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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 pow = require( '@stdlib/math/base/special/pow' );
var ln = require( '@stdlib/math/base/special/ln' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var NINF = require( '@stdlib/constants/float64/ninf' );
var FLOAT64_SMALLEST_NORMAL = require( '@stdlib/constants/float64/smallest-normal' );
// MAIN //
/**
* Evaluates the natural logarithm of the probability density function (PDF) for a Rayleigh distribution with scale parameter `sigma` at a value `x`.
*
* @param {number} x - input value
* @param {NonNegativeNumber} sigma - scale parameter
* @returns {number} evaluated logPDF
*
* @example
* var y = logpdf( 0.3, 1.0 );
* // returns ~-1.249
*
* @example
* var y = logpdf( 2.0, 0.8 );
* // returns ~-1.986
*
* @example
* var y = logpdf( -1.0, 0.5 );
* // returns -Infinity
*
* @example
* var y = logpdf( 0.0, NaN );
* // returns NaN
*
* @example
* var y = logpdf( NaN, 2.0 );
* // returns NaN
*
* @example
* // Negative scale parameter:
* var y = logpdf( 2.0, -1.0 );
* // returns NaN
*/
function logpdf( x, sigma ) {
var s2i;
var lq;
var s2;
var x2;
var r;
var q;
if (
isnan( x ) ||
isnan( sigma ) ||
sigma < 0.0
) {
return NaN;
}
if ( sigma === 0.0 ) {
return ( x === 0.0 ) ? PINF : NINF;
}
if ( x < 0.0 || x === PINF ) {
return NINF;
}
s2 = pow( sigma, 2.0 );
s2i = 1.0 / s2;
x2 = pow( x, 2.0 );
q = s2i * x;
if (
s2 >= FLOAT64_SMALLEST_NORMAL &&
2.0 * s2 < PINF &&
x2 < PINF &&
q >= FLOAT64_SMALLEST_NORMAL
) {
return ln( q ) - ( x2 / ( 2.0 * s2 ) );
}
// For `sigma` or `x` far from `1`, forming `sigma^2` or `x^2` overflows or underflows, so evaluate via the ratio `x/sigma` instead. `ln( x/sigma^2 )` is taken directly while that ratio is normal, as subtracting two large logarithms cancels when `x` is close to `sigma^2`...
r = x / sigma;
q = r / sigma;
if ( q >= FLOAT64_SMALLEST_NORMAL && q < PINF ) {
lq = ln( q );
} else if ( r >= FLOAT64_SMALLEST_NORMAL ) {
lq = ln( r ) - ln( sigma );
} else {
lq = ( ln( x ) - ln( sigma ) ) - ln( sigma );
}
return lq - ( 0.5 * r * r );
}
// EXPORTS //
module.exports = logpdf;
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