Home Manual Packages Global Index Keywords Quick Reference ``` /* RANDOM.I Random numbers with various distributions. \$Id\$ */ /* Copyright (c) 1996. The Regents of the University of California. All rights reserved. */ /* Contents: random_x - avoids the 2.e9 bins of the random function at a cost of calling random twice, can be used as a drop-in replacement for random random_u - convenience routine to give uniform deviate on an interval other than (0,1) random_n - return gaussian deviate random_ipq - arbitrary piecewise linear deviate, with optional power law or exponential tails random_rej - generic implementation of rejection method, can be used either in conjunction with a piecewise linear bounding function, or an arbitrary bounding function (in the latter case, the inverse of the integral of the bounding function must be supplied as well) In all cases, these routines accept a dimlist or arguments to determine the dimensions of the returned random deviates. Furthermore, in all cases you will get back the same sequence of deviates no matter what the dimensionality of the calls -- for example, if at some point the call to random_n(5) returns [.3,-.1,-.8,1.1,-.9], then if instead random_n(3) followed by random_n(2) had been called, the return values would have been [.3,-.1,-.8] and [1.1,-.9]. */ /* ------------------------------------------------------------------------ */ func build_dimlist (&dimlist, arg) /* DOCUMENT build_dimlist, dimlist, next_argument build a DIMLIST, as used in the array function. Use like this: func your_function(arg1, arg2, etc, dimlist, ..) { while (more_args()) build_dimlist, dimlist, next_arg(); ... } After this, DIMLIST will be an array of the form [#dims, dim1, dim2, ...], compounded from the multiple arguments in the same way as the array function. If no DIMLIST arguments given, DIMLIST will be [] instead of [0], which will act the same in most situations. If that possibility is unacceptible, you may add if (is_void(dimlist)) dimlist= [0]; after the while loop. */ { if (is_void(dimlist)) dimlist= [0]; else if (!dimsof(dimlist)(1)) dimlist= [1,dimlist]; if (is_void(arg)) return; if (!dimsof(arg)(1)) { grow, dimlist, arg; dimlist(1)+= 1; } else { n= arg(1); grow, dimlist, arg(2:1+n); dimlist(1)+= n; } } /* ------------------------------------------------------------------------ */ func random_x (dimlist, ..) /* DOCUMENT random_x(dimlist) same as random(DIMLIST), except that random_x calls random twice at each point, to avoid the defect that random only can produce about 2.e9 numbers on the interval (0.,1.) (see random for an explanation of these bins). You may set random=random_x to get these "better" random numbers in every call to random. Unlike random, there is a chance in 1.e15 or so that random_x may return exactly 1.0 or 0.0 (the latter may not be possible with IEEE standard arithmetic, while the former apparently is). Since cosmic rays are far more likely, you may as well not worry about this. Also, because of rounding errors, some bit patterns may still be more likely than others, but the 0.5e-9 wide bins of random will be absent. SEE ALSO: random */ { while (more_args()) build_dimlist, dimlist, next_arg(); if (is_void(dimlist)) { dimlist= [1,2]; } else if (!dimsof(dimlist)(1)) { dimlist= [2,2,dimlist]; } else { dimlist= grow([dimlist(1)+1],dimlist); dimlist(2)= 2; } r= random_0(dimlist); return r(1,..) + (r(2,..)-0.5)/2147483562. } if (is_void(random_0)) random_0= random; /* ------------------------------------------------------------------------ */ func random_u (a, b, dimlist, ..) /* DOCUMENT random_u(a, b, dimlist) return uniformly distributed random numbers between A and B. (Will never exactly equal A or B.) The DIMLIST is as for the array function. Same as (b-a)*random(dimlist)+a. If A==0, you are better off just writing B*random(dimlist). SEE ALSO: random, random_x, random_n, random_ipq, random_rej */ { while (more_args()) build_dimlist, dimlist, next_arg(); return (b-a)*random(dimlist)+a; } /* ------------------------------------------------------------------------ */ func random_n (dimlist, ..) /* DOCUMENT random_n(dimlist) returns an array of normally distributed random double values with the given DIMLIST (see array function, nil for a scalar result). The mean is 0.0 and the standard deviation is 1.0. The algorithm follows the Box-Muller method (see Numerical Recipes by Press et al.). SEE ALSO: random, random_x, random_u, random_ipq, random_rej */ { while (more_args()) build_dimlist, dimlist, next_arg(); a= array(0.0, dimlist); scalar= !dimsof(a)(1); if (scalar) a= [a]; /* work around long-standing Yorick bug */ /* crucial feature of this algorithm is that the same sequence * of random numbers is returned independent of the number requested * each time, just like random itself */ na= numberof(a); np= numberof(_random_n_prev); n= (na-np+1)/2; nr= 2*n; if (n) { x= random(2,n); r= sqrt(-2.0*log(x(1,))); theta= 2.0*pi*x(2,); x(1,)= r*cos(theta); x(2,)= r*sin(theta); a(np+1:na)= x(1:na-np); } if (np) { a(1)= _random_n_prev; _random_n_prev= []; } if (na-np=0 of same number and dimensionality as x, normalized so that the integral of target_dist(x) from -infinity to +infinity is 1.0. The BOUNDING_DIST function must have the same calling sequence as TARGET_DIST: func bounding_dist(x) returning b(x)>=u(x) everywhere. Since u(x) is normalized, the integral of b(x) must be >=1.0. Finally, BOUNDING_RAND is a function which converts an array of uniformly distributed random numbers on (0,1) -- as returned by random -- into an array distributed according to BOUNDING_DIST: func bounding_rand(uniform_x_01) Mathematically, BOUNDING_RAND is the inverse of the integral of BOUNDING_DIST from -infinity to x, with its input scaled to (0,1). If BOUNDING_DIST is not a function, then it must be an IPQ_MODEL returned by the ipq_setup function. In this case BOUNDING_RAND is omitted -- ipq_compute will be used automatically. SEE ALSO: random, random_x, random_u, random_n, random_ipq, ipq_setup */ { if (is_func(bound)) { brand= dimlist; dimlist= more_args()? next_arg() : []; } while (more_args()) build_dimlist, dimlist, next_arg(); if (!is_func(target) || (!is_void(brand) && !is_func(brand)) || (!is_func(bound) && structof(bound)!=pointer)) error, "improper calling sequence, try help,random_rej"; if (!is_func(bound)) ymax= (*bound(4))(1); /* build result to requested shape */ x= array(0.0, dimlist); nreq= nx= numberof(x); ix= 1; do { /* get 25% more pairs of random numbers than nreq in order * to allow for some to be rejected -- should actually go for * integral(bounding_dist) times nreq, but don't know what * that is -- could refine the estimate as each pass gets a * better notion of the fraction rejected, but don't bother */ r= random(2, max(nreq+nreq/4,10)); /* first get xx distributed according to bounding_rand, * then accept according to the second random number * continue until at least one is accepted */ for (xx=[] ; !numberof(xx) ; xx=xx(list)) { if (is_func(bound)) { xx= brand(r(1,..)); list= where(bound(xx)*r(2,..) <= target(xx)); } else { xx= ipq_compute(bound, ymax*r(1,..)); list= where(ipq_function(bound,xx)*r(2,..) <= target(xx)); } } nxx= numberof(xx); if (nxx>nreq) { xx= xx(1:nreq); nxx= nreq; } nreq-= nxx; x(ix:nx-nreq)= xx; ix+= nxx; } while (nreq); return x; } func ipq_setup (x,u,power=,slope=) /* DOCUMENT model= ipq_setup(x, u) or model= ipq_setup(x, u, power=[pleft,prght]) or model= ipq_setup(x, u, power=[pleft,prght], slope=[sleft,srght]) compute a model for the ipq_compute function, which computes the inverse of a piecewise quadratic function. This function occurs when computing random numbers distributed according to a piecewise linear function. The piecewise linear function is u(x), determined by the discrete points X and U input to ipq_setup. None of the values of U may be negative, and X must be strictly increasing, X(i)0 while SRGHT<0. If either power is greater than or equal to 100, an exponential tail will be used. As a convenience, you may also specify PLEFT or PRGHT of 0 to get an exponential tail. Note: ipq_function(model, xp) returns the function values u(xp) at the points xp, including the tails (if any). ipq_compute(model, yp) returns the xp for which (integral from -infinity to xp) of u(x) equals yp; i.e.- the inverse of the piecewise quadratic. SEE ALSO: random_ipq, random_rej */ { x= double(x); u= double(u); if (dimsof(x)(1)!=1 || numberof(x)<2 || dimsof(u)(1)!=1 || numberof(u)!=numberof(x) || anyof(u<0.)) error, "bad U or X arrays"; /* compute the integral of u(x), starting from x(1), * both at the given points x and at the midpoints of the intervals * integ(u,x,xx) is the basic piecewise quadratic function */ bins= (u(zcen)*x(dif))(cum); cens= x(pcen); /* right shape, wrong values */ yc= integ(u,x, x(zcen)); dy= bins(dif); /* note that these cens are constrained to lie between -1 and +1 */ cens(2:-1)= 4.