<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Find Scale (Standard Deviation) Example</title> <link rel="stylesheet" href="../../../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../../../index.html" title="Math Toolkit"> <link rel="up" href="../find_eg.html" title="Find Location and Scale Examples"> <link rel="prev" href="find_location_eg.html" title="Find Location (Mean) Example"> <link rel="next" href="find_mean_and_sd_eg.html" title="Find mean and standard deviation example"> </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 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name="math_toolkit.dist.stat_tut.weg.find_eg.find_scale_eg"></a><a class="link" href="find_scale_eg.html" title="Find Scale (Standard Deviation) Example"> Find Scale (Standard Deviation) Example</a> </h6></div></div></div> <p> </p> <p> First we need some includes to access the <a class="link" href="../../../dist_ref/dists/normal_dist.html" title="Normal (Gaussian) Distribution">Normal Distribution</a>, the algorithms to find scale (and some std output of course). </p> <p> </p> <p> </p> <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> <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><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> <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> <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> <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> <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> <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> <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> <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> <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">limits</span><span class="special">></span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span> </pre> <p> </p> <p> </p> <p> For this example, we will use the standard <a class="link" href="../../../dist_ref/dists/normal_dist.html" title="Normal (Gaussian) Distribution">Normal Distribution</a>, with location (mean) zero and standard deviation (scale) unity. Conveniently, this is also the default for this implementation's constructor. </p> <p> </p> <p> </p> <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><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></pre> <p> </p> <p> </p> <p> Suppose we want to find a different normal distribution with standard deviation so that only fraction p (here 0.001 or 0.1%) are below a certain chosen limit (here -2. standard deviations). </p> <p> </p> <p> </p> <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><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> <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> <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> <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> <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> <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="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="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> <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></pre> <p> </p> <p> </p> <pre class="programlisting">Normal distribution with mean = 0 has fraction <= -2, p = 0.0227501 Normal distribution with mean = 0 has fraction > -2, p = 0.97725 </pre> <p> </p> <p> Noting that p = 0.02 instead of our target of 0.001, we can now use <code class="computeroutput"><span class="identifier">find_scale</span></code> to give a new standard deviation. </p> <p> </p> <p> </p> <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> <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> <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></pre> <p> </p> <p> </p> <p> that outputs: </p> <p> </p> <pre class="programlisting">scale (standard deviation) = 0.647201 </pre> <p> </p> <p> showing that we need to reduce the standard deviation from 1. to 0.65. </p> <p> </p> <p> Then we can check that we have achieved our objective by constructing a new distribution with the new standard deviation (but same zero mean): </p> <p> </p> <p> </p> <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></pre> <p> </p> <p> </p> <p> And re-calculating the fraction below (and above) our chosen limit. </p> <p> </p> <p> </p> <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> <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> <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> <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> <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">", 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></pre> <p> </p> <p> </p> <pre class="programlisting">Normal distribution with mean = 0 has fraction <= -2, p = 0.001 Normal distribution with mean = 0 has fraction > -2, p = 0.999 </pre> <p> <a name="math_toolkit.dist.stat_tut.weg.find_eg.find_scale_eg.controlling_how_errors_from_find_scale_are_handled"></a> </p> <h5> <a name="id1000236"></a> <a class="link" href="find_scale_eg.html#math_toolkit.dist.stat_tut.weg.find_eg.find_scale_eg.controlling_how_errors_from_find_scale_are_handled">Controlling how Errors from find_scale are handled</a> </h5> <p> </p> <p> We can also control the policy for handling various errors. For example, we can define a new (possibly unwise) policy to ignore domain errors ('bad' arguments). </p> <p> </p> <p> Unless we are using the boost::math namespace, we will need: </p> <p> </p> <p> </p> <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> <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> <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></pre> <p> </p> <p> </p> <p> Using a typedef is convenient, especially if it is re-used, although it is not required, as the various examples below show. </p> <p> </p> <p> </p> <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> <span class="comment">// find_scale with new policy, using typedef. </span><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> <span class="comment">// Default policy policy<>, needs using boost::math::policies::policy; </span> <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="comment">// Default policy, fully specified. </span><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> <span class="comment">// New policy, without typedef. </span><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></pre> <p> </p> <p> </p> <p> If we want to express a probability, say 0.999, that is a complement, <code class="computeroutput"><span class="number">1</span> <span class="special">-</span> <span class="identifier">p</span></code> we should not even think 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> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">l</span><span class="special">)</span></code>, but <a class="link" href="../../overview/complements.html#why_complements">instead</a>, use the <a class="link" href="../../overview.html#complements">complements</a> version. </p> <p> </p> <p> </p> <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> <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><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> <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><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> <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> <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> <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></pre> <p> </p> <p> </p> <p> Sadly, it is all too easy to get probabilities the wrong way round, when you may get a warning like this: </p> <p> </p> <pre class="programlisting">Message from thrown exception was: Error in function boost::math::find_scale<Dist, Policy>(complement(double, double, double, Policy)): Computed scale (-0.48043523852179076) is <= 0! Was the complement intended? </pre> <p> </p> <p> The default error handling policy is to throw an exception with this message, but if you chose a policy to ignore the error, the (impossible) negative scale is quietly returned. </p> <p> See <a href="../../../../../../../../example/find_scale_example.cpp" target="_top">find_scale_example.cpp</a> for full source code: the program output looks like this: </p> <pre class="programlisting">Example: Find scale (standard deviation). Normal distribution with mean = 0, standard deviation 1, has fraction <= -2, p = 0.0227501 Normal distribution with mean = 0, standard deviation 1, has fraction > -2, p = 0.97725 scale (standard deviation) = 0.647201 Normal distribution with mean = 0 has fraction <= -2, p = 0.001 Normal distribution with mean = 0 has fraction > -2, p = 0.999 Normal distribution with mean = 0.946339 has fraction <= -2 = 0.001 Normal distribution with mean = 0.946339 has fraction > -2 = 0.999 </pre> <p> </p> </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 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and 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>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="find_location_eg.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../find_eg.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="find_mean_and_sd_eg.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>