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# -*- coding: utf-8 -*- # Natural Language Toolkit: Probability and Statistics # # Copyright (C) 2001-2012 NLTK Project # Author: Edward Loper <edloper@gradient.cis.upenn.edu> # Steven Bird <sb@csse.unimelb.edu.au> (additions) # Trevor Cohn <tacohn@cs.mu.oz.au> (additions) # Peter Ljunglöf <peter.ljunglof@heatherleaf.se> (additions) # Liang Dong <ldong@clemson.edu> (additions) # Geoffrey Sampson <sampson@cantab.net> (additions) # # URL: <http://www.nltk.org/> # For license information, see LICENSE.TXT
Classes for representing and processing probabilistic information.
The ``FreqDist`` class is used to encode "frequency distributions", which count the number of times that each outcome of an experiment occurs.
The ``ProbDistI`` class defines a standard interface for "probability distributions", which encode the probability of each outcome for an experiment. There are two types of probability distribution:
- "derived probability distributions" are created from frequency distributions. They attempt to model the probability distribution that generated the frequency distribution. - "analytic probability distributions" are created directly from parameters (such as variance).
The ``ConditionalFreqDist`` class and ``ConditionalProbDistI`` interface are used to encode conditional distributions. Conditional probability distributions can be derived or analytic; but currently the only implementation of the ``ConditionalProbDistI`` interface is ``ConditionalProbDist``, a derived distribution.
"""
##////////////////////////////////////////////////////// ## Frequency Distributions ##//////////////////////////////////////////////////////
# [SB] inherit from defaultdict? # [SB] for NLTK 3.0, inherit from collections.Counter?
""" A frequency distribution for the outcomes of an experiment. A frequency distribution records the number of times each outcome of an experiment has occurred. For example, a frequency distribution could be used to record the frequency of each word type in a document. Formally, a frequency distribution can be defined as a function mapping from each sample to the number of times that sample occurred as an outcome.
Frequency distributions are generally constructed by running a number of experiments, and incrementing the count for a sample every time it is an outcome of an experiment. For example, the following code will produce a frequency distribution that encodes how often each word occurs in a text:
>>> from nltk.tokenize import word_tokenize >>> from nltk.probability import FreqDist >>> sent = 'This is an example sentence' >>> fdist = FreqDist() >>> for word in word_tokenize(sent): ... fdist.inc(word.lower())
An equivalent way to do this is with the initializer:
>>> fdist = FreqDist(word.lower() for word in word_tokenize(sent))
""" """ Construct a new frequency distribution. If ``samples`` is given, then the frequency distribution will be initialized with the count of each object in ``samples``; otherwise, it will be initialized to be empty.
In particular, ``FreqDist()`` returns an empty frequency distribution; and ``FreqDist(samples)`` first creates an empty frequency distribution, and then calls ``update`` with the list ``samples``.
:param samples: The samples to initialize the frequency distribution with. :type samples: Sequence """
""" Increment this FreqDist's count for the given sample.
:param sample: The sample whose count should be incremented. :type sample: any :param count: The amount to increment the sample's count by. :type count: int :rtype: None :raise NotImplementedError: If ``sample`` is not a supported sample type. """
""" Set this FreqDist's count for the given sample.
:param sample: The sample whose count should be incremented. :type sample: any hashable object :param count: The new value for the sample's count :type count: int :rtype: None :raise TypeError: If ``sample`` is not a supported sample type. """
# Invalidate the caches
""" Return the total number of sample outcomes that have been recorded by this FreqDist. For the number of unique sample values (or bins) with counts greater than zero, use ``FreqDist.B()``.
:rtype: int """
""" Return the total number of sample values (or "bins") that have counts greater than zero. For the total number of sample outcomes recorded, use ``FreqDist.N()``. (FreqDist.B() is the same as len(FreqDist).)
:rtype: int """
""" Return a list of all samples that have been recorded as outcomes by this frequency distribution. Use ``fd[sample]`` to determine the count for each sample.
:rtype: list """ return self.keys()
""" Return a list of all samples that occur once (hapax legomena)
:rtype: list """ return [item for item in self if self[item] == 1]
""" Return the number of samples with count r.
:type r: int :param r: A sample count. :type bins: int :param bins: The number of possible sample outcomes. ``bins`` is used to calculate Nr(0). In particular, Nr(0) is ``bins-self.B()``. If ``bins`` is not specified, it defaults to ``self.B()`` (so Nr(0) will be 0). :rtype: int """ if r < 0: raise IndexError('FreqDist.Nr(): r must be non-negative')
# Special case for Nr(0): if r == 0: if bins is None: return 0 else: return bins-self.B()
# We have to search the entire distribution to find Nr. Since # this is an expensive operation, and is likely to be used # repeatedly, cache the results. if self._Nr_cache is None: self._cache_Nr_values()
if r >= len(self._Nr_cache): return 0 return self._Nr_cache[r]
Nr = [0] for sample in self: c = self.get(sample, 0) if c >= len(Nr): Nr += [0]*(c+1-len(Nr)) Nr[c] += 1 self._Nr_cache = Nr
""" Return the cumulative frequencies of the specified samples. If no samples are specified, all counts are returned, starting with the largest.
:param samples: the samples whose frequencies should be returned. :type sample: any :rtype: list(float) """ cf = 0.0 if not samples: samples = self.keys() for sample in samples: cf += self[sample] yield cf
# slightly odd nomenclature freq() if FreqDist does counts and ProbDist does probs, # here, freq() does probs """ Return the frequency of a given sample. The frequency of a sample is defined as the count of that sample divided by the total number of sample outcomes that have been recorded by this FreqDist. The count of a sample is defined as the number of times that sample outcome was recorded by this FreqDist. Frequencies are always real numbers in the range [0, 1].
:param sample: the sample whose frequency should be returned. :type sample: any :rtype: float """ return 0
""" Return the sample with the greatest number of outcomes in this frequency distribution. If two or more samples have the same number of outcomes, return one of them; which sample is returned is undefined. If no outcomes have occurred in this frequency distribution, return None.
:return: The sample with the maximum number of outcomes in this frequency distribution. :rtype: any or None """ raise ValueError('A FreqDist must have at least one sample before max is defined.')
""" Plot samples from the frequency distribution displaying the most frequent sample first. If an integer parameter is supplied, stop after this many samples have been plotted. If two integer parameters m, n are supplied, plot a subset of the samples, beginning with m and stopping at n-1. For a cumulative plot, specify cumulative=True. (Requires Matplotlib to be installed.)
