When the state equation of a model is an autoregressive process with unknown constant coefficients, EnKF can be easily modified to include Monte Carlo estimation of the constant coefficients. We recall that the standard particle filtering method suffers from weight degeneracy especially when the estimation includes static parameters. EnKF, however, does not suffer from degeneracy and provides a good estimate of the coefficients by integrating a sample minimum mean square estimate (MMSE) of the coefficients within the EnKF algorithm.
Without loss of generality, let us consider a first-order wide-sense stationary autoregressive state equation given by
egin{equation}
x_n = a x_{n-1} + u_n label{firstOrder} end{equation} where $a$ is a constant …show more content…
label{SimModel} end{eqnarray} where $x_n$ and $y_n$ denote state and measurement variables respectively. The state noise $u_n$ and the measurement noise $w_n$ have zero-mean Gaussian distributions with variances $sigma_u^2=10$ and $sigma_w=1$.
We have run the EnKF algorithm provided in Table ( ef{tSISp}) for ensemble sizes of $10$, $25$ and $50$, and compared it to a particle filter having a particle (ensemble) size of $50$ $100$ and $500$ and $1000$ by computing the mean square error (MSE) of the estimate of $x_n$ as
egin{equation}
MSE = frac{1}{M}sumlimits_{n=1}^{M}(x_n-{hat x}_n)^2 label{MSE} end{equation}
Figure ( ef{figure1}) shows the plot of one realization of the estimate obtained from EnKF of ensemble size $50$ and particle filter of ensemble size $1000$.
egin{figure}[htpb]
%epsfxsize=linewidth epsffile{figure.eps, …show more content…
We simulated the first-order autoregressive model with $a=0.62$. Figure ( ef{figure3}) shows the plot of the the estimate at different instants of time. As seen from the figure the estimates converges to close to the actual value in about the first $10$ time instants and the final estimate $0.6272$ is close to the actual value.
egin{figure}[htpb]
%epsfxsize=linewidth epsffile{figure.eps, width=15.5cm} centerline{epsfig{figure=ConstCoeff.eps,width=9.5cm}} caption{Estimate of constant coefficient $a$ as a function of time.} label{figure3} end{figure}
section{Conclusions} label{Conc} In this paper, we have presented that ensemble Kalman filter can be an alternative method to particle filter algorithm for the estimation of state vectors of a low-order nonlinear state-space models. Previous work has established that ensemble Kalman filter is more suited than Particle filter for high-order models mainly because particle suffers weight degeneracy. In this paper, we argue, through simulations, that even in low-order models ensemble Kalman filter can be an attractive alternative to particle filter for computationally constrained implementation. Furthermore, we discussed methods to improve the ensemble Kalman filter and its adaptation to estimating constant coefficients of an autoregressive state
Without loss of generality, let us consider a first-order wide-sense stationary autoregressive state equation given by
egin{equation}
x_n = a x_{n-1} + u_n label{firstOrder} end{equation} where $a$ is a constant …show more content…
label{SimModel} end{eqnarray} where $x_n$ and $y_n$ denote state and measurement variables respectively. The state noise $u_n$ and the measurement noise $w_n$ have zero-mean Gaussian distributions with variances $sigma_u^2=10$ and $sigma_w=1$.
We have run the EnKF algorithm provided in Table ( ef{tSISp}) for ensemble sizes of $10$, $25$ and $50$, and compared it to a particle filter having a particle (ensemble) size of $50$ $100$ and $500$ and $1000$ by computing the mean square error (MSE) of the estimate of $x_n$ as
egin{equation}
MSE = frac{1}{M}sumlimits_{n=1}^{M}(x_n-{hat x}_n)^2 label{MSE} end{equation}
Figure ( ef{figure1}) shows the plot of one realization of the estimate obtained from EnKF of ensemble size $50$ and particle filter of ensemble size $1000$.
egin{figure}[htpb]
%epsfxsize=linewidth epsffile{figure.eps, …show more content…
We simulated the first-order autoregressive model with $a=0.62$. Figure ( ef{figure3}) shows the plot of the the estimate at different instants of time. As seen from the figure the estimates converges to close to the actual value in about the first $10$ time instants and the final estimate $0.6272$ is close to the actual value.
egin{figure}[htpb]
%epsfxsize=linewidth epsffile{figure.eps, width=15.5cm} centerline{epsfig{figure=ConstCoeff.eps,width=9.5cm}} caption{Estimate of constant coefficient $a$ as a function of time.} label{figure3} end{figure}
section{Conclusions} label{Conc} In this paper, we have presented that ensemble Kalman filter can be an alternative method to particle filter algorithm for the estimation of state vectors of a low-order nonlinear state-space models. Previous work has established that ensemble Kalman filter is more suited than Particle filter for high-order models mainly because particle suffers weight degeneracy. In this paper, we argue, through simulations, that even in low-order models ensemble Kalman filter can be an attractive alternative to particle filter for computationally constrained implementation. Furthermore, we discussed methods to improve the ensemble Kalman filter and its adaptation to estimating constant coefficients of an autoregressive state