﻿ :: Industrial Engineering & Management Systems :: • Editorial Board +
• For Contributors +
• Journal Search +
Journal Search Engine
ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.21 No.1 pp.128-136
DOI : https://doi.org/10.7232/iems.2022.21.1.128

# Rare Event Simulation by Combining Trend and Probability: An Application to Naval Ship Critical Failures

Jinwoo Choi*, Seongam Moon
Defense Management, Korea National Defense University, Republic of Korea
*Corresponding Author, E-mail: chlwlsdn8570@gmail.com
August 25, 2021 October 13, 2021 February 7, 2022

## ABSTRACT

Although the Rare event has a very wide application range, it has limitations in that it requires mathematical knowledge and the calculation process is complicated for the Bayesian approach, which was mainly used in previous studies. Therefore, rare events need to be easily analyzed. In this study, the critical failures of the ROK Naval ship were analyzed. Bayesian inference was not applied. The long-term trend (B-spline) and short-term probability (phase-type distribution) were fitted and combined. Phase-type distribution is a useful technique that can estimate the probability distribution of a desired shape with one parameter. This study is meaningful in that only an easy methodology was used, Phase-type distribution was applied to the analysis of rare events, and only a very small number of data were analyzed.

## 1. INTRODUCTION

ROK Naval ships are sometimes in a situation where they have to return to their home port during their mission due to failure. In case of general failure, immediate maintenance is performed using the spare-part provided by the ship. A failure that must return to the home port belongs to a critical failure that cannot normally perform its mission. ROK Navy performs periodic maintenance to prevent failure. Nevertheless, failures do occur. Naval ship crews who have served on ships for more than 20 years range from those who have never experienced a critical failure to those who have experienced a dozen or so. This means that critical failures occur very rarely. Let's take 10 failures as an example. 20 years is about 7,300 days, of which 10 failures are equivalent to 0.0014. Rubino and Tuffin (2009) stated that the probability criterion for defining a rare event has not been determined, and that it should be judged whether it is rare based on domain knowledge. The total life of ROK Naval ship is about 31 years. In 31 years, all major and minor engine system failures are about 1,050 (Moon and Choi, 2021). The subsystem of the Naval ship can be classified into 4 types including the engine system (Tinga et al., 2017). It can be assumed that the total number of failures of the Naval ship is about 4,200 (1,050 * 4) cases. Assuming that failures occur uniformly over 31 years, the total number of failures in 20 years is 2,709, so 10 failures occurring in 20 years corresponds to about 0.0036. 10 failures are fatal as they lead to mission failure. Considering the ratio of the number of failures (0.0014) and the ratio of critical failures among total failures (0.0036), the critical failure of a Naval ship is sufficient to classify it as a rare event.

Rare events generally include human casualties due to failure of transportation means, and failure of security systems in nuclear plants. Although the incidence is low, it is fatal. Therefore, the probability is small but cannot be ignored (Rubino and Tuffin, 2009). Attempts have been made to estimate the probability of occurrence of a rare event as a distribution. If the Rare event probability distribution is known, computational based Bayesian inference can be performed (Straub et al., 2016;Wahal and Biros, 2019;Rao et al., 2020). Straub et al. (2016) performed general Bayesian inference to derive the posterior through prior and data. Rare event probabilities were derived through Bayesian update. In order for the posterior to become a rare event distribution, a large amount of data is needed to reduce the influence of the prior and increase the influence of the likelihood (Gajewski et al., 2008;Kruschke, 2014;Jiang et al., 2015). In the study of Straub et al. (2016), general Bayesian inference was possible because it was analyzed based on a large amount of data obtained from monitor and sensor technology.

However, since rare events have a low probability of occurrence, data are often small. In such cases, assuming the distribution obtained from the observed data as posterior, a marginal rare event distribution is generated by solving the Bayesian inverse problem (Wagner et al., 2021;Wahal and Biros, 2019;Rao et al., 2020). In order to increase the accuracy of the generated marginal distribution, it is necessary to fit the posterior with high reliability. However, since the observed data distribution, posterior, is a distribution with a reduced prior range, it is difficult to explain it in the form of a general distribution (normal, gamma, etc.). In previous studies, the observed data was generally fitted with a biasing distribution through importance sampling. Wagner et al. (2021) solved the inverse Bayesian problem by fitting the observed data (posterior) using the Ensemble Kalman filter applied with importance sampling. Rao et al. (2020) studied importance sampling based on machine learning, but fundamental limitations of machine learning that require a lot of learning data remain. The Bayesian inverse problem through importance sampling used in past studies not only requires a high level of mathematical knowledge, but also has expensive computational cost in the sampling process.

