Application of VPRS model with enhanced threshold parameter selection mechanism to automatic stock market forecasting and portfolio selection

Expert Systems with Applications 36 (2009) 11652–11661

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Application of VPRS model with enhanced threshold parameter selection mechanism to automatic stock market forecasting and portfolio selection Kuang Yu Huang* Department of Information Management, Ling Tung University, #1 Ling Tung Road, Taichung City 408, Taiwan

a r t i c l e

i n f o

Keywords: Variable Precision Rough Set Fuzzy theory Fuzzy C-Means ARX model Grey relational analysis Stock portfolio

a b s t r a c t This study proposes a technique based upon Fuzzy C-Means (FCM) classification theory and related fuzzy theories for choosing an appropriate value of the Variable Precision Rough Set (VPRS) threshold parameter (b) when applied to the classification of continuous information systems. The VPRS model is then combined with a moving Average Autoregressive Exogenous (ARX) prediction model and Grey Systems theory to create an automatic stock market forecasting and portfolio selection mechanism. In the proposed mechanism, financial data are collected automatically every quarter and are input to an ARX prediction model to forecast the future trends of the collected data over the next quarter or half-year period. The forecast data are then reduced using a GM(1, N) model, classified using a FCM clustering algorithm, and then supplied to a VPRS classification module which selects appropriate investment stocks in accordance with a pre-determined set of decision-making rules. Finally, a grey relational analysis technique is employed to weight the selected stocks in such a way as to maximize the rate of return of the stock portfolio. The validity of the proposed approach is demonstrated using electronic stock data extracted from the financial database maintained by the Taiwan Economic Journal (TEJ). The portfolio results obtained using the proposed hybrid model are compared with those obtained using a Rough Set (RS) selection model. The effects of the number of attributes of the RS lower approximation set and VPRS b-lower approximation set on the classification are systematically examined and compared. Overall, the results show that the proposed stock forecasting and stock selection mechanism not only yields a greater number of selected stocks in the b-lower approximation set than in the RS approximation set, but also yields a greater rate of return. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Predicting stock prices in today’s volatile markets is notoriously difficult and represents a major challenge for traditional time-series-based forecasting mechanisms. Consequently, a requirement exists for robust forecasting schemes capable of accurately predicting the future behavior of the stock market in order to provide investors with a reliable indication as to when and where they should invest their money in order to maximize their rate of return. Many applications have been presented in recent decades for predicting market trends. Typical mechanisms include the use of genetic algorithms (GAs) to choose optimal portfolios (Bauer, 1994; Hassan, Nath, & Kirley, 2007), the application of neural networks to predict real-world stock trends (Aiken & Bsat, 1999; Cao, Leggio, & Schniederjans, 2005; Kuo, Chen, & Hwang, 2001), the integration of fuzzy logic and forecasting techniques to create artificial intelligence systems for market tracking and forecasting * Tel.: +886 922621030. E-mail address: [email protected] 0957-4174/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2009.03.028

purposes (Abraham, Nath, & Mohanthi, 2001; Romahi & Shen, 2000), the use of statistical approaches for the forecasting of economic indicators (Box & Jenkins, 1976; Engle, 1982; Kolarik & Rudorfer, 1994; Pankratz, 1983; Tse, 1997), the application of Rough Set (RS) theory to predict behavior of stock market performance indices (Skalko, 1996), and so on. Rough Set theory was introduced more than 20 years ago (Pawlak, 1982) and has emerged as a powerful technique for the automatic classification of objects (Tsumato, Slowinski, Komorowsk, & Grzymala-Busse, 2004). The literature contains many examples of the application of RS theory in a diverse range of fields, including machine learning, forecasting, knowledge acquisition, decision analysis, knowledge discovery, pattern recognition, and data mining (Beynon & Peel, 2001; Li & Wang, 2004; Pawlak, 1994; Pawlak, Wang, & Ziarko, 1988; Slowinski, 1992). Most RS applications are designed to deal with classification problems of one form or another. In constructing such applications, RS theory is generally integrated with other theories such as Grey Systems theory, for example (Huang & Jane, 2007). Typical applications include multi-criteria classification schemes (Blaszczyn´ski, Greco, & Slowin´ski, 2007), stock market

K.Y. Huang / Expert Systems with Applications 36 (2009) 11652–11661

analysis algorithms (Bazan & Szczuka, 2000; Shen & Loh, 2004; Wang, 2003), Variable Precision Rough Set models (Beynona & Dri1eldb, 2005; Huang & Jane, 2009), and so forth. However, the success of RS techniques in correctly classifying a dataset relies upon all the collected data being correct and certain. In other words, performing a classification function with a controlled degree of uncertainty or misclassification error falls outside the realm of the RS approach (Ziarko, 1993). In an attempt to extend the applicability of the RS method, Ziarko developed the Variable Precision Rough Set (VPRS) theory (Ziarko, 1993). In contrast to the original RS model, the aim of the VPRS approach is to analyze and identify data patterns which represent statistical trends rather than functional patterns (Ziarko, 2001). VPRS deals with uncertain classification problems through the use of a precision parameter, b. Essentially, b represents a threshold value which determines the portion of the objects in a particular conditional class which are assigned to the same decision class. (Note that in conventional RS theory, b has a value of one.) Thus, determining an appropriate value of b is essential in deriving knowledge from partially-related data objects. Many VPRS studies presented in the literature discuss the problem of b-reducts or approximate reducts. Ziarko (1993) stated that b-reducts, i.e. a subset P of the set of conditional attributes C with respect to a set of decision attributes D, must satisfy the following conditions: (i) the subset P offers the same quality of classification (subject to the same b value) as that achieved using the whole set of conditional attributes C; and (ii) no attribute can be eliminated from the subset P without affecting the classification quality (subject to the same b value). Despite the fundamental role played by the b-reducts in determining the VPRS classification results, the literature lacks any formal empirical evidence to support the choice of any particular method of b-reducts selection (Beynon, 2000). Ziarko (1993) suggested that the b value could simply be determined at the discretion of the decision maker. Beynon (2000) proposed two methods for selecting an appropriate b-reducts based upon the assumption that the b value fell within an interval whose range was determined by the desired level of classification performance (Beynon, 2001). Beynon (2002) introduced the concept of a ðl; uÞ-quality graph to map the quality of the VPRS classification performance to the values assigned to the l and u bounds. Su and Hsu (2006) proposed a method for determining a suitable value of the precision parameter based on an assessment of the minimum acceptable upper bound of the misclassification error. Ziarko (1993) argued that the b value represented a classification error and suggested that its value should be confined to the domain [0.0, 0.5). Conversely, An, Shan, Chan, Cercone, and Ziarko (1996) and Beynon (2001) used the b parameter to denote the proportion of correct classifications, and argued that its value should fall within the range (0.5, 1.0). Many researchers have investigated the problem of assigning an appropriate value to b using some form of b-reducts approach (Beynon, 2000, 2001; Mi, Wu, & Zhang, 2004). Basically, these b-reducts methods utilize the principle of the ‘‘extent of classification correctness” to determine an appropriate b value under the premise that changing the number of attributes has no effect on the classification results. However, such approaches have the major drawback that the value of b can only be determined once the classification results have been obtained. In general, information systems are characterized by continuous attribute values, and thus their processing requires some form of clustering technique. However, when clustering continuous data, errors are liable to arise as a result of the fuzzy nature of the data. As a result, selecting an appropriate value of b for the classification of continuous datasets using a b-reducts approach is problematic. Accordingly, the current study presents a method for determining b using the Fuzzy C-Means (FCM) clustering method and related fuzzy set theories.

