9 static const double int_step_size = 50.0;
10 static const double pdf_step_size = 10.0;
12 // rating constant (see below)
13 static const double rating_constant = 455.0;
17 double prob_score(double a, double rd);
18 double prob_score_real(double a, double prodai, double rd_norm);
19 double prodai(double a);
21 // probability of match ending 10-a when winnerR - loserR = RD
26 // | Poisson[lambda1, t](a) * Erlang[lambda2, 10](t) dt
31 // where lambda1 = 1.0, lambda2 = 2^(rd/455)
33 // The constant of 455 is chosen carefully so to match with the
34 // Glicko/Bradley-Terry assumption that a player rated 400 points over
35 // his/her opponent will win with a probability of 10/11 =~ 0.90909.
37 double prob_score(double a, double rd)
39 return prob_score_real(a, prodai(a), rd/rating_constant);
42 // Same, but takes in Product(a+i, i=1..9) as an argument in addition to a. Faster
43 // if you already have that precomputed, and assumes rd is already divided by 455.
44 double prob_score_real(double a, double prodai, double rd_norm)
47 pow(2.0, -a*rd_norm) * pow(2.0, 10.0*rd_norm) * pow(pow(2.0, -rd_norm) + 1.0, -a)
49 double denom = 362880 * pow(1.0 + pow(2.0, rd_norm), 10.0);
53 // Calculates Product(a+i, i=1..9) (see above).
54 double prodai(double a)
56 return (a+1)*(a+2)*(a+3)*(a+4)*(a+5)*(a+6)*(a+7)*(a+8)*(a+9);
60 // Computes the integral
65 // | ProbScore[a] (r2-r1) Gaussian[mu2, sigma2] (dr2) dr2
70 // For practical reasons, -inf and +inf are replaced by 0 and 3000, which
71 // is reasonable in the this context.
73 // The Gaussian is not normalized.
75 // Set the last parameter to 1.0 if player 1 won, or -1.0 if player 2 won.
76 // In the latter case, ProbScore will be given (r1-r2) instead of (r2-r1).
78 static inline double evaluate_int_point(double a, double prodai_precompute, double r1, double mu2, double sigma2, double winfac, double x);
80 double opponent_rating_pdf(double a, double r1, double mu2, double sigma2, double winfac)
82 double prodai_precompute = prodai(a);
83 winfac /= rating_constant;
85 int n = int(3000.0 / int_step_size + 0.5);
86 double h = 3000.0 / double(n);
87 double sum = evaluate_int_point(a, prodai_precompute, r1, mu2, sigma2, winfac, 0.0);
89 for (int i = 1; i < n; i += 2) {
90 sum += 4.0 * evaluate_int_point(a, prodai_precompute, r1, mu2, sigma2, winfac, i * h);
92 for (int i = 2; i < n; i += 2) {
93 sum += 2.0 * evaluate_int_point(a, prodai_precompute, r1, mu2, sigma2, winfac, i * h);
95 sum += evaluate_int_point(a, prodai_precompute, r1, mu2, sigma2, winfac, 3000.0);
100 static inline double evaluate_int_point(double a, double prodai_precompute, double r1, double mu2, double sigma2, double winfac, double x)
102 double probscore = prob_score_real(a, prodai_precompute, (r1 - x)*winfac);
103 double z = (x - mu2)/sigma2;
104 double gaussian = exp(-(z*z/2.0));
105 return probscore * gaussian;
108 // normalize the curve so we know that A ~= 1
109 void normalize(vector<pair<double, double> > &curve)
