Joint Return Research Articles

Influence of Variations of Hydrothermal Conditions on Normalized Difference Vegetation Index in Typical Temperature Zones Along the East Coast of China

The eastern coastal areas of China span multiple climatic zones, and the impacts of climate warming on their ecological environment show regional differences. This dissertation used the Normalized Difference Vegetation Index(NDVI) as the indicator to characterize the ecological environment, and selected Guangdong, Jiangsu and Liaoning as its typical research areas. In this dissertation, the author selected the NDVI, average temperature and precipitation data of the yearly growth season respectively from 1982 to 2016, and adopted the copula functions model based on Markov Chain Monte Carlo to carry out the research of bivariate joint distribution so as to calculate the joint probability, the joint exceedance probability, the joint return period and the co-occurrence return period. The results showed that: (1) the temperature and precipitation in the three regions were respectively related to the NDVI sequence showing the characteristic that was correlated at the upper tail and asymptotically independent at the lower tail, which demonstrated that the temperature and precipitation had little effect on NDVI when they reached their minimum values, and the temperature and precipitation had obvious effect on NDVI when they reached their maximum values. (2) The shorter the return period was, the wider the ranges of the climate factor and the NDVI were, showing that when the climate factor was constant, the probability of the NDVI having a shorter return period was higher. The greater the climate factor was, the longer the return period was, indicating that the probability of plant growth inhibition was higher when the climate factor exceeded a certain threshold. (3) The suitable temperature and precipitation for vegetation growth in the three regions gradually decreased from south to north. In addition, this research can provide theoretical guidance and scientific foundation for the protection of regional ecological environment and enhance the understanding of the impact of climate change on the ecosystem.

A multi-load joint distribution model to estimate environmental design parameters for floating structures

Floating structure designs require an environmental condition simulation and corresponding design parameter determination. Typically, the design values are defined by individually calculating the N-yr return period; however, this not only results in overestimation, but it also reduces the accuracy of the structural reliability analysis. Therefore, a new environmental design parameter estimation method which is based on the joint distribution of multiple loads and conditional joint probability models was developed. A multivariate joint model for the metoceanic parameters of the South China Sea based on 30 yrs of annual extremum hindcast data and conditional probability models with one predominant factor were established; the most likely pair of other variables were also considered because of the complexity and variability of the environmental conditions. Environmental parameters with N-yr joint return period, as well as the structural response surface, corresponding to the most unsafe conditions were determined to derive design parameters. The possible conservatism of the conventional design standard was assessed by applying this joint design criterion to a semi-submersible platform example. The results indicate the design response is much more conservative than those obtained using proposed criteria, neither multivariate nor conditional joint models. The proposed design criteria can significantly reduce construction cost.

Copula based analysis of meteorological drought and catchment resilience across Indian river basins

AbstractDrought analysis is an important part of risk management plan. Drought is usually characterized by variables such as severity and duration. Using standardized precipitation index (SPI) at an aggregated scale of 12 months, we construct different copula models for different river basins of India. Based on goodness of fit tests, suitable distributions are selected for duration and severity. These marginal distributions are then used to construct copula models from amongst‐Frank, Gumbel, Clayton and Student's t copula. It is found that for some river basins Frank copula can capture the dependence structure between duration and severity whereas for others Gumbel copula is effective. Exceedance probability, conditional probability and joint return period of different drought events are calculated which allude to differing drought resilience and persistence conditions in river basins. The river basins in the western India have a longer joint return period and smaller exceedance probability compared to the river basins in south India. We explore the conjunctive use for joint return period and exceedance probability to qualitatively assess the resilience of river basins to meteorological drought. The results suggest that the drought events in the south Indian river basins are less severe and more frequent whereas the ones in Central and Western India are severe and longer. The results of this study can provide useful information for drought mitigation strategies at a national scale.

