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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 sqrt = require( '@stdlib/math/base/special/sqrt' );
// MAIN //
/**
* Evaluates the quantile function for a normal distribution with mean `mu` and standard deviation `sigma` at a probability `p`.
*
* ## Method
*
* The quantile function is
*
* ```tex
* Q(p) = \mu + \sigma \sqrt{2} \operatorname{erf}^{-1}(2p-1) = \mu - \sigma \sqrt{2} \operatorname{erfc}^{-1}(2p)
* ```
*
* The function evaluates the second form. In the first, \\(2p-1\\) loses the low-order digits of small \\(p\\) and rounds to \\(-1\\) for \\(p < 2^{-54}\\), which returns \\(-\infty\\).
*
* @param {Probability} p - input value
* @param {number} mu - mean
* @param {NonNegativeNumber} sigma - standard deviation
* @returns {number} evaluated quantile function
*
* @example
* var y = quantile( 0.8, 0.0, 1.0 );
* // returns ~0.842
*
* @example
* var y = quantile( 0.5, 4.0, 2.0 );
* // returns 4.0
*
* @example
* var y = quantile( 1.1, 0.0, 1.0 );
* // returns NaN
*
* @example
* var y = quantile( -0.2, 0.0, 1.0 );
* // returns NaN
*
* @example
* var y = quantile( NaN, 0.0, 1.0 );
* // returns NaN
*
* @example
* var y = quantile( 0.0, NaN, 1.0 );
* // returns NaN
*
* @example
* var y = quantile( 0.0, 0.0, NaN );
* // returns NaN
*
* @example
* // Negative standard deviation:
* var y = quantile( 0.5, 0.0, -1.0 );
* // returns NaN
*/
function quantile( p, mu, sigma ) {
var A;
var B;
if (
isnan( mu ) ||
isnan( sigma ) ||
isnan( p ) ||
sigma < 0.0 ||
p < 0.0 ||
p > 1.0
) {
return NaN;
}
if ( sigma === 0.0 ) {
return mu;
}
A = mu;
B = sigma * sqrt( 2.0 );
return A - ( B * erfcinv( 2.0*p ) );
}
// EXPORTS //
module.exports = quantile;
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