Partial entropy of uncertain random variables

Abstract

In this paper, we mainly propose a definition of partial entropy for uncertain random variables. In fact, partial entropy is a tool to characterize how much of entropy of an uncertain random variable belongs to the uncertain variable. Furthermore, some properties of partial quadratic entropy are investigated such as positive linearity. Finally, some other types of partial entropies are studied.

Keywords

Chance theory uncertain random variable entropy partial entropy

1 Introduction

In real life, we are always around indeterminacy which means that the outcomes of events cannot be exactly predicted in advance, such as rolling a die, stock price, coal reserve, and strength of bridge. For describing indeterminate phenomena, we introduce probability theory as a commonly used tool. However, probability theory is valid under the assumption that the estimated probability distribution is close enough to the real frequency. For obtaining estimated probability distribution by statistic method, we should possess lots of observed data. However, it is not easy for us to obtain the observed data due to economic, technical or some other reasons. In this case, we have to invite some domain experts to estimate their belief degrees that possible events may occur. While Nobelist Kahneman and Tversky [15] asserted that human beings tend to overweight the unlikely events. That is, belief degrees has a larger range of values than true frequencies. In this circumstance, probability theory is not enough to model human beings’ belief degrees. Then Zadeh [36] founded fuzzy set theory to model fuzziness, however a series of paradoxes presented by Liu [21], which implies that fuzzy set theory is not suitable to model this type of uncertain phenomena.

In order to model human uncertainty, another type of indeterminacy, uncertainty theory was introduced by Liu [18] and refined by Liu [19], which is a branch of mathematics on the basis of the normality, duality, subadditivity, and product axioms. Then some concepts were proposed in [18], such as uncertain measure for modeling belief degrees, uncertain variable for describing uncertain quantities, uncertainty distribution for describing uncertain variables and expected value for ranking uncertain variables. Nowadays uncertainty theory is almost complete in theoretical aspect. In fact, it is also well developed in many fields and many scholars have done lost of work such as [5 , 35].

Entropy is a commonly used tool to measure the degree of uncertainty in information sciences. It was first proposed by Shannon [29] for random variables in 1948. In real world, there is a large number of probability distributions with respect to one variable due to lack of information. For solving this problem, Jaynes [12] presented maximum entropy principle, that is, we choose the proper one from all the probability distributions with common expected value and variance by using maximum entropy. Inspired by the Shannon entropy of random variables, Zadeh [36] introduced fuzzy entropy to quantify the degree of fuzziness of a fuzzy variable in 1968. After that, it has been studied by many researchers such as De Luca and Termini [4], Kaufmann [13], Yager [33], Kosko [14], Bhandari and Pal [1]. In 2009, Liu [19] defined entropy for uncertain variable. Then some properties of entropy for uncertain variables were investigated by Dai and Chen [3], and the maximum entropy principle for uncertain variable was proposed by Chen and Dai [2]. Following that, Tang and Gao [32] presented triangular entropy and Ning et al. [26] employed such entropy to portfolio selection. In addition, Yao, Gao and Dai [34] proposed sine entropy for uncertain variable and studied its main properties such as positive linearity.

However, in many cases, randomness and uncertainty exist simultaneously in a complex system. Inspired by Kwakernaak [16, 17], Puri and Ralescu [27], Kruse and Meyer [28], and Liu and Liu [24, 25], Liu [22] first introduced chance theory for modeling the complex phenomena including uncertainty and randomness. Then some basic concepts were presented in [22] such as chance measure, uncertain random variable, chance distribution, and expected value. Following that, Liu [23] provided the operational law for calculating a monotone function of uncertain random variables, and gave the formula to calculate expected value. Sheng et al. [30] introduced the concept of entropy of uncertain random variables with application to minimum spanning tree problem. As contributions to chance theory, many scholars done a lot of work such as [6 , 9].

