12 static const double int_step_size = 75.0;
14 // rating constant (see below)
15 static const double rating_constant = 455.0;
19 static double prob_score(int k, int a, double rd);
20 static double prob_score_real(int k, int a, double binomial, double rd_norm);
21 static double prodai(int k, int a);
22 static double fac(int x);
25 // probability of match ending k-a (k>a) when winnerR - loserR = RD
30 // | Poisson[lambda1, t](a) * Erlang[lambda2, k](t) dt
35 // where lambda1 = 1.0, lambda2 = 2^(rd/455)
37 // The constant of 455 is chosen carefully so to match with the
38 // Glicko/Bradley-Terry assumption that a player rated 400 points over
39 // his/her opponent will win with a probability of 10/11 =~ 0.90909.
41 static double prob_score(int k, int a, double rd)
43 return prob_score_real(k, a, prodai(k, a) / fac(k-1), rd/rating_constant);
46 // computes x^a, probably more efficiently than pow(x, a) (but requires that a
47 // is n unsigned integer)
48 static double intpow(double x, unsigned a)
63 // Same, but takes in binomial(a+k-1, k-1) as an argument in
64 // addition to a. Faster if you already have that precomputed, and assumes rd
65 // is already divided by 455.
66 static double prob_score_real(int k, int a, double binomial, double rd_norm)
68 double nom = binomial * intpow(pow(2.0, rd_norm), a);
69 double denom = intpow(1.0 + pow(2.0, rd_norm), k+a);
73 // Calculates Product(a+i, i=1..k-1) (see above).
74 static double prodai(int k, int a)
77 for (int i = 1; i < k; ++i)
82 static double fac(int x)
85 for (int i = 2; i <= x; ++i)
90 static void compute_opponent_rating_pdf(int k, int a, double mu2, double sigma2, double winfac, vector<pair<double, double> > &result)
92 double binomial_precompute = prodai(k, a) / fac(k-1);
93 winfac /= rating_constant;
95 int sz = (6000.0 - 0.0) / int_step_size;
96 double h = (6000.0 - 0.0) / sz;
99 complex<double> *func1, *func2, *res;
101 func1 = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
102 func2 = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
103 res = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
104 f1 = fftw_plan_dft_1d(sz*2,
105 reinterpret_cast<fftw_complex*>(func1),
106 reinterpret_cast<fftw_complex*>(func1),
109 f2 = fftw_plan_dft_1d(sz*2,
110 reinterpret_cast<fftw_complex*>(func2),
111 reinterpret_cast<fftw_complex*>(func2),
114 b = fftw_plan_dft_1d(sz*2,
115 reinterpret_cast<fftw_complex*>(res),
116 reinterpret_cast<fftw_complex*>(res),
121 for (int i = 0; i < sz*2; ++i) {
122 func1[i].real() = func1[i].imag() = func2[i].real() = func2[i].imag() = 0.0;
125 for (int i = 0; i < sz; ++i) {
126 double x1 = 0.0 + h*i;
127 double z = (x1 - mu2)/sigma2;
128 func1[i].real() = exp(-(z*z/2.0));
130 double x2 = -3000.0 + h*i;
131 func2[(i - sz/2 + sz*2)%(sz*2)].real() = prob_score_real(k, a, binomial_precompute, x2*winfac);
134 result.reserve(sz*2);
139 for (int i = 0; i < sz*2; ++i) {
140 res[i] = func1[i] * func2[i];
143 for (int i = 0; i < sz; ++i) {
145 result.push_back(make_pair(r1, abs(res[i])));
149 // normalize the curve so we know that A ~= 1
150 static void normalize(vector<pair<double, double> > &curve)
153 for (vector<pair<double, double> >::const_iterator i = curve.begin(); i != curve.end(); ++i) {
154 peak = max(peak, i->second);
157 double invpeak = 1.0 / peak;
158 for (vector<pair<double, double> >::iterator i = curve.begin(); i != curve.end(); ++i) {
159 i->second *= invpeak;
163 // computes matA^T * matB
164 static void mat_mul_trans(double *matA, unsigned ah, unsigned aw,
165 double *matB, unsigned bh, unsigned bw,
169 for (unsigned y = 0; y < bw; ++y) {
170 for (unsigned x = 0; x < aw; ++x) {
172 for (unsigned c = 0; c < ah; ++c) {
173 sum += matA[x*ah + c] * matB[y*bh + c];
175 result[y*bw + x] = sum;
180 // solves Ax = B by Gauss-Jordan elimination, where A is a 3x3 matrix,
181 // x is a column vector of length 3 and B is a row vector of length 3.
182 // Destroys its input in the process.
