"""

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 count, groupby, repeat

from bisect import bisect_left, bisect_right

from math import hypot, sqrt, fabs, exp, erf, tau, log, fsum, sumprod

from math import isfinite, isinf, pi, cos, sin, tan, cosh, asin, atan, acos

from functools import reduce

from operator import itemgetter

from collections import Counter, namedtuple, defaultdict

_SQRT2 = sqrt(2.0)

_random = random

# === Exceptions ===

class StatisticsError(ValueError):

pass

# === Private utilities ===

def _sum(data):

"""_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.

"""

count = 0

types = set()

types_add = types.add

partials = {}

partials_get = partials.get

for typ, values in groupby(data, type):

types_add(typ)

for n, d in map(_exact_ratio, values):

count += 1

partials[d] = partials_get(d, 0) + n

if None in partials:

# The sum will be a NAN or INF. We can ignore all the finite

# partials, and just look at this special one.

total = partials[None]

assert not _isfinite(total)

else:

# Sum all the partial sums using builtin sum.

total = sum(Fraction(n, d) for d, n in partials.items())

T = reduce(_coerce, types, int) # or raise TypeError

return (T, total, count)

def _ss(data, c=None):

"""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.

"""

if c is not None:

T, ssd, count = _sum((d := x - c) * d for x in data)

return (T, ssd, c, count)

count = 0

types = set()

types_add = types.add

sx_partials = defaultdict(int)

sxx_partials = defaultdict(int)

for typ, values in groupby(data, type):

types_add(typ)

for n, d in map(_exact_ratio, values):

count += 1

sx_partials[d] += n

sxx_partials[d] += n * n

if not count:

ssd = c = Fraction(0)

elif None in sx_partials:

# The sum will be a NAN or INF. We can ignore all the finite

# partials, and just look at this special one.

ssd = c = sx_partials[None]

assert not _isfinite(ssd)

else:

sx = sum(Fraction(n, d) for d, n in sx_partials.items())

sxx = sum(Fraction(n, d*d) for d, n in sxx_partials.items())

# This formula has poor numeric properties for floats,

# but with fractions it is exact.

ssd = (count * sxx - sx * sx) / count

c = sx / count

T = reduce(_coerce, types, int) # or raise TypeError

return (T, ssd, c, count)

def _isfinite(x):

try:

return x.is_finite() # Likely a Decimal.

except AttributeError:

return math.isfinite(x) # Coerces to float first.

def _coerce(T, S):

"""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.

"""

# See http://bugs.python.org/issue24068.

assert T is not bool, "initial type T is bool"

# If the types are the same, no need to coerce anything. Put this

# first, so that the usual case (no coercion needed) happens as soon

# as possible.

if T is S: return T

# Mixed int & other coerce to the other type.

if S is int or S is bool: return T

if T is int: return S

# If one is a (strict) subclass of the other, coerce to the subclass.

if issubclass(S, T): return S

if issubclass(T, S): return T

# Ints coerce to the other type.

if issubclass(T, int): return S

if issubclass(S, int): return T

# Mixed fraction & float coerces to float (or float subclass).

if issubclass(T, Fraction) and issubclass(S, float):

return S

if issubclass(T, float) and issubclass(S, Fraction):

return T

# Any other combination is disallowed.

msg = "don't know how to coerce %s and %s"

raise TypeError(msg % (T.__name__, S.__name__))

def _exact_ratio(x):

"""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.

"""

# XXX We should revisit whether using fractions to accumulate exact

# ratios is the right way to go.

# The integer ratios for binary floats can have numerators or

# denominators with over 300 decimal digits. The problem is more

# acute with decimal floats where the default decimal context

# supports a huge range of exponents from Emin=-999999 to

# Emax=999999. When expanded with as_integer_ratio(), numbers like

# Decimal('3.14E+5000') and Decimal('3.14E-5000') have large

# numerators or denominators that will slow computation.

# When the integer ratios are accumulated as fractions, the size

# grows to cover the full range from the smallest magnitude to the

# largest. For example, Fraction(3.14E+300) + Fraction(3.14E-300),

# has a 616 digit numerator. Likewise,

# Fraction(Decimal('3.14E+5000')) + Fraction(Decimal('3.14E-5000'))

# has 10,003 digit numerator.

# This doesn't seem to have been problem in practice, but it is a

# potential pitfall.

try:

return x.as_integer_ratio()

except AttributeError:

pass

except (OverflowError, ValueError):

# float NAN or INF.

assert not _isfinite(x)

return (x, None)

try:

# x may be an Integral ABC.

return (x.numerator, x.denominator)

except AttributeError:

msg = f"can't convert type '{type(x).__name__}' to numerator/denominator"

raise TypeError(msg)

def _convert(value, T):

"""Convert value to given numeric type T."""

if type(value) is T:

# This covers the cases where T is Fraction, or where value is

# a NAN or INF (Decimal or float).

return value

if issubclass(T, int) and value.denominator != 1:

T = float

try:

# FIXME: what do we do if this overflows?

return T(value)

except TypeError:

if issubclass(T, Decimal):

return T(value.numerator) / T(value.denominator)

else:

raise

def _fail_neg(values, errmsg='negative value'):

"""Iterate over values, failing if any are less than zero."""

for x in values:

if x < 0:

raise StatisticsError(errmsg)

yield x

def _rank(data, /, *, key=None, reverse=False, ties='average', start=1) -> list[float]:

