Example #1
0
double qpois(double p, double lambda, int lower_tail, int log_p)
{
    double mu, sigma, gamma, z, y;
#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(lambda))
	return p + lambda;
#endif
    if(!R_FINITE(lambda))
	ML_ERR_return_NAN;
    if(lambda < 0) ML_ERR_return_NAN;
    R_Q_P01_check(p);
    if(lambda == 0) return 0;
    if(p == R_DT_0) return 0;
    if(p == R_DT_1) return ML_POSINF;

    mu = lambda;
    sigma = sqrt(lambda);
    /* gamma = sigma; PR#8058 should be kurtosis which is mu^-0.5 */
    gamma = 1.0/sigma;

    /* Note : "same" code in qpois.c, qbinom.c, qnbinom.c --
     * FIXME: This is far from optimal [cancellation for p ~= 1, etc]: */
    if(!lower_tail || log_p) {
	p = R_DT_qIv(p); /* need check again (cancellation!): */
	if (p == 0.) return 0;
	if (p == 1.) return ML_POSINF;
    }
    /* temporary hack --- FIXME --- */
    if (p + 1.01*DBL_EPSILON >= 1.) return ML_POSINF;

    /* y := approx.value (Cornish-Fisher expansion) :  */
    z = qnorm(p, 0., 1., /*lower_tail*/TRUE, /*log_p*/FALSE);
#ifdef HAVE_NEARBYINT
    y = nearbyint(mu + sigma * (z + gamma * (z*z - 1) / 6));
#else
    y = round(mu + sigma * (z + gamma * (z*z - 1) / 6));
#endif

    z = ppois(y, lambda, /*lower_tail*/TRUE, /*log_p*/FALSE);

    /* fuzz to ensure left continuity; 1 - 1e-7 may lose too much : */
    p *= 1 - 64*DBL_EPSILON;

    /* If the mean is not too large a simple search is OK */
    if(lambda < 1e5) return do_search(y, &z, p, lambda, 1);
    /* Otherwise be a bit cleverer in the search */
    {
	double incr = floor(y * 0.001), oldincr;
	do {
	    oldincr = incr;
	    y = do_search(y, &z, p, lambda, incr);
	    incr = fmax2(1, floor(incr/100));
	} while(oldincr > 1 && incr > lambda*1e-15);
	return y;
    }
}
Example #2
0
double qunif(double p, double a, double b, int lower_tail, int log_p)
{
#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(a) || ISNAN(b))
        return p + a + b;
#endif
    R_Q_P01_check(p);
    if (b <= a ) ML_ERR_return_NAN;

    return a + R_DT_qIv(p) * (b - a);
}
Example #3
0
double qwilcox(double x, double m, double n, int lower_tail, int log_p)
{
    double c, p;

#ifdef IEEE_754
    if (ISNAN(x) || ISNAN(m) || ISNAN(n))
	return(x + m + n);
#endif
    if(!R_FINITE(x) || !R_FINITE(m) || !R_FINITE(n))
	ML_ERR_return_NAN;
    R_Q_P01_check(x);

    m = R_forceint(m);
    n = R_forceint(n);
    if (m <= 0 || n <= 0)
	ML_ERR_return_NAN;

    if (x == R_DT_0)
	return(0);
    if (x == R_DT_1)
	return(m * n);

    if(log_p || !lower_tail)
	x = R_DT_qIv(x); /* lower_tail,non-log "p" */

    int mm = (int) m, nn = (int) n;
    w_init_maybe(mm, nn);
    c = choose(m + n, n);
    p = 0;
    int q = 0;
    if (x <= 0.5) {
	x = x - 10 * DBL_EPSILON;
	for (;;) {
	    p += cwilcox(q, mm, nn) / c;
	    if (p >= x)
		break;
	    q++;
	}
    }
    else {
	x = 1 - x + 10 * DBL_EPSILON;
	for (;;) {
	    p += cwilcox(q, mm, nn) / c;
	    if (p > x) {
		q = (int) (m * n - q);
		break;
	    }
	    q++;
	}
    }

    return(q);
}
Example #4
0
double qsignrank(double x, double n, int lower_tail, int log_p)
{
    double f, p;

#ifdef IEEE_754
    if (ISNAN(x) || ISNAN(n))
	return(x + n);
#endif
    if (!R_FINITE(x) || !R_FINITE(n))
	ML_ERR_return_NAN;
    R_Q_P01_check(x);

    n = floor(n + 0.5);
    if (n <= 0)
	ML_ERR_return_NAN;

    if (x == R_DT_0)
	return(0);
    if (x == R_DT_1)
	return(n * (n + 1) / 2);

    if(log_p || !lower_tail)
	x = R_DT_qIv(x); /* lower_tail,non-log "p" */

    int nn = (int) n;
    w_init_maybe(nn);
    f = exp(- n * M_LN2);
    p = 0;
    int q = 0;
    if (x <= 0.5) {
	x = x - 10 * DBL_EPSILON;
	for (;;) {
	    p += csignrank(q, nn) * f;
	    if (p >= x)
		break;
	    q++;
	}
    }
    else {
	x = 1 - x + 10 * DBL_EPSILON;
	for (;;) {
	    p += csignrank(q, nn) * f;
	    if (p > x) {
		q = (int)(n * (n + 1) / 2 - q);
		break;
	    }
	    q++;
	}
    }

    return(q);
}
Example #5
0
double qnbinom(double p, double size, double prob, int lower_tail, int log_p)
{
    double P, Q, mu, sigma, gamma, z, y;

#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(size) || ISNAN(prob))
	return p + size + prob;
#endif
    if (prob <= 0 || prob > 1 || size <= 0) ML_ERR_return_NAN;
    /* FIXME: size = 0 is well defined ! */
    if (prob == 1) return 0;

    R_Q_P01_boundaries(p, 0, ML_POSINF);

    Q = 1.0 / prob;
    P = (1.0 - prob) * Q;
    mu = size * P;
    sigma = sqrt(size * P * Q);
    gamma = (Q + P)/sigma;

    /* Note : "same" code in qpois.c, qbinom.c, qnbinom.c --
     * FIXME: This is far from optimal [cancellation for p ~= 1, etc]: */
    if(!lower_tail || log_p) {
	p = R_DT_qIv(p); /* need check again (cancellation!): */
	if (p == R_DT_0) return 0;
	if (p == R_DT_1) return ML_POSINF;
    }
    /* temporary hack --- FIXME --- */
    if (p + 1.01*DBL_EPSILON >= 1.) return ML_POSINF;

    /* y := approx.value (Cornish-Fisher expansion) :  */
    z = qnorm(p, 0., 1., /*lower_tail*/TRUE, /*log_p*/FALSE);
    y = floor(mu + sigma * (z + gamma * (z*z - 1) / 6) + 0.5);

    z = pnbinom(y, size, prob, /*lower_tail*/TRUE, /*log_p*/FALSE);

    /* fuzz to ensure left continuity: */
    p *= 1 - 64*DBL_EPSILON;

    /* If the C-F value is not too large a simple search is OK */
    if(y < 1e5) return do_search(y, &z, p, size, prob, 1);
    /* Otherwise be a bit cleverer in the search */
    {
	double incr = floor(y * 0.001), oldincr;
	do {
	    oldincr = incr;
	    y = do_search(y, &z, p, size, prob, incr);
	    incr = fmax2(1, floor(incr/100));
	} while(oldincr > 1 && incr > y*1e-15);
	return y;
    }
}
Example #6
0
double qcauchy(double p, double location, double scale,
               int lower_tail, int log_p)
{
#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(location) || ISNAN(scale))
        return p + location + scale;
#endif
    if(!R_FINITE(p) || !R_FINITE(location) || !R_FINITE(scale))
        ML_ERR_return_NAN;
    R_Q_P01_check(p);
    if (scale <= 0) ML_ERR_return_NAN;

    return location + scale * tan(M_PI * (R_DT_qIv(p) - 0.5));
}
Example #7
0
File: qnt.c Project: csilles/cxxr
double qnt(double p, double df, double ncp, int lower_tail, int log_p)
{
    const static double accu = 1e-13;
    const static double Eps = 1e-11; /* must be > accu */

    double ux, lx, nx, pp;

#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(df) || ISNAN(ncp))
	return p + df + ncp;
#endif
    if (!R_FINITE(df)) ML_ERR_return_NAN;

