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329 lines
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<title>Description of DiffuseLikelihoodH3</title>
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<div><a href="../index.html">Home</a> > <a href="index.html">.</a> > DiffuseLikelihoodH3.m</div>
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<!--<table width="100%"><tr><td align="left"><a href="../index.html"><img alt="<" border="0" src="../left.png"> Master index</a></td>
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<td align="right"><a href="index.html">Index for . <img alt=">" border="0" src="../right.png"></a></td></tr></table>-->
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<h1>DiffuseLikelihoodH3
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</h1>
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<h2><a name="_name"></a>PURPOSE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="box"><strong>M. Ratto added lik in output [October 2005]</strong></div>
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<h2><a name="_synopsis"></a>SYNOPSIS <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="box"><strong>function [LIK, lik] = DiffuseLikelihoodH3(T,R,Q,H,Pinf,Pstar,Y,trend,start) </strong></div>
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<h2><a name="_description"></a>DESCRIPTION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="fragment"><pre class="comment"> M. Ratto added lik in output [October 2005]
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changes by M. Ratto
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introduced new global variable id_ for termination of DKF
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introduced a persistent fmax, in order to keep track the max order of
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magnitude of the 'zero' values in Pinf at DKF termination
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new icc counter for Finf steps in DKF
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new termination for DKF
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likelihood terms for Fstar must be cumulated in DKF also when Pinf is non
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zero. this bug is fixed.
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stephane.adjemian@cepremap.cnrs.fr [07-19-2004]
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See "Filtering and Smoothing of State Vector for Diffuse State Space
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Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
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Analysis, vol. 24(1), pp. 85-98).
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Case where F_{\infty,t} is singular ==> Univariate treatment of multivariate
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time series.
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THE PROBLEM:
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y_t = Z_t * \alpha_t + \varepsilon_t
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\alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t
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with:
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\alpha_1 = a + A*\delta + R_0*\eta_0
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m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their
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columns constitue all the columns of the m*m identity matrix) so that
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A'*R_0 = 0 and A'*\alpha_1 = \delta
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We assume that the vector \delta is distributed as a N(0,\kappa*I_q)
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for a given \kappa > 0. So that the expectation of \alpha_1 is a and
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its variance is P, with
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P = \kappa*P_{\infty} + P_{\star}
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P_{\infty} = A*A'
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P_{\star} = R_0*Q_0*R_0'
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P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.
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and where:
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y_t is a pp*1 vector
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\alpha_t is a mm*1 vector
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\varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))
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\eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))
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a_1 is a mm*1 vector
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Z_t is a pp*mm matrix
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T_t is a mm*mm matrix
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H_t is a pp*pp matrix
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R_t is a mm*rr matrix
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Q_t is a rr*rr matrix
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P_1 is a mm*mm matrix
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FILTERING EQUATIONS:
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v_t = y_t - Z_t* a_t
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F_t = Z_t * P_t * Z_t' + H_t
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K_t = T_t * P_t * Z_t' * F_t^{-1}
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L_t = T_t - K_t * Z_t
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a_{t+1} = T_t * a_t + K_t * v_t
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P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'
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DIFFUSE FILTERING EQUATIONS:
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a_{t+1} = T_t*a_t + K_{\ast,t}v_t
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P_{\infty,t+1} = T_t*P_{\infty,t}*T_t'
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P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\ast,t}' + R_t*Q_t*R_t'
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K_{\ast,t} = T_t*P_{\ast,t}*Z_t'*F_{\ast,t}^{-1}
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v_t = y_t - Z_t*a_t
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L_{\ast,t} = T_t - K_{\ast,t}*Z_t
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F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t</pre></div>
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<!-- crossreference -->
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<h2><a name="_cross"></a>CROSS-REFERENCE INFORMATION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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This function calls:
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<ul style="list-style-image:url(../matlabicon.gif)">
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</ul>
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This function is called by:
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<ul style="list-style-image:url(../matlabicon.gif)">
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<li><a href="DsgeLikelihood.html" class="code" title="function [fval,cost_flag,ys,trend_coeff,info] = DsgeLikelihood(xparam1,gend,data)">DsgeLikelihood</a> stephane.adjemian@cepremap.cnrs.fr [09-07-2004]</li></ul>
