Hyperbolic entropy of uncertain sets and its applications

Abstract

As a powerful tool to model some unsharp concepts in real life, uncertain sets have been studied by more and more scholars. In order to characterize the degree of difficulty of uncertain sets, the hyperbolic entropy of an uncertain set and the hyperbolic relative entropy of uncertain sets are introduced in this paper. After that, this paper derived a key formula to calculate the hyperbolic entropy of an uncertain set via membership function, and some mathematical properties of hyperbolic entropy are also investigated in this paper. Finally, the hyperbolic entropy is applied in some research fields such as uncertain learning curve, clustering of rare books and portfolio selection of collecting rare books.

Keywords

Uncertainty theory uncertain set hyperbolic entropy uncertain learning curve

1 Introduction

Decision making in real life with uncertain information or uncertain data is very difficult for individuals. In order to quantify this uncertain information or uncertain data for input into the decision-making system, the study of entropy was started by Shannon [21] in 1949 in which the Shannon entropy was proposed to model the signal for a random variable. To this day, entropy has become a powerful tool for characterizing the uncertainty of a random variable in real life, and the entropy theory has been developed by many scholars and has been applied in many science fileds. For example, Kullback and Leibler [4] studied the relative entropy of random variables to characterize the differences between two random variables, Vaida [23] proposed the quadratic entropy of random variables to describe the degree of predicting the realization of a random variable, and Li et al. [6] investigated the slope entropy and applied the slope entropy in underwater acoustic signal processing.

However, the data in reality are often not random or fuzzy, but uncertain. In order to illustrate this conclusion, a large number of scholars have carried out many empirical studies such as epidemic spread (Chen et al. [1], Jia and Chen [3], Lio and Liu [7]), grain yield (Liu [17]), natural gas price (Mehrdoust et al. [19]), stock price (Liu and Liu [16]), currency exchange rate (Ye and Liu [28]), electric circuit (Liu [15]), China’s population (Liu [18]) and so on, and their research results all illustrate that real-world data should be modeled as uncertain variables rather than random variables. To model uncertain data, Liu [8] founded uncertainty theory in 2007 which was perfected by Liu [9] subsequently. Up to now, uncertainty theory has become a branch of mathematics under the researches of many scholars such as Liu and Lio [14], and Sheng and Kar [20].

Among various theoretical branches of uncertainty theory, uncertain set is an important concept for modeling unclear concepts in real life, which was proposed by Liu [10] in 2010. Later, in order to facilitate the calculation of the intersection, union, complement and other operations between uncertain sets, Liu [12] proposed the concept of membership function of an uncertain set to describe the degree to which a point belongs to an uncertain set. However, some uncertain sets have no membership function. In order to figure out what properties an uncertain set has to have a membership function, Liu [13] proved the existence theorem of membership function for an uncertain set, which shows that the totally ordered uncertain sets on continuous uncertainty space always have membership functions. Note that this theorem is only a sufficient condition for the existence of the membership function of an uncertain set. Since the membership function of an uncertain set was proposed, the operation of uncertain sets is no longer a difficult problem. For the sake of characterizing the degree of difficulty of predicting the realization of an uncertain set, the study of the entropy of uncertain sets was started by Liu [11] based on the membership function. Following that, the entropy theory of uncertain sets was developed by many scholars such as Yao and Ke [26] derived some formulas to calculate the value of the entropy of an uncertain set, Yao [27] investigated the sine entropy of an uncertain set, Wang and Ha [24] presented the quadratic entropy of an uncertain set, Gao and Ralescu [2] considered the elliptic entropy for uncertain sets and Tan and Yu [22] proposed the arc entropy of an uncertain set. Nowadays, entropy theory of uncertain sets has been proved to be an effective tool for dealing with uncertain data or uncertain phenomena.

However, the existing logarithmic entropy proposed for handling uncertain sets fails to measure the degrees of uncertainty with respect to some uncertain sets. As an effective supplement for the entropy theory of uncertain sets, the hyperbolic entropy will be introduced in this paper, and the contributions of this paper are as follows:

The hyperbolic entropy is first proposed to measure the degree of difficulty of predicting the realization of an uncertain set.

The concept of hyperbolic relative entropy is introduced to characterize the degree of differences between two uncertain sets.

Some applications of the hyperbolic entropy such as uncertain learning curve, clustering of rare books, and portfolio selection of rare books are investigated.

This paper is divided into the following ten major parts. The related concepts and theorems about uncertainty theory are introduced in Section 2. The definition of hyperbolic entropy is presented in Section 3, and a key hyperbolic entropy formula via inverse functions of an uncertain set is provided in Section 4 to calculate the hyperbolic entropy quickly. After that, a general hyperbolic entropy is introduced in Section 5, and the application of the general hyperbolic entropy for uncertain learning curve is provided in Section 6. Section 7 proposes the concept of hyperbolic relative entropy as an extension of the hyperbolic entropy. Then the application of hyperbolic relative entropy in clustering of rare books is investigated in Section 8, and another application in portfolio selection of rare books is also investigated in Section 9. Finally, a concise conclusion is given in Section 10.

2 Preliminaries

In this section, we will review some basic definitions and theorems in uncertainty theory including uncertain measure and uncertain set.

Definition 2.1. (Liu [8]) Assume that Γ is a universal set and Ł is a σ-algebra over Γ, M is a measurable set function on the σ-algebra Ł by following three axioms:

Axiom 1: (Normality Axiom) scripfontM {Γ} =1.

Axiom 2: (Duality Axiom) scripfontM {Λ} + scripfontM {Λ^c} =1, $\forall Λ \in L$ .

Axiom 3: (Subadditivity Axiom) For any countable sequence {Λ_i}, we always have $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}} .$ Then the set function M is called an uncertain measure, and the triplet (Γ, Ł , M) is called an uncertainty space.

For the purpose of obtaining the uncertain measure of composite event, the product uncertain measure M on the product σ-algebra Ł was defined by Liu [9] by the following product axiom.

Axiom 4. (Product Axiom) Assume (Γ_i, Ł _i, _Mi) are uncertainty spaces for i = 1, 2, ⋯ M is an uncertain measure on the σ-algebra satisfying

$M {\prod_{i = 1}^{\infty} Λ_{i}} = ⋀_{i = 1}^{\infty} M_{i} {Λ_{i}}$ where Λ_i are arbitrarily chosen events from Ł_i for i = 1, 2, ⋯, respectively. Then the uncertain measure M is called a product uncertain measure.

Definition 2.2. (Liu [10]) uncertain set An uncertain set is a measurable set-valued function ξ from an uncertainty space (Γ, Ł , M) to a collection of sets of real numbers, i.e., both ${ξ \in B} = {γ \in Γ | ξ (γ) \in B}$ and ${B \in ξ} = {γ \in Γ | B \in ξ (γ)}$ are events for any Borel set B of real numbers.

