Covariance and Pseudo-Covariance of Complex Uncertain Variables

1 Introduction

The complex number lets us to model many phenomena that traditionally cannot model by invoking the real number, such as periodic signal, alternating current in electricity and two-dimensional potential flow in fluid mechanics. For modeling such phenomena, the concept of complex normal random variable was proposed by Wooding [16]. After that, the characteristic function of such random variable was studied by Turin [15]. Furthermore, Goodman [9] established the statistical properties of complex variables.

It is mentioned that probability theory is a tool to model randomness related to historical data(frequency). The frequencies are collected from samples. However, in many situations we have no sample in the real world. Therefore, in these situations, we should invoke to expert’s belief degrees. Thus, uncertainty theory was proposed by Liu [10] as a branch of mathematics. Then some basic concepts were proposed in Liu [10] such as uncertain measure, uncertain variable, and uncertainty distribution. As we all know, randomness and uncertainty usually appear in a same system simultaneously. For model this hybrid phenomena, Liu [13] introduced chance theory with proposing the concept of uncertain random variable. As the contribution to the development of uncertainty theory and chance theory, some of recent works were studied, such as [1, 8].

In order to model complex phenomena involving uncertainty, Peng [14] proposed the concept of complex uncertain variables. Furthermore, the concepts of uncertainty distribution and expected value of a complex uncertain variable were proposed. After that, Chen et al. [4] established several convergence theorems for complex uncertain variables. Furthermore, the concepts of variance and pseudo variance for a complex uncertain variable were presented by Chen et al. [5]. And, several formulas for calculating these concepts were provided. As an extension of complex uncertain variables, Gao et al. [6] presented the concept of complex uncertain random variables and studied several properties of these variables such as chance distribution, expected value and variance. After that Ahmadzade et al. [3] discussed about convergence of complex uncertain random sequences. In order to measure the association between two complex uncertain variables, we introduce the concepts of covariance and pseudo covariance for complex uncertain variables. Also, by using inverse uncertainty distribution, we provide several formulas for calculating covariance and pseudo-covariance of complex uncertain variables, in this paper. The rest of this paper is organized as follows. In Section 2, some basic concepts of uncertainty theory are provided as they are needed. In Section 3, by invoking inverse uncertainty distribution, several formulas for calculating covariance and pseudo-covariance of complex uncertain variables are derived. Also, two inequalities about covariance and pseudo-covariance of complex uncertain variables are stated and proved. Finally, some conclusions are derived in Section 4.

2 Preliminaries

In this section, we review some concepts in uncertainty theory, including uncertain variable, complex uncertain variable, operational law, expected value and variance.

Let eulerL be a σ-algebra on a nonempty set Γ. A set function M : eulerL → [0, 1] is called an uncertain measure if it satisfies the following axioms:

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

Definition 1. (Liu [10]) An uncertain variable ξ is a function from an uncertainty space (Γ, eulerL, M) to the set of real numbers such that {ξ ∈ B} is an event for any Borel set B of real numbers.

Definition 2. (Liu [10]) The uncertain variables ξ₁, ξ₂, ⋯, ξ _n are said to be independent if M { ⋂ i = 1 n { ξ i ∈ B i } } = ⋀ i = 1 n M { ξ i ∈ B i } for any Borel sets B₁, B₂, ⋯ , B _n of real numbers.

Theorem 1. (Liu [10]) Let ξ₁, ξ₂, ⋯ , ξ _n be independent uncertain variables, and f₁, f₂, ⋯, f _n be measurable functions. Then f₁ (ξ₁) , f₂ (ξ₂) , ⋯, f _n  (ξ _n ) are independent uncertain variables.

Definition 3. (Liu [11]) The uncertainty distribution Φ of an uncertain variable ξ is defined by Φ ( x ) = M { ξ ≤ x } for any real number x.

Definition 4. (Liu [11]) An uncertainty distribution Φ (x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < Φ (x) <1, and lim x → - ∞ Φ ( x ) = 0 , lim x → ∞ Φ ( x ) = 1 .

