13 #define MAX_PLAYERS 4096
15 float mu[MAX_PLAYERS];
16 float sigma[MAX_PLAYERS];
21 * L(mu_vec, sigma_vec, matches) = product[ L(mu_A, sigma_A, mu_B, sigma_B, score_AB - score_BA) ]
22 * log-likelihood = sum[ log( L(mu_A, sigma_A, mu_B, sigma_B, score_AB - score_BA) ) ]
24 * L(mu1, sigma1, mu2, sigma2, score2 - score1) = sigmoid(mu2 - mu1, sqrt(sigma1² + sigma2²), (score2 - score1))
26 * pdf := 1/(sigma * sqrt(2*Pi)) * exp(-(x - mu)^2 / (2 * sigma^2));
27 * pdfs := subs({ mu = mu1 - mu2, sigma = sqrt(sigma1^2 + sigma2^2) }, pdf);
28 * diff(log(pdfs), mu1);
36 map<int, vector<match> > matches_for_player;
38 void dump_scores(const vector<string> &players, const float *mu, const float *sigma, int num_players)
41 for (int i = 0; i < num_players; ++i) {
42 printf("%s=[%5.1f, %4.1f] ", players[i].c_str(), mu[i], sigma[i]);
46 for (int i = 0; i < num_players; ++i) {
47 printf("%5.1f ", mu[i]);
51 for (int i = 0; i < num_players; ++i) {
52 printf("%5.1f %s\n", mu[i], players[i].c_str());
59 * diff(logL, mu1) = -w * (mu1 - mu2 - x) / sigma_c^2
60 * maximizer for mu1 is given by: sum_i[ (w_i/sigma_c_i)^2 (mu1 - mu2_i - x_i) ] = 0
61 * sum_i[ (w_i/sigma_c_i)^2 mu1 ] = sum_i [ (w_i/sigma_c_i)^2 ( mu2_i + x_i ) ]
62 * mu1 = sum_i [ (w_i/sigma_c_i)^2 ( mu2_i + x_i ) ] / sum_i[ (w_i/sigma_c_i)^2 ]
64 void update_mu(float *mu, float *sigma, int player_num, const vector<match> &matches)
66 if (matches.empty()) {
70 float nom = 0.0f, denom = 0.0f;
71 for (unsigned i = 0; i < matches.size(); ++i) {
72 float sigma1 = sigma[player_num];
73 float sigma2 = sigma[matches[i].other_player];
74 float inv_sigma_c2 = matches[i].weight / (sigma1 * sigma1 + sigma2 * sigma2);
75 float x = matches[i].margin; // / 70.0f;
77 nom += (mu[matches[i].other_player] + x) * inv_sigma_c2;
78 denom += inv_sigma_c2;
80 mu[player_num] = nom / denom;
84 * diff(logL, sigma1) = sigma1 (-sigma1² - sigma2² + (x - mu)²) / sigma_c²
85 * maximizer for sigma1 is given by: sum_i[ (1/sigma_c_i)² sigma1 ((x - mu)² - (sigma1² + sigma2²) ] = 0
86 * sum_i[ (x - mu)² - sigma1² - sigma2² ] = 0 |: sigma1 != 0, sigma2 != 0
87 * sum_i[ (x - mu)² - sigma2² ] = sum[ sigma1² ]
88 * sigma1 = sqrt( sum_i[ (x - mu)² - sigma2² ] / N )
90 void update_sigma(float *mu, float *sigma, int player_num, const vector<match> &matches)
92 if (matches.size() < 2) {
97 for (unsigned i = 0; i < matches.size(); ++i) {
98 float mu1 = mu[player_num];
99 float mu2 = mu[matches[i].other_player];
100 float mu = mu1 - mu2;
101 float sigma2 = sigma[matches[i].other_player];
102 float x = matches[i].margin;
104 //fprintf(stderr, "x=%f mu=%f sigma2=%f add %f-%f = %f\n", x, mu, sigma2, (x-mu)*(x-mu), sigma2*sigma2, (x - mu) * (x - mu) - sigma2 * sigma2);
105 sum += (x - mu) * (x - mu) - sigma2 * sigma2;
111 //fprintf(stderr, "sum=%f\n", sum);
112 sigma[player_num] = sqrt(sum / matches.size());
115 void renormalize(float *mu, float *sigma, int num_players)
118 for (int i = 0; i < num_players; ++i) {
121 float corr = 1500.0f - avg / num_players;
122 for (int i = 0; i < num_players; ++i) {
128 * Compute Fisher information matrix of the log-likelihood, evaluated at the MLE,
130 * ie. M_ij = E[ (D_i logL) (D_j logL) ] = - sum( ( x - (mu_1 - mu_2) )² / sigma_c⁴ ) for i != j
131 * = - sum( 1 / sigma_c² ) for i == j
133 * The Hessian matrix is generally zero and thus not as interesting.
