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 double invsq2sigma2 = 1.0 / (sqrt(2.0) * sigma2);
126 for (int i = 0; i < sz; ++i) {
127 double x1 = 0.0 + h*i;
128 double z = (x1 - mu2) * invsq2sigma2;
129 func1[i].real() = exp(-z*z);
131 double x2 = -3000.0 + h*i;
132 func2[(i - sz/2 + sz*2)%(sz*2)].real() = prob_score_real(k, a, binomial_precompute, x2*winfac);
135 result.reserve(sz*2);
140 for (int i = 0; i < sz*2; ++i) {
141 res[i] = func1[i] * func2[i];
146 for (int i = 0; i < sz; ++i) {
148 result.push_back(make_pair(r1, abs(res[i])));
152 // normalize the curve so we know that A ~= 1
153 static void normalize(vector<pair<double, double> > &curve)
156 for (vector<pair<double, double> >::const_iterator i = curve.begin(); i != curve.end(); ++i) {
157 peak = max(peak, i->second);
160 double invpeak = 1.0 / peak;
161 for (vector<pair<double, double> >::iterator i = curve.begin(); i != curve.end(); ++i) {
162 i->second *= invpeak;
166 // computes matA^T * matB
167 static void mat_mul_trans(double *matA, unsigned ah, unsigned aw,
168 double *matB, unsigned bh, unsigned bw,
172 for (unsigned y = 0; y < bw; ++y) {
173 for (unsigned x = 0; x < aw; ++x) {
175 for (unsigned c = 0; c < ah; ++c) {
176 sum += matA[x*ah + c] * matB[y*bh + c];
178 result[y*bw + x] = sum;
183 // solves Ax = B by Gauss-Jordan elimination, where A is an NxN matrix,
184 // x is a column vector of length N and B is a row vector of length N.
185 // Destroys its input in the process.
187 static void solve_matrix(double *A, double *x, double *B)
189 for (int i = 0; i < N; ++i) {
190 for (int j = i+1; j < N; ++j) {
191 // row j -= row i * (a[i,j] / a[i,i])
192 double f = A[j+i*N] / A[i+i*N];
195 for (int k = i+1; k < N; ++k) {
196 A[j+k*N] -= A[i+k*N] * f;
204 for (int i = N; i --> 0; ) {
205 for (int j = i; j --> 0; ) {
206 // row j -= row i * (a[j,j] / a[j,i])
207 double f = A[i+j*N] / A[j+j*N];
215 for (int i = 0; i < N; ++i) {
216 x[i] = B[i] / A[i+i*N];
220 // Give an OK starting estimate for the least squares, by numerical integration
221 // of statistical moments.
222 static void estimate_musigma(vector<pair<double, double> > &curve, double &mu_result, double &sigma_result)
224 double h = (curve.back().first - curve.front().first) / (curve.size() - 1);
226 double area = curve.front().second;
227 double ex = curve.front().first * curve.front().second;
228 double ex2 = curve.front().first * curve.front().first * curve.front().second;
230 for (unsigned i = 1; i < curve.size() - 1; i += 2) {
231 double x = curve[i].first;
232 double y = curve[i].second;
235 ex2 += 4.0 * x * x * y;
237 for (unsigned i = 2; i < curve.size() - 1; i += 2) {
238 double x = curve[i].first;
239 double y = curve[i].second;
242 ex2 += 2.0 * x * x * y;
245 area += curve.back().second;
246 ex += curve.back().first * curve.back().second;
247 ex2 += curve.back().first * curve.back().first * curve.back().second;
249 area = (h/3.0) * area;
250 ex = (h/3.0) * ex / area;
251 ex2 = (h/3.0) * ex2 / area;
254 sigma_result = sqrt(ex2 - ex * ex);
257 // Find best fit of the data in curves to a Gaussian pdf, based on the
258 // given initial estimates. Works by nonlinear least squares, iterating
259 // until we're below a certain threshold.
