"""

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.

"""

__all__ = [

'NormalDist',

'StatisticsError',

'correlation',

'covariance',

'fmean',

'geometric_mean',

'harmonic_mean',

'kde',

'kde_random',

'linear_regression',

'mean',

'median',

'median_grouped',

'median_high',

'median_low',

'mode',

'multimode',

'pstdev',

'pvariance',

'quantiles',

'stdev',

'variance',

]

import math

import numbers

import random

import sys

from fractions import Fraction

from decimal import Decimal

from itertools import compress, count, groupby, repeat

from bisect import bisect_left, bisect_right

from math import hypot, sqrt, fabs, exp, erfc, log, fsum, sumprod

from math import isfinite, isinf, pi, sin, cosh

from math import sinpi, cospi, tanpi, asinpi, acospi, atanpi

from functools import reduce

from operator import itemgetter

from collections import Counter, namedtuple, defaultdict

_SQRT2 = sqrt(2.0)

_SQRT2PI = float.fromhex('0x1.40d931ff62706p+1') # Correctly rounded sqrt(2*pi)

_random = random

## Exceptions ##############################################################

class StatisticsError(ValueError):

pass

## Measures of central tendency (averages) #################################

def mean(data):

"""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.

"""

T, total, n = _sum(data)

if n < 1:

raise StatisticsError('mean requires at least one data point')

return _convert(total / n, T)

def fmean(data, weights=None):

"""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

"""

if weights is None:

try:

n = len(data)

except TypeError:

# Handle iterators that do not define __len__().

counter = count(1)

total = fsum(compress(data, counter))

n = next(counter) - 1

else:

total = fsum(data)

if not n:

raise StatisticsError('fmean requires at least one data point')

return total / n

if not isinstance(weights, (list, tuple)):

weights = list(weights)

try:

num = sumprod(data, weights)

except ValueError:

raise StatisticsError('data and weights must be the same length')

den = fsum(weights)

if not den:

raise StatisticsError('sum of weights must be non-zero')

return num / den

def geometric_mean(data):

"""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

"""

n = 0

found_zero = False

def count_positive(iterable):

nonlocal n, found_zero

for n, x in enumerate(iterable, start=1):

if x > 0.0 or math.isnan(x):

yield x

elif x == 0.0:

found_zero = True

else:

raise StatisticsError(f'No negative inputs allowed: {x!r}')

total = fsum(map(log, count_positive(data)))

if not n:

raise StatisticsError('Must have a non-empty dataset')

if math.isnan(total):

return math.nan

if found_zero:

return math.nan if total == math.inf else 0.0

return exp(total / n)

def harmonic_mean(data, weights=None):

"""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``.

"""

if iter(data) is data:

data = list(data)

errmsg = 'harmonic mean does not support negative values'

n = len(data)

if n < 1:

raise StatisticsError('harmonic_mean requires at least one data point')

elif n == 1 and weights is None:

x = data[0]

if isinstance(x, (numbers.Real, Decimal)):

if x < 0:

raise StatisticsError(errmsg)

return x

else:

raise TypeError('unsupported type')

if weights is None:

weights = repeat(1, n)

sum_weights = n

else:

if iter(weights) is weights:

weights = list(weights)

if len(weights) != n:

raise StatisticsError('Number of weights does not match data size')

_, sum_weights, _ = _sum(w for w in _fail_neg(weights, errmsg))

try:

data = _fail_neg(data, errmsg)

T, total, count = _sum(w / x if w else 0 for w, x in zip(weights, data))

except ZeroDivisionError:

return 0

if total <= 0:

raise StatisticsError('Weighted sum must be positive')

return _convert(sum_weights / total, T)

def median(data):

"""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

"""

data = sorted(data)

n = len(data)

if n == 0:

raise StatisticsError("no median for empty data")

if n % 2 == 1:

return data[n // 2]

else:

i = n // 2

return (data[i - 1] + data[i]) / 2

def median_low(data):

"""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

"""

# Potentially the sorting step could be replaced with a quickselect.

# However, it would require an excellent implementation to beat our

# highly optimized builtin sort.

data = sorted(data)

n = len(data)

if n == 0:

raise StatisticsError("no median for empty data")

if n % 2 == 1:

return data[n // 2]

else:

return data[n // 2 - 1]

def median_high(data):

"""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

"""

data = sorted(data)

n = len(data)

if n == 0:

raise StatisticsError("no median for empty data")

return data[n // 2]

def median_grouped(data, interval=1.0):

"""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.

"""

data = sorted(data)

n = len(data)

if not n:

raise StatisticsError("no median for empty data")

# Find the value at the midpoint. Remember this corresponds to the

# midpoint of the class interval.

x = data[n // 2]

# Using O(log n) bisection, find where all the x values occur in the data.

# All x will lie within data[i:j].

i = bisect_left(data, x)

j = bisect_right(data, x, lo=i)

# Coerce to floats, raising a TypeError if not possible

try:

interval = float(interval)

x = float(x)

except ValueError:

raise TypeError(f'Value cannot be converted to a float')

# Interpolate the median using the formula found at:

# https://www.cuemath.com/data/median-of-grouped-data/

L = x - interval / 2.0 # Lower limit of the median interval

cf = i # Cumulative frequency of the preceding interval

f = j - i # Number of elements in the median internal

return L + interval * (n / 2 - cf) / f

def mode(data):

"""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.