    /* Was
     * df = floor(df + 0.5);
     * if (df < 1 || ncp < 0) ML_ERR_return_NAN;
     */
    if (df <= 0.0) ML_ERR_return_NAN;

    if(ncp == 0.0 && df >= 1.0) return qt(p, df, lower_tail, log_p);

    R_Q_P01_boundaries(p, ML_NEGINF, ML_POSINF);

    p = R_DT_qIv(p);

    /* Invert pnt(.) :
     * 1. finding an upper and lower bound */
    if(p > 1 - DBL_EPSILON) return ML_POSINF;
    pp = fmin2(1 - DBL_EPSILON, p * (1 + Eps));
    for(ux = fmax2(1., ncp);
	ux < DBL_MAX && pnt(ux, df, ncp, TRUE, FALSE) < pp;
	ux *= 2);
    pp = p * (1 - Eps);
    for(lx = fmin2(-1., -ncp);
	lx > -DBL_MAX && pnt(lx, df, ncp, TRUE, FALSE) > pp;
	lx *= 2);

    /* 2. interval (lx,ux)  halving : */
    do {
	nx = 0.5 * (lx + ux);
	if (pnt(nx, df, ncp, TRUE, FALSE) > p) ux = nx; else lx = nx;
    }
    while ((ux - lx) / fabs(nx) > accu);

    return 0.5 * (lx + ux);
}
Example #8
0
double qnbeta(double p, double a, double b, double ncp,
	      int lower_tail, int log_p)
{
    const static double accu = 1e-15;
    const static double Eps = 1e-14; /* must be > accu */

    double ux, lx, nx, pp;

#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(a) || ISNAN(b) || ISNAN(ncp))
	return p + a + b + ncp;
#endif
    if (!R_FINITE(a)) ML_ERR_return_NAN;

    if (ncp < 0. || a <= 0. || b <= 0.) ML_ERR_return_NAN;

    R_Q_P01_boundaries(p, 0, 1);

    p = R_DT_qIv(p);

    /* Invert pnbeta(.) :
     * 1. finding an upper and lower bound */
    if(p > 1 - DBL_EPSILON) return 1.0;
    pp = fmin2(1 - DBL_EPSILON, p * (1 + Eps));
    for(ux = 0.5;
	ux < 1 - DBL_EPSILON && pnbeta(ux, a, b, ncp, TRUE, FALSE) < pp;
	ux = 0.5*(1+ux));
    pp = p * (1 - Eps);
    for(lx = 0.5;
	lx > DBL_MIN && pnbeta(lx, a, b, ncp, TRUE, FALSE) > pp;
	lx *= 0.5);

    /* 2. interval (lx,ux)  halving : */
    do {
	nx = 0.5 * (lx + ux);
	if (pnbeta(nx, a, b, ncp, TRUE, FALSE) > p) ux = nx; else lx = nx;
    }
    while ((ux - lx) / nx > accu);

    return 0.5 * (ux + lx);
}
Example #9
0
// Returns both qbeta() and its "mirror" 1-qbeta(). Useful notably when qbeta() ~= 1
attribute_hidden void
qbeta_raw(double alpha, double p, double q, int lower_tail, int log_p,
	  int swap_01, // {TRUE, NA, FALSE}: if NA, algorithm decides swap_tail
	  double log_q_cut, /* if == Inf: return log(qbeta(..));
			       otherwise, if finite: the bound for
			       switching to log(x)-scale; see use_log_x */
	  int n_N,  // number of "unconstrained" Newton steps before switching to constrained
	  double *qb) // = qb[0:1] = { qbeta(), 1 - qbeta() }
{
    Rboolean
	swap_choose = (swap_01 == MLOGICAL_NA),
	swap_tail,
	log_, give_log_q = (log_q_cut == ML_POSINF),
	use_log_x = give_log_q, // or u < log_q_cut  below
	warned = FALSE, add_N_step = TRUE;
    int i_pb, i_inn;
    double a, la, logbeta, g, h, pp, p_, qq, r, s, t, w, y = -1.;
    volatile double u, xinbta;

    // Assuming p >= 0, q >= 0  here ...

    // Deal with boundary cases here:
    if(alpha == R_DT_0) {
#define return_q_0						\
	if(give_log_q) { qb[0] = ML_NEGINF; qb[1] = 0; }	\
	else {           qb[0] = 0;         qb[1] = 1; }	\
	return

	return_q_0;
    }
    if(alpha == R_DT_1) {
#define return_q_1						\
	if(give_log_q) { qb[0] = 0; qb[1] = ML_NEGINF; }	\
	else {           qb[0] = 1; qb[1] = 0;         }	\
	return

	return_q_1;
    }

    // check alpha {*before* transformation which may all accuracy}:
    if((log_p && alpha > 0) ||
       (!log_p && (alpha < 0 || alpha > 1))) { // alpha is outside
	R_ifDEBUG_printf("qbeta(alpha=%g, %g, %g, .., log_p=%d): %s%s\n",
			 alpha, p,q, log_p, "alpha not in ",
			 log_p ? "[-Inf, 0]" : "[0,1]");
	// ML_ERR_return_NAN :
	ML_ERROR(ME_DOMAIN, "");
	qb[0] = qb[1] = ML_NAN; return;
    }

    //  p==0, q==0, p = Inf, q = Inf  <==> treat as one- or two-point mass
    if(p == 0 || q == 0 || !R_FINITE(p) || !R_FINITE(q)) {
	// We know 0 < T(alpha) < 1 : pbeta() is constant and trivial in {0, 1/2, 1}
	R_ifDEBUG_printf(
	    "qbeta(%g, %g, %g, lower_t=%d, log_p=%d): (p,q)-boundary: trivial\n",
	    alpha, p,q, lower_tail, log_p);
	if(p == 0 && q == 0) { // point mass 1/2 at each of {0,1} :
	    if(alpha < R_D_half) { return_q_0; }
	    if(alpha > R_D_half) { return_q_1; }
	    // else:  alpha == "1/2"
#define return_q_half					\
	    if(give_log_q) qb[0] = qb[1] = -M_LN2;	\
	    else	   qb[0] = qb[1] = 0.5;		\
	    return

	    return_q_half;
	} else if (p == 0 || p/q == 0) { // point mass 1 at 0 - "flipped around"
	    return_q_0;
	} else if (q == 0 || q/p == 0) { // point mass 1 at 0 - "flipped around"
	    return_q_1;
	}
	// else:  p = q = Inf : point mass 1 at 1/2
	return_q_half;
    }

    /* initialize */
    p_ = R_DT_qIv(alpha);/* lower_tail prob (in any case) */
    // Conceptually,  0 < p_ < 1  (but can be 0 or 1 because of cancellation!)
    logbeta = lbeta(p, q);

    swap_tail = (swap_choose) ? (p_ > 0.5) : swap_01;
    // change tail; default (swap_01 = NA): afterwards 0 < a <= 1/2
    if(swap_tail) { /* change tail, swap  p <-> q :*/
	a = R_DT_CIv(alpha); // = 1 - p_ < 1/2
	/* la := log(a), but without numerical cancellation: */
	la = R_DT_Clog(alpha);
	pp = q; qq = p;
    }
    else {
	a = p_;
	la = R_DT_log(alpha);
	pp = p; qq = q;
    }

    /* calculate the initial approximation */

    /* Desired accuracy for Newton iterations (below) should depend on  (a,p)
     * This is from Remark .. on AS 109, adapted.
     * However, it's not clear if this is "optimal" for IEEE double prec.

     * acu = fmax2(acu_min, pow(10., -25. - 5./(pp * pp) - 1./(a * a)));

     * NEW: 'acu' accuracy NOT for squared adjustment, but simple;
     * ---- i.e.,  "new acu" = sqrt(old acu)
    */
    double acu = fmax2(acu_min, pow(10., -13. - 2.5/(pp * pp) - 0.5/(a * a)));
    // try to catch  "extreme left tail" early
    double tx, u0 = (la + log(pp) + logbeta) / pp; // = log(x_0)
    static const double
	log_eps_c = M_LN2 * (1. - DBL_MANT_DIG);// = log(DBL_EPSILON) = -36.04..
    r = pp*(1.-qq)/(pp+1.);

    t = 0.2;
    // FIXME: Factor 0.2 is a bit arbitrary;  '1' is clearly much too much.

