8 // integration step size
9 static const double step_size = 10.0;
11 // rating constant (see below)
12 static const double rating_constant = 455.0;
16 double prob_score(double a, double rd);
17 double prob_score_real(double a, double prodai, double rd_norm);
18 double prodai(double a);
20 // probability of match ending 10-a when winnerR - loserR = RD
25 // | Poisson[lambda1, t](a) * Erlang[lambda2, 10](t) dt
30 // where lambda1 = 1.0, lambda2 = 2^(rd/455)
32 // The constant of 455 is chosen carefully so to match with the
33 // Glicko/Bradley-Terry assumption that a player rated 400 points over
34 // his/her opponent will win with a probability of 10/11 =~ 0.90909.
36 double prob_score(double a, double rd)
38 return prob_score_real(a, prodai(a), rd/rating_constant);
41 // Same, but takes in Product(a+i, i=1..9) as an argument in addition to a. Faster
42 // if you already have that precomputed, and assumes rd is already divided by 455.
43 double prob_score_real(double a, double prodai, double rd_norm)
46 pow(2.0, -a*rd_norm) * pow(2.0, 10.0*rd_norm) * pow(pow(2.0, -rd_norm) + 1.0, -a)
48 double denom = 362880 * pow(1.0 + pow(2.0, rd_norm), 10.0);
52 // Calculates Product(a+i, i=1..9) (see above).
53 double prodai(double a)
55 return (a+1)*(a+2)*(a+3)*(a+4)*(a+5)*(a+6)*(a+7)*(a+8)*(a+9);
59 // Computes the integral
64 // | ProbScore[a] (r2-r1) Gaussian[mu2, sigma2] (dr2) dr2
69 // For practical reasons, -inf and +inf are replaced by 0 and 3000, which
70 // is reasonable in the this context.
72 // The Gaussian is not normalized.
74 // Set the last parameter to 1.0 if player 1 won, or -1.0 if player 2 won.
75 // In the latter case, ProbScore will be given (r1-r2) instead of (r2-r1).
77 double opponent_rating_pdf(double a, double r1, double mu2, double sigma2, double winfac)
80 double prodai_precompute = prodai(a);
81 winfac /= rating_constant;
82 for (double r2 = 0.0; r2 < 3000.0; r2 += step_size) {
83 double x = r2 + step_size*0.5;
84 double probscore = prob_score_real(a, prodai_precompute, (r1 - x)*winfac);
85 double z = (x - mu2)/sigma2;
86 double gaussian = exp(-(z*z/2.0));
87 sum += step_size * probscore * gaussian;
92 // normalize the curve so we know that A ~= 1
93 void normalize(vector<pair<double, double> > &curve)
96 for (vector<pair<double, double> >::const_iterator i = curve.begin(); i != curve.end(); ++i) {
97 peak = max(peak, i->second);
100 double invpeak = 1.0 / peak;
101 for (vector<pair<double, double> >::iterator i = curve.begin(); i != curve.end(); ++i) {
102 i->second *= invpeak;
106 // computes matA * matB
107 void mat_mul(double *matA, unsigned ah, unsigned aw,
108 double *matB, unsigned bh, unsigned bw,
112 for (unsigned y = 0; y < bw; ++y) {
113 for (unsigned x = 0; x < ah; ++x) {
115 for (unsigned c = 0; c < aw; ++c) {
116 sum += matA[c*ah + x] * matB[y*bh + c];
118 result[y*bw + x] = sum;
123 // computes matA^T * matB
124 void mat_mul_trans(double *matA, unsigned ah, unsigned aw,
125 double *matB, unsigned bh, unsigned bw,
129 for (unsigned y = 0; y < bw; ++y) {
130 for (unsigned x = 0; x < aw; ++x) {
132 for (unsigned c = 0; c < ah; ++c) {
133 sum += matA[x*ah + c] * matB[y*bh + c];
135 result[y*bw + x] = sum;
140 void print3x3(double *M)
142 printf("%f %f %f\n", M[0], M[3], M[6]);
143 printf("%f %f %f\n", M[1], M[4], M[7]);
144 printf("%f %f %f\n", M[2], M[5], M[8]);
147 void print3x1(double *M)
149 printf("%f\n", M[0]);
150 printf("%f\n", M[1]);
151 printf("%f\n", M[2]);
154 // solves Ax = B by Gauss-Jordan elimination, where A is a 3x3 matrix,
155 // x is a column vector of length 3 and B is a row vector of length 3.
156 // Destroys its input in the process.