Definition 2.3. (Liu [12]) membership function An uncertain set ξ is said to have a membership function μ if $M {B \subset ξ} = inf_{x \in B} μ (x)$ and $M {ξ \subset B} = 1 - sup_{x \in B^{c}} μ (x)$ hold for any Borel set B of real numbers. The above equations are called measure inversion formulas.

The inverse membership function of ξ (denoted by ξ ∼ (ξ, μ^-1)) is defined as $μ^{- 1} (α) = {x \in R | μ (x) \geq α}, \forall α \in (0, 1] .$ For each given constant α ∈ (0, 1], the set μ^-1 (α) has left and right inverse functions of ξ, i.e., $μ_{l}^{- 1} (α) = inf μ^{- 1} (α)$ and $μ_{r}^{- 1} (α) = sup μ^{- 1} (α),$ respectively, which can be denoted by $(μ_{l}^{- 1}, μ_{r}^{- 1})$ . Note that μ^-1, $μ_{l}^{- 1}$ and $μ_{r}^{- 1}$ are useful tools to investigate uncertain sets.

Example 2.1. (Liu [10]) A rectangular uncertain set ξ denoted by (a, b) has a membership function as $μ (x) = {\begin{matrix} 1, & if a \leq x \leq b \\ 0, & otherwise \end{matrix}$ (1) and an inverse membership function as [a, b].

Example 2.2. (Liu [10]) A triangular uncertain set ξ denoted by (a, b, c) has a membership function as $μ (x) = {\begin{matrix} (x - a) / (b - a), & if a \leq x \leq b \\ (x - c) / (b - c), & if b < x \leq c \end{matrix}$ (2) and an inverse membership function as $[a + (b - a) α, c + (b - c) α] .$ (3)

Example 2.3. (Liu [10]) A trapezoidal uncertain set ξ denoted by (a, b, c, d) has a membership function as $μ (x) = {\begin{matrix} (x - a) / (b - a), & if a \leq x \leq b \\ 1, & if b < x \leq c \\ (x - d) / (c - d), & if c < x \leq d \end{matrix}$ (4) and an inverse membership function as $[a + (b - a) α, d + (c - d) α] .$ (5)

Theorem 2.1. (Liu [12])Let ξ_i, i = 1, 2, ⋯ , n be independent uncertain sets with inverse membership functions $μ_{i}^{- 1}$ , i = 1, 2, ⋯ , n, respectively. If g (x₁, ⋯ , x_n) is continuous, strictly increasing with respect to x₁, ⋯ , x_m and strictly decreasing with respect to x_m+1, ⋯ , x_n, then $ξ = g (ξ_{1}, \dots, ξ_{n})$ has a left membership function as $λ_{l}^{- 1} = g (μ_{1 l}^{- 1}, \dots, μ_{ml}^{- 1}, μ_{m + 1, r}^{- 1}, \dots, μ_{nr}^{- 1}),$ and a right membership function as $λ_{r}^{- 1} = g (μ_{1 r}^{- 1}, \dots, μ_{mr}^{- 1}, μ_{m + 1, l}^{- 1}, \dots, μ_{nl}^{- 1}),$ where the left inverse membership functions of ξ_i, i = 1, 2, ⋯ , n are expressed as $μ_{l}^{- 1}, μ_{1 l}^{- 1}, \dots, μ_{nl}^{- 1}$ , and the right inverse membership functions are expressed as $μ_{r}^{- 1}, μ_{1 r}^{- 1}, \dots, μ_{nr}^{- 1} .$

Theorem 2.2. (Liu [12]) Let ξ ∼ (ξ, μ) be an uncertain set with inverse membership function $(μ_{l}^{- 1}, μ_{r}^{- 1})$ . Then the expected value of ξ is defined as $E [ξ] = \frac{1}{2} \int_{0}^{1} (μ_{l}^{- 1} + μ_{r}^{- 1}) d x .$

In order to measure the degree of difficulty of predicting the realization of an uncertain set, the entropy of an uncertain set was proposed by Liu [11] in 2011. Next we will introduce some concepts about the entropy of uncertain set.

Theorem 2.3. (Liu [11]) An uncertain set ξ with membership function μ has Liu’s entropy as follows, $\begin{matrix} L [ξ] \\ = & \int_{- \infty}^{+ \infty} - μ (x) ln (μ (x)) - (1 - μ (x)) ln (1 - μ (x)) d x . \end{matrix}$

Definition 2.4. (Yao [26]) An uncertain set ξ with membership function μ has Yao’s entropy as $S [ξ] = \int_{- \infty}^{+ \infty} \sin (π μ (x)) d x .$

Definition 2.5. (Wang and Ha [24]) The quadratic entropy of an uncertain set ξ with membership function μ is given by $Q [ξ] = \int_{- \infty}^{+ \infty} [1 - | 1 - 2 μ (x) |] dx .$

Definition 2.6. (Gao and Ralescu [2]) An uncertain set ξ with membership function μ has an elliptic entropy as $E [ξ] = \int_{- \infty}^{+ \infty} 2 k \sqrt{μ (x) (1 - μ (x))} d x .$

Definition 2.7. (Li et al. [5]) The uncertain cost of the Ath production is defined as $C = {KA}^{β}$ where K is a constant representing the cost of the first unit with K ∈ R⁺, and β ∈ [-1, 0] is an uncertain learning parameter with uncertainty distribution Φ (x).

3 Hyperbolic entropy of uncertain sets

For the purpose of charactering the degree of difficulty of an uncertain set, this section will introduce the hyperbolic entropy of an uncertain set with a membership function. The complement equality and translation invariance of the hyperbolic entropy will be proved, and the principles of minimum and maximum hyperbolic entropy will also be proposed.

Definition 3.1. Let ξ be an uncertain set with membership function μ. Then the hyperbolic entropy of ξ is defined as $\begin{matrix} HY [ξ] & = \int_{- \infty}^{\infty} hy (μ (x)) d x \\ = \int_{- \infty}^{\infty} \sqrt{0.5} - \sqrt{0.5 + μ^{2} (x) - μ (x)} d x . \end{matrix}$ (6)

It is easy to find that $hy (μ (x)) = \sqrt{0.5} - \sqrt{0.5 + μ (x)^{2} - μ (x)}$ is a hyperbolic function opening downward, and is a symmetric function about μ (x) =0.5. Notably, it has its unique maximum $\sqrt{0.5} - 0.5$ at μ = 0.5. See Fig. 1.

Fig. 1

The graph of hyperbolic entropy.

By using Definition 3.1, we can calculate the hyperbolic entropies of some special uncertain sets.

Example 3.1. Let ξ be a crisp set [a, b] whose membership function is $μ (x) = {\begin{matrix} 1, & a \leq x \leq b \\ 0, & otherwise . \end{matrix}$ Then it follows from (6) that the hyperbolic entropy of ξ can be calculated as $\begin{matrix} HY [ξ] & = \int_{- \infty}^{a} \sqrt{0.5} - \sqrt{0.5 - 0^{2} + 0} d x \\ + \int_{a}^{b} \sqrt{0.5} - \sqrt{0.5 + 1^{2} - 1} d x \\ + \int_{b}^{+ \infty} \sqrt{0.5} - \sqrt{0.5 + 0^{2} - 0} d x \\ = 0 . \end{matrix}$

This implies that the hyperbolic entropy of any crisp set is always 0.

