Some concepts and properties of uncertain fields

Abstract

Uncertain field is virtually an extension of uncertain process with the index space changing from a totally ordered set into a partially ordered one such as time-space or a surface. For describing the uncertain field, this paper introduces several concepts of uncertainty distribution and inverse uncertainty distribution. In addition, a sufficient and necessary condition is proved for the uncertainty distribution of an uncertain field. Then a concept of independence of uncertain fields is introduced and the operational law is derived for a strictly monotone function of independent uncertain fields. Furthermore, another two concepts of uncertain stationary independent increment field and Liu field are proposed, and meanwhile some of their properties are investigated.

Keywords

Uncertainty theory uncertain process uncertain field uncertainty distribution

1 Introduction

Liu [9] asserted that there exist two types of mathematical systems for modeling indeterminacy. One type is probability theory which is used to model randomness associated with frequency. While, the other type is uncertainty theory which is applied to describing uncertain phenomena associated with belief degree.

Uncertainty theory was established by Liu [4] in 2007 for modeling belief degrees via uncertain measure, and refined by Liu [6] with presenting product uncertain measure to deal with the operations between uncertain variables in different uncertainty spaces. Then some fundamental concepts were put forward such as uncertain variable for modeling an uncertain quantity, uncertainty distribution and inverse uncertainty distribution for describing an uncertain variable, and expected value for ranking uncertain variables. Additionally, Liu [4] defined uncertain vector and its joint uncertainty distribution, and Liu [11] gave a concept of independence of uncertain vectors. So far, uncertainty theory has became an almost complete mathematical system and been studied by large numbers of experts, such as Gao [2], Liu [14], Liu and Ha [15], Peng and Iwamura [16], Sheng and Kar [17], Wang and Peng [18] and Yao [19].

Uncertain process was initialized by Liu [5] in 2008 for modeling the evolution of uncertain phenomena. For describing the uncertain process, Liu [13] introduced some concepts of uncertainty distribution and inverse uncertainty distribution. Besides, a concept of independence of uncertain processes was proposed by Liu [13] using uncertain vectors. After that, Liu provided the operational law for calculating the inverse uncertainty distribution of a strictly monotone function of independent uncertain processes. Furthermore, Liu [5] proposed a concept of uncertain independent increment process and proved a sufficient and necessary condition for its inverse uncertainty distribution. Then Liu [12] gave the uncertainty distributions of extreme values and the first hitting time of uncertain independent increment processes. After that, a concept of uncertain stationary independent increment process was put forward by Liu [5]. Meanwhile, its inverse uncertainty distribution was studied by Liu [13] which is linear function of time. Additionally, Liu [8] proved that the expected value is also a linear function of time, and Chen [1] verified that the variance is proportional to the square of time.

We all know that uncertain process is a sequence of uncertain variables indexed by time which is a totally ordered set. If the totally ordered set becomes partially ordered one, what does the sequence of uncertain variables become? To answer this question, Liu [13] presented a new concept of uncertain field. This paper mainly study some concepts and properties about the uncertain field. The remainder of the paper is arranged as follows. In Section 2, we will review some fundamental concepts and properties concerning uncertain variables and uncertain processes. In Section 3, we will retrospect the concept of uncertain field and take some examples to illustrate what is the uncertain field in reality. In Section 4, we will introduce a concept of uncertainty distribution of an uncertain field and give a sufficient and necessary condition for it. In Section 5, we will present a concept of inverse uncertainty distribution and give a sufficient and necessary condition for it. In Section 6, we will define independence of uncertain fields and provide the operational law for a strictly monotone function of independent uncertain fields. Then the uncertain field with stationary independent increments will be studied in Section 7. And Section 8 will be devoted to initializing Liu field that is a type of uncertain field with stationary and independent increments. Some properties of Liu field will also be investigated in this section. Finally, we make a summary of this paper in Section 9.

2 Preliminaries

In this section, some fundamental concepts and properties will be reviewed concerning uncertain variables and uncertain processes.

2.1 Uncertain variable

Let Γ be a nonempty set, and ℒ a σ-algebra over Γ. Each element Λ in ℒ is called an event and assigned a number ℳ {Λ} to indicate the belief degree that we believe Λ will happen. In order to deal with belief degrees rationally in mathematics, Liu [4] suggested the following three axioms:

Axiom 1. (Normality Axiom) ℳ {Γ} =1 for the universal set Γ;

Axiom 2. (Duality Axiom) ℳ {Λ} + ℳ {Λ^c} =1 for any event Λ;

Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, ⋯, we have $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}} .$

Definition 2.1. (Liu [4]) The set function ℳ is called an uncertain measure if it satisfies the normality, duality, and subadditivity axioms.

The triplet (Γ, ℒ, ℳ) is called an uncertainty space. Furthermore, the product uncertain measure on the product σ-algebra ℒ was defined by Liu [6] as follows:

Axiom 4. (Product Axiom) Let (Γ_k, ℒ_k, ℳ_k) be uncertainty spaces for k = 1, 2, ⋯. The product uncertain measure ℳ is an uncertain measure satisfying $M {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} M_{k} {Λ_{k}}$ where Λ_k are arbitrary events chosen from ℒ_k for k = 1, 2, ⋯, respectively.

Definition 2.2. (Liu [4]) An uncertain variable is a measurable function ξ from an uncertainty space (Γ, ℒ, ℳ) to the set of real numbers, i.e., for any Borel set B of real numbers, the set ${ξ \in B} = {γ \in Γ | ξ (γ) \in B}$ is an event.