It is clear that a regular uncertainty distribution Φ (x) has an inverse function on the range of x with 0 < Φ (x) <1, and the inverse function Φ^-1 (α) exists on the open interval (0, 1).

Definition 5. (Liu [11]) Let ξ be 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. (Liu [11]) Let ξ₁, ⋯ , ξ _n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ _n , respectively. If f is a strictly increasing function, then ξ = f (ξ₁, ξ₂, ⋯ , ξ _n ) is an uncertain variable with inverse uncertainty distribution Ψ - 1 ( α ) = f ( Φ 1 - 1 ( α ) , ⋯ , Φ n - 1 ( α ) ) .

Definition 6. (Liu [10]) The expected value of an uncertain variable ξ is defined by E [ ξ ] = ∫ 0 + ∞ M { ξ ≥ x } x - ∫ - ∞ 0 M { ξ ≤ x } x provided that at least one of the two integrals is finite.

Theorem 3. (Liu [10]) Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected value exists, then E [ ξ ] = ∫ 0 + ∞ ( 1 - Φ ( x ) ) x - ∫ - ∞ 0 Φ ( x ) x .

Liu and Ha [12] proposed a generalized formula for expected value by inverse uncertainty distribution.

Theorem 4. (Liu and Ha [12]) Let ξ₁, ξ₂, ⋯ , ξ _n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ _n , respectively. If f (ξ₁, ξ₂, ⋯ , ξ _n ) is strictly increasing with respect to ξ₁, ξ₂, ⋯ , ξ _m and strictly decreasing with respect to ξ_m+1, ξ_m+2, ⋯ , ξ _n , then the uncertain variable ξ = f (ξ₁, ξ₂, ⋯ , ξ _n ) has an expected value E [ ξ ] = ∫ 0 1 f ( Φ 1 - 1 ( α ) , ⋯ , Φ m - 1 ( α ) , Φ m + 1 - 1 ( 1 - α ) , ⋯ , Φ n - 1 ( 1 - α ) ) α .

It is mentioned that the expected value operator has property of linearity. On the other hand, let ξ and η be two independent uncertain variables, then we have E [aξ + bη] = aE [ξ] + bE [η] where a and b are real numbers, for more details see [11].

Definition 7. (Liu [10]) If τ is an uncertain variable with finite expected value E [τ], then the variance of τ is defined by Var ( τ ) = E [ ( τ - E [ τ ] ) 2 ] .

Theorem 5. (Yao [17]) If τ is an uncertain variable with finite expected value E [τ], then the variance of τ is Var ( τ ) = ∫ 0 1 ( Φ - 1 ( α ) - E [ τ ] ) 2 d α .

Definition 8. (Zhao et al. [18]) Let τ₁ and τ₂ be two uncertain variables with the expected values E [τ₁] and E [τ₂], respectively. The covariance of τ₁ and τ₂ is defined by Cov ( τ 1 , τ 2 ) = E [ ( τ 1 - E [ τ 1 ] ) ( τ 2 - E [ τ 2 ] ) ] .

Since the uncertain measure is a subadditive measure, Zhao et al. [18] established the following stipulation for calculating the covariance of two uncertain variables as a linear function of expected values.

Stipulation 1. (Zhao et al. [18]) Let τ₁ and τ₂ be two uncertain variables with finite expected values E [τ₁] and E [τ₂], respectively. Then the covariance of τ₁ and τ₂ is Cov ( τ 1 , τ 2 ) = ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) d α , where, Φ 1 - 1 ( α ) and Φ 2 - 1 ( α ) are the inverse uncertainty distributions of τ₁ and τ₂, respectively.

Example 1. (Zhao et al. [18]) Suppose that τ₁ and τ₂ are two uncertain variables τ 1 ∼ L ( a 1 , b 1 ) and τ 2 ∼ L ( a 2 , b 2 ) . Then, Stipulation 1 implies that Cov ( τ 1 , τ 2 ) = ( b 2 - a 2 ) ( b 1 - a 1 ) 12 , and Var ( τ 1 ) = Cov ( τ 1 , τ 1 ) = ( b 1 - a 1 ) 2 12 .