Parametric Vine Copula Construction for Flood Analysis for Kelantan River Basin in Malaysia

The multivariate approach of flood characteristics such as flood peak flow (P), volume (V), and duration (D) is much beneficial in recognizing the critical behaviour of flood episodes at a river basin scale. The incorporation of 2-dimensional copulas for establishing bivariate flood dependency frequently appears, but it could be more comprehensive if we focus all the three flood characteristic simultaneously. In such circumstances, incorporation of vine or Pair-Copula Construction (PCC) could produce a better approximation of joint probability density and much practical approach in the uncertainty analysis, in comparison with conventional trivariate copula distribution. This study demonstrated the efficacy of parametric vine copula in the modelling of trivariate flood characteristics for the Kelantan River basin in Malaysia. The D-vine tree structure is selected where the Gaussian and Frank copula is recognized for bivariate flood pairs (P-V) and (P-D) pairs in the first stage, using the maximum-pseudo-likelihood (MPL) estimation procedure. Similarly, the Gumbel copula is selected in the modelling of conditioned data obtained through the conditional distribution function of bivariate copulas selected in the previous stage based on the partial differentiation, also called h-function. Finally, the full density function of the 3-dimension structure is derived and compared with the observed flood characteristics. Furthermore, tail dependence properties and behaviour of D-vine copula are also investigated, which reveals for well capturing the general behaviour of Gaussian and Frank copula fitted to flood pair (P-V) and (V-D) and reproduces the overall flood correlation structure fairely well. Both the primary ‘OR’ and ‘AND’ joint return periods for trivariate flood events are estimated which pointing that ‘AND’ joint case produces higher return value than ‘OR’ case.

Change in the heatwave statistical characteristics over China during the climate warming slowdown

Heatwaves are multi-day periods of extremely high-temperature weather relative to a region's average conditions and are among the deadliest natural disasters. This study was designed to evaluate changes in the characteristics of heatwaves during the periods of warming acceleration versus warming slowdown using the multivariate joint probability distributions of heatwave duration and severity. The results revealed that: (1) From the warming acceleration period (1961–1997) to the warming slowdown period (1998–2017), the return levels of heatwave duration increased by <1.0 d as the return period increased. As the return period lengthened, the return level of heatwave severity rose as well; (2) The joint occurrence probability of heatwaves with both variables exceeding the 20-yr, 50-yr, and 100-yr return levels exhibited relative increases of 3%, 8%, and 16% across China, respectively. There were greater than the joint probability of heatwaves with only 1 variable exceeding the return levels; (3) The number of heatwaves with both duration and severity exceeding the 50-yr joint return period increased in China, and this increase was dominated by heatwaves with joint return periods of 50–100 years, leading to a strengthening of heatwave severity.

Drought Risk Analysis in the Eastern Cape Province of South Africa: The Copula Lens

This research study was carried out to investigate the characteristics of drought based on the joint distribution of two dependent variables, the duration and severity, in the Eastern Cape Province, South Africa. The drought variables were computed from the Standardized Precipitation Index for 6- and 12-month accumulation period (hereafter SPI-6 and SPI-12) time series calculated from the monthly rainfall data spanning the last five decades. In this context, the characteristics of climatological drought duration and severity were based on multivariate copula analysis. Five copula functions (from the Archimedean and Elliptical families) were selected and fitted to the drought duration and severity series in order to assess the dependency measure of the two variables. In addition, Joe and Gaussian copula functions were considered and fitted to the drought duration and severity to assess the joint return periods for the dual and cooperative cases. The results indicate that the dependency measure of drought duration and severity are best described by Tawn copula families. The dependence structure results suggest that the study area exhibited low probability of drought duration and high probability of drought severity. Furthermore, the multivariate return period for the dual case is found to be always longer across all the selected univariate return periods. Based on multivariate analysis, the study area (particularly Buffalo City, OR Tambo and Alfred Zoo regions) is determined to have higher/lower risks in terms of the conjunctive/cooperative multivariate drought risk (copula) probability index. The results of the present study could contribute towards policy and decision making through e.g., formulation of the forward-looking contingent plans for sustainable management of water resources and the consequent applications in the preparedness for and adaptation to the drought risks in the water-linked sectors of the economy.

Analysis and Application of Drought Characteristics Based on Theory of Runs and Copulas in Yunnan, Southwest China.