Example 3.2. Let ξ be a triangular uncertain set whose membership function is showed in (2). Then the hyperbolic entropy of ξ is $\begin{matrix} HY [ξ] & = \int_{a}^{b} \sqrt{0.5} - \sqrt{0.5 + {(\frac{x - a}{b - a})}^{2} - \frac{x - a}{b - a}} d x \\ + \int_{b}^{c} \sqrt{0.5} - \sqrt{0.5 + {(\frac{x - c}{b - c})}^{2} - \frac{x - c}{b - c}} d x \\ = \frac{\sqrt{2} - ln (\sqrt{2} + 1)}{4} (c - a) . \end{matrix}$

Example 3.3. Let ξ be a trapezoidal uncertain set whose membership function is showed in (4). Then the hyperbolic entropy of ξ is $\begin{matrix} HY [ξ] \\ = & \int_{a}^{b} \sqrt{0.5} - \sqrt{0.5 + {(\frac{x - a}{b - a})}^{2} - \frac{x - a}{b - a}} d x \\ + \int_{b}^{c} \sqrt{0.5} - \sqrt{0.5 + 1^{2} - 1} d x \\ + \int_{c}^{d} \sqrt{0.5} - \sqrt{0.5 + {(\frac{x - d}{c - d})}^{2} - \frac{x - d}{c - d}} d x \\ = & \frac{\sqrt{2} - ln (\sqrt{2} + 1)}{4} (d - c + b - a) . \end{matrix}$

Theorem 3.1. (Complement Equality of Hyperbolic Entropy) Let ξ be an uncertain set whose membership function is μ. Then we have $HY [ξ^{c}] = HY [ξ] .$

Proof. It is easy to infer that ξ^c has a membership function 1 - μ. Then it follows from (6) that $\begin{matrix} HY [ξ^{c}] \\ = & \int_{- \infty}^{+ \infty} \sqrt{0.5} - \sqrt{0.5 + (1 - μ (x))^{2} - (1 - μ (x))} d x \\ = & \int_{- \infty}^{+ \infty} \sqrt{0.5} - \sqrt{0.5 + μ (x)^{2} - μ (x)} d x \\ = & HY [ξ] . \end{matrix}$

Theorem 3.2. (Principle of Minimum Hyperbolic Entropy) Let ξ be an uncertain set whose membership function is μ. Then we have HY [ξ] ≥0. In addition, we have HY [ξ] =0 if and only if ξ is a crisp set.

Proof. Since the membership function μ always takes value between 0 and 1, we get $hy (μ (x)) = \sqrt{0.5} - \sqrt{0.5 + μ (x)^{2} - μ (x)} \geq 0$ for all x ∈ R. Thus the integral of hy (μ (x)) over the real number field must be non-negative. That is, we always have $\begin{matrix} HY [ξ] \geq 0 . \end{matrix}$ Specially, HY [ξ] =0 holds if and only if μ (x) =0 or μ (x) =1. Thus the proof is completed.

Theorem 3.3. (Principle of Maximum Hyperbolic Entropy) Let ξ be an uncertain set whose membership function is μ. Then we have

$HY [ξ] \leq (\sqrt{0.5} - 0.5) (b - a)$ and the equality holds if and only if μ ≡ 0.5, where a is the lower bound of μ and b is the upper bound of μ.

Proof. Since the function $hy (μ (x)) = \sqrt{0.5} - \sqrt{0.5 + (μ (x))^{2} - μ (x)}$ is a hyperbolic function opening downward, it reaches its maximum value $\sqrt{0.5} - 0.5$ at μ (x) =0.5 . Then we have $\begin{matrix} HY [ξ] & = \int_{a}^{b} hy (μ (x)) d x \\ = \int_{a}^{b} \sqrt{0.5} - \sqrt{0.5 + (μ (x))^{2} - μ (x)} d x \\ \leq \int_{a}^{b} \sqrt{0.5} - 0.5 d x \\ = (\sqrt{0.5} - 0.5) (b - a) . \end{matrix}$ Thus the proof is completed.

Theorem 3.4. (Translation Invariance) Let ξ be an uncertain set whose membership function is μ. Then we have

$HY [ξ + k] = HY [ξ]$ for any $k \in R$ .

Proof. Note that the uncertain set ξ + k has a membership function μ (x + k). Then it follows from Definition 3.1 that $\begin{matrix} HY [ξ + k] \\ = & \int_{- \infty}^{+ \infty} hy (μ (x + k)) d x \\ = & \int_{- \infty}^{+ \infty} \sqrt{0.5} - \sqrt{0.5 + (μ (x + k))^{2} - μ (x + k)} d (x + k) \\ = & \int_{- \infty}^{+ \infty} hy (μ (x)) d x \\ = & HY [ξ] . \end{matrix}$ Thus the proof is completed.

The hyperbolic entropy of an uncertain set is proposed via the hyperbolic function, which provides a powerful tool for studying the degrees of difficulty of an uncertain set.

4 Hyperbolic entropy formula

For the purpose of investigating the properties of hyperbolic entropy and calculating the hyperbolic entropy quickly, a key formula is derived by using the inverse membership function ( $μ_{l}^{- 1} (x), μ_{r}^{- 1} (x)$ ) of an uncertain set ξ in this section. Based on the formula, some mathematical properties of hyperbolic entropy are studied, and some basic theorems are also provided.

Theorem 4.1. Let ξ be an uncertain set whose membership function is μ. If the hyperbolic entropy HY [ξ] exists, then we have $HY [ξ] = \int_{0}^{1} (μ_{r}^{- 1} (α) - μ_{l}^{- 1} (α)) \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α .$

Proof. Let us write $f (α) = \sqrt{0.5} - \sqrt{0.5 + α^{2} - α} .$ It is easy to infer that the derivable function of f (α) is $f^{'} (α) = \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} .$

Let x₀ be the point satisfying μ (x₀) =1 . Then according to Definition 3.1, we have $\begin{matrix} HY [ξ] & = \int_{- \infty}^{+ \infty} f (μ (x)) d x \\ = \int_{- \infty}^{x_{0}} \int_{0}^{μ (x)} f^{'} (α) d α d x \\ + \int_{x_{0}}^{+ \infty} \int_{0}^{μ (x)} f^{'} (α) d α d x . \end{matrix}$ It follows from the Fubini theorem that $\begin{matrix} HY [ξ] \\ = \int_{0}^{1} \int_{μ_{l}^{- 1} (α)}^{x_{0}} f^{'} (α) d x d α \\ + \int_{0}^{1} \int_{x_{0}}^{μ_{r}^{- 1} (α)} f^{'} (α) d x d α \\ = \int_{0}^{1} (x_{0} - μ_{l}^{- 1} (α)) f^{'} (α) d α \end{matrix}$ $\begin{matrix} + \int_{0}^{1} (μ_{r}^{- 1} (α) - x_{0}) f^{'} (α) d α \\ = \int_{0}^{1} (μ_{r}^{- 1} (α) - μ_{l}^{- 1} (α)) f^{'} (α) d α \\ = \int_{0}^{1} (μ_{r}^{- 1} (α) - μ_{l}^{- 1} (α)) \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α . \end{matrix}$ The theorem is thus verified.

