double
CDF (double x, double mu, double sigma)
{
double t;
x -= mu;
x /= sigma;
if (x < 0) return 1 - CDF (-x, 0, 1);
t = 1 / (1 + b0 * x);
return 1 - PDF_0 (x) * (b1 * t + b2 * t * t + b3 * t * t * t + b4 * t * t * t * t + b5 * t * t * t * t * t);
}
vector<double> Rico::Pr(vector<double> x) const {
int N = x.size();
vector<double> p;
vector<double> P = CDF(x);
vector<double> Q = Util::diff(P);
p.push_back(P[0]);
for(int i = 0; i < N-1; i++) p.push_back(Q[i]);
p.push_back(1 - P[N-1]);
return p;
}
// generate samples from a distribution using inverse CDF sampling
// pdf input is expected to be an Nx2 matrix with col 1 = values and col 2 = probabilities of those values
// step size scales pdf to equal 1, set equal
Matrix invCDFsample(const int& samples, const double& step,
const Matrix& pdf)
{
Matrix samp(samples,1);
double tempSamp = 0.;
int j = 0;
// integrate to generate CDF
Matrix CDF(pdf.rows(),1);
CDF[0][0] = pdf[0][1]*step;
for(int i=0;i<CDF.rows()-1;i++)
{
//cout << "CDF[i] = " << CDF[i][0] << " pdf[i] = " << pdf[i][1] << endl;
CDF[i+1][0] = CDF[i][0] + pdf[i+1][1];
//cout << "CDF[i+1] = " << CDF[i+1][0] << endl;
}
// multiply by step size to scale integral correctly
CDF = CDF*step;
//CDF.print();
// choose sample from uniform generator according to CDF
for(int i=0;i<samples;i++)
{
j = 0;
tempSamp = rvStdUniform();
while( CDF[j][0] < tempSamp )
{
j++;
}
if( j == 0 )
samp[i][0] = pdf[j][0];
else
samp[i][0] = (pdf[j-1][0]+pdf[j][0])/2;
}
return samp;
}
/**
* Computes the Lomb-Scargle periodogram of the matrix "data". "data" should contain at least three
* columns: time, measurement and measurement error. The periodogram is calculated in "samples" intervals
* between "Pmin" and "Pmax", spaced logarithmically.
*
* The function returns a matrix of "samples" rows and several columns, including period, power (z) and
* an estimation of the upper bound for the false alarm probability. The estimation is calculated using
* the method of Baluev, 2008 (Baluev08). The column PS_Z_LS contains the unnormalized LS periodogram
* (z = 1/2 * (Chi^2_0 - Chi^2_SC)), while the column PS_Z contains z_1 = 1/2 * N_H * z / Chi^2_0 (z_1 in Baluev08).
* The FAP upper bound is estimated as ~ tau(z_1). (Another estimate of the FAP can be calculated by
* estimating the indep. frequencies through your own algorithm, or using the ok_periodogram_boot routine.)
*
* @param data Input data containing the data; each row containing (t_i, x_i, sigma_i)
* @param samples Number of frequencies sampled
* @param Pmin Minimum period sampled
* @param Pmax Maximum period sampled
* @param method Method to compute periodogram (ignored)
* @param timecol Time column (e.g. 0) in the matrix data
* @param valcol Value column (e.g. 1) in the matrix data
* @param sigmacol Sigma column (e.g. 2) in the matrix data
* @param p If not NULL, it is used to return additional info for the periodogram and reuse matrices to save space/speed. If you pass
* a value different than NULL, you are responsible for deallocating the workspace and its fields. p->buf is an array of
* gsl_matrix*, sized the same as the value of omp_get_max_threads().
* @return A matrix containing: {PS_TIME, PS_Z, PS_FAP, PS_Z_LS} (period, power, FAP upper limit, unnormalized
* LS power). You are responsible for deallocating it.
*/
gsl_matrix* ok_periodogram_ls(const gsl_matrix* data, const unsigned int samples, const double Pmin, const double Pmax, const int method,
unsigned int timecol, unsigned int valcol, unsigned int sigcol, ok_periodogram_workspace* p) {
gsl_matrix* ret = NULL;
gsl_matrix* buf = NULL;
gsl_vector* bufv = gsl_vector_alloc(data->size1);
int ndata = data->size1;
// If no pre-allocated buffers are passed through p, or p is null,
// allocate those buffers.
