CHAPTER VI FUZZY MODEL

Transcription

1 CHAPTER VI This chapter covers development of a model using fuzzy logic to relate degree of recording/evaluation with the effectiveness of financial decision making and validate the model vis-à-vis results derived through primary survey using statistical techniques Implementation of Fuzzy Logic Fuzzy ideas and fuzzy logic are so often utilized in our routine life that nobody even pays attention to them. For instance, take a case where a person is totally ignorant, ignorance brings uncertainty and risk to the decision making; on the other side, other person has the complete knowledge, so he is certain and decision making will be deterministic and easy. But in real world, we don't always fall in these two categories. We may be knowledgeable but to a certain extent and that knowledge may be imprecise or incomplete so decision making may not be possible with mathematical reasoning. Risk Ignorance (0) Complete Knowledge (1) Fuzzy Logic Uncertainty/ Complexity Probability Certain/ Simple FIGURE 6.1: FROM UNCERTAINTY TO CERTAINTY Fuzzy logic idea is similar to the human being s feeling and inference process. Unlike classical control strategy, which is a point-to-point control, fuzzy logic control is a range-to-point or range-to-range control. The output of a fuzzy controller is derived from fuzzification of both inputs and outputs using the associated membership functions. Fuzzy logic implementation requires the following three steps : 144

2 1. Fuzzification derives the membership functions for input and output variables and represents them with linguistic variables. 2. Fuzzy Inference Process combines membership functions with the fuzzy rules to derive the fuzzy output. The fuzzy output is still a linguistic variable 3. Defuzzification converts back the linguistic variable (fuzzy output) to the crisp or classical output Fuzzification The crisp input and output must be converted to linguistic variables with fuzzy components. For instance, to control Personal Financial Management System, the input Recording and Evaluation of financial transaction must be converted to the associated linguistic variables such as, 'Very Low (VL), 'Low' (L), 'Medium Low' (ML), 'Medium High' (MH), 'High' (H), or 'Very High' (VH). The output control variable Decision Making must be converted to the associated linguistic variables such as 'Poor', 'Fair', 'Average', 'Good' or 'Excellent'. Membership Function of the input Fuzzy Logic uses fuzzy sets; combine them with fuzzy rules to model the world and make decisions. Fuzzy set is a collection of related items which belong to that set to different degrees. Five types of fuzzy sets are used in the study as described below: Income Set Income set is collection of categories of income which belong to different degrees of recording transactions. There are 15 members in this set. Income = { Salary, Income from Profession, Income from Business, Agriculture income, Bonus/ Incentives, Dividend, Interest Received, Rent received, Gain / Loss in Share, Investment Income, Sale of Assets, Cash Back, Policy Maturity, Tax refund} Expenditure Set Expenditure set is a collection of categories of expenditure which belong to different degrees of recording transactions. There are 18 members in this set 145

3 Expenditure = {Utilities, Education, Household, Medical, Vehicle, Dividend, House, Wages, Insurance, Recreation, Social, Holiday Vacation, Business Travel, Personal Care, Shopping, Tax, Interest payment, Share Purchase / Sale, Other Deduction from Salary} Asset Set Asset set is a collection of categories of assets which belong to different degrees of recording transactions. There are 11 members in this set Asset = {Bank account, Cash in hand, House/Land/Agriculture/ Property, Shares Purchase, Fixed deposits, Mutual fund, Gold stock, Receivables, EPF, PPF, Equipment} Liability Set Liability set is a collection of categories of liabilities which belong to different degrees of recording transactions. There are 3 members in this set Liability = {Payables, Credit card, EMI/Principal Payment} Evaluation Set Evaluation set is a collection of financial transactions which belong to different degrees of evaluating transactions. There are 6 members in this set Evaluation = {Income, Expenditure, Assets, Liability, Savings, Net worth} Where Saving = Income- expenditure, Net Worth = Assets Liabilities Fuzzification makes use of fuzzy logic and integrates financial, accounting, decision making aspects, to process both quantitative and qualitative information. 53 value drivers are explicitly taken into account and combined together via if-then rules to produce an output. The output is a real number in the interval [0, 1], representing the economic worth creation power of an individual. Recording and evaluation of financial transactions for various heads related to income, expense, assets & liability is considered for input variables in financial decisions like budgeting, Income Statement, Cash Flow and Net Worth creation. There will be other factors for economic worth but are not suited for mathematical tractability e.g. financial intelligence, financial behaviour, financial literacy or financial health. 146