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<!-- crossreference -->
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<h2><a name="_source"></a>SOURCE CODE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="fragment"><pre>0001 <a name="_sub0" href="#_subfunctions" class="code">function [LIK, lik] = DiffuseLikelihoodH3(T,R,Q,H,Pinf,Pstar,Y,trend,start)</a>
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0002 <span class="comment">% M. Ratto added lik in output [October 2005]</span>
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0003 <span class="comment">% changes by M. Ratto</span>
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0004 <span class="comment">% introduced new global variable id_ for termination of DKF</span>
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0005 <span class="comment">% introduced a persistent fmax, in order to keep track the max order of</span>
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0006 <span class="comment">% magnitude of the 'zero' values in Pinf at DKF termination</span>
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0007 <span class="comment">% new icc counter for Finf steps in DKF</span>
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0008 <span class="comment">% new termination for DKF</span>
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0009 <span class="comment">% likelihood terms for Fstar must be cumulated in DKF also when Pinf is non</span>
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0010 <span class="comment">% zero. this bug is fixed.</span>
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0011 <span class="comment">%</span>
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0012 <span class="comment">% stephane.adjemian@cepremap.cnrs.fr [07-19-2004]</span>
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0013 <span class="comment">%</span>
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0014 <span class="comment">% See "Filtering and Smoothing of State Vector for Diffuse State Space</span>
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0015 <span class="comment">% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series</span>
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0016 <span class="comment">% Analysis, vol. 24(1), pp. 85-98).</span>
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0017 <span class="comment">%</span>
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0018 <span class="comment">% Case where F_{\infty,t} is singular ==> Univariate treatment of multivariate</span>
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0019 <span class="comment">% time series.</span>
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0020 <span class="comment">%</span>
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0021 <span class="comment">% THE PROBLEM:</span>
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0022 <span class="comment">%</span>
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0023 <span class="comment">% y_t = Z_t * \alpha_t + \varepsilon_t</span>
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0024 <span class="comment">% \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t</span>
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0025 <span class="comment">%</span>
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0026 <span class="comment">% with:</span>
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0027 <span class="comment">%</span>
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0028 <span class="comment">% \alpha_1 = a + A*\delta + R_0*\eta_0</span>
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0029 <span class="comment">%</span>
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0030 <span class="comment">% m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their</span>
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0031 <span class="comment">% columns constitue all the columns of the m*m identity matrix) so that</span>
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0032 <span class="comment">%</span>
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0033 <span class="comment">% A'*R_0 = 0 and A'*\alpha_1 = \delta</span>
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0034 <span class="comment">%</span>
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0035 <span class="comment">% We assume that the vector \delta is distributed as a N(0,\kappa*I_q)</span>
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0036 <span class="comment">% for a given \kappa > 0. So that the expectation of \alpha_1 is a and</span>
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0037 <span class="comment">% its variance is P, with</span>
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0038 <span class="comment">%</span>
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0039 <span class="comment">% P = \kappa*P_{\infty} + P_{\star}</span>
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0040 <span class="comment">%</span>
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0041 <span class="comment">% P_{\infty} = A*A'</span>
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0042 <span class="comment">% P_{\star} = R_0*Q_0*R_0'</span>
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0043 <span class="comment">%</span>
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0044 <span class="comment">% P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.</span>
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0045 <span class="comment">%</span>
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0046 <span class="comment">%</span>
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0047 <span class="comment">% and where:</span>
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0048 <span class="comment">%</span>
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0049 <span class="comment">% y_t is a pp*1 vector</span>
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0050 <span class="comment">% \alpha_t is a mm*1 vector</span>
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0051 <span class="comment">% \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))</span>
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0052 <span class="comment">% \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))</span>
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0053 <span class="comment">% a_1 is a mm*1 vector</span>
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0054 <span class="comment">%</span>
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0055 <span class="comment">% Z_t is a pp*mm matrix</span>
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0056 <span class="comment">% T_t is a mm*mm matrix</span>
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0057 <span class="comment">% H_t is a pp*pp matrix</span>
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0058 <span class="comment">% R_t is a mm*rr matrix</span>
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0059 <span class="comment">% Q_t is a rr*rr matrix</span>
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0060 <span class="comment">% P_1 is a mm*mm matrix</span>
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0061 <span class="comment">%</span>
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0062 <span class="comment">%</span>
