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  27 <a name="math_toolkit.stat_tut.weg.find_eg.find_location_eg"></a><a class="link" href="find_location_eg.html" title="Find Location (Mean) Example">Find
  28           Location (Mean) Example</a>
  29 </h5></div></div></div>
  30 <p>
  31             First we need some includes to access the normal distribution, the algorithms
  32             to find location (and some std output of course).
  33           </p>
  34 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span> <span class="comment">// for normal_distribution</span>
  35   <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>
  36 <span class="preprocessor">#include</span> <span class="special">&lt;</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">cauchy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for cauchy_distribution</span>
  37   <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">cauchy</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
  38 <span class="preprocessor">#include</span> <span class="special">&lt;</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_location</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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">find_location</span><span class="special">;</span> <span class="comment">// for mean</span>
  40 <span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
  41   <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="comment">// for standard devation</span>
  42   <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>
  43   <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>
  44
  45 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
  46   <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>
  47 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
  48   <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>
  49 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">limits</span><span class="special">&gt;</span>
  50   <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span>
  51 </pre>
  52 <p>
  53             For this example, we will use the standard normal distribution, with
  54             mean (location) zero and standard deviation (scale) unity. This is also
  55             the default for this implementation.
  56           </p>
  57 <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 and</span>
  58 <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">// normal default standard deviation is 1.</span>
  59 </pre>
  60 <p>
  61             Suppose we want to find a different normal distribution whose mean is
  62             shifted so that only fraction p (here 0.001 or 0.1%) are below a certain
  63             chosen limit (here -2, two standard deviations).
  64           </p>
  65 <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>
  66 <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</span>
  67
  68 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">location</span><span class="special">()</span>
  69   <span class="special">&lt;&lt;</span> <span class="string">", standard deviation "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">scale</span><span class="special">()</span>
  70   <span class="special">&lt;&lt;</span> <span class="string">", has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
  71   <span class="special">&lt;&lt;</span> <span class="string">", p = "</span>  <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  72 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">location</span><span class="special">()</span>
  73   <span class="special">&lt;&lt;</span> <span class="string">", standard deviation "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N01</span><span class="special">.</span><span class="identifier">scale</span><span class="special">()</span>
  74   <span class="special">&lt;&lt;</span> <span class="string">", has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
  75   <span class="special">&lt;&lt;</span> <span class="string">", p = "</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">N01</span><span class="special">,</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Note: uses complement.</span>
  76 </pre>
  77 <pre class="programlisting">Normal distribution with mean = 0, standard deviation 1, has fraction &lt;= -2, p = 0.0227501
  78 Normal distribution with mean = 0, standard deviation 1, has fraction &gt; -2, p = 0.97725
  79 </pre>
  80 <p>
  81             We can now use ''find_location'' to give a new offset mean.
  82           </p>
  83 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">);</span>
  84 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"offset location (mean) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  85 </pre>
  86 <p>
  87             that outputs:
  88           </p>
  89 <pre class="programlisting">offset location (mean) = 1.09023
  90 </pre>
  91 <p>
  92             showing that we need to shift the mean just over one standard deviation
  93             from its previous value of zero.
  94           </p>
  95 <p>
  96             Then we can check that we have achieved our objective by constructing
  97             a new distribution with the offset mean (but same standard deviation):
  98           </p>
  99 <pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">np001pc</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 (location) shifted.</span>
 100 </pre>
 101 <p>
 102             And re-calculating the fraction below our chosen limit.
 103           </p>
 104 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span>
 105     <span class="special">&lt;&lt;</span> <span class="string">" has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
 106     <span class="special">&lt;&lt;</span> <span class="string">", p = "</span>  <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 107   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span>
 108     <span class="special">&lt;&lt;</span> <span class="string">" has "</span> <span class="special">&lt;&lt;</span> <span class="string">"fraction &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span>
 109     <span class="special">&lt;&lt;</span> <span class="string">", p = "</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">np001pc</span><span class="special">,</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 110 </pre>
 111 <pre class="programlisting">Normal distribution with mean = 1.09023 has fraction &lt;= -2, p = 0.001
 112 Normal distribution with mean = 1.09023 has fraction &gt; -2, p = 0.999
 113 </pre>
 114 <h5>
 115 <a name="math_toolkit.stat_tut.weg.find_eg.find_location_eg.h0"></a>
 116             <span class="phrase"><a name="math_toolkit.stat_tut.weg.find_eg.find_location_eg.controlling_error_handling_from_"></a></span><a class="link" href="find_location_eg.html#math_toolkit.stat_tut.weg.find_eg.find_location_eg.controlling_error_handling_from_">Controlling
 117             Error Handling from find_location</a>
 118           </h5>
 119 <p>
 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
 122             ('bad' arguments).
