# The concept of diagonal approximated signature: new surrogate modeling approach for continuous-state systems in the context of resilience optimization

*Dis Prev Res*2023;2:4.

## Abstract

The increasing size and complexity of modern systems presents engineers with the inevitable challenge of developing more efficient yet comprehensive computational tools that enable sound analyses and ensure stable system operation. The previously introduced resilience framework for complex and sub-structured systems provides a solid foundation for comprehensive stakeholder decision-making, taking into account limited resources. In their work, a survival function approach based on the concept of survival signature models the reliability of system components and subsystems. However, it is limited to a binary component and system state consideration. This limitation needs to be overcome to ensure comprehensive resilience analyses of real world systems. An extension is needed that guarantees both maintaining the existing advantages of the original resilience framework, yet enables continuous performance consideration. This work introduces the continuous-state survival function and concept of the Diagonal Approximated Signature (DAS) as a corresponding surrogate model. The proposed concept is based on combinatorial decomposition adapted from the concept of survival signature. This allows for the advantageous property of separating topological and probabilistic information. Potentially high-dimensional coherent structure functions are the foundation. A stochastic process models the time-dependent degradation of the continuous-state components. The proposed approach enables direct computation of the continuous-state survival function by means of an explicit formula and a stored DAS, avoiding costly online Monte Carlos Simulation (MCS) and overcoming the limitation of a binary component and system state consideration during resilience optimization for sub-structured systems. A proof of concept is provided for multi-dimensional systems and an arbitrary infrastructure system.

## Keywords

*,*continuous-state system

*,*survival function

*,*coherent structure function

*,*resilience optimization

*,*system reliability

*,*monte carlo simulation

## INTRODUCTION

Engineering systems, such as infrastructure networks and complex machines, are ubiquitous worldwide and form the backbone of modern societies. As societies grow, these systems become increasingly sophisticated in size and complexity. Evidently, the stable operation of such systems is crucial for the economy and an undisturbed and safe everyday life of civilians. This challenge is exacerbated by exposure to an increasingly inhospitable, changing and uncertain environment. It is evident that it is exceedingly difficult if not impossible to identify and prevent all potential adverse impacts. The focus in design and maintenance of complex systems has to be extended from a pure failure prevention and failure persistence strategy to the capabilities of adaptation and recovery. The concept of resilience meets exactly these needs both from a technical and economic point of view and ensures steady functioning ^{[1–3]}. Consequently, there is an increasing need for sophisticated and efficient computational tools that adapt this perspective in order to exploit the potential emerging benefits in engineering practice.

A fundamental precondition for the assessment of resilience of complex systems is an appropriate quantitative resilience metric. In ^{[4–6]}, the authors present a broad review of current resilience metrics. In ^{[7]}, Linkov and Trump provided a critical analysis of resilience definitions and metrics found in literature, their practical application and specifically compare them to the concept of the traditional notion of risk. Hosseini *et al.* presented in ^{[5]} a categorization scheme for resilience quantification approaches. Among these, performance-based resilience metrics are the most common and are based on comparing the performance of a system before and after an adverse event. Theoretically, such an adverse event could correspond to rare shock events on a large time scale or persistent degrading effects on an infinitesimally small time scale. Further subcategories distinguish between time in-/dependence and characterization as deterministic or probabilistic. As motivated in ^{[5]} and ^{[8]}, it is assumed that a performance-based and time-dependent metric is capable of considering the following system states before and after a disruptive event:

● The initial state that remains unchanged until the occurrence of an effectively disruptive event, characterized by system reliability, that is interpreted as the ability of the system to sustain typical performance prior to a disruptive event ^{[5, 9]}.

● The disrupted state, determined by the system robustness, i.e., the ability of the system to mitigate an effectively disruptive event and its counterpart, vulnerability, represented by a potential loss of performance after the occurrence of a disruptive event ^{[10, 11]}.

● The recoverability of the system characterizes the duration of the degraded state and the recovery to a new stable state ^{[8, 10]}.

