# Complex Risk Statistics with Scenario Analysis

Authors: Fei Sun; Yichuan Dong
Journal: Complexity
 (2021)

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## 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.


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## Body

## 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 Section2, 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. Letℝd be the d-dimensional Euclidean space, d≥1. For any x=x1,…,xd, y=y1,…,yd∈ℝd, x≤y means xi≤yi, 1≤i≤d. For any positive integer ki, the element X in product Euclidean space ℝk1×ℝk2×⋯×ℝkd is denoted by X≔X11,…,Xk11,X12,…,Xk22,……X1d,…,Xkdd. For any X,Y∈ℝk1×ℝk2×⋯×ℝkd and 1≤i≤d, X≽Y means ∑j=1kiXji≤∑j=1kiYji. From now on, the addition and multiplication are all defined pointwise, X,Y=∑i=1d∑j=1kiXji∑j=1kiYji, x,y=∑i=1dxiyi. For any X∈ℝk1×ℝk2×⋯×ℝkd, Xki≔0,……,0,X1i,…,Xkii,0,……,0∈0k1×⋯×0ki−1×ℝki×0ki+1×⋯×0kd.Definition 1.
A simple risk statistic is a functionϱ:ℝd⟶ℝ∪+∞ that satisfies the following properties:(A1)
Monotonicity: for anyx,y∈ℝd, x≥y implies ϱx≥ϱy;(A2)
Convexity: for anyx,y∈ℝd and λ∈0,1, ϱλx+1−λy≤λϱx+1−λϱy.Remark 1.
The properties A1–A2 are very well known and have been studied in detail in the study of risk statistics.Definition 2.
A clustering function is a functionϕ:ℝk1×ℝk2×⋯×ℝkd⟶ℝd that satisfies the following properties:(B1)
Monotonicity: for anyX,Y∈ℝk1×ℝk2×⋯×ℝkd, X≽Y implies ϕX≥ϕY;(B2)
Convexity: for anyX,Y∈ℝk1×ℝk2×⋯×ℝkd and λ∈0,1, ϕλX+1−λY≤λϕX+1−λϕY;(B3)
Correlation: for anyX∈ℝk1×ℝk2×⋯×ℝkd, there exists a simple risk statistic ϱ such that ϱ∘ϕXk1,ϱ∘ϕXk2,…,ϱ∘ϕXkd=ϕX.Definition 3.
A complex risk statistic is a functionρ:ℝk1×ℝk2×⋯×ℝkd⟶ℝ that satisfies the following properties:(C1)
Monotonicity: for anyX,Y∈ℝk1×ℝk2×⋯×ℝkd, X≽Y implies ρX≥ρY;(C2)
Convexity: for anyX,Y∈ℝk1×ℝk2×⋯×ℝkd and λ∈0,1, ρλX+1−λY≤λρX+1−λρY;(C3)
Statistical convexity: for anyX,Y,Z∈ℝk1×ℝk2×⋯×ℝkd, λ∈0,1, and 1≤i≤d, if ρZki=λρXki+1−λρYki, then ρZ≤λρX+1−λρY.


