#### Abstract

Complex risk is a critical factor for both intelligent systems and risk management. In this paper, we consider a special class of risk statistics, named complex risk statistics. Our result provides a new approach for addressing complex risk, especially in deep neural networks. By further developing the properties related to complex risk statistics, we are able to derive dual representations for such risk.

#### 1. Introduction

Research on complex risk is a popular topic in both intelligent systems and theoretical research, and complex risk models have attracted considerable attention, especially in deep neural networks. The quantitative calculation of risk involves two problems: choosing an appropriate complex risk model and allocating complex risk to individual components. This has led to further research on complex risk.

In a seminal paper, Artzner et al. [1, 2] first introduced the class of coherent risk measures. Later, Sun et al. [3] and Sun and Hu [4] focused on set-valued risk measures. However, traditional risk measures may fail to describe the characteristics of complex risk. This concept has promoted the study of complex risk measures. Systemic risk measures were axiomatically introduced by Chen et al. [5]. Other studies of complex risk measures include those of Acharya et al. [6], Armenti et al. [7], Biagini et al. [8], Brunnermeier and Cheridito [9], Feinstein et al. [10], Gauthier et al. [11], Tarashev et al. [12], and the references therein.

From the statistical point of view, the behaviour of a random variable can be characterized by its observations, the samples of the random variable. Heyde et al. [13, 14] first introduced the class of natural risk statistics, the corresponding dual representations are also derived. An alternative proof of the dual representations of the natural risk statistics was also derived by [15]. Later, Tian and Suo [16] obtained dual representations for convex risk statistics, and the corresponding results for quasiconvex risk statistics were obtained by [17]. Deng and Sun [18] focused on the regulator-based risk statistics for portfolios. However, all of these risk statistics are designed to quantify risk of simple component (i.e., a random variable) by its samples. A natural question is determining how to quantify complex risk by its samples.

The main focus of this paper is a new class of risk statistics, named complex risk statistics. In this context, we divide the measurement of complex risk into two steps. Our results illustrate that each complex risk statistic can be decomposed into a clustering function and a simple risk statistic, which provides a new approach for addressing complex risk. By further developing the axioms related to complex risk statistics, we are able to derive their dual representations.

The remainder of this paper is organized as follows. In Section 2, we derive the definitions related to complex risk statistics. Section 3 discusses a new measurement of complex risk statistics. Finally, in Section 4, we consider the dual representations of complex risk statistics.

#### 2. The Definition of Complex Risk Statistics

In this section, we state the definitions related to complex risk statistics. Let be the -dimensional Euclidean space, . For any , , means , . For any positive integer , the element in product Euclidean space is denoted by . For any and , means . From now on, the addition and multiplication are all defined pointwise, , . For any , .

*Definition 1. *A simple risk statistic is a function : that satisfies the following properties:(A1)Monotonicity: for any , implies ;(A2)Convexity: for any and , .

*Remark 1. *The properties A1â€“A2 are very well known and have been studied in detail in the study of risk statistics.

*Definition 2. *A clustering function is a function that satisfies the following properties:(B1)Monotonicity: for any , implies ;(B2)Convexity: for any and , ;(B3)Correlation: for any , there exists a simple risk statistic such that .

*Definition 3. *A complex risk statistic is a function : that satisfies the following properties:(C1)Monotonicity: for any , implies ;(C2)Convexity: for any and , ;(C3)Statistical convexity: for any , , and , if , then .

#### 3. How to Measure Complex Risk

In this section, we derive a new approach to measure complex risk in intelligent systems. To this end, we show that each complex risk statistic can be decomposed into a simple risk statistic and a clustering function . In other words, the measurement of complex risk statistics can be simplified into two steps.

Theorem 1. *A function : is a complex risk statistic in the case of there exists a clustering function and a simple risk statistic : such that is the composition of and , i.e.,*

