Partial similarity measure of uncertain random variables and its application to portfolio selection

Abstract

A similarity measure determines the similarity between two objects. As important roles of similarity measure in chance theory, this paper introduces the concept of partial similarity measure for two uncertain random variables. Based on maximum similarity principle, partial similarity measure are used to recognize pattern problems. As an application in finance, partial similarity measure is applied to optimize portfolio selection of uncertain random returns via Monte-Carlo simulation and craw search algorithm.

Keywords

Chance theory uncertain random variable partial similarity measure portfolio selection pattern recognition

1 Introduction

A similarity measure characterizes the similarity between two objects and can be formulated as inverse of distance measure. In general, the similarity measures can be considered as the suitable decreasing functions of distance measures. In order to characterize similarity measure of two indeterminate phenomena, Zwich et al. [31] proposed several similarity measures of fuzzy sets and compared them in numerical examples. As an application of similarity measure in pattern recognition, Li and Cheng [17] presented the concept of similarity measure for intuitionistic fuzzy sets. After that Li et al. [16] established some similarity measures between two fuzzy variables via credibility measure. Also, it is mentioned several authors devoted their works to the topic of similarity measure, for instance [7 , 28].

As another type of modeling indeterminacy, Liu [20] presented uncertainty theory as a branch of mathematics. After that, Liu [20] presented the important concepts of uncertain variable and uncertainty distribution. Then, a sufficient and necessary condition for a function being an uncertainty distribution was established by Peng and Iwamura [26]. Moreover, In order to rank uncertain variables, Liu [22] proposed the concept of expected value. In order to measure similarity between two uncertain variables, Li and liu [19] proposed several similarity measures and applied them to pattern recognition.

In order to model phenomena including randomness and uncertainty simultaneously, Liu [23] presented the concept of uncertain random variables, chance distributions, expected values and variances. Following that, Liu [24] proved the operational law of uncertain random variable and the formula of expected value. Ahmadzade et al. [5] studied variance of an uncertain random variable through inverse uncertainty distribution. In order to measure the indeterminacy of an uncertain random variable, Ahmadzade et al. [3, 4] proposed the concepts of partial and quadratic partial entropies for uncertain random variables, respectively. After that, Ahmadzade and Gao [1] introduced the concept of covariance for two uncertain random variables and obtained a formula for calculating it via inverse uncertainty distributions. They proved that the variance of sum of uncertain random variables can be written as a linear function of variance and covariance function of uncertain random variables. By invoking this major, they optimize the portfolio selection problem of uncertain random returns via mean-variance-covariance model easily with Lingo software. But, in many real-world optimization problems, the objective function may be very complex and may have a large number of local optima.

In such cases, classical optimization methods are usually failed and may be trapped at local optima. Hence, due to the significant performance that new metaheuristic algorithms have shown in solving such problems, these algorithms are considered by many researchers. Some of the well-known new metaheuristic algorithms are as follows: Genetic Algorithms (GA) [13], Particle Swarm Optimization (PSO) [15], cuckoo search algorithm [29], Firefly Algorithm (FA) [30], etc. Recently, Based on the intelligent behavior of crows, a new metaheuristic algorithm called Crow Search Algorithm (CSA) [6] has been introduced for solving constrained optimization problems. CSA is a population-based optimization algorithm, and includes only two adjustable parameters (flight length and awareness probability) that makes it very attractive for various applications. Therefore, in this paper, we propose the concept of partial similarity measure and applied to portfolio selection model of uncertain random returns. Since the similarity-mean-variance model is very complex in optimization. The CSA algorithm is used to solve the similarity-mean-variance portfolio selection model.

The rest of this paper is organized as follows. In Section 2, some basic concepts in uncertainty theory and chance theory are reviewed. In Section 3, a definition of partial similarity measure of two uncertain random variables is presented and several examples are derived. Based on maximum similarity principle, we apply the partial similarity measure to the case of pattern recognition, in Section 4. Based on similarity-mean-variance model, we optimize the portfolio selection problem by Monte-Carlo simulation and craw search algorithm, in Section 5. Finally, some conclusions are provided in Section 6.

2 Preliminaries

In this section, we review some concepts of uncertain variables and uncertain random variables. And some relative properties are also reviewed.

2.1 Uncertain variables

In this subsection, we provide several definitions and elementary concepts of uncertainty theory that will be used in the next sections. For more details, the reader refers to [20, 21].

Let ℒ be a σ-algebra on a nonempty set Γ. A set function ℳ: ℒ → [0, 1] is called an uncertain measure if it satisfies three axioms (i), (ii), (iii):

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

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

(iii) (Subadditivity) For every countable sequence of events Λ₁, Λ₂, ⋯ , we have $\begin{matrix} M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}} . \end{matrix}$ Then the triple (Γ, ℒ, ℳ) is called uncertainty space. Next, the product uncertain measure was proposed by Liu [21] via the following axiom.

(iv) (Product Axiom) Let (Γ_k, ℒ_k, _ℳk) 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 arbitrarily chosen events from ℒ_k for k = 1, 2, ⋯ , respectively.

Definition 1. 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.

Definition 2. 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.

Theorem 1. 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 [21]) Let ξ be an uncertain random variable. Then its chance distribution is defined as $Φ (x) = M {ξ \leq x}$ for any x ∈ ℛ .

Definition 4.(Liu [21]) Let ξ be an uncertain variable with regular uncertainty distribution Φ (x). Then the inverse function Φ^-1 (x) is called the inverse uncertainty distribution of ξ.

Theorem 2. (Liu [21]) Let ξ₁, ⋯ , ξ_n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. If f (x₁, ⋯ , x_n) is strictly increasing with respect to x₁, ⋯ , x_m and decreasing with respect to x_m+1, ⋯ , x_n then ξ = f (ξ₁, ξ₂, ⋯ , ξ_n) is an uncertain variable with inverse uncertainty distribution $\begin{matrix} Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α) . \end{matrix}$

For characterize difference between two uncertain variables, Li and Liu [19] presented the concept of distance for two uncertain variables.

Definition 5. (Li and Liu [19]) Let τ₁ and τ₂ be two uncertain variable with uncertainty distributions Φ₁ and Φ2, respectively. The distance of τ₁ and τ₂ is defined by $\begin{matrix} D (τ_{1}, τ_{2}) = \int_{ℝ} | Φ_{1} (x) - Φ_{2} (x) | d x . \end{matrix}$

Since inverse uncertainty distributions play important roles in uncertainty theory, Gao et al. [11] presented the distance of two uncertain variables based on their inverse uncertainty distributions in the following theorem.

