202 lines
14 KiB
HTML
202 lines
14 KiB
HTML
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
|
<title>Complements are supported too - and when to use them</title>
|
|
<link rel="stylesheet" href="../../../math.css" type="text/css">
|
|
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
|
<link rel="home" href="../../../index.html" title="Math Toolkit 2.11.0">
|
|
<link rel="up" href="../overview.html" title="Overview of Statistical Distributions">
|
|
<link rel="prev" href="generic.html" title="Generic operations common to all distributions are non-member functions">
|
|
<link rel="next" href="parameters.html" title="Parameters can be calculated">
|
|
</head>
|
|
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
|
<table cellpadding="2" width="100%"><tr>
|
|
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td>
|
|
<td align="center"><a href="../../../../../../../index.html">Home</a></td>
|
|
<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td>
|
|
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
|
|
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
|
<td align="center"><a href="../../../../../../../more/index.htm">More</a></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="generic.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../overview.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="parameters.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h4 class="title">
|
|
<a name="math_toolkit.stat_tut.overview.complements"></a><a class="link" href="complements.html" title="Complements are supported too - and when to use them">Complements
|
|
are supported too - and when to use them</a>
|
|
</h4></div></div></div>
|
|
<p>
|
|
Often you don't want the value of the CDF, but its complement, which is
|
|
to say <code class="computeroutput"><span class="number">1</span><span class="special">-</span><span class="identifier">p</span></code> rather than <code class="computeroutput"><span class="identifier">p</span></code>.
|
|
It is tempting to calculate the CDF and subtract it from <code class="computeroutput"><span class="number">1</span></code>, but if <code class="computeroutput"><span class="identifier">p</span></code>
|
|
is very close to <code class="computeroutput"><span class="number">1</span></code> then cancellation
|
|
error will cause you to lose accuracy, perhaps totally.
|
|
</p>
|
|
<p>
|
|
<a class="link" href="complements.html#why_complements">See below <span class="emphasis"><em>"Why and when
|
|
to use complements?"</em></span></a>
|
|
</p>
|
|
<p>
|
|
In this library, whenever you want to receive a complement, just wrap all
|
|
the function arguments in a call to <code class="computeroutput"><span class="identifier">complement</span><span class="special">(...)</span></code>, for example:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
|
|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"CDF at t = 1 is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="number">1.0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Complement of CDF at t = 1 is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="number">1.0</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
|
|
</pre>
|
|
<p>
|
|
But wait, now that we have a complement, we have to be able to use it as
|
|
well. Any function that accepts a probability as an argument can also accept
|
|
a complement by wrapping all of its arguments in a call to <code class="computeroutput"><span class="identifier">complement</span><span class="special">(...)</span></code>,
|
|
for example:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
|
|
|
|
<span class="keyword">for</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="number">1e10</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">*=</span> <span class="number">10</span><span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="comment">// Calculate the quantile for a 1 in i chance:</span>
|
|
<span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="number">1</span><span class="special">/</span><span class="identifier">i</span><span class="special">));</span>
|
|
<span class="comment">// Print it out:</span>
|
|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Quantile of students-t with 5 degrees of freedom\n"</span>
|
|
<span class="string">"for a 1 in "</span> <span class="special"><<</span> <span class="identifier">i</span> <span class="special"><<</span> <span class="string">" chance is "</span> <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
</pre>
|
|
<div class="tip"><table border="0" summary="Tip">
|
|
<tr>
|
|
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td>
|
|
<th align="left">Tip</th>
|
|
</tr>
|
|
<tr><td align="left" valign="top">
|
|
<p>
|
|
<span class="bold"><strong>Critical values are just quantiles</strong></span>
|
|
</p>
|
|
<p>
|
|
Some texts talk about quantiles, or percentiles or fractiles, others
|
|
about critical values, the basic rule is:
|
|
</p>
|
|
<p>
|
|
<span class="emphasis"><em>Lower critical values</em></span> are the same as the quantile.
|
|
</p>
|
|
<p>
|
|
<span class="emphasis"><em>Upper critical values</em></span> are the same as the quantile
|
|
from the complement of the probability.
