
Args:
|
|
ps: NumPy array of numbers in the range [0, 1]
|
|
|
|
Returns:
|
|
NumPy array of values
|
|
"""
|
|
ps = np.asarray(ps)
|
|
if np.any(ps < 0) or np.any(ps > 1):
|
|
raise ValueError('Probability p must be in range [0, 1]')
|
|
|
|
index = np.searchsorted(self.ps, ps, side='left')
|
|
return self.xs[index]
|
|
|
|
def Percentile(self, p):
|
|
"""Returns the value that corresponds to percentile p.
|
|
|
|
Args:
|
|
p: number in the range [0, 100]
|
|
|
|
Returns:
|
|
number value
|
|
"""
|
|
return self.Value(p / 100.0)
|
|
|
|
def PercentileRank(self, x):
|
|
"""Returns the percentile rank of the value x.
|
|
|
|
x: potential value in the CDF
|
|
|
|
returns: percentile rank in the range 0 to 100
|
|
"""
|
|
return self.Prob(x) * 100.0
|
|
|
|
def Random(self):
|
|
"""Chooses a random value from this distribution."""
|
|
return self.Value(random.random())
|
|
|
|
def Sample(self, n):
|
|
"""Generates a random sample from this distribution.
|
|
|
|
n: int length of the sample
|
|
returns: NumPy array
|
|
"""
|
|
ps = np.random.random(n)
|
|
return self.ValueArray(ps)
|
|
|
|
def Mean(self):
|
|
"""Computes the mean of a CDF.
|
|
|
|
Returns:
|
|
float mean
|
|
"""
|
|
old_p = 0
|
|
total = 0.0
|
|
for x, new_p in zip(self.xs, self.ps):
|
|
p = new_p - old_p
|
|
total += p * x
|
|
old_p = new_p
|
|
return total
|
|
|
|
def CredibleInterval(self, percentage=90):
|
|
"""Computes the central credible interval.
|
|
|
|
If percentage=90, computes the 90% CI.
|
|
|
|
Args:
|
|
percentage: float between 0 and 100
|
|
|
|
Returns:
|
|
sequence of two floats, low and high
|
|
"""
|
|
prob = (1 - percentage / 100.0) / 2
|
|
interval = self.Value(prob), self.Value(1 - prob)
|
|
return interval
|
|
|
|
ConfidenceInterval = CredibleInterval
|
|
|
|
def _Round(self, multiplier=1000.0):
|
|
"""
|
|
An entry is added to the cdf only if the percentile differs
|
|
from the previous value in a significant digit, where the number
|
|
of significant digits is determined by multiplier. The
|
|
default is 1000, which keeps log10(1000) = 3 significant digits.
|
|
"""
|
|
# TODO(write this method)
|
|
raise UnimplementedMethodException()
|
|
|
|
def Render(self, **options):
|
|
"""Generates a sequence of points suitable for plotting.
|
|
|
|
An empirical CDF is a step function; linear interpolation
|
|
can be misleading.
|
|
|
|
Note: options are ignored
|
|
|
|
Returns:
|
|
tuple of (xs, ps)
|
|
"""
|
|
def interleave(a, b):
|
|
c = np.empty(a.shape[0] + b.shape[0])
|
|
c[::2] = a
|
|
c[1::2] = b
|
|
return c
|
|
|
|
a = np.array(self.xs)
|
|
xs = interleave(a, a)
|
|
shift_ps = np.roll(self.ps, 1)
|
|
shift_ps[0] = 0
|
|
ps = interleave(shift_ps, self.ps)
|
|
return xs, ps
|
|
|
|
def Max(self, k):
|
|
"""Computes the CDF of the maximum of k selections from this dist.
|
|
|
|
k: int
|
|
|
|
returns: new Cdf
|
|
"""
|
|
cdf = self.Copy()
|
|
cdf.ps **= k
|
|
return cdf
|
|
|
|
|
|
def MakeCdfFromItems(items, label=None):
|
|
"""Makes a cdf from an unsorted sequence of (value, frequency) pairs.
|
|
|
|
Args:
|
|
items: unsorted sequence of (value, frequency) pairs
|
|
label: string label for this CDF
|
|
|
|
Returns:
|
|
cdf: list of (value, fraction) pairs
|
|
"""
|
|
return Cdf(dict(items), label=label)
|
|
|
|
|
|
def MakeCdfFromDict(d, label=None):
|
|
"""Makes a CDF from a dictionary that maps values to frequencies.
|
|
|
|
Args:
|
|
d: dictionary that maps values to frequencies.
|
|
label: string label for the data.
|
|
|
|
Returns:
|
|
Cdf object
|
|
"""
|
|
return Cdf(d, label=label)
|
|
|
|
|
|
def MakeCdfFromList(seq, label=None):
|
|
"""Creates a CDF from an unsorted sequence.
|
|
|
|
Args:
|
|
seq: unsorted sequence of sortable values
|
|
label: string label for the cdf
|
|
|
|
Returns:
|
|
Cdf object
|
|
"""
|
|
return Cdf(seq, label=label)
|
|
|
|
|
|
def MakeCdfFromHist(hist, label=None):
|
|
"""Makes a CDF from a Hist object.
|
|
|
|
Args:
|
|
hist: Pmf.Hist object
|
|
label: string label for the data.
|
|
|
|
Returns:
|
|
Cdf object
|
|
"""
|
|
if label is None:
|
|
label = hist.label
|
|
|
|
return Cdf(hist, label=label)
|
|
|
|
|
|
def MakeCdfFromPmf(pmf, label=None):
|
|
"""Makes a CDF from a Pmf object.
|
|
|
|
Args:
|
|
pmf: Pmf.Pmf object
|
|
label: string label for the data.
|
|
|
|
Returns:
|
|
Cdf object
|
|
"""
|
|
if label is None:
|
|
label = pmf.label
|
|
|
|
return Cdf(pmf, label=label)
|
|
|
|
|
|
class UnimplementedMethodException(Exception):
|
|
"""Exception if someone calls a method that should be overridden."""
|
|
|
|
|
|
class Suite(Pmf):
|
|
"""Represents a suite of hypotheses and their probabilities."""
|
|
|
|
def Update(self, data):
|
|
"""Updates each hypothesis based on the data.
|
|
|
|
data: any representation of the data
|
|
|
|
returns: the normalizing constant
|
|
"""
|
|
for hypo in self.Values():
|
|
like = self.Likelihood(data, hypo)
|
|
self.Mult(hypo, like)
|
|
return self.Normalize()
|
|
|
|
def LogUpdate(self, data):
|
|
"""Updates a suite of hypotheses based on new data.
|
|
|
|
Modifies the suite directly; if you want to keep the original, make
|
|
a copy.
