README.md (4332B)
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 # Cumulative Distribution Function 22 23 > [Raised cosine][cosine-distribution] distribution [cumulative distribution function][cdf]. 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/cdf/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 cdf = require( '@stdlib/stats/base/dists/cosine/cdf' ); 50 ``` 51 52 #### cdf( x, mu, s ) 53 54 Evaluates 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 = cdf( 2.0, 0.0, 3.0 ); 58 // returns ~0.971 59 60 y = cdf( 0.0, 0.0, 1.0 ); 61 // returns ~0.5 62 63 y = cdf( -1.0, 4.0, 2.0 ); 64 // returns 0.0 65 ``` 66 67 If provided `NaN` as any argument, the function returns `NaN`. 68 69 ```javascript 70 var y = cdf( NaN, 0.0, 1.0 ); 71 // returns NaN 72 73 y = cdf( 0.0, NaN, 1.0 ); 74 // returns NaN 75 76 y = cdf( 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 = cdf( 2.0, 0.0, -1.0 ); 84 // returns NaN 85 ``` 86 87 If provided `s = 0`, the function evaluates the [CDF][cdf] of a [degenerate distribution][degenerate-distribution] centered at `mu`. 88 89 ```javascript 90 var y = cdf( 2.0, 8.0, 0.0 ); 91 // returns 0.0 92 93 y = cdf( 8.0, 8.0, 0.0 ); 94 // returns 1.0 95 96 y = cdf( 10.0, 8.0, 0.0 ); 97 // returns 1.0 98 ``` 99 100 #### cdf.factory( mu, s ) 101 102 Returns a function for evaluating 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 mycdf = cdf.factory( 10.0, 2.0 ); 106 107 var y = mycdf( 10.0 ); 108 // returns 0.5 109 110 y = mycdf( 8.0 ); 111 // returns 0.0 112 113 y = mycdf( 12.0 ); 114 // returns 1.0 115 ``` 116 117 </section> 118 119 <!-- /.usage --> 120 121 <section class="examples"> 122 123 ## Examples 124 125 <!-- eslint no-undef: "error" --> 126 127 ```javascript 128 var randu = require( '@stdlib/random/base/randu' ); 129 var cdf = require( '@stdlib/stats/base/dists/cosine/cdf' ); 130 131 var mu; 132 var s; 133 var x; 134 var y; 135 var i; 136 137 for ( i = 0; i < 10; i++ ) { 138 x = randu() * 10.0; 139 mu = randu() * 10.0; 140 s = randu() * 10.0; 141 y = cdf( x, mu, s ); 142 console.log( 'x: %d, µ: %d, s: %d, F(x;µ,s): %d', x, mu, s, y ); 143 } 144 ``` 145 146 </section> 147 148 <!-- /.examples --> 149 150 <section class="links"> 151 152 [cosine-distribution]: https://en.wikipedia.org/wiki/Raised_cosine_distribution 153 154 [cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function 155 156 [degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution 157 158 </section> 159 160 <!-- /.links -->