Theorem 3. (Gao et al. [12]) Let τ₁ and τ₂2 be two uncertain variable with uncertainty distributions Φ₁ and Φ₂, respectively. The distance of τ₁ and τ₂ is $\begin{matrix} D (τ_{1}, τ_{2}) = \int_{0}^{1} | Φ_{1}^{- 1} (α) - Φ_{2}^{- 1} (α) | d α . \end{matrix}$

In order to characterize the similarity of two uncertain variables via uncertainty distributions, Li and Liu [18] introduced the concept of similarity measure of two uncertain variables.

Definition 6. (Li and Liu [19]) Let X be a set of uncertain variables on the uncertainty space (Γ, ℒ, ℳ). A real valued function s (τ₁, τ₂) on the Cartesian product X × X is a similarity measure if, for any τ₁, τ₂ ; τ₃ ∈ X, it satisfies the following conditions:

i) 0 ≤ s (τ₁, τ₂) ≤1,

ii) s (τ₁, τ₂) =1 iff τ₁ = τ₂,

iii) s (τ₁, τ₂) = s (τ₂, τ₁) ,

iv) $M {τ_{1} \leq x} \leq M {τ_{2} \leq x} \leq M {τ_{3} \leq x} \forall x \in ℝ$ , then s (τ₂, τ₃) ≤ s (τ₁, τ₃) and s (τ₁, τ₂) ≤ s (τ₁, τ₃).

It is clear that we can calculate a similarity of two uncertain variables via their distance. Thus, we can express the relationship between similarity and distance by invoking several functions as follows: $\begin{matrix} g_{1} (t) & = exp (t), \\ g_{2} (t) & = \frac{1}{1 + t}, \\ g_{3} (t) & = 1 - \frac{1 - exp (- t)}{1 + exp (- t)}, \end{matrix}$ which, they are all strictly decreasing functions such that 0 ≤ g_i (t) ≤1, i = 1, 2, 3, ∀ t ≥ 0 . We can call g_i’s generator functions.

2.2 Uncertain random variable

The chance space is refer to the product (Γ, ℒ, ℳ) × (Ω, $A$ , Pr), in which (Γ, ℒ, ℳ) is an uncertainty space and (Ω, $A$ , Pr) is a probability space, respectively.

Definition 7. (Liu [23]) Let (Γ, ℒ, ℳ) × (Ω, $A$ , Pr) be a chance space, and Θ ∈ ℒ × $A$ be an uncertain random event. Then the chance measure of Θ is defined as $\begin{matrix} Ch {Θ} = \int_{0}^{1} \Pr {ω \in Ω ∣ M {γ \in Γ | (γ, ω) \in Θ} \\ \geq r} d r . \end{matrix}$

Liu [23] proved that a chance measure satisfies normality, duality, and monotonicity properties, that is (i) Ch {Γ × Ω} =1; (ii) Ch {Θ} + Ch {Θ^c} =1 for any event Θ; (iii) Ch {Θ₁} ≤ Ch {Θ₂} for any real number set Θ₁ ⊂ Θ₂ . Besides, Hou [14] proved the subadditivity of chance measure, that is, $Ch {⋃_{i = 1}^{\infty} Θ_{i}} \leq \sum_{i = 1}^{\infty} Ch {Θ_{i}}$ for a sequence of events Θ₁, Θ₂, ⋯ .

Definition 8. (Liu [23]) An uncertain random variable is a measurable function ξ from a chance space (Γ, ℒ, ℳ) × (Ω, $A$ , Pr) to the set of real numbers, i . e . , {ξ ∈ B} is an event for any Borel set B of real numbers.

To calculate the chance measure, Liu [24] presented a definition of chance distribution.

Definition 9. (Liu [24]) Let ξ be an uncertain random variable. Then its chance distribution is defined by $Φ (x) = Ch {ξ \leq x}$ for any x ∈ ℛ .

When an uncertain random variable degenerates into a random variable, the chance distribution becomes probability distribution and when an uncertain random variable degenerates into an uncertain variable, the chance distribution becomes uncertainty distribution.

Theorem 4. (Liu [24]) Let η₁, η₂, ⋯ , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, ⋯ , Ψ_m, respectively, and let τ₁, τ₂, ⋯ , τ_n be uncertain variables. Then the uncertain random variable $ξ = f (η_{1}, η_{2}, \dots, η_{m}, τ_{1}, τ_{2}, \dots, τ_{n})$ has a chance distribution $Φ (x) = \int_{R^{m}} F (x, y_{1}, \dots, y_{m}) d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m})$ where F (x, y₁, ⋯ , y_m) is the uncertainty distribution of uncertain variable f (η₁, η₂, ⋯ , η_m, τ₁, τ₂, ⋯, τ_n) for any real numbers y₁, y₂, ⋯ , y_m .

Definition 10. (Liu [24]) Let ξ be an uncertain random variable. Then its expected value is defined by $E [ξ] = \int_{0}^{+ \infty} Ch {ξ \geq r} d r - \int_{- \infty}^{0} Ch {ξ \leq r} d r$ provided that at least one of the two integrals is finite.

Let Φ denote the chance distribution of ξ. Liu [24] proved a formula to calculate the expected value of uncertain random variable with chance distribution, that is, $E [ξ] = \int_{0}^{+ \infty} (1 - Φ (x)) d x - \int_{- \infty}^{0} Φ (x) d x .$

Theorem 5. (Liu [24]) Let η₁, η₂, ⋯ , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, ⋯ , Ψ_m, respectively, and τ₁, τ₂, ⋯ , τ_n be independent uncertain variables (not necessarily independent), then the uncertain random variable ξ = f (η₁, ⋯ , η_m, τ₁, ⋯ , τ_n) has an expected value $\begin{matrix} E [ξ] = \int_{ℝ^{m}} E [f (y_{1}, \dots, y_{m}, τ_{1}, \dots, τ_{n})] d Ψ_{1} \dots d Ψ_{m} \end{matrix}$ -3ptwhere E [f (y₁, ⋯ , y_m, τ₁, ⋯ , τ_n)] is the expected value of the uncertain variable f (y₁, ⋯ , y_m, τ₁, ⋯ , τ_n) for any real numbers y₁, ⋯ , y_m .

