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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 erfcinv = require( '@stdlib/math/base/special/erfcinv' );
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var round = require( '@stdlib/math/base/special/round' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
var cdf = require( '@stdlib/stats/base/dists/poisson/cdf' );
var SQRT2 = require( '@stdlib/constants/float64/sqrt-two' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var search = require( './search.js' );
// MAIN //
/**
* Evaluates the quantile function for a Poisson distribution with mean parameter `lambda` at a probability `p`.
*
* @param {Probability} p - input value
* @param {NonNegativeNumber} lambda - mean parameter
* @returns {NonNegativeInteger} evaluated quantile function
*
* @example
* var y = quantile( 0.5, 2.0 );
* // returns 2
*
* @example
* var y = quantile( 0.9, 4.0 );
* // returns 7
*
* @example
* var y = quantile( 0.1, 200.0 );
* // returns 182
*
* @example
* var y = quantile( 1.1, 0.0 );
* // returns NaN
*
* @example
* var y = quantile( -0.2, 0.0 );
* // returns NaN
*
* @example
* var y = quantile( NaN, 0.5 );
* // returns NaN
*
* @example
* var y = quantile( 0.0, NaN );
* // returns NaN
*
* @example
* var y = quantile( 0.5, Infinity );
* // returns NaN
*/
function quantile( p, lambda ) {
var sigmaInv;
var guess;
var sigma;
var corr;
var x2;
var x;
if ( isnan( lambda ) || lambda < 0.0 || lambda === PINF ) {
return NaN;
}
if ( isnan( p ) || p < 0.0 || p > 1.0 ) {
return NaN;
}
if ( lambda === 0.0 ) {
return 0.0;
}
if ( p === 0.0 ) {
return 0.0;
}
if ( p === 1.0 ) {
return PINF;
}
// Cornish-Fisher expansion:
sigma = sqrt( lambda );
sigmaInv = 1.0 / sigma;
if ( p < 0.5 ) {
x = -erfcinv( 2.0 * p ) * SQRT2;
} else {
x = erfcinv( 2.0 * ( 1.0 - p ) ) * SQRT2;
}
x2 = x * x;
// Skewness correction:
corr = x + (sigmaInv * ( x2 - 1.0 ) / 6.0);
guess = round( lambda + (sigma * corr) );
return ( cdf( guess, lambda ) >= p ) ?
search.left( guess, p, lambda ) :
search.right( guess, p, lambda );
}
// EXPORTS //
module.exports = quantile;
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