Figure 1 illustrates these system states and their transitions simplified for a single effectively disruptive event and its potentially infinitesimal small period. Note that the terminologies concerning the governing properties, phases and states presented here, although in their physical interpretation perceived alike or at least similarly, are discussed in literature partly controversially. Thus, for example, what is described here, and, e.g., in ^{[11]}, as system robustness is referred to as resistance of a system, as in ^{[12]}. In fact, the boundaries between the interpretations of reliability and robustness are fluid when extending the conventional perspective as shall be seen in the further course of this work. For the developments subsequently proposed, it is critical to define a concise interpretation of reliability from a probabilistic perspective. In accordance with ^{[13]}, let reliability refer to the probability of a system or some entity under consideration to uninterruptedly perform a certain specified function during a stated interval of a life variable, e.g., time, within a certain specified environment.

In the field of engineering, resilience as a concept has consistently gained popularity in recent years ^{[4, 14]}. There are numerous ways to improve the resilience of systems. However, there are limits to available resources, and resilience cannot be increased indefinitely. Therefore, it is important not only to be able to differentiate and balance between different resilience-enhancing measures, but also to take into account their monetary aspects ^{[15, 16]}. In ^{[17]}, Salomon *et al.* present a method for determining the most cost-efficient allocation of resilience-enhancing investments. Further, current research related to resilience focuses on improved metrics for quantifying resilience, such as those proposed in ^{[18]}, and overarching frameworks for stakeholder decision-making, such as for transportation networks in the presence of seismic hazards ^{[19]}. Other recent studies have examined the complexity of real-world infrastructure systems, the consequences of failures, recovery sequences, and various externalities. For instance, in ^{[20]}, the authors demonstrated the tremendous complexity of modern critical infrastructures and their multifactorial nature as cyber-human-physical systems, and explored appropriate modeling and resilience analysis techniques. Moreover, the studies ^{[21]} and ^{[22]} address the implications for decision-making considering stakeholder priorities and enhancement or recovery strategies. Climate change challenges have been explored in the context of resilience, e.g., in ^{[23]}. A comprehensive literature review of resilience assessment frameworks balancing both resources and performance can be found in ^{[24]}.

Salomon *et al.* recently introduced in ^{[25]} an efficient resilience framework for large, complex and sub-structured systems, providing a solid foundation for comprehensive stakeholder decision-making, taking into account limited resources. In their work, a survival function approach based on the concept of survival signature, first introduced in ^{[26]}, models the reliability of system components and subsystems of investigated systems. This reliability approach separates information on the topological (sub)system reliability and the component failure time behavior. Thereby, the survival signature captures the topological information in an efficient manner ^{[27]} and thus, can be seen as a type of surrogate modeling technique. This allows for significantly reduced computational effort when it comes to repeated model evaluations, as the demanding evaluation of the topological system model is circumvented ^{[28]}. This is all the more relevant the larger and more complex the system under consideration is. The repeated model evaluations are of crucial importance when the parameters examined during the resilience optimization affect the probability structure of the system components. This results in a high number of changes in the probability structure during the resilience analysis, which can be ideally covered by the separation property of the survival signature with minimal computational effort.

A major restriction of the survival signature in its original form is the limitation to a binary component and system state consideration. Consequently, the resilience framework for complex and sub-structured systems in ^{[25]} is subject to the same constraints during resilience optimization. However, for a comprehensive resilience analysis of real world systems, a continuous component and system performance state consideration is an indispensable prerequisite. Therefore, an extension is needed that guarantees both the already existing advantages of the resilience framework in ^{[25]} based on the original form of the survival signature, yet enables continuous performance consideration.

The most widespread reliability assessment methods follow a binary-state consideration, i.e., reducing the consideration of system performance to the set of the two states of either perfect functioning or complete failure, compare ^{[29]}. Jain *et al.* states that the "Majority of the existing models have computed system reliability at a holistic level but fail to consider the interactions at component and sub-system levels [...]." In ^{[30]}, Yang & Xue highlight the importance of a continuous-state consideration in reliability analysis. It is evident that the consideration of continuous component and (sub)system states is equally important for resilience analysis and thus indispensable for realistic resilience optimization. In the last years several researchers proposed various concepts that bring the survival signature to a multi-state consideration, e.g., see ^{[31–34]}, which can be seen as a first step in development, towards continuous consideration and potential implementation into proposed resilience framework for sub-structured complex systems ^{[25]}.

