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Code Editor : statistics.pyc
+ [clj&� � �Z a � R` th0 t R t. RaOt^ RIt^ RIt^ RIt^ RIt^ RIHt ^ RI H t ^ RI H t H t Ht ^ RIHtHt ^ RIHtHtHtHtHtHtHtHtHt ^ R IHtH t H t H t H t H t H!t!H"t"H#t#H$t$ ^ R I%H&t& ^ R I'H(t( ^ R I)H*t*H+t+H,t, ]! R 4 t-]t. ! R R]/4 t0R t1RbR lt2R t3RbR lt4R t5R t6R t7RcR lt8R t9R t:RbR lt;RbR lt<RbR lt=RbR lt>R t?R R /R lt@]+! R!Rd4 tAR"R#/R$ ltB/ tCR% tD]D! R&R'4 R( 4 tE]D! R)4 R* 4 tF]D! R+4 R, 4 tG]D! R-R.4 R/ 4 tH]D! R04 R1 4 tI]D! R2R34 R4 4 tJReR5 ltKR6 tL]D! R7R84 R9 4 tMR: tN]D! R;4 R< 4 tO]D! R=4 R> 4 tPADAEAFAGAHAIAJAMAOAPRfR?R#/R@ lltQRfRAR/RB lltRRC^R RD/RE ltS ! RF R4 tTRG tURbRH ltVRI tWRJ tXRK tYRL tZRgRM lt[RNRROR#RPRQRR^/RS RT llt\RU RV lt]^]P� P� , ^, t`] ^ k RW RX ltaRY RZ ltbR[ tcR\ R] ltdR^ te ^ R_IfHete R# ]g d R# i ; i)ha� 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. � NormalDist�StatisticsErrorN��Fraction)�Decimal)�count�groupby�repeat)� bisect_left� bisect_right) �hypot�sqrt�fabs�exp�erfc�tau�log�fsum�sumprod) �isfinite�isinf�pi�cos�sin�tan�cosh�asin�atan�acos)�reduce)� itemgetter)�Counter� namedtuple� defaultdict� @c � � ] tR t^�tRtR# )r � N)�__name__� __module__� __qualname__�__firstlineno__�__static_attributes__r& � � ../lib/python3.14/statistics.pyr r � s � �r, c �f � \ V 4 w rpV^8 d \ R4 h\ W#, V4 # )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. z%mean requires at least one data point)�_sumr �_convert)�data�T�total�ns & r- �meanr5 � s3 � �" �t�*�K�A�a��1�u� � E�F�F� �E�I�q� !�!r, c �� � Vf3 \ V 4 p\ V 4 pV' g \ R4 hW2, # \ V\ \ 34 ' g \ V4 p \ W4 p\ T4 pT' g \ R4 hYV, # \ d@ \ 4 p\ \ \ ^ 4 \ Y4 4 4 p\ T4 p L�i ; i \ d \ R4 hi ; i)z�Convert 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 z&fmean requires at least one data pointz(data and weights must be the same lengthz sum of weights must be non-zero)�lenr � TypeErrorr �mapr �zip�nextr � isinstance�list�tupler � ValueError)r1 �weightsr4 r3 �counter�num�dens && r- �fmeanrD � s� � � �� ��D� �A� ��J�E��!�"J�K� K��y�� �g��e�}� -� -��w�-��J��d�$�� �w�-�C� � � ?�@�@� �9���5 � ��g�G���Z��]�C��,>�?�@�E��W� �A� ��$ � J� � H�I�I�J�s � B � C �AC�C� C4c �Z aa� ^ oRoVV3R lp\ \ \ V! V 4 4 4 pS' g \ R4 h\ P ! V4 ' d \ P # S' d( V\ P 8X d \ P # R# \ VS, 4 # )aV Convert data to floats and compute the geometric mean. Raises a StatisticsError if the input dataset is empty or if it contains a negative value. Returns zero if the product of inputs is zero. 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 Fc 3 � <" � \ V ^R7 FD w opVR8� g \ P ! V4 ' d Vx � K/ VR8X d RoK: \ RV4 h R# 5i)� ��start� TzNo negative inputs allowedN)� enumerate�math�isnanr )�iterable�x� found_zeror4 s & ��r- �count_positive�&geometric_mean.<locals>.count_positive� sM �� � � �h�a�0�D�A�q��3�w�$�*�*�Q�-�-����c�� !� �%�&B�A�F�F� 1�s �6A�!Az Must have a non-empty datasetrJ ) r r9 r r rL rM �nan�infr )r1 rQ r3 rP r4 s & @@r- �geometric_meanrU � s �� � �A��J�G� ��S�.��.�/� 0�E� � � =�>�>� �z�z�%����x�x��� �D�H�H�,�t�x�x�5�#�5� �u�q�y�>�r, c � � \ V 4 V J d \ V 4 p Rp\ V 4 pV^8 d \ R4 hV^8X dS VfO V ^ , p\ V\ P \ 34 ' d V^ 8 d \ V4 hV# \ R4 hVf \ ^V4 pTpMS\ V4 VJ d \ V4 p\ V4 V8w d \ R4 h\ R \ W4 4 4 w rep \ W4 p \ R \ W4 4 4 w rxp T^ 8: d \ R4 h\ YX, T4 # \ d ^ # i ; i)a� Return the harmonic mean of data. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the data. It can be used for averaging ratios or rates, for example speeds. Suppose a car travels 40 km/hr for 5 km and then speeds-up to 60 km/hr for another 5 km. What is the average speed? >>> harmonic_mean([40, 60]) 48.0 Suppose a car travels 40 km/hr for 5 km, and when traffic clears, speeds-up to 60 km/hr for the remaining 30 km of the journey. What is the average speed? >>> harmonic_mean([40, 60], weights=[5, 30]) 56.0 If ``data`` is empty, or any element is less than zero, ``harmonic_mean`` will raise ``StatisticsError``. z.harmonic mean does not support negative valuesz.harmonic_mean requires at least one data pointzunsupported typez*Number of weights does not match data sizec 3 �$ " � T F qx � K R # 5i�Nr& )�.0�ws & r- � <genexpr>� harmonic_mean.<locals>.<genexpr>9 s � � � G�,F�q��,F�s �c 3 �J " � T F w rV' d W, M^ x � K R# 5i)� Nr& )rY rZ rO s & r- r[ r\ = s � � � P�=O�T�Q��q�u�q� 0�=O�� �!#z Weighted sum must be positive)�iterr= r7 r r< �numbers�Realr r8 r r/ � _fail_negr: �ZeroDivisionErrorr0 ) r1 r@ �errmsgr4 rO � sum_weights�_r2 r3 r s && r- � harmonic_meanrh sD � �0 �D�z�T���D�z�� =�F� �D� �A��1�u� � N�O�O� �a��G�O� ��G�� �a�'�,�,��0� 1� 1��1�u�%�f�-�-��H�� .�/� /�����A�,��� � ��=�G� #��7�m�G� �w�<�1� �!