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<!-- doxytag: class="QuantLib::OneFactorCopula" --><!-- doxytag: inherits="QuantLib::LazyObject" -->
<p>Abstract base class for one-factor copula models.
<a href="class_quant_lib_1_1_one_factor_copula.html#details">More...</a></p>
<p><code>#include <ql/experimental/credit/onefactorcopula.hpp></code></p>
<div class="dynheader">
Inheritance diagram for OneFactorCopula:</div>
<div class="dyncontent">
<div class="center"><img src="class_quant_lib_1_1_one_factor_copula__inherit__graph.png" border="0" usemap="#_one_factor_copula_inherit__map" alt="Inheritance graph"/></div>
<map name="_one_factor_copula_inherit__map" id="_one_factor_copula_inherit__map">
<area shape="rect" id="node5" href="class_quant_lib_1_1_one_factor_gaussian_copula.html" title="One-factor Gaussian Copula." alt="" coords="335,5,513,35"/><area shape="rect" id="node7" href="class_quant_lib_1_1_one_factor_gaussian_student_copula.html" title="One-factor Gaussian-Student t-Copula." alt="" coords="313,58,535,89"/><area shape="rect" id="node9" href="class_quant_lib_1_1_one_factor_student_copula.html" title="One-factor Double Student t-Copula." alt="" coords="340,111,508,142"/><area shape="rect" id="node11" href="class_quant_lib_1_1_one_factor_student_gaussian_copula.html" title="One-factor Student t - Gaussian Copula." alt="" coords="313,165,535,195"/><area shape="rect" id="node2" href="class_quant_lib_1_1_lazy_object.html" title="Framework for calculation on demand and result caching." alt="" coords="5,85,93,115"/></map>
<center><span class="legend">[<a href="graph_legend.html">legend</a>]</span></center></div>
<p><a href="class_quant_lib_1_1_one_factor_copula-members.html">List of all members.</a></p>
<table class="memberdecls">
<tr><td colspan="2"><h2><a name="pub-methods"></a>
Public Member Functions</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="af4fe1d014da6e427fc150e9b79864621"></a><!-- doxytag: member="QuantLib::OneFactorCopula::OneFactorCopula" ref="af4fe1d014da6e427fc150e9b79864621" args="(const Handle< Quote > &correlation, Real maximum=5.0, Size integrationSteps=50, Real minimum=-5.0)" -->
 </td><td class="memItemRight" valign="bottom"><b>OneFactorCopula</b> (const <a class="el" href="class_quant_lib_1_1_handle.html">Handle</a>< <a class="el" href="class_quant_lib_1_1_quote.html">Quote</a> > &<a class="el" href="class_quant_lib_1_1_one_factor_copula.html#ac63b6b972e9461ed9c0de2899f6fdc45">correlation</a>, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> maximum=5.0, <a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> integrationSteps=50, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> minimum=-5.0)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a150d8a5373b942a7411cad42857ac0c2">density</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> m) const =0</td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Density function of M. <a href="#a150d8a5373b942a7411cad42857ac0c2"></a><br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top">virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a4d78de201751058a580a4104b569252f">cumulativeZ</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> z) const =0</td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Cumulative distribution of Z. <a href="#a4d78de201751058a580a4104b569252f"></a><br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top">virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_copula.html#aa5a31f6c22dc65de5eefc911e33d230e">cumulativeY</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> y) const </td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Cumulative distribution of Y. <a href="#aa5a31f6c22dc65de5eefc911e33d230e"></a><br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top">virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a37961e30f2d8786a4e4e6d1ef65516ed">inverseCumulativeY</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> p) const </td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Inverse cumulative distribution of Y. <a href="#a37961e30f2d8786a4e4e6d1ef65516ed"></a><br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="ac63b6b972e9461ed9c0de2899f6fdc45"></a><!