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.
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 R n . 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.
Preliminary
In this section, we reintroduce some important concepts related to intuitive fuzzy sets (IFS).
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
(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
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.
1./ Similarity measure proposed by [29].
2./ Similarity measure proposed by [30]
Entropy of IFSs used to measure the intuition of them. In 2001, Szmidt et al. [16] introduced the concept of entropy of IFSs.
(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).
(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.
1. From the similarity measure of Song et al. (2019) in the Equation (2) we get
- Entropy of [15]:
- Entropy of [17]:
and
- Entropy of [19]:
for all i = 1, 2, . . . , n.
and
A0.5 = {(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 Am+1 ⊂ A
m
(m > 0) and
Comparison of entropies in example 3.2
“bold” is reasonable
A0.5 = {(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 Am+1 ⊂ A
m
(m > 0) and
Comparison of entropies in example 3.3
“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.
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
ER: the expected return of asset i is calculated by
Std: the standard deviation of asset i is calculated by
SR: the sharpe ratio of asset i is calculated by
PR: the positive rate of asset i is calculated by
ML: the maximum loss of asset i is calculated by
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.
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 xi,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:
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
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.
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
, 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.Dataset descriptions
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.
FSTE 100 Dataset ranking by Similarity
STE 100 Dataset ranking by SR
NASDAQ Dataset ranking by Similarity
NASDAQ Dataset ranking by SR
MIBTEL 295 Dataset ranking by Similarity
MIBTEL 295 Dataset ranking by SR
EuroStoxx 50 Dataset ranking by Similarity
EuroStoxx 50 Dataset ranking by SR
S&P 500 Dataset ranking by Similarity
S&P 500 Dataset ranking by SR
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.
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
Footnotes
Acknowledgment
This research is funded by International School, Vietnam National University, Hanoi (VNU-IS) under project number CS.NNC/2020-02.