    R_ifDEBUG_printf(
	"qbeta(%g, %g, %g, lower_t=%d, log_p=%d):%s\n"
	"  swap_tail=%d, la=%g, u0=%g (bnd: %g (%g)) ",
	alpha, p,q, lower_tail, log_p,
	(log_p && (p_ == 0. || p_ == 1.)) ? (p_==0.?" p_=0":" p_=1") : "",
	swap_tail, la, u0,
	(t*log_eps_c - log(fabs(pp*(1.-qq)*(2.-qq)/(2.*(pp+2.)))))/2.,
	 t*log_eps_c - log(fabs(r))
	);

    if(M_LN2 * DBL_MIN_EXP < u0 && // cannot allow exp(u0) = 0 ==> exp(u1) = exp(u0) = 0
       u0 < -0.01 && // (must: u0 < 0, but too close to 0 <==> x = exp(u0) = 0.99..)
       // qq <= 2 && // <--- "arbitrary"
       // u0 <  t*log_eps_c - log(fabs(r)) &&
       u0 < (t*log_eps_c - log(fabs(pp*(1.-qq)*(2.-qq)/(2.*(pp+2.)))))/2.)
    {
// TODO: maybe jump here from below, when initial u "fails" ?
// L_tail_u:
	// MM's one-step correction (cheaper than 1 Newton!)
	r = r*exp(u0);// = r*x0
	if(r > -1.) {
	    u = u0 - log1p(r)/pp;
	    R_ifDEBUG_printf("u1-u0=%9.3g --> choosing u = u1\n", u-u0);
	} else {
	    u = u0;
	    R_ifDEBUG_printf("cannot cheaply improve u0\n");
	}
	tx = xinbta = exp(u);
	use_log_x = TRUE; // or (u < log_q_cut)  ??
	goto L_Newton;
    }

    // y := y_\alpha in AS 64 := Hastings(1955) approximation of qnorm(1 - a) :
    r = sqrt(-2 * la);
    y = r - (const1 + const2 * r) / (1. + (const3 + const4 * r) * r);

    if (pp > 1 && qq > 1) { // use  Carter(1947), see AS 109, remark '5.'
	r = (y * y - 3.) / 6.;
	s = 1. / (pp + pp - 1.);
	t = 1. / (qq + qq - 1.);
	h = 2. / (s + t);
	w = y * sqrt(h + r) / h - (t - s) * (r + 5. / 6. - 2. / (3. * h));
	R_ifDEBUG_printf("p,q > 1 => w=%g", w);
	if(w > 300) { // exp(w+w) is huge or overflows
	    t = w+w + log(qq) - log(pp); // = argument of log1pexp(.)
	    u = // log(xinbta) = - log1p(qq/pp * exp(w+w)) = -log(1 + exp(t))
		(t <= 18) ? -log1p(exp(t)) : -t - exp(-t);
	    xinbta = exp(u);
	} else {
	    xinbta = pp / (pp + qq * exp(w + w));
	    u = // log(xinbta)
		- log1p(qq/pp * exp(w+w));
	}
    } else { // use the original AS 64 proposal, Scheffé-Tukey (1944) and Wilson-Hilferty
	r = qq + qq;
	/* A slightly more stable version of  t := \chi^2_{alpha} of AS 64
	 * t = 1. / (9. * qq); t = r * R_pow_di(1. - t + y * sqrt(t), 3);  */
	t = 1. / (3. * sqrt(qq));
	t = r * R_pow_di(1. + t*(-t + y), 3);// = \chi^2_{alpha} of AS 64
	s = 4. * pp + r - 2.;// 4p + 2q - 2 = numerator of new t = (...) / chi^2
	R_ifDEBUG_printf("min(p,q) <= 1: t=%g", t);
	if (t == 0 || (t < 0. && s >= t)) { // cannot use chisq approx
	    // x0 = 1 - { (1-a)*q*B(p,q) } ^{1/q}    {AS 65}
	    // xinbta = 1. - exp((log(1-a)+ log(qq) + logbeta) / qq);
	    double l1ma;/* := log(1-a), directly from alpha (as 'la' above):
			 * FIXME: not worth it? log1p(-a) always the same ?? */
	    if(swap_tail)
		l1ma = R_DT_log(alpha);
	    else
		l1ma = R_DT_Clog(alpha);
	    R_ifDEBUG_printf(" t <= 0 : log1p(-a)=%.15g, better l1ma=%.15g\n", log1p(-a), l1ma);
	    double xx = (l1ma + log(qq) + logbeta) / qq;
	    if(xx <= 0.) {
		xinbta = -expm1(xx);
		u = R_Log1_Exp (xx);// =  log(xinbta) = log(1 - exp(...A...))
	    } else { // xx > 0 ==> 1 - e^xx < 0 .. is nonsense
		R_ifDEBUG_printf(" xx=%g > 0: xinbta:= 1-e^xx < 0\n", xx);
		xinbta = 0; u = ML_NEGINF; /// FIXME can do better?
	    }
	} else {
	    t = s / t;
	    R_ifDEBUG_printf(" t > 0 or s < t < 0:  new t = %g ( > 1 ?)\n", t);
	    if (t <= 1.) { // cannot use chisq, either
		u = (la + log(pp) + logbeta) / pp;
		xinbta = exp(u);
	    } else { // (1+x0)/(1-x0) = t,  solved for x0 :
		xinbta = 1. - 2. / (t + 1.);
		u = log1p(-2. / (t + 1.));
	    }
	}
    }

    // Problem: If initial u is completely wrong, we make a wrong decision here
    if(swap_choose &&
       (( swap_tail && u >= -exp(  log_q_cut)) || // ==> "swap back"
	(!swap_tail && u >= -exp(4*log_q_cut) && pp / qq < 1000.))) { // ==> "swap now" (much less easily)
	// "revert swap" -- and use_log_x
	swap_tail = !swap_tail;
	R_ifDEBUG_printf(" u = %g (e^u = xinbta = %.16g) ==> ", u, xinbta);
	if(swap_tail) {
	    a = R_DT_CIv(alpha); // needed ?
	    la = R_DT_Clog(alpha);
	    pp = q; qq = p;
	}
	else {
	    a = p_;
	    la = R_DT_log(alpha);
	    pp = p; qq = q;
	}
	R_ifDEBUG_printf("\"%s\"; la = %g\n",
			 (swap_tail ? "swap now" : "swap back"), la);
	// we could redo computations above, but this should be stable
	u = R_Log1_Exp(u);
	xinbta = exp(u);

/* Careful: "swap now"  should not fail if
   1) the above initial xinbta is "completely wrong"
   2) The correction step can go outside (u_n > 0 ==>  e^u > 1 is illegal)
   e.g., for
	qbeta(0.2066, 0.143891, 0.05)
*/
    }

    if(!use_log_x)
	use_log_x = (u < log_q_cut);//(per default) <==> xinbta = e^u < 4.54e-5
    Rboolean
	bad_u = !R_FINITE(u),
	bad_init = bad_u || xinbta > p_hi;

    R_ifDEBUG_printf(" -> u = %g, e^u = xinbta = %.16g, (Newton acu=%g%s)\n",
	     u, xinbta, acu,
	     (bad_u ? ", ** bad u **" :
	      (use_log_x ? ", on u = log(x) scale" : "")));

    double u_n = 1.; // -Wall
    tx = xinbta; // keeping "original initial x" (for now)

    if(bad_u || u < log_q_cut) { /* e.g.
		    qbeta(0.21, .001, 0.05)
		    try "left border" quickly, i.e.,
		    try at smallest positive number: */
	w = pbeta_raw(DBL_very_MIN, pp, qq, TRUE, log_p);
	if(w > (log_p ? la : a)) {
	    R_ifDEBUG_printf(" quantile is left of smallest positive number; \"convergence\"\n");
	    if(log_p || fabs(w - a) < fabs(0 - a)) { // DBL_very_MIN is better than 0
		tx   = DBL_very_MIN;
		u_n  = DBL_log_v_MIN;// = log(DBL_very_MIN)
	    } else {
		tx   = 0.;
		u_n  = ML_NEGINF;
	    }
	    use_log_x = log_p; add_N_step = FALSE; goto L_return;
	}
	else {
	    R_ifDEBUG_printf(" pbeta(smallest pos.) = %g <= %g  --> continuing\n",
		     w, (log_p ? la : a));
	    if(u  < DBL_log_v_MIN) {
		u = DBL_log_v_MIN;// = log(DBL_very_MIN)
		xinbta = DBL_very_MIN;
	    }
	}
    }


