README.md (4104B)
1 <!-- 2 3 @license Apache-2.0 4 5 Copyright (c) 2018 The Stdlib Authors. 6 7 Licensed under the Apache License, Version 2.0 (the "License"); 8 you may not use this file except in compliance with the License. 9 You may obtain a copy of the License at 10 11 http://www.apache.org/licenses/LICENSE-2.0 12 13 Unless required by applicable law or agreed to in writing, software 14 distributed under the License is distributed on an "AS IS" BASIS, 15 WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 16 See the License for the specific language governing permissions and 17 limitations under the License. 18 19 --> 20 21 # Logarithm of Probability Density Function 22 23 > [Lévy][levy-distribution] distribution logarithm of [probability density function (PDF)][pdf]. 24 25 <section class="intro"> 26 27 The [probability density function][pdf] (PDF) for a [Lévy][levy-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:levy_pdf" align="center" raw="f(x;\mu,c)=\begin{cases} \sqrt{\frac{c}{2\pi}}~~\frac{e^{ -\frac{c}{2(x-\mu)}}} {(x-\mu)^{3/2}} & \text{ for } x > \mu \\ 0 & \text{ otherwise} \end{cases}" alt="Probability density function (PDF) for a Lévy distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="f(x;\mu,c)=\begin{cases} \sqrt{\frac{c}{2\pi}}~~\frac{e^{ -\frac{c}{2(x-\mu)}}} {(x-\mu)^{3/2}} & \text{ for } x > \mu \\ 0 & \text{ otherwise} \end{cases}" data-equation="eq:levy_pdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/levy/logpdf/docs/img/equation_levy_pdf.svg" alt="Probability density function (PDF) for a Lévy distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `μ` is the location parameter and `c > 0` is the scale parameter. 39 40 </section> 41 42 <!-- /.intro --> 43 44 <section class="usage"> 45 46 ## Usage 47 48 ```javascript 49 var logpdf = require( '@stdlib/stats/base/dists/levy/logpdf' ); 50 ``` 51 52 #### logpdf( x, mu, c ) 53 54 Evaluates the logarithm of the [probability density function][pdf] (PDF) for a [Lévy][levy-distribution] distribution with parameters `mu` (location parameter) and `c` (scale parameter). 55 56 ```javascript 57 var y = logpdf( 2.0, 0.0, 1.0 ); 58 // returns ~-2.209 59 60 y = logpdf( -1.0, 4.0, 4.0 ); 61 // returns -Infinity 62 ``` 63 64 If provided `NaN` as any argument, the function returns `NaN`. 65 66 ```javascript 67 var y = logpdf( NaN, 0.0, 1.0 ); 68 // returns NaN 69 70 y = logpdf( 0.0, NaN, 1.0 ); 71 // returns NaN 72 73 y = logpdf( 0.0, 0.0, NaN ); 74 // returns NaN 75 ``` 76 77 If provided `c <= 0`, the function returns `NaN`. 78 79 ```javascript 80 var y = logpdf( 2.0, 0.0, -1.0 ); 81 // returns NaN 82 83 y = logpdf( 2.0, 0.0, 0.0 ); 84 // returns NaN 85 ``` 86 87 #### logpdf.factory( mu, c ) 88 89 Returns a function for evaluating the logarithm of the [probability density function][pdf] (PDF) of a [Lévy][levy-distribution] distribution with parameters `mu` (location parameter) and `c` (scale parameter). 90 91 ```javascript 92 var mylogpdf = logpdf.factory( 10.0, 2.0 ); 93 94 var y = mylogpdf( 11.0 ); 95 // returns ~-1.572 96 97 y = mylogpdf( 20.0 ); 98 // returns ~-4.126 99 ``` 100 101 </section> 102 103 <!-- /.usage --> 104 105 <section class="notes"> 106 107 ## Notes 108 109 - In virtually all cases, using the `logpdf` or `logcdf` functions is preferable to manually computing the logarithm of the `pdf` or `cdf`, respectively, since the latter is prone to overflow and underflow. 110 111 </section> 112 113 <!-- /.notes --> 114 115 <section class="examples"> 116 117 ## Examples 118 119 <!-- eslint no-undef: "error" --> 120 121 ```javascript 122 var randu = require( '@stdlib/random/base/randu' ); 123 var EPS = require( '@stdlib/constants/float64/eps' ); 124 var logpdf = require( '@stdlib/stats/base/dists/levy/logpdf' ); 125 126 var mu; 127 var c; 128 var x; 129 var y; 130 var i; 131 132 for ( i = 0; i < 10; i++ ) { 133 mu = randu() * 10.0; 134 x = ( randu()*10.0 ) + mu; 135 c = ( randu()*10.0 ) + EPS; 136 y = logpdf( x, mu, c ); 137 console.log( 'x: %d, µ: %d, c: %d, ln(f(x;µ,c)): %d', x, mu, c, y ); 138 } 139 ``` 140 141 </section> 142 143 <!-- /.examples --> 144 145 <section class="links"> 146 147 [levy-distribution]: https://en.wikipedia.org/wiki/L%C3%A9vy_distribution 148 149 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function 150 151 </section> 152 153 <!-- /.links -->