4 Factors Value Drivers Income 15 Expenditure 18 Asset 11 Liability 3 Evaluation 6 Total 53 TABLE 6.1: VALUE DRIVERS FOR FINANCIAL DECISION MAKING The x-axis of Input Membership Function collects all possible numerical values for the above mentioned financial decision, whose unit of measure is the extent of recording transactions. The y-axis collects the degrees at which a linguistic attribute is activated (membership degrees). The Very Low attribute is represented by a trapezium (basis ranges from 0 to 20%) and the others are depicted as triangles (bases range, respectively, from 10% to 40%, from 30% to 60%, 45% to 80%, 60% to 100%, 85% to 100%). Recording/ Evaluation Membership Degree Core Very Low 0% to 20% 0-10 Low 10% to 40% 20 Medium Low 30% to 60% 40 Medium high 45% to 80% 65 High 60% to 100% 85 Very High 85% to 100% 100 TABLE 6.2: INPUT MEMBERSHIP FUNCTIONS The core of a fuzzy set is the set of elements whose degree of membership in that set is equal to 1, which is equivalent to a crisp set. The boundary of a fuzzy set indicates the range in which all elements whose degree of membership in that set is between 0 and 1. The support of a fuzzy set, says LOW, is the set of elements whose degree of membership in LOW is greater than 0. This support can be expressed in a function form as Support (LOW) = { x ϵ T µlow(x) > 0 } (1.1) It can be shown that the support of a fuzzy set is a classical set. 147

5 FIGURE 6.2: INPUT MEMBERSHIP FUNCTIONS For example, a degree of recording of 37 is Low at a degree of 17%, Medium Low at a degree of 66%, and zero degree otherwise. Similarly, a degree of Evaluation of 40 is Medium Low at a degree of 100% and zero degree otherwise. Membership Function of the output Factors underlying various financial decisions like Budgeting, Income Statement, Cash Flow and Net Worth are Poor, Average or Good. The x-axis collects all possible numerical values for the above mentioned financial decision. The y-axis collects the degrees at which a linguistic attribute is activated (membership degrees). The Poor and Excellent attributes are represented by a trapezium (its basis ranges from 0 to 30% and 70% to 100%) and the Fair, Average and Good is depicted as triangle respectively (their bases range from 10% to 50%, 30% to 70% and 50% to 90%). 148

6 Decision Making Membership Degree Core Poor 0% to 30% 0-10 Fair 10% to 50% 30 Average 30% to 70% 50 Good 50% to 90% 70 Excellent 70% to 100% TABLE 6.3: OUTPUT MEMBERSHIP FUNCTIONS The support of a fuzzy set, says Poor is the set of elements whose degree of membership in POOR is greater than 0. This support can be expressed in a function form as Support (POOR) = { x ϵ T µpoor(x) > 0 } (1.2) FIGURE 6.3: OUTPUT MEMBERSHIP FUNCTIONS For example, a degree of recording of 32 is Fair at degree of 95%, Average at a degree of 10%, and Poor, Good & Excellent at a zero degree. 149

7 Fuzzy Inference Process After the membership functions are defined for input and output, the next step is to define the fuzzy control rule. This IF-THEN rule is widely used by the fuzzy inference system to compute the degree to which the input data matches the condition of a rule. The fuzzy rule is represented by a sequence of the form IF THEN, leading to algorithms describing what action or output should be taken in terms of the inputs. The rows represent first input i.e. recording of financial transactions and the columns represent second input i.e. evaluation of financial transactions, and those inputs are related to IF parts in IF THEN rules. The conclusion or output can be considered as a third dimensional variable that is located at the cross point of each row (recording of financial transactions) and each column (evaluation of financial transactions), and that conclusion (effectiveness of financial decision making) is associated with the THEN part. In this process, membership functions derived through fuzzification are combined with the fuzzy control rules to derive the fuzzy output. FIGURE 6.4: DERIVATION OF FUZZY CONTROL RULES 150