Example 2. (Zhao et al. [18]) Suppose that τ₁ and τ₂ are two uncertain variables τ 1 ∼ N ( e 1 , σ 1 ) and τ 2 ∼ N ( e 2 , σ 2 ) . Then, Stipulation 1 implies that Cov ( τ 1 , τ 2 ) = σ 1 σ 2 , and Var ( τ 1 ) = Cov ( τ 1 , τ 1 ) = σ 1 2 .

Definition 9. (Peng [14]) A complex uncertain variable is a measurable function τ from an uncertainty space (Γ, eulerL, M) to the set of complex numbers, i.e., for any Borel set B of complex numbers, the set { τ ∈ B } = { γ ∈ Γ | τ ( γ ) ∈ B } , is an event.

Definition 10. (Peng [14]) The complex uncertainty distribution Φ (x) of a complex uncertain variable ξ is a function from ℂ to [0, 1] defined by Φ ( z ) = M { Re ( ξ ) ≤ Re ( z ) , Im ( ξ ) ≤ Im ( z ) } for any complex z.

In order to model a complex uncertain variable, the expected value is proposed as below.

Definition 11. (Peng [14]) Let ξ be a complex uncertain variable. The expected value of ξ is defined by E [ξ] = E [Re (ξ)] + iE [Im (ξ)] provided that E [Re (ξ)] and E [Im (ξ)] are finite, where E [Re (ξ)] and E [Im (ξ)] are expected values of uncertain variables Re (ξ) and Im (ξ), respectively.

Definition 12. (Peng [14]) Suppose that ξ is a complex uncertain variable with expected value E [ξ]. Then the variance of ξ is defined by Var ( ξ ) = E [ | | ξ - E [ ξ ] | | 2 ] . Since the uncertain measure is a subadditivity measure, the variance of complex uncertain variable ξ cannot be derived by the uncertainty distribution. A stipulation of variance of ξ with inverse uncertainty distribution of the real and imaginary parts of ξ is presented as follows.

Stipulation 2. (Chen et al. [5]) Let ξ = τ₁ + iτ₂ be a complex uncertain variable with the real part τ₁ and imaginary part τ₂. The expected value of ξ exists and E [ ξ ] = E [ τ 1 ] + iE [ τ 2 ] . Assume τ₁ and τ₂ are independent uncertain variables with regular uncertainty distributions Φ₁ and Φ₂, respectively. Then the variance of ξ is

Var ( ξ ) = ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) 2 d α + ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) 2 d α .

Definition 13. Let ξ be a complex uncertain variable with expected value E [ξ]. Then the pseudo-variance is defined by V ˜ ar [ ξ ] = E [ ( ξ - E [ ξ ] ) 2 ] . Stipulation 3. (Chen et al. [5]) Let ξ = τ₁ + iτ₂ be a complex uncertain variable with the real part τ₁ and imaginary part τ₂. The expected value of ξ exists and E [ξ] = E [τ₁] + iE [τ₂]. Assume τ₁ and τ₂ are independent uncertain variables with uncertainty distributions Φ₁ and Φ₂, respectively. Then pseudo variance of ξ is V ˜ ar ( ξ ) = ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) 2 d α - ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) 2 d α + 2 i ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) × ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) d α .

3 Covariance and pseudo-covariance of complex uncertain variables

In this section, we derive a formula for calculating the expected value of a complex uncertain random variable. In addition, in order to calculate variance of a complex uncertain random variable, a stipulation. For better illustration of main results, several examples are explained.

Definition 14. Let τ₁ and τ₂ be two complex uncertain variables with expected value E [τ₁] and E [τ₂], respectively. Then the covariance is defined by Cov ( τ 1 , τ 2 ) = E [ ( τ 1 - E [ τ 1 ] ) ( τ 2 - E [ τ 2 ] ) * ] where (τ₂ - E [τ₂]) ^* is the conjugate of the complex uncertain variable (τ₂ - E [τ₂]) .

Remark 1. If complex uncertain variables degenerate to uncertain ones in above definition, the result of Definition 8 is concluded.

