Improving tail accuracy of the predicted cumulative distribution function of time of failure

Prognostic information is used to make decisions such as when to perform maintenance or - in time sensitive and safety critical applications - when to change operational settings. Where distributions about expected end of life (EOL) are available, these decisions are often based on risk-informed thresholds, for example a 2 sigma or 3 sigma criterion which considers the probability of making a bad decision at 5% or 0.3%, respectively, as tolerable. Sampling-based techniques such as Monte Carlo Sampling (MCS) and Latin Hypercube Sampling (LHS) can provide effective approaches to the propagation and analysis of uncertainty. Due to its efficient manner of stratifying across the range of each sampled variable, LHS requires less computational effort than MCS and is therefore more often used. However, since the focus is placed on accurately predicting the tails of the Cumulative Distribution Function (CDF) of Time of Failure (ToF) sampling-base techniques may not properly represent these areas. Although one might be tempted to use a brute force approach and simply increase the number of samples, some safety-critical applications may be computationally constrained. Such applications include electric UAV where the decision making process has to be fast in order to take action as soon as possible. This paper explores the ability of MCS and LHS to perform tail prediction with small sample sizes. The results show that LHS does not provide a significant advantage over MCS in terms of characterizing the tails of the CDF of the battery End of Discharge (EOD) prediction. Then, a methodology that combines MCS and Kernel Density Estimation (KDE) is investigated. The advantages of KDE in terms of reducing sample size while improving tail accuracy are demonstrated on battery end-of-discharge data.