8 Let Fuzzy set A (Antecedent 1 : Fuzzy set Recording) represents certainty/uncertainty which progresses from Very Low to Very High. Fuzzy set B (Antecedent 2 : Fuzzy set Evaluation) represents simplicity/complexity which also progresses from Very Low to Very High. Union of fuzzy set A and fuzzy set B is defined as the resultant set (Consequent : Effectiveness of financial decisions) containing both A and B. For example, if A = {fully certain} and B = {simple} Then union of both A and B is AUB = {fully certain or simple}; the resultant set is represented by Excellent financial decision making. Similarly if A = {fully uncertain} and B = {complex} Then union of both A and B is AUB = {fully uncertain or complex}; the resultant set is represented by Poor financial decision making. All other rules follow a similar strategy, which is very similar to a human being s intuition. 36 rules are developed for the study. For high accuracy, the input and output have been divided into small segments, and fuzzy rules have been applied. Evaluation Transactions Very High Poor Average Average Good Good Excellent High Poor Fair Average Good Good Good Med. High Poor Fair Average Average Good Good Med. Low Poor Fair Fair Average Average Average Low Poor Poor Fair Fair Fair Average Very Low Poor Poor Poor Poor Poor Poor E R Very Low Low Med. Low Med. High High Very High Recording Transactions FIGURE 6.5: FUZZY CONTROL RULES 151

9 For example, when the evaluation of financial transaction is Low, and the recording of financial transactions is also Medium Low, the decision making will be Fair. This can be represented by the IF-THEN rule as IF recording of financial transactions is Medium Low at a degree of x AND evaluation of financial transactions is Low at a degree of y THEN the Financial decision making is Fair at a degree of z With x, y, z being real number in [0,1]. The value of z is obtained through aggregation of the membership degrees x and y of the antecedent variables Defuzzification The defuzzification process is meant to convert the fuzzy output back to the crisp or classical output to the control objective. Remember, the fuzzy conclusion or output is still a linguistic variable, and this linguistic variable needs to be converted to the crisp variable via the defuzzification process. Three defuzzification techniques are commonly used, which are: Mean of Maximum method, Center of Gravity method and the Height method. The Center of Gravity method (COG) is the most popular defuzzification technique and is widely utilized in actual applications. It returns the center of area under the curve i.e. the gravity of the area bounded by the membership function curve is computed to be the most crisp value of the fuzzy quantity. Centroid of fuzzy output membership function (Poor) can be calculated as: FIGURE 6.6: CALCULATION OF CENTROID FOR POOR MEMBERSHIP FUNCTION 152

10 The above figure is divided into two parts. First part is rectangle and second is triangle. Area and X coordinate of both the parts is calculated separately. Ax is calculated as area multiplied by x. COG is calculated by dividing summation of Ax by summation of A. Let s calculate COG of this figure, Area (A) X coordinate (x) Ax (Area * X coord) Part Part Summation COG ( Ax / A) TABLE 6.4: COG CALCULATION FOR POOR MEMBERSHIP FUNCTION Similarly, to explain the calculation of centroid of the fuzzy output membership function Excellent, the following figure is used. FIGURE 6.7: CALCULATION OF CENTROID FOR EXCELLENT MEMBERSHIP FUNCTION COG of the fuzzy output membership function excellent shown in above figure is calculated as: Area Centre of X axis Ax (Area * X coord) Part Part Summation COG ( Ax / A) TABLE 6.5: COG CALCULATION FOR EXCELLENT MEMBERSHIP FUNCTION 153

11 Similarly, Centroid of remaining fuzzy output membership functions (Fair, Average and Good) is calculated. These figures are triangle shaped so centroid is simple to calculate and shown in the following figure. FIGURE 6.8: POSITION OF CENTROID The above rules are designed to decide effectiveness of the decision making which might result in "Poor (10.83%), Fair (30%), Average (50%), Good (70%) and Excellent (88.33%) decision making. Decision Class COG Poor Fair 30 Average 50 Good 70 Excellent TABLE 6.6: COG FOR DIFFERENT DECISION CLASS 154

12 Derivation of Fuzzy Model The terminal product of defuzzification is the lookup table. Defuzzification needs to be performed for each subset of a membership function, both inputs and outputs. For instance, in the decision making system, one needs to perform defuzzification for each subset of recording/evaluation transactions input such as Very Low, Low, Medium Low, Medium High, High and Very High based on the associated fuzzy rules. The defuzzification result for each subset needs to be stored in the associated location in the lookup table according to the current recording of transactions and evaluating of transactions. In the following an example has been used to illustrate the defuzzification process and creation of lookup table. FIGURE 6.9: DEGREE OF ACTUAL RECORDING AND DEGREE OF ACTUAL EVALUATION As an example, consider the recording of financial transactions is 17 and evaluation of financial transaction is 50. Recording score of financial transactions i.e. 17 belongs to Very Low membership function at a degree of 0.33 and Low membership function at a degree of Similarly and evaluation score of financial transaction i.e. 50 belongs to Medium Low membership function at a degree of 0.50 and Medium High membership function at a degree of To make this illustration simple, four combinations of rules are possible and have been applied to this personal financial decision making system, which are 155