Remark 2. Since probability measure is additive, the covariance of two complex random variables can be written as follows: Cov ( τ 1 , τ 2 ) = E [ τ 1 τ 2 * ] - E [ τ 1 ] E [ τ 2 * ] . Remark 3. It is mentioned that the subadditivity of uncertain measure concludes the difficulty of calculating of the covariance of two uncertain variables based on Definition 14, i.e. we can not express the covariance as a linear function of expected values. Thus, by inception of Stipulation 1 and 3, we present a formula for calculating the covariance of two uncertain variables as follows. Stipulation 4. Suppose that ξ₁ = τ₁ + iτ₂ and ξ₂ = τ₃ + iτ₄ are two complex uncertain variables, such that τ₁, τ₂, τ₃ and τ₄ are independent uncertain variables with uncertainty distributions Φ₁, Φ₂, Φ₃ and Φ₄, respectively. Then we have Cov ( ξ 1 , ξ 2 ) = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - E [ τ 1 ] - iE [ τ 2 ] ) ( Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) - E [ τ 3 ] - iE [ τ 4 ] ) * d α , where Φ i - 1 ( α ) is the inverse uncertainty distribution of the uncertain variable τ _i , i = 1, 2, 3, 4. As an extension of complex uncertain variables, Gao et al. [6] proposed the concepts of complex uncertain random variables as follows.

Definition 15. (Gao et al. [6]) A complex uncertain random variable is a function ξ from a chance space (Γ × Ω, eulerL × eulerA, M × Pr) to the set of complex numbers such that {ξ ∈ B} = {(γ, ω) ∈ Γ × Ω|ξ (γ, ω) ∈ B} is an event in Γ × Ω for any Borel Set B of complex numbers.

By inception of Stipulation 4, we can present the following formula for calculating of two complex uncertain random variables. Stipulation 5. Let η₁, ⋯ , η₄ be independent random variables with probability distributions Ψ₁, ⋯ , Ψ₄, respectively, and let τ₁, ⋯ , τ₄ be independent uncertain variables with uncertainty distributions ϒ₁, ⋯ , ϒ₄, respectively. Set ξ _i  = f _i  (η _i , τ _i ) , i = 1, ⋯ , 4 . Consider ζ₁ = ξ₁ + iξ₂ and ζ₂ = ξ₃ + iξ₄ as two complex uncertain variables, then we have Cov ( ζ 1 , ζ 2 ) = ∫ 0 1 ( F 1 - 1 ( α , y 1 ) + iF 2 - 1 ( α , y 2 ) - E [ τ 1 ] - iE [ τ 2 ] ) ( F 3 - 1 ( α , y 3 ) + iF 4 - 1 ( α , y 4 ) - E [ τ 3 ] - iE [ τ 4 ] ) * d α d Ψ 1 ( y 1 ) ⋯ d Ψ 4 ( y 4 ) , where F^-1 (α, y _i ) is the inverse uncertainty distribution of the uncertain variable f (y _i , τ _i ) and is determined by ϒ _i .

Remark 4. If a complex uncertain random variable reduces to a complex uncertain variable in above stipulation, then the results of Stipulation 4 are concluded.

Theorem 6. Suppose that ξ = τ₁ + iτ₂ is a complex uncertain variables, such that τ₁ and τ₂ are independent uncertain variables with uncertainty distributions Φ₁ and Φ₂, respectively. Then the variance of ξ is Var ( ξ ) = Cov ( ξ , ξ ) .

Proof. By invoking Stipulations 2 and 4, we have Cov ( ξ , ξ ) ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - ( E [ τ 1 ] + iE [ τ 2 ] ) ) ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - ( E [ τ 1 ] + iE [ τ 2 ] ) ) * d α ∫ 0 1 ( ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) + i ( Φ 2 - 1 ( α - E [ τ 2 ] ) ) ( ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) + i ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) ) * d α ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) 2 d α + ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) 2 ) d α Var ( ξ ) .

Theorem 7. If ξ₁ = τ₁ + iτ₂ and ξ₂ = τ₃ + iτ₄ are two complex uncertain variables, such that τ₁, τ₂, τ₃ and τ₄ are independent uncertain variables with uncertainty distributions Φ₁, Φ₂, Φ₃ and Φ₄, respectively, then we have Cov ( ξ 1 , ξ 2 ) = ( Cov ( ξ 2 , ξ 1 ) ) * .

