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upload trueskill for testing purposes

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data analysis/analysis/trueskill/__about__.py Normal file
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# -*- coding: utf-8 -*-
"""
trueskill.__about__
~~~~~~~~~~~~~~~~~~~
"""
__version__ = '0.4.5'
__license__ = 'BSD'
__author__ = 'Heungsub Lee'
__author_email__ = 'sub@subl.ee'
__url__ = 'http://trueskill.org/'
__description__ = 'The video game rating system'

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data analysis/analysis/trueskill/__init__.py Normal file
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# -*- coding: utf-8 -*-
"""
trueskill
~~~~~~~~~
The video game rating system.
:copyright: (c) 2012-2016 by Heungsub Lee
:license: BSD, see LICENSE for more details.
"""
from __future__ import absolute_import
from itertools import chain
import math
from six import iteritems
from six.moves import map, range, zip
from .__about__ import __version__ # noqa
from .backends import choose_backend
from .factorgraph import (LikelihoodFactor, PriorFactor, SumFactor,
TruncateFactor, Variable)
from .mathematics import Gaussian, Matrix
__all__ = [
# TrueSkill objects
'TrueSkill', 'Rating',
# functions for the global environment
'rate', 'quality', 'rate_1vs1', 'quality_1vs1', 'expose', 'setup',
'global_env',
# default values
'MU', 'SIGMA', 'BETA', 'TAU', 'DRAW_PROBABILITY',
# draw probability helpers
'calc_draw_probability', 'calc_draw_margin',
# deprecated features
'transform_ratings', 'match_quality', 'dynamic_draw_probability',
]
#: Default initial mean of ratings.
MU = 25.
#: Default initial standard deviation of ratings.
SIGMA = MU / 3
#: Default distance that guarantees about 76% chance of winning.
BETA = SIGMA / 2
#: Default dynamic factor.
TAU = SIGMA / 100
#: Default draw probability of the game.
DRAW_PROBABILITY = .10
#: A basis to check reliability of the result.
DELTA = 0.0001
def calc_draw_probability(draw_margin, size, env=None):
"""Calculates a draw-probability from the given ``draw_margin``.
:param draw_margin: the draw-margin.
:param size: the number of players in two comparing teams.
:param env: the :class:`TrueSkill` object. Defaults to the global
environment.
"""
if env is None:
env = global_env()
return 2 * env.cdf(draw_margin / (math.sqrt(size) * env.beta)) - 1
def calc_draw_margin(draw_probability, size, env=None):
"""Calculates a draw-margin from the given ``draw_probability``.
:param draw_probability: the draw-probability.
:param size: the number of players in two comparing teams.
:param env: the :class:`TrueSkill` object. Defaults to the global
environment.
"""
if env is None:
env = global_env()
return env.ppf((draw_probability + 1) / 2.) * math.sqrt(size) * env.beta
def _team_sizes(rating_groups):
"""Makes a size map of each teams."""
team_sizes = [0]
for group in rating_groups:
team_sizes.append(len(group) + team_sizes[-1])
del team_sizes[0]
return team_sizes
def _floating_point_error(env):
if env.backend == 'mpmath':
msg = 'Set "mpmath.mp.dps" to higher'
else:
msg = 'Cannot calculate correctly, set backend to "mpmath"'
return FloatingPointError(msg)
class Rating(Gaussian):
"""Represents a player's skill as Gaussian distrubution.
The default mu and sigma value follows the global environment's settings.
If you don't want to use the global, use :meth:`TrueSkill.create_rating` to
create the rating object.
:param mu: the mean.
:param sigma: the standard deviation.
"""
def __init__(self, mu=None, sigma=None):
if isinstance(mu, tuple):
mu, sigma = mu
elif isinstance(mu, Gaussian):
mu, sigma = mu.mu, mu.sigma
if mu is None:
mu = global_env().mu
if sigma is None:
sigma = global_env().sigma
super(Rating, self).__init__(mu, sigma)
def __int__(self):
return int(self.mu)
def __long__(self):
return long(self.mu)
def __float__(self):
return float(self.mu)
def __iter__(self):
return iter((self.mu, self.sigma))
def __repr__(self):
c = type(self)
args = ('.'.join([c.__module__, c.__name__]), self.mu, self.sigma)
return '%s(mu=%.3f, sigma=%.3f)' % args
class TrueSkill(object):
"""Implements a TrueSkill environment. An environment could have
customized constants. Every games have not same design and may need to
customize TrueSkill constants.
For example, 60% of matches in your game have finished as draw then you
should set ``draw_probability`` to 0.60::
env = TrueSkill(draw_probability=0.60)
For more details of the constants, see `The Math Behind TrueSkill`_ by
Jeff Moser.