|
|
|
|
Note: unlike Update, LogUpdate does not normalize.
|
|
|
|
Args:
|
|
data: any representation of the data
|
|
"""
|
|
for hypo in self.Values():
|
|
like = self.LogLikelihood(data, hypo)
|
|
self.Incr(hypo, like)
|
|
|
|
def UpdateSet(self, dataset):
|
|
"""Updates each hypothesis based on the dataset.
|
|
|
|
This is more efficient than calling Update repeatedly because
|
|
it waits until the end to Normalize.
|
|
|
|
Modifies the suite directly; if you want to keep the original, make
|
|
a copy.
|
|
|
|
dataset: a sequence of data
|
|
|
|
returns: the normalizing constant
|
|
"""
|
|
for data in dataset:
|
|
for hypo in self.Values():
|
|
like = self.Likelihood(data, hypo)
|
|
self.Mult(hypo, like)
|
|
return self.Normalize()
|
|
|
|
def LogUpdateSet(self, dataset):
|
|
"""Updates each hypothesis based on the dataset.
|
|
|
|
Modifies the suite directly; if you want to keep the original, make
|
|
a copy.
|
|
|
|
dataset: a sequence of data
|
|
|
|
returns: None
|
|
"""
|
|
for data in dataset:
|
|
self.LogUpdate(data)
|
|
|
|
def Likelihood(self, data, hypo):
|
|
"""Computes the likelihood of the data under the hypothesis.
|
|
|
|
hypo: some representation of the hypothesis
|
|
data: some representation of the data
|
|
"""
|
|
raise UnimplementedMethodException()
|
|
|
|
def LogLikelihood(self, data, hypo):
|
|
"""Computes the log likelihood of the data under the hypothesis.
|
|
|
|
hypo: some representation of the hypothesis
|
|
data: some representation of the data
|
|
"""
|
|
raise UnimplementedMethodException()
|
|
|
|
def Print(self):
|
|
"""Prints the hypotheses and their probabilities."""
|
|
for hypo, prob in sorted(self.Items()):
|
|
print(hypo, prob)
|
|
|
|
def MakeOdds(self):
|
|
"""Transforms from probabilities to odds.
|
|
|
|
Values with prob=0 are removed.
|
|
"""
|
|
for hypo, prob in self.Items():
|
|
if prob:
|
|
self.Set(hypo, Odds(prob))
|
|
else:
|
|
self.Remove(hypo)
|
|
|
|
def MakeProbs(self):
|
|
"""Transforms from odds to probabilities."""
|
|
for hypo, odds in self.Items():
|
|
self.Set(hypo, Probability(odds))
|
|
|
|
|
|
def MakeSuiteFromList(t, label=None):
|
|
"""Makes a suite from an unsorted sequence of values.
|
|
|
|
Args:
|
|
t: sequence of numbers
|
|
label: string label for this suite
|
|
|
|
Returns:
|
|
Suite object
|
|
"""
|
|
hist = MakeHistFromList(t, label=label)
|
|
d = hist.GetDict()
|
|
return MakeSuiteFromDict(d)
|
|
|
|
|
|
def MakeSuiteFromHist(hist, label=None):
|
|
"""Makes a normalized suite from a Hist object.
|
|
|
|
Args:
|
|
hist: Hist object
|
|
label: string label
|
|
|
|
Returns:
|
|
Suite object
|
|
"""
|
|
if label is None:
|
|
label = hist.label
|
|
|
|
# make a copy of the dictionary
|
|
d = dict(hist.GetDict())
|
|
return MakeSuiteFromDict(d, label)
|
|
|
|
|
|
def MakeSuiteFromDict(d, label=None):
|
|
"""Makes a suite from a map from values to probabilities.
|
|
|
|
Args:
|
|
d: dictionary that maps values to probabilities
|
|
label: string label for this suite
|
|
|
|
Returns:
|
|
Suite object
|
|
"""
|
|
suite = Suite(label=label)
|
|
suite.SetDict(d)
|
|
suite.Normalize()
|
|
return suite
|
|
|
|
|
|
class Pdf(object):
|
|
"""Represents a probability density function (PDF)."""
|
|
|
|
def Density(self, x):
|
|
"""Evaluates this Pdf at x.
|
|
|
|
Returns: float or NumPy array of probability density
|
|
"""
|
|
raise UnimplementedMethodException()
|
|
|
|
def GetLinspace(self):
|
|
"""Get a linspace for plotting.
|
|
|
|
Not all subclasses of Pdf implement this.
|
|
|
|
Returns: numpy array
|
|
"""
|
|
raise UnimplementedMethodException()
|
|
|
|
def MakePmf(self, **options):
|
|
"""Makes a discrete version of this Pdf.
|
|
|
|
options can include
|
|
label: string
|
|
low: low end of range
|
|
high: high end of range
|
|
n: number of places to evaluate
|
|
|
|
Returns: new Pmf
|
|
"""
|
|
label = options.pop('label', '')
|
|
xs, ds = self.Render(**options)
|
|
return Pmf(dict(zip(xs, ds)), label=label)
|
|
|
|
def Render(self, **options):
|
|
"""Generates a sequence of points suitable for plotting.
|
|
|
|
If options includes low and high, it must also include n;
|
|
in that case the density is evaluated an n locations between
|
|
low and high, including both.
|
|
|
|
If options includes xs, the density is evaluate at those location.
|
|
|
|
Otherwise, self.GetLinspace is invoked to provide the locations.
|
|
|
|
Returns:
|
|
tuple of (xs, densities)
|
|
"""
|
|
low, high = options.pop('low', None), options.pop('high', None)
|
|
if low is not None and high is not None:
|
|
n = options.pop('n', 101)
|
|
xs = np.linspace(low, high, n)
|
|
else:
|
|
xs = options.pop('xs', None)
|
|
if xs is None:
|
|
xs = self.GetLinspace()
|
|
|
|
ds = self.Density(xs)
|
|
return xs, ds
|
|
|
|
def Items(self):
|
|
"""Generates a sequence of (value, probability) pairs.
|
|
"""
|
|
return zip(*self.Render())
|
|
|
|
|
|
class NormalPdf(Pdf):
|
|
"""Represents the PDF of a Normal distribution."""
|
|
|
|
def __init__(self, mu=0, sigma=1, label=None):
|
|
"""Constructs a Normal Pdf with given mu and sigma.
|
|
|
|
mu: mean
|
|
sigma: standard deviation
|
|
label: string
|
|
"""
|
|
self.mu = mu
|
|
self.sigma = sigma
|
|
self.label = label if label is not None else '_nolegend_'
|
|
|
|
def __str__(self):
|
|
return 'NormalPdf(%f, %f)' % (self.mu, self.sigma)
|
|
|
|
def GetLinspace(self):
|
|
"""Get a linspace for plotting.