Proof. By invoking Stipulation 4, we obtain

Cov ( ξ 1 , ξ 2 ) U = ∫ 0 1 [ Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - E [ τ 1 ] - iE [ τ 2 ] ] [ Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) - E [ τ 3 ] - iE [ τ 4 ] ] * d α = ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) ( Φ 3 - 1 ( α ) - E [ τ 3 ] ) d α + ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) ( Φ 4 - 1 ( α ) - E [ τ 4 ] ) d α + i ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) ( Φ 3 - 1 ( α ) - E [ τ 3 ] ) d α - i ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) ( Φ 4 - 1 ( α ) - E [ τ 4 ] ) d α .(1) Similarly,

Cov ( ξ 2 , ξ 1 ) = ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) ( Φ 3 - 1 ( α ) - E [ τ 3 ] ) d α + ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) ( Φ 4 - 1 ( α ) - E [ τ 4 ] ) d α - i ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) ( Φ 3 - 1 ( α ) - E [ τ 3 ] ) d α + i ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) ( Φ 4 - 1 ( α ) - E [ τ 4 ] ) d α .(2) Relations (1) and (2) imply that Cov ( ξ 1 , ξ 2 ) = ( Cov ( ξ 2 , ξ 1 ) ) * .

Theorem 8. If ξ₁ = τ₁ + iτ₂ and ξ₂ = τ₃ + iτ₄ are two complex uncertain variables, such that τ₁, τ₂, τ₃ and τ₄ are independent uncertain variables with uncertainty distributions Φ₁, Φ₂, Φ₃ and Φ₄, respectively, then we have Cov ( ξ 1 , ξ 2 ) = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) ) × ( Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) ) * α - E [ ξ 1 ] E [ ξ 2 * ] .

Proof. By using Stipulation 4, we have Cov ( ξ 1 , ξ 2 ) = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - ( E [ τ 1 ] + iE [ τ 2 ] ) ) ( Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) - ( E [ τ 3 ] + iE [ τ 4 ] ) ) * d α = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - ( E [ τ 1 ] + iE [ τ 2 ] ) ) ( Φ 3 - 1 ( α ) - i Φ 4 - 1 ( α ) - ( E [ τ 3 ] - iE [ τ 4 ] ) ) d α = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) ) ( Φ 3 - 1 ( α ) - i Φ 4 - 1 ( α ) ) d α - ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) ) d α ( E [ τ 3 ] - iE [ τ 4 ] ) - ( E [ τ 1 ] + iE [ τ 2 ] ) ∫ 0 1 ( Φ 3 - 1 ( α ) - i Φ 4 - 1 ( α ) ) d α + ( E [ τ 1 ] + iE [ τ 2 ] ) ( E [ τ 3 ] - iE [ τ 4 ] ) = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) ) ( Φ 3 - 1 ( α ) - i Φ 4 - 1 ( α ) ) d α - ( E [ τ 1 ] + iE [ τ 2 ] ) ( E [ τ 3 ] - iE [ τ 4 ] ) = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) ) ( Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) ) * d α - ( E [ τ 1 ] + iE [ τ 2 ] ) ( E [ τ 3 ] + iE [ τ 4 ] ) * = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) ) ( Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) ) * d α - E [ ξ 1 ] E [ ξ 2 * ] .

Theorem 9. If ξ₁ = τ₁ + iτ₂ and ξ₂ = τ₃ + iτ₄ are two complex uncertain variables, such that τ₁, τ₂, τ₃ and τ₄ are independent uncertain variables with uncertainty distributions Φ₁, Φ₂, Φ₃ and Φ₄, respectively, then Cov ( ξ 1 , ξ 2 ) = Cov ( τ 1 , τ 3 ) - iCov ( τ 1 , τ 4 ) + iCov ( τ 2 , τ 4 ) + Cov ( τ 2 , τ 4 ) .