Theorem 2.1. Let ξ₁, ξ₂, ⋯ , ξ_n be uncertain variables, and f a real-valued measurable function. Then ξ = f (ξ₁, ξ₂, ⋯ , ξ_n) is an uncertain variable defined by $ξ (γ) = f (ξ_{1} (γ), ξ_{2} (γ), \dots, ξ_{n} (γ)), \forall γ \in Γ .$

For describing an uncertain variable, Liu defined uncertainty distribution as follows.

Definition 2.3. (Liu [4]) Suppose ξ is an uncertain variable. Then the uncertainty distribution of ξ is defined by $Φ (x) = M {ξ \leq x}$ for any real number x .

Theorem 2.2. (Peng-Iwamura Theorem [16]) A function $Φ (x) : R \to [0, 1]$ is the uncertainty distribution of an uncertain variable if and only if it is a monotone increasing function except Φ (x) ≡0 and Φ (x) ≡1.

An uncertainty distribution Φ (x) is said to be regular if its inverse function Φ⁻¹ (α) exists and is unique for each α ∈ (0, 1). Inverse uncertainty distribution plays an important role in the operation of independent uncertain variables, thus the concept of inverse uncertainty distribution is presented in the following.

Definition 2.4. (Liu [8]) Suppose ξ is an uncertain variable with regular uncertainty distribution Φ (x). Then the inverse function Φ⁻¹ (α) is called the inverse uncertainty distribution of ξ.

Theorem 2.3. (Liu [10]) A function $Φ^{- 1} (α) : (0, 1) \to R$ is the inverse uncertainty distribution of an uncertain variable if and only if it is a continuous and strictly increasing function with respect α.

The operational law of independent uncertain variables was given by Liu [8] in order to calculate the uncertainty distribution of strictly increasing or decreasing function of uncertain variables. Before introducing the operational law, a concept of independence of uncertain variables is presented as follows.

Definition 2.5. (Liu [6]) The uncertain variables ξ₁, ξ₂, ⋯ , ξ_n are said to be independent if $M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋀_{i = 1}^{n} M {ξ_{i} \in B_{i}}$ for any Borel sets B₁, B₂, ⋯ , B_n of real numbers.

Theorem 2.4. (Liu [6]) Let ξ₁, ξ₂, ⋯ , ξ_n be independent uncertain variables, and let f₁, f₂, ⋯ , f_n be measurable functions. Then f₁ (ξ₁) , f₂ (ξ₂) , ⋯ , f_n (ξ_n) are independent uncertain variables.

Theorem 2.5. (Liu [8]) Let ξ₁, ξ₂, ⋯ , ξ_n be independent uncertain variables with continuous uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. If the function f (x₁, x₂, ⋯ , x_n) is strictly increasing with respect to x₁, x₂, ⋯ , x_m and strictly decreasing with respect to x_m+1, x_m+2, ⋯ , x_n, then the uncertain variable $ξ = f (ξ_{1}, ξ_{2}, \dots, ξ_{n})$ has an uncertainty distribution $\begin{matrix} Φ (x) = sup_{f (x_{1}, x_{2}, \dots, x_{n}) = x} (min_{1 \leq i \leq m} Φ_{i} (x_{i}) \land \\ min_{m + 1 \leq i \leq n} (1 - Φ_{i} (x_{i}))) . \end{matrix}$ Moreover, if Φ₁, Φ₂, ⋯ , Φ_n are regular, then ξ has an inverse uncertainty distribution $\begin{matrix} Φ^{- 1} (α) = f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (α) \dots Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), Φ_{m + 2}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) . \end{matrix}$

For ranking uncertain variables, the concept of expected value was proposed by Liu [4] in the following.

Definition 2.6. (Liu [4]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by $E [ξ] = \int_{0}^{+ \infty} M {ξ \geq x} d x - \int_{- \infty}^{0} M {ξ \leq x} d x$ provided that at least one of the two integrals is finite.

Theorem 2.6. (Liu [4]) Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected value exists, then $E [ξ] = \int_{- \infty}^{+ \infty} x d Φ (x) .$

2.2 Uncertain process

An uncertain process is essentially a sequence of uncertain variables indexed by time. The study of uncertain process was started by Liu in 2008.

Definition 2.7. (Liu [5]) Let T be a totally ordered set (e.g. time) and let (Γ, ℒ, ℳ) be an uncertainty space. An uncertain process is a measurable function from T × (Γ, ℒ, ℳ) to the set of real numbers such that {X_t ∈ B} is an event for any Borel set B for each t.

For describing uncertain process well, Liu proposed a concept of uncertainty distribution. In fact, an uncertainty distribution of an uncertain process is a sequence of uncertainty distributions of uncertain variables indexed by time.

Definition 2.8. (Liu [13]) An uncertain process X_t is said to have an uncertainty distribution Φ_t (x) if the uncertain variable X_t has the uncertainty distribution Φ_t (x) at each time t.

From the definition, it is clear that the uncertainty distribution of uncertain process is a surface instead of a curve.

