Covariance and Pseudo-Covariance of Complex Uncertain Variables

Transcript

1 Covariance and Pseudo-Covariance of Complex Uncertain Variables Rong Gao 1, Hamed Ahmadzade 2, Mojtaba Esfahani 3 1. School of Economics and Management, Hebei University of Technology, Tianjin 341, China rgao@hebut.edu.cn 2. Department of Mathematical Sciences, University of Sistan and Baluchestan, Zahedan, Iran 3. Department of Mathematics, Velayat University, Iranshahr, Iran June 28, 218 Abstract Covariance is a measure to characterize the joint variability of two complex uncertain variables. Since, calculating covariance is not easy based on uncertain measure, we present two formulas for covariance and pseudo covariance of complex uncertain variables. For calculating the covariance of complex uncertain variables, some theorems are proved and several formulas are provided by using the inverse uncertainty distribution. The main results are explained by using several examples. Keywords: uncertainty theory, complex uncertain variable, covariance, pseudo covariance 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 [9]. After that, the characteristic function of such random variable was studied by Turin [8]. Furthermore, Goodman [3] established the statistical properties of complex variables. It is mentioned that probability theory is a tool to model randomness related to historical datafrequency). 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 degree. Thus, uncertainty theory was proposed by Liu [4] as Corresponding author. 1

2 a branch of mathematics. In order to model complex phenomena involving uncertainty, Peng [7] 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. [1] 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. [2]. And, several formulas for calculating these concepts were provided. 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 L be a σ-algebra on a nonempty set Γ. A set function M : L [, 1] is called an uncertain measure if it satisfies the following axioms: i) Normality Axiom) M{Γ} 1 for the universal set Γ. ii) Duality Axiom) M{Λ} + M{Λ c } 1 for any event Λ. iii) Subadditivity Axiom) For every countable sequence of events Λ 1, Λ 2,, we have { } M Λ i M {Λ i }. i1 iv) 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{ Λ k } M k {Λ k } k1 where Λ k are arbitrarily chosen events from L k for k 1, 2,, respectively. Definition 1 Liu [4]) An uncertain variable ξ is a function from an uncertainty space Γ, L, M) to the set of real numbers such that {ξ B} is an event for any Borel set B of real numbers. 2 i1 k1

3 Definition 2 Liu [4]) The uncertain variables ξ 1, ξ 2,, ξ n are said to be independent if { n } n M {ξ i B i } M {ξ i B i } i1 i1 for any Borel sets B 1, B 2,, B n of real numbers. Theorem 1 Liu [4]) Let ξ 1, ξ 2,, ξ n be independent uncertain variables, and f 1, f 2,, f n be measurable functions. Then f 1 ξ 1 ), f 2 ξ 2 ),, f n ξ n ) are independent uncertain variables. Definition 3 Liu [5]) The uncertainty distribution of an uncertain variable ξ is defined by x) M{ξ x} for any real number x. Definition 4 Liu [5]) An uncertainty distribution x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which < x) < 1, and lim x), lim x x) 1. x It is clear that a regular uncertainty distribution x) has an inverse function on the range of x with < x) < 1, and the inverse function α) exists on the open interval, 1). Definition 5 Liu [5]) Let ξ be an uncertain variable with regular uncertainty distribution x). Then the inverse function α) is called the inverse uncertainty distribution of ξ. Theorem 2 Liu [5]) Let ξ 1,, ξ n be independent uncertain variables with regular uncertainty distributions 1, 2,, n, respectively. If f is a strictly increasing function, then ξ fξ 1, ξ 2,, ξ n ) is an uncertain variable with inverse uncertainty distribution Ψ α) f 1 α),, α)). Definition 6 Liu [4]) The expected value of an uncertain variable ξ is defined by E[ξ] + M{ξ x}dx M{ξ x}dx provided that at least one of the two integrals is finite. n 3

