As a demonstrative example, we consider the following stochastic oscillator system
x˙ | ˙ | (6) | ||
1 = αx2 + σ Wx_{1} | ||||
x˙ | _{2} = 4 | ˙ | , | |
− 4x_{1} + σ W_{x}_{2} |
where α = 1 is an arbitrary parameter, W^{˙} is white noise with unit variance and σ is the level of system noise. Our assump-tion is that only observations of x_{1} sampled at rate dt = 0.2, corrupted by Gaussian observational noise, are available. These observations though are restricted in that any measurement below a value of 0 is censored, implying a censored interval of [−∞, 0].
Given the noisy, censored observations, our goal is to estimate x_{1} and x_{2} as well as parameter α using the proposed non-linear censored filter. In conducting our estimation, we assume the following deterministic oscillator model for the filter
x˙_{1} = αx_{2} | (7) |
x˙_{2} = 4 − 4x_{1}.
Notice in this example, the system from which the observations are from,is different than the model used by the filter,
This model mismatch serves as a proxy for the model error we will encounter later when considering real experimental data analysis.
The use of Kalman filtering for parameter estimation has received considerable attention. A popular approach is the so-called dual estimation method which treats the model parameters q as auxiliary state variables that evolve slowly over time. In this article, we assume trivial dynamics, namely q˙ = 0. Using this approach, we assume α˙ = 0 and form an augmented state vector consisting of the original state variables x_{1} and x_{2} and now α, allowing for simultaneous state and parameter estimation.
shows the state and parameter estimation results in the stochastic oscillatory system with system noise variance
- ^{2} = 0.01 (total system noise variance of 0.02 since there are two independent noise variables) and observational noise variance of 0.3. Black circles indicate the noisy observations, black lines denote the true trajectory of the variables and parameters and solid grey curves reflect the filter estimate. In the estimation results for parameter α, we also include the filter estimated 95% confidence interval (dashed grey curves). shows the estimation results with a higher system
noise variance σ ^{2} = 0.1 (total system noise variance of 0.2). After an initial transient period, the filter is able to estimate the system variables and parameter with great accuracy despite the model error introduced by the stochastic noise term. Of particular importance, we notice the fidelity of the variable reconstruction during the periods of censored data.
As previously mentioned, one of the main advantages of using the Kalman filter for estimation is that it is a sequential estimator. While this means that new observations can be processed online without re-analzying the entire dataset, the more important implication is that it allows for the tracking of parameters whose values may drift over time. To simulate
Fig. 1. State and parameter estimation in stochastic oscillator system when α is constant over time. System noise variance of (a) σ ^{2} = 0.01 (total system noise variance of 0.02) and (b) σ ^{2} = 0.1 (total system noise variance of 0.2) considered. Observations (black circles) of the x_{1} variable are corrupted by observational noise with variance of 0.3 and censored below a value of 0. Solid black lines denote the true variable/parameter trajectory and solid grey lines the filter estimates. Dashed grey lines denote the filter estimated 95% confidence region. Despite the presence of censored data, the filter is able to accurately estimate both state variables as well as the unknown parameter. In particular, we note the fidelity of the reconstruction during censored regions of the data.scenario, we considered the estimation problem in the above oscillator system when α changes over time. Namely, its value changes from 1 to 0.5 after 15 units of time. shows the resulting estimation in this nonstationary case. Once again after the initial transient period of the filter we see convergence of α to its correct value and accurate estimation of the state variables. As α drifts, the filter loses track of the x_{1} and x_{2} variables but is able to recover after a suﬃcient amount of data has been observed. Furthermore, the filter is able to accurately track the drift in α, despite the presence of censored data.
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