#include <complex>
#include <fftw3.h>
+#define USE_LOGISTIC_DISTRIBUTION 0
+
// step sizes
static const double int_step_size = 75.0;
// rating constant (see below)
static const double rating_constant = 455.0;
+#if USE_LOGISTIC_DISTRIBUTION
+// constant used in the logistic pdf
+static const double l_const = M_PI / (2.0 * sqrt(3.0));
+#endif
+
using namespace std;
static double prob_score(int k, int a, double rd);
static double prodai(int k, int a);
static double fac(int x);
+#if USE_LOGISTIC_DISTRIBUTION
+// sech²(x)
+static double sech2(double x)
+{
+ double c = cosh(x);
+ return 1.0 / (c*c);
+}
+#endif
// probability of match ending k-a (k>a) when winnerR - loserR = RD
//
func1[i].real() = func1[i].imag() = func2[i].real() = func2[i].imag() = 0.0;
}
+#if USE_LOGISTIC_DISTRIBUTION
+ double invsigma2 = 1.0 / sigma2;
+#else
+ double invsq2sigma2 = 1.0 / (sqrt(2.0) * sigma2);
+#endif
for (int i = 0; i < sz; ++i) {
double x1 = 0.0 + h*i;
- double z = (x1 - mu2)/sigma2;
- func1[i].real() = exp(-(z*z/2.0));
+
+ // opponent's pdf
+#if USE_LOGISTIC_DISTRIBUTION
+ double z = (x1 - mu2) * invsigma2;
+ double ch = cosh(l_const * z);
+ func1[i].real() = 1.0 / (ch * ch);
+#else
+ double z = (x1 - mu2) * invsq2sigma2;
+ func1[i].real() = exp(-z*z);
+#endif
double x2 = -3000.0 + h*i;
func2[(i - sz/2 + sz*2)%(sz*2)].real() = prob_score_real(k, a, binomial_precompute, x2*winfac);
res[i] = func1[i] * func2[i];
}
fftw_execute(b);
+
+ result.reserve(sz);
for (int i = 0; i < sz; ++i) {
double r1 = i*h;
result.push_back(make_pair(r1, abs(res[i])));
}
}
-// solves Ax = B by Gauss-Jordan elimination, where A is a 3x3 matrix,
-// x is a column vector of length 3 and B is a row vector of length 3.
+// solves Ax = B by Gauss-Jordan elimination, where A is an NxN matrix,
+// x is a column vector of length N and B is a row vector of length N.
// Destroys its input in the process.
-static void solve3x3(double *A, double *x, double *B)
+template<int N>
+static void solve_matrix(double *A, double *x, double *B)
{
- // row 1 -= row 0 * (a1/a0)
- {
- double f = A[1] / A[0];
- A[1] = 0.0;
- A[4] -= A[3] * f;
- A[7] -= A[6] * f;
-
- B[1] -= B[0] * f;
- }
-
- // row 2 -= row 0 * (a2/a0)
- {
- double f = A[2] / A[0];
- A[2] = 0.0;
- A[5] -= A[3] * f;
- A[8] -= A[6] * f;
-
- B[2] -= B[0] * f;
- }
-
- // row 2 -= row 1 * (a5/a4)
- {
- double f = A[5] / A[4];
- A[5] = 0.0;
- A[8] -= A[7] * f;
-
- B[2] -= B[1] * f;
- }
-
- // back substitute:
-
- // row 1 -= row 2 * (a7/a8)
- {
- double f = A[7] / A[8];
- A[7] = 0.0;
-
- B[1] -= B[2] * f;
- }
-
- // row 0 -= row 2 * (a6/a8)
- {
- double f = A[6] / A[8];
- A[6] = 0.0;
+ for (int i = 0; i < N; ++i) {
+ for (int j = i+1; j < N; ++j) {
+ // row j -= row i * (a[i,j] / a[i,i])
+ double f = A[j+i*N] / A[i+i*N];
+
+ A[j+i*N] = 0.0;
+ for (int k = i+1; k < N; ++k) {
+ A[j+k*N] -= A[i+k*N] * f;
+ }
- B[0] -= B[2] * f;
+ B[j] -= B[i] * f;
+ }
}
- // row 0 -= row 1 * (a3/a4)
- {
- double f = A[3] / A[4];
- A[3] = 0.0;
-
- B[0] -= B[1] * f;
+ // back-substitute
+ for (int i = N; i --> 0; ) {
+ for (int j = i; j --> 0; ) {
+ // row j -= row i * (a[j,j] / a[j,i])
+ double f = A[i+j*N] / A[j+j*N];
+
+ // A[j+i*N] = 0.0;
+ B[j] -= B[i] * f;
+ }
}
// normalize
- x[0] = B[0] / A[0];
- x[1] = B[1] / A[4];
- x[2] = B[2] / A[8];
+ for (int i = 0; i < N; ++i) {
+ x[i] = B[i] / A[i+i*N];
+ }
}
// Give an OK starting estimate for the least squares, by numerical integration