.. _The Math Behind TrueSkill:: http://bit.ly/trueskill-math
:param mu: the initial mean of ratings.
:param sigma: the initial standard deviation of ratings. The recommended
value is a third of ``mu``.
:param beta: the distance which guarantees about 76% chance of winning.
The recommended value is a half of ``sigma``.
:param tau: the dynamic factor which restrains a fixation of rating. The
recommended value is ``sigma`` per cent.
:param draw_probability: the draw probability between two teams. It can be
a ``float`` or function which returns a ``float``
by the given two rating (team performance)
arguments and the beta value. If it is a
``float``, the game has fixed draw probability.
Otherwise, the draw probability will be decided
dynamically per each match.
:param backend: the name of a backend which implements cdf, pdf, ppf. See
:mod:`trueskill.backends` for more details. Defaults to
``None``.
"""
def __init__(self, mu=MU, sigma=SIGMA, beta=BETA, tau=TAU,
draw_probability=DRAW_PROBABILITY, backend=None):
self.mu = mu
self.sigma = sigma
self.beta = beta
self.tau = tau
self.draw_probability = draw_probability
self.backend = backend
if isinstance(backend, tuple):
self.cdf, self.pdf, self.ppf = backend
else:
self.cdf, self.pdf, self.ppf = choose_backend(backend)
def create_rating(self, mu=None, sigma=None):
"""Initializes new :class:`Rating` object, but it fixes default mu and
sigma to the environment's.
>>> env = TrueSkill(mu=0, sigma=1)
>>> env.create_rating()
trueskill.Rating(mu=0.000, sigma=1.000)
"""
if mu is None:
mu = self.mu
if sigma is None:
sigma = self.sigma
return Rating(mu, sigma)
def v_win(self, diff, draw_margin):
"""The non-draw version of "V" function. "V" calculates a variation of
a mean.
"""
x = diff - draw_margin
denom = self.cdf(x)
return (self.pdf(x) / denom) if denom else -x
def v_draw(self, diff, draw_margin):
"""The draw version of "V" function."""
abs_diff = abs(diff)
a, b = draw_margin - abs_diff, -draw_margin - abs_diff
denom = self.cdf(a) - self.cdf(b)
numer = self.pdf(b) - self.pdf(a)
return ((numer / denom) if denom else a) * (-1 if diff < 0 else +1)
def w_win(self, diff, draw_margin):
"""The non-draw version of "W" function. "W" calculates a variation of
a standard deviation.
"""
x = diff - draw_margin
v = self.v_win(diff, draw_margin)
w = v * (v + x)
if 0 < w < 1:
return w
raise _floating_point_error(self)
def w_draw(self, diff, draw_margin):
"""The draw version of "W" function."""
abs_diff = abs(diff)
a, b = draw_margin - abs_diff, -draw_margin - abs_diff
denom = self.cdf(a) - self.cdf(b)
if not denom:
raise _floating_point_error(self)
v = self.v_draw(abs_diff, draw_margin)
return (v ** 2) + (a * self.pdf(a) - b * self.pdf(b)) / denom
def validate_rating_groups(self, rating_groups):
"""Validates a ``rating_groups`` argument. It should contain more than
2 groups and all groups must not be empty.
>>> env = TrueSkill()
>>> env.validate_rating_groups([])
Traceback (most recent call last):
...
ValueError: need multiple rating groups
>>> env.validate_rating_groups([(Rating(),)])
Traceback (most recent call last):
...
ValueError: need multiple rating groups
>>> env.validate_rating_groups([(Rating(),), ()])
Traceback (most recent call last):
...
ValueError: each group must contain multiple ratings
>>> env.validate_rating_groups([(Rating(),), (Rating(),)])
... #doctest: +ELLIPSIS
[(truekill.Rating(...),), (trueskill.Rating(...),)]
"""
# check group sizes
if len(rating_groups) < 2:
raise ValueError('Need multiple rating groups')
elif not all(rating_groups):
raise ValueError('Each group must contain multiple ratings')
# check group types
group_types = set(map(type, rating_groups))
if len(group_types) != 1:
raise TypeError('All groups should be same type')
elif group_types.pop() is Rating:
raise TypeError('Rating cannot be a rating group')
# normalize rating_groups
if isinstance(rating_groups[0], dict):
dict_rating_groups = rating_groups
rating_groups = []
keys = []
for dict_rating_group in dict_rating_groups:
rating_group, key_group = [], []
for key, rating in iteritems(dict_rating_group):
rating_group.append(rating)
key_group.append(key)
rating_groups.append(tuple(rating_group))
keys.append(tuple(key_group))
else:
rating_groups = list(rating_groups)
keys = None
return rating_groups, keys
def validate_weights(self, weights, rating_groups, keys=None):
if weights is None:
weights = [(1,) * len(g) for g in rating_groups]
elif isinstance(weights, dict):
weights_dict, weights = weights, []
for x, group in enumerate(rating_groups):
w = []
weights.append(w)
for y, rating in enumerate(group):
if keys is not None:
y = keys[x][y]
w.append(weights_dict.get((x, y), 1))
return weights
def factor_graph_builders(self, rating_groups, ranks, weights):
"""Makes nodes for the TrueSkill factor graph.