4 Theorem 3 Liu [4]) Let ξ be an uncertain variable with uncertainty distribution. If the expected value exists, then E[ξ] + 1 x))dx x)dx. Liu and Ha [6] proposed a generalized formula for expected value by inverse uncertainty distribution. Theorem 4 Liu and Ha [6]) Let ξ 1, ξ 2,, ξ n be independent uncertain variables with regular uncertainty distributions 1, 2,, n, respectively. If fξ 1, ξ 2,, ξ n ) is strictly increasing with respect to ξ 1, ξ 2,, ξ m and strictly decreasing with respect to ξ m+1, ξ m+2,, ξ n, then the uncertain variable ξ fξ 1, ξ 2,, ξ n ) has an expected value E[ξ] f 1 α),, m α), m+1 1 α),, n 1 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 [5]. Definition 7 Liu [4]) If τ is an uncertain variable with finite expected value E[τ], then the variance of τ is defined by V arτ) E[τ E[τ]) 2 ]. Theorem 5 Yao [1]) If τ is an uncertain variable with finite expected value E[τ], then the variance of τ is V arτ) α) E[τ] ) 2. Definition 8 Zhao et al. [11]) Let τ 1 and τ 2 be two uncertain variables with the expected values E[τ 1 ] and E[τ 2 ], respectively. The covariance of τ 1 and τ 2 is defined by Covτ 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. [11] established the following stipulation for calculating the covariance of two uncertain variables as a linear function of expected values. Stipulation 1Zhao et al. [11]) Let τ 1 and τ 2 be two uncertain variables with finite expected values E[τ 1 ] and E[τ 2 ], respectively. Then the covariance of τ 1 and τ 2 is Covτ 1, τ 2 ) 1 α) E[τ 1] ) 2 α) E[τ 2] ), where, 1 α) and 2 α) are the inverse uncertainty distributions of τ 1 and τ 2, respectively. 4

5 Example 1 Zhao et al. [11]) Suppose that τ 1 and τ 2 are two uncertain variables τ 1 La 1, b 1 ) and τ 2 La 2, b 2 ). Then, Stipulation 1 implies that and Covτ 1, τ 2 ) b 2 a 2 )b 1 a 1 ), 12 V arτ 1 ) Covτ 1, τ 1 ) b 1 a 1 ) Example 2 Zhao et al. [11]) Suppose that τ 1 and τ 2 are two uncertain variables τ 1 N e 1, σ 1 ) and τ 2 N e 2, σ 2 ). Then, Stipulation 1 implies that Covτ 1, τ 2 ) σ 1 σ 2, and V arτ 1 ) Covτ 1, τ 1 ) σ 2 1. Definition 9 Peng [7]) A complex uncertain variable is a measurable function τ from an uncertainty space Γ, L, M) to the set of complex numbers, i.e., for any Borel set B of complex numbers, the set {τ B} {γ Γ τγ) B}, is an event. Definition 1 Peng [7]) The complex uncertainty distribution x) of a complex uncertain variable ξ is a function from C to [, 1] defined by for any complex z. z) M{Reξ) Rez), Imξ) Imz)} In order to model a complex uncertain variable, the expected value is proposed as below. Definition 11 Peng [7]) Let ξ be a complex uncertain variable. The expected value of ξ is defined by E[ξ] E[Reξ)] + ie[imξ)] provided that E[Reξ)] and E[Imξ)] are finite, where E[Reξ)] and E[Imξ)] are expected values of uncertain variables Reξ) and Imξ), respectively. Definition 12 Peng [7]) Suppose that ξ is a complex uncertain variable with expected value E[ξ]. Then the variance of ξ is defined by V arξ) E[ ξ E[ξ] 2 ]. 5

