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statistics.cpython-312.pyc
CRh3 U d Z g dZddlZddlZddlZddlZddlmZ ddlm Z ddl m Z m Z m Z ddlmZmZ ddlmZmZmZmZmZmZmZmZmZ dd lmZ dd l m Z dd l m Z m Z m!Z! ed Z" G d de# Z$d Z%dCdZ&d Z'd Z(d Z)d Z*dDdZ+dddddde,e- fdZ.d e/d e/de/fd Z0d ejb jd z d z Z3e/e4d!< d e/d e/de-fd"Z5d e/d e/de fd#Z6d$ Z7dCd%Z8d& Z9dCd'Z:d( Z;d) Z<d* Z=dEd+Z>d, Z?d- Z@d.d/d0d1ZAdCd2ZBdCd3ZCdCd4ZDdCd5ZEd6 ZFd7 ZGd8d9d:ZH e d;d< ZIdd=d>ZJd? ZK dd@lLmKZK G dA dB ZNy# eM$ r Y w xY w)Fa Basic statistics module. This module provides functions for calculating statistics of data, including averages, variance, and standard deviation. Calculating averages -------------------- ================== ================================================== Function Description ================== ================================================== mean Arithmetic mean (average) of data. fmean Fast, floating-point arithmetic mean. geometric_mean Geometric mean of data. harmonic_mean Harmonic mean of data. median Median (middle value) of data. median_low Low median of data. median_high High median of data. median_grouped Median, or 50th percentile, of grouped data. mode Mode (most common value) of data. multimode List of modes (most common values of data). quantiles Divide data into intervals with equal probability. ================== ================================================== Calculate the arithmetic mean ("the average") of data: >>> mean([-1.0, 2.5, 3.25, 5.75]) 2.625 Calculate the standard median of discrete data: >>> median([2, 3, 4, 5]) 3.5 Calculate the median, or 50th percentile, of data grouped into class intervals centred on the data values provided. E.g. if your data points are rounded to the nearest whole number: >>> median_grouped([2, 2, 3, 3, 3, 4]) #doctest: +ELLIPSIS 2.8333333333... This should be interpreted in this way: you have two data points in the class interval 1.5-2.5, three data points in the class interval 2.5-3.5, and one in the class interval 3.5-4.5. The median of these data points is 2.8333... Calculating variability or spread --------------------------------- ================== ============================================= Function Description ================== ============================================= pvariance Population variance of data. variance Sample variance of data. pstdev Population standard deviation of data. stdev Sample standard deviation of data. ================== ============================================= Calculate the standard deviation of sample data: >>> stdev([2.5, 3.25, 5.5, 11.25, 11.75]) #doctest: +ELLIPSIS 4.38961843444... If you have previously calculated the mean, you can pass it as the optional second argument to the four "spread" functions to avoid recalculating it: >>> data = [1, 2, 2, 4, 4, 4, 5, 6] >>> mu = mean(data) >>> pvariance(data, mu) 2.5 Statistics for relations between two inputs ------------------------------------------- ================== ==================================================== Function Description ================== ==================================================== covariance Sample covariance for two variables. correlation Pearson's correlation coefficient for two variables. linear_regression Intercept and slope for simple linear regression. ================== ==================================================== Calculate covariance, Pearson's correlation, and simple linear regression for two inputs: >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9] >>> y = [1, 2, 3, 1, 2, 3, 1, 2, 3] >>> covariance(x, y) 0.75 >>> correlation(x, y) #doctest: +ELLIPSIS 0.31622776601... >>> linear_regression(x, y) #doctest: LinearRegression(slope=0.1, intercept=1.5) Exceptions ---------- A single exception is defined: StatisticsError is a subclass of ValueError. ) NormalDistStatisticsError correlation covariancefmeangeometric_mean harmonic_meanlinear_regressionmeanmedianmedian_grouped median_high median_lowmode multimodepstdev pvariance quantilesstdevvariance NFraction)Decimal)countgroupbyrepeat) bisect_left bisect_right) hypotsqrtfabsexperftaulogfsumsumprod)reduce) itemgetter)Counter namedtuple defaultdict @c e Zd Zy)r N)__name__ __module__ __qualname__ !