Drought is a complex natural disaster phenomenon. It is of great significance to analyze the occurrence and development of drought events for drought prevention. In this study, two drought characteristic variables (the drought duration and severity) were extracted by using the Theory of Runs based on four drought indexes (i.e., the percentage of precipitation anomaly, the standardized precipitation index, the standardized precipitation evapotranspiration index and the improved comprehensive meteorological drought index). The joint distribution model of drought characteristic variables was built based on four types of Archimedean copulas. The joint cumulative probability and the joint return period of drought events were analyzed and the relationship between the drought characteristics and the actual crop drought reduction area was also studied. The results showed that: (1) The area of the slight drought and the extreme drought were both the zonal increasing distribution from northeast to southwest in Yunnan Province from 1960 to 2015. The area of the high frequency middle drought was mainly distributed in Huize and Zhanyi in Northeast Yunnan, Kunming in Central Yunnan and some areas of Southwest Yunnan, whereas the severe drought was mainly occurred in Deqin, Gongshan and Zhongdian in Northwest Yunnan; (2) The drought duration and severity were fitted the Weibull and Gamma distribution, respectively and the Frank copula function was the optimal joint distribution function. The Drought events were mostly short duration and high severity, long duration and low severity and short duration and low severity. The joint cumulative probability and joint return period were increased with the increase of drought duration and severity; (3) The error range between the theoretical return period and the actual was 0.1–0.4 a. The year of the agricultural disaster can be accurately reflected by the combined return period in Yunnan Province. The research can provide guidelines for the agricultural management in the drought area.

Dry and wet combination dynamics and their possible driving forces in a changing environment

Previous studies on dry and wet combinations had focused on their evolution characteristics in a single season (i.e. spring or summer), while the dynamics and driving forces between the adjacent seasons remain unsolved, which is important for water resources planning and management in a changing environment. To this end, the dry and wet events in adjacent seasons in the Yellow River Basin (YRB) were evaluated based on the standardized precipitation index (SPI), and the joint return periods of various combinations of dry and wet conditions under two scenarios (moderate and severe) were calculated using a copula function. The Mann-Kendall test was used to explore the joint return period dynamics with a 31-year sliding window. The Copula-based likelihood-ratio method and cross-wavelet transform were adopted to explore the driving forces of dry and wet combination dynamics. The results indicate that (1) spring-summer and summer-autumn are prone to continuous dry (wet) events, while the probabilities of the combinations of dry and wet conditions occurring in autumn–winter and winter-spring are almost the same; (2) continuous wet events frequently occur above Longyang gorge and below Huayuankou, while the Inner flow area is prone to continuous dry events; (3) the risk of dry and wet combinations with high frequency decreases, whereas the risk of compound events with low frequency increases; (4) the dependency structure dynamics of the SPI series between adjacent seasons show the strongest correlation with the El Niño-Southern Oscillation (ENSO), followed by the Arctic Oscillation and sunspot activities, implying that the ENSO has a dominant control on the dry and wet combination dynamics in the YRB. This study brings new insights into dry and wet combination dynamics and driving forces in a changing environment and provides a scientific basis for the sustainable utilization of water resources in the YRB.

A Joint Return Policy for a Multi-Item Perishable Inventory Model with Deterministic Demands, Return and All-Units Discount

In this paper, we develop a multi-item perishable inventory model with deterministic demands, return and all-units discount. We consider a situation where a retailer sells several products to the customer and orders the products from one supplier. Demands are assumed to be deterministic following an inventory-dependent demand, and the supplier offers all-units discount to the retailer who has an opportunity to return unsold or deteriorated products to the supplier at a certain cost. In order to minimize the total cost for the retailer, the decision variables are the optimal return time and the optimal ordering quantity. Considering a multi-item case as an extension of the model by Setiawan et al. (2018) is the main contribution of this paper. We also develop an algorithm to find the optimal solution of the model. Numerical examples for three items are given to illustrate the model and a sensitivity analysis is performed to study the effect of the changes in parameter values on the optimal solution. We consider two scenarios, one with all-units discount and one with no discount. Within these two scenarios, we consider conditions of individual or joint return time for these three items. It is found that the individual return time with no discount gives the least total inventory cost in the numerical examples. Also, in general increasing the value of holding cost, deteriorating rate, return cost per unit and backorder cost will increase the total inventory cost in all scenarios.

