A common strategy for the modeling of stochastic loads in time-dependent reliability analysis is to describe the loads as independent Gaussian stochastic processes. This assumption does not hold for many engineering applications. This paper proposes a Vine-autoregressive-moving average (Vine-ARMA) load model for time-dependent reliability analysis, in problems with a vector of correlated non-Gaussian stochastic loads. The marginal stochastic processes are modeled as univariate ARMA models. The correlations among different univariate ARMA models are captured using the Vine copula. The ARMA model maintains the correlation over time. The Vine copula represents not only the correlation among different ARMA models but also the tail dependence of different ARMA models. Therefore, the developed Vine-ARMA model can flexibly model a vector of high-dimensional correlated non-Gaussian stochastic processes with the consideration of tail dependence. Due to the complicated structure of the Vine-ARMA model, new challenges are introduced in time-dependent reliability analysis. In order to overcome these challenges, the Vine-ARMA model is integrated with a single-loop Kriging (SILK) surrogate modeling method. A hydrokinetic turbine blade subjected to a vector of correlated river flow loads is used to demonstrate the effectiveness of the proposed method.
In problems with stochastic load histories and time-dependent factor, such as material degradation, wear, and corrosion, the reliability is also time-dependent [1–3]. Time-dependent reliability has been investigated intensively during the past decades using upcrossing rate methods [4–7], sampling-based methods [8–11], and surrogate modeling methods [12,13]. A common strategy in current time-dependent reliability analysis methods is to assume the stochastic loads as Gaussian stochastic processes  with precisely known mean, variance, and correlation functions. Based on the representation of Gaussian stochastic processes, the non-Gaussian stochastic processes are simulated based on probability transformations . The most commonly used methods for the simulation of Gaussian stochastic processes include Karhunen–Loève (KL) expansion , orthogonal series expansion (OSE) , and polynomial chaos expansion .
Along with the above methods, ARMA modeling, which is the focus of this paper, has also been used in the time-dependent reliability analysis to describe the loading time histories based on the available experimental or history data. For instance, Singh and Mourelatos described the roughness of the road as an ARMA model for the time-dependent reliability analysis of automotive vehicles ; Hu et al. [17,18] used a Bayesian ARMA model to represent the stochastic load history in the time-dependent reliability analysis in the presence of parametric uncertainty; Ling et al. [19,20] described the fatigue load history as an ARMA and autoregressive-integrated moving average (ARIMA) model to perform fatigue life prognosis; and Wang and Billinton applied the ARMA model to the reliability analysis of an electric power system under the stochastic wind speed .
The above methods and applications of ARMA models mainly focus on a single Gaussian stochastic process or multiple independent Gaussian stochastic processes. In practical situations, however, it is quite possible that the stochastic processes are correlated and non-Gaussian. For example, the ocean waves acting on a jack-up platform are correlated at different locations of the platform , and the wind pressures on a building are correlated at different locations and directions . A possible way of describing the high-dimensional correlated Gaussian stochastic processes is to use the multivariate ARMA models [24,25]. The basic principle of the multivariate ARMA model is to represent the noise terms as a multivariate normal distribution at each time instant . The correlations of the stochastic processes over time are captured by the marginal ARMA model, and the correlations among different ARMA models are described by the multivariate normal distribution. The employment of the multivariate normal distribution in multivariate ARMA models, however, has two main limitations. First, the multivariate normal distribution is applicable only for noise terms with Gaussian marginal distribution. Second, the multivariate normal distribution cannot capture the tail dependence among different ARMA models . Tail dependence refers to the phenomenon that two correlated random variables also have dependence between their extreme values. For example, there is a high probability that the rain takes its extreme value when the wind takes its extreme value during a hurricane. Accounting for the tail dependence is important in reliability analysis, because failures are usually caused by extreme events.