/usr/lib/python3.12/statistics.pyr r s r3 r c d}t }|j }i }|j }t | t D ]9 \ }} || t t | D ] \ }} |dz } || d |z || < ; d|v r|d } t | r"J t d |j D } t t |t } | | |fS )a _sum(data) -> (type, sum, count) Return a high-precision sum of the given numeric data as a fraction, together with the type to be converted to and the count of items. Examples -------- >>> _sum([3, 2.25, 4.5, -0.5, 0.25]) (<class 'float'>, Fraction(19, 2), 5) Some sources of round-off error will be avoided: # Built-in sum returns zero. >>> _sum([1e50, 1, -1e50] * 1000) (<class 'float'>, Fraction(1000, 1), 3000) Fractions and Decimals are also supported: >>> from fractions import Fraction as F >>> _sum([F(2, 3), F(7, 5), F(1, 4), F(5, 6)]) (<class 'fractions.Fraction'>, Fraction(63, 20), 4) >>> from decimal import Decimal as D >>> data = [D("0.1375"), D("0.2108"), D("0.3061"), D("0.0419")] >>> _sum(data) (<class 'decimal.Decimal'>, Fraction(6963, 10000), 4) Mixed types are currently treated as an error, except that int is allowed. r Nc 3 : K | ] \ }}t || y wNr .0dns r4 <genexpr>z_sum.<locals>.<genexpr> s @/?tq!HQN/? ) setaddgetr typemap _exact_ratio _isfinitesumitemsr( _coerceint) datar types types_addpartials partials_gettypvaluesr< r; totalTs r4 _sumrS s @ E EE IH<<L tT* V# f-DAq QJE&q!,q0HQK . + x U### @x~~/?@@ws#A ue r3 c t fd| D \ }}}|||fS d}t }|j }t t }t t }t | t D ]G \ } } || t t | D ]' \ } |dz }|xx | z cc< |xx | | z z cc< ) I |st d x}nkd|v r|d x}t | rUJ t d |j D } t d |j D } || z | | z z |z }| |z t t |t }|||fS )a3 Return the exact mean and sum of square deviations of sequence data. Calculations are done in a single pass, allowing the input to be an iterator. If given *c* is used the mean; otherwise, it is calculated from the data. Use the *c* argument with care, as it can lead to garbage results. Nc 3 2 K | ] }|z xz y wr8 r2 )r: xcr; s r4 r= z_ss.<locals>.<genexpr> s <t!1q5jaA -ts r r6 c 3 : K | ] \ }}t || y wr8 r r9 s r4 r= z_ss.<locals>.<genexpr> s @,?DAq!Q ,?r> c 3 @ K | ] \ }}t |||z y wr8 r r9 s r4 r= z_ss.<locals>.<genexpr> s" D/Ctq!(1ac"/Cs )rS r? r@ r, rI r rB rC rD r rE rF rG r( rH )rJ rW rR ssdr rK rL sx_partials sxx_partialsrO rP r< sxsxxr; s ` @r4 _ssr_ s\ } <t << 335!! E EE I c"K s#L tT* V# f-DAq QJE Na N Oq1u $O . + 1+ a d##a S>!! @K,=,=,?@ @D|/A/A/CDD s{R"W$- Jws#A sAu r3 c l | j S # t $ r t j | cY S w xY wr8 ) is_finiteAttributeErrormathisfinite)rV s r4 rE rE s1 {{} }}Q s 33c | t usJ d | |u r| S |t u s|t u r| S | t u r|S t || r|S t | | r| S t | t r|S t |t r| S t | t rt |t r|S t | t rt |t r| S d}t || j |j fz )zCoerce types T and S to a common type, or raise TypeError. Coercion rules are currently an implementation detail. See the CoerceTest test class in test_statistics for details. zinitial type T is boolz"don't know how to coerce %s and %s)boolrI issubclassr float TypeErrorr/ )rR Smsgs r4 rH rH s D=222= AvqCx19axCx(!Q(!Q(!S1H!S1H!X :a#7!U 1h 7 .C C1::qzz22 33r3 c | j S # t $ r Y n%t t f$ r t | rJ | dfcY S w xY w | j | j fS # t $ r% dt | j d}t | w xY w)zReturn Real number x to exact (numerator, denominator) pair. >>> _exact_ratio(0.25) (1, 4) x is expected to be an int, Fraction, Decimal or float. Nzcan't convert type 'z' to numerator/denominator) as_integer_ratiorb OverflowError ValueErrorrE numerator denominatorrB r/ ri )rV rk s r4 rD rD s <!!## : & Q< 4y Q]]++ $T!W%5%5$66PQn s ? ??A .B c t | |u r| S t |t r| j dk7 rt } || S # t $ r9 t |t r' || j || j z cY S w xY w)z&Convert value to given numeric type T.r6 )rB rg rI rq rh ri r rp )valuerR s r4 _convertrt M sz E{a !Se / /1 4 x a !U__%%*;*;(<< < s > >B >B c # B K | D ] }|dk r t | | yw)z7Iterate over values, failing if any are less than zero.r N)r )rP errmsgrV s r4 _fail_negrw _ s' q5!