Definition 2.9. (Liu [13]) Uncertain processes X_1t, X_2t, ⋯ , X_nt are said to be independent if for any positive integer k and any times t₁, t₂, ⋯ , t_k, the uncertain vectors $ξ_{i} = (X_{{it}_{1}}, X_{{it}_{2}}, \dots, X_{{it}_{k}}), i = 1, 2, \dots, n$ are independent, i.e., for any k-dimensional Borel sets B₁, B₂, ⋯ , B_n of real numbers, we have $M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋀_{i = 1}^{n} M {ξ_{i} \in B_{i}} .$

Theorem 2.7. (Liu [13]) Let X_1t, X_2t, ⋯ , X_nt be independent uncertain processes with regular uncertainty distributions Φ_1t, Φ_2t, ⋯ , Φ_nt, respectively. If the function f (x₁, x₂, ⋯ , x_n) is strictly increasing with respect to x₁, x₂, ⋯ , x_m and strictly decreasing with respect to x_m+1, x_m+2, ⋯ , x_n, then the uncertain process $X_{t} = f (X_{1 t}, X_{2 t}, \dots, X_{nt})$ has an inverse uncertainty distribution $\begin{matrix} Φ_{t}^{- 1} (α) = f (Φ_{1 t}^{- 1} (α), \dots, Φ_{mt}^{- 1} (α), \\ Φ_{m + 1, t}^{- 1} (1 - α), \dots, Φ_{nt}^{- 1} (1 - α)) . \end{matrix}$

Definition 2.10. (Liu [5]) An uncertain process X_t is said to have independent increments if $X_{t_{0}}, X_{t_{1}} - X_{t_{0}}, X_{t_{2}} - X_{t_{1}}, \dots, X_{t_{k}} - X_{t_{k - 1}}$ are independent uncertain variables where t₀ is initial time and t₁, t₂, ⋯ , t_k are any times with t₁ < t₂ < ⋯ < t_k.

Definition 2.11. (Liu [5]) An uncertain process X_t is said to have stationary increments if for any given t > 0, the increments X_s+t − X_s are identically distributed uncertain variables for all s > 0.

Definition 2.12. (Liu [5]) An uncertain process X_t is said to be a stationary independent increment process if it has not only stationary increments but also independent increments.

Definition 2.13. (Liu [6]) An uncertain process C_t is said to be a Liu process if

C₀ = 0 and almost all sample paths are Lipschitz continuous,

C_t has stationary and independent increments,

every increment C_s+t − C_s is a normal uncertain variable with expected value 0 and variance t², whose uncertainty distribution is $Φ (x) = {(1 + exp (\frac{- π x}{\sqrt{3} t}))}^{- 1}, x \in R .$

3 Uncertain field

As a generalization of the uncertain process, uncertain field was presented by Liu [13] whose index space is a partially ordered set such as a surface or a three-dimensional space. There are many uncertain fields in real world. For example, suppose we have m types of new rice cultivars and n experimental fields, then we combine new types and fields arbitrarily and next wait for the yields of all the combinations. That is an uncertain field, because all the rice cultivars are new without the known yields. Consider the wave height of one point at some time and location on a sea, then it is also an uncertain field because of lacking history data under the case of changeable weather.

Definition 3.1. (Liu [13]) Let (Γ, ℒ, ℳ) be an uncertainty space and let T be a partially ordered set. An uncertain field is a function X_t (γ) from T × (Γ, ℒ, ℳ) to the set of real numbers such that {X_t ∈ B} is an event for any Borel set B for each t.

Remark 3.1. If X_t is an uncertain field, then X_t is an uncertain variable for each given t.

Example 3.1. Let a and b be real numbers with a < b and let f be a nonnegative continuous function of t. Assume X_t is a linear uncertain variable for each t, i.e., $X_{t} \sim L (af (t), bf (t)), \forall t .$ Then X_t is an uncertain field.

Example 3.2. Let a, b, c be real numbers with a < b < c and let f be a nonnegative continuous function with respect to t in a partially ordered set T. Assume X_t is a zigzag uncertain variable for each t, i.e., $X_{t} \sim Z (af (t), bf (t), cf (t)), \forall t .$ Then X_t is an uncertain field.

Example 3.3. Let e and σ be real numbers with σ > 0 and let f be a nonnegative continuous function with respect to t. Assume X_t is a normal uncertain variable for each t, i.e., $X_{t} \sim N (ef (t), σ f (t)), \forall t .$ Then X_t is an uncertain field.

Example 3.4. Let e and σ be real numbers with σ > 0 and let f be a nonnegative continuous function with respect to t. Assume X_t is a lognormal uncertain variable for each t, i.e., $X_{t} \sim L O G N (ef (t), σ f (t)), \forall t .$ Then X_t is an uncertain field.

Definition 3.2. Assume X_t is an uncertain field. Then for each γ ∈ Γ, the function X_t (γ) is called a sample path of X_t.

In fact X_t (γ) is a real-valued function of t for any given γ. Hence, an uncertain field may also be considered as a function from an uncertainty space to a collection of sample paths.

Definition 3.3. An uncertain field X_t is said to be sample-continuous if almost all sample paths are continuous functions with respect to t.

Fig.1

A sample path of an uncertain field.

4 Uncertainty distribution

As we all know, the uncertainty distribution of an uncertain process is a sequence of uncertainty distributions of uncertain variables indexed by time. In fact, the uncertainty distribution of an uncertain field is an extension of the uncertainty distribution of an uncertain process with the index space changing from a totally ordered set into a partially ordered one. Thus an uncertainty distribution of an uncertain field is a hyperplane instead of surface or a curve. Now we give a formal mathematical definition as follows.

Definition 4.1. An uncertain field X_t is said to have an uncertainty distribution Φ_t (x), if for each t, the uncertain variable X_t has an uncertainty distribution Φ_t (x).

