#include <complex>
#include <fftw3.h>
+#define USE_LOGISTIC_DISTRIBUTION 0
+
// step sizes
static const double int_step_size = 75.0;
using namespace std;
-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);
+#if USE_LOGISTIC_DISTRIBUTION
+// sech²(x)
+static double sech2(double x)
+{
+ double e = exp(2.0 * x);
+ return 4.0 * e / ((e+1.0) * (e+1.0));
+}
+#endif
+#if 0
// probability of match ending k-a (k>a) when winnerR - loserR = RD
//
// +inf
{
return prob_score_real(k, a, prodai(k, a) / fac(k-1), rd/rating_constant);
}
+#endif
// computes x^a, probably more efficiently than pow(x, a) (but requires that a
// is n unsigned integer)
return prod;
}
-static 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;
int sz = (6000.0 - 0.0) / int_step_size;
double h = (6000.0 - 0.0) / sz;
- fftw_plan f1, f2, b;
- complex<double> *func1, *func2, *res;
-
- func1 = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
- func2 = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
- res = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
- f1 = fftw_plan_dft_1d(sz*2,
- reinterpret_cast<fftw_complex*>(func1),
- reinterpret_cast<fftw_complex*>(func1),
- FFTW_FORWARD,
- FFTW_MEASURE);
- f2 = fftw_plan_dft_1d(sz*2,
- reinterpret_cast<fftw_complex*>(func2),
- reinterpret_cast<fftw_complex*>(func2),
- FFTW_FORWARD,
- FFTW_MEASURE);
- b = fftw_plan_dft_1d(sz*2,
- reinterpret_cast<fftw_complex*>(res),
- reinterpret_cast<fftw_complex*>(res),
- FFTW_BACKWARD,
- FFTW_MEASURE);
+ static bool inited = false;
+ static fftw_plan f1, f2, b;
+ static complex<double> *func1, *func2, *res;
+
+ if (!inited) {
+ func1 = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
+ func2 = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
+ res = reinterpret_cast<complex<double> *>(fftw_malloc(sz*2*sizeof(complex<double>)));
+ f1 = fftw_plan_dft_1d(sz*2,
+ reinterpret_cast<fftw_complex*>(func1),
+ reinterpret_cast<fftw_complex*>(func1),
+ FFTW_FORWARD,
+ FFTW_MEASURE);
+ f2 = fftw_plan_dft_1d(sz*2,
+ reinterpret_cast<fftw_complex*>(func2),
+ reinterpret_cast<fftw_complex*>(func2),
+ FFTW_FORWARD,
+ FFTW_MEASURE);
+ b = fftw_plan_dft_1d(sz*2,
+ reinterpret_cast<fftw_complex*>(res),
+ reinterpret_cast<fftw_complex*>(res),
+ FFTW_BACKWARD,
+ FFTW_MEASURE);
+ inited = true;
+ }
// start off by zero
for (int i = 0; i < sz*2; ++i) {
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;
+
+ // opponent's pdf
+#if USE_LOGISTIC_DISTRIBUTION
double z = (x1 - mu2) * invsigma2;
- func1[i].real() = exp(-(z*z/2.0));
+ func1[i].real() = sech2(0.5 * z);
+#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);
}
- result.reserve(sz*2);
+ result->reserve(sz*2);
// convolve
fftw_execute(f1);
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])));
+ result->push_back(make_pair(r1, abs(res[i])));
}
}
// normalize the curve so we know that A ~= 1
-static 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) {
+ for (vector<pair<double, double> >::const_iterator i = curve->begin(); i != curve->end(); ++i) {
peak = max(peak, i->second);
}
double invpeak = 1.0 / peak;
- for (vector<pair<double, double> >::iterator i = curve.begin(); i != curve.end(); ++i) {
+ for (vector<pair<double, double> >::iterator i = curve->begin(); i != curve->end(); ++i) {
i->second *= invpeak;
}
}
// Give an OK starting estimate for the least squares, by numerical integration
// of statistical moments.