&) ) s Faverager6 )keyreversetiesstartreturnc T |dk7 rt d| | t || } t t | t | }|dz }dgt | z }t |t d D ]: \ }} t | } t | } || dz dz z } | D ] \ } }| ||< || z }< |S )a Rank order a dataset. The lowest value has rank 1. Ties are averaged so that equal values receive the same rank: >>> data = [31, 56, 31, 25, 75, 18] >>> _rank(data) [3.5, 5.0, 3.5, 2.0, 6.0, 1.0] The operation is idempotent: >>> _rank([3.5, 5.0, 3.5, 2.0, 6.0, 1.0]) [3.5, 5.0, 3.5, 2.0, 6.0, 1.0] It is possible to rank the data in reverse order so that the highest value has rank 1. Also, a key-function can extract the field to be ranked: >>> goals = [('eagles', 45), ('bears', 48), ('lions', 44)] >>> _rank(goals, key=itemgetter(1), reverse=True) [2.0, 1.0, 3.0] Ranks are conventionally numbered starting from one; however, setting *start* to zero allows the ranks to be used as array indices: >>> prize = ['Gold', 'Silver', 'Bronze', 'Certificate'] >>> scores = [8.1, 7.3, 9.4, 8.3] >>> [prize[int(i)] for i in _rank(scores, start=0, reverse=True)] ['Bronze', 'Certificate', 'Gold', 'Silver'] rx z Unknown tie resolution method: )rz r6 r )ry ) ro rC sortedzipr lenr r) list)rJ ry rz r{ r| val_posiresult_ggroupsizerankrs orig_poss r4 _rankr g s J y:4(CDD 3~Suw'9G AS3w< FZ]31Q5zD1H>! $OE8 #F8 % T 4 Mr3 r< mc N t j | |z }|||z |z | k7 z S )zFSquare root of n/m, rounded to the nearest integer using round-to-odd.)rc isqrt)r< r as r4 _integer_sqrt_of_frac_rtor s- 16A !A r3 r _sqrt_bit_widthc | j |j z t z dz }|dk\ r t | |d|z z |z }d}||z S t | d|z z | }d| z }||z S )z1Square root of n/m as a float, correctly rounded.r r r6 ) bit_lengthr r )r< r qrp rq s r4 _float_sqrt_of_fracr s !,,. (? :q@AAv-aa!e<A { "" .a26k1= A2g { ""r3 c | dk r| s t d S | | }} t | t | z j }|j \ }}|j }|j \ }}d| z ||z dz z |||z ||z z dz z kD r|S |j }|j \ } } d| z || z dz z ||| z | |z z dz z k r|S |S )z3Square root of n/m as a Decimal, correctly rounded.r z0.0 r )r r rm next_plus next_minus) r< r rootnrdrplusnpdpminusnmdms r4 _decimal_sqrt_of_fracr s Av5> !rA21 AJ # ) ) +D " " $FB >> D " " $FB1u2zABB 2 22 OO E # # %FB1u2zABB 2 22 Kr3 c ^ t | \ }}}|dk r t d t ||z | S )a Return the sample arithmetic mean of data. >>> mean([1, 2, 3, 4, 4]) 2.8 >>> from fractions import Fraction as F >>> mean([F(3, 7), F(1, 21), F(5, 3), F(1, 3)]) Fraction(13, 21) >>> from decimal import Decimal as D >>> mean([D("0.5"), D("0.75"), D("0.625"), D("0.375")]) Decimal('0.5625') If ``data`` is empty, StatisticsError will be raised. r6 z%mean requires at least one data point)rS r rt )rJ rR rQ r< s r4 r r s7 t*KAua1u EFF EAIq !!r3 c \ |) t | t | }s t d |z S t |t t f s t | } t | | }t | }|s t d ||z S # t $ r dfd} || } Y w xY w# t $ r t d w xY w)zConvert data to floats and compute the arithmetic mean. This runs faster than the mean() function and it always returns a float. If the input dataset is empty, it raises a StatisticsError. >>> fmean([3.5, 4.0, 5.25]) 4.25 r c 3 @ K t | d D ] \ }| y w)Nr6 r| ) enumerate)iterablerV r< s r4 r zfmean.<locals>.count s! %ha 8DAqG 9s z&fmean requires at least one data pointz(data and weights must be the same lengthz sum of weights must be non-zero) r ri r& r isinstancer tupler' ro )rJ weightsr rQ numdenr< s @r4 r r s D A T !"JK Kqy ge} -w-JdG$ w-C ?@@ 9+ A ;D J HIIJs A8 B 8BBB+c z t t t t | S # t $ r t d dw xY w)aY Convert data to floats and compute the geometric mean. Raises a StatisticsError if the input dataset is empty, if it contains a zero, or if it contains a negative value. No special efforts are made to achieve exact results. (However, this may change in the future.) >>> round(geometric_mean([54, 24, 36]), 9) 36.0 zGgeometric mean requires a non-empty dataset containing positive numbersN)r" r rC r% ro r )rJ s r4 r r sE G5S$ ()) G < =BF GGs !$ :c x t | | u r t | } d}t | }|dk r t d |dk( rD|B| d }t |t j t f r|dk r t | |S t d |t d| }|}nQt | |u r t | }t | |k7 r t d t d t || D \ }}} t | | } t d t || D \ }}} |dk r t d t ||z | S # t $ r Y yw xY w) a Return the harmonic mean of data. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the data. It can be used for averaging ratios or rates, for example speeds. Suppose a car travels 40 km/hr for 5 km and then speeds-up to 60 km/hr for another 5 km. What is the average speed? >>> harmonic_mean([40, 60]) 48.0 Suppose a car travels 40 km/hr for 5 km, and when traffic clears, speeds-up to 60 km/hr for the remaining 30 km of the journey. What is the average speed? >>> harmonic_mean([40, 60], weights=[5, 30]) 56.0 If ``data`` is empty, or any element is less than zero, ``harmonic_mean`` will raise ``StatisticsError``. z.harmonic mean does not support negative valuesr6 z.harmonic_mean requires at least one data pointr zunsupported typez*Number of weights does not match data sizec 3 K | ] }| y wr8 r2 )r: ws r4 r= z harmonic_mean.