Example 4.1. D-linear The linear uncertain field $X_{t} \sim L (af (t),$ bf (t)) has an uncertainty distribution $Φ_{t} (x) = {\begin{matrix} 0, if x \leq af (t) \\ \frac{x - af (t)}{(b - a) f (t)}, if af (t) \leq x \leq bf (t) \\ 1, if x \geq bf (t) . \end{matrix}$

Example 4.2. The zigzag uncertain field $X_{t} \sim Z (af (t),$ bf (t) , cf (t)) has an uncertainty distribution $Φ_{t} (x) = {\begin{matrix} 0, if x \leq af (t) \\ \frac{x - af (t)}{2 (b - a) f (t)}, if af (t) \leq x \leq bf (t) \\ \frac{x + cf (t) - 2 bf (t)}{2 (c - b) f (t)}, if bf (t) \leq x \leq cf (t) \\ 1, if x \geq cf (t) . \end{matrix}$

Example 4.3. The normal uncertain field $X_{t} \sim N$ (ef (t) , σf (t)) has an uncertainty distribution $Φ_{t} (x) = {(1 + exp (\frac{π (ef (t) - x)}{\sqrt{3} σ f (t)}))}^{- 1}$ where σf (t) is a positive function.

Example 4.4. The normal uncertain field $X_{t} \sim L O G N$ (ef (t) , σf (t)) has an uncertainty distribution $Φ_{t} (x) = {(1 + exp (\frac{π (ef (t) - ln x)}{\sqrt{3} σ f (t)}))}^{- 1}$ where σf (t) is a positive function.

Theorem 4.1. (Sufficient and Necessary Condition) A function $Φ_{t} (x) : T \times R \to [0, 1]$ is an uncertainty distribution of uncertain field if and only if for each t ∈ T, it is a monotone increasing function with respect to x except Φ_t (x) ≡0 and Φ_t (x) ≡1.

Proof. If Φ_t (x) is an uncertainty distribution of some uncertain field X_t, then for any given t, Φ_t is the uncertainty distribution of uncertain variable X_t. It follows from Peng-Iwamura Theorem that Φ_t (x) is a monotone increasing function in regard to x and Φ_t (x) ≢ 0, Φ_t (x) ≢ 1.

Conversely, if for any t, Φ_t (x) is a monotone increasing function with respect to x expect Φ _t (x) ≡ 0 and Φ _t (x) ≡1, it follows from Peng-Iwamura Theorem that there exists an uncertain variable ξ_t whose uncertainty distribution is just Φ_t (x). Define $X_{t} = ξ_{t}, \forall t \in T .$ Then X_t is an uncertain field and has the uncertainty distribution Φ_t (x).

Thus the theorem is derived. □

Theorem 4.2. Let X_t be an uncertain field with uncertainty distribution Φ_t (x), and let f (x) be a measurable function. Then f (X_t) is an uncertain field. Moreover, if f (x) is a strictly increasing function, then f (X_t) has an uncertainty distribution $Ψ_{t} (x) = Φ_{t} (f^{- 1} (x)),$ (1) and if f (x) is a strictly decreasing function, then f (X_t) has an uncertainty distribution $Ψ_{t} (x) = 1 - Φ_{t} (f^{- 1} (x)) .$ (2)

Proof. Since X_t is an uncertain variable for any given t, it follows from Theorem 2.1 that f (X_t) is also an uncertain variable. Thus f (X_t) is an uncertain field. Then equations (1) and (2) may be derived from Theorem 2.5 immediately. □

Definition 4.2. An uncertainty distribution Φ_t (x) is said to be regular if for each t, it is a continuous and strictly increasing function with respect to x such that 0 < Φ_t (x) <1, and $lim_{x \to - \infty} Φ_{t} (x) = 0, lim_{x \to \infty} Φ_{t} (x) = 1 .$

Note that we have stipulated that constant variable has an regular uncertainty distribution. That is we allow a regular uncertain field to be a constant at some t₀ whose uncertainty is $Φ_{t_{0}} (x) = {\begin{matrix} 1, if x \geq X_{t} \\ 0, if x < X_{t} \end{matrix}$ and say Φ_{t
₀} (x) is a continuous and strictly increasing function with respect to x with 0 < Φ_t (x) <1 even though it is discontinuous at t₀.

5 Inverse uncertainty distribution

Definition 5.1. Let X_t be an uncertain field with regular uncertainty distribution Φ_t (x). Then the inverse function $Φ_{t}^{- 1} (α)$ is called the inverse uncertainty distribution of X_t.

Example 5.1. The linear uncertain field $X_{t} \sim L (af (t),$ bf (t)) has an inverse uncertainty distribution $Φ_{t}^{- 1} (α) = (1 - α) af (t) .$

Example 5.2. The zigzag uncertain field $X_{t} \sim Z (af (t),$ bf (t) , cf (t)) has an uncertainty distribution $Φ_{t}^{- 1} (α) = {\begin{matrix} (1 - 2 α) af (t) + 2 α bf (t), if α < 0.5 \\ (1 - 2 α) af (t) + (2 α - 1) bf (t), if α \geq 0.5 . \end{matrix}$

Example 5.3. The normal uncertain field $X_{t} \sim N$ (ef (t) , σf (t)) has an uncertainty distribution $Φ_{t}^{- 1} (α) = et + \frac{σ t \sqrt{3}}{π} ln \frac{α}{a - α}$ where σf (t) is a positive function.

Example 5.4. The normal uncertain field $X_{t} \sim L O G N$ (ef (t) , σf (t)) has an uncertainty distribution $Φ_{t}^{- 1} (α) = exp (et + \frac{σ t \sqrt{3}}{π} ln \frac{α}{1 - α})$ where σf (t) is a positive function.

Theorem 5.1. (Sufficient and Necessary Condition) A function $Φ_{t}^{- 1} (α) : T \times (0, 1) \to R$ is an inverse uncertainty distribution of uncertain field if and only if for each t ∈ T, it is a monotone increasing function with respect to α.

