README.md (4249B)
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 > [Rayleigh][rayleigh-distribution] distribution logarithm of [probability density function][pdf] (PDF). 24 25 <section class="intro"> 26 27 The [probability density function][pdf] (PDF) for a [Rayleigh][rayleigh-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:rayleigh_pdf" align="center" raw="f(x;\sigma) = \begin{cases} \frac{x}{\sigma^2} e^{-x^2/(2\sigma^2)} & \text{ for } x \ge 0 \\ 0 & \text{ otherwise } \end{cases}" alt="Probability density function (PDF) for a Rayleigh distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="f(x;\sigma) = \begin{cases} \frac{x}{\sigma^2} e^{-x^2/(2\sigma^2)} &amp; \text{ for } x \ge 0 \\ 0 & \text{ otherwise } \end{cases}" data-equation="eq:rayleigh_pdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/rayleigh/logpdf/docs/img/equation_rayleigh_pdf.svg" alt="Probability density function (PDF) for a Rayleigh distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `sigma > 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/rayleigh/logpdf' ); 50 ``` 51 52 #### logpdf( x, sigma ) 53 54 Evaluates the logarithm of the [probability density function][pdf] for a [Rayleigh][rayleigh-distribution] distribution with scale parameter `sigma`. 55 56 ```javascript 57 var y = logpdf( 0.3, 1.0 ); 58 // returns ~-1.249 59 60 y = logpdf( 2.0, 0.8 ); 61 // returns ~-1.986 62 63 y = logpdf( -1.0, 0.5 ); 64 // returns -Infinity 65 ``` 66 67 If provided `NaN` as any argument, the function returns `NaN`. 68 69 ```javascript 70 var y = logpdf( NaN, 1.0 ); 71 // returns NaN 72 73 y = logpdf( 0.0, NaN ); 74 // returns NaN 75 ``` 76 77 If provided `sigma < 0`, the function returns `NaN`. 78 79 ```javascript 80 var y = logpdf( 2.0, -1.0 ); 81 // returns NaN 82 ``` 83 84 If provided `sigma = 0`, the function evaluates the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `0`. 85 86 ```javascript 87 var y = logpdf( -2.0, 0.0 ); 88 // returns -Infinity 89 90 y = logpdf( 0.0, 0.0 ); 91 // returns +Infinity 92 93 y = logpdf( 2.0, 0.0 ); 94 // returns -Infinity 95 ``` 96 97 #### logpdf.factory( sigma ) 98 99 Returns a function for evaluating the logarithm of the [probability density function][pdf] (PDF) of a [Rayleigh][rayleigh-distribution] distribution with parameter `sigma` (scale parameter). 100 101 ```javascript 102 var mylogpdf = logpdf.factory( 4.0 ); 103 104 var y = mylogpdf( 6.0 ); 105 // returns ~-2.106 106 107 y = mylogpdf( 4.0 ); 108 // returns ~-1.886 109 ``` 110 111 </section> 112 113 <!-- /.usage --> 114 115 <section class="notes"> 116 117 ## Notes 118 119 - 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. 120 121 </section> 122 123 <!-- /.notes --> 124 125 <section class="examples"> 126 127 ## Examples 128 129 <!-- eslint no-undef: "error" --> 130 131 ```javascript 132 var randu = require( '@stdlib/random/base/randu' ); 133 var logpdf = require( '@stdlib/stats/base/dists/rayleigh/logpdf' ); 134 135 var sigma; 136 var x; 137 var y; 138 var i; 139 140 for ( i = 0; i < 10; i++ ) { 141 x = randu() * 10.0; 142 sigma = randu() * 10.0; 143 y = logpdf( x, sigma ); 144 console.log( 'x: %d, σ: %d, f(x;σ): %d', x.toFixed( 4 ), sigma.toFixed( 4 ), y.toFixed( 4 ) ); 145 } 146 ``` 147 148 </section> 149 150 <!-- /.examples --> 151 152 <section class="links"> 153 154 [degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution 155 156 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function 157 158 [rayleigh-distribution]: https://en.wikipedia.org/wiki/Rayleigh_distribution 159 160 </section> 161 162 <!-- /.links -->