:param title: The title for the graph :type title: str :param cumulative: A flag to specify whether the plot is cumulative (default = False) :type title: bool """ try: import pylab except ImportError: raise ValueError('The plot function requires the matplotlib package (aka pylab). ' 'See http://matplotlib.sourceforge.net/')
if len(args) == 0: args = [len(self)] samples = list(islice(self, *args))
cumulative = _get_kwarg(kwargs, 'cumulative', False) if cumulative: freqs = list(self._cumulative_frequencies(samples)) ylabel = "Cumulative Counts" else: freqs = [self[sample] for sample in samples] ylabel = "Counts" # percents = [f * 100 for f in freqs] only in ProbDist?
pylab.grid(True, color="silver") if not "linewidth" in kwargs: kwargs["linewidth"] = 2 if "title" in kwargs: pylab.title(kwargs["title"]) del kwargs["title"] pylab.plot(freqs, **kwargs) pylab.xticks(range(len(samples)), [unicode(s) for s in samples], rotation=90) pylab.xlabel("Samples") pylab.ylabel(ylabel) pylab.show()
""" Tabulate the given samples from the frequency distribution (cumulative), displaying the most frequent sample first. If an integer parameter is supplied, stop after this many samples have been plotted. If two integer parameters m, n are supplied, plot a subset of the samples, beginning with m and stopping at n-1. (Requires Matplotlib to be installed.)
:param samples: The samples to plot (default is all samples) :type samples: list """ if len(args) == 0: args = [len(self)] samples = list(islice(self, *args))
cumulative = _get_kwarg(kwargs, 'cumulative', False) if cumulative: freqs = list(self._cumulative_frequencies(samples)) else: freqs = [self[sample] for sample in samples] # percents = [f * 100 for f in freqs] only in ProbDist?
for i in range(len(samples)): print("%4s" % str(samples[i]), end=' ') print() for i in range(len(samples)): print("%4d" % freqs[i], end=' ') print()
""" Return the samples sorted in decreasing order of frequency.
:rtype: list(any) """ # this will return iterator under python 3
""" Return the samples sorted in decreasing order of frequency.
:rtype: list(any) """ self._sort_keys_by_value() # this will return iterator under python 3 return map(itemgetter(1), self._item_cache)
""" Return the items sorted in decreasing order of frequency.
:rtype: list(tuple) """
""" Return the samples sorted in decreasing order of frequency.
:rtype: iter """
""" Return the samples sorted in decreasing order of frequency.
:rtype: iter """ return iter(self.keys())
""" Return the values sorted in decreasing order.
:rtype: iter """ return iter(self.values())
""" Return the items sorted in decreasing order of frequency.
:rtype: iter of any """
""" Create a copy of this frequency distribution.
:rtype: FreqDist """ return self.__class__(self)
""" Update the frequency distribution with the provided list of samples. This is a faster way to add multiple samples to the distribution.
:param samples: The samples to add. :type samples: list """
self._N -= 1 self._reset_caches() return dict.pop(self, other)
self._N -= 1 self._reset_caches() return dict.popitem(self)
self._N = 0 self._reset_caches() dict.clear(self)
clone = self.copy() clone.update(other) return clone
if not isinstance(other, FreqDist): return False return set(self).issubset(other) and all(self[key] <= other[key] for key in self) if not isinstance(other, FreqDist): return False return self <= other and self != other if not isinstance(other, FreqDist): return False return other <= self if not isinstance(other, FreqDist): return False return other < self
""" Return a string representation of this FreqDist.
:rtype: string """
""" Return a string representation of this FreqDist.
:rtype: string """ items = ['%r: %r' % (s, self[s]) for s in self.keys()[:10]] if len(self) > 10: items.append('...') return '<FreqDist: %s>' % ', '.join(items)
##////////////////////////////////////////////////////// ## Probability Distributions ##//////////////////////////////////////////////////////
""" A probability distribution for the outcomes of an experiment. A probability distribution specifies how likely it is that an experiment will have any given outcome. For example, a probability distribution could be used to predict the probability that a token in a document will have a given type. Formally, a probability distribution can be defined as a function mapping from samples to nonnegative real numbers, such that the sum of every number in the function's range is 1.0. A ``ProbDist`` is often used to model the probability distribution of the experiment used to generate a frequency distribution. """ """True if the probabilities of the samples in this probability distribution will always sum to one."""
if self.__class__ == ProbDistI: raise NotImplementedError("Interfaces can't be instantiated")
""" Return the probability for a given sample. Probabilities are always real numbers in the range [0, 1].
:param sample: The sample whose probability should be returned. :type sample: any :rtype: float """ raise NotImplementedError()
""" Return the base 2 logarithm of the probability for a given sample.
:param sample: The sample whose probability should be returned. :type sample: any :rtype: float """ # Default definition, in terms of prob() # Use some approximation to infinity. What this does # depends on your system's float implementation. return _NINF else:
""" Return the sample with the greatest probability. If two or more samples have the same probability, return one of them; which sample is returned is undefined.
:rtype: any """ raise NotImplementedError()
""" Return a list of all samples that have nonzero probabilities. Use ``prob`` to find the probability of each sample.
:rtype: list """ raise NotImplementedError()
# cf self.SUM_TO_ONE """ Return the ratio by which counts are discounted on average: c*/c
:rtype: float """ return 0.0
# Subclasses should define more efficient implementations of this, # where possible. """ Return a randomly selected sample from this probability distribution. The probability of returning each sample ``samp`` is equal to ``self.prob(samp)``. """ p = random.random() for sample in self.samples(): p -= self.prob(sample) if p <= 0: return sample # allow for some rounding error: if p < .0001: return sample # we *should* never get here if self.SUM_TO_ONE: warnings.warn("Probability distribution %r sums to %r; generate()" " is returning an arbitrary sample." % (self, 1-p)) return random.choice(list(self.samples()))
""" A probability distribution that assigns equal probability to each sample in a given set; and a zero probability to all other samples. """ """ Construct a new uniform probability distribution, that assigns equal probability to each sample in ``samples``.
:param samples: The samples that should be given uniform probability. :type samples: list :raise ValueError: If ``samples`` is empty. """ if len(samples) == 0: raise ValueError('A Uniform probability distribution must '+ 'have at least one sample.') self._sampleset = set(samples) self._prob = 1.0/len(self._sampleset) self._samples = list(self._sampleset)
if sample in self._sampleset: return self._prob else: return 0 return '<UniformProbDist with %d samples>' % len(self._sampleset)
""" A probability distribution whose probabilities are directly specified by a given dictionary. The given dictionary maps samples to probabilities. """ """ Construct a new probability distribution from the given dictionary, which maps values to probabilities (or to log probabilities, if ``log`` is true). If ``normalize`` is true, then the probability values are scaled by a constant factor such that they sum to 1.
If called without arguments, the resulting probability distribution assigns zero probabiliy to all values. """ self._prob_dict = {} else:
# Normalize the distribution, if requested. for x in prob_dict: self._prob_dict[x] = logp else: else: p = 1.0/len(prob_dict) for x in prob_dict: self._prob_dict[x] = p else:
else:
return self._prob_dict.get(sample, _NINF) else:
return '<ProbDist with %d samples>' % len(self._prob_dict)
""" The maximum likelihood estimate for the probability distribution of the experiment used to generate a frequency distribution. The "maximum likelihood estimate" approximates the probability of each sample as the frequency of that sample in the frequency distribution. """ """ Use the maximum likelihood estimate to create a probability distribution for the experiment used to generate ``freqdist``.