In a complex system, the combination of uncertain variables and random variables always has a complex way. For describe the uncertainty of information taken by a function of uncertain variables and random variables, a new concept of partial entropy is put forward. And we provide the formula for calculating the partial entropy by using inverse uncertainty distribution. The rest of this paper is organized as follows. In Section 2, some concepts of uncertainty variables and uncertain random variables are reviewed. In Section 3, a definition of partial entropy of uncertain random variable is proposed and some properties are presented. As supplements, other types of partial entropies are studied in Section 4. Finally, some conclusions are given in Section 5.

2 Preliminaries

In this section, we will review some concepts and properties with respect to uncertain variables and uncertain random variables.

2.1 Uncertain variables

In this subsection, we will introduce several elementary concepts and properties concerning uncertain variables.

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 the following three axioms:

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

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

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

Then we use (Γ, ℒ, ℳ) to represent an uncertainty space. As an essential difference compared with probability theory, the product axiom was put forward by Liu [19] on a product σ-algebra ℒ as follows:

(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 arbitrarily chosen events from ℒ_k for k = 1, 2, ⋯, respectively.

Definition 2.1. (Liu [18]) 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 2.2. (Liu [19]) 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 2.1. (Liu [19]) 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 2.3. (Liu [19]) 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 2.2. (Liu [19]) Let ξ₁, ⋯, ξ_n be independent uncertain variables with regular 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, respectively, then ξ = f (ξ₁, ξ₂, ⋯, ξ_n) is an uncertain variable with inverse uncertainty distribution $\begin{matrix} Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) . \end{matrix}$

Definition 2.4. (Liu [19]) Suppose that ξ is an uncertain variable with uncertainty distribution Φ. Then its entropy is defined by $\begin{matrix} H [ξ] = \int_{- \infty}^{\infty} S (Φ (x)) d x, \end{matrix}$ where S (t) = - t ln t - (1 - t) ln(1 - t).

Theorem 2.3. (Dai and Chen [3]) Let ξ be an uncertain variable with regular uncertainty distribution Φ. Then $\begin{matrix} H [ξ] = \int_{0}^{1} Φ^{- 1} (α) ln \frac{α}{1 - α} d α . \end{matrix}$

2.2 Uncertain random variables

The chance space refers 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.

Definition 2.5. (Liu [22]) 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 [22] 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 [11] 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 2.6. (Liu [22]) 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.

Definition 2.7. (Liu [23]) Let ξ be an uncertain random variable. Then its chance distribution is defined by $Φ (x) = Ch {ξ \leq x}$ for any x ∈ ℛ.

Remark 2.1. When an uncertain random variable degenerates to an uncertain variable, the chance distribution becomes uncertainty distribution of an uncertain variable. When an uncertain random variable degenerates to a random variable, the chance distribution becomes probability distribution of a random variable.

Theorem 2.4. (Liu [23]) Let η₁, η₂, ⋯, η_m be independent random variables with probability distributions Ψ₁, Ψ₂, ⋯, Ψ_m, respectively, and τ₁, τ₂, ⋯, τ_n be uncertain variables. Then the uncertain random variable ξ = f (η₁, η₂, ⋯, η_m, τ₁, τ₂, ⋯, 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 2.8. (Liu [23]) 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.

3 Partial entropy of uncertain random variables

Sheng et al. [30] introduced the concept of entropy for uncertain random variables and applied this concept to minimum spanning tree problem. First, we review their definition for uncertain random variables.

Definition 3.1. [30] Let ξ be an uncertain random variable with chance distribution Φ (x). Then its entropy is defined by $\begin{matrix} H [ξ] = \int_{- \infty}^{\infty} S (Φ (x)) d x, \end{matrix}$ where S (t) = - t ln t - (1 - t) ln(1 - t).

However, one question may arise. How much of entropy of uncertain random variable associated to uncertain variable? For solving this problem, we introduce the concept of partial entropy.

