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26 <div class=
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27 <a name=
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28 Scale (Standard Deviation) Example
</a>
29 </h5></div></div></div>
31 First we need some includes to access the
<a class=
"link" href=
"../../../dist_ref/dists/normal_dist.html" title=
"Normal (Gaussian) Distribution">Normal
32 Distribution
</a>, the algorithms to find scale (and some std output
35 <pre class=
"programlisting"><span class=
"preprocessor">#include
</span> <span class=
"special"><</span><span class=
"identifier">boost
</span><span class=
"special">/
</span><span class=
"identifier">math
</span><span class=
"special">/
</span><span class=
"identifier">distributions
</span><span class=
"special">/
</span><span class=
"identifier">normal
</span><span class=
"special">.
</span><span class=
"identifier">hpp
</span><span class=
"special">></span> <span class=
"comment">// for normal_distribution
</span>
36 <span class=
"keyword">using
</span> <span class=
"identifier">boost
</span><span class=
"special">::
</span><span class=
"identifier">math
</span><span class=
"special">::
</span><span class=
"identifier">normal
</span><span class=
"special">;
</span> <span class=
"comment">// typedef provides default type is double.
</span>
37 <span class=
"preprocessor">#include
</span> <span class=
"special"><</span><span class=
"identifier">boost
</span><span class=
"special">/
</span><span class=
"identifier">math
</span><span class=
"special">/
</span><span class=
"identifier">distributions
</span><span class=
"special">/
</span><span class=
"identifier">find_scale
</span><span class=
"special">.
</span><span class=
"identifier">hpp
</span><span class=
"special">></span>
38 <span class=
"keyword">using
</span> <span class=
"identifier">boost
</span><span class=
"special">::
</span><span class=
"identifier">math
</span><span class=
"special">::
</span><span class=
"identifier">find_scale
</span><span class=
"special">;
</span>
39 <span class=
"keyword">using
</span> <span class=
"identifier">boost
</span><span class=
"special">::
</span><span class=
"identifier">math
</span><span class=
"special">::
</span><span class=
"identifier">complement
</span><span class=
"special">;
</span> <span class=
"comment">// Needed if you want to use the complement version.
</span>
40 <span class=
"keyword">using
</span> <span class=
"identifier">boost
</span><span class=
"special">::
</span><span class=
"identifier">math
</span><span class=
"special">::
</span><span class=
"identifier">policies
</span><span class=
"special">::
</span><span class=
"identifier">policy
</span><span class=
"special">;
</span> <span class=
"comment">// Needed to specify the error handling policy.
</span>
42 <span class=
"preprocessor">#include
</span> <span class=
"special"><</span><span class=
"identifier">iostream
</span><span class=
"special">></span>
43 <span class=
"keyword">using
</span> <span class=
"identifier">std
</span><span class=
"special">::
</span><span class=
"identifier">cout
</span><span class=
"special">;
</span> <span class=
"keyword">using
</span> <span class=
"identifier">std
</span><span class=
"special">::
</span><span class=
"identifier">endl
</span><span class=
"special">;
</span>
44 <span class=
"preprocessor">#include
</span> <span class=
"special"><</span><span class=
"identifier">iomanip
</span><span class=
"special">></span>
45 <span class=
"keyword">using
</span> <span class=
"identifier">std
</span><span class=
"special">::
</span><span class=
"identifier">setw
</span><span class=
"special">;
</span> <span class=
"keyword">using
</span> <span class=
"identifier">std
</span><span class=
"special">::
</span><span class=
"identifier">setprecision
</span><span class=
"special">;
</span>
46 <span class=
"preprocessor">#include
</span> <span class=
"special"><</span><span class=
"identifier">limits
</span><span class=
"special">></span>
47 <span class=
"keyword">using
</span> <span class=
"identifier">std
</span><span class=
"special">::
</span><span class=
"identifier">numeric_limits
</span><span class=
"special">;
</span>
50 For this example, we will use the standard
<a class=
"link" href=
"../../../dist_ref/dists/normal_dist.html" title=
"Normal (Gaussian) Distribution">Normal
51 Distribution
</a>, with location (mean) zero and standard deviation
52 (scale) unity. Conveniently, this is also the default for this implementation's
55 <pre class=
"programlisting"><span class=
"identifier">normal
</span> <span class=
"identifier">N01
</span><span class=
"special">;
</span> <span class=
"comment">// Default 'standard' normal distribution with zero mean
</span>
56 <span class=
"keyword">double
</span> <span class=
"identifier">sd
</span> <span class=
"special">=
</span> <span class=
"number">1.
