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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.dist_ref.nmp"></a><a class="link" href="nmp.html" title="Non-Member Properties">Non-Member Properties</a> |
| </h3></div></div></div> |
| <p> |
| Properties that are common to all distributions are accessed via non-member |
| getter functions: non-membership allows more of these functions to be added |
| over time, as the need arises. Unfortunately the literature uses many different |
| and confusing names to refer to a rather small number of actual concepts; |
| refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.concept_index">concept |
| index</a> to find the property you want by the name you are most familiar |
| with. Or use the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.function_index">function |
| index</a> to go straight to the function you want if you already know |
| its name. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h0"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.function_index"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.function_index">Function |
| Index</a> |
| </h5> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution |
| Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of the Cumulative |
| Distribution Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a> |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">range</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile from the |
| complement of the probability</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>. |
| </li> |
| </ul></div> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h1"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.concept_index"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.concept_index">Conceptual |
| Index</a> |
| </h5> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of the Cumulative |
| Distribution Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution |
| Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf_inv">Inverse Cumulative |
| Distribution Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival_inv">Inverse Survival |
| Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a> |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.lower_critical">Lower Critical |
| Value</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a> |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">P</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.percent">Percent Point Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pmf">Probability Mass Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">range</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">Q</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile from the |
| complement of the probability</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a> |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival">Survival Function</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.upper_critical">Upper Critical |
| Value</a>. |
| </li> |
| <li class="listitem"> |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>. |
| </li> |
| </ul></div> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h2"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative |
| Distribution Function</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">cdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">x</span><span class="special">);</span> |
| </pre> |
| <p> |
| The <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution |
| Function</a> is the probability that the variable takes a value less than |
| or equal to x. It is equivalent to the integral from -infinity to x of the |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the random variable is outside the defined range for the distribution. |
| </p> |
| <p> |
| For example, the following graph shows the cdf for the normal distribution: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/cdf.png"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h3"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.ccdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement |
| of the Cumulative Distribution Function</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Distribution</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">cdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Unspecified-Complement-Type</em></span><span class="special"><</span><span class="identifier">Distribution</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">>&</span> <span class="identifier">comp</span><span class="special">);</span> |
| </pre> |
| <p> |
| The complement of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative |
| Distribution Function</a> is the probability that the variable takes a |
| value greater than x. It is equivalent to the integral from x to infinity |
| of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density |
| Function</a>, or 1 minus the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative |
| Distribution Function</a> of x. |
| </p> |
| <p> |
| This is also known as the survival function. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the random variable is outside the defined range for the distribution. |
| </p> |
| <p> |
| In this library, it is obtained by wrapping the arguments to the <code class="computeroutput"><span class="identifier">cdf</span></code> function in a call to <code class="computeroutput"><span class="identifier">complement</span></code>, for example: |
| </p> |
| <pre class="programlisting"><span class="comment">// standard normal distribution object:</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="identifier">norm</span><span class="special">;</span> |
| <span class="comment">// print survival function for x=2.0:</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</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">norm</span><span class="special">,</span> <span class="number">2.0</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| For example, the following graph shows the __complement of the cdf for the |
| normal distribution: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/survival.png"></span> |
| </p> |
| <p> |
| See <a class="link" href="../stat_tut/overview/complements.html#why_complements">why complements?</a> for why the complement |
| is useful and when it should be used. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h4"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.hazard"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard |
| Function</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">hazard</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">x</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a> |
| of <span class="emphasis"><em>x</em></span> and distibution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the random variable is outside the defined range for the distribution. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/hazard.svg"></span> |
| </p> |
| <div class="caution"><table border="0" summary="Caution"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td> |
| <th align="left">Caution</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| Some authors refer to this as the conditional failure density function |
| rather than the hazard function. |
| </p></td></tr> |
| </table></div> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h5"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.chf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative |
| Hazard Function</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">chf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">x</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard |
| Function</a> of <span class="emphasis"><em>x</em></span> and distibution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the random variable is outside the defined range for the distribution. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/chf.svg"></span> |