:type freqdist: FreqDist :param freqdist: The frequency distribution that the probability estimates should be based on. """
""" Return the frequency distribution that this probability distribution is based on.
:rtype: FreqDist """ return self._freqdist
return self._freqdist.max()
""" :rtype: str :return: A string representation of this ``ProbDist``. """ return '<MLEProbDist based on %d samples>' % self._freqdist.N()
""" The Lidstone estimate for the probability distribution of the experiment used to generate a frequency distribution. The "Lidstone estimate" is paramaterized by a real number *gamma*, which typically ranges from 0 to 1. The Lidstone estimate approximates the probability of a sample with count *c* from an experiment with *N* outcomes and *B* bins as ``c+gamma)/(N+B*gamma)``. This is equivalant to adding *gamma* to the count for each bin, and taking the maximum likelihood estimate of the resulting frequency distribution. """ """ Use the Lidstone estimate to create a probability distribution for the experiment used to generate ``freqdist``.
:type freqdist: FreqDist :param freqdist: The frequency distribution that the probability estimates should be based on. :type gamma: float :param gamma: A real number used to paramaterize the estimate. The Lidstone estimate is equivalant to adding *gamma* to the count for each bin, and taking the maximum likelihood estimate of the resulting frequency distribution. :type bins: int :param bins: The number of sample values that can be generated by the experiment that is described by the probability distribution. This value must be correctly set for the probabilities of the sample values to sum to one. If ``bins`` is not specified, it defaults to ``freqdist.B()``. """ name = self.__class__.__name__[:-8] raise ValueError('A %s probability distribution ' % name + 'must have at least one bin.') name = self.__class__.__name__[:-8] raise ValueError('\nThe number of bins in a %s distribution ' % name + '(%d) must be greater than or equal to\n' % bins + 'the number of bins in the FreqDist used ' + 'to create it (%d).' % freqdist.N())
# In extreme cases we force the probability to be 0, # which it will be, since the count will be 0: self._gamma = 0 self._divisor = 1
""" Return the frequency distribution that this probability distribution is based on.
:rtype: FreqDist """ return self._freqdist
# For Lidstone distributions, probability is monotonic with # frequency, so the most probable sample is the one that # occurs most frequently. return self._freqdist.max()
""" Return a string representation of this ``ProbDist``.
:rtype: str """ return '<LidstoneProbDist based on %d samples>' % self._freqdist.N()
""" The Laplace estimate for the probability distribution of the experiment used to generate a frequency distribution. The "Laplace estimate" approximates the probability of a sample with count *c* from an experiment with *N* outcomes and *B* bins as *(c+1)/(N+B)*. This is equivalant to adding one to the count for each bin, and taking the maximum likelihood estimate of the resulting frequency distribution. """ """ Use the Laplace estimate to create a probability distribution for the experiment used to generate ``freqdist``.
:type freqdist: FreqDist :param freqdist: The frequency distribution that the probability estimates should be based on. :type bins: int :param bins: The number of sample values that can be generated by the experiment that is described by the probability distribution. This value must be correctly set for the probabilities of the sample values to sum to one. If ``bins`` is not specified, it defaults to ``freqdist.B()``. """ LidstoneProbDist.__init__(self, freqdist, 1, bins)
""" :rtype: str :return: A string representation of this ``ProbDist``. """ return '<LaplaceProbDist based on %d samples>' % self._freqdist.N()
""" The expected likelihood estimate for the probability distribution of the experiment used to generate a frequency distribution. The "expected likelihood estimate" approximates the probability of a sample with count *c* from an experiment with *N* outcomes and *B* bins as *(c+0.5)/(N+B/2)*. This is equivalant to adding 0.5 to the count for each bin, and taking the maximum likelihood estimate of the resulting frequency distribution. """ """ Use the expected likelihood estimate to create a probability distribution for the experiment used to generate ``freqdist``.
:type freqdist: FreqDist :param freqdist: The frequency distribution that the probability estimates should be based on. :type bins: int :param bins: The number of sample values that can be generated by the experiment that is described by the probability distribution. This value must be correctly set for the probabilities of the sample values to sum to one. If ``bins`` is not specified, it defaults to ``freqdist.B()``. """
""" Return a string representation of this ``ProbDist``.
:rtype: str """ return '<ELEProbDist based on %d samples>' % self._freqdist.N()
""" The heldout estimate for the probability distribution of the experiment used to generate two frequency distributions. These two frequency distributions are called the "heldout frequency distribution" and the "base frequency distribution." The "heldout estimate" uses uses the "heldout frequency distribution" to predict the probability of each sample, given its frequency in the "base frequency distribution".
In particular, the heldout estimate approximates the probability for a sample that occurs *r* times in the base distribution as the average frequency in the heldout distribution of all samples that occur *r* times in the base distribution.
This average frequency is *Tr[r]/(Nr[r].N)*, where:
- *Tr[r]* is the total count in the heldout distribution for all samples that occur *r* times in the base distribution. - *Nr[r]* is the number of samples that occur *r* times in the base distribution. - *N* is the number of outcomes recorded by the heldout frequency distribution.
In order to increase the efficiency of the ``prob`` member function, *Tr[r]/(Nr[r].N)* is precomputed for each value of *r* when the ``HeldoutProbDist`` is created.
:type _estimate: list(float) :ivar _estimate: A list mapping from *r*, the number of times that a sample occurs in the base distribution, to the probability estimate for that sample. ``_estimate[r]`` is calculated by finding the average frequency in the heldout distribution of all samples that occur *r* times in the base distribution. In particular, ``_estimate[r]`` = *Tr[r]/(Nr[r].N)*. :type _max_r: int :ivar _max_r: The maximum number of times that any sample occurs in the base distribution. ``_max_r`` is used to decide how large ``_estimate`` must be. """ """ Use the heldout estimate to create a probability distribution for the experiment used to generate ``base_fdist`` and ``heldout_fdist``.
:type base_fdist: FreqDist :param base_fdist: The base frequency distribution. :type heldout_fdist: FreqDist :param heldout_fdist: The heldout frequency distribution. :type bins: int :param bins: The number of sample values that can be generated by the experiment that is described by the probability distribution. This value must be correctly set for the probabilities of the sample values to sum to one. If ``bins`` is not specified, it defaults to ``freqdist.B()``. """
self._base_fdist = base_fdist self._heldout_fdist = heldout_fdist
# The max number of times any sample occurs in base_fdist. self._max_r = base_fdist[base_fdist.max()]
# Calculate Tr, Nr, and N. Tr = self._calculate_Tr() Nr = [base_fdist.Nr(r, bins) for r in range(self._max_r+1)] N = heldout_fdist.N()
# Use Tr, Nr, and N to compute the probability estimate for # each value of r. self._estimate = self._calculate_estimate(Tr, Nr, N)
""" Return the list *Tr*, where *Tr[r]* is the total count in ``heldout_fdist`` for all samples that occur *r* times in ``base_fdist``.