Here's an example of a TrueSkill factor graph when 1 vs 2 vs 1 match::
rating_layer: O O O O (PriorFactor)
| | | |
| | | |
perf_layer: O O O O (LikelihoodFactor)
| \ / |
| | |
team_perf_layer: O O O (SumFactor)
\ / \ /
| |
team_diff_layer: O O (SumFactor)
| |
| |
trunc_layer: O O (TruncateFactor)
"""
flatten_ratings = sum(map(tuple, rating_groups), ())
flatten_weights = sum(map(tuple, weights), ())
size = len(flatten_ratings)
group_size = len(rating_groups)
# create variables
rating_vars = [Variable() for x in range(size)]
perf_vars = [Variable() for x in range(size)]
team_perf_vars = [Variable() for x in range(group_size)]
team_diff_vars = [Variable() for x in range(group_size - 1)]
team_sizes = _team_sizes(rating_groups)
# layer builders
def build_rating_layer():
for rating_var, rating in zip(rating_vars, flatten_ratings):
yield PriorFactor(rating_var, rating, self.tau)
def build_perf_layer():
for rating_var, perf_var in zip(rating_vars, perf_vars):
yield LikelihoodFactor(rating_var, perf_var, self.beta ** 2)
def build_team_perf_layer():
for team, team_perf_var in enumerate(team_perf_vars):
if team > 0:
start = team_sizes[team - 1]
else:
start = 0
end = team_sizes[team]
child_perf_vars = perf_vars[start:end]
coeffs = flatten_weights[start:end]
yield SumFactor(team_perf_var, child_perf_vars, coeffs)
def build_team_diff_layer():
for team, team_diff_var in enumerate(team_diff_vars):
yield SumFactor(team_diff_var,
team_perf_vars[team:team + 2], [+1, -1])
def build_trunc_layer():
for x, team_diff_var in enumerate(team_diff_vars):
if callable(self.draw_probability):
# dynamic draw probability
team_perf1, team_perf2 = team_perf_vars[x:x + 2]
args = (Rating(team_perf1), Rating(team_perf2), self)
draw_probability = self.draw_probability(*args)
else:
# static draw probability
draw_probability = self.draw_probability
size = sum(map(len, rating_groups[x:x + 2]))
draw_margin = calc_draw_margin(draw_probability, size, self)
if ranks[x] == ranks[x + 1]: # is a tie?
v_func, w_func = self.v_draw, self.w_draw
else:
v_func, w_func = self.v_win, self.w_win
yield TruncateFactor(team_diff_var,
v_func, w_func, draw_margin)
# build layers
return (build_rating_layer, build_perf_layer, build_team_perf_layer,
build_team_diff_layer, build_trunc_layer)
def run_schedule(self, build_rating_layer, build_perf_layer,
build_team_perf_layer, build_team_diff_layer,
build_trunc_layer, min_delta=DELTA):
"""Sends messages within every nodes of the factor graph until the
result is reliable.
