tra-analysis/data analysis/analysis/trueskill/mathematics.py

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2019-10-04 14:02:46 +00:00
# -*- coding: utf-8 -*-
"""
trueskill.mathematics
~~~~~~~~~~~~~~~~~~~~~
This module contains basic mathematics functions and objects for TrueSkill
algorithm. If you have not scipy, this module provides the fallback.
:copyright: (c) 2012-2016 by Heungsub Lee.
:license: BSD, see LICENSE for more details.
"""
from __future__ import absolute_import
import copy
import math
try:
from numbers import Number
except ImportError:
Number = (int, long, float, complex)
from six import iterkeys
__all__ = ['Gaussian', 'Matrix', 'inf']
inf = float('inf')
class Gaussian(object):
"""A model for the normal distribution."""
#: Precision, the inverse of the variance.
pi = 0
#: Precision adjusted mean, the precision multiplied by the mean.
tau = 0
def __init__(self, mu=None, sigma=None, pi=0, tau=0):
if mu is not None:
if sigma is None:
raise TypeError('sigma argument is needed')
elif sigma == 0:
raise ValueError('sigma**2 should be greater than 0')
pi = sigma ** -2
tau = pi * mu
self.pi = pi
self.tau = tau
@property
def mu(self):
"""A property which returns the mean."""
return self.pi and self.tau / self.pi
@property
def sigma(self):
"""A property which returns the the square root of the variance."""
return math.sqrt(1 / self.pi) if self.pi else inf
def __mul__(self, other):
pi, tau = self.pi + other.pi, self.tau + other.tau
return Gaussian(pi=pi, tau=tau)
def __truediv__(self, other):
pi, tau = self.pi - other.pi, self.tau - other.tau
return Gaussian(pi=pi, tau=tau)
__div__ = __truediv__ # for Python 2
def __eq__(self, other):
return self.pi == other.pi and self.tau == other.tau
def __lt__(self, other):
return self.mu < other.mu
def __le__(self, other):
return self.mu <= other.mu
def __gt__(self, other):
return self.mu > other.mu
def __ge__(self, other):
return self.mu >= other.mu
def __repr__(self):
return 'N(mu={:.3f}, sigma={:.3f})'.format(self.mu, self.sigma)
def _repr_latex_(self):
latex = r'\mathcal{{ N }}( {:.3f}, {:.3f}^2 )'.format(self.mu, self.sigma)
return '$%s$' % latex
class Matrix(list):
"""A model for matrix."""
def __init__(self, src, height=None, width=None):
if callable(src):
f, src = src, {}
size = [height, width]
if not height:
def set_height(height):
size[0] = height
size[0] = set_height
if not width:
def set_width(width):
size[1] = width
size[1] = set_width
try:
for (r, c), val in f(*size):
src[r, c] = val
except TypeError:
raise TypeError('A callable src must return an interable '
'which generates a tuple containing '
'coordinate and value')
height, width = tuple(size)
if height is None or width is None:
raise TypeError('A callable src must call set_height and '
'set_width if the size is non-deterministic')
if isinstance(src, list):
is_number = lambda x: isinstance(x, Number)
unique_col_sizes = set(map(len, src))
everything_are_number = filter(is_number, sum(src, []))
if len(unique_col_sizes) != 1 or not everything_are_number:
raise ValueError('src must be a rectangular array of numbers')
two_dimensional_array = src
elif isinstance(src, dict):
if not height or not width:
w = h = 0
for r, c in iterkeys(src):
if not height:
h = max(h, r + 1)
if not width:
w = max(w, c + 1)
if not height:
height = h
if not width:
width = w
two_dimensional_array = []
for r in range(height):
row = []
two_dimensional_array.append(row)
for c in range(width):
row.append(src.get((r, c), 0))
else:
raise TypeError('src must be a list or dict or callable')
super(Matrix, self).__init__(two_dimensional_array)
@property
def height(self):
return len(self)
@property
def width(self):
return len(self[0])
def transpose(self):
height, width = self.height, self.width
src = {}
for c in range(width):
for r in range(height):
src[c, r] = self[r][c]
return type(self)(src, height=width, width=height)
def minor(self, row_n, col_n):
height, width = self.height, self.width
if not (0 <= row_n < height):
raise ValueError('row_n should be between 0 and %d' % height)
elif not (0 <= col_n < width):
raise ValueError('col_n should be between 0 and %d' % width)
two_dimensional_array = []
for r in range(height):
if r == row_n:
continue
row = []
two_dimensional_array.append(row)
for c in range(width):
if c == col_n:
continue
row.append(self[r][c])
return type(self)(two_dimensional_array)
def determinant(self):
height, width = self.height, self.width
if height != width:
raise ValueError('Only square matrix can calculate a determinant')
tmp, rv = copy.deepcopy(self), 1.
for c in range(width - 1, 0, -1):
pivot, r = max((abs(tmp[r][c]), r) for r in range(c + 1))
pivot = tmp[r][c]
if not pivot:
return 0.
tmp[r], tmp[c] = tmp[c], tmp[r]
if r != c:
rv = -rv
rv *= pivot
fact = -1. / pivot
for r in range(c):
f = fact * tmp[r][c]
for x in range(c):
tmp[r][x] += f * tmp[c][x]
return rv * tmp[0][0]
def adjugate(self):
height, width = self.height, self.width
if height != width:
raise ValueError('Only square matrix can be adjugated')
if height == 2:
a, b = self[0][0], self[0][1]
c, d = self[1][0], self[1][1]
return type(self)([[d, -b], [-c, a]])
src = {}
for r in range(height):
for c in range(width):
sign = -1 if (r + c) % 2 else 1
src[r, c] = self.minor(r, c).determinant() * sign
return type(self)(src, height, width)
def inverse(self):
if self.height == self.width == 1:
return type(self)([[1. / self[0][0]]])
return (1. / self.determinant()) * self.adjugate()
def __add__(self, other):
height, width = self.height, self.width
if (height, width) != (other.height, other.width):
raise ValueError('Must be same size')
src = {}
for r in range(height):
for c in range(width):
src[r, c] = self[r][c] + other[r][c]
return type(self)(src, height, width)
def __mul__(self, other):
if self.width != other.height:
raise ValueError('Bad size')
height, width = self.height, other.width
src = {}
for r in range(height):
for c in range(width):
src[r, c] = sum(self[r][x] * other[x][c]
for x in range(self.width))
return type(self)(src, height, width)
def __rmul__(self, other):
if not isinstance(other, Number):
raise TypeError('The operand should be a number')
height, width = self.height, self.width
src = {}
for r in range(height):
for c in range(width):
src[r, c] = other * self[r][c]
return type(self)(src, height, width)
def __repr__(self):
return '{}({})'.format(type(self).__name__, super(Matrix, self).__repr__())
def _repr_latex_(self):
rows = [' && '.join(['%.3f' % cell for cell in row]) for row in self]
latex = r'\begin{matrix} %s \end{matrix}' % r'\\'.join(rows)
return '$%s$' % latex