| </p> |
| <div class="caution"><table border="0" summary="Caution"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td> |
| <th align="left">Caution</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| Some authors refer to this as simply the "Hazard Function". |
| </p></td></tr> |
| </table></div> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h6"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.mean"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the mean of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the distribution does not have a defined mean (for example the Cauchy |
| distribution). |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h7"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.median"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">median</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the median of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h8"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.mode"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">mode</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the mode of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the distribution does not have a defined mode. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h9"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.pdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability |
| Density Function</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">pdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">x</span><span class="special">);</span> |
| </pre> |
| <p> |
| For a continuous function, the probability density function (pdf) returns |
| the probability that the variate has the value x. Since for continuous distributions |
| the probability at a single point is actually zero, the probability is better |
| expressed as the integral of the pdf between two points: see the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative |
| Distribution Function</a>. |
| </p> |
| <p> |
| For a discrete distribution, the pdf is the probability that the variate |
| takes the value x. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the random variable is outside the defined range for the distribution. |
| </p> |
| <p> |
| For example, for a standard normal distribution the pdf looks like this: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/pdf.png"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h10"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.range"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">Range</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">></span> <span class="identifier">range</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the valid range of the random variable over distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h11"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.quantile"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">quantile</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">p</span><span class="special">);</span> |
| </pre> |
| <p> |
| The quantile is best viewed as the inverse of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative |
| Distribution Function</a>, it returns a value <span class="emphasis"><em>x</em></span> such |
| that <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> |
| <span class="identifier">p</span></code>. |
| </p> |
| <p> |
| This is also known as the <span class="emphasis"><em>percent point function</em></span>, or |
| <span class="emphasis"><em>percentile</em></span>, or <span class="emphasis"><em>fractile</em></span>, it is |
| also the same as calculating the <span class="emphasis"><em>lower critical value</em></span> |
| of a distribution. |
| </p> |
| <p> |
| This function returns a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the probability lies outside [0,1]. The function may return an <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a> |
| if there is no finite value that has the specified probability. |
| </p> |
| <p> |
| The following graph shows the quantile function for a standard normal distribution: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/quantile.png"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h12"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.quantile_c"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile |
| from the complement of the probability.</a> |
| </h5> |
| <p> |
| See also <a class="link" href="../stat_tut/overview/complements.html" title="Complements are supported too - and when to use them">complements</a>. |
| </p> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Distribution</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">quantile</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Unspecified-Complement-Type</em></span><span class="special"><</span><span class="identifier">Distribution</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">>&</span> <span class="identifier">comp</span><span class="special">);</span> |
| </pre> |
| <p> |
| This is the inverse of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement |
| of the Cumulative Distribution Function</a>. It is calculated by wrapping |
| the arguments in a call to the quantile function in a call to <span class="emphasis"><em>complement</em></span>. |
| For example: |
| </p> |
| <pre class="programlisting"><span class="comment">// define a standard normal distribution:</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="identifier">norm</span><span class="special">;</span> |
| <span class="comment">// print the value of x for which the complement</span> |
| <span class="comment">// of the probability is 0.05:</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">norm</span><span class="special">,</span> <span class="number">0.05</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| The function computes a value <span class="emphasis"><em>x</em></span> such that <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> |
| <span class="identifier">x</span><span class="special">))</span> <span class="special">==</span> <span class="identifier">q</span></code> where |
| <span class="emphasis"><em>q</em></span> is complement of the probability. |
| </p> |
| <p> |
| <a class="link" href="../stat_tut/overview/complements.html#why_complements">Why complements?</a> |
| </p> |
| <p> |
| This function is also called the inverse survival function, and is the same |
| as calculating the <span class="emphasis"><em>upper critical value</em></span> of a distribution. |
| </p> |
| <p> |
| This function returns a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the probablity lies outside [0,1]. The function may return an <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a> |
| if there is no finite value that has the specified probability. |
| </p> |
| <p> |
| The following graph show the inverse survival function for the normal distribution: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/survival_inv.png"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h13"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.sd"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">Standard |
| Deviation</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the standard deviation of distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the distribution does not have a defined standard deviation. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h14"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.support"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">></span> <span class="identifier">support</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the supported range of random variable over the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| The distribution is said to be 'supported' over a range that is <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">"the |
| smallest closed set whose complement has probability zero"</a>. |