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As discussed above, a requirement exists for reliable forecasting schemes capable of predicting future stock market trends in order to assist investors in determining where best to invest their funds so as to increase their rate of return on investment. However, the use of a VPRS approach to extract information from the stock market is problematic since financial data and market indices vary sequentially over time and have the form of a correlated signal system. In other words, the stock market is essentially a dynamic process. In general, such processes can best be described using a discrete-time series model such as an autoregressive (AR) model, a moving average (MA) model, or an autoregressive with exogenous inputs (ARX) model (Bhardwaj & Swanson, 2006; Ljung, 1999; Marcellino, Stock, & Watson, 2006; Sohn & Lim, 2007). Accordingly, the present study develops an algorithm for predicting the future behavior of the stock market and selecting an appropriate stock portfolio based upon an ARX prediction model, the multivariate GM(1, N) model (Deng, 1989), the FCM clustering technique, VPRS theory, grey relational analysis, and the investment guidelines prescribed by Buffett (Hagstrom, Miller, & Fisher, 2005). The feasibility and effectiveness of the proposed approach are demonstrated using the case of electronic stocks for illustration purposes. The performance of the proposed algorithm is verified by comparing the rate of return on the selected stock portfolio with that of a portfolio chosen using a conventional RS-based hybrid model (Huang & Jane, 2009). The remainder of this paper is organized as follows. Section 2 presents the fundamental principles of the ARX model, VPRS theory and Grey theory, respectively. Section 3 describes the integration of these concepts to create an automatic stock market forecasting and portfolio selection scheme. Section 4 compares the performance of the proposed scheme with that of the conventional RS-based model. Finally, Section 5 presents some brief concluding remarks and indicates the intended direction of future research.

2. Review of related methodologies 2.1. ARX model In general, the objective of ARX prediction models is to minimize a positive function of the prediction error. The predictions of an ARX model are based on the assumption that the system changes only gradually over time and can be described by a set of parameters. When constructing the ARX model, these parameters are generally estimated using a system identification process. The basic ARX model has the form

yðtÞ þ a1 yðt  1Þ þ    þ ana yðt  na Þ ¼ b1 uðt  1Þ þ    þ bnb uðt  nb Þ þ eðtÞ; where yðtÞ is the output, uðtÞ is the input and eðtÞ is a white-noise term. In the application considered in the present study, i.e. the forecasting of time-series financial data, the most important aspect of the ARX model is its one-step-ahead predictor, which has the form ^ðt; hÞ ¼ hT uðtÞ, where h ¼ ½a1 ; . . . ana ; b1 ; . . . ; bnb T is obtained using y P ^ðt; hÞÞ2 , the least squares method and yields minh N1 Nt¼1 ðyðtÞ  y T and u ðtÞ ¼ ½yðt  1Þ; . . . ; yðt  na Þ; uðt  1Þ; . . . ; uðt  nb Þ. This one-step-ahead predictor is essentially the scalar product of a known data vector uðtÞ and the parameter vector h. In statistics, such a model is known as a linear regression model and vector uðtÞ is the regression vector. The prediction error of the ARX model ^ðt; hÞ and can be computed given a knowlis given by eðtÞ ¼ yðtÞ  y edge of yðtÞ and uðtÞ. Note that for convenience, each instance of ðyðtÞ; uðtÞÞ is generally referred to as a data point.

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2.2. Variable Precision Rough Set (VPRS) theory VPRS theory operates on what may be described as a knowledge-representation system or an information system (Su & Hsu, 2006). The basic principles of information systems and the application of VPRS theory to the processing of information systems are described in the following sections. (1) Information systems A typical information system has the form S ¼ ðU; A; V q ; fq Þ, where U is a non-empty finite set of objects and A is a non-empty finite set of attributes describing each object. Assume that the attributes in set A can be partitioned into a set of conditional attributes C – / and another set of decisional attributes D – /, A ¼ C [ D and C \ D ¼ /. If one lists the values of all the attributes (both conditional and decisive) and makes a table, such a table is known as a decision table. For each attribute q 2 A; V q represents S the domain of q, i.e. V ¼ V q . Finally, fq : U  A ! V is the information function such that f ðx; qÞ 2 V q for 8q 2 A and 8x 2 U. For every subset P, where P # C and P – /, the equivalence relation is given by I ¼ fðx; yÞ 2 U  U : f ðx; qÞ ¼ f ðy; qÞ8q 2 Pg. The corresponding equivalence class is denoted by IðPÞ and indicates the objects in a particular conditional class which are assigned to the same decisional class. In VPRS theory, the value assigned to b represents a threshold value which governs the proportion of objects in a particular conditional class which fall within the same equivalent class. (2) b-Lower and b-upper approximation sets In processing the information system S ¼ ðU; A; V q ; fq Þ, A ¼ C [ D, X # U, P # C using a VPRS model with 0:5 < b 6 1, the data analysis procedure hinges on two basic concepts, namely the b-lower and b-upper approximations of a set. The b-lower approximation of sets X # U and P # C can be expressed as follows:

  jIðPÞ \ Xj RbP ðXÞ ¼ [ IðPÞ : Pb : jIðPÞj Similarly, the b-upper approximation of sets X # U and P # C is given by

