在本研究中，我們考慮兩種時間序列模型來配適衰變資料，並預測產品的首次到達時間（First Hitting Time）或剩餘使用壽命（Remaining Useful Life）。其一為具漂移項之差分整合移動平均（Integrated Moving Average with a Drift）模型；此外，為捕捉不同衰變路徑間的異質性，我們將漂移項視為隨機效應，進而建構出具隨機漂移項之差分整合移動平均（Integrated Moving Average with a Random Drift）模型。本研究推導了這兩種模型的概似函數，並求得對應之最大概似估計量。基於所提出之模型，我們進一步推導出首次到達時間與剩餘使用壽命的機率分配，並透過蒙地卡羅模擬（Monte Carlo simulations）驗證了模型的有效性。在實證資料應用方面，本文分析了疲勞裂縫成長資料與NASA電池衰變資料。首先，我們配適具隨機漂移項之差分整合移動平均模型來進行一步與兩步向前之預測，並估計相應的首次到達時間分配。接著，考量到NASA電池資料集中存在的容量回覆跳躍（regeneration-jumps）現象，我們結合時間序列之干預分析（intervention analysis），進而完善了電池的首次到達時間與剩餘使用壽命分配預測。

In this study, we consider two time series models to fit degradation data and predict the product’s first hitting time (FHT) or remaining useful life (RUL). One is the integrated moving average with a drift term (IMAD) model. Alternatively, to capture the heterogeneity within the degradation paths, we treat the drift term as a random effect and then construct the integrated moving average with a random drift (IMADR) model. The likelihood functions of both proposed models are derived, and the corresponding maximum likelihood estimators are obtained. Based on the proposed models, the FHT and RUL distributions are analytically derived. Monte Carlo simulations confirm the validity of the proposed models. For real data applications, the fatigue crack growth data and NASA battery data are analyzed. First, we fit the IMADR model to perform one- and two-step-ahead predictions, as well as to estimate the corresponding FHT distribution. Next, consider the regeneration-jumps in the NASA dataset, we use the intervention analysis of time series model, and then complete the FHT and RUL distributions of battery.

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