�"N�O� O� � G�I�g�,F� G� G�������&�� � P�S��=O� P�P���%� ��z� � =�>�>� �K�'�� +�+�� � ���s �:)E � E�Ec �� � \ V 4 p \ V 4 pV^ 8X d \ R4 hV^, ^8X d W^, , # V^, pW^, , W, , ^, # )a" 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 �no median for empty data��sortedr7 r )r1 r4 �is & r- �medianrn G sa � � �$�<�D� �D� �A��A�v� � 8�9�9��1�u��z���F�|�� ��F����U� �d�g�%��*�*r, c � � \ V 4 p \ V 4 pV^ 8X d \ R4 hV^, ^8X d W^, , # W^, ^, , # )z�Return the low median of numeric data. When the number of data points is odd, the middle value is returned. When it is even, the smaller of the two middle values is returned. >>> median_low([1, 3, 5]) 3 >>> median_low([1, 3, 5, 7]) 3 rj rk �r1 r4 s & r- � median_lowrq _ sQ � � �$�<�D� �D� �A��A�v� � 8�9�9��1�u��z���F�|����F�Q�J� � r, c �p � \ V 4 p \ V 4 pV^ 8X d \ R4 hW^, , # )z�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 rj rk rp s & r- � median_highrs x s5 � � �$�<�D� �D� �A��A�v� � 8�9�9� �Q��<�r, c �~ � \ V 4 p \ V 4 pV' g \ R4 hW^, , p\ W4 p\ WVR7 p \ V4 p\ V4 pY1R, , pTpYT, pYaT^, T, , T, , # \ d \ R4 hi ; i)a Estimates the median for numeric data binned around the midpoints of consecutive, fixed-width intervals. The *data* can be any iterable of numeric data with each value being exactly the midpoint of a bin. At least one value must be present. The *interval* is width of each bin. For example, demographic information may have been summarized into consecutive ten-year age groups with each group being represented by the 5-year midpoints of the intervals: >>> demographics = Counter({ ... 25: 172, # 20 to 30 years old ... 35: 484, # 30 to 40 years old ... 45: 387, # 40 to 50 years old ... 55: 22, # 50 to 60 years old ... 65: 6, # 60 to 70 years old ... }) The 50th percentile (median) is the 536th person out of the 1071 member cohort. That person is in the 30 to 40 year old age group. The regular median() function would assume that everyone in the tricenarian age group was exactly 35 years old. A more tenable assumption is that the 484 members of that age group are evenly distributed between 30 and 40. For that, we use median_grouped(). >>> data = list(demographics.elements()) >>> median(data) 35 >>> round(median_grouped(data, interval=10), 1) 37.5 The caller is responsible for making sure the data points are separated by exact multiples of *interval*. This is essential for getting a correct result. The function does not check this precondition. Inputs may be any numeric type that can be coerced to a float during the interpolation step. rj )�loz$Value cannot be converted to a floatr$ )rl r7 r r r �floatr? r8 ) r1 �intervalr4 rO rm �j�L�cf�fs && r- �median_groupedr| � s� � �V �$�<�D� �D� �A� � � 8�9�9� �!�V� �A� �D� �A��T��#�A�A���?�� �!�H�� �s�N��A� �B� ��A� �1�q�5�2�:�&��*� *�*�� � A��>�@�@�A�s �B% �%B<c � � \ \ V 4 4 P ^4 p V^ , ^ , # \ d \ R4 Rhi ; i)aD Return the most common data point from discrete or nominal data. ``mode`` assumes discrete data, and returns a single value. This is the standard treatment of the mode as commonly taught in schools: >>> mode([1, 1, 2, 3, 3, 3, 3, 4]) 3 This also works with nominal (non-numeric) data: >>> mode(["red", "blue", "blue", "red", "green", "red", "red"]) 'red' If there are multiple modes with same frequency, return the first one encountered: >>> mode(['red', 'red', 'green', 'blue', 'blue']) 'red' If *data* is empty, ``mode``, raises StatisticsError. zno mode for empty dataN)r! r` � most_common� IndexErrorr )r1 �pairss & r- �moder� � sP � �. �D��J� � +� +�A� .�E�B��Q�x��{��� � B� � 6�7�T�A�B�s �5 �A c �� � \ \ V 4 4 pV' g . # \ VP 4 4 pVP 4 UUu. uF w r4WB8X g K VNK upp# u uppi )a Return a list of the most frequently occurring values. Will return more than one result if there are multiple modes or an empty list if *data* is empty. >>> multimode('aabbbbbbbbcc') ['b'] >>> multimode('aabbbbccddddeeffffgg') ['b', 'd', 'f'] >>> multimode('') [] )r! r` �max�values�items)r1 �counts�maxcount�valuer s & r- � multimoder� � sQ � � �T�$�Z� �F� �� ��6�=�=�?�#�H�&,�l�l�n� J�n�l�e��8I�E�E�n� J�J�� Js � A$�A$c �t � \ W4 w r#rEV^8 d \ R4 h\ W5^, , V4 # )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) z*variance requires at least two data points��_ssr r0 )r1 �xbarr2 �ss�cr4 s && r- �variancer� s8 � �P �d�/�K�A�1��1�u� � J�K�K� �B�a�%�L�!� $�$r, c �f � \ W4 w r#rEV^8 d \ R4 h\ W5, V4 # )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) z*pvariance requires at least one data pointr� )r1 �mur2 r� r� r4 s && r- � pvariancer� 6 s4 � �J �d�-�K�A�1��1�u� � J�K�K� �B�F�A� � r, c � � \ W4 w r#rEV^8 d \ R4 hW5^, , p VP pVP p\ T\ 4 ' d \ Yx4 # \ Yx4 # \ d \ R4 hi ; i)z�Return 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 �'stdev requires at least two data points� inf or nan encountered in data� r� r � numerator� denominator�AttributeErrorr? � issubclassr �_decimal_sqrt_of_frac�_float_sqrt_of_frac) r1 r� r2 r� r� r4 �mss� mss_numerator�mss_denominators && r- �stdevr� a s� � � �d�/�K�A�1��1�u� � G�H�H� �A��,�C�;�� � � � �/�/�� �!