Supplementary Material for: Approximation error approach in spatiotemporally chaotic models with application to Kuramoto-Sivashinsky equation J.M.J. Huttunen, J.P. Kaipio and H. Haario Corresponding author: Janne M.J. Huttunen,
[email protected] S.1
Performance with different values of η and γ (P1)
We studied the performance of the approach for different values of the model parameters η and γ in problem P1. Generation of simulated data and computation of estimates were carried out as described section 4.1 and 4.2, but in some cases larger samples sizes (Ns = 5000) were required. First row in figure S1 shows results of three simulations in which the model parameters are randomly drawn from the distribution of the parameters in P2 (η, γ ∼ N (1, 0.052 )). The results are similar to the original experiment P1 for such smaller deviations, but also the EKF-LO estimate is feasible for first simulation run. The last two rows in figure S1 show result for a larger deviation of either η or γ. We can see that both EKF-LO and EKF-BAE estimate may fail when either η or γ is made significantly smaller. On the other hand, for the larger values of η or γ, all estimates (including EKF-LO) turn out to be sufficient. The reason for such behaviour is that (chaotic) spatial-temporal behaviour depends strongly on the model parameters η and γ (as noted in section 2 in the paper). This has also an effect to accuracy of the numerical solution: for lower values of η and γ, the chosen number of basis functions in the low-dimensional model (N = 32) is insufficient and also the BAE approach fails to handle the error due to the discretization. Increasing accuracy of the low–dimensional model (N ) provides sufficient estimates, as can be seen in Fig S2. On contrary, for larger η and γ, the low-dimensional model can be sufficiently accurate even without the BAE approach. We note also that even the EKF-HI estimate fails if the parameters are small enough (e.g. when η = 0.77 or γ = 0.6). This indicates that the measurement setup is insufficient for the problem and it should be adjusted for wider ranges of parameters. η = 1.075, γ = 1.013 10
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Figure S1: The relative norm of the errors for estimates computed using different values of η and γ in P1. The high-dimensional EKF-HI estimate (thick blue line), the EKF-BAE estimate (medium green line) and the EKF-LO estimate (thin red line).
S.2
Performance with the different choices of initial distributions (P1)
In order to study the effect of the initial distribution of the state variable, we have computed solutions for P1 using different values of the variances of A and φ. The simulated data is same as used for experiments in section 4.3. The estimates were computed as described in section 4.2. For the EKF-BAE approach,
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Figure S2: The relative norm of the errors for estimates for cases (η, γ) = (0.91, 1) and (η, γ) = (1, 0.9) in P1 with different number of basis function pairs in the discretization of the low–dimensional model (N ). The high-dimensional EKF-HI estimate (thick blue line), the EKF-BAE estimate (medium green line) and the EKF-LO estimate (thin red line). The case N = 32 corresponds to the original discretization in P1. we also computed the new set of samples (Ns = 2000 for each) corresponding the chosen values of the variances. Fig. S3 shows results for the altering the variance of A. The results shows that reducing the variance of A makes the BAE estimate infeasible. This is probable due to the fact that the true initial value is not covered by the reduced variance and therefore samples representing the error processes are farther from the true errors. This increases the effect of the data thinning and reduces accuracy of the conditional mean and covariance computed using the importance sampling approach. On the contrary, the effect of increasing the variance did not have that large effect to the results in this case, but there is some level of loss in accuracy in the BAE estimate. However, we note that the wider distribution can require more samples of {εk } and {νk } to be included to approximate the conditional statistics accurately without data thinning. Similar results can also be observed when the variance of φ or the variances of the both A and φ are changed. 2 × 1/42 σA 10 0
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Figure S3: The relative norm of the errors for estimates computed using decreased and increased levels of the variance of A (Problem P1). The high-dimensional EKF-HI estimate (thick blue line), the EKF-BAE estimate (medium green line) and the EKF-LO estimate (thin red line).
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The effect of observation noise level (P1)
We studied the effect of the observation noise level σvk to the performance by scaling the noise in the original simulated data (P1) to correspond increased or decreased noise levels. The results are shown in Fig. S4. As can be seen, the problem is clearly sensitive to the noise level. With 1/4× noise level (compared to the original noise level in P1), the EKF-LO estimate is also feasible over the whole time domain. On the contrary, increased noise levels will make also EKF-BAE estimate infeasible (2× noise level compare to the original). It is to be noted that, with 4× noise level, the reference estimate (EKF-HI) also becomes useless. σvk × 1/4 38 10
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Figure S4: The effect of the noise level in P1: The figures shows the relative norm of the errors (left) ˆ k for the two points x (middle and right). The high-dimensional and trajectories of the filter estimate X EKF-HI estimate (thick blue line), the EKF-BAE estimate (medium green line) and the EKF-LO estimate (thin red line). The dash-dotted lines are 2×SD error estimates. The true solution is shown as a black line.
S.4
Maximum likelihood estimate for η and γ (P2)
We have computed the maximum likelihood estimate (MLE) for the parameters η and γ by maximazing the EKF likelihood corresponding to the BAE error model (10): L(y1 , . . . , yk |η, γ) = −
K 1 X T y −1 rk (Σk ) rk + log det(Σyk ) 2 k=1
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Figure S5: MLE estimate (green cross) with 95%-confidence region (red line represents boundaries), the EKF-BAE estimate (blue plus-sign) and the true values of the parameters (black circle). The box on top shows a zoom-in picture of the confidence region. As we can see, the MLE estimate is near the EKF-BAE estimate (also shown in Fig. S5). Furthermore, it is to be noted that 95% confidence interval based on the likelihood does not include the true value. This is probable due to the uncertainties and approximations in the EKF that are not covered in the solution. It is to be noted that the likelihood is rather “noisy” due to the chaotic behaviour with limitations in numerical accuracy. This makes optimization, for example, using the Newton-Rhapson method impossible or very difficult. Similar behaviour was also observed in the reference [26]. We wish to note that the computation of the MLE estimate is computationally a very tedios task (at least when solved using a grid). Better results are obtained with less computational expense by employing the high dimensional model with a traditional EKF problem (as our reference estimate).
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