"background": "Water treatment systems in many regions face persistent challenges in operational efficiency, leading to resource wastage and service shortfalls. Diagnostic tools for long-term performance evaluation are often lacking, particularly for infrastructure with limited historical monitoring data. ", "purpose and objectives": "This case study develops and validates a time-series forecasting model to diagnose efficiency trends in water treatment facilities. The objective is to provide a replicable methodological framework for quantifying efficiency gains or losses over extended operational periods. ", "methodology": "A case study methodology was employed, analysing historical operational data from multiple facilities. The core analytical tool was an Autoregressive Integrated Moving Average (ARIMA) model, specified as \ᵈ yt = c + \1 \ᵈ y{t-1 +. . . + \ \ᵈ yt-p + \1 -1 +. . . + \ -q + \, where yₜ is the efficiency metric. Model parameters were estimated using maximum likelihood, and forecasts were generated with 95% confidence intervals to assess diagnostic reliability. ", "findings": "The model application revealed a significant upward trend in the diagnostic efficiency index for the studied systems, with an average annual improvement of approximately 2. 3%. Forecasts indicated a high likelihood (p < 0. 05) of continued positive trajectory under current management practices, though with widening confidence intervals in later projection periods. ", "conclusion": "The time-series forecasting model proved effective as a diagnostic tool for long-term efficiency assessment, translating complex operational data into clear performance trajectories. It provides a quantitative basis for distinguishing between random fluctuation and systemic change. ", "recommendations": "Adopt the forecasting model for routine efficiency diagnostics across similar infrastructure portfolios. Integrate model outputs with preventative maintenance scheduling. Future work should incorporate exogenous variables, such as energy costs and raw water quality, into the forecasting framework. ", "key words": "Operational efficiency, AR
Kariuki et al. (Thu,) studied this question.