Repository for a workshop on Bayesian statistics
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"Think Bayes", both by Allen B. Downey, available from greenteapress.com
Copyright 2014 Allen B. Downey
License: GNU GPLv3 http://www.gnu.org/licenses/gpl.html
� )�print_function�divisionN)�Counter)�
itemgetter)�stats)�special)�ndimage)�gamma)�open� c C s! t j | � t j j | � d S)zEInitialize the random and np.random generators.
x: int seed
N)�random�seed�np)�x� r �)/home/user/BayesMadeSimple/thinkbayes2.py�
RandomSeed2 s
r c C s" | d k r t d � S| d | S)aa Computes odds for a given probability.
Example: p=0.75 means 75 for and 25 against, or 3:1 odds in favor.
Note: when p=1, the formula for odds divides by zero, which is
normally undefined. But I think it is reasonable to define Odds(1)
to be infinity, so that's what this function does.
p: float 0-1
Returns: float odds
� �inf)�float)�pr r r �Odds; s
r c C s | | d S)z�Computes the probability corresponding to given odds.
Example: o=2 means 2:1 odds in favor, or 2/3 probability
o: float odds, strictly positive
Returns: float probability
r r )�or r r � ProbabilityM s r c C s | | | S)z�Computes the probability corresponding to given odds.
Example: yes=2, no=1 means 2:1 odds in favor, or 2/3 probability.
yes, no: int or float odds in favor
r )�yes�nor r r � Probability2Y s r c @ sF e Z d Z d Z d d � Z d d � Z d d � Z d d � Z d
S) � Interpolatorz�Represents a mapping between sorted sequences; performs linear interp.
Attributes:
xs: sorted list
ys: sorted list
c C s | | _ | | _ d S)N)�xs�ys)�selfr r r r r �__init__k s zInterpolator.__init__c C s | j | | j | j � S)z4Looks up x and returns the corresponding value of y.)�_Bisectr r )r r r r r �Lookupo s zInterpolator.Lookupc C s | j | | j | j � S)z4Looks up y and returns the corresponding value of x.)r" r r )r �yr r r �Reverses s zInterpolator.Reversec C s� | | d k r | d S| | d k r0 | d St j | | � } d | | | d | | | | d } | | d | d | | | | d } | S)zHelper function.r r g �?�����r& )�bisect)r r r r �i�fracr$ r r r r" w s **zInterpolator._BisectN)�__name__�
__module__� __qualname__�__doc__r! r# r% r" r r r r r c s
r �
_nolegend_c @ s� e Z d Z d Z d d d d � Z d d � Z d d � Z d d
� Z d d � Z d
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d d � Z d d � Z d d � Z
d d � Z d d � Z d d d � Z d d � Z d d d � Z d d! d" � Z d# d$ � Z d% d& � Z d' d( � Z d) d* � Z d+ d, � Z d- d. � Z d d/ d0 � Z d1 d2 � Z d3 d4 d5 � Z d6 d7 d8 � Z d9 d: � Z d; d< � Z d= d> � Z d? d@ � Z! dA dB dC � Z" dA dD dE � Z# d S)F� _DictWrapperz%An object that contains a dictionary.Nc C sB | d k r | n t | _ i | _ d | _ | d k r= d St | t t t f � rs | d k rg | n | j | _ t | t � r� | j j | j
� � nx t | t t t f � r� | j j | j � � nG t | t j
� r� | j j | j � j � � n | j j t | � � t | � d k r>t | t � r>| j � d S)z�Initializes the distribution.
obj: Hist, Pmf, Cdf, Pdf, dict, pandas Series, list of pairs
label: string label
NFr )�
DEFAULT_LABEL�label�d�log�
isinstancer/ �Cdf�Pdf�dict�update�items�Items�pandas�SeriesZ value_counts� iteritemsr �len�Pmf� Normalize)r �objr1 r r r r! � s !z_DictWrapper.__init__c C s
t | � S)N)�id)r r r r �__hash__� s z_DictWrapper.__hash__c C s= | j j } | j t k r2 d | t | j � f S| j Sd S)Nz%s(%s))� __class__r* r1 r0 �strr2 )r �clsr r r �__str__� s z_DictWrapper.__str__c C sY | j j } | j t k r2 d | t | j � f Sd | t | j � t | j � f Sd S)Nz%s(%s)z
%s(%s, %s))rD r* r1 r0 �reprr2 )r rF r r r �__repr__� s z_DictWrapper.__repr__c C s1 y | j | j k SWn t k
r, d SYn Xd S)NF)r2 �AttributeError)r �otherr r r �__eq__� s
z_DictWrapper.__eq__c C s
t | j � S)N)r> r2 )r r r r �__len__� s z_DictWrapper.__len__c C s
t | j � S)N)�iterr2 )r r r r �__iter__� s z_DictWrapper.__iter__c C s
t | j � S)z Returns an iterator over keys.)rN r2 )r r r r �iterkeys� s z_DictWrapper.iterkeysc C s
| | j k S)N)r2 )r �valuer r r � __contains__� s z_DictWrapper.__contains__c C s | j j | d � S)Nr )r2 �get)r rQ r r r � __getitem__� s z_DictWrapper.__getitem__c C s | | j | <d S)N)r2 )r rQ �probr r r � __setitem__� s z_DictWrapper.__setitem__c C s | j | =d S)N)r2 )r rQ r r r � __delitem__� s z_DictWrapper.__delitem__c C sF t j | � } t j | j � | _ | d k r6 | n | j | _ | S)z�Returns a copy.
Make a shallow copy of d. If you want a deep copy of d,
use copy.deepcopy on the whole object.
label: string label for the new Hist
returns: new _DictWrapper with the same type
N)�copyr2 r1 )r r1 �newr r r �Copy� s
z_DictWrapper.Copyc C sN | j � } | j j � x. | j � D] \ } } | j | | | � q& W| S)zmMultiplies the values by a factor.
factor: what to multiply by
Returns: new object
)rZ r2 �clearr: �Set)r �factorrY �valrU r r r �Scale� s
z_DictWrapper.Scalec C s� | j r t d � � d | _ | d k r6 | j � } xP | j j � D]? \ } } | rx | j | t j | | � � qF | j | � qF Wd S)z�Log transforms the probabilities.
Removes values with probability 0.
Normalizes so that the largest logprob is 0.
z&Pmf/Hist already under a log transformTN)r3 �
ValueError�MaxLiker2 r9 r\ �math�Remove)r �mr r r r r �Log� s z_DictWrapper.Logc C sw | j s t d � � d | _ | d k r6 | j � } x: | j j � D]) \ } } | j | t j | | � � qF Wd S)z�Exponentiates the probabilities.
m: how much to shift the ps before exponentiating
If m is None, normalizes so that the largest prob is 1.
z"Pmf/Hist not under a log transformFN)r3 r` ra r2 r9 r\ rb �exp)r rd r r r r r �Exp s z_DictWrapper.Expc C s | j S)zGets the dictionary.)r2 )r r r r �GetDict s z_DictWrapper.GetDictc C s
| | _ d S)zSets the dictionary.N)r2 )r r2 r r r �SetDict s z_DictWrapper.SetDictc C s
| j j � S)z�Gets an unsorted sequence of values.
Note: one source of confusion is that the keys of this
dictionary are the values of the Hist/Pmf, and the
values of the dictionary are frequencies/probabilities.
)r2 �keys)r r r r �Values! s z_DictWrapper.Valuesc C s
| j j � S)z6Gets an unsorted sequence of (value, freq/prob) pairs.)r2 r9 )r r r r r: * s z_DictWrapper.Itemsc s� d d � � t � f d d � | j � D� � rD d } t j | � y t | j j � � SWn t k
r| | j j � SYn Xd S)zvGets a sorted sequence of (value, freq/prob) pairs.
