README.md (4837B)
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 > Evaluate the natural logarithm of the [cumulative distribution function][cdf] (CDF) for a [raised cosine][cosine-distribution] distribution. 24 25 <section class="intro"> 26 27 The [cumulative distribution function][cdf] for a [raised cosine][cosine-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:cosine_cdf" align="center" raw="F(x;\mu ,s)=\begin{cases} 0 & \text{ for } x < \mu - s \\ {\frac {1}{2}}\left[1\!+\!{\frac {x\!-\!\mu }{s}}\!+\!{\frac {1}{\pi }}\sin \left({\frac {x\!-\!\mu }{s}}\,\pi \right)\right] & \text{ for } \mu - s \le x \le \mu + s \\ 1 & \text{ for } x > \mu + s \end{cases}" alt="Cumulative distribution function for a raised cosine distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="F(x;\mu ,s)=\begin{cases} 0 & \text{ for } x < \mu - s \\ {\frac {1}{2}}\left[1\!+\!{\frac {x\!-\!\mu }{s}}\!+\!{\frac {1}{\pi }}\sin \left({\frac {x\!-\!\mu }{s}}\,\pi \right)\right] & \text{ for } \mu - s \le x \le \mu + s \\ 1 & \text{ for } x > \mu + s \end{cases}" data-equation="eq:cosine_cdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/cosine/logcdf/docs/img/equation_cosine_cdf.svg" alt="Cumulative distribution function for a raised cosine distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `μ` is the location parameter and `s > 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/cosine/logcdf' ); 50 ``` 51 52 #### logcdf( x, mu, s ) 53 54 Evaluates the natural logarithm of the [cumulative distribution function][cdf] (CDF) for a [raised cosine][cosine-distribution] distribution with parameters `mu` (location parameter) and `s` (scale parameter). 55 56 ```javascript 57 var y = logcdf( 2.0, 0.0, 3.0 ); 58 // returns ~-0.029 59 60 y = logcdf( 0.0, 0.0, 1.0 ); 61 // returns ~-0.693 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, 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 `s < 0`, the function returns `NaN`. 81 82 ```javascript 83 var y = logcdf( 2.0, 0.0, -1.0 ); 84 // returns NaN 85 ``` 86 87 If provided `s = 0`, the function evaluates the logarithm of the [CDF][cdf] for a [degenerate distribution][degenerate-distribution] centered at `mu`. 88 89 ```javascript 90 var y = logcdf( 2.0, 8.0, 0.0 ); 91 // returns -Infinity 92 93 y = logcdf( 8.0, 8.0, 0.0 ); 94 // returns 0.0 95 96 y = logcdf( 10.0, 8.0, 0.0 ); 97 // returns 0.0 98 ``` 99 100 #### logcdf.factory( mu, s ) 101 102 Returns a function for evaluating the natural logarithm of the [cumulative distribution function][cdf] of a [raised cosine][cosine-distribution] distribution with parameters `mu` (location parameter) and `s` (scale parameter). 103 104 ```javascript 105 var mylogcdf = logcdf.factory( 10.0, 2.0 ); 106 107 var y = mylogcdf( 10.0 ); 108 // returns ~-0.693 109 110 y = mylogcdf( 8.0 ); 111 // returns -Infinity 112 113 y = mylogcdf( 12.0 ); 114 // returns 0.0 115 ``` 116 117 </section> 118 119 <!-- /.usage --> 120 121 <section class="notes"> 122 123 ## Notes 124 125 - 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. 126 127 </section> 128 129 <!-- /.notes --> 130 131 <section class="examples"> 132 133 ## Examples 134 135 <!-- eslint no-undef: "error" --> 136 137 ```javascript 138 var randu = require( '@stdlib/random/base/randu' ); 139 var logcdf = require( '@stdlib/stats/base/dists/cosine/logcdf' ); 140 141 var mu; 142 var s; 143 var x; 144 var y; 145 var i; 146 147 for ( i = 0; i < 10; i++ ) { 148 x = randu() * 10.0; 149 mu = randu() * 10.0; 150 s = randu() * 10.0; 151 y = logcdf( x, mu, s ); 152 console.log( 'x: %d, µ: %d, s: %d, ln(F(x;µ,s)): %d', x, mu, s, y ); 153 } 154 ``` 155 156 </section> 157 158 <!-- /.examples --> 159 160 <section class="links"> 161 162 [cosine-distribution]: https://en.wikipedia.org/wiki/Raised_cosine_distribution 163 164 [cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function 165 166 [degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution 167 168 </section> 169 170 <!-- /.links -->