|
|
|
|
Returns: numpy array
|
|
"""
|
|
low, high = self.mu-3*self.sigma, self.mu+3*self.sigma
|
|
return np.linspace(low, high, 101)
|
|
|
|
def Density(self, xs):
|
|
"""Evaluates this Pdf at xs.
|
|
|
|
xs: scalar or sequence of floats
|
|
|
|
returns: float or NumPy array of probability density
|
|
"""
|
|
return stats.norm.pdf(xs, self.mu, self.sigma)
|
|
|
|
|
|
class ExponentialPdf(Pdf):
|
|
"""Represents the PDF of an exponential distribution."""
|
|
|
|
def __init__(self, lam=1, label=None):
|
|
"""Constructs an exponential Pdf with given parameter.
|
|
|
|
lam: rate parameter
|
|
label: string
|
|
"""
|
|
self.lam = lam
|
|
self.label = label if label is not None else '_nolegend_'
|
|
|
|
def __str__(self):
|
|
return 'ExponentialPdf(%f)' % (self.lam)
|
|
|
|
def GetLinspace(self):
|
|
"""Get a linspace for plotting.
|
|
|
|
Returns: numpy array
|
|
"""
|
|
low, high = 0, 5.0/self.lam
|
|
return np.linspace(low, high, 101)
|
|
|
|
def Density(self, xs):
|
|
"""Evaluates this Pdf at xs.
|
|
|
|
xs: scalar or sequence of floats
|
|
|
|
returns: float or NumPy array of probability density
|
|
"""
|
|
return stats.expon.pdf(xs, scale=1.0/self.lam)
|
|
|
|
|
|
class EstimatedPdf(Pdf):
|
|
"""Represents a PDF estimated by KDE."""
|
|
|
|
def __init__(self, sample, label=None):
|
|
"""Estimates the density function based on a sample.
|
|
|
|
sample: sequence of data
|
|
label: string
|
|
"""
|
|
self.label = label if label is not None else '_nolegend_'
|
|
self.kde = stats.gaussian_kde(sample)
|
|
low = min(sample)
|
|
high = max(sample)
|
|
self.linspace = np.linspace(low, high, 101)
|
|
|
|
def __str__(self):
|
|
return 'EstimatedPdf(label=%s)' % str(self.label)
|
|
|
|
def GetLinspace(self):
|
|
"""Get a linspace for plotting.
|
|
|
|
Returns: numpy array
|
|
"""
|
|
return self.linspace
|
|
|
|
def Density(self, xs):
|
|
"""Evaluates this Pdf at xs.
|
|
|
|
returns: float or NumPy array of probability density
|
|
"""
|
|
return self.kde.evaluate(xs)
|
|
|
|
|
|
def CredibleInterval(pmf, percentage=90):
|
|
"""Computes a credible interval for a given distribution.
|
|
|
|
If percentage=90, computes the 90% CI.
|
|
|
|
Args:
|
|
pmf: Pmf object representing a posterior distribution
|
|
percentage: float between 0 and 100
|
|
|
|
Returns:
|
|
sequence of two floats, low and high
|
|
"""
|
|
cdf = pmf.MakeCdf()
|
|
prob = (1 - percentage / 100.0) / 2
|
|
interval = cdf.Value(prob), cdf.Value(1 - prob)
|
|
return interval
|
|
|
|
|
|
def PmfProbLess(pmf1, pmf2):
|
|
"""Probability that a value from pmf1 is less than a value from pmf2.
|
|
|
|
Args:
|
|
pmf1: Pmf object
|
|
pmf2: Pmf object
|
|
|
|
Returns:
|
|
float probability
|
|
"""
|
|
total = 0.0
|
|
for v1, p1 in pmf1.Items():
|
|
for v2, p2 in pmf2.Items():
|
|
if v1 < v2:
|
|
total += p1 * p2
|
|
return total
|
|
|
|
|
|
def PmfProbGreater(pmf1, pmf2):
|
|
"""Probability that a value from pmf1 is less than a value from pmf2.
|
|
|
|
Args:
|
|
pmf1: Pmf object
|
|
pmf2: Pmf object
|
|
|
|
Returns:
|
|
float probability
|
|
"""
|
|
total = 0.0
|
|
for v1, p1 in pmf1.Items():
|
|
for v2, p2 in pmf2.Items():
|
|
if v1 > v2:
|
|
total += p1 * p2
|
|
return total
|
|
|
|
|
|
def PmfProbEqual(pmf1, pmf2):
|
|
"""Probability that a value from pmf1 equals a value from pmf2.
|
|
|
|
Args:
|
|
pmf1: Pmf object
|
|
pmf2: Pmf object
|
|
|
|
Returns:
|
|
float probability
|
|
"""
|
|
total = 0.0
|
|
for v1, p1 in pmf1.Items():
|
|
for v2, p2 in pmf2.Items():
|
|
if v1 == v2:
|
|
total += p1 * p2
|
|
return total
|
|
|
|
|
|
def RandomSum(dists):
|
|
"""Chooses a random value from each dist and returns the sum.
|
|
|
|
dists: sequence of Pmf or Cdf objects
|
|
|
|
returns: numerical sum
|
|
"""
|
|
total = sum(dist.Random() for dist in dists)
|
|
return total
|
|
|
|
|
|
def SampleSum(dists, n):
|
|
"""Draws a sample of sums from a list of distributions.
|
|
|
|
dists: sequence of Pmf or Cdf objects
|
|
n: sample size
|
|
|
|
returns: new Pmf of sums
|
|
"""
|
|
pmf = Pmf(RandomSum(dists) for i in range(n))
|
|
return pmf
|
|
|
|
|
|
def EvalNormalPdf(x, mu, sigma):
|
|
"""Computes the unnormalized PDF of the normal distribution.
|
|
|
|
x: value
|
|
mu: mean
|
|
sigma: standard deviation
|
|
|
|
returns: float probability density
|
|
"""
|
|
return stats.norm.pdf(x, mu, sigma)
|
|
|
|
|
|
def MakeNormalPmf(mu, sigma, num_sigmas, n=201):
|
|
"""Makes a PMF discrete approx to a Normal distribution.
|
|
|
|
mu: float mean
|
|
sigma: float standard deviation
|
|
num_sigmas: how many sigmas to extend in each direction
|
|
n: number of values in the Pmf
|
|
|
|
returns: normalized Pmf
|
|
"""
|
|
pmf = Pmf()
|
|
low = mu - num_sigmas * sigma
|
|
high = mu + num_sigmas * sigma
|
|
|
|
for x in np.linspace(low, high, n):
|
|
p = EvalNormalPdf(x, mu, sigma)
|
|
pmf.Set(x, p)
|
|
pmf.Normalize()
|
|
return pmf
|
|
|
|
|
|
def EvalBinomialPmf(k, n, p):
|
|
"""Evaluates the binomial PMF.