Proof. If $Φ_{t}^{- 1} (α)$ is an inverse uncertainty distribution of some uncertain field X_t, then for any given t, Φ_t is the inverse uncertainty distribution of uncertain variable X_t. It follows from Theorem 2.3 that $Φ_{t}^{- 1} (α)$ is a continuous and strictly increasing function in regard to α ∈ (0, 1).

Conversely, if for any t, $Φ_{t}^{- 1} (α)$ is a continuous and strictly increasing function in regard to α ∈ (0, 1), then from Theorem 2.4, there exists an uncertain variable ξ_t whose inverse uncertainty distribution is just $Φ_{t}^{- 1} (α)$ . Define $X_{t} = ξ_{t}, \forall t \in T .$ Then X_t is an uncertain field and has the inverse uncertainty distribution $Φ_{t}^{- 1} (α)$ . Thus the theorem is obtained. □

We stipulate that if an uncertain field X_t is a constant at t₀, we say X_{t
₀} has an inverse uncertainty distribution $Φ_{t_{0}}^{- 1} (α) \equiv X_{t_{0}}$ and $Φ_{t_{0}}^{- 1} (α) \equiv X_{t_{0}}$ is a continuous and strictly increasing function in regard to α ∈ (0, 1) even though it is not.

6 Independence of uncertain fields

The concept of independence of uncertain variables was defined by Liu [6] as a relationship between uncertain variables. And the uncertain measure of the intersection or union of a sequence of finite independent uncertain events is the minimum or maximum of respective uncertain measures of these uncertain events. As an extension of uncertain variable, Liu [4] presented a concept of uncertain vector whose components are uncertain variables. Then the independence of these uncertain vectors was proposed by Liu [10]. Based on independence of uncertain vectors, we give a concept of independence of uncertain fields.

Definition 6.1. Uncertain fields X_1t, X_2t, ⋯ , X_nt are said to be independent if for any positive integer k and any points t₁, t₂, ⋯ , t_k, the uncertain vectors $ξ_{i} = (X_{i t_{1}}, X_{i t_{2}}, \dots, X_{i t_{k}}), i = 1, 2, \dots, n$ are independent, i.e., for any k-dimensional Borel sets B₁, B₂, ⋯ , B_n of real numbers, there is $M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋀_{i = 1}^{n} M {ξ_{i} \in B_{i}} .$

Theorem 6.1. Uncertain fields X_1t, X_2t, ⋯ , X_nt are independent if and only if for any positive integer k, any indices t₁, t₂, ⋯ , t_k, and any k-dimensional Borel sets B₁, B₂, ⋯ , B_n of real numbers, there is $M {⋃_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋁_{i = 1}^{n} M {ξ_{i} \in B_{i}},$ (3) where $ξ_{i} = (X_{i t_{1}}, X_{i t_{2}}, \dots, X_{i t_{k}}), i = 1, 2, \dots, n .$

Proof. If X_1t, X_2t, ⋯ , X_nt are independent, then it follows from Definition 6.1 that $\begin{matrix} M {⋃_{i = 1}^{n} (ξ_{i} \in B_{i})} & = 1 - M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})^{c}} \\ = 1 - M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i}^{c})} \\ = 1 - ⋀_{i = 1}^{n} M {ξ_{i} \in B_{i}^{c}} \\ = 1 - ⋀_{i = 1}^{n} (1 - M {ξ_{i} \in B_{i}}) \\ = ⋁_{i = 1}^{n} M {ξ_{i} \in B_{i}} . \end{matrix}$ If Equation (3) holds, then we have $\begin{matrix} M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} & = & 1 - M {⋃_{i = 1}^{n} (ξ_{i} \in B_{i})^{c}} \\ = & 1 - M {⋃_{i = 1}^{n} (ξ_{i} \in B_{i}^{c})} \\ = & 1 - ⋁_{i = 1}^{n} M {ξ_{i} \in B_{i}^{c}} \\ = & 1 - ⋁_{i = 1}^{n} (1 - M {ξ_{i} \in B_{i}}) \\ = & ⋀_{i = 1}^{n} M {ξ_{i} \in B_{i}} . \end{matrix}$ It follows from Definition 6.1 that X_1t, X_2t, ⋯ , X_nt are independent. Thus the proof is finished. □

Theorem 6.2. Let X_1t, X_2t, ⋯ , X_nt be independent uncertain fields with regular uncertainty distributions Φ_1t, Φ_2t, ⋯ , Φ_nt. If the function f (x₁, x₂, ⋯ , x_n) is strictly increasing with respect to x₁, x₂, ⋯ , x_m and is strictly decreasing with respect to x_m+1, x_m+2, ⋯ , x_n, then the uncertain field $X_{t} = f (X_{1 t}, X_{2 t}, \dots, X_{nt})$ has an inverse uncertainty distribution $\begin{matrix} Φ_{t}^{- 1} (α) = f (Φ_{1 t}^{- 1} (α), \dots, Φ_{mt}^{- 1} (α), \\ Φ_{m + 1, t}^{- 1} (1 - α), \dots, Φ_{nt}^{- 1} (1 - α)) . \end{matrix}$

Proof. For each given point t, X_1t, X_2t, ⋯ , X_nt are independent uncertain variables with inverse uncertainty distributions $Φ_{1 t}^{- 1} (α), Φ_{2 t}^{- 1} (α), \dots, Φ_{nt}^{- 1} (α)$ , respectively. Hence, this theorem follows from Theorem 2.5 immediately. □

7 Stationary independent increment field

Let a = (a₁, a₂, ⋯ , a_m), b = (b₁, b₂, ⋯ , b_m) with a_i ≺ b_i, i = 1, 2, ⋯ , m, and let an interval (a, b] = {(x₁, x₂, ⋯ , x_m) ∣ a_j < x_j ≤ b_j, j = 1, 2, ⋯ , m}.

