Comparison of MAEs of various HR estimation methods for the ISPC dataset.

Out-of-sample traditional and rolling window forecasts accuracy measures MAE, RMSE, MAPE, and MDA for MKSARMAX, SARMA, SARMAX, additive Holt-Winters, and KARMA models forecast of the Caconde UV dataset. The best value for each measure is shaded light gray.

Parameter setting situation.

Comparison of four algorithm prediction indicators.

DF11 locomotive wheel diameter measurement report (five times).

Each result was obtained from all datasets (n = 47): ISPC (n = 23) and BAMI (n = 24) datasets. The performance was evaluated on the basis of the mean absolute error (MAE).

Comparison of the interpolation accuracies achieved using MSE, MAE and MRE based on a cross-validation analysis.

Cascaded tanks 5-fold cross validated accuracies: Derivatives, mean absolute error (MAE).

Performance comparison of the DFDF, FSM-DFDF, and FSM-SGPS HR estimation methods.

Performance metrics (R2, RMSE, MSE, MAE) for eight machine learning algorithms in predicting economic factors related to rice production.

Database statistics for day (07:00–19:00) and night (19:00–07:00) periods.

The optimal MAE values and corresponding hyperparameters (Gaussian noise with σ = 10−3).

A key advantage of polarizable force fields is their ability to model the atomic polarization effects that play key roles in the atomic many-body interactions. In this work, we assessed the accuracy of the recently developed polarizable Gaussian Multipole (pGM) models in reproducing quantum mechanical (QM) interaction energies, many-body interaction energies, as well as the nonadditive and additive contributions to the many-body interactions for peptide main-chain hydrogen-bonding conformers, using glycine dipeptide oligomers as the model systems. Two types of pGM models were considered, including that with (pGM-perm) and without (pGM-ind) permanent atomic dipoles. The performances of the pGM models were compared with several widely used force fields, including two polarizable (Amoeba13 and ff12pol) and three additive (ff19SB, ff15ipq, and ff03) force fields. Encouragingly, the pGM models outperform all other force fields in terms of reproducing QM interaction energies, many-body interaction energies, as well as the nonadditive and additive contributions to the many-body interactions, as measured by the root-mean-square errors (RMSEs) and mean absolute errors (MAEs). Furthermore, we tested the robustness of the pGM models against polarizability parameterization errors by employing alternative polarizabilities that are either scaled or obtained from other force fields. The results show that the pGM models with alternative polarizabilities exhibit improved accuracy in reproducing QM many-body interaction energies as well as the nonadditive and additive contributions compared with other polarizable force fields, suggesting that the pGM models are robust against the errors in polarizability parameterizations. This work shows that the pGM models are capable of accurately modeling polarization effects and have the potential to serve as templates for developing next-generation polarizable force fields for modeling various biological systems.