|
|
|
|
Returns the probabily of k successes in n trials with probability p.
|
|
"""
|
|
return stats.binom.pmf(k, n, p)
|
|
|
|
|
|
def EvalHypergeomPmf(k, N, K, n):
|
|
"""Evaluates the hypergeometric PMF.
|
|
|
|
Returns the probabily of k successes in n trials from a population
|
|
N with K successes in it.
|
|
"""
|
|
return stats.hypergeom.pmf(k, N, K, n)
|
|
|
|
|
|
def EvalPoissonPmf(k, lam):
|
|
"""Computes the Poisson PMF.
|
|
|
|
k: number of events
|
|
lam: parameter lambda in events per unit time
|
|
|
|
returns: float probability
|
|
"""
|
|
# don't use the scipy function (yet). for lam=0 it returns NaN;
|
|
# should be 0.0
|
|
# return stats.poisson.pmf(k, lam)
|
|
return lam ** k * math.exp(-lam) / special.gamma(k+1)
|
|
|
|
|
|
def MakePoissonPmf(lam, high, step=1):
|
|
"""Makes a PMF discrete approx to a Poisson distribution.
|
|
|
|
lam: parameter lambda in events per unit time
|
|
high: upper bound of the Pmf
|
|
|
|
returns: normalized Pmf
|
|
"""
|
|
pmf = Pmf()
|
|
for k in range(0, high + 1, step):
|
|
p = EvalPoissonPmf(k, lam)
|
|
pmf.Set(k, p)
|
|
pmf.Normalize()
|
|
return pmf
|
|
|
|
|
|
def EvalExponentialPdf(x, lam):
|
|
"""Computes the exponential PDF.
|
|
|
|
x: value
|
|
lam: parameter lambda in events per unit time
|
|
|
|
returns: float probability density
|
|
"""
|
|
return lam * math.exp(-lam * x)
|
|
|
|
|
|
def EvalExponentialCdf(x, lam):
|
|
"""Evaluates CDF of the exponential distribution with parameter lam."""
|
|
return 1 - math.exp(-lam * x)
|
|
|
|
|
|
def MakeExponentialPmf(lam, high, n=200):
|
|
"""Makes a PMF discrete approx to an exponential distribution.
|
|
|
|
lam: parameter lambda in events per unit time
|
|
high: upper bound
|
|
n: number of values in the Pmf
|
|
|
|
returns: normalized Pmf
|
|
"""
|
|
pmf = Pmf()
|
|
for x in np.linspace(0, high, n):
|
|
p = EvalExponentialPdf(x, lam)
|
|
pmf.Set(x, p)
|
|
pmf.Normalize()
|
|
return pmf
|
|
|
|
|
|
def StandardNormalCdf(x):
|
|
"""Evaluates the CDF of the standard Normal distribution.
|
|
|
|
See http://en.wikipedia.org/wiki/Normal_distribution
|
|
#Cumulative_distribution_function
|
|
|
|
Args:
|
|
x: float
|
|
|
|
Returns:
|
|
float
|
|
"""
|
|
return (math.erf(x / ROOT2) + 1) / 2
|
|
|
|
|
|
def EvalNormalCdf(x, mu=0, sigma=1):
|
|
"""Evaluates the CDF of the normal distribution.
|
|
|
|
Args:
|
|
x: float
|
|
|
|
mu: mean parameter
|
|
|
|
sigma: standard deviation parameter
|
|
|
|
Returns:
|
|
float
|
|
"""
|
|
return stats.norm.cdf(x, loc=mu, scale=sigma)
|
|
|
|
|
|
def EvalNormalCdfInverse(p, mu=0, sigma=1):
|
|
"""Evaluates the inverse CDF of the normal distribution.
|
|
|
|
See http://en.wikipedia.org/wiki/Normal_distribution#Quantile_function
|
|
|
|
Args:
|
|
p: float
|
|
|
|
mu: mean parameter
|
|
|
|
sigma: standard deviation parameter
|
|
|
|
Returns:
|
|
float
|
|
"""
|
|
return stats.norm.ppf(p, loc=mu, scale=sigma)
|
|
|
|
|
|
def EvalLognormalCdf(x, mu=0, sigma=1):
|
|
"""Evaluates the CDF of the lognormal distribution.
|
|
|
|
x: float or sequence
|
|
mu: mean parameter
|
|
sigma: standard deviation parameter
|
|
|
|
Returns: float or sequence
|
|
"""
|
|
return stats.lognorm.cdf(x, loc=mu, scale=sigma)
|
|
|
|
|
|
def RenderExpoCdf(lam, low, high, n=101):
|
|
"""Generates sequences of xs and ps for an exponential CDF.
|
|
|
|
lam: parameter
|
|
low: float
|
|
high: float
|
|
n: number of points to render
|
|
|
|
returns: numpy arrays (xs, ps)
|
|
"""
|
|
xs = np.linspace(low, high, n)
|
|
ps = 1 - np.exp(-lam * xs)
|
|
#ps = stats.expon.cdf(xs, scale=1.0/lam)
|
|
return xs, ps
|
|
|
|
|
|
def RenderNormalCdf(mu, sigma, low, high, n=101):
|
|
"""Generates sequences of xs and ps for a Normal CDF.
|
|
|
|
mu: parameter
|
|
sigma: parameter
|
|
low: float
|
|
high: float
|
|
n: number of points to render
|
|
|
|
returns: numpy arrays (xs, ps)
|
|
"""
|
|
xs = np.linspace(low, high, n)
|
|
ps = stats.norm.cdf(xs, mu, sigma)
|
|
return xs, ps
|
|
|
|
|
|
def RenderParetoCdf(xmin, alpha, low, high, n=50):
|
|
"""Generates sequences of xs and ps for a Pareto CDF.
|
|
|
|
xmin: parameter
|
|
alpha: parameter
|
|
low: float
|
|
high: float
|
|
n: number of points to render
|
|
|
|
returns: numpy arrays (xs, ps)
|
|
"""
|
|
if low < xmin:
|
|
low = xmin
|
|
xs = np.linspace(low, high, n)
|
|
ps = 1 - (xs / xmin) ** -alpha
|
|
#ps = stats.pareto.cdf(xs, scale=xmin, b=alpha)
|
|
return xs, ps
|
|
|
|
|
|
class Beta(object):
|
|
"""Represents a Beta distribution.
|
|
|
|
See http://en.wikipedia.org/wiki/Beta_distribution
|
|
"""
|
|
def __init__(self, alpha=1, beta=1, label=None):
|
|
"""Initializes a Beta distribution."""
