A novel entropy of intuitionistic fuzzy sets based on similarity and its application in finance

Abstract

In this paper, we propose a new formula for the entropy based on similarity measures of intuitionistic fuzzy sets (IFS). The contribution of this work is the proof that the new formula satisfies all the conditions of entropy. The experimentation on some examples shows that the new entropy is useful. Besides, we use the new entropy and similarity measures to design an algorithm for ranking assets in stock markets. The numerical results on 5 benchmark data sets were reported. It points out that the entropy and the similarity measures of IFS may provide an alternative tool for solving portfolio selection problems.

Keywords

Intuitionistic fuzzy set similarity measures stock markets assets ranking entropy

1 Introduction

In the real-world multi-criteria decision making (MCDM) problems, decision-making information provided by decision-makers is often inaccurate, ambiguous and uncertain causing by time pressure and lack of data, or ability limitation of the decision-makers to process information. To solve these problems, one of the solutions is applying the fuzzy set. However, there are still some limitations of the fuzzy set approach due to not representing much information. The IFS have been shown to have higher accuracy and increased ambiguity than fuzzy sets [1] thanks to adding a membership function compared to the usual fuzzy set [2] and the sum of two functions in the IFS [3] not greater than 1. Recently, the research in similarity and entropy of the IFS topic has been paid much attention [4]. Similarity has been widely used in many fields of real problems namely instance pattern recognition [5], selecting target market [6], green supplier selection [7], medical diagnosis [4], etc.

In the form of the MCDM problem, alternatives are evaluated under multiple attributes (or criteria). Determining the relative importance of the properties and priority of alternatives with respect to criteria are the two main tasks. Entropy plays an important role in assessing the uncertainty of information in fuzzy environments [8 –13]. It is often used to evaluate the weights of attributes or so-called criteria in MCDM problems; the similarity, meanwhile, is used to rank the alternatives [1, 14]. In the entropy measures of the intuitively fuzzy set, the previous researches formulated the direct entropy formula based on the axioms from the entropy definition [15 –19], {sine/cosine function or exponent function as in [20 –22]. However, this method is exhausting. Otherwise, using the existing measures to build other measures is natural and easy to practice. Therefore, in this work, we have suggested the new entropy measures of intuitionistic fuzzy sets based on the existing similarity measure.

Portfolio selection is an important problem in finance management. Investors want to select good assets in stock markets. In literature, portfolio selection can be formulated as an optimization problem in which one maximizes the return (with a fixed level of risk) or minimizes the risk (with a fixed level of return). The idea is to use a set of assets to avoid systematic risk. The first optimization model was given by Markowitz in 1952 [23]. In the Markowitz’s model, the variance is used to evaluate the risk. There have been some work to measure the risk of an asset or a portfolio using other indicators. In [24], the author used the value at risk as a measurement of risk. The work in [25] was to maximize the Sharpe ratio that is a combination of return and standard deviation. In [26, 27], the authors considered cardinality constraints to diversify the portfolio. Besides, the maximum loss or the percentage of periods with negative returns can be used as a risk indicator. The mentioned models must combine assets to form a portfolio and avoid systematic risk. In some markets, such as Vietnam, the investors just want to invest in a very few assets namely one or two assets. Thus, it is hard to apply optimization models to build a good strategic investment. Normally, the investor wants to choose good assets. A good asset must provide a high return and a low risk while there are many indicators that can be used to measure the risk and some other indicators represent the profit/return. Hence, we need to rank the assets and select the best ones.

The MCDM models allow to rank alternatives based on the combination of multiple criteria in which each criterion impact the ranking result by its weight in the model. This weight can be defined by the entropy of each criterion. Along with that, a similarity measure are used to rank alternatives. Therefore, it motivates us to use the MCDM model in order to rank assets. There are some publications using a single criterion or multi-criteria to rank assets. The existing approaches include Simple Additive Weight (SAW) method, AHP method, p-TOPSIS method, p-VIKOR method, Estimation of Weight method [28]. These methods use distances on Rⁿ. Although the indicators of assets are not certainty, there are no methods based on IFS for ranking assets. In this work, our approach is to use the similarity on IFSs for ranking assets. The historical returns are used to compute indicators for assets. The set of criteria was considered as a universal set in which an asset is an intuitionistic fuzzy set. The similarity between the assets and a perfect element is calculated. Afterward, the assets ranking is generated by using the similarities.

The contributions of this paper are:

Construct a new entropy of IFS based on the similarity measure.

Compare the new proposed entropy with some existing entropies.

Propose a MCDM model based on the new entropy and similarity measures

The application of the similarity on IFS for ranking assets in stock markets and solving portfolio selection problems. It is a new approach for ranking assets.

The paper is organized as follows: Section 2 briefly presents preliminary; Section 3 introduces a new entropy of IFS; applications are presented in Section 4 while Section 5 is reserved to conclusions.

2 Preliminary

In this section, we reintroduce some important concepts related to intuitive fuzzy sets (IFS).