Example 4.1. Let ξ be a triangular uncertain set whose inverse membership function is showed in (3). Then the hyperbolic entropy of ξ can be calculated as $\begin{matrix} HY [ξ] \\ = \int_{0}^{1} (α b + (1 - α) c - (1 - α) a - α b]) \\ \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = (c - a) \int_{0}^{1} \frac{(1 - α) (1 - 2 α)}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = \frac{\sqrt{2} - ln (\sqrt{2} + 1)}{4} (c - a) . \end{matrix}$

Example 4.2. Let ξ be a trapezoidal uncertain set whose inverse membership function is showed in (5). Then the hyperbolic entropy of ξ can be calculated as $\begin{matrix} HY [ξ] \\ = \int_{0}^{1} (α c + (1 - α) d - (1 - α) a - α b) \\ \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = (c - b) \int_{0}^{1} \frac{α (1 - 2 α)}{2 \sqrt{0.5 + α^{2} - α}} d α \\ + (d - a) \int_{0}^{1} \frac{(1 - α) (1 - 2 α)}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = \frac{\sqrt{2} - ln (\sqrt{2} + 1)}{4} (d - c + b - a) . \end{matrix}$

Theorem 4.2. Let ξ_i, i = 1, 2, ⋯ , n be independent uncertain sets whose inverse membership functions are $(μ_{il}^{- 1}, μ_{ir}^{- 1})$ , i = 1, 2, ⋯ , n, respectively. If g (x₁, ⋯ , x_n) is continuous, strictly increasing with respect to x₁, ⋯ , x_m and strictly decreasing with respect to x_m+1, ⋯ , x_n, then the hyperbolic entropy of ξ = g (ξ₁, ξ₂, ·· · , ξ_n) can be calculated as $\begin{matrix} HY [ξ] \\ = \int_{0}^{1} (g (μ_{1 r}^{- 1}, μ_{mr}^{- 1}, \dots, μ_{m + 1, l}^{- 1}, μ_{nl}^{- 1}) \\ - g (μ_{1 l}^{- 1}, μ_{ml}^{- 1}, \dots, μ_{m + 1, r}^{- 1}, μ_{nr}^{- 1})) \\ \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α . \end{matrix}$

Proof. It follows from Theorem 2 that the left and right inverse functions of ξ = g (ξ₁, ξ₂, ·· · , ξ_n) are given by $μ_{l}^{- 1} = g (μ_{1 l}^{- 1}, \cdot \cdot \cdot, μ_{ml}^{- 1}, μ_{m + 1, r}^{- 1}, \cdot \cdot \cdot, μ_{nr}^{- 1})$ and $μ_{r}^{- 1} = g (μ_{1 r}^{- 1}, \cdot \cdot \cdot, μ_{mr}^{- 1}, μ_{m + 1, l}^{- 1}, \cdot \cdot \cdot, μ_{nl}^{- 1}),$ respectively. According to Theorem 4, the above theorem is verified.

Corollary 4.1. Let ξ_i, i = 1, 2, ⋯ , n be independent uncertain sets whose inverse membership functions are $(μ_{il}^{- 1}, μ_{ir}^{- 1})$ , i = 1, 2, ⋯ , n, respectively. If g (x₁, ⋯ , x_n) is continuous and strictly increasing with respect to x₁, x₂, ⋯ , x_n, then the hyperbolic entropy of ξ = g (ξ₁, ξ₂, ·· · , ξ_n) can be calculated as $\begin{matrix} HY [ξ] & = \int_{0}^{1} (g (μ_{1 r}^{- 1}, \cdot \cdot \cdot, μ_{mr}^{- 1}, \cdot \cdot \cdot, μ_{nr}^{- 1}) \\ - g (μ_{1 l}^{- 1}, \cdot \cdot \cdot, μ_{ml}^{- 1}, \cdot \cdot \cdot, μ_{nl}^{- 1})) \\ \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α . \end{matrix}$

Corollary 4.2. Let ξ_i, i = 1, 2, ⋯ , n be independent uncertain sets whose inverse membership functions are $(μ_{il}^{- 1}, μ_{ir}^{- 1})$ , i = 1, 2, ⋯ , n, respectively. If g (x₁, ⋯ , x_n) is continuous and strictly decreasing with respect to x₁, x₂, ⋯ , x_n, then the hyperbolic entropy of ξ = g (ξ₁, ξ₂, ·· · , ξ_n) can be calculated as $\begin{matrix} HY [ξ] & = \int_{0}^{1} (g (μ_{1 l}^{- 1}, \cdot \cdot \cdot, μ_{ml}^{- 1}, \cdot \cdot \cdot, μ_{nl}^{- 1}) \\ - g (μ_{1 r}^{- 1}, \cdot \cdot \cdot, μ_{mr}^{- 1}, \cdot \cdot \cdot, μ_{nr}^{- 1})) \\ \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α . \end{matrix}$

Example 4.3. Let ξ be a triangular uncertain set whose inverse membership function is showed in (3) and k be a real number.

If k > 0, then the inverse functions of the multiplication kξ are $μ_{l}^{- 1} (α) = (1 - α) (ka) + α (kb)$ and $μ_{r}^{- 1} (α) = (1 - α) (kc) + α (kb),$ respectively. Then the hyperbolic entropy of kξ is $\begin{matrix} HY [k ξ] \\ = \int_{0}^{1} (α (kb) + (1 - α) (kc) - (1 - α) (ka) - α (kb)) \\ \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = kHY [ξ] . \end{matrix}$ If k < 0, then the inverse functions of the multiplication kξ are $μ_{l}^{- 1} (α) = (1 - α) (kc) + α (kb)$ and $μ_{r}^{- 1} (α) = (1 - α) (ka) + α (kb),$ respectively. The hyperbolic entropy of kξ is the same as the case of k > 0.

Theorem 4.3. Let ξ and η be two independent uncertain sets with membership functions μ and ν, respectively. Then we have

$HY [a ξ + b η] = | a | HY [ξ] + | b | HY [η]$ for any real numbers a, b ∈ R .

Proof. Let λ denote the membership function of aξ. We first prove that HY [aξ] = |a|HY [ξ] .

