#include <algorithm>
// step sizes
-static const double int_step_size = 50.0;
-static const double pdf_step_size = 10.0;
+static const double int_step_size = 75.0;
+static const double pdf_step_size = 15.0;
// rating constant (see below)
static const double rating_constant = 455.0;
using namespace std;
double prob_score(int k, double a, double rd);
-double prob_score_real(int k, double a, double prodai, double kfac, double rd_norm);
+double prob_score_real(int k, double a, double binomial, double rd_norm);
double prodai(int k, double a);
double fac(int x);
+// Numerical integration using Simpson's rule
+template<class T>
+double simpson_integrate(const T &evaluator, double from, double to, double step)
+{
+ int n = int((to - from) / step + 0.5);
+ double h = (to - from) / n;
+ double sum = evaluator(from);
+
+ for (int i = 1; i < n; i += 2) {
+ sum += 4.0 * evaluator(from + i * h);
+ }
+ for (int i = 2; i < n; i += 2) {
+ sum += 2.0 * evaluator(from + i * h);
+ }
+ sum += evaluator(to);
+
+ return (h/3.0) * sum;
+}
+
// probability of match ending k-a (k>a) when winnerR - loserR = RD
//
// +inf
//
double prob_score(int k, double a, double rd)
{
- return prob_score_real(k, a, prodai(k, a), fac(k-1), rd/rating_constant);
+ return prob_score_real(k, a, prodai(k, a) / fac(k-1), rd/rating_constant);
}
-// Same, but takes in Product(a+i, i=1..k-1) and (k-1)! as an argument in
+// Same, but takes in binomial(a+k-1, k-1) as an argument in
// addition to a. Faster if you already have that precomputed, and assumes rd
// is already divided by 455.
-double prob_score_real(int k, double a, double prodai, double kfac, double rd_norm)
+double prob_score_real(int k, double a, double binomial, double rd_norm)
{
- double nom = prodai * pow(2.0, -rd_norm * a);
- double denom = kfac * pow(1.0 + pow(2.0, -rd_norm), k+a);
+ double nom = binomial * pow(2.0, rd_norm * a);
+ double denom = pow(1.0 + pow(2.0, rd_norm), k+a);
return nom/denom;
}
// +inf
// /
// |
-// | ProbScore[a] (r2-r1) Gaussian[mu2, sigma2] (dr2) dr2
+// | ProbScore[a] (r1-r2) Gaussian[mu2, sigma2] (dr2) dr2
// |
// /
// -inf
// Set the last parameter to 1.0 if player 1 won, or -1.0 if player 2 won.
// In the latter case, ProbScore will be given (r1-r2) instead of (r2-r1).
//
-static inline double evaluate_int_point(int k, double a, double prodai_precompute, double kfac_precompute, double r1, double mu2, double sigma2, double winfac, double x);
+class ProbScoreEvaluator {
+private:
+ int k;
+ double a;
+ double binomial_precompute, r1, mu2, sigma2, winfac;
+
+public:
+ ProbScoreEvaluator(int k, double a, double binomial_precompute, double r1, double mu2, double sigma2, double winfac)
+ : k(k), a(a), binomial_precompute(binomial_precompute), r1(r1), mu2(mu2), sigma2(sigma2), winfac(winfac) {}
+ inline double operator() (double x) const
+ {
+ double probscore = prob_score_real(k, a, binomial_precompute, (r1 - x)*winfac);
+ double z = (x - mu2)/sigma2;
+ double gaussian = exp(-(z*z/2.0));
+ return probscore * gaussian;
+ }
+};
double opponent_rating_pdf(int k, double a, double r1, double mu2, double sigma2, double winfac)
{
- double prodai_precompute = prodai(k, a);
- double kfac_precompute = fac(k-1);
+ double binomial_precompute = prodai(k, a) / fac(k-1);
winfac /= rating_constant;
- int n = int(3000.0 / int_step_size + 0.5);
- double h = 3000.0 / double(n);
- double sum = evaluate_int_point(k, a, prodai_precompute, kfac_precompute, r1, mu2, sigma2, winfac, 0.0);
-
- for (int i = 1; i < n; i += 2) {
- sum += 4.0 * evaluate_int_point(k, a, prodai_precompute, kfac_precompute, r1, mu2, sigma2, winfac, i * h);
- }
- for (int i = 2; i < n; i += 2) {
- sum += 2.0 * evaluate_int_point(k, a, prodai_precompute, kfac_precompute, r1, mu2, sigma2, winfac, i * h);
- }
- sum += evaluate_int_point(k, a, prodai_precompute, kfac_precompute, r1, mu2, sigma2, winfac, 3000.0);