-- doxytag: member="QuantLib::OneFactorCopula::correlation" ref="ac63b6b972e9461ed9c0de2899f6fdc45" args="() const " -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_copula.html#ac63b6b972e9461ed9c0de2899f6fdc45">correlation</a> () const </td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Single correlation parameter. <br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a2f12fdc46df584c77c692449afb0f70f">conditionalProbability</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> prob, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> m) const </td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Conditional probability. <a href="#a2f12fdc46df584c77c692449afb0f70f"></a><br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top">std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a43021a316d68b67fe3baacd8b41060ec">conditionalProbability</a> (const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > &prob, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> m) const </td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">Vector of conditional probabilities. <a href="#a43021a316d68b67fe3baacd8b41060ec"></a><br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a3466ad62259b66bd721d275d6f37b3b5">integral</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> p) const </td></tr>
<tr><td class="memTemplParams" colspan="2">template<class F > </td></tr>
<tr><td class="memTemplItemLeft" align="right" valign="top"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_copula.html#ab0919dadc85dad55932aae6f76a373a5">integral</a> (const F &f, std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > &probabilities) const </td></tr>
<tr><td class="memTemplParams" colspan="2">template<class F > </td></tr>
<tr><td class="memTemplItemLeft" align="right" valign="top">Distribution </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_copula.html#aa05d4372c6545648d67b4feac3ea358c">integral</a> (const F &f, const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > &nominals, const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > &probabilities) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a3a370ec01879c1aafe26f2b74c305205">checkMoments</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> tolerance) const </td></tr>
<tr><td colspan="2"><h2><a name="pro-methods"></a>
Protected Member Functions</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="ae973b574644545ebcb1aa2a5a3aa6e5b"></a><!-- doxytag: member="QuantLib::OneFactorCopula::steps" ref="ae973b574644545ebcb1aa2a5a3aa6e5b" args="() const " -->
<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> </td><td class="memItemRight" valign="bottom"><b>steps</b> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="acccb56f699284835bfef4b5b46fa7b5d"></a><!-- doxytag: member="QuantLib::OneFactorCopula::dm" ref="acccb56f699284835bfef4b5b46fa7b5d" args="(Size i) const " -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>dm</b> (<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> i) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a68322c1f689d52daee478f92cfb267c1"></a><!-- doxytag: member="QuantLib::OneFactorCopula::m" ref="a68322c1f689d52daee478f92cfb267c1" args="(Size i) const " -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>m</b> (<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> i) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a40441224027fda24cf5336b87a2803f5"></a><!-- doxytag: member="QuantLib::OneFactorCopula::densitydm" ref="a40441224027fda24cf5336b87a2803f5" args="(Size i) const " -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>densitydm</b> (<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> i) const </td></tr>
<tr><td colspan="2"><h2><a name="pro-attribs"></a>
Protected Attributes</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a0bbb8459d54c70260f5dc06646c622fe"></a><!-- doxytag: member="QuantLib::OneFactorCopula::correlation_" ref="a0bbb8459d54c70260f5dc06646c622fe" args="" -->