It items are unsortable, the result is unsorted.
c S s. y t j | � SWn t k
r) d SYn Xd S)NF)rb �isnan� TypeError)r r r r rl 3 s
z'_DictWrapper.SortedItems.<locals>.isnanc s g | ] } � | � � q Sr r )�.0r )rl r r �
<listcomp>9 s z,_DictWrapper.SortedItems.<locals>.<listcomp>z)Keys contain NaN, may not sort correctly.N)�anyrk �logging�warning�sortedr2 r9 rm )r �msgr )rl r � SortedItems. s %
z_DictWrapper.SortedItemsc K s t | j � � S)z�Generates a sequence of points suitable for plotting.
Note: options are ignored
Returns:
tuple of (sorted value sequence, freq/prob sequence)
)�zipru )r �optionsr r r �RenderB s z_DictWrapper.Renderc C s+ | d k r | n | j } t | d | �S)z Makes a Cdf.Nr1 )r1 r5 )r r1 r r r �MakeCdfL s z_DictWrapper.MakeCdfc C s. x' | j � D] \ } } t | | � q
Wd S)z5Prints the values and freqs/probs in ascending order.N)ru �print)r r^ rU r r r �PrintQ s z_DictWrapper.Printr c C s | | j | <d S)z�Sets the freq/prob associated with the value x.
Args:
x: number value
y: number freq or prob
N)r2 )r r r$ r r r r\ V s z_DictWrapper.Setr c C s$ | j j | d � | | j | <d S)z�Increments the freq/prob associated with the value x.
Args:
x: number value
term: how much to increment by
r N)r2 rS )r r �termr r r �Incr_ s z_DictWrapper.Incrc C s$ | j j | d � | | j | <d S)z�Scales the freq/prob associated with the value x.
Args:
x: number value
factor: how much to multiply by
r N)r2 rS )r r r] r r r �Multh s z_DictWrapper.Multc C s | j | =d S)zRemoves a value.
Throws an exception if the value is not there.
Args:
x: value to remove
N)r2 )r r r r r rc q s z_DictWrapper.Removec C s t | j j � � } | S)z>Returns the total of the frequencies/probabilities in the map.)�sumr2 �values)r �totalr r r �Total{ s z_DictWrapper.Totalc C s t | j j � � S)z5Returns the largest frequency/probability in the map.)�maxr2 r� )r r r r ra � s z_DictWrapper.MaxLike�
c C s# t | j j � d d �d | � S)zhReturns the largest n values, with frequency/probability.
n: number of items to return
�reverseTN)rs r2 r9 )r �nr r r �Largest� s z_DictWrapper.Largestc C s# t | j j � d d �d | � S)ziReturns the smallest n values, with frequency/probability.
n: number of items to return
r� FN)rs r2 r9 )r r� r r r �Smallest� s z_DictWrapper.Smallest)$r* r+ r, r- r! rC rG rI rL rM rO rP rR rT rV rW rZ r_ re rg rh ri rk r: ru rx ry r{ r\ r} r~ rc r� ra r� r� r r r r r/ � sB
r/ c @ sF e Z d Z d Z d d � Z d d � Z d d � Z d d � Z d
S) �Histz�Represents a histogram, which is a map from values to frequencies.
Values can be any hashable type; frequencies are integer counters.
c C s | j j | d � S)z�Gets the frequency associated with the value x.
Args:
x: number value
Returns:
int frequency
r )r2 rS )r r r r r �Freq� s z Hist.Freqc s � f d d � | D� S)z*Gets frequencies for a sequence of values.c s g | ] } � j | � � q Sr )r� )rn r )r r r ro � s z Hist.Freqs.<locals>.<listcomp>r )r r r )r r �Freqs� s z
Hist.Freqsc C s: x3 | j � D]% \ } } | | j | � k r
d Sq
Wd S)zfChecks whether the values in this histogram are a subset of
the values in the given histogram.FT)r: r� )r rK r^ �freqr r r �IsSubset� s z
Hist.IsSubsetc C s2 x+ | j � D] \ } } | j | | � q
Wd S)z@Subtracts the values in the given histogram from this histogram.N)r: r} )r rK r^ r� r r r �Subtract� s z
Hist.SubtractN)r* r+ r, r- r� r� r� r� r r r r r� � s
r� c @ s� e Z d Z d Z d d d � Z d d � Z d d � Z d d
� Z d d � Z d
d � Z d d d � Z
d d � Z d d � Z d d � Z
d d � Z d d d � Z d d � Z d d d � Z d! d" � Z e Z e Z d# d$ d% � Z d& d' � Z e Z d( d) � Z d* d+ � Z d, d- � Z d. d/ � Z d0 d1 � Z d2 d3 � Z d4 d5 � Z d6 d7 � Z e Z d8 d9 � Z! d: d; � Z" d S)<r? z�Represents a probability mass function.
Values can be any hashable type; probabilities are floating-point.
Pmfs are not necessarily normalized.
r c C s | j j | | � S)z�Gets the probability associated with the value x.
Args:
x: number value
default: value to return if the key is not there
Returns:
float probability
)r2 rS )r r �defaultr r r �Prob� s
zPmf.Probc s � f d d � | D� S)z,Gets probabilities for a sequence of values.c s g | ] } � j | � � q Sr )r� )rn r )r r r ro � s z Pmf.Probs.<locals>.<listcomp>r )r r r )r r �Probs� s z Pmf.Probsc C sQ | d } d } x: t | j � � D]& \ } } | | 7} | | k r# | Sq# Wd S)z�Computes a percentile of a given Pmf.
Note: this is not super efficient. If you are planning
to compute more than a few percentiles, compute the Cdf.
percentage: float 0-100
returns: value from the Pmf
�d r N)rs r: )r �
percentager r� r^ rU r r r �
Percentile� s
zPmf.Percentilec sL t � t � r t | � � S� f d d � | j j � D� } t | � Sd S)zrProbability that a sample from this Pmf exceeds x.
x: number
returns: float probability
c s( g | ] \ } } | � k r | � q Sr r )rn r^ rU )r r r ro � s z#Pmf.ProbGreater.<locals>.<listcomp>N)r4 r/ �PmfProbGreaterr2 r9 r )r r �tr )r r � ProbGreater� s
"zPmf.ProbGreaterc sL t � t � r t | � � S� f d d � | j j � D� } t | � Sd S)zwProbability that a sample from this Pmf is less than x.
x: number
returns: float probability
c s( g | ] \ } } | � k r | � q Sr r )rn r^ rU )r r r ro � s z Pmf.ProbLess.<locals>.<listcomp>N)r4 r/ � PmfProbLessr2 r9 r )r r r� r )r r �ProbLess� s
"z Pmf.ProbLessc C s( t | t � r t | | � S| | Sd S)zuProbability that a sample from this Pmf is exactly x.
x: number
returns: float probability
N)r4 r/ � PmfProbEqual)r r r r r � ProbEqual� s
z
Pmf.ProbEqualr c C sl | j r t d � � | j � } | d k r9 t d � � | | } x" | j D] } | j | | 9<qM W| S)z�Normalizes this PMF so the sum of all probs is fraction.
Args:
fraction: what the total should be after normalization
Returns: the total probability before normalizing
z'Normalize: Pmf is under a log transformr z%Normalize: total probability is zero.)r3 r` r� r2 )r �fractionr� r] r r r r r@ s
z
Pmf.Normalizec C s\ t j � } d } x7 | j j � D]& \ } } | | 7} | | k r" | Sq" Wt d � � d S)z�Chooses a random element from this PMF.
Note: this is not very efficient. If you plan to call
this more than a few times, consider converting to a CDF.
Returns:
float value from the Pmf
r z$Random: Pmf might not be normalized.N)r r2 r9 r` )r �targetr� r r r r r �Random s
z
Pmf.Randomc C s | j � j | � S)z�Generates a random sample from this distribution.
n: int length of the sample
returns: NumPy array
)ry �Sample)r r� r r r r� / s z
Pmf.Samplec C s t d d � | j � D� � S)zMComputes the mean of a PMF.