  jIðPÞ \ Xj RbP ðXÞ ¼ [ IðPÞ : >1b : jIðPÞj In the case where b ¼ 1, RbP ðXÞ and RbP ðXÞ are equivalent to the lower and upper approximation sets in RS theory, respectively, i.e. the VPRS model reverts to the original RS model. Ziarko (1993) defined the following alternative expressions for the b-negative region and b-boundary region of X in S:

to facilitate the application of RS theory and the VPRS model to data mining and rule construction applications. 2.2.1. Determination of suitable threshold value b This study proposes a mechanism for determining a suitable value of the VPRS precision parameter b based on the Fuzzy C-Means clustering method and the general principles of fuzzy algorithms and fuzzy set theory. The various components of the proposed mechanism are introduced in the sections below. (1) Fuzzy C-Means (FCM) clustering method (Cox, 2005; Glackina, Maguirea, McIvorb, Humphreysb, & Hermana, 2007; Ramze Rezaee, Lelieveldt, & Reiber, 1998). FCM, first developed by Dunn (1973) and later refined by Bezdek (1981), is an unsupervised clustering algorithm with multiple applications, ranging from feature analysis, to clustering and classifier design. Fuzzy clustering techniques differ from hard clustering algorithms such as the K-means scheme in that a single data object may be mapped simultaneously to multiple clusters rather than being assigned exclusively to a single cluster. FCM clustering consists of two basic procedures, namely (1) calculating the cluster centers and assigning the data points to these centers on the basis of their Euclidean distance, and (2) determining the cluster memberships of each sample point. The first procedure is repeated iteratively until the cluster centers remain stable from one iteration to the next. However, before the first iteration can be performed, it is necessary to select an initial set of membership values. In doing so, FCM imposes the following direct constraint on the fuzzy membership function associated with each point: p X

lj ðxi Þ ¼ 1; i ¼ 1; 2; 3 . . . ; k;

j¼1

where p is the number of specified clusters, k is the number of objects, xi is the ith object, and lj ðxi Þ returns the membership value of xi in the jth cluster, i.e. the degree of belongingness of xi to the jth cluster. Clearly, the sum of the cluster membership values of each object from one specific attribute must be equal to one (Bezdek, 1981). The objective of FCM is to minimize a standard loss function expressed as the weighted sum of the squared error within each cluster, i.e.

l¼

p X n X 0 ½lj ðxi Þm kxi  cj k2 ; j¼1

1 < m0 < 1;

i¼1

NEGbP ðXÞ ¼

where p is the number of specified clusters, n is the number of objects, lj ðxi Þ is the membership value of object xi in the jth cluster, xi is the ith object, m0 is the fuzzification parameter, and cj is the center of the jth cluster. It has been shown by Bezdek (1981) that if jjxi  cj jj > 0 for all i and j, then the loss function is minimized when m0 > 1. Under this condition, the corresponding cluster center value can be computed in accordance with

where j    j denotes the cardinality of a set. In VPRS theory, the quality of the classification result is quantified using the metric

P m0 i ½lj ðxi Þ xi cj ¼ P m0 i ½lj ðxi Þ

 [ jIðPÞ \ Xj IðPÞ : 61b ; jIðPÞj  [ jIðPÞ \ Xj b IðPÞ : 1  b < BNDP ðXÞ ¼ 6b ; jIðPÞj

  

cb ðP; DÞ ¼ [ IðPÞ :

  jX \ IðPÞj P b  jUj: jIðPÞj

Essentially, the value of cb ðP; DÞ indicates the proportion of objects whose probability of belonging to equivalence class IðPÞ is greater than or equal to b. In other words, the cb ðP; DÞ involves combining all the b-positive regions and aggregating the number of objects involved within each of the combined region. In practice, the cb ðP; DÞ is used operationally to define and extract suitable reducts

for 1 6 j 6 p:

Having calculated the cluster centers, the second step in the FCM procedure is to determine the cluster memberships of each sample point. To do so, it is necessary to determine the distance from each point xi to each of the cluster centers ðc1 ; c2 ; . . . ; cp Þ. In practice, this is achieved by computing the Euclidean distance between the point and the cluster center in accordance with

dji ¼ kxi  cj k2 ; where dji is the distance of xi from the center of cluster cj .

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Since the FCM algorithm constrains the total cluster membership value of a single point to one, one can recalculate the membership of a point to a particular cluster is expressed as a fraction of all the total possible memberships assigned to that point. In other words, the membership of a particular object xi to the jth cluster is given by

 

lj ðxi Þ ¼

1 dji

Pp  1 m011 k¼1 dki

¼

Pp

1  

in which the linguistic variables x and y take values of A and B, respectively. The underlying analytical form of this rule is given by the following implication relation:

dji k¼1 dki

Z

lðx; yÞ=ðx; yÞ;

ðx;yÞ 1 m0 1

for 1 6 j 6 p;

1 6 i 6 n;

where dji is the distance metric of xi from the center of cluster cj , m0 is the fuzzification parameter, p is the number of specified clusters, and dki is the distance metric of xi from the center of cluster ck . Having computed the value of lj ðxi Þ, it is used in place of the original value of lj ðxi Þ in the first step of the FCM procedure. This two-step procedure is repeated iteratively until the centers of all the clusters within the dataset converge. (2) Index function Imax . Suppose that there exists only one conditional attribute for object xi . Furthermore, assume that this attribute can be divided into p groups such that each object owns p membership functions lj ðxi Þ; j ¼ 1; 2; . . . ; p. The index function Imax is defined as

Imax ðlj ðxi ÞÞ ¼ Indexðmaxðlj ðxi ÞÞÞ ¼ Cðxi Þ ¼ k;

where lðx; yÞ is the membership function of the implication relation Rðx; yÞ (Tsoukalas & Uhrig, 1997). Several options exist for acquiring the membership function of an implication relation. For the rule given above, the implication operator / takes the membership functions of the antecedent and consequent parts, i.e. lA ðxÞ and lB ðyÞ, respectively, and generates an output lðx; yÞ, i.e.

lðx; yÞ ¼ /½lA ðxÞ; lB ðyÞ: Typical implication operators include: (a) Zadeh max–min implication operator

/m ½lA ðxÞ; lB ðyÞ ¼ lðx; yÞ ¼ ðlA ðxÞ ^ lB ðyÞÞ _ ð1  lA ðxÞÞ: (b) Mamdani min implication operator