:rtype: list(float) """ Tr = [0.0] * (self._max_r+1) for sample in self._heldout_fdist: r = self._base_fdist[sample] Tr[r] += self._heldout_fdist[sample] return Tr
""" Return the list *estimate*, where *estimate[r]* is the probability estimate for any sample that occurs *r* times in the base frequency distribution. In particular, *estimate[r]* is *Tr[r]/(N[r].N)*. In the special case that *N[r]=0*, *estimate[r]* will never be used; so we define *estimate[r]=None* for those cases.
:rtype: list(float) :type Tr: list(float) :param Tr: the list *Tr*, where *Tr[r]* is the total count in the heldout distribution for all samples that occur *r* times in base distribution. :type Nr: list(float) :param Nr: The list *Nr*, where *Nr[r]* is the number of samples that occur *r* times in the base distribution. :type N: int :param N: The total number of outcomes recorded by the heldout frequency distribution. """ estimate = [] for r in range(self._max_r+1): if Nr[r] == 0: estimate.append(None) else: estimate.append(Tr[r]/(Nr[r]*N)) return estimate
""" Return the base frequency distribution that this probability distribution is based on.
:rtype: FreqDist """ return self._base_fdist
""" Return the heldout frequency distribution that this probability distribution is based on.
:rtype: FreqDist """ return self._heldout_fdist
return self._base_fdist.keys()
# Use our precomputed probability estimate. r = self._base_fdist[sample] return self._estimate[r]
# Note: the Heldout estimation is *not* necessarily monotonic; # so this implementation is currently broken. However, it # should give the right answer *most* of the time. :) return self._base_fdist.max()
raise NotImplementedError()
""" :rtype: str :return: A string representation of this ``ProbDist``. """ s = '<HeldoutProbDist: %d base samples; %d heldout samples>' return s % (self._base_fdist.N(), self._heldout_fdist.N())
""" The cross-validation estimate for the probability distribution of the experiment used to generate a set of frequency distribution. The "cross-validation estimate" for the probability of a sample is found by averaging the held-out estimates for the sample in each pair of frequency distributions. """ """ Use the cross-validation estimate to create a probability distribution for the experiment used to generate ``freqdists``.
:type freqdists: list(FreqDist) :param freqdists: A list of the frequency distributions generated by the experiment. :type bins: int :param bins: The number of sample values that can be generated by the experiment that is described by the probability distribution. This value must be correctly set for the probabilities of the sample values to sum to one. If ``bins`` is not specified, it defaults to ``freqdist.B()``. """ self._freqdists = freqdists
# Create a heldout probability distribution for each pair of # frequency distributions in freqdists. self._heldout_probdists = [] for fdist1 in freqdists: for fdist2 in freqdists: if fdist1 is not fdist2: probdist = HeldoutProbDist(fdist1, fdist2, bins) self._heldout_probdists.append(probdist)
""" Return the list of frequency distributions that this ``ProbDist`` is based on.
:rtype: list(FreqDist) """ return self._freqdists
# [xx] nb: this is not too efficient return set(sum([fd.keys() for fd in self._freqdists], []))
# Find the average probability estimate returned by each # heldout distribution. prob = 0.0 for heldout_probdist in self._heldout_probdists: prob += heldout_probdist.prob(sample) return prob/len(self._heldout_probdists)
raise NotImplementedError()
""" Return a string representation of this ``ProbDist``.
:rtype: str """ return '<CrossValidationProbDist: %d-way>' % len(self._freqdists)
""" The Witten-Bell estimate of a probability distribution. This distribution allocates uniform probability mass to as yet unseen events by using the number of events that have only been seen once. The probability mass reserved for unseen events is equal to *T / (N + T)* where *T* is the number of observed event types and *N* is the total number of observed events. This equates to the maximum likelihood estimate of a new type event occurring. The remaining probability mass is discounted such that all probability estimates sum to one, yielding:
- *p = T / Z (N + T)*, if count = 0 - *p = c / (N + T)*, otherwise """
""" Creates a distribution of Witten-Bell probability estimates. This distribution allocates uniform probability mass to as yet unseen events by using the number of events that have only been seen once. The probability mass reserved for unseen events is equal to *T / (N + T)* where *T* is the number of observed event types and *N* is the total number of observed events. This equates to the maximum likelihood estimate of a new type event occurring. The remaining probability mass is discounted such that all probability estimates sum to one, yielding:
- *p = T / Z (N + T)*, if count = 0 - *p = c / (N + T)*, otherwise
The parameters *T* and *N* are taken from the ``freqdist`` parameter (the ``B()`` and ``N()`` values). The normalising factor *Z* is calculated using these values along with the ``bins`` parameter.
:param freqdist: The frequency counts upon which to base the estimation. :type freqdist: FreqDist :param bins: The number of possible event types. This must be at least as large as the number of bins in the ``freqdist``. If None, then it's assumed to be equal to that of the ``freqdist`` :type bins: int """ assert bins is None or bins >= freqdist.B(),\ 'Bins parameter must not be less than freqdist.B()' if bins is None: bins = freqdist.B() self._freqdist = freqdist self._T = self._freqdist.B() self._Z = bins - self._freqdist.B() self._N = self._freqdist.N() # self._P0 is P(0), precalculated for efficiency: if self._N==0: # if freqdist is empty, we approximate P(0) by a UniformProbDist: self._P0 = 1.0 / self._Z else: self._P0 = self._T / float(self._Z * (self._N + self._T))
# inherit docs from ProbDistI c = self._freqdist[sample] if c == 0: return self._P0 else: return c / float(self._N + self._T)
return self._freqdist.max()
return self._freqdist.keys()
return self._freqdist
raise NotImplementedError()
""" Return a string representation of this ``ProbDist``.