|
|
self.alpha = alpha
|
|
self.beta = beta
|
|
self.label = label if label is not None else '_nolegend_'
|
|
|
|
def Update(self, data):
|
|
"""Updates a Beta distribution.
|
|
|
|
data: pair of int (heads, tails)
|
|
"""
|
|
heads, tails = data
|
|
self.alpha += heads
|
|
self.beta += tails
|
|
|
|
def Mean(self):
|
|
"""Computes the mean of this distribution."""
|
|
return self.alpha / (self.alpha + self.beta)
|
|
|
|
def Random(self):
|
|
"""Generates a random variate from this distribution."""
|
|
return random.betavariate(self.alpha, self.beta)
|
|
|
|
def Sample(self, n):
|
|
"""Generates a random sample from this distribution.
|
|
|
|
n: int sample size
|
|
"""
|
|
size = n,
|
|
return np.random.beta(self.alpha, self.beta, size)
|
|
|
|
def EvalPdf(self, x):
|
|
"""Evaluates the PDF at x."""
|
|
return x ** (self.alpha - 1) * (1 - x) ** (self.beta - 1)
|
|
|
|
def MakePmf(self, steps=101, label=None):
|
|
"""Returns a Pmf of this distribution.
|
|
|
|
Note: Normally, we just evaluate the PDF at a sequence
|
|
of points and treat the probability density as a probability
|
|
mass.
|
|
|
|
But if alpha or beta is less than one, we have to be
|
|
more careful because the PDF goes to infinity at x=0
|
|
and x=1. In that case we evaluate the CDF and compute
|
|
differences.
|
|
"""
|
|
if self.alpha < 1 or self.beta < 1:
|
|
cdf = self.MakeCdf()
|
|
pmf = cdf.MakePmf()
|
|
return pmf
|
|
|
|
xs = [i / (steps - 1.0) for i in range(steps)]
|
|
probs = [self.EvalPdf(x) for x in xs]
|
|
pmf = Pmf(dict(zip(xs, probs)), label=label)
|
|
return pmf
|
|
|
|
def MakeCdf(self, steps=101):
|
|
"""Returns the CDF of this distribution."""
|
|
xs = [i / (steps - 1.0) for i in range(steps)]
|
|
ps = [special.betainc(self.alpha, self.beta, x) for x in xs]
|
|
cdf = Cdf(xs, ps)
|
|
return cdf
|
|
|
|
|
|
class Dirichlet(object):
|
|
"""Represents a Dirichlet distribution.
|
|
|
|
See http://en.wikipedia.org/wiki/Dirichlet_distribution
|
|
"""
|
|
|
|
def __init__(self, n, conc=1, label=None):
|
|
"""Initializes a Dirichlet distribution.
|
|
|
|
n: number of dimensions
|
|
conc: concentration parameter (smaller yields more concentration)
|
|
label: string label
|
|
"""
|
|
if n < 2:
|
|
raise ValueError('A Dirichlet distribution with '
|
|
'n<2 makes no sense')
|
|
|
|
self.n = n
|
|
self.params = np.ones(n, dtype=np.float) * conc
|
|
self.label = label if label is not None else '_nolegend_'
|
|
|
|
def Update(self, data):
|
|
"""Updates a Dirichlet distribution.
|
|
|
|
data: sequence of observations, in order corresponding to params
|
|
"""
|
|
m = len(data)
|
|
self.params[:m] += data
|
|
|
|
def Random(self):
|
|
"""Generates a random variate from this distribution.
|
|
|
|
Returns: normalized vector of fractions
|
|
"""
|
|
p = np.random.gamma(self.params)
|
|
return p / p.sum()
|
|
|
|
def Likelihood(self, data):
|
|
"""Computes the likelihood of the data.
|
|
|
|
Selects a random vector of probabilities from this distribution.
|
|
|
|
Returns: float probability
|
|
"""
|
|
m = len(data)
|
|
if self.n < m:
|
|
return 0
|
|
|
|
x = data
|
|
p = self.Random()
|
|
q = p[:m] ** x
|
|
return q.prod()
|
|
|
|
def LogLikelihood(self, data):
|
|
"""Computes the log likelihood of the data.
|
|
|
|
Selects a random vector of probabilities from this distribution.
|
|
|
|
Returns: float log probability
|
|
"""
|
|
m = len(data)
|
|
if self.n < m:
|
|
return float('-inf')
|
|
|
|
x = self.Random()
|
|
y = np.log(x[:m]) * data
|
|
return y.sum()
|
|
|
|
def MarginalBeta(self, i):
|
|
"""Computes the marginal distribution of the ith element.
|
|
|
|
See http://en.wikipedia.org/wiki/Dirichlet_distribution
|
|
#Marginal_distributions
|
|
|
|
i: int
|
|
|
|
Returns: Beta object
|
|
"""
|
|
alpha0 = self.params.sum()
|
|
alpha = self.params[i]
|
|
return Beta(alpha, alpha0 - alpha)
|
|
|
|
def PredictivePmf(self, xs, label=None):
|
|
"""Makes a predictive distribution.
|
|
|
|
xs: values to go into the Pmf
|
|
|
|
Returns: Pmf that maps from x to the mean prevalence of x
|
|
"""
|
|
alpha0 = self.params.sum()
|
|
ps = self.params / alpha0
|
|
return Pmf(zip(xs, ps), label=label)
|
|
|
|
|
|
def BinomialCoef(n, k):
|
|
"""Compute the binomial coefficient "n choose k".
|
|
|
|
n: number of trials
|
|
k: number of successes
|
|
|
|
Returns: float
|
|
"""
|
|
return scipy.misc.comb(n, k)
|
|
|
|
|
|
def LogBinomialCoef(n, k):
|
|
"""Computes the log of the binomial coefficient.
|
|
|
|
http://math.stackexchange.com/questions/64716/
|
|
approximating-the-logarithm-of-the-binomial-coefficient
|
|
|
|
n: number of trials
|
|
k: number of successes
|
|
|
|
Returns: float
|
|
"""
|
|
return n * math.log(n) - k * math.log(k) - (n - k) * math.log(n - k)
|
|
|
|
|
|
def NormalProbability(ys, jitter=0.0):
|
|
"""Generates data for a normal probability plot.
|
|
|
|
ys: sequence of values
|
|
jitter: float magnitude of jitter added to the ys
|
|
|
|
returns: numpy arrays xs, ys
|
|
"""
|
|
n = len(ys)
|
|
xs = np.random.normal(0, 1, n)
|
|
xs.sort()
|
|
|
|
if jitter:
|
|
ys = Jitter(ys, jitter)
|
|
else:
|
|
ys = np.array(ys)
|
|
ys.sort()
|
|
|
|
return xs, ys
|
|
|
|
|
|
def Jitter(values, jitter=0.5):
|
|
"""Jitters the values by adding a uniform variate in (-jitter, jitter).