Theorem 6. (Liu [24], Linearity of Expected Value Operator) Assume η₁ and η₂ are random variables (not necessarily independent), τ₁ and τ₂ are independent uncertain variables, and f₁ and f₂ are measurable functions. Then $E [f_{1} (η_{1}, τ_{1}) + f_{2} (η_{2}, τ_{2})] = E [f_{1} (η_{1}, τ_{1})] + E [f_{2} (η_{2}, τ_{2})] .$

Definition 11. (Liu [24]) Let ξ be an uncertain random variable with a finite expected value E [ξ]. Then the variance of ξ is $V [ξ] = E [(ξ - E [ξ])^{2}] .$

Ahmadzade et al. [5] derived two formulas for calculating variance and moment of uncertain random variables via inverse uncertainty distribution.

Theorem 7. (Ahmadzade et al. [5]) Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions ϒ₁, ϒ₂, . . . , ϒ_n, respectively. Then $ξ = f (η_{1}, . . ., η_{m}, τ_{1}, . . ., τ_{n})$ has a variance $\begin{matrix} V [ξ] = \int_{ℝ^{m}} \int_{0}^{1} (F^{- 1} (α, y_{1}, . . ., y_{m}) - E [ξ])^{2} \\ d α d Ψ_{1} (y_{1}) . . . d Ψ_{m} (y_{m}) \end{matrix}$ where F^-1 (x, y₁, . . . , y_m) is the inverse uncertainty distribution of the uncertain variable f (y₁, . . . , y_m, τ₁, . . . , τ_n) and is determined by ϒ₁, ϒ₂, . . . , ϒ_n.

Theorem 8. (Ahmadzade et al. [5]) Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions ϒ₁, ϒ₂, . . . , ϒ_n, respectively. Suppose ξ = f (η₁, . . . , η_m, τ₁, . . . , τ_n). Then $E [ξ^{k}] = \int_{ℝ^{m}} \int_{0}^{1} (F^{- 1} (α, y_{1}, . . ., y_{m}))^{k} d α d Ψ_{1} (y_{1}) . . . d Ψ_{m} (y_{m}) .$

Theorem 9. (Ahmadzade et al. [5]) Let η₁, η₂, ⋯, η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, ⋯ , τ_n be independent uncertain variables with uncertainty distributions ϒ₁, ϒ₂, . . . , ϒ_n respectively. Then the uncertain random variable $ξ = f (η_{1}, \dots, η_{m}, τ_{1}, \dots, τ_{n})$ has a variance $\begin{matrix} V [ξ] = \int_{ℝ^{m}} \int_{0}^{1} (F^{- 1} (α, y_{1}, . . ., y_{m}))^{k} d α d Ψ_{1} (y_{1}) . . . d Ψ_{m} (y_{m}), \end{matrix}$ where F^-1 (α, y₁, ⋯ , y_m) is the inverse uncertainty distribution of the uncertain random variable ξ and is determined by ϒ₁, ϒ₂, . . . , ϒ_n.

In order to characterize the joint variability of two uncertain random variable, Ahmadzade and Gao [1] presented the concept of covariance of two uncertain random variables.

Definition 12. (Ahmadzade and Gao [1]) Let ξ₁ and ξ₂ be two uncertain random variables. Then the covariance of ξ₁ and ξ₂ is defined by $\begin{matrix} Cov (ξ_{1}, ξ_{2}) = E [(ξ_{1} - E [ξ_{1}]) (ξ_{2} - E [ξ_{2}])] . \end{matrix}$

Stipulation 3 (Ahmadzade and Gao [1]) Let η₁ and η₂ be independent random variables with probability distributions Ψ₁ and Ψ₂, and let τ₁ and τ₂ be independent uncertain variables with uncertainty distributions ϒ₁ and ϒ₂ respectively. Suppose $ξ_{1} = f_{1} (η_{1}, τ_{1}), ξ_{2} = f_{2} (η_{2}, τ_{2}) .$ Then the covariance of ξ₂ and ξ₂ is $\begin{matrix} Cov (ξ_{1}, ξ_{2}) = \int_{ℝ^{2}} \int_{0}^{1} (F_{1}^{- 1} (α, y_{1}) - E [ξ_{1}]) \\ (F_{2}^{- 1} (α, y_{2}) - E [ξ_{2}]) d α d Ψ_{1} (y_{1}) d Ψ_{2} (y_{2}) . \end{matrix}$ Theorem 10. (Ahmadzade and Gao [1]) Let τ₂ and τ₂ be two uncertain variables with uncertainty distribution function Φ₁ and Φ₂, respectively. Let η₁ and η₂ be two random variables (not necessary independent) with joint distribution function Ψ (. ,.). If ξ₁ = η₁ + τ₁ and ξ₂ = η₂ + τ₂ then Cov (ξ₁, ξ₂) = Cov (η₁, η₂) + Cov (τ₁, τ₂) .

Theorem 11. (Ahmadzade and Gao [1]) Let τ₁, τ₂, ⋯ , τ_n be independent uncertain variables, and let eta₁, η₂, ⋯ , η_n be independent random variables with probability distribution functions Φ₁, Φ₂, ⋯ , Φ_n, respectively. Suppose $ξ_{1} = f_{1} (η_{1}, τ_{1}), ξ_{2} = f_{2} (η_{2}, τ_{2}), \dots, ξ_{n} = f_{n} (η_{n}, τ_{n}) .$ Then $\begin{matrix} V [ξ_{1} + \dots + ξ_{n}] = \sum_{i = 1}^{n} V [ξ_{i}] + 2 \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} Cov (ξ_{i}, ξ_{j}) . \end{matrix}$

3 Partial similarity measure of uncertain random variables

In this section, we want to review some distance measures in uncertainty theory and probability theory. By inception of Kullback Leibler divergence measure, Chen et al. presented the concept of cross(relative) entropy for two uncertain variables to characterize difference between two uncertain variables as follows.

Definition 13. (Chen et al. [8]) Let τ₁ and τ₂ be two uncertain variables with uncertainty distributions Φ₁ and Φ₂, respectively. As a distance measure, the cross entropy distance of τ₁ and τ₂ is defined by $\begin{matrix} D (τ_{1}, τ_{2}) = \int_{ℝ} (Φ_{1} (x) ln (\frac{Φ_{1} (x)}{Φ_{2} (x)}) + (1 - Φ_{1} (x)) \\ ln (\frac{1 - Φ_{1} (x)}{1 - Φ_{2} (x)})) d x . \end{matrix}$

Also, based on L_p measure, Gao et al. [12] proposed a distance measure for two uncertain variables as follows.

