README.md (4505B)
1 <!-- 2 3 @license Apache-2.0 4 5 Copyright (c) 2021 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 > [Gamma][gamma-distribution] distribution logarithm of cumulative distribution function (CDF). 24 25 <section class="intro"> 26 27 The [cumulative distribution function][cdf] for a [gamma][gamma-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:gamma_cdf" align="center" raw="F(x;\alpha,\beta) = \int_0^x f(u;\alpha,\beta)\,du= \frac{\gamma(\alpha, \beta x)}{\Gamma(\alpha)}" alt="Cumulative distribution function for a Gamma distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="F(x;\alpha,\beta) = \int_0^x f(u;\alpha,\beta)\,du= \frac{\gamma(\alpha, \beta x)}{\Gamma(\alpha)}" data-equation="eq:gamma_cdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@9dcb6eb6ab19f2ea81f3bcab5344b29961028a0c/lib/node_modules/@stdlib/stats/base/dists/gamma/logcdf/docs/img/equation_gamma_cdf.svg" alt="Cumulative distribution function for a Gamma distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `alpha` is the shape parameter and `beta` is the rate parameter of the distribution. `gamma` is the lower [incomplete gamma function][@stdlib/math/base/special/gammainc]. 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/gamma/logcdf' ); 50 ``` 51 52 #### logcdf( x, alpha, beta ) 53 54 Evaluates the natural logarithm of the [cumulative distribution function][cdf] (CDF) for a [gamma][gamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter). 55 56 ```javascript 57 var y = logcdf( 2.0, 0.5, 1.0 ); 58 // returns ~-0.047 59 60 y = logcdf( 0.1, 1.0, 1.0 ); 61 // returns ~-2.352 62 63 y = logcdf( -1.0, 4.0, 2.0 ); 64 // returns -Infinity 65 ``` 66 67 If provided `NaN` as any argument, the function returns `NaN`. 68 69 ```javascript 70 var y = logcdf( NaN, 1.0, 1.0 ); 71 // returns NaN 72 73 y = logcdf( 0.0, NaN, 1.0 ); 74 // returns NaN 75 76 y = logcdf( 0.0, 1.0, NaN ); 77 // returns NaN 78 ``` 79 80 If provided `alpha < 0`, the function returns `NaN`. 81 82 ```javascript 83 var y = logcdf( 2.0, -0.5, 1.0 ); 84 // returns NaN 85 ``` 86 87 If provided `alpha = 0`, the function evaluates the logarithm of the [CDF][cdf] for a [degenerate distribution][degenerate-distribution] centered at `0`. 88 89 ```javascript 90 var y = logcdf( 2.0, 0.0, 2.0 ); 91 // returns 0.0 92 93 y = logcdf( -2.0, 0.0, 2.0 ); 94 // returns -Infinity 95 96 y = logcdf( 0.0, 0.0, 2.0 ); 97 // returns 0.0 98 ``` 99 100 If provided `beta <= 0`, the function returns `NaN`. 101 102 ```javascript 103 var y = logcdf( 2.0, 1.0, 0.0 ); 104 // returns NaN 105 106 y = logcdf( 2.0, 1.0, -1.0 ); 107 // returns NaN 108 ``` 109 110 #### logcdf.factory( alpha, beta ) 111 112 Returns a `function` for evaluating the natural logarithm of the [CDF][cdf] for a [gamma][gamma-distribution] distribution with parameters `alpha` (shape parameter) and `beta` (rate parameter). 113 114 ```javascript 115 var mylogcdf = logcdf.factory( 3.0, 1.5 ); 116 117 var y = mylogcdf( 1.0 ); 118 // returns ~-1.655 119 120 y = mylogcdf( 4.0 ); 121 // returns ~-0.064 122 ``` 123 124 </section> 125 126 <!-- /.usage --> 127 128 <section class="examples"> 129 130 ## Examples 131 132 <!-- eslint no-undef: "error" --> 133 134 ```javascript 135 var randu = require( '@stdlib/random/base/randu' ); 136 var logcdf = require( '@stdlib/stats/base/dists/gamma/logcdf' ); 137 138 var alpha; 139 var beta; 140 var x; 141 var y; 142 var i; 143 144 for ( i = 0; i < 10; i++ ) { 145 x = randu() * 3.0; 146 alpha = randu() * 5.0; 147 beta = randu() * 5.0; 148 y = logcdf( x, alpha, beta ); 149 console.log( 'x: %d, α: %d, β: %d, ln(F(x;α,β)): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) ); 150 } 151 ``` 152 153 </section> 154 155 <!-- /.examples --> 156 157 <section class="links"> 158 159 [gamma-distribution]: https://en.wikipedia.org/wiki/Gamma_distribution 160 161 [cdf]: https://en.wikipedia.org/wiki/Cumulative_Distribution_Function 162 163 [degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution 164 165 [@stdlib/math/base/special/gammainc]: https://www.npmjs.com/package/@stdlib/math-base-special-gammainc 166 167 </section> 168 169 <!-- /.links -->