if (p != NULL) {
if (p->per != NULL && MROWS(p->per) == samples && MCOLS(p->per) == PS_SIZE)
ret = p->per;
if (p->buf != NULL && MROWS(p->buf) == ndata && MCOLS(p->per) == 5)
ret = p->buf;
}
ret = (ret != NULL ? ret : gsl_matrix_alloc(samples, PS_SIZE));
buf = (buf != NULL ? buf : gsl_matrix_alloc(ndata, 5));
double fmin = 1. / Pmax;
double fmax = 1. / Pmin;
double df = (fmax - fmin) / (double) samples;
gsl_matrix_get_col(bufv, data, timecol);
double W = 2. * M_PI * gsl_stats_sd(bufv->data, 1, ndata) / Pmin;
gsl_matrix_get_col(bufv, data, valcol);
double avg = gsl_stats_mean(bufv->data, 1, ndata);
double z1_max = 0.;
double xa[ndata];
// pre-calculate cdf, sdf
for (int i = 0; i < ndata; i++) {
double t = MGET(data, i, timecol) - MGET(data, 0, timecol);
MSET(buf, i, BUF_CDF, cos(2 * M_PI * df * t));
MSET(buf, i, BUF_SDF, sin(2 * M_PI * df * t));
MSET(buf, i, BUF_C, cos(2 * M_PI * fmin * t));
MSET(buf, i, BUF_S, sin(2 * M_PI * fmin * t));
MSET(buf, i, BUF_SIG, 1. / (MGET(data, i, sigcol) * MGET(data, i, sigcol)));
xa[i] = MGET(data, i, valcol) - avg;
}
// Calculate periodogram by looping over all angular frequencies
for (int i = 0; i < samples; i++) {
// Current frequency
double f = fmin + df * i;
double w = 2 * M_PI*f;
// Calculate tau(w)
double s_2wt = 0.;
double c_2wt = 0.;
for (int j = 0; j < ndata; j++) {
double cos_wt = C(j);
double sin_wt = S(j);
c_2wt += (1. - 2. * sin_wt * sin_wt) * SIG(j);
s_2wt += (2. * sin_wt * cos_wt) * SIG(j);
}
double tau = atan2(s_2wt, c_2wt) / (2. * w);
double numa = 0.;
double numb = 0.;
double dena = 0.;
double denb = 0.;
double numa_w = 0.;
double numb_w = 0.;
double dena_w = 0.;
double denb_w = 0.;
double coswtau = cos(w * tau);
double sinwtau = sin(w * tau);
double chi2_h = 0.;
double chi2_h_w = 0;
for (int j = 0; j < ndata; j++) {
double sig = SIG(j);
const double cos_wt = C(j);
const double sin_wt = S(j);
double cos_wdf = CDF(j);
double sin_wdf = SDF(j);
double c = cos_wt * coswtau + sin_wt * sinwtau;
double s = sin_wt * coswtau - cos_wt * sinwtau;
double x = xa[j];
MSET(buf, j, BUF_C, cos_wt * cos_wdf - sin_wt * sin_wdf);
MSET(buf, j, BUF_S, sin_wt * cos_wdf + cos_wt * sin_wdf);
numa += x * c * sig;
numb += x * s * sig;
dena += c * c * sig;
denb += s * s * sig;
chi2_h += x * x * sig;
numa_w += c;
numb_w += s;
dena_w += c*c;
denb_w += s*s;
chi2_h_w += 1;
}
double z = 0.5 * (numa * numa / dena + numb * numb / denb);
double z_1 = z * ndata / chi2_h;
double w_1 = 0.5 * (numa_w * numa_w / dena_w + numb_w * numb_w / denb_w) * ndata / chi2_h_w;
double fap_single = pow(1. - 2. * z_1 / (double) ndata, 0.5 * (double) (ndata - 3.));
double tau_z = W * fap_single * sqrt(z_1);
MSET(ret, samples - i - 1, PS_TIME, 1. / f);
MSET(ret, samples - i - 1, PS_Z, z_1);
MSET(ret, samples - i - 1, PS_Z_LS, z);
MSET(ret, samples - i - 1, PS_FAP, MIN(fap_single + tau_z, 1.));
MSET(ret, samples - i - 1, PS_TAU, tau);
MSET(ret, samples - i - 1, PS_WIN, w_1);
z1_max = MAX(z1_max, z_1);
}
if (p != NULL && p->calc_z_fap) {
gsl_root_fsolver * s = gsl_root_fsolver_alloc(gsl_root_fsolver_brent);
double pars[3];
pars[0] = ndata;
pars[1] = W;
pars[2] = 0.;
gsl_function F;
F.function = _baluev_tau;
F.params = pars;
double zz = z1_max;
while (_baluev_tau(zz, pars) > 1e-3)
zz *= 2;
p->z_fap_3 = _find_z(s, &F, 1e-3, 0.1, zz);
p->z_fap_2 = _find_z(s, &F, 1e-2, 0.1, p->z_fap_3);
p->z_fap_1 = _find_z(s, &F, 1e-1, 0.1, p->z_fap_2);
gsl_root_fsolver_free(s);
p->calc_z_fap = false;
}
if (p == NULL) {
gsl_matrix_free(buf);
} else {
p->per = ret;
p->buf = buf;
p->zmax = z1_max;
};
gsl_vector_free(bufv);
return ret;
}
float V(float t) {
float cdf = CDF(t);
return (cdf == 0) ? 0 : PDF(t) / cdf;
}
vector<double> Rico::CDF(vector<double> X) const {
vector<double> P;
for(int i = 0; i < X.size(); i++) P.push_back(CDF(X[i]));
return P;
}