Definition 14. (Gao et al. [11]) Let ξ₁ and ξ₂ be two uncertain variable with chance distributions Φ₁ and Φ₂, respectively. The distance of ξ₁ and ξ₂ is defined by $\begin{matrix} D (ξ_{1}, ξ_{2}) = (\int_{ℝ} | Φ_{1} (x) - Φ_{2} (x) |^{p} d x)^{\frac{1}{p}} . \end{matrix}$

In many situations, we want to characterize indeterminacy associated to uncertain variables with the presence of random variables. Thus, we can present the concept of partial distance as follows.

Definition 15. Suppose that η₁ and η₂ are independent random variables, and suppose that τ₁ and η₂ are independent uncertain variables. Set $ξ_{1} = f_{1} (η_{1}, τ_{1}), ξ_{2} = f_{2} (η_{2}, τ_{2})$ which are two uncertain random variables. Then partial distance of two uncertain random variables ξ₁ and ξ₂ is $\begin{matrix} PD (ξ_{1}, ξ_{2}) = \int_{ℝ^{3}} (F_{1} (x, y_{1}) ln (\frac{F_{1} (x, y_{1})}{F_{2} (x, y_{2})}) \\ + (1 - F_{1} (x, y_{1})) ln (\frac{1 - F_{1} (x, y_{1})}{1 - F_{2} (x, y_{2})})) \\ d x d Ψ_{1} (y_{1}) d Ψ_{2} (y_{2}), \end{matrix}$ where F₁ (x, y₁) and F₂ (x, y₂) are the uncertainty distributions of uncertain variables f₁ (y₁, τ₁) and f₂ (y₂, τ₂) for any real numbers y₁ and y₂ respectively.

Remark 1. In above definition, if uncertain random variables reduce to uncertain ones, Definition 13 concludes.

Definition 16. Suppose that η₁ and η₂ are independent random variables, and suppose that τ₁ and η₂ are independent uncertain variables. Set $ξ_{1} = f_{1} (η_{1}, τ_{1}), ξ_{2} = f_{2} (η_{2}, τ_{2})$ which are two uncertain random variables. Then partial distance of two uncertain random variables ξ₁ and ξ₂ is $\begin{matrix} PD (ξ_{1}, ξ_{2}) = (\int_{ℝ^{3}} | F_{1} (x, y_{1}) - F_{2} (x, y_{2}) |^{p} \\ d x d Ψ_{1} (y_{1}) d Ψ_{2} (y_{2}))^{\frac{1}{p}}, \end{matrix}$ where F₁ (x, y₁) and F₂ (x, y₂) are the uncertainty distributions of uncertain variables f₁ (y₁, τ₁) and f₂ (y₂, τ₂) for any real numbers y₁ and y₂ respectively.

Remark 2. In above definition, if uncertain random variables reduce to uncertain ones, Definition 14 concludes.

Definition 17. (Ahmadzade et al. [2]) Suppose that η₁, η₂, ⋯ , η_m, η_m+1, ⋯ , η_n are independent random variables, and suppose that η₁, η₂, ⋯ , η_m, η_m+1, ⋯ , η_n are independent uncertain variables. Set $\begin{matrix} ξ_{1} = f_{1} (η_{1}, \dots, η_{m}, τ_{1}, \dots, τ_{m}), \\ ξ_{2} = f_{2} (η_{m + 1}, \dots, η_{n}, τ_{m + 1}, \dots, τ_{n}) \end{matrix}$ which are two uncertain random variables. Then partial distance of two uncertain random variables ξ₁ and ξ₂ is $\begin{matrix} PD [ξ_{1}, ξ_{2}] = \int_{ℝ^{n}} \int_{- \infty}^{\infty} | F_{1} (x, y_{1}, \dots, y_{m}) \\ - F_{2} (x, y_{m + 1}, \dots, y_{n}) | d x d Ψ_{1} (y_{1}) \dots d Ψ_{n} (y_{n}), \end{matrix}$ where F₁ (x, y₁, ⋯ , y_m) and F₂ (x, y_m+1, ⋯ , y_n) are the uncertainty distributions of uncertain variables f₁ (y₁, ⋯ , y_m, τ₁, ⋯ , τ_m) and f₂ (y_m+1, ⋯, y_n, τ_m+1, ⋯ , τ_n) for any real numbers y₁, ⋯, y_n, respectively.

Theorem 12. (Ahmadzade et al. [2]) Suppose that η₁, η₂, ⋯ , η_m, η_m+1, ⋯ , η_n are independent random variables, and suppose that τ₁, η₂, ⋯ , η_m, η_m+1, ⋯ , η_n are uncertain variables. Set $\begin{matrix} ξ_{1} = f_{1} (ξ_{1}, \dots, ξ_{m}, τ_{1}, \dots, τ_{m}), \\ ξ_{2} = f_{2} (η_{m + 1}, \dots, η_{n}, τ_{m + 1}, \dots, τ_{n}) \end{matrix}$ which are two uncertain random variables. Then partial distance of two uncertain random variables ξ₁ and ξ₂ is $\begin{matrix} PD [ξ_{1}, ξ_{2}] = \int_{ℝ^{n}} \int_{0}^{1} | F_{1}^{- 1} (α, y_{1}, \dots, y_{m}) - \\ F_{2}^{- 1} (α, y_{m + 1}, \dots, y_{n}) | d α d Ψ_{1} (y_{1}) \dots d Ψ_{n} (y_{n}) . \end{matrix}$

In order to characterize the similarity of two uncertain random variables via chance distributions, we introduce the concept of similarity measure of two uncertain variables.

Definition 18. Let X be a set of uncertain variables on the uncertainty space. A real valued function s (ξ₁, ξ₂) on the Cartesian product X × X is a similarity measure if, for any ξ₁, ξ₂ ; ξ₃ ∈ X, it satisfies the following conditions:

i) 0 ≤ s (ξ₁, ξ₂) ≤1,

ii) s (ξ₁, ξ₂) =1 iff ξ₁ = ξ₂,

iii) s (ξ₁, ξ₂) = s (ξ₂, ξ₁) ,

iv) $Ch {ξ_{1} \leq x} \leq Ch {ξ_{2} \leq x} \leq Ch {ξ_{3} \leq x} \forall x \in ℝ$ , then s (ξ₂, ξ₃) ≤ s (ξ₁, ξ₃) and s (ξ₁, ξ₂) ≤ s (ξ₁, ξ₃).

