A Genetic Type 2 Fuzzy Logic Based System for Financial

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uncertainty levels have been employed for the generation of is a crisp number in 0 1 The membership functions of. classification models 15 16 However the existing type 2 interval type 2 fuzzy sets include a Footprint of Uncertainty. fuzzy classification systems are not suited for the financial FOU which provide additional degrees of freedom that make. domain where such type 2 FLSs generate big rule bases it possible to directly model and handle uncertainties In the. besides they make the assumption that all the possible rules interval type 2 fuzzy sets all the third dimension values are. are represented in the existing models which is impossible for equal to one. systems with big number of inputs where the generated model The proposed system in the paper is a type 2 fuzzy. will only cover a small subset of the search space This means classification system and hence it does not follow the structure. that FLSs make the assumption that the obtained rule bases of the type 2 FLSs where the classification system process is. are always able to cover the decision space which may not be summarized in the following section. always true An interval type 2 fuzzy set denoted as is written as. In this paper we will present a genetic type 2 FLS for the follows. modelling and prediction of financial applications The 1. proposed system avoids the drawbacks of the existing type 2. fuzzy classification systems where the proposed system is able represent the upper and lower membership. to carry prediction based on a relativity small pre specified. rule base size even if the incoming data vector does not match functions respectively of the interval type 2 fuzzy set. any rules in the FLS rule base The proposed type 2 FLS aims III A BRIEF OVERVIEW ON FUZZY LOGIC CLASSIFICATION. to increase the understandability of the generated model by SYSTEMS. achieving the best performance with a limited and summarized. In fuzzy logic classification systems for a given c class. number of rules in order to achieve simplicity and. pattern classification problem with n attributes or features a. comprehensibility for the user We have carried various. given rule in the FLS rule base could be written as follows. evaluations where we are going to present in this paper results. from two distinctive financial domains one for the prediction Rule Rj If x1 is and and xn is then Class Cj with. of good bad customers for a credit card approval application CFj j 1 2 N 2. and the other domain was in the prediction of arbitrage. opportunities in the stock markets The proposed system was Where x1 xn represent the n dimensional pattern vector. able to use the generated optimised summarized models for is the fuzzy set representing the linguistic label for the. the prediction within financial applications The proposed antecedent pattern i Cj is a consequent class which could be. system outperformed white box financial models like the one of the possible c classes N is the number of fuzzy IF. Evolving Decision Rule EDR procedure which is a white THEN rules in the FLS rule base CFj is a certainty grade of. box model based on Genetic Programming GP and decision rule j i e rule weight In case each input pattern is. trees 14 and gave a comparable performance to black box represented by K fuzzy sets and given that we have n input. models like neural networks while the proposed system patterns the possible number of rules that will cover the whole. provided a white box model which is easy to understand and search space is Kn However in the vast majority of financial. analyse by the lay user It should be noted that due to the applications we do not have enough data to generate this huge. limited space we will not be able to compare the performance number of rules Hence there will be various cases where the. of the type 2 and type 1 fuzzy based systems The superiority incoming input vector will not fire any rule in the FLS rule. of type 2 based classification systems over their type 1 base In financial applications we cannot discharge a given. counterparts have been shown in 15 and 16 input pattern because there are no matching rules in the rule. Section II will provide a brief overview on type 2 FLS base A technique to resolve this problem was proposed in. while Section III will present a brief overview on fuzzy logic 12 13 and 14 this technique keeps in a rule repository all. based classification systems Section IV will present the the rules for the minority class in unbalanced data sets All the. proposed genetic type 2 FLS for financial applications inputs that do not match any rule in the repository are. modelling and prediction Section V will present the considered belonging to the majority class This technique can. experiments and results The conclusions and future work will work in unbalanced data set but might not work in all cases. be presented in Section VI,IV THE PROPOSED GENTEIC TYPE 2 FUZZY LOGIC BASED. II A BRIEF OVERVIEW ON INTERVAL TYPE 2 FUZZY SYSTEM FOR FINANCIAL APPLICATIONS MODELLING AND. LOGIC SYSTEMS PREDCITION, Interval type 2 FLSs employ type 2 fuzzy sets where an In FLSs the choice of the appropriate parameters of the. interval type 2 fuzzy set is characterized by a fuzzy fuzzy sets poses a major challenge to the design of a FLS By. membership function i e the membership value or simply changing the fuzzy sets parameters it is possible to. membership grade for each element of this set is a fuzzy set change the behaviour of a FLS for example in the field of. in 0 1 unlike a type 1 fuzzy set where the membership grade managing risk in financial systems it is possible to build. riskier or risk averse fuzzy systems by changing the A Employing Genetic Algorithms to determine the type 2. parameters of the fuzzy sets to make the FLS passing more or Fuzzy Sets Parameters. less customers It is extremely difficult though to find the The GAs implementation of 17 18 was used to encode. optimal configuration using a simple manual or heuristic the chromosomes using real numbers. approach because of the number of the variables to be The GA objective function is to optimize the Recall of the. optimised and the interaction of these variables In our work classifier on each class in a weighted way The Recall is. Genetic Algorithms GAs were used to tune the parameters of defined as the proportion of the class cases that were correctly. the type 2 fuzzy sets of the FLS identified We employ a weight wi to give a weighted. The GA uses a population where each chromosome importance to a given class i hence the GA fitness could be. describe a fuzzy set space in other words the position of each written as follows. membership function for each input In simple terms the GA. is positioned before the rule extraction algorithm and aim to 3. find the best membership function configuration for describing C is the number of classes for the given problem. in the best way the problem The GA basically tries different. configurations that through the specified rule extraction By default in a standard configuration all weights are set to. algorithm produce the best performances 1 in order to give the same importance to all classes. In the GA each instance of the FLS is created by using each Each input for the FLS is represented by five type 2 fuzzy. individual of the population and each instance generates a sets which need 17 parameters to be represented as shown in. different fitness value As shown in Fig 1 the steps followed Fig 2 To fully build the fuzzy sets needed by the system the. by the proposed genetic type 2 fuzzy system can be total number of parameters genes to be optimised can be. summarised as found as follows, 1 Initialize randomly the first generation to generate the Number of parameters 17 x F 4. type 2 fuzzy sets parameters, 2 Build a rule base for each parameter configuration of the Where F is the number of inputs or features For a dataset. type 2 fuzzy sets as provided by a given chromosome As with 7 input features the total number of parameter to tune will. the matter of the fact each chromosome describes the be 119 parameters thus creating a chromosome composed of. fuzzy membership functions configuration and this in 119 genes. conjunction with the training data is used to build the An example of the Membership Functions MFs that can. rule base the rule base generation process is discussed in be generated by the GA is shown in Fig 3 As can be seen the. Section IV B membership functions produced are not equally spaced and. 3 Evaluate the classification ability of the generated type 2 they are also asymmetric thus the use of the GA allows to. FLS over a validation data set and produce the fitness better model the uncertainity available in the data set. value for each individual, 4 If an individual reaches desired fitness value or the max.
