upload trueskill for testing purposes

data analysis/analysis/trueskill/__about__.py

@@ -0,0 +1,11 @@
+# -*- 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'

data analysis/analysis/trueskill/__init__.py

@@ -0,0 +1,726 @@
+# -*- 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)

data analysis/analysis/trueskill/backends.py

@@ -0,0 +1,135 @@
+# -*- 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

data analysis/analysis/trueskill/deprecated.py

@@ -0,0 +1,134 @@
+# -*- 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)

data analysis/analysis/trueskill/factorgraph.py

@@ -0,0 +1,199 @@
+# -*- 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)

data analysis/analysis/trueskill/mathematics.py

@@ -0,0 +1,260 @@
+# -*- 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