Definition 7.1. An uncertain field X_t is said to have independent increments if for any integer n and s₁, s₂, ⋯ , s_n, t₁, t₂, ⋯ , t_n, $X_{s_{1}}, X_{t_{1}} - X_{s_{1}}, X_{t_{2}} - X_{s_{2}}, \dots, X_{t_{n}} - X_{s_{n}}$ are independent uncertain variables where s₁ is initial point, s_i ≺ t_i and (s_i, t_i] , i = 1, 2, ⋯ , n are pairwise disjoint.

Theorem 7.1. IIP Let X_t be an uncertain independent increment field. Then for any real numbers a and b, the uncertain field $Y_{t} = {aX}_{t} + b$ is also an uncertain independent increment field.

Proof. Since X_t is an uncertain independent increment field, the uncertain variables $X_{t_{1}}, X_{t_{2}} - X_{t_{1}}, \dots, X_{t_{n}} - X_{t_{n - 1}}$ are independent. It follows from Y_t = aX_t + b and Theorem 2.4 that $Y_{t_{1}}, Y_{t_{2}} - Y_{t_{1}}, \dots, Y_{t_{n}} - Y_{t_{n - 1}}$ are also independent. That is, Y_t is an uncertain independent increment field. □

Definition 7.2. An uncertain field X_t is said to have stationary increments if the increments X_s+t − X_s are identically distributed uncertain variables for any s and any given t.

Definition 7.3. An uncertain field X_t is said to be a stationary independent increment field if it has not only stationary increments but also independent increments.

Theorem 7.2. Let X_t be an uncertain stationary independent increment field. Then for any real numbers a and b, the uncertain field $Y_{t} = {aX}_{t} + b$ is also an uncertain stationary independent increment field.

Proof. Since X_t is an uncertain independent increment field, it follows from Theorem 7.1 that Y_t is also an uncertain independent increment field. On the other hand, since X_t is an uncertain stationary increment field, the increments $X_{s + t} - X_{s}$ are identically distributed uncertain variables for all s. Thus $Y_{s + t} - Y_{s} = a (X_{s + t} - X_{s})$ are identically distributed uncertain variables for all s, and Y_t is an uncertain stationary increment field. Hence the theorem is proved. □

8 Liu field

Liu field is a type of an uncertain field with zero initial value whose increments are stationary and independent normal uncertain variables. And it is an extension of Liu process with the index space changing form a totally ordered set into a partially ordered one.

Definition 8.1. An uncertain field C_t is said to be a Liu field with parameters λ₁, λ₂, ⋯ , λ_m (all are nonnegative) if

C₀ = 0 and almost all sample paths are Lipschitz continuous;

C_t has stationary and independent increments;

for any t = (t₁, t₂, ⋯ , t_m) ⪰0, every increment C_t+s − C_s is a normal uncertain variable with expected value 0 and standard variance $\sum_{i = 1}^{m} λ_{i} t_{i}$ .

Remark 8.1. Liu field is an uncertain stationary independent increment field whose increments have a normal uncertainty distribution with expected value 0 and standard variance ∑λ_it_i. The uncertainty distribution of C_t is $Φ_{t} (x) = {(1 + exp (- \frac{π x}{\sqrt{3} \sum λ_{i} t_{i}}))}^{- 1}$ (4) and the inverse uncertainty distribution is $Φ_{t}^{- 1} (α) = \frac{\sqrt{3} \sum λ_{i} t_{i}}{π} ln \frac{α}{1 - α} .$

Theorem 8.1. Let C_t be a Liu field with parameters λ₁, λ₂, ⋯ , λ_m. Then for any point t ⪰ 0, the ratio $C_{t} / \sum_{i = 1}^{m} λ_{i} t_{i}$ is a normal uncertain variable with expected value 0 and variance 1. That is, $\frac{C_{t}}{\sum_{i = 1}^{m} λ_{i} t_{i}} \sim N (0, 1)$ for any point t ⪰ 0.

Proof. Since C_t is a normal uncertain variable $N (0,$ ∑λ_it_i) for each t, that is, C_t has the uncertainty distribution $Φ_{t} (x) = {(1 + exp (- \frac{π x}{\sqrt{3} \sum_{i = 1}^{m} λ_{i} t_{i}}))}^{- 1}$ for any point t, we obtain from operational law that $C_{t} / \sum_{i = 1}^{m} λ_{i} t_{i}$ has an uncertainty distribution $Ψ (x) = Φ_{t} (x \sum_{i = 1}^{m} λ_{i} t_{i}) = {(1 + exp (- \frac{π x}{\sqrt{3}}))}^{- 1} .$ Hence $C_{t} / \sum_{i = 1}^{m} λ_{i} t_{i}$ is a normal uncertain variable with expected value 0 and variance 1. The proof is finished. □

Theorem 8.2. Let C_t be a Liu field with parameters λ₁, λ₂, ⋯ , λ_m. Then for any point t ⪰ 0 and each i, i = 1, 2, ⋯ , m, the ratio ∂C_t/∂t_i is a normal uncertain variable with expected value 0 and variance $λ_{i}^{2}$ . That is, $\frac{\partial C_{t}}{\partial t_{i}} \sim N (0, λ_{i})$ for any point t ⪰ 0 and each i, i = 1, 2, ⋯ , m.