<locals>.<genexpr>N s G,Fq,Fs c 3 4 K | ] \ }}|r||z nd yw)r Nr2 )r: r rV s r4 r= z harmonic_mean.<locals>.<genexpr>Q s" P=OTQq1uq 0=Os z Weighted sum must be positive)iterr r r r numbersRealr ri r rS rw r ZeroDivisionErrorrt ) rJ r rv r< rV sum_weightsr rR rQ r s r4 r r ! sM . DzTDz =F D A1u NOO aGO G a',,0 11u%f--H ./ /A, =G #7mG w<1 !"NO O GIgv,F G G;v& PS$=O PP5% z =>> K%' ++ s ",D- - D98D9c t | } t | }|dk( r t d |dz dk( r| |dz S |dz }| |dz | | z dz S )aB Return the median (middle value) of numeric data. When the number of data points is odd, return the middle data point. When the number of data points is even, the median is interpolated by taking the average of the two middle values: >>> median([1, 3, 5]) 3 >>> median([1, 3, 5, 7]) 4.0 r no median for empty datar r6 r r r )rJ r< r s r4 r r Y sg $<D D AAv 8991uzAF| FQU d1g%**r3 c t | } t | }|dk( r t d |dz dk( r| |dz S | |dz dz S )a Return the low median of numeric data. When the number of data points is odd, the middle value is returned. When it is even, the smaller of the two middle values is returned. >>> median_low([1, 3, 5]) 3 >>> median_low([1, 3, 5, 7]) 3 r r r r6 r rJ r< s r4 r r q sU $<D D AAv 8991uzAF|AFQJ r3 c ^ t | } t | }|dk( r t d | |dz S )a Return the high median of data. When the number of data points is odd, the middle value is returned. When it is even, the larger of the two middle values is returned. >>> median_high([1, 3, 5]) 3 >>> median_high([1, 3, 5, 7]) 5 r r r r r s r4 r r s7 $<D D AAv 899 Q<r3 c * t | } t | }|s t d | |dz }t | | }t | || } t | }t | }||dz z }|}||z }|||dz |z z |z z S # t $ r t d w xY w)a Estimates the median for numeric data binned around the midpoints of consecutive, fixed-width intervals. The *data* can be any iterable of numeric data with each value being exactly the midpoint of a bin. At least one value must be present. The *interval* is width of each bin. For example, demographic information may have been summarized into consecutive ten-year age groups with each group being represented by the 5-year midpoints of the intervals: >>> demographics = Counter({ ... 25: 172, # 20 to 30 years old ... 35: 484, # 30 to 40 years old ... 45: 387, # 40 to 50 years old ... 55: 22, # 50 to 60 years old ... 65: 6, # 60 to 70 years old ... }) The 50th percentile (median) is the 536th person out of the 1071 member cohort. That person is in the 30 to 40 year old age group. The regular median() function would assume that everyone in the tricenarian age group was exactly 35 years old. A more tenable assumption is that the 484 members of that age group are evenly distributed between 30 and 40. For that, we use median_grouped(). >>> data = list(demographics.elements()) >>> median(data) 35 >>> round(median_grouped(data, interval=10), 1) 37.5 The caller is responsible for making sure the data points are separated by exact multiples of *interval*. This is essential for getting a correct result. The function does not check this precondition. Inputs may be any numeric type that can be coerced to a float during the interpolation step. r r )loz$Value cannot be converted to a floatr- )r r r r r rh ro ri ) rJ intervalr< rV r jLcffs r4 r r s V $<D D A 899 Q!V A D! AT1#AA? !H HsNA B AA x1q52:&* ** A>@@As A= =Bc t t | j d } |d d S # t $ r t d dw xY w)ax Return the most common data point from discrete or nominal data. ``mode`` assumes discrete data, and returns a single value. This is the standard treatment of the mode as commonly taught in schools: >>> mode([1, 1, 2, 3, 3, 3, 3, 4]) 3 This also works with nominal (non-numeric) data: >>> mode(["red", "blue", "blue", "red", "green", "red", "red"]) 'red' If there are multiple modes with same frequency, return the first one encountered: >>> mode(['red', 'red', 'green', 'blue', 'blue']) 'red' If *data* is empty, ``mode``, raises StatisticsError. r6 r zno mode for empty dataN)r* r most_common IndexErrorr )rJ pairss r4 r r sP . DJ + +A .EBQx{ B 67TABs - Ac t t | }|sg S t |j }|j D cg c] \ }}||k( s | c}}S c c}}w )a. Return a list of the most frequently occurring values. Will return more than one result if there are multiple modes or an empty list if *data* is empty. >>> multimode('aabbbbbbbbcc') ['b'] >>> multimode('aabbbbccddddeeffffgg') ['b', 'd', 'f'] >>> multimode('') [] )r* r maxrP rG )rJ countsmaxcountrs r s r4 r r sS T$Z F 6==?