However, one question may arise. How much of similarity measure of uncertain random variables belong to uncertain variables? For this purpose, we introduce the concept of partial similarity measure for uncertain random variables.

By inception of the method in [19], we can characterize the similarity of two uncertain random variables via decreasing functions of distance of two uncertain random variables.

Definition 19. Suppose that ξ₁ and ξ₂ are two uncertain random variables with partial distance PD (ξ₁, ξ₂). Then partial similarity measures of ξ₁ and ξ₂ are defined by ${PS}_{i} (ξ_{1}, ξ_{2}) = g_{i} (PD [ξ_{1}, ξ_{2}]),$ where g₁ (t) = exp (- t) , $g_{2} (t) = \frac{1}{1 + t},$ and $g_{3} (t) = 1 - \frac{1 - exp (- t)}{1 + exp (- t)} .$

We prefer and select the distance measure which introduced in Definition 17 rather than the partial distances in Definitions 15 and 16. Since, the partial distance in Definition 17 can be written as an expectation of functions of random variables, we can compute this measure via Monte Carlo simulation. However, the partial distance in Definition 15 and 16 can not be expressed as an expectation of functions of random variables. Thus, we can not invoke Monte Carlo simulation. Therefor, computations of these partial distances are very difficult. Now, we want to explain Monte Carlo simulation for numerical computation of partial distance.

Suppose that η₁ and η₂ are independent random variables and suppose that τ₁ and τ₂ are independent uncertain variables. Consider ξ₁ = f₁ (η₁, τ₁) and ξ₂ = f₂ (η₂, τ₂) as two uncertain random variables. By Theorem 12, we have $\begin{matrix} PD [ξ_{1}, ξ_{2}] = \int_{ℝ^{2}} \int_{0}^{1} | F_{1}^{- 1} (α, y_{1}) - F_{2}^{- 1} (α, y_{2}) | \\ d α d Ψ_{1} (y_{1}) d Ψ_{2} (y_{2}) = E [| F_{1}^{- 1} (U, Z) - F_{2}^{- 1} (U, W) |], \end{matrix}$ -3ptwhere, U ∼ U (0, 1) and Z and W follow from probability distributions Ψ₁ and Ψ₂. In oreder to invoke Monte Carlo simulation, we act as follows.

First, we generate 3 samples (u₁, u₂, ⋯ , u_N), (z₁, z₂, ⋯ , z_ℳ) and (w₁, w₂, ⋯ , w_K) from probability distributions U (0, 1), Ψ₁ and Ψ₂ .

Second, compute $| F_{1}^{- 1} (u_{i}, z_{j}) - F_{2}^{- 1} (u_{i}, w_{l}) |$ , for i = 1, 2, ⋯ , N, j = 1, 2, ⋯ , ℳ and l = 1, 2, ⋯, K .

Finally, consider $\frac{1}{NMK} \sum_{i = 1}^{N} \sum_{j = 1}^{ℳ} \sum_{l = 1}^{K} | F_{1}^{- 1} (u_{i},$ $z_{j}) - F_{2}^{- 1} (u_{i}, w_{l}) |$ as an approximation for partial distance.

Example 1. Suppose that τ₁ and τ₂ are two independent uncertain variables with uncertainty distributions N (e₁, σ₁) and N (e₂, σ₂), respectively. Assume that η₁ and η₂ are two independent exponential random variable with probability distribution Exp (λ₁) and Exp (λ₂), respectively. Then the partial divergence of ξ₁ = η₁ + τ₁ and ξ₂ = η₂ + τ₂ is $\begin{matrix} PD [ξ_{1}, ξ_{2}] = \int_{ℝ^{2}} \int_{0}^{1} | F_{1}^{- 1} (α, y_{1}) - F_{2}^{- 1} (α, y_{2}) | d α \\ d Ψ_{1} (y_{1}) d Ψ_{2} (y_{2}) = \int_{ℝ^{2}} \int_{0}^{1} | y_{2} + e_{2} + \frac{σ_{2} \sqrt{3}}{π} \\ ln \frac{α}{1 - α} - y_{1} - e_{1} - \frac{σ_{1} \sqrt{3}}{π} ln \frac{α}{1 - α} | d α d y_{1} d y_{2} . \end{matrix}$ In special case, consider e₁ = 2, e₂ = 4, σ₁ = 2, σ₂ = 3, λ₁ = 4, λ₂ = 5. By using Monte-Carlo simulation, we obtain PD [ξ₁, ξ₂] =5.6 and consequently PS₁ (ξ₁, ξ₂) = g₁ (PD [ξ₁, ξ₂]) =0.003, PS₂ (ξ₁, ξ₂) = g₂ (PD [ξ₁, ξ₂]) =0.15, PS₃ (ξ₁, ξ₂) = g₃ (PD [ξ₁, ξ₂]) =0.007 .

4 Application of partial similarity measure to pattern recognition

In this section, we obtain several examples to characterize pattern recognition via partial similarity measure. Suppose that we have m patterns which be represented by uncertain random variables ξ_i, i = 1, 2, ⋯ , m . We want to recognize a sample ξ belongs to which patterns ξ_i ; i = 1, 2, ⋯ , m . Thus, we calculate partial similarities of ξ and ξ_i, i = 1, 2, ⋯ , m . By taking maximum, we have $PS (ξ_{i 0}, ξ) = max_{1 \leq i \leq m} PS (ξ_{i}, ξ) .$ Therefore, we can decide that the sample ξ belongs to the pattern ξ_i0 .

