Example of ARX model estimation

Here, we use the ARX class to estimate parameters and structure. ARX model is defined as follows:

$y_t = \theta' \psi_t + \rho e_t$

where $y_t$ is the system output, $[\theta,\rho]$ is vector of unknown parameters, $\psi_t$ is an vector of data-dependent regressors, and noise $e_t$ is assumed to be Normal distributed $\mathcal{N}(0,1)$ .

Special cases include:...

Mathematical background:

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.

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 $\psi$ are redundant. The number of possible hypotheses is then the number of all possible combinations of all regressors.

Software implementation:

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.

Structure estimation is implemented in method ARX::structure_est() which uses brute force tree search approach.

Examples of Use:

There are many ways how to use the object.

Pure C++, as it is used in unit testing of the class arx, arx_test.cpp
C++ application with UI configuration file, arx_test_ui
Matlab interface, arx_matlab