It is difficult to apply Bayesian inference to the ROK Naval ship failure data in this study. Critical failures are analyzed on a daily basis. If applied to Bayesian inference, it should be possible to obtain the daily failure probability. In the case of Naval ship data, the total life cycle of ship is long, the number of ships is small, and the data period is short according to the security policy of the military. In particular, it is more difficult to estimate the probability of critical failure that can occur in ships of a specific age because the failure probability of each age is different (Moon and Choi, 2021). Naval ship critical failures are rare event, especially when there are little data. Assuming that the occurrence of critical failure is 1, even if all data are aggregated, daily failures are a majority of 0s and a very small fraction of 1. In this case, it is difficult to determine the prior or posterior (for Bayesian inverse problem). Therefore, Bayesian was not used in this study. Also, Rare event analysis is generally considered a difficult field. On the other hand, there are countless fields of application, such as equipment failures, weather phenomena, and human behavior in the industrial field. Therefore, rare event analysis should be easy to use. In this study, a rare event simulation was performed using only easy methodologies (Linear and Polynomial regression, B-spline, and Phase-type distribution). In particular, it is difficult to find that phase-type distribution is applied to rare event analysis. Phase-type distribution enables flexible fitting of the distribution shape, so it is convenient to apply expert opinions over Bayesian prior.

The goal of this study is to reduce the mathematical knowledge and computational cost by maximizing the use of small data, and to estimate the rare event probability with high reliability. In this study, it was assumed that the critical failure of the Naval ship, a rare event, would have an occurrence pattern according to age. If there is a pattern over time, the Fourier decomposition used in Prophet can be used (Taylor and Letham, 2018). However, because Fourier decomposition requires a lot of data, it is not suitable for this study. In this study, the concept of time series unit division of Fourier decomposition was used. Naval ship failure data can be divided into 6-month intervals for a total lifetime. The ship’s operating pattern at 6-month intervals is repeated for 31 years. Therefore, the failure trend of 31 years per year and the failure probability of 6 months were separately estimated and combined. The 6-monthly probabilistic pattern repeats in a trend change over 31 years of the entire lifecycle. For example, if the expected number of failures in year 1 is 36, it can be estimated that 18 failures will occur in 0.5 years, and about 0.1 failures per day (a year is assumed to be 360 ​​days). If the number of ships in year 1 failure is 10, the number of failures per day per ship is 0.01. If the total number of naval ships was 100, and 10 of them had a failure, it can be estimated that one failure per day is 0.001. 0.001 is used as the standard probability of Monte-Carlo simulation. If a random number is generated from uniform (0, 1) and a number less than 0.001 is generated, it is counted as a failure. The performance of the simulation was confirmed by checking whether the observed data can be restored as a result of the simulation range. A summary of such a workflow is shown in Fig.1.

The remainder of this paper consists of 5 sections. Chapter 2 confirmed the characteristics of naval failure data. The curve fitting methods (Simple linear and polynomial regression, B-spline, Phase-type distribution) used to estimate the yearly failure rate and the daily failure rate are reviewed. Section 3 describes the curve fitting process and compares the results. By combining the yearly failure trend and the failure ratio within one year, the daily failure ratio of the total life cycle was derived. Chapter 4 checks whether observed data is restored through Monte-Carlo simulation. and lastly, conclusions are presented in section 5. Note that failure in this paper refers to a critical failure (rare event) that makes a mission impossible.

## 2. BACKGROUND

This study analyzed the data of 191 failures that occurred between 2012 and 2019 in 66 ROK Naval ships. This was rearranged by age of the ship and the location of the failure was confirmed in units of years and days. The phase-type distribution (PH) fitting, which is not widely known compared to the linear and polynomial regression and B-spline used for data fitting, was checked in detail. PH is a probability distribution of a finite continuous-time absorption Markov chain, and a desired distribution can be fitted with only one parameter.