Proof. By using Stipulations 4, we can obtain Cov ( ξ 1 , ξ 2 ) = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - E [ τ 1 ] - iE [ τ 2 ] ) ( Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) - E [ τ 3 ] - iE [ τ 4 ] ) * d α = ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] + i ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) ) ( ( Φ 3 - 1 ( α ) - E [ τ 3 ] ) - i ( Φ 4 - 1 ( α ) - E [ τ 4 ] ) ) d α ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) ( Φ 3 - 1 ( α ) - E [ τ 3 ] ) d α i ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) ( Φ 4 - 1 ( α ) - E [ τ 4 ] ) d α + i ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) ( Φ 3 - 1 ( α ) - E [ τ 3 ] ) d α - i 2 ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) ( Φ 4 - 1 ( α ) - E [ τ 4 ] ) d α Cov ( τ 1 , τ 3 ) - iCov ( τ 1 , τ 4 ) τ 3 ) + Cov ( τ 2 , τ 4 ) , where, the last equality are concluded by Stipulation 1.

Example 3. Consider the complex linear uncertain variables ξ₁ = τ₁ + iτ₂ and ξ₂ = τ₃ + iτ₄ such that τ₁, τ₂, τ₃ and τ₄ are independent uncertain variables with τ i ∼ L ( a i , b i ) , i = 1 , 2 , 3 , 4 . By using Theorem 9, we have Cov ( ξ 1 , ξ 2 ) = ( b 1 - a 1 ) ( b 3 - a 3 ) 12 - i ( b 1 - a 1 ) ( b 4 - a 4 ) 12 + i ( b 2 - a 2 ) ( b 3 - a 3 ) 12 + ( b 2 - a 2 ) ( b 4 - a 4 ) 12 .

Theorem 10. If ξ₁ = τ₁ + iτ₂, ξ₂ = τ₃ + iτ₄ and ξ₃ = τ₅ + iτ₆ are complex linear uncertain variables such that τ₁, τ₂, ⋯ , τ₆ are independent uncertain variables, then we obtain Cov ( ξ 1 + ξ 3 , ξ 2 ) = Cov ( ξ 1 , ξ 2 ) + Cov ( ξ 3 , ξ 2 ) .

Proof. Stipulation 4 implies that Cov ( ξ 1 + ξ 3 , ξ 2 ) = ∫ 0 1 [ ( ( Φ 1 - 1 ( α ) + Φ 3 - 1 ( α ) ) + i ( Φ 2 - 1 ( α ) + Φ 4 - 1 ( α ) ) ) - ( E [ τ 1 ] + E [ τ 3 ] + iE [ τ 2 ] + iE [ τ 4 ] ) ] ( Φ 5 - 1 ( α ) + i Φ 6 - 1 ( α ) - ( E [ τ 5 ] + iE [ τ 6 ] ) ) * d α = ∫ 0 1 [ ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) + i ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) + ( ( Φ 3 - 1 ( α ) - E [ τ 3 ] ) + i ( Φ 4 - 1 ( α ) - E [ τ 4 ] ) ) ] ( ( Φ 5 - 1 ( α ) - E [ τ 5 ] ) + i ( Φ 6 - 1 ( α ) - E [ τ 6 ] ) ) * d α = ∫ 0 1 ( ( Φ 1 - 1 ( α ) - E [ τ 1 ] + i ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) ) ( ( Φ 5 - 1 ( α ) - E [ τ 5 ] ) + i ( Φ 6 - 1 ( α ) - E [ τ 6 ] ) ) * d α + ∫ 0 1 [ Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) - ( E [ τ 3 ] + iE [ τ 4 ] ) ] [ Φ 5 - 1 ( α ) + i Φ 6 - 1 ( α ) - ( E [ τ 5 ] + iE [ τ 6 ] ) ] * d α = Cov ( ξ 1 , ξ 2 ) + Cov ( ξ 2 , ξ 3 ) .

Theorem 11. If ξ + 1 = τ₁ + iτ₂ and ξ₂ = τ₃ + iτ₄ are two complex uncertain random variables such that τ₁, τ₂, τ₃ and τ₄ are two independent uncertain variables, then we have | | Cov ( ξ 1 , ξ 2 ) | | ≤ Var ( ξ 1 ) Var ( ξ 2 ) .