Definition 2.1 (Atanassov). An IFS on the universal set X has form $A = {(x, μ_{A} (x), ν_{A} (x)) | x \in X}$ where μ_A (x) ∈ [0, 1], ν_A (x) ∈ [0, 1] are the membership and non-membership functions of x in A, respectively, such that $μ_{A} (x) + ν_{A} (x)) \leq 1$ for all x ∈ X

The degree of hesitant of x in X is π_A (x) =1 - μ_A (x) - ν_A (x).

We denote IFS (X) is the collection of IFS in the universal X.

For two IFS A, B ∈ IFS (X), we have:

•A ⊂ B iff μ_A (x) ≤ μ_B (x) and ν_A (x) ≥ ν_B (x) for all x ∈ X.

•A = B iff μ_A (x) = μ_B (x) and ν_A (x) = ν_B (x) for all x ∈ X.

•The complement of A ∈ IFS (X) is $A^{C} = {(x, ν_{A} (x), μ_{A} (x)) | x \in X}$ •For all λ > 0, we have $A = {(x, (μ_{A} (x))^{λ}, 1 - (1 - ν_{A} (x))^{λ}) | x \in X}$ (1)

Definition 2.2. A mapping S : IFS (X) × IFS (X) → [0, 1] is a similarity measure of the IFSs if it fulfills the following axioms:

(Sim1) 0 ≤ S (A, B) ≤1 for all A, B ∈ IFS (X),

(Sim2) S (A, A^C) =0 if A is crisp set,

(Sim3) S (A, A) =1 for all A ∈ IFS (X),

(Sim4) S (A, B) = S (B, A) for all A, B ∈ IFS (X),

(Sim5) S (A, C) ≤ min {S (A, B) , S (B, C)} for all A, B, C ∈ IFS (X) such that A ⊂ B ⊂ C.

In this paper we only consider the universal set X = {x₁, x₂, . . . , x_n} is a finite set. For each element x_i, we assign with a real number ω_i ∈ [0, 1], such that $\sum_{i = 1}^{n} ω_{i} = 1$ , called the weight of x_i, for all i = 1, 2, . . . , n.

Let A = {(x, μ_A (x) , ν_A (x)) |x ∈ X} , B = {(x, μ_B (x) , ν_B (x)) |x ∈ X} be two IFSs on X.

Example 2.1. Some existing similarity measures:

1./ Similarity measure proposed by [29].

$\begin{matrix} S_{S} (A, B) = \sum_{i = 1}^{n} \frac{ω_{i}}{3} {2 \sqrt{μ_{A} (x_{i}) μ_{B} (x_{i})} \\ + 2 \sqrt{ν_{A} (x_{i}) ν_{B} (x_{i})} + \sqrt{π_{A} (x_{i}) π_{B} (x_{i})} \\ + 2 \sqrt{(1 - μ_{A} (x_{i})) (1 - μ_{B} (x_{i}))} \\ + 2 \sqrt{(1 - ν_{A} (x_{i})) (1 - ν_{B} (x_{i}))}} \end{matrix}$ (2)

2./ Similarity measure proposed by [30] $S_{Q} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} S_{Q}^{i} (A, B)$ (3) where $S_{Q}^{i} (A, B) = {\begin{matrix} 1, if μ_{A} (x_{i}) = μ_{B} (x_{i}) and ν_{A} (x_{i}) = ν_{B} (x_{i}) \\ \frac{3 + \min {μ_{A} (x_{i}), μ_{B} (x_{i})} - \max {ν_{A} (x_{i}), ν_{B} (x_{i})} - | μ_{A} (x_{i}) - μ_{B} (x_{i}) | - | ν_{A} (x_{i}) - ν_{B} (x_{i}) |}{4}, & other cases \end{matrix}$ for all i = 1, 2, . . . , n.

3 New entropy of IFSs based on similarity measures

Entropy of IFSs used to measure the intuition of them. In 2001, Szmidt et al. [16] introduced the concept of entropy of IFSs.

Definition 3.1 (Szmidt and Kacprzyk). A function E : IFS (X) → [0, 1] is an entropy on IFS (X) if it satisfies axioms:

(En1) E (A) =0 if A having μ_A (x_i) =1 or ν_A (x_i) =1,∀ x_i ∈ X,

(En2) E (A) =1 iff μ_A (x_i) = ν_A (x_i) for all x_i ∈ X,

(En3) E (A) = E (A^C) for all A ∈ IFS (X),

(En4) For all A, B ∈ IFS (X), if μ_A (X_i) ≤ μ_B (x_i) ≤ ν_B (x_i) ≤ ν_A (x_i) or μ_A (X_i) ≥ μ_B (x_i) ≥ ν_B (x_i) ≥ ν_A (x_i) for all x_i ∈ X, then E (A) ≤ E (B).

Theorem 3.1. The function E : IFS (X) → [0, 1] defined by E (A) = S (A, A^C) (A ∈ IFS (X)) is an entropy on IFS (X).

Proof. Based on condition (Sim1) of similarity measures, we have E (A) = S (A, A^C) ∈ [0, 1].

(En1) If A is a crisp set then E (A) = S (A, A^C) =0.

(En2) It is obviously.

(En3) It is obviously.