183 static void solve3x3(double *A, double *x, double *B)
185 // row 1 -= row 0 * (a1/a0)
187 double f = A[1] / A[0];
195 // row 2 -= row 0 * (a2/a0)
197 double f = A[2] / A[0];
205 // row 2 -= row 1 * (a5/a4)
207 double f = A[5] / A[4];
216 // row 1 -= row 2 * (a7/a8)
218 double f = A[7] / A[8];
224 // row 0 -= row 2 * (a6/a8)
226 double f = A[6] / A[8];
232 // row 0 -= row 1 * (a3/a4)
234 double f = A[3] / A[4];
246 // Give an OK starting estimate for the least squares, by numerical integration
247 // of statistical moments.
248 static void estimate_musigma(vector<pair<double, double> > &curve, double &mu_result, double &sigma_result)
250 double h = (curve.back().first - curve.front().first) / (curve.size() - 1);
252 double area = curve.front().second;
253 double ex = curve.front().first * curve.front().second;
254 double ex2 = curve.front().first * curve.front().first * curve.front().second;
256 for (unsigned i = 1; i < curve.size() - 1; i += 2) {
257 double x = curve[i].first;
258 double y = curve[i].second;
261 ex2 += 4.0 * x * x * y;
263 for (unsigned i = 2; i < curve.size() - 1; i += 2) {
264 double x = curve[i].first;
265 double y = curve[i].second;
268 ex2 += 2.0 * x * x * y;
271 area += curve.back().second;
272 ex += curve.back().first * curve.back().second;
273 ex2 += curve.back().first * curve.back().first * curve.back().second;
275 area = (h/3.0) * area;
276 ex = (h/3.0) * ex / area;
277 ex2 = (h/3.0) * ex2 / area;
280 sigma_result = sqrt(ex2 - ex * ex);
283 // Find best fit of the data in curves to a Gaussian pdf, based on the
284 // given initial estimates. Works by nonlinear least squares, iterating
285 // until we're below a certain threshold.
287 // Note that the algorithm blows up quite hard if the initial estimate is
288 // not good enough. Use estimate_musigma to get a reasonable starting
290 static void least_squares(vector<pair<double, double> > &curve, double mu1, double sigma1, double &mu_result, double &sigma_result)
294 double sigma = sigma1;
297 double matA[curve.size() * 3]; // N x 3
298 double dbeta[curve.size()]; // N x 1
300 // A^T * A: 3xN * Nx3 = 3x3
303 // A^T * dβ: 3xN * Nx1 = 3x1
309 //printf("A=%f mu=%f sigma=%f\n", A, mu, sigma);
311 // fill in A (depends only on x_i, A, mu, sigma -- not y_i)
312 for (unsigned i = 0; i < curve.size(); ++i) {
313 double x = curve[i].first;
316 matA[i + 0 * curve.size()] =
317 exp(-(x-mu)*(x-mu)/(2.0*sigma*sigma));
320 matA[i + 1 * curve.size()] =
321 A * (x-mu)/(sigma*sigma) * matA[i + 0 * curve.size()];
324 matA[i + 2 * curve.size()] =
325 matA[i + 1 * curve.size()] * (x-mu)/sigma;
329 for (unsigned i = 0; i < curve.size(); ++i) {
330 double x = curve[i].first;
331 double y = curve[i].second;
333 dbeta[i] = y - A * exp(- (x-mu)*(x-mu)/(2.0*sigma*sigma));
337 mat_mul_trans(matA, curve.size(), 3, matA, curve.size(), 3, matATA);
338 mat_mul_trans(matA, curve.size(), 3, dbeta, curve.size(), 1, matATdb);
341 solve3x3(matATA, dlambda, matATdb);
347 // terminate when we're down to three digits
348 if (fabs(dlambda[0]) <= 1e-3 && fabs(dlambda[1]) <= 1e-3 && fabs(dlambda[2]) <= 1e-3)
353 sigma_result = sigma;
356 static void compute_new_rating(double mu1, double sigma1, double mu2, double sigma2, int score1, int score2, double &mu, double &sigma)
358 vector<pair<double, double> > curve;
360 if (score1 > score2) {
361 compute_opponent_rating_pdf(score1, score2, mu2, sigma2, -1.0, curve);
363 compute_opponent_rating_pdf(score2, score1, mu2, sigma2, 1.0, curve);
366 // multiply in the gaussian
367 for (unsigned i = 0; i < curve.size(); ++i) {
368 double r1 = curve[i].first;
369 double z = (r1 - mu1) / sigma1;
370 double gaussian = exp(-(z*z/2.0));
371 curve[i].second *= gaussian;
374 double mu_est, sigma_est;
376 estimate_musigma(curve, mu_est, sigma_est);
377 least_squares(curve, mu_est, sigma_est, mu, sigma);
380 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)
382 vector<pair<double, double> > curve, newcurve;
383 double mu_t = mu3 + mu4;