Wind-ice Joint Probability Distribution Analysis based on Copula Function

Taking the wind speed and ice thickness field measurement data in Southwest China during November 2016 to March 2018 as analytical sample, the probability distribution of the wind speed and ice intensity were analyzed and fitted by using Gumbel distribution model, Weibull distribution model and Generalized extreme value distribution (GEV) model, respectively. The goodness of fitting results had been compared. After the determination of marginal distribution functions for wind speed and ice intensity, 500-year pseudo wind speed and ice intensity samples were generated based on the Monto Carlo method. Five Copula functions were employed in the construction of joint probability distribution function and the most suitable Copula function was chosen for building the wind-ice joint probability distribution model. On the basis of the model, the return periods for 30 years, 50 years and 100 years had been calculated, respectively. The analysis results show that the GEV model well matches the characteristics of wind speed and ice intensity. As for binary Copula function, the Frank Copula function has the best goodness for the joint probability distribution of wind speed and ice intensity. The return period calculations show that the design value of wind-ice joint return period (JRP) is larger than that of co-occurrence return period (CRP) and the value of Kendall return period (KRP) is between that of JRP and CRP. Considering that the dangerous areas of return periods are different, the return period for design should be determined by the combination analysis of different kinds of the return periods. Meanwhile, the importance and the failure results of the project should be taken into account.

Risk Analysis of Marine Environmental Elements Based on Kendall Return Period

Against the background of global warming and rising sea levels, the threat of typhoon disasters to marine engineering structures has become increasingly serious. Therefore, the research on the design standards of marine environmental elements has become an important topic. In this study, two-dimensional joint distributions of wave height and surge height, surge height and wind speed, and wave height and wind speed were constructed based on the Gumbel–Hougaard (G-H) Copula function according to the data of marine environmental elements under extreme sea conditions from Naozhou observation station in the sea waters of western Guangdong, and the Kendall return period is introduced. The joint return periods, co-occurrence return periods, and Kendall return periods of joint distributions were calculated, along with the latter’s corresponding design values of environmental elements. The results showed that the Kendall return periods and the corresponding design values are between two kinds of traditional return periods. After analysis, the conclusion is that the Kendall return period can reflect the occurrence regularity of marine hydrological events more accurately, and the design values of marine environmental elements calculated under this standard can reasonably lower the investment under the premise of ensuring structural safety. Therefore, the Kendall return period can serve as the new selection for marine engineering design and risk management.

Predicting Future Flood Frequency Under Climate Change Using Copula Function

AbstractThe joint behaviour of flood variables under climate change is of high importance for the economics of projects and risk reduction. This study investigates the implications of climate change using Gumbel–Hougaard copula function for future bivariate of flood peak and volume variables, in Azarshahr chay watershed. Canadian Earth system model (CanESM2) under three Representative Concentration Pathways (RCPs) along with statistical downscaling method (SDSM) and soil and water assessment tool (SWAT) were adopted to assess both baseline (1976–2005) and future (2030–2059) periods. Bivariate analysis of Copula improved the accuracy of model with an average NSE of 0.97 for all scenarios. Joint return period for severe floods has declined in the future, especially in RCP8.5. For a constant discharge and volume, joint return periods at the base period, RCP2.6, RCP4.5 and RCP8.5 were 24, 10, 13 and 9 years, respectively. Multivariate analysis may also provide useful information for flood risk assessment.

Climate change impact assessment on mild and extreme drought events using copulas over Ankara, Turkey

Climate change, one of the major environmental challenges facing mankind, has caused intermittent droughts in many regions resulting in reduced water resources. This study investigated the impact of climate change on the characteristics (occurrence, duration, and severity) of meteorological drought across Ankara, Turkey. To this end, the observed monthly rainfall series from five meteorology stations scattered across Ankara Province as well as dynamically downscaled outputs of three global climate models that run under RCP 4.5 and RCP 8.5 scenarios was used to attain the well-known SPI series during the reference period of 1986–2018 and the future period of 2018–2050, respectively. Analyzing drought features in two time periods generally indicated the higher probability of occurrence of drought in the future period. The results showed that the duration of mild droughts may increase, and extreme droughts will occur with longer durations and larger severities. Moreover, joint return period analysis through different copula functions revealed that the return period of mild droughts will remain the same in the near future, while it declines by 12% over extreme droughts in the near future.