This paper investigates the integration of Vine copula concepts with the ARMA models to develop a Vine-ARMA model to overcome the above-mentioned limitations. Vine is a decomposition strategy, which decomposes a high-dimensional copula into bivariate copulas . A reason Vine is very useful is that current multivariate copula is available only for Gaussian copula or Gaussian-related copula. When a Gaussian copula is not applicable, other copula families (such as Clayton, Gumbel, and Frank) are well studied only for bivariate cases. The decomposition using Vine enables us to flexibly model a high-dimensional copula based on currently available bivariate non-Gaussian copulas. An advantage of using the non-Gaussian bivariate copula to model the high-dimensional copula is that the tail dependence among different variables can be captured using the non-Gaussian copula. Since the first construction  and formal definition  of Vines in the 1990s, they have been intensively studied in many areas . Recently, Jiang et al.  applied Vine copula to reliability analysis by integrating Vine copula with MCS and FORM. In this work, the Vine copula is integrated with ARMA models to model high-dimensional correlated non-Gaussian load processes.
The developed Vine-ARMA model is then applied to time-dependent reliability analysis. Since current time-dependent reliability analysis methods are mainly developed based on the Gaussian and independent assumptions of stochastic loads, the introduction of the Vine-ARMA model into time-dependent reliability analysis raises new challenges. During the past years, a group of time-dependent reliability analysis methods have been developed to improve the accuracy of reliability analysis. These methods, however, cannot be directly applied to problems with Vine-ARMA models. How to accurately perform time-dependent reliability analysis is the first challenge. A surrogate modeling approach might be a promising way for problems with loads described as Vine-ARMA models. Current surrogate model-based time-dependent reliability analysis methods, however, mainly focus on problems without stochastic processes [12,13]. How to integrate the Vine-ARMA model with the surrogate model-based methods to efficiently perform time-dependent reliability analysis is the second challenge. In order to overcome these challenges, the Vine-ARMA model is combined with a recently developed SILK surrogate modeling method . The combination of the Vine-ARMA model and the SILK method enables us to accurately and efficiently evaluate the time-dependent reliability. The main contributions of this paper are, therefore, summarized as: (1) the application of Vine copula in time-dependent reliability analysis, (2) the integration of Vine copula and ARMA techniques for the modeling of high-dimensional correlated non-Gaussian stochastic processes, and (3) the synthesis of Vine-ARMA and SILK for efficient and accurate time-dependent reliability analysis.
The remainder of the paper is organized as follows: Section 2 provides background review of the time-dependent reliability analysis methods and multivariate ARMA modeling. Section 3 develops a Vine-ARMA model for the modeling of high-dimensional non-Gaussian stochastic loads. Section 4 develops a method for time-dependent reliability analysis by using the Vine-ARMA model and SILK method. A turbine blade example is used to demonstrate the proposed method in Sec. 5, and concluding remarks are given in Sec. 6.
In practical applications, the stochastic processes may be correlated. Since the focus of this paper is to use time series models to represent stochastic processes, next, we briefly review the multivariate ARMA model which can be applied to the modeling of multiple correlated stochastic processes.
Multivariate Autoregressive-Moving Average Model.
Data-driven time series models have been widely used in many areas for modeling of stochastic processes and performing predictions based on modeling. The commonly used time series models include the autoregressive (AR) model, moving average (MA) model, and ARMA model. The AR, MA, and ARMA are for stationary stochastic processes. When the stochastic process is nonstationary, the ARIMA model is employed . In this work, the ARMA model is used to illustrate the proposed method. The proposed method, however, can be easily extended to ARIMA models.
Even though multivariate ARMA models can effectively model and simulate the correlated stochastic processes, their application is limited by two main drawbacks: (1) the marginal distribution of the noise term is limited to Gaussian distribution since the Gaussian distribution can significantly facilitate the regression of ARMA models and (2) the multivariate Gaussian distribution cannot capture the tail dependence between different load processes. In Sec. 3, the Vine copula is integrated with ARMA models to overcome the drawbacks of multivariate ARMA models.
Vine-ARMA Model for High-Dimensional Stochastic Load Modeling
This section first introduces the concepts of tail dependence and Vine copula, followed by development of the proposed Vine-ARMA model.