:rtype: str """ return '<WittenBellProbDist based on %d samples>' % self._freqdist.N()
##////////////////////////////////////////////////////// ## Good-Turing Probablity Distributions ##//////////////////////////////////////////////////////
# Good-Turing frequency estimation was contributed by Alan Turing and # his statistical assistant I.J. Good, during their collaboration in # the WWII. It is a statistical technique for predicting the # probability of occurrence of objects belonging to an unknown number # of species, given past observations of such objects and their # species. (In drawing balls from an urn, the 'objects' would be balls # and the 'species' would be the distinct colors of the balls (finite # but unknown in number). # # The situation frequency zero is quite common in the original # Good-Turing estimation. Bill Gale and Geoffrey Sampson present a # simple and effective approach, Simple Good-Turing. As a smoothing # curve they simply use a power curve: # # Nr = a*r^b (with b < -1 to give the appropriate hyperbolic # relationsihp) # # They estimate a and b by simple linear regression technique on the # logarithmic form of the equation: # # log Nr = a + b*log(r) # # However, they suggest that such a simple curve is probably only # appropriate for high values of r. For low values of r, they use the # measured Nr directly. (see M&S, p.213) # # Gale and Sampson propose to use r while the difference between r and # r* is 1.96 greather than the standar deviation, and switch to r* if # it is less or equal: # # |r - r*| > 1.96 * sqrt((r + 1)^2 (Nr+1 / Nr^2) (1 + Nr+1 / Nr)) # # The 1.96 coefficient correspond to a 0.05 significance criterion, # some implementations can use a coefficient of 1.65 for a 0.1 # significance criterion. #
""" The Good-Turing estimate of a probability distribution. This method calculates the probability mass to assign to events with zero or low counts based on the number of events with higher counts. It does so by using the smoothed count *c\**:
- *c\* = (c + 1) N(c + 1) / N(c)* for c >= 1 - *things with frequency zero in training* = N(1) for c == 0
where *c* is the original count, *N(i)* is the number of event types observed with count *i*. We can think the count of unseen as the count of frequency one (see Jurafsky & Martin 2nd Edition, p101). """
""" :param freqdist: The frequency counts upon which to base the estimation. :type freqdist: FreqDist :param bins: The number of possible event types. This must be at least as large as the number of bins in the ``freqdist``. If None, then it's assumed to be equal to that of the ``freqdist`` :type bins: int """ assert bins is None or bins >= freqdist.B(),\ 'Bins parameter must not be less than freqdist.B()' if bins is None: bins = freqdist.B() self._freqdist = freqdist self._bins = bins
count = self._freqdist[sample]
# unseen sample's frequency (count zero) uses frequency one's if count == 0 and self._freqdist.N() != 0: p0 = 1.0 * self._freqdist.Nr(1) / self._freqdist.N() if self._bins == self._freqdist.B(): p0 = 0.0 else: p0 = p0 / (1.0 * self._bins - self._freqdist.B())
nc = self._freqdist.Nr(count) ncn = self._freqdist.Nr(count + 1)
# avoid divide-by-zero errors for sparse datasets if nc == 0 or self._freqdist.N() == 0: return 0
return 1.0 * (count + 1) * ncn / (nc * self._freqdist.N())
return self._freqdist.max()
return self._freqdist.keys()
""" :return: The probability mass transferred from the seen samples to the unseen samples. :rtype: float """ return 1.0 * self._freqdist.Nr(1) / self._freqdist.N()
return self._freqdist
""" Return a string representation of this ``ProbDist``.
:rtype: str """ return '<GoodTuringProbDist based on %d samples>' % self._freqdist.N()
##////////////////////////////////////////////////////// ## Simple Good-Turing Probablity Distributions ##//////////////////////////////////////////////////////
""" SimpleGoodTuring ProbDist approximates from frequency to freqency of frequency into a linear line under log space by linear regression. Details of Simple Good-Turing algorithm can be found in:
- Good Turing smoothing without tears" (Gale & Sampson 1995), Journal of Quantitative Linguistics, vol. 2 pp. 217-237. - "Speech and Language Processing (Jurafsky & Martin), 2nd Edition, Chapter 4.5 p103 (log(Nc) = a + b*log(c)) - http://www.grsampson.net/RGoodTur.html
Given a set of pair (xi, yi), where the xi denotes the freqency and yi denotes the freqency of freqency, we want to minimize their square variation. E(x) and E(y) represent the mean of xi and yi.
- slope: b = sigma ((xi-E(x)(yi-E(y))) / sigma ((xi-E(x))(xi-E(x))) - intercept: a = E(y) - b.E(x) """ """ :param freqdist: The frequency counts upon which to base the estimation. :type freqdist: FreqDist :param bins: The number of possible event types. This must be larger than the number of bins in the ``freqdist``. If None, then it's assumed to be equal to ``freqdist``.B() + 1 :type bins: int """ assert bins is None or bins > freqdist.B(),\ 'Bins parameter must not be less than freqdist.B() + 1' if bins is None: bins = freqdist.B() + 1 self._freqdist = freqdist self._bins = bins r, nr = self._r_Nr() self.find_best_fit(r, nr) self._switch(r, nr) self._renormalize(r, nr)
""" Split the frequency distribution in two list (r, Nr), where Nr(r) > 0 """ r, nr = [], [] b, i = 0, 0 while b != self._freqdist.B(): nr_i = self._freqdist.Nr(i) if nr_i > 0: b += nr_i r.append(i) nr.append(nr_i) i += 1 return (r, nr)
""" Use simple linear regression to tune parameters self._slope and self._intercept in the log-log space based on count and Nr(count) (Work in log space to avoid floating point underflow.) """ # For higher sample frequencies the data points becomes horizontal # along line Nr=1. To create a more evident linear model in log-log # space, we average positive Nr values with the surrounding zero # values. (Church and Gale, 1991)
if not r or not nr: # Empty r or nr? return
zr = [] for j in range(len(r)): if j > 0: i = r[j-1] else: i = 0 if j != len(r) - 1: k = r[j+1] else: k = 2 * r[j] - i zr_ = 2.0 * nr[j] / (k - i) zr.append(zr_)
log_r = [math.log(i) for i in r] log_zr = [math.log(i) for i in zr]
xy_cov = x_var = 0.0 x_mean = 1.0 * sum(log_r) / len(log_r) y_mean = 1.0 * sum(log_zr) / len(log_zr) for (x, y) in zip(log_r, log_zr): xy_cov += (x - x_mean) * (y - y_mean) x_var += (x - x_mean)**2 if x_var != 0: self._slope = xy_cov / x_var else: self._slope = 0.0 self._intercept = y_mean - self._slope * x_mean
""" Calculate the r frontier where we must switch from Nr to Sr when estimating E[Nr]. """ for i, r_ in enumerate(r): if len(r) == i + 1 or r[i+1] != r_ + 1: # We are at the end of r, or there is a gap in r self._switch_at = r_ break
Sr = self.smoothedNr smooth_r_star = (r_ + 1) * Sr(r_+1) / Sr(r_) unsmooth_r_star = 1.0 * (r_ + 1) * nr[i+1] / nr[i]
std = math.sqrt(self._variance(r_, nr[i], nr[i+1])) if abs(unsmooth_r_star-smooth_r_star) <= 1.96 * std: self._switch_at = r_ break
r = float(r) nr = float(nr) nr_1 = float(nr_1) return (r + 1.0)**2 * (nr_1 / nr**2) * (1.0 + nr_1 / nr)
""" It is necessary to renormalize all the probability estimates to ensure a proper probability distribution results. This can be done by keeping the estimate of the probability mass for unseen items as N(1)/N and renormalizing all the estimates for previously seen items (as Gale and Sampson (1995) propose). (See M&S P.213, 1999) """ prob_cov = 0.0 for r_, nr_ in zip(r, nr): prob_cov += nr_ * self._prob_measure(r_) if prob_cov: self._renormal = (1 - self._prob_measure(0)) / prob_cov
""" Return the number of samples with count r.