#H&,lln JnleU8IEn JJ Js AAr exclusive)r< methodc ( |dk r t d t | } t | }|dk r t d |dk( rW|dz }g }t d| D ]? }t ||z | \ }}| | ||z z | |dz |z z |z } |j | A |S |dk( rn|dz }g }t d| D ]V }||z |z }|dk rdn||dz kD r|dz n|}||z ||z z }| |dz ||z z | | |z z |z } |j | X |S t d| )a Divide *data* into *n* continuous intervals with equal probability. Returns a list of (n - 1) cut points separating the intervals. Set *n* to 4 for quartiles (the default). Set *n* to 10 for deciles. Set *n* to 100 for percentiles which gives the 99 cuts points that separate *data* in to 100 equal sized groups. The *data* can be any iterable containing sample. The cut points are linearly interpolated between data points. If *method* is set to *inclusive*, *data* is treated as population data. The minimum value is treated as the 0th percentile and the maximum value is treated as the 100th percentile. r6 zn must be at least 1r z"must have at least two data points inclusiver Unknown method: )r r r rangedivmodappendro ) rJ r< r ldr r r r delta interpolateds r4 r r 9 sa 1u 455 $<D TB Av BCC Fq!A a!eQ'HAu Gq5y 1DQK%4G G1LL MM, ' Fq!AA AUBqD1aAaC!A#IE QK1u9 5Q% G1LL MM, ' 'z2 33r3 c h t | | \ }}}}|dk r t d t ||dz z | S )a Return the sample variance of data. data should be an iterable of Real-valued numbers, with at least two values. The optional argument xbar, if given, should be the mean of the data. If it is missing or None, the mean is automatically calculated. Use this function when your data is a sample from a population. To calculate the variance from the entire population, see ``pvariance``. Examples: >>> data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5] >>> variance(data) 1.3720238095238095 If you have already calculated the mean of your data, you can pass it as the optional second argument ``xbar`` to avoid recalculating it: >>> m = mean(data) >>> variance(data, m) 1.3720238095238095 This function does not check that ``xbar`` is actually the mean of ``data``. Giving arbitrary values for ``xbar`` may lead to invalid or impossible results. Decimals and Fractions are supported: >>> from decimal import Decimal as D >>> variance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")]) Decimal('31.01875') >>> from fractions import Fraction as F >>> variance([F(1, 6), F(1, 2), F(5, 3)]) Fraction(67, 108) r z*variance requires at least two data pointsr6 r_ r rt )rJ xbarrR ssrW r< s r4 r r j s@ L dD/KAr1a1u JKK B!a%L! $$r3 c b t | | \ }}}}|dk r t d t ||z | S )a, Return the population variance of ``data``. data should be a sequence or iterable of Real-valued numbers, with at least one value. The optional argument mu, if given, should be the mean of the data. If it is missing or None, the mean is automatically calculated. Use this function to calculate the variance from the entire population. To estimate the variance from a sample, the ``variance`` function is usually a better choice. Examples: >>> data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25] >>> pvariance(data) 1.25 If you have already calculated the mean of the data, you can pass it as the optional second argument to avoid recalculating it: >>> mu = mean(data) >>> pvariance(data, mu) 1.25 Decimals and Fractions are supported: >>> from decimal import Decimal as D >>> pvariance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")]) Decimal('24.815') >>> from fractions import Fraction as F >>> pvariance([F(1, 4), F(5, 4), F(1, 2)]) Fraction(13, 72) r6 z*pvariance requires at least one data pointr )rJ murR r rW r< s r4 r r s< F dB-KAr1a1u JKK BFA r3 c t | | \ }}}}|dk r t d ||dz z }t |t r t |j |j S t |j |j S )zReturn the square root of the sample variance. See ``variance`` for arguments and other details. >>> stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75]) 1.0810874155219827 r 'stdev requires at least two data pointsr6 r_ r rg r r rp rq r )rJ r rR r rW r< msss r4 r r sk dD/KAr1a1u GHH A,C!W $S]]COODD s}}coo >>r3 c t | | \ }}}}|dk r t d ||z }t |t r t |j |j S t |j |j S )zReturn the square root of the population variance. See ``pvariance`` for arguments and other details. >>> pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75]) 0.986893273527251 r6 z'pstdev requires at least one data pointr )rJ r rR r rW r< r s r4 r r sg dB-KAr1a1u GHH q&C!W $S]]COODD s}}coo >>r3 c t | \ }}}}|dk r t d ||dz z } t | t |j |j fS # t $ r% t | t | t | z fcY S w xY w)zFIn one pass, compute the mean and sample standard deviation as floats.r r r6 )r_ r rh r rp rq rb )rJ rR r r r< r s r4 _mean_stdevr s YNAr41u GHH A,C4T{ / s OOO 4T{E$K%) 3334s *A +BBc t | }t | |k7 r t d |dk r t d t | |z t | |z t fd| D fd|D }||dz z S )ap Covariance Return the sample covariance of two inputs *x* and *y*. Covariance is a measure of the joint variability of two inputs. >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9] >>> y = [1, 2, 3, 1, 2, 3, 1, 2, 3] >>> covariance(x, y) 0.75 >>> z = [9, 8, 7, 6, 5, 4, 3, 2, 1] >>> covariance(x, z) -7.5 >>> covariance(z, x) -7.5 zDcovariance requires that both inputs have same number of data pointsr z,covariance requires at least two data pointsc 3 ( K | ] }|z y wr8 r2 )r: xir s r4 r= z covariance.<locals>.<genexpr> s )q29q c 3 ( K | ] }|z y wr8 r2 r: yiybars r4 r= z covariance.<locals>.<genexpr> s +B"BIr r6 )r r r& r' )rV yr< sxyr r s @@r4 r r sx " AA 1v{ dee1u LMM 7Q;D 7Q;D )q)+B+B CC !a%=r3 linear)r c t | }t | |k7 r t d |dk r t d |dvrt d| |dk( r#|dz dz }t | | } t || }n@t | |z }t | |z }| D cg c] }||z } }|D cg c] }||z }}t | | } t | | } t || } | t | | z z S c c}w c c}w # t $ r t d w xY w) al Pearson's correlation coefficient Return the Pearson's correlation coefficient for two inputs. Pearson's correlation coefficient *r* takes values between -1 and +1. It measures the strength and direction of a linear relationship. >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9] >>> y = [9, 8, 7, 6, 5, 4, 3, 2, 1] >>> correlation(x, x) 1.0 >>> correlation(x, y) -1.0 If *method* is "ranked", computes Spearman's rank correlation coefficient for two inputs. The data is replaced by ranks. Ties are averaged so that equal values receive the same rank. The resulting coefficient measures the strength of a monotonic relationship. Spearman's rank correlation coefficient is appropriate for ordinal data or for continuous data that doesn't meet the linear proportion requirement for Pearson's correlation coefficient. zEcorrelation requires that both inputs have same number of data pointsr z-correlation requires at least two data points> r rankedr r r6 r r z&at least one of the inputs is constant)r r ro r r& r' r r ) rV r r r< r| r r r r r r^ syys r4 r r s+ . AA 1v{ eff1u MNN ))+F:677 Q" !5 ! !5 !Aw{Aw{!" #2R$Y #!" #2R$Y # !Q-C !Q-C !Q-CHT#)_$$ $ # H FGGHs C%! C*C/ /DLinearRegressionslope intercept) proportionalc t | }t | |k7 r t d |dk r t d |s9t | |z }t | |z | D cg c] }||z } } fd|D }t | | dz }t | | } ||z }|rdn |z z } t || S c c}w # t $ r t d w xY w)a Slope and intercept for simple linear regression. Return the slope and intercept of simple linear regression parameters estimated using ordinary least squares. Simple linear regression describes relationship between an independent variable *x* and a dependent variable *y* in terms of a linear function: y = slope * x + intercept + noise where *slope* and *intercept* are the regression parameters that are estimated, and noise represents the variability of the data that was not explained by the linear regression (it is equal to the difference between predicted and actual values of the dependent variable). The parameters are returned as a named tuple. >>> x = [1, 2, 3, 4, 5] >>> noise = NormalDist().samples(5, seed=42) >>> y = [3 * x[i] + 2 + noise[i] for i in range(5)] >>> linear_regression(x, y) #doctest: +ELLIPSIS LinearRegression(slope=3.09078914170..., intercept=1.75684970486...) If *proportional* is true, the independent variable *x* and the dependent variable *y* are assumed to be directly proportional. The data is fit to a line passing through the origin. Since the *intercept* will always be 0.0, the underlying linear function simplifies to: y = slope * x + noise >>> y = [3 * x[i] + noise[i] for i in range(5)] >>> linear_regression(x, y, proportional=True) #doctest: +ELLIPSIS LinearRegression(slope=3.02447542484..., intercept=0.0) zKlinear regression requires that both inputs have same number of data pointsr z3linear regression requires at least two data pointsc 3 ( K | ] }|z y wr8 r2 r s r4 r= z$linear_regression.