<a class="el" href="class_quant_lib_1_1_handle.html">Handle</a>< <a class="el" href="class_quant_lib_1_1_quote.html">Quote</a> > </td><td class="memItemRight" valign="bottom"><b>correlation_</b></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a5b014cf28588b976cafd9ef99bd90bdd"></a><!-- doxytag: member="QuantLib::OneFactorCopula::max_" ref="a5b014cf28588b976cafd9ef99bd90bdd" args="" -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>max_</b></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="ad193bceb88346d2755789a09afa91d86"></a><!-- doxytag: member="QuantLib::OneFactorCopula::steps_" ref="ad193bceb88346d2755789a09afa91d86" args="" -->
<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> </td><td class="memItemRight" valign="bottom"><b>steps_</b></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a0109cc1a8c0dd24c6bac204843ca21ce"></a><!-- doxytag: member="QuantLib::OneFactorCopula::min_" ref="a0109cc1a8c0dd24c6bac204843ca21ce" args="" -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>min_</b></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a71a6c36122b9ef005c4c52bda3890e24"></a><!-- doxytag: member="QuantLib::OneFactorCopula::y_" ref="a71a6c36122b9ef005c4c52bda3890e24" args="" -->
std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > </td><td class="memItemRight" valign="bottom"><b>y_</b></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a6704dd8a7ed41ffcfd146c09c417b028"></a><!-- doxytag: member="QuantLib::OneFactorCopula::cumulativeY_" ref="a6704dd8a7ed41ffcfd146c09c417b028" args="" -->
std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > </td><td class="memItemRight" valign="bottom"><b>cumulativeY_</b></td></tr>
</table>
<hr/><a name="details" id="details"></a><h2>Detailed Description</h2>
<div class="textblock"><p>Abstract base class for one-factor copula models. </p>
<p>Reference: John Hull and Alan White, The Perfect Copula, June 2006</p>
<p>Let <img class="formulaInl" alt="$Q_i(t)$" src="form_87.png"/> be the cumulative probability of default of counterparty i before time t.</p>
<p>In a one-factor model, consider random variables </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ Y_i = a_i\,M+\sqrt{1-a_i^2}\:Z_i \]" src="form_88.png"/>
</p>
<p> where <img class="formulaInl" alt="$M$" src="form_89.png"/> and <img class="formulaInl" alt="$Z_i$" src="form_90.png"/> have independent zero-mean unit-variance distributions and <img class="formulaInl" alt="$-1\leq a_i \leq 1$" src="form_91.png"/>. The correlation between <img class="formulaInl" alt="$Y_i$" src="form_92.png"/> and <img class="formulaInl" alt="$Y_j$" src="form_93.png"/> is then <img class="formulaInl" alt="$a_i a_j$" src="form_94.png"/>.</p>
<p>Let <img class="formulaInl" alt="$F_Y(y)$" src="form_95.png"/> be the cumulative distribution function of <img class="formulaInl" alt="$Y_i$" src="form_92.png"/>. <img class="formulaInl" alt="$y$" src="form_96.png"/> is mapped to <img class="formulaInl" alt="$t$" src="form_97.png"/> such that percentiles match, i.e. <img class="formulaInl" alt="$F_Y(y)=Q_i(t)$" src="form_98.png"/> or <img class="formulaInl" alt="$y=F_Y^{-1}(Q_i(t))$" src="form_99.png"/>.</p>
<p>Now let <img class="formulaInl" alt="$F_Z(z)$" src="form_100.png"/> be the cumulated distribution function of <img class="formulaInl" alt="$Z_i$" src="form_90.png"/>. For given realization of <img class="formulaInl" alt="$M$" src="form_89.png"/>, this determines the distribution of <img class="formulaInl" alt="$y$" src="form_96.png"/>: </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ Prob \,(Y_i < y|M) = F_Z \left( \frac{y-a_i\,M}{\sqrt{1-a_i^2}}\right) \qquad \mbox{or} \qquad Prob \,(t_i < t|M) = F_Z \left( \frac{F_Y^{-1}(Q_i(t))-a_i\,M} {\sqrt{1-a_i^2}} \right) \]" src="form_101.png"/>
</p>
<p>The distribution functions of <img class="formulaInl" alt="$ M, Z_i $" src="form_102.png"/> are specified in derived classes. The distribution function of <img class="formulaInl" alt="$ Y $" src="form_103.png"/> is then given by the convolution </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ F_Y(y) = Prob\,(Y<y) = \int_{-\infty}^\infty\,\int_{-\infty}^{\infty}\: D_Z(z)\,D_M(m) \quad \Theta \left(y - a\,m - \sqrt{1-a^2}\,z\right)\,dm\,dz, \qquad \Theta (x) = \left\{ \begin{array}{ll} 1 & x \geq 0 \\ 0 & x < 0 \end{array}\right. \]" src="form_104.png"/>
</p>
