12 static const double int_step_size = 75.0;
14 // rating constant (see below)
15 static const double rating_constant = 455.0;
19 double prob_score(int k, int a, double rd);
20 double prob_score_real(int k, int a, double binomial, double rd_norm);
21 double prodai(int k, int a);
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 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 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 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 double prodai(int k, int a)
77 for (int i = 1; i < k; ++i)
85 for (int i = 2; i <= x; ++i)
90 void convolve(int size)
94 void compute_opponent_rating_pdf(int k, int a, double mu2, double sigma2, double winfac, vector<pair<double, double> > &result)
96 double binomial_precompute = prodai(k, a) / fac(k-1);
97 winfac /= rating_constant;
99 int sz = (6000.0 - 0.0) / int_step_size;
100 double h = (6000.0 - 0.0) / sz;
103 complex<double> *func1, *func2, *res;
105 func1 = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
106 func2 = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
107 res = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
108 f1 = fftw_plan_dft_1d(sz*2,
109 reinterpret_cast<fftw_complex*>(func1),
110 reinterpret_cast<fftw_complex*>(func1),
113 f2 = fftw_plan_dft_1d(sz*2,
114 reinterpret_cast<fftw_complex*>(func2),
115 reinterpret_cast<fftw_complex*>(func2),
118 b = fftw_plan_dft_1d(sz*2,
119 reinterpret_cast<fftw_complex*>(res),
120 reinterpret_cast<fftw_complex*>(res),
125 for (int i = 0; i < sz*2; ++i) {
126 func1[i].real() = func1[i].imag() = func2[i].real() = func2[i].imag() = 0.0;
129 for (int i = 0; i < sz; ++i) {
130 double x1 = 0.0 + h*i;
131 double z = (x1 - mu2)/sigma2;
132 func1[i].real() = exp(-(z*z/2.0));
134 double x2 = -3000.0 + h*i;
135 func2[(i - sz/2 + sz*2)%(sz*2)].real() = prob_score_real(k, a, binomial_precompute, x2*winfac);
138 result.reserve(sz*2);
143 for (int i = 0; i < sz*2; ++i) {
144 res[i] = func1[i] * func2[i];
147 for (int i = 0; i < sz; ++i) {
149 result.push_back(make_pair(r1, abs(res[i])));
153 // normalize the curve so we know that A ~= 1
154 void normalize(vector<pair<double, double> > &curve)
157 for (vector<pair<double, double> >::const_iterator i = curve.begin(); i != curve.end(); ++i) {
158 peak = max(peak, i->second);
161 double invpeak = 1.0 / peak;
162 for (vector<pair<double, double> >::iterator i = curve.begin(); i != curve.end(); ++i) {
163 i->second *= invpeak;
167 // computes matA * matB
168 void mat_mul(double *matA, unsigned ah, unsigned aw,
169 double *matB, unsigned bh, unsigned bw,
173 for (unsigned y = 0; y < bw; ++y) {
174 for (unsigned x = 0; x < ah; ++x) {
176 for (unsigned c = 0; c < aw; ++c) {
177 sum += matA[c*ah + x] * matB[y*bh + c];
179 result[y*bw + x] = sum;
184 // computes matA^T * matB
185 void mat_mul_trans(double *matA, unsigned ah, unsigned aw,
186 double *matB, unsigned bh, unsigned bw,
190 for (unsigned y = 0; y < bw; ++y) {
191 for (unsigned x = 0; x < aw; ++x) {
193 for (unsigned c = 0; c < ah; ++c) {
194 sum += matA[x*ah + c] * matB[y*bh + c];
196 result[y*bw + x] = sum;
201 void print3x3(double *M)
203 printf("%f %f %f\n", M[0], M[3], M[6]);
204 printf("%f %f %f\n", M[1], M[4], M[7]);
205 printf("%f %f %f\n", M[2], M[5], M[8]);
208 void print3x1(double *M)
210 printf("%f\n", M[0]);
211 printf("%f\n", M[1]);
212 printf("%f\n", M[2]);
215 // solves Ax = B by Gauss-Jordan elimination, where A is a 3x3 matrix,
216 // x is a column vector of length 3 and B is a row vector of length 3.
217 // Destroys its input in the process.
218 void solve3x3(double *A, double *x, double *B)
220 // row 1 -= row 0 * (a1/a0)
222 double f = A[1] / A[0];
230 // row 2 -= row 0 * (a2/a0)
232 double f = A[2] / A[0];
240 // row 2 -= row 1 * (a5/a4)
242 double f = A[5] / A[4];
251 // row 1 -= row 2 * (a7/a8)
253 double f = A[7] / A[8];
259 // row 0 -= row 2 * (a6/a8)
261 double f = A[6] / A[8];
267 // row 0 -= row 1 * (a3/a4)
269 double f = A[3] / A[4];
281 // Give an OK starting estimate for the least squares, by numerical integration
282 // of statistical moments.