## 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 statisticvarrho and a clustering function ϕ. In other words, the measurement of complex risk statistics can be simplified into two steps.Theorem 1.
A functionρ:ℝk1×ℝk2×⋯×ℝkd⟶ℝ is a complex risk statistic in the case of there exists a clustering function ϕ:ℝk1×ℝk2×⋯×ℝkd⟶ℝd and a simple risk statistic ϱ:ℝd⟶ℝ such that ρ is the composition of ϱ and ϕ, i.e.,(1)ρX=ϱ∘ϕX,for allX∈ℝk1×ℝk2×⋯×ℝkd.Proof.
We first derive the “only if“ part. We suppose thatρ is a complex risk statistic and define a function ϕ by(2)ϕX≔ρXk1,ρXk2,…,ρXkd,for any X∈ℝk1×ℝk2×⋯×ℝkd. Since ρ satisfies the convexity C2, it follows(3)ϕλX+1−λY=ρλXk1+1−λYk1,…,ρλXkd+1−λYkd≤λXk1,…,ρXkd+1−λρYk1,…,ρYkd·λϕX+1−λϕY,for any X,Y∈ℝk1×ℝk2×⋯×ℝkd and λ∈0,1. Thus, ϕ satisfies the convexity B2. Similarly, the monotonicity B1 of ϕ can also be implied by the monotonicity C1 of ρ. Next, we consider a function ϱ:ϕℝk1×ℝk2×⋯×ℝkd⟶ℝ that is defined by(4)ϱx≔ρX,whereX∈ℝk1×ℝk2×⋯×ℝkdwithϕX=x.
Thus, we immediately know thatϕ satisfies the correlation B3, which means ϕ defined above is a clustering function. Next, we illustrate that the ϱ defined above is a simple risk statistic. Suppose x,y∈ϕℝk1×ℝk2×⋯×ℝkd with x≥y, there exists X,Y∈ℝk1×ℝk2×⋯×ℝkd such that ϕX=x, ϕY=y. Then, we have ϕX≥ϕY, which means X≥Y by the monotonicity of ϕ. Thus, it follows from the property C1 of ρ that(5)ϱx=ρX≥ρY=ϱy,which implies ϱ satisfies the monotonicity A1. Let x,y∈ϕℝk1×ℝk2×⋯×ℝkd with ϕX=x, ϕY=y, for any X,Y∈ℝk1×ℝk2×⋯×ℝkd, which implies ϱx=ρX and ϱy=ρY. We also consider z≔λx+1−λy for any λ∈0,1. Thus, from the definition of ϱ, there exists a Z∈ℝk1×ℝk2×⋯×ℝkd such that(6)ϱλx+1−λy=ρZ,with ϕZ=λx+1−λy. Hence, from the statistic convexity C3 of ρ, we know that(7)ϱλx+1−λy=ρZ≤λρX+1−λρY=λϱx+1−λϱy,which implies the convexity A2 of ϱ. Thus, ϱ is a simple risk statistic and from (2) and (3), we have ρ=ϱ°ϕ. Next, we derive the “if” part. We suppose that ϕ is a clustering function and ϱ is a simple risk statistic. Furthermore, define ρ=ϱ°ϕ. Since ϱ and ϕ are monotone and convex, it is relatively easy to check that ρ satisfies monotonicity C1 and convexity C2. We now suppose that X,Y,Z∈ℝk1×ℝk2×⋯×ℝkd which satisfies(8)ρZki=λρXki+1−λρYki,for any λ∈0,1. Then, the property B3 of ϕ implies(9)ϕZ=λϕX+1−λϕY.
Thus, we have(10)ρZ=ϱλϕX+1−λϕY≤λρX+1−λρY,which indicates ρ satisfies the property C3. Thus, the ρ defined above is a complex risk statistic.Remark 2.
Theorem1 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 ϕ 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 ofϕ and ϱ.