<p> where <img class="formulaInl" alt="$ D_Z(z) $" src="form_105.png"/> and <img class="formulaInl" alt="$ D_M(m) $" src="form_106.png"/> are the probability densities of <img class="formulaInl" alt="$ Z$" src="form_107.png"/> and <img class="formulaInl" alt="$ M, $" src="form_108.png"/> respectively.</p>
<p>This convolution can also be written </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ F(y) = Prob \,(Y < y) = \int_{-\infty}^\infty D_M(m)\,dm\: \int_{-\infty}^{g(y,a,m)} D_Z(z)\,dz, \qquad g(y,a,m) = \frac{y - a\cdot m}{\sqrt{1-a^2}}, \qquad a < 1 \]" src="form_109.png"/>
</p>
<p>or</p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ F(y) = Prob \,(Y < y) = \int_{-\infty}^\infty D_Z(z)\,dz\: \int_{-\infty}^{h(y,a,z)} D_M(m)\,dm, \qquad h(y,a,z) = \frac{y - \sqrt{1 - a^2}\cdot z}{a}, \qquad a > 0. \]" src="form_110.png"/>
</p>
<p>In general, <img class="formulaInl" alt="$ F_Y(y) $" src="form_111.png"/> needs to be computed numerically.</p>
<dl class="todo"><dt><b><a class="el" href="todo.html#_todo000007">Possible enhancements:</a></b></dt><dd>Improve on simple Euler integration </dd></dl>
</div><hr/><h2>Member Function Documentation</h2>
<a class="anchor" id="a150d8a5373b942a7411cad42857ac0c2"></a><!-- doxytag: member="QuantLib::OneFactorCopula::density" ref="a150d8a5373b942a7411cad42857ac0c2" args="(Real m) const =0" -->
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<td class="memname">virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a150d8a5373b942a7411cad42857ac0c2">density</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>m</em></td><td>)</td>
<td> const<code> [pure virtual]</code></td>
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<p>Density function of M. </p>
<p>Derived classes must override this method and ensure zero mean and unit variance. </p>
<p>Implemented in <a class="el" href="class_quant_lib_1_1_one_factor_student_gaussian_copula.html#a87748e5305096b30afeeaa93ce59137c">OneFactorStudentGaussianCopula</a>, <a class="el" href="class_quant_lib_1_1_one_factor_gaussian_student_copula.html#a87748e5305096b30afeeaa93ce59137c">OneFactorGaussianStudentCopula</a>, <a class="el" href="class_quant_lib_1_1_one_factor_student_copula.html#a87748e5305096b30afeeaa93ce59137c">OneFactorStudentCopula</a>, and <a class="el" href="class_quant_lib_1_1_one_factor_gaussian_copula.html#a87748e5305096b30afeeaa93ce59137c">OneFactorGaussianCopula</a>.</p>
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<a class="anchor" id="a4d78de201751058a580a4104b569252f"></a><!-- doxytag: member="QuantLib::OneFactorCopula::cumulativeZ" ref="a4d78de201751058a580a4104b569252f" args="(Real z) const =0" -->
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<td class="memname">virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a4d78de201751058a580a4104b569252f">cumulativeZ</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>z</em></td><td>)</td>
<td> const<code> [pure virtual]</code></td>
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<p>Cumulative distribution of Z. </p>
<p>Derived classes must override this method and ensure zero mean and unit variance. </p>
<p>Implemented in <a class="el" href="class_quant_lib_1_1_one_factor_student_gaussian_copula.html#a7f12eff80e70545a68af09aaab9a5bfa">OneFactorStudentGaussianCopula</a>, <a class="el" href="class_quant_lib_1_1_one_factor_gaussian_student_copula.html#a7f12eff80e70545a68af09aaab9a5bfa">OneFactorGaussianStudentCopula</a>, <a class="el" href="class_quant_lib_1_1_one_factor_student_copula.html#a7f12eff80e70545a68af09aaab9a5bfa">OneFactorStudentCopula</a>, and <a class="el" href="class_quant_lib_1_1_one_factor_gaussian_copula.html#a7f12eff80e70545a68af09aaab9a5bfa">OneFactorGaussianCopula</a>.</p>
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<a class="anchor" id="aa5a31f6c22dc65de5eefc911e33d230e"></a><!-- doxytag: member="QuantLib::OneFactorCopula::cumulativeY" ref="aa5a31f6c22dc65de5eefc911e33d230e" args="(Real y) const " -->
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<td class="memname">virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_one_factor_copula.html#aa5a31f6c22dc65de5eefc911e33d230e">cumulativeY</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>y</em></td><td>)</td>