Example 2. Suppose that τ₁, τ₂, τ₃ and τ are independent uncertain variables with τ₁ ∼ N (2, 2) , τ₂ ∼ N (3 ;3) , τ₃ ∼ N (4, 4) and τ ∼ N (3, 2). Also, η₁, η₂, η₃ and η are independent random variables such that η₁ ∼ N (0, 4), η₂ ∼ N (0, 9) , η₃ ∼ N (0, 16) and η ∼ N (0, 25) . Consider three patterns which represented by the uncertain random variables ξ_i = τ_i + η_i, i = 1, 2, 3 . We want to recognize the sample ξ = τ + η. By using Monte-Carlo simulation, we have $\begin{matrix} PD [ξ_{1}, ξ] = \frac{1}{2 π \times 2 \times 5} \int_{ℝ^{2}} \int_{0}^{1} | (y_{1} + 2 + \frac{2 \sqrt{3}}{π} \\ ln \frac{α}{1 - α}) - (y + 3 + \frac{2 \sqrt{3}}{π} ln \frac{α}{1 - α}) | \\ exp (- \frac{1}{2 \times 4} y_{1}^{2} - \frac{1}{2 \times 25} y^{2}) d α d y_{1} d y = 0.295, \end{matrix}$ and consequently, PS₁ (ξ₁, ξ) = g₁ (PD [ξ₁, ξ]) =0.744, PS₂ (ξ₁, ξ) = g₂ (PD [ξ₁, ξ]) =0.77 and PS₃ (ξ₁, ξ) = g₃ (PD [ξ₁, ξ]) =0.85 . $\begin{matrix} PD [ξ_{2}, ξ] = \frac{1}{2 π \times 3 \times 5} \int_{ℝ^{2}} \int_{0}^{1} | (y_{2} + 3 + \frac{3 \sqrt{3}}{π} \\ ln \frac{α}{1 - α}) - (y + 3 + \frac{2 \sqrt{3}}{π} ln \frac{α}{1 - α}) | \\ exp (- \frac{1}{2 \times 9} y_{1}^{2} - \frac{1}{2 \times 25} y^{2}) d α d y_{2} d y = 0.2158, \end{matrix}$ and consequently, PS₁ (ξ₂, ξ) = g₁ (PD [ξ₂, ξ]) =0.80, PS₂ (ξ₂, ξ) = g₂ (PD [ξ₂, ξ]) =0.82 and PS₃ (ξ₂, ξ) = g₃ (PD (ξ₂, ξ]) =0.89 . $\begin{matrix} PD [ξ_{3}, ξ] = \frac{1}{2 π \times 4 \times 5} \int_{ℝ^{2}} \int_{0}^{1} | (y_{3} + 4 + \frac{4 \sqrt{3}}{π} \\ ln \frac{α}{1 - α}) - (y + 3 + \frac{2 \sqrt{3}}{π} ln \frac{α}{1 - α}) | \\ exp (- \frac{1}{2 \times 16} y_{3}^{2} - \frac{1}{2 \times 25} y^{2}) d α d y_{3} d y = 0.19, \end{matrix}$ and consequently, PS₁ (ξ₃, ξ) = g₁ (PD [ξ₃, ξ]) =0.82, PS₂ (ξ₃, ξ) = g₂ (PD [ξ₃, ξ]) =0.83 and PS₃ (ξ₃, ξ) = g₃ (PD [ξ₃, ξ]) =0.90 . Thus, we conclude that ξ should be belonged to ξ₃.

Example 3. Suppose that τ₁, τ₂, τ₃ and τ are independent uncertain variables with τ₁ ∼ LOGN (2, 2), τ₂ ∼ LOGN (3, 3) , τ₃ ∼ LOGN (4, 4) and τ ∼ LOGN (3, 2). Also, η₁, η₂, η₃ and η are independent random variables such that η₁ ∼ N (0, 4) , η₂ ∼ N (0, 9) , η₃ ∼ N (0, 16) and η ∼ N (0, 25) . Consider three patterns which represented by the uncertain random variables ξ_i = τ_iη_i, i = 1, 2, 3 . We want to recognize the sample ξ = ητ . By using Monte-Carlo simulation, we have $\begin{matrix} PD [ξ_{1}, ξ] = \frac{1}{2 π \times 2 \times 5} \int_{ℝ^{2}} \int_{0}^{1} | y_{1} (2 + \frac{2 \sqrt{3}}{π} ln \\ \frac{α}{1 - α}) - y (3 + \frac{2 \sqrt{3}}{π} ln \frac{α}{1 - α}) | exp (- \frac{1}{2 \times 4} \\ y_{1}^{2} - \frac{1}{2 \times 25} y^{2}) d α d y_{1} d y = 2.983624 e + 01, \end{matrix}$ and consequently, PS₁ (ξ₁, ξ) = g₁ (PD [ξ₁, ξ] ) = 1.102264e - 13, PS₂ (ξ₁, ξ) = g₂ (PD [ξ₁, ξ] ) = 3.242938e - 02 and PS₃ (ξ₁, ξ) = g₃ (PD [ξ₁, ξ] ) = 2.203793e - 13 . $\begin{matrix} PD [ξ_{2}, ξ] = \frac{1}{2 π \times 3 \times 5} \int_{ℝ^{2}} \int_{0}^{1} | (y_{2} (3 + \frac{3 \sqrt{3}}{π} ln \\ \frac{α}{1 - α}) - y (3 + \frac{2 \sqrt{3}}{π} ln \frac{α}{1 - α}) | exp (- \frac{1}{2 \times 9} \\ y_{1}^{2} - \frac{1}{2 \times 25} y^{2}) d α d y_{2} d y = 3.801694 e + 01, \end{matrix}$ and consequently, PS₁ (ξ₂, ξ) = g₁ (PD [ξ₂, ξ] ) = 3.086397e - 17, PS₂ (ξ₂, ξ) = g₂ (PD [ξ₂, ξ]) =2.562989e - 02 and PS₃ (ξ₂, ξ) = g₃ (PD [ξ₂, ξ] )= 0.000000e + 00 . $\begin{matrix} PD [ξ_{3}, ξ] = \frac{1}{2 π \times 4 \times 5} \int_{ℝ^{2}} \int_{0}^{1} | y_{3} (4 + \frac{4 \sqrt{3}}{π} ln \\ \frac{α}{1 - α}) - y (3 + \frac{2 \sqrt{3}}{π} ln \frac{α}{1 - α}) | exp (- \frac{1}{2 \times 16} \\ y_{3}^{2} - \frac{1}{2 \times 25} y^{2}) d α d y_{3} d y = 3.801694 e + 01, \end{matrix}$ and consequently, PS₁ (ξ₃, ξ) = g₁ (PD [ξ₃, ξ] ) = 3.086397e - 17, PS₂ (ξ₃, ξ) = g₂ (PD [ξ₃, ξ] ) = 2.562989e - 02 and PS₃ (ξ₃, ξ) = g₃ (PD [ξ₃, ξ] )= 0.000000e + 00 . Thus, we conclude that ξ should be belonged to ξ₁.