�W� � �$�]�D�D� �}� >�>�� � ;��9�:�:�;�s �A5 �5B c � � \ W4 w r#rEV^8 d \ R4 hW5, p VP pVP p\ T\ 4 ' d \ Yx4 # \ Yx4 # \ d \ R4 hi ; i)z�Return the square root of the population variance. See ``pvariance`` for arguments and other details. >>> pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75]) 0.986893273527251 z'pstdev requires at least one data pointr� r� ) r1 r� r2 r� r� r4 r� r� r� s && r- �pstdevr� x s� � � �d�-�K�A�1��1�u� � G�H�H� �&�C�;�� � � � �/�/�� �!�W� � �$�]�D�D� �}� >�>�� � ;��9�:�:�;�s �A. �.Bc � aa� \ V 4 p\ V4 V8w d \ R4 hV^8 d \ R4 h\ V 4 V, o\ V4 V, o\ V3R lV 4 V3R lV 4 4 pW2^, , # )a@ 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 pointsz,covariance requires at least two data pointsc 3 �4 <" � T F qS, x � K R # 5irX r& )rY �xir� s & �r- r[ � covariance.<locals>.<genexpr>� s �� � �)�q���9�9�q�� �c 3 �4 <" � T F qS, x � K R # 5irX r& �rY �yi�ybars & �r- r[ r� � s �� � �+B��"��I�I��r� )r7 r r r )rO �yr4 �sxyr� r� s "" @@r- � covariancer� � sv �� �$ �A��A� �1�v��{� � d�e�e��1�u� � L�M�M� ��7�Q�;�D� ��7�Q�;�D� �)�q�)�+B��+B� C�C� �a�%�=�r, �method�linearc �R � \ V 4 p\ V4 V8w d \ R4 hV^8 d \ R4 hVR9 d \ RV: 24 hVR8X d* V^, R, p\ WR7 p \ WR7 pMR\ V 4 V, p\ V4 V, pV Uu. uF qwV, NK p pV Uu. uF q�V, NK pp\ W4 p \ W 4 p \ W4 p V \ W�4 , # u upi u upi \ d \ R4 hi ; i) a) 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 pointsz-correlation requires at least two data points�ranked�Unknown method: rH z&at least one of the inputs is constant> r� r� ���)r7 r r? �_rankr r � _sqrtprodrd ) rO r� r� r4 rI r� r� r� r� r� �sxx�syys ""$ r- � correlationr� � s � �4 �A��A� �1�v��{� � e�f�f��1�u� � M�N�N� �)�)��+�F�:�6�7�7� ����Q��"� �� �!� !�� �!� !�� �A�w��{���A�w��{��!"� #��2�$�Y�Y��� #�!"� #��2�$�Y�Y��� #� �!�-�C� �!�-�C� �!�-�C�H��Y�s�(�(�(�� $�� #�� � H� � F�G�G�H�s �'D�>D �3D �D&�LinearRegression� proportionalFc �� a � \ V 4 p\ V4 V8w d \ R4 hV^8 d \ R4 hV' gH \ V 4 V, p\ V4 V, o V Uu. uF qUV, NK p pV 3R lV 4 p\ W4 R, p\ W 4 p Wg, pT' d RMS TX, , p \ Y�R7 # u upi \ d \ R4 hi ; i)aO Slope and intercept for simple linear regression. Return the slope and intercept of simple linear regression parameters estimated using ordinary least squares. Simple linear regression describes relationship between an independent variable *x* and a dependent variable *y* in terms of a linear function: y = slope * x + intercept + noise where *slope* and *intercept* are the regression parameters that are estimated, and noise represents the variability of the data that was not explained by the linear regression (it is equal to the difference between predicted and actual values of the dependent variable). The parameters are returned as a named tuple. >>> x = [1, 2, 3, 4, 5] >>> noise = NormalDist().samples(5, seed=42) >>> y = [3 * x[i] + 2 + noise[i] for i in range(5)] >>> linear_regression(x, y) #doctest: +ELLIPSIS LinearRegression(slope=3.17495..., intercept=1.00925...) If *proportional* is true, the independent variable *x* and the dependent variable *y* are assumed to be directly proportional. The data is fit to a line passing through the origin. Since the *intercept* will always be 0.0, the underlying linear function simplifies to: y = slope * x + noise >>> y = [3 * x[i] + noise[i] for i in range(5)] >>> linear_regression(x, y, proportional=True) #doctest: +ELLIPSIS LinearRegression(slope=2.90475..., intercept=0.0) zKlinear regression requires that both inputs have same number of data pointsz3linear regression requires at least two data pointsc 3 �4 <" � T F qS, x � K R # 5irX r& r� s & �r- r[ �$linear_regression.<locals>.<genexpr> s �� � � #��2�$�Y�Y��r� rJ z x is constant��slope� intercept)r7 r r r rd r� ) rO r� r� r4 r� r� r� r� r� r� r� s ""$ @r- �linear_regressionr� � s� �� �N �A��A� �1�v��{� � k�l�l��1�u� � S�T�T� ��A�w��{���A�w��{��!"� #��2�$�Y�Y��� #� #�� #�� �!�-�#� �C� �!�-�C�/�� �� $����� �)<�I� �%� =�=�� $�� � /� �o�.�.�/�s �*C�'C �C1c � a � V 3R lpV# )zCLoad the kernel's pdf, cdf, invcdf, and support into _kernel_specs.c �` <� \ \ RV ! 4 4 4 pS F pV\ V& K V # )�pdf�r� �cdf�invcdf�support)�dictr: � _kernel_specs)�builder�spec�kernel�kernelss & �r- �deco�register.<locals>.deco- s0 �� ��C�;�W�Y�G�H�� �F�$(�M�&� !� ��r, r& )r� r� s j r- �registerr� + s �� �� �Kr, �normal�gaussc �t aa� \ ^\ , 4 o\ ^4 ) oV3R lp V3R lpR pRpWW#3# )� c �D <� \ RV , V , 4 S, # )rG � r )�t�sqrt2pis &�r- �<lambda>� normal_kernel.<locals>.<lambda>8 s �� �C��q��1� �%��/r, c �6 <� R\ V S, 4 , # �rG � �?)r )r� � neg_sqrt2s &�r- r� r� 9 s �� �C�$�q�9�}�-�-r, c � � \ V R R4 # �rJ � �?��_normal_dist_inv_cdf�r� s &r- r� r� : s � �+�A�s�C�8r, N)r r )r� r� r� r� r� r� s @@r- � normal_kernelr� 4 s: �� ��1�r�6�l�G��a���I� /�C� -�C� 8�F��G� �V� $�$r, �logisticc � � R p R pR pRpWW#3# )c �4 � RR\ V 4 , , # )rG r� r� �r r� s &r- r� �!logistic_kernel.<locals>.<lambda>A s � �C�3��a��=�)r, c �B � R R \ V 4 R , , , # �r� r� r� s &r- r� r� B s � �C�#��Q��#� �.