Proof. By invoking Stipulation 2, we have Var ( ξ 1 ) = ∫ 0 1 ∥ Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - ( E [ τ 1 ] + iE [ τ 2 ] ) ∥ 2 d α = ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) 2 d α + ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) 2 d α . and Var ( ξ 2 ) = ∫ 0 1 | | Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) - ( E [ τ 3 ] + iE [ τ 4 ] ) | | 2 d α = ∫ 0 1 ( Φ 3 - 1 ( α ) - E [ τ 3 ] ) 2 d α + ∫ 0 1 ( Φ 4 - 1 ( α ) - E [ τ 4 ] ) 2 d α . Cauchy Schwarz inequality of complex valued functions implies that | | Cov ( ξ 1 , ξ 2 ) | | = | | ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - ( E [ τ 1 ] + iE [ τ 2 ] ) ) ( Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) - ( E [ τ 3 ] + iE [ τ 4 ] ) ) * d α | | ≤ ( ∫ 0 1 | | Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - ( E [ τ 1 ] + iE [ τ 2 ] ) | | 2 d α ) 1 2 ( ∫ 0 1 | | Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) - ( E [ τ 3 ] + iE [ τ 4 ] ) | | 2 d α ) 1 2 = Var ( ξ 1 ) Var ( ξ 2 ) .

Example 4. Suppose that τ₁, τ₂, τ₃ and τ₄ are independent uncertain variables such that τ i ∼ N ( e i , σ i ) , i = 1, 2, 3, 4 . By invoking Theorem 9, we have Cov ( ξ 1 , ξ 2 ) = Cov ( τ 1 , τ 3 ) + Cov ( τ 2 , τ 4 ) + i ( Cov ( τ 2 , τ 3 ) - Cov ( τ 1 , τ 4 ) ) = σ 1 σ 3 + σ 2 σ 4 + i ( σ 2 σ 3 - σ 1 σ 4 ) . Also, Stipulation 2 and Theorem 5 imply that Var ( ξ 1 ) = Var ( τ 1 ) + Var ( τ 2 ) = σ 1 2 + σ 2 2 Var ( ξ 2 ) = Var ( τ 3 ) + Var ( τ 4 ) = σ 3 2 + σ 4 2 . By taking norm, we have | | C ˜ ov ( ξ 1 , ξ 2 ) | | = [ ( σ 1 σ 3 + σ 2 σ 4 ) 2 + ( σ 2 σ 3 - σ 1 σ 4 ) 2 ] 1 2 = [ ( σ 1 2 σ 3 2 + σ 2 2 σ 4 2 ) + ( σ 2 2 σ 3 2 + σ 1 2 σ 4 2 ) ] 1 2 = [ ( σ 1 2 + σ 2 2 ) ( σ 3 2 + σ 4 2 ) ] 1 2 = Var ( ξ 1 ) Var ( ξ 2 ) .

Definition 16. Let τ₁ and τ₂ be two complex uncertain variables with expected value E [τ₁] and E [τ₂], respectively. Then the pseudo covariance is defined by C ˜ ov ( τ 1 , τ 2 ) = E [ ( τ 1 - E [ τ 1 ] ) ( τ 2 - E [ τ 2 ] ) ] .

Remark 5. If complex uncertain variables degenerate to uncertain ones in above definition, the result of Definition 8 is concluded.

Remark 6. It is mentioned that the subadditivity of uncertain measure concludes the difficulty of calculating of the covariance of two uncertain variables based on Definition 16. Thus, by inception of Stipulation 1 and 3, we present a formula for calculating the covariance of two uncertain variables as follows.

Stipulation 5. Suppose that ξ₁ = τ₁ + iτ₂ and ξ₂ = τ₃ + iτ₄ are two complex uncertain variables, such that τ₁, τ₂, τ₃ and τ₄ are independent uncertain variables with uncertainty distributions Φ₁, Φ₂, Φ₃ and Φ₄, respectively. C ˜ ov ( ξ 1 , ξ 2 ) = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - E [ τ 1 ] - iE [ τ 2 ] ) ( Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) - E [ τ 3 ] - iE [ τ 4 ] ) d α , where Φ i - 1 ( α ) is the inverse uncertainty distribution of the uncertain variables τ _i , i = 1, 2, 3, 4.