"""
if min_delta <= 0:
raise ValueError('min_delta must be greater than 0')
layers = []
def build(builders):
layers_built = [list(build()) for build in builders]
layers.extend(layers_built)
return layers_built
# gray arrows
layers_built = build([build_rating_layer,
build_perf_layer,
build_team_perf_layer])
rating_layer, perf_layer, team_perf_layer = layers_built
for f in chain(*layers_built):
f.down()
# arrow #1, #2, #3
team_diff_layer, trunc_layer = build([build_team_diff_layer,
build_trunc_layer])
team_diff_len = len(team_diff_layer)
for x in range(10):
if team_diff_len == 1:
# only two teams
team_diff_layer[0].down()
delta = trunc_layer[0].up()
else:
# multiple teams
delta = 0
for x in range(team_diff_len - 1):
team_diff_layer[x].down()
delta = max(delta, trunc_layer[x].up())
team_diff_layer[x].up(1) # up to right variable
for x in range(team_diff_len - 1, 0, -1):
team_diff_layer[x].down()
delta = max(delta, trunc_layer[x].up())
team_diff_layer[x].up(0) # up to left variable
# repeat until to small update
if delta <= min_delta:
break
# up both ends
team_diff_layer[0].up(0)
team_diff_layer[team_diff_len - 1].up(1)
# up the remainder of the black arrows
for f in team_perf_layer:
for x in range(len(f.vars) - 1):
f.up(x)
for f in perf_layer:
f.up()
return layers
def rate(self, rating_groups, ranks=None, weights=None, min_delta=DELTA):
"""Recalculates ratings by the ranking table::
env = TrueSkill() # uses default settings
# create ratings
r1 = env.create_rating(42.222)
r2 = env.create_rating(89.999)
# calculate new ratings
rating_groups = [(r1,), (r2,)]
rated_rating_groups = env.rate(rating_groups, ranks=[0, 1])
# save new ratings
(r1,), (r2,) = rated_rating_groups
``rating_groups`` is a list of rating tuples or dictionaries that
represents each team of the match. You will get a result as same
structure as this argument. Rating dictionaries for this may be useful
to choose specific player's new rating::
# load players from the database
p1 = load_player_from_database('Arpad Emrick Elo')
p2 = load_player_from_database('Mark Glickman')
p3 = load_player_from_database('Heungsub Lee')
# calculate new ratings
rating_groups = [{p1: p1.rating, p2: p2.rating}, {p3: p3.rating}]
rated_rating_groups = env.rate(rating_groups, ranks=[0, 1])
# save new ratings
for player in [p1, p2, p3]:
player.rating = rated_rating_groups[player.team][player]
:param rating_groups: a list of tuples or dictionaries containing
:class:`Rating` objects.
:param ranks: a ranking table. By default, it is same as the order of
the ``rating_groups``.
:param weights: weights of each players for "partial play".
:param min_delta: each loop checks a delta of changes and the loop
will stop if the delta is less then this argument.
:returns: recalculated ratings same structure as ``rating_groups``.
:raises: :exc:`FloatingPointError` occurs when winners have too lower
rating than losers. higher floating-point precision couls
solve this error. set the backend to "mpmath".
.. versionadded:: 0.2
"""
rating_groups, keys = self.validate_rating_groups(rating_groups)
weights = self.validate_weights(weights, rating_groups, keys)
group_size = len(rating_groups)
if ranks is None:
ranks = range(group_size)
elif len(ranks) != group_size:
raise ValueError('Wrong ranks')
# sort rating groups by rank
by_rank = lambda x: x[1][1]
sorting = sorted(enumerate(zip(rating_groups, ranks, weights)),
key=by_rank)
sorted_rating_groups, sorted_ranks, sorted_weights = [], [], []
for x, (g, r, w) in sorting:
sorted_rating_groups.append(g)
sorted_ranks.append(r)
# make weights to be greater than 0
sorted_weights.append(max(min_delta, w_) for w_ in w)
# build factor graph
args = (sorted_rating_groups, sorted_ranks, sorted_weights)
builders = self.factor_graph_builders(*args)
args = builders + (min_delta,)
layers = self.run_schedule(*args)
# make result
rating_layer, team_sizes = layers[0], _team_sizes(sorted_rating_groups)
transformed_groups = []
for start, end in zip([0] + team_sizes[:-1], team_sizes):
group = []
for f in rating_layer[start:end]:
group.append(Rating(float(f.var.mu), float(f.var.sigma)))
transformed_groups.append(tuple(group))
by_hint = lambda x: x[0]
unsorting = sorted(zip((x for x, __ in sorting), transformed_groups),
key=by_hint)
if keys is None:
return [g for x, g in unsorting]
# restore the structure with input dictionary keys
return [dict(zip(keys[x], g)) for x, g in unsorting]
def quality(self, rating_groups, weights=None):
"""Calculates the match quality of the given rating groups. A result
is the draw probability in the association::
env = TrueSkill()
if env.quality([team1, team2, team3]) < 0.50:
print('This match seems to be not so fair')
:param rating_groups: a list of tuples or dictionaries containing
:class:`Rating` objects.
:param weights: weights of each players for "partial play".
.. versionadded:: 0.2
"""
rating_groups, keys = self.validate_rating_groups(rating_groups)
weights = self.validate_weights(weights, rating_groups, keys)
flatten_ratings = sum(map(tuple, rating_groups), ())
flatten_weights = sum(map(tuple, weights), ())
length = len(flatten_ratings)
# a vector of all of the skill means
mean_matrix = Matrix([[r.mu] for r in flatten_ratings])