13 1) IF recording of financial transactions is VERY LOW, and evaluation of financial transaction is MEDIUM LOW, THEN decision making should be POOR 2) IF recording of financial transactions is VERY LOW, and evaluation of financial transaction is MEDIUM HIGH, THEN decision making should be POOR 3) IF recording of financial transactions is LOW, and evaluation of financial transaction is MEDIUM LOW, THEN decision making should be FAIR 4) IF recording of financial transactions is LOW, and evaluation of financial transaction is MEDIUM HIGH, THEN decision making should be FAIR Based on the assumption made for membership function and fuzzy rules, four fuzzy rules can be interpreted as functional diagrams, as shown in Figure 9. FIGURE 6.10: FUZZY OUTPUT CALCULATION It can be found that the points of intersection between recording of financial transactions of 17 and the graph in the first column (recording of financial transactions input µc) have the membership functions of 0.33, 0.33, 0.67 and Likewise, the second column (evaluation of financial transaction) shows that the evaluation of financial transaction of 50 has the membership functions of 0.50, 0.26, 0.50 and

14 The fuzzy output for the four rules is the intersection of the paired values obtained from the graph, then the AND result (Implication operator) between the recording of financial transactions and the evaluation of financial transaction. AND result should be: min (0.33, 0.50), min (0.33, 0.26), min (0.67, 0.50) and min (0.67, 0.26), which produces to 0.33, 0.26, 0.50 and 0.26, respectively. Aggregate all outputs Aggregation is the process by which the fuzzy sets that represent the outputs of each rule are combined into a single fuzzy set. Aggregation only occurs once for each output variable, just prior to the final step, defuzzification. The input of the aggregation process is the list of truncated output functions returned by the implication process for each rule. The output of the aggregation process is one fuzzy set for each output variable. FIGURE 6.11: DETERMINATION OF FUZZY OUTPUT BY CGD METHOD So all four rules have been placed together to show how the output of each rule is combined, or aggregated, into a single fuzzy set whose membership function assigns a weighting for every output (financial decision making) value. 157

15 Defuzzification The input for the defuzzification process is a aggregate output fuzzy set and the output is a single number. However, the aggregate of a fuzzy set encompasses a range of output values, and must be defuzzified in order to resolve a single output value from the set. There are five built-in methods supported: centroid (CGD), bisector, middle of maximum (MOM), largest of maximum, and smallest of maximum. In the study, the most popular CGD method is used which returns the center of area under the curve. For instance, for recording of financial transactions of 17 and evaluation of financial transaction of 50, the fuzzy output element y for this input pair is Y = (0.33 X X X X 30) = 21 ( ) This defuzzified fuzzy output i.e. 21 is a crisp or classical value, and should be entered into a certain location in a table called the lookup table. VH H MH ML L VL E R VL L ML MH H VH FIGURE 6.12: LOOK UP TABLE WITH CALCULATED VALUE FROM EXAMPLE Since this fuzzy output element is associated with a recording of financial transactions of 17 (belonging to VERY LOW and LOW in the recording) and evaluation of financial transaction of 50 (belonging to MEDIUM LOW and MEDIUM HIGH in evaluation), thus output value should be located in the cross point between the VERY LOW & LOW recording of financial transaction and the MEDIUM LOW & MEDIUM HIGH evaluation of financial transaction. 158

16 The Lookup Table The terminal product of defuzzification is the lookup table. Defuzzification needs to be performed for each subset of a membership function, both inputs and outputs. The defuzzification result for each subset needs to be stored in the associated location in the lookup table according to the recording transaction and evaluating transaction. Defuzzification technique is used to calculate all other fuzzy output values and locate them in the associated positions in the lookup table. Thus fuzzy model is derived that gives effectiveness of financial decision making related to recording/evaluation of financial transactions. FIGURE 6.13: FOR EFFECTIVENESS OF FINANCIAL DECISION MAKING This lookup table gives an insight into potential for improving Net Worth. Similarly we can calculate the count and percentage of respondents in each cell. 159