Definition 3.2. Suppose that η₁, η₂, ⋯, η_m are independent random variables and τ₁, τ₂, ⋯, τ_m are uncertain variables, also ξ = f (η₁, ⋯, η_m, τ₁, ⋯, τ_m) is an uncertain random variable. Partial entropy of uncertain random variable ξ is defined as following $\begin{matrix} PH [ξ] \\ = \int_{ℝ^{m}} \int_{- \infty}^{+ \infty} S (F (x, y_{1}, \dots, y_{m})) d x d Ψ_{1} (y_{1}) \\ \dots d Ψ_{m} (y_{m}), \end{matrix}$ where S (t) = - t ln t - (1 - t) ln(1 - t) and F (x, y₁, ⋯, y_m) is the uncertainty distribution of uncertain variable f (y₁, ⋯, y_m, τ₁, ⋯, τ_m) for any real numbers y₁, ⋯, y_m.

Remark 3.1. Partial entropy measures how much the entropy of an uncertain random variable belongs to the uncertain variable. Furthermore, we know that a random variable has no term of uncertain variable, so when the uncertain random variable degenerates to random variable, the partial entropy is zero. When the uncertain random variables degenerate to uncertain variables, the partial entropy becomes the entropy of uncertain variables (Definition 2.4).

Theorem 3.1. Let η₁, η₂, ⋯, η_m be independent random variables with probability distributions Ψ₁, Ψ₂, ⋯, Ψ_m, and τ₁, τ₂, ⋯, τ_n be independent uncertain variables with uncertainty distributions ϒ₁, ϒ₂, ⋯, ϒ_n, respectively, and let f be a measurable function. Then $\begin{matrix} ξ = f (η_{1}, η_{2}, \dots, η_{m}, τ_{1}, τ_{2}, \dots, τ_{n}) \end{matrix}$ has partial entropy $\begin{matrix} PH [ξ] & = & \int_{ℝ^{m}} \int_{0}^{1} F^{- 1} (α, y_{1}, \dots, y_{m}) ln \frac{α}{1 - α} d α \\ d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}) . \end{matrix}$

Proof. It is clear that S (α) is a derivable function with $S^{'} (α) = - ln \frac{α}{1 - α}$ . Since $\begin{matrix} S (F (x, y_{1}, \dots, y_{m})) & = & \int_{0}^{F (x, y_{1}, \dots, y_{m})} S^{'} (α) d α \\ = & - \int_{F (x, y_{1}, \dots, y_{m})}^{1} S^{'} (α) d α, \end{matrix}$ we have $\begin{matrix} PH [ξ] \\ = \int_{ℝ^{m}} \int_{- \infty}^{+ \infty} S (F (x, y_{1}, \dots, y_{m})) d x d Ψ_{1} (y_{1}) \\ \dots d Ψ_{m} (y_{m}) \\ = \int_{ℝ^{m}} \int_{- \infty}^{0} S (F (x, y_{1}, \dots, y_{m})) d x d Ψ_{1} (y_{1}) \\ \dots d Ψ_{m} (y_{m}) \\ + \int_{ℝ^{m}} \int_{0}^{+ \infty} S (F (x, y_{1}, \dots, y_{m})) d x d Ψ_{1} (y_{1}) \\ \dots d Ψ_{m} (y_{m}) \\ = \int_{ℝ^{m}} \int_{- \infty}^{0} \int_{0}^{F (x, y_{1}, \dots, y_{m})} S^{'} (α) d α d x d Ψ_{1} (y_{1}) \\ \dots d Ψ_{m} (y_{m}) \\ - \int_{ℝ^{m}} \int_{0}^{+ \infty} \int_{F (x, y_{1}, \dots, y_{m})}^{1} S^{'} (α) d α d x d Ψ_{1} (y_{1}) \\ \dots d Ψ_{m} (y_{m}) . \end{matrix}$

