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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 isnan = require( '@stdlib/math/base/assert/is-nan' );
var atan2 = require( '@stdlib/math/base/special/atan2' );
// VARIABLES //
var ONE_OVER_PI = 0.3183098861837907;
// MAIN //
/**
* Evaluates the cumulative distribution function (CDF) for a Cauchy distribution with location parameter `x0` and scale parameter `gamma` at a value `x`.
*
* @param {number} x - input value
* @param {number} x0 - location parameter
* @param {PositiveNumber} gamma - scale parameter
* @returns {Probability} evaluated CDF
*
* @example
* var y = cdf( 4.0, 0.0, 2.0 );
* // returns ~0.852
*
* @example
* var y = cdf( 1.0, 0.0, 2.0 );
* // returns ~0.648
*
* @example
* var y = cdf( 1.0, 3.0, 2.0 );
* // returns 0.25
*
* @example
* var y = cdf( NaN, 0.0, 2.0 );
* // returns NaN
*
* @example
* var y = cdf( 1.0, 2.0, NaN );
* // returns NaN
*
* @example
* var y = cdf( 1.0, NaN, 3.0 );
* // returns NaN
*/
function cdf( x, x0, gamma ) {
if (
isnan( x ) ||
isnan( x0 ) ||
isnan( gamma ) ||
gamma <= 0.0
) {
return NaN;
}
// Below `z = -1`, `0.5 + atan(z)/pi` cancels more and more as `z` goes to `-infinity`; `atan2( gamma, x0-x )/pi` is the same quantity evaluated without the subtraction. Nearer the median the sum does not cancel and is the more accurate of the two...
if ( x-x0 < -gamma ) {
return ONE_OVER_PI * atan2( gamma, x0-x );
}
return ( ONE_OVER_PI * atan2( x-x0, gamma ) ) + 0.5;
}
// EXPORTS //
module.exports = cdf;
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