It has been shown that both UTDC and LTDC are zero for the bivariate normal distribution . Since the multivariate normal distribution can be represented as combinations of bivariate normal distributions or a multivariate Gaussian copula, this means that the tail dependence between individual ARMA models cannot be captured in the multivariate ARMA models presented in Sec. 2.2. Tail dependence is important for both time-independent and time-dependent reliability analyses, as failure events are often caused by extreme (rare) events. The non-Gaussian copulas are able to capture the tail dependence between two random variables. But current non-Gaussian copulas are well studied only for bivariate distributions. In order to represent the correlations over time and between individual ARMA models and at the same time capture their tail dependences, a promising way is to integrate Vine copula with ARMA models. Note that Levy process has also been used to capture the anomalous tail behaviors and correlation structure of a stochastic process [38,39]. However, the focus of this paper is to capture the tail dependence between different stochastic processes. The Vine-ARMA model presented in this paper can be extended to capture the tail dependence between different Levy processes by developing a Vine-Levy model.
During the past decades, numerous methods have been developed to model the extreme value distributions and perform load combination of extreme value distributions [40–42]. Most of these methods, however, are based on certain assumptions and simplifications. In this paper, the Vine copula is employed to develop a more rigorous way of modeling the extreme values of loads. As discussed in the above sections, using a multivariate Gaussian distribution cannot capture the tail dependence between load processes. In this situation, the high-dimensional copula can facilitate the correlation representation and tail dependence modeling of high-dimensional variables, as the marginal distribution of the correlated random variables is independent of the copula function. However, most of the copula families, which can capture the tail dependence, are well studied only for bivariate cases.
We use a four-variable problem below to illustrate the differences among different Vines. Figure 1 shows the D-Vine, C-Vine, and a regular Vine decomposition of the joint PDF of four random variables.
The previous discussion indicates that a high-dimensional joint distribution can be decomposed into bivariate copulas using Vines. This enables us to use non-Gaussian bivariate copulas to model complicated high-dimensional dependences instead of using multivariate Gaussian distributions. Next, we will develop the proposed integration of Vine and ARMA models.
Vine-ARMA Model for Correlated Load Process Modeling.
The basic principle for the integration of Vine copula with ARMA models is to implement a top-down procedure during the simulation process of the multivariate ARMA model. It is different from the conventional multivariate ARMA models that implement a bottom-up procedure. More specifically, during the conventional simulation of multivariate ARMA models, realizations of the noise terms, , , are generated first. Based on realizations of the noise terms, realizations of the stochastic processes are obtained. However, in the proposed Vine-ARMA model, samples are generated at the stochastic process response level (physical quantity level) first. The generated samples are then transformed to the noise terms to simulate the ARMA models.
Equation (25) indicates that the copula at the stochastic process response level (physical quantity level) is the same as the copula at the noise-term level in the conventional ARMA models. This property significantly facilitates the integration of the Vine and ARMA models.
In order to use the Vine copula to capture the tail dependence among different ARMA models, we build a Vine copula model to capture the dependence at any given time instant and use the univariate ARMA models to maintain the correlation over time. The Vine copula and the univariate ARMA model are connected through the CDF values in the copula function as indicated in Eq. (25). Next, we will first briefly discuss how to build such a Vine-ARMA model based on data. Following that, we will explain the simulation of the Vine-ARMA model.
Construction of the Vine-ARMA Model.
Since the Vine copula is a function of CDF values and is independent from the marginal distribution types of the univariate ARMA models and the noise terms, the Vine copula and the univariate ARMA models can be built independently.
During the construction of the Vine-ARMA model, the experimental data on are first transformed into data on CDF values and data on the associated standard Gaussian stochastic processes based on the marginal distributions of the stochastic processes. The data on the CDF values are then used to build the Vine copula, and the data on the associated Gaussian stochastic processes are used to build the individual ARMA models. The construction of the Vine copula involves the selection of Vine structure and copula types. Once the Vine structure and copula type are determined, parameters of the bivariate copulas need to be estimated. The construction of the Vine copula has been well studied in the literature and is not the focus of this paper; we direct the readers to Refs. [43,46,47] for details. The methods for the construction of univariate ARMA models are also well-documented, such as the Yule-Walker method, Burg method, covariance method, and maximum likelihood estimation [34,48]. In this work, we, therefore, do not give details about how to construct the ARMA models. Note that the Vine tree structure, the copula type, and copula parameters may affect the reliability analysis results. As mentioned above, the selection of Vine copula has been studied in Refs. [43,46,47,49] using either maximum likelihood-based method or Bayesian methods. More details about the selection of Vine copula can be found in Refs. [43,46,47,49]. The Vine-ARMA model and the associated reliability analysis method developed in the subsequent sections are based on the assumption that the Vine copula is already selected based on available data.
