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<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">&lt;&lt;</span> <span class="string">"CDF at t = 1 is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Complement of CDF at t = 1 is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">i</span> <span class="special">&lt;&lt;</span> <span class="string">" chance is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">t</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
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<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>
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<tr>
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</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>
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      Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
      Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
      Benjamin Sobotta, 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>)
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	24	</div>
	25	<div class="section">
	26	<div class="titlepage"><div><div><h4 class="title">
	27	<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
	28	are supported too - and when to use them</a>
	29	</h4></div></div></div>
	30	<p>
	31	Often you don't want the value of the CDF, but its complement, which is
	32	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>.
	33	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>
	34	is very close to <code class="computeroutput"><span class="number">1</span></code> then cancellation
	35	error will cause you to lose accuracy, perhaps totally.
	36	</p>
	37	<p>
	38	<a class="link" href="complements.html#why_complements">See below <span class="emphasis"><em>"Why and when
	39	to use complements?"</em></span></a>
	40	</p>
	41	<p>
	42	In this library, whenever you want to receive a complement, just wrap all
	43	the function arguments in a call to <code class="computeroutput"><span class="identifier">complement</span><span class="special">(...)</span></code>, for example:
	44	</p>
	45	<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>
	46	<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>
	47	<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>
	48	</pre>
	49	<p>
	50	But wait, now that we have a complement, we have to be able to use it as
	51	well. Any function that accepts a probability as an argument can also accept
	52	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>,
	53	for example:
	54	</p>
	55	<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>
	56
	57	<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>
	58	<span class="special">{</span>
	59	<span class="comment">// Calculate the quantile for a 1 in i chance:</span>
	60	<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>
	61	<span class="comment">// Print it out:</span>
	62	<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Quantile of students-t with 5 degrees of freedom\n"</span>
	63	<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>
	64	<span class="special">}</span>
65	</pre>
66	<div class="tip"><table border="0" summary="Tip">
67	<tr>
68	<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td>
69	<th align="left">Tip</th>
70	</tr>
71	<tr><td align="left" valign="top">
72	<p>
73	<span class="bold"><strong>Critical values are just quantiles</strong></span>
74	</p>
75	<p>
76	Some texts talk about quantiles, or percentiles or fractiles, others
77	about critical values, the basic rule is:
78	</p>
79	<p>
80	<span class="emphasis"><em>Lower critical values</em></span> are the same as the quantile.
81	</p>
82	<p>
83	<span class="emphasis"><em>Upper critical values</em></span> are the same as the quantile
84	from the complement of the probability.
85	</p>
86	<p>
87	For example, suppose we have a Bernoulli process, giving rise to a binomial
88	distribution with success ratio 0.1 and 100 trials in total. The <span class="emphasis"><em>lower
89	critical value</em></span> for a probability of 0.05 is given by:
90	</p>
91	<p>
92	<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>
93	</p>
94	<p>
95	and the <span class="emphasis"><em>upper critical value</em></span> is given by:
96	</p>
97	<p>
98	<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>
99	</p>
100	<p>
101	which return 4.82 and 14.63 respectively.
102	</p>
103	</td></tr>
104	</table></div>
105	<a name="why_complements"></a><div class="tip"><table border="0" summary="Tip">
106	<tr>
107	<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td>
108	<th align="left">Tip</th>
109	</tr>
110	<tr><td align="left" valign="top">
111	<p>
112	<span class="bold"><strong>Why bother with complements anyway?</strong></span>
113	</p>
114	<p>
115	It's very tempting to dispense with complements, and simply subtract
116	the probability from 1 when required. However, consider what happens
117	when the probability is very close to 1: let's say the probability expressed
118	at float precision is <code class="computeroutput"><span class="number">0.999999940f</span></code>,
119	then <code class="computeroutput"><span class="number">1</span> <span class="special">-</span>
120	<span class="number">0.999999940f</span> <span class="special">=</span>
121	<span class="number">5.96046448e-008</span></code>, but the result
122	is actually accurate to just <span class="emphasis"><em>one single bit</em></span>: the
123	only bit that didn't cancel out!
124	</p>
125	<p>
126	Or to look at this another way: consider that we want the risk of falsely
127	rejecting the null-hypothesis in the Student's t test to be 1 in 1 billion,
128	for a sample size of 10,000. This gives a probability of 1 - 10<sup>-9</sup>, which
129	is exactly 1 when calculated at float precision. In this case calculating
130	the quantile from the complement neatly solves the problem, so for example:
131	</p>
132	<p>
133	<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>
134	</p>
135	<p>
136	returns the expected t-statistic <code class="computeroutput"><span class="number">6.00336</span></code>,
137	where as:
138	</p>
139	<p>
140	<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>
141	</p>
142	<p>
143	raises an overflow error, since it is the same as:
144	</p>
145	<p>
146	<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>
147	</p>
148	<p>
149	Which has no finite result.
150	</p>
151	<p>
152	With all distributions, even for more reasonable probability (unless
153	the value of p can be represented exactly in the floating-point type)
154	the loss of accuracy quickly becomes significant if you simply calculate
155	probability from 1 - p (because it will be mostly garbage digits for
156	p ~ 1).
157	</p>
158	<p>
159	So always avoid, for example, using a probability near to unity like
160	0.99999
161	</p>
162	<p>
163	<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">my_distribution</span><span class="special">,</span>
164	<span class="number">0.99999</span><span class="special">)</span></code>
165	</p>
166	<p>
167	and instead use
168	</p>
169	<p>
170	<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>
171	<span class="number">0.00001</span><span class="special">))</span></code>
172	</p>
173	<p>
174	since 1 - 0.99999 is not exactly equal to 0.00001 when using floating-point
175	arithmetic.
176	</p>
177	<p>
178	This assumes that the 0.00001 value is either a constant, or can be computed
179	by some manner other than subtracting 0.99999 from 1.
180	</p>
181	</td></tr>
182	</table></div>
183	</div>
184	<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
185	<td align="left"></td>
186	<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
187	Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
188	Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
189	Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
190	Distributed under the Boost Software License, Version 1.0. (See accompanying
191	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>)
192	</p>
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