If a > 0, then the two inverse functions of aξ are $λ_{r}^{- 1} = a μ_{r}^{- 1}, λ_{l}^{- 1} = a μ_{l}^{- 1} .$ By using Theorem 4, we have $\begin{matrix} HY [a ξ] \\ = \int_{0}^{1} (λ_{r}^{- 1} (α) - λ_{l}^{- 1} (α)) \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = a \int_{0}^{1} (μ_{r}^{- 1} (α) - μ_{l}^{- 1} (α)) \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = | a | HY [ξ] . \end{matrix}$ If a < 0, then the two inverse functions of aξ are $λ_{r}^{- 1} = - a μ_{l}^{- 1}, λ_{l}^{- 1} = - a μ_{r}^{- 1} .$ It follows from Theorem 2 that $\begin{matrix} HY [a ξ] \\ = \int_{0}^{1} (λ_{r}^{- 1} (α) - λ_{l}^{- 1} (α)) \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = - a \int_{0}^{1} (μ_{r}^{- 1} (α) - μ_{l}^{- 1} (α)) \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = | a | HY [ξ] . \end{matrix}$ If a = 0, then it is easy to infer that $HY [a ξ] = | a | HY [ξ] = 0 .$ Thus $HY [a ξ] = | a | HY [ξ]$ always holds.

Secondly, we prove that $HY [ξ + η] = HY [ξ] + HY [η] .$ Note that the inverse functions of ξ + η are $λ_{r}^{- 1} = μ_{r}^{- 1} + ν_{r}^{- 1}, λ_{l}^{- 1} = μ_{l}^{- 1} + ν_{l}^{- 1} .$ It follows from Theorem 4 that $\begin{matrix} HY [ξ + η] \\ = \int_{0}^{1} (λ_{r}^{- 1} (α) - λ_{l}^{- 1} (α)) \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = \int_{0}^{1} (μ_{r}^{- 1} (α) + ν_{r}^{- 1} (α) - μ_{l}^{- 1} (α) - ν_{l}^{- 1} (α)) \\ \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = \int_{0}^{1} (μ_{r}^{- 1} (α) - μ_{l}^{- 1} (α)) \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ + \int_{0}^{1} (ν_{r}^{- 1} (α) - ν_{l}^{- 1} (α)) \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = HY [ξ] + HY [η] . \end{matrix}$ Therefore we can obtain $HY [a ξ + b η] = | a | HY [ξ] + | b | HY [η], \forall a, b \in R .$ The proof is thus completed.

Example 4.4. Let ξ and η be two independent triangular uncertain sets with membership functions (a, b, c) and (e, f, g), respectively. By using Theorem 4, we have $\begin{matrix} HY [2 ξ_{1} + 3 ξ_{2}] \\ = 2 HY [ξ_{1}] + 3 HY [ξ_{2}] \\ = \frac{\sqrt{2} - ln (\sqrt{2} + 1)}{4} (2 c - 2 a + 3 g - 3 e) . \end{matrix}$

5 A general hyperbolic entropy

A generalized concept of the hyperbolic entropy for an uncertain set will be presented in this section, and another definition of the hyperbolic entropy via inverse functions will be also investigated.

Definition 5.1. Let ξ be an uncertain set with membership function μ. Then a general hyperbolic entropy is defined as $\begin{matrix} HY [ξ] & = k \int_{- \infty}^{\infty} hy (μ (x)) d x \\ = k \int_{- \infty}^{\infty} \sqrt{0.5} - \sqrt{0.5 + μ^{2} (x) - μ (x)} d x . \end{matrix}$ where k > 0 is a parameter.

It is clear that the function $hy (t, k) = k (\sqrt{0.5} - \sqrt{0.5 + t^{2} - t})$ with 0 ≤ t ≤ 1, and k > 0 is a symmetric function about t = 0.5 as shown in Fig. 2.

Fig. 2

$h (t, k) = k (\sqrt{0.5} - \sqrt{0.5 + t^{2} - t})$ .

Corollary 5.1. Let ξ be an uncertain set with membership function μ. Then the two general hyperbolic entropy of ξ can be calculated by $\begin{matrix} HY [ξ] \\ = k \int_{0}^{1} (μ_{r}^{- 1} (α) - μ_{l}^{- 1} (α)) \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α . \end{matrix}$

6 Uncertain learning curve

Uncertain learning curve is a key tool to predict the future cost based on uncertainty theory. The study of learning curves was started by Wright [25] in 1936. For deep learning applications, the learning parameter will be treated as uncertain sets in this section.

Definition 6.1. (Uncertain Learning Curve Model) Let ξ be a set of production rate events representing the uncertain average rate per unit demanded for producing R units with 0 ≤ R ≤ 1, and let constant K (K > 0) be the rate of the first unit of production. Assume the uncertain set η is the collection of the learning curve variables with η ⊂ [-1, 0]. Then the uncertain learning curve model is defined as $ξ = {KR}^{η} .$

Note that the uncertain set η tend to {-1} representing a high learning rate, and tend to {0} representing a low learning rate.

Theorem 6.1. Let ξ be a set of production rate events, and let η be an uncertain learning set with inverse functions $μ_{r}^{- 1}$ and $μ_{l}^{- 1}$ . Then the hyperbolic entropy of the uncertain learning set is $\begin{matrix} HY [ξ] \\ = K \int_{0}^{1} R^{(μ_{l}^{- 1} (α) - μ_{r}^{- 1} (α))} \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \end{matrix}$

Proof. Note that the function $ξ = {KR}^{η}$ is strictly decreasing with respect to η. Since the uncertain set η has inverse functions $μ_{r}^{- 1}$ and $μ_{l}^{- 1}$ , it follows from Corollary 4 that $\begin{matrix} HY [ξ] \\ = K \int_{0}^{1} R^{(μ_{l}^{- 1} (α) - μ_{r}^{- 1} (α))} \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α . \end{matrix}$ Thus the proof is completed.

Example 6.1. Let ξ be a triangular uncertain set (-0.7, - 0.4, - 0.2) with inverse functions $μ_{l}^{- 1} = - 0.7 + 0.3 α$ and $μ_{r}^{- 1} = - 0.2 - 0.2 α .$ Then it follows from Theorem 6 that the hyperbolic entropy of ξ can be obtained by $\begin{matrix} HY [ξ] \\ = K \int_{0}^{1} R^{(μ_{l}^{- 1} (α) - μ_{r}^{- 1} (α))} \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = K \int_{0}^{1} R^{- 0.5 + 0.5 α} \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α . \end{matrix}$ If K = 8 and R = 0.3, then we have HY [ξ] ≈0.87 .

Example 6.2. Let ξ be a trapezoidal uncertain set (-0.9, - 0.7, - 0.3, - 0.2) with inverse functions $μ_{l}^{- 1} = - 0.9 + 0.2 α$ and $μ_{r}^{- 1} = - 0.2 - 0.1 α .$ Then it follows from Theorem 6 that the hyperbolic entropy of ξ can be obtained by $\begin{matrix} HY [ξ] \\ = K \int_{0}^{1} R^{(μ_{l}^{- 1} (α) - μ_{r}^{- 1} (α))} \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α \\ = K \int_{0}^{1} R^{- 0.7 + 0.3 α} \frac{1 - 2 α}{2 \sqrt{0.5 + α^{2} - α}} d α . \end{matrix}$ If K = 0.8 and R = 0.2, then we have HY [ξ] ≈0.75 .