6 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. [2]) Let ξ τ 1 + iτ 2 be a complex uncertain variable with the real part τ 1 and imaginary part τ 2. The expected value of ξ exists and E[ξ] E[τ 1 ] + ie[τ 2 ]. Assume τ 1 and τ 2 are independent uncertain variables with regular uncertainty distributions 1 and 2, respectively. Then the variance of ξ is V arξ) [ 1 α) E[τ 1]) α) E[τ 2]) 2 ]. Definition 13 Let ξ be a complex uncertain variable with expected value E[ξ]. Then the pseudo-variance is defined by Ṽ ar[ξ] E[ξ E[ξ]) 2 ]. Stipulation 3 Chen et al. [2]) Let ξ τ 1 + iτ 2 be a complex uncertain variable with the real part τ 1 and imaginary part τ 2. The expected value of ξ exists and E[ξ] E[τ 1 ]+ie[τ 2 ]. Assume τ 1 and τ 2 are independent uncertain variables with uncertainty distributions 1 and 2, respectively. Then pseudo variance of ξ is Ṽ arξ) +2i 1 α) E[τ 1]) 2 2 α) E[τ 2]) 2) 1 α) E[τ 1]) 2 α) E[τ 2]). 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 τ 1 and τ 2 be two complex uncertain variables with expected value E[τ 1 ] and E[τ 2 ], respectively. Then the covariance is defined by Covτ 1, τ 2 ) E[τ 1 E[τ 1 ])τ 2 E[τ 2 ]) ] where τ 2 E[τ 2 ]) is the conjugate of the complex uncertain variable τ 2 E[τ 2 ]). 6

7 Remark 1 Since probability measure is additive, the covariance of two complex random variables can be written as follows: Covτ 1, τ 2 ) E[τ 1 τ 2 ] E[τ 1 ]E[τ 2 ]. For calculation the covariance as a linear of expected values, we present the following stipulation for the covariance of two complex uncertain variables. Stipulation 4 Suppose that ξ 1 τ 1 + iτ 2 and ξ 2 τ 3 + iτ 4 are two complex uncertain variables, such that τ 1, τ 2, τ 3 and τ 4 are independent uncertain variables with uncertainty distributions 1, 2, 3 and 4, respectively. Then we have Covξ 1, ξ 2 ) 1 α) + i 2 α) E[τ 1] ie[τ 2 ] ) 3 α) + i 4 α) E[τ 3] ie[τ 4 ] ), where i α) is the inverse uncertainty distribution of the uncertain variable τ i, i 1, 2, 3, 4. Theorem 6 Suppose that ξ τ 1 +iτ 2 is a complex uncertain variables, such that τ 1 and τ 2 are independent uncertain variables with uncertainty distributions 1, and 2, respectively. Then the variance of ξ is V arξ) Covξ, ξ). Proof: By invoking Stipulations 4 and 2, we have Covξ, ξ) 1 α) + i 2 α E[τ 1] + ie[τ 2 ])) ) 1 α) + i 2 α E[τ 1] + ie[τ 2 ])) ) 1 α) E[τ 1]) + i 2 α E[τ 2]) ) 1 α) E[τ 1]) + i 2 α) E[τ 2]) ) V arξ). 1 α) E[τ 1]) α) E[τ 2]) 2) 7

8 Theorem 7 If ξ 1 τ 1 + iτ 2 and ξ 2 τ 3 + iτ 4 are two complex uncertain variables, such that τ 1, τ 2, τ 3 and τ 4 are independent uncertain variables with uncertainty distributions 1, 2, 3 and 4, respectively, then we have Covξ 1, ξ 2 ) Covξ 2, ξ 1 )). Proof: By invoking Stipulation 4, we obtain Covξ 1, ξ 2 ) Similarly, Covξ 2, ξ 1 ) + + i i + i + i 1 α) + i 2 α) E[τ 1] ie[τ 2 ] ) 3 α) + i 4 α) E[τ 3] ie[τ 4 ] ) 1 α) E[τ 1] ) 3 α) E[τ 3] ) 2 α) E[τ 2] ) 4 α) E[τ 4] ) 2 α) E[τ 2] ) 3 α) E[τ 3] ) 1 α) E[τ 1] ) 4 α) E[τ 4] ). 1) 1 α) E[τ 1] ) 3 α) E[τ 3] ) Relations 1) and 2) imply that Covξ 1, ξ 2 ) Covξ 2, ξ 1 )). 2 α) E[τ 2] ) 4 α) E[τ 4] ) 2 α) E[τ 2] ) 3 α) E[τ 3] ) 1 α) E[τ 1] ) 4 α) E[τ 4] ). 2) 8