|
|
|
|
values: sequence
|
|
jitter: scalar magnitude of jitter
|
|
|
|
returns: new numpy array
|
|
"""
|
|
n = len(values)
|
|
return np.random.uniform(-jitter, +jitter, n) + values
|
|
|
|
|
|
def NormalProbabilityPlot(sample, fit_color='0.8', **options):
|
|
"""Makes a normal probability plot with a fitted line.
|
|
|
|
sample: sequence of numbers
|
|
fit_color: color string for the fitted line
|
|
options: passed along to Plot
|
|
"""
|
|
xs, ys = NormalProbability(sample)
|
|
mean, var = MeanVar(sample)
|
|
std = math.sqrt(var)
|
|
|
|
fit = FitLine(xs, mean, std)
|
|
thinkplot.Plot(*fit, color=fit_color, label='model')
|
|
|
|
xs, ys = NormalProbability(sample)
|
|
thinkplot.Plot(xs, ys, **options)
|
|
|
|
|
|
def Mean(xs):
|
|
"""Computes mean.
|
|
|
|
xs: sequence of values
|
|
|
|
returns: float mean
|
|
"""
|
|
return np.mean(xs)
|
|
|
|
|
|
def Var(xs, mu=None, ddof=0):
|
|
"""Computes variance.
|
|
|
|
xs: sequence of values
|
|
mu: option known mean
|
|
ddof: delta degrees of freedom
|
|
|
|
returns: float
|
|
"""
|
|
xs = np.asarray(xs)
|
|
|
|
if mu is None:
|
|
mu = xs.mean()
|
|
|
|
ds = xs - mu
|
|
return np.dot(ds, ds) / (len(xs) - ddof)
|
|
|
|
|
|
def Std(xs, mu=None, ddof=0):
|
|
"""Computes standard deviation.
|
|
|
|
xs: sequence of values
|
|
mu: option known mean
|
|
ddof: delta degrees of freedom
|
|
|
|
returns: float
|
|
"""
|
|
var = Var(xs, mu, ddof)
|
|
return math.sqrt(var)
|
|
|
|
|
|
def MeanVar(xs, ddof=0):
|
|
"""Computes mean and variance.
|
|
|
|
Based on http://stackoverflow.com/questions/19391149/
|
|
numpy-mean-and-variance-from-single-function
|
|
|
|
xs: sequence of values
|
|
ddof: delta degrees of freedom
|
|
|
|
returns: pair of float, mean and var
|
|
"""
|
|
xs = np.asarray(xs)
|
|
mean = xs.mean()
|
|
s2 = Var(xs, mean, ddof)
|
|
return mean, s2
|
|
|
|
|
|
def Trim(t, p=0.01):
|
|
"""Trims the largest and smallest elements of t.
|
|
|
|
Args:
|
|
t: sequence of numbers
|
|
p: fraction of values to trim off each end
|
|
|
|
Returns:
|
|
sequence of values
|
|
"""
|
|
n = int(p * len(t))
|
|
t = sorted(t)[n:-n]
|
|
return t
|
|
|
|
|
|
def TrimmedMean(t, p=0.01):
|
|
"""Computes the trimmed mean of a sequence of numbers.
|
|
|
|
Args:
|
|
t: sequence of numbers
|
|
p: fraction of values to trim off each end
|
|
|
|
Returns:
|
|
float
|
|
"""
|
|
t = Trim(t, p)
|
|
return Mean(t)
|
|
|
|
|
|
def TrimmedMeanVar(t, p=0.01):
|
|
"""Computes the trimmed mean and variance of a sequence of numbers.
|
|
|
|
Side effect: sorts the list.
|
|
|
|
Args:
|
|
t: sequence of numbers
|
|
p: fraction of values to trim off each end
|
|
|
|
Returns:
|
|
float
|
|
"""
|
|
t = Trim(t, p)
|
|
mu, var = MeanVar(t)
|
|
return mu, var
|
|
|
|
|
|
def CohenEffectSize(group1, group2):
|
|
"""Compute Cohen's d.
|
|
|
|
group1: Series or NumPy array
|
|
group2: Series or NumPy array
|
|
|
|
returns: float
|
|
"""
|
|
diff = group1.mean() - group2.mean()
|
|
|
|
n1, n2 = len(group1), len(group2)
|
|
var1 = group1.var()
|
|
var2 = group2.var()
|
|
|
|
pooled_var = (n1 * var1 + n2 * var2) / (n1 + n2)
|
|
d = diff / math.sqrt(pooled_var)
|
|
return d
|
|
|
|
|
|
def Cov(xs, ys, meanx=None, meany=None):
|
|
"""Computes Cov(X, Y).
|
|
|
|
Args:
|
|
xs: sequence of values
|
|
ys: sequence of values
|
|
meanx: optional float mean of xs
|
|
meany: optional float mean of ys
|
|
|
|
Returns:
|
|
Cov(X, Y)
|
|
"""
|
|
xs = np.asarray(xs)
|
|
ys = np.asarray(ys)
|
|
|
|
if meanx is None:
|
|
meanx = np.mean(xs)
|
|
if meany is None:
|
|
meany = np.mean(ys)
|
|
|
|
cov = np.dot(xs-meanx, ys-meany) / len(xs)
|
|
return cov
|
|
|
|
|
|
def Corr(xs, ys):
|
|
"""Computes Corr(X, Y).
|
|
|
|
Args:
|
|
xs: sequence of values
|
|
ys: sequence of values
|
|
|
|
Returns:
|
|
Corr(X, Y)
|
|
"""
|
|
xs = np.asarray(xs)
|
|
ys = np.asarray(ys)
|
|
|
|
meanx, varx = MeanVar(xs)
|
|
meany, vary = MeanVar(ys)
|
|
|
|
corr = Cov(xs, ys, meanx, meany) / math.sqrt(varx * vary)
|
|
|
|
return corr
|
|
|
|
|
|
def SerialCorr(series, lag=1):
|
|
"""Computes the serial correlation of a series.
|
|
|
|
series: Series
|
|
lag: integer number of intervals to shift
|
|
|
|
returns: float correlation
|
|
"""
|
|
xs = series[lag:]
|
|
ys = series.shift(lag)[lag:]
|
|
corr = Corr(xs, ys)
|
|
return corr
|
|
|
|
|
|
def SpearmanCorr(xs, ys):
|
|
"""Computes Spearman's rank correlation.
|
|
|
|
Args:
|
|
xs: sequence of values
|
|
ys: sequence of values
|
|
|
|
Returns:
|
|
float Spearman's correlation
|
|
"""
|
|
xranks = pandas.Series(xs).rank()
|
|
yranks = pandas.Series(ys).rank()
|
|
return Corr(xranks, yranks)
|
|
|
|
|
|
def MapToRanks(t):
|
|
"""Returns a list of ranks corresponding to the elements in t.