In the current work, theoretical fundamentals are first summarized. Then the concept of the DAS is introduced as a new surrogate modeling approach, based on the concept of survival signature and potentially high-dimensional coherent structure functions describing the relationship between degrading components and corresponding continuous-state system performance. The proposed approach enables direct computation of continuous-state survival function by means of an explicit formula and a stored DAS, thus avoiding a costly online MCS and overcoming the limitation of a binary component and system state consideration. A proof of concept is provided for multi-dimensional systems consisting of min- and max-operators, where exact results are obtained. Further, the applicability of the concept is investigated for an arbitrary infrastructure system. Finally, a conclusions and outlook are presented.

## THEORETICAL FUNDAMENTALS

### Structure function

According to ^{[35]}, the performance of a system depends only on its components, i.e., their states, and their interactions. Then, a vector

#### Binary-state structure function

The structure function of a system is a fundamental concept to represent system topology in reliability analysis. For a binary-state system the structure function can be defined as follows. Let a system consist of

as proposed, e.g., in ^{[26]} Accordingly,

Let a system consist of components of different types, i.e.,

#### Multi-state structure function

Analogously, the structure function can be defined for a discrete multi-state consideration. Then, the system and component states degrade from a perfect state over a set of intermediate states to the state of complete failure:

compare ^{[36]}.

#### Continuous-state structure function

When following a continuous multi-state consideration, the set of possible system and component states are all elements of the interval between ^{[37]}:

### Coherent system

A special case of the general system is the class of coherent systems. Note that binary-state, discrete multi-state, as well as continuous multi-state structure functions can be coherent. In accordance with Hudson & Kapur ^{[38]}, this class can be defined as follows. A (discrete or continuous multi-state) system is defined to be coherent if the three subsequent conditions are fulfilled:

●

●

● The set

### Concept of binary-state survival signature

The concept of the survival signature is a promising approach for a more efficient evaluation of system reliability, especially when it comes to repeated model evaluations. Introduced in ^{[26]}, this concept enables to compute the survival function of a system. The approach attracted increasing attention over the last decade due to its advantageous features compared to traditional methods ^{[27]}. One of its benefits is the efficiency in repeated model evaluations due to a separation of the probability structure of system components and the topological system reliability. In addition, the survival signature significantly condenses information on the topological reliability for systems with multiple component types. Components are of the same type if their failure times are independent and identically distributed (^{[39]}. For more information on claimed exchangeability in practice, see ^{[28, 39]}. In the following the derivation of the concept of survival signature is shown for a binary-state system with a single component type and multiple component types, respectively, based on Coolen *et al.*^{[26]}. More detailed information about further applications and the derivation of the concept can be found in ^{[26, 39, 40]}.

Consider a coherent system with a given structure function. Given a binary-state vector specifying the state of

where ^{[41]}.

The probability structure of system components specifies the probability that a certain number of components of type

The topological reliability described by Eqn. (4) and the probability structure characterizing the component failure times can be brought together to obtain the survival function as

where ^{[27]}.

The survival function ^{[42, 43]}. It is typically interpreted as the mathematical formalization of the definition of reliability provided in section "INTRODUCTION" and quantifies the system failure time to be greater or equal to

It is also possible to define the concept of survival signature for ^{[26]} for more details.

### Concept of continuous-state survival signature

The original concept of survival signature achieves considerable efficiency advantages when computing system reliability but is limited to a binary-state consideration. However, a multi-state or even continuous-state consideration might be beneficial for the assessment of most real-world systems in terms of safety and cost efficiency. In the last years several researchers proposed various concepts that bring the survival signature to a multi-state consideration, see ^{[31, 33, 34]}.

In ^{[32]}, Liu *et al.* introduced an approach for the concept of survival signature in the context of continuous-state systems, for which the component functionality is characterized by a stress-strength relation. The strength of the components are assumed to be ^{[31]} in a combinatorial manner but directly based on the number of path sets. Analogously, the survival signature for continuous multi-state systems is given as

with

that is the number of components in state

where ^{[32]}.