Deal Protection Devices

In mergers and acquisitions transactions, a buyer and a seller will often agree to contractual mechanisms (deal protection devices) to deter third parties from jumping the deal and to compensate a disappointed buyer. This paper analyzes various deal protection devices, with a focus on two most commonly used mechanisms: target termination fees and match rights. A match right gives the buyer a right to “match” a third party’s offer so as to prevent the third party from snatching the target away, while a termination fee compensates the buyer when a third party acquires the target. Such mechanisms raise a number of important corporate and contract law questions. How effective are they in preventing third parties from competing for the target? Do they steer the target to be sold to the buyer who values the target less? Are the devices harmful to the target shareholders? To what extent can the negotiated deal price represent the target’s “fair value” when such devices reduce or eliminate the competition? The paper attempts to answer these questions with the help of auction theory. The paper shows, foremost, that these devices can actually increase the target and the buyer’s joint return and possibly the target’s stand-alone return. Termination fees and match rights function quite differently, however. While a larger termination fee reduces the target’s stand-alone return and can lead to allocative inefficiency, unlimited match right increases the target’s stand-alone return and promotes allocative efficiency. The paper argues that answering the corporate law questions ultimately turns on the question of how and why the target directors are utilizing the devices. If the devices are being deployed with the objective of maximizing the target shareholders’ return, not only can they be beneficial for the target shareholders, but their presence can also make the deal price a more reliable indicator of target’s fair value. With an improper objective, not only do the devices undermine target shareholders’ return, but the court also should not use the deal price as evidence of fair value. The paper also analyzes stock and asset lockups and examines deal protection devices through the lens of contract law.

Oil rig protection against wind and wave in Lebanon

The oil rigs that will be installed in the Lebanese offshore are floating rigs. This type can be affected by environmental risks such as extreme wind speed and wave height. We study these risks in Beddawi region in the northern Lebanon during the winter season by applying the extreme value theory to real data. We estimate at a first stage the return levels associated to return periods of 50, 100, and 500 years for each risk separately using the classical univariate theory. At a second stage, we use the logistic parametric model and the bivariate copula functions in multivariate extreme value theory to estimate the dependence between extreme wind speed and wave height as well as joint exceedance probabilities and joint return levels to take into consideration the risk of these two environmental factors simultaneously.

Projected changes of bivariate flood quantiles and estimation uncertainty based on multi-model ensembles over China

As atmospheric moisture capacity is highly sensitive to rising temperatures, precipitation extremes are widely projected to intensify with a warming climate and thus altering the flooding generation regime. Previous works seldomly focused on bivariate flood quantiles under climate change at a national scale, and fewer flooding projections quantified the estimation uncertainty sourced from sample size limitation. This study systematically investigates the changes in bivariate quantiles of flood peak and volume with incorporation of sampling uncertainty for 151 catchments over China, with climate trajectories projected by a set of multi-model ensemble under representative concentration pathway (RCP) 8.5. After correcting the systematical biases of eight CMIP5 GCM outputs, four state-of-the-art hydrological models are driven and validated for each catchment, and the best-simulation model is selected to project future streamflow scenarios. The copula function is employed to construct the joint distribution of flood peak and volume, and then the most likely realizations of bivariate quantiles are derived under different Joint Return Periods (JRPs), with the uncertainty envelope quantified with the area of 90% confidence ellipse by a copula-based parametric bootstrapping uncertainty (C-PBU) approach. Our results project an overall ascending trend of temperature and precipitation over China, and the bivariate flood quantiles and corresponding estimation uncertainty of most catchments in the future period (2056–2100) are much larger than the baseline (1961–2005), despite accompanied by substantial climate model uncertainty and spatial variability in magnitude. Many basins would be subjected to a dramatic increase of flood magnitude by over 50%, while only few basins are projected to experience a decreasing flood risk, suggesting an urgent need to increase societal resilience to a warming climate over China.

Measuring compound flood potential from river discharge and storm surge extremes at the global scale

Abstract. The interaction between physical drivers from oceanographic, hydrological, and meteorological processes in coastal areas can result in compound flooding. Compound flood events, like Cyclone Idai and Hurricane Harvey, have revealed the devastating consequences of the co-occurrence of coastal and river floods. A number of studies have recently investigated the likelihood of compound flooding at the continental scale based on simulated variables of flood drivers, such as storm surge, precipitation, and river discharges. At the global scale, this has only been performed based on observations, thereby excluding a large extent of the global coastline. The purpose of this study is to fill this gap and identify regions with a high compound flooding potential from river discharge and storm surge extremes in river mouths globally. To do so, we use daily time series of river discharge and storm surge from state-of-the-art global models driven with consistent meteorological forcing from reanalysis datasets. We measure the compound flood potential by analysing both variables with respect to their timing, joint statistical dependence, and joint return period. Our analysis indicates many regions that deviate from statistical independence and could not be identified in previous global studies based on observations alone, such as Madagascar, northern Morocco, Vietnam, and Taiwan. We report possible causal mechanisms for the observed spatial patterns based on existing literature. Finally, we provide preliminary insights on the implications of the bivariate dependence behaviour on the flood hazard characterisation using Madagascar as a case study. Our global and local analyses show that the dependence structure between flood drivers can be complex and can significantly impact the joint probability of discharge and storm surge extremes. These emphasise the need to refine global flood risk assessments and emergency planning to account for these potential interactions.