</span><span class=
"special">;
</span> <span class=
"comment">// and standard deviation is
1.
</span>
59 Suppose we want to find a different normal distribution with standard
60 deviation so that only fraction p (here
0.001 or
0.1%) are below a certain
61 chosen limit (here -
2. standard deviations).
63 <pre class=
"programlisting"><span class=
"keyword">double
</span> <span class=
"identifier">z
</span> <span class=
"special">=
</span> <span class=
"special">-
</span><span class=
"number">2.
</span><span class=
"special">;
</span> <span class=
"comment">// z to give prob p
</span>
64 <span class=
"keyword">double
</span> <span class=
"identifier">p
</span> <span class=
"special">=
</span> <span class=
"number">0.001</span><span class=
"special">;
</span> <span class=
"comment">// only
0.1% below z = -
2</span>
66 <span class=
"identifier">cout
</span> <span class=
"special"><<</span> <span class=
"string">"Normal distribution with mean = "</span> <span class=
"special"><<</span> <span class=
"identifier">N01
</span><span class=
"special">.
</span><span class=
"identifier">location
</span><span class=
"special">()
</span> <span class=
"comment">// aka N01.mean()
</span>
67 <span class=
"special"><<</span> <span class=
"string">", standard deviation "</span> <span class=
"special"><<</span> <span class=
"identifier">N01
</span><span class=
"special">.
</span><span class=
"identifier">scale
</span><span class=
"special">()
</span> <span class=
"comment">// aka N01.standard_deviation()
</span>
68 <span class=
"special"><<</span> <span class=
"string">", has "</span> <span class=
"special"><<</span> <span class=
"string">"fraction <= "</span> <span class=
"special"><<</span> <span class=
"identifier">z
</span>
69 <span class=
"special"><<</span> <span class=
"string">", p = "</span> <span class=
"special"><<</span> <span class=
"identifier">cdf
</span><span class=
"special">(
</span><span class=
"identifier">N01
</span><span class=
"special">,
</span> <span class=
"identifier">z
</span><span class=
"special">)
</span> <span class=
"special"><<</span> <span class=
"identifier">endl
</span><span class=
"special">;
</span>
70 <span class=
"identifier">cout
</span> <span class=
"special"><<</span> <span class=
"string">"Normal distribution with mean = "</span> <span class=
"special"><<</span> <span class=
"identifier">N01
</span><span class=
"special">.
</span><span class=
"identifier">location
</span><span class=
"special">()
</span>
71 <span class=
"special"><<</span> <span class=
"string">", standard deviation "</span> <span class=
"special"><<</span> <span class=
"identifier">N01
</span><span class=
"special">.
</span><span class=
"identifier">scale
</span><span class=
"special">()
</span>
72 <span class=
"special"><<</span> <span class=
"string">", has "</span> <span class=
"special"><<</span> <span class=
"string">"fraction > "</span> <span class=
"special"><<</span> <span class=
"identifier">z
</span>
73 <span class=
"special"><<</span> <span class=
"string">", p = "</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">N01
</span><span class=
"special">,
</span> <span class=
"identifier">z
</span><span class=
"special">))
</span> <span class=
"special"><<</span> <span class=
"identifier">endl
</span><span class=
"special">;
</span> <span class=
"comment">// Note: uses complement.
</span>
75 <pre class=
"programlisting">Normal distribution with mean =
0 has fraction
<= -
2, p =
0.0227501
76 Normal distribution with mean =
0 has fraction
> -
2, p =
0.97725
79 Noting that p =
0.02 instead of our target of
0.001, we can now use
80 <code class=
"computeroutput"><span class=
"identifier">find_scale
</span></code> to give a
81 new standard deviation.
83 <pre class=
"programlisting"><span class=
"keyword">double
</span> <span class=
"identifier">l
</span> <span class=
"special">=
</span> <span class=
"identifier">N01
</span><span class=
"special">.