### 2.1 Naval Ship Failure Data

Data are 191 failures that occurred in 66 ships between 2012 and 2019. Since this is a fatal event that leads to mission failure, all are reported. That is, there are no additional failures that are missing or unrecorded. In the data, the date of failure and the name of the ship are recorded. 66 ships had at least 1 to 9 failures during the period. Moon and Choi (2021) stated that the failure rate of ROK Naval ships varies according to the age of the ship, and the number of failures increases with age. Therefore, information on age was added through the data of the year of construction of the ship. For example, a 2012 failure of a ship built in 2012 corresponds to the first year of life. The 2012 failure of a ship built in 2010 corresponds to the third age. As such, the 66 ships had different construction years, so the data were sorted by age. Figure 2 is failure data according to age. No failure occurred at age 8, 12, or 21 years. It is not that there were no ships corresponding to the ages of 8, 12, and 21 during the period from 2012 to 2019. As in the study of Moon and Choi (2021), it was confirmed that failures occur a lot in old ships. y-axis in Figure 2 and total number of ships per year are expressed as scaled values due to ROK Naval security.

On the other hand, ROK Naval ships typically repeat a 4.5 month mission and 1.5 months of preventive maintenance (ROK Navy, 2018). In other words, it can be said that the ship is operated every 6 months. However, due to delay (training, etc.), maintenance does not start exactly after 4.5 months (about 139 days). It was checked how many days after the end of preventive maintenance the failures occurred. As a results, failures were distributed from 2 days to 191 days. In this study, a year was assumed to be 360 days. In other words, it was assumed that failure beyond 180 days was not a normal ship operation situation. One data corresponding to day 191 was removed. The daily distribution of 190 final data is shown in Figure 3. Figure 3 has 5 major peaks. This can be explained through a general ship operation pattern. Where there is a peak, it can be understood as a section where a ship performs its mission at sea. After preventive maintenance, ships tend to perform about three or four drills or missions until the next preventive maintenance. ① ~ ④ are peaks that can occur during drills or missions. ⑤ is special. Normal ship service is 4.5 months (139 days). ⑤ is a peak that appears near 160 days. Even for ships that have reached the maintenance cycle, there are cases where they are not delivered on time due to various reasons. In this case, the ship tends to perform additional missions rather than waiting for maintenance. ⑤ is the peak that appears in this case. The size of the peak suggests that the risk of failure is greater if the ship with delayed maintenance performs its mission.

Figure 2 and Figure 3 represents critical failures differently. Figure 2 shows the amount of failure according to age by year. Figure 3 shows the amount of failure occurring in one ship operation cycle (180 days). Figure 2 is the trend of failure rate through curve fitting according to age. However, Figure 2 is discrete data for years, so it should be considered discrete even after curve fitting. It cannot be said that monthly or daily is interpolated. For interpolates less than yearly, the fitting results in Figure 3 should be used. Figure 3 shows the interpolate effect by fitting the histogram to a continuous probability distribution. That is, the daily failure rate can be estimated. Note that, as mentioned in section 1, since the number of data is very small, it was judged that it is meaningless to graph the failure rate on a daily basis. The bin of the histogram is a value that can represent the mission pattern of the ROK Naval ships and was determined by reflecting the opinions of experts. Since a year is assumed to be 360 ​​days, it is assumed that the pattern of Figure 3 occurs twice in the same way. If the two patterns in Figure 3 are fitted with probability distributions, it becomes the failure probability for each day that can occur during the year (See section 3.2 for details).

### 2.2 Curve and Probability Distribution Fitting

There are several methods for fitting Figure 2, 3. ROK Navy selects preventive maintenance intervals based on MTBF (Mean Time Between Failure) (Choi et al., 2020, 2021). 1/MTBF is the average probability of failure. In Fig.2 and 3, the average failure probability for each unit time can be obtained. However, since the average failure rate is expressed as a single constant, it is not suitable for fitting the failure trend in Figure 2 or the probability distribution in Figure 3. Linear and polynomial regression has the advantage of being able to derive a trend simply (Faraway, 2004). However, linear regression has a limitation in that it cannot express the curves in Figure 2, 3. Faraway (2004) and McElreath (2000) stated that the degree of curvature can be controlled through the knot of B-spline curve fitting, so that the desired shape of the curve can be fitted. Moon and Choi (2021) set the B-spline knot suitable for the overhaul cycle of the ROK Naval ship and applied it to the hierarchical Bayesian model. B-spline fitting has the advantage of being able to apply domain knowledge about the curve that the analyst has by using a knot.