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0063 <span class="comment">% FILTERING EQUATIONS:</span>
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0064 <span class="comment">%</span>
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0065 <span class="comment">% v_t = y_t - Z_t* a_t</span>
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0066 <span class="comment">% F_t = Z_t * P_t * Z_t' + H_t</span>
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0067 <span class="comment">% K_t = T_t * P_t * Z_t' * F_t^{-1}</span>
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0068 <span class="comment">% L_t = T_t - K_t * Z_t</span>
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0069 <span class="comment">% a_{t+1} = T_t * a_t + K_t * v_t</span>
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0070 <span class="comment">% P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'</span>
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0071 <span class="comment">%</span>
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0072 <span class="comment">%</span>
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0073 <span class="comment">% DIFFUSE FILTERING EQUATIONS:</span>
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0074 <span class="comment">%</span>
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0075 <span class="comment">% a_{t+1} = T_t*a_t + K_{\ast,t}v_t</span>
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0076 <span class="comment">% P_{\infty,t+1} = T_t*P_{\infty,t}*T_t'</span>
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0077 <span class="comment">% P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\ast,t}' + R_t*Q_t*R_t'</span>
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0078 <span class="comment">% K_{\ast,t} = T_t*P_{\ast,t}*Z_t'*F_{\ast,t}^{-1}</span>
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0079 <span class="comment">% v_t = y_t - Z_t*a_t</span>
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0080 <span class="comment">% L_{\ast,t} = T_t - K_{\ast,t}*Z_t</span>
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0081 <span class="comment">% F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t</span>
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0082 <span class="keyword">global</span> bayestopt_ options_
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0083
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0084 mf = bayestopt_.mf;
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0085 pp = size(Y,1);
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0086 mm = size(T,1);
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0087 smpl = size(Y,2);
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0088 a = zeros(mm,1);
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0089 QQ = R*Q*transpose(R);
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0090 t = 0;
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0091 lik = zeros(smpl+1,1);
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0092 lik(smpl+1) = smpl*pp*log(2*pi); <span class="comment">%% the constant of minus two times the log-likelihood</span>
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0093 notsteady = 1;
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0094 crit = options_.kalman_tol;
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0095 crit1 = 1.e-6;
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0096 newRank = rank(Pinf,crit1);
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0097 icc = 0;
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0098 <span class="keyword">while</span> newRank & t < smpl <span class="comment">%% Matrix Finf is assumed to be zero</span>
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0099 t = t+1;
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0100 <span class="keyword">for</span> i=1:pp
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0101 v(i) = Y(i,t)-a(mf(i))-trend(i,t);
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0102 Fstar = Pstar(mf(i),mf(i))+H(i,i);
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0103 Finf = Pinf(mf(i),mf(i));
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0104 Kstar = Pstar(:,mf(i));
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0105 <span class="keyword">if</span> Finf > crit & newRank
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0106 icc = icc + 1;
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0107 Kinf = Pinf(:,mf(i));
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0108 a = a + Kinf*v(i)/Finf;
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0109 Pstar = Pstar + Kinf*transpose(Kinf)*Fstar/(Finf*Finf) - <span class="keyword">...</span>
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0110 (Kstar*transpose(Kinf)+Kinf*transpose(Kstar))/Finf;
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0111 Pinf = Pinf - Kinf*transpose(Kinf)/Finf;
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0112 lik(t) = lik(t) + log(Finf);
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0113 <span class="comment">% start new termination criterion for DKF</span>
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0114 <span class="keyword">if</span> ~isempty(options_.diffuse_d),
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0115 newRank = (icc<options_.diffuse_d);
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0116 <span class="comment">%if newRank & any(diag(Pinf(mf,mf))>crit)==0; % M. Ratto this line is BUGGY</span>
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0117 <span class="keyword">if</span> newRank & (any(diag(Pinf(mf,mf))>crit)==0 & rank(Pinf,crit1)==0);
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0118 options_.diffuse_d = icc;
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0119 newRank=0;
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0120 disp(<span class="string">'WARNING: Change in OPTIONS_.DIFFUSE_D in univariate DKF'</span>)
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0121 disp([<span class="string">'new OPTIONS_.DIFFUSE_D = '</span>,int2str(icc)])
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0122 disp(<span class="string">'You may have to reset the optimisation'</span>)
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0123 <span class="keyword">end</span>
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0124 <span class="keyword">else</span>
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0125 <span class="comment">%newRank = any(diag(Pinf(mf,mf))>crit); % M. Ratto this line is BUGGY</span>
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0126 newRank = (any(diag(Pinf(mf,mf))>crit) | rank(Pinf,crit1));
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0127 <span class="keyword">if</span> newRank==0,
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0128 P0= T*Pinf*transpose(T);
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0129 <span class="comment">%newRank = any(diag(P0(mf,mf))>crit); % M. Ratto this line is BUGGY</span>
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0130 newRank = (any(diag(P0(mf,mf))>crit) | rank(P0,crit1)); <span class="comment">% M. Ratto 10 Oct 2005</span>
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0131 <span class="keyword">if</span> newRank==0,
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0132 options_.diffuse_d = icc;
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0133 <span class="keyword">end</span>
