-#include <stdio.h>
-#include <math.h>
-#include <assert.h>
+#include <cstdio>
+#include <cmath>
+#include <cassert>
#include <vector>
#include <algorithm>
+#include <complex>
+#include <fftw3.h>
+
// 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;
// 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 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);
+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);
- 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
//
// Glicko/Bradley-Terry assumption that a player rated 400 points over
// his/her opponent will win with a probability of 10/11 =~ 0.90909.
//
-double prob_score(int k, double a, double rd)
+static double prob_score(int k, int a, double rd)
{
return prob_score_real(k, a, prodai(k, a) / fac(k-1), rd/rating_constant);
}
+// computes x^a, probably more efficiently than pow(x, a) (but requires that a
+// is n unsigned integer)
+static double intpow(double x, unsigned a)
+{
+ double result = 1.0;
+
+ while (a > 0) {
+ if (a & 1) {
+ result *= x;
+ }
+ a >>= 1;
+ x *= x;
+ }
+
+ return result;
+}
+
// 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 binomial, double rd_norm)
+static double prob_score_real(int k, int a, double binomial, double rd_norm)
{
- double nom = binomial * pow(2.0, rd_norm * a);
- double denom = pow(1.0 + pow(2.0, rd_norm), k+a);
+ double nom = binomial * intpow(pow(2.0, rd_norm), a);
+ double denom = intpow(1.0 + pow(2.0, rd_norm), k+a);
return nom/denom;
}
// Calculates Product(a+i, i=1..k-1) (see above).
-double prodai(int k, double a)
+static double prodai(int k, int a)
{
double prod = 1.0;
for (int i = 1; i < k; ++i)
return prod;
}
-double fac(int x)
+static double fac(int x)
{
double prod = 1.0;
for (int i = 2; i <= x; ++i)
return prod;
}
-//
-// Computes the integral
-//
-// +inf
-// /
-// |
-// | ProbScore[a] (r2-r1) Gaussian[mu2, sigma2] (dr2) dr2
-// |
-// /
-// -inf
-//
-// For practical reasons, -inf and +inf are replaced by 0 and 3000, which
-// is reasonable in the this context.
-//
-// The Gaussian is not normalized.
-//
-// 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).
-//
-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, (x - r1)*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)
+static void compute_opponent_rating_pdf(int k, int a, double mu2, double sigma2, double winfac, vector<pair<double, double> > &result)
{
double binomial_precompute = prodai(k, a) / fac(k-1);
winfac /= rating_constant;
- return simpson_integrate(ProbScoreEvaluator(k, a, binomial_precompute, r1, mu2, sigma2, winfac), 0.0, 3000.0, int_step_size);
+ int sz = (6000.0 - 0.0) / int_step_size;
+ double h = (6000.0 - 0.0) / sz;
+
+ fftw_plan f1, f2, b;
+ complex<double> *func1, *func2, *res;
+
+ func1 = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
+ func2 = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
+ res = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
+ f1 = fftw_plan_dft_1d(sz*2,
+ reinterpret_cast<fftw_complex*>(func1),
+ reinterpret_cast<fftw_complex*>(func1),
+ FFTW_FORWARD,
+ FFTW_MEASURE);
+ f2 = fftw_plan_dft_1d(sz*2,
+ reinterpret_cast<fftw_complex*>(func2),
+ reinterpret_cast<fftw_complex*>(func2),
+ FFTW_FORWARD,
+ FFTW_MEASURE);
+ b = fftw_plan_dft_1d(sz*2,
+ reinterpret_cast<fftw_complex*>(res),
+ reinterpret_cast<fftw_complex*>(res),
+ FFTW_BACKWARD,
+ FFTW_MEASURE);
+
+ // start off by zero
+ for (int i = 0; i < sz*2; ++i) {
+ func1[i].real() = func1[i].imag() = func2[i].real() = func2[i].imag() = 0.0;
+ }
+
+ for (int i = 0; i < sz; ++i) {
+ double x1 = 0.0 + h*i;
+ double z = (x1 - mu2)/sigma2;
+ func1[i].real() = exp(-(z*z/2.0));
+
+ double x2 = -3000.0 + h*i;
+ func2[(i - sz/2 + sz*2)%(sz*2)].real() = prob_score_real(k, a, binomial_precompute, x2*winfac);
+ }
+
+ result.reserve(sz*2);
+
+ // convolve
+ fftw_execute(f1);
+ fftw_execute(f2);
+ for (int i = 0; i < sz*2; ++i) {
+ res[i] = func1[i] * func2[i];
+ }
+ fftw_execute(b);
+ for (int i = 0; i < sz; ++i) {
+ double r1 = i*h;
+ result.push_back(make_pair(r1, abs(res[i])));
+ }
}
// normalize the curve so we know that A ~= 1
-void normalize(vector<pair<double, double> > &curve)
+
static void normalize(vector<pair<double, double> > &curve)
{
double peak = 0.0;
for (vector<pair<double, double> >::const_iterator i = curve.begin(); i != curve.end(); ++i) {
}
}
-// computes matA * matB
-void mat_mul(double *matA, unsigned ah, unsigned aw,
- double *matB, unsigned bh, unsigned bw,
- double *result)
-{
- assert(aw == bh);
- for (unsigned y = 0; y < bw; ++y) {
- for (unsigned x = 0; x < ah; ++x) {
- double sum = 0.0;
- for (unsigned c = 0; c < aw; ++c) {
- sum += matA[c*ah + x] * matB[y*bh + c];
- }
- result[y*bw + x] = sum;
- }
- }
-}
-
// computes matA^T * matB
-void mat_mul_trans(double *matA, unsigned ah, unsigned aw,
- double *matB, unsigned bh, unsigned bw,
- double *result)
+static void mat_mul_trans(double *matA, unsigned ah, unsigned aw,
+ double *matB, unsigned bh, unsigned bw,
+ double *result)
{
assert(ah == bh);
for (unsigned y = 0; y < bw; ++y) {
}
}
-void print3x3(double *M)
-{
- printf("%f %f %f\n", M[0], M[3], M[6]);
- printf("%f %f %f\n", M[1], M[4], M[7]);
- printf("%f %f %f\n", M[2], M[5], M[8]);
-}
-
-void print3x1(double *M)
-{
- printf("%f\n", M[0]);
- printf("%f\n", M[1]);
- printf("%f\n", M[2]);
-}
-
// solves Ax = B by Gauss-Jordan elimination, where A is a 3x3 matrix,
// x is a column vector of length 3 and B is a row vector of length 3.
