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| <div class="titlepage"><div><div><h6 class="title"> |
| <a 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="id991616"></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, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
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