    /* Sometimes the approximation is negative (and == 0 is also not "ok") */
    if (bad_init && !(use_log_x && tx > 0)) {
	if(u == ML_NEGINF) {
	    R_ifDEBUG_printf("  u = -Inf;");
	    u = M_LN2 * DBL_MIN_EXP;
	    xinbta = DBL_MIN;
	} else {
	    R_ifDEBUG_printf(" bad_init: u=%g, xinbta=%g;", u,xinbta);
	    xinbta = (xinbta > 1.1) // i.e. "way off"
		? 0.5 // otherwise, keep the respective boundary:
		: ((xinbta < p_lo) ? exp(u) : p_hi);
	    if(bad_u)
		u = log(xinbta);
	    // otherwise: not changing "potentially better" u than the above
	}
	R_ifDEBUG_printf(" -> (partly)new u=%g, xinbta=%g\n", u,xinbta);
    }

L_Newton:
    /* --------------------------------------------------------------------

     * Solve for x by a modified Newton-Raphson method, using pbeta_raw()
     */
    r = 1 - pp;
    t = 1 - qq;
    double wprev = 0., prev = 1., adj = 1.; // -Wall

    if(use_log_x) { // find  log(xinbta) -- work in  u := log(x) scale
	// if(bad_init && tx > 0) xinbta = tx;// may have been better

	for (i_pb=0; i_pb < 1000; i_pb++) {
	    // using log_p == TRUE  unconditionally here
	    // FIXME: if exp(u) = xinbta underflows to 0, like different formula pbeta_log(u, *)
	    y = pbeta_raw(xinbta, pp, qq, /*lower_tail = */ TRUE, TRUE);

	    /* w := Newton step size for   L(u) = log F(e^u)  =!= 0;   u := log(x)
	     *   =  (L(.) - la) / L'(.);  L'(u)= (F'(e^u) * e^u ) / F(e^u)
	     *   =  (L(.) - la)*F(.) / {F'(e^u) * e^u } =
	     *   =  (L(.) - la) * e^L(.) * e^{-log F'(e^u) - u}
	     *   =  ( y   - la) * e^{ y - u -log F'(e^u)}
		and  -log F'(x)= -log f(x) =  + logbeta + (1-p) log(x) + (1-q) log(1-x)
			       = logbeta + (1-p) u + (1-q) log(1-e^u)
	     */
	    w = (y == ML_NEGINF) // y = -Inf  well possible: we are on log scale!
		? 0. : (y - la) * exp(y - u + logbeta + r * u + t * R_Log1_Exp(u));
	    if(!R_FINITE(w))
		break;
	    if (i_pb >= n_N && w * wprev <= 0.)
		prev = fmax2(fabs(adj),fpu);
	    R_ifDEBUG_printf("N(i=%2d): u=%#20.16g, pb(e^u)=%#12.6g, w=%#15.9g, %s prev=%11g,",
			     i_pb, u, y, w, (w * wprev <= 0.) ? "new" : "old", prev);
	    g = 1;
	    for (i_inn=0; i_inn < 1000; i_inn++) {
		adj = g * w;
		// take full Newton steps at the beginning; only then safe guard:
		if (i_pb < n_N || fabs(adj) < prev) {
		    u_n = u - adj; // u_{n+1} = u_n - g*w
		    if (u_n <= 0.) { // <==> 0 <  xinbta := e^u  <= 1
			if (prev <= acu || fabs(w) <= acu) {
			    /* R_ifDEBUG_printf(" -adj=%g, %s <= acu  ==> convergence\n", */
			    /*	 -adj, (prev <= acu) ? "prev" : "|w|"); */
			    R_ifDEBUG_printf(" it{in}=%d, -adj=%g, %s <= acu  ==> convergence\n",
					     i_inn, -adj, (prev <= acu) ? "prev" : "|w|");
			    goto L_converged;
			}
			// if (u_n != ML_NEGINF && u_n != 1)
			break;
		    }
		}
		g /= 3;
	    }
	    // (cancellation in (u_n -u) => may differ from adj:
	    double D = fmin2(fabs(adj), fabs(u_n - u));
	    /* R_ifDEBUG_printf(" delta(u)=%g\n", u_n - u); */
	    R_ifDEBUG_printf(" it{in}=%d, delta(u)=%9.3g, D/|.|=%.3g\n",
			     i_inn, u_n - u, D/fabs(u_n + u));
	    if (D <= 4e-16 * fabs(u_n + u))
		goto L_converged;
	    u = u_n;
	    xinbta = exp(u);
	    wprev = w;
	} // for(i )

    } else

    for (i_pb=0; i_pb < 1000; i_pb++) {
	y = pbeta_raw(xinbta, pp, qq, /*lower_tail = */ TRUE, log_p);
	// delta{y} :   d_y = y - (log_p ? la : a);
#ifdef IEEE_754
	if(!R_FINITE(y) && !(log_p && y == ML_NEGINF))// y = -Inf  is ok if(log_p)
#else
	if (errno)
#endif
	{ // ML_ERR_return_NAN :
	    ML_ERROR(ME_DOMAIN, "");
	    qb[0] = qb[1] = ML_NAN; return;
	}


	/* w := Newton step size  (F(.) - a) / F'(.)  or,
	 * --   log: (lF - la) / (F' / F) = exp(lF) * (lF - la) / F'
	 */
	w = log_p
	    ? (y - la) * exp(y + logbeta + r * log(xinbta) + t * log1p(-xinbta))
	    : (y - a)  * exp(    logbeta + r * log(xinbta) + t * log1p(-xinbta));
	if (i_pb >= n_N && w * wprev <= 0.)
	    prev = fmax2(fabs(adj),fpu);
	R_ifDEBUG_printf("N(i=%2d): x0=%#17.15g, pb(x0)=%#17.15g, w=%#17.15g, %s prev=%g,",
			 i_pb, xinbta, y, w, (w * wprev <= 0.) ? "new" : "old", prev);
	g = 1;
	for (i_inn=0; i_inn < 1000;i_inn++) {
	    adj = g * w;
	    // take full Newton steps at the beginning; only then safe guard:
	    if (i_pb < n_N || fabs(adj) < prev) {
		tx = xinbta - adj; // x_{n+1} = x_n - g*w
		if (0. <= tx && tx <= 1.) {
		    if (prev <= acu || fabs(w) <= acu) {
			R_ifDEBUG_printf(" it{in}=%d, delta(x)=%g, %s <= acu  ==> convergence\n",
					 i_inn, -adj, (prev <= acu) ? "prev" : "|w|");
			goto L_converged;
		    }
		    if (tx != 0. && tx != 1)
			break;
		}
	    }
	    g /= 3;
	}
	R_ifDEBUG_printf(" it{in}=%d, delta(x)=%g\n", i_inn, tx - xinbta);
	if (fabs(tx - xinbta) <= 4e-16 * (tx + xinbta)) // "<=" : (.) == 0
	    goto L_converged;
	xinbta = tx;
	if(tx == 0) // "we have lost"
	    break;
	wprev = w;
    }

    /*-- NOT converged: Iteration count --*/
    warned = TRUE;
    ML_ERROR(ME_PRECISION, "qbeta");