283 void estimate_musigma(vector<pair<double, double> > &curve, double &mu_result, double &sigma_result)
285 double h = (curve.back().first - curve.front().first) / (curve.size() - 1);
287 double area = curve.front().second;
288 double ex = curve.front().first * curve.front().second;
289 double ex2 = curve.front().first * curve.front().first * curve.front().second;
291 for (unsigned i = 1; i < curve.size() - 1; i += 2) {
292 double x = curve[i].first;
293 double y = curve[i].second;
296 ex2 += 4.0 * x * x * y;
298 for (unsigned i = 2; i < curve.size() - 1; i += 2) {
299 double x = curve[i].first;
300 double y = curve[i].second;
303 ex2 += 2.0 * x * x * y;
306 area += curve.back().second;
307 ex += curve.back().first * curve.back().second;
308 ex2 += curve.back().first * curve.back().first * curve.back().second;
310 area = (h/3.0) * area;
311 ex = (h/3.0) * ex / area;
312 ex2 = (h/3.0) * ex2 / area;
315 sigma_result = sqrt(ex2 - ex * ex);
318 // Find best fit of the data in curves to a Gaussian pdf, based on the
319 // given initial estimates. Works by nonlinear least squares, iterating
320 // until we're below a certain threshold.
322 // Note that the algorithm blows up quite hard if the initial estimate is
323 // not good enough. Use estimate_musigma to get a reasonable starting
325 void least_squares(vector<pair<double, double> > &curve, double mu1, double sigma1, double &mu_result, double &sigma_result)
329 double sigma = sigma1;
332 double matA[curve.size() * 3]; // N x 3
333 double dbeta[curve.size()]; // N x 1
335 // A^T * A: 3xN * Nx3 = 3x3
338 // A^T * dβ: 3xN * Nx1 = 3x1
344 //printf("A=%f mu=%f sigma=%f\n", A, mu, sigma);
346 // fill in A (depends only on x_i, A, mu, sigma -- not y_i)
347 for (unsigned i = 0; i < curve.size(); ++i) {
348 double x = curve[i].first;
351 matA[i + 0 * curve.size()] =
352 exp(-(x-mu)*(x-mu)/(2.0*sigma*sigma));
355 matA[i + 1 * curve.size()] =
356 A * (x-mu)/(sigma*sigma) * matA[i + 0 * curve.size()];
359 matA[i + 2 * curve.size()] =
360 matA[i + 1 * curve.size()] * (x-mu)/sigma;
364 for (unsigned i = 0; i < curve.size(); ++i) {
365 double x = curve[i].first;
366 double y = curve[i].second;
368 dbeta[i] = y - A * exp(- (x-mu)*(x-mu)/(2.0*sigma*sigma));
372 mat_mul_trans(matA, curve.size(), 3, matA, curve.size(), 3, matATA);
373 mat_mul_trans(matA, curve.size(), 3, dbeta, curve.size(), 1, matATdb);
376 solve3x3(matATA, dlambda, matATdb);
382 // terminate when we're down to three digits
383 if (fabs(dlambda[0]) <= 1e-3 && fabs(dlambda[1]) <= 1e-3 && fabs(dlambda[2]) <= 1e-3)
388 sigma_result = sigma;
391 void compute_new_rating(double mu1, double sigma1, double mu2, double sigma2, int score1, int score2, double &mu, double &sigma)
393 vector<pair<double, double> > curve;
395 if (score1 > score2) {
396 compute_opponent_rating_pdf(score1, score2, mu2, sigma2, -1.0, curve);
398 compute_opponent_rating_pdf(score2, score1, mu2, sigma2, 1.0, curve);
401 // multiply in the gaussian
402 for (unsigned i = 0; i < curve.size(); ++i) {
403 double r1 = curve[i].first;
404 double z = (r1 - mu1) / sigma1;
405 double gaussian = exp(-(z*z/2.0));
406 curve[i].second *= gaussian;
409 double mu_est, sigma_est;
411 estimate_musigma(curve, mu_est, sigma_est);
412 least_squares(curve, mu_est, sigma_est, mu, sigma);
415 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)
417 vector<pair<double, double> > curve, newcurve;
418 double mu_t = mu3 + mu4;
419 double sigma_t = sqrt(sigma3*sigma3 + sigma4*sigma4);
421 if (score1 > score2) {
422 compute_opponent_rating_pdf(score1, score2, mu_t, sigma_t, -1.0, curve);
424 compute_opponent_rating_pdf(score2, score1, mu_t, sigma_t, 1.0, curve);
428 double h = 3000.0 / curve.size();
429 for (unsigned i = 0; i < curve.size(); ++i) {