/c ½lA ðxÞ; lB ðyÞ ¼ ðlA ðxÞ ^ lB ðyÞÞ:

1 6 k 6 p;

where Imax ðlj ðxi ÞÞ is a clustering index in which the maximum membership value of point xi corresponds to the fuzzy cluster; and Cðxi Þ is determined by the maximum membership value and indicates the cluster to which the object xi belongs. For example, suppose that the attribute can be divided into three clusters and the values of membership functions of the first object x1 are given by l1 ðx1 Þ ¼ 0:35, l2 ðx1 Þ ¼ 0:63 and l3 ðx1 Þ ¼ 0:02, respectively. By definition, Cðx1 Þ ¼ Imax ðlj ðx1 ÞÞ ¼ 2, and thus the first object belongs to the second cluster. The example described above is easily extended to the case of multiple attributes. For example, if every object has m conditional attributes and the lth attribute al can be divided into pl clusters, then C l ðxi Þ gives the index of the cluster to which the lth attribute al of object xi belongs. Here C l ðxi Þ is given by

C l ðxi Þ ¼ Imax ðlj ðxi ðal ÞÞÞ ¼ Indexðmaxðlj ðxi ðal ÞÞÞÞ for 1 6 l 6 m;

if x is A then y is B;

Rðx; yÞ ¼

1 m0 1

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1 6 i 6 n:

where Imax ðlj ðxi ðal ÞÞÞ index of the cluster corresponding to the maximum value of the membership functions xi associated with the lth attribute. (3) Implication relations (Tsoukalas & Uhrig, 1997). Fuzzy if =then rules are conditional statements describing the dependence of one (or more) linguistic variable(s) on another variable. The underlying analytical form of an if =then rule is a fuzzy relationship known as an implication relation. More than 40 different forms of implication relation have been reported in the literature (Lee, 1990a, 1990b). Typically, these implication relations are acquired by inputting the left-hand side (LHS) and right-hand side (RHS) of a rule into an implication operator, /. The choice of an appropriate implication operator is a significant step in the overall development of a fuzzy linguistic description and reflects both the application-specific criteria and the logical and intuitive considerations relating to the interpretation of the connectives AND, OR and ELSE. Note that an extensive discussion of the most common implication relations can be found in Refs. Lee (1990a, 1990b), Mizumoto (1988) & Ruan & Kerre (1993). Consider a generic rule involving two linguistic variables, one on each side of the rule expression, i.e.

(4) Fuzzy algorithms. Fuzzy algorithms are essentially an automated procedure for interpreting a linguistic statement formulated as a collection of fuzzy rules. All of the rules within the collection are formed by the same attributes and are connected by the connective ELSE. The final result may be interpreted as either a union or an intersection of the rules depending on the implication operator used in the individual rules. For example, consider the following set of rules:

if x is A1 then y is B1 ELSE if x is A2 then y is B2 ELSE ... if x is An then y is Bn Recall that each rule above is represented analytically by an implication relation Rðx; yÞ, where the form of Rðx; yÞ depends on the choice of implication operator. In this study, the interpretations of the connective ELSE are given as follows: Implication: /m (Zadeh), Interpretation of ELSE : ANDð^Þ Implication: /m (Mamdani), Interpretation of ELSE : ORð_Þ The relation describing the entire collection of rules given above is known as the algorithmic relation and has the form

Ra ðx; yÞ ¼

Z ðx;yÞ

la ðx; yÞ=ðx; yÞ:

Note that Ra is either a union (_) or an intersection (^) of the implication relations of the individual rules depending on the chosen implication operator. In an elementary fuzzy algorithm, each implication rule has just one variable in the antecedent side and one variable in the consequent side. However, in general applications, the linguistic descriptions have more than one variable on either side and the corresponding algorithm is referred to as a multivariate fuzzy algorithm. The interpretation of the connective ELSE in a multivariate fuzzy algorithm is the same as that in an elementary fuzzy algorithm. Consider an rule of the form

if x1 is A1 AND x2 is A2 . . . AND xm is Am then y is B;

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where x1 ; . . . ; xm are the antecedent linguistic variables and have A1 ; . . . ; Am as their corresponding fuzzy values, and y is the consequent linguistic variable with a fuzzy value B. The connectives AND in the LHS of the rule given above can be modeled analytically as either min (^) or product (). In such a case, the proposition in the LHS of the rule can be combined through min (^) and an appropriate implication operator / can then be applied to acquire the membership function of the implication relation. Thus, it can be shown that

lðx1 ; x2 ; . . . ; xm ; yÞ ¼ /½lA1 ðx1 Þ ^ lA2 ðx2 Þ ^    ^ lAm ðxm Þ; lB ðyÞ; in which / is an appropriate implication operator (as described in the previous section). 2.2.2. Detailed procedure for determining b value It seems reasonable to speculate that the occurrence of errors when classifying continuous systems can be attributed at least in part to inaccuracies in the fuzzy clustering phase performed before the actual classification procedure. Accordingly, this study presents a five-step procedure for determining an appropriate value of the threshold parameter b in the VPRS model. Step 1: Fuzzify the attributes of the information system using the FCM method. A continuous-value information system can only be converted into an equivalent fuzzy information system under the condition once a classified fuzzy set has been provided. Assume that each attribute of the information system can be divided into an arbitrary e 11 ; A e 12 g indicates that number of clusters, e.g. the notation a1 ¼ f A the attribute a1 has a certain degree of belonging to two different clusters. In the FCM clustering method, the interval values ½a; b of the lth attribute al can be divided into pl fuzzy clusters, and the continuous-value information system ðU; A; V q ; fq Þ can then be converted e lj jl 6 into the fuzzy information system ðU; A; U; dÞ, where U ¼ f A e lj ¼ l ðxi ðal ÞÞ. Here, l ðxi ðal ÞÞ represents the m; j 6 pl g, in which A j j values of the membership functions associated with the lth conditional attribute al of the ith object. Step 2: Find the cluster(s) conditional attribute. Utilizing the index function Imax ðlj ðxi ðal ÞÞÞ ¼ Indexðmaxðlj ðxi ðal ÞÞÞÞ for 1 6 l 6 m; 1 6 i 6 n, identify the cluster(s) corresponding to the lth attribute ðal Þ of every object xi ðal Þ. Step 3: Arrange the implication relations. As discussed above, the use of different implication operators results in the construction of different implication relations. The present study utilizes the Zadeh max–min implication operator, which yields the following implication relation:

