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| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.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> |
| </h5></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> |
| </p> |
| <p> |
| <span class="bold"><strong>Critical values are just quantiles</strong></span> |
| </p> |
| <p> |
| Some texts talk about quantiles, or percentiles, 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><p> |
| </p> |
| <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> |
| </p> |
| <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> |
| <p> |
| </p> |
| </td></tr> |
| </table></div> |
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| Thijs van den Berg<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>) |
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