:param r: The amount of freqency. :type r: int :rtype: float """
# Nr = a*r^b (with b < -1 to give the appropriate hyperbolic # relationship) # Estimate a and b by simple linear regression technique on # the logarithmic form of the equation: log Nr = a + b*log(r)
return math.exp(self._intercept + self._slope * math.log(r))
""" Return the sample's probability.
:param sample: sample of the event :type sample: str :rtype: float """ count = self._freqdist[sample] p = self._prob_measure(count) if count == 0: if self._bins == self._freqdist.B(): p = 0.0 else: p = p / (1.0 * self._bins - self._freqdist.B()) else: p = p * self._renormal return p
if count == 0 and self._freqdist.N() == 0 : return 1.0 elif count == 0 and self._freqdist.N() != 0: return 1.0 * self._freqdist.Nr(1) / self._freqdist.N()
if self._switch_at > count: Er_1 = 1.0 * self._freqdist.Nr(count+1) Er = 1.0 * self._freqdist.Nr(count) else: Er_1 = self.smoothedNr(count+1) Er = self.smoothedNr(count)
r_star = (count + 1) * Er_1 / Er return r_star / self._freqdist.N()
prob_sum = 0.0 for i in range(0, len(self._Nr)): prob_sum += self._Nr[i] * self._prob_measure(i) / self._renormal print("Probability Sum:", prob_sum) #assert prob_sum != 1.0, "probability sum should be one!"
""" This function returns the total mass of probability transfers from the seen samples to the unseen samples. """ return 1.0 * self.smoothedNr(1) / self._freqdist.N()
return self._freqdist.max()
return self._freqdist.keys()
return self._freqdist
""" Return a string representation of this ``ProbDist``.
:rtype: str """ return '<SimpleGoodTuringProbDist based on %d samples>'\ % self._freqdist.N()
""" An mutable probdist where the probabilities may be easily modified. This simply copies an existing probdist, storing the probability values in a mutable dictionary and providing an update method. """
""" Creates the mutable probdist based on the given prob_dist and using the list of samples given. These values are stored as log probabilities if the store_logs flag is set.
:param prob_dist: the distribution from which to garner the probabilities :type prob_dist: ProbDist :param samples: the complete set of samples :type samples: sequence of any :param store_logs: whether to store the probabilities as logarithms :type store_logs: bool """ try: import numpy except ImportError: print("Error: Please install numpy; for instructions see http://www.nltk.org/") exit() self._samples = samples self._sample_dict = dict((samples[i], i) for i in range(len(samples))) self._data = numpy.zeros(len(samples), numpy.float64) for i in range(len(samples)): if store_logs: self._data[i] = prob_dist.logprob(samples[i]) else: self._data[i] = prob_dist.prob(samples[i]) self._logs = store_logs
# inherit documentation return self._samples
# inherit documentation i = self._sample_dict.get(sample) if i is not None: if self._logs: return 2**(self._data[i]) else: return self._data[i] else: return 0.0
# inherit documentation i = self._sample_dict.get(sample) if i is not None: if self._logs: return self._data[i] else: return math.log(self._data[i], 2) else: return float('-inf')
""" Update the probability for the given sample. This may cause the object to stop being the valid probability distribution - the user must ensure that they update the sample probabilities such that all samples have probabilities between 0 and 1 and that all probabilities sum to one.
:param sample: the sample for which to update the probability :type sample: any :param prob: the new probability :type prob: float :param log: is the probability already logged :type log: bool """ i = self._sample_dict.get(sample) assert i is not None if self._logs: if log: self._data[i] = prob else: self._data[i] = math.log(prob, 2) else: if log: self._data[i] = 2**(prob) else: self._data[i] = prob
##////////////////////////////////////////////////////// ## Probability Distribution Operations ##//////////////////////////////////////////////////////
if (not isinstance(test_pdist, ProbDistI) or not isinstance(actual_pdist, ProbDistI)): raise ValueError('expected a ProbDist.') # Is this right? return sum(actual_pdist.prob(s) * math.log(test_pdist.prob(s), 2) for s in actual_pdist)
##////////////////////////////////////////////////////// ## Conditional Distributions ##//////////////////////////////////////////////////////
""" A collection of frequency distributions for a single experiment run under different conditions. Conditional frequency distributions are used to record the number of times each sample occurred, given the condition under which the experiment was run. For example, a conditional frequency distribution could be used to record the frequency of each word (type) in a document, given its length. Formally, a conditional frequency distribution can be defined as a function that maps from each condition to the FreqDist for the experiment under that condition.
Conditional frequency distributions are typically constructed by repeatedly running an experiment under a variety of conditions, and incrementing the sample outcome counts for the appropriate conditions. For example, the following code will produce a conditional frequency distribution that encodes how often each word type occurs, given the length of that word type:
>>> from nltk.probability import ConditionalFreqDist >>> from nltk.tokenize import word_tokenize >>> sent = "the the the dog dog some other words that we do not care about" >>> cfdist = ConditionalFreqDist() >>> for word in word_tokenize(sent): ... condition = len(word) ... cfdist[condition].inc(word)
An equivalent way to do this is with the initializer:
>>> cfdist = ConditionalFreqDist((len(word), word) for word in word_tokenize(sent))
The frequency distribution for each condition is accessed using the indexing operator:
>>> cfdist[3] <FreqDist with 6 outcomes> >>> cfdist[3].freq('the') 0.5 >>> cfdist[3]['dog'] 2
When the indexing operator is used to access the frequency distribution for a condition that has not been accessed before, ``ConditionalFreqDist`` creates a new empty FreqDist for that condition.
""" """ Construct a new empty conditional frequency distribution. In particular, the count for every sample, under every condition, is zero.
:param cond_samples: The samples to initialize the conditional frequency distribution with :type cond_samples: Sequence of (condition, sample) tuples """
""" Return a list of the conditions that have been accessed for this ``ConditionalFreqDist``. Use the indexing operator to access the frequency distribution for a given condition. Note that the frequency distributions for some conditions may contain zero sample outcomes.
:rtype: list """ return sorted(self.keys())
""" Return the total number of sample outcomes that have been recorded by this ``ConditionalFreqDist``.
:rtype: int """ return sum(fdist.N() for fdist in compat.itervalues(self))
""" Plot the given samples from the conditional frequency distribution. For a cumulative plot, specify cumulative=True. (Requires Matplotlib to be installed.)