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0134 <span class="keyword">end</span>
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0135 <span class="keyword">end</span>,
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0136 <span class="comment">% end new termination and checks for DKF and fmax</span>
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0137 <span class="keyword">elseif</span> Finf > crit
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0138 <span class="comment">%% Note that : (1) rank(Pinf)=0 implies that Finf = 0, (2) outside this loop (when for some i and t the condition</span>
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0139 <span class="comment">%% rank(Pinf)=0 is satisfied we have P = Pstar and F = Fstar and (3) Finf = 0 does not imply that</span>
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0140 <span class="comment">%% rank(Pinf)=0. [stphane,11-03-2004].</span>
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0141 <span class="comment">%if rank(Pinf) == 0</span>
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0142 <span class="comment">% the likelihood terms should alwasy be cumulated, not only</span>
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0143 <span class="comment">% when Pinf=0, otherwise the lik would depend on the ordering</span>
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0144 <span class="comment">% of observed variables</span>
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0145 lik(t) = lik(t) + log(Fstar) + v(i)*v(i)/Fstar;
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0146 <span class="comment">%end</span>
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0147 a = a + Kstar*v(i)/Fstar;
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0148 Pstar = Pstar - Kstar*transpose(Kstar)/Fstar;
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0149 <span class="keyword">else</span>
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0150 <span class="comment">% disp(['zero F term in DKF for observed ',int2str(i),' ',num2str(Fi)])</span>
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0151 <span class="keyword">end</span>
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0152 <span class="keyword">end</span>
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0153 <span class="keyword">if</span> newRank
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0154 oldRank = rank(Pinf,crit1);
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0155 <span class="keyword">else</span>
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0156 oldRank = 0;
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0157 <span class="keyword">end</span>
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0158 a = T*a;
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0159 Pstar = T*Pstar*transpose(T)+QQ;
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0160 Pinf = T*Pinf*transpose(T);
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0161 <span class="keyword">if</span> newRank
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0162 newRank = rank(Pinf,crit1);
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0163 <span class="keyword">end</span>
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0164 <span class="keyword">if</span> oldRank ~= newRank
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0165 disp(<span class="string">'DiffuseLiklihoodH3 :: T does influence the rank of Pinf!'</span>)
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0166 <span class="keyword">end</span>
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0167 <span class="keyword">end</span>
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0168 <span class="keyword">if</span> t == smpl
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0169 error([<span class="string">'There isn''t enough information to estimate the initial'</span> <span class="keyword">...</span><span class="comment"> </span>
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0170 <span class="string">' conditions of the nonstationary variables'</span>]);
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0171 <span class="keyword">end</span>
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0172 <span class="keyword">while</span> notsteady & t < smpl
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0173 t = t+1;
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0174 <span class="keyword">for</span> i=1:pp
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0175 v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
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0176 Fi = Pstar(mf(i),mf(i))+H(i,i);
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0177 <span class="keyword">if</span> Fi > crit
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0178 Ki = Pstar(:,mf(i));
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0179 a = a + Ki*v(i)/Fi;
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0180 Pstar = Pstar - Ki*transpose(Ki)/Fi;
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0181 lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
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0182 <span class="keyword">end</span>
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0183 <span class="keyword">end</span>
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0184 oldP = Pstar;
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0185 a = T*a;
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0186 Pstar = T*Pstar*transpose(T) + QQ;
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0187 notsteady = ~(max(max(abs(Pstar-oldP)))<crit);
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0188 <span class="keyword">end</span>
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0189 <span class="keyword">while</span> t < smpl
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0190 t = t+1;
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0191 <span class="keyword">for</span> i=1:pp
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0192 v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
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0193 Fi = Pstar(mf(i),mf(i))+H(i,i);
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0194 <span class="keyword">if</span> Fi > crit
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0195 Ki = Pstar(:,mf(i));
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0196 a = a + Ki*v(i)/Fi;
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0197 Pstar = Pstar - Ki*transpose(Ki)/Fi;
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0198 lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
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0199 <span class="keyword">end</span>
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0200 <span class="keyword">end</span>
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0201 a = T*a;
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0202 <span class="keyword">end</span>
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0203 LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);
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0204</pre></div>
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<hr><address>Generated on Fri 16-Jun-2006 09:09:06 by <strong><a href="http://www.artefact.tk/software/matlab/m2html/">m2html</a></strong> © 2003</address>
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