261 // Note that the algorithm blows up quite hard if the initial estimate is
262 // not good enough. Use estimate_musigma to get a reasonable starting
264 static void least_squares(vector<pair<double, double> > &curve, double mu1, double sigma1, double &mu_result, double &sigma_result)
268 double sigma = sigma1;
271 double matA[curve.size() * 3]; // N x 3
272 double dbeta[curve.size()]; // N x 1
274 // A^T * A: 3xN * Nx3 = 3x3
277 // A^T * dβ: 3xN * Nx1 = 3x1
283 //printf("A=%f mu=%f sigma=%f\n", A, mu, sigma);
285 // fill in A (depends only on x_i, A, mu, sigma -- not y_i)
286 for (unsigned i = 0; i < curve.size(); ++i) {
287 double x = curve[i].first;
290 matA[i + 0 * curve.size()] =
291 exp(-(x-mu)*(x-mu)/(2.0*sigma*sigma));
294 matA[i + 1 * curve.size()] =
295 A * (x-mu)/(sigma*sigma) * matA[i + 0 * curve.size()];
298 matA[i + 2 * curve.size()] =
299 matA[i + 1 * curve.size()] * (x-mu)/sigma;
303 for (unsigned i = 0; i < curve.size(); ++i) {
304 double x = curve[i].first;
305 double y = curve[i].second;
307 dbeta[i] = y - A * exp(- (x-mu)*(x-mu)/(2.0*sigma*sigma));
311 mat_mul_trans(matA, curve.size(), 3, matA, curve.size(), 3, matATA);
312 mat_mul_trans(matA, curve.size(), 3, dbeta, curve.size(), 1, matATdb);
315 solve_matrix<3>(matATA, dlambda, matATdb);
321 // terminate when we're down to three digits
322 if (fabs(dlambda[0]) <= 1e-3 && fabs(dlambda[1]) <= 1e-3 && fabs(dlambda[2]) <= 1e-3)
327 sigma_result = sigma;
330 static void compute_new_rating(double mu1, double sigma1, double mu2, double sigma2, int score1, int score2, double &mu, double &sigma)
332 vector<pair<double, double> > curve;
334 if (score1 > score2) {
335 compute_opponent_rating_pdf(score1, score2, mu2, sigma2, -1.0, curve);
337 compute_opponent_rating_pdf(score2, score1, mu2, sigma2, 1.0, curve);
340 // multiply in the gaussian
341 for (unsigned i = 0; i < curve.size(); ++i) {
342 double r1 = curve[i].first;
343 double z = (r1 - mu1) / sigma1;
344 double gaussian = exp(-(z*z/2.0));
345 curve[i].second *= gaussian;
348 double mu_est, sigma_est;
350 estimate_musigma(curve, mu_est, sigma_est);
351 least_squares(curve, mu_est, sigma_est, mu, sigma);
354 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)
356 vector<pair<double, double> > curve, newcurve;
357 double mu_t = mu3 + mu4;
358 double sigma_t = sqrt(sigma3*sigma3 + sigma4*sigma4);
360 if (score1 > score2) {
361 compute_opponent_rating_pdf(score1, score2, mu_t, sigma_t, -1.0, curve);
363 compute_opponent_rating_pdf(score2, score1, mu_t, sigma_t, 1.0, curve);
366 newcurve.reserve(curve.size());
369 double h = 3000.0 / curve.size();
370 for (unsigned i = 0; i < curve.size(); ++i) {
373 // could be anything, but this is a nice start
374 //double r1 = curve[i].first;
378 double invsq2sigma2 = 1.0 / (sqrt(2.0) * sigma2);
379 for (unsigned j = 0; j < curve.size(); ++j) {
380 double r1plusr2 = curve[j].first;
381 double r2 = r1plusr2 - r1;
383 double z = (r2 - mu2) * invsq2sigma2;
384 double gaussian = exp(-z*z);
385 sum += curve[j].second * gaussian;
388 double z = (r1 - mu1) / sigma1;
389 double gaussian = exp(-(z*z/2.0));
390 newcurve.push_back(make_pair(r1, gaussian * sum));
394 double mu_est, sigma_est;