�.r, c �4 � \ V R V , , 4 # r� )r ��ps &r- r� r� C s � �s�1��a��=�)r, Nr& r� s r- �logistic_kernelr� > s# � � *�C� .�C� )�F��G� �V� $�$r, �sigmoidc � aaa� ^\ , o^\ , o\ ^, oV3R lp V3R lpV3R lpRpWW#3# )rG c �( <� S\ V 4 , # rX r� )r� �c1s &�r- r� � sigmoid_kernel.<locals>.<lambda>M s �� �B��a��Lr, c �: <� S\ \ V 4 4 , # rX )r r �r� �c2s &�r- r� r� N s �� �B��c�!�f� �%r, c �: <� \ \ V S, 4 4 # rX )r r )r� �c3s &�r- r� r� O s �� �s�3�q�2�v�;�'r, N�r )r� r� r� r� r� r� r� s @@@r- �sigmoid_kernelr G s? �� � �R��B� �R��B� �a��B� �C� %�C� '�F��G� �V� $�$r, � rectangular�uniformc � � R p R pR pRpWW#3# )c � � R# r� r& r� s &r- r� �$rectangular_kernel.<locals>.<lambda>U s � �Cr, c �" � RV , R, # r� r& r� s &r- r� r V s � �C�!�G�c�Mr, c �" � R V , R, # �r$ r� r& r� s &r- r� r W s � �s�Q�w��}r, r� r& r� s r- �rectangular_kernelr S s! � � �C� !�C� $�F��G� �V� $�$r, � triangularc � � R p R pR pRpWW#3# )c �& � R \ V 4 , # r� ��absr� s &r- r� �#triangular_kernel.<locals>.<lambda>] s � �C�#�a�&�Lr, c �N � W , V R 8 d RMR, V , R, # )rJ r� r� r& r� s &r- r� r ^ s � �A�C�!�c�'�3�t�4�q�8�3�>r, c � � V R8 d \ RV , 4 R, # R\ RRV , , 4 , # )rG r$ r� r� )r r� s &r- r� r _ s1 � �A��G�t�C��E�{�S�(�P��t�C�#�a�%�K�?P�9P�Pr, r� r& r� s r- �triangular_kernelr [ s! � � �C� >�C� P�F��G� �V� $�$r, � parabolic� epanechnikovc � � R p R pR pRpWW#3# )c �. � RRW , , , # )� r� � �?r& r� s &r- r� �"parabolic_kernel.<locals>.<lambda>e s � �C�3���;�'r, c �. � \ RV ^, V R34 # )rG r� )g пr r� �r r� s &r- r� r f s � �G� ,�q�!�t�Q��n�=r, c �x � R \ \ R V , R, 4 \ , R, 4 , # )r$ r� g @)r r r r� s &r- r� r g s% � �s�S�$�s�1�u�s�{�"3�b�"8�C�!?� @�@r, r� r& r� s r- �parabolic_kernelr c s! � � '�C� =�C� @�F��G� �V� $�$r, c � a aaa� VV VV3R lpV# )c � <� S! V 4 p\ S! V4 V , ;p4 S8� d WS! V4 , , pK6 V# )u= Return x such that f(x) ≈ y within the specified tolerance.r )r� rO �diffr{ �f_inv_estimate�f_prime� tolerances & ����r- �f_inv� _newton_raphson.<locals>.f_invl s= �� � �1� ���!�A�$��(�"�$�#�i�/� ��� �"� "�A��r, r& )r! r{ r" r# r$ s ffff r- �_newton_raphsonr& k s �� �� � �Lr, c � � V R 8: d RV 3M R RV , 3w rV R8 d RV , R, R, V, # RV , R, R, pV R8 d( VR\ RV , R , 4 , , pW!, # ) rG r� g�g��s��?r$ gq �-��?g��鼹A�?�V-����?gM�p�^v�?g��$2h @g_���@r� � �r �r� �signrO s & r- �_quartic_invcdf_estimater- t s| � ��s�(�s�A�h��s�Q�w��G�D��6�z��q��V�#�c�)�T�1�1� �q��_�$�s�*�A��5�y� �[�3�{�Q��1A� A�B� B�B�� �8�Or, �quartic�biweightc �< � R p R p\ \ WR7 pRpWW#3# )c �< � RRW , , ^, , # )� r� � �?r& r� s &r- r� � quartic_kernel.<locals>.<lambda>� � � �E�S�1�5�[�Q� .�.r, c �> � \ RV ^, V ^, V R34 # )r r� )g �?g �r3 r� r r� s &r- r� r4 � s � �G� 4� ��T�1�a�4��C� 0�2r, �r{ r" r� )r& r- r� s r- �quartic_kernelr8 ~ s+ � � .�C� 2�C� � 5�� J�F��G� �V� $�$r, c � � V R 8: d RV 3M R RV , 3w rRV , R, R, pRT u;8 d R8 d= M W!, # VR\ R\ , V R, , 4 , , pW!, # ) rG r� r$ g�c���?g�h㈵��>r( gL7�A`�?g ��Q� �?g�Q� ��?r� r) )r r r+ s & r- �_triweight_invcdf_estimater: � so � ��s�(�s�A�h��s�Q�w��G�D� �q��'�'�#�-�A����U�� �8�O� �U�S����q�5�y�1�2� 2�2�� �8�Or, � triweightc �< � R p R p\ \ WR7 pRpWW#3# )c �< � RRW , , ^, , # )�# r� � ��?r& r� s &r- r� �"triweight_kernel.<locals>.<lambda>� r5 r, c �N � \ RV ^, V ^, V ^, V R34 # )� r� )g Ŀg �?g ��r? r� r r� s &r- r� r@ � s& � �G� >� ��T�1�a�4��A��q�#� 6�8r, r7 r� )r& r: r� s r- �triweight_kernelrC � s+ � � .�C� 8�C� � 7�3� L�F��G� �V� $�$r, �cosinec �f aa� \ ^, o\ ^, oVV3R lp V3R lpR pRpWW#3# )� c �6 <� S\ SV , 4 , # rX )r )r� r� r� s &��r- r� � cosine_kernel.<locals>.<lambda>� s �� �B��R�!�V��$r, c �D <� R\ SV , 4 , R, # r� r* r� s &�r- r� rH � s �� �C�#�b�1�f�+�%��+r, c �X � R \ R V , R, 4 , \ , # r )r r r� s &r- r� rH � s � �s�T�#��'�C�-� 0�0�2�5r, r� r� )r� r� r� r� r� r� s @@r- � cosine_kernelrK � s4 �� � �a��B� �a��B� $�C� +�C� 5�F��G� �V� $�$r, � cumulativec �, a aaa a a a � \ S 4 o S ' g \ R4 h\ S ^ , \ \ 34 ' g \ R4 hSR8: d \ RS: 24 h\ P V4 pVf \ RV: 24 hVR, oVR, o VR, pVf VV V3R lpV V V3R lpM*\ S 4 o SV, o VV V VV V 3R lpV V V VV V 3R lpV' d R S: RV: 2Vn V# RS: RV: 2Vn V# )a� Kernel Density Estimation: Create a continuous probability density function or cumulative distribution function from discrete samples. The basic idea is to smooth the data using a kernel function to help draw inferences about a population from a sample. The degree of smoothing is controlled by the scaling parameter h which is called the bandwidth. Smaller values emphasize local features while larger values give smoother results. The kernel determines the relative weights of the sample data points. Generally, the choice of kernel shape does not matter as much as the more influential bandwidth smoothing parameter. Kernels that give some weight to every sample point: normal (gauss) logistic sigmoid Kernels that only give weight to sample points within the bandwidth: rectangular (uniform) triangular parabolic (epanechnikov) quartic (biweight) triweight cosine If *cumulative* is true, will return a cumulative distribution function. A StatisticsError will be raised if the data sequence is empty. Example ------- Given a sample of six data points, construct a continuous function that estimates the underlying probability density: >>> sample = [-2.1, -1.3, -0.4, 1.9, 5.1, 6.2] >>> f_hat = kde(sample, h=1.5) Compute the area under the curve: >>> area = sum(f_hat(x) for x in range(-20, 20)) >>> round(area, 4) 1.0 Plot the estimated probability density function at evenly spaced points from -6 to 10: >>> for x in range(-6, 11): ... density = f_hat(x) ... plot = ' ' * int(density * 400) + 'x' ... print(f'{x:2}: {density:.3f} {plot}') ... -6: 0.002 x -5: 0.009 x -4: 0.031 x -3: 0.070 x -2: 0.111 x -1: 0.125 x 0: 0.110 x 1: 0.086 x 2: 0.068 x 3: 0.059 x 4: 0.066 x 5: 0.082 x 6: 0.082 x 7: 0.058 x 8: 0.028 x 9: 0.009 x 10: 0.002 x Estimate P(4.5 < X <= 7.5), the probability that a new sample value will be between 4.5 and 7.5: >>> cdf = kde(sample, h=1.5, cumulative=True) >>> round(cdf(7.5) - cdf(4.5), 2) 0.22 References ---------- Kernel density estimation and its application: https://www.itm-conferences.org/articles/itmconf/pdf/2018/08/itmconf_sam2018_00037.pdf Kernel functions in common use: https://en.wikipedia.org/wiki/Kernel_(statistics)#kernel_functions_in_common_use Interactive graphical demonstration and exploration: https://demonstrations.wolfram.com/KernelDensityEstimation/ Kernel estimation of cumulative distribution function of a random variable with bounded support https://www.econstor.eu/bitstream/10419/207829/1/10.21307_stattrans-2016-037.pdf �Empty data sequence�)Data sequence must contain ints or floatsrJ �$Bandwidth h must be positive, not h=�Unknown kernel name: r� r� r� c �b <a � \ VVV 3R lS 4 4 \ S4 S, , # )c 3 �P <" � T F pS! SV, S, 4 x � K R # 5irX r& �rY �x_i�K�hrO s & ���r- r[ �#kde.<locals>.pdf.<locals>.<genexpr> �! �� � �8�4�C�q�!�c�'�Q� �'�'�4�� �#&��sumr7 )rO rV r1 rW s f���r- r� �kde.<locals>.pdf s! �� ��8�4�8�8�C��I��M�J� Jr, c �T <a � \ VVV 3R lS 4 4 \ S4 , # )c 3 �P <" � T F pS! SV, S, 4 x � K R # 5irX r& �rY rU �WrW rO s & ���r- r[ �#kde.<locals>.cdf.<locals>.<genexpr>" rY rZ r[ )rO ra r1 rW s f���r- r� �kde.<locals>.cdf! s �� ��8�4�8�8�3�t�9�D� Dr, c �� <a � \ S4 S8w d \ S4 o \ S4 o\ S S S, 4 p\ S S S, 4 pS W p\ VVV 3R lV 4 4 SS, , # )c 3 �P <" � T F pS! SV, S, 4 x � K R # 5irX r& rT s & ���r- r[ rX 1 s! �� � �=�9�C�q�!�c�'�Q� �'�'�9�rZ �r7 rl r r r\ ) rO rm rx � supportedrV � bandwidthr1 rW r4 �samples f ������r- r� r] ) sc �� ��4�y�A�~� �� ����I���F�A� �M�2�A� �V�Q��]�3�A� �q� �I��=�9�=�=��Q��G� Gr, c �� <a � \ S4 S8w d \ S4 o \ S4 o\ S S S, 4 p\ S S S, 4 pS W p\ VVV 3R lV 4 V4 S, # )c 3 �P <" � T F pS! SV, S, 4 x � K R # 5irX r& r` s & ���r- r[ rb ; s! �� � �>�I�S��1�s�7�a�-�(�(�I�rZ rf ) rO rm rx rg ra rh r1 rW r4 ri s f ������r- r� rc 3 sa �� ��4�y�A�~� �� ����I���F�A� �M�2�A� �V�Q��]�3�A� �q� �I��>�I�>��B�Q�F� Fr, zCDF estimate with h=� and kernel=zPDF estimate with h=) r7 r r< �intrv r8 r� �getrl �__doc__) r1 rW r� rL � kernel_specr� r� r� rV ra rh r4 ri s ff&$ @@@@@r- �kderq � s �� �H �D� �A� � � 3�4�4� �d�1�g��U�|� ,� ,��C�D�D��C�x� � E�1�&� I�J�J� �#�#�F�+�K��� � 5�f�Z� @�A�A��E��A��E��A��)�$�G��� K� E� E� �� ����K� � H� H� G� G� �-�1�&� �f�[�A�� �� � .�1�&� �f�[�A�� �� r, �seedc �� a aaa a � \ S 4 pV' g \ R4 h\ S ^ , \ \ 34 ' g \ R4 hSR8: d \ RS: 24 h\ P V4 pVf \ RV: 24 hVR, o \ P V4 pVP o VP oVV VV V 3R lpRS: R V: 2Vn V# ) a< Return a function that makes a random selection from the estimated probability density function created by kde(data, h, kernel). Providing a *seed* allows reproducible selections within a single thread. The seed may be an integer, float, str, or bytes. A StatisticsError will be raised if the *data* sequence is empty. Example: >>> data = [-2.1, -1.3, -0.4, 1.9, 5.1, 6.2] >>> rand = kde_random(data, h=1.5, seed=8675309) >>> new_selections = [rand() for i in range(10)] >>> [round(x, 1) for x in new_selections] [0.7, 6.2, 1.2, 6.9, 7.0, 1.8, 2.5, -0.5, -1.8, 5.6] rN rO rJ rP rQ r� c �F <� S ! S4 SS! S! 4 4 , , # rX r& )�choicer1 rW r� �randoms �����r- �rand�kde_random.<locals>.randk s �� ��d�|�a�&���"2� 2�2�2r, z Random KDE selection with h=rl ) r7 r r< rm rv r8 r� rn �_random�Randomrv ru ro ) r1 rW r� rr r4 rp �prngrw ru r� rv s ff&$ @@@r- � kde_randomr| F s� �� �$ �D� �A� � � 3�4�4� �d�1�g��U�|� ,� ,��C�D�D��C�x� � E�1�&� I�J�J� �#�#�F�+�K��� � 5�f�Z� @�A�A� �� "�F� �>�>�$� �D� �[�[�F� �[�[�F�3� 3� 3��v�]�6�+�F�D�L� �Kr, r4 � exclusivec �. � V^8 d \ R4 h\ V 4 p \ V 4 pV^8 d" V^8X d W^, , # \ R4 hVR8X d} V^, p. p\ ^V4 F_ p\ Wd, V4 w rxW, W, , W^, , V, , V, p VP V 4 Ka V# VR8X d� V^, p. p\ ^V4 F� pWd, V, pV^8 d ^MWs^, 8� d V^, MTpWd, Wq, , pW^, , W, , W, V, , V, p VP V 4 K� V# \ RV: 24 h)af 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. zn must be at least 1z!must have at least one data point� inclusiver} r� )r rl r7 �range�divmod�appendr? ) r1 r4 r� �ld�m�resultrm rx �delta� interpolateds &$$ r- � quantilesr� � s\ � �" �1�u� � 4�5�5� �$�<�D� �T��B� �A�v� ��7��q�5�>� !