*(yc-bins(1:-1))/(dy+!dy) - 2.; ymax= bins(0); if (!is_void(power)) { if (dimsof(power)(1)!=1 || numberof(power)!=2 || anyof(power<=1.&power!=0.)) error, "illegal power= keyword"; if (!power(1) || !u(1)) power(1)= 100.; if (!power(0) || !u(0)) power(0)= 100.; if (is_void(slope)) slope= [(u(2)-u(1))/(x(2)-x(1)), (u(0)-u(-1))/(x(0)-x(-1))]; if (dimsof(slope)(1)!=1 || numberof(slope)!=2 || slope(1)<0. || slope(0)>0.) error, "illegal slope= keyword, or upward slope at endpoint"; cens(1)= u(1)? slope(1)/u(1) : 1000./(x(2)-x(1)); cens(0)= u(0)? -slope(0)/u(1) : 1000./(x(0)-x(-1)); yi= u(1)/cens(1); if (power(1)<100.) yi*= power(1)/(power(1)-1.); ymax+= yi; bins+= yi; yi= u(0)/cens(0); if (power(0)<100.) yi*= power(0)/(power(0)-1.); ymax+= yi; } else { power= [100.,100.]; cens(1)= 1000./(x(2)-x(1)); cens(0)= 1000./(x(0)-x(-1)); } parm= [ymax, power(1), power(0)]; return [&bins, &x, &cens, &parm, &u]; } /* ------------------------------------------------------------------------ */ func ipq_compute (model, y) { /* * model= [&bins, &vals, &cens, &parm] where: * bins values of y, a piecewise quadratic function of x * vals values of x that go with bins * cens 4*(yc-y0)/(y1-y0) - 2 where yc is value of y at * x=vals(pcen), except for first and last points * which are du/dx / u0 for the extrapolation model * parm [ymax, left_power, right_power] * maximum possible value of y, and * [left,right] powers (>1.0) of x for extrap. model */ local bins, vals, cens, parm; eq_nocopy, bins, *model(1); eq_nocopy, vals, *model(2); eq_nocopy, cens, *model(3); eq_nocopy, parm, *model(4); i= digitize(y, bins); mask0= (i>1); list= where(mask0); if (numberof(list)) { yy= y(list); ii= i(list); mask= (ii<=numberof(bins)); list= where(mask); if (numberof(list)) { /* handle piecewise quadratic part */ j= ii(list); yb= bins(j); xb= vals(j); aa= cens(j); /* 4*(yc-y0)/(y1-y0) - 2 */ j-= 1; ya= bins(j); xa= vals(j); bb= 0.5*(1.+aa); yq= (yy(list)-ya)/(yb-ya); xq= bb+sqrt(bb*bb-aa*yq); xq= xa + (xb-xa)*( yq/(xq+!xq) ); } list= where(!mask); if (numberof(list)) { /* handle right tail */ ymax= parm(1); if (ymax > bins(0)) { yy= (ymax - yy(list))/(ymax - bins(0)); xa= vals(0); aa= cens(0); /* du/dx / u0 */ pp= parm(2); /* power */ yy0= yy<=0.0; yy= max(yy,0.0)+yy0; if (pp>=100.) xt= -log(yy)/aa; else xt= (pp/aa) * (yy^(1./(1.-pp)) - 1.0); xt+= xa + 1.e9*yy0*(xa-vals(1)); } else { xt= array(vals(0), numberof(list)); } } xq= merge(xq, xt, mask); } list= where(!mask0); if (numberof(list)) { /* handle left tail */ if (bins(1)) { yy= y(list)/bins(1); xa= vals(1); aa= cens(1); /* du/dx / u0 */ pp= parm(3); /* power */ yy0= yy<=0.0; yy= max(yy,0.0)+yy0; if (pp>=100.) xt= log(yy)/aa; else xt= (pp/aa) * (1.0 - yy^(1./(1.-pp))); xt+= xa - 1.e9*yy0*(vals(0)-xa); } else { xt= array(vals(1), numberof(list)); } } else { xt= []; } return merge(xq, xt, mask0); } func ipq_function (model, x) { /* * model= [&bins, &vals, &cens, &parm, &valu] where: * bins values of y, a piecewise quadratic function of x * vals values of x that go with bins * cens 4*(yc-y0)/(y1-y0) - 2 where yc is value of y at * x=vals(pcen), except for first and last points * which are du/dx / u0 for the extrapolation model * parm [ymax, left_power, right_power] * maximum possible value of y, and * [left,right] powers (>1.0) of x for extrap. model * valu values of u that go with x */ local vals, cens, parm, valu; eq_nocopy, vals, *model(2); eq_nocopy, cens, *model(3); eq_nocopy, parm, *model(4); eq_nocopy, valu, *model(5); i= digitize(x, vals); mask0= (i>1); list= where(mask0); if (numberof(list)) { xx= x(list); ii= i(list); mask= (ii<=numberof(vals)); list= where(mask); if (numberof(list)) { /* handle piecewise linear part */ uq= interp(valu, vals, xx(list)); } list= where(!mask); if (numberof(list)) { /* handle right tail */ if (valu(0)) { xx= xx(list) - vals(0); aa= cens(0); /* du/dx / u0 */ pp= parm(2); /* power */ if (pp>=100.) ut= valu(0)*exp(-aa*xx); else ut= valu(0) / (1. + (aa/pp)*xx)^pp; } else { ut= array(0.0, numberof(list)); } } uq= merge(uq, ut, mask); } list= where(!mask0); if (numberof(list)) { /* handle left tail */ if (valu(1)) { xx= vals(1) - x(list); aa= cens(1); /* du/dx / u0 */ pp= parm(3); /* power */ if (pp>=100.) ut= valu(1)*exp(-aa*xx); else ut= valu(1) / (1. + (aa/pp)*xx)^pp; } else { ut= array(0.0, numberof(list)); } } else { ut= []; } return merge(uq, ut, mask0); } /* ------------------------------------------------------------------------ */ ```