(En4) For all A, B ∈ IFS (X), if μ_A (X_i) ≤ μ_B (x_i) ≤ ν_B (x_i) ≤ ν_A (x_i), for all x_i ∈ X, then A ⊆ B ⊆ B^C ⊆ A^C;if μ_A (X_i) ≥ μ_B (x_i) ≥ ν_B (x_i) ≥ ν_A (x_i), for all x_i ∈ X, then A^C ⊆ B^C ⊆ B ⊆ A. Depend on (Sim5), we have E (A) = S (A, A^C) ≤ S (A, B^C) = E (B). □

From Theorem 1, we can determine some new entropy of IFS as shown in example 2.

Example 3.1. Example 2. Given weight of x_i is $ω_{i} = \frac{1}{n}$ , for all i = 1, 2, . . . , n. We have:

1. From the similarity measure of Song et al. (2019) in the Equation (2) we get $\begin{matrix} E_{S} (A) = \frac{1}{3 n} \sum_{i = 1}^{n} {4 \sqrt{μ_{A} (x_{i}) ν_{A} (x_{i})} \\ + 1 - μ_{A} (x_{i}) - ν_{A} (x_{i}) + 2 \sqrt{(1 - μ_{A} (x_{i})) (1 - ν_{A} (x_{i}))}} \end{matrix}$ (4) 2. From the similarity measure of [30] in the Equation (3) we get $E_{Q} (A) = \sum_{i = 1}^{n} E_{Q}^{i} (A)$ (5) where $E_{Q}^{i} (A) = {\begin{matrix} 1, if μ_{A} (x_{i}) = ν_{A} (x_{i}) \\ \frac{3 + \min {μ_{A} (x_{i}), ν_{A} (x_{i})} - \max {μ_{A} (x_{i}), ν_{A} (x_{i})} - 2 | μ_{A} (x_{i}) - ν_{A} (x_{i}) |}{4}, & other cases \end{matrix}$ for all i = 1, 2, . . . , n.

Compare to some existing other entropies .

- Entropy of [15]: $E_{BB} (A) = \frac{1}{n} \sum_{i = 1}^{n} π_{A} (x_{i})$ (6) - Entropy of [16]: $E_{SK} (A) = \frac{1}{n} \sum_{i = 1}^{n} \frac{maxcount (A (x_{i}) \cap A^{C} (x_{i}))}{maxcount (A (x_{i}) \cup A^{C} (x_{i}))}$ (7) where maxcountA (x_i) = μ_A (x_i) + π_A (x_i) for all i = 1, 2, . . . , n.

- Entropy of [17]: $E_{H} (A) = \frac{1}{n (β - 1)} \sum_{i = 1}^{n} \log (μ_{A}^{β} (x_{i}) + ν_{A}^{β} (x_{i}) + π_{A}^{β} (x_{i}))$ (8) where 0 < β < 1

and

$\begin{matrix} E_{HY} (A) = \frac{1}{n} \sum_{i = 1}^{n} [μ_{A} (x_{i}) \log (μ_{A} (x_{i})) \\ + ν_{A} (x_{i}) \log (ν_{A} (x_{i})) + π_{A} (x_{i}) \log (π_{A} (x_{i}))] \end{matrix}$ (9) for all i = 1, 2, . . . , n.

- Entropy of [19]:

$\begin{matrix} E_{JK 1} (A) = \frac{1}{n} \sum_{i = 1}^{n} [μ_{A} (x_{i}) \log (μ_{A} (x_{i})) + ν_{A} (x_{i}) \log (ν_{A} (x_{i})) \\ - (1 - π_{A} (x_{i})) \log (1 - π_{A} (x_{i})) - π_{A} (x_{i})] \end{matrix}$ (10)

for all i = 1, 2, . . . , n.

and

$\begin{matrix} E_{JK 2} (A) = \frac{1}{n (\sqrt{2} - 1)} \\ \sum_{i = 1}^{n} ((μ_{A}^{\frac{1}{2}} (x_{i}) + ν_{A}^{\frac{1}{2}} (x_{i})) ((μ_{A} (x_{i}) \\ + ν_{A} (x_{i}))^{\frac{1}{2}} + \sqrt{2} π_{A} (x_{i}) - 1) \end{matrix}$ (11) - Entropy of [18]:

$\begin{matrix} E_{ZL} (A) = \frac{1}{4 n} \\ \sum_{i = 1}^{n} ((2 - μ_{A} (x_{i}) - ν_{A} (x_{i})) (1 - | μ_{A} (x_{i}) \\ - ν_{A} (x_{i}) | + π_{A} (x_{i}))) \end{matrix}$ (12) - Entropy of [20]:

$\begin{matrix} E_{Y} (A) = \frac{1}{n (\sqrt{2} - 1)} \\ \sum_{i = 1}^{n} (\sin \frac{π (1 + μ_{A} (x_{i}) - ν_{A} (x_{i}))}{4} \\ + \sin \frac{π (1 - μ_{A} (x_{i}) + ν_{A} (x_{i}))}{4} - 1) \end{matrix}$ (13) To verify the feasibility of the proposed entropy measures, we consider the comparison of ten entropies via two examples as follows.