� � A�B�B� �� � ��F�����q�!��A� �a�e�Q�'�H�A� �G�q�y� 1�D�Q��K�%�4G� G�1�L�L� �M�M�,� '� � � � �� � ��F�����q�!��A���� �A���U���q�D���1��a�A��C�!�#�I�E� �Q��K�1�9� 5���%�� G�1�L�L� �M�M�,� '� � � � �'��z�2� 3�3r, c � a � ] tR tRt o RtRRRR/tR$R lt]R 4 tR R /R lt R t R t R t R%R lt R tR t]R 4 t]R 4 t]R 4 t]R 4 t]R 4 tR tR tR tR tR tR t]t R t ]t R t R t R t!R! t"R" t#R#t$V t%R # )&r i� z(Normal distribution of a random variable�_muz(Arithmetic mean of a normal distribution�_sigmaz+Standard deviation of a normal distributionc �j � VR8 d \ R4 h\ V4 V n \ V4 V n R# )zDNormalDist where mu is the mean and sigma is the standard deviation.rJ zsigma must be non-negativeN)r rv r� r� )�selfr� �sigmas &&&r- �__init__�NormalDist.__init__� s+ � � �3�;�!�">�?� ?���9����E�l�� r, c � � V ! \ V4 ! # )z5Make a normal distribution instance from sample data.)� _mean_stdev)�clsr1 s &&r- � from_samples�NormalDist.from_samples� s � � �K��%�&�&r, rr Nc � � Vf \ P M \ P ! V4 P p\ pV P pV P p\ RV4 Uu. uF qt! V! 4 WV4 NK up# u upi )z=Generate *n* samples for a given mean and standard deviation.N)rv rz r� r� r� r )r� r4 rr �rnd�inv_cdfr� r� rg s &&$ r- �samples�NormalDist.samples� s^ � � #�|�f�m�m�� � �t�1D�1K�1K��&�� �X�X��� � ��39�$��?�C�?�a����r�)�?�C�C��Cs �!A:c � � V P V P , pV' g \ R4 hWP , p\ W3, RV, , 4 \ \ V, 4 , # )z4Probability density function. P(x <= X < x+dx) / dxz$pdf() not defined when sigma is zerog �)r� r r� r r r )r� rO r� r s && r- r� �NormalDist.pdf� sR � ��;�;����,���!�"H�I� I��8�8�|���4�;�$��/�2�3�d�3��>�6J�J�Jr, c � � V P ' g \ R4 hR\ V P V, V P \ , , 4 , # )z,Cumulative distribution function. P(X <= x)z$cdf() not defined when sigma is zeror� )r� r r r� �_SQRT2�r� rO s &&r- r� �NormalDist.cdf� s= � ��{�{�{�!�"H�I� I��T�4�8�8�a�<�D�K�K�&�,@�A�B�B�Br, c �t � VR8: g VR8� d \ R4 h\ WP V P 4 # )a# 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. rJ r� z$p must be in the range 0.0 < p < 1.0)r r� r� r� )r� r� s &&r- r� �NormalDist.inv_cdf� s2 � � ��8�q�C�x�!�"H�I� I�#�A�x�x����=�=r, c �l � \ ^V4 Uu. uF q P W!, 4 NK up# u upi )aF 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. )r� r� )r� r4 rm s && r- r� �NormalDist.quantiles s+ � � .3�1�a�[�9�[�� � �Q�U�#�[�9�9��9s � 1c �� � \ V\ 4 ' g \ R4 hYr2VP VP 3VP VP 38 d Y2r2VP VP rTV' d V' g \ R4 hWT, p\ VP VP , 4 pV' g/ \ VRVP , \ , , 4 # VP V, VP V, , pVP VP , \ Ww, V\ WT, 4 , , 4 , p W�, V, p W�, V, p R\ VP V 4 VP V 4 , 4 \ VP V 4 VP V 4 , 4 , , # )az 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 r8 r� r� r� r r r r� r r r� ) r� �other�X�Y�X_var�Y_var�dv�dm�a�b�x1�x2s && r- �overlap�NormalDist.overlap sI � � �%��,�,�� B�C� C��1� �H�H�a�e�e� ����!�%�%� 0� 0��q��z�z�1�:�:�u��E�!�"L�M� M� �]�� �!�%�%�!�%�%�-� �����c�A�H�H�n�v� 5�6�7� 7� �E�E�E�M�A�E�E�E�M� )�� �H�H�q�x�x� �$�r�w��c�%�-�6H�1H�'H�"I� I���e�r�\���e�r�\���d�1�5�5��9�q�u�u�R�y�0�1�D����r��Q�U�U�2�Y�9N�4O�O�P�Pr, c � � V P ' g \ R4 hWP , V P , # )z�Compute 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 &&r- �zscore�NormalDist.zscore3 s. � � �{�{�{�!�"K�L� L��H�H� �� � �+�+r, c � � V P # )z+Arithmetic mean of the normal distribution.�r� �r� s &r- r5 �NormalDist.mean> � � � �x�x�r, c � � V P # )z,Return the median of the normal distributionr� r� s &r- rn �NormalDist.medianC r� r, c � � V P # )z�Return the mode of the normal distribution The mode is the value x where which the probability density function (pdf) takes its maximum value. r� r� s &r- r� �NormalDist.modeH s � � �x�x�r, c � � V P # )z.Standard deviation of the normal distribution.�r� r� s &r- r� �NormalDist.stdevQ s � � �{�{�r, c �< � V P V P , # )z!Square of the standard deviation.r� r� s &r- r� �NormalDist.varianceV s � � �{�{�T�[�[�(�(r, c � � \ V\ 4 ' dF \ V P VP , \ V P VP 4 4 # \ V P V, V P 4 # )a: 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 &&r- �__add__�NormalDist.__add__[ �R � � �b�*� %� %� �b�f�f�r�v�v�o�u�R�Y�Y�� � �/J�K� K��"�&�&�2�+�r�y�y�1�1r, c � � \ V\ 4 ' dF \ V P VP , \ V P VP 4 4 # \ V P V, V P 4 # )aC 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. r� r� s &&r- �__sub__�NormalDist.__sub__i r� r, c �p � \ V P V, V P \ V4 , 4 # )z�Multiply 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 r� s &&r- �__mul__�NormalDist.__mul__w �& � � �"�&�&�2�+�r�y�y�4��8�';�<�<r, c �p � \ V P V, V P \ V4 , 4 # )z�Divide 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� s &&r- � __truediv__�NormalDist.__truediv__ r� r, c �B � \ V P V P 4 # )z Return a copy of the instance.�r r� r� �r� s &r- �__pos__�NormalDist.