9 Theorem 8 If ξ 1 τ 1 + iτ 2 and ξ 2 τ 3 + iτ 4 are two complex uncertain variables, such that τ 1, τ 2, τ 3 and τ 4 are independent uncertain variables with uncertainty distributions 1, 2, 3 and 4, respectively, then we have Covξ 1, ξ 2 ) 1 α) + i 2 α)) 3 α) + i 4 α)) E[ξ1 ]E[ξ2]. 9

10 Proof: By using Stipulation 4, we have Covξ 1, ξ 2 ) 1 α) + i 2 α) E[τ 1] + ie[τ 2 ]) ) 3 α) + i 4 α) E[τ 3] + ie[τ 4 ]) ) 1 α) + i 2 α) E[τ 1] + ie[τ 2 ]) ) 3 α) i 4 α) E[τ 3] ie[τ 4 ]) ) 1 α) + i 2 α)) 3 α) i 4 α)) 1 α) + i 2 α)) E[τ 3 ] ie[τ 4 ]) E[τ 1 ] + ie[τ 2 ]) 3 + E[τ 1 ] + ie[τ 2 ])E[τ 3 ] ie[τ 4 ]) α) i 4 α) 1 α) + i 2 α)) 3 α) i 4 α)) E[τ 1 ] + ie[τ 2 ])E[τ 3 ] ie[τ 4 ]) 1 α) + i 2 α)) 3 α) + i 4 α)) E[τ 1 ] + ie[τ 2 ])E[τ 3 ] + ie[τ 4 ]) 1 α) + i 2 α)) 3 α) + i 4 α)) E[ξ 1 ]E[ξ 2]. 1

11 Theorem 9 If ξ 1 τ 1 + iτ 2 and ξ 2 τ 3 + iτ 4 are two complex uncertain variables, such that τ 1, τ 2, τ 3 and τ 4 are independent uncertain variables with uncertainty distributions 1, 2, 3 and 4, respectively, then Covξ 1, ξ 2 ) Covτ 1, τ 3 ) icovτ 1, τ 4 ) + icovτ 2, τ 4 ) + Covτ 2, τ 4 ). Proof: By using Stipulations 4 and 1, we can obtain Covξ 1, ξ 2 ) i + i 1 α) + i 2 α) E[τ 1] ie[τ 2 ] ) 3 α) + i 4 α) E[τ 3] ie[τ 4 ] ) 1 α) E[τ 1]) + i 2 α) E[τ 2]) ) 3 α) E[τ 3]) i 4 α) E[τ 4]) ) 1 α) E[τ 1]) 3 α) E[τ 3]) i 2 1 α) E[τ 1]) 4 α) E[τ 4]) 2 α) E[τ 2]) 3 α) E[τ 3]) 2 α) E[τ 2]) 4 α) E[τ 4]) Covτ 1, τ 3 ) icovτ 1, τ 4 ) + icovτ 2, τ 3 ) + Covτ 2, τ 4 ). Example 3 Consider the complex linear uncertain variables ξ 1 τ 1 +iτ 2 and ξ 2 τ 3 +iτ 4 such that τ 1, τ 2, τ 3 and τ 4 are independent uncertain variables with τ i La i, b i ), i 1, 2, 3, 4. By using Theorem 9, we have Covξ 1, ξ 2 ) b 1 a 1 )b 3 a 3 ) 12 +i b 2 a 2 )b 3 a 3 ) i b 1 a 1 )b 4 a 4 ) 12 + b 2 a 2 )b 4 a 4 ). 12