<locals>.<genexpr>u s #2R$Yr z x is constantr )r r r& r' r r ) rV r r r< r r r r^ r r r s @r4 r r F s L AA 1v{ kll1u STT Aw{Aw{!" #2R$Y # # # !Q-# C !Q-C/c $ )<I %9 == $ / o../s B+ B0 0Cc | dz }t | dk rpd||z z }d|z dz |z dz |z dz |z dz |z d z |z d z |z d z |z }d |z d z |z dz |z dz |z dz |z dz |z dz |z dz }||z }|||z z S |dk r| nd| z }t t | }|dk r^|dz }d|z dz |z dz |z dz |z dz |z d z |z d z |z d z }d |z d z |z d!z |z d"z |z d#z |z d$z |z d%z |z dz }n]|dz }d&|z d'z |z d(z |z d)z |z d*z |z d+z |z d,z |z d-z }d.|z d/z |z d0z |z d1z |z d2z |z d3z |z d4z |z dz }||z }|dk r| }|||z z S )5N ?g333333?g Q ?g^}o)@gE.kR@g U l@g*u>l@g N@g"]Ξ@gnC`@gu @giK~j@gv|E@gd|1@gfRr@gu.2@g~y@gn8(E@ ?r g @g?g鬷ZaI?ggElD?g7\?guSS?g=. @gj%b@gHw@gjRe?g9dh? >g('߿A?g~z ?g@3?gɅ3?g3fRx?gIFl @gt>g*Yn>gESB\T?gN;A+?gUR1?gE F?gPn@g&>@gi<g@F>gtcI,\>gŝI?g*F2v?gC4?gO1?)r! r r% )pr sigmar rr r rV s r4 _normal_dist_inv_cdfr s CA Aw% q1u 0140145601456 11 566 1 1 56 6 1 1 56 6 1 1 56 6 11 566 1140145601456 11 566 1 1 56 6 1 1 56 6 1 1 56 6 #IQY #X37A c!fW ACx G1A51256712567 22 677 2 2 67 7 2 2 67 7 2 2 67 7 22 2A51256712567 22 677 2 2 67 7 2 2 67 7 2 2 67 7 G1A51256712567 22 677 2 2 67 7 2 2 67 7 2 2 67 7 22 3Q61256712567 22 677 2 2 67 7 2 2 67 7 2 2 67 7 c A3w B U r3 )r c e Zd ZdZdddZd!dZed Zddd Zd Z d Z d Z d"d Z d Z d Zed Zed Zed Zed Zed Zd Zd Zd Zd Zd Zd ZeZd Z eZ d Z d Z d Z d Z!d Z"y)#r z(Normal distribution of a random variablez(Arithmetic mean of a normal distributionz+Standard deviation of a normal distribution_mu_sigmac d |dk r t d t | | _ t | | _ y)zDNormalDist where mu is the mean and sigma is the standard deviation.r zsigma must be non-negativeN)r rh r r )selfr r s r4 __init__zNormalDist.__init__ s+ 3;!">? ?9El r3 c | t | S )z5Make a normal distribution instance from sample data.)r )clsrJ s r4 from_sampleszNormalDist.from_samples s K%&&r3 N)seedc |t j n t j | j }| j | j }}t d| D cg c] } ||| c}S c c}w )z=Generate *n* samples for a given mean and standard deviation.N)randomgaussRandomr r r )r r< r r r r r s r4 sampleszNormalDist.samples sV $ &--2E2K2KHHdkkE*0q/:/Qb% /:::s A+c | j | j z }|s t d || j z }t ||z d|z z t t |z z S )z4Probability density function. P(x <= X < x+dx) / dxz$pdf() not defined when sigma is zerog )r r r r" r r$ )r rV r diffs r4 pdfzNormalDist.pdf sV ;;,!"HI I488|4$;$/23d3>6JJJr3 c | j s t d ddt || j z | j t z z z z S )z,Cumulative distribution function. P(X <= x)z$cdf() not defined when sigma is zeror r )r r r# r _SQRT2r rV s r4 cdfzNormalDist.cdf s@ {{!"HI IcCTXX $++2F G HHIIr3 c n |dk s|dk\ r t d t || j | j S )aS Inverse cumulative distribution function. x : P(X <= x) = p Finds the value of the random variable such that the probability of the variable being less than or equal to that value equals the given probability. This function is also called the percent point function or quantile function. r r z$p must be in the range 0.0 < p < 1.0)r r r r )r r s r4 inv_cdfzNormalDist.inv_cdf s4 8qCx!"HI I#Atxx==r3 c d t d| D cg c] }| j ||z c}S c c}w )an Divide into *n* continuous intervals with equal probability. Returns a list of (n - 1) cut points separating the intervals. Set *n* to 4 for quartiles (the default). Set *n* to 10 for deciles. Set *n* to 100 for percentiles which gives the 99 cuts points that separate the normal distribution in to 100 equal sized groups. r6 )r r )r r< r s r4 r zNormalDist.quantiles s/ .31a[9[ QU#[999s -c t |t s t d | |}}|j |j f|j |j fk r||}}|j |j }}|r|s t d ||z }t |j |j z }|s%dt |d|j z t z z z S |j |z |j |z z }|j |j z t ||z |t ||z z z z } || z |z } || z |z } dt |j | |j | z t |j | |j | z z z S )a Compute the overlapping coefficient (OVL) between two normal distributions. Measures the agreement between two normal probability distributions. Returns a value between 0.0 and 1.0 giving the overlapping area in the two underlying probability density functions. >>> N1 = NormalDist(2.4, 1.6) >>> N2 = NormalDist(3.2, 2.0) >>> N1.overlap(N2) 0.8035050657330205 z$Expected another NormalDist instancez(overlap() not defined when sigma is zeror r- ) r r ri r r r r r! r# r r r% r ) r otherXYX_varY_vardvr r bx1x2s r4 overlapzNormalDist.overlap s_ %, BC CU1 HHaee !