/M ½lA ðxÞ; lB ðyÞ ¼ lðx; yÞ ¼ ðlA ðxÞ ^ lB ðyÞÞ _ ð1  lA ðxÞÞ: Suppose that there exist m conditional attributes and one decision attribute for each object. The implication relation for the ith object ðxi Þ is therefore given as

/M ½la1 ðxi ða1 ÞÞ ^ la2 ðxi ða2 ÞÞ    ^ lam ðxi ðam ÞÞ; ld ðxi ðdÞÞ ¼ lðxi ða1 Þ; xi ða2 Þ; . . . ; xi ðam Þ; xi ðdÞÞ; where lj ðxi ðal ÞÞ represents the jth membership function of the lth conditional attribute ðal Þ of the ith object ðxi Þ, and lj ðxi ðdÞÞ is the jth membership function of the decision attribute (d) of the ith object ðxi Þ. If the decision attribute is crisp (i.e. it has a value of 0 or 1), the implication relation given above has the form /M ½la1 ðxi ða1 ÞÞ ^ la2 ðxi ða2 ÞÞ    ^ lam ðxi ðam ÞÞ; 1 ¼ lðxi ða1 Þ; xi ða2 Þ; . . . ; xi ðam Þ; 1Þ. For example, suppose that two conditional attributes ða1 ; a2 Þ and one decision attribute (d) are to be classified using the FCM approach. Suppose further that the objective of the clustering process is to divide the conditional attributes into three clusters and the decision attribute into two clusters. Finally, suppose that the ith

object’s class data set is given by (1, 3, 2), where (1, 3) denotes the clustering indexes of the two conditional attributes ða1 ; a2 Þ, respectively, and (2) denotes the clustering index of the decision attribute (d). In general, there exist a total of m classes for classified objects for which the decision attribute index has a value of (2). Suppose that there exist three objects complying with the class (1, 3, 2). Suppose further that the serial numbers of these three objects are 4, 7 and 9, respectively. The implication relation of the object with a serial number of ‘‘4” can be expressed as

/M ½la1 ðx4 ða1 ÞÞ ^ la2 ðx4 ða2 ÞÞ; ld ðx4 ðdÞÞ ¼ lðx4 ða1 Þ; x4 ða2 Þ; x4 ðdÞÞ: Step 4: Determine the value of the VPRS b parameter using a fuzzy algorithm. Taking the implication relation example given in Step 3 for illustration purposes, the objects, which belong to the same lower approximation of RS, can be related to the linguistic descriptions in a fuzzy algorithm by means of the connective operator ELSE, i.e.

l1 ðx4 ða1 ÞÞ and a2 is l3 ðx4 ða2 ÞÞ then d is l2 ðx4 ðdÞÞ ELSE; l1 ðx7 ða1 ÞÞ and a2 is l3 ðx7 ða2 ÞÞ then d is l2 ðx7 ðdÞÞ ELSE; if a1 is l1 ðx9 ða1 ÞÞ and a2 is l3 ðx9 ða2 ÞÞ then d is l2 ðx9 ðdÞÞ; if a1 is

if a1 is

where lj ðxi ðal ÞÞ represents the jth membership function of the lth conditional attribute ðal Þ of the ith object ðxi Þ, and lj ðxi ðdÞÞ represents the jth membership function value of the decision attribute (d) of the ith object ðxi Þ. Note that as described above, the interpretation of the connective operator ELSE in the fuzzy algorithm varies in accordance with the particular choice of implication operator, i.e. ANDð^Þ for the Zadeh implication operator or ORð_Þ for the Mamdani implication operator. Finally, the b value can be obtained directly using a fuzzy algorithm as follows:

b ¼ lq ðxi ða1 Þ; xi ða2 Þ; . . . ; xi ðam Þ; xi ðdÞÞ where xi 2 X q : Here, lq ðxi ða1 Þ; xi ða2 Þ; . . . ; xi ðam Þ; xi ðdÞÞ is the threshold value obtained from the fuzzy algorithm for a certain q classified objects comprising a set X q given a fixed decision attribute. In other words, a unique value of b exists for each set of classified objects. Step 5: Identify VPRS regions. Having determined the value of b using the procedure outlined above, it is then possible to determine whether the classified objects belong to the b-lower approximation of the set, the b-upper approximation of the set, or the b-boundary regions as defined by RbP ðXÞ, RbP ðXÞ, and BNDbP ðXÞ, respectively (see Section 2.2). 2.3. GM(1, N) model ð0Þ

Imagine a system described by the sequences xi ðkÞ; i ¼ ð0Þ 1; 2; 3; . . . ; n, in which x1 ðkÞ describes the main factor of interest ð0Þ ð0Þ ð0Þ and sequences x2 ðkÞ; x3 ðkÞ; . . . ; xn ðkÞ are the factors which affect this main factor. Such a system can be analyzed using the following multivariate GM(1, N) Grey model: ð0Þ

x1 ðkÞ þ azð1Þ ðkÞ ¼

N X

ð1Þ

bj xj ðkÞ;

where k ¼ 2; 3; . . . ; n;

j¼2

P ð1Þ ð0Þ ð1Þ ð1Þ ð1Þ in which xj ðkÞ ¼ ki¼1 xj ðiÞ and z1 ðkÞ ¼ 0:5x1 ðkÞ þ 0:5x1 ðk  1Þ; k P 2. ð1Þ Substituting all possible xj ðkÞ terms into above equation yields a matrix of the form

32 3 a 7 6 76 7 6 ð0Þ 7 6 x ð3Þ 7 6 zð1Þ ð3Þxð1Þ ð3Þ . . . xð1Þ b2 7 ð3Þ n 2 7 6 1 76 6 1 7 ¼ Ba ^: XN ¼ 6 . 7 ¼ 6 76 . 6 . 74 .. 7 6 . 7 6 .. 5 5 4 . 5 4 2