In this study, phase type distribution (PH) was applied in addition to the above four well-known fitting methods (average of failure rate, linear and polynomial regression, B-spline). PH is the probability distribution of a finite continuous-time absorption Markov chain, and is an improved form of the erlang distribution (Neuts, 1981;Bhat, 1990). PH can approximate a continuous probability distribution according to the setting of the phase (Faddy, 1995) and enables analytical tractability (Osogami and Harchol-Balter, 2006). According to these characteristics, PH has been used as an approximation to the general distribution in quantitative reliability and performance evaluation (Okamura et al., 2011). In the past, it was widely used in research on queues, but recently it is also used in finance, insurance, and health fields (Asmussen et al., 1996;Asmussen et al., 2019). Asmussen et al. (1996) proposed a computational approach using EM algorithms, overcoming the limitation that the amount of computation dramatically increases when the phase increases.

Since PH is a probability distribution, it is inappropriate for fitting in Figure 2. On the other hand, the fitting in Figure 3 can be applied more simply than the above popular 4 fitting methods. This is because, in the computational approach, the parameter of PH is only one, and it is possible to fit the distribution intuitively without mathematical knowledge. Therefore, the fitting in Figure 3 included PH and compared with other fitting methods.

## 3. FITTING AND RESULTS

In this section, the curve and distribution fitting process of Figure 2 and 3 are described. Fit using the 4 popular methods mentioned in Section 2.2 and PH, and compare the RMSE (Root mean square error) to select the best fitting method for performance. Combining the results of the two fittings, the daily unit failure probability for 31 years of the total life cycle is estimated.

### 3.1 Failure Trend Fitting of Total Life Cycle (Figure 2 Fitting)

Figure 2 indicates the number of failures per year. There are no records of failures at 8, 12, or 21 years, but these are not missing values. This is because critical failures are reported and recorded whenever they occur. Therefore, the number of failures in years 8, 12, and 21 was corrected to 0. Failure data by age were compared by fitting four methods (mean of failure rate, linear and polynomial regression, B-spline). In order to evaluate the failure trend fitting results, train and test sets were divided. Generally, train : test = 8 : 2. From a total of 31 data, 6 data were randomly sampled and used as a test set, and the remaining 25 data were used as a train set. Since the number of data is small, the evaluation result may vary depending on which test set is selected. Therefore, the average accuracy (RMSE) was used as a model evaluation index by repeating all processes from train and test classification to fitting multiple times (1,000 times).

Four models were compared.

• ⦁ Mod. 1-1. Average critical failures model

• ⦁ Mod. 1-2. Linear regression model (intercept: -3.98, slope: 0.62)

• ⦁ Mod. 1-3. Polynomial regression model (dimension: 5)

• ⦁ Mod. 1-4. Cubic B-spline model (knot ages: 6, 11, 16, 21, 26)

Mod.1-1 is simply calculated as ‘sum of 25 train set failure rate / number of training set data (25)’. Mod.1-2 and Mod.1-3 used statsmodel, a library of programming language python. In the case of Mod.1-2, the variance was large depending on the test set, but on average, -3.98 (intercept) and 0.62 (slope) were suitable. Polynomial can be overfitted to train set in higher dimension. The average dimension when the RMSE was the smallest was calculated while increasing the dimension from 2 to 10 for each random sampled test set. The optimal dimension of Mod.1-3 was 5. The location of the B-spline knot was determined according to the maintenance, increase, and decrease patterns of failures by age. Visually, it was confirmed that failures were maintained, increased, or decreased starting at 6, 11, 16, 21, and 26 years. Therefore, knot ages of Mod.1-4 were determined to be 6, 11, 16, 21, and 26

Figure 4 shows the results of 4 curve fittings. The train and test set data in Figure 4 is a representation of one of the random sample sets. Table 1 compares the RMSE of the models. Since Mod.1-1 is a constant, it has a constant value. RMSE is the largest as it does not reflect a change in trend. Mod.1-2 has a lower RMSE than Mod.1-1 because it reflects relatively trend changes as a slope. The optimal dimension of polynomial regression was degree 5. At this time, Mod.1-3 reflects the descent and increase of data. Mod.1-4 also reflects the small amplitude of the data according to the position of the knot. As a result, Mod.1-4 had the lowest RMSE. Therefore, failure trend fitting of total life cycle was selected as Mod.1-4.

