Forecasting occupational accidents in Turkey using multivariate ARMAX and NLARX models

Abstract

Occupational accidents remain a critical issue in Turkey, with significant social and economic consequences, and understanding accident trends is essential for developing effective prevention strategies. This study employed both linear AutoRegressive Moving Average with Exogenous Input (ARMAX) and Nonlinear AutoRegressive with Exogenous Input (NLARX) models to forecast future occupational accidents using four accident-related populations (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_1$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_2$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_3$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_4$$\end{document}) derived from official insurance records. Due to the lack of consistently reported monthly data from the Social Security Institution (SSI), exogenous variables such as sectoral, economic, or demographic indicators were not incorporated, and the models were therefore identified based solely on the endogenous accident dynamics. The ARMAX identification process yielded a relatively large set of candidate parameters, which were subsequently evaluated using statistical significance criteria; only coefficients with p values below 0.05 and confidence intervals excluding zero were retained for interpretation. Model performance was evaluated using the normalized mean squared error (NMSE), which was computed separately for the training period, the test (out-of-sample forecasting) period, and the full dataset for each model. This multi-level evaluation enabled a consistent comparison of in-sample fitting accuracy, out-of-sample generalization capability, and overall predictive performance across ARMAX and NLARX models. The significance-based analysis revealed distinct linear dynamic structures across the output groups, with the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y_1,y_2)$$\end{document} populations characterized by a larger number of moderate-magnitude significant coefficients, whereas \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y_3,y_4)$$\end{document} exhibited fewer but more dominant linear effects. The ARMAX model produced the lowest NMSE values across the training, test, and full datasets for most populations, demonstrating particularly strong and stable predictive accuracy for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_2$$\end{document}. The NLARX model yielded the best performance for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_1$$\end{document} and showed comparable NMSE values to ARMAX for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_2$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_4$$\end{document}, although it exhibited higher forecasting errors for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_3$$\end{document}, especially in the test period. Overall, the results indicated that while NLARX was capable of capturing nonlinear patterns in specific cases, the ARMAX framework provided a more robust, interpretable, and consistently generalizable representation of the dominant temporal dynamics governing occupational accident trends. These findings highlighted the potential of multivariate time series models to support evidence-based decision-making in occupational safety planning and policy development.