<td> const<code> [virtual]</code></td>
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<p>Cumulative distribution of Y. </p>
<p>This is the default implementation based on tabulated data. The table needs to be filled by derived classes. If analytic calculation is feasible, this method can also be overridden. </p>
<p>Reimplemented in <a class="el" href="class_quant_lib_1_1_one_factor_gaussian_copula.html#a712150bf4c40b6443a311f2569117760">OneFactorGaussianCopula</a>.</p>
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<a class="anchor" id="a37961e30f2d8786a4e4e6d1ef65516ed"></a><!-- doxytag: member="QuantLib::OneFactorCopula::inverseCumulativeY" ref="a37961e30f2d8786a4e4e6d1ef65516ed" args="(Real p) const " -->
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<td class="memname">virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a37961e30f2d8786a4e4e6d1ef65516ed">inverseCumulativeY</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>p</em></td><td>)</td>
<td> const<code> [virtual]</code></td>
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<p>Inverse cumulative distribution of Y. </p>
<p>This is the default implementation based on tabulated data. The table needs to be filled by derived classes. If analytic calculation is feasible, this method can also be overridden. </p>
<p>Reimplemented in <a class="el" href="class_quant_lib_1_1_one_factor_gaussian_copula.html#a309f9b6f2c8476ddc7d4d750a0f7d75c">OneFactorGaussianCopula</a>.</p>
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<a class="anchor" id="a2f12fdc46df584c77c692449afb0f70f"></a><!-- doxytag: member="QuantLib::OneFactorCopula::conditionalProbability" ref="a2f12fdc46df584c77c692449afb0f70f" args="(Real prob, Real m) const " -->
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<td class="memname"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a2f12fdc46df584c77c692449afb0f70f">conditionalProbability</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>prob</em>, </td>
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<td></td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>m</em> </td>
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<td></td>
<td>)</td>
<td></td><td> const</td>
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<p>Conditional probability. </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \hat p(m) = F_Z \left( \frac{F_Y^{-1}(p)-a\,m}{\sqrt{1-a^2}}\right) \]" src="form_112.png"/>
</p>
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<a class="anchor" id="a43021a316d68b67fe3baacd8b41060ec"></a><!-- doxytag: member="QuantLib::OneFactorCopula::conditionalProbability" ref="a43021a316d68b67fe3baacd8b41060ec" args="(const std::vector< Real > &prob, Real m) const " -->
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<td class="memname">std::vector<<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a>> <a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a2f12fdc46df584c77c692449afb0f70f">conditionalProbability</a> </td>
<td>(</td>
<td class="paramtype">const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > & </td>
<td class="paramname"><em>prob</em>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>m</em> </td>
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<td></td>
<td>)</td>
<td></td><td> const</td>
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<p>Vector of conditional probabilities. </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \hat p_i(m) = F_Z \left( \frac{F_Y^{-1}(p_i)-a\,m}{\sqrt{1-a^2}} \right) \]" src="form_113.png"/>
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<a class="anchor" id="a3466ad62259b66bd721d275d6f37b3b5"></a><!-- doxytag: member="QuantLib::OneFactorCopula::integral" ref="a3466ad62259b66bd721d275d6f37b3b5" args="(Real p) const " -->
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<td class="memname"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a3466ad62259b66bd721d275d6f37b3b5">integral</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>p</em></td><td>)</td>
<td> const</td>