### 3.2 Failure Ratio Fitting between Preventive Maintenance (Figure 3 Fitting)

Figure 3 shows how many days the failure occurred after the completion of preventive maintenance regardless of the age of the ships where failures occurred. Although it is possible to estimate the failure rate yearly through Figure 4, there is a limit to estimating the failure probability on a daily basis. The models in Figure 4 are continuous, and it is possible to estimate the amount of failure on a daily basis through interpolation, but it is not possible to grasp the detailed pattern that appears for each day. Figure 3 was fitted with a probability distribution as shown in Figure 5 to interpolate the daily failure probability.

As in section 3.1, there are 4 models, and the optimal model was determined through RMSE.

• ⦁ Mod. 2-1. Average critical failures model

• ⦁ Mod. 2-2. Polynomial regression model (dimension: 8)

• ⦁ Mod. 2-3. Cubic B-spline model (knot ages: 25, 50, 70, 90, 130, 160)

• ⦁ Mod. 2-4. Phase-type distribution model (phase: 80)

The model construction method of from Mod.2-1 to Mod.2-3 is as in section 3.1. However, the polynomial dimension was increased to 8, and 6 knots of Mod.2-3 were selected in consideration of the 5 peaks of the data. The programming language Julia's package EMPht provides the computational solution of PH. This was reported by Asmussen et al. (1996) implemented a computational approach using EM algorithms. Therefore, the desired probability distribution can be obtained by adjusting only one parameter called phase through EMPht package. In PH, phase can be understood as the degree of curvature of the probability distribution. If the probability distribution has severe curvature, set the phase to be large. Phase 80 was suitable for expressing 5 peaks.

The polynomial regression in Figure 5 has a dimension of 8, but there is a limit to expressing all the peaks. B-spline also set 6 knots, but there was a limit in peak expression. Since the peak is related to the operation of the ship, it must be fitted. However, it is judged that the peaks of ④ and ⑤ are not relevant even if they are expressed as one. As in section 2.1, the normal maintenance cycle for a Naval ship is 139 days, which is the peak location in ④. This is because the failure ratio may increase if a ship that has passed the maintenance period performs additional missions. Through the RMSE comparison in Table 2, the failure ratio fitting between preventive maintenance was selected as Mod.2-4.

### 3.3 Critical Failure Function of Total Life Cycle

As in section 2.2, PH is a probability distribution. Mod.2-4 is PH unscaled for comparison between models. And Figure 5 is the fitting result for 0.5 years (180 days), so the same distribution should be repeated once more for 1 year. Figure 4 shows the discrete failure trend on an annual basis. Figure 5 was repeated twice to divide the failure amount in one year probabilistically. Figure 6 shows the probability distribution PH concatenated twice. Between 180 days and 181 days is unnatural, and there is a limit that the probability temporarily becomes 0. On the 180th day, the planned maintenance is finished. If a failure occurs on the date of completion of planned maintenance, it is not recorded as a failure and is judged as incomplete maintenance. Therefore, the probability of failure on the 0th day and the 180th day becomes 0. However, additional research is needed on the possibility of failure if not incomplete maintenance. PH in Figure 6 is a continuous probability distribution. However, since the time unit of the rare event simulation was set to daily, the failure ratio was indicated by dots for visual convenience.

In the probability distribution of 0.5 years, failures occur more frequently during the initial mission, and failures decrease again in the middle. As the preventive maintenance period arrives, failures increase again. The purple dot in Figure 6 is the approximate average of each peak. Each peak is the mission period of the Naval ship. The purple line connects the dots. The change in failure rate that occurs during a mission is similar to the bathtub shape. In past studies, it is known that the failure rate shows a bathtub shape during the total life cycle. The failure function estimation result of Moon and Choi (2021) also showed a distorted bathtub shape. Although the time unit is changed from the total life to the preventive maintenance cycle, additional research is needed on the bathtub shape.

Selected Mod.1-4 and Mod.2-4 should be combined. As in section 2.1, the failure rate of Mod.1-4 can be divided according to the probability of Mod.2-4. For example, the estimated failure rate at year 1 is 3.94. The failures in year 1 can be divided so that the sum of the probability distributions in Mod.2-4 is 3.94. The 50-day failure probability of Mod.2-4 is 0.00425. At this time, the expected failure rate at 50 days in year 1 is 0.0169 (3.94 * 0.0043). However, Mod.1-4 is the failure rate when a rare event occurs. In the case of the first year, a total of 4 failures in 4 ships were fitted. The proportion of 4 ships out of all ships corresponding to the first year should be reflected. The total number of ships in Year 1 is OO. That is, the expected failure rate at 50 days in the first year is 0.0169 / OO. In this way, the failure rate of a total of 11,160 days (31 * 360) in the 31st year is recalculated as shown in Figure 7. The total number of ships per year is not included in this study because it is a security matter of military data.