Example 3.2. Let A = {(x₆, 0.1, 0.8) , (x₇, 0.3, 0.6) , (x₈, 0.5, 0.4) , (x₉, 0.8, 0.1) , (x₁₀, 1, 0)} be a IFS on the universal set X = {x₆, x₇, x₈, x₉, x₁₀}, where x_i is an apartment having i rooms, i = 6, 7, 8, 9, 10. In the environment of the linguistic variables, we remark A as “Large”, then based on the Equation (1), we remark A^0.5,A², A³ and A⁴ as “Quite Large”, “Very Large”, “Quite Very Large” and “Very Very Large”, respectively, in which

A^0.5 = {(x₆, 0.3126, 0.5528) , (x₇, 0.5477, 0.3675) , (x₈, 0.7071, 0.2254) , (x₉, 0.8944, 0.0513) , (x₁₀, 1, 0)}

A² = {(x₆, 0.01, 0.96) , (x₇, 0.09, 0.84) , (x₈, 0.25, 0.64) , (x₉, 0.64, 0.19) , (x₁₀, 1, 0)}

A³ = {(x₆, 0.001, 0.992) , (x₇, 0.027, 0.936) , (x₈, 0.125, 0.784) , (x₉, 0.512, 0.271) , (x₁₀, 1, 0)}

A⁴ = {(x₆, 0.0001, 0.9984) , (x₇, 0.0081, 0.9974) , (x₈, 0.0625, 0.8704) , (x₉, 0.4096, 0.3439) , (x₁₀, 1, 0)}

According to the mathematical view and intuitive of human, we easy to see that A^m+1 ⊂ A^m (m > 0) and $lim_{m \to \infty}$ ∅ = {(x₆, 0, 1) , (x₇, 0, 1) , (x₈, 0, 1) , (x₉, 1, 0) , (x₁₀, 1, 0)}. So that, the entropy measure E must be gratified E (A^0.5) > E (A) > E (A²) > E (A³) > E (A⁴). According to Table 1, we find that four entropy measures of IFSs satisfy this statement, that is E_Y, E_JK2, and two proposed E_S and E_Q, meanwhile other entropies are not feasible in this case.

Table 1

Comparison of entropies in example 3.2

	E _Q	E _T	E _BB	E _Y	E _SK	E _H	E _HY	E _JK1	E _JK2	E _ZL
A ^0.5	0.3388	0.6680	0.0675	0.5888	0.3717	0.7198	0.6076	0.5762	0.6727	0.1421
A	0.3000	0.6666	0.0800	0.5778	0.3698	0.7378	0.6239	0.5955	0.6702	0.4130
A ²	0.2190	0.5431	0.0760	0.4295	0.2281	0.6430	0.5034	0.4834	0.5439	0.1033
A ³	0.1800	0.4398	0.0704	0.3350	0.1980	0.5401	0.4056	0.3990	0.4382	0.0904
A ⁴	0.1743	0.3669	0.0665	0.2794	0.2131	0.4588	0.3401	0.3446	0.3639	0.0902

“bold” is reasonable

Example 3.3. Let A = {(x₆, 0.15, 0.8) , (x₇, 0.3, 0.65) , (x₈, 0.5, 0.45) , (x₉, 0.8, 0.1) , (x₁₀, 1, 0)} be a IFS on the universal set X = {x₆, x₇, x₈, x₉, x₁₀}, where x_i is an apartment having i rooms, i = 6, 7, 8, 9, 10. In the environment of the linguistic variables, we remark A as “Large”, then based on the Equation (1), we remark A^0.5,A², A³ and A⁴ as “Quite Large”, “Very Large”, “Quite Very Large”,and “Very Very Large”, respectively, in which

A^0.5 = {(x₆, 0.3873, 0.5528) , (x₇, 0.5477, 0.4084) , (x₈, 0.7071, 0.2584) , (x₉, 0.8944, 0.0513) , (x₁₀, 1, 0)}

A² = {(x₆, 0.0225, 0.96) , (x₇, 0.09, 0.8775) , (x₈, 0.25, 0.6975) , (x₉, 0.64, 0.19) , (x₁₀, 1, 0)}

A³ = {(x₆, 0.0034, 0.992) , (x₇, 0.027, 0.9572) , (x₈, 0.125, 0.8336) , (x₉, 0.512, 0.271) , (x₁₀, 1, 0)}

A⁴ = {(x₆, 0.0005, 0.9984) , (x₇, 0.0081, 0.9850) , (x₈, 0.0625, 0.9085) , (x₉, 0.4096, 0.3439) , (x₁₀, 1, 0)}

According to the mathematical view and intuitive of human, we easy to see that A^m+1 ⊂ A^m (m > 0) and $lim_{m \to \infty}$ ∅ = {(x₆, 0, 1) , (x₇, 0, 1) , (x₈, 0, 1) , (x₉, 1, 0) , (x₁₀, 1, 0)}. So that, the entropy measure E must be gratified E (A^0.5) > E (A) > E (A²) > E (A³) > E (A⁴). According to Table 1, we find that four entropy measures of IFSs satisfy this statement, that is E_Y, and two proposed E_S and E_Q, meanwhile other entropies are not feasible in this case.