L_converged:
    log_ = log_p || use_log_x; // only for printing
    R_ifDEBUG_printf(" %s: Final delta(y) = %g%s\n",
	     warned ? "_NO_ convergence" : "converged",
	     y - (log_ ? la : a), (log_ ? " (log_)" : ""));
    if((log_ && y == ML_NEGINF) || (!log_ && y == 0)) {
	// stuck at left, try if smallest positive number is "better"
	w = pbeta_raw(DBL_very_MIN, pp, qq, TRUE, log_);
	if(log_ || fabs(w - a) <= fabs(y - a)) {
	    tx  = DBL_very_MIN;
	    u_n = DBL_log_v_MIN;// = log(DBL_very_MIN)
	}
	add_N_step = FALSE; // not trying to do better anymore
    }
    else if(!warned && (log_ ? fabs(y - la) > 3 : fabs(y - a) > 1e-4)) {
	if(!(log_ && y == ML_NEGINF &&
	    // e.g. qbeta(-1e-10, .2, .03, log=TRUE) cannot get accurate ==> do NOT warn
	     pbeta_raw(DBL_1__eps, // = 1 - eps
		       pp, qq, TRUE, TRUE) > la + 2))
	    MATHLIB_WARNING2( // low accuracy for more platform independent output:
    "qbeta(a, *) =: x0 with |pbeta(x0,*%s) - alpha| = %.5g is not accurate",
	    (log_ ? ", log_" : ""), fabs(y - (log_ ? la : a)));
    }
L_return:
    if(give_log_q) { // ==> use_log_x , too
	if(!use_log_x) // (see if claim above is true)
	    MATHLIB_WARNING(
		"qbeta() L_return, u_n=%g;  give_log_q=TRUE but use_log_x=FALSE -- please report!",
		u_n);
	double r = R_Log1_Exp(u_n);
	if(swap_tail) {
	    qb[0] = r;	 qb[1] = u_n;
	} else {
	    qb[0] = u_n; qb[1] = r;
	}
    } else {
	if(use_log_x) {
	    if(add_N_step) {
		/* add one last Newton step on original x scale, e.g., for
		   qbeta(2^-98, 0.125, 2^-96) */
		xinbta = exp(u_n);
		y = pbeta_raw(xinbta, pp, qq, /*lower_tail = */ TRUE, log_p);
		w = log_p
		    ? (y - la) * exp(y + logbeta + r * log(xinbta) + t * log1p(-xinbta))
		    : (y - a)  * exp(    logbeta + r * log(xinbta) + t * log1p(-xinbta));
		tx = xinbta - w;
		R_ifDEBUG_printf(
		    "Final Newton correction(non-log scale): xinbta=%.16g, y=%g, w=%g. => new tx=%.16g\n",
		    xinbta, y, w, tx);
	    } else {
		if(swap_tail) {
		    qb[0] = -expm1(u_n); qb[1] =  exp  (u_n);
		} else {
		    qb[0] =  exp  (u_n); qb[1] = -expm1(u_n);
		}
		return;
	    }
	}
	if(swap_tail) {
	    qb[0] = 1 - tx;	qb[1] = tx;
	} else {
	    qb[0] = tx;	qb[1] = 1 - tx;
	}
    }
    return;
}
Example #10
0
double qbinom(double p, double n, double pr, int lower_tail, int log_p)
{
    double q, mu, sigma, gamma, z, y;

#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(n) || ISNAN(pr))
	return p + n + pr;
#endif
    if(!R_FINITE(n) || !R_FINITE(pr))
	ML_ERR_return_NAN;
    /* if log_p is true, p = -Inf is a legitimate value */
    if(!R_FINITE(p) && !log_p)
	ML_ERR_return_NAN;

    if(n != floor(n + 0.5)) ML_ERR_return_NAN;
    if (pr < 0 || pr > 1 || n < 0)
	ML_ERR_return_NAN;

    R_Q_P01_boundaries(p, 0, n);

    if (pr == 0. || n == 0) return 0.;

    q = 1 - pr;
    if(q == 0.) return n; /* covers the full range of the distribution */
    mu = n * pr;
    sigma = sqrt(n * pr * q);
    gamma = (q - pr) / sigma;

#ifdef DEBUG_qbinom
    REprintf("qbinom(p=%7g, n=%g, pr=%7g, l.t.=%d, log=%d): sigm=%g, gam=%g\n",
	     p,n,pr, lower_tail, log_p, sigma, gamma);
#endif
    /* Note : "same" code in qpois.c, qbinom.c, qnbinom.c --
     * FIXME: This is far from optimal [cancellation for p ~= 1, etc]: */
    if(!lower_tail || log_p) {
	p = R_DT_qIv(p); /* need check again (cancellation!): */
	if (p == 0.) return 0.;
	if (p == 1.) return n;
    }
    /* temporary hack --- FIXME --- */
    if (p + 1.01*DBL_EPSILON >= 1.) return n;

    /* y := approx.value (Cornish-Fisher expansion) :  */
    z = qnorm(p, 0., 1., /*lower_tail*/TRUE, /*log_p*/FALSE);
    y = floor(mu + sigma * (z + gamma * (z*z - 1) / 6) + 0.5);

    if(y > n) /* way off */ y = n;

#ifdef DEBUG_qbinom
    REprintf("  new (p,1-p)=(%7g,%7g), z=qnorm(..)=%7g, y=%5g\n", p, 1-p, z, y);
#endif
    z = pbinom(y, n, pr, /*lower_tail*/TRUE, /*log_p*/FALSE);

    /* fuzz to ensure left continuity: */
    p *= 1 - 64*DBL_EPSILON;

    if(n < 1e5) return do_search(y, &z, p, n, pr, 1);
    /* Otherwise be a bit cleverer in the search */
    {
	double incr = floor(n * 0.001), oldincr;
	do {
	    oldincr = incr;
	    y = do_search(y, &z, p, n, pr, incr);
	    incr = fmax2(1, floor(incr/100));
	} while(oldincr > 1 && incr > n*1e-15);
	return y;
    }
}
Example #11
0
double qbinom(double p, double n, double pr, int lower_tail, int log_p)
{
    double q, mu, sigma, gamma, z, y;

#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(n) || ISNAN(pr))
	return p + n + pr;
#endif
    if(!R_FINITE(p) || !R_FINITE(n) || !R_FINITE(pr))
	ML_ERR_return_NAN;
    R_Q_P01_check(p);

    if(n != floor(n + 0.5)) ML_ERR_return_NAN;
    if (pr < 0 || pr > 1 || n < 0)
	ML_ERR_return_NAN;

    if (pr == 0. || n == 0) return 0.;
    if (p == R_DT_0) return 0.;
    if (p == R_DT_1) return n;

    q = 1 - pr;
    if(q == 0.) return n; /* covers the full range of the distribution */
    mu = n * pr;
    sigma = sqrt(n * pr * q);
    gamma = (q - pr) / sigma;

#ifdef DEBUG_qbinom
    REprintf("qbinom(p=%7g, n=%g, pr=%7g, l.t.=%d, log=%d): sigm=%g, gam=%g\n",
	     p,n,pr, lower_tail, log_p, sigma, gamma);
#endif
    /* Note : "same" code in qpois.c, qbinom.c, qnbinom.c --
     * FIXME: This is far from optimal [cancellation for p ~= 1, etc]: */
    if(!lower_tail || log_p) {
	p = R_DT_qIv(p); /* need check again (cancellation!): */
	if (p == 0.) return 0.;
	if (p == 1.) return n;
    }
    /* temporary hack --- FIXME --- */
    if (p + 1.01*DBL_EPSILON >= 1.) return n;

    /* y := approx.value (Cornish-Fisher expansion) :  */
    z = qnorm(p, 0., 1., /*lower_tail*/TRUE, /*log_p*/FALSE);
    y = floor(mu + sigma * (z + gamma * (z*z - 1) / 6) + 0.5);
    if(y > n) /* way off */ y = n;

#ifdef DEBUG_qbinom
    REprintf("  new (p,1-p)=(%7g,%7g), z=qnorm(..)=%7g, y=%5g\n", p, 1-p, z, y);
#endif
    z = pbinom(y, n, pr, /*lower_tail*/TRUE, /*log_p*/FALSE);

    /* fuzz to ensure left continuity: */
    p *= 1 - 64*DBL_EPSILON;

/*-- Fixme, here y can be way off --
  should use interval search instead of primitive stepping down or up */

#ifdef maybe_future
    if((lower_tail && z >= p) || (!lower_tail && z <= p)) {
#else
    if(z >= p) {
#endif
			/* search to the left */
#ifdef DEBUG_qbinom
	REprintf("\tnew z=%7g >= p = %7g  --> search to left (y--) ..\n", z,p);
#endif
	for(;;) {
	    if(y == 0 ||
	       (z = pbinom(y - 1, n, pr, /*l._t.*/TRUE, /*log_p*/FALSE)) < p)
		return y;
	    y = y - 1;
	}
    }
    else {		/* search to the right */
#ifdef DEBUG_qbinom
	REprintf("\tnew z=%7g < p = %7g  --> search to right (y++) ..\n", z,p);
#endif
	for(;;) {
	    y = y + 1;
	    if(y == n ||
	       (z = pbinom(y, n, pr, /*l._t.*/TRUE, /*log_p*/FALSE)) >= p)
		return y;
	}
    }
}
Example #12
0
double qnorm5(double p, double mu, double sigma, int lower_tail, int log_p)
{
    double p_, q, r, val;

#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(mu) || ISNAN(sigma))
	return p + mu + sigma;
#endif
    R_Q_P01_boundaries(p, ML_NEGINF, ML_POSINF);

    if(sigma  < 0)	ML_ERR_return_NAN;
    if(sigma == 0)	return mu;

    p_ = R_DT_qIv(p);/* real lower_tail prob. p */
    q = p_ - 0.5;

#ifdef DEBUG_qnorm
    REprintf("qnorm(p=%10.7g, m=%g, s=%g, l.t.= %d, log= %d): q = %g\n",
	     p,mu,sigma, lower_tail, log_p, q);
#endif


/*-- use AS 241 --- */
/* double ppnd16_(double *p, long *ifault)*/
/*      ALGORITHM AS241  APPL. STATIST. (1988) VOL. 37, NO. 3

        Produces the normal deviate Z corresponding to a given lower
        tail area of P; Z is accurate to about 1 part in 10**16.