384 double sigma_t = sqrt(sigma3*sigma3 + sigma4*sigma4);
386 if (score1 > score2) {
387 compute_opponent_rating_pdf(score1, score2, mu_t, sigma_t, -1.0, curve);
389 compute_opponent_rating_pdf(score2, score1, mu_t, sigma_t, 1.0, curve);
393 double h = 3000.0 / curve.size();
394 for (unsigned i = 0; i < curve.size(); ++i) {
397 // could be anything, but this is a nice start
398 //double r1 = curve[i].first;
402 for (unsigned j = 0; j < curve.size(); ++j) {
403 double r1plusr2 = curve[j].first;
404 double r2 = r1plusr2 - r1;
406 double z = (r2 - mu2) / sigma2;
407 double gaussian = exp(-(z*z/2.0));
408 sum += curve[j].second * gaussian;
411 double z = (r1 - mu1) / sigma1;
412 double gaussian = exp(-(z*z/2.0));
413 newcurve.push_back(make_pair(r1, gaussian * sum));
417 double mu_est, sigma_est;
419 estimate_musigma(newcurve, mu_est, sigma_est);
420 least_squares(newcurve, mu_est, sigma_est, mu, sigma);
423 int main(int argc, char **argv)
425 FILE *fp = fopen("fftw-wisdom", "rb");
427 fftw_import_wisdom_from_file(fp);
431 double mu1 = atof(argv[1]);
432 double sigma1 = atof(argv[2]);
433 double mu2 = atof(argv[3]);
434 double sigma2 = atof(argv[4]);
437 double mu3 = atof(argv[5]);
438 double sigma3 = atof(argv[6]);
439 double mu4 = atof(argv[7]);
440 double sigma4 = atof(argv[8]);
441 int score1 = atoi(argv[9]);
442 int score2 = atoi(argv[10]);
444 compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, score1, score2, mu, sigma);
445 printf("%f %f\n", mu, sigma);
446 } else if (argc > 8) {
447 double mu3 = atof(argv[5]);
448 double sigma3 = atof(argv[6]);
449 double mu4 = atof(argv[7]);
450 double sigma4 = atof(argv[8]);
451 int k = atoi(argv[9]);
453 // assess all possible scores
454 for (int i = 0; i < k; ++i) {
455 double newmu1_1, newmu1_2, newmu2_1, newmu2_2;
456 double newsigma1_1, newsigma1_2, newsigma2_1, newsigma2_2;
457 compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, k, i, newmu1_1, newsigma1_1);
458 compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, k, i, newmu1_2, newsigma1_2);
459 compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, i, k, newmu2_1, newsigma2_1);
460 compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, i, k, newmu2_2, newsigma2_2);
461 printf("%u-%u,%f,%+f,%+f,%+f,%+f\n",
462 k, i, prob_score(k, i, mu3+mu4-(mu1+mu2)), newmu1_1-mu1, newmu1_2-mu2,
463 newmu2_1-mu3, newmu2_2-mu4);
465 for (int i = k; i --> 0; ) {
466 double newmu1_1, newmu1_2, newmu2_1, newmu2_2;
467 double newsigma1_1, newsigma1_2, newsigma2_1, newsigma2_2;
468 compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, i, k, newmu1_1, newsigma1_1);
469 compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, i, k, newmu1_2, newsigma1_2);
470 compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, k, i, newmu2_1, newsigma2_1);
471 compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, k, i, newmu2_2, newsigma2_2);
472 printf("%u-%u,%f,%+f,%+f,%+f,%+f\n",
473 i, k, prob_score(k, i, mu1+mu2-(mu3+mu4)), newmu1_1-mu1, newmu1_2-mu2,
474 newmu2_1-mu3, newmu2_2-mu4);
476 } else if (argc > 6) {
477 int score1 = atoi(argv[5]);
478 int score2 = atoi(argv[6]);
480 compute_new_rating(mu1, sigma1, mu2, sigma2, score1, score2, mu, sigma);
481 printf("%f %f\n", mu, sigma);
483 int k = atoi(argv[5]);
485 // assess all possible scores
486 for (int i = 0; i < k; ++i) {
487 double newmu1, newmu2, newsigma1, newsigma2;
488 compute_new_rating(mu1, sigma1, mu2, sigma2, k, i, newmu1, newsigma1);
489 compute_new_rating(mu2, sigma2, mu1, sigma1, i, k, newmu2, newsigma2);
490 printf("%u-%u,%f,%+f,%+f\n",
491 k, i, prob_score(k, i, mu2-mu1), newmu1-mu1, newmu2-mu2);
493 for (int i = k; i --> 0; ) {
494 double newmu1, newmu2, newsigma1, newsigma2;
495 compute_new_rating(mu1, sigma1, mu2, sigma2, i, k, newmu1, newsigma1);
496 compute_new_rating(mu2, sigma2, mu1, sigma1, k, i, newmu2, newsigma2);
497 printf("%u-%u,%f,%+f,%+f\n",
498 i, k, prob_score(k, i, mu1-mu2), newmu1-mu1, newmu2-mu2);
502 fp = fopen("fftw-wisdom", "wb");
504 fftw_export_wisdom_to_file(fp);