Table 2

Comparison of entropies in example 3.3

	E _Q	E _T	E _BB	E _Y	E _SK	E _H	E _HY	E _JK1	E _JK2	E _ZL
A ^0.5	0.3605	0.6766	0.0385	0.6039	0.4001	0.6833	0.5635	0.5236	0.6812	0.1361
A	0.3375	0.6762	0.0500	0.5861	0.3733	0.7036	0.5797	0.5420	0.6828	0.1323
A ²	0.2066	0.5430	0.0545	0.4127	0.2047	0.6114	0.4586	0.4365	0.5540	0.0907
A ³	0.1697	0.4318	0.0558	0.3148	0.1823	0.5081	0.3658	0.3612	0.4411	0.0819
A ⁴	0.1670	0.3560	0.0567	0.2633	0.2028	0.4272	0.3084	0.3172	0.3626	0.0847

“bold” is reasonable

We easily find that the example 3 and example 4 have a slight variation in the rating of apartments with 6, 7, and 8 rooms, respectively, the results of example 3 and example 4 show that, our proposed entropy measures are robust. That is shown the reasonable of two new entropy measures.

4 Application of similarity measures and entropy in portfolio selection

4.1 Problem statement

We consider a portfolio selection problem in finance management. We suppose that there are m assets and the historical data of m assets are collected in time horizon [0, T]. The price of asset i at time period t is p_it. The returns r_it of asset i at period t is calculated by $r_{it} = \frac{p_{it} - p_{i (t - 1)}}{P_{i (t - 1)}}$ . The investor wants to select an asset to invest in. Each asset has several financial indicators which can be separated into two groups. The first group represents the profitability of the asset and the second group represents the risk. We suppose that five indicators are considered

ER: the expected return of asset i is calculated by ${ER}_{i} = \frac{1}{T} \sum_{t = 1}^{T} r_{it}$

Std: the standard deviation of asset i is calculated by ${Std}_{i} = \sqrt{\frac{1}{T} \sum_{t = 1}^{T} (r_{it} - {ER}_{i})^{2}}$

SR: the sharpe ratio of asset i is calculated by ${SR}_{i} = \frac{{ER}_{i}}{{Std}_{i}}$

PR: the positive rate of asset i is calculated by ${PR}_{i} = \frac{1}{T} \sum_{t = 1}^{T} η (r_{it})$ where η (r_it) =1 if r_it > 0, 0 otherwise

ML: the maximum loss of asset i is calculated by ${ML}_{i} = {\begin{matrix} | min_{t} r_{it} |, if there is r_{it} < 0 \\ 0, otherwise \end{matrix}$

Based on the indicators of assets, the investor wants to rank the assets, which help the investor to select good assets to invest in. The task is very hard if we work on the space of real values. Our idea is to convert the vector of the indicator of each asset to an IFS and rank the asset by using similarity measures.

4.2 Solution method

We consider m asset A_i, A₂, . . . , A_m and n financial indicators C₁, C₂, . . . , C_n. From the historical data, we compute the indicator j for asset i. The value is denoted by x_i,j. We consider the universal set C = {C₁, C₂, . . . , C_n}. The membership and non membership function for asset A_i can be defined as follows:

Membership function: $μ_{A_{i}} (C_{j}) = μ_{ij} = \frac{x_{ij} - {lb}_{j}}{{ub}_{j} - {lb}_{j} + ɛ}$ (14) Non-membership function: $ν_{A_{i}} (C_{j}) = ν_{ij} = \frac{{ub}_{j} - x_{ij}}{{ub}_{j} - {lb}_{j} + ɛ}$ (15) Where ${lb}_{j} = min_{i} x_{ij}, {ub}_{j} = max_{i} x_{ij}$ and ɛ is a positive constant. It is easy to see that function μ and ν defined as above satisfy the conditions of membership and non-membership function.

In this part, a multi-criteria decision making model (MCDM model), namely the intuitionistic fuzzy portfolio selection model is proposed for ranking assets. The method uses the new entropy and IFS similarity measure.

The new model has seven steps as follows:

1. Determine the criteria for assessment C = {C₁, C₂, . . . , C_n} .

2. Identify different alternatives for assessment A = {A₁, A₂, . . . , A_m}

3. Determine the IFSs A_i = {(C_j, μ_ij, ν_ij) |C_j ∈ C} for all i = 1, 2, . . . , m. We consider that A_i (i = 1, 2, . . . , m) is an IFS on the universal set C = {C₁, C₂, . . . , C_n}.

4. Determine the weight ω_j of the criteria C_j (j = 1, 2, . . . , n). In this step, we consider C_j = {(A_i, μ_ij, ν_ij) |A_i ∈ A} is an intuitionistic set on the universal set of alternatives A. We denote e_j = E (C_j), then we can determine the weight $ω_{j} = \frac{1 - e_{j}}{n - \sum_{i = 1}^{n} e_{j}}$ (16) for all C_j (j = 1, 2, . . . , n).

5. Choose the perfect alternative A_p: cost criteria are (0, 1) and benefit cri-teria are (1,0). We consider that A_p is an IFS on the universal set C = {C₁, C₂, . . . , C_n}.

6. Calculate the similarity measure S (A_b, A_i) of each option A_i i = 1, 2, . . . , m with the perfect alternative A_b.

7. Rank options based on the similarity measure. The alternative A_i is better than the alternative A_k (we denote A_i ≻ A_k) if S (A_i, A_b) > S (A_k, A_b), for all i, k = 1, 2, . . . , m.

