// rating constant (see below)
static const double rating_constant = 455.0;
-#if USE_LOGISTIC_DISTRIBUTION
-// constant used in the logistic pdf
-static const double l_const = M_PI / (2.0 * sqrt(3.0));
-#endif
-
using namespace std;
-static double prob_score(int k, int a, double rd);
static double prob_score_real(int k, int a, double binomial, double rd_norm);
static double prodai(int k, int a);
static double fac(int x);
// sech²(x)
static double sech2(double x)
{
- double c = cosh(x);
- return 1.0 / (c*c);
+ double e = exp(2.0 * x);
+ return 4.0 * e / ((e+1.0) * (e+1.0));
}
#endif
+#if 0
// probability of match ending k-a (k>a) when winnerR - loserR = RD
//
// +inf
{
return prob_score_real(k, a, prodai(k, a) / fac(k-1), rd/rating_constant);
}
+#endif
// computes x^a, probably more efficiently than pow(x, a) (but requires that a
// is n unsigned integer)
// opponent's pdf
#if USE_LOGISTIC_DISTRIBUTION
double z = (x1 - mu2) * invsigma2;
- double ch = cosh(l_const * z);
- func1[i].real() = 1.0 / (ch * ch);
+ func1[i].real() = sech2(0.5 * z);
#else
double z = (x1 - mu2) * invsq2sigma2;
func1[i].real() = exp(-z*z);
#if USE_LOGISTIC_DISTRIBUTION
// df/dA(x_i)
- matA[i + 0 * curve.size()] = sech2(l_const * (x-mu)/sigma);
+ matA[i + 0 * curve.size()] = sech2(0.5 * (x-mu)/sigma);
// df/dµ(x_i)
- matA[i + 1 * curve.size()] = 2.0 * l_const * A * matA[i + 0 * curve.size()]
- * tanh(l_const * (x-mu)/sigma) / sigma;
+ matA[i + 1 * curve.size()] = A * matA[i + 0 * curve.size()]
+ * tanh(0.5 * (x-mu)/sigma) / sigma;
// df/dσ(x_i)
matA[i + 2 * curve.size()] =
double y = curve[i].second;
#if USE_LOGISTIC_DISTRIBUTION
- dbeta[i] = y - A * sech2(l_const * (x-mu)/sigma);
+ dbeta[i] = y - A * sech2(0.5 * (x-mu)/sigma);
#else
dbeta[i] = y - A * exp(- (x-mu)*(x-mu)/(2.0*sigma*sigma));
#endif
sigma_result = sigma;
}
-static void compute_new_rating(double mu1, double sigma1, double mu2, double sigma2, int score1, int score2, double &mu, double &sigma)
+static void compute_new_rating(double mu1, double sigma1, double mu2, double sigma2, int score1, int score2, double &mu, double &sigma, double &probability)
{
vector<pair<double, double> > curve;
// my pdf
double z = (r1 - mu1) / sigma1;
#if USE_LOGISTIC_DISTRIBUTION
- double ch = cosh(l_const * z);
- curve[i].second /= (ch * ch);
+ curve[i].second *= sech2(0.5 * z);
#else
double gaussian = exp(-(z*z/2.0));
curve[i].second *= gaussian;
#endif
}
+
+ // Compute the overall probability of the given result, by integrating
+ // the entire resulting pdf. Note that since we're actually evaluating
+ // a double integral, we'll need to multiply by h² instead of h.
