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| 23 | <h1><a class="anchor" name="arx">Example of ARX model estimation </a></h1>Here, we use the <code>ARX</code> class to estimate parameters and structure. ARX model is defined as follows: <p class="formulaDsp"> |
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| 24 | <img class="formulaDsp" alt="\[ y_t = \theta' \psi_t + \rho e_t \]" src="form_101.png"> |
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| 25 | <p> |
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| 26 | where <img class="formulaInl" alt="$y_t$" src="form_94.png"> is the system output, <img class="formulaInl" alt="$[\theta,\rho]$" src="form_95.png"> is vector of unknown parameters, <img class="formulaInl" alt="$\psi_t$" src="form_102.png"> is an vector of data-dependent regressors, and noise <img class="formulaInl" alt="$e_t$" src="form_54.png"> is assumed to be Normal distributed <img class="formulaInl" alt="$\mathcal{N}(0,1)$" src="form_97.png">.<p> |
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| 27 | Special cases include:...<h2><a class="anchor" name="math"> |
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| 28 | Mathematical background:</a></h2> |
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| 29 | This particular model belongs to the exponential family, hence it has conjugate distribution of the Gauss-inverse-Wishart form (class egiw). See, [reference] for details.<p> |
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| 30 | For this model, structure estimation is a form of model selection procedure. Specifically, we compare hypotheses that the data were generated by the full model with hypotheses that some regressors in vector <img class="formulaInl" alt="$\psi$" src="form_67.png"> are redundant. The number of possible hypotheses is then the number of all possible combinations of all regressors.<h2><a class="anchor" name="soft"> |
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| 31 | Software implementation:</a></h2> |
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| 32 | Estimation with this class of model is perfromed by class ARX which is derived from class BMEF (estimation of exponential family). The posterior density ( ARX::_epdf() ) is class egiw, which represents Gauss-inverse-Wishart density.<p> |
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| 33 | Structure estimation is implemented in method ARX::structure_est() which uses brute force tree search approach.<h2><a class="anchor" name="exa"> |
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| 34 | Examples of Use:</a></h2> |
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| 35 | There are many ways how to use the object.<ul> |
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| [271] | 36 | <li>Pure C++, see below, or unit testing file of class ARX, <a class="el" href="arx__test_8cpp.html">arx_test.cpp</a>.</li><li>C++ application <code>estimator</code> with UI configuration file, <a class="el" href="arx_ui.html">Running experiment <code>estimator</code> with ARX data fields</a>.</li><li>Matlab .mex file <code>estimator</code> with UI configuration file, <a class="el" href="arx_ui.html">Running experiment <code>estimator</code> with ARX data fields</a>.</li></ul> |
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| 37 | <p> |
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| 38 | The easiest way how to use class ARX is: <div class="fragment"><pre class="fragment"><span class="preprocessor">#include <<a class="code" href="arx_8h.html" title="Bayesian Filtering for generalized autoregressive (ARX) model.">estim/arx.h</a>></span> |
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| 39 | <span class="keyword">using namespace </span>bdm; |
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| 40 | |
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| 41 | <span class="comment">// estimation of AR(0) model</span> |
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| 42 | <span class="keywordtype">int</span> main() { |
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| 43 | <span class="comment">//prior </span> |
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| 44 | mat V0 = 0.00001*eye(2); V0(0,0)= 0.1; <span class="comment">//</span> |
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| 45 | ARX Ar; |
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| 46 | Ar.set_statistics(1, V0); <span class="comment">//nu is default (set to have finite moments)</span> |
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| 47 | <span class="comment">// forgetting is default: 1.0</span> |
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| 48 | mat Data = concat_vertical( randn(1,100), ones(1,100) ); |
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| 49 | Ar.bayesB( Data); |
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| 50 | |
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| 51 | cout << <span class="stringliteral">"Expected value of Theta is: "</span> << Ar.posterior().mean() <<endl; |
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| 52 | } |
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| 53 | </pre></div><p> |
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| 54 | Here, it is assumed that </div> |
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| 55 | <hr size="1"><address style="text-align: right;"><small>Generated on Sun Feb 15 21:43:51 2009 for mixpp by |
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