Despite an extension to discrete and continuous multi-state consideration, the authors limited their considerations in ^{[32]} to a time-independent reliability analyses. Thereby,

## PROPOSED METHODOLOGY

In this section, the continuous-state survival function is defined. In contrast to the previously outlined approaches that are either probability measures of state

### Continuous-state survival function

In this work, the probability

Thereby, the continuous-state survival function constitutes a time-dependent probability measure that characterizes the distribution of performance states of the considered entity over time. From another perspective, the continuous-state survival function can be interpreted as

where

In the context of systems that consist of components facing disruptive events, the entity under consideration may correspond to either a system or one of its components. Thereby,

Consider a system with a coherent and time-invariant structure function

Figure 2. Examples for

where

In fact, the identification of

where

### Surrogate model: the concept of diagonal approximated signature

The concept of the DAS is introduced as a surrogate modeling approach that enables the computation of the true continuous-state survival function or at least an approximation of it depending on the characteristics of

#### Fundamental statement

With regard to the current developments, several categorizations for three properties of a coherent system structure function are introduced. As first property, the diagonal state sign can be defined: A coherent structure function is referred to as diagonally state positive if it holds that *et al.* also investigated ^{[32]}. The min- and max-operators can be interpreted as analogy of series and parallel operators known from the binary-state consideration, as stated in ^{[30]}. The binary operators often appear in reliability block diagrams.

Assume a coherent structure function to be diagonally state neutral or at least positive and at least diagonally state constant. Then, the basic concept of the DAS can be stated as

where

#### Derivation of the fundamental statement

The derivation of Equation(14) can be given as follows. Let

where

Secondly, the set-theoretical decomposition

The claim that the coherent structure function is diagonally state neutral or positive and at least diagonally state constant implies that

The expression proposed in Equation (14) involving the time-dependent state probability distribution results from Equation (17) when considering two reformulations: At first, note the simplification

Then, Equation (16) and Equation (18) are brought together to finally obtain the expression presented in Equation (14):

Note that the topological information captured beforehand in

#### Basic algorithm for evaluating the DAS

At a first attempt, the approximation of

Algorithm 1 Evaluation of | |

function evaluateDiagonalApproximatedSignature( | |

▷ fixed point evaluation | |

ifthen | |

ifthen | |

return s | |

end if | |

end if | |

▷ start iteration | |

| ▷ initialize auxiliary iteration variables |

| ▷ initialize iteration counter |

| |

whiledo | |

ifthen | |

| |

else ifthen | |

| |

| |

end if | |

| |

| |

end while | |

return | |

end function | |

▷ auxiliary function | |

function DetermineStateVector | |

return | |

end function |

Considering Algorithm 1,

#### Extended statements

Equation (14) and Equation (19), respectively, as well as the Algorithm 1 form the basis for all further developments of the concept of DAS. However, the established expression still appears to be computationally expensive, as the sum over all permutations becomes increasingly demanding for systems comprising a large number of components. Therefore, a naive approach is introduced based on counting the occurrences of equal values of

For most systems, the application of Equation (20) will lead to a tremendous reduction of computational cost since typically

For systems with high

where

## CASE STUDIES

In this section, various system models are established that are designed for a proof of concept and a test of applicability of the developed approaches. Subsequently, the numerical results are presented.

### System structure functions

Here, the structure functions are presented that will be studied to achieve a proof of concept and test the applicability of the approach. Note that the structure functions model the system topology, i.e., the functional interaction of components with each other.

#### Proof of concept: Min- and Max-Systems

The min- and the max-operator are crucial in the context of continuous-state system reliability as these correspond to the fundamental series- and parallel-operator well-known from the binary-state consideration of system functionality. Typically, they appear in the context of reliability block diagrams. Several systems composed by these operators are established in order to proof the fundamental methodologies proposed in section "PROPOSED METHODOLOGY". The following coherent structure functions composed by min- and max-operators are considered:

● 2-Component-Min-System

The system is composed by two continuous-state components with

This structure function is diagonally state neutral, consequently, also diagonally state invariant, and diagonally state extreme. A graphical representation is given in Figure 3.

● 2-Component-Max-System

The system is composed by two continuous-state components with

This structure function is diagonally state neutral, consequently, also diagonally state invariant, and diagonally state extreme. A graphical representation is given in Figure 4.

● 8-Component-MinMax-System

The system is composed by eight continuous-state components with

This structure function is diagonally state neutral, consequently, also diagonally state invariant, and diagonally state extreme. A graphical representation is given in Figure 5.

● 21-Component-MinMax-System

The system is composed by 21 continuous-state components with

This structure function is diagonally state neutral, consequently, also diagonally state invariant, and diagonally state extreme. A graphical representation is given in Figure 6.