-static void estimate_musigma(vector<pair<double, double> > &curve, double &mu_result, double &sigma_result)
+static void estimate_musigma(const vector<pair<double, double> > &curve, double *mu_result, double *sigma_result)
{
double h = (curve.back().first - curve.front().first) / (curve.size() - 1);
ex = (h/3.0) * ex / area;
ex2 = (h/3.0) * ex2 / area;
- mu_result = ex;
- sigma_result = sqrt(ex2 - ex * ex);
+ *mu_result = ex;
+ *sigma_result = sqrt(ex2 - ex * ex);
}
// Find best fit of the data in curves to a Gaussian pdf, based on the
// 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.
-static void least_squares(vector<pair<double, double> > &curve, double mu1, double sigma1, double &mu_result, double &sigma_result)
+static void least_squares(const vector<pair<double, double> > &curve, double mu1, double sigma1, double *mu_result, double *sigma_result)
{
double A = 1.0;
double mu = mu1;
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(0.5 * (x-mu)/sigma);
+
+ // df/dµ(x_i)
+ matA[i + 1 * curve.size()] = A * matA[i + 0 * curve.size()]
+ * tanh(0.5 * (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(0.5 * (x-mu)/sigma);
+#else
dbeta[i] = y - A * exp(- (x-mu)*(x-mu)/(2.0*sigma*sigma));
+#endif
}
// compute a and b
break;
}
- mu_result = mu;
- sigma_result = sigma;
+ *mu_result = mu;
+ *sigma_result = sigma;
}
-static void compute_new_rating(double mu1, double sigma1, double mu2, double sigma2, int score1, int score2, double &mu, double &sigma)
+void compute_new_rating(double mu1, double sigma1, double mu2, double sigma2, int score1, int score2, double *mu, double *sigma, double *probability)
{
vector<pair<double, double> > curve;
if (score1 > score2) {
- compute_opponent_rating_pdf(score1, score2, mu2, sigma2, -1.0, curve);
+ compute_opponent_rating_pdf(score1, score2, mu2, sigma2, -1.0, &curve);
} else {
- compute_opponent_rating_pdf(score2, score1, mu2, sigma2, 1.0, curve);
+ compute_opponent_rating_pdf(score2, score1, mu2, sigma2, 1.0, &curve);
}
// 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
+ curve[i].second *= sech2(0.5 * z);
+#else
double gaussian = exp(-(z*z/2.0));
curve[i].second *= gaussian;
+#endif
+ }
+
+ // Compute the overall probability of the given result, by integrating
+ // the entire resulting pdf. Note that since we're actually evaluating
+ // a double integral, we'll need to multiply by h² instead of h.
+ {
+ double h = (curve.back().first - curve.front().first) / (curve.size() - 1);
+ double sum = curve.front().second;
+ for (unsigned i = 1; i < curve.size() - 1; i += 2) {
+ sum += 4.0 * curve[i].second;
+ }
+ for (unsigned i = 2; i < curve.size() - 1; i += 2) {
+ sum += 2.0 * curve[i].second;
+ }
+ sum += curve.back().second;
+ sum *= h * h / 3.0;
+
+ // FFT convolution multiplication factor (FFTW computes unnormalized
+ // transforms)
+ sum /= (curve.size() * 2);
+
+ // pdf normalization factors
+#if USE_LOGISTIC_DISTRIBUTION
+ sum /= (sigma1 * 4.0);
+ sum /= (sigma2 * 4.0);
+#else
+ sum /= (sigma1 * sqrt(2.0 * M_PI));
+ sum /= (sigma2 * sqrt(2.0 * M_PI));
+#endif
+
+ *probability = sum;
}
double mu_est, sigma_est;
- normalize(curve);
- estimate_musigma(curve, mu_est, sigma_est);
+ normalize(&curve);
+ estimate_musigma(curve, &mu_est, &sigma_est);
least_squares(curve, mu_est, sigma_est, mu, 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)
+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, double *probability)
{
vector<pair<double, double> > curve, newcurve;
double mu_t = mu3 + mu4;
double sigma_t = sqrt(sigma3*sigma3 + sigma4*sigma4);
if (score1 > score2) {
- compute_opponent_rating_pdf(score1, score2, mu_t, sigma_t, -1.0, curve);
+ compute_opponent_rating_pdf(score1, score2, mu_t, sigma_t, -1.0, &curve);
} else {
- compute_opponent_rating_pdf(score2, score1, mu_t, sigma_t, 1.0, curve);
+ compute_opponent_rating_pdf(score2, score1, mu_t, sigma_t, 1.0, &curve);
}
newcurve.reserve(curve.size());
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;
+#if USE_LOGISTIC_DISTRIBUTION
double z = (r2 - mu2) * invsigma2;
- double gaussian = exp(-(z*z/2.0));
+ double gaussian = sech2(0.5 * 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(0.5 * z);
+#else
double z = (r1 - mu1) / sigma1;
double gaussian = exp(-(z*z/2.0));
+#endif
newcurve.push_back(make_pair(r1, gaussian * sum));
}
+ // Compute the overall probability of the given result, by integrating
+ // the entire resulting pdf. Note that since we're actually evaluating
+ // a triple integral, we'll need to multiply by 4h³ (no idea where the
+ // 4 factor comes from, probably from the 0..6000 range somehow) instead
+ // of h.