|
|
</p>
|
|
<p>
|
|
For example, suppose we have a Bernoulli process, giving rise to a binomial
|
|
distribution with success ratio 0.1 and 100 trials in total. The <span class="emphasis"><em>lower
|
|
critical value</em></span> for a probability of 0.05 is given by:
|
|
</p>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">100</span><span class="special">,</span> <span class="number">0.1</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span></code>
|
|
</p>
|
|
<p>
|
|
and the <span class="emphasis"><em>upper critical value</em></span> is given by:
|
|
</p>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">100</span><span class="special">,</span> <span class="number">0.1</span><span class="special">),</span> <span class="number">0.05</span><span class="special">))</span></code>
|
|
</p>
|
|
<p>
|
|
which return 4.82 and 14.63 respectively.
|
|
</p>
|
|
</td></tr>
|
|
</table></div>
|
|
<a name="why_complements"></a><div class="tip"><table border="0" summary="Tip">
|
|
<tr>
|
|
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td>
|
|
<th align="left">Tip</th>
|
|
</tr>
|
|
<tr><td align="left" valign="top">
|
|
<p>
|
|
<span class="bold"><strong>Why bother with complements anyway?</strong></span>
|
|
</p>
|
|
<p>
|
|
It's very tempting to dispense with complements, and simply subtract
|
|
the probability from 1 when required. However, consider what happens
|
|
when the probability is very close to 1: let's say the probability expressed
|
|
at float precision is <code class="computeroutput"><span class="number">0.999999940f</span></code>,
|
|
then <code class="computeroutput"><span class="number">1</span> <span class="special">-</span>
|
|
<span class="number">0.999999940f</span> <span class="special">=</span>
|
|
<span class="number">5.96046448e-008</span></code>, but the result
|
|
is actually accurate to just <span class="emphasis"><em>one single bit</em></span>: the
|
|
only bit that didn't cancel out!
|
|
</p>
|
|
<p>
|
|
Or to look at this another way: consider that we want the risk of falsely
|
|
rejecting the null-hypothesis in the Student's t test to be 1 in 1 billion,
|
|
for a sample size of 10,000. This gives a probability of 1 - 10<sup>-9</sup>, which
|
|
is exactly 1 when calculated at float precision. In this case calculating
|
|
the quantile from the complement neatly solves the problem, so for example:
|
|
</p>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">students_t</span><span class="special">(</span><span class="number">10000</span><span class="special">),</span> <span class="number">1e-9</span><span class="special">))</span></code>
|
|
</p>
|
|
<p>
|
|
returns the expected t-statistic <code class="computeroutput"><span class="number">6.00336</span></code>,
|
|
where as:
|
|
</p>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">students_t</span><span class="special">(</span><span class="number">10000</span><span class="special">),</span> <span class="number">1</span><span class="special">-</span><span class="number">1e-9f</span><span class="special">)</span></code>
|
|
</p>
|
|
<p>
|
|
raises an overflow error, since it is the same as:
|
|
</p>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">students_t</span><span class="special">(</span><span class="number">10000</span><span class="special">),</span> <span class="number">1</span><span class="special">)</span></code>
|
|
</p>
|
|
<p>
|
|
Which has no finite result.
|
|
</p>
|
|
<p>
|
|
With all distributions, even for more reasonable probability (unless
|
|
the value of p can be represented exactly in the floating-point type)
|
|
the loss of accuracy quickly becomes significant if you simply calculate
|
|
probability from 1 - p (because it will be mostly garbage digits for
|
|
p ~ 1).
|
|
</p>
|
|
<p>
|
|
So always avoid, for example, using a probability near to unity like
|
|
0.99999
|
|
</p>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">my_distribution</span><span class="special">,</span>
|
|
<span class="number">0.99999</span><span class="special">)</span></code>
|
|
</p>
|
|
<p>
|
|
and instead use
|
|
</p>
|
|
<p>
|
|
<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">my_distribution</span><span class="special">,</span>
|
|
<span class="number">0.00001</span><span class="special">))</span></code>
|
|
</p>
|
|
<p>
|
|
since 1 - 0.99999 is not exactly equal to 0.00001 when using floating-point
|
|
arithmetic.
|
|
</p>
|
|
<p>
|
|
This assumes that the 0.00001 value is either a constant, or can be computed
|
|
by some manner other than subtracting 0.99999 from 1.
|
|
</p>
|
|
</td></tr>
|
|
</table></div>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
|
Daryle Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="generic.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../overview.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="parameters.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
</body>
|
|
</html>
|