time-to-botec

Benchmark sampling in different programming languages
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README.md (4115B)

      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
     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 > Evaluate the natural logarithm of the [cumulative distribution function][cdf] (CDF) for a [Student's t][t-distribution] distribution.
     24 
     25 <section class="intro">
     26 
     27 The [cumulative distribution function][cdf] (CDF) for a [t distribution][t-distribution] random variable is
     28 
     29 <!-- <equation class="equation" label="eq:t_cdf" align="center" raw="F(x;\nu) = 1 - \frac{1}{2} \frac{\operatorname{Beta}(\tfrac{\nu}{\nu + x^2};\,\tfrac{\nu}{2},\tfrac{1}{2})}{\operatorname{Beta}(\tfrac{\nu}{2}, \tfrac{1}{2})}" alt="Cumulative distribution function (CDF) for a Student's t distribution."> -->
     30 
     31 <div class="equation" align="center" data-raw-text="F(x;\nu) = 1 - \frac{1}{2} \frac{\operatorname{Beta}(\tfrac{\nu}{\nu + x^2};\,\tfrac{\nu}{2},\tfrac{1}{2})}{\operatorname{Beta}(\tfrac{\nu}{2}, \tfrac{1}{2})}" data-equation="eq:t_cdf">
     32     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/t/logcdf/docs/img/equation_t_cdf.svg" alt="Cumulative distribution function (CDF) for a Student's t distribution.">
     33     <br>
     34 </div>
     35 
     36 <!-- </equation> -->
     37 
     38 where `v > 0` is the degrees of freedom. In the definition, `Beta( x; a, b )` denotes the lower incomplete beta function and `Beta( a, b )` the [beta function][beta-function].
     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/t/logcdf' );
     50 ```
     51 
     52 #### logcdf( x, v )
     53 
     54 Evaluates the natural logarithm of the [cumulative distribution function][cdf] (CDF) for a [Student's t][t-distribution] distribution with degrees of freedom `v`.
     55 
     56 ```javascript
     57 var y = logcdf( 2.0, 0.1 );
     58 // returns ~-0.493
     59 
     60 y = logcdf( 1.0, 2.0 );
     61 // returns ~-0.237
     62 
     63 y = logcdf( -1.0, 4.0 );
     64 // returns ~-1.677
     65 ```
     66 
     67 If provided `NaN` as any argument, the function returns `NaN`.
     68 
     69 ```javascript
     70 var y = logcdf( NaN, 1.0 );
     71 // returns NaN
     72 
     73 y = logcdf( 0.0, NaN );
     74 // returns NaN
     75 ```
     76 
     77 If provided `v <= 0`, the function returns `NaN`.
     78 
     79 ```javascript
     80 var y = logcdf( 2.0, -1.0 );
     81 // returns NaN
     82 
     83 y = logcdf( 2.0, 0.0 );
     84 // returns NaN
     85 ```
     86 
     87 #### logcdf.factory( v )
     88 
     89 Returns a `function` for evaluating the natural logarithm of the [CDF][cdf] of a [Student's t][t-distribution] distribution with degrees of freedom `v`.
     90 
     91 ```javascript
     92 var mylogcdf = logcdf.factory( 0.5 );
     93 var y = mylogcdf( 3.0 );
     94 // returns ~-0.203
     95 
     96 y = mylogcdf( 1.0 );
     97 // returns ~-0.358
     98 ```
     99 
    100 </section>
    101 
    102 <!-- /.usage -->
    103 
    104 <section class="notes">
    105 
    106 ## Notes
    107 
    108 -   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.
    109 
    110 </section>
    111 
    112 <!-- /.notes -->
    113 
    114 <section class="examples">
    115 
    116 ## Examples
    117 
    118 <!-- eslint no-undef: "error" -->
    119 
    120 ```javascript
    121 var randu = require( '@stdlib/random/base/randu' );
    122 var logcdf = require( '@stdlib/stats/base/dists/t/logcdf' );
    123 
    124 var v;
    125 var x;
    126 var y;
    127 var i;
    128 
    129 for ( i = 0; i < 10; i++ ) {
    130     x = (randu() * 6.0) - 3.0;
    131     v = randu() * 10.0;
    132     y = logcdf( x, v );
    133     console.log( 'x: %d, v: %d, ln(F(x;v)): %d', x.toFixed( 4 ), v.toFixed( 4 ), y.toFixed( 4 ) );
    134 }
    135 ```
    136 
    137 </section>
    138 
    139 <!-- /.examples -->
    140 
    141 <section class="links">
    142 
    143 [beta-function]: https://en.wikipedia.org/wiki/Beta_function
    144 
    145 [cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function
    146 
    147 [t-distribution]: https://en.wikipedia.org/wiki/Student%27s_t-distribution
    148 
    149 </section>
    150 
    151 <!-- /.links -->