Figure 7 has two meanings. First, the expected failure rate for each day that can occur during the total life cycle of the ship was calculated. The expected failure rate for each ship was calculated using the failure probability that changes according to the mission cycle of the ship, the failure rate that occurred during the total life cycle, and the total number of ships by age. Second, the data for the 8th, 12th, and 21st years were interpolated in the process of calculating the failure rate by age.

### 3.4 Rare Event simulation (Restoring Failure Data)

In the past rare event simulation, the Monte-Carlo method was mainly used. For Monte-Carlo simulation, it is necessary to know the probability of failure and the distribution in which this failure can occur. Bayesian inference has been widely used for distribution estimation. According to the random number extracted from the estimated distribution and the failure probability, the number of failure occurrences can be simulated. In this study, Figure 7 can be applied as the failure probability of Monte-Carlo simulation. However, since the Bayesian approach is not used, the space of the distribution where the failure occurs cannot be determined. Therefore, the space of Uniform (0, 1) is assumed as the probability occurrence distribution. If the random number generated in Uniform (0, 1) is less than the failure probability, it can be said that a failure has occurred. For example, the 200th day failure probability in Figure 7 is 0.000586. If the random number is smaller than this value, it can be considered that a failure occurred on the 200th day.

Whether the failure probability in Figure 7 is properly derived can be confirmed through simulation. Data is the failure rate by age. When the simulation is repeated as many as the number of Naval ships for each age, the failure ratio can be verified through whether the failure rate of each age generated in the simulation matches the failure rate of the data. Figure 8 is the simulation result.

The green lines in Figure 8 are the failures derived when the simulation is repeated as many as the number of Naval ships per year. The red line is the average value of the simulation results for each year, and the blue line is the amount of data failure. It can be confirmed that the failure rate of the data is included in the range of failures that occur through simulation. Therefore, it can be confirmed that the failure ratio of this study appropriately estimates the critical failure ratio of the Naval ship.

## 4. CONCLUSION AND FUTURE STUDY

Naval ships sometimes return to their home port due to failure during mission. This is a catastrophic failure that corresponds to a mission failure. The fatality is high, but the probability of occurrence is very low, so the failure of the Naval ship is a rare event. In past studies, efforts have been made to estimate and simulate rare event probability. These studies required mathematical knowledge, and the computational cost for simulation was expensive. A rare event is used in a wide field, such as equipment failure or human behavior. Therefore, rare events should be easily analyzed.

Therefore, in this study, rare events were analyzed using only easy methodologies. Failure data of ROK Naval ships are very few and are time series data. The time series of the data was reconstructed. Yearly failure trend curve was fitted, and the distribution of failure probability throughout the year was fitted. Among the comparative models, the B-spline model (failure trend) and the phase type distribution model (failure ratio), which had the lowest RMSE, were estimated. By combining the two models, the probability of failure on a daily basis was derived for the total life of the ship. In general, the change in failure probability according to age is called a failure function. In this study, a failure function for a rare event, Naval ship critical failure, was derived. It was verified that the failure ratio was properly derived by confirming that the observed data was within the possible range through Monte-Carlo simulation.

This study is meaningful in that it performed rare event simulation more easily than previous studies through the combination of trend and distribution fitting. In the process of fitting and combining, it is interpolated for the part without data, so it can be used as a simulation of fake data generation for future research. It can be said that it is an empirical study in that it used the actual failure data of the ROK Naval ship. In addition, there is a difference in that a phase-type distribution that has not been used for rare events is applied.

Further research on the following is needed. First, the random number generating distribution of Monte-Carlo simulation is assumed to be uniform in this study. If sufficient data are secured in the future, it is likely that the range of occurrence of failure probability can be estimated through Bayesian inference as in previous studies. Second, an unnatural curve was estimated while reconstructing the failure ratio distribution generated every 0.5 year by one year, and there is a part with zero probability. It is necessary to estimate the failure ratio by separating the time series units differently. Finally, it is necessary to study whether the distribution of the failure ratio generated every 0.5 year can be said to follow the bathtub shape. It is generally known that failures follow the bathtub throughout its life cycle. An in-depth study is needed on the occurrence of bathtub shape between repeated preventive maintenance cycles.

