Economic Modelling Using Computational Intelligence Techniques

Attempting to successfully and accurately predict thefinancial market has long attracted the interests and attention ofeconomists, bankers, mathematicians and scientists alike. Thefinancial markets form the bedrock of any economy. There are alarge number of factors and parameters that influence thedirection, volume, price and flow of traded stocks. This coupledwith the markets’ vulnerability to external and non-financerelated factors and the resulting intrinsic volatility makes thedevelopment of a robust and accurate financial marketprediction model an interesting research and engineeringproblem. In an attempt to solve this engineering problem, theauthors of this paper present a rough set theory based predictivemodel for the financial markets. Rough set theory has, as its base,imperfect data analysis and approximation. The theory is used toextract a set of reducts and a set of trading rules based on tradingdata of the Johannesburg Stock Exchange (JSE) for the period 1April 2006 to 1 April 2011. To increase the efficiency of themodel four of dicretization algorithms were used on the data set,namely (Equal Frequency Binning (EFB), Boolean Reasoning,Entropy and the Naïve Algorithm. The EFB algorithm gives theleast number of rules and highest accuracy. Next, the reducts areextracted using the Genetic Algorithm and finally the set ofdependency rules are generated from the set of reducts. A roughset confusion matrix is used to assess the accuracy of the model.The model gave a prediction accuracy of 80.4% using theStandard Voting classifier.Keywords-rough set theory; financial market modelling, neuralnetworks, discretization, classification.INTRODUCTIONAround the world, trading in the stock market has gainedenormous popularity as a means through which one can reaphuge profits. Attempting to successfully and accurately predictthe financial market has long attracted the interests andattention of economists, bankers, mathematicians andscientists alike. Thus, stock price movement prediction haslong been a cherished desire of investors, speculators andindustries [1].For a long time statistical techniques such as Bayesian models,regression and some econometric techniques have dominatedresearch activities in prediction [2]. The primary approach tofinancial forecasting has been the identification of a stockprice trend and continuation of the investment strategy untilevidence suggests that the trend has reversed [3]. One of thebiggest problems with the use of regression methods is thatthey fail to give satisfactory forecasting results for someseries’ because of their linear structure and other inherentlimitations [8, 9]. However the emergence of computational intelligencetechniques as a viable alternative to the “traditional” statisticalmodels that have dominated this area since the 1930’s [2,3]has given impetus to the increasing usage of these techniquesin fields such as economics and finance [3]. Apart from these,there have been many other successful applications ofintelligent systems to decision support and complexautomation tasks [4 6]. Since the year of its inception in1982, rough set theory has been extensively used as aneffective data mining and knowledge discovery technique innumerous applications in the finance, investment and bankingfields [4, 7, 11, 12, 20].Data mining is a discipline in computational intelligence thatdeals with knowledge discovery, data analysis, and full andsemi-autonomous decision making [7]. It entails the analysisof data sets such that unsuspected relationships among dataobjects are found. Predictive modeling is the practice ofderiving future inferences based on these relationships. Thefinancial markets form the bedrock of any economy. There area large number of factors and parameters that influence thedirection, volume, price and flow of traded stocks. Thiscoupled with the markets’ vulnerability to external and non-finance related factors and the resulting intrinsic volatilitymakes the development of a robust and accurate financialmarket prediction model an interesting research andengineering problem. This paper presents a generic stock priceprediction model based on rough set theory. The model isderived on data from the daily movements of theJohannesburg Stock Exchange’s All Share Index. The dataused was collected over a five year period from 1 April 2006to 1 April 2011. The methodology used in this paper is as follows: data pre-processing, data splitting, data discreetization, redundantattribute elimination, reduct generation, rule generation andprediction. 71The rest of this paper is organized as follows: Section IIcovers the theoretical foundations of rough set theory, SectionIII gives the design of the proposed prediction model while ananalysis of the results and conclusions are presented inSections IV and V respectively. THEORETICAL FOUNDATIONS OF ROUGH SETS Rough set theory (RST) was introduced by Pawlak in 1982.The theory can be regarded as a mathematical tool used forimperfect data analysis [10]. Thus RST has proved useful inapplications spanning the engineering, financial and decisionsupport domains to mention but a few. It is based on theassumption that “with every object in the universe of discourse,some information (data or knowledge) is associated” [10]. Inpractical applications, the “universe of discourse” described in[10] is usually a table called the decision table in which therows are objects or data elements and the columns are attributesand the entries are called the attribute values [3].The objects or data elements described by the sameattributes are said to be indiscernible (indistinguishable) by theattribute set. Any set of indiscernible data elements forms agranule or atom of knowledge about the entire “universe ofdiscourse” (information system framework) [10]. A union ofthese elementary sets (granules) is said to be a precise or crispset, other-wise the set is said to be rough [1,2,7,10]. Everyrough set will have boundary cases i.e data objects whichcannot certainly be classified as belonging to the set or itscomplement when using the available information [10].Associated with every rough set is a pair of sets called thelower and upper approximation of the rough set. The lowerapproximation consists of those objects which one candefinitively say belong to the target set. The upperapproximation consists of those objects which possibly belongto the target set. The difference between the two sets is theboundary region. The decision rule derived specifies anoutcome based on certain conditions. Where the derived ruleuniquely identifies outcomes based on some conditions the ruleis said to be certain else it is uncertain. Every decision rule hasa pair of probabilities associated with it, the certainty andcoverage coefficients [3].These conditional probabilities also satisfy Bayes’ theorem[7,10]. The certainty coefficient is the conditional probabilitythat an object that belongs to the decision class outlined by therule given that it satisfies the conditions of the rule. Thecoverage coefficient on other hand expresses the conditionalprobability of reasons given some decision [10]. Clearly RSTcan be seen to overlap with many other theories in the realm ofimperfect knowledge analysis such as evidence theory,Bayesian inference, fuzzy sets etc [1, 3, 4, 10, 11, 12].To define rough sets mathematically, we begin by defining aninformation system S = (U,A), where U and A are finite andnon-empty sets that represent the data objects and attributesrespectively. Every attribute has a set of possible valuesVa. Va is called the domain of a. A subset of A say B willdetermine a binary relation I(B) on U, which is called theindiscerniblity relation. The relation is defined as follows:( ) ( ) if and only if a(x) = a(y) for every a in B, wherea(x) denotes the value of attribute a for data object x [10]. I(B)is an equivalence relation. All equivalence classes of I(B) asU/I(B). An equivalence class of I(B) containing x is denoted asB(x). If (x,y) belong to I(B) they are said to be indiscerniblewith respect to B. All equivalence classes of theindescernibility relation, I(B), are referred to as B-granules orB-elementary sets [10].In the information system defined above, we define as in [10]:(1)And,(2) We now define the two operators assigned to every (1) two setscalled the upper and lower approximation of X. The two setsare defined as follows [10]:( ) ⋃ { ( ) ( ) }(3)And,( ) ⋃ { ( ) ( ) }(4) Thus, the lower approximation is the union of all B-elementary sets that are included in the target set, whilst theupper approximation is the union of all B-elementary sets thathave a non-empty intersection with the target set. Thedifference between the two sets is called the boundary ofregion of X.