An Integrated Framework for Extreme Drought Assessments Using the Natural Drought Index, Copula and Gi* Statistic

This study proposes a framework to evaluate extreme drought using the natural drought index, copula and hot spot analysis. The study area was South Korea. Data were used from 59 automatic synoptic observing system stations; the variable infiltration capacity model was used for the period from 1981 to 2016. The natural drought index was constructed from precipitation, runoff and soil moisture data, which reflect the water cycle. The average interval, duration and severity of extreme drought events were determined following Runs theory. The most extreme drought period occurred in 2014–2016, with 46 of the 59 weather stations exhibiting drought conditions and 78% exhibiting extreme drought conditions. The Inje and Seosan stations exhibited the longest drought duration of 6 months, the most severe drought was 5 times higher than the extreme drought severity threshold. The periods and locations of extreme drought were identified using the natural drought index corresponded to the historical droughts in South Korea. Furthermore, the joint return period of 50 years indicated the northeastern area of South Korea suffered from long-lasting and severe drought. Meanwhile, hot spot analysis was used to explore the extreme drought conditions and showed an increasing trend in the middle and northeastern parts of South Korea. Overall, this framework provides water resource managers with essential information about extreme events, extreme joint return periods, locations and significant trends of natural drought that have not been considered before.

Observed and projected trends of extreme precipitation and maximum temperature during 1992–2100 in Isfahan province, Iran using REMO model and copula theory

AbstractMeteorological extreme events have a major impact on water resources, economic development, and ecosystem health. In this study, maximum precipitation and maximum temperature indices were derived for Isfahan province, in central Iran, over the historical (1992–2017) and future (2020–2100) periods. Precipitation and maximum temperature data from the REMO model under RCP4.5 scenario were used to investigate changes in extreme values over the future period. The results showed that extreme precipitation in the historical and future periods has respectively a decreasing and increasing trend. Based on the extreme indices, temperature in the study area has a significant increasing trend in the baseline and future period. Various combinations of extreme precipitation indicators were created for joint modeling by copula theory. Copula modeling for the three weather stations for which REMO had satisfactory performance in simulating extremes over the historic period showed that the average return period of extreme precipitation combinations will be reduced in the future period compared to the historical period at Daran and Shahreza, while the average return period of combinations will have both increasing and decreasing trends at Naeen.Recommendations for Resource Managers Knowing information about the probability of occurrence of extreme precipitation with a certain value that exceeds a certain threshold will help planning for water resource systems under drought conditions and future increasing temperature. The joint return period of extreme precipitation can help to know the return period of extreme events such as floods and droughts. The findings of this study are important to assess the prediction of climate extreme. Also, these results can be useful to provide the appropriate strategies for water resources managers in drought conditions under future increasing temperature.

A factorial Bayesian copula framework for partitioning uncertainties in multivariate risk inference

In this study, a factorial Bayesian copula (FBC) method is proposed to quantify parameter uncertainties in copula-based models and then reveal their impacts on hydrologic risk inferences within a multivariate context. In detail, Bayesian inference and factorial analysis are integrated into copula-based multivariate risk models to (1) quantify parameter uncertainties, (ii) reveal their individual and interactive effects, and (iii) identify their detailed contributions to uncertain risk inferences. Streamflow observations at Xiangxi and Wei River basins is China are used to illustrate the applicability of FBC. The results indicate that imprecise parameters in marginal distributions and the dependence structure would lead to extensive uncertainties in predictive joint return periods and failure probabilities. Also, individual and interactive effects of parameters are well revealed through multilevel factorial analysis, and the detailed contributions of one parameter to different failure probabilities under different service time scenarios are identified.