# a matrix whose diagonal values are the variances (sigma ** 2) of each
# of the players.
def variance_matrix(height, width):
variances = (r.sigma ** 2 for r in flatten_ratings)
for x, variance in enumerate(variances):
yield (x, x), variance
variance_matrix = Matrix(variance_matrix, length, length)
# the player-team assignment and comparison matrix
def rotated_a_matrix(set_height, set_width):
t = 0
for r, (cur, _next) in enumerate(zip(rating_groups[:-1],
rating_groups[1:])):
for x in range(t, t + len(cur)):
yield (r, x), flatten_weights[x]
t += 1
x += 1
for x in range(x, x + len(_next)):
yield (r, x), -flatten_weights[x]
set_height(r + 1)
set_width(x + 1)
rotated_a_matrix = Matrix(rotated_a_matrix)
a_matrix = rotated_a_matrix.transpose()
# match quality further derivation
_ata = (self.beta ** 2) * rotated_a_matrix * a_matrix
_atsa = rotated_a_matrix * variance_matrix * a_matrix
start = mean_matrix.transpose() * a_matrix
middle = _ata + _atsa
end = rotated_a_matrix * mean_matrix
# make result
e_arg = (-0.5 * start * middle.inverse() * end).determinant()
s_arg = _ata.determinant() / middle.determinant()
return math.exp(e_arg) * math.sqrt(s_arg)
def expose(self, rating):
"""Returns the value of the rating exposure. It starts from 0 and
converges to the mean. Use this as a sort key in a leaderboard::
leaderboard = sorted(ratings, key=env.expose, reverse=True)
.. versionadded:: 0.4
"""
k = self.mu / self.sigma
return rating.mu - k * rating.sigma
def make_as_global(self):
"""Registers the environment as the global environment.
>>> env = TrueSkill(mu=50)
>>> Rating()
trueskill.Rating(mu=25.000, sigma=8.333)
>>> env.make_as_global() #doctest: +ELLIPSIS
trueskill.TrueSkill(mu=50.000, ...)
>>> Rating()
trueskill.Rating(mu=50.000, sigma=8.333)
But if you need just one environment, :func:`setup` is better to use.
"""
return setup(env=self)
def __repr__(self):
c = type(self)
if callable(self.draw_probability):
f = self.draw_probability
draw_probability = '.'.join([f.__module__, f.__name__])
else:
draw_probability = '%.1f%%' % (self.draw_probability * 100)
if self.backend is None:
backend = ''
elif isinstance(self.backend, tuple):
backend = ', backend=...'
else:
backend = ', backend=%r' % self.backend
args = ('.'.join([c.__module__, c.__name__]), self.mu, self.sigma,
self.beta, self.tau, draw_probability, backend)
return ('%s(mu=%.3f, sigma=%.3f, beta=%.3f, tau=%.3f, '
'draw_probability=%s%s)' % args)
def rate_1vs1(rating1, rating2, drawn=False, min_delta=DELTA, env=None):
"""A shortcut to rate just 2 players in a head-to-head match::
alice, bob = Rating(25), Rating(30)
alice, bob = rate_1vs1(alice, bob)
alice, bob = rate_1vs1(alice, bob, drawn=True)
:param rating1: the winner's rating if they didn't draw.
:param rating2: the loser's rating if they didn't draw.
:param drawn: if the players drew, set this to ``True``. Defaults to
``False``.
:param min_delta: will be passed to :meth:`rate`.
:param env: the :class:`TrueSkill` object. Defaults to the global
environment.
:returns: a tuple containing recalculated 2 ratings.
.. versionadded:: 0.2
"""
if env is None:
env = global_env()
ranks = [0, 0 if drawn else 1]
teams = env.rate([(rating1,), (rating2,)], ranks, min_delta=min_delta)
return teams[0][0], teams[1][0]
def quality_1vs1(rating1, rating2, env=None):
"""A shortcut to calculate the match quality between just 2 players in
a head-to-head match::
if quality_1vs1(alice, bob) < 0.50:
print('This match seems to be not so fair')
:param rating1: the rating.
:param rating2: the another rating.
:param env: the :class:`TrueSkill` object. Defaults to the global
environment.
.. versionadded:: 0.2
"""
if env is None:
env = global_env()
return env.quality([(rating1,), (rating2,)])
def global_env():
"""Gets the :class:`TrueSkill` object which is the global environment."""
try:
global_env.__trueskill__
except AttributeError:
# setup the default environment
setup()
return global_env.__trueskill__
def setup(mu=MU, sigma=SIGMA, beta=BETA, tau=TAU,
draw_probability=DRAW_PROBABILITY, backend=None, env=None):
"""Setups the global environment.
:param env: the specific :class:`TrueSkill` object to be the global
environment. It is optional.
>>> Rating()
trueskill.Rating(mu=25.000, sigma=8.333)
>>> setup(mu=50) #doctest: +ELLIPSIS
trueskill.TrueSkill(mu=50.000, ...)