Theorem 12. Suppose that ξ = τ₁ + iτ₂ is a complex uncertain variables, such that τ₁ and τ₂ are independent uncertain variables with uncertainty distributions Φ₁ and Φ₂, respectively. Then the variance of ξ is V ˜ ar ( ξ ) = C ˜ ov ( ξ , ξ ) .

Proof. By invoking Stipulations 5 and 3, we have C ˜ ov ( ξ , ξ ) = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - ( E [ τ 1 ] + iE [ τ 2 ] ) ) ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) - ( E [ τ 1 ] + iE [ τ 2 ] ) ) d α = ∫ 0 1 ( ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) + i ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) ) ( ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) + i ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) ) d α = ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) 2 d α - ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) 2 d α + 2 i ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) d α = V ˜ ar ( ξ ) .

Theorem 13. Suppose that ξ = τ₁ + iτ₂ is a complex uncertain variable such that τ₁ and τ₂ are independent uncertain variables with uncertainty distributions Φ₁ and Φ₂, respectively. Then we have V ˜ ar ( ξ ) = Var ( τ 1 ) - Var ( τ 2 ) + 2 iCov ( τ 1 , τ 2 ) .

Proof. By using Stipulations 3 and 5, we have V ˜ ar ( ξ ) = ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) 2 d α - ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) 2 d α + 2 i ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) d α = Var ( τ 1 ) - Var ( τ 2 ) + 2 iCov ( τ 1 , τ 2 ) .

Example 5. Suppose that τ 1 ∼ N ( e 1 , σ 1 ) and τ 2 ∼ N ( e 2 , σ 2 ) are independent uncertain variables. Consider ξ = τ₁ + iτ₂ as a complex uncertain variable. By using Stipulations 3, we have

V ˜ ar ( ξ ) = ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) 2 d α - ∫ 0 1 ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) 2 d α + 2 i ∫ 0 1 ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) d α = ∫ 0 1 ( e 1 + 3 σ 1 π ln α 1 - α - e 1 ) 2 d α + ∫ 0 1 ( e 2 + 3 σ 2 π ln α 1 - α - e 2 ) 2 d α + 2 i ∫ 0 1 ( e 1 + 3 σ 1 π ln α 1 - α - e 1 ) ( e 2 + 3 σ 2 π ln α 1 - α - e 2 ) d α = 3 σ 1 2 π 2 ∫ 0 1 ( ln α 1 - α ) 2 d α - 3 σ 2 2 π 2 ∫ 0 1 ( ln α 1 - α ) 2 d α + 2 i 3 σ 1 σ 2 π 2 ∫ 0 1 ( ln α 1 - α ) 2 d α = σ 1 2 - σ 2 2 + 2 i σ 1 σ 2(3) Besides, by Example 2, we have

Var ( τ 1 ) = σ 1 2 , Var ( τ 2 ) = σ 2 2 , Cov ( τ 1 , τ 2 ) = σ 1 σ 2 .(4) By using relations (3) and (4), we conclude that V ˜ ar ( ξ ) = Var ( τ 1 ) - Var ( τ 2 ) + 2 iCov ( τ 1 , τ 2 ) .

Theorem 14. Suppose that ξ₁ = τ₁ + iτ₂ and ξ₂ = τ₃ + iτ₄ are two complex uncertain variables, such that τ₁, τ₂, τ₃ and τ₄ are independent uncertain variables with uncertainty distributions Φ₁, Φ₂, Φ₃ and Φ₄, respectively. Then we obtain C ˜ ov ( ξ 1 , ξ 2 ) = ∫ 0 1 ( Φ 1 - 1 ( α ) + i Φ 2 - 1 ( α ) ) × ( Φ 3 - 1 ( α ) + i Φ 4 - 1 ( α ) ) d α - E [ ξ 1 ] E [ ξ 2 ] .