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* @license Apache-2.0
*
* Copyright (c) 2018 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
'use strict';
// MODULES //
var floor = require( '@stdlib/math/base/special/floor' );
var sign = require( '@stdlib/math/base/special/signum' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
var abs = require( '@stdlib/math/base/special/abs' );
var ln = require( '@stdlib/math/base/special/ln' );
var correction = require( './correction.js' );
// VARIABLES //
var ONE_SIXTH = 1.0 / 6.0;
// MAIN //
/**
* Generates a binomially distributed pseudorandom number.
*
* ## References
*
* - Hörmann, Wolfgang. 1993. "The generation of binomial random variates." _Journal of Statistical Computation and Simulation_ 46 (1-2): 101–10. doi:[10.1080/00949659308811496][@hormann:1993a].
*
* [@hormann:1993a]: http://dx.doi.org/10.1080/00949659308811496
*
* @private
* @param {PRNG} rand - PRNG for uniformly distributed numbers
* @param {PositiveInteger} n - number of trials
* @param {Probability} p - success probability
* @returns {NonNegativeInteger} pseudorandom number
*/
function sample( rand, n, p ) {
var alpha;
var urvr;
var snpq;
var npq;
var rho;
var tmp;
var nm;
var nr;
var us;
var km;
var nk;
var vr;
var a;
var b;
var c;
var f;
var h;
var i;
var k;
var m;
var q;
var r;
var t;
var u;
var v;
var x;
m = floor( (n + 1) * p );
nm = n - m + 1;
q = 1.0 - p;
r = p / q;
nr = (n + 1) * r;
npq = n * p * q;
snpq = sqrt( npq );
b = 1.15 + (2.53 * snpq);
a = -0.0873 + (0.0248*b) + (0.01*p);
c = (n*p) + 0.5;
alpha = (2.83 + (5.1/b)) * snpq;
vr = 0.92 - (4.2/b);
urvr = 0.86 * vr;
h = (m + 0.5) * ln( (m+1) / (r*nm) );
h += correction( m ) + correction( n-m );
while ( true ) {
v = rand();
if ( v <= urvr ) {
u = (v/vr) - 0.43;
r = (u * ( (2.0*a / (0.5 - abs(u))) + b )) + c;
return floor( r );
}
if ( v >= vr ) {
u = rand() - 0.5;
} else {
u = (v/vr) - 0.93;
u = (sign( u ) * 0.5) - u;
v = vr * rand();
}
us = 0.5 - abs(u);
k = floor( (u * ( (2.0*a/us) + b )) + c );
if ( k < 0 || k > n ) {
// Try again...
continue;
}
v = v * alpha / ( (a/(us*us)) + b );
km = abs( k - m );
if ( km > 15 ) {
v = ln( v );
rho = km / npq;
tmp = ( (km/3) + 0.625 ) * km;
tmp += ONE_SIXTH;
tmp /= npq;
rho *= tmp + 0.5;
t = -(km * km) / (2.0 * npq);
if ( v < t - rho ) {
return k;
}
if ( v <= t + rho ) {
nk = n - k + 1;
x = h + ( (n+1)*ln( nm/nk ) );
x += (k+0.5) * ln( nk*r/(k+1) );
x += -(correction( k ) + correction( n-k ));
if ( v <= x ) {
return k;
}
}
} else {
f = 1.0;
if ( m < k ) {
for ( i = m; i <= k; i++ ) {
f *= (nr/i) - r;
}
} else if ( m > k ) {
for ( i = k; i <= m; i++ ) {
v *= (nr/i) - r;
}
}
if ( v <= f ) {
return k;
}
}
}
}
// EXPORTS //
module.exports = sample;
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