>>> Rating()
trueskill.Rating(mu=50.000, sigma=8.333)
"""
if env is None:
env = TrueSkill(mu, sigma, beta, tau, draw_probability, backend)
global_env.__trueskill__ = env
return env
def rate(rating_groups, ranks=None, weights=None, min_delta=DELTA):
"""A proxy function for :meth:`TrueSkill.rate` of the global environment.
.. versionadded:: 0.2
"""
return global_env().rate(rating_groups, ranks, weights, min_delta)
def quality(rating_groups, weights=None):
"""A proxy function for :meth:`TrueSkill.quality` of the global
environment.
.. versionadded:: 0.2
"""
return global_env().quality(rating_groups, weights)
def expose(rating):
"""A proxy function for :meth:`TrueSkill.expose` of the global environment.
.. versionadded:: 0.4
"""
return global_env().expose(rating)
# Append deprecated methods into :class:`TrueSkill` and :class:`Rating`
from . import deprecated # noqa
from .deprecated import ( # noqa
dynamic_draw_probability, match_quality, transform_ratings)
deprecated.ensure_backward_compatibility(TrueSkill, Rating)

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# -*- coding: utf-8 -*-
"""
trueskill.backends
~~~~~~~~~~~~~~~~~~
Provides mathematical statistics backend chooser.
:copyright: (c) 2012-2016 by Heungsub Lee.
:license: BSD, see LICENSE for more details.
"""
from __future__ import absolute_import
import math
from six.moves import range
__all__ = ['available_backends', 'choose_backend', 'cdf', 'pdf', 'ppf']
def _gen_erfcinv(erfc, math=math):
"""Generates the inverse function of erfc by the given erfc function and
math module.
"""
def erfcinv(y):
"""The inverse function of erfc."""
if y >= 2:
return -100.
elif y <= 0:
return 100.
zero_point = y < 1
if not zero_point:
y = 2 - y
t = math.sqrt(-2 * math.log(y / 2.))
x = -0.70711 * \
((2.30753 + t * 0.27061) / (1. + t * (0.99229 + t * 0.04481)) - t)
for i in range(2):
err = erfc(x) - y
x += err / (1.12837916709551257 * math.exp(-(x ** 2)) - x * err)
return x if zero_point else -x
return erfcinv
def _gen_ppf(erfc, math=math):
"""ppf is the inverse function of cdf. This function generates cdf by the
given erfc and math module.
"""
erfcinv = _gen_erfcinv(erfc, math)
def ppf(x, mu=0, sigma=1):
"""The inverse function of cdf."""
return mu - sigma * math.sqrt(2) * erfcinv(2 * x)
return ppf
def erfc(x):
"""Complementary error function (via `http://bit.ly/zOLqbc`_)"""
z = abs(x)
t = 1. / (1. + z / 2.)
r = t * math.exp(-z * z - 1.26551223 + t * (1.00002368 + t * (
0.37409196 + t * (0.09678418 + t * (-0.18628806 + t * (
0.27886807 + t * (-1.13520398 + t * (1.48851587 + t * (
-0.82215223 + t * 0.17087277
)))
)))
)))
return 2. - r if x < 0 else r
def cdf(x, mu=0, sigma=1):
"""Cumulative distribution function"""
return 0.5 * erfc(-(x - mu) / (sigma * math.sqrt(2)))
def pdf(x, mu=0, sigma=1):
"""Probability density function"""
return (1 / math.sqrt(2 * math.pi) * abs(sigma) *
math.exp(-(((x - mu) / abs(sigma)) ** 2 / 2)))
ppf = _gen_ppf(erfc)
def choose_backend(backend):
"""Returns a tuple containing cdf, pdf, ppf from the chosen backend.
>>> cdf, pdf, ppf = choose_backend(None)
>>> cdf(-10)
7.619853263532764e-24
>>> cdf, pdf, ppf = choose_backend('mpmath')
>>> cdf(-10)
mpf('7.6198530241605255e-24')
.. versionadded:: 0.3
"""
if backend is None: # fallback
return cdf, pdf, ppf
elif backend == 'mpmath':
try:
import mpmath
except ImportError:
raise ImportError('Install "mpmath" to use this backend')
return mpmath.ncdf, mpmath.npdf, _gen_ppf(mpmath.erfc, math=mpmath)
elif backend == 'scipy':
try:
from scipy.stats import norm
except ImportError:
raise ImportError('Install "scipy" to use this backend')
return norm.cdf, norm.pdf, norm.ppf
raise ValueError('%r backend is not defined' % backend)
def available_backends():
"""Detects list of available backends. All of defined backends are
``None`` -- internal implementation, "mpmath", "scipy".