396 estimate_musigma(newcurve, mu_est, sigma_est);
397 least_squares(newcurve, mu_est, sigma_est, mu, sigma);
400 int main(int argc, char **argv)
402 FILE *fp = fopen("fftw-wisdom", "rb");
404 fftw_import_wisdom_from_file(fp);
408 double mu1 = atof(argv[1]);
409 double sigma1 = atof(argv[2]);
410 double mu2 = atof(argv[3]);
411 double sigma2 = atof(argv[4]);
414 double mu3 = atof(argv[5]);
415 double sigma3 = atof(argv[6]);
416 double mu4 = atof(argv[7]);
417 double sigma4 = atof(argv[8]);
418 int score1 = atoi(argv[9]);
419 int score2 = atoi(argv[10]);
421 compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, score1, score2, mu, sigma);
422 printf("%f %f\n", mu, sigma);
423 } else if (argc > 8) {
424 double mu3 = atof(argv[5]);
425 double sigma3 = atof(argv[6]);
426 double mu4 = atof(argv[7]);
427 double sigma4 = atof(argv[8]);
428 int k = atoi(argv[9]);
430 // assess all possible scores
431 for (int i = 0; i < k; ++i) {
432 double newmu1_1, newmu1_2, newmu2_1, newmu2_2;
433 double newsigma1_1, newsigma1_2, newsigma2_1, newsigma2_2;
434 compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, k, i, newmu1_1, newsigma1_1);
435 compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, k, i, newmu1_2, newsigma1_2);
436 compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, i, k, newmu2_1, newsigma2_1);
437 compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, i, k, newmu2_2, newsigma2_2);
438 printf("%u-%u,%f,%+f,%+f,%+f,%+f\n",
439 k, i, prob_score(k, i, mu3+mu4-(mu1+mu2)), newmu1_1-mu1, newmu1_2-mu2,
440 newmu2_1-mu3, newmu2_2-mu4);
442 for (int i = k; i --> 0; ) {
443 double newmu1_1, newmu1_2, newmu2_1, newmu2_2;
444 double newsigma1_1, newsigma1_2, newsigma2_1, newsigma2_2;
445 compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, i, k, newmu1_1, newsigma1_1);
446 compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, i, k, newmu1_2, newsigma1_2);
447 compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, k, i, newmu2_1, newsigma2_1);
448 compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, k, i, newmu2_2, newsigma2_2);
449 printf("%u-%u,%f,%+f,%+f,%+f,%+f\n",
450 i, k, prob_score(k, i, mu1+mu2-(mu3+mu4)), newmu1_1-mu1, newmu1_2-mu2,
451 newmu2_1-mu3, newmu2_2-mu4);
453 } else if (argc > 6) {
454 int score1 = atoi(argv[5]);
455 int score2 = atoi(argv[6]);
457 compute_new_rating(mu1, sigma1, mu2, sigma2, score1, score2, mu, sigma);
458 printf("%f %f\n", mu, sigma);
460 int k = atoi(argv[5]);
462 // assess all possible scores
463 for (int i = 0; i < k; ++i) {
464 double newmu1, newmu2, newsigma1, newsigma2;
465 compute_new_rating(mu1, sigma1, mu2, sigma2, k, i, newmu1, newsigma1);
466 compute_new_rating(mu2, sigma2, mu1, sigma1, i, k, newmu2, newsigma2);
467 printf("%u-%u,%f,%+f,%+f\n",
468 k, i, prob_score(k, i, mu2-mu1), newmu1-mu1, newmu2-mu2);
470 for (int i = k; i --> 0; ) {
471 double newmu1, newmu2, newsigma1, newsigma2;
472 compute_new_rating(mu1, sigma1, mu2, sigma2, i, k, newmu1, newsigma1);
473 compute_new_rating(mu2, sigma2, mu1, sigma1, k, i, newmu2, newsigma2);
474 printf("%u-%u,%f,%+f,%+f\n",
475 i, k, prob_score(k, i, mu1-mu2), newmu1-mu1, newmu2-mu2);
479 fp = fopen("fftw-wisdom", "wb");
481 fftw_export_wisdom_to_file(fp);