5 Application of partial similarity measure to portfolio selection

In this section, the Kapur distance minimization model is considered under uncertain situation with controllable random variables. And consequently, we can consider partial similarity maximization model for uncertain random returns. Suppose that we have n securities whose returns are uncertain variables ξ₁, ξ₂, ⋯ , ξ_n, respectively. Let ξ_i denote the investment proportions in security i, i = 1, 2, ⋯ , n. Then, the total return from the investment is p₁ξ₁ + p₂ξ₂ + ⋯ + p_nξ_n, which is an uncertain random variable. In addition, a priori uncertain random return ζ is available for an investor. Now, the partial similarity is maximized by the total return from the investment p₁ξ₁ + p₂ξ₂ + ⋯ + p_nξ_n from the priori uncertain random return ζ. To obtain best portfolio, a large expected value should be obtained, i.e. $E [p_{1} ξ_{1} + p_{2} ξ_{2} + \dots + p_{n} ξ_{n}] > β .$ To obtain under control risk, we should derive a small variance. Therefore, the portfolio selection model is presented as follows: ${\begin{matrix} max_{p_{i}} {PS}_{j} (p_{1} ξ_{1} + p_{2} ξ_{2} + \dots + p_{n} ξ_{n}, ζ) \\ subject to : \\ V [p_{1} ξ_{1} + p_{2} ξ_{2} + \dots + p_{n} ξ_{n}] \leq δ, \\ E [p_{1} ξ_{1} + p_{2} ξ_{2} + \dots + p_{n} ξ_{n}] \geq β, \\ p_{1} + p_{2} + . . . + p_{n} = 1, p_{i} \geq 0, i = 1, 2, . . ., n, \end{matrix}$ where, δ and β are predetermined parameters.

Suppose that η₁, η₂, ⋯ , η_n are independent random variables with joint probability distributions Ψ (y₁, ⋯ , y_n) , and let τ₁, τ₂, ⋯ , τ_n be independent uncertain variables with uncertainty distributions ϒ₁, ϒ₂, ⋯ , ϒ_n, respectively. Consider ξ₌f_i (η_i, τ_i) , i = 1, 2, ⋯ , n as uncertain random variables. Furthermore, consider $F_{i}^{- 1} (α, y_{i})$ is the inverse uncertainty distribution of the uncertain variable f_i (y_i, τ_i) and is determined by ϒ_i. By invoking Theorem 7, we can write $\begin{matrix} E [p_{1} ξ_{1} + \dots + p_{n} ξ_{n}] = \sum_{i = 1}^{n} p_{i} \int_{ℝ^{n}} \int_{0}^{1} F_{i}^{- 1} (α, y_{i}) \\ d α d Ψ (y_{1}, \dots, y_{n}) = \sum_{i = 1}^{n} p_{i} E [ξ_{i}] . \end{matrix}$ Furthermore, Theorem 10 implies that $\begin{matrix} V [p_{1} ξ_{1} + \dots + p_{n} ξ_{n}] = \sum_{i = 1}^{n} p_{i}^{2} V [ξ_{i}] \\ + 2 \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} p_{i} p_{j} Cov (ξ_{i}, ξ_{j}) . \end{matrix}$

Therefor, the above similarity-mean-variance problems is converted to ordinary(crisp) optimization problems as follows: ${\begin{matrix} max_{p_{i}} {PS}_{j} (p_{1} ξ_{1} + p_{2} ξ_{2} + \dots + p_{n} ξ_{n}, ζ) \\ subject to : \\ \sum_{i = 1}^{n} p_{i}^{2} V [ξ_{i}] + 2 \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} p_{i} p_{j} Cov (ξ_{i}, ξ_{j}) \leq δ, \\ \sum_{i = 1}^{n} p_{i} E [ξ_{i}] \geq β, \\ p_{1} + p_{2} + . . . + p_{n} = 1, p_{i} \geq 0, i = 1, 2, . . ., n, \end{matrix}$

As an especial case, suppose that we have n securities as a mixture of historical markets with random returns η_i and new markets with uncertain returns τ_i. In fact, we have n securities with uncertain random returns ξ_i = η_i + τ_i, i = 1, ⋯ , n. Also, assume that η₁, η₂, ⋯ , η_n are random variables with expected value vector μ′ = (e₁, ⋯ , e_n) and variance-covariance matrix Σ₁ as follows $Σ_{1} = [\begin{matrix} σ_{1}^{2} & σ_{12} & \dots & σ_{1 n} \\ σ_{12} & σ_{2}^{2} & \dots & σ_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ σ_{1 n} & σ_{2 n} & \dots & σ_{n}^{2} \end{matrix}]$ and let τ₁, τ₂, ⋯ , τ_n be independent uncertain variables with expected value vector μ′ = (μ₁, ⋯ , μ_n) and variance-covariance matrix Σ₂ as follows $Σ_{2} = [\begin{matrix} ϱ_{1}^{2} & ϱ_{12} & \dots & ϱ_{1 n} \\ ϱ_{12} & ϱ_{2}^{2} & \dots & ϱ_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ ϱ_{1 n} & ϱ_{2 n} & \dots & ϱ_{n}^{2} \end{matrix}]$

Consider ξ_i = η_i + τ_i, i = 1, 2, ⋯ , n, as uncertain random variables. Theorems 7 and 9 imply that the uncertain random variables ξ₁, ⋯ , ξ_n have expected value vector μ′ = (e₁ + μ₁, ⋯ , e_n + μ_n) and the variance-covariance matrix Σ as follows: $Σ_{2} = [\begin{matrix} σ_{1}^{2} + ϱ_{1}^{2} & σ_{12} + ϱ_{12} & \dots & σ_{1 n} + ϱ_{1 n} \\ σ_{12} + ϱ_{12} & σ_{2}^{2} + ϱ_{2}^{2} & \dots & σ_{2 n} + ϱ_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ σ_{1 n} + ϱ_{1 n} & σ_{2 n} + ϱ_{2 n} & \dots & σ_{n}^{2} + ϱ_{n}^{2} \end{matrix}]$

Therefore, the mean variance models can be written as follows: ${\begin{matrix} max_{p_{i}} {PS}_{j} (p_{1} ξ_{1} + p_{2} ξ_{2} + \dots + p_{n} ξ_{n}, ζ) \\ subject to : \\ \sum_{i = 1}^{n} p_{i}^{2} (σ_{i}^{2} + ϱ_{i}^{2}) + 2 \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} p_{i} p_{j} (σ_{ij} + ϱ_{ij}) \leq δ, \\ \sum_{i = 1}^{n} p_{i} (μ_{i} + e_{i}) \geq β, \\ p_{1} + p_{2} + . . . + p_{n} = 1, p_{i} \geq 0, i = 1, 2, . . ., n, \end{matrix}$