Assume that from the construction, we obtain the parameters of the bivariate copula function as (the parameters given in Eqs. (17) and (18)) and the parameters of the univaraite ARMA models as and (the coefficients given in Eq. (5)). Next, we will discuss how to simulate the Vine-ARMA model based on , , and .
Simulation of Vine-ARMA Model.
At a given time instant , there are two widely used methods that can be employed for the simulation of correlated random variables: the Rosenblatt transformation  and the Nataf transformation . The Nataf transformation is basically the multivariate Gaussian copula, which cannot capture the tail dependence. The Rosenblatt transformation is accurate but requires the joint CDF of the correlated random variables. Since the joint CDF and tail dependence of the high-dimensional load processes are accurately described by the Vine copula, the Rosenblatt transformation can be used to simulate the load processes at the given time instant.
Note that the inverse -function (i.e., ) is required to generate the CDF samples. The inverse -function for commonly used Gaussian, Clayton, Student’s , and Gumbel copulas can be found in Ref. . For different Vine structures, the sampling orders of the CDF values are different. Typically, the C-Vine is computationally more efficient than the D-Vine, as the C-Vine has fewer numbers of conditional distribution functions to be computed in the simulation than the D-Vine. In this work, the C-Vine is used in the Vine-ARMA model.
Based on the samples of , samples of and at time instant conditioned on realizations before this time instant are then obtained using Eqs. (21)–(23) and the coefficients of the ARMA models. Table 1 gives a simplified algorithm for the simulation of the Vine-ARMA model based on the C-Vine structure. In the algorithm, the simulation algorithm of the Vine structure is adopted from Ref. . The discussions presented in Secs. 3.3–3.5 show that no Gaussian assumption is made regarding the noise term of the ARMA models. Therefore, the developed Vine-ARMA model is also applicable to multivariate ARMA models with non-Gaussian noise terms. This advantage comes from the distribution-type independence property of the copula function.
Next, we will develop our proposed approach to apply the Vine-ARMA model to time-dependent reliability analysis.
Time-Dependent Reliability Analysis Using the Vine-ARMA Model
The Vine-ARMA model is able to flexibly capture the tail dependence and correlations among different stochastic processes and over time. The complicated Vine copula structure, however, creates new challenges for time-dependent reliability analysis when the Vine-ARMA model is substituted in Eq. (2). In order to overcome the difficulty introduced by the Vine-ARMA model, a SILK surrogate modeling method recently developed by the authors is employed to perform time-dependent reliability analysis . In this section, we first briefly review the SILK method. Based on that, we discuss time-dependent reliability analysis based on the combination of the SILK and Vine-ARMA models.
SILK Method for Time-Dependent Reliability Analysis.
During the past decades, a group of time-dependent reliability analysis methods have been developed, such as the upcrossing rate method [6,52], composite limit-state function method , and extreme value surrogate modeling-based methods [12,13]. Even though the upcrossing rate methods can perform time-dependent reliability analysis for problems with stochastic processes, currently available methods cannot be applied to problems with Vine-ARMA models considering that the complicated Vine copula structure brings significant challenges to the correlation computation of the system response. The surrogate modeling-based method is a potential solution to this. Current extreme value surrogate modeling-based methods, however, have low computational efficiency for problems with stochastic processes. Recently, a SILK surrogate modeling method has been developed by the authors for the reliability analysis of problems with stochastic processes by converting the double-loop procedure in the extreme value surrogate modeling methods into a single-loop procedure. The SILK method has been found to significantly improve the efficiency of time-dependent reliability analysis without sacrificing the accuracy. In this paper, we therefore combine SILK with the Vine-ARMA model to overcome the challenges created by introducing the Vine-ARMA model. A very brief summary of the SILK method is provided here; refer to  for details.
Time-Dependent Reliability Analysis Based on SILK and Vine-ARMA.