+ {
+ double h = (newcurve.back().first - newcurve.front().first) / (newcurve.size() - 1);
+ double sum = newcurve.front().second;
+ for (unsigned i = 1; i < newcurve.size() - 1; i += 2) {
+ sum += 4.0 * newcurve[i].second;
+ }
+ for (unsigned i = 2; i < newcurve.size() - 1; i += 2) {
+ sum += 2.0 * newcurve[i].second;
+ }
+ sum += newcurve.back().second;
+
+ sum *= 4.0 * h * h * h / 3.0;
+
+ // FFT convolution multiplication factor (FFTW computes unnormalized
+ // transforms)
+ sum /= (newcurve.size() * 2);
+
+ // pdf normalization factors
+#if USE_LOGISTIC_DISTRIBUTION
+ sum /= (sigma1 * 4.0);
+ sum /= (sigma2 * 4.0);
+ sum /= (sigma_t * 4.0);
+#else
+ sum /= (sigma1 * sqrt(2.0 * M_PI));
+ sum /= (sigma2 * sqrt(2.0 * M_PI));
+ sum /= (sigma_t * sqrt(2.0 * M_PI));
+#endif
+
+ *probability = sum;
+ }
double mu_est, sigma_est;
- normalize(newcurve);
- estimate_musigma(newcurve, mu_est, sigma_est);
+ normalize(&newcurve);
+ estimate_musigma(newcurve, &mu_est, &sigma_est);
least_squares(newcurve, mu_est, sigma_est, mu, sigma);
}
double sigma4 = atof(argv[8]);
int score1 = atoi(argv[9]);
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);
+ double mu, sigma, probability;
+ compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, score1, score2, &mu, &sigma, &probability);
+ printf("%f %f %f\n", mu, sigma, probability);
} else if (argc > 8) {
double mu3 = atof(argv[5]);
double sigma3 = atof(argv[6]);
double mu4 = atof(argv[7]);
double sigma4 = atof(argv[8]);
int k = atoi(argv[9]);
-
+
// assess all possible scores
for (int i = 0; i < k; ++i) {
double newmu1_1, newmu1_2, newmu2_1, newmu2_2;
double newsigma1_1, newsigma1_2, newsigma2_1, newsigma2_2;
- compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, k, i, newmu1_1, newsigma1_1);
- compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, k, i, newmu1_2, newsigma1_2);
- compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, i, k, newmu2_1, newsigma2_1);
- compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, i, k, newmu2_2, newsigma2_2);
+ double probability;
+ compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, k, i, &newmu1_1, &newsigma1_1, &probability);
+ compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, k, i, &newmu1_2, &newsigma1_2, &probability);
+ compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, i, k, &newmu2_1, &newsigma2_1, &probability);
+ compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, i, k, &newmu2_2, &newsigma2_2, &probability);
printf("%u-%u,%f,%+f,%+f,%+f,%+f\n",
- k, i, prob_score(k, i, mu3+mu4-(mu1+mu2)), newmu1_1-mu1, newmu1_2-mu2,
+ k, i, probability, newmu1_1-mu1, newmu1_2-mu2,
newmu2_1-mu3, newmu2_2-mu4);
}
for (int i = k; i --> 0; ) {