%% 0 0aqAzz1::uE!"LM M U] !%%!%%- R3>F#: ;<< < EEEMAEEEM ) HHqxx $rBwc%%-6H1H'H"I I!er\!er\d1559quuRy01DrQUU2Y9N4OOPPr3 c h | j s t d || j z | j z S )zCompute the Standard Score. (x - mean) / stdev Describes *x* in terms of the number of standard deviations above or below the mean of the normal distribution. z'zscore() not defined when sigma is zero)r r r r s r4 zscorezNormalDist.zscore9 s. {{!"KL LDHH ++r3 c | j S )z+Arithmetic mean of the normal distribution.r r s r4 r zNormalDist.meanD xxr3 c | j S )z,Return the median of the normal distributionr/ r0 s r4 r zNormalDist.medianI r1 r3 c | j S )zReturn the mode of the normal distribution The mode is the value x where which the probability density function (pdf) takes its maximum value. r/ r0 s r4 r zNormalDist.modeN s xxr3 c | j S )z.Standard deviation of the normal distribution.r r0 s r4 r zNormalDist.stdevW s {{r3 c 4 | j | j z S )z!Square of the standard deviation.r5 r0 s r4 r zNormalDist.variance\ s {{T[[((r3 c t |t rAt | j |j z t | j |j S t | j |z | j S )aj Add a constant or another NormalDist instance. If *other* is a constant, translate mu by the constant, leaving sigma unchanged. If *other* is a NormalDist, add both the means and the variances. Mathematically, this works only if the two distributions are independent or if they are jointly normally distributed. r r r r r r) r* s r4 __add__zNormalDist.__add__a O b* % bffrvvouRYY /JK K"&&2+ryy11r3 c t |t rAt | j |j z t | j |j S t | j |z | j S )as Subtract a constant or another NormalDist instance. If *other* is a constant, translate by the constant mu, leaving sigma unchanged. If *other* is a NormalDist, subtract the means and add the variances. Mathematically, this works only if the two distributions are independent or if they are jointly normally distributed. r8 r9 s r4 __sub__zNormalDist.__sub__o r; r3 c ` t | j |z | j t | z S )zMultiply both mu and sigma by a constant. Used for rescaling, perhaps to change measurement units. Sigma is scaled with the absolute value of the constant. r r r r! r9 s r4 __mul__zNormalDist.__mul__} & "&&2+ryy48';<<r3 c ` t | j |z | j t | z S )zDivide both mu and sigma by a constant. Used for rescaling, perhaps to change measurement units. Sigma is scaled with the absolute value of the constant. r? r9 s r4 __truediv__zNormalDist.__truediv__ rA r3 c B t | j | j S )z Return a copy of the instance.r r r r) s r4 __pos__zNormalDist.__pos__ s "&&")),,r3 c D t | j | j S )z(Negates mu while keeping sigma the same.rE rF s r4 __neg__zNormalDist.__neg__ s 266'299--r3 c | |z S )z<Subtract a NormalDist from a constant or another NormalDist.r2 r9 s r4 __rsub__zNormalDist.__rsub__ s bzr3 c t |t st S | j |j k( xr | j |j k( S )zFTwo NormalDist objects are equal if their mu and sigma are both equal.)r r NotImplementedr r r9 s r4 __eq__zNormalDist.__eq__ s7 "j)! !vv :BII$::r3 c D t | j | j f S )zCNormalDist objects hash equal if their mu and sigma are both equal.)hashr r r0 s r4 __hash__zNormalDist.__hash__ s TXXt{{+,,r3 c f t | j d| j d| j dS )Nz(mu=z, sigma=))rB r/ r r r0 s r4 __repr__zNormalDist.__repr__ s. t*%%&d488,ht{{oQOOr3 c 2 | j | j fS r8 r r0 s r4 __getstate__zNormalDist.__getstate__ s xx$$r3 c " |\ | _ | _ y r8 r )r states r4 __setstate__zNormalDist.__setstate__ s % $+r3 )r r )r )#r/ r0 r1 __doc__ __slots__r classmethodr r r r r r r+ r- propertyr r r r r r: r= r@ rC rG rI __radd__rK __rmul__rN rQ rT rV rY r2 r3 r4 r r s . :?I # ' ' "& ; KJ > : QD , ) ) 2 2 ==-. H H; -P%&r3 r r8 )znegative value)r )OrZ __all__rc r r sys fractionsr decimalr itertoolsr r r bisectr r r r r! r" r# r$ r% r& r' functoolsr( operatorr) collectionsr* r+ r, r ro r rS r_ rE rH rD rt rw r rh r rI r float_infomant_digr __annotations__r r r r r r r r r r r r r r r r r r r r r r r _statistics ImportErrorr r2 r3 r4 <module>rn s hT . , , , E E E 8 8 c j 3 l& R 4>+ \$ IQ 34PU; 3l 3>>222Q6 6 #3 #3 #5 #S S W <",!HG&5,p+0 ,&E+PB<Kr ; (4b)%X& R?$?$ 4(8 $, -H` 02HI 05 7>zG V 0 Z& Z& s 1E E E