#### Test of applicability: infrastructure system

In today's highly developed world, complex systems such as infrastructure networks and industrial plants are omnipresent and of vital importance to the functioning of modern societies. Consequently, the resilience of these systems is of utmost importance as well. Therefore, in the following, an arbitrarily chosen infrastructure network, represented by a graph, is considered. Figure 7 illustrates the graph of this exemplary system. Hereafter, This system is referred to as 18-Component-Infrastructure-System.

The graph consists of 15 nodes (capital letters, e.g.,

As, e.g., in ^{[44]}, ^{[17]} and ^{[25]}, for the analysis of this infrastructure system it is assumed that it has a performance function defined by the so-called network efficiency. According to Latora and Marchiori ^{[45]}, the network efficiency

with ^{[46]} and ^{[47]}. Furthermore, the authors in ^{[45]} and ^{[48]} proposed the utilization of a normalized network efficiency

### Stochastic modeling of the component degradation process

As fundamental step for computing the continuous-state survival function via a structure function

The stochastic degradation of components is modeled by combining an inverse Gamma process and a Gamma process. These types of processes are widely spread in stochastic degradation modeling ^{[49–51]}. Correspondingly, let *Gamma**Gamma*

see ^{[52]}.

MCS is applied to obtain a true solution estimate. In this case, the

In the case of the DAS, a continuous-state survival function describes the probabilistic characteristics of a component. Accordingly, the continuous-state survival function of a component can be established by solving the integral

where

As exemplary parameters, *InverseGamma*

### Numerical results

In this section all computed results are presented. Convergence studies for the number of samples as well as studies concerning the computation time with respect to the number of samples and the number of states were conducted. Further, contour plots of the continuous-state survival function approximated by the DAS and contour plots depicting the corresponding error are provided. Note that the code utilized to compute the following numerical results was not optimized in terms of computational efficiency for the DAS and included *print* statements for computations based on MCS and DAS. Further, the code was not parallelized and variations in the capacity of working memory were unavoidable during the studies concerning convergence and computation time. Besides the study of computation time in terms of the number of considered states, all plots were generated with this number set to

#### 2-Component-Min-System

At first, consider the results computed for the continuous-state survival function of the 2-Component-Min-System. In Figure 9, the approximation of the continuous-state survival function by means of the concept of DAS is depicted. The contour plot shows

Figure 9. 2-Component-Min-System: DAS condensed approximation of continuous-state survival function and the corresponding error.

This becomes even more evident when considering Figure 10. The convergence study was conducted for sample sizes in the interval

Figure 10. 2-Component-Min-System: Convergence study of MCS true solution estimate vs. DAS approximation of the continuous-state survival function with MAE, MSE and RMSE as error measures in terms of sample size

#### 2-Component-Max-System

Secondly, consider the computed results for the continuous-state survival function of the 2-Component-Max-System. Again, Figure 12 shows the approximation of the continuous-state survival function by means of the DASC while Figure 12b depicts the corresponding error. Considering Figure 12a, the continuous-state survival function indicates higher reliability and robustness of the 2-Component-Max-System compared to the 2-Component-Min-System as expected. Not only is the domain for which

Figure 12. 2-Component-Max-System: DAS condensed approximation of continuous-state survival function and the corresponding error.

Considering Figure 13 it becomes evident that also for this case study

Figure 13. 2-Component-Max-System: Convergence study of MCS true solution estimate vs. DAS approximation of the continuous-state survival function with MAE, MSE and RMSE as error measures in terms of sample size

#### 8-Component-MinMax-System

Again, Figure 15 verifies the expected behavior of the DAS and the DASC. In this example, the sample size

Figure 15. 8-Component-MinMax-System: DAS condensed approximation of continuous-state survival function and the corresponding error.

When considering Figure 16, the behavior of all three error measures appears similar to the previous examples. This is counterintuitive as one would expect an increasing error when sampling in higher dimensions, compare ^{[53]}. But this seems not to hold true for diagonally state invariant structure functions. The ranges of sample sizes for both studies of computation time shown in Figure 17 are the same as in the previous example. Analogously to the previous examples, the computation times of the MCS are characterized by a similar linear relation with respect to both sample size and number of considered states. In contrast, DAS and DASC are constant in their relation with respect to the sample size. In terms of the number of considered state, both DAS and DASC follow linear relations. It is noteworthy, that the factor of the linear relation of the DAS seems significantly larger than before. Also in terms of the sample size the computational time during the online phase significantly increased for the DAS. It can be observed that for

Figure 16. 8-Component-MinMax-System: Convergence study of MCS true solution estimate vs. DAS approximation of the continuous-state survival function with MAE, MSE and RMSE as error measures in terms of sample size

#### 21-Component-MinMax-System

For this case study, the fundamental concept of DAS was omitted due to the combinatorial complexity of

Figure 18. 21-Component-MinMax-System: DAS condensed approximation of continuous-state survival function and the corresponding error.