+ {
+ double h = (curve.back().first - curve.front().first) / (curve.size() - 1);
+ double sum = curve.front().second;
+ for (unsigned i = 1; i < curve.size() - 1; i += 2) {
+ sum += 4.0 * curve[i].second;
+ }
+ for (unsigned i = 2; i < curve.size() - 1; i += 2) {
+ sum += 2.0 * curve[i].second;
+ }
+ sum += curve.back().second;
+ sum *= h * h / 3.0;
+
+ // FFT convolution multiplication factor (FFTW computes unnormalized
+ // transforms)
+ sum /= (curve.size() * 2);
+
+ // pdf normalization factors
+#if USE_LOGISTIC_DISTRIBUTION
+ sum /= (sigma1 * 4.0);
+ sum /= (sigma2 * 4.0);
+#else
+ sum /= (sigma1 * sqrt(2.0 * M_PI));
+ sum /= (sigma2 * sqrt(2.0 * M_PI));
+#endif
+
+ probability = sum;
+ }
double mu_est, sigma_est;
normalize(curve);
least_squares(curve, mu_est, sigma_est, mu, sigma);
}
-static void compute_new_double_rating(double mu1, double sigma1, double mu2, double sigma2, double mu3, double sigma3, double mu4, double sigma4, int score1, int score2, double &mu, double &sigma)
+static void compute_new_double_rating(double mu1, double sigma1, double mu2, double sigma2, double mu3, double sigma3, double mu4, double sigma4, int score1, int score2, double &mu, double &sigma, double &probability)
{
vector<pair<double, double> > curve, newcurve;
double mu_t = mu3 + mu4;
#if USE_LOGISTIC_DISTRIBUTION
double z = (r2 - mu2) * invsigma2;
- double gaussian = sech2(l_const * z);
+ double gaussian = sech2(0.5 * z);
#else
double z = (r2 - mu2) * invsq2sigma2;
double gaussian = exp(-z*z);
#if USE_LOGISTIC_DISTRIBUTION
double z = (r1 - mu1) / sigma1;
- double gaussian = sech2(l_const * z);
+ double gaussian = sech2(0.5 * z);
#else
double z = (r1 - mu1) / sigma1;
double gaussian = exp(-(z*z/2.0));
newcurve.push_back(make_pair(r1, gaussian * sum));
}
+ // Compute the overall probability of the given result, by integrating
+ // the entire resulting pdf. Note that since we're actually evaluating
+ // a triple integral, we'll need to multiply by 4h³ (no idea where the
+ // 4 factor comes from, probably from the 0..6000 range somehow) instead
+ // of h.
+ {
+ double h = (newcurve.back().first - newcurve.front().first) / (newcurve.size() - 1);
+ double sum = newcurve.front().second;
+ for (unsigned i = 1; i < newcurve.size() - 1; i += 2) {
+ sum += 4.0 * newcurve[i].second;
+ }
+ for (unsigned i = 2; i < newcurve.size() - 1; i += 2) {
+ sum += 2.0 * newcurve[i].second;
+ }
+ sum += newcurve.back().second;
+
+ sum *= 4.0 * h * h * h / 3.0;
+
+ // FFT convolution multiplication factor (FFTW computes unnormalized
+ // transforms)
+ sum /= (newcurve.size() * 2);
+
+ // pdf normalization factors
+#if USE_LOGISTIC_DISTRIBUTION
+ sum /= (sigma1 * 4.0);
+ sum /= (sigma2 * 4.0);
+ sum /= (sigma_t * 4.0);
+#else
+ sum /= (sigma1 * sqrt(2.0 * M_PI));
+ sum /= (sigma2 * sqrt(2.0 * M_PI));
+ sum /= (sigma_t * sqrt(2.0 * M_PI));
+#endif
+
+ probability = sum;
+ }
double mu_est, sigma_est;
normalize(newcurve);
double sigma4 = atof(argv[8]);
int score1 = atoi(argv[9]);
int score2 = atoi(argv[10]);
- double mu, sigma;
- compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, score1, score2, mu, sigma);
- printf("%f %f\n", mu, sigma);
+ double mu, sigma, probability;
+ compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, score1, score2, mu, sigma, probability);
+ printf("%f %f %f\n", mu, sigma, probability);
} else if (argc > 8) {
double mu3 = atof(argv[5]);
double sigma3 = atof(argv[6]);
double mu4 = atof(argv[7]);
double sigma4 = atof(argv[8]);
int k = atoi(argv[9]);
-
+
// assess all possible scores
for (int i = 0; i < k; ++i) {
double newmu1_1, newmu1_2, newmu2_1, newmu2_2;
double newsigma1_1, newsigma1_2, newsigma2_1, newsigma2_2;
- compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, k, i, newmu1_1, newsigma1_1);
- compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, k, i, newmu1_2, newsigma1_2);
- compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, i, k, newmu2_1, newsigma2_1);
- compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, i, k, newmu2_2, newsigma2_2);
+ double probability;
+ compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, k, i, newmu1_1, newsigma1_1, probability);
+ compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, k, i, newmu1_2, newsigma1_2, probability);
+ compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, i, k, newmu2_1, newsigma2_1, probability);
+ compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, i, k, newmu2_2, newsigma2_2, probability);
printf("%u-%u,%f,%+f,%+f,%+f,%+f\n",
- k, i, prob_score(k, i, mu3+mu4-(mu1+mu2)), newmu1_1-mu1, newmu1_2-mu2,
+ k, i, probability, newmu1_1-mu1, newmu1_2-mu2,
newmu2_1-mu3, newmu2_2-mu4);
}
for (int i = k; i --> 0; ) {
double newmu1_1, newmu1_2, newmu2_1, newmu2_2;
double newsigma1_1, newsigma1_2, newsigma2_1, newsigma2_2;
- compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, i, k, newmu1_1, newsigma1_1);
- compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, i, k, newmu1_2, newsigma1_2);
- compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, k, i, newmu2_1, newsigma2_1);
- compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, k, i, newmu2_2, newsigma2_2);
+ double probability;
+ compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, k, i, newmu1_1, newsigma1_1, probability);
+ compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, i, k, newmu1_1, newsigma1_1, probability);
+ compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, i, k, newmu1_2, newsigma1_2, probability);
+ compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, k, i, newmu2_1, newsigma2_1, probability);
+ compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, k, i, newmu2_2, newsigma2_2, probability);
printf("%u-%u,%f,%+f,%+f,%+f,%+f\n",
- i, k, prob_score(k, i, mu1+mu2-(mu3+mu4)), newmu1_1-mu1, newmu1_2-mu2,
+ i, k, probability, newmu1_1-mu1, newmu1_2-mu2,
newmu2_1-mu3, newmu2_2-mu4);
}
} else if (argc > 6) {
int score1 = atoi(argv[5]);
int score2 = atoi(argv[6]);
- double mu, sigma;
- compute_new_rating(mu1, sigma1, mu2, sigma2, score1, score2, mu, sigma);
- printf("%f %f\n", mu, sigma);
+ double mu, sigma, probability;
+ compute_new_rating(mu1, sigma1, mu2, sigma2, score1, score2, mu, sigma, probability);
+
+ printf("%f %f %f\n", mu, sigma, probability);
} else {
int k = atoi(argv[5]);
// assess all possible scores
for (int i = 0; i < k; ++i) {
- double newmu1, newmu2, newsigma1, newsigma2;
- compute_new_rating(mu1, sigma1, mu2, sigma2, k, i, newmu1, newsigma1);
- compute_new_rating(mu2, sigma2, mu1, sigma1, i, k, newmu2, newsigma2);
+ double newmu1, newmu2, newsigma1, newsigma2, probability;
+ compute_new_rating(mu1, sigma1, mu2, sigma2, k, i, newmu1, newsigma1, probability);
+ compute_new_rating(mu2, sigma2, mu1, sigma1, i, k, newmu2, newsigma2, probability);
printf("%u-%u,%f,%+f,%+f\n",
- k, i, prob_score(k, i, mu2-mu1), newmu1-mu1, newmu2-mu2);
+ k, i, probability, newmu1-mu1, newmu2-mu2);
}
for (int i = k; i --> 0; ) {
- double newmu1, newmu2, newsigma1, newsigma2;
- compute_new_rating(mu1, sigma1, mu2, sigma2, i, k, newmu1, newsigma1);
- compute_new_rating(mu2, sigma2, mu1, sigma1, k, i, newmu2, newsigma2);
+ double newmu1, newmu2, newsigma1, newsigma2, probability;
+ compute_new_rating(mu1, sigma1, mu2, sigma2, i, k, newmu1, newsigma1, probability);
+ compute_new_rating(mu2, sigma2, mu1, sigma1, k, i, newmu2, newsigma2, probability);
printf("%u-%u,%f,%+f,%+f\n",
- i, k, prob_score(k, i, mu1-mu2), newmu1-mu1, newmu2-mu2);
+ i, k, probability, newmu1-mu1, newmu2-mu2);
}
}
projects / foosball / blobdiff