7 Hyperbolic relative entropy

Relative entropy was first introduced by Kullback and Leibler [4] to study the distribution divergence between variables. This section aims to present a new definition of hyperbolic relative entropy for uncertain sets and study its properties.

Definition 7.1. Let ξ and η be two independent uncertain sets with membership functions μ and ν, respectively. Then the hyperbolic relative entropy between ξ and η is defined as $\begin{matrix} HR [ξ, η] \\ = \int_{- \infty}^{+ \infty} (μ (x) - ν (x)) (\sqrt{0.5 + {(\frac{ν (x)}{2})}^{2} - \frac{ν (x)}{2}} \\ - \sqrt{0.5 + {(\frac{μ (x)}{2})}^{2} - \frac{μ (x)}{2}}) d x \end{matrix}$

Example 7.1. Let ξ = scripfontR (m, n) and η = scripfontR (p, t) be two independent rectangular uncertain sets. If m ≤ n ≤ p ≤ t, then the hyperbolic relative entropy between ξ and η is $\begin{matrix} HR [ξ, η] \\ \int_{m}^{n} (1 - 0) (\sqrt{0.5 + 0^{2} - 0} \\ - \sqrt{0.5 + {(\frac{1}{2})}^{2} - \frac{1}{2}}) d x \\ + \int_{p}^{t} (0 - 1) (\sqrt{0.5 + {(\frac{1}{2})}^{2} - \frac{1}{2}} \\ - \sqrt{0.5 + 0^{2} - 0}) d x \\ = (\sqrt{0.5} - 0.5) (n - m + t - p) . \end{matrix}$ If m ≤ p ≤ n ≤ t, then the hyperbolic relative entropy between ξ and η is $\begin{matrix} HR [ξ, η] \\ = \int_{m}^{p} (1 - 0) (\sqrt{0.5 + 0^{2} - 0} \\ - \sqrt{0.5 + {(\frac{1}{2})}^{2} - \frac{1}{2}}) d x \\ + \int_{n}^{t} (0 - 1) (\sqrt{0.5 + {(\frac{1}{2})}^{2} - \frac{1}{2}} \\ - \sqrt{0.5 + 0^{2} - 0}) d x \\ = (\sqrt{0.5} - 0.5) (p - m + t - n) . \end{matrix}$

Theorem 7.1. Let ξ and η be two independent uncertain sets with membership functions μ and ν, respectively. Then we have

$HR [ξ, η] \geq 0$ and equality holds if and only if μ (x) = ν (x).

Proof. Let us write a function as $\begin{matrix} hr (s, t) & = (s - t) (\sqrt{0.5 + {(\frac{t}{2})}^{2} - \frac{t}{2}} \\ - \sqrt{0.5 + {(\frac{s}{2})}^{2} - \frac{s}{2}}) . \end{matrix}$ It is easy to infer that $hr (s, t) \geq 0$ for all s, t ∈ [0, 1] . That is, $(μ - ν) (\sqrt{0.5 + {(\frac{ν}{2})}^{2} - \frac{ν}{2}} - \sqrt{0.5 + {(\frac{μ}{2})}^{2} - \frac{μ}{2}})$ is always nonnegative. Thus we have $HR [ξ, η] \geq 0 .$ Notably, HR [ξ, η] =0 if and only if hr (μ (x) , ν (x)) =0 for all $x \in R$ . Thus the proof is completed.

Corollary 7.1. Let ξ and η be two independent uncertain sets with membership functions μ and ν, respectively. Then the general hyperbolic relative entropy of ξ and η is defined by

$\begin{matrix} HR [ξ, η] \\ = k \int_{- \infty}^{+ \infty} (μ (x) - ν (x)) (\sqrt{a^{2} + {(\frac{ν (x)}{2})}^{2} - \frac{ν (x)}{2}} \\ - \sqrt{a^{2} + {(\frac{μ (x)}{2})}^{2} - \frac{μ (x)}{2}}) d x \end{matrix}$ where k > 0 and a > 0.5 are uncertain parameters.

8 Clustering of rare books

This section will investigate the application of hyperbolic relative entropy in clustering rare books, which have been widely used in the rare book markets. Suppose that we have five levels, denoted by A, B, C, D and E, respectively. The corresponding membership functions are given in Fig. 3.

Fig. 3

Membership function of rare books.

If a type of rare books is described by an uncertain set ξ = (96, 156, 192), then how do we know which type it belongs to? First, it is easy to see that the book does not belong to the “A" and “E" types. In order to find the proper cluster for uncertain set ξ, let us calculate the hyperbolic relative entropies between ξ and B, C, D. Based on the definition of hyperbolic relative entropy, we have $\begin{matrix} HR [ξ, B] \\ = \int_{0}^{64} \frac{x}{64} (\sqrt{0.5} - \sqrt{0.5 + (\frac{x}{128})^{2} - \frac{x}{128}}) d x \\ + \int_{64}^{96} \frac{128 - x}{64} (\sqrt{0.5} \\ - \sqrt{0.5 + (\frac{128 - x}{128})^{2} - \frac{128 - x}{128}}) d x \\ + \int_{96}^{128} (\frac{128 - x}{64} - \frac{x - 96}{60}) \\ (\sqrt{0.5 + (\frac{x - 96}{120})^{2} - \frac{x - 96}{120}} \\ - \sqrt{0.5 + (\frac{128 - x}{128})^{2} - \frac{128 - x}{128}}) d x \\ + \int_{128}^{156} \frac{x - 96}{60} (\sqrt{0.5} \\ - \sqrt{0.5 + (\frac{x - 96}{120})^{2} + \frac{x - 96}{120}}) d x \end{matrix}$ $\begin{matrix} + \int_{156}^{192} \frac{192 - x}{36} (\sqrt{0.5} \\ - \sqrt{0.5 + (\frac{192 - x}{72})^{2} - \frac{192 - x}{72}}) d x \\ \approx 22.14 . \end{matrix}$

$\begin{matrix} HR [ξ, C] \\ = \int_{64}^{96} \frac{x - 64}{64} (\sqrt{0.5} \\ - \sqrt{0.5 + (\frac{x - 64}{128})^{2} - \frac{x - 64}{128}}) d x \\ + \int_{96}^{128} (\frac{x - 64}{64} - \frac{x - 96}{60}) \\ (\sqrt{0.5 + (\frac{x - 96}{120})^{2} - \frac{x - 96}{120}} \\ - \sqrt{0.5 + (\frac{x - 64}{128})^{2} - \frac{x - 64}{128}}) d x \\ + \int_{128}^{156} (\frac{192 - x}{64} - \frac{x - 96}{60}) \\ (\sqrt{0.5 + (\frac{x - 96}{120})^{2} - \frac{x - 96}{120}} \\ - \sqrt{0.5 + (\frac{192 - x}{128})^{2} - \frac{192 - x}{128}}) d x \\ + \int_{156}^{192} (\frac{192 - x}{64} - \frac{192 - x}{36}) \\ (\sqrt{0.5 + (\frac{192 - x}{72})^{2} - \frac{192 - x}{72}} \\ - \sqrt{0.5 + (\frac{192 - x}{128})^{2} - \frac{192 - x}{128}}) d x \\ \approx 3.09 . \end{matrix}$