|
|
|
|
Args:
|
|
t: sequence of numbers
|
|
|
|
Returns:
|
|
list of integer ranks, starting at 1
|
|
"""
|
|
# pair up each value with its index
|
|
pairs = enumerate(t)
|
|
|
|
# sort by value
|
|
sorted_pairs = sorted(pairs, key=itemgetter(1))
|
|
|
|
# pair up each pair with its rank
|
|
ranked = enumerate(sorted_pairs)
|
|
|
|
# sort by index
|
|
resorted = sorted(ranked, key=lambda trip: trip[1][0])
|
|
|
|
# extract the ranks
|
|
ranks = [trip[0]+1 for trip in resorted]
|
|
return ranks
|
|
|
|
|
|
def LeastSquares(xs, ys):
|
|
"""Computes a linear least squares fit for ys as a function of xs.
|
|
|
|
Args:
|
|
xs: sequence of values
|
|
ys: sequence of values
|
|
|
|
Returns:
|
|
tuple of (intercept, slope)
|
|
"""
|
|
meanx, varx = MeanVar(xs)
|
|
meany = Mean(ys)
|
|
|
|
slope = Cov(xs, ys, meanx, meany) / varx
|
|
inter = meany - slope * meanx
|
|
|
|
return inter, slope
|
|
|
|
|
|
def FitLine(xs, inter, slope):
|
|
"""Fits a line to the given data.
|
|
|
|
xs: sequence of x
|
|
|
|
returns: tuple of numpy arrays (sorted xs, fit ys)
|
|
"""
|
|
fit_xs = np.sort(xs)
|
|
fit_ys = inter + slope * fit_xs
|
|
return fit_xs, fit_ys
|
|
|
|
|
|
def Residuals(xs, ys, inter, slope):
|
|
"""Computes residuals for a linear fit with parameters inter and slope.
|
|
|
|
Args:
|
|
xs: independent variable
|
|
ys: dependent variable
|
|
inter: float intercept
|
|
slope: float slope
|
|
|
|
Returns:
|
|
list of residuals
|
|
"""
|
|
xs = np.asarray(xs)
|
|
ys = np.asarray(ys)
|
|
res = ys - (inter + slope * xs)
|
|
return res
|
|
|
|
|
|
def CoefDetermination(ys, res):
|
|
"""Computes the coefficient of determination (R^2) for given residuals.
|
|
|
|
Args:
|
|
ys: dependent variable
|
|
res: residuals
|
|
|
|
Returns:
|
|
float coefficient of determination
|
|
"""
|
|
return 1 - Var(res) / Var(ys)
|
|
|
|
|
|
def CorrelatedGenerator(rho):
|
|
"""Generates standard normal variates with serial correlation.
|
|
|
|
rho: target coefficient of correlation
|
|
|
|
Returns: iterable
|
|
"""
|
|
x = random.gauss(0, 1)
|
|
yield x
|
|
|
|
sigma = math.sqrt(1 - rho**2)
|
|
while True:
|
|
x = random.gauss(x * rho, sigma)
|
|
yield x
|
|
|
|
|
|
def CorrelatedNormalGenerator(mu, sigma, rho):
|
|
"""Generates normal variates with serial correlation.
|
|
|
|
mu: mean of variate
|
|
sigma: standard deviation of variate
|
|
rho: target coefficient of correlation
|
|
|
|
Returns: iterable
|
|
"""
|
|
for x in CorrelatedGenerator(rho):
|
|
yield x * sigma + mu
|
|
|
|
|
|
def RawMoment(xs, k):
|
|
"""Computes the kth raw moment of xs.
|
|
"""
|
|
return sum(x**k for x in xs) / len(xs)
|
|
|
|
|
|
def CentralMoment(xs, k):
|
|
"""Computes the kth central moment of xs.
|
|
"""
|
|
mean = RawMoment(xs, 1)
|
|
return sum((x - mean)**k for x in xs) / len(xs)
|
|
|
|
|
|
def StandardizedMoment(xs, k):
|
|
"""Computes the kth standardized moment of xs.
|
|
"""
|
|
var = CentralMoment(xs, 2)
|
|
std = math.sqrt(var)
|
|
return CentralMoment(xs, k) / std**k
|
|
|
|
|
|
def Skewness(xs):
|
|
"""Computes skewness.
|
|
"""
|
|
return StandardizedMoment(xs, 3)
|
|
|
|
|
|
def Median(xs):
|
|
"""Computes the median (50th percentile) of a sequence.
|
|
|
|
xs: sequence or anything else that can initialize a Cdf
|
|
|
|
returns: float
|
|
"""
|
|
cdf = Cdf(xs)
|
|
return cdf.Value(0.5)
|
|
|
|
|
|
def IQR(xs):
|
|
"""Computes the interquartile of a sequence.
|
|
|
|
xs: sequence or anything else that can initialize a Cdf
|
|
|
|
returns: pair of floats
|
|
"""
|
|
cdf = Cdf(xs)
|
|
return cdf.Value(0.25), cdf.Value(0.75)
|
|
|
|
|
|
def PearsonMedianSkewness(xs):
|
|
"""Computes the Pearson median skewness.
|
|
"""
|
|
median = Median(xs)
|
|
mean = RawMoment(xs, 1)
|
|
var = CentralMoment(xs, 2)
|
|
std = math.sqrt(var)
|
|
gp = 3 * (mean - median) / std
|
|
return gp
|
|
|
|
|
|
class FixedWidthVariables(object):
|
|
"""Represents a set of variables in a fixed width file."""
|
|
|
|
def __init__(self, variables, index_base=0):
|
|
"""Initializes.
|
|
|
|
variables: DataFrame
|
|
index_base: are the indices 0 or 1 based?
|
|
|
|
Attributes:
|
|
colspecs: list of (start, end) index tuples
|
|
names: list of string variable names
|
|
"""
|
|
self.variables = variables
|
|
|
|
# note: by default, subtract 1 from colspecs
|
|
self.colspecs = variables[['start', 'end']] - index_base
|
|
|
|
# convert colspecs to a list of pair of int
|
|
self.colspecs = self.colspecs.astype(np.int).values.tolist()
|
|
self.names = variables['name']
|
|
|
|
def ReadFixedWidth(self, filename, **options):
|
|
"""Reads a fixed width ASCII file.
|
|
|
|
filename: string filename
|
|
|
|
returns: DataFrame
|
|
"""
|
|
df = pandas.read_fwf(filename,
|
|
colspecs=self.colspecs,
|
|
names=self.names,
|
|
**options)
|
|
return df
|
|
|
|
|
|
def ReadStataDct(dct_file, **options):
|
|
"""Reads a Stata dictionary file.