:param samples: The samples to plot :type samples: list :param title: The title for the graph :type title: str :param conditions: The conditions to plot (default is all) :type conditions: list """ try: import pylab except ImportError: raise ValueError('The plot function requires the matplotlib package (aka pylab).' 'See http://matplotlib.sourceforge.net/')
cumulative = _get_kwarg(kwargs, 'cumulative', False) conditions = _get_kwarg(kwargs, 'conditions', self.conditions()) title = _get_kwarg(kwargs, 'title', '') samples = _get_kwarg(kwargs, 'samples', sorted(set(v for c in conditions for v in self[c]))) # this computation could be wasted if not "linewidth" in kwargs: kwargs["linewidth"] = 2
for condition in conditions: if cumulative: freqs = list(self[condition]._cumulative_frequencies(samples)) ylabel = "Cumulative Counts" legend_loc = 'lower right' else: freqs = [self[condition][sample] for sample in samples] ylabel = "Counts" legend_loc = 'upper right' # percents = [f * 100 for f in freqs] only in ConditionalProbDist? kwargs['label'] = str(condition) pylab.plot(freqs, *args, **kwargs)
pylab.legend(loc=legend_loc) pylab.grid(True, color="silver") pylab.xticks(range(len(samples)), [unicode(s) for s in samples], rotation=90) if title: pylab.title(title) pylab.xlabel("Samples") pylab.ylabel(ylabel) pylab.show()
""" Tabulate the given samples from the conditional frequency distribution.
:param samples: The samples to plot :type samples: list :param title: The title for the graph :type title: str :param conditions: The conditions to plot (default is all) :type conditions: list """
cumulative = _get_kwarg(kwargs, 'cumulative', False) conditions = _get_kwarg(kwargs, 'conditions', self.conditions()) samples = _get_kwarg(kwargs, 'samples', sorted(set(v for c in conditions for v in self[c]))) # this computation could be wasted
condition_size = max(len(str(c)) for c in conditions) print(' ' * condition_size, end=' ') for s in samples: print("%4s" % str(s), end=' ') print() for c in conditions: print("%*s" % (condition_size, str(c)), end=' ') if cumulative: freqs = list(self[c]._cumulative_frequencies(samples)) else: freqs = [self[c][sample] for sample in samples]
for f in freqs: print("%4d" % f, end=' ') print()
if not isinstance(other, ConditionalFreqDist): return False return set(self.conditions()).issubset(other.conditions()) \ and all(self[c] <= other[c] for c in self.conditions()) if not isinstance(other, ConditionalFreqDist): return False return self <= other and self != other if not isinstance(other, ConditionalFreqDist): return False return other <= self if not isinstance(other, ConditionalFreqDist): return False return other < self
""" Return a string representation of this ``ConditionalFreqDist``.
:rtype: str """ return '<ConditionalFreqDist with %d conditions>' % len(self)
""" A collection of probability distributions for a single experiment run under different conditions. Conditional probability distributions are used to estimate the likelihood of each sample, given the condition under which the experiment was run. For example, a conditional probability distribution could be used to estimate the probability of each word type in a document, given the length of the word type. Formally, a conditional probability distribution can be defined as a function that maps from each condition to the ``ProbDist`` for the experiment under that condition. """ raise NotImplementedError("Interfaces can't be instantiated")
""" Return a list of the conditions that are represented by this ``ConditionalProbDist``. Use the indexing operator to access the probability distribution for a given condition.
:rtype: list """ return self.keys()
""" Return a string representation of this ``ConditionalProbDist``.
:rtype: str """ return '<%s with %d conditions>' % (type(self).__name__, len(self))
""" A conditional probability distribution modelling the experiments that were used to generate a conditional frequency distribution. A ConditionalProbDist is constructed from a ``ConditionalFreqDist`` and a ``ProbDist`` factory:
- The ``ConditionalFreqDist`` specifies the frequency distribution for each condition. - The ``ProbDist`` factory is a function that takes a condition's frequency distribution, and returns its probability distribution. A ``ProbDist`` class's name (such as ``MLEProbDist`` or ``HeldoutProbDist``) can be used to specify that class's constructor.
The first argument to the ``ProbDist`` factory is the frequency distribution that it should model; and the remaining arguments are specified by the ``factory_args`` parameter to the ``ConditionalProbDist`` constructor. For example, the following code constructs a ``ConditionalProbDist``, where the probability distribution for each condition is an ``ELEProbDist`` with 10 bins:
>>> from nltk.probability import ConditionalProbDist, ELEProbDist >>> cpdist = ConditionalProbDist(cfdist, ELEProbDist, 10) >>> print(cpdist['run'].max()) 'NN' >>> print(cpdist['run'].prob('NN')) 0.0813 """ *factory_args, **factory_kw_args): """ Construct a new conditional probability distribution, based on the given conditional frequency distribution and ``ProbDist`` factory.
:type cfdist: ConditionalFreqDist :param cfdist: The ``ConditionalFreqDist`` specifying the frequency distribution for each condition. :type probdist_factory: class or function :param probdist_factory: The function or class that maps a condition's frequency distribution to its probability distribution. The function is called with the frequency distribution as its first argument, ``factory_args`` as its remaining arguments, and ``factory_kw_args`` as keyword arguments. :type factory_args: (any) :param factory_args: Extra arguments for ``probdist_factory``. These arguments are usually used to specify extra properties for the probability distributions of individual conditions, such as the number of bins they contain. :type factory_kw_args: (any) :param factory_kw_args: Extra keyword arguments for ``probdist_factory``. """ # self._probdist_factory = probdist_factory # self._cfdist = cfdist # self._factory_args = factory_args # self._factory_kw_args = factory_kw_args
*factory_args, **factory_kw_args) *factory_args, **factory_kw_args)
""" An alternative ConditionalProbDist that simply wraps a dictionary of ProbDists rather than creating these from FreqDists. """
""" :param probdist_dict: a dictionary containing the probdists indexed by the conditions :type probdist_dict: dict any -> probdist """ defaultdict.__init__(self, DictionaryProbDist) self.update(probdist_dict)
##////////////////////////////////////////////////////// ## Adding in log-space. ##//////////////////////////////////////////////////////
# If the difference is bigger than this, then just take the bigger one:
""" Given two numbers ``logx`` = *log(x)* and ``logy`` = *log(y)*, return *log(x+y)*. Conceptually, this is the same as returning ``log(2**(logx)+2**(logy))``, but the actual implementation avoids overflow errors that could result from direct computation. """ return logx
# Use some approximation to infinity. What this does # depends on your system's float implementation. else:
##////////////////////////////////////////////////////// ## Probabilistic Mix-in ##//////////////////////////////////////////////////////
""" A mix-in class to associate probabilities with other classes (trees, rules, etc.). To use the ``ProbabilisticMixIn`` class, define a new class that derives from an existing class and from ProbabilisticMixIn. You will need to define a new constructor for the new class, which explicitly calls the constructors of both its parent classes. For example:
>>> from nltk.probability import ProbabilisticMixIn >>> class A: ... def __init__(self, x, y): self.data = (x,y) ... >>> class ProbabilisticA(A, ProbabilisticMixIn): ... def __init__(self, x, y, **prob_kwarg): ... A.__init__(self, x, y) ... ProbabilisticMixIn.__init__(self, **prob_kwarg)
See the documentation for the ProbabilisticMixIn ``constructor<__init__>`` for information about the arguments it expects.