ð0Þ

x1 ð2Þ

ð0Þ

x1 ðnÞ

3

2

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

z1 ð2Þx2 ð2Þ . . . xn ð2Þ

z1 ðnÞx2 ðnÞ . . . xn ðnÞ

bn

K.Y. Huang / Expert Systems with Applications 36 (2009) 11652–11661

The values of bj ; j ¼ 2; 3; . . . ; N can be found by applying the matrix ^ ¼ ðBT BÞ1 BT X N . The relative influence exerted on the operation a major sequence by each influencing sequence can then be determined by inspecting the bj value of the corresponding sequence. 2.4. Grey relational analysis In grey relational analysis (GRA), data characterized by the same set of features are regarded as belonging to the same series. The relationship between two series can be determined by evaluating the differences between them and assigning an appropriate grey relational grade. According to Huang and Jane (2008), the , where x0 and attribute impulse factor has the form doi ðkÞ ¼ jxjx0i ðkÞj ðkÞj xi are the reference object and the inspected object, respectively. Furthermore, the grey relational grade is defined as

C0i ¼

D0i  Dmin ; Dmax  Dmin

where i ¼ 1; 2; . . . ; m; x0 ðkÞ is the 1reference value, xi ðkÞ is the comQm m parative value, D0i ¼ k¼1 d0i ðkÞ , Dmin ¼ min8i fD0i g, and Dmax ¼ max8i fD0i g. Having calculated the grey relational grades amongst the various sequences, they are ranked using a so-called grey relational ranking procedure. For example, for the case of a reference sequence x0 ðkÞ, the grey relational rank of xi ðkÞ is greater than that of xj ðkÞ if cðx0 ; xi Þ > cðx0 ; xj Þ, and thus the corresponding ranking is denoted as xi  xj . 3. Hybrid model for stock market forecasting and portfolio selection 3.1. Use of ARX prediction model in preparing data set for VPRS processing Assuming that U is the domain of discourse and R is the set of equivalences of U, then the VPRS problem can be formulated as

X # U is : ðRbP ðXÞ; RbP ðXÞÞ; BNDbP ðXÞ; where X is the set of elements; U=INDðRÞ is the equivalence of R; INDðRÞ is the indiscernibility of R; / is the zero set; R is the attribute set of X which includes the condition set(C) and the decision-making set ðDÞ; P # C; 0:5 < b 6 1; RbP ðXÞ is the b-lower approximation of X; RbP ðXÞ is the b-upper approximation of X; and BNDbP ðXÞ is the boundary of X. Every element in the domain of discourse UðX # UÞ has an attribute set(R), which describes the particular value of X. In the hybrid stock prediction and portfolio selection model developed in this study, each element ðX i Þ in U is processed by the ARX prediction model and is assigned appropriate values of the conditional attributes ðC 1  C n Þ and decision-making attributes ðD1  Dm Þ in accordance with their trends over the previous quarter. The resulting attributes are then processed using VPRS theory and a Grey relational analysis technique to determine an optimal stock portfolio. 3.2. Selection of decision-making attributes In the proposed model, the forecast data generated by the ARX model are processed in accordance with a set of decision-making attributes which dictate those stocks which should at least be considered by the VPRS model and those which can be immediately discounted. In the current study, these decision-making attributes are selected in accordance with the investment principles advocated by Buffett (Hagstrom et al., 2005). Buffet asserted that enterprises with ‘‘low costs”, ‘‘a high profit margin”, and ‘‘a high inventory turnover” will enjoy financial suc-

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cess and hold the lion’s share of the consumer market for its product line. Further, he argued that reducing costs was the key factor in enabling an enterprise to improve its competitive ability and to rival its competitors in terms of price, while high profit margins and a high inventory turnover were both reliable indicators of its financial well-being. Buffet asserted that only companies with all three attributes could be certain of survival and could earn profit for their shareholders by improving the manufacturing process, developing new products, acquiring or merging with other enterprises, and so forth. In accordance with these basic investment principles, the hybrid stock market prediction and stock selection mechanism developed in this study utilizes the following decision-making attributes: D1: D2: D3: D4: D5: D6: D7:

return on asset (after tax) greater than zero, return on equity greater than zero, gross profit ratio greater than zero, equity growth rate greater than zero, quick ratio greater than industry median, rate of inventory turnover greater than industry median, constant EPS greater than zero.

Note that to ensure that all enterprises with a profit-making potential are included within the VPRS/ Grey relational classification process, the threshold values of decision-making attributes D1, D2, D3, D4 and D7 are deliberately stated as ‘‘greater than 0” rather than being assigned a particular threshold value. 3.3. Detailed processing steps in hybrid forecasting and stock selection model The detailed processing steps in the proposed forecasting and stock selection model are illustrated in flow chart form in Fig. 1 and can be summarized as follows: Step 1: Data collection and attribute determination In the current application, the conditional attributes should reflect the financial quality of a company. Therefore, the proposed model specifies the following attributes: the profitability, the capitalized cost ratio, the individual share ratio, the growth rate, the debt ratio, the operational leverage and the statutory financial ratios. The decision as to which of the selected companies should actually be processed by the VPRS portfolio selection mechanism is made by processing the forecast data generated by the ARX model in accordance with the seven decision-making attributes specified in the previous section. Step 2: Data preprocessing Having collected the relevant financial data for each quarterly period, a basic preprocessing operation is performed to improve the efficiency of the ARX prediction model. For example, any data records containing missing fields (i.e. attributes) are immediately rejected. In addition, the problem of data outliers is addressed by using the Box Plots method (Chakravarti, Laha, & Roy, 1967) to establish an inter-quartile range such that any data falling outside this range can be automatically assigned a default value depending on the interval within which it falls. Step 3: ARX prediction As discussed in the Introduction section, the stock market is a sequentially correlated signal system, and hence the current model uses an ARX forecasting model to predict the future trends of the financial variables of each of the selected companies. In the ARX model, the forecasting activity is deliberately restricted to a one-step-ahead mode in order to prevent an accumulation of the prediction errors from previous forecasting periods. Step 4: Information reduction using GM(1, N) multivariate model To improve the efficiency of the VPRS/Grey relational analysis procedure, certain conditional attributes are removed if they are found to have little effect on the decision-making attributes. In the

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K.Y. Huang / Expert Systems with Applications 36 (2009) 11652–11661

Start

attributes. However, if the rate of return is deemed unacceptable, the suitability of the conditional attributes is reviewed and the choice of attributes amended as appropriate.