It follows from Fubini’s theorem that $\begin{matrix} PH [ξ] = \int_{ℝ^{m}} \int_{0}^{F (0, y_{1}, \dots, y_{m})} \\ \int_{F^{- 1} (α, y_{1}, \dots, y_{m})}^{0} S^{'} (α) d x d α \\ d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}) - \int_{ℝ^{m}} \int_{F (0, y_{1}, \dots, y_{m})}^{1} \\ \int_{0}^{F^{- 1} (α, y_{1}, \dots, y_{m})} S^{'} (α) d x d α d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}) \\ = - \int_{ℝ^{m}} \int_{0}^{F (0, y_{1}, \dots, y_{m})} F^{- 1} (α, y_{1}, \dots, y_{m}) S^{'} (α) d α \\ d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}) - \int_{ℝ^{m}} \int_{F (0, y_{1}, \dots, y_{m})}^{1} S^{'} (α) \\ F^{- 1} (α, y_{1}, \dots, y_{m}) d α d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}) \\ = - \int_{ℝ^{m}} \int_{0}^{1} F^{- 1} (α, y_{1}, \dots, y_{m}) S^{'} (α) d α d Ψ_{1} (y_{1}) \\ \dots d Ψ_{m} (y_{m}) \\ = \int_{ℝ^{m}} \int_{0}^{1} F^{- 1} (α, y_{1}, \dots, y_{m}) ln \frac{α}{1 - α} d α d Ψ_{1} (y_{1}) \\ \dots d Ψ_{m} (y_{m}) . \end{matrix}$

Thus the proof is finished. □

Theorem 3.2. Let τ be an uncertain variable with uncertainty distribution function Φ and η be a random variable with probability distribution function Ψ. If ξ = η + τ, then $PH [ξ] = H [τ] .$

Proof. It is obvious that F^-1 (α, y) = Φ^-1 (α) + y, therefore by using Theorem 3.1, we obtain $\begin{matrix} PH [ξ] & = & \int_{ℝ} \int_{0}^{1} F^{- 1} (α, y) ln \frac{α}{1 - α} d α d Ψ (y) \\ = & \int_{ℝ} \int_{0}^{1} (Φ^{- 1} (α) + y) ln \frac{α}{1 - α} d α d Ψ (y) \\ = & \int_{ℝ} \int_{0}^{1} Φ^{- 1} (α) ln \frac{α}{1 - α} d α d Ψ (y) \\ + \int_{ℝ} \int_{0}^{1} y ln \frac{α}{1 - α} d α d Ψ (y) \\ = & H [τ] . □ \end{matrix}$

Theorem 3.3. Let τ be an uncertain variable with uncertainty distribution function Φ and η be a random variable with probability distribution function Ψ. If ξ = τη, then PH [ξ] = H [τ] E [η].

Proof. It is obvious that F^-1 (α, y) = Φ^-1 (α) y, therefore by using Theorem 3.1, we obtain $\begin{matrix} PH [ξ] & = & \int_{ℝ} \int_{0}^{1} F^{- 1} (α, y) ln \frac{α}{1 - α} d α d Ψ (y) \\ = & \int_{ℝ} \int_{0}^{1} y Φ^{- 1} (α) ln \frac{α}{1 - α} d α d Ψ (y) \\ = & \int_{0}^{1} Φ^{- 1} (α) ln \frac{α}{1 - α} d α \int_{ℝ} y d Ψ (y) \\ = & H [τ] E [η] . □ \end{matrix}$

Theorem 3.4. Let η₁, η₂, ⋯, η_n be independent random variables and τ₁, η₂, ⋯, η_n be independent uncertain variables. Also, suppose that $\begin{matrix} ξ_{1} & = & f_{1} (η_{1}, τ_{1}), ξ_{2} = f_{2} (η_{2}, τ_{2}), \\ \dots, ξ_{n} = f_{n} (η_{n}, τ_{n}) . \end{matrix}$

