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trueskill.py
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from math import sqrt
from scipy.special import erfinv
from scipy.special import log_ndtr
from numpy import logaddexp
import math
import scipy.stats
import numpy as np
def logdiffexp(a, b):
if a == b: return -10000000
return a + math.log(-np.expm1(b-a))
class TrueSkill:
def __init__(self, beta2, tau2, prob_draw, var0):
self.beta2 = beta2 # rating class width
self.tau2 = tau2 # additive dynamics
self.draw_margin = TrueSkill.inv_cdf(0.5*(prob_draw + 1.0))*sqrt(2*beta2 + 2*var0) # the draw margin
#print self.draw_margin
def update_rating(self, winner, loser, isDraw):
# unpack data
mu_winner, var_winner = winner
mu_loser, var_loser = loser
# before updates perform additive dynamics
var_winner += self.tau2
var_loser += self.tau2
# precompute some values
c = sqrt(2*self.beta2 + var_winner + var_loser)
t = (mu_winner - mu_loser)/c
eps = self.draw_margin/c
logpdfmm = TrueSkill.logpdf(-eps-t)
logpdfpp = logpdfmm
logpdfpm = TrueSkill.logpdf(eps-t)
logpdfmp = logpdfpm
logcdfmm = TrueSkill.logcdf(-eps-t)
logcdfpp = TrueSkill.logcdf(eps+t)
logcdfpm = TrueSkill.logcdf(eps-t)
logcdfmp = TrueSkill.logcdf(-eps+t)
if isDraw:
v = self.v_draw(t, logpdfmm, logpdfpm, logcdfmm, logcdfpm, logpdfmp, logpdfpp, logcdfmp, logcdfpp)
w = self.w_draw(t, eps, logpdfmm, logpdfpm, logcdfmm, logcdfpm, logpdfmp, logpdfpp, logcdfmp, logcdfpp)
else:
v = self.v_nondraw(logpdfmp, logcdfmp)
w = self.w_nondraw(t, eps, logpdfmp, logcdfmp)
mu_winner += var_winner*v/c
mu_loser -= var_loser*v/c
var_winner *= (1.0 - var_winner*w/(c**2))
var_loser *= (1.0 - var_loser*w/(c**2))
if var_winner < 0 or var_loser < 0: print(mu_winner, mu_loser)
return ((mu_winner, var_winner), (mu_loser, var_loser))
@staticmethod
def inv_cdf(prob):
return sqrt(2.0)*erfinv(2*prob - 1.0)
@staticmethod
def logpdf(x):
return 0.5*(-x**2 - math.log(2*math.pi))
@staticmethod
def logcdf(x):
#if x > 0: return math.log1p(-math.exp(TrueSkill.logcdf(-x)))
return log_ndtr(x)
@staticmethod
def v_nondraw(logpdfmp, logcdfmp):
return math.exp(logpdfmp - logcdfmp)
@staticmethod
def v_draw(t, logpdfmm, logpdfpm, logcdfmm, logcdfpm, logpdfmp, logpdfpp, logcdfmp, logcdfpp):
if t<0: return -TrueSkill.v_draw(-t, logpdfmp, logpdfpp, logcdfmp, logcdfpp, logpdfmm, logpdfpm, logcdfmm, logcdfpm)
mul = 1.0
if logpdfmm < logpdfpm:
logpdfmm, logpdfpm = logpdfpm, logpdfmm
mul = -mul
if logcdfpm < logcdfmm:
logcdfpm, logcdfmm = logcdfmm, logcdfpm
mul = -mul
return mul*math.exp(logdiffexp(logpdfmm, logpdfpm) - logdiffexp(logcdfpm, logcdfmm))
@staticmethod
def w_nondraw(t, eps, logpdfmp, logcdfmp):
vnondraw = TrueSkill.v_nondraw(logpdfmp, logcdfmp)
return vnondraw*(vnondraw + t - eps)
@staticmethod