157 void solve3x3(double *A, double *x, double *B)
159 // row 1 -= row 0 * (a1/a0)
161 double f = A[1] / A[0];
169 // row 2 -= row 0 * (a2/a0)
171 double f = A[2] / A[0];
179 // row 2 -= row 1 * (a5/a4)
181 double f = A[5] / A[4];
190 // row 1 -= row 2 * (a7/a8)
192 double f = A[7] / A[8];
198 // row 0 -= row 2 * (a6/a8)
200 double f = A[6] / A[8];
206 // row 0 -= row 1 * (a3/a4)
208 double f = A[3] / A[4];
220 // Give an OK starting estimate for the least squares, by numerical integration
221 // of statistical moments.
222 void estimate_musigma(vector<pair<double, double> > &curve, double &mu_result, double &sigma_result)
224 double sum_area = 0.0;
228 for (unsigned i = 1; i < curve.size(); ++i) {
229 double x1 = curve[i].first;
230 double x0 = curve[i-1].first;
231 double y1 = curve[i].second;
232 double y0 = curve[i-1].second;
233 double xm = 0.5 * (x0 + x1);
234 double ym = 0.5 * (y0 + y1);
235 sum_area += (x1-x0) * ym;
236 ex += (x1-x0) * xm * ym;
237 ex2 += (x1-x0) * xm * xm * ym;
244 sigma_result = sqrt(ex2 - ex * ex);
247 // Find best fit of the data in curves to a Gaussian pdf, based on the
248 // given initial estimates. Works by nonlinear least squares, iterating
249 // until we're below a certain threshold.
251 // Note that the algorithm blows up quite hard if the initial estimate is
252 // not good enough. Use estimate_musigma to get a reasonable starting
254 void least_squares(vector<pair<double, double> > &curve, double mu1, double sigma1, double &mu_result, double &sigma_result)
258 double sigma = sigma1;
261 double matA[curve.size() * 3]; // N x 3
262 double dbeta[curve.size()]; // N x 1
264 // A^T * A: 3xN * Nx3 = 3x3
267 // A^T * dβ: 3xN * Nx1 = 3x1
273 //printf("A=%f mu=%f sigma=%f\n", A, mu, sigma);
275 // fill in A (depends only on x_i, A, mu, sigma -- not y_i)
276 for (unsigned i = 0; i < curve.size(); ++i) {
277 double x = curve[i].first;
280 matA[i + 0 * curve.size()] =
281 exp(-(x-mu)*(x-mu)/(2.0*sigma*sigma));
284 matA[i + 1 * curve.size()] =
285 A * (x-mu)/(sigma*sigma) * matA[i + 0 * curve.size()];
288 matA[i + 2 * curve.size()] =
289 matA[i + 1 * curve.size()] * (x-mu)/sigma;
293 for (unsigned i = 0; i < curve.size(); ++i) {
294 double x = curve[i].first;
295 double y = curve[i].second;
297 dbeta[i] = y - A * exp(- (x-mu)*(x-mu)/(2.0*sigma*sigma));
301 mat_mul_trans(matA, curve.size(), 3, matA, curve.size(), 3, matATA);
302 mat_mul_trans(matA, curve.size(), 3, dbeta, curve.size(), 1, matATdb);
305 solve3x3(matATA, dlambda, matATdb);
311 // terminate when we're down to three digits
312 if (fabs(dlambda[0]) <= 1e-3 && fabs(dlambda[1]) <= 1e-3 && fabs(dlambda[2]) <= 1e-3)
317 sigma_result = sigma;
320 int main(int argc, char **argv)
322 double mu1 = atof(argv[1]);
323 double sigma1 = atof(argv[2]);
324 double mu2 = atof(argv[3]);
325 double sigma2 = atof(argv[4]);
326 int score1 = atoi(argv[5]);
327 int score2 = atoi(argv[6]);
328 vector<pair<double, double> > curve;
331 for (double r1 = 0.0; r1 < 3000.0; r1 += step_size) {
332 double z = (r1 - mu1) / sigma1;
333 double gaussian = exp(-(z*z/2.0));
334 curve.push_back(make_pair(r1, gaussian * opponent_rating_pdf(score2, r1, mu2, sigma2, 1.0)));
337 for (double r1 = 0.0; r1 < 3000.0; r1 += step_size) {
338 double z = (r1 - mu1) / sigma1;
339 double gaussian = exp(-(z*z/2.0));
340 curve.push_back(make_pair(r1, gaussian * opponent_rating_pdf(score1, r1, mu2, sigma2, -1.0)));
344 double mu_est, sigma_est, mu, sigma;
346 estimate_musigma(curve, mu_est, sigma_est);
347 least_squares(curve, mu_est, sigma_est, mu, sigma);
348 printf("%f %f\n", mu, sigma);