time-to-botec

Benchmark sampling in different programming languages
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README.md (4111B)

      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
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     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 > [Cauchy][cauchy-distribution] distribution logarithm of [cumulative distribution function][cdf].
     24 
     25 <section class="intro">
     26 
     27 The [cumulative distribution function][cdf] for a [Cauchy][cauchy-distribution] random variable is
     28 
     29 <!-- <equation class="equation" label="eq:cauchy_cdf" align="center" raw="F(x; x_0,\gamma)=\frac{1}{\pi} \operatorname{arctan} \left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}" alt="Cumulative distribution function for a Cauchy distribution."> -->
     30 
     31 <div class="equation" align="center" data-raw-text="F(x; x_0,\gamma)=\frac{1}{\pi} \operatorname{arctan} \left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}" data-equation="eq:cauchy_cdf">
     32     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/docs/img/equation_cauchy_cdf.svg" alt="Cumulative distribution function 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 logcdf = require( '@stdlib/stats/base/dists/cauchy/logcdf' );
     50 ```
     51 
     52 #### logcdf( x, x0, gamma )
     53 
     54 Evaluates the natural logarithm of the [cumulative distribution function][cdf] (CDF) for a [Cauchy][cauchy-distribution] distribution with parameters `x0` (location parameter) and `gamma > 0` (scale parameter).
     55 
     56 ```javascript
     57 var y = logcdf( 4.0, 0.0, 2.0 );
     58 // returns ~-0.16
     59 
     60 y = logcdf( 1.0, 0.0, 2.0 );
     61 // returns ~-0.435
     62 
     63 y = logcdf( 1.0, 3.0, 2.0 );
     64 // returns ~-1.386
     65 ```
     66 
     67 If provided `NaN` as any argument, the function returns `NaN`.
     68 
     69 ```javascript
     70 var y = logcdf( NaN, 0.0, 2.0 );
     71 // returns NaN
     72 
     73 y = logcdf( 1.0, 2.0, NaN );
     74 // returns NaN
     75 
     76 y = logcdf( 1.0, NaN, 3.0 );
     77 // returns NaN
     78 ```
     79 
     80 If provided `gamma <= 0`, the function returns `NaN`.
     81 
     82 ```javascript
     83 var y = logcdf( 2.0, 0.0, -1.0 );
     84 // returns NaN
     85 
     86 y = logcdf( 2.0, 0.0, 0.0 );
     87 // returns NaN
     88 ```
     89 
     90 #### logcdf.factory( x0, gamma )
     91 
     92 Returns a function for evaluating the natural logarithm of the [cumulative distribution function][cdf] of a [Cauchy][cauchy-distribution] distribution with parameters  `x0` (location parameter) and `gamma > 0` (scale parameter).
     93 
     94 ```javascript
     95 var mylogcdf = logcdf.factory( 10.0, 2.0 );
     96 
     97 var y = mylogcdf( 10.0 );
     98 // returns ~-0.693
     99 
    100 y = mylogcdf( 12.0 );
    101 // returns ~-0.288
    102 ```
    103 
    104 </section>
    105 
    106 <!-- /.usage -->
    107 
    108 <section class="notes">
    109 
    110 ## Notes
    111 
    112 -   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.
    113 
    114 </section>
    115 
    116 <!-- /.notes -->
    117 
    118 <section class="examples">
    119 
    120 ## Examples
    121 
    122 <!-- eslint no-undef: "error" -->
    123 
    124 ```javascript
    125 var randu = require( '@stdlib/random/base/randu' );
    126 var EPS = require( '@stdlib/constants/float64/eps' );
    127 var logcdf = require( '@stdlib/stats/base/dists/cauchy/logcdf' );
    128 
    129 var gamma;
    130 var x0;
    131 var x;
    132 var y;
    133 var i;
    134 
    135 for ( i = 0; i < 10; i++ ) {
    136     x = randu() * 10.0;
    137     x0 = randu() * 10.0;
    138     gamma = ( randu()*10.0 ) + EPS;
    139     y = logcdf( x, x0, gamma );
    140     console.log( 'x: %d, x0: %d, γ: %d, ln(F(x;x0,γ)): %d', x, x0, gamma, y );
    141 }
    142 ```
    143 
    144 </section>
    145 
    146 <!-- /.examples -->
    147 
    148 <section class="links">
    149 
    150 [cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function
    151 
    152 [cauchy-distribution]: https://en.wikipedia.org/wiki/Cauchy_distribution
    153 
    154 </section>
    155 
    156 <!-- /.links -->