Proof. Since C_{t₁,t₂,⋯,t_m} − C_{t₁,⋯,t_i-1,0,t_i+1,⋯,t_m} is a normal uncertain variable $N (0, λ_{i} t_{i})$ , that is, C_{t₁ ,t₂, ⋯ ,t_m} −C_{t₁,⋯,t_i−1,0,t_i+1,⋯,t_m} has the uncertainty distribution $Φ_{t_{i}} (x) = {(1 + exp (- \frac{π x}{\sqrt{3} λ_{i} t_{i}}))}^{- 1},$ we obtain from operational law that (C_{t₁,t₂,⋯,t_m} − C_{t₁,⋯,t_i−1,0,t_i+1,⋯,t_m})/t_i has an uncertainty distribution $Ψ (x) = Φ_{t_{i}} ({xt}_{i}) = {(1 + exp (- \frac{π x}{\sqrt{3} λ_{i}}))}^{- 1} .$ Hence ∂C_t/∂t_i is a normal uncertain variable with expected value 0 and variance $λ_{i}^{2}$ . The proof is finished. □

Next will give the upper and lower bounds for the expected value and variance of the square of Liu field.

Theorem 8.3. Let C_t be a Liu field with parameters λ₁, λ₂, ⋯ , λ_m. Then for each point t ⪰ 0, we have $\frac{1}{2} {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2} \leq E [C_{t}^{2}] \leq {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2} .$

Proof. Since C_t is a normal uncertain variable and has an uncertainty distribution Φ_t (x) in (4) for any point t. It follows the definition of expected value that $\begin{matrix} E [C_{t}^{2}] & = & \int_{0}^{\infty} M {C_{t}^{2} \geq x} d x \\ = & \int_{0}^{\infty} M {(C_{t} \geq \sqrt{x}) \cup (C_{t} \leq - \sqrt{x})} d x . \end{matrix}$ On the one hand, we have $\begin{matrix} E [C_{t}^{2}] & \leq & \int_{0}^{\infty} [M {C_{t} \geq \sqrt{x}} + M {C_{t} \leq - \sqrt{x}}] d x \\ = & \int_{0}^{\infty} (1 - Φ_{t} (\sqrt{x})) d x + \int_{0}^{\infty} Φ_{t} (- \sqrt{x}) d x \\ = & {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2} . \end{matrix}$ On the other hand, we have $\begin{matrix} E [C_{t}^{2}] & \leq & \int_{0}^{\infty} [M {C_{t} \geq \sqrt{x}} d x \\ = & \int_{0}^{\infty} (1 - Φ_{t} (\sqrt{x})) = \frac{1}{2} {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2} . \end{matrix}$ Thus the proof is finished. □

Theorem 8.4. Let C_t be a Liu field with parameters λ₁, λ₂, ⋯ , λ_m. Then for any point t ⪰ 0, we have $1.24 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} < V [C_{t}^{2}] < 4.31 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} .$