112 for (vector<pair<double, double> >::const_iterator i = curve.begin(); i != curve.end(); ++i) {
113 peak = max(peak, i->second);
116 double invpeak = 1.0 / peak;
117 for (vector<pair<double, double> >::iterator i = curve.begin(); i != curve.end(); ++i) {
118 i->second *= invpeak;
122 // computes matA * matB
123 void mat_mul(double *matA, unsigned ah, unsigned aw,
124 double *matB, unsigned bh, unsigned bw,
128 for (unsigned y = 0; y < bw; ++y) {
129 for (unsigned x = 0; x < ah; ++x) {
131 for (unsigned c = 0; c < aw; ++c) {
132 sum += matA[c*ah + x] * matB[y*bh + c];
134 result[y*bw + x] = sum;
139 // computes matA^T * matB
140 void mat_mul_trans(double *matA, unsigned ah, unsigned aw,
141 double *matB, unsigned bh, unsigned bw,
145 for (unsigned y = 0; y < bw; ++y) {
146 for (unsigned x = 0; x < aw; ++x) {
148 for (unsigned c = 0; c < ah; ++c) {
149 sum += matA[x*ah + c] * matB[y*bh + c];
151 result[y*bw + x] = sum;
156 void print3x3(double *M)
158 printf("%f %f %f\n", M[0], M[3], M[6]);
159 printf("%f %f %f\n", M[1], M[4], M[7]);
160 printf("%f %f %f\n", M[2], M[5], M[8]);
163 void print3x1(double *M)
165 printf("%f\n", M[0]);
166 printf("%f\n", M[1]);
167 printf("%f\n", M[2]);
170 // solves Ax = B by Gauss-Jordan elimination, where A is a 3x3 matrix,
171 // x is a column vector of length 3 and B is a row vector of length 3.
172 // Destroys its input in the process.
173 void solve3x3(double *A, double *x, double *B)
175 // row 1 -= row 0 * (a1/a0)
177 double f = A[1] / A[0];
185 // row 2 -= row 0 * (a2/a0)
187 double f = A[2] / A[0];
195 // row 2 -= row 1 * (a5/a4)
197 double f = A[5] / A[4];
206 // row 1 -= row 2 * (a7/a8)
208 double f = A[7] / A[8];
214 // row 0 -= row 2 * (a6/a8)
216 double f = A[6] / A[8];
222 // row 0 -= row 1 * (a3/a4)
224 double f = A[3] / A[4];
236 // Give an OK starting estimate for the least squares, by numerical integration
237 // of statistical moments.
238 void estimate_musigma(vector<pair<double, double> > &curve, double &mu_result, double &sigma_result)
240 double h = (curve.back().first - curve.front().first) / (curve.size() - 1);
242 double area = curve.front().second;
243 double ex = curve.front().first * curve.front().second;
244 double ex2 = curve.front().first * curve.front().first * curve.front().second;
246 for (unsigned i = 1; i < curve.size() - 1; i += 2) {
247 double x = curve[i].first;
248 double y = curve[i].second;
251 ex2 += 4.0 * x * x * y;
253 for (unsigned i = 2; i < curve.size() - 1; i += 2) {
254 double x = curve[i].first;
255 double y = curve[i].second;
258 ex2 += 2.0 * x * x * y;
261 area += curve.back().second;
262 ex += curve.back().first * curve.back().second;
263 ex2 += curve.back().first * curve.back().first * curve.back().second;
265 area = (h/3.0) * area;
266 ex = (h/3.0) * ex / area;
267 ex2 = (h/3.0) * ex2 / area;
270 sigma_result = sqrt(ex2 - ex * ex);
273 // Find best fit of the data in curves to a Gaussian pdf, based on the
274 // given initial estimates. Works by nonlinear least squares, iterating
275 // until we're below a certain threshold.