135 void construct_fim(const float *mu, const float *sigma, int num_players)
137 float fim[MAX_PLAYERS][MAX_PLAYERS];
138 memset(fim, 0, sizeof(fim));
140 for (int i = 0; i < num_players; ++i) {
142 float sigma1 = sigma[i];
144 for (unsigned k = 0; k < matches_for_player[i].size(); ++k) {
145 int j = matches_for_player[i][k].other_player;
147 float sigma2 = sigma[j];
149 float x = matches_for_player[i][k].margin;
150 float sigma_sq = sqrt(sigma1 * sigma1 + sigma2 * sigma2);
152 fprintf(stderr, "exp_diff_sq=%f sigma_sq=%f\n", (x - (mu1 - mu2)) * (x - (mu1 - mu2)), sigma_sq * sigma_sq);
155 fim[i][i] += (x - (mu1 - mu2)) * (x - (mu1 - mu2)) / (sigma_sq * sigma_sq);
156 fim[i][j] -= (x - (mu1 - mu2)) * (x - (mu1 - mu2)) / (sigma_sq * sigma_sq);
158 fim[i][i] -= 1.0f / sigma_sq;
159 fim[i][j] += 1.0f / sigma_sq;
163 for (int j = 0; j < num_players; ++j) {
164 printf("%f ", fim[i][j]);
170 int main(int argc, char **argv)
173 if (scanf("%d", &num_players) != 1) {
174 fprintf(stderr, "Could't read number of players\n");
178 if (num_players > MAX_PLAYERS) {
179 fprintf(stderr, "Max %d players supported\n", MAX_PLAYERS);
183 vector<string> players;
184 map<string, int> player_map;
186 for (int i = 0; i < num_players; ++i) {
188 if (scanf("%s", buf) != 1) {
189 fprintf(stderr, "Couldn't read player %d\n", i);
193 players.push_back(buf);
199 char pl1[256], pl2[256];
203 if (scanf("%s %s %d %d %f", pl1, pl2, &score1, &score2, &weight) != 5) {
204 fprintf(stderr, "Read %d matches.\n", num_matches);
210 if (player_map.count(pl1) == 0) {
211 fprintf(stderr, "Unknown player '%s'\n", pl1);
214 if (player_map.count(pl2) == 0) {
215 fprintf(stderr, "Unknown player '%s'\n", pl2);
220 m1.other_player = player_map[pl2];
221 m1.margin = score1 - score2;
223 matches_for_player[player_map[pl1]].push_back(m1);
226 m2.other_player = player_map[pl1];
227 m2.margin = score2 - score1;
229 matches_for_player[player_map[pl2]].push_back(m2);
232 float mu[MAX_PLAYERS];
233 float sigma[MAX_PLAYERS];
235 for (int i = 0; i < num_players; ++i) {
237 sigma[i] = 70.0f / sqrt(2.0f);
239 renormalize(mu, sigma, num_players);
241 for (int j = 0; j < 1000; ++j) {
242 float old_mu[MAX_PLAYERS];
243 float old_sigma[MAX_PLAYERS];
244 memcpy(old_mu, mu, sizeof(mu));
245 memcpy(old_sigma, sigma, sizeof(sigma));
246 for (int i = 0; i < num_players; ++i) {
247 update_mu(mu, sigma, i, matches_for_player[i]);
248 renormalize(mu, sigma, num_players);
250 /* for (int i = 0; i < num_players; ++i) {
251 update_sigma(mu, sigma, i, matches_for_player[i]);
252 dump_scores(players, mu, sigma, num_players);
255 float sumdiff = 0.0f;
256 for (int i = 0; i < num_players; ++i) {
257 sumdiff += (mu[i] - old_mu[i]) * (mu[i] - old_mu[i]);
258 sumdiff += (sigma[i] - old_sigma[i]) * (sigma[i] - old_sigma[i]);
260 if (sumdiff < EPSILON) {
261 fprintf(stderr, "Converged after %d iterations. Stopping.\n", j);
265 dump_scores(players, mu, sigma, num_players);
267 // construct_fim(mu, sigma, num_players);