Renew the Attributes Determination

4. Evaluation of proposed hybrid model using electronic stock data

Data collection and attribute determination

4.1. Data extraction

Data preprocessing

The feasibility of the proposed forecasting and stock selection mechanism was evaluated using electronic stock data extracted from the New Taiwan Economy database (TEJ). The data collection period extended from the first quarter of 2003 to the fourth quarter of 2006, giving a total of 16 quarters in all. Meanwhile, the forecasting period extended from the first quarter in 2003 to the first quarter in 2007, giving a total of 17 quarters. In general, the financial statements for a particular accounting period are subject to a considerable delay before they are actually published. For example, annual reports are published after four months, half-yearly reports after two months, and first and third quarterly reports (without notarization) after around one month. The submission deadlines for the financial statements maintained in the TEJ database are as follows:

ARX prediction

Information reduction using GM(1,N) multivariate model

Fuzzy C-means

Selection of approximate sets

Fund allocation

Modeling applicable

No

Yes To Proceed next quarter investment Yes

To continue investment No End

Fig. 1. Flow chart of proposed forecasting and stock selection model.

proposed model, this pruning operation is performed by using a GM(1, N) multivariate scheme to identify the top ten influential sequences (i.e. conditional attributes) and then eliminating the remainder. Step 5: Fuzzy C-Means clustering Prior to submission to the VPRS stock selection mechanism, the forecast values of the conditional attributes ðC 1  C n Þ are clustered into three groups using a FCM clustering algorithm. Step 6: Selection of approximate sets Having clustered the forecast data, the VPRS method is applied to determine the b-lower approximate set. The appropriate investment stocks are derived as the b-lower approximate set is acquired. Step 7: Fund allocation Having identified suitable stocks in which to invest, appropriate weights should be assigned to each stock in order to optimize the overall rate of return on the portfolio. In the proposed hybrid model, this fund allocation problem is processed using a Grey Relation niþ1 , Sequence algorithm based on the metric stock weightðiÞ ¼ P i i¼n

where i is the grey relation order of each stock item and n is the total number of invested stocks. Having completed all of the steps described above, a check is made on the overall rate of return on the investment. If the rate of return is acceptable, a decision is made as to whether or not the model should be run for a further quarter using the existing

(1) Annual report: the submission deadline laid down by the Security Superintendence Commission is 4 months after the closing balance day. However, companies listed in previous years (TSE and OTC) are permitted to delay filing until 5/ 31. (2) Half-yearly report: the submission deadline laid down by the Security Superintendence Commission is 2 months after the closing balance day. However, companies listed in previous years (TSE and OTC) can delay filing until 9/21. (3) First-quarter report: the submission deadline laid down by the Security Superintendence Commission is 1 month after the closing balance day. However, companies listed in previous years (TSE and OTC) can delay filing until 5/31. (4) Third-quarter report: the submission deadline laid down by the Security Superintendence Commission is 1 month after the closing balance day. However, companies listed in previous years (TSE and OTC) are permitted to delay filing until 11/15. Since the financial report for the final quarter of any year is not available until May 31st the following year, the financial data can not be used by the ARX model to predict the financial trends over the first quarter of the year. In other words, the forecasting and stock selection process proposed in this study can only be performed three times every year, i.e. 5/31–09/22, 9/22–11/15 and 11/15–05/31 the following year. In addition, in the decision-making rules used in the VPRS stock selection process, the Return on Equity (ROE) and constant EPS indicators are based on the full 12 months of the previous year. Thus, the forecasting period for investment purposes is reduced from the initial time frame of the first quarter in 2003 to the first quarter in 2007 to the second quarter in 2004 to the fourth quarter in 2006. 4.2. Performance evaluation of VPRS-based stock forecasting and selection mechanism In this study, the performance of the proposed VPRS-based stock forecasting and selection mechanism was benchmarked against that of a RS-based selection scheme. Tables 1a and 1b summarizes the number of indiscernible classes and the number of companies selected by the RS- and VPRS-based models as a function of the

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K.Y. Huang / Expert Systems with Applications 36 (2009) 11652–11661

Table 1a Comparison of number of indiscernible classes and selected companies for different numbers of conditional attributes and a single decision attribute. Note that each conditional attribute is associated with three fuzzy clusters and the cluster code of the decision attribute is specified as ‘‘1”(P0). Number of conditional attributes

2

3

4

5

9

10

Number of indiscernible classes as computed using RS model (a) Number of companies selected by RS model

5 22

10 23

17 29

21 34

31 35

33 35

Number of indiscernible classes as computed using VPRS model (b) Number of companies selected by VPRS model

7 39

11 33

18 35

22 37

31 35

33 35

(b)–(a)

17

10

6

3

0

0

Table 1b Comparison of code(s) of indiscernible classes and selected companies for different numbers of conditional attributes and a single decision attribute. Note that each conditional attribute is associated with three fuzzy clusters and the cluster code of the decision attribute is specified as ‘‘1”(P0). Two conditional attributes

Three conditional attributes

Codes of the conditional attributes

Company code(s) of indiscernible classes as computed using RS model

Company code(s) of indiscernible classes as computed using VPRS model

Codes of the conditional attributes

Company code(s) of indiscernible classes as computed using RS model

Company code(s) of indiscernible classes as computed using VPRS model

{2, 2}

{22, 58, 250, 272}

{22, 58, 250, 272}

{1, 2}

{75, 77, 95, 99, 116, 134, 171, 201, 246, 262, 269, 275}

{75, 77, 95, 99, 116, 134, 171, 201, 246, 262, 269, 275}

{2, {2, {2, {1,

2, 2, 2, 2,

3} 2} 1} 2}

{1, 1} {3, 1}

{132, 188} {133, 243}

{132, 188} {133, 243}

{2, 3} {3, 3} a {3, 3} {2, 1}

{242, 259}

{242, 259} {49, 61, 179, 239, 251} {54, 81} {55, 126, 137, 153, 192, 225, 234, 257, 263} {156}

{1, {1, {3, {3, {2, {3,

2, 1, 1, 1, 3, 3,

1} 1} 1} 3} 3} 1}

{22} {58, 250} {272} {75, 77, 99, 116, 134, 171, 201, 246, 262, 269, 275} {95} {132, 188} {133} {243} {242, 259} {61}

{22} {58, 250} {272} {75, 77, 99, 116, 134, 171, 201, 246, 262, 269, 275} {95} {132, 188} {133} {243} {242, 259} {61}

a

{2, 1} a

{2, 1, 1}

{55, 126, 137, 153, 192, 225, 234, 257, 263} {156}

a

{2, 1, 1}

Indicates the cluster code of the decision attribute is specified as ‘‘2”(<0).