Returns:
float mean
c s s | ] \ } } | | Vq d S)Nr )rn r r r r r � <genexpr>= s zPmf.Mean.<locals>.<genexpr>)r r: )r r r r �Mean7 s zPmf.Meanc C s | j � j d � S)zQComputes the median of a PMF.
Returns:
float median
�2 )ry r� )r r r r �Median? s z
Pmf.MedianNc s; � d k r | j � � t � f d d � | j � D� � S)z�Computes the variance of a PMF.
mu: the point around which the variance is computed;
if omitted, computes the mean
returns: float variance
Nc 3 s' | ] \ } } | | � d Vq d S)r Nr )rn r r )�mur r r� R s zPmf.Var.<locals>.<genexpr>)r� r r: )r r� r )r� r �VarG s zPmf.Varc s& t j � f d d � | j � D� � S)zWComputes the expectation of func(x).
Returns:
expectation
c 3 s% | ] \ } } | � | � Vq d S)Nr )rn r r )�funcr r r� Z s z Pmf.Expect.<locals>.<genexpr>)r r r: )r r� r )r� r �ExpectT s z
Pmf.Expectc C s | j | � } t j | � S)z�Computes the standard deviation of a PMF.
mu: the point around which the variance is computed;
if omitted, computes the mean
returns: float standard deviation
)r� rb �sqrt)r r� �varr r r �Std\ s zPmf.Stdc C s) t d d � | j � D� � \ } } | S)z\Returns the value with the highest probability.
Returns: float probability
c s s! | ] \ } } | | f Vq d S)Nr )rn r^ rU r r r r� l s zPmf.Mode.<locals>.<genexpr>)r� r: )r �_r^ r r r �Modeg s %zPmf.Mode�Z c C s | j � } | j | � S)z�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
)ry �CredibleInterval)r r� �cdfr r r r� v s zPmf.CredibleIntervalc C s7 y | j | � SWn t k
r2 | j | � SYn Xd S)z�Computes the Pmf of the sum of values drawn from self and other.
other: another Pmf or a scalar
returns: new Pmf
N)�AddPmfrJ � AddConstant)r rK r r r �__add__� s
z Pmf.__add__c C s] t � } xM | j � D]? \ } } x0 | j � D]" \ } } | | | | | 7<q/ Wq W| S)zComputes the Pmf of the sum of values drawn from self and other.
other: another Pmf
returns: new Pmf
)r? r: )r rK �pmf�v1�p1�v2�p2r r r r� � s
z
Pmf.AddPmfc C sT | d k r | j � St � } x. | j � D] \ } } | j | | | � q, W| S)zxComputes the Pmf of the sum a constant and values from self.
other: a number
returns: new Pmf
r )rZ r? r: r\ )r rK r� r� r� r r r r� � s
zPmf.AddConstantc C s8 y | j | � SWn t k
r3 | j | � SYn Xd S)z�Computes the Pmf of the diff of values drawn from self and other.
other: another Pmf
returns: new Pmf
N)�SubPmfrJ r� )r rK r r r �__sub__� s
z Pmf.__sub__c C s_ t � } xO | j � D]A \ } } x2 | j � D]$ \ } } | j | | | | � q/ Wq W| S)z�Computes the Pmf of the diff of values drawn from self and other.
other: another Pmf
returns: new Pmf
)r? r: r} )r rK r� r� r� r� r� r r r r� � s
z
Pmf.SubPmfc C s7 y | j | � SWn t k
r2 | j | � SYn Xd S)z�Computes the Pmf of the product of values drawn from self and other.
other: another Pmf
returns: new Pmf
N)�MulPmfrJ � MulConstant)r rK r r r �__mul__� s
z Pmf.__mul__c C s_ t � } xO | j � D]A \ } } x2 | j � D]$ \ } } | j | | | | � q/ Wq W| S)z�Computes the Pmf of the diff of values drawn from self and other.
other: another Pmf
returns: new Pmf
)r? r: r} )r rK r� r� r� r� r� r r r r� � s
z
Pmf.MulPmfc C s> t � } x. | j � D] \ } } | j | | | � q W| S)zComputes the Pmf of the product of a constant and values from self.
other: a number
returns: new Pmf
)r? r: r\ )r rK r� r� r� r r r r� � s zPmf.MulConstantc C s; y | j | � SWn# t k
r6 | j d | � SYn Xd S)z�Computes the Pmf of the ratio of values drawn from self and other.
other: another Pmf
returns: new Pmf
r N)�DivPmfrJ r� )r rK r r r �__div__� s
z Pmf.__div__c C s_ t � } xO | j � D]A \ } } x2 | j � D]$ \ } } | j | | | | � q/ Wq W| S)z�Computes the Pmf of the ratio of values drawn from self and other.
other: another Pmf
returns: new Pmf
)r? r: r} )r rK r� r� r� r� r� r r r r� � s
z
Pmf.DivPmfc C s | j � } | j | C_ | S)zrComputes the CDF of the maximum of k selections from this dist.
k: int
returns: new Cdf
)ry �ps)r �kr� r r r �Max s zPmf.Max)#r* r+ r, r- r� r� r� r� r� r� r@ r� r� r� r� r� r� r� r� �MAPZMaximumLikelihoodr� r� �__radd__r� r� r� r� r� r� r� r� � __truediv__r� r� r r r r r? � s@
r? c @ sC e Z d Z d Z d d d � Z d d d � Z d d d � Z d S)
�JointzTRepresents a joint distribution.
The values are sequences (usually tuples)
Nc C sD t d | � } x. | j � D] \ } } | j | | | � q W| S)z�Gets the marginal distribution of the indicated variable.
i: index of the variable we want
Returns: Pmf
r1 )r? r: r} )r r( r1 r� �vsrU r r r �Marginal s zJoint.Marginalc C sa t d | � } xA | j � D]3 \ } } | | | k r; q | j | | | � q W| j � | S)a! 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
r1 )r? r: r} r@ )r r( �jr^ r1 r� r� rU r r r � Conditional% s
zJoint.Conditionalr� c C sx g } d } d d � | j � D� } | j d d � x< | D]4 \ } } | j | � | | 7} | | d k r< Pq< W| S)a 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
r c S s" g | ] \ } } | | f � q Sr r )rn r^ rU r r r ro F s z)Joint.MaxLikeInterval.<locals>.<listcomp>r� Tr� )r: �sort�append)r r� �intervalr� r� rU r^ r r r �MaxLikeInterval9 s
zJoint.MaxLikeInterval)r* r+ r, r- r� r� r� r r r r r� s r� c C sa t � } xQ | j � D]C \ } } x4 | j � D]& \ } } | j | | f | | � q/ Wq W| S)z�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
)r� r: r\ )�pmf1�pmf2Zjointr� r� r� r� r r r � MakeJointR s
"r� c C s t | d | �S)z�Makes a histogram from an unsorted sequence of values.
Args:
t: sequence of numbers
label: string label for this histogram
Returns:
Hist object
r1 )r� )r� r1 r r r �MakeHistFromListe s
r� c C s
t | | � S)z�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
)r� )r2 r1 r r r �MakeHistFromDictr s
r� c C s t | d | �S)z�Makes a PMF from an unsorted sequence of values.
Args:
t: sequence of numbers
label: string label for this PMF
Returns:
Pmf object
r1 )r? )r� r1 r r r �MakePmfFromList s
r� c C s t | d | �S)z�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
r1 )r? )r2 r1 r r r �MakePmfFromDict� s
r� c C s t t | � d | �S)z�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
r1 )r? r7 )r� r1 r r r �MakePmfFromItems� s
r� c C s% | d k r | j } t | d | �S)z�Makes a normalized PMF from a Hist object.