If 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 ξ = f (ξ₁, ξ₂, ⋯, ξ_n) has partial entropy $\begin{matrix} PH [ξ] = \int_{ℝ^{n}} \int_{0}^{1} f (F_{1}^{- 1} (α, y_{1}), \dots, F_{m}^{- 1} (α, y_{m}), \\ F_{m + 1}^{- 1} (1 - α, y_{m + 1}), \dots, F_{n}^{- 1} (1 - α, y_{n})) ln \frac{α}{1 - α} \\ d α d Ψ_{1} (y_{1}) \dots d Ψ_{n} (y_{n}) . \end{matrix}$ where $F_{i}^{- 1} (α, y_{i})$ is the inverse uncertainty distribution of uncertain variable f_i (y_i, τ_i) for any real number y_i, i = 1, 2, ⋯, n.

Proof. Since 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, it follows from Theorem 2.2 that $\begin{matrix} F^{- 1} (α, y_{1}, \dots, y_{n}) = f (F_{1}^{- 1} (α, y_{1}), \dots, \\ F_{m}^{- 1} (α, y_{m}), F_{m + 1}^{- 1} (1 - α, y_{m + 1}), \dots, \\ F_{n}^{- 1} (1 - α, y_{n})) . \end{matrix}$

By invoking Theorem 3.1, the proof is complete. □

Theorem 3.5. Let η₁ and η₂ be random variables with probability distribution functions Ψ₁ and Ψ₂ respectively, and τ₁ and τ₂ be uncertain variables with uncertainty distribution functions Φ₁ and Φ₂, respectively. If ξ₁ = η₁ + τ₁ and ξ₂ = η₂ + τ₂, then $\begin{matrix} PH [ξ_{1} ξ_{2}] = H [τ_{1} τ_{2}] + E [η_{1}] H [τ_{2}] + E [η_{2}] H [τ_{1}] . \end{matrix}$

Proof. It is clear that $F_{1}^{- 1} (α, y_{1}) = y_{1} + Φ_{1}^{- 1} (α)$ and $F_{2}^{- 1} (α, y_{2}) = y_{2} + Φ_{2}^{- 1} (α)$ . By using Theorem 3.4, we have $\begin{matrix} PH [ξ_{1} ξ_{2}] \\ = \int_{ℝ^{2}} \int_{0}^{1} F^{- 1} (α, y_{1}, y_{2}) ln \frac{α}{1 - α} d α d Ψ_{1} (y_{1}) d Ψ_{2} (y_{2}) \\ = \int_{ℝ^{2}} \int_{0}^{1} F_{1}^{- 1} (α, y_{1}) F_{2}^{- 1} (α, y_{2}) ln \frac{α}{1 - α} d α d Ψ_{1} (y_{1}) \\ d Ψ_{2} (y_{2}) \\ = \int_{ℝ^{2}} \int_{0}^{1} (y_{1} + Φ_{1}^{- 1} (α)) (y_{2} + Φ_{2}^{- 1} (α)) ln \frac{α}{1 - α} d α \\ d Ψ_{1} (y_{1}) d Ψ_{2} (y_{2}) \\ = \int_{0}^{1} Φ_{1}^{- 1} (α) Φ_{2}^{- 1} (α) ln \frac{α}{1 - α} d α \\ + \int_{ℝ} \int_{0}^{1} y_{1} Φ_{2}^{- 1} (α) ln \frac{α}{1 - α} d α d Ψ_{1} (y_{1}) \\ + \int_{ℝ} \int_{0}^{1} y_{2} Φ_{1}^{- 1} (α) ln \frac{α}{1 - α} d α d Ψ_{2} (y_{2}) \\ = H [τ_{1} τ_{2}] + E [η_{1}] H [τ_{2}] + E [η_{2}] H [τ_{1}] . □ \end{matrix}$