## 4. Dual Representations

Before we study the dual representations of complex risk statistics onℝk1×ℝk2×⋯×ℝkd, the acceptance sets should be defined. Since each complex risk statistic ρ can be decomposed into a clustering function ϕ and a simple risk statistic ϱ, we need only to define the acceptance sets of ϕ and ϱ, i.e.,(11)Aϱ≔c,x∈ℝ×ℝd:ϱx≤c,(12)Aϕ≔y,X∈ℝd×ℝk1×ℝk2×⋯×ℝkd:ϕX≤y.We will see later on that these acceptance sets can be used to provide complex risk statistics onℝk1×ℝk2×⋯×ℝkd. The following properties are needed in the subsequent study.Definition 4.
LetM and N be two ordered linear spaces. A set A⊂M×N satisfies f-monotonicity if m,n∈A, q∈N and n≥q implies m,q∈A. A set A⊂M×N satisfies b-monotonicity if m,n∈A, p∈M, and p≥m implies p,n∈A.Proposition 1.
We suppose thatρ=ϱ∘ϕ is a complex risk statistic with a clustering function ϕ:ℝk1×ℝk2×⋯×ℝkd⟶ℝd and a simple risk statistic ϱ: ℝd⟶ℝ. The corresponding acceptance sets Aϱ and Aϕ are defined by (11) and (12). Then, Aϕ and Aϱ are convex sets and they satisfy the f-monotonicity and b-monotonicity.Proof.
It is easy to check the abovementioned properties from definitions ofϕ and ϱ.
The next proposition provides the primal representation of complex risk statistics onℝk1×ℝk2×⋯×ℝkd, considering the acceptance sets.Proposition 2.
We suppose thatρ=ϱ∘ϕ is a complex risk statistic with a clustering function ϕ:ℝk1×ℝk2×⋯×ℝkd⟶ℝd and a simple risk statistic ϱ: ℝd⟶ℝ. The corresponding acceptance sets Aϱ and Aϕ are defined by (11) and (12). Then, for any X∈ℝk1×ℝk2×⋯×ℝkd,(13)ρX=infc∈ℝ:c,x∈Aϱ,x,X∈Aϕ,where we setinf∅=+∞.Proof.
Sinceρ=ϱ∘ϕ, we have(14)ρX=infc∈ℝ:ϱ∘ϕX≤c.
Using the definition ofAϱ, we know that(15)ϱx=infc∈ℝ:c,x∈Aϱ,for any x∈ℝd. Then, from (14) and (15),(16)ρX=infc∈ℝ:c,ϕX∈Aϱ.
It is easy to check that(17)c∈ℝ:c,ϕX∈Aϱ=c∈ℝ:c,x∈Aϱ,x,X∈Aϕ.
Thus,(18)ρX=infc∈ℝ:c,x∈Aϱ,x,X∈Aϕ.
With Proposition2, we now introduce the main result of this section: the dual representations of complex risk statistics on ℝk1×ℝk2×⋯×ℝkd.Theorem 2.
We suppose thatρ=ϱ°ϕ is a complex risk statistic characterized by a continue clustering function ϕ and a continue simple risk statistic ϱ. Then, for any X∈ℝk1×ℝk2×⋯×ℝkd, ρX has the following form:(19)ρX=supy^,X^∈PX^,X−αy^,X^,where α:ℝd×ℝk1×ℝk2×⋯×ℝkd⟶ℝ is defined by(20)αy^,X^≔supc,x∈Aϱy,Y∈Aϕ−c−y^,y−x+X^,Y,P≔y^,X^∈ℝd×ℝk1×ℝk2×⋯×ℝkd,αy^,X^<∞.Proof.
Using Proposition2, we have(21)ρX=infc∈ℝ:c,x∈Aϱ,x,X∈Aϕ,for any X∈ℝk1×ℝk2×⋯×ℝkd. Furthermore, we can rewrite this formula as(22)ρX=infc,x∈ℝ×ℝdc+IAϱc,x+IAϕx,X,where the indicator function of a set A∈X×Y is defined by(23)IAa,b≔0,a,b∈X×Y,∞,otherwise.
From Proposition1, we know that Aϱ and Aϕ are convex sets. Thus,(24)IAϱ′c^,x^=supc¯,x¯∈Aϱc^c¯+x^,x¯,c^∈ℝ,x^∈ℝd,IAϕ′y^,X^=supy¯,X¯∈Aϕy^y¯+X^,X¯,y^∈ℝd,X^∈ℝk1×ℝk2×⋯×ℝkd.
Next, sinceϱ is continue, it follows that Aϱ is closed. Thus, by the duality theorem for conjugate functions, we have(25)IAϱc,x=IAϱ″c,x=supc^,x^∈ℝ×ℝdc^c+x^,x−IAϱ′c^,x^=supc^,x^∈ℝ×ℝdc^c+x^,x−supc¯,x¯∈Aϱc^c¯+x^,x¯.
Similarly, we have(26)IAϕx,X=IAϕ″x,X=supy^,X^∈ℝd×ℝk1×ℝk2×⋯×ℝkdy^,x+X^,X−IAϕ′y^,X^=supy^,X^∈ℝd×ℝk1×ℝk2×⋯×ℝkdy^,x+X^,X−supy¯,X¯∈Aϕy^,y¯+X^,X¯.
Thus, we know that(27)ρX=infc,x∈ℝ×ℝdc+IAϱc,x+IAϕx,X=infc,x∈ℝ×ℝdsupc^,x^∈ℝ×ℝdy^,X^∈ℝd×ℝk1×ℝk2×⋯×ℝkdc1+c^+x^+y^,x+X^,X−IAϱ′c^,x^−IAϕ′y^,X^.
From the continuity ofϱ and the continuity of ϕ, we can interchange the supremum and the infimum above, i.e.,(28)ρX=supc^,x^∈ℝ×ℝdy^,X^∈ℝd×ℝk1×ℝk2×⋯×ℝkdinfc,x∈ℝ×ℝdc1+c^+x^+y^,x+X^,X−IAϱ′c^,x^−IAϕ′y^,X^=supy^,X^∈ℝd×ℝk1×ℝk2×⋯×ℝkdX^,X−supc¯,x¯∈Aϱy¯,X¯∈Aϕ−c¯−y^,y¯−x¯+X^,X¯.
Withαy^,X^ defined by(29)αy^,X^≔supc¯,x¯∈Aϱy¯,X¯∈Aϕ−c¯−y^,y¯−x¯+X^,X¯=supc,x∈Aϱy,Y∈Aϕ−c−y^,y−x+X^,Y,P≔y^,X^∈ℝd×ℝk1×ℝk2×⋯×ℝkd,αy^,X^<∞,it immediately follows that(30)ρX=supy^,X^∈PX^,X−αy^,X^.Remark 3.
Note that the proof of Theorem1 utilized the primal representation of complex risk statistics in Proposition 2, which indicates that the acceptance sets Aϱ and Aϕ play a vital role. Thus, the dual representations of complex risk statistics ρ still dependent on the clustering function ϕ and the simple risk statistic ϱ.


## 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.



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