 123           </p>
 124 <p>
 125             Unless we are using the boost::math namespace, we will need:
 126           </p>
 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>
 130 </pre>
 131 <p>
 132             Using a typedef is often convenient, especially if it is re-used, although
 133             it is not required, as the various examples below show.
 134           </p>
 135 <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">domain_error</span><span class="special">&lt;</span><span class="identifier">ignore_error</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="identifier">ignore_domain_policy</span><span class="special">;</span>
 136 <span class="comment">// find_location with new policy, using typedef.</span>
 137 <span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">,</span> <span class="identifier">ignore_domain_policy</span><span class="special">());</span>
 138 <span class="comment">// Default policy policy&lt;&gt;, needs "using boost::math::policies::policy;"</span>
 139 <span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">,</span> <span class="identifier">policy</span><span class="special">&lt;&gt;());</span>
 140 <span class="comment">// Default policy, fully specified.</span>
 141 <span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">sd</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">&lt;&gt;());</span>
 142 <span class="comment">// A new policy, ignoring domain errors, without using a typedef.</span>
 143 <span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">sd</span><span class="special">,</span> <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">domain_error</span><span class="special">&lt;</span><span class="identifier">ignore_error</span><span class="special">&gt;</span> <span class="special">&gt;());</span>
 144 </pre>
 145 <p>
 146             If we want to use a probability that is the <a class="link" href="../../overview/complements.html" title="Complements are supported too - and when to use them">complements</a>
 147             of our probability, we should not even think of writing <code class="computeroutput"><span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</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">sd</span><span class="special">)</span></code>, but use the complement version, see
 148             <a class="link" href="../../overview/complements.html#why_complements">why complements?</a>.
 149           </p>
 150 <pre class="programlisting"><span class="identifier">z</span> <span class="special">=</span> <span class="number">2.</span><span class="special">;</span>
 151 <span class="keyword">double</span> <span class="identifier">q</span> <span class="special">=</span> <span class="number">0.95</span><span class="special">;</span> <span class="comment">// = 1 - p; // complement.</span>
 152 <span class="identifier">l</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</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">sd</span><span class="special">));</span>
 153
 154 <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(location) shifted</span>
 155 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span> <span class="special">&lt;&lt;</span> <span class="string">" has "</span>
 156   <span class="special">&lt;&lt;</span> <span class="string">"fraction &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span>  <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 157 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Normal distribution with mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">l</span> <span class="special">&lt;&lt;</span> <span class="string">" has "</span>
 158   <span class="special">&lt;&lt;</span> <span class="string">"fraction &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">z</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</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">np95pc</span><span class="special">,</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 159 </pre>
 160 <p>
 161             See <a href="../../../../../../example/find_location_example.cpp" target="_top">find_location_example.cpp</a>
 162             for full source code: the program output looks like this:
 163           </p>
 164 <pre class="programlisting">Example: Find location (mean).
 165 Normal distribution with mean = 0, standard deviation 1, has fraction &lt;= -2, p = 0.0227501
 166 Normal distribution with mean = 0, standard deviation 1, has fraction &gt; -2, p = 0.97725
 167 offset location (mean) = 1.09023
 168 Normal distribution with mean = 1.09023 has fraction &lt;= -2, p = 0.001
 169 Normal distribution with mean = 1.09023 has fraction &gt; -2, p = 0.999
 170 Normal distribution with mean = 0.355146 has fraction &lt;= 2 = 0.95
 171 Normal distribution with mean = 0.355146 has fraction &gt; 2 = 0.05
 172 </pre>
 173 </div>
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