tra-analysis/analysis-master/analysis/trueskill.py

from __future__ import absolute_import

from itertools import chain
import math

from six import iteritems
from six.moves import map, range, zip
from six import iterkeys

import copy
try:
    from numbers import Number
except ImportError:
    Number = (int, long, float, complex)

inf = float('inf')

class Gaussian(object):
    #: 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):
        return self.pi and self.tau / self.pi

    @property
    def sigma(self):
        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):
    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

def _gen_erfcinv(erfc, math=math):
    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):
    erfcinv = _gen_erfcinv(erfc, math)
    def ppf(x, mu=0, sigma=1):
        return mu - sigma * math.sqrt(2) * erfcinv(2 * x)
    return ppf

def erfc(x):
    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):
    return 0.5 * erfc(-(x - mu) / (sigma * math.sqrt(2)))


def pdf(x, mu=0, sigma=1):
    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):
    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():
    backends = [None]
    for backend in ['mpmath', 'scipy']:
        try:
            __import__(backend)
        except ImportError:
            continue
        backends.append(backend)
    return backends

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)

#: 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):
    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):
    if env is None:
        env = global_env()
    return env.ppf((draw_probability + 1) / 2.) * math.sqrt(size) * env.beta


def _team_sizes(rating_groups):
    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):
    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):
    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):
        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):
        x = diff - draw_margin
        denom = self.cdf(x)
        return (self.pdf(x) / denom) if denom else -x

    def v_draw(self, diff, draw_margin):
        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):
        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):
        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):
        # 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):
        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):
        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):
        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):
        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):
        k = self.mu / self.sigma
        return rating.mu - k * rating.sigma

    def make_as_global(self):
        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):
    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):
    if env is None:
        env = global_env()
    return env.quality([(rating1,), (rating2,)])


def global_env():
    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):
    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):
    return global_env().rate(rating_groups, ranks, weights, min_delta)


def quality(rating_groups, weights=None):
    return global_env().quality(rating_groups, weights)


def expose(rating):
    return global_env().expose(rating)