double newmu1_1, newmu1_2, newmu2_1, newmu2_2;
double newsigma1_1, newsigma1_2, newsigma2_1, newsigma2_2;
- compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, i, k, newmu1_1, newsigma1_1);
- compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, i, k, newmu1_2, newsigma1_2);
- compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, k, i, newmu2_1, newsigma2_1);
- compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, k, i, newmu2_2, newsigma2_2);
+ double probability;
+ compute_new_double_rating(mu1, sigma1, mu2, sigma2, mu3, sigma3, mu4, sigma4, i, k, &newmu1_1, &newsigma1_1, &probability);
+ compute_new_double_rating(mu2, sigma2, mu1, sigma1, mu3, sigma3, mu4, sigma4, i, k, &newmu1_2, &newsigma1_2, &probability);
+ compute_new_double_rating(mu3, sigma3, mu4, sigma4, mu1, sigma1, mu2, sigma2, k, i, &newmu2_1, &newsigma2_1, &probability);
+ compute_new_double_rating(mu4, sigma4, mu3, sigma3, mu1, sigma1, mu2, sigma2, k, i, &newmu2_2, &newsigma2_2, &probability);
printf("%u-%u,%f,%+f,%+f,%+f,%+f\n",
- i, k, prob_score(k, i, mu1+mu2-(mu3+mu4)), newmu1_1-mu1, newmu1_2-mu2,
+ i, k, probability, newmu1_1-mu1, newmu1_2-mu2,
newmu2_1-mu3, newmu2_2-mu4);
}
} else if (argc > 6) {
int score1 = atoi(argv[5]);
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);
+ double mu, sigma, probability;
+ compute_new_rating(mu1, sigma1, mu2, sigma2, score1, score2, &mu, &sigma, &probability);
+
+ printf("%f %f %f\n", mu, sigma, probability);
} else {
int k = atoi(argv[5]);
// assess all possible scores
for (int i = 0; i < k; ++i) {
- double newmu1, newmu2, newsigma1, newsigma2;
- compute_new_rating(mu1, sigma1, mu2, sigma2, k, i, newmu1, newsigma1);
- compute_new_rating(mu2, sigma2, mu1, sigma1, i, k, newmu2, newsigma2);
+ double newmu1, newmu2, newsigma1, newsigma2, probability;
+ compute_new_rating(mu1, sigma1, mu2, sigma2, k, i, &newmu1, &newsigma1, &probability);
+ compute_new_rating(mu2, sigma2, mu1, sigma1, i, k, &newmu2, &newsigma2, &probability);
printf("%u-%u,%f,%+f,%+f\n",
- k, i, prob_score(k, i, mu2-mu1), newmu1-mu1, newmu2-mu2);
+ k, i, probability, newmu1-mu1, newmu2-mu2);
}
for (int i = k; i --> 0; ) {
- double newmu1, newmu2, newsigma1, newsigma2;
- compute_new_rating(mu1, sigma1, mu2, sigma2, i, k, newmu1, newsigma1);
- compute_new_rating(mu2, sigma2, mu1, sigma1, k, i, newmu2, newsigma2);
+ double newmu1, newmu2, newsigma1, newsigma2, probability;
+ compute_new_rating(mu1, sigma1, mu2, sigma2, i, k, &newmu1, &newsigma1, &probability);
+ compute_new_rating(mu2, sigma2, mu1, sigma1, k, i, &newmu2, &newsigma2, &probability);
printf("%u-%u,%f,%+f,%+f\n",
- i, k, prob_score(k, i, mu1-mu2), newmu1-mu1, newmu2-mu2);
+ i, k, probability, newmu1-mu1, newmu2-mu2);
}
}
projects / foosball / blobdiff