"""

pairs = Counter(iter(data)).most_common(1)

try:

return pairs[0][0]

except IndexError:

raise StatisticsError('no mode for empty data') from None

def multimode(data):

"""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('')

[]

"""

counts = Counter(iter(data))

if not counts:

return []

maxcount = max(counts.values())

return [value for value, count in counts.items() if count == maxcount]

## Measures of spread ######################################################

def variance(data, xbar=None):

"""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)

"""

# http://mathworld.wolfram.com/SampleVariance.html

T, ss, c, n = _ss(data, xbar)

if n < 2:

raise StatisticsError('variance requires at least two data points')

return _convert(ss / (n - 1), T)

def pvariance(data, mu=None):

"""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)

"""

# http://mathworld.wolfram.com/Variance.html

T, ss, c, n = _ss(data, mu)

if n < 1:

raise StatisticsError('pvariance requires at least one data point')

return _convert(ss / n, T)

def stdev(data, xbar=None):

"""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

"""

T, ss, c, n = _ss(data, xbar)

if n < 2:

raise StatisticsError('stdev requires at least two data points')

mss = ss / (n - 1)

try:

mss_numerator = mss.numerator

mss_denominator = mss.denominator

except AttributeError:

raise ValueError('inf or nan encountered in data')

if issubclass(T, Decimal):

return _decimal_sqrt_of_frac(mss_numerator, mss_denominator)

return _float_sqrt_of_frac(mss_numerator, mss_denominator)

def pstdev(data, mu=None):

"""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

"""

T, ss, c, n = _ss(data, mu)

if n < 1:

raise StatisticsError('pstdev requires at least one data point')

mss = ss / n

try:

mss_numerator = mss.numerator

mss_denominator = mss.denominator

except AttributeError:

raise ValueError('inf or nan encountered in data')

if issubclass(T, Decimal):

return _decimal_sqrt_of_frac(mss_numerator, mss_denominator)

return _float_sqrt_of_frac(mss_numerator, mss_denominator)

## Statistics for relations between two inputs #############################

def covariance(x, y, /):

"""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

"""

# https://en.wikipedia.org/wiki/Covariance

n = len(x)

if len(y) != n:

raise StatisticsError('covariance requires that both inputs have same number of data points')

if n < 2:

raise StatisticsError('covariance requires at least two data points')

xbar = fsum(x) / n

ybar = fsum(y) / n

sxy = sumprod((xi - xbar for xi in x), (yi - ybar for yi in y))

return sxy / (n - 1)

def correlation(x, y, /, *, method='linear'):

"""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.

"""

# https://en.wikipedia.org/wiki/Pearson_correlation_coefficient

# https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient

n = len(x)

if len(y) != n:

raise StatisticsError('correlation requires that both inputs have same number of data points')

if n < 2:

raise StatisticsError('correlation requires at least two data points')

if method not in {'linear', 'ranked'}:

raise ValueError(f'Unknown method: {method!r}')

if method == 'ranked':

start = (n - 1) / -2 # Center rankings around zero

x = _rank(x, start=start)

y = _rank(y, start=start)

else:

xbar = fsum(x) / n

ybar = fsum(y) / n

x = [xi - xbar for xi in x]

y = [yi - ybar for yi in y]

sxy = sumprod(x, y)

sxx = sumprod(x, x)

syy = sumprod(y, y)

try:

return sxy / _sqrtprod(sxx, syy)

except ZeroDivisionError:

raise StatisticsError('at least one of the inputs is constant')

LinearRegression = namedtuple('LinearRegression', ('slope', 'intercept'))

def linear_regression(x, y, /, *, proportional=False):

"""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)

"""

# https://en.wikipedia.org/wiki/Simple_linear_regression

n = len(x)

if len(y) != n:

raise StatisticsError('linear regression requires that both inputs have same number of data points')

if n < 2:

raise StatisticsError('linear regression requires at least two data points')

if not proportional:

xbar = fsum(x) / n

ybar = fsum(y) / n

x = [xi - xbar for xi in x] # List because used three times below

y = (yi - ybar for yi in y) # Generator because only used once below

sxy = sumprod(x, y) + 0.0 # Add zero to coerce result to a float

sxx = sumprod(x, x)

try:

slope = sxy / sxx # equivalent to: covariance(x, y) / variance(x)

except ZeroDivisionError:

raise StatisticsError('x is constant')

intercept = 0.0 if proportional else ybar - slope * xbar

return LinearRegression(slope=slope, intercept=intercept)

## Kernel Density Estimation ###############################################

_kernel_specs = {}

def register(*kernels):

"Load the kernel's pdf, cdf, invcdf, and support into _kernel_specs."

def deco(builder):

spec = dict(zip(('pdf', 'cdf', 'invcdf', 'support'), builder()))

for kernel in kernels:

