Fig 1.
Outputs from the true (or reference) function (in red). and those from the biased one (in green).
For better visualization, we re-plot the reference and biased outputs over the input interval [−2, 2] in a separate, zoomed-in subplot.
Fig 2.
Histograms of the unimodal inputs and noisy residuals (as labels), with respect to the training and cross-validation (CV) datasets, respectively.
Fig 3.
Box plots of data mismatch at different iteration steps, with respect to the (a) training and (b) CV datasets in case of unimodal inputs.
Fig 4.
Error-bar plots in case of unimodal training inputs, in the form of ensemble mean ± ensemble std, with respect to the initial (in blue) and final (in red) ensembles of scale (Panel (a)) and weight (Panel (b)) parameters, respectively, associated with 200 center points that are evenly distributed over the interval [−6, 6).
Fig 5.
Red (reference curve) and green (biased curve) curves in all panels of the current figure are the same as those in Fig 1.
Panels (a) and (c) show the initial and final ensembles of predictions (with respect to the case of unimodal training inputs), obtained by adding to the biased curve the corresponding ensembles of residual terms, which are computed using Eqs (18) through (20), wherein the kernel parameters correspond to the initial and final ensembles of scale and weight parameters, respectively. Panels (b) and (d) also report the means of the initial and final ensembles of predictions, respectively.
Fig 6.
Histograms of the multi-modal inputs and noisy residuals, with respect to the training and CV datasets, respectively.
Fig 7.
As in Fig 5, but for the case with multi-modal inputs, for which no multi-modal learning strategy (MMLS) is adopted.
For better visualization, in Panel (d) we re-plot the reference, the biased and the mean corrected curves over the input interval [−2, 2] in a separate, zoomed-in subplot.
Table 1.
Number of training data points associated with each GMM component (cluster), and the corresponding parameters estimated using the training data.
Fig 8.
Box plots of data mismatch at different iteration steps, with respect to the (a) training and (b) CV datasets in case of multi-modal inputs.
Fig 9.
Similar to Fig 4, but for multi-modal training inputs.
For visualization, we plot scale (left column) and weight (right column) parameters associated with different clusters separately.
Fig 10.
Similar to Fig 7, but for the case in which the multi-modal learning strategy (MMLS) is adopted.
In the experiment, the number Ncl of clusters is 3, the same as the number of modes in the training inputs. Note that the learning process is carried out cluster by cluster.
Fig 11.
Similar to Fig 10, but for the final prediction results after all the training data in different clusters are used to learn kernel parameters.
Presented here are the results with respect to of the choices of using 2, 4, 6 and 8 clusters to fit the GMM (from top to bottom), respectively.
Fig 12.
Reference and mean models in the perfect scenario.
Top row: Reference model (Panel (a)) used to generate observations (cf. Fig 13(a)), and the mean model (Panel (b)) of the initial ensemble. Bottom row: mean of the final ensemble obtained through data assimilation without any model-error correction (MEC) (Panel (c)), and the corresponding mean when MEC is still adopted (Panel (d)) even though the forward simulator is perfect.
Fig 13.
Real and simulated observations in the perfect scenario.
Top row: Real observations (Panel (a)) generated using the reference model in Fig 12(a), and the mean of simulated observations obtained by applying the forward simulator to the initial ensemble of model variables (Panel (b)). Bottom row: As in Panel (b), but for the mean of simulated observations with respect to the final ensemble obtained without (Panel (c)) and with (Panel (d)) MEC in data assimilation, respectively.
Fig 14.
Box plots of data mismatch (top) and RMSE (bottom) with respect to the ensembles at different iteration steps in the perfect scenario.
Results in Panels (a) and (c) correspond to the case without MEC adopted in data assimilation, whereas those in Panels (b) and (d) to the case with MEC. Unless otherwise stated, data mismatch in the experiment with MEC is always calculated using the modified forward simulator with a residual term, as in Eq (33). Note that in Panels (a) and (c), the iES terminates at the iteration step 7, due to the stopping criterion that the average data mismatch at this step is less than four times the number of observations (which is 4 × 12, 000 in this case) for the first time.
Table 2.
Means and STDs of data mismatch and RMSE with respect to the initial ensemble, and the final ensembles with or without model-error correction (MEC), in the perfect scenario.
Fig 15.
Box plots of data mismatch differences at different iteration steps, for the experiment with MEC in the perfect scenario.
At a given iteration step, these differences are derived using data matching values that are calculated with the residual term excluded from Eq (33), minus data matching values that are computed with the residual term included in Eq (33), with respect to the corresponding ensemble of model variables at that iteration step. Therefore, positive data mismatch differences indicate that including the residual term helps match real observations better. For better visualization, we show the box plots from iteration steps 2 to 10 in a separate, zoomed-in subplot in the upper right corner.
Table 3.
Means and STDs of data mismatch and RMSE with respect to the initial ensemble, and the final ensembles with or without model-error correction (MEC), in the imperfect scenario.
Fig 16.
As in Fig 12, but for the experiment results in the imperfect scenario (Ncl = 1).
Fig 17.
As in Fig 13, but for the experiment results in the imperfect scenario (Ncl = 1).
Fig 18.
As in Fig 14, but for the experiment results in the imperfect scenario (Ncl = 1).
Fig 19.
As in Fig 15, but for the experiment results in the imperfect scenario (Ncl = 1).
Fig 20.
Experiment results with bias-based MEC in the imperfect scenario.
Fig 21.
Histogram of the mean of the residuals with respect to the initial ensemble.
Fig 22.
Box plots of RMSEs of the final ensembles, obtained with different numbers Ncl of clusters.