12 Theorem 1 If ξ 1 τ 1 + iτ 2, ξ 2 τ 3 + iτ 4 and ξ 3 τ 5 + iτ 6 are complex linear uncertain variables such that τ 1, τ 2,, τ 6 are independent uncertain variables, then we obtain Proof: Stipulation 4 implies that Covξ 1 + ξ 3, ξ 2 ) 1 α) + 3 Covξ 1 + ξ 3, ξ 2 ) Covξ 1, ξ 2 ) + Covξ 3, ξ 2 ). α)) + i 2 α) + 4 α)) E[τ 1] + E[τ 3 ] + ie[τ 2 ] + ie[τ 4 ]) ) 5 α) + i 6 α) E[τ 5] + ie[τ 6 ]) ) + 1 α) E[τ 1]) + i 2 α) E[τ 2]) + 3 α) E[τ 3]) + i 4 α) E[τ 4])) ) 5 α) E[τ 5]) + i 6 α) E[τ 6]) ) 1 α) E[τ 1] + i 2 α) E[τ 2])) ) 5 α) E[τ 5]) + i 6 α) E[τ 6]) ) 3 α) + i 4 α) E[τ 3] + ie[τ 4 ]) ) 5 α) + i 6 α) E[τ 5] + ie[τ 6 ]) ) Covξ 1, ξ 2 ) + Covξ 2, ξ 3 ). Theorem 11 If ξ + 1 τ 1 + iτ 2 and ξ 2 τ 3 + iτ 4 are two complex uncertain random variables such that τ 1, τ 2, τ 3 and τ 4 are two independent uncertain variables, then we have Covξ 1, ξ 2 ) V arξ 1 )V arξ 2 ). 12

13 Proof: By invoking Stipulation 2, we have V arξ 1 ) 1 α) + i 2 α) E[τ 1] + ie[τ 2 ]) 2 and V arξ 2 ) 1 α) E[τ 1]) α) E[τ 2]) 2). 3 α) + i 4 α) E[τ 3] + ie[τ 4 ]) 2 3 α) E[τ 3]) α) E[τ 4]) 2). Cauchy Schwarz inequality of complex valued functions implies that Covξ 1, ξ 2 ) 1 α) + i 2 α) E[τ 1] + ie[τ 2 ]) ) 3 α) + i 4 α) E[τ 3] + ie[τ 4 ]) ) ) 1 1 α) + i 2 α) E[τ 1] + ie[τ 2 ]) 2 2 V arξ 1 )V arξ 2 ). ) 1 3 α) + i 4 α) E[τ 3] + ie[τ 4 ]) 2 2 Example 4 Suppose that τ 1, τ 2, τ 3 and τ 4 are independent uncertain variables such that τ i N e i, σ i ), i 1, 2, 3, 4. By invoking Theorem 9, we have Covξ 1, ξ 2 ) Covτ 1, τ 3 ) + Covτ 2, τ 4 ) + icovτ 2, τ 3 ) Covτ 1, τ 4 )) σ 1 σ 3 + σ 2 σ 4 + iσ 2 σ 3 σ 1 σ 4 ). Also, Stipulation 2 and Theorem 5 imply that V arξ 1 ) V arτ 1 ) + Varτ 2 ) σ1 2 + σ2 2 V arξ 2 ) V arτ 3 ) + Varτ 4 ) σ3 2 + σ

14 By taking norm, we have Covξ 1, ξ 2 ) σ 1 σ 3 + σ 2 σ 4 ) 2 + σ 2 σ 3 σ 1 σ 4 ) 2) 1 2 σ 2 1σ σ 2 2σ 2 4) + σ 2 2σ σ 2 1σ 2 4) ) 1 2 σ σ 2 2)σ σ 2 4) ) 1 2 V arξ 1 )V arξ 2 ). Definition 15 Let τ 1 and τ 2 be two complex uncertain variables with expected value E[τ 1 ] and E[τ 2 ], respectively. Then the pseudo covariance is defined by Covτ 1, τ 2 ) E[τ 1 E[τ 1 ])τ 2 E[τ 2 ])]. Stipulation 5 Suppose that ξ 1 τ 1 + iτ 2 and ξ 2 τ 3 + iτ 4 are two complex uncertain variables, such that τ 1, τ 2, τ 3 and τ 4 are independent uncertain variables with uncertainty distributions 1, 2, 3 and 4, respectively. Covξ 1, ξ 2 ) 1 α) + i 2 α) E[τ 1] ie[τ 2 ]) 3 α) + i 4 α) E[τ 3] ie[τ 4 ]), where i α) is the inverse uncertainty distribution of the uncertain variables τ i, i 1, 2, 3, 4. Theorem 12 Suppose that ξ τ 1 + iτ 2 is a complex uncertain variables, such that τ 1 and τ 2 are independent uncertain variables with uncertainty distributions 1, and 2, respectively. Then the variance of ξ is Ṽ arξ) Covξ, ξ). 14