5 Experimental settings and results

To demonstrate our approaches, we use 5 datasets: FSTE 100 (UK), NASDAQ (USA), MIBTEL 295 (Italy), EuroStoxx 50 (Europe) and S&P 500 (USA) 1

¹
http://host.uniroma3.it/docenti/cesarone/datasetsw3_tardella.html

, obtained from [31]. The data recorded the returns from Yahoo Finance over time in 264 weeks from 10-Mar-2003 to 24-Mar-2008. Each data has different number of stocks. As in [31], the stocks with more than two consecutive missing values were disregarded and for the remaining stock, the missing values were interpolated. The Table 3 describes the dataset information.

Table 3

Dataset descriptions

Name	No of stock	Time	Missing values	Market/Source
FSTE 100	79	10-Mar-2003 to 24-Mar-2008	No	UK/Yahoo Finance
NASDAQ	2196	10-Mar-2003 to 24-Mar-2008	No	USA/Yahoo Finance
MIBTEL 295	226	10-Mar-2003 to 24-Mar-2008	No	Italy/Yahoo Finance
EuroStoxx50	48	10-Mar-2003 to 24-Mar-2008	No	Europe/Yahoo Finance
S&P 500	476	10-Mar-2003 to 24-Mar-2008	No	USA/Yahoo Finance

We consider 5 indicators: Expected Return (ER), Standard Deviation (Std), Positive Rate (PR), Sharpe Ratio (SR), Max Loss (ML). The expressions for those metrics are described in the previous section.

The results for 5 datasets are presented in Tables 4 to 13 which are ranked by Similarity and SR. We use the ratio SR because SR is an important indicator which balances return and standard deviation. In the tables, we only report the top ten stock prices ranking on similarities and SR, and the epsilon value is chosen to be 0.000001.

Table 4

FSTE 100 Dataset ranking by Similarity

Name	ER	PR	SR	STD	ML	Similarity	Similarity	SR
						(10^-3)	Ranking	Ranking
SSE.L	0.00322	0.56818	0.15438	0.02088	0.05741	180.33044	1	6
BATS.L	0.00430	0.53788	0.18018	0.02387	0.07236	179.74047	2	3
TSCO.L	0.00368	0.55682	0.14870	0.02472	0.06891	178.96875	3	11
BG.L	0.00595	0.59848	0.18085	0.03292	0.08890	178.25076	4	2
DGE.L	0.00196	0.53030	0.10078	0.01950	0.06116	177.92841	5	29
SCTN.L	0.00348	0.53030	0.12660	0.02748	0.07007	177.75803	6	18
IMT.L	0.00329	0.54545	0.14059	0.02337	0.08078	177.67859	7	13
IPR.L	0.00652	0.57955	0.18728	0.03479	0.10196	177.19817	8	1
JMAT.L	0.00368	0.54924	0.12876	0.02857	0.07906	177.12129	9	17
HSBA.L	0.00217	0.51515	0.10834	0.02004	0.07504	176.89598	10	25

Table 5

STE 100 Dataset ranking by SR

Name	ER	PR	SR	STD	ML	Similarity	Similarity	SR
						(10^-3)	Ranking	Ranking
IPR.L	0.00652	0.57955	0.18728	0.03479	0.10196	177.19817	8	1
BG.L	0.00595	0.59848	0.18085	0.03292	0.08890	178.25076	4	2
BATS.L	0.00430	0.53788	0.18018	0.02387	0.07236	179.74047	2	3
SAB.L	0.00558	0.57197	0.16948	0.03290	0.14407	173.00140	20	4
BLT.L	0.00687	0.59470	0.15725	0.04367	0.15551	171.43674	28	5
SSE.L	0.00322	0.56818	0.15438	0.02088	0.05741	180.33044	1	6
TLW.L	0.00768	0.56818	0.15317	0.05014	0.17037	169.10570	37	7
CPW.L	0.00651	0.54545	0.15071	0.04321	0.09986	175.33875	12	8
BA.L	0.00542	0.60227	0.15003	0.03611	0.13822	173.11366	19	9
ANTO.L	0.01515	0.60606	0.14941	0.10140	0.18713	165.71333	52	10

Table 6

NASDAQ Dataset ranking by Similarity

Name	ER	PR	SR	STD	ML	Similarity	Similarity	SR
						(10^-3)	Ranking	Ranking
HCBK	0.00454	0.54924	0.16860	0.02694	0.08545	171.00318	1	23
SIAL	0.00404	0.56818	0.16714	0.02417	0.08696	170.89390	2	25
UMBF	0.00321	0.54167	0.12211	0.02629	0.06201	170.66426	3	141
KTII	0.00801	0.54545	0.20695	0.03873	0.10649	170.51224	4	4
PTNR	0.00812	0.53409	0.19839	0.04095	0.10718	169.89223	5	6
NRGY	0.00336	0.55303	0.12335	0.02727	0.07235	169.72177	6	136
NTRS	0.00330	0.55682	0.11446	0.02883	0.06894	169.44506	7	171
TAXI	0.00472	0.53409	0.12703	0.03719	0.07450	169.33184	8	121
BOKF	0.00213	0.51515	0.09429	0.02258	0.06134	169.04731	9	298
TROW	0.00531	0.56439	0.14562	0.03644	0.09136	168.91783	10	67