Example 4. Assume there are five securities in the market and their returns are uncertain random variables given in Table 1, where ξ_i = η_i + τ_i, i = 1, 2, ⋯ , n . Suppose the priori uncertain random return ζ = η_* + τ_* in Table 2. the investor sets δ = 5 for the variance of the total return, and β = 0.2 for the expected value. We want to optimize the portfolio selection model by maximizing the partial similarity measure as follows: ${\begin{matrix} max_{p_{i}} {PS}_{j} (p_{1} ξ_{1} + p_{2} ξ_{2} + \dots + p_{n} ξ_{n}, ζ) \\ subject to : \\ 4.16 p_{1}^{2} + 9.25 p_{2}^{2} + 1.26 p_{3}^{2} + 9.04 p_{4}^{2} + 4.09 p_{5}^{2} + 12 p_{1} p_{2} + 6 p_{1} p_{3} \\ + 12 p_{1} p_{4} + 6 p_{1} p_{5} + 9 p_{2} p_{3} + 18 p_{2} p_{4} + 12 p_{2} p_{5} + 9 p_{3} p_{4} + 6 p_{3} p_{5} + 12 p_{4} p_{5} \leq δ, \\ 1.3 p_{1} + 2.4 p_{2} + 1.6 p_{3} + 1.1 p_{4} + 1.4 p_{5} \geq β, \\ p_{1} + p_{2} + . . . + p_{n} = 1, p_{i} \geq 0, i = 1, 2, . . ., n, j = 1, 2, 3 . \end{matrix}$

Table 1
Uncertain random returns

No Uncertain term Random term Inverse of uncertainty distribution

1 $τ_{1} \sim N (1, 2)$ $η_{1} \sim N (0.3, 0.16)$ $F_{1}^{- 1} (α, y_{1}) = y_{1} +$ $(1 + \frac{2 \sqrt{3}}{π} ln (\frac{α}{1 - α}))$

2 $τ_{2} \sim N (2, 3)$ $η_{2} \sim N (0.4, 0.25)$ $F_{2}^{- 1} (α, y_{2}) = y_{2} +$ $(2 + \frac{3 \sqrt{3}}{π} ln (\frac{α}{1 - α}))$

3 $τ_{3} \sim N (1.5, 1.5)$ $η_{3} \sim N (0.1, 0.01)$ $F_{3}^{- 1} (α, y_{3}) = y_{3} +$ $(1.5 + \frac{1.5 \sqrt{3}}{π} ln (\frac{α}{1 - α}))$

4 $τ_{4} \sim N (1, 3)$ $η_{4} \sim N (0.1, 0.04)$ $F_{4}^{- 1} (α, y_{4}) = y_{4} +$ $(1 + \frac{3 \sqrt{3}}{π} ln (\frac{α}{1 - α}))$

5 $τ_{5} \sim N (1.2, 2)$ $η_{5} \sim N (0.2, 0.09)$ $F_{5}^{- 1} (α, y_{5}) = y_{5} +$ $(1.2 + \frac{2 \sqrt{3}}{π} ln (\frac{α}{1 - α}))$

Table 2

Priori uncertain random return

No	Uncertain term	Random term	Inverse of uncertainty distribution
1	$τ_{*} \sim N (0, 1)$	$η_{*} \sim N (0, 1)$	$F_{}^{- 1} (α, y_{}) = y_{*} + \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}$

Now, we want to solve the portfolio selection model based on PS₁ = g₁ (PD). Hence, the optimal solutions are shown in Table 3. And the partial similarity measure of the total return with the priori return is 0.326071. Second, the optimal solutions based on PS₂ = g₂ (PD) are shown in Table 4. And the partial measure of the total return with the priori return is 0.486053 . Finally, based on PS₃ = g₃ (PD) the optimal solutions are shown in Table 5. And the partial measure of the total return with the priori return is 0.472281 . Comparison of different similarities is shown in Table 6. Thus, we can conclude the optimal solutions based on g₂ are near to ideal prior return.

Table 3

Proportion of portfolio on securities based on PS₁

No	1	2	3	4	5
Proportion of portfolio	0.55	0.02	0.06	0.07	0.3

Table 4

Proportion of portfolio on securities based on PS₂

No	1	2	3	4	5
Proportion of portfolio	0.54	0.02	0.23	0.05	0.16

Table 5

Proportion of portfolio on securities based on PS₃

No	1	2	3	4	5
Proportion of portfolio	0.27	0	0.06	0.2	0.47

Table 6

Similarity measures based on generator functions

Generator function	g ₁	g ₂	g ₃
Partial similarity	0.326071	0.486053	0.472281

Figure 1 displays the convergence curves of the goal function by invoking g₁, g₂ and g₃. As the iteration increases, PS₂ (based on g₂) converges faster than PS₁ and PS₃. Furthermore, since the partial similarity of the portfolio with the priori return based on g₂ is greater than the other than the other partial similarity measures based on g₁ and g₃, we prefer the partial similarity measure (PS₂) to optimize any similar portfolio selection model.

Fig. 1

Convergence curve of different similarity measures.

6 Conclusions

This paper proposed a new definition of partial similarity measure for uncertain random variables. Based on this definition, several properties of this concept were derived. Theoretically, the results in this paper extend the existing results in [19]. As an application, we planed to investigate portfolio selection of uncertain random variables involving uncertain factors(new markets) and under control random variables (historical markets) via mean-variance constraint model. As a future work, we can optimize portfolio selection of uncertain random returns based on partial similarity measure as a goal function and other types of constraints such as skewness, entropy and expectation. Also, future researches will cover the applications of similarity measure of uncertain random variables in the fields of image compression, clustering analysis, uncertain machine scheduling problem and so on.

Footnotes

Acknowledgments

This work was supported by the Natural Science Foundation of Hebei Province (No. F2020202056) and Key Project of Hebei Education Department (No. ZD2020125). The authors wish to thank the Editor-in-Chief, Associate Editor and anonymous referees for their constructive comments and insightful suggestions that aided in the improvement of this article. All computations were carried out by MATLAB R2015a in Windows Server 2016 Standard of desktop PC machine with Intel(R) Xeon(R) CPU E7-8890 v4 @ 2.20GHz, 2195 MHz, 24 Core(s), 24 Logical Processor(s) and 16.0 GB RAM. The computer program is available from the third author upon request.

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