Args:
hist: Hist object
label: string label
Returns:
Pmf object
Nr1 )r1 r? )�histr1 r r r �MakePmfFromHist� s
r� �mixc C s_ t d | � } xI | j � D]; \ } } x, | j � D] \ } } | | | | 7<q5 Wq W| S)z�Make a mixture distribution.
Args:
metapmf: Pmf that maps from Pmfs to probs.
label: string label for the new Pmf.
Returns: Pmf object.
r1 )r? r: )Zmetapmfr1 r� r� r� r r� r r r � MakeMixture� s
r� c C sG t � } x- t j | | | � D] } | j | d � q W| j � | S)zwMake a uniform Pmf.
low: lowest value (inclusive)
high: highest value (inclusize)
n: number of values
r )r? r �linspacer\ r@ )�low�highr� r� r r r r �MakeUniformPmf� s
r� c @ s� e Z d Z d Z d d d d d � Z d d � Z d d � Z d d
� Z d d � Z d
d � Z d d � Z
d d � Z d d � Z d d d � Z
d d d � Z d d � Z d d � Z d d � Z d d � Z d! d" � Z e Z d# d$ � Z d d% d& � Z e Z d' d( � Z d) d* � Z d+ d, � Z d- d. � Z d/ d0 � Z d1 d2 � Z d3 d4 � Z d5 d6 d7 � Z e Z d8 d9 d: � Z! d; d< � Z" d= d> � Z# d S)?r5 z�Represents a cumulative distribution function.
Attributes:
xs: sequence of values
ps: sequence of probabilities
label: string used as a graph label.
Nc C s� | d k r | n t | _ t | t t t f � rW | sW | d k rK | n | j | _ | d k r� t j g � | _ t j g � | _ | d k r� t
j d � d S| d k r� t | t � r� t
j d � t j | � | _ t j | � | _ d St | t � r1t
j
| j � | _ t
j
| j � | _ d St | t � rI| } n t | � } t | � d k r�t j g � | _ t j g � | _ d St t | j � � � \ } } t j | � | _ t j | d t j �| _ | j | j d _ d S)z�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
Nz+Cdf: can't pass ps without also passing xs.zCdf: ps can't be a stringr �dtyper r& )r0 r1 r4 r/ r5 r6 r �asarrayr r� rq rr rE rX r� r> rv rs r: �cumsumr )r rA r� r1 �dwr �freqsr r r r! � s>
z Cdf.__init__c C sI | j j } | j t k r> d | t | j � t | j � f S| j Sd S)Nz
%s(%s, %s))rD r* r1 r0 rE r r� )r rF r r r rG s #z Cdf.__str__c C sq | j j } | j t k r> d | t | j � t | j � f Sd | t | j � t | j � t | j � f Sd S)Nz
%s(%s, %s)z%s(%s, %s, %s))rD r* r1 r0 rE r r� rH )r rF r r r rI s
# z Cdf.__repr__c C s
t | j � S)N)r> r )r r r r rM s z Cdf.__len__c C s
| j | � S)N)r� )r r r r r rT # s zCdf.__getitem__c C s
t � � d S)N)� UnimplementedMethodException)r r r r rV & s zCdf.__setitem__c C s
t � � d S)N)r� )r r r r rW ) s zCdf.__delitem__c C s4 t j | j | j k � o3 t j | j | j k � S)N)r �allr r� )r rK r r r rL , s z
Cdf.__eq__c C s7 x0 t | j | j � D] \ } } t | | � q Wd S)z5Prints the values and freqs/probs in ascending order.N)rv r r� rz )r r^ rU r r r r{ / s "z Cdf.Printc C s: | d k r | j } t t | j � t | j � d | �S)zQReturns a copy of this Cdf.
label: string label for the new Cdf
Nr1 )r1 r5 �listr r� )r r1 r r r rZ 4 s zCdf.Copyc C s% | d k r | j } t | d | �S)z Makes a Pmf.Nr1 )r1 r? )r r1 r r r �MakePmf= s z Cdf.MakePmfc C s9 | j } t j | d � } d | d <t | j | | � S)zqReturns a sorted sequence of (value, probability) pairs.
Note: in Python3, returns an iterator.
r r )r� r �rollrv r )r �a�br r r r: C s
z Cdf.Itemsc C s | j � } | j | | _ | S)z>Adds a term to the xs.
term: how much to add
)rZ r )r r| rY r r r �ShiftM s z Cdf.Shiftc C s | j � } | j | | _ | S)zLMultiplies the xs by a factor.
factor: what to multiply by
)rZ r )r r] rY r r r r_ W s z Cdf.Scalec C sA | | j d k r d St j | j | � } | j | d } | S)z�Returns CDF(x), the probability that corresponds to value x.
Args:
x: number
Returns:
float probability
r r )r r' r� )r r �indexr r r r r� a s
zCdf.Probc C sV t j | � } t j | j | d d �} | j | d } d | | | j d k <| S)z�Gets probabilities for a sequence of values.
xs: any sequence that can be converted to NumPy array
returns: NumPy array of cumulative probabilities
�side�rightr r )r r� � searchsortedr r� )r r r� r� r r r r� p s
z Cdf.Probsc C sD | d k s | d k r$ t d � � t j | j | � } | j | S)z�Returns InverseCDF(p), the value that corresponds to probability p.
Args:
p: number in the range [0, 1]
Returns:
number value
r r z%Probability p must be in range [0, 1])r` r' � bisect_leftr� r )r r r� r r r �Value s z Cdf.Valuec C s~ | d k r | j St j | � } t j | d k � sL t j | d k � rX t d � � t j | j | d d �} | j | S)z�Returns InverseCDF(p), the value that corresponds to probability p.
If ps is not provided, returns all values.
Args:
ps: NumPy array of numbers in the range [0, 1]
Returns:
NumPy array of values
Nr r z%Probability p must be in range [0, 1]r� �left)r r r� rp r` r� r� )r r� r� r r r rk � s * z
Cdf.Valuesc C s | j | d � S)z�Returns the value that corresponds to percentile p.
Args:
p: number in the range [0, 100]
Returns:
number value
r� )r� )r r r r r r� � s zCdf.Percentilec C s t j | � } | j | d � S)z�Returns the value that corresponds to percentiles ps.
Args:
ps: numbers in the range [0, 100]
Returns:
array of values
r� )r r� rk )r r� r r r � Percentiles� s zCdf.Percentilesc C s | j | � d S)z�Returns the percentile rank of the value x.
x: potential value in the CDF
returns: percentile rank in the range 0 to 100
r� )r� )r r r r r �PercentileRank� s zCdf.PercentileRankc C s | j t � d S)z�Returns the percentile ranks of the values in xs.
xs: potential value in the CDF
returns: array of percentile ranks in the range 0 to 100
r� )r� r )r r r r r �PercentileRanks� s zCdf.PercentileRanksc C s | j t j � � S)z.Chooses a random value from this distribution.)r� r )r r r r r� � s z
Cdf.Randomc C s t j j | � } | j | � S)z�Generates a random sample from this distribution.
n: int length of the sample
returns: NumPy array
)r r �
ValueArray)r r� r� r r r r� � s z
Cdf.Samplec C sT d } d } xA t | j | j � D]* \ } } | | } | | | 7} | } q" W| S)zMComputes the mean of a CDF.
Returns:
float mean
r )rv r r� )r Zold_pr� r Znew_pr r r r r� � s "
zCdf.Meanr� c C s8 d | d d } | j | � | j d | � f } | S)z�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
r r� r )r� )r r� rU r� r r r r� � s "zCdf.CredibleIntervali� c C s
t � � d S)a
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.