You can check if the backend is available in the current environment with
this function::
if 'mpmath' in available_backends():
# mpmath can be used in the current environment
setup(backend='mpmath')
.. versionadded:: 0.3
"""
backends = [None]
for backend in ['mpmath', 'scipy']:
try:
__import__(backend)
except ImportError:
continue
backends.append(backend)
return backends

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# -*- coding: utf-8 -*-
"""
trueskill.deprecated
~~~~~~~~~~~~~~~~~~~~
Deprecated features.
:copyright: (c) 2012-2016 by Heungsub Lee
:license: BSD, see LICENSE for more details.
"""
from __future__ import absolute_import
import warnings
from . import DELTA, expose, global_env, quality_1vs1, rate_1vs1, Rating
__all__ = ['transform_ratings', 'match_quality', 'dynamic_draw_probability',
'ensure_backward_compatibility']
# deprecated functions
def transform_ratings(rating_groups, ranks=None, min_delta=DELTA):
return global_env().transform_ratings(rating_groups, ranks, min_delta)
def match_quality(rating_groups):
return global_env().match_quality(rating_groups)
def dynamic_draw_probability(rating1, rating2, env=None):
"""Deprecated. It was an approximation for :func:`quality_1vs1`.
.. deprecated:: 0.4.1
Use :func:`quality_1vs1` instead.
"""
warnings.warn('Use quality_1vs1() instead', DeprecationWarning)
return quality_1vs1(rating1, rating2, env=env)
# deprecated methods
def addattr(obj, attr, value):
if hasattr(obj, attr):
raise AttributeError('The attribute already exists')
return setattr(obj, attr, value)
def ensure_backward_compatibility(TrueSkill, Rating):
addattr(TrueSkill, 'Rating', TrueSkill_Rating)
addattr(TrueSkill, 'transform_ratings', TrueSkill_transform_ratings)
addattr(TrueSkill, 'match_quality', TrueSkill_match_quality)
addattr(TrueSkill, 'rate_1vs1', TrueSkill_rate_1vs1)
addattr(TrueSkill, 'quality_1vs1', TrueSkill_quality_1vs1)
addattr(Rating, 'exposure', Rating_exposure)
def TrueSkill_Rating(self, mu=None, sigma=None):
"""Deprecated. Used to create a :class:`Rating` object.
.. deprecated:: 0.2
Override :meth:`create_rating` instead.
"""
warnings.warn('Use TrueSkill.create_rating() instead', DeprecationWarning)
return self.create_rating(mu, sigma)
def TrueSkill_transform_ratings(self, rating_groups, ranks=None,
min_delta=DELTA):
"""Deprecated. Used to rate the given ratings.
.. deprecated:: 0.2
Override :meth:`rate` instead.
"""
warnings.warn('Use TrueSkill.rate() instead', DeprecationWarning)
rating_groups = [(r,) if isinstance(r, Rating) else r
for r in rating_groups]
return self.rate(rating_groups, ranks, min_delta=min_delta)
def TrueSkill_match_quality(self, rating_groups):
"""Deprecated. Used to calculate a match quality.
.. deprecated:: 0.2
Override :meth:`quality` instead.
"""
warnings.warn('Use TrueSkill.quality() instead', DeprecationWarning)
rating_groups = [(r,) if isinstance(r, Rating) else r
for r in rating_groups]
return self.quality(rating_groups)
def TrueSkill_rate_1vs1(self, rating1, rating2, drawn=False, min_delta=DELTA):
"""Deprecated. Used to rate just a head-to-haed match.
.. deprecated:: 0.4
Use :func:`rate_1vs1` instead.
"""
warnings.warn('Use global function rate_1vs1() instead',
DeprecationWarning)
return rate_1vs1(rating1, rating2, drawn, min_delta, self)
def TrueSkill_quality_1vs1(self, rating1, rating2):
"""Deprecated. Used to calculate a match quality for a head-to-haed match.
.. deprecated:: 0.4
Use :func:`quality_1vs1` instead.
"""
warnings.warn('Use global function quality_1vs1() instead',
DeprecationWarning)
return quality_1vs1(rating1, rating2, self)
@property
def Rating_exposure(self):
"""Deprecated. Used to get a value that will go up on the whole.
.. deprecated:: 0.4
Use :meth:`TrueSkill.expose` instead.
"""
warnings.warn('Use TrueSkill.expose() instead', DeprecationWarning)
return expose(self)

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# -*- coding: utf-8 -*-
"""
trueskill.factorgraph
~~~~~~~~~~~~~~~~~~~~~
This module contains nodes for the factor graph of TrueSkill algorithm.
:copyright: (c) 2012-2016 by Heungsub Lee.
:license: BSD, see LICENSE for more details.