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<p>Integral over the density <img class="formulaInl" alt="$ \rho(m) $" src="form_114.png"/> of M and the conditional probability related to p:</p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \int_{-\infty}^\infty\,dm\,\rho(m)\, F_Z \left( \frac{F_Y^{-1}(p)-a\,m}{\sqrt{1-a^2}}\right) \]" src="form_115.png"/>
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<a class="anchor" id="ab0919dadc85dad55932aae6f76a373a5"></a><!-- doxytag: member="QuantLib::OneFactorCopula::integral" ref="ab0919dadc85dad55932aae6f76a373a5" args="(const F &f, std::vector< Real > &probabilities) const " -->
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<td class="memname"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> <a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a3466ad62259b66bd721d275d6f37b3b5">integral</a> </td>
<td>(</td>
<td class="paramtype">const F & </td>
<td class="paramname"><em>f</em>, </td>
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<td></td>
<td class="paramtype">std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > & </td>
<td class="paramname"><em>probabilities</em> </td>
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<td>)</td>
<td></td><td> const</td>
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<p>Integral over the density <img class="formulaInl" alt="$ \rho(m) $" src="form_114.png"/> of M and a one-dimensional function <img class="formulaInl" alt="$ f $" src="form_37.png"/> of conditional probabilities related to the input vector of probabilities p:</p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \int_{-\infty}^\infty\,dm\,\rho(m)\, f (\hat p_1, \hat p_2, \dots, \hat p_N), \qquad \hat p_i (m) = F_Z \left( \frac{F_Y^{-1}(p_i)-a\,m}{\sqrt{1-a^2}} \right) \]" src="form_116.png"/>
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<a class="anchor" id="aa05d4372c6545648d67b4feac3ea358c"></a><!-- doxytag: member="QuantLib::OneFactorCopula::integral" ref="aa05d4372c6545648d67b4feac3ea358c" args="(const F &f, const std::vector< Real > &nominals, const std::vector< Real > &probabilities) const " -->
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<td class="memname">Distribution <a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a3466ad62259b66bd721d275d6f37b3b5">integral</a> </td>
<td>(</td>
<td class="paramtype">const F & </td>
<td class="paramname"><em>f</em>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > & </td>
<td class="paramname"><em>nominals</em>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > & </td>
<td class="paramname"><em>probabilities</em> </td>
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<td></td>
<td>)</td>
<td></td><td> const</td>
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<p>Integral over the density <img class="formulaInl" alt="$ \rho(m) $" src="form_114.png"/> of M and a multi-dimensional function <img class="formulaInl" alt="$ f $" src="form_37.png"/> of conditional probabilities related to the input vector of probabilities p:</p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \int_{-\infty}^\infty\,dm\,\rho(m)\, f (\hat p_1, \hat p_2, \dots, \hat p_N), \qquad \hat p_i = F_Z \left( \frac{F_Y^{-1}(p_i)-a\,m}{\sqrt{1-a^2}}\right) \]" src="form_117.png"/>
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<a class="anchor" id="a3a370ec01879c1aafe26f2b74c305205"></a><!-- doxytag: member="QuantLib::OneFactorCopula::checkMoments" ref="a3a370ec01879c1aafe26f2b74c305205" args="(Real tolerance) const " -->
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<td class="memname">int <a class="el" href="class_quant_lib_1_1_one_factor_copula.html#a3a370ec01879c1aafe26f2b74c305205">checkMoments</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>tolerance</em></td><td>)</td>
<td> const</td>
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<p>Check moments (unit norm, zero mean and unit variance) of the distributions of M, Z, and Y by numerically integrating the respective density. <a class="el" href="class_quant_lib_1_1_parameter.html" title="Base class for model arguments.">Parameter</a> tolerance is the maximum tolerable absolute error. </p>
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