N)r� )r Z
multiplierr r r �_Round� s z
Cdf._Roundc K sh d d � } t j | j � } | | | � } t j | j d � } d | d <| | | j � } | | f S)z�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)
c S sK t j | j d | j d � } | | d d d � <| | d d d � <| S)Nr r r )r �empty�shape)r� r� �cr r r �
interleave s !z Cdf.Render.<locals>.interleaver r )r �arrayr r� r� )r rw r� r� r Zshift_psr� r r r rx s
z
Cdf.Renderc C s | j � } | j | C_ | S)zrComputes the CDF of the maximum of k selections from this dist.
k: int
returns: new Cdf
)rZ r� )r r� r� r r r r� s zCdf.Max)$r* r+ r, r- r! rG rI rM rT rV rW rL r{ rZ r� r: r� r_ r� r� Z ProbArrayr� rk r� r� r� r� r� r� r� r� r� ZConfidenceIntervalr� rx r� r r r r r5 � sB 5
r5 c C s t t | � d | �S)z�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
r1 )r5 r7 )r9 r1 r r r �MakeCdfFromItems( s
r� c C s t | d | �S)z�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
r1 )r5 )r2 r1 r r r �MakeCdfFromDict5 s
r� c C s t | d | �S)z�Creates a CDF from an unsorted sequence.
Args:
seq: unsorted sequence of sortable values
label: string label for the cdf
Returns:
Cdf object
r1 )r5 )�seqr1 r r r �MakeCdfFromListB s
r� c C s% | d k r | j } t | d | �S)z�Makes a CDF from a Hist object.
Args:
hist: Pmf.Hist object
label: string label for the data.
Returns:
Cdf object
Nr1 )r1 r5 )r� r1 r r r �MakeCdfFromHistO s
r c C s% | d k r | j } t | d | �S)z�Makes a CDF from a Pmf object.
Args:
pmf: Pmf.Pmf object
label: string label for the data.
Returns:
Cdf object
Nr1 )r1 r5 )r� r1 r r r �MakeCdfFromPmf_ s
r c @ s e Z d Z d Z d S)r� z>Exception if someone calls a method that should be overridden.N)r* r+ r, r- r r r r r� o s r� c @ s� e Z d Z d Z d d � Z d d � Z d d � Z d d � Z d
d � Z d d
� Z d d � Z
d d � Z d d � Z d S)�Suitez9Represents a suite of hypotheses and their probabilities.c C sC x6 | j � D]( } | j | | � } | j | | � q
W| j � S)z�Updates each hypothesis based on the data.
data: any representation of the data
returns: the normalizing constant
)rk �
Likelihoodr~ r@ )r �data�hypo�liker r r �Updatev s z Suite.Updatec C s= x6 | j � D]( } | j | | � } | j | | � q
Wd S)a 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
N)rk �
LogLikelihoodr} )r r r r r r r � LogUpdate� s zSuite.LogUpdatec C sT xG | D]? } x6 | j � D]( } | j | | � } | j | | � q Wq W| j � S)aW 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
)rk r r~ r@ )r �datasetr r r r r r � UpdateSet� s
zSuite.UpdateSetc C s" x | D] } | j | � q Wd S)z�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
N)r )r r
r r r r � LogUpdateSet� s
zSuite.LogUpdateSetc C s
t � � d S)z�Computes the likelihood of the data under the hypothesis.
hypo: some representation of the hypothesis
data: some representation of the data
N)r� )r r r r r r r � s zSuite.Likelihoodc C s
t � � d S)z�Computes the log likelihood of the data under the hypothesis.
hypo: some representation of the hypothesis
data: some representation of the data
N)r� )r r r r r r r � s zSuite.LogLikelihoodc C s4 x- t | j � � D] \ } } t | | � q Wd S)z.Prints the hypotheses and their probabilities.N)rs r: rz )r r rU r r r r{ � s z Suite.Printc C sM xF | j � D]8 \ } } | r8 | j | t | � � q
| j | � q
Wd S)zXTransforms from probabilities to odds.
Values with prob=0 are removed.
N)r: r\ r rc )r r rU r r r �MakeOdds� s zSuite.MakeOddsc C s7 x0 | j � D]" \ } } | j | t | � � q
Wd S)z&Transforms from odds to probabilities.N)r: r\ r )r r Zoddsr r r � MakeProbs� s zSuite.MakeProbsN)
r* r+ r, r- r r r r r r r{ r
r r r r r r s s
r c C s( t | d | �} | j � } t | � S)z�Makes a suite from an unsorted sequence of values.
Args:
t: sequence of numbers
label: string label for this suite
Returns:
Suite object
r1 )r� rh �MakeSuiteFromDict)r� r1 r� r2 r r r �MakeSuiteFromList� s
r c C s4 | d k r | j } t | j � � } t | | � S)z�Makes a normalized suite from a Hist object.
Args:
hist: Hist object
label: string label
Returns:
Suite object
N)r1 r7 rh r )r� r1 r2 r r r �MakeSuiteFromHist� s
r c C s* t d | � } | j | � | j � | S)z�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
r1 )r ri r@ )r2 r1 �suiter r r r � s
r c @ sR e Z d Z d Z d d � Z d d � Z d d � Z d d � Z d
d � Z d S)
r6 z0Represents a probability density function (PDF).c C s
t � � d S)z_Evaluates this Pdf at x.
Returns: float or NumPy array of probability density
N)r� )r r r r r �Density s z Pdf.Densityc C s
t � � d S)zvGet a linspace for plotting.
Not all subclasses of Pdf implement this.
Returns: numpy array
N)r� )r r r r � GetLinspace s zPdf.GetLinspacec K sF | j d d � } | j | � \ } } t t t | | � � d | �S)z�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
r1 � )�poprx r? r7 rv )r rw r1 r �dsr r r r� s z Pdf.MakePmfc K s� | j d d � | j d d � } } | d k rg | d k rg | j d d � } t j | | | � } n* | j d d � } | d k r� | j � } | j | � } | | f S)a� 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)
r� Nr� r� �e r )r r r� r r )r rw r� r� r� r r r r r rx * s % z
Pdf.Renderc C s t | j � � S)z<Generates a sequence of (value, probability) pairs.
)rv rx )r r r r r: D s z Pdf.ItemsN) r* r+ r, r- r r r� rx r: r r r r r6 s r6 c @ sO e Z d Z d Z d d d d d � Z d d � Z d d
� Z d d � Z d S)
� NormalPdfz,Represents the PDF of a Normal distribution.r r Nc C s1 | | _ | | _ | d k r$ | n d | _ d S)z�Constructs a Normal Pdf with given mu and sigma.
mu: mean
sigma: standard deviation
label: string
Nr. )r� �sigmar1 )r r� r r1 r r r r! M s zNormalPdf.__init__c C s d | j | j f S)NzNormalPdf(%f, %f))r� r )r r r r rG X s zNormalPdf.__str__c C s< | j d | j | j d | j } } t j | | d � S)zCGet a linspace for plotting.
Returns: numpy array
� r )r� r r r� )r r� r� r r r r [ s )zNormalPdf.GetLinspacec C s t j j | | j | j � S)z�Evaluates this Pdf at xs.
xs: scalar or sequence of floats
returns: float or NumPy array of probability density
)r �norm�pdfr� r )r r r r r r c s zNormalPdf.Density)r* r+ r, r- r! rG r r r r r r r J s
r c @ sL e Z d Z d Z d d d d � Z d d � Z d d � Z d
d � Z d S) �ExponentialPdfz2Represents the PDF of an exponential distribution.r Nc C s( | | _ | d k r | n d | _ d S)zoConstructs an exponential Pdf with given parameter.
lam: rate parameter
label: string
Nr. )�lamr1 )r r r1 r r r r! p s zExponentialPdf.__init__c C s d | j S)NzExponentialPdf(%f))r )r r r r rG y s zExponentialPdf.__str__c C s' d d | j } } t j | | d � S)zCGet a linspace for plotting.