| Non-mathematicians might say it means the 'interesting' smallest range of |
| random variate x that has the cdf going from zero to unity. Outside are uninteresting |
| zones where the pdf is zero, and the cdf zero or unity. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h15"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.variance"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">Variance</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the variance of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the distribution does not have a defined variance. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h16"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.skewness"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">Skewness</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">skewness</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the skewness of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the distribution does not have a defined skewness. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h17"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.kurtosis"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">Kurtosis</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">kurtosis</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the 'proper' kurtosis (normalized fourth moment) of the distribution |
| <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| kertosis = β<sub>2</sub>  = μ<sub>4</sub>   / μ<sub>2</sub><sup>2</sup> |
| </p> |
| <p> |
| Where μ<sub>i</sub>   is the i'th central moment of the distribution, and in particular |
| μ<sub>2</sub>   is the variance of the distribution. |
| </p> |
| <p> |
| The kurtosis is a measure of the "peakedness" of a distribution. |
| </p> |
| <p> |
| Note that the literature definition of kurtosis is confusing. The definition |
| used here is that used by for example <a href="http://mathworld.wolfram.com/Kurtosis.html" target="_top">Wolfram |
| MathWorld</a> (that includes a table of formulae for kurtosis excess |
| for various distributions) but NOT the definition of <a href="http://en.wikipedia.org/wiki/Kurtosis" target="_top">kurtosis |
| used by Wikipedia</a> which treats "kurtosis" and "kurtosis |
| excess" as the same quantity. |
| </p> |
| <pre class="programlisting"><span class="identifier">kurtosis_excess</span> <span class="special">=</span> <span class="char">'proper'</span> <span class="identifier">kurtosis</span> <span class="special">-</span> <span class="number">3</span> |
| </pre> |
| <p> |
| This subtraction of 3 is convenient so that the <span class="emphasis"><em>kurtosis excess</em></span> |
| of a normal distribution is zero. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the distribution does not have a defined kurtosis. |
| </p> |
| <p> |
| 'Proper' kurtosis can have a value from zero to + infinity. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h18"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.kurtosis_excess"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">Kurtosis |
| excess</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="identifier">RealType</span> <span class="identifier">kurtosis_excess</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the kurtosis excess of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| kurtosis excess = γ<sub>2</sub>  = μ<sub>4</sub>   / μ<sub>2</sub><sup>2</sup>  - 3 = kurtosis - 3 |
| </p> |
| <p> |
| Where μ<sub>i</sub>   is the i'th central moment of the distribution, and in particular |
| μ<sub>2</sub>   is the variance of the distribution. |
| </p> |
| <p> |
| The kurtosis excess is a measure of the "peakedness" of a distribution, |
| and is more widely used than the "kurtosis proper". It is defined |
| so that the kurtosis excess of a normal distribution is zero. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if the distribution does not have a defined kurtosis excess. |
| </p> |
| <p> |
| Kurtosis excess can have a value from -2 to + infinity. |
| </p> |
| <pre class="programlisting"><span class="identifier">kurtosis</span> <span class="special">=</span> <span class="identifier">kurtosis_excess</span> <span class="special">+</span><span class="number">3</span><span class="special">;</span> |
| </pre> |
| <p> |
| The kurtosis excess of a normal distribution is zero. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h19"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdfPQ"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">P |
| and Q</a> |
| </h5> |
| <p> |
| The terms P and Q are sometimes used to refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative |
| Distribution Function</a> and its <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">complement</a> |
| respectively. Lowercase p and q are sometimes used to refer to the values |
| returned by these functions. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h20"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.percent"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.percent">Percent |
| Point Function or Percentile</a> |
| </h5> |
| <p> |
| The percent point function, also known as the percentile, is the same as |
| the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h21"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdf_inv"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf_inv">Inverse |
| CDF Function.</a> |
| </h5> |
| <p> |
| The inverse of the cumulative distribution function, is the same as the |
| <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h22"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.survival_inv"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival_inv">Inverse |
| Survival Function.</a> |
| </h5> |
| <p> |
| The inverse of the survival function, is the same as computing the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">quantile from the complement |
| of the probability</a>. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h23"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.pmf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pmf">Probability |
| Mass Function</a> |
| </h5> |
| <p> |
| The Probability Mass Function is the same as the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability |
| Density Function</a>. |
| </p> |
| <p> |
| The term Mass Function is usually applied to discrete distributions, while |
| the term <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density |
| Function</a> applies to continuous distributions. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h24"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.lower_critical"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.lower_critical">Lower |
| Critical Value.</a> |
| </h5> |
| <p> |
| The lower critical value calculates the value of the random variable given |
| the area under the left tail of the distribution. It is equivalent to calculating |
| the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h25"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.upper_critical"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.upper_critical">Upper |
| Critical Value.</a> |
| </h5> |
| <p> |
| The upper critical value calculates the value of the random variable given |
| the area under the right tail of the distribution. It is equivalent to calculating |
| the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">quantile from the |
| complement of the probability</a>. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.nmp.h26"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.nmp.survival"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival">Survival |
| Function</a> |
| </h5> |
| <p> |
| Refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of |
| the Cumulative Distribution Function</a>. |
| </p> |
| </div> |
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| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
| </p> |
| </div></td> |
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