def w_draw(t, eps, logpdfmm, logpdfpm, logcdfmm, logcdfpm, logpdfmp, logpdfpp, logcdfmp, logcdfpp):
if t < 0: return TrueSkill.w_draw(-t, eps, logpdfmp, logpdfpp, logcdfmp, logcdfpp, logpdfmm, logpdfpm, logcdfmm, logcdfpm)
sign = 1.0
if eps - t > 0:
logpm = math.log(eps - t)
else:
logpm = math.log(t-eps)
sign = -sign
logpp = math.log(eps + t)
if sign == 1: loga = logaddexp(logpm + logpdfpm, logpp + logpdfpp)
else:
if logpm + logpdfpm > logpp + logpdfpp:
loga = logdiffexp(logpm + logpdfpm, logpp + logpdfpp)
sign = -1.0
else:
loga = logdiffexp(logpp + logpdfpp, logpm + logpdfpm)
sign = 1.0
logb = logdiffexp(logcdfpm, logcdfmm)
vdraw = TrueSkill.v_draw(t, logpdfmm, logpdfpm, logcdfmm, logcdfpm, logpdfmp, logpdfpp, logcdfmp, logcdfpp)
return vdraw**2 + sign*math.exp(loga - logb)
def precompute(eps, t):
logpdfmm = TrueSkill.logpdf(-eps-t)
logpdfpp = logpdfmm
logpdfpm = TrueSkill.logpdf(eps-t)
logpdfmp = logpdfpm
logcdfmm = TrueSkill.logcdf(-eps-t)
logcdfpp = TrueSkill.logcdf(eps+t)
logcdfpm = TrueSkill.logcdf(eps-t)
logcdfmp = TrueSkill.logcdf(-eps+t)
return logpdfmm, logpdfpp, logpdfpm, logpdfmp, logcdfmm, logcdfpp, logcdfpm, logcdfmp
'''
from pylab import *
import numpy as np
x = np.linspace(-50,50,1000)
epss = (0.0001,0.1,1, 4, 10)
subplot(2,2,1)
for eps in epss:
y = []
for t in x:
logpdfmm, logpdfpp, logpdfpm, logpdfmp, logcdfmm, logcdfpp, logcdfpm, logcdfmp = precompute(eps, t)
#if cdfmp == 0: cdfmp = 1E-17
y.append(TrueSkill.v_nondraw(logpdfmp, logcdfmp))
plot(x,y)
subplot(2,2,2)
for eps in epss:
y = []
for t in x:
logpdfmm, logpdfpp, logpdfpm, logpdfmp, logcdfmm, logcdfpp, logcdfpm, logcdfmp = precompute(eps, t)
y.append(TrueSkill.v_draw(t, logpdfmm, logpdfpm, logcdfmm, logcdfpm, logpdfmp, logpdfpp, logcdfmp, logcdfpp))
plot(x,y)
subplot(2,2,3)
for eps in epss:
y = []
for t in x:
logpdfmm, logpdfpp, logpdfpm, logpdfmp, logcdfmm, logcdfpp, logcdfpm, logcdfmp = precompute(eps, t)
y.append(TrueSkill.w_nondraw(t, eps, logpdfmp, logcdfmp))
plot(x,y)
subplot(2,2,4)
for eps in epss:
y = []
for t in x:
logpdfmm, logpdfpp, logpdfpm, logpdfmp, logcdfmm, logcdfpp, logcdfpm, logcdfmp = precompute(eps, t)
y.append(TrueSkill.w_draw(t, eps, logpdfmm, logpdfpm, logcdfmm, logcdfpm, logpdfmp, logpdfpp, logcdfmp, logcdfpp))
plot(x,y)
show()
# testing TrueSkill with recommended values
mu0 = 25.0
var0 = (mu0/3.0)**2
beta2 = var0/4.0
tau2 = var0*100
prob_draw = 0.1
ts = TrueSkill(beta2, tau2, prob_draw, var0)
#ts = TrueSkill(1,0,0)
mu1, var1, mu2, var2 = 25, 8, 20, 5
print ts.update_rating((mu1, var1), (mu2, var2), False)
print ts.update_rating((mu2, var2), (mu1, var1), False)
print ts.update_rating((mu1, var1), (mu2, var2), True)
#for i in xrange(100):
# print i, mu1, var1, mu2, var2
# mu1, var1, mu2, var2 = ts.update_rating(mu1, var1, mu2, var2, True)
'''