Returns: numpy array
r g @r )r r r� )r r� r� r r r r | s zExponentialPdf.GetLinspacec C s t j j | d d | j �S)z�Evaluates this Pdf at xs.
xs: scalar or sequence of floats
returns: float or NumPy array of probability density
�scaleg �?)r Zexponr r )r r r r r r � s zExponentialPdf.Density)r* r+ r, r- r! rG r r r r r r r m s
r c @ sU e Z d Z d Z d d d � Z d d � Z d d � Z d d
� Z d d � Z d S)
� EstimatedPdfz"Represents a PDF estimated by KDE.Nc C sa | d k r | n d | _ t j | � | _ t | � } t | � } t j | | d � | _ d S)zrEstimates the density function based on a sample.
sample: sequence of data
label: string
Nr. r )r1 r Z gaussian_kde�kde�minr� r r� )r �sampler1 r� r� r r r r! � s
zEstimatedPdf.__init__c C s d t | j � S)NzEstimatedPdf(label=%s))rE r1 )r r r r rG � s zEstimatedPdf.__str__c C s | j S)zCGet a linspace for plotting.
Returns: numpy array
)r� )r r r r r � s zEstimatedPdf.GetLinspacec C s | j j | � S)z`Evaluates this Pdf at xs.
returns: float or NumPy array of probability density
)r" �evaluate)r r r r r r � s zEstimatedPdf.Densityc C s | j j | � j � S)zUGenerates a random sample from the estimated Pdf.
n: size of sample
)r" �resample�flatten)r r� r r r r� � s zEstimatedPdf.Sample) r* r+ r, r- r! rG r r r� r r r r r! � s r! r� c C sD | j � } d | d d } | j | � | j d | � f } | S)a 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
r r� r )ry r� )r� r� r� rU r� r r r r� � s "r� c C s^ d } xQ | j � D]C \ } } x4 | j � D]& \ } } | | k r, | | | 7} q, Wq W| S)z�Probability that a value from pmf1 is less than a value from pmf2.
Args:
pmf1: Pmf object
pmf2: Pmf object
Returns:
float probability
r )r: )r� r� r� r� r� r� r� r r r r� � s
r� c C s^ d } xQ | j � D]C \ } } x4 | j � D]& \ } } | | k r, | | | 7} q, Wq W| S)z�Probability that a value from pmf1 is less than a value from pmf2.
Args:
pmf1: Pmf object
pmf2: Pmf object
Returns:
float probability
r )r: )r� r� r� r� r� r� r� r r r r� � s
r� c C s^ d } xQ | j � D]C \ } } x4 | j � D]& \ } } | | k r, | | | 7} q, Wq W| S)z�Probability that a value from pmf1 equals a value from pmf2.
Args:
pmf1: Pmf object
pmf2: Pmf object
Returns:
float probability
r )r: )r� r� r� r� r� r� r� r r r r� � s
r� c C s t d d � | D� � } | S)z�Chooses a random value from each dist and returns the sum.
dists: sequence of Pmf or Cdf objects
returns: numerical sum
c s s | ] } | j � Vq d S)N)r� )rn �distr r r r� s z RandomSum.<locals>.<genexpr>)r )�distsr� r r r � RandomSum s r* c s) t � f d d � t | � D� � } | S)z�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
c 3 s | ] } t � � Vq d S)N)r* )rn r( )r) r r r� s z SampleSum.<locals>.<genexpr>)r? �range)r) r� r� r )r) r � SampleSum s %r, c C s t j j | | | � S)z�Computes the unnormalized PDF of the normal distribution.
x: value
mu: mean
sigma: standard deviation
returns: float probability density
)r r r )r r� r r r r �
EvalNormalPdf s r- �� c C su t � } | | | } | | | } x? t j | | | � D]( } t | | | � } | j | | � q; W| j � | S)z�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
)r? r r� r- r\ r@ ) r� r Z
num_sigmasr� r� r� r� r r r r r �
MakeNormalPmf# s
r/ c C s t j j | | | � S)zjEvaluates the binomial PMF.
Returns the probabily of k successes in n trials with probability p.
)r �binomr� )r� r� r r r r �EvalBinomialPmf8 s r1 c C sD t � } x4 t | d � D]" } t j j | | | � | | <q W| S)zkEvaluates the binomial PMF.
Returns the distribution of successes in n trials with probability p.
r )r? r+ r r0 r� )r� r r� r� r r r �MakeBinomialPmf@ s r2 c C s$ | | d t j | � t | � S)z�Computes the Gamma PDF.
x: where to evaluate the PDF
a: parameter of the gamma distribution
returns: float probability
r )r rf r )r r� r r r � EvalGammaPdfK s r3 c C sG t j | � } t | | � } t t t | | � � � } | j � | S)z�Makes a PMF discrete approx to a Gamma distribution.
lam: parameter lambda in events per unit time
xs: upper bound of the Pmf
returns: normalized Pmf
)r r� r3 r? r7 rv r@ )r r� r� r� r r r � MakeGammaPmfV s
r4 c C s t j j | | d | �S)z�Evaluates the geometric PMF.
With loc=0: Probability of `k` trials to get one success.
With loc=-1: Probability of `k` trials before first success.
k: number of trials
p: probability of success on each trial
�loc)r �geomr� )r� r r5 r r r �EvalGeometricPmfe s r7 r� c C sM t � } x3 t | � D]% } t j j | | d | �| | <q W| j � | S)z�Evaluates the binomial PMF.
With loc=0: PMF of trials to get one success.
With loc=-1: PMF of trials before first success.
p: probability of success
high: upper bound where PMF is truncated
r5 )r? r+ r r6 r� r@ )r r5 r� r� r� r r r �MakeGeometricPmfq s
#
r8 c C s t j j | | | | � S)z�Evaluates the hypergeometric PMF.
Returns the probabily of k successes in n trials from a population
N with K successes in it.
)r Z hypergeomr� )r� �N�Kr� r r r �EvalHypergeomPmf� s r; c C s t j j | | � S)z�Computes the Poisson PMF.
k: number of events
lam: parameter lambda in events per unit time
returns: float probability
)r �poissonr� )r� r r r r �EvalPoissonPmf� s r= r c C s] t � } xC t d | d | � D]+ } t j j | | � } | j | | � q W| j � | S)z�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
r r )r? r+ r r< r� r\ r@ )r r� �stepr� r� r r r r �MakePoissonPmf� s
r? c C s | t j | | � S)z�Computes the exponential PDF.
x: value
lam: parameter lambda in events per unit time
returns: float probability density
)rb rf )r r r r r �EvalExponentialPdf� s r@ c C s d t j | | � S)zAEvaluates CDF of the exponential distribution with parameter lam.r )rb rf )r r r r r �EvalExponentialCdf� s rA �� c C sV t � } x< t j d | | � D]% } t | | � } | j | | � q W| j � | S)z�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
r )r? r r� r@ r\ r@ )r r� r� r� r r r r r �MakeExponentialPmf� s
rC c C s0 | | } | | | | d t j | | � S)z�Computes the Weibull PDF.
x: value
lam: parameter lambda in events per unit time
k: parameter
returns: float probability density
r )r rf )r r r� �argr r r �EvalWeibullPdf� s
rE c C s | | } d t j | | � S)z*Evaluates CDF of the Weibull distribution.r )r rf )r r r� rD r r r �EvalWeibullCdf� s
rF c C s@ t j d | | � } t | | | � } t t t | | � � � S)z�Makes a PMF discrete approx to a Weibull distribution.
lam: parameter lambda in events per unit time
k: parameter
high: upper bound
n: number of values in the Pmf
returns: normalized Pmf
r )r r� rE r? r7 rv )r r� r� r� r r� r r r �MakeWeibullPmf� s
rG c C s t j j | | d | �S)z�Computes the Pareto.
xm: minimum value (scale parameter)
alpha: shape parameter
returns: float probability density
r )r �paretor )r �xm�alphar r r �
EvalParetoPdf� s rK r c C sO t j | | | � } t j j | | d | �} t t t | | � � � } | S)z�Makes a PMF discrete approx to a Pareto distribution.
xm: minimum value (scale parameter)
alpha: shape parameter
high: upper bound value
num: number of values
returns: normalized Pmf
r )r r� r rH r r? r7 rv )rI rJ r� �numr r� r� r r r �
MakeParetoPmf� s
rM c C s t j | t � d d S)z�Evaluates the CDF of the standard Normal distribution.