</span><span class=
"identifier">location
</span><span class=
"special">();
</span>
84 <span class=
"keyword">double
</span> <span class=
"identifier">s
</span> <span class=
"special">=
</span> <span class=
"identifier">find_scale
</span><span class=
"special"><</span><span class=
"identifier">normal
</span><span class=
"special">>(
</span><span class=
"identifier">z
</span><span class=
"special">,
</span> <span class=
"identifier">p
</span><span class=
"special">,
</span> <span class=
"identifier">l
</span><span class=
"special">);
</span>
85 <span class=
"identifier">cout
</span> <span class=
"special"><<</span> <span class=
"string">"scale (standard deviation) = "</span> <span class=
"special"><<</span> <span class=
"identifier">s
</span> <span class=
"special"><<</span> <span class=
"identifier">endl
</span><span class=
"special">;
</span>
90 <pre class=
"programlisting">scale (standard deviation) =
0.647201
93 showing that we need to reduce the standard deviation from
1. to
0.65.
96 Then we can check that we have achieved our objective by constructing
97 a new distribution with the new standard deviation (but same zero mean):
99 <pre class=
"programlisting"><span class=
"identifier">normal
</span> <span class=
"identifier">np001pc
</span><span class=
"special">(
</span><span class=
"identifier">N01
</span><span class=
"special">.
</span><span class=
"identifier">location
</span><span class=
"special">(),
</span> <span class=
"identifier">s
</span><span class=
"special">);
</span>
102 And re-calculating the fraction below (and above) our chosen limit.
104 <pre class=
"programlisting"><span class=
"identifier">cout
</span> <span class=
"special"><<</span> <span class=
"string">"Normal distribution with mean = "</span> <span class=
"special"><<</span> <span class=
"identifier">l
</span>
105 <span class=
"special"><<</span> <span class=
"string">" has "</span> <span class=
"special"><<</span> <span class=
"string">"fraction <= "</span> <span class=
"special"><<</span> <span class=
"identifier">z
</span>
106 <span class=
"special"><<</span> <span class=
"string">", p = "</span> <span class=
"special"><<</span> <span class=
"identifier">cdf
</span><span class=
"special">(
</span><span class=
"identifier">np001pc
</span><span class=
"special">,
</span> <span class=
"identifier">z
</span><span class=
"special">)
</span> <span class=
"special"><<</span> <span class=
"identifier">endl
</span><span class=
"special">;
</span>
107 <span class=
"identifier">cout
</span> <span class=
"special"><<</span> <span class=
"string">"Normal distribution with mean = "</span> <span class=
"special"><<</span> <span class=
"identifier">l
</span>
108 <span class=
"special"><<</span> <span class=
"string">" has "</span> <span class=
"special"><<</span> <span class=
"string">"fraction > "</span> <span class=
"special"><<</span> <span class=
"identifier">z
</span>
109 <span class=
"special"><<</span> <span class=
"string">", p = "</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">np001pc
</span><span class=
"special">,
</span> <span class=
"identifier">z
</span><span class=
"special">))
</span> <span class=
"special"><<</span> <span class=
"identifier">endl
</span><span class=
"special">;
</span>
111 <pre class=
"programlisting">Normal distribution with mean =
0 has fraction
<= -
2, p =
0.001
112 Normal distribution with mean =
0 has fraction
> -
2, p =
0.999
115 <a name=
"math_toolkit.stat_tut.weg.find_eg.find_scale_eg.h0"></a>
116 <span class=
"phrase"><a name=
"math_toolkit.stat_tut.weg.find_eg.find_scale_eg.controlling_how_errors_from_find"></a></span><a class=
"link" href=
"find_scale_eg.html#math_toolkit.stat_tut.weg.find_eg.find_scale_eg.controlling_how_errors_from_find">Controlling
117 how Errors from find_scale are handled
</a>
120 We can also control the policy for handling various errors. For example,
121 we can define a new (possibly unwise) policy to ignore domain errors
125 Unless we are using the boost::math namespace, we will need:
127 <pre class=
"programlisting"><span class=
"keyword">using
</span> <span class=
"identifier">boost
</span><span class=
"special">::
</span><span class=
"identifier">math
</span><span class=
"special">::
</span><span class=
"identifier">policies
</span><span class=
"special">::
</span><span class=
"identifier">policy
</span><span class=
"special">;
</span>
128 <span class=
"keyword">using
</span> <span class=
"identifier">boost
</span><span class=
"special">::
</span><span class=
"identifier">math
</span><span class=
"special">::
</span><span class=
"identifier">policies
</span><span class=
"special">::
</span><span class=
"identifier">domain_error
</span><span class=
"special">;
</span>
129 <span class=
"keyword">using
</span> <span class=
"identifier">boost
</span><span class=
"special">::
</span><span class=
"identifier">math
</span><span class=
"special">::
</span><span class=
"identifier">policies
</span><span class=
"special">::
</span><span class=
"identifier">ignore_error
</span><span class=
"special">;
</span>
132 Using a typedef is convenient, especially if it is re-used, although
133 it is not required, as the various examples below show.