"""
from __future__ import absolute_import
import math
from six.moves import zip
from .mathematics import Gaussian, inf
__all__ = ['Variable', 'PriorFactor', 'LikelihoodFactor', 'SumFactor',
'TruncateFactor']
class Node(object):
pass
class Variable(Node, Gaussian):
def __init__(self):
self.messages = {}
super(Variable, self).__init__()
def set(self, val):
delta = self.delta(val)
self.pi, self.tau = val.pi, val.tau
return delta
def delta(self, other):
pi_delta = abs(self.pi - other.pi)
if pi_delta == inf:
return 0.
return max(abs(self.tau - other.tau), math.sqrt(pi_delta))
def update_message(self, factor, pi=0, tau=0, message=None):
message = message or Gaussian(pi=pi, tau=tau)
old_message, self[factor] = self[factor], message
return self.set(self / old_message * message)
def update_value(self, factor, pi=0, tau=0, value=None):
value = value or Gaussian(pi=pi, tau=tau)
old_message = self[factor]
self[factor] = value * old_message / self
return self.set(value)
def __getitem__(self, factor):
return self.messages[factor]
def __setitem__(self, factor, message):
self.messages[factor] = message
def __repr__(self):
args = (type(self).__name__, super(Variable, self).__repr__(),
len(self.messages), '' if len(self.messages) == 1 else 's')
return '<%s %s with %d connection%s>' % args
class Factor(Node):
def __init__(self, variables):
self.vars = variables
for var in variables:
var[self] = Gaussian()
def down(self):
return 0
def up(self):
return 0
@property
def var(self):
assert len(self.vars) == 1
return self.vars[0]
def __repr__(self):
args = (type(self).__name__, len(self.vars),
'' if len(self.vars) == 1 else 's')
return '<%s with %d connection%s>' % args
class PriorFactor(Factor):
def __init__(self, var, val, dynamic=0):
super(PriorFactor, self).__init__([var])
self.val = val
self.dynamic = dynamic
def down(self):
sigma = math.sqrt(self.val.sigma ** 2 + self.dynamic ** 2)
value = Gaussian(self.val.mu, sigma)
return self.var.update_value(self, value=value)
class LikelihoodFactor(Factor):
def __init__(self, mean_var, value_var, variance):
super(LikelihoodFactor, self).__init__([mean_var, value_var])
self.mean = mean_var
self.value = value_var
self.variance = variance
def calc_a(self, var):
return 1. / (1. + self.variance * var.pi)
def down(self):
# update value.
msg = self.mean / self.mean[self]
a = self.calc_a(msg)
return self.value.update_message(self, a * msg.pi, a * msg.tau)
def up(self):
# update mean.
msg = self.value / self.value[self]
a = self.calc_a(msg)
return self.mean.update_message(self, a * msg.pi, a * msg.tau)
class SumFactor(Factor):
def __init__(self, sum_var, term_vars, coeffs):
super(SumFactor, self).__init__([sum_var] + term_vars)
self.sum = sum_var
self.terms = term_vars
self.coeffs = coeffs
def down(self):
vals = self.terms
msgs = [var[self] for var in vals]
return self.update(self.sum, vals, msgs, self.coeffs)
def up(self, index=0):
coeff = self.coeffs[index]
coeffs = []
for x, c in enumerate(self.coeffs):
try:
if x == index:
coeffs.append(1. / coeff)
else:
coeffs.append(-c / coeff)
except ZeroDivisionError:
coeffs.append(0.)
vals = self.terms[:]
vals[index] = self.sum
msgs = [var[self] for var in vals]
return self.update(self.terms[index], vals, msgs, coeffs)
def update(self, var, vals, msgs, coeffs):
pi_inv = 0
mu = 0
for val, msg, coeff in zip(vals, msgs, coeffs):
div = val / msg
mu += coeff * div.mu
if pi_inv == inf:
continue
try:
# numpy.float64 handles floating-point error by different way.
# For example, it can just warn RuntimeWarning on n/0 problem
# instead of throwing ZeroDivisionError. So div.pi, the
# denominator has to be a built-in float.
pi_inv += coeff ** 2 / float(div.pi)
except ZeroDivisionError:
pi_inv = inf
pi = 1. / pi_inv
tau = pi * mu
return var.update_message(self, pi, tau)
class TruncateFactor(Factor):
def __init__(self, var, v_func, w_func, draw_margin):
super(TruncateFactor, self).__init__([var])
self.v_func = v_func
self.w_func = w_func
self.draw_margin = draw_margin
def up(self):
val = self.var
msg = self.var[self]
div = val / msg
sqrt_pi = math.sqrt(div.pi)
args = (div.tau / sqrt_pi, self.draw_margin * sqrt_pi)
v = self.v_func(*args)
w = self.w_func(*args)
denom = (1. - w)
pi, tau = div.pi / denom, (div.tau + sqrt_pi * v) / denom
return val.update_value(self, pi, tau)

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# -*- 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