See http://en.wikipedia.org/wiki/Normal_distribution
#Cumulative_distribution_function
Args:
x: float
Returns:
float
r r )rb �erf�ROOT2)r r r r �StandardNormalCdf s rP c C s t j j | d | d | �S)z�Evaluates the CDF of the normal distribution.
Args:
x: float
mu: mean parameter
sigma: standard deviation parameter
Returns:
float
r5 r )r r r� )r r� r r r r �
EvalNormalCdf s
rQ c C s t j j | d | d | �S)a! 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
r5 r )r r Zppf)r r� r r r r �EvalNormalCdfInverse! s rR c C s t j j | d | d | �S)z�Evaluates the CDF of the lognormal distribution.
x: float or sequence
mu: mean parameter
sigma: standard deviation parameter
Returns: float or sequence
r5 r )r Zlognormr� )r r� r r r r �EvalLognormalCdf3 s rS c C s7 t j | | | � } d t j | | � } | | f S)z�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)
r )r r� rf )r r� r� r� r r� r r r �
RenderExpoCdf? s
rT c C s7 t j | | | � } t j j | | | � } | | f S)z�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)
)r r� r r r� )r� r r� r� r� r r� r r r �RenderNormalCdfO s rU r� c C sD | | k r | } t j | | | � } d | | | } | | f S)z�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)
r )r r� )�xminrJ r� r� r� r r� r r r �RenderParetoCdf_ s
rW c @ s� e Z d Z d Z d d d d d � Z d d � Z d d � Z d
d � Z d d
� Z d d � Z d d � Z
d d d d � Z d d d � Z d d � Z
d S)�Betaz\Represents a Beta distribution.
See http://en.wikipedia.org/wiki/Beta_distribution
r Nc C s1 | | _ | | _ | d k r$ | n d | _ d S)z Initializes a Beta distribution.Nr. )rJ �betar1 )r rJ rY r1 r r r r! w s z
Beta.__init__c C s. | \ } } | j | 7_ | j | 7_ d S)zOUpdates a Beta distribution.
data: pair of int (heads, tails)
N)rJ rY )r r �headsZtailsr r r r } s z Beta.Updatec C s | j | j | j S)z'Computes the mean of this distribution.)rJ rY )r r r r r� � s z Beta.Meanc C s& | j d } | j d } | | | S)z7Computes the value with maximum a posteori probability.r )rJ rY )r r� r� r r r r� � s
zBeta.MAPc C s t j | j | j � S)z2Generates a random variate from this distribution.)r � betavariaterJ rY )r r r r r� � s z Beta.Randomc C s% | f } t j j | j | j | � S)zVGenerates a random sample from this distribution.
n: int sample size
)r r rY rJ )r r� �sizer r r r� � s z Beta.Samplec C s" | | j d d | | j d S)zEvaluates the PDF at x.r )rJ rY )r r r r r �EvalPdf� s z Beta.EvalPdfr c s� | d k r$ � j d k r$ � j } � j d k sB � j d k r^ � j � } | j � } | S� f d d � t � � D� } � f d d � | D� } t t t | | � � d | �} | S)a� 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.
The result is a little funny, because the values at 0 and 1
are not symmetric. Nevertheless, it is a reasonable discrete
model of the continuous distribution, and behaves well as
the number of values increases.
Nr c s g | ] } | � d � q S)g �?r )rn r( )�stepsr r ro � s z Beta.MakePmf.<locals>.<listcomp>c s g | ] } � j | � � q Sr )r] )rn r )r r r ro � s r1 ) r1 rJ rY ry r� r+ r? r7 rv )r r^ r1 r� r� r Zprobsr )r r^ r r� � s !z Beta.MakePmfc sM � f d d � t � � D� } t j | j | j | � } t | | � } | S)z%Returns the CDF of this distribution.c s g | ] } | � d � q S)g �?r )rn r( )r^ r r ro � s z Beta.MakeCdf.<locals>.<listcomp>)r+ r ZbetaincrJ rY r5 )r r^ r r� r� r )r^ r ry � s z Beta.MakeCdfc C s2 t j | � d } t j | j | j | � } | S)zmReturns the given percentiles from this distribution.
ps: scalar, array, or list of [0-100]
r� )r r� r Z
betaincinvrJ rY )r r� r r r r r� � s zBeta.Percentile)r* r+ r, r- r! r r� r� r� r� r] r� ry r� r r r r rX r s rX c @ ss e Z d Z d Z d d d d � Z d d � Z d d � Z d
d � Z d d
� Z d d � Z d d d � Z
d S)� DirichletzfRepresents a Dirichlet distribution.
See http://en.wikipedia.org/wiki/Dirichlet_distribution
r Nc C s_ | d k r t d � � | | _ t j | d t j �| | _ | d k rR | n d | _ d S)z�Initializes a Dirichlet distribution.
n: number of dimensions
conc: concentration parameter (smaller yields more concentration)
label: string label
r z0A Dirichlet distribution with n<2 makes no senser� Nr. )r` r� r �onesr �paramsr1 )r r� Zconcr1 r r r r! � s
zDirichlet.__init__c C s' t | � } | j d | � | 7<d S)ztUpdates a Dirichlet distribution.
data: sequence of observations, in order corresponding to params
N)r> ra )r r rd r r r r � s zDirichlet.Updatec C s# t j j | j � } | | j � S)zlGenerates a random variate from this distribution.
Returns: normalized vector of fractions
)r r r ra r )r r r r r r� � s zDirichlet.Randomc C sO t | � } | j | k r d S| } | j � } | d | � | } | j � S)z�Computes the likelihood of the data.
Selects a random vector of probabilities from this distribution.
Returns: float probability
r N)r> r� r� �prod)r r rd r r �qr r r r � s zDirichlet.Likelihoodc C sX t | � } | j | k r% t d � S| j � } t j | d | � � | } | j � S)z�Computes the log likelihood of the data.
Selects a random vector of probabilities from this distribution.
Returns: float log probability
z-infN)r> r� r r� r r3 r )r r rd r r$ r r r r s
zDirichlet.LogLikelihoodc C s- | j j � } | j | } t | | | � S)z�Computes the marginal distribution of the ith element.
See http://en.wikipedia.org/wiki/Dirichlet_distribution
#Marginal_distributions
i: int
Returns: Beta object
)ra r rX )r r( �alpha0rJ r r r � MarginalBeta s
zDirichlet.MarginalBetac C s5 | j j � } | j | } t t | | � d | �S)z�Makes a predictive distribution.
xs: values to go into the Pmf
Returns: Pmf that maps from x to the mean prevalence of x
r1 )ra r r? rv )r r r1 rd r� r r r �
PredictivePmf! s
zDirichlet.PredictivePmf) r* r+ r, r- r! r r� r r re rf r r r r r_ � s r_ c C s t j j | | � S)z{Compute the binomial coefficient "n choose k".
n: number of trials
k: number of successes
Returns: float
)�scipy�miscZcomb)r� r� r r r � BinomialCoef- s ri c C s; | t j | � | t j | � | | t j | | � S)z�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
)rb r3 )r� r� r r r �LogBinomialCoef8 s rj c C si t | � } t j j d d | � } | j � | rF t | | � } n t j | � } | j � | | f S)z�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
r r )r> r r �normalr� �Jitterr� )r �jitterr� r r r r �NormalProbabilityF s
rn g �?c C s& t | � } t j j d | | � | S)z�Jitters the values by adding a uniform variate in (-jitter, jitter).