135 <pre class=
"programlisting"><span class=
"keyword">typedef
</span> <span class=
"identifier">policy
</span><span class=
"special"><</span><span class=
"identifier">domain_error
</span><span class=
"special"><</span><span class=
"identifier">ignore_error
</span><span class=
"special">></span> <span class=
"special">></span> <span class=
"identifier">ignore_domain_policy
</span><span class=
"special">;
</span>
136 <span class=
"comment">// find_scale with new policy, using typedef.
</span>
137 <span class=
"identifier">l
</span> <span class=
"special">=
</span> <span class=
"identifier">find_scale
</span><span class=
"special"><</span><span class=
"identifier">normal
</span><span class=
"special">>(
</span><span class=
"identifier">z
</span><span class=
"special">,
</span> <span class=
"identifier">p
</span><span class=
"special">,
</span> <span class=
"identifier">l
</span><span class=
"special">,
</span> <span class=
"identifier">ignore_domain_policy
</span><span class=
"special">());
</span>
138 <span class=
"comment">// Default policy policy
<>, needs using boost::math::policies::policy;
</span>
140 <span class=
"identifier">l
</span> <span class=
"special">=
</span> <span class=
"identifier">find_scale
</span><span class=
"special"><</span><span class=
"identifier">normal
</span><span class=
"special">>(
</span><span class=
"identifier">z
</span><span class=
"special">,
</span> <span class=
"identifier">p
</span><span class=
"special">,
</span> <span class=
"identifier">l
</span><span class=
"special">,
</span> <span class=
"identifier">policy
</span><span class=
"special"><>());
</span>
141 <span class=
"comment">// Default policy, fully specified.
</span>
142 <span class=
"identifier">l
</span> <span class=
"special">=
</span> <span class=
"identifier">find_scale
</span><span class=
"special"><</span><span class=
"identifier">normal
</span><span class=
"special">>(
</span><span class=
"identifier">z
</span><span class=
"special">,
</span> <span class=
"identifier">p
</span><span class=
"special">,
</span> <span class=
"identifier">l
</span><span class=
"special">,
</span> <span class=
"identifier">boost
</span><span class=
"special">::
</span><span class=
"identifier">math
</span><span class=
"special">::
</span><span class=
"identifier">policies
</span><span class=
"special">::
</span><span class=
"identifier">policy
</span><span class=
"special"><>());
</span>
143 <span class=
"comment">// New policy, without typedef.
</span>
144 <span class=
"identifier">l
</span> <span class=
"special">=
</span> <span class=
"identifier">find_scale
</span><span class=
"special"><</span><span class=
"identifier">normal
</span><span class=
"special">>(
</span><span class=
"identifier">z
</span><span class=
"special">,
</span> <span class=
"identifier">p
</span><span class=
"special">,
</span> <span class=
"identifier">l
</span><span class=
"special">,
</span> <span class=
"identifier">policy
</span><span class=
"special"><</span><span class=
"identifier">domain_error
</span><span class=
"special"><</span><span class=
"identifier">ignore_error
</span><span class=
"special">></span> <span class=
"special">>());
</span>
147 If we want to express a probability, say
0.999, that is a complement,
148 <code class=
"computeroutput"><span class=
"number">1</span> <span class=
"special">-
</span>
149 <span class=
"identifier">p
</span></code> we should not even think
150 of writing
<code class=
"computeroutput"><span class=
"identifier">find_scale
</span><span class=
"special"><</span><span class=
"identifier">normal
</span><span class=
"special">>(
</span><span class=
"identifier">z
</span><span class=
"special">,
</span> <span class=
"number">1</span> <span class=
"special">-
</span>
151 <span class=
"identifier">p
</span><span class=
"special">,
</span>
152 <span class=
"identifier">l
</span><span class=
"special">)
</span></code>,
153 but use the
<a class=
"link" href=
"../../overview/complements.html" title=
"Complements are supported too - and when to use them">complements
</a>
154 version (see
<a class=
"link" href=
"../../overview/complements.html#why_complements">why complements?
</a>).
156 <pre class=
"programlisting"><span class=
"identifier">z
</span> <span class=
"special">=
</span> <span class=
"special">-
</span><span class=
"number">2.
