A Residual Exogenous–Autoregressive Gated Forecasting Framework for Nonlinear Dynamic Time Series: Application to Hydrogen Sulfide Prediction

Abstract

Multi-horizon forecasting of nonlinear dynamic time series with exogenous inputs is challenging when the target variable exhibits strong temporal persistence and the exogenous variables provide horizon-dependent corrective information. Direct forecasting models must learn both the carry-forward behavior of the target and the nonlinear deviations caused by changes in the process inputs. This study proposes a residual exogenous–autoregressive gated forecasting framework for nonlinear dynamic prediction. The proposed model decomposes the forecasting operator into a persistence component and a learnable residual correction term. Historical target dynamics and exogenous input dynamics are encoded using two dedicated CNN-LSTM branches, and their latent representations are combined through a sample-dependent sigmoid gating mechanism. The final prediction is obtained by adding the learned correction to the most recent target observation. The framework is evaluated on a benchmark sulfur recovery unit dataset for multi-horizon hydrogen sulfide H2S concentration forecasting using a leakage-aware nested blocked hyperparameter selection and evaluation protocol. Three forecasting horizons are considered: one-step, five-step, and ten-step ahead prediction. The proposed method achieved the lowest RMSE at the one-step and five-step horizons and remained highly competitive at the ten-step horizon, where its RMSE was nearly identical to the best PatchTST baseline. Across the three horizons, the proposed model obtained RMSE values of 0.0096±0.0020, 0.0436±0.0097, and 0.0521±0.0138, corresponding to RMSE reductions over the persistence baseline of 39.7%, 10.0%, and 13.7%, respectively. The model also maintained a compact parameter count and sub-millisecond inference latency, supporting its feasibility for online soft-sensing applications. Regression, time-series, error distribution, Taylor diagram, and SHAP analyses show that the residual gated formulation is particularly effective for short- and medium-horizon forecasting, while longer-horizon prediction remains more difficult because of increasing temporal uncertainty. The SHAP results indicate that historical H2S dominates short-horizon prediction, whereas airflow-related variables become more influential at the longer horizon. The results demonstrate that the proposed framework provides an interpretable and computationally compact learning approach for residual forecasting in persistent nonlinear dynamic systems.