Table 7

NASDAQ Dataset ranking by SR

Name	ER	PR	SR	STD	ML	Similarity	Similarity	SR
						(10^-3)	Ranking	Ranking
AMOV	0.01014	0.57197	0.22763	0.04454	0.15862	166.32549	38	1
MICC	0.01659	0.60606	0.22666	0.07320	0.26082	155.63782	386	2
RIMM	0.01523	0.57197	0.21928	0.06945	0.16938	164.15792	83	3
KTII	0.00801	0.54545	0.20695	0.03873	0.10649	170.51224	4	4
AAPL	0.01129	0.60985	0.20191	0.05592	0.21603	159.70515	216	5
PTNR	0.00812	0.53409	0.19839	0.04095	0.10718	169.89223	5	6
ISRG	0.01383	0.55682	0.19569	0.07067	0.16991	162.56124	119	7
HANS	0.01606	0.53409	0.19497	0.08236	0.34224	139.20903	1088	8
HUBG	0.01182	0.54924	0.19460	0.06076	0.14197	165.63203	48	9
CETV	0.01050	0.56061	0.19366	0.05421	0.15147	164.95943	64	10

Table 8

MIBTEL 295 Dataset ranking by Similarity

Name	ER	PR	SR	STD	ML	Similarity	Similarity	SR
						(10^-3)	Ranking	Ranking
ISPR.MI	0.00481	0.57955	0.16535	0.02908	0.07220	199.95435	1	7
ISP.MI	0.00519	0.54167	0.16787	0.03093	0.07166	199.93424	2	5
ACE.MI	0.00559	0.54545	0.17863	0.03132	0.07808	199.69471	3	3
CB.MI	0.00338	0.50379	0.13446	0.02512	0.06543	199.05530	4	24
ENEL.MI	0.00288	0.56439	0.14417	0.01999	0.07806	198.57450	5	14
IMA.MI	0.00197	0.51136	0.07982	0.02471	0.05550	197.04235	6	77
ENI.MI	0.00283	0.54545	0.11902	0.02379	0.07882	196.91118	7	32
A2A.MI	0.00403	0.45076	0.13535	0.02975	0.07696	196.90091	8	21
UBI.MI	0.00302	0.54545	0.09387	0.03216	0.06349	196.83271	9	57
IP.MI	0.00315	0.50758	0.10802	0.02918	0.07784	195.81195	10	39

Table 9

MIBTEL 295 Dataset ranking by SR

Name	ER	PR	SR	STD	ML	Similarity	Similarity	SR
						(10^-3)	Ranking	Ranking
TFI.MI	0.01143	0.56061	0.21296	0.05366	0.21858	183.15183	92	1
DANR.MI	0.00966	0.54924	0.20414	0.04731	0.15866	190.49829	48	2
ACE.MI	0.00559	0.54545	0.17863	0.03132	0.07808	199.69471	3	3
DAN.MI	0.00903	0.54167	0.17642	0.05120	0.13525	191.83193	38	4
ISP.MI	0.00519	0.54167	0.16787	0.03093	0.07166	199.93424	2	5
TEN.MI	0.00811	0.56818	0.16768	0.04837	0.16319	188.63801	60	6
ISPR.MI	0.00481	0.57955	0.16535	0.02908	0.07220	199.95435	1	7
SIR.MI	0.00544	0.46970	0.16148	0.03370	0.12516	191.50238	43	8
FSAR.MI	0.00535	0.58712	0.16147	0.03312	0.12487	193.83051	25	9
MIR.MI	0.00432	0.53030	0.16138	0.02675	0.11085	194.92761	16	10

Table 10

EuroStoxx 50 Dataset ranking by Similarity

Name	ER	PR	SR	STD	ML	Similarity	Similarity	SR
						(10^-3)	Ranking	Ranking
AABA.AS	0.00496	0.54545	0.17656	0.02809	0.06982	203.64622	1	2
ISP.MI	0.00514	0.54167	0.16640	0.03090	0.07166	202.96673	2	4
ELE.MC	0.00527	0.58333	0.20556	0.02564	0.08971	202.76123	3	1
ENEL.MI	0.00288	0.56439	0.14417	0.01999	0.07806	202.31502	4	7
ENI.MI	0.00283	0.54545	0.11888	0.02382	0.07928	200.96115	5	14
BBVA.MC	0.00320	0.55682	0.11526	0.02777	0.07739	200.95898	6	15
SAN.MC	0.00598	0.58712	0.17497	0.03416	0.09712	200.81799	7	3
VIV.PA	0.00330	0.54545	0.10656	0.03098	0.07575	200.51113	8	18
TEF.MC	0.00334	0.57197	0.12676	0.02634	0.09137	200.10007	9	11
SZE.PA	0.00519	0.56818	0.15196	0.03418	0.10219	199.30218	10	6

