I am investigating orbit determination for planetary exploration. I would like to try different kinds of estimation methods such as batch-based estimation and EKF. Up till now, I have found that the accuracy of the batch-based method is higher than the sequential filter. I would like to know why this happens because the principle of the sequential filter also indicates that all the previous measurements contribute to the current estimation.
Since no more information is added, why is the accuracy of two methods different? Could anyone explain?
Dear Zheng-shi Yu,
If I understood your question appropriately, the batch-based methods meant for Moving horizon estimators (I hope that I am correct), where you need to process the larger amount of data at one particular time instant when compared with the previous time instant. These approaches often perform better than conventional estimation algorithms, which make use of probability density function(s). You have correctly noticed that, processing large amount of data (previous time instants) at present time instant, is the key advantage of these algorithms. Bemporad Morari, and Rawlings, have specified this aspect in their research work.
Coming to sequential filters (assuming that you have used particle filters), performance of these sampling based filters is often not consistent due to several aspects. Choice of samples is not consistent for the nonlinear dynamical systems and degeneracy as well as choice of importance function, are the critical issues. As mentioned earlier all these parameters were can result in inconsistency in the filter performance. You can also check the algorithm (sequential filter) with EKF or UKF as importance functions. Please refer to the following paper titled as, "Practical Issues in State Estimation using Particle Filters" (Shah et al., Journal of Process Control).
I hope that I have addressed your query.
Hi Kottakki! Thank you for your answer. The batch-based methods is just like the Moving horizon estimator, only the different is that the cost function is about the residual of observations. The sequential filter I used is EKF. Theoretically, the EKF is a optimal estimator and the previous observations are used for the estimation of state at current time. However, in reality, the performance may not be perfect if the nonlinearity is high.
Hi Zheng-shi Yu, this is an interesting question! The point is that the estimate at a given time step may be improved by data obtained at a later time step (Drummond, O. E., SPIE 1997, "Target Tracking with Retrodicted Discrete Probabilities") Insofar, the EKF does not provide the optimal estimate (non-linearities not considered), but the optimal estimate at the time of the measurement.
The batch estimator may use information contained in measurements that are obtained later in time. Thus, it should rather be compared to the EKF with retrodiction. There is a concept that combines both approaches: the accumulated state densities. These contain a batch of estimates, with an estimated covariance that contains the correlations between all state estimates. For ASDs, a recursive filtering algorithm is available, which unifies the Kalman-filtering and -retrodiction steps. (Koch, W.; Govaers, F., IEEE T-AES 2011, "On Accumulated State Densities with Applications to Out-of-Sequence Measurement Processing").
The filter algorithm is better in the on-line estimation procedure. Batch-based estimation could give better results in the off-line estimation procedure.
Since you are using MHE and EKF, the system is clearly non-linear. With non-linear systems, there is no optimal filter - this is because the assumption of Gaussian pdf of the states does not hold true right from the second observation (since a non-linear function of Gaussian is not Gaussian). So, with every time step, one loses information - inevitably - due to the inherent non-linearity of the system.
One way to improve the performance would be to include all data available at any time (and not only just the previous instant's data) to estimate the state - the full information problem. Since this cannot be implemented, we turn to MHE. But at this point, it is fairly intuitive that the estimator that uses more data points will give perform better.
Hence, the batch batch based algorithm (MHE) outperforms the sequential algorithm (EKF), although Markov property holds in both cases. (It is just that since we cannot explain the entire non-Gaussian pdf with finite moments, we lose information at every step and hence, the more data we use for estimation, the more accurate we are).
Hope this clarifies.
@Laya Das Thank yo so much for your explnation. I am laso studying this issue and try to find the reason why the MHE is better. I'll update this question if I have something valuable.
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You may want to consider the recursive least squares algorithm (RLS). RLS is the recursive application of least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. The correlation parameters are updated gradually. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables.
The RLS algorithm with forgetting factor (RLS-FF) allows acquired data to be weighted according to its age, with increased weight given to the most recent data. ― Application example ― I have applied RLS-FF to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer:
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