You should generally also redefine the string representation methods, the comparison methods, and the hashing method. """ """ Initialize this object's probability. This initializer should be called by subclass constructors. ``prob`` should generally be the first argument for those constructors.
:param prob: The probability associated with the object. :type prob: float :param logprob: The log of the probability associated with the object. :type logprob: float """ raise TypeError('Must specify either prob or logprob ' '(not both)') else: ProbabilisticMixIn.set_logprob(self, kwargs['logprob']) else:
""" Set the probability associated with this object to ``prob``.
:param prob: The new probability :type prob: float """
""" Set the log probability associated with this object to ``logprob``. I.e., set the probability associated with this object to ``2**(logprob)``.
:param logprob: The new log probability :type logprob: float """ self.__logprob = logprob self.__prob = None
""" Return the probability associated with this object.
:rtype: float """ self.__prob = 2**(self.__logprob)
""" Return ``log(p)``, where ``p`` is the probability associated with this object.
:rtype: float """ if self.__logprob is None: if self.__prob is None: return None self.__logprob = math.log(self.__prob, 2) return self.__logprob
raise ValueError('%s is immutable' % self.__class__.__name__) raise ValueError('%s is immutable' % self.__class__.__name__)
## Helper function for processing keyword arguments
if key in kwargs: arg = kwargs[key] del kwargs[key] else: arg = default return arg
##////////////////////////////////////////////////////// ## Demonstration ##//////////////////////////////////////////////////////
""" Create a new frequency distribution, with random samples. The samples are numbers from 1 to ``numsamples``, and are generated by summing two numbers, each of which has a uniform distribution. """ import random fdist = FreqDist() for x in range(numoutcomes): y = (random.randint(1, (1+numsamples)/2) + random.randint(0, numsamples/2)) fdist.inc(y) return fdist
""" Return the true probability distribution for the experiment ``_create_rand_fdist(numsamples, x)``. """ fdist = FreqDist() for x in range(1, (1+numsamples)/2+1): for y in range(0, numsamples/2+1): fdist.inc(x+y) return MLEProbDist(fdist)
""" A demonstration of frequency distributions and probability distributions. This demonstration creates three frequency distributions with, and uses them to sample a random process with ``numsamples`` samples. Each frequency distribution is sampled ``numoutcomes`` times. These three frequency distributions are then used to build six probability distributions. Finally, the probability estimates of these distributions are compared to the actual probability of each sample.
:type numsamples: int :param numsamples: The number of samples to use in each demo frequency distributions. :type numoutcomes: int :param numoutcomes: The total number of outcomes for each demo frequency distribution. These outcomes are divided into ``numsamples`` bins. :rtype: None """
# Randomly sample a stochastic process three times. fdist1 = _create_rand_fdist(numsamples, numoutcomes) fdist2 = _create_rand_fdist(numsamples, numoutcomes) fdist3 = _create_rand_fdist(numsamples, numoutcomes)
# Use our samples to create probability distributions. pdists = [ MLEProbDist(fdist1), LidstoneProbDist(fdist1, 0.5, numsamples), HeldoutProbDist(fdist1, fdist2, numsamples), HeldoutProbDist(fdist2, fdist1, numsamples), CrossValidationProbDist([fdist1, fdist2, fdist3], numsamples), GoodTuringProbDist(fdist1), SimpleGoodTuringProbDist(fdist1), SimpleGoodTuringProbDist(fdist1, 7), _create_sum_pdist(numsamples), ]
# Find the probability of each sample. vals = [] for n in range(1,numsamples+1): vals.append(tuple([n, fdist1.freq(n)] + [pdist.prob(n) for pdist in pdists]))
# Print the results in a formatted table. print(('%d samples (1-%d); %d outcomes were sampled for each FreqDist' % (numsamples, numsamples, numoutcomes))) print('='*9*(len(pdists)+2)) FORMATSTR = ' FreqDist '+ '%8s '*(len(pdists)-1) + '| Actual' print(FORMATSTR % tuple(repr(pdist)[1:9] for pdist in pdists[:-1])) print('-'*9*(len(pdists)+2)) FORMATSTR = '%3d %8.6f ' + '%8.6f '*(len(pdists)-1) + '| %8.6f' for val in vals: print(FORMATSTR % val)
# Print the totals for each column (should all be 1.0) zvals = list(zip(*vals)) sums = [sum(val) for val in zvals[1:]] print('-'*9*(len(pdists)+2)) FORMATSTR = 'Total ' + '%8.6f '*(len(pdists)) + '| %8.6f' print(FORMATSTR % tuple(sums)) print('='*9*(len(pdists)+2))
# Display the distributions themselves, if they're short enough. if len(repr(str(fdist1))) < 70: print(' fdist1:', str(fdist1)) print(' fdist2:', str(fdist2)) print(' fdist3:', str(fdist3)) print()
print('Generating:') for pdist in pdists: fdist = FreqDist(pdist.generate() for i in range(5000)) print('%20s %s' % (pdist.__class__.__name__[:20], str(fdist)[:55])) print()
from nltk import corpus emma_words = corpus.gutenberg.words('austen-emma.txt') fd = FreqDist(emma_words) gt = GoodTuringProbDist(fd) sgt = SimpleGoodTuringProbDist(fd) katz = SimpleGoodTuringProbDist(fd, 7) print('%18s %8s %12s %14s %12s' \ % ("word", "freqency", "GoodTuring", "SimpleGoodTuring", "Katz-cutoff" )) for key in fd: print('%18s %8d %12e %14e %12e' \ % (key, fd[key], gt.prob(key), sgt.prob(key), katz.prob(key)))
demo(6, 10) demo(5, 5000) gt_demo()
'ConditionalProbDistI', 'CrossValidationProbDist', 'DictionaryConditionalProbDist', 'DictionaryProbDist', 'ELEProbDist', 'FreqDist', 'GoodTuringProbDist', 'SimpleGoodTuringProbDist', 'HeldoutProbDist', 'ImmutableProbabilisticMixIn', 'LaplaceProbDist', 'LidstoneProbDist', 'MLEProbDist', 'MutableProbDist', 'ProbDistI', 'ProbabilisticMixIn', 'UniformProbDist', 'WittenBellProbDist', 'add_logs', 'log_likelihood', 'sum_logs', 'entropy'] |