using namespace std;
-double prob_score(int k, int a, double rd);
-double prob_score_real(int k, int a, double binomial, double rd_norm);
-double prodai(int k, int a);
-double fac(int x);
+static double prob_score(int k, int a, double rd);
+static double prob_score_real(int k, int a, double binomial, double rd_norm);
+static double prodai(int k, int a);
+static double fac(int x);
// probability of match ending k-a (k>a) when winnerR - loserR = RD
// Glicko/Bradley-Terry assumption that a player rated 400 points over
// his/her opponent will win with a probability of 10/11 =~ 0.90909.
//
-double prob_score(int k, int a, double rd)
+static double prob_score(int k, int a, double rd)
{
return prob_score_real(k, a, prodai(k, a) / fac(k-1), rd/rating_constant);
}
// computes x^a, probably more efficiently than pow(x, a) (but requires that a
// is n unsigned integer)
-double intpow(double x, unsigned a)
+static double intpow(double x, unsigned a)
{
double result = 1.0;
// Same, but takes in binomial(a+k-1, k-1) as an argument in
// addition to a. Faster if you already have that precomputed, and assumes rd
// is already divided by 455.
-double prob_score_real(int k, int a, double binomial, double rd_norm)
+static double prob_score_real(int k, int a, double binomial, double rd_norm)
{
double nom = binomial * intpow(pow(2.0, rd_norm), a);
double denom = intpow(1.0 + pow(2.0, rd_norm), k+a);
}
// Calculates Product(a+i, i=1..k-1) (see above).
-double prodai(int k, int a)
+static double prodai(int k, int a)
{
double prod = 1.0;
for (int i = 1; i < k; ++i)
return prod;
}
-double fac(int x)
+static double fac(int x)
{
double prod = 1.0;
for (int i = 2; i <= x; ++i)
return prod;
}
-void convolve(int size)
-{
-}
-
-void compute_opponent_rating_pdf(int k, int a, double mu2, double sigma2, double winfac, vector<pair<double, double> > &result)
+static void compute_opponent_rating_pdf(int k, int a, double mu2, double sigma2, double winfac, vector<pair<double, double> > &result)
{
double binomial_precompute = prodai(k, a) / fac(k-1);
winfac /= rating_constant;
func1[i].real() = func1[i].imag() = func2[i].real() = func2[i].imag() = 0.0;
}
+ double invsq2sigma2 = 1.0 / (sqrt(2.0) * sigma2);
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));
+ double z = (x1 - mu2) * invsq2sigma2;
+ func1[i].real() = exp(-z*z);
double x2 = -3000.0 + h*i;
func2[(i - sz/2 + sz*2)%(sz*2)].real() = prob_score_real(k, a, binomial_precompute, x2*winfac);
}
// normalize the curve so we know that A ~= 1
-void normalize(vector<pair<double, double> > &curve)
+static void normalize(vector<pair<double, double> > &curve)
{
double peak = 0.0;
for (vector<pair<double, double> >::const_iterator i = curve.begin(); i != curve.end(); ++i) {
}
}
-// computes matA * matB
-void mat_mul(double *matA, unsigned ah, unsigned aw,
- double *matB, unsigned bh, unsigned bw,
- double *result)
-{
- assert(aw == bh);
- for (unsigned y = 0; y < bw; ++y) {
- for (unsigned x = 0; x < ah; ++x) {
- double sum = 0.0;
- for (unsigned c = 0; c < aw; ++c) {
- sum += matA[c*ah + x] * matB[y*bh + c];
- }
- result[y*bw + x] = sum;
- }
- }
-}
-
// computes matA^T * matB
-void mat_mul_trans(double *matA, unsigned ah, unsigned aw,
- double *matB, unsigned bh, unsigned bw,
- double *result)
+static void mat_mul_trans(double *matA, unsigned ah, unsigned aw,
+ double *matB, unsigned bh, unsigned bw,
+ double *result)
{
assert(ah == bh);
for (unsigned y = 0; y < bw; ++y) {
}
}
-void print3x3(double *M)
-{
- printf("%f %f %f\n", M[0], M[3], M[6]);
- printf("%f %f %f\n", M[1], M[4], M[7]);
- printf("%f %f %f\n", M[2], M[5], M[8]);
-}
-
-void print3x1(double *M)
-{
- printf("%f\n", M[0]);
- printf("%f\n", M[1]);
- printf("%f\n", M[2]);
-}
-
-// 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.
-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
// of statistical moments.
-void estimate_musigma(vector<pair<double, double> > &curve, double &mu_result, double &sigma_result)
+static void estimate_musigma(vector<pair<double, double> > &curve, double &mu_result, double &sigma_result)
{
double h = (curve.back().first - curve.front().first) / (curve.size() - 1);
// Note that the algorithm blows up quite hard if the initial estimate is
// not good enough. Use estimate_musigma to get a reasonable starting
// estimate.
-void least_squares(vector<pair<double, double> > &curve, double mu1, double sigma1, double &mu_result, double &sigma_result)
+static void least_squares(vector<pair<double, double> > &curve, double mu1, double sigma1, double &mu_result, double &sigma_result)
{
double A = 1.0;
double mu = mu1;
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];
sigma_result = sigma;
}
-void compute_new_rating(double mu1, double sigma1, double mu2, double sigma2, int score1, int score2, double &mu, double &sigma)
+static void compute_new_rating(double mu1, double sigma1, double mu2, double sigma2, int score1, int score2, double &mu, double &sigma)
{
vector<pair<double, double> > curve;
least_squares(curve, mu_est, sigma_est, mu, sigma);
}
-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)
+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)
{
vector<pair<double, double> > curve, newcurve;
double mu_t = mu3 + mu4;
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
+ double invsq2sigma2 = 1.0 / (sqrt(2.0) * sigma2);
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));
+ double z = (r2 - mu2) * invsq2sigma2;
+ double gaussian = exp(-z*z);
sum += curve[j].second * gaussian;
}