15 Proof: By invoking Stipulations 5 and 3, we have Covξ, ξ) 1 α) + i 2 α E[τ 1] + ie[τ 2 ])) ) 1 α) + i 2 α E[τ 1] + ie[τ 2 ])) ) 1 α) E[τ 1]) + i 2 α E[τ 2]) ) 1 α) E[τ 1]) + i 2 α) E[τ 2]) ) 1 α) E[τ 1]) 2 2 α) E[τ 2]) 2) + 2i Ṽ arξ). 1 α) E[τ 1]) 2 α) E[τ 2]) Theorem 13 Suppose that ξ τ 1 +iτ 2 is a complex uncertain variable such that τ 1 and τ 2 are independent uncertain variables with uncertainty distributions 1 and 2, respectively. Then we have Ṽ arξ) V arτ 1 ) + V arτ 2 ) + 2iCovτ 1, τ 2 ). Proof: By using Stipulations 3 and 5, we have Ṽ arξ) 1 α) E[τ 1]) 2 2 α) E[τ 2]) 2 + 2i 1 α) E[τ 1]) 2 α) E[τ 2]) V arτ 1 ) V arτ 2 ) + 2iCovτ 1, τ 2 ). Theorem 14 Suppose that ξ 1 τ 1 + iτ 2 and ξ 2 τ 3 + iτ 4 are two complex uncertain variables, such that τ 1, τ 2, τ 3 and τ 4 are independent uncertain variables with uncertainty 15

16 distributions 1, 2, 3 and 4, respectively. Then we obtain Covξ 1, ξ 2 ) Proof: Stipulation 5 implies that Covξ 1, ξ 2 ) 1 α) + i 2 α)) 3 α) + i 4 α)) E[ξ 1 ]E[ξ 2 ]. 1 α) + i 2 α) E[τ 1] + ie[τ 2 ]) ) 3 α) + i 4 α) E[τ 3] + ie[τ 4 ]) ) 1 α) + i 2 α)) 3 α) + i 4 α)) 1 α) + i 2 α)) E[τ 3 ] + ie[τ 4 ]) E[τ 1 ] + ie[τ 2 ]) 3 + E[τ 1 ] + ie[τ 2 ])E[τ 3 ] + ie[τ 4 ]) α) + i 4 α) 1 α) + i 2 α)) 3 α) + i 4 α)) E[τ 1 ] + ie[τ 2 ])E[τ 3 ] + ie[τ 4 ]) 1 α) + i 2 α)) 3 α) + i 4 α)) E[ξ 1 ]E[ξ 2 ]. Theorem 15 If ξ 1 τ 1 + iτ 2 and ξ 2 τ 3 + iτ 4 are two complex uncertain variables, such that τ 1, τ 2, τ 3 and τ 4 are independent uncertain variables with uncertainty distributions 1, 2, 3 and 4, respectively, then we have Covξ 1, ξ 2 ) Covτ 1, τ 3 ) + icovτ 1, τ 4 ) + icovτ 2, τ 4 ) Covτ 2, τ 4 ). 16

17 Proof: By invoking Stipulations 5 and 1, we have Covξ 1, ξ 2 ) + i + i 1 α) + i 2 α) E[τ 1] ie[τ 2 ] ) 3 α) + i 4 α) E[τ 3] ie[τ 4 ] ) 1 α) E[τ 1]) + i 2 α) E[τ 2]) ) 3 α) E[τ 3]) + i 4 α) E[τ 4]) ) 1 α) E[τ 1]) 3 α) E[τ 3]) + i 2 1 α) E[τ 1]) 4 α) E[τ 4]) 2 α) E[τ 2]) 3 α) E[τ 3]) 2 α) E[τ 2]) 4 α) E[τ 4]) Covτ 1, τ 3 ) + icovτ 1, τ 4 ) + icovτ 2, τ 3 ) Covτ 2, τ 4 ). Theorem 16 If ξ 1 τ 1 +iτ 2 and ξ 2 τ 3 +iτ 4 are two complex uncertain random variables such that τ 1, τ 2, τ 3 and τ 4 are two independent uncertain variables, then we obtain Proof: Stipulation 2 implies that V arξ 1 ) Covξ 1, ξ 2 ) V arξ 1 )V arξ 2 ). 1 α) + i 2 α) E[τ 1] + ie[τ 2 ]) 2 1 α) E[τ 1]) α) E[τ 2]) 2). 17