values: sequence
jitter: scalar magnitude of jitter
returns: new numpy array
r )r> r r rk )r� rm r� r r r rl [ s rl z0.8c K s� t | � \ } } t | � \ } } t j | � } t | | | � } t j | d | d d � t | � \ } } t j | | | � d S)z�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
�colorr1 �modelN)rn �MeanVarrb r� �FitLine� thinkplot�Plot) r$ Z fit_colorrw r r �meanr� �std�fitr r r �NormalProbabilityPlotg s rx c C s
t j | � S)zHComputes mean.
xs: sequence of values
returns: float mean
)r ru )r r r r r� y s r� c C sO t j | � } | d k r' | j � } | | } t j | | � t | � | S)z�Computes variance.
xs: sequence of values
mu: option known mean
ddof: delta degrees of freedom
returns: float
N)r r� ru �dotr> )r r� �ddofr r r r r� � s
r� c C s t | | | � } t j | � S)z�Computes standard deviation.
xs: sequence of values
mu: option known mean
ddof: delta degrees of freedom
returns: float
)r� rb r� )r r� rz r� r r r r� � s r� c C s7 t j | � } | j � } t | | | � } | | f S)z�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
)r r� ru r� )r rz ru �s2r r r rq � s rq g{�G�z�?c C s1 t | t | � � } t | � | | � } | S)z�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
)�intr> rs )r� r r� r r r �Trim� s
r} c C s t | | � } t | � S)z�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
)r} r� )r� r r r r � TrimmedMean� s
r~ c C s+ t | | � } t | � \ } } | | f S)z�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
)r} rq )r� r r� r� r r r �TrimmedMeanVar� s r c C sx | j � | j � } t | � t | � } } | j � } | j � } | | | | | | } | t j | � } | S)zpCompute Cohen's d.
group1: Series or NumPy array
group2: Series or NumPy array
returns: float
)ru r> r� rb r� ) Zgroup1Zgroup2�diff�n1�n2Zvar1Zvar2Z
pooled_varr2 r r r �CohenEffectSize� s r� c C s| t j | � } t j | � } | d k r9 t j | � } | d k rT t j | � } t j | | | | � t | � } | S)z�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)
N)r r� ru ry r> )r r �meanx�meany�covr r r �Cov� s $r� c C sl t j | � } t j | � } t | � \ } } t | � \ } } t | | | | � t j | | � } | S)z�Computes Corr(X, Y).
Args:
xs: sequence of values
ys: sequence of values
Returns:
Corr(X, Y)
)r r� rq r� rb r� )r r r� �varxr� Zvary�corrr r r �Corr
s
&r� c C s< | | d � } | j | � | d � } t | | � } | S)z�Computes the serial correlation of a series.
series: Series
lag: integer number of intervals to shift
returns: float correlation
N)�shiftr� )�series�lagr r r� r r r �
SerialCorr!
s r� c C s7 t j | � j � } t j | � j � } t | | � S)z�Computes Spearman's rank correlation.
Args:
xs: sequence of values
ys: sequence of values
Returns:
float Spearman's correlation
)r; r< �rankr� )r r ZxranksZyranksr r r � SpearmanCorr/
s
r� c C s_ t | � } t | d t d � �} t | � } t | d d d � �} d d � | D� } | S)z�Returns a list of ranks corresponding to the elements in t.
Args:
t: sequence of numbers
Returns:
list of integer ranks, starting at 1
�keyr c S s | d d S)Nr r r )�tripr r r �<lambda>Q
s z MapToRanks.<locals>.<lambda>c S s g | ] } | d d � q S)r r r )rn r� r r r ro T
s z MapToRanks.<locals>.<listcomp>)� enumeraters r )r� �pairsZ sorted_pairsZrankedZresortedZranksr r r �
MapToRanks>
s
r� c C sO t | � \ } } t | � } t | | | | � | } | | | } | | f S)z�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)
)rq r� r� )r r r� r� r� �slope�interr r r � LeastSquaresX
s
r� c C s' t j | � } | | | } | | f S)zrFits a line to the given data.
xs: sequence of x
returns: tuple of numpy arrays (sorted xs, fit ys)
)r r� )r r� r� Zfit_xsZfit_ysr r r rr k
s rr c C s4 t j | � } t j | � } | | | | } | S)z�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
)r r� )r r r� r� �resr r r � Residualsw
s r� c C s d t | � t | � S)z�Computes the coefficient of determination (R^2) for given residuals.
Args:
ys: dependent variable
res: residuals
Returns:
float coefficient of determination
r )r� )r r� r r r �CoefDetermination�
s
r� c c sT t j d d � } | Vt j d | d � } x t j | | | � } | Vq1 Wd S)z�Generates standard normal variates with serial correlation.
rho: target coefficient of correlation
Returns: iterable
r r r N)r �gaussrb r� )�rhor r r r r �CorrelatedGenerator�
s r� c c s( x! t | � D] } | | | Vq
Wd S)z�Generates normal variates with serial correlation.
mu: mean of variate
sigma: standard deviation of variate
rho: target coefficient of correlation
Returns: iterable
N)r� )r� r r� r r r r �CorrelatedNormalGenerator�
s r� c s' t � f d d � | D� � t | � S)z'Computes the kth raw moment of xs.
c 3 s | ] } | � Vq d S)Nr )rn r )r� r r r� �
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s r� c s9 t | d � � t � � f d d � | D� � t | � S)z+Computes the kth central moment of xs.
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s z CentralMoment.<locals>.<genexpr>)r� r r> )r r� r )r� ru r �
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s r� c C s3 t | d � } t j | � } t | | � | | S)z0Computes the kth standardized moment of xs.
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s r� c C s t | � } | j d � S)z�Computes the median (50th percentile) of a sequence.
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s r� c C s( t | � } | j d � | j d � f S)z�Computes the interquartile of a sequence.
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returns: pair of floats
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s r� c C sO t | � } t | d � } t | d � } t j | � } d | | | } | S)z*Computes the Pearson median skewness.
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s r� c @ s1 e Z d Z d Z d d d � Z d d � Z d S)�FixedWidthVariablesz4Represents a set of variables in a fixed width file.r c C sR | | _ | d d g | | _ | j j t j � j j � | _ | d | _ d S)z�Initializes.
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n: sample size (default: len(xs))
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r� T)r� r> )r� r r r � ResampleRowsT s r� Zfinalwgtc C sB | | } t t | � � } | j t | � � } | j | } | S)z�Resamples a DataFrame using probabilities proportional to given column.
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r� c C s. | j \ } } t | | d � } | | f S)z|Selects the row from a sorted array that maps to percentile p.
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PercentileRowm s r� c s� t | � } t | d � } t j | | f � � x' t | � D] \ } } | � | f <q> Wt j � d d �� � f d d � | D� } | S)a� Given a collection of lines, selects percentiles along vertical axis.
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ys_seq: sequence of lines (y values)
percents: list of percentiles (0-100) to select
returns: list of NumPy arrays, one for each percentile
r �axisc s g | ] } t � | � � q Sr )r� )rn r )r� r r ro � s z"PercentileRows.<locals>.<listcomp>)r> r �zerosr� r� )Zys_seq�percentsr� �ncolsr( r r� r )r� r �PercentileRowsy s r� c K s t j j | | | � S)zpSmooths a NumPy array with a Gaussian filter.
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returns: float p-value
c s% g | ] } � j � j � � � q Sr )r� �RunModel)rn r� )r r r ro � s z)HypothesisTest.PValue.<locals>.<listcomp>c 3 s$ | ] } | � j k r d Vq d S)r N)r� )rn r )r r r r� � s z(HypothesisTest.PValue.<locals>.<genexpr>)r+ r� r5 r� r )r �iters�countr )r r �PValue� s
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t | j � S)zDReturns the largest test statistic seen during simulations.
)r� r� )r r r r � MaxTestStat� s zHypothesisTest.MaxTestStatNc C s3 d d � } | | j � t j | j d | �d S)zCDraws a Cdf with vertical lines at the observed test stat.
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