Fig 1.
Example of Lorenz-63 realization.
500 samples, or 25 units of time, of noisy training data (grey circles) are available for (a) x, (b) y and (c) z. Note, we have only shown the last 5 units of time worth of training data for visualization purposes. From the end of the training data (indicated by dashed black line), we want to accurately predict the system dynamics within the next unit of time (solid black line).
Fig 2.
Comparison of the prediction methods in the Lorenz-63 system.
Results of predicting the Lorenz-63 (a) x, (b) y and (c) z variables averaged over 500 realizations. Error bars denote standard error over the 500 realizations. Training data consists of 500 data points generated from Eq 6 with σ = 10, ρ = 28 and β = 8/3 with sample rate h = 0.05. Data are corrupted by Gaussian observational noise with mean 0 and variance of 4. Parametric (black), nonparametric (blue) and hybrid (red) prediction accuracy with parameter uncertainty of 80% (solid line) plotted as a function of forecast horizon. Hybrid prediction, which utilizes mechanistic equations in describing x and z but nonparametrically represents y, offers an improvement in short-term prediction accuracy over standalone nonparametric prediction. Parametric prediction at this uncertainty level performs poorly in predicting all three variables and in the case of (b) and (c) is not seen due to the scale of the error. For comparison purposes, the parametric method with 50% (dotted line) and 20% (dashed line) uncertainty is also considered. As the uncertainty shrinks, performance of the parametric method improves. However, only at a small uncertainty level does it outperform the short-term improvement in prediction afforded by the hybrid method.
Table 1.
Summary of Lorenz-63 parameter estimation results.
Mean and standard deviation calculated over 500 realizations. The hybrid method, which only needs to estimate σ and β, is robust to a large initial parameter uncertainty. The parametric method on the other hand is unable to obtain reliable estimates of the Lorenz-63 parameters unless the uncertainty is small enough.
Fig 3.
Predicting Lorenz-63 x dynamics at three different noise levels.
Training data corrupted by noise with variance of (a) 1, (b) 4 and (c) 16. Error bars denote standard error over 500 realizations. Hybrid prediction with 80% uncertainty (solid red line), which utilizes mechanistic equations in describing x only and nonparametrically represents y and z, offers an improvement in prediction accuracy over standalone nonparametric prediction (solid blue line). While the accuracy of both methods decreases as the noise level increases, the hybrid method offers improved prediction accuracy.
Fig 4.
Predicting neuron potential x3 in random 3-neuron Hindmarsh-Rose networks.
(a) 3000 samples (or 240 ms) of training data (grey circles) are available from each neuron in the network. From the end of the training data (indicated by dashed black line), we want to accurately predict the next 8 ms of x3 (solid black line). (b) Forecast accuracy in predicting 8 ms of x3 when using parametric (black), nonparametric (blue) and hybrid (red) methods. Results averaged over 200 randomly generated 3-neuron Hindmarsh-Rose network realizations and error bars, shown only for every tenth forecast point, denote standard error. At 80% uncertainty (solid line), the hybrid method outperforms both parametric and nonparametric methods. When considering the parametric method with 50% uncertainty, prediction accuracy between it and the hybrid method is comparable over the first 2 ms.
Table 2.
Summary of neuron 3 parameter estimation results.
Mean and standard deviation calculated over 200 realizations. The hybrid method once again is robust to a large initial parameter uncertainty. The parametric method on the other hand is unable to obtain reliable estimates of the neuron parameters with large uncertainty.
Fig 5.
Example data set from T. castaneum experiment presented in [30].
37 observations, or 74 weeks, of training data (grey circles) are available for (a) larvae, (b) pupae and (c) adult population levels. From the end of the training data (indicated by dashed black line), we want to accurately predict the next 8 weeks of population dynamics (solid black line).
Fig 6.
Results for predicting population levels of T. castaneum.
Average SRMSE over 21 experimental datasets when using parametric (black curve), nonparametric (blue curve) and hybrid (red curve) methods for predicting (a) larvae, (b) pupae and (c) adult population levels with uncertainty of 80% (solid line) and 50% (dashed-dotted line). Error bars correspond to standard error over the 21 datasets. Hybrid prediction with 80% uncertainty offers improved prediction over both nonparametric and parametric with 80% uncertainty (not visible due to scale of error), and comparable performance to parametric with 50% uncertainty.