$\begin{matrix} HR [ξ, D] \\ = \int_{96}^{128} \frac{x - 96}{60} (\sqrt{0.5} \end{matrix}$

$\begin{matrix} - \sqrt{0.5 + (\frac{x - 96}{120})^{2} - \frac{x - 96}{120}}) d x \\ + \int_{128}^{156} (\frac{x - 128}{64} - \frac{x - 96}{60}) \\ (\sqrt{0.5 + (\frac{x - 96}{120})^{2} - \frac{x - 96}{120}} \end{matrix}$

$\begin{matrix} - \sqrt{0.5 + (\frac{x - 128}{128})^{2} - \frac{x - 128}{128}}) d x \\ + \int_{156}^{192} (\frac{x - 128}{64} - \frac{192 - x}{36}) \\ (\sqrt{0.5 + (\frac{192 - x}{72})^{2} - \frac{192 - x}{72}} \\ - \sqrt{0.5 + (\frac{x - 128}{128})^{2} - \frac{x - 128}{128}}) d x \\ + \int_{192}^{256} \frac{256 - x}{64} (\sqrt{0.5} \\ - \sqrt{0.5 + (\frac{256 - x}{128})^{2} - \frac{256 - x}{128}}) d x \\ \approx 16.2 . \end{matrix}$ Therefore, we have $HR [ξ, C] < HR [ξ, B] < HR [ξ, D],$ and then the given rare book is suggested to belong to color type “C".

9 Portfolio selection of rare books

The expected payoffs of stock models based on uncertain variables were studied by Yu [29] and Yu [30] in 2012. Next we consider the problem of portfolio selection in uncertain rare book markets. Assume there are n rare books in a financial market. In order to describe the portfolio selection problem clearly, we first introduce the hyperbolic relative entropy model.

Let ξ₁, ξ₂, ⋯ , ξ_n be the rare books of securities, and let t₁, t₂, ⋯ , t_n be the portfolio numbers of ξ₁, ξ₂, ⋯ , ξ_n, respectively. Assume that λ denotes the predetermined risk of investment, and M₀ represents the predetermined level of an investor.

Note that each investor expects to make the greatest profit with the least risk. In other words, the goal is to minimize the hyperbolic relative entropy $HR [t_{1} ξ_{1} + t_{2} ξ + \cdot \cdot \cdot + t_{n} ξ_{n}]$ while maximizing the hyperbolic expected value $E [M (t_{1} ξ_{1} + t_{2} ξ + \cdot \cdot \cdot + t_{n} ξ_{n})] .$ Thus the hyperbolic relative entropy model is as follows, ${\begin{matrix} min_{ξ} HR [t_{1} ξ_{1} + t_{2} ξ + \cdot \cdot \cdot + t_{n} ξ_{n} | λ] \\ max_{m} E [M (t_{1} ξ_{1} + t_{2} ξ + \cdot \cdot \cdot + t_{n} ξ_{n})] \\ subject to : \\ t_{1} + t_{2} + \cdot \cdot \cdot + t_{n} = 1 \\ t_{i} \geq 0, i = 1, 2, \cdot \cdot \cdot, n, \end{matrix}$ where t = (t₁, t₂, ·· · , t_n) is the portfolio set that we are searching for. In order to solve the above optimization problem, we transform the portfolio selection problem into an objective programming problem which can be expressed as ${\begin{matrix} min_{t} HR [t_{1} ξ_{1} + t_{2} ξ + \cdot \cdot \cdot + t_{n} ξ_{n} | λ] \\ subject to : \\ E [M (t_{1} ξ_{1} + t_{2} ξ + \cdot \cdot \cdot + t_{n} ξ_{n})] \geq M_{0} \\ t_{1} + t_{2} + \cdot \cdot \cdot + t_{n} = 1 \\ t_{i} \geq 0, i = 1, 2, \cdot \cdot \cdot, n . \end{matrix}$

Let μ₁, μ₂, ⋯ , μ_n be the membership functions of the uncertain rare book sets ξ₁, ξ₂, ⋯ , ξ_n, respectively. Then $t_{1} ξ_{1} + t_{2} ξ + \cdot \cdot \cdot + t_{n} ξ_{n}$ is a portfolio function whose inverse functions are $μ_{l}^{- 1} (α) = t_{1} μ_{1 l}^{- 1} (α) + t_{2} μ_{2 l}^{- 1} (α) + \cdot \cdot \cdot + t_{n} μ_{nl}^{- 1} (α)$ and $μ_{r}^{- 1} (α) = t_{1} μ_{1 r}^{- 1} (α) + t_{2} μ_{2 r}^{- 1} (α) + \cdot \cdot \cdot + t_{n} μ_{nr}^{- 1} (α) .$ Since the function M is increasing, we have $\begin{matrix} E [M (t_{1} ξ_{1} + t_{2} ξ + \cdot \cdot \cdot + t_{n} ξ_{n})] \\ = & \frac{1}{2} \int_{0}^{1} (M (μ_{l}^{- 1}) + M (μ_{r}^{- 1})) d α . \end{matrix}$ Hence this portfolio model can be formulated as ${\begin{matrix} min_{t} HR [t_{1} ξ_{1} + t_{2} ξ + \cdot \cdot \cdot + t_{n} ξ_{n} | λ] \\ subject to : \\ \frac{1}{2} \int_{0}^{1} (M (μ_{l}^{- 1}) + M (μ_{r}^{- 1})) d α \geq M_{0} \\ t_{1} + t_{2} + \cdot \cdot \cdot + t_{n} = 1 \\ t_{i} \geq 0, i = 1, 2, \cdot \cdot \cdot, n . \end{matrix}$ Without losing generality, suppose that n = 3 and there are three rare books to be selected in the financial market. Consider the uncertain sets $ξ_{1} = (1, 3), ξ_{2} = (1, 2, 5), ξ_{3} = (1, 3, 5) .$ Let the portfolio selection be $M (ξ_{1} + ξ_{2} + ξ_{3}) = ξ_{1} + ξ_{2} + ξ_{3} .$ Then the inverse functions of M (t₁ ξ₁ + t₂ ξ₂ + t₃ ξ) are $μ_{l}^{- 1} (α) = t_{1} + (1 + α) t_{2} + (1 + 2 α) t_{3}$ and $μ_{r}^{- 1} (α) = 3 t_{1} + (5 - 3 α) t_{2} + (5 - 2 α) t_{3} .$ Taking the uncertain risk be λ = (2, 3, 4) and M₀ = 1.5. Then we have ${\begin{matrix} min_{t} (\underset{x \in [1, 2]}{{HR}_{1}} [μ_{1}], \underset{x \in [2, 3]}{{HR}_{2}} [μ_{2}, x - 2], \\ \underset{x \in [3, 4]}{{HR}_{2}} [μ_{3}, 4 - x], \underset{x \in [4, 5]}{{HR}_{2}} [μ_{3}]) \\ subject to : \\ 4 t_{1} + 5 t_{2} + 6 t_{3} \geq 3 \\ t_{1} + t_{2} + t_{3} = 1 \\ t_{1} \geq 0, t_{2} \geq 0, t_{3} \geq 0, \end{matrix}$ where $μ_{1} = t_{1} + (x - 1) t_{2} + \frac{x - 1}{2} t_{3}, x \in [1, 2],$ $μ_{2} = t_{1} + \frac{5 - x}{3} t_{2} + \frac{x - 1}{2} t_{3}, x \in [2, 3]$ and $μ_{3} = \frac{5 - x}{3} t_{2} + \frac{5 - x}{2} t_{3}, x \in [3, 5] .$ By solving the above objective programming problem by MATLAB 1 , we obtain $t = (0.2129, 0.3948, 0.3923)$ and the greatest profit is $(0.2129 ξ_{1}, 0.3948 ξ_{2}, 0.3923 ξ_{3}),$ the minimal hyperbolic relative entropy is $HR [t_{1} ξ_{1} + t_{2} ξ + t_{3} ξ_{3} | λ] = 0.0077 .$