277 // Note that the algorithm blows up quite hard if the initial estimate is
278 // not good enough. Use estimate_musigma to get a reasonable starting
280 void least_squares(vector<pair<double, double> > &curve, double mu1, double sigma1, double &mu_result, double &sigma_result)
284 double sigma = sigma1;
287 double matA[curve.size() * 3]; // N x 3
288 double dbeta[curve.size()]; // N x 1
290 // A^T * A: 3xN * Nx3 = 3x3
293 // A^T * dβ: 3xN * Nx1 = 3x1
299 //printf("A=%f mu=%f sigma=%f\n", A, mu, sigma);
301 // fill in A (depends only on x_i, A, mu, sigma -- not y_i)
302 for (unsigned i = 0; i < curve.size(); ++i) {
303 double x = curve[i].first;
306 matA[i + 0 * curve.size()] =
307 exp(-(x-mu)*(x-mu)/(2.0*sigma*sigma));
310 matA[i + 1 * curve.size()] =
311 A * (x-mu)/(sigma*sigma) * matA[i + 0 * curve.size()];
314 matA[i + 2 * curve.size()] =
315 matA[i + 1 * curve.size()] * (x-mu)/sigma;
319 for (unsigned i = 0; i < curve.size(); ++i) {
320 double x = curve[i].first;
321 double y = curve[i].second;
323 dbeta[i] = y - A * exp(- (x-mu)*(x-mu)/(2.0*sigma*sigma));
327 mat_mul_trans(matA, curve.size(), 3, matA, curve.size(), 3, matATA);
328 mat_mul_trans(matA, curve.size(), 3, dbeta, curve.size(), 1, matATdb);
331 solve3x3(matATA, dlambda, matATdb);
337 // terminate when we're down to three digits
338 if (fabs(dlambda[0]) <= 1e-3 && fabs(dlambda[1]) <= 1e-3 && fabs(dlambda[2]) <= 1e-3)
343 sigma_result = sigma;
346 void compute_new_rating(double mu1, double sigma1, double mu2, double sigma2, int score1, int score2, double &mu, double &sigma)
348 vector<pair<double, double> > curve;
351 for (double r1 = 0.0; r1 < 3000.0; r1 += pdf_step_size) {
352 double z = (r1 - mu1) / sigma1;
353 double gaussian = exp(-(z*z/2.0));
354 curve.push_back(make_pair(r1, gaussian * opponent_rating_pdf(score2, r1, mu2, sigma2, 1.0)));
357 for (double r1 = 0.0; r1 < 3000.0; r1 += pdf_step_size) {
358 double z = (r1 - mu1) / sigma1;
359 double gaussian = exp(-(z*z/2.0));
360 curve.push_back(make_pair(r1, gaussian * opponent_rating_pdf(score1, r1, mu2, sigma2, -1.0)));
364 double mu_est, sigma_est;
366 estimate_musigma(curve, mu_est, sigma_est);
367 least_squares(curve, mu_est, sigma_est, mu, sigma);
370 int main(int argc, char **argv)
372 double mu1 = atof(argv[1]);
373 double sigma1 = atof(argv[2]);
374 double mu2 = atof(argv[3]);
375 double sigma2 = atof(argv[4]);
378 int score1 = atoi(argv[5]);
379 int score2 = atoi(argv[6]);
381 compute_new_rating(mu1, sigma1, mu2, sigma2, score1, score2, mu, sigma);
382 printf("%f %f\n", mu, sigma);
384 // assess all possible scores
385 for (int i = 0; i <= 9; ++i) {
386 double newmu1, newmu2, newsigma1, newsigma2;
387 compute_new_rating(mu1, sigma1, mu2, sigma2, 10, i, newmu1, newsigma1);
388 compute_new_rating(mu2, sigma2, mu1, sigma1, i, 10, newmu2, newsigma2);
389 printf("10-%u,%f,%+f,%+f\n",
390 i, prob_score(i, mu1-mu2), newmu1-mu1, newmu2-mu2);
392 for (int i = 10; i --> 0; ) {
393 double newmu1, newmu2, newsigma1, newsigma2;
394 compute_new_rating(mu1, sigma1, mu2, sigma2, i, 10, newmu1, newsigma1);
395 compute_new_rating(mu2, sigma2, mu1, sigma1, 10, i, newmu2, newsigma2);
396 printf("%u-10,%f,%+f,%+f\n",
397 i, prob_score(i, mu2-mu1), newmu1-mu1, newmu2-mu2);