        (original fortran code used PARAMETER(..) for the coefficients
         and provided hash codes for checking them...)
*/
    if (fabs(q) <= .425) {/* 0.075 <= p <= 0.925 */
        r = .180625 - q * q;
	val =
            q * (((((((r * 2509.0809287301226727 +
                       33430.575583588128105) * r + 67265.770927008700853) * r +
                     45921.953931549871457) * r + 13731.693765509461125) * r +
                   1971.5909503065514427) * r + 133.14166789178437745) * r +
                 3.387132872796366608)
            / (((((((r * 5226.495278852854561 +
                     28729.085735721942674) * r + 39307.89580009271061) * r +
                   21213.794301586595867) * r + 5394.1960214247511077) * r +
                 687.1870074920579083) * r + 42.313330701600911252) * r + 1.);
    }
    else { /* closer than 0.075 from {0,1} boundary */

	/* r = min(p, 1-p) < 0.075 */
	if (q > 0)
	    r = R_DT_CIv(p);/* 1-p */
	else
	    r = p_;/* = R_DT_Iv(p) ^=  p */

	r = sqrt(- ((log_p &&
		     ((lower_tail && q <= 0) || (!lower_tail && q > 0))) ?
		    p : /* else */ log(r)));
        /* r = sqrt(-log(r))  <==>  min(p, 1-p) = exp( - r^2 ) */
#ifdef DEBUG_qnorm
	REprintf("\t close to 0 or 1: r = %7g\n", r);
#endif

        if (r <= 5.) { /* <==> min(p,1-p) >= exp(-25) ~= 1.3888e-11 */
            r += -1.6;
            val = (((((((r * 7.7454501427834140764e-4 +
                       .0227238449892691845833) * r + .24178072517745061177) *
                     r + 1.27045825245236838258) * r +
                    3.64784832476320460504) * r + 5.7694972214606914055) *
                  r + 4.6303378461565452959) * r +
                 1.42343711074968357734)
                / (((((((r *
                         1.05075007164441684324e-9 + 5.475938084995344946e-4) *
                        r + .0151986665636164571966) * r +
                       .14810397642748007459) * r + .68976733498510000455) *
                     r + 1.6763848301838038494) * r +
                    2.05319162663775882187) * r + 1.);
        }
        else { /* very close to  0 or 1 */
            r += -5.;
            val = (((((((r * 2.01033439929228813265e-7 +
                       2.71155556874348757815e-5) * r +
                      .0012426609473880784386) * r + .026532189526576123093) *
                    r + .29656057182850489123) * r +
                   1.7848265399172913358) * r + 5.4637849111641143699) *
                 r + 6.6579046435011037772)
                / (((((((r *
                         2.04426310338993978564e-15 + 1.4215117583164458887e-7)*
                        r + 1.8463183175100546818e-5) * r +
                       7.868691311456132591e-4) * r + .0148753612908506148525)
                     * r + .13692988092273580531) * r +
                    .59983220655588793769) * r + 1.);
        }

	if(q < 0.0)
	    val = -val;
        /* return (q >= 0.)? r : -r ;*/
    }
    return mu + sigma * val;
}
Example #13
0
File: qt.c Project: 6e441f9c/julia
double qt(double p, double ndf, int lower_tail, int log_p)
{
    const static double eps = 1.e-12;

    double P, q;
    Rboolean neg;

#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(ndf))
	return p + ndf;
#endif

    R_Q_P01_boundaries(p, ML_NEGINF, ML_POSINF);

    if (ndf <= 0) ML_ERR_return_NAN;

    if (ndf < 1) { /* based on qnt */
	const static double accu = 1e-13;
	const static double Eps = 1e-11; /* must be > accu */

	double ux, lx, nx, pp;
	
	int iter = 0;

	p = R_DT_qIv(p);

	/* Invert pt(.) :
	 * 1. finding an upper and lower bound */
	if(p > 1 - DBL_EPSILON) return ML_POSINF;
	pp = fmin2(1 - DBL_EPSILON, p * (1 + Eps));
	for(ux = 1.; ux < DBL_MAX && pt(ux, ndf, TRUE, FALSE) < pp; ux *= 2);
	pp = p * (1 - Eps);
	for(lx =-1.; lx > -DBL_MAX && pt(lx, ndf, TRUE, FALSE) > pp; lx *= 2);

	/* 2. interval (lx,ux)  halving
	   regula falsi failed on qt(0.1, 0.1)
	 */
	do {
	    nx = 0.5 * (lx + ux);
	    if (pt(nx, ndf, TRUE, FALSE) > p) ux = nx; else lx = nx;
	} while ((ux - lx) / fabs(nx) > accu && ++iter < 1000);

	if(iter >= 1000) ML_ERROR(ME_PRECISION, "qt");

	return 0.5 * (lx + ux);
    }

    /* Old comment:
     * FIXME: "This test should depend on  ndf  AND p  !!
     * -----  and in fact should be replaced by
     * something like Abramowitz & Stegun 26.7.5 (p.949)"
     *
     * That would say that if the qnorm value is x then
     * the result is about x + (x^3+x)/4df + (5x^5+16x^3+3x)/96df^2
     * The differences are tiny even if x ~ 1e5, and qnorm is not
     * that accurate in the extreme tails.
     */
    if (ndf > 1e20) return qnorm(p, 0., 1., lower_tail, log_p);

    P = R_D_qIv(p); /* if exp(p) underflows, we fix below */

    neg = (!lower_tail || P < 0.5) && (lower_tail || P > 0.5);
    if(neg)
	P = 2 * (log_p ? (lower_tail ? P : -expm1(p)) : R_D_Lval(p));
    else
	P = 2 * (log_p ? (lower_tail ? -expm1(p) : P) : R_D_Cval(p));
    /* 0 <= P <= 1 ; P = 2*min(P', 1 - P')  in all cases */

/* Use this if(log_p) only : */
#define P_is_exp_2p (lower_tail == neg) /* both TRUE or FALSE == !xor */

     if (fabs(ndf - 2) < eps) {	/* df ~= 2 */
	if(P > DBL_MIN) {
	    if(3* P < DBL_EPSILON) /* P ~= 0 */
		q = 1 / sqrt(P);
	    else if (P > 0.9)	   /* P ~= 1 */
		q = (1 - P) * sqrt(2 /(P * (2 - P)));
	    else /* eps/3 <= P <= 0.9 */
		q = sqrt(2 / (P * (2 - P)) - 2);
	}
	else { /* P << 1, q = 1/sqrt(P) = ... */
	    if(log_p)
		q = P_is_exp_2p ? exp(- p/2) / M_SQRT2 : 1/sqrt(-expm1(p));
	    else
		q = ML_POSINF;
	}
    }
    else if (ndf < 1 + eps) { /* df ~= 1  (df < 1 excluded above): Cauchy */
	if(P > 0)
	    q = 1/tan(P * M_PI_2);/* == - tan((P+1) * M_PI_2) -- suffers for P ~= 0 */

	else { /* P = 0, but maybe = 2*exp(p) ! */
	    if(log_p) /* 1/tan(e) ~ 1/e */
		q = P_is_exp_2p ? M_1_PI * exp(-p) : -1./(M_PI * expm1(p));
	    else
		q = ML_POSINF;
	}
    }
    else {		/*-- usual case;  including, e.g.,  df = 1.1 */
	double x = 0., y, log_P2 = 0./* -Wall */,
	    a = 1 / (ndf - 0.5),
	    b = 48 / (a * a),
	    c = ((20700 * a / b - 98) * a - 16) * a + 96.36,
	    d = ((94.5 / (b + c) - 3) / b + 1) * sqrt(a * M_PI_2) * ndf;