"""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']

"""

# If this function becomes public at some point, more thought

# needs to be given to the signature. A list of ints is

# plausible when ties is "min" or "max". When ties is "average",

# either list[float] or list[Fraction] is plausible.

# Default handling of ties matches scipy.stats.mstats.spearmanr.

if ties != 'average':

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

if key is not None:

data = map(key, data)

val_pos = sorted(zip(data, count()), reverse=reverse)

i = start - 1

result = [0] * len(val_pos)

for _, g in groupby(val_pos, key=itemgetter(0)):

group = list(g)

size = len(group)

rank = i + (size + 1) / 2

for value, orig_pos in group:

result[orig_pos] = rank

i += size

return result

def _integer_sqrt_of_frac_rto(n: int, m: int) -> int:

"""Square root of n/m, rounded to the nearest integer using round-to-odd."""

# Reference: https://www.lri.fr/~melquion/doc/05-imacs17_1-expose.pdf

a = math.isqrt(n // m)

return a | (a*a*m != n)

# For 53 bit precision floats, the bit width used in

# _float_sqrt_of_frac() is 109.

_sqrt_bit_width: int = 2 * sys.float_info.mant_dig + 3

def _float_sqrt_of_frac(n: int, m: int) -> float:

"""Square root of n/m as a float, correctly rounded."""

# See principle and proof sketch at: https://bugs.python.org/msg407078

q = (n.bit_length() - m.bit_length() - _sqrt_bit_width) // 2

if q >= 0:

numerator = _integer_sqrt_of_frac_rto(n, m << 2 * q) << q

denominator = 1

else:

numerator = _integer_sqrt_of_frac_rto(n << -2 * q, m)

denominator = 1 << -q

return numerator / denominator # Convert to float

def _decimal_sqrt_of_frac(n: int, m: int) -> Decimal:

"""Square root of n/m as a Decimal, correctly rounded."""

# Premise: For decimal, computing (n/m).sqrt() can be off

# by 1 ulp from the correctly rounded result.

# Method: Check the result, moving up or down a step if needed.

if n <= 0:

if not n:

return Decimal('0.0')

n, m = -n, -m

root = (Decimal(n) / Decimal(m)).sqrt()

nr, dr = root.as_integer_ratio()

plus = root.next_plus()

np, dp = plus.as_integer_ratio()

# test: n / m > ((root + plus) / 2) ** 2

if 4 * n * (dr*dp)**2 > m * (dr*np + dp*nr)**2:

return plus

minus = root.next_minus()

nm, dm = minus.as_integer_ratio()

# test: n / m < ((root + minus) / 2) ** 2

if 4 * n * (dr*dm)**2 < m * (dr*nm + dm*nr)**2:

return minus

return root

# === 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__().

n = 0

def count(iterable):

nonlocal n

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

yield x

data = count(data)

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('No negative inputs allowed', x)

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)

# FIXME: investigate ways to calculate medians without sorting? Quickselect?

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

"""

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]

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

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

"""

n = len(data)

if not n:

raise StatisticsError('Empty data sequence')

if not isinstance(data[0], (int, float)):

raise TypeError('Data sequence must contain ints or floats')

if h <= 0.0:

raise StatisticsError(f'Bandwidth h must be positive, not {h=!r}')

match kernel:

case 'normal' | 'gauss':

sqrt2pi = sqrt(2 * pi)

sqrt2 = sqrt(2)

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

W = lambda t: 1/2 * (1.0 + erf(t / sqrt2))

support = None

case 'logistic':

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

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

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

support = None

case 'sigmoid':

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

c1 = 1 / pi

c2 = 2 / pi

K = lambda t: c1 / cosh(t)

W = lambda t: c2 * atan(exp(t))

support = None

case 'rectangular' | 'uniform':

K = lambda t: 1/2

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

support = 1.0

case 'triangular':

K = lambda t: 1.0 - abs(t)

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

support = 1.0

case 'parabolic' | 'epanechnikov':

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

W = lambda t: -1/4 * t**3 + 3/4 * t + 1/2

support = 1.0

case 'quartic' | 'biweight':

K = lambda t: 15/16 * (1.0 - t * t) ** 2

W = lambda t: 3/16 * t**5 - 5/8 * t**3 + 15/16 * t + 1/2

support = 1.0

case 'triweight':

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

W = lambda t: 35/32 * (-1/7*t**7 + 3/5*t**5 - t**3 + t) + 1/2

support = 1.0

case 'cosine':

c1 = pi / 4

c2 = pi / 2

K = lambda t: c1 * cos(c2 * t)

W = lambda t: 1/2 * sin(c2 * t) + 1/2

support = 1.0

case _:

raise StatisticsError(f'Unknown kernel name: {kernel!r}')

if support is None:

def pdf(x):

n = len(data)

return sum(K((x - x_i) / h) for x_i in data) / (n * h)

def cdf(x):

n = len(data)

return sum(W((x - x_i) / h) for x_i in data) / n

else:

sample = sorted(data)

bandwidth = h * support

def pdf(x):

nonlocal n, sample

if len(data) != n:

sample = sorted(data)

n = len(data)

i = bisect_left(sample, x - bandwidth)

j = bisect_right(sample, x + bandwidth)

supported = sample[i : j]

return sum(K((x - x_i) / h) for x_i in supported) / (n * h)