|
|
|
|
dct_file: string filename
|
|
options: dict of options passed to open()
|
|
|
|
returns: FixedWidthVariables object
|
|
"""
|
|
type_map = dict(byte=int, int=int, long=int, float=float, double=float)
|
|
|
|
var_info = []
|
|
for line in open(dct_file, **options):
|
|
match = re.search( r'_column\(([^)]*)\)', line)
|
|
if match:
|
|
start = int(match.group(1))
|
|
t = line.split()
|
|
vtype, name, fstring = t[1:4]
|
|
name = name.lower()
|
|
if vtype.startswith('str'):
|
|
vtype = str
|
|
else:
|
|
vtype = type_map[vtype]
|
|
long_desc = ' '.join(t[4:]).strip('"')
|
|
var_info.append((start, vtype, name, fstring, long_desc))
|
|
|
|
columns = ['start', 'type', 'name', 'fstring', 'desc']
|
|
variables = pandas.DataFrame(var_info, columns=columns)
|
|
|
|
# fill in the end column by shifting the start column
|
|
variables['end'] = variables.start.shift(-1)
|
|
variables.loc[len(variables)-1, 'end'] = 0
|
|
|
|
dct = FixedWidthVariables(variables, index_base=1)
|
|
return dct
|
|
|
|
|
|
def Resample(xs, n=None):
|
|
"""Draw a sample from xs with the same length as xs.
|
|
|
|
xs: sequence
|
|
n: sample size (default: len(xs))
|
|
|
|
returns: NumPy array
|
|
"""
|
|
if n is None:
|
|
n = len(xs)
|
|
return np.random.choice(xs, n, replace=True)
|
|
|
|
|
|
def SampleRows(df, nrows, replace=False):
|
|
"""Choose a sample of rows from a DataFrame.
|
|
|
|
df: DataFrame
|
|
nrows: number of rows
|
|
replace: whether to sample with replacement
|
|
|
|
returns: DataDf
|
|
"""
|
|
indices = np.random.choice(df.index, nrows, replace=replace)
|
|
sample = df.loc[indices]
|
|
return sample
|
|
|
|
|
|
def ResampleRows(df):
|
|
"""Resamples rows from a DataFrame.
|
|
|
|
df: DataFrame
|
|
|
|
returns: DataFrame
|
|
"""
|
|
return SampleRows(df, len(df), replace=True)
|
|
|
|
|
|
def ResampleRowsWeighted(df, column='finalwgt'):
|
|
"""Resamples a DataFrame using probabilities proportional to given column.
|
|
|
|
df: DataFrame
|
|
column: string column name to use as weights
|
|
|
|
returns: DataFrame
|
|
"""
|
|
weights = df[column]
|
|
cdf = Cdf(dict(weights))
|
|
indices = cdf.Sample(len(weights))
|
|
sample = df.loc[indices]
|
|
return sample
|
|
|
|
|
|
def PercentileRow(array, p):
|
|
"""Selects the row from a sorted array that maps to percentile p.
|
|
|
|
p: float 0--100
|
|
|
|
returns: NumPy array (one row)
|
|
"""
|
|
rows, cols = array.shape
|
|
index = int(rows * p / 100)
|
|
return array[index,]
|
|
|
|
|
|
def PercentileRows(ys_seq, percents):
|
|
"""Given a collection of lines, selects percentiles along vertical axis.
|
|
|
|
For example, if ys_seq contains simulation results like ys as a
|
|
function of time, and percents contains (5, 95), the result would
|
|
be a 90% CI for each vertical slice of the simulation results.
|
|
|
|
ys_seq: sequence of lines (y values)
|
|
percents: list of percentiles (0-100) to select
|
|
|
|
returns: list of NumPy arrays, one for each percentile
|
|
"""
|
|
nrows = len(ys_seq)
|
|
ncols = len(ys_seq[0])
|
|
array = np.zeros((nrows, ncols))
|
|
|
|
for i, ys in enumerate(ys_seq):
|
|
array[i,] = ys
|
|
|
|
array = np.sort(array, axis=0)
|
|
|
|
rows = [PercentileRow(array, p) for p in percents]
|
|
return rows
|
|
|
|
|
|
def Smooth(xs, sigma=2, **options):
|
|
"""Smooths a NumPy array with a Gaussian filter.
|
|
|
|
xs: sequence
|
|
sigma: standard deviation of the filter
|
|
"""
|
|
return ndimage.filters.gaussian_filter1d(xs, sigma, **options)
|
|
|
|
|
|
class HypothesisTest(object):
|
|
"""Represents a hypothesis test."""
|
|
|
|
def __init__(self, data):
|
|
"""Initializes.
|
|
|
|
data: data in whatever form is relevant
|
|
"""
|
|
self.data = data
|
|
self.MakeModel()
|
|
self.actual = self.TestStatistic(data)
|
|
self.test_stats = None
|
|
self.test_cdf = None
|
|
|
|
def PValue(self, iters=1000):
|
|
"""Computes the distribution of the test statistic and p-value.
|
|
|
|
iters: number of iterations
|
|
|
|
returns: float p-value
|
|
"""
|
|
self.test_stats = [self.TestStatistic(self.RunModel())
|
|
for _ in range(iters)]
|
|
self.test_cdf = Cdf(self.test_stats)
|
|
|
|
count = sum(1 for x in self.test_stats if x >= self.actual)
|
|
return count / iters
|
|
|
|
def MaxTestStat(self):
|
|
"""Returns the largest test statistic seen during simulations.
|
|
"""
|
|
return max(self.test_stats)
|
|
|
|
def PlotCdf(self, label=None):
|
|
"""Draws a Cdf with vertical lines at the observed test stat.
|
|
"""
|
|
def VertLine(x):
|
|
"""Draws a vertical line at x."""
|
|
thinkplot.Plot([x, x], [0, 1], color='0.8')
|
|
|
|
VertLine(self.actual)
|
|
thinkplot.Cdf(self.test_cdf, label=label)
|
|
|
|
def TestStatistic(self, data):
|
|
"""Computes the test statistic.
|
|
|
|
data: data in whatever form is relevant
|
|
"""
|
|
raise UnimplementedMethodException()
|
|
|
|
def MakeModel(self):
|
|
"""Build a model of the null hypothesis.
|
|
"""
|
|
pass
|
|
|
|
def RunModel(self):
|
|
"""Run the model of the null hypothesis.
|
|
|
|
returns: simulated data
|
|
"""
|
|
raise UnimplementedMethodException()
|
|
|
|
|
|
def main():
|
|
pass
|
|
|
|
|
|
if __name__ == '__main__':
|
|
main()
|