	Rboolean P_ok1 = P > DBL_MIN || !log_p,  P_ok = P_ok1;
	if(P_ok1) {
	    y = pow(d * P, 2 / ndf);
	    P_ok = (y >= DBL_EPSILON);
	}
	if(!P_ok) { /* log_p && P very small */
	    log_P2 = P_is_exp_2p ? p : R_Log1_Exp(p); /* == log(P / 2) */
	    x = (log(d) + M_LN2 + log_P2) / ndf;
	    y = exp(2 * x);
	}

	if ((ndf < 2.1 && P > 0.5) || y > 0.05 + a) { /* P > P0(df) */
	    /* Asymptotic inverse expansion about normal */
	    if(P_ok)
		x = qnorm(0.5 * P, 0., 1., /*lower_tail*/TRUE,  /*log_p*/FALSE);
	    else /* log_p && P underflowed */
		x = qnorm(log_P2,  0., 1., lower_tail,	        /*log_p*/ TRUE);

	    y = x * x;
	    if (ndf < 5)
		c += 0.3 * (ndf - 4.5) * (x + 0.6);
	    c = (((0.05 * d * x - 5) * x - 7) * x - 2) * x + b + c;
	    y = (((((0.4 * y + 6.3) * y + 36) * y + 94.5) / c
		  - y - 3) / b + 1) * x;
	    y = expm1(a * y * y);
	    q = sqrt(ndf * y);
	} else { /* re-use 'y' from above */

	    if(!P_ok && x < - M_LN2 * DBL_MANT_DIG) {/* 0.5* log(DBL_EPSILON) */
		/* y above might have underflown */
		q = sqrt(ndf) * exp(-x);
	    }
	    else {
		y = ((1 / (((ndf + 6) / (ndf * y) - 0.089 * d - 0.822)
			   * (ndf + 2) * 3) + 0.5 / (ndf + 4))
		     * y - 1) * (ndf + 1) / (ndf + 2) + 1 / y;
		q = sqrt(ndf * y);
	    }
	}


	/* Now apply 2-term Taylor expansion improvement (1-term = Newton):
	 * as by Hill (1981) [ref.above] */

	/* FIXME: This can be far from optimal when log_p = TRUE
	 *      but is still needed, e.g. for qt(-2, df=1.01, log=TRUE).
	 *	Probably also improvable when  lower_tail = FALSE */

	if(P_ok1) {
	    int it=0;
	    while(it++ < 10 && (y = dt(q, ndf, FALSE)) > 0 &&
		  R_FINITE(x = (pt(q, ndf, FALSE, FALSE) - P/2) / y) &&
		  fabs(x) > 1e-14*fabs(q))
		/* Newton (=Taylor 1 term):
		 *  q += x;
		 * Taylor 2-term : */
		q += x * (1. + x * q * (ndf + 1) / (2 * (q * q + ndf)));
	}
    }
    if(neg) q = -q;
    return q;
}
Example #14
0
double qbeta(double alpha, double p, double q, int lower_tail, int log_p)
{
    int swap_tail, i_pb, i_inn;
    double a, adj, logbeta, g, h, pp, p_, prev, qq, r, s, t, tx, w, y, yprev;
    double acu;
    volatile double xinbta;

    /* test for admissibility of parameters */

    if (isnan(p) || isnan(q) || isnan(alpha)){
        return p + q + alpha;
    }

    if(p < 0. || q < 0.){
      report_error("shape parameters for qbeta must be > 0.");
    }

    R_Q_P01_boundaries(alpha, 0, 1);

    p_ = R_DT_qIv(alpha);/* lower_tail prob (in any case) */

    if(log_p && (p_ == 0. || p_ == 1.))
        return p_; /* better than NaN or infinite loop;
                      FIXME: suboptimal, since -Inf < alpha ! */

    /* initialize */
    logbeta = lbeta(p, q);

    /* change tail if necessary;  afterwards   0 < a <= 1/2      */
    if (p_ <= 0.5) {
        a = p_; pp = p; qq = q; swap_tail = 0;
    } else { /* change tail, swap  p <-> q :*/
        a = (!lower_tail && !log_p)? alpha : 1 - p_;
        pp = q; qq = p; swap_tail = 1;
    }

    /* calculate the initial approximation */

    /* y := {fast approximation of} qnorm(1 - a) :*/
    r = sqrt(-2 * log(a));
    y = r - (const1 + const2 * r) / (1. + (const3 + const4 * r) * r);
    if (pp > 1 && qq > 1) {
        r = (y * y - 3.) / 6.;
        s = 1. / (pp + pp - 1.);
        t = 1. / (qq + qq - 1.);
        h = 2. / (s + t);
        w = y * sqrt(h + r) / h - (t - s) * (r + 5. / 6. - 2. / (3. * h));
        xinbta = pp / (pp + qq * exp(w + w));
    } else {
        r = qq + qq;
        t = 1. / (9. * qq);
        t = r * pow(1. - t + y * sqrt(t), 3.0);
        if (t <= 0.)
            xinbta = 1. - exp((log1p(-a)+ log(qq) + logbeta) / qq);
        else {
            t = (4. * pp + r - 2.) / t;
            if (t <= 1.)
                xinbta = exp((log(a * pp) + logbeta) / pp);
            else
                xinbta = 1. - 2. / (t + 1.);
        }
    }

    /* solve for x by a modified newton-raphson method, */
    /* using the function pbeta_raw */

    r = 1 - pp;
    t = 1 - qq;
    yprev = 0.;
    adj = 1;
    /* Sometimes the approximation is negative! */
    if (xinbta < lower)
        xinbta = 0.5;
    else if (xinbta > upper)
        xinbta = 0.5;

    /* Desired accuracy should depend on  (a,p)
     * This is from Remark .. on AS 109, adapted.
     * However, it's not clear if this is "optimal" for IEEE double prec.

     * acu = std::max<double>(acu_min, pow(10., -25. - 5./(pp * pp) - 1./(a * a)));

     * NEW: 'acu' accuracy NOT for squared adjustment, but simple;
     * ---- i.e.,  "new acu" = sqrt(old acu)

    */
    acu = std::max<double>(acu_min, pow(10., -13 - 2.5/(pp * pp) - 0.5/(a * a)));
    tx = prev = 0.;     /* keep -Wall happy */

    for (i_pb=0; i_pb < 1000; i_pb++) {
        y = pbeta_raw(xinbta, pp, qq, /*lower_tail = */ true, false);
        if(!std::isfinite(y)){
          report_error("algorithm blew up ni qbeta");
        }

        y = (y - a) *
            exp(logbeta + r * log(xinbta) + t * log1p(-xinbta));
        if (y * yprev <= 0.)
            prev = std::max<double>(fabs(adj),fpu);
        g = 1;
        for (i_inn=0; i_inn < 1000;i_inn++) {
            adj = g * y;
            if (fabs(adj) < prev) {
                tx = xinbta - adj; /* trial new x */
                if (tx >= 0. && tx <= 1) {
                    if (prev <= acu)    goto L_converged;
                    if (fabs(y) <= acu) goto L_converged;
                    if (tx != 0. && tx != 1)
                        break;
                }
            }
            g /= 3;
        }
        if (fabs(tx - xinbta) < 1e-15*xinbta) goto L_converged;
        xinbta = tx;
        yprev = y;
    }
    /*-- NOT converged: Iteration count --*/
    report_error("algorithm did not converge in qbeta");

L_converged:
    return swap_tail ? 1 - xinbta : xinbta;
}
Example #15
0
double qgamma(double p, double alpha, double scale, int lower_tail, int log_p)
/*			shape = alpha */
{
#define C7	4.67
#define C8	6.66
#define C9	6.73
#define C10	13.32

#define EPS1 1e-2
#define EPS2 5e-7/* final precision */
#define MAXIT 1000/* was 20 */

#define pMIN 1e-100    /* was 0.000002 = 2e-6 */
#define pMAX (1-1e-12)/* was 0.999998 = 1 - 2e-6 */

    const double
	i420  = 1./ 420.,
	i2520 = 1./ 2520.,
	i5040 = 1./ 5040;

    double p_, a, b, c, ch, g, p1, v;
    double p2, q, s1, s2, s3, s4, s5, s6, t, x;
    int i;

    /* test arguments and initialise */

#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(alpha) || ISNAN(scale))
	return p + alpha + scale;
#endif
    R_Q_P01_check(p);
    if (alpha <= 0) ML_ERR_return_NAN;