Theorem 3.6. Let η₁ and η₂ be random variables with probability distribution functions Ψ₁ and Ψ₂ respectively, and τ₁ and τ₂ be uncertain variables with uncertainty distribution functions Φ₁ and Φ₂, respectively. If ξ₁ = η₁τ₁ and ξ₂ = η₂τ₂, then $\begin{matrix} PH [\frac{ξ_{1}}{ξ_{2}}] = H [\frac{τ_{1}}{τ_{2}}] E [η_{1}] E [\frac{1}{η_{2}}] . \end{matrix}$

Proof. By using a similar method of Theorem 3.5 and independence of random variables, the proof is straightforward. □

Theorem 3.7. Let η₁ and η₂ be independent random variables and τ₁ and τ₂ be independent uncertain variables. Also suppose that ξ₁ = f₁ (η₁, τ₁) and ξ₂ = f₂ (η₂, τ₂). Then for any real numbers a and b, we have $\begin{matrix} PH [a ξ_{1} + b ξ_{2}] = | a | PH [ξ_{1}] + | b | PH [ξ_{2}] . \end{matrix}$

Proof. STEP 1: We prove PH [aξ₁] = |a|PH [ξ₁]. If a > 0, then the inverse uncertainty distribution of af₁ (τ₁, y₁) is $\begin{matrix} F^{- 1} (α, y_{1}) = {aF}_{1}^{- 1} (α, y_{1}), \end{matrix}$ where $F_{1}^{- 1} (α, y_{1})$ is the inverse uncertainty distribution of f₁ (τ₁, y₁). It follows from Theorem 3.4 that $\begin{matrix} PH [a ξ_{1}] & = & \int_{ℝ} \int_{0}^{1} a F_{1}^{- 1} (α, y_{1}) d α d Ψ_{1} (y_{1}) \\ = & a \int_{ℝ} \int_{0}^{1} F_{1}^{- 1} (α, y_{1}) ln \frac{α}{1 - α} d α d Ψ_{1} (y_{1}) \\ = & | a | PH [ξ_{1}] . \end{matrix}$

If a < 0, then the inverse uncertainty distribution of af₁ (τ₁, y₁) is $\begin{matrix} F^{- 1} (α, y_{1}) = {aF}_{1}^{- 1} (1 - α, y_{1}) . \end{matrix}$

It follows from Theorem 3.4 that $\begin{matrix} PH [a ξ_{1}] \\ = a \int_{ℝ} \int_{0}^{1} F_{1}^{- 1} (1 - α, y_{1}) ln \frac{α}{1 - α} d α d Ψ_{1} (y_{1}) \\ = a \int_{ℝ} \int_{1}^{0} F_{1}^{- 1} (α, y_{1}) ln \frac{1 - α}{α} d (- α) d Ψ_{1} (y_{1}) \\ = - a \int_{ℝ} \int_{0}^{1} F_{1}^{- 1} (α, y_{1}) ln \frac{α}{1 - α} d α d Ψ_{1} (y_{1}) \\ = | a | PH [ξ_{1}] . \end{matrix}$

STEP 2: We prove PH [ξ₁ + ξ₂] = PH [ξ₁] + PH [ξ₂]. Note that the inverse uncertainty distribution of f₁ (τ₁, y₁) + f₂ (τ₂, y₂) is $\begin{matrix} F^{- 1} (α, y_{1}, y_{2}) = F_{1}^{- 1} (α, y_{1}) + F_{2}^{- 1} (α, y_{2}) . \end{matrix}$

It follows from Theorem 3.4 that $\begin{matrix} PH [ξ_{1} + ξ_{2}] & = & \int_{ℝ^{2}} \int_{0}^{1} (F_{1}^{- 1} (α, y_{1}) + F_{2}^{- 1} (α, y_{2})) \\ ln \frac{α}{1 - α} d α d Ψ_{1} (y_{1}) d Ψ_{2} (y_{2}) \\ = & PH [ξ_{1}] + PH [ξ_{2}] . \end{matrix}$

STEP 3: Finally, for any real numbers a and b, it follows from Steps 1 and 2 that $\begin{matrix} PH [a ξ_{1} + b ξ_{2}] & = & PH [a ξ_{1}] + PH [b ξ_{2}] \\ = & | a | PH [ξ_{1}] + | b | PH [ξ_{2}] . \end{matrix}$

The theorem is proved. □

4 Other types of partial entropies

In this section, we define other types of entropies for an uncertain random variable.