</span><span class=
"special">;
</span>
157 <span class=
"keyword">double
</span> <span class=
"identifier">q
</span> <span class=
"special">=
</span> <span class=
"number">0.999</span><span class=
"special">;
</span> <span class=
"comment">// =
1 - p; // complement of
0.001.
</span>
158 <span class=
"identifier">sd
</span> <span class=
"special">=
</span> <span class=
"identifier">find_scale
</span><span class=
"special"><</span><span class=
"identifier">normal
</span><span class=
"special">>(
</span><span class=
"identifier">complement
</span><span class=
"special">(
</span><span class=
"identifier">z
</span><span class=
"special">,
</span> <span class=
"identifier">q
</span><span class=
"special">,
</span> <span class=
"identifier">l
</span><span class=
"special">));
</span>
160 <span class=
"identifier">normal
</span> <span class=
"identifier">np95pc
</span><span class=
"special">(
</span><span class=
"identifier">l
</span><span class=
"special">,
</span> <span class=
"identifier">sd
</span><span class=
"special">);
</span> <span class=
"comment">// Same standard_deviation (scale) but with mean(scale) shifted
</span>
161 <span class=
"identifier">cout
</span> <span class=
"special"><<</span> <span class=
"string">"Normal distribution with mean = "</span> <span class=
"special"><<</span> <span class=
"identifier">l
</span> <span class=
"special"><<</span> <span class=
"string">" has "</span>
162 <span class=
"special"><<</span> <span class=
"string">"fraction <= "</span> <span class=
"special"><<</span> <span class=
"identifier">z
</span> <span class=
"special"><<</span> <span class=
"string">" = "</span> <span class=
"special"><<</span> <span class=
"identifier">cdf
</span><span class=
"special">(
</span><span class=
"identifier">np95pc
</span><span class=
"special">,
</span> <span class=
"identifier">z
</span><span class=
"special">)
</span> <span class=
"special"><<</span> <span class=
"identifier">endl
</span><span class=
"special">;
</span>
163 <span class=
"identifier">cout
</span> <span class=
"special"><<</span> <span class=
"string">"Normal distribution with mean = "</span> <span class=
"special"><<</span> <span class=
"identifier">l
</span> <span class=
"special"><<</span> <span class=
"string">" has "</span>
164 <span class=
"special"><<</span> <span class=
"string">"fraction > "</span> <span class=
"special"><<</span> <span class=
"identifier">z
</span> <span class=
"special"><<</span> <span class=
"string">" = "</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">np95pc
</span><span class=
"special">,
</span> <span class=
"identifier">z
</span><span class=
"special">))
</span> <span class=
"special"><<</span> <span class=
"identifier">endl
</span><span class=
"special">;
</span>
167 Sadly, it is all too easy to get probabilities the wrong way round, when
168 you may get a warning like this:
170 <pre class=
"programlisting">Message from thrown exception was:
171 Error in function boost::math::find_scale
<Dist, Policy
>(complement(double, double, double, Policy)):
172 Computed scale (-
0.48043523852179076) is
<=
0! Was the complement intended?
175 The default error handling policy is to throw an exception with this
176 message, but if you chose a policy to ignore the error, the (impossible)
177 negative scale is quietly returned.
180 See
<a href=
"../../../../../../example/find_scale_example.cpp" target=
"_top">find_scale_example.cpp
</a>
181 for full source code: the program output looks like this:
183 <pre class=
"programlisting">Example: Find scale (standard deviation).
184 Normal distribution with mean =
0, standard deviation
1, has fraction
<= -
2, p =
0.0227501
185 Normal distribution with mean =
0, standard deviation
1, has fraction
> -
2, p =
0.97725
186 scale (standard deviation) =
0.647201
187 Normal distribution with mean =
0 has fraction
<= -
2, p =
0.001
188 Normal distribution with mean =
0 has fraction
> -
2, p =
0.999
189 Normal distribution with mean =
0.946339 has fraction
<= -
2 =
0.001
190 Normal distribution with mean =
0.946339 has fraction
> -
2 =
0.999
193 <table xmlns:
rev=
"http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width=
"100%"><tr>
194 <td align=
"left"></td>
195 <td align=
"right"><div class=
"copyright-footer">Copyright
© 2006-
2010,
2012-
2014 Nikhar Agrawal,
196 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
197 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R
åde, Gautam Sewani,
198 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang
<p>
199 Distributed under the Boost Software License, Version
1.0. (See accompanying
200 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>)
205 <div class=
"spirit-nav">
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"p" href=
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