We now summarize the general procedure for time-dependent reliability analysis using the SILK method and Vine-ARMA models. The Vine-ARMA model is used to accurately generate samples for the stochastic process, and SILK is used to identify training points from the generated samples to efficiently perform time-dependent reliability analysis. The Vine-ARMA model guarantees the accuracy of extreme value modeling of stochastic load and SILK reduces the computational effort of reliability analysis. Table 2 gives the generalized algorithm for time-dependent reliability analysis using the SILK and Vine-ARMA models.
Overall, the proposed method consists of mainly two parts. The Vine-ARMA model accurately captures the tail dependence between high-dimensional correlated stochastic loads, which ensures the accuracy of time-dependent reliability analysis. The combination of Vine-ARMA model and SILK method makes the time-dependent reliability analysis with Vine-ARMA models possible and efficient. The developed method is able to perform time-dependent reliability analysis accurately and efficiently. In Table 2, 5% is used as the error requirement for the surrogate modeling; this is only for the sake of illustration. The actual value is chosen by the decision-maker based on the risk tolerance for the particular problem. However, when the number is decreased (i.e., higher requirement of accuracy), the reliability estimate will become accurate but the number of training points for the surrogate model will increase. In the Section 5, we use a numerical example with a hydrokinetic turbine blade to illustrate the proposed Vine-ARMA model and the SILK-Vine-ARMA approach for time-dependent reliability analysis.
Numerical Example: A Hydrokinetic Turbine Blade
A hydrokinetic turbine as shown in Fig. 2 is a mechanism to extract energy from river water flow [6,56]. During its operation, the turbine is subjected to stochastic flow loads. At different locations (stations) of the turbine blade, the stochastic river loads are different and correlated. In this section, the time-dependent reliability analysis of a hydrokinetic turbine blade is used to illustrate the proposed Vine-ARMA load model and the combination of SILK and Vine-ARMA methods. This example is modified from Ref. .
A three-dimensional model of the hydrokinetic turbine blade is shown in Fig. 3. The hydrokinetic turbine blade is 1 m long and made of steel. The turbine blade is twisted and has variable chord length along the radial direction. The hydrofoil of the turbine blade is NREL S809. The lift and drag coefficients of the hydrofoil are available in Ref. . Figures 4 and 5 show the chord length and twist angle distribution of the turbine blade along the radial direction. Failure of the turbine blade is defined to occur when the maximum stress on the turbine blade is larger than the strength of the material. Note that rotation of the turbine blade is an important issue for the safety of the hydrokinetic turbine blade, because rotation will change the load conditions of the turbine system. There are also uncertainty sources in the rotation model of the blade. Considering rotation and fluid-structure interactions of the turbine system will make the problem more practical and complicated. Since the turbine blade is just an illustrative example to show the effectiveness of the proposed reliability analysis method and load model, we did not consider the effect of rotation to simplify the problem.
After the discretization, the edgewise force and flapwise force at station are computed using the blade element momentum (BEM) theory  based on the geometry of the turbine blade, local pitch, and the river velocity at the station. More details about the load analysis of the turbine blade are available in Ref. . After the forces at stations of the turbine blade are obtained, they are input into finite element analysis (FEA) to get the stress response of the blade. Figure 6 gives the flowchart of the stress analysis. Figure 7 plots a snapshot of the stress analysis results of the turbine blade. In the stress analysis, nonlinearity is not considered. The maximum stress of the turbine blade is therefore independent from the strength of the material.
The stress response of the turbine blade is governed by the river flow velocity at different stations. In this paper, velocities at different stations are modeled using the Vine-ARMA model to account for the correlation and tail dependence over time and space. In addition, for the sake of illustration, it is assumed that the velocities at every four stations are the same. There are therefore totally 12 correlated stochastic processes (as indicated in Fig. 3(c)) that need to be modeled using the Vine-ARMA approach.
Simulation of River Velocity Using the Vine-ARMA Model.