_kernel_specs[kernel] = spec

return builder

return deco

@register('normal', 'gauss')

def normal_kernel():

sqrt2pi = _SQRT2PI

neg_sqrt2 = -_SQRT2

pdf = lambda t: exp(-1/2 * t * t) / sqrt2pi

cdf = lambda t: 1/2 * erfc(t / neg_sqrt2)

invcdf = lambda t: _normal_dist_inv_cdf(t, 0.0, 1.0)

support = None

return pdf, cdf, invcdf, support

@register('logistic')

def logistic_kernel():

# 1.0 / (exp(t) + 2.0 + exp(-t))

pdf = lambda t: 1/2 / (1.0 + cosh(t))

cdf = lambda t: 1.0 - 1.0 / (exp(t) + 1.0)

invcdf = lambda p: log(p / (1.0 - p))

support = None

return pdf, cdf, invcdf, support

@register('sigmoid')

def sigmoid_kernel():

# (2/pi) / (exp(t) + exp(-t))

recip_pi = 1 / pi

pdf = lambda t: recip_pi / cosh(t)

cdf = lambda t: 2.0 * atanpi(exp(t))

invcdf = lambda p: log(tanpi(p * 0.5))

support = None

return pdf, cdf, invcdf, support

@register('rectangular', 'uniform')

def rectangular_kernel():

pdf = lambda t: 1/2

cdf = lambda t: 1/2 * t + 1/2

invcdf = lambda p: 2.0 * p - 1.0

support = 1.0

return pdf, cdf, invcdf, support

@register('triangular')

def triangular_kernel():

pdf = lambda t: 1.0 - abs(t)

cdf = lambda t: t*t * (1/2 if t < 0.0 else -1/2) + t + 1/2

invcdf = lambda p: sqrt(2.0*p) - 1.0 if p < 1/2 else 1.0 - sqrt(2.0 - 2.0*p)

support = 1.0

return pdf, cdf, invcdf, support

@register('parabolic', 'epanechnikov')

def parabolic_kernel():

pdf = lambda t: 3/4 * (1.0 - t * t)

cdf = lambda t: sumprod((-1/4, 3/4, 1/2), (t**3, t, 1.0))

invcdf = lambda p: 2.0 * cospi((acospi(2.0 * p - 1.0) + 1.0) / 3.0)

support = 1.0

return pdf, cdf, invcdf, support

def _newton_raphson(f_inv_estimate, f, f_prime, tolerance=1e-12):

def f_inv(y):

"Return x such that f(x) ≈ y within the specified tolerance."

x = f_inv_estimate(y)

while abs(diff := f(x) - y) > tolerance:

x -= diff / f_prime(x)

return x

return f_inv

def _quartic_invcdf_estimate(p):

# A handrolled piecewise approximation. There is no magic here.

sign, p = (1.0, p) if p <= 1/2 else (-1.0, 1.0 - p)

if p < 0.0106:

return ((2.0 * p) ** 0.3838 - 1.0) * sign

x = (2.0 * p) ** 0.4258865685331 - 1.0

if p < 0.499:

x += 0.026818732 * sin(7.101753784 * p + 2.73230839482953)

return x * sign

@register('quartic', 'biweight')

def quartic_kernel():

pdf = lambda t: 15/16 * (u := 1.0 - t * t) * u

cdf = lambda t: sumprod((3/16, -5/8, 15/16, 1/2),

(t**5, t**3, t, 1.0))

invcdf = _newton_raphson(_quartic_invcdf_estimate, f=cdf, f_prime=pdf)

support = 1.0

return pdf, cdf, invcdf, support

def _triweight_invcdf_estimate(p):

# A handrolled piecewise approximation. There is no magic here.

sign, p = (1.0, p) if p <= 1/2 else (-1.0, 1.0 - p)

x = (2.0 * p) ** 0.3400218741872791 - 1.0

if 0.00001 < p < 0.499:

x -= 0.033 * sinpi(2.14 * (p - 0.035))

return x * sign

@register('triweight')

def triweight_kernel():

pdf = lambda t: 35/32 * (1.0 - t * t) ** 3

cdf = lambda t: sumprod((-5/32, 21/32, -35/32, 35/32, 1/2),

(t**7, t**5, t**3, t, 1.0))

invcdf = _newton_raphson(_triweight_invcdf_estimate, f=cdf, f_prime=pdf)

support = 1.0

return pdf, cdf, invcdf, support

@register('cosine')

def cosine_kernel():

c1 = pi / 4

pdf = lambda t: c1 * cospi(0.5 * t)

cdf = lambda t: 1/2 * sinpi(0.5 * t) + 1/2

invcdf = lambda p: 2.0 * asinpi(2.0 * p - 1.0)

support = 1.0

return pdf, cdf, invcdf, support

del register, normal_kernel, logistic_kernel, sigmoid_kernel

del rectangular_kernel, triangular_kernel, parabolic_kernel

del quartic_kernel, triweight_kernel, cosine_kernel

def kde(data, h, kernel='normal', *, cumulative=False):

"""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