18 and V arξ 2 ) 3 α) + i 4 α) E[τ 3] + ie[τ 4 ]) 2 3 α) E[τ 3]) α) E[τ 4]) 2). By using Cauchy Schwarz inequality of complex valued functions, we have Covξ 1, ξ 2 ) 1 α) + i 2 α) E[τ 1] + ie[τ 2 ]) ) 3 α) + i 4 α) E[τ 3] + ie[τ 4 ]) ) ) 1 1 α) + i 2 α) E[τ 1] + ie[τ 2 ]) 2 2 V arξ 1 )V arξ 2 ). ) 1 3 α) + i 4 α) E[τ 3] + ie[τ 4 ]) 2 2 Example 5 Suppose that τ 1, τ 2, τ 3 and τ 4 are independent uncertain variables such that τ i N e i, σ i ), i 1, 2, 3, 4. By invoking Theorem 15, we have Covξ 1, ξ 2 ) Covτ 1, τ 3 ) Covτ 2, τ 4 ) + icovτ 2, τ 3 ) + Covτ 1, τ 4 )) σ 1 σ 3 σ 2 σ 4 + iσ 2 σ 3 + σ 1 σ 4 ). Also, using Stipulation 2 and Theorem 5 conclude that V arξ 1 ) V arτ 1 ) + Varτ 2 ) σ σ 2 2 V arξ 2 ) V arτ 3 ) + Varτ 4 ) σ σ

19 By taking norm, we have Covξ 1, ξ 2 ) σ 1 σ 3 σ 2 σ 4 ) 2 + σ 2 σ 3 + σ 1 σ 4 ) 2) 1 2 σ 2 1σ σ 2 2σ 2 4) + σ 2 2σ σ 2 1σ 2 4) ) 1 2 σ σ 2 2)σ σ 2 4) ) 1 2 V arξ 1 )V arξ 2 ). 4 Conclusions In this paper, the covariance and pseudo covariance of two complex uncertain variables were studied. Also, by using inverse uncertainty distributions, we presented two stipulations for calculating the covariance and pseudo covariance of two complex uncertain variables. Furthermore, the relationships among covariance, pseudo covariance and variance were investigated. References [1] Chen X.M., Ning Y.F., Wang X., Convergence of complex uncertain sequences, Journal of Intelligent and Fuzzy Systems, Vol. 6, , 216. [2] Chen X.M., Ning Y.F., Wang X. Formulas to calculate the variance and Pseudo-variance of complex uncertain variable, Proceedings of the fourth international forum on decision sciences, , 217. [3] Goodman N.R., Statistial analysis based on a certain multivariate complex Gaussian distribution, Ann Math Stat, Vol. 34, No. 1, , [4] Liu B., Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 27. [5] Liu B., Some research problems in uncertainty theory, Journal of Uncertain Systems, Vol.3, No.1, 3-1, 29. [6] Liu Y.H. and Ha M.H., Expected value of function of uncertain variables, Journal of Uncertain Systems, Vol.4, No.3, , 21. [7] Peng Z.X., Complex uncertain variables, Doctoral Dissertation, Tsinghua University, 212. [8] Turin G.L., The characteristic function of hermitian quadratic forms in complex normal variable, Biometrika, Vol. 47, , 196. [9] Wooding R.A., The multivariate distribution of complex normal variable, Biometrica, Vol. 43, ,

20 [1] Yao K., A formula to calculate the variance of uncertain variable, Soft Computing, Vol. 19, No 1., , 215. [11] Zhao M., Liu Y., Ralescu Dan A., Zhou J., The Covariance of Uncertain Variables: Definition and Calculation Formulae, Fuzzy Optimization and Decision Making, Vol. 27, No. 2, ,