Parameter estimation of time varying mixed AR model (Q2726060)

The authors consider parameter estimation of the following linear time-varying system: NEWLINE\[NEWLINEY_{k-N+j}=\sum_{i=1}^4 a_i(k) t_j^{i-1}+e_{k-N+j}, \quad j=1,\ldots, N,NEWLINE\]NEWLINE NEWLINE\[NEWLINEe_{k-N+j}=\varphi_1(k)e_{k-N+j-1} +\cdots+\varphi_p(k)e_{k-N+j-p}+\varepsilon_{k-N+j},NEWLINE\]NEWLINE where \(k\geq N\), \(t_j=j\Delta t\), \(\Delta t\) is the sampling interval, \(\{ Y_{j}\}\) is the measured data, and \(\{\varepsilon_{k-N+j}\}\) is the white noise sequence, which is called the time varying mixed AR model by them. Using a two-steps least squares method and the restricted memory least squares method, they present an adaptive algorithm for estimating the time-varying parameters \(a_i(k)\), \(i=1,\ldots,4,\) and \(\varphi_i(k)\), \(i=1,\ldots,p\). A numerical simulation example is given to illustrate their algorithm and its effectiveness.