README.md (4196B)
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 Cumulative Distribution Function 22 23 > [Lévy][levy-distribution] distribution logarithm of [cumulative distribution function][cdf]. 24 25 <section class="intro"> 26 27 The [cumulative distribution function][cdf] for a [Lévy][levy-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:levy_cdf" align="center" raw="F(x;\mu,b) = \begin{cases} \operatorname{erfc}\left(\sqrt{\frac{c}{2(x-\mu)}}\right) & \text{ for } x > \mu \\ 0 & \text{ otherwise } \end{cases}" alt="Cumulative distribution function for a Lévy distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="F(x;\mu,b) = \begin{cases} \operatorname{erfc}\left(\sqrt{\frac{c}{2(x-\mu)}}\right) & \text{ for } x > \mu \\ 0 & \text{ otherwise } \end{cases}" data-equation="eq:levy_cdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/levy/logcdf/docs/img/equation_levy_cdf.svg" alt="Cumulative distribution function for a Lévy distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `mu` is the location parameter and `b > 0` is the scale parameter. 39 40 </section> 41 42 <!-- /.intro --> 43 44 <section class="usage"> 45 46 ## Usage 47 48 ```javascript 49 var logcdf = require( '@stdlib/stats/base/dists/levy/logcdf' ); 50 ``` 51 52 #### logcdf( x, mu, c ) 53 54 Evaluates the logarithm of the [cumulative distribution function][cdf] (CDF) for a [Lévy][levy-distribution] distribution with parameters `mu` (location parameter) and `c > 0` (scale parameter). 55 56 ```javascript 57 var y = logcdf( 2.0, 0.0, 1.0 ); 58 // returns ~-0.735 59 60 y = logcdf( 12.0, 10.0, 3.0 ); 61 // returns ~-1.51 62 63 y = logcdf( 9.0, 10.0, 3.0 ); 64 // returns -Infinity 65 ``` 66 67 If provided `NaN` as any argument, the function returns `NaN`. 68 69 ```javascript 70 var y = logcdf( NaN, 0.0, 1.0 ); 71 // returns NaN 72 73 y = logcdf( 0.0, NaN, 1.0 ); 74 // returns NaN 75 76 y = logcdf( 0.0, 0.0, NaN ); 77 // returns NaN 78 ``` 79 80 If provided `c <= 0`, the function returns `NaN`. 81 82 ```javascript 83 var y = logcdf( 2.0, 0.0, -1.0 ); 84 // returns NaN 85 86 y = logcdf( 2.0, 0.0, 0.0 ); 87 // returns NaN 88 ``` 89 90 #### logcdf.factory( mu, c ) 91 92 Returns a function for evaluating the logarithm of the [cumulative distribution function][cdf] of a [Lévy][levy-distribution] distribution with parameters `mu` (location parameter) and `c > 0` (scale parameter). 93 94 ```javascript 95 var mylogcdf = logcdf.factory( 3.0, 1.5 ); 96 97 var y = mylogcdf( 4.0 ); 98 // returns ~-1.511 99 100 y = mylogcdf( 2.0 ); 101 // returns -Infinity 102 ``` 103 104 </section> 105 106 <!-- /.usage --> 107 108 <section class="notes"> 109 110 ## Notes 111 112 - 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. 113 114 </section> 115 116 <!-- /.notes --> 117 118 <section class="examples"> 119 120 ## Examples 121 122 <!-- eslint no-undef: "error" --> 123 124 ```javascript 125 var randu = require( '@stdlib/random/base/randu' ); 126 var EPS = require( '@stdlib/constants/float64/eps' ); 127 var logcdf = require( '@stdlib/stats/base/dists/levy/logcdf' ); 128 129 var mu; 130 var c; 131 var x; 132 var y; 133 var i; 134 135 for ( i = 0; i < 100; i++ ) { 136 mu = randu() * 10.0; 137 x = ( randu()*10.0 ) + mu; 138 c = ( randu()*10.0 ) + EPS; 139 y = logcdf( x, mu, c ); 140 console.log( 'x: %d, µ: %d, c: %d, ln(F(x;µ,c)): %d', x.toFixed( 4 ), mu.toFixed( 4 ), c.toFixed( 4 ), y.toFixed( 4 ) ); 141 } 142 ``` 143 144 </section> 145 146 <!-- /.examples --> 147 148 <section class="links"> 149 150 [cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function 151 152 [levy-distribution]: https://en.wikipedia.org/wiki/L%C3%A9vy_distribution 153 154 </section> 155 156 <!-- /.links -->