for (unsigned i = 0; i < curve.size(); ++i) {
double x = curve[i].first;
+#if USE_LOGISTIC_DISTRIBUTION
+ // df/dA(x_i)
+ matA[i + 0 * curve.size()] = sech2(l_const * (x-mu)/sigma);
+
+ // df/dµ(x_i)
+ matA[i + 1 * curve.size()] = 2.0 * l_const * A * matA[i + 0 * curve.size()]
+ * tanh(l_const * (x-mu)/sigma) / sigma;
+
+ // df/dσ(x_i)
+ matA[i + 2 * curve.size()] =
+ matA[i + 1 * curve.size()] * (x-mu)/sigma;
+#else
// df/dA(x_i)
matA[i + 0 * curve.size()] =
exp(-(x-mu)*(x-mu)/(2.0*sigma*sigma));
// df/dµ(x_i)
- matA[i + 1 * curve.size()] =
+ matA[i + 1 * curve.size()] =
A * (x-mu)/(sigma*sigma) * matA[i + 0 * curve.size()];
// df/dσ(x_i)
matA[i + 2 * curve.size()] =
matA[i + 1 * curve.size()] * (x-mu)/sigma;
+#endif
}
// find dβ
double x = curve[i].first;
double y = curve[i].second;
+#if USE_LOGISTIC_DISTRIBUTION
+ dbeta[i] = y - A * sech2(l_const * (x-mu)/sigma);
+#else
dbeta[i] = y - A * exp(- (x-mu)*(x-mu)/(2.0*sigma*sigma));
+#endif
}
// compute a and b
mat_mul_trans(matA, curve.size(), 3, dbeta, curve.size(), 1, matATdb);
// solve
- solve3x3(matATA, dlambda, matATdb);
+ solve_matrix<3>(matATA, dlambda, matATdb);
A += dlambda[0];
mu += dlambda[1];
// multiply in the gaussian
for (unsigned i = 0; i < curve.size(); ++i) {
double r1 = curve[i].first;
+
+ // my pdf
double z = (r1 - mu1) / sigma1;
+#if USE_LOGISTIC_DISTRIBUTION
+ double ch = cosh(l_const * z);
+ curve[i].second /= (ch * ch);
+#else
double gaussian = exp(-(z*z/2.0));
curve[i].second *= gaussian;
+#endif
}
double mu_est, sigma_est;
compute_opponent_rating_pdf(score2, score1, mu_t, sigma_t, 1.0, curve);
}
+ newcurve.reserve(curve.size());
+
// iterate over r1
double h = 3000.0 / curve.size();
for (unsigned i = 0; i < curve.size(); ++i) {
double r1 = i * h;
// iterate over r2
+#if USE_LOGISTIC_DISTRIBUTION
+ double invsigma2 = 1.0 / sigma2;
+#else
+ double invsq2sigma2 = 1.0 / (sqrt(2.0) * sigma2);
+#endif
for (unsigned j = 0; j < curve.size(); ++j) {
double r1plusr2 = curve[j].first;
double r2 = r1plusr2 - r1;
- double z = (r2 - mu2) / sigma2;
- double gaussian = exp(-(z*z/2.0));
+#if USE_LOGISTIC_DISTRIBUTION
+ double z = (r2 - mu2) * invsigma2;
+ double gaussian = sech2(l_const * z);
+#else
+ double z = (r2 - mu2) * invsq2sigma2;
+ double gaussian = exp(-z*z);
+#endif
sum += curve[j].second * gaussian;
}
+#if USE_LOGISTIC_DISTRIBUTION
+ double z = (r1 - mu1) / sigma1;
+ double gaussian = sech2(l_const * z);
+#else
double z = (r1 - mu1) / sigma1;
double gaussian = exp(-(z*z/2.0));
+#endif
newcurve.push_back(make_pair(r1, gaussian * sum));
}
int score2 = atoi(argv[10]);
double mu, sigma;
compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, score1, score2, mu, sigma);
- printf("%f %f\n", mu, sigma);
+ if (score1 > score2) {
+ printf("%f %f %f\n", mu, sigma, prob_score(score1, score2, mu3+mu4-(mu1+mu2)));
+ } else {
+ printf("%f %f %f\n", mu, sigma, prob_score(score2, score1, mu1+mu2-(mu3+mu4)));
+ }
} else if (argc > 8) {
double mu3 = atof(argv[5]);
double sigma3 = atof(argv[6]);
int score2 = atoi(argv[6]);
double mu, sigma;
compute_new_rating(mu1, sigma1, mu2, sigma2, score1, score2, mu, sigma);
- printf("%f %f\n", mu, sigma);
+
+ if (score1 > score2) {
+ printf("%f %f %f\n", mu, sigma, prob_score(score1, score2, mu2-mu1));
+ } else {
+ printf("%f %f %f\n", mu, sigma, prob_score(score2, score1, mu1-mu2));
+ }
} else {
int k = atoi(argv[5]);
projects / foosball / blobdiff