[261] | 1 | /*! |
---|
| 2 | \page arx Example of ARX model estimation |
---|
| 3 | |
---|
| 4 | Here, we use the \c ARX class to estimate parameters and structure. |
---|
| 5 | ARX model is defined as follows: |
---|
| 6 | \f[ |
---|
| 7 | y_t = \theta' \psi_t + \rho e_t |
---|
| 8 | \f] |
---|
| 9 | where \f$y_t\f$ is the system output, \f$[\theta,\rho]\f$ is vector of unknown parameters, \f$\psi_t\f$ is an |
---|
| 10 | vector of data-dependent regressors, and noise \f$e_t\f$ is assumed to be Normal distributed \f$\mathcal{N}(0,1)\f$. |
---|
| 11 | |
---|
| 12 | Special cases include:... |
---|
| 13 | |
---|
| 14 | \section math Mathematical background: |
---|
| 15 | 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. |
---|
| 16 | |
---|
| 17 | For this model, structure estimation is a form of model selection procedure. |
---|
| 18 | Specifically, we compare hypotheses that the data were generated by the full model with hypotheses that some regressors in vector \f$\psi\f$ are redundant. The number of possible hypotheses is then the number of all possible combinations of all regressors. |
---|
| 19 | |
---|
| 20 | \section soft Software implementation: |
---|
| 21 | Estimation with this class of model is perfromed by class ARX which is derived from class BMEF (estimation of exponential family). |
---|
| 22 | The posterior density ( ARX::_epdf() ) is class egiw, which represents Gauss-inverse-Wishart density. |
---|
| 23 | |
---|
| 24 | Structure estimation is implemented in method ARX::structure_est() which uses brute force tree search approach. |
---|
| 25 | |
---|
| 26 | \section exa Examples of Use: |
---|
| 27 | |
---|
| 28 | There are many ways how to use the object. |
---|
| 29 | - Pure C++, as it is used in unit testing of the class arx, \subpage arx_test.cpp |
---|
| 30 | - C++ application with UI configuration file, \subpage arx_test_ui |
---|
| 31 | - Matlab interface, \subpage arx_matlab |
---|
| 32 | |
---|
| 33 | */ |
---|