For the convergence study depicted in Figure 19, the number of samples was increased for the entire range. The evaluated sample sizes lie in the interval

Figure 19. 21-Component-MinMax-System: Convergence study of MCS true solution estimate vs. DAS approximation of the continuous-state survival function with MAE, MSE and RMSE as error measures in terms of sample size

#### 18-Component-Infrastructure-System

For this example, solely the DASC and the rounded DASC (referred to as DASCR) were considered. To compute the underlying DAS for this example the maximum number of iteration steps

The continuous-state survival function obtained by means of the DASC is depicted in Figure 21a. The contour plot appears reasonable. As expected, the DASC achieves an approximation that in the worst case underestimates the true solution but never overestimates it. The theoretical findings can be verified when considering Figure 21b. The contour plot of the error between the MCS true solution estimate and the approximation is positive over the entire domain. Dark blue indicates an error magnitude of zero while dark purple represents magnitudes in the scale of machine precision.

Figure 21. 18-Component-Infrastructure-System: Contiunous-state survival functions computed by means of DASC and the corresponding error.

The applied scheme already yields satisfying results taking into account that it is only a first-order scheme for at least diagonally state neutral structure functions. However, a higher-order implementation could significantly decrease the error in the remaining domain. In general, the proposed methodology is also applicable to diagonally state negative structure functions when adjusting the corresponding formula. For this example, it was ensured that the structure function is at least diagonally state neutral by accordingly specifying the exponential transformation function mapping component degradation to travel time. Figure 22 shows the true solution estimate of the continuous-state survival function obtained by means of MCS. The region of significant magnitudes of the error between the MCS and the DASC occurs as the underlying structure function is no longer diagonally state constant. As the structure function is still at least diagonally state neutral this is the only source for errors besides the natural variance of the stochastic degradation process.

The DASCR was applied to further increase the computational efficiency. In the following, the potential decrease of accuracy is studied. The proposed methodology still ensures pure underestimation of the true solution. For

23. 18-Component-Infrastructure-System: Continuous-state survival functions computed by means of DASCR with

For

Figure 24. 18-Component-Infrastructure-System: Continuous-state survival functions computed by means of DASCR with

## DISCUSSION

### Case studies

The case studies show that the DAS converges to the true solution of the continuous-state survival function for all MinMax-Systems regardless of their dimensionality. The global error vanishes for

The DASC was introduced and investigated as a naive solution to achieve increased computational efficiency also for larger systems that are characterized by an at least diagonally state constant and at least diagonally state neutral coherent structure function. The performance enhancement is achieved by condensing the DAS in terms of all possible permutations depending on

The application of the DASCR is not required for diagonally state extreme structure functions. In contrast, it is particularly useful when this criterion is not fulfilled. In the case of a diagonally state constant or higher order structure function an iteration has to be performed in order to approximate the DAS for each combination of

In the test of applicability for an arbitrary infrastructure system, the first-order DASC and DASCR5, 3, 2 perform well and underestimate the true solution of the continuous-state survival function as expected. Depending on the parameter

In its basic form, the DAS requires

### Comparison with related research

Subsequently, the developed concept of DAS is compared to approaches based on the concept of survival signature with regard to the properties of diagonal state sign, order, and variance and also based on the findings of the case studies. In ^{[31]}, Eryilmaz & Tuncel introduced an explicit formula from a combinatorial perspective to compute a multi-state survival signature based on multiple path-wise binary-state structure functions to model the discrete multi-state perspective. The fundamental decomposition is based on the number of components in ^{[34]} by Qin & Coolen exhibits similar properties to those of the concept developed by Eryilmaz & Tuncel. The authors investigate discrete multi-state systems with multi-state components based on rule-based structure functions. In comparison with ^{[31]}, Qin & Coolen developed a refined notation. The researchers based the combinatorial decomposition on the number of components working in state ^{[33]}, Yi *et al.* proposed a fundamentally different approach on how to establish the discrete multi-state survival signature values. The authors adopt an probabilistic and conditional interpretation of the survival signature and further establish transformation relations ^{[54]}.