number of iterations are reached the algorithm terminates FOU. 5 The GA uses the population and their fitness values to. evolve and produce a new population,6 Go to step 2. Fig 2 The number of type 2 fuzzy sets 17 parameters to be tuned for each. Fig 3 The MFs learnt by the GA for MonNess input of arbitrage dataset. Fig 1 An overview of the proposed genetic type 2 fuzzy logic system. Fig 3 shows the learnt membership functions for the. MonNess feature in the arbitrage dataset used in this paper. This input feature is defined as the division between strike average 2 amongst these rules. price and the underlying index level It is interesting to note. The financial data is usually highly imbalanced for. that the shapes of the produced membership functions are. asymmetric and they are actually different from each other in example in a lending application it is expected that the. the same dimension This degree of freedom allows having majority of people will be good customers and a minority. more fine grained rules in areas of the decision space where is being bad customers and usually the interesting class is the. more needed minority class Hence we will present a new approach called. weighted scaled dominance which is an extension of our. previous work on scaled dominance and the weighted. B Rule generation in the proposed Genetic Type 2 FLS confidence work introduced by Ishibuchi in 19 This. This subsection will show how the rules of the type 2 FLS method tries to handle imbalanced data by trying to give. are modelled taking as an input a dataset and the fuzzy sets minority classes a fair chance when competing with the. whose parameters were optimised by the GA This is called majority class In order to compute the scaled dominance for a. modelling phase In the modelling phase the rule base of the given rule having a consequent Class Cj we divide the firing. type 2 fuzzy classification system is constructed from the strength of this rule by the summation of the firing strengths of. existing training dataset Once the model has been built the all the rules which had Cj as the consequent class This allows. FLS can be used to predict new inputs This is called handling the imbalance of data towards a given class We. prediction phase In the prediction phase the generated type 2 scale the firing strength by scaling the upper and lower bounds. FLS is used to predict the incoming input vectors Fig 4 of the firing strengths as follows. shows an overview on the modelling and prediction phases. 1 The modelling phase 8, The modelling phase operates according to the following. steps as shown in Fig 4 9, Step 1 Raw Rule Extraction For a fixed input output. pair x t C t in the dataset t 1 T T is the total number of Step 2 Scaled Support and Scaled Confidence. data training instances available for the modelling phase Calculation Many of the generated rules will share the same. compute the upper and lower membership values for antecedents but different consequents To resolve this conflict. each antecedent fuzzy set q 1 K K is the total number of we will calculate the scaled confidence and scaled support. fuzzy sets representing the input pattern s where s 1 n which are calculated by grouping the rules that have the same. antecedents and conflicting classes For given m rules having. Generate all rules combining the matched fuzzy sets i e. the same antecedents and conflicting classes the scaled. either 0 or 0 for all s 1 n Thus the rules, confidence Cq defined by its upper bound and lower. generated by x t C t will have different antecedents and the bound it is scaled as it involves the scaled firing strengths. same consequent class C t Thus each of the extracted raw. mentioned in the step above that class is the consequent. rules by x t C t could be written as follows, class for the antecedents where there are m conflicting.
Rj IF x1 is and and xn is rules with the same antecedents and conflicting consequents. could be written as follows,THEN Class Ct t 1 2 T 5. For each generated rule we calculate the firing strength Ft 10. This firing strength measures the strength of the points x t. belonging to the fuzzy region covered by the rule Ft is defined. in terms of the lower and upper bounds of the firing strength. of this rule which are calculated as follows 11, The scaled confidence can be viewed as measuring the. 7 validity of Aq Cq The confidence can be viewed as a. numerical approximation of the conditional probability 20. The denotes the minimum or product t norm Step 1 is The scaled support defined by its upper bound and lower. repeated for all the t data points from 1 to T to obtain bound it is scaled as it involves the sc. A Genetic Type 2 Fuzzy Logic Based System for Financial Applications Modelling and Prediction Dario Bernardo Hani Hagras Edward Tsang The Computational

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