432 // could be anything, but this is a nice start
433 //double r1 = curve[i].first;
437 for (unsigned j = 0; j < curve.size(); ++j) {
438 double r1plusr2 = curve[j].first;
439 double r2 = r1plusr2 - r1;
441 double z = (r2 - mu2) / sigma2;
442 double gaussian = exp(-(z*z/2.0));
443 sum += curve[j].second * gaussian;
446 double z = (r1 - mu1) / sigma1;
447 double gaussian = exp(-(z*z/2.0));
448 newcurve.push_back(make_pair(r1, gaussian * sum));
452 double mu_est, sigma_est;
454 estimate_musigma(newcurve, mu_est, sigma_est);
455 least_squares(newcurve, mu_est, sigma_est, mu, sigma);
458 int main(int argc, char **argv)
460 FILE *fp = fopen("fftw-wisdom", "rb");
462 fftw_import_wisdom_from_file(fp);
466 double mu1 = atof(argv[1]);
467 double sigma1 = atof(argv[2]);
468 double mu2 = atof(argv[3]);
469 double sigma2 = atof(argv[4]);
472 double mu3 = atof(argv[5]);
473 double sigma3 = atof(argv[6]);
474 double mu4 = atof(argv[7]);
475 double sigma4 = atof(argv[8]);
476 int score1 = atoi(argv[9]);
477 int score2 = atoi(argv[10]);
479 compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, score1, score2, mu, sigma);
480 printf("%f %f\n", mu, sigma);
481 } else if (argc > 8) {
482 double mu3 = atof(argv[5]);
483 double sigma3 = atof(argv[6]);
484 double mu4 = atof(argv[7]);
485 double sigma4 = atof(argv[8]);
486 int k = atoi(argv[9]);
488 // assess all possible scores
489 for (int i = 0; i < k; ++i) {
490 double newmu1_1, newmu1_2, newmu2_1, newmu2_2;
491 double newsigma1_1, newsigma1_2, newsigma2_1, newsigma2_2;
492 compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, k, i, newmu1_1, newsigma1_1);
493 compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, k, i, newmu1_2, newsigma1_2);
494 compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, i, k, newmu2_1, newsigma2_1);
495 compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, i, k, newmu2_2, newsigma2_2);
496 printf("%u-%u,%f,%+f,%+f,%+f,%+f\n",
497 k, i, prob_score(k, i, mu3+mu4-(mu1+mu2)), newmu1_1-mu1, newmu1_2-mu2,
498 newmu2_1-mu3, newmu2_2-mu4);
500 for (int i = k; i --> 0; ) {
501 double newmu1_1, newmu1_2, newmu2_1, newmu2_2;
502 double newsigma1_1, newsigma1_2, newsigma2_1, newsigma2_2;
503 compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, i, k, newmu1_1, newsigma1_1);
504 compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, i, k, newmu1_2, newsigma1_2);
505 compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, k, i, newmu2_1, newsigma2_1);
506 compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, k, i, newmu2_2, newsigma2_2);
507 printf("%u-%u,%f,%+f,%+f,%+f,%+f\n",
508 i, k, prob_score(k, i, mu1+mu2-(mu3+mu4)), newmu1_1-mu1, newmu1_2-mu2,
509 newmu2_1-mu3, newmu2_2-mu4);
511 } else if (argc > 6) {
512 int score1 = atoi(argv[5]);
513 int score2 = atoi(argv[6]);
515 compute_new_rating(mu1, sigma1, mu2, sigma2, score1, score2, mu, sigma);
516 printf("%f %f\n", mu, sigma);
518 int k = atoi(argv[5]);
520 // assess all possible scores
521 for (int i = 0; i < k; ++i) {
522 double newmu1, newmu2, newsigma1, newsigma2;
523 compute_new_rating(mu1, sigma1, mu2, sigma2, k, i, newmu1, newsigma1);
524 compute_new_rating(mu2, sigma2, mu1, sigma1, i, k, newmu2, newsigma2);
525 printf("%u-%u,%f,%+f,%+f\n",
526 k, i, prob_score(k, i, mu2-mu1), newmu1-mu1, newmu2-mu2);
528 for (int i = k; i --> 0; ) {
529 double newmu1, newmu2, newsigma1, newsigma2;
530 compute_new_rating(mu1, sigma1, mu2, sigma2, i, k, newmu1, newsigma1);
531 compute_new_rating(mu2, sigma2, mu1, sigma1, k, i, newmu2, newsigma2);
532 printf("%u-%u,%f,%+f,%+f\n",
533 i, k, prob_score(k, i, mu1-mu2), newmu1-mu1, newmu2-mu2);
537 fp = fopen("fftw-wisdom", "wb");
539 fftw_export_wisdom_to_file(fp);