Proof. Let e be the expected value of $C_{t}^{2}$ . It follows the definition of expected value that $\begin{matrix} V [C_{t}^{2}] = \int_{0}^{\infty} M {(C_{t}^{2} - e)^{2} \geq x} d x \\ = \int_{0}^{\infty} M {(C_{t}^{2} - e \geq \sqrt{x}) \cup (C_{t}^{2} - e \geq - \sqrt{x})} d x \\ = \int_{0}^{\infty} M {(C_{t} \geq \sqrt{e + \sqrt{x}}) \cup (C_{t} \leq - \sqrt{e + \sqrt{x}}) \\ \cup (- \sqrt{e - \sqrt{x}} \leq C_{t} \leq \sqrt{e - \sqrt{x}})} d x . \end{matrix}$ On the one hand, we have $\begin{matrix} V [C_{t}^{2}] \\ = \int_{0}^{\infty} M {(C_{t}^{2} - e)^{2} \geq x} d x \\ \leq \int_{0}^{\infty} M {(C_{t} \geq \sqrt{e + \sqrt{x}})} d x \\ + \int_{0}^{\infty} M {(C_{t} \leq - \sqrt{e + \sqrt{x}})} d x \\ + \int_{0}^{\infty} M {(- \sqrt{e - \sqrt{x}} \leq C_{t} \leq \sqrt{e - \sqrt{x}})} d x \end{matrix}$ Since $\frac{1}{2} {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2} \leq E [C_{t}^{2}] \leq {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2}$ , we have $\begin{matrix} \int_{0}^{\infty} M {(C_{t} \geq \sqrt{e + \sqrt{x}})} d x \\ \leq \int_{0}^{\infty} M {(C_{t} \geq \sqrt{(\frac{1}{2} \sum_{i = 1}^{m} λ_{i} t_{i})^{2} + \sqrt{x}})} d x \\ = \int_{0}^{\infty} [1 - (1 + \\ exp (- \frac{π \sqrt{(\frac{1}{2} \sum_{i = 1}^{m} λ_{i} t_{i})^{2} + \sqrt{x}}}{\sqrt{3} \sum_{i = 1}^{m} λ_{i} t_{i}}))^{- 1}] d x \\ Letting y = \sqrt{{(\frac{1}{2} \sum_{i = 1}^{m} λ_{i} t_{i})}^{2} + \sqrt{x}} \\ = \int_{\frac{\sum_{i = 1}^{m} λ_{i} t_{i}}{\sqrt{2}}}^{\infty} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \\ = \int_{0}^{\infty} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \\ - \int_{0}^{\frac{\sum_{i = 1}^{m} λ_{i} t_{i}}{\sqrt{2}}} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \\ \leq \int_{0}^{\infty} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \end{matrix}$ $\begin{matrix} + \int_{0}^{\frac{\sum_{i = 1}^{m} λ_{i} t_{i}}{\sqrt{2}}} ({yt}^{2} - 2 y^{3}) d y \\ = 1.6 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} + 0.125 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} \\ = 1.725 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4}, \end{matrix}$ $\begin{matrix} \int_{0}^{\infty} M {(C_{t} \leq - \sqrt{e + \sqrt{x}})} d x \\ \leq \int_{0}^{\infty} M {(C_{t} \leq - \sqrt{(\frac{1}{2} \sum_{i = 1}^{m} λ_{i} t_{i})^{2} + \sqrt{x}})} d x \\ = \int_{0}^{\infty} {(1 + exp (\frac{π \sqrt{{(\frac{1}{2} \sum_{i = 1}^{m} λ_{i} t_{i})}^{2} + \sqrt{x}}}{\sqrt{3} \sum_{i = 1}^{m} λ_{i} t_{i}}))}^{- 1} d x \\ Letting y = \sqrt{{(\frac{1}{2} \sum_{i = 1}^{m} λ_{i} t_{i})}^{2} + \sqrt{x}} \\ = \int_{\frac{\sum_{i = 1}^{m} λ_{i} t_{i}}{\sqrt{2}}}^{\infty} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \\ = \int_{0}^{\infty} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \\ - \int_{0}^{\frac{\sum_{i = 1}^{m} λ_{i} t_{i}}{\sqrt{2}}} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \\ \leq \int_{0}^{\infty} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \end{matrix}$ $\begin{matrix} + \int_{0}^{\frac{\sum_{i = 1}^{m} λ_{i} t_{i}}{\sqrt{2}}} ({yt}^{2} - 2 y^{3}) d y \\ = 1.6 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} + 0.125 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} \\ = 1.725 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4}, \end{matrix}$ and $\begin{matrix} \int_{0}^{\infty} M {(- \sqrt{e - \sqrt{x}} \leq C_{t} \leq \sqrt{e - \sqrt{x}})} d x \\ \leq \int_{0}^{\infty} M {(C_{t} \leq \sqrt{(\sum_{i = 1}^{m} λ_{i} t_{i})^{2} - \sqrt{x}})} d x \\ = \int_{0}^{\infty} {(1 + exp (- \frac{π \sqrt{{(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2} + \sqrt{x}}}{\sqrt{3} \sum_{i = 1}^{m} λ_{i} t_{i}}))}^{- 1} d x \\ Letting y = \sqrt{{(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2} - \sqrt{x}} \\ = \int_{0}^{\sum_{i = 1}^{m} λ_{i} t_{i}} \frac{4 {yt}^{2} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{3}}{1 + exp (- \frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \\ < \int_{0}^{\sum_{i = 1}^{m} λ_{i} t_{i}} \frac{4 {yt}^{2} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{3}}{1 + exp (- \frac{π}{\sqrt{3}})} d y \\ = \frac{t^{4}}{1 + exp (- \frac{π}{\sqrt{3}})} = 0.86 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} . \end{matrix}$ Thus we have $\begin{matrix} V [C_{t}^{2}] & \leq & 1.725 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} + 1.725 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} \\ + 0.86 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} = 4.31 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} \end{matrix}$ On the other hand, since $\frac{1}{2} {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2} \leq E [C_{t}^{2}] \leq {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2}$ we have $\begin{matrix} V [C_{t}^{2}] = \int_{0}^{\infty} M {(C_{t}^{2} - e)^{2} \geq x} d x \\ \geq \int_{0}^{\infty} M {(C_{t} \geq \sqrt{e + \sqrt{x}})} d x \\ \geq \int_{0}^{\infty} M {(C_{t} \geq \sqrt{(\sum_{i = 1}^{m} λ_{i} t_{i})^{2} + \sqrt{x}})} d x \\ = \int_{0}^{\infty} [1 - (1 + \\ exp (- \frac{π \sqrt{{(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2} + \sqrt{x}}}{\sqrt{3} \sum_{i = 1}^{m} λ_{i} t_{i}}))^{- 1}] d x \\ Letting y = \sqrt{{(\sum_{i = 1}^{m} λ_{i} t_{i})}^{2} + \sqrt{x}} \\ = \int_{\sum_{i = 1}^{m} λ_{i} t_{i}}^{\infty} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \\ = \int_{0}^{\infty} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \\ - \int_{0}^{\sum_{i = 1}^{m} λ_{i} t_{i}} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \\ > \int_{0}^{\infty} \frac{4 y^{3} - 4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2}}{1 + exp (\frac{π y}{\sqrt{3} (\sum_{i = 1}^{m} λ_{i} t_{i})})} d y \\ + \int_{0}^{\sum_{i = 1}^{m} λ_{i} t_{i}} \frac{4 y (\sum_{i = 1}^{m} λ_{i} t_{i})^{2} - 4 y^{3}}{1 + exp (\frac{π}{\sqrt{3}})} d y \\ = 1.1 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} + 0.14 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} \\ = 1.24 {(\sum_{i = 1}^{m} λ_{i} t_{i})}^{4} . \end{matrix}$ Thus the proof is finished. □

9 Conclusion

For describing the sequence of uncertain variables indexed by a partially ordered set, Liu [5] proposed a concept of uncertain field and this paper took some examples of uncertain fields for an intuitive understanding. In order to describe an uncertain field, this paper proposed concepts of uncertainty distribution and inverse uncertainty distribution of uncertain fields, respectively. In addition, we put up a concept of independence of uncertain fields, and provided the operational law of a strictly monotone function of independent uncertain fields. Furthermore, we studied the uncertain stationary independent increment field. Finally, we defined Liu field, and derived the upper and lower bounds for the expected value and variance of the square of Liu field, respectively.

Acknowledgments

This work was supported by National Natural Science Foundation of China Grant Nos. 61573210 and 61304182.

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