Definition 4.1. Suppose that η₁, η₂, ⋯, η_m are independent random variables and τ₁, τ₂, ⋯, τ_m are uncertain variables, ξ = f (η₁, η₂, ⋯, η_m, τ₁, τ₂, ⋯, τ_m) is also an uncertain random variable. Partial sine entropy of uncertain random variable ξ is defined as following $\begin{matrix} PS [ξ] = \int_{ℝ^{m}} \int_{- \infty}^{\infty} sin (π F (x, y_{1}, \dots, y_{m})) d x \\ d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}), \end{matrix}$ where F (x, y₁, ⋯, y_m) is the uncertainty distribution of uncertain variable f (y₁, ⋯, y_m, τ₁, ⋯, τ_m) for any real numbers y₁, ⋯, y_m.

Definition 4.2. Suppose that η₁, η₂, ⋯, η_m are independent random variables and τ₁, τ₂, ⋯, τ_m are uncertain variables, ξ = f (η₁, η₂, ⋯, η_m, τ₁, τ₂, ⋯, τ_m) is also an uncertain random variable. Partial quadratic entropy of uncertain random variable ξ is defined as following $\begin{matrix} PQ [ξ] \\ = \int_{ℝ^{m}} \int_{- \infty}^{\infty} F (x, y_{1}, \dots, y_{m}) \\ (1 - F (x, y_{1}, \dots, y_{m})) d x d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}), \end{matrix}$ where F (x, y₁, ⋯, y_m) is the uncertainty distribution of uncertain variable f (y₁, ⋯, y_m, τ₁, ⋯, τ_m) for any real numbers y₁, ⋯, y_m.

Definition 4.3. Suppose that η₁, η₂, ⋯, η_m are independent random variables and τ₁, τ₂, ⋯, τ_m are uncertain variables, also ξ = f (η₁, η₂, ⋯, η_m, τ₁, τ₂, ⋯, τ_m) is an uncertain random variable. Partial circle entropy of uncertain random variable ξ is defined as following $\begin{matrix} PC [ξ] = \int_{ℝ^{m}} \int_{- \infty}^{\infty} \sqrt{1 - {(F (x, y_{1}, \dots, y_{m}) - \frac{1}{2})}^{2}} \\ d x d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}), \end{matrix}$ where F (x, y₁, ⋯, y_m) is the uncertainty distribution of uncertain variable f (y₁, ⋯, y_m, τ₁, ⋯, τ_m) for any real numbers y₁, ⋯, y_m.

5 Conclusion

This paper studied some properties of partial entropy of uncertain random variables. We first introduced a definition of partial entropy for uncertain random variables. Based on this definition, several properties were derived. Finally, other types of partial entropies were investigated. The study of properties of other partial entropies are potential works for future research.

Entropy is a measurement of the degree of uncertainty concerning the information. The concept of entropy can be used to provide a quantitative measurement of the uncertainty associated with variables. The larger the entropy of variables is, the larger the uncertainty of variables is. That is, we need more information to make clear the variables. And the entropy can describe the ordered degree of a system. Higher entropy implies a more confused system, and lower entropy implies a more ordered system. Hence, we can apply the entropy into the process of information transmission. And we can reduce noise interference in acquisition and processing of information by the entropy. Additionally, entropy can be applied into portfolio selection and clustering.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant Nos. 61573210, 61462086, 61563050).

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