    /* FIXME: This (cutoff to {0, +Inf}) is far from optimal when log_p: */
    p_ = R_DT_qIv(p);/* lower_tail prob (in any case) */
    if (/* 0 <= */ p_ < pMIN) return 0;
    if (/* 1 >= */ p_ > pMAX) return BOOM::infinity();

    v = 2*alpha;

    c = alpha-1;
    g = lgammafn(alpha);/* log Gamma(v/2) */


/*----- Phase I : Starting Approximation */

#ifdef DEBUG_qgamma
    REprintf("qgamma(p=%7g, alpha=%7g, scale=%7g, l.t.=%2d, log_p=%2d): ",
	     p,alpha,scale, lower_tail, log_p);
#endif

    if(v < (-1.24)*R_DT_log(p)) {	/* for small chi-squared */

#ifdef DEBUG_qgamma
	REprintf(" small chi-sq.\n");
#endif
	/* FIXME: Improve this "if (log_p)" :
	 *	  (A*exp(b)) ^ 1/al */
	ch = pow(p_* alpha*exp(g+alpha*M_LN2), 1/alpha);
	if(ch < EPS2) {/* Corrected according to AS 91; MM, May 25, 1999 */
	    goto END;
	}

    } else if(v > 0.32) {	/*  using Wilson and Hilferty estimate */

	x = qnorm(p, 0, 1, lower_tail, log_p);
	p1 = 0.222222/v;
	ch = v*pow(x*sqrt(p1)+1-p1, 3);

#ifdef DEBUG_qgamma
	REprintf(" v > .32: Wilson-Hilferty; x = %7g\n", x);
#endif
	/* starting approximation for p tending to 1 */

	if( ch > 2.2*v + 6 )
	    ch = -2*(R_DT_Clog(p) - c*log(0.5*ch) + g);

    } else { /* for v <= 0.32 */

	ch = 0.4;
	a = R_DT_Clog(p) + g + c*M_LN2;
#ifdef DEBUG_qgamma
	REprintf(" v <= .32: a = %7g\n", a);
#endif
	do {
	    q = ch;
	    p1 = 1. / (1+ch*(C7+ch));
	    p2 = ch*(C9+ch*(C8+ch));
	    t = -0.5 +(C7+2*ch)*p1 - (C9+ch*(C10+3*ch))/p2;
	    ch -= (1- exp(a+0.5*ch)*p2*p1)/t;
	} while(fabs(q - ch) > EPS1*fabs(ch));
    }

#ifdef DEBUG_qgamma
    REprintf("\t==> ch = %10g:", ch);
#endif

/*----- Phase II: Iteration
 *	Call pgamma() [AS 239]	and calculate seven term taylor series
 */
    for( i=1 ; i <= MAXIT ; i++ ) {
	q = ch;
	p1 = 0.5*ch;
	p2 = p_ - pgamma(p1, alpha, 1, /*lower_tail*/true, /*log_p*/false);
#ifdef IEEE_754
	if(!R_FINITE(p2))
#else
	if(errno != 0)
#endif
		return numeric_limits<double>::quiet_NaN();

	t = p2*exp(alpha*M_LN2+g+p1-c*log(ch));
	b = t/ch;
	a = 0.5*t - b*c;
	s1 = (210+a*(140+a*(105+a*(84+a*(70+60*a))))) * i420;
	s2 = (420+a*(735+a*(966+a*(1141+1278*a)))) * i2520;
	s3 = (210+a*(462+a*(707+932*a))) * i2520;
	s4 = (252+a*(672+1182*a)+c*(294+a*(889+1740*a))) * i5040;
	s5 = (84+2264*a+c*(1175+606*a)) * i2520;
	s6 = (120+c*(346+127*c)) * i5040;
	ch += t*(1+0.5*t*s1-b*c*(s1-b*(s2-b*(s3-b*(s4-b*(s5-b*s6))))));
	if(fabs(q - ch) < EPS2*ch)
	    goto END;
    }
    ML_ERROR(ME_PRECISION);/* no convergence in MAXIT iterations */
 END:
    return 0.5*scale*ch;
}
Example #16
0
double qtukey(double p, double rr, double cc, double df,
	      int lower_tail, int log_p)
{
    const static double eps = 0.0001;
    const int maxiter = 50;

    double ans = 0.0, valx0, valx1, x0, x1, xabs;
    int iter;

#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(rr) || ISNAN(cc) || ISNAN(df)) {
	ML_ERROR(ME_DOMAIN, "qtukey");
	return p + rr + cc + df;
    }
#endif

    /* df must be > 1 ; there must be at least two values */
    if (df < 2 || rr < 1 || cc < 2) ML_ERR_return_NAN;

    R_Q_P01_boundaries(p, 0, ML_POSINF);

    p = R_DT_qIv(p); /* lower_tail,non-log "p" */

    /* Initial value */

    x0 = qinv(p, cc, df);

    /* Find prob(value < x0) */

    valx0 = ptukey(x0, rr, cc, df, /*LOWER*/TRUE, /*LOG_P*/FALSE) - p;

    /* Find the second iterate and prob(value < x1). */
    /* If the first iterate has probability value */
    /* exceeding p then second iterate is 1 less than */
    /* first iterate; otherwise it is 1 greater. */

    if (valx0 > 0.0)
	x1 = fmax2(0.0, x0 - 1.0);
    else
	x1 = x0 + 1.0;
    valx1 = ptukey(x1, rr, cc, df, /*LOWER*/TRUE, /*LOG_P*/FALSE) - p;

    /* Find new iterate */

    for(iter=1 ; iter < maxiter ; iter++) {
	ans = x1 - ((valx1 * (x1 - x0)) / (valx1 - valx0));
	valx0 = valx1;

	/* New iterate must be >= 0 */

	x0 = x1;
	if (ans < 0.0) {
	    ans = 0.0;
	    valx1 = -p;
	}
	/* Find prob(value < new iterate) */

	valx1 = ptukey(ans, rr, cc, df, /*LOWER*/TRUE, /*LOG_P*/FALSE) - p;
	x1 = ans;

	/* If the difference between two successive */
	/* iterates is less than eps, stop */

	xabs = fabs(x1 - x0);
	if (xabs < eps)
	    return ans;
    }

    /* The process did not converge in 'maxiter' iterations */
    ML_ERROR(ME_NOCONV, "qtukey");
    return ans;
}
Example #17
0
double qnbinom(double p, double n, double pr, int lower_tail, int log_p)
{
    double P, Q, mu, sigma, gamma, z, y;

#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(n) || ISNAN(pr))
	return p + n + pr;
#endif
    R_Q_P01_check(p);
    if (pr <= 0 || pr >= 1 || n <= 0) ML_ERR_return_NAN;

    if (p == R_DT_0) return 0;
    if (p == R_DT_1) return ML_POSINF;
    Q = 1.0 / pr;
    P = (1.0 - pr) * Q;
    mu = n * P;
    sigma = sqrt(n * P * Q);
    gamma = (Q + P)/sigma;

    /* Note : "same" code in qpois.c, qbinom.c, qnbinom.c --
     * FIXME: This is far from optimal [cancellation for p ~= 1, etc]: */
    if(!lower_tail || log_p) {
	p = R_DT_qIv(p); /* need check again (cancellation!): */
	if (p == R_DT_0) return 0;
	if (p == R_DT_1) return ML_POSINF;
    }
    /* temporary hack --- FIXME --- */
    if (p + 1.01*DBL_EPSILON >= 1.) return ML_POSINF;

    /* y := approx.value (Cornish-Fisher expansion) :  */
    z = qnorm(p, 0., 1., /*lower_tail*/TRUE, /*log_p*/FALSE);
    y = floor(mu + sigma * (z + gamma * (z*z - 1) / 6) + 0.5);

    z = pnbinom(y, n, pr, /*lower_tail*/TRUE, /*log_p*/FALSE);

    /* fuzz to ensure left continuity: */
    p *= 1 - 64*DBL_EPSILON;

/*-- Fixme, here y can be way off --
  should use interval search instead of primitive stepping down or up */

#ifdef maybe_future
    if((lower_tail && z >= p) || (!lower_tail && z <= p)) {
#else
    if(z >= p) {
#endif
			/* search to the left */
	for(;;) {
	    if(y == 0 ||
	       (z = pnbinom(y - 1, n, pr, /*l._t.*/TRUE, /*log_p*/FALSE)) < p)
		return y;
	    y = y - 1;
	}
    }
    else {		/* search to the right */

	for(;;) {
	    y = y + 1;
	    if((z = pnbinom(y, n, pr, /*l._t.*/TRUE, /*log_p*/FALSE)) >= p)
		return y;
	}
    }
}