## Figure Workflow. Critical failures occurring over the total life cycle. Critical failures occurring between preventive Results of fitting critical failures of the total life cycle. Results of fitting critical failures between preventive maintenance Failure ratio in year Expected number of failures per day in the total life cycle of one ship Results of rare event simulation.

## Table

RMSE of models (1-1 ~ 1-4)

RMSE of models (2-1 ∿ 2-4)

## REFERENCES

1. Asmussen, S. , Laub, P. J. , and Yang, H. (2019), Phase-type models in life insurance: Fitting and valuation of equity-linked benefits, Risks, 7(1), 17. 2. Asmussen, S. , Nerman, O. , and Olsson, M. (1996), Fitting phase-type distributions via the EM algorithm, Scandinavian Journal of Statistics, 23(4), 419-441.
3. Bhat, U. N. (1990), Structured stochastic matrices of M/G/1 type and their applications, Journal of the American Statistical Association, 85(412), 1176-1178. 4. Choi, J. -W. , Moon, H. -J. , and Jo, W. -Y. (2020), Preventive maintenance interval optimization based on lifecycle failure prediction, Korean Journal of Logistics, 28(6), 57-70. 5. Choi, J. -W. , Moon, S. -A. , and Jo, W. -Y. (2021), A study on the probability-based planned maintenance effectiveness of naval combat ships, Journal of Korea Institute of Industrial Engineers, 47(2), 190-198. 6. Faddy, M. J. (1995), Phase-type distributions for failure times, Mathematical and Computer Modelling, 22(10-12), 63-70. 7. Faraway, J. J. (2004), Linear Models with R, Chapman and Hall/CRC Press, New York.
8. Gajewski, B. J. , Simon, S. D. , and Carlson, S. E. (2008), Predicting accrual in clinical trials with Bayesian posterior predictive distributions, Statistics in medicine, 27(13), 2328-2340.  9. Jiang, Y. , Simon, S. , Mayo, M. S. , and Gajewski, B. J. (2015), Modeling and validating Bayesian accrual models on clinical data and simulations using adaptive priors, Statistics in medicine, 34(4), 613-629.   10. Kruschke, J. (2014), Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan, Elsevier, London. 11. McElreath, R. (2020), Statistical Rethinking: A Bayesian Course with Examples in R and Stan, CRC Press, Florida. 12. Moon, H. and Choi, J. (2021), Hierarchical spline for time series prediction: An application to naval ship engine failure rate, Applied AI Letters, 2(1), e22. 13. Neuts, M. F. and Meier, K. S. (1981), On the use of phase type distributions in reliability modelling of systems with two components, Operations-Research-Spektrum, 2(4), 227-234. 14. Okamura, H. , Dohi, T. , and Trivedi, K. S. (2011), A refined EM algorithm for PH distributions, Performance Evaluation, 68(10), 938-954. 15. Osogami, T. and Harchol-Balter, M. (2006), Closed form solutions for mapping general distributions to quasi-minimal PH distributions, Performance Evaluation, 63(6), 524-552. 16. Rao, V. , Maulik, R. , Constantinescu, E. , and Anitescu, M. (2020), A machine-learning-based importance sampling method to compute rare event probabilities, In International Conference on Computational Science, Springer, Cham, 169-182.  17. ROK Navy (2018), Ship's Maintenance System, 9-10, Republic of Korea.
18. Rubino, G. and Tuffin, B. (2009), Rare Event Simulation using Monte Carlo Methods, John Wiley & Sons, Chichester. 19. Straub, D. , Papaioannou, I. , and Betz, W. (2016), Bayesian analysis of rare events, Journal of Computational Physics, 314, 538-556. 20. Taylor, S. J. and Letham, B. (2018), Forecasting at scale, The American Statistician, 72(1), 37-45. 21. Tinga, T. , Tiddens, W. W. , Amoiralis, F. , and Politis, M. (2017), Predictive maintenance of maritime systems: models and challenges, In European Safety and Reliability Conference (ESREL). 22. Wagner, F. , Papaioannou, I. , and Ullmann, E. (2021), The ensemble Kalman filter for rare event estimation, 10(1), SIAM/ASA Journal on Uncertainty Quantification, 23. Wahal, S. and Biros, G. (2019), BIMC: The Bayesian Inverse Monte Carlo method for goal-oriented uncertainty quantification, Part I, Do not open for a day Close