__pos__� s � ��"�&�&�"�)�)�,�,r, c �D � \ V P ) V P 4 # )z(Negates mu while keeping sigma the same.r� r� s &r- �__neg__�NormalDist.__neg__� s � ��2�6�6�'�2�9�9�-�-r, c � � W, ) # )z<Subtract a NormalDist from a constant or another NormalDist.r& r� s &&r- �__rsub__�NormalDist.__rsub__� s � ���z�r, c � � \ V\ 4 ' g \ # V P VP 8H ;' d V P VP 8H # )zFTwo NormalDist objects are equal if their mu and sigma are both equal.)r< r �NotImplementedr� r� r� s &&r- �__eq__�NormalDist.__eq__� s= � ��"�j�)�)�!� !��v�v���� �:�:�B�I�I����$:�:r, c �D � \ V P V P 34 # )zCNormalDist objects hash equal if their mu and sigma are both equal.)�hashr� r� r� s &r- �__hash__�NormalDist.__hash__� s � ��T�X�X�t�{�{�+�,�,r, c �j � \ V 4 P R V P : RV P : R2# )z(mu=z, sigma=�))�typer' r� r� r� s &r- �__repr__�NormalDist.__repr__� s. � ��t�*�%�%�&�d�4�8�8�,�h�t�{�{�o�Q�O�Or, c �2 � V P V P 3# rX �r� r� r� s &r- � __getstate__�NormalDist.__getstate__� s � ��x�x����$�$r, c �$ � Vw V n V n R # rX r� )r� �states &&r- � __setstate__�NormalDist.__setstate__� s � � %� ���$�+r, r� r� )rF )&r' r( r) r* ro � __slots__r� � classmethodr� r� r� r� r� r� r� r� �propertyr5 rn r� r� r� r� r� r� r� r� r� �__radd__r� �__rmul__r� r� r� r� r� r+ �__classdictcell__)� __classdict__s @r- r r � s �� � �.� �9��?��I� #� �'� �'�D�� D�K�C� >� :� Q�D ,� �� �� �� �� �� �� �� �� �)� �)� 2� 2� =�=�-�.� �H�� �H�;� -�P�%�&� &r, c � � ^ p\ 4 pVP p/ pVP p\ V \ 4 FA w rgV! V4 \ \ V4 F w r�V^, pV! V ^ 4 V, WI& K" KC RV9 d VR, p \ V 4 ' d Q hM \ R VP 4 4 4 p \ \ V\ 4 p W�V3# )aU _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. Nc 3 �< " � T F w r\ W!4 x � K R # 5irX r �rY �dr4 s & r- r[ �_sum.<locals>.<genexpr>� s � � �@�/?�t�q�H�Q�N�N�/?�� � ) �set�addrn r r� r9 � _exact_ratio� _isfiniter\ r� r �_coercerm ) r1 r �types� types_add�partials� partials_get�typr� r4 r� r3 r2 s & r- r/ r/ � s� � �B �E� �E�E�� � �I��H��<�<�L� �t�T�*� ���#��� �f�-�D�A� �Q�J�E�&�q�!�,�q�0�H�K� .� +� �x�� ���� �U�#�#�#�#�#� �@�x�~�~�/?�@�@���w��s�#�A� �e� � r, c � aa� Se \ VV3R lV 4 4 w r#pW#SV3# ^ p\ 4 pVP p\ \ 4 p\ \ 4 p\ V \ 4 F` w r�V! V 4 \ \ V 4 F? w p oV^, pVS;;, V , uu& VS;;, W�, , uu& KA Kb V' g \ ^ 4 ;poM�RV9 d VR, ;po\ V4 ' d Q hMd\ R VP 4 4 4 p \ R VP 4 4 4 p WM, W�, , V, pW�, o\ \ V\ 4 pW#SV3# )a# 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 �F <" � T F qS, ;oS, x � K R # 5irX r& )rY rO r� r� s & ��r- r[ �_ss.<locals>.<genexpr>� s �� � � <�t�!�q�5�j�a�A� -� -�t�s � !c 3 �< " � T F w r\ W!4 x � K R # 5irX r r� s & r- r[ r s � � �@�,?�D�A��!� � �,?�r c 3 �J " � T F w r\ W!V, 4 x � K R # 5irX r r� s & r- r[ r s � � �D�/C�t�q�(�1��c�"�"�/C�r_ )r/ r r r# rm r r� r9 r r r r\ r� r r )r1 r� r2 �ssdr r r � sx_partials� sxx_partialsr r� r4 �sxr� r� s &f @r- r� r� � sR �� � �}� � <�t� <�<� �����5�!�!� �E� �E�E�� � �I� �c�"�K� �s�#�L� �t�T�*� ���#��� �f�-�D�A�q� �Q�J�E� ��N�a� �N� ��O�q�u� $�O� .� +� ��1�+� ��a� �� � �d�#�#��a� �S�>�>�!�!�>� �@�K�,=�,=�,?�@� @���D�|�/A�/A�/C�D�D�� �{�R�W�$��-�� �J���w��s�#�A� �A�u� � r, c �r � V P 4 # \ d \ P ! T 4 u # i ; irX )� is_finiter� rL r )rO s &r- r r s1 � � ��{�{�}� �� � ��}�}�Q� � � �s � �!6�6c �, � V \ Jg Q R4 hWJ d V # V\ J g V\ J d V # V \ J d V# \ W4 ' d V# \ W4 ' d V # \ V \ 4 ' d V# \ V\ 4 ' d V # \ V \ 4 ' d \ V\ 4 ' d V# \ V \ 4 ' d \ V\ 4 ' d V # Rp\ W P VP 3, 4 h)z�Coerce 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)�boolrm r� r rv r8 r' )r2 �S�msgs && r- r r s� � � �D�=�2�2�2�=� �v�q���C�x�1��9�a�x��C�x��(��!����(��!����(��!�S���1�H��!�S���1�H��!�X� � �:�a��#7�#7����!�U��� �1�h� 7� 7��� .�C� �C�:�:�q�z�z�2�2� 3�3r, c �0 � V P 4 # \ d M-\ \ 3 d \ T 4 ' d Q hT R3u # i ; i T P T P 3# \ d& R\ T 4 P R2p\ T4 hi ; i)z�Return 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_ratior� � OverflowErrorr? r r� r� r� r' r8 )rO r s & r- r r = s� � ���!�!�#�#�� � � � �:� &� � �Q�<�<� � ��4�y���� �� � �Q�]�]�+�+�� � �$�T�!�W�%5�%5�$6�6P�Q����n� � �s% � � A �A �A �A � A% �%0Bc �2 � \ V 4 VJ d V # \ V\ 4 ' d V P ^8w d \ p V! V 4 # \ dD \ T\ 4 ' d, T! T P 4 T! T P 4 , u # h i ; i)z&Convert value to given numeric type T.)r� r� rm r� rv r8 r r� )r� r2 s &&r- r0 r0 W s� � � �E�{�a�� � ��!�S���e� /� /�1� 4� �����x��� � � �a�� !� !��U�_�_�%��%�*;�*;�(<�<� <� � �s � A �A B�Bc # �L " � V F pV^ 8 d \ V4 hVx � K R# 5i)z7Iterate over values, failing if any are less than zero.N)r )r� re rO s && r- rc rc k s& � � � �� �q�5�!�&�)� )��� �s �"$�key�reverse�ties�averagerI c �: � V ^8� d QhR\ \ , /# )r� �return)r= rv )�formats "r- � __annotate__r' s s � � 3� 3�4�PU�;� 3r, c � � VR8w d \ RV: 24 hVe \ W4 p \ \ V \ 4 4 VR7 pV^, p^ .\ V4 , p\ V\ ^ 4 R7 FH w r�\ V 4 p \ V 4 p Wk^, ^, , p V F w r�W�V& K Wk, pKJ V# )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'] r# z Unknown tie resolution method: )r! )r ) r? r9 rl r: r r7 r r r= )r1 r r! r" rI �val_posrm r� rg �g�group�size�rankr� �orig_poss "$$$$ r- r� r� s s� � �J �y���:�4�(�C�D�D� ���3�~���S��u�w�'��9�G� �� �A��S�3�w�<� �F���Z��]�3����Q����5�z���1�H��>�!