time-to-botec

Benchmark sampling in different programming languages
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README.md (4574B)

      1 <!--
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      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 > [Fréchet][frechet-distribution] distribution logarithm of [probability density function][pdf].
     24 
     25 <section class="intro">
     26 
     27 The [probability density function][pdf] for a [Fréchet][frechet-distribution] random variable is
     28 
     29 <!-- <equation class="equation" label="eq:frechet_pdf" align="center" raw="f\left( x; \mu, \beta \right ) = {\frac{\alpha }{s}}\;\left({\frac{x-m}{s}}\right)^{{-1-\alpha }}\;e^{{-({\frac{x-m}{s}})^{-\alpha}}}" alt="Probability density function for a Fréchet distribution."> -->
     30 
     31 <div class="equation" align="center" data-raw-text="f\left( x; \mu, \beta \right ) = {\frac{\alpha }{s}}\;\left({\frac{x-m}{s}}\right)^{{-1-\alpha }}\;e^{{-({\frac{x-m}{s}})^{-\alpha}}}" data-equation="eq:frechet_pdf">
     32     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/frechet/logpdf/docs/img/equation_frechet_pdf.svg" alt="Probability density function for a Fréchet distribution.">
     33     <br>
     34 </div>
     35 
     36 <!-- </equation> -->
     37 
     38 where `α > 0` is the shape, `s > 0` the scale and `m` the location 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/frechet/logpdf' );
     50 ```
     51 
     52 #### logpdf( x, alpha, s, m )
     53 
     54 Evaluates the logarithm of the [probability density function][pdf] (PDF) for a [Fréchet][frechet-distribution] distribution with shape `alpha`, scale `s`, and location `m` at a value `x`.
     55 
     56 ```javascript
     57 var y = logpdf( 10.0, 2.0, 3.0, 5.0 );
     58 // returns ~-2.298
     59 
     60 y = logpdf( -3.0, 1.0, 2.0, -4.0 );
     61 // returns ~-1.307
     62 
     63 y = logpdf( 0.0, 2.0, 1.0, -1.0 );
     64 // returns ~-0.307
     65 ```
     66 
     67 If provided `x <= m`, the function returns `-Infinity`.
     68 
     69 ```javascript
     70 y = logpdf( -2.0, 2.0, 1.0, -1.0 );
     71 // returns -Infinity
     72 ```
     73 
     74 If provided `NaN` as any argument, the function returns `NaN`.
     75 
     76 ```javascript
     77 var y = logpdf( NaN, 1.0, 1.0, 0.0 );
     78 // returns NaN
     79 
     80 y = logpdf( 0.0, NaN, 1.0, 0.0 );
     81 // returns NaN
     82 
     83 y = logpdf( 0.0, 1.0, NaN, 0.0);
     84 // returns NaN
     85 
     86 y = logpdf( 0.0, 1.0, 1.0, NaN );
     87 // returns NaN
     88 ```
     89 
     90 If provided `alpha <= 0`, the function returns `NaN`.
     91 
     92 ```javascript
     93 var y = logpdf( 2.0, -0.1, 1.0, 1.0 );
     94 // returns NaN
     95 
     96 y = logpdf( 2.0, 0.0, 1.0, 1.0 );
     97 // returns NaN
     98 ```
     99 
    100 If provided `s <= 0`, the function returns `NaN`.
    101 
    102 ```javascript
    103 var y = logpdf( 2.0, 1.0, -1.0, 1.0 );
    104 // returns NaN
    105 
    106 y = logpdf( 2.0, 1.0, 0.0, 1.0 );
    107 // returns NaN
    108 ```
    109 
    110 #### logpdf.factory( alpha, s, m )
    111 
    112 Returns a function for evaluating the logarithm of the [probability density function][pdf] of a [Fréchet][frechet-distribution] distribution with shape `alpha`, scale `s`, and location `m`.
    113 
    114 ```javascript
    115 var mylogpdf = logpdf.factory( 3.0, 3.0, 5.0 );
    116 
    117 var y = mylogpdf( 10.0 );
    118 // returns ~-2.259
    119 
    120 y = mylogpdf( 7.0 );
    121 // returns ~-1.753
    122 ```
    123 
    124 </section>
    125 
    126 <!-- /.usage -->
    127 
    128 <section class="notes">
    129 
    130 ## Notes
    131 
    132 -   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.
    133 
    134 </section>
    135 
    136 <!-- /.notes -->
    137 
    138 <section class="examples">
    139 
    140 ## Examples
    141 
    142 <!-- eslint no-undef: "error" -->
    143 
    144 ```javascript
    145 var randu = require( '@stdlib/random/base/randu' );
    146 var logpdf = require( '@stdlib/stats/base/dists/frechet/logpdf' );
    147 
    148 var alpha;
    149 var m;
    150 var s;
    151 var x;
    152 var y;
    153 var i;
    154 
    155 for ( i = 0; i < 100; i++ ) {
    156     alpha = randu() * 10.0;
    157     x = randu() * 10.0;
    158     s = randu() * 10.0;
    159     m = randu() * 10.0;
    160     y = logpdf( x, alpha, s, m );
    161     console.log( 'x: %d, α: %d, s: %d, m: %d, ln(f(x;α,s,m)): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), s.toFixed( 4 ), m.toFixed( 4 ), y.toFixed( 4 ) );
    162 }
    163 ```
    164 
    165 </section>
    166 
    167 <!-- /.examples -->
    168 
    169 <section class="links">
    170 
    171 [frechet-distribution]: https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution
    172 
    173 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function
    174 
    175 </section>
    176 
    177 <!-- /.links -->