Recent developments show that the survival signature finds increased attention in the field of stress-strength reliability. The works ^{[55, 56]} investigate approaches for statistical inference based on the concept of survival signature for multi-state system with multi-state components in this context. In ^{[32]}, Liu *et al.* proposed an approach to compute

In contrast to the approaches presented above, the DAS was developed to evaluate the continuous-state survival function

### Contextualization in terms of resilience

Three different approaches to determine ^{[17]}. To show this theoretically, suppose that the performance deterioration over time being investigated empirically by exposing the entity to a certain environment in which potentially damaging effects or events occur in some frequency. Suppose the measurement only observes the state

In the context of the multidimensional and sub-structured resilience framework established in ^{[25]}, the fact that ^{[25]} after establishing a monotone sampling procedure based on

## CONCLUSIONS & OUTLOOK

In this work, the notion of the continuous-state survival function was presented and the concept of DAS was introduced as a corresponding surrogate modeling procedure. Thereby, the continuous-state survival function is defined as a time-dependent probability measure that characterizes the distribution of performance states of the considered system over time. This consideration gives engineers a new perspective when faced with the challenge of maintaining system performance in the face of disruptive events in a hostile environment. In light of the theoretical proof and the results in the case studies, the concept of DAS appears to be a solid foundation for more sophisticated surrogate modeling techniques. The relations to the phases characterized by reliability and robustness when quantifying system resilience were identified and discussed. The proposed methodology appears as an adequate approach to integrate a continuous-state consideration into a sub-structured resilience framework, as presented in ^{[25]}.

In the course of this work, three different variants of the concept of DAS were established: At first, the fundamental statement Equation (14) was introduced to provide a comprehensive proof that DAS yields exact results for diagonal extremal and constant structure functions. For systems with a small number of components the DAS outperforms the MCS in terms of both computational time and accuracy. Secondly, the DASC Equation (20) was developed to overcome the limitations for larger systems. Moreover, DASCR was defined in Equation (21) to consider structure functions with a diagonal state order higher than constant. Thus, the current methodology extends the range of application of the separation property inherited by the concept of survival signature. It should be noted that the code can be further optimized, e.g., by integrating parallel computing. This leads to an additional increase in computational efficiency. In summary, the concepts of DAS developed in the current work show good results and open a rich and promising research topic.

The following items can be listed as critical developments concerning the concept of DAS as an autonomous surrogate model but also in particular its integration into the resilience framework for complex and sub-structured systems ^{[25]}.

● Integration into the resilience framework: The behavior of the DAS when integrated to the multidimensional and sub-structured resilience decision-making framework should be investigated in detail. The relationship between the endowment properties and the continuous-state survival function should also be explored.

● Broadening the range of application: Higher-order schemes should be addressed to reduce the approximation error for structure functions that are not diagonally state constant. Further, the DAS formulas should be extended for diagonally state negative structure functions and multiple component types.

● Consideration of uncertainties: Extension of the DAS towards a consideration of uncertainties based on the proposed approach in ^{[28]} and integration into the multidimensional resilience decision-making framework for complex and sub-structured systems ^{[25]}. Approaches to reduce the storage requirements and to further condensate the developed formulas for enhanced efficiency during the online phase are of great interest.

## DECLARATIONS

### Authors' contributions

Made substantial contributions to conception and design of the work: Winnewisser NR, Salomon J, Broggi M

Performed data analysis and interpretation: Winnewisser NR

Provided administrative and technical support: Broggi M, Beer M

### Availability of data and materials

Not applicable.

### Financial support and sponsorship

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) SPP 2388 501624329 and the "Reliability and Safety Engineering and Technology for large maritime engineering systems" (RESET) programme 730888.

### Conflicts of interest

All authors declared that there are no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Copyright

© The Authors 2023.

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## Cite This Article

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Winnewisser, N. R.; Salomon J.; Broggi M.; Beer M. The concept of diagonal approximated signature: new surrogate modeling approach for continuous-state systems in the context of resilience optimization. *Dis. Prev. Res.* **2023**, *2*, 4. http://dx.doi.org/10.20517/dpr.2023.03

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