// Destroys its input in the process.
-void solve3x3(double *A, double *x, double *B)
+static void solve3x3(double *A, double *x, double *B)
{
// row 1 -= row 0 * (a1/a0)
{
// Give an OK starting estimate for the least squares, by numerical integration
// of statistical moments.
-void estimate_musigma(vector<pair<double, double> > &curve, double &mu_result, double &sigma_result)
+
static void estimate_musigma(vector<pair<double, double> > &curve, double &mu_result, double &sigma_result)
{
double h = (curve.back().first - curve.front().first) / (curve.size() - 1);
// Note that the algorithm blows up quite hard if the initial estimate is
// not good enough. Use estimate_musigma to get a reasonable starting
// estimate.
-void least_squares(vector<pair<double, double> > &curve, double mu1, double sigma1, double &mu_result, double &sigma_result)
+
static void least_squares(vector<pair<double, double> > &curve, double mu1, double sigma1, double &mu_result, double &sigma_result)
{
double A = 1.0;
double mu = mu1;
sigma_result = sigma;
}
-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)
{
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 * opponent_rating_pdf(score1, score2, r1, mu2, sigma2, 1.0)));
- }
+ compute_opponent_rating_pdf(score1, score2, mu2, sigma2, -1.0, curve);
} 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)));
- }
+ compute_opponent_rating_pdf(score2, score1, mu2, sigma2, 1.0, curve);
+ }
+
+ // multiply in the gaussian
+ for (unsigned i = 0; i < curve.size(); ++i) {
+ double r1 = curve[i].first;
+ double z = (r1 - mu1) / sigma1;
+ double gaussian = exp(-(z*z/2.0));
+ curve[i].second *= gaussian;
}
double mu_est, sigma_est;
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)
+{
+ vector<pair<double, double> > curve, newcurve;
+ double mu_t = mu3 + mu4;
+ double sigma_t = sqrt(sigma3*sigma3 + sigma4*sigma4);
+
+ if (score1 > score2) {
+ compute_opponent_rating_pdf(score1, score2, mu_t, sigma_t, -1.0, curve);
+ } else {
+ compute_opponent_rating_pdf(score2, score1, mu_t, sigma_t, 1.0, curve);
+ }
+
+ // iterate over r1
+ double h = 3000.0 / curve.size();
+ for (unsigned i = 0; i < curve.size(); ++i) {
+ double sum = 0.0;
+
+ // could be anything, but this is a nice start
+ //double r1 = curve[i].first;
+ double r1 = i * h;
+
+ // iterate over r2
+ for (unsigned j = 0; j < curve.size(); ++j) {
+ double r1plusr2 = curve[j].first;
+ double r2 = r1plusr2 - r1;
+
+ double z = (r2 - mu2) / sigma2;
+ double gaussian = exp(-(z*z/2.0));
+ sum += curve[j].second * gaussian;
+ }
+
+ double z = (r1 - mu1) / sigma1;
+ double gaussian = exp(-(z*z/2.0));
+ newcurve.push_back(make_pair(r1, gaussian * sum));
+ }
+
+
+ double mu_est, sigma_est;
+ normalize(newcurve);
+ estimate_musigma(newcurve, mu_est, sigma_est);
+ least_squares(newcurve, mu_est, sigma_est, mu, sigma);
+}
+
int main(int argc, char **argv)
{
+ FILE *fp = fopen("fftw-wisdom", "rb");
+ if (fp != NULL) {
+ fftw_import_wisdom_from_file(fp);
+ fclose(fp);
+ }
+
double mu1 = atof(argv[1]);
double sigma1 = atof(argv[2]);
double mu2 = atof(argv[3]);
double sigma2 = atof(argv[4]);
- if (argc > 6) {
+ if (argc > 10) {
+ 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 > 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);
+ 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,
+ 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);
+ 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,
+ newmu2_1-mu3, newmu2_2-mu4);
+ }
+ } else if (argc > 6) {
int score1 = atoi(argv[5]);
int score2 = atoi(argv[6]);
double mu, sigma;
i, k, prob_score(k, i, mu1-mu2), newmu1-mu1, newmu2-mu2);
}
}
+
+ fp = fopen("fftw-wisdom", "wb");
+ if (fp != NULL) {
+ fftw_export_wisdom_to_file(fp);
+ fclose(fp);
+ }
}
projects / foosball / blobdiff