10 Conclusion

For the purpose of describing the degree of difficulty of uncertain sets, this paper proposed two important concepts: hyperbolic entropy and hyperbolic relative entropy. Afterwards, a key formula was introduced to calculate the hyperbolic entropy and some mathematical properties of hyperbolic entropy were also studied. Moreover, some applications of hyperbolic entropy were investigated including uncertain learning curve, clustering of rare books and portfolio selection of collecting rare books. The future research direction can consider the statistical inference of membership function of uncertain set based on the hyperbolic entropy.

Footnotes

MATLAB R2022a, 9.12.0.1884302, maci64, Optimization Toolbox, “fminsearch" function.

References

Chen

, Li

, Xiao

and Yang

, Numerical solution andparameter estimation for uncertain SIR model with application toCOVID-19, Fuzzy Optimization and Decision Making 20(2) (2021), 189–208.

Gao

and Ralescu

, Elliptic entropy of uncertain set and itsapplications, International Journal of Intelligent Systems 33(4) (2018), 836–857.

Jia

and Chen

, Uncertain SEIAR model for COVID-19 cases inChina, Fuzzy Optimization and Decision Making 20(2) (2021), 243–259.

Kullback

and Leibler

, On information and sufficiency, TheAnnals of Mathematical Statistics 22(1) (1951), 79–86.

, Yang

and Yang

, Uncertain learning curve and itsapplication in scheduling, Computers and IndustrialEngineering 131 (2019), 534–541.

, Tang

and Yi

, A novel complexity-based mode featurerepresentation for feature extraction of ship-radiated noise usingVMD and slope entropy, Applied Acoustics 196 (2022), 108899.

Lio

and Liu

, Initial value estimation of uncertaindifferential equations and zero-day of COVID-19 spread in China, Fuzzy Optimization and Decision Making 20(2) (2021), 177–188.

Liu

, Uncertainty Theory (2nd ed.). Berlin: Springer; (2007).

Liu

, Some research problems in uncertainty theory, Journalof Uncertain Systems 3(1) (2009), 3–10.

10.

Liu

, Uncertain set theory and uncertain inference rule withapplication to uncertain control, Journal of Uncertain Systems 4(2) (2010a), 83–98.

11.

Liu

, Uncertain logic for modeling human language, Journal of Uncertain Systems 5(1) (2011), 3–20.

12.

Liu

, Membership functions and operational law of uncertain sets, Fuzzy Optimization and Decision Making 11(4) (2012), 387–410.

13.

Liu

, Totally ordered uncertain sets, Fuzzy Optimization andDecision Making 17(1) (2018), 1–11.

14.

Liu

and Lio

, A revision of sufficient and necessary condition of uncertainty distribution, Journal of Intelligent & Fuzzy Systems 38(4) (2020), 4845–4854.

15.

Liu

, Uncertain circuit equation, Journal of Uncertain Systems 14(3) (2021), Article 2150018.

16.

Liu

and Liu

, Residual analysis and parameter estimation of uncertain differential equations, Fuzzy Optimization and Decision Making 21(4) (2022), 513–530.

17.

Liu

, Moment estimation for uncertain regression model with application to factors analysis of grain yield, Communications in Statistics – Simulation and Computation (2022). DOI: 10.1080/03610918.2022.2160461.

18.

Liu

, Analysis of China’s population with uncertain statistics, Journal of Uncertain Systems 15(4) (2022), Article 2243001.

19.

Mehrdoust

, Noorani

and Xu

, Uncertain energy model for electricity and gas futures with application in spark-spread option price, Fuzzy Optimization and Decision Making (2022). DOI: 10.1007/s10700-022-09386-z.

20.

Sheng

and Kar

, Some results of moments of uncertain variable through inverse uncertainty distribution, Fuzzy Optimization and Decision Making 14(1) (2015), 57–76.

21.

Shnnon

and Weaver

, The Mathematical Theory of communications, The Univeraity of Illinois Press, Urbana, (1949).

22.

Tan

and Yu

, Arc entropy of uncertain sets and its applications, Journal of Intelligent & Fuzzy Systems 43(5) (2022), 6561–6574.

23.

Vaida

, Bounds of the minimal error probability on checking afinite or countable number of hypotheses, Problemy Peredachi Informatsii 4(1) (1968), 9–19.

24.

Wang

and Ha

, Quadratic entropy of uncertain sets, Fuzzy Optimization and Decision Making 12(1) (2013), 99–109.

25.

Wright

, Factors affecting the cost of airplanes, Journal ofthe Aeronautical Sciences 3(4) (1936), 122–128.

26.

Yao

and Ke

, Entropy operator for membership function of uncertain set, Applied Mathematics and Computation 242(2014), 898–906.

27.

Yao

, Sine entropy of uncertain sets and its applications, Applied Soft Computing 22 (2014), 432–442.

28.

and Liu

, Uncertain hypothesis test for uncertain differential equations, Fuzzy Optimization and Decision Making (2022). DOI: 10.1007/s10700-022-09389-w

29.

, Expected payoff of trading strategies for uncertain financial markets, Information: An International InterdisciplinaryJournal 14(10) (2012), 3175–3186.

30.

, A stock model with jumps for uncertain markets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 20(3) (2012), 421–432.