time-to-botec

Benchmark sampling in different programming languages
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README.md (4070B)

      1 <!--
      2 
      3 @license Apache-2.0
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      5 Copyright (c) 2018 The Stdlib Authors.
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      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
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     11    http://www.apache.org/licenses/LICENSE-2.0
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     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
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     19 -->
     20 
     21 # Logarithm of Probability Density Function
     22 
     23 > [Cauchy][cauchy-distribution] distribution logarithm of probability density function (logPDF).
     24 
     25 <section class="intro">
     26 
     27 The [probability density function][pdf] (PDF) for a [Cauchy][cauchy-distribution] random variable is
     28 
     29 <!-- <equation class="equation" label="eq:cauchy_cauchy_pdf" align="center" raw="f(x;\gamma,x_0)=\frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\!" alt="Probability density function (PDF) for a Cauchy distribution."> -->
     30 
     31 <div class="equation" align="center" data-raw-text="f(x;\gamma,x_0)=\frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\!" data-equation="eq:cauchy_cauchy_pdf">
     32     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/cauchy/logpdf/docs/img/equation_cauchy_cauchy_pdf.svg" alt="Probability density function (PDF) for a Cauchy distribution.">
     33     <br>
     34 </div>
     35 
     36 <!-- </equation> -->
     37 
     38 where `x0` is the location parameter and `gamma > 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/cauchy/logpdf' );
     50 ```
     51 
     52 #### logpdf( x, x0, gamma )
     53 
     54 Evaluates the natural logarithm of the [probability density function][pdf] (PDF) for a [Cauchy][cauchy-distribution] distribution with parameters `x0` (location parameter) and `gamma > 0` (scale parameter).
     55 
     56 ```javascript
     57 var y = logpdf( 2.0, 1.0, 1.0 );
     58 // returns ~-1.838
     59 
     60 y = logpdf( 4.0, 3.0, 0.1 );
     61 // returns ~-3.457
     62 
     63 y = logpdf( 4.0, 3.0, 3.0 );
     64 // returns ~-2.349
     65 ```
     66 
     67 If provided `NaN` as any argument, the function returns `NaN`.
     68 
     69 ```javascript
     70 var y = logpdf( NaN, 1.0, 1.0 );
     71 // returns NaN
     72 
     73 y = logpdf( 2.0, NaN, 1.0 );
     74 // returns NaN
     75 
     76 y = logpdf( 2.0, 1.0, NaN );
     77 // returns NaN
     78 ```
     79 
     80 If provided `gamma <= 0`, the function returns `NaN`.
     81 
     82 ```javascript
     83 var y = logpdf( 2.0, 0.0, -1.0 );
     84 // returns NaN
     85 ```
     86 
     87 #### logpdf.factory( x0, gamma )
     88 
     89 Returns a `function` for evaluating the natural logarithm of the [PDF][pdf] of a [Cauchy][cauchy-distribution] distribution with location parameter `x0` and scale parameter `gamma`.
     90 
     91 ```javascript
     92 var mylogpdf = logpdf.factory( 10.0, 2.0 );
     93 
     94 var y = mylogpdf( 10.0 );
     95 // returns ~-1.838
     96 
     97 y = mylogpdf( 5.0 );
     98 // returns ~-3.819
     99 ```
    100 
    101 </section>
    102 
    103 <!-- /.usage -->
    104 
    105 <section class="notes">
    106 
    107 ## Notes
    108 
    109 -   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.
    110 
    111 </section>
    112 
    113 <!-- /.notes -->
    114 
    115 <section class="examples">
    116 
    117 ## Examples
    118 
    119 <!-- eslint no-undef: "error" -->
    120 
    121 ```javascript
    122 var randu = require( '@stdlib/random/base/randu' );
    123 var EPS = require( '@stdlib/constants/float64/eps' );
    124 var logpdf = require( '@stdlib/stats/base/dists/cauchy/logpdf' );
    125 
    126 var gamma;
    127 var x0;
    128 var x;
    129 var y;
    130 var i;
    131 
    132 for ( i = 0; i < 10; i++ ) {
    133     x = randu() * 10.0;
    134     x0 = ( randu()*10.0 ) - 5.0;
    135     gamma = ( randu()*20.0 ) + EPS;
    136     y = logpdf( x, gamma, x0 );
    137     console.log( 'x: %d, x0: %d, γ: %d, ln(f(x;x0,γ)): %d', x.toFixed(4), x0.toFixed(4), gamma.toFixed(4), y.toFixed(4) );
    138 }
    139 ```
    140 
    141 </section>
    142 
    143 <!-- /.examples -->
    144 
    145 <section class="links">
    146 
    147 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function
    148 
    149 [cauchy-distribution]: https://en.wikipedia.org/wiki/Cauchy_distribution
    150 
    151 </section>
    152 
    153 <!-- /.links -->