�� $�O�E� #�8� � %� � �� 4� �Mr, c �< � V ^8� d QhR\ R\ R\ /# �r� r4 r� r% )rm )r& s "r- r' r' � s! � � � �� �� �� r, c �l � \ P ! W, 4 pW"V, V, V 8g , # )zFSquare root of n/m, rounded to the nearest integer using round-to-odd.)rL �isqrt)r4 r� r� s && r- �_integer_sqrt_of_frac_rtor3 � s) � � � � �1�6��A� �!��A��� � �r, c �< � V ^8� d QhR\ R\ R\ /# r0 )rm rv )r& s "r- r' r' � s! � � #� #�3� #�3� #�5� #r, c �* � V P 4 VP 4 , \ , ^, pV^ 8� d+ \ W^V, , 4 V, p^pW4, # \ V RV, , V4 p^V) , pW4, # )z1Square root of n/m as a float, correctly rounded.r� )� bit_length�_sqrt_bit_widthr3 )r4 r� �qr� r� s && r- r� r� � s| � � � � � �!�,�,�.� (�?� :�q�@�A��A�v�-�a�a�!�e��<��A� �� � � "�"� .�a�2��6�k�1�=� ��A�2�g� � � "�"r, c �< � V ^8� d QhR\ R\ R\ /# r0 )rm r )r& s "r- r' r' � s! � � � �S� �S� �W� r, c �V � V ^ 8: d V ' g \ R4 # V ) V) r\ V 4 \ V4 , P 4 pVP 4 w r4VP 4 pVP 4 w rg^V , WG, ^, , WV, Ws, , ^, , 8� d V# VP 4 pVP 4 w r�^V , WJ, ^, , WV , W�, , ^, , 8 d V# V# )z3Square root of n/m as a Decimal, correctly rounded.z0.0)r r r � next_plus� next_minus) r4 r� �root�nr�dr�plus�np�dp�minus�nmr� s && r- r� r� � s� � � �A�v���5�>� !��r�A�2�1� �A�J���� #� )� )� +�D� � "� "� $�F�B� �>�>� �D� � "� "� $�F�B��1�u����z��A�B���� �� 2� 2�2�� � �O�O� �E� � #� #� %�F�B��1�u����z��A�B���� �� 2� 2�2�� � �Kr, c �& � \ V 4 w rr4V^8 d \ R4 hW$^, , p \ V4 \ VP VP 4 3# \ d* \ T4 \ T4 \ T4 , 3u # i ; i)zFIn one pass, compute the mean and sample standard deviation as floats.r� )r� r rv r� r� r� r� )r1 r2 r� r� r4 r� s & r- r� r� � s| � ���Y�N�A�4��1�u� � G�H�H� �A��,�C�4��T�{� /�� � �s��� O�O�O�� � 4��T�{�E�$�K�%��)� 3�3�3�4�s �*A � 1B�Bc �< � V ^8� d QhR\ R\ R\ /# )r� rO r� r% )rv )r& s "r- r' r' � s! � � � �� �5� �U� r, c � � \ W, 4 p\ V4 ' gW \ V4 ' dD \ V 4 ' g3 \ V4 ' g" Rp\ W0, W1, 4 V, # V# V' g4 V ' d* V' d" Rp\ W0, W1, 4 V, # V# \ W3W) 34 pW$RV, , , # )zRReturn sqrt(x * y) computed with improved accuracy and without overflow/underflow.r$ g � g �a)r r r r� r )rO r� rW �scaler� s && r- r� r� � s� � � �Q�U� �A� �A�;�;� ��8�8�E�!�H�H�U�1�X�X� �E� �U�Y�� �2�U�:� :��� � �� �E� �U�Y�� �2�U�:� :��� ����B�� �A� �C�!�G�}� � r, c � � V R , p\ V4 R8: d� RW3, , pRV, R, V, R, V, R, V, R, V, R, V, R , V, R , V, pR V, R , V, R , V, R, V, R, V, R, V, R, V, R, pWV, pWV, , # VR8: d T MRV , p\ \ V4 ) 4 pVR8: d� VR, pRV, R, V, R, V, R, V, R, V, R, V, R , V, R , pR V, R , V, R , V, R!, V, R", V, R#, V, R$, V, R, pM�VR, pR%V, R&, V, R', V, R(, V, R), V, R*, V, R+, V, R,, pR-V, R., V, R/, V, R0, V, R1, V, R2, V, R3, V, R, pWV, pVR8 d V) pWV, , # )4r� g333333�?g ��Q� �?g^�}o)��@g�E.k�R�@g ��U l�@g*u��>l�@g �N����@g�"]Ξ@gnC���`@gu� � @giK��~j�@gv��|E�@g��d�|1�@gfR��r��@g��u.2�@g���~y�@g�n8(E@r� rJ g @g�������?g鬷�ZaI?gg�El�D�?g7\�����?g�uS�S�?g�=�. @gj%b�@g���Hw�@gjR�e�?g�9dh? >g('߿��A?g��~z �?g@�3��?gɅ3��?g3fR�x�?gI�F��l @g����t��>g*�Y��n�>gESB\T?g�N;A+�?g�UR1��?gE� F���?gP�n��@g&�>���@g����i�<g�@�F�>g�tcI,\�>g�ŝ���I?g*F2�v�?g�C4�?g��O�1�?)r r r )r� r� r� r8 �rrB rC rO s &&& r- r� r� s� � � �C��A� �A�w�%�� �q�u� ��0�1�4�0�1�45�6�0�1�45�6� 1�1� 56�6� 1� 1� 56� 6� 1� 1� 56� 6� 1� 1� 56� 6� 1�1� 56�6�� 1�1�4�0�1�45�6�0�1�45�6� 1�1� 56�6� 1� 1� 56� 6� 1� 1� 56� 6� 1� 1� 56� 6� ��� �I����Y� � � �#�X��3��7�A� �c�!�f�W� �A��C�x� ��G��1�A�5�1�2�56�7�1�2�56�7� 2�2� 67�7� 2� 2� 67� 7� 2� 2� 67� 7� 2� 2� 67� 7� 2�2�� 2�A�5�1�2�56�7�1�2�56�7� 2�2� 67�7� 2� 2� 67� 7� 2� 2� 67� 7� 2� 2� 67� 7� ��� ��G��1�A�5�1�2�56�7�1�2�56�7� 2�2� 67�7� 2� 2� 67� 7� 2� 2� 67� 7� 2� 2� 67� 7� 2�2�� 3�Q�6�1�2�56�7�1�2�56�7� 2�2� 67�7� 2� 2� 67� 7� 2� 2� 67� 7� 2� 2� 67� 7� ��� � �A��3�w� �B�� �U�� �r, r� c �@ � V ^8� d Qh/ ^ \ 9 d \ ;R& # )r� r7 )�__conditional_annotations__rm )r& s "r- r' r' s# � � h� h�b5 7� 6�� 6�c5hr, )r r r� r� rD rU rh rq r| r� r5 rn r| rs rq r� r� r� r� r� r� r� rX r� r� )g�-���q=)r� )znegative value)irL ro �__all__rL ra rv �sys� fractionsr �decimalr � itertoolsr r r �bisectr r r r r r r r r r r r r r r r r r r r r � functoolsr �operatorr � collectionsr! r" r# r� ry r? r r5 rD rU rh rn rq rs r| r� r� r� r� r� r� r� r� r� r� r� r� r� r� r r r r r& r- r8 r: rC rK rq r| r� r r/ r� r r r r0 rc r� r3 � float_info�mant_digr7 r� r� r� r� r� � _statistics� ImportErrorr' )rL s @r- �<module>rZ s� ���h�T ��2 � � � � � � ,� ,� ,� F� F� F� K� K� K� � � 8� 8� �c��� �� �j� � "�.(�V%�P<,�~+�0 �2�&E+�PB�<K�.+%�\( �V?�.?�2�:4H�8� 4H�n � 0�2H�I� �<>�u� <>�B � �� �(�G� �%� �%� �*��%� �%� �)�� %� � %� �-��#�%� $�%� �,��%� �%� �+�~�&�%� '�%��� �)�Z� �%� !�%�� �+��%� �%� �(��%� �%� �m�_�n��)�+;��$�m�\�� \�~*�� *�b04�� 04�;� 04�j\&� \&�B7 �t, �^ � 4�@ �4�(�3�$� 3�� 3�I� 3�Q� 3�l � �3�>�>�2�2�2�Q�6�� 6� #��8 4� �4K �^ �0��� �� �s �H � H*�)H*
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