Table 2a Rate of return in 4th quarter 2006 using proposed VPRS-based model. Listed Company Code

Call(11/15)

Number

Put(5/31) next year

Return rate (%)

2368 3037 6192 2483 2423 Total return rate: 30.11%

19.93 36.20 38.82 23.15 16.76

16 7 5 5 3

24.15 45.58 59.11 22.63 32.99

21.17 25.91 52.27 2.25 96.83

Table 2b Rate of return in 4th quarter 2006 using RS-based model. Listed Company Code

Call(11/15)

Number

Put(5/31) next year

Return rate (%)

2368 3003 3037 6166 3042 Total return rate: 20.81%

19.93 27.36 36.20 29.45 46.63

16 9 5 3 1

24.15 31.17 45.58 36.55 58.99

21.17 13.92 25.91 24.10 26.51

number of conditional attributes. As shown, the number of indiscernible classes increases with an increasing number of conditional attributes in both the RS- and the VPRS-based models. However, in both models, the number of objects within each indiscernible class reduces as the number of conditional attributes increases. From inspection, it is evident that for a given number of conditional attributes, the number of indiscernible classes in the VPRS-based scheme is either higher than or equal to that in the RS-based model. VPRS performs a classification function with a controlled degree of misclassification error falling outside the realm of the RS approach.

For the case where the number of conditional attributes is specified as {2, 3, 4, 5}, the VPRS model increases the number of indiscernible classes by {2, 1, 1, 1} relative to that in the RS scheme. However, when a greater number of conditional attributes are considered, i.e. {9, 10}, the VPRS and RS models yield an identical numbers of indiscernible classes. Furthermore, it is observed that for a lower number of conditional attributes, i.e. {2, 3, 4, 5}, the VPRS model yields a greater number of selected companies, while for a higher number of attributes, i.e. {9, 10}, the two schemes select an identical number of companies. The probability of which two objects are

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K.Y. Huang / Expert Systems with Applications 36 (2009) 11652–11661

Table 3 Comparison of rates of return obtained using proposed VPRS-based model and RS-based model. Investment period

Forecasting method RS

VPRS

Quarter

Yearly

Quarter

Yearly

2.66%

13.68% 1.36% 9.66% 14.44% 3.34% 35.39% 3.97% 11.33% 30.11% 29.33% 87.98%

2.66%

Rate of return 2004

2005

2006

Second quarter Third quarter Fourth quarter Second quarter Third quarter Fourth quarter Second quarter Third quarter Fourth quarter

Average yearly rate of return Accumulated rate of return over 3 years

13.68% 1.36% 9.66% 14.44% 3.34% 38.51% 3.97% 11.33% 20.81% 27.27% 81.80%

indiscernible due to identical clustering indexes would be reduced by increasing the number of conditional attributes or increasing the number of clusters in each attribute. Tables 2a and 2b summarize the rates of return in the 4th quarter of 2006 obtained from the stock companies selected using the RS-based model and the proposed VPRS-based model, respectively. It can be seen that the two models recommend a slightly different selection of companies for investment purposes. The VPRS model yields a greater number of selected companies and results in a slightly different selection of companies. It can also be seen that the rate of return provided by the companies selected using the VPRS-based model is higher than that obtained from the RS-based model. Table 3 summarizes the quarterly and yearly rates of return obtained over the nine investment periods between 2004 and 2006 using the proposed VPRS-based fusion model and the RS-based model, respectively. Note that in both models, the results are obtained using five conditional attributes and one decision attribute. As shown, the fusion model achieves a higher average yearly rate of return than the RS-based model. 5. Conclusions and discussions This study has presented a mechanism for predicting stock market trends and selecting the optimal stock portfolio based upon an ARX prediction model, Grey Systems theory and VPRS theory. The performance of the VPRS-based approach has been compared with that of a RS-based method using electronic stock data extracted from the New Taiwan Economy database (TEJ). The major findings and contributions of this study can be summarized as follows: (1) In contrast to the b-reducts method used in previous studies, this study determines an appropriate value of the VPRS threshold parameter b using a Fuzzy C-Means clustering approach. Suppose that each object possesses a number of attributes which can each be assigned to multiple clusters during the fuzzy classification process. According to the definition of the index function Imax , the membership function of a particular attribute is represented by the value of the maximum membership function of the attribute amongst all of the clusters with which it is associated. The value of the VPRS threshold parameter b is then derived using the Zadeh implication operator and the Zadeh fuzzy algorithm operator (AND). (2) The probability of two objects being indiscernible as the result of having an identical clustering index reduces as the number of conditional attributes considered in the clus-

56.29%

28.17%

53.17%

37.46%

tering process increases. The number of objects in any indiscernible class of the RS lower approximation set may to be reduced to one. For the case where the number of objects in a particular classified group is equal to one, the lower RS approximation for that classification is identical to the b-lower approximation obtained using the VPRS model irrespective of the value assigned to the b threshold parameter. (3) The VRPS b-lower approximation set contains a greater number of selected stocks than the RS lower approximation set, and therefore results in a different ranking of the selected stocks by the GRA scheme. The portfolio recommendations obtained using the proposed VPRS-based model differ from those obtained from a RS-based hybrid model and yield an improved rate of return. (4) The generalized rules extracted by the b-lower approximate set are all found to be recognized rules or relationships in the investment industry, which indicates that the VPRS method not only provides a feasible means of identifying top stock performers by classifying the contributions of the attributes, but is also helpful in constructing decision rules with which to evaluate new stocks. In other words, the VPRS model removes the requirement for the blind, haphazard stock selection methods typically used by investors in the past. VPRS is essentially an extension of RS in that setting an appropriate value of the threshold value b eases the strict definition of the approximation boundary in the RS model. As a result, the indiscernible classes in the VPRS model include all of the indiscernible classes in the RS model. However, as the number of conditional attributes or number of clusters associated with each attribute increases, the number of objects in each class decreases to one. Under this condition, both the number of indiscernible classes and the object in each class are identical in the VPRS and RS models. Overall, the results presented in this study have confirmed that the proposed VPRS-based model provides a promising tool for stock portfolio management purposes. Furthermore, the structure of the proposed model represents a suitable foundation for a broad range of derivatives in other research fields. In the current model, the fund allocation task is performed using the Grey relational analysis method proposed by Huang and Jane (2008). However, a future study will investigate the feasibility of improving the rate of return obtained from the selected stocks by integrating the fusion model developed in this study with a heuristic scheme designed to optimize the fund weight distribution subject to the constraint of minimizing the portfolio risk.

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