*Proof. *We first derive the â€śonly ifâ€ś part. We suppose that is a complex risk statistic and define a function byfor any . Since satisfies the convexity , it followsfor any and . Thus, satisfies the convexity . Similarly, the monotonicity of can also be implied by the monotonicity of . Next, we consider a function : that is defined byThus, we immediately know that satisfies the correlation , which means defined above is a clustering function. Next, we illustrate that the defined above is a simple risk statistic. Suppose with , there exists such that , . Then, we have , which means by the monotonicity of . Thus, it follows from the property of thatwhich implies satisfies the monotonicity . Let with , , for any , which implies and . We also consider for any . Thus, from the definition of , there exists a such thatwith . Hence, from the statistic convexity of , we know thatwhich implies the convexity of . Thus, is a simple risk statistic and from (2) and (3), we have . Next, we derive the â€śifâ€ť part. We suppose that is a clustering function and is a simple risk statistic. Furthermore, define . Since and are monotone and convex, it is relatively easy to check that satisfies monotonicity and convexity . We now suppose that which satisfiesfor any . Then, the property of impliesThus, we havewhich indicates satisfies the property . Thus, the defined above is a complex risk statistic.

*Remark 2. *Theorem 1 not only provides a decomposition result for complex risk statistics but also proposes an approach to deal with complex risk, especially in large scale integration. Notably, we first use the clustering function to convert the complex system risk into simple, and then we quantify the simplified risk by the simple risk statistic. Therefore, an engineer who deals with the measurement of complex risk in large-scale integration can construct a reasonable complex risk statistic by choosing an appropriate clustering function and an appropriate simple risk statistic. The clustering function should reflect his preferences regarding the uncertainty of large scale integration.

In the following section, we derive the dual representations of complex risk statistics with the acceptance sets of and .

#### 4. Dual Representations

Before we study the dual representations of complex risk statistics on , the acceptance sets should be defined. Since each complex risk statistic can be decomposed into a clustering function and a simple risk statistic , we need only to define the acceptance sets of and , i.e.,

We will see later on that these acceptance sets can be used to provide complex risk statistics on . The following properties are needed in the subsequent study.

*Definition 4. *Let and be two ordered linear spaces. A set satisfies f-monotonicity if , and implies . A set satisfies b-monotonicity if , , and implies .

Proposition 1. *We suppose that is a complex risk statistic with a clustering function and a simple risk statistic : . The corresponding acceptance sets and are defined by (11) and (12). Then, and are convex sets and they satisfy the f-monotonicity and b-monotonicity.*

*Proof. *It is easy to check the abovementioned properties from definitions of and .

The next proposition provides the primal representation of complex risk statistics on , considering the acceptance sets.

Proposition 2. *We suppose that is a complex risk statistic with a clustering function and a simple risk statistic : . The corresponding acceptance sets and are defined by (11) and (12). Then, for any , where we set.*

*Proof. *Since , we haveUsing the definition of , we know thatfor any . Then, from (14) and (15),It is easy to check thatThus,With Proposition 2, we now introduce the main result of this section: the dual representations of complex risk statistics on .

Theorem 2. *We suppose that is a complex risk statistic characterized by a continue clustering function and a continue simple risk statistic . Then, for any , has the following form:where is defined by*

*Proof. *Using Proposition 2, we havefor any . Furthermore, we can rewrite this formula aswhere the indicator function of a set is defined byFrom Proposition 1, we know that and are convex sets. Thus,Next, since is continue, it follows that is closed. Thus, by the duality theorem for conjugate functions, we haveSimilarly, we haveThus, we know thatFrom the continuity of and the continuity of , we can interchange the supremum and the infimum above, i.e.,With defined byit immediately follows that

*Remark 3. *Note that the proof of Theorem 1 utilized the primal representation of complex risk statistics in Proposition 2, which indicates that the acceptance sets and play a vital role. Thus, the dual representations of complex risk statistics still dependent on the clustering function and the simple risk statistic .

#### 5. Conclusions

In this paper, we derive a new class of risk statistics in intelligent systems, especially in deep neural networks, named complex risk statistics. Our results illustrate that an engineer who deals with the measurement of complex risk in intelligent systems can construct a reasonable complex risk statistic by choosing an appropriate clustering function and an appropriate simple risk statistic.

#### Data Availability

No data, code, were generated or used during the study.

#### Conflicts of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as potential conflicts of interest.

#### Authorsâ€™ Contributions

F. Sun and Y. Dong conceptualized the study, did formal analysis, wrote the original draft, and reviewed and edited the article; F. Sun was responsible for funding acquisition.

#### Acknowledgments

This work was supported by Guangdong Basic and Applied Basic Research Foundation (2020A1515110671) and Young Innovative Talents Project of Guangdong Province (2019KQNCX156) Jiangmen Basic and Applied Basic Research Foundation (2021030100070004859).