-
- return (h/3.0) * sum;
-}
-
-static inline double evaluate_int_point(int k, double a, double prodai_precompute, double kfac_precompute, double r1, double mu2, double sigma2, double winfac, double x)
-{
- double probscore = prob_score_real(k, a, prodai_precompute, kfac_precompute, (r1 - x)*winfac);
- double z = (x - mu2)/sigma2;
- double gaussian = exp(-(z*z/2.0));
- return probscore * gaussian;
+ return simpson_integrate(ProbScoreEvaluator(k, a, binomial_precompute, r1, mu2, sigma2, winfac), 0.0, 6000.0, int_step_size);
}
// normalize the curve so we know that A ~= 1
for (double r1 = 0.0; r1 < 3000.0; r1 += pdf_step_size) {
double z = (r1 - mu1) / sigma1;
double gaussian = exp(-(z*z/2.0));
- curve.push_back(make_pair(r1, gaussian * opponent_rating_pdf(score1, score2, r1, mu2, sigma2, 1.0)));
+ curve.push_back(make_pair(r1, gaussian * opponent_rating_pdf(score1, score2, r1, mu2, sigma2, -1.0)));
+ }
+ } else {
+ for (double r1 = 0.0; r1 < 3000.0; r1 += pdf_step_size) {
+ double z = (r1 - mu1) / sigma1;
+ double gaussian = exp(-(z*z/2.0));
+ curve.push_back(make_pair(r1, gaussian * opponent_rating_pdf(score2, score1, r1, mu2, sigma2, 1.0)));
+ }
+ }
+
+ double mu_est, sigma_est;
+ normalize(curve);
+ estimate_musigma(curve, mu_est, sigma_est);
+ least_squares(curve, mu_est, sigma_est, mu, sigma);
+}
+
+// int(normpdf[mu2, sigma2](t2) * ..., t2=0..3000);
+class OuterIntegralEvaluator {
+private:
+ double theta1, mu2, sigma2, mu_t, sigma_t;
+ int score1, score2;
+ double winfac;
+
+public:
+ OuterIntegralEvaluator(double theta1, double mu2, double sigma2, double mu3, double sigma3, double mu4, double sigma4, int score1, int score2, double winfac)
+ : theta1(theta1), mu2(mu2), sigma2(sigma2), mu_t(mu3 + mu4), sigma_t(sqrt(sigma3*sigma3 + sigma4*sigma4)), score1(score1), score2(score2), winfac(winfac) {}
+
+ double operator() (double theta2) const
+ {
+ double z = (theta2 - mu2) / sigma2;
+ double gaussian = exp(-(z*z/2.0));
+ double r1 = theta1 + theta2;
+ return gaussian * opponent_rating_pdf(score1, score2, r1, mu_t, sigma_t, winfac);
+ }
+};
+
+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)
+{
+ vector<pair<double, double> > curve;
+
+ if (score1 > score2) {
+ for (double r1 = 0.0; r1 < 3000.0; r1 += pdf_step_size) {
+ double z = (r1 - mu1) / sigma1;
+ double gaussian = exp(-(z*z/2.0));
+ curve.push_back(make_pair(r1, gaussian * simpson_integrate(OuterIntegralEvaluator(r1,mu2,sigma2,mu3,sigma3,mu4,sigma4,score1,score2,-0.5), 0.0, 3000.0, int_step_size)));
}
} else {
for (double r1 = 0.0; r1 < 3000.0; r1 += pdf_step_size) {
double z = (r1 - mu1) / sigma1;
double gaussian = exp(-(z*z/2.0));
- curve.push_back(make_pair(r1, gaussian * opponent_rating_pdf(score2, score1, r1, mu2, sigma2, -1.0)));
+ curve.push_back(make_pair(r1, gaussian * simpson_integrate(OuterIntegralEvaluator(r1,mu2,sigma2,mu3,sigma3,mu4,sigma4,score2,score1,0.5), 0.0, 3000.0, int_step_size)));
}
}
double mu2 = atof(argv[3]);
double sigma2 = atof(argv[4]);
- if (argc > 6) {
+ if (argc > 8) {
+ double mu3 = atof(argv[5]);
+ double sigma3 = atof(argv[6]);
+ double mu4 = atof(argv[7]);
+ 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);
+ } else if (argc > 6) {
int score1 = atoi(argv[5]);
int score2 = atoi(argv[6]);
double mu, sigma;
compute_new_rating(mu1, sigma1, mu2, sigma2, k, i, newmu1, newsigma1);
compute_new_rating(mu2, sigma2, mu1, sigma1, i, k, newmu2, newsigma2);
printf("%u-%u,%f,%+f,%+f\n",
- k, i, prob_score(k, i, mu1-mu2), newmu1-mu1, newmu2-mu2);
+ k, i, prob_score(k, i, mu2-mu1), 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);
printf("%u-%u,%f,%+f,%+f\n",
- i, k, prob_score(k, i, mu2-mu1), newmu1-mu1, newmu2-mu2);
+ i, k, prob_score(k, i, mu1-mu2), newmu1-mu1, newmu2-mu2);
}
}
}
projects / foosball / blobdiff