Assume that the Vine-ARMA model is trained based on experimental data. Since there are 12 stochastic processes, there are totally 66 copulas in the Vine structure (C-Vine is used in this example). Tables 3 and 4 give the families and parameters of the 66 bivariate copulas. All the marginal distributions of the 12 stochastic processes are assumed to be two-parameter Weibull distribution. Table 5 gives the parameters of the 12 Weibull distributions. For the 12 ARMA models of the associated Gaussian stochastic processes, the noise terms are assumed to be zero-mean Gaussian distributions with standard deviations as . Tables 6 and 7 present the AR and MA coefficients of the 12 ARMA models.
Based on the parameters of the Vine copula and ARMA models, the river flow velocities at locations 1 to 12 as indicated in Fig. 3(c) are simulated using the algorithm given in Table 1. Figure 8 gives one realization of the stochastic processes at four of the locations, and Fig. 9 depicts the velocity samples at several locations at a given time instant. It shows that the correlation over time and locations as well as the tail-dependence are maintained in the Vine-ARMA model as can be seen in Fig. 9.
Time-Dependent Reliability Analysis Using SILK-Vine-ARMA Model.
We then perform time-dependent reliability analysis based on the simulated river velocities using the SILK method. In time-dependent reliability analysis, the Poisson ratio and the strength of the material are assumed to be random variables given by and , where is the lognormal distribution with mean and standard deviation . In order to verify the accuracy of the proposed method, a surrogate model is trained first for using the FEA model given in Sec. 5.2 with a large number of training points (i.e., 900). The trained surrogate model is then used to substitute the original model, as directly performing MCS on the original model is computationally prohibitive. Hereinafter, we call it the “high-fidelity surrogate model.” We first compare the developed Vine-ARMA model and the multivariate ARMA model by performing MCS on the high-fidelity surrogate model. The multivariate ARMA model is trained based on the same sample as the Vine-ARMA model. Table 8 gives the results comparison between the Vine-ARMA model and multivariate ARMA (MARMA). It shows that the MARMA model overestimates the probability of failure which is consistent with what is expected because the Clayton copulas which have lower-tail dependences are used in the Vine-ARMA models (as shown in Table 3).
We then perform time-dependent reliability analysis using the proposed Vine-ARMA-SILK method. Table 9 gives the results comparison of time-dependent reliability analysis obtained from the proposed method and MCS based on the high-fidelity surrogate model for the time duration of interest (0 to 150 mins). Note that we can compare the proposed method only with MCS, as MCS is the only currently available method for this kind of reliability analysis problem (as discussed in Sec. 4). The results show that the proposed method is able to perform time-dependent reliability analysis accurately and efficiently.
In practical applications, structures are often subjected to high-dimensional correlated and non-Gaussian stochastic load processes. Current methods for the modeling of such processes mainly rely on multivariate ARMA models. The multivariate ARMA model with Gaussian noise assumption has difficulty in capturing the tail dependence among different stochastic processes. In this paper, the Vine copula is combined with univariate ARMA models to model the correlated stochastic processes and accurately capture their tail dependence. In order to overcome the computational challenges in time-dependent reliability analysis caused by the introduction of Vine-ARMA model, the Vine-ARMA model is integrated with SILK surrogate modeling to efficiently and accurately estimate the time-dependent failure probability. A hydrokinetic turbine blade subjected to correlated river flow loads demonstrated the effectiveness of the proposed method.
Even though the Vine copula is able to flexibly model high-dimensional non-Gaussian copula and capture the tail dependence, the number of conditional distribution functions in the simulation of Vines will increase significantly with the dimension of the problem. For example, there are conditional distribution functions in the simulation of an -variable D-Vine. Simulation of the Vine-ARMA model for a high-dimensional problem is computationally expensive. Improving the efficiency of the Vine-ARMA model simulation needs to be investigated in the future. In addition, the surrogate modeling-based method is used in this paper to substitute the computationally expensive simulation model. However, construction effort and accuracy of the surrogate model may suffer from the curse of dimensionality, as the number of variables and the nonlinearity in the problem increases. How to efficiently construct surrogate models for high-dimensional problems is an active research issue in the literature.
The research reported in this paper was supported by the Air Force Office of Scientific Research (Grant No. FA9550-15-1-0018, Technical Monitor: Dr. David Stargel). The support is gratefully acknowledged.