17 FIGURE 6.14: COUNT AND PERCENTAGE OF RESPONDENTS The model reflects the degree of recording and evaluation of financial transactions on financial decision making. There is a positive impact of recording and evaluation of financial transactions on effectiveness of personal financial decisions. More the recording and evaluation of financial transactions, more effective are the financial decisions Application of the Fuzzy Model on the survey results Based on the fuzzy model, survey respondents are grouped as below: Personal financial decision making classes Average Effectiveness of decision classes Percentage of respondents Poor 17% 7% Fair 38% 19% Average 53% 34% Good 66% 38% Excellent 78% 2% TABLE 6.7: AVERAGE EFFECTIVENESS OF DIFFERENT DECISION CLASSES 1. It is found that people who fall in Poor decision class will have 17% of their decisions effective and Excellent decision class will have 78% of their decisions effective from the perspective of long term wealth creation. 160

18 2. It has been observed that majority of the salaried and self employed class people fall in average or below decision class when looking from the angle of personal financial management. This means that 7% of people fall in Poor decision class and 19% of people fall in Fair decision class. This point to low level of financial literacy amongst these people. Such people need to upgrade their financial literacy level. 3. It is observed that about 34% of people fall in Average decision class. As a result they are at best, an average decision maker. They need to improve on their financial behaviour a) Tracking financial transactions through categorizing and recording income/ expenditure/ asset/ liability into various heads. b) Evaluation of financial flows i.e. inflows & outflows of money. c) Awareness about their financial worth and changes in it from time to time. d) Personal financial decisions linked to financial goals 4. It has been observed that about 40% of the people are in Good or Excellent decision class. For improving the quality of personal financial decisions, an individual needs to shift to Good or Excellent decision class. This means an individual needs to consistently, persistently and meticulously follow the personal financial management process for effective decision making Validation of the Fuzzy Model The results derived through fuzzy model are validated against the results derived through primary survey. The model is validated by administering an additional question related to decision making. To test the association between effectiveness of decision making using fuzzy model and effectiveness of actual decision making, following hypothesis is designed. Hypothesis 6: There is a correlation between effectiveness of decision making derived through fuzzy model and effectiveness of actual financial decisions Test Statistic: T test and Karl Pearson Coefficient of Correlation at1% level. 161

19 To test the hypothesis about association between effectiveness of financial decision making derived through fuzzy model and effectiveness of actual financial decisions, following null hypothesis is designed H 06 : There is no correlation between effectiveness of decision making derived through fuzzy model and effectiveness of actual financial decisions H 16 : There is a correlation between effectiveness of decision making derived through fuzzy model and effectiveness of actual financial decisions To study significance and impact of decision making by fuzzy logic and actual financial decisions, descriptive statistics is obtained. Results are as follows: Valid N Minimum Maximum Mean Std. Deviation Decision making fuzzy score Decision making actual score TABLE 6.8: DESCRIPTIVE STATISTICS OF FUZZY SCORE AND ACTUAL SCORE Above results indicate that mean score for decision making by fuzzy logic is and by actual financial decisions is To verify these scores significantly differ or not, T test is applied. Results of T test are as follows. Decision making Paired Differences t-cal Df t-table Diff of Mean Std. Deviation Std. Error Mean Pair 1 fuzzy score - actual score TABLE 6.9: PAIRED T-TEST OF FUZZY SCORE AND ACTUAL SCORE Above result indicate that calculated T value is which is less than table value 1.96 therefore T-test is accepted. So there is no significant difference between mean scores of decision making by fuzzy logic and actual decision making. To study association between two decision making factors, Pearson s coefficient of correlation is calculated. Results are as follows. 162

20 Decision making fuzzy score Decision making actual score Decision Pearson Correlation 1.45 ** making fuzzy Sig. (2-tailed).000 score N Decision Pearson Correlation.45 ** 1 making actual Sig. (2-tailed).000 score N **. Correlation is significant at the 0.01 level (2-tailed). TABLE 6.10: CORRELATION BETWEEN FUZZY SCORE AND ACTUAL SCORE Above table indicate that coefficient of correlation is 0.45 which is positive and significant at 1% level. There is a correlation between effectiveness of financial decision making derived through fuzzy model and effectiveness of actual financial decisions. Above information can be shown in following diagram. FIGURE 6.15: SCATTER DIAGRAM USING CORRELATION There is a positive correlation between uncertainty and complexity of financial decision making. In other words, more uncertain is the financial information, more complex is financial decision making. 163