Table 11

EuroStoxx 50 Dataset ranking by SR

Name	ER	PR	SR	STD	ML	Similarity	Similarity	SR
						(10^-3)	Ranking	Ranking
ELE.MC	0.00527	0.58333	0.20556	0.02564	0.08971	202.76123	3	1
AABA.AS	0.00496	0.54545	0.17656	0.02809	0.06982	203.64622	1	2
SAN.MC	0.00598	0.58712	0.17497	0.03416	0.09712	200.81799	7	3
ISP.MI	0.00514	0.54167	0.16640	0.03090	0.07166	202.96673	2	4
RWE.DE	0.00545	0.57955	0.16359	0.03331	0.12513	197.38495	19	5
SZE.PA	0.00519	0.56818	0.15196	0.03418	0.10219	199.30218	10	6
ENEL.MI	0.00288	0.56439	0.14417	0.01999	0.07806	202.31502	4	7
BAY.DE	0.00590	0.56818	0.13758	0.04285	0.16827	190.87250	36	8
IBE.MC	0.01156	0.48106	0.12750	0.09066	0.06538	197.88707	15	9
BAS.DE	0.00413	0.55682	0.12710	0.03247	0.09709	198.95134	13	10

Table 12

S&P 500 Dataset ranking by Similarity

Name	ER	PR	SR	STD	ML	Similarity	Similarity	SR
						(10^-3)	Ranking	Ranking
MCD	0.00597	0.56818	0.20631	0.02894	0.06601	169.76733	1	6
K	0.00274	0.55682	0.15440	0.01777	0.05658	169.13840	2	39
BCR	0.00468	0.58712	0.17812	0.02629	0.06720	168.83401	3	13
XEL	0.00307	0.55682	0.14451	0.02126	0.05710	168.54611	4	63
EIX	0.00553	0.62500	0.21313	0.02594	0.09352	168.29846	5	4
PG	0.00242	0.57197	0.13526	0.01790	0.05926	168.20865	6	84
PCG	0.00430	0.57576	0.16827	0.02553	0.07183	168.20062	7	22
PCP	0.00835	0.59470	0.22568	0.03701	0.09413	168.14915	8	2
DUK	0.00416	0.55303	0.14888	0.02792	0.06493	167.81050	9	51
RTN	0.00399	0.61364	0.16315	0.02447	0.07820	167.68670	10	29

Table 13

S&P 500 Dataset ranking by SR

Name	ER	PR	SR	STD	ML	Similarity	Similarity	SR
						(10^-3)	Ranking	Ranking
MON	0.01035	0.60985	0.23961	0.04321	0.13159	165.84942	43	1
PCP	0.00835	0.59470	0.22568	0.03701	0.09413	168.14915	8	2
RRC	0.01097	0.59091	0.21462	0.05110	0.14929	163.25929	94	3
EIX	0.00553	0.62500	0.21313	0.02594	0.09352	168.29846	5	4
AYE	0.00828	0.56439	0.20756	0.03988	0.17172	161.45951	147	5
MCD	0.00597	0.56818	0.20631	0.02894	0.06601	169.76733	1	6
AAPL	0.01129	0.60985	0.20191	0.05592	0.21603	157.93240	239	7
CME	0.00933	0.57576	0.19767	0.04719	0.16851	161.19502	155	8
AMT	0.00778	0.56818	0.19285	0.04035	0.10461	166.03450	38	9
TIE	0.01291	0.55682	0.18151	0.07112	0.18907	157.35830	260	10

From the results, we can see that:

The expected return of every top ten assets is positive

The positive rate of every top ten assets is higher than 50%

The similarity ranking differentiates from SR ranking. But, top ten assets by similarity ranking are highly ranked by Sharpe ratio. Hence, the Sharpe ratio has strong influences on the results by similarity. In our point of view, results by using similarity is better because the similarity combines the indicators and the membership, non-membership reflect the comparison of assets with a specific indicator.

The proposed ranking is a good combination of indicators in finance to evaluate assets in stock markets. Thus, it is very useful in the case where investor want to ranking assets and select several assets to invest. Ranking assets by using the similarity of IFSs may be a new alternative methods in portfolio selection.

6 Conclusions

In this paper, we proposed a new entropy on IFSs. The new entropy was compared to existing entropies to show its usefulness and then the new entropy was applied to calculate the weight in the procedure for solving multi-criteria decision making problems. We proposed an application of the new entropy and similarity measure for ranking assets in stock markets as well. The numerical results on 5 benchmark data sets have showed that the entropy and similarity measures of IFSs can be used for solving portfolio selection problems.

For the future work, we will continue studying this approach for other real problems. Additionally, we find new measures of IFSs, Pythagorean fuzzy sets, picture fuzzy sets, and neutrosophic sets and apply them to real problems as clustering, medical diagnosis, stock segment, decision support system, etc.

The source code for this work is shared publicly on github. 2

https://github.com/doandongnguyen/ifsdata

Footnotes

Acknowledgment

This research is funded by International School, Vietnam National University, Hanoi (VNU-IS) under project number CS.NNC/2020-02.

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A novel entropy of intuitionistic fuzzy sets based on similarity and its application in finance

Abstract

Keywords

1 Introduction

2 Preliminary

4.1 Problem statement

4.2 Solution method

1 http://host.uniroma3.it/docenti/cesarone/datasetsw3_tardella.html

Footnotes

Acknowledgment

References

¹
http://host.uniroma3.it/docenti/cesarone/datasetsw3_tardella.html