A Statistical approach for Analyse the Causes of School Dropout Using. Triangular Fuzzy Cognitive Maps and Combined Effect Time Dependent

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1 Malaya J. Mat. S(1)(2015) A Statistical approach for Analyse the Causes of School Dropout Using Triangular Fuzzy Cognitive Maps and Combined Effect Time Dependent Data matrix (STrFCMs&CETDM) P. Selvam a, a Department of Mathematics, SVS college of Engineering, Coimbatore , Tamil Nadu, India. Abstract Dropouts are defined as young children, who enroll in schools and for some reason other than death leaves school before completing the grade without transferring to another school. It may be termed as dropouts. Dropouts rate is the percentage of dropouts in a given year out of the total number of those enrolled in a programme in the same year. There are so many causes for school dropout and also have been reports from many areas about the increase of dropout. Therefore, the causes of school dropout can be analyzed by using STrFCM&CETDM. Charles Spearman s coefficients, appropriate for both continuous and discrete variable, including ordinal variable give the Pearson correlation coefficient between the ranked variable. Keywords: Dropout,Fuzzy Cognitive Map, Triangular fuzzy number, ATD Matrix, RTD Matrix, CETD Matrix, Rank correlation coefficient MSC: 54A10, 54A20. c 2012 MJM. All rights reserved. 1 Introduction Lotfi. A. Zadeh (1965) has introduced a mathematical model called Fuzzy Cognitive Maps. After a decade, Political scientist Axelord (1976) used this fuzzy model to study decision making in social and political systems. Then Kosko (1986, 1988 and 1997) enhanced the power of cognitive maps considering fuzzy values for the concepts of the cognitive map and fuzzy degrees of Corresponding author. address: thiruvibhaa@gmail.com (P. Selvam).

2 P. Selvam / STrFCMs&CETDM 253 interrelationships between concepts. FCMs can successfully represent knowledge and human experience, introduced concepts to represent the essential elements and the cause and effect relationships among the concepts to model the behaviour of any system. It is a very convenient, simple and powerful tool, which is used in numerous fields such as social, economical and medical etc. Usually, we analyze the number of attributes ON-OFF position. But the thing is here, this causes gives the weightage of the attributes, which is called the ranking of the attributes. Now, we see the basic definitions for FCMs to develop the Triangular Fuzzy Cognitive Maps (TrFCM). 2 Degrees of the Triangular Fuzzy Number The linguistic values of the triangular fuzzy numbers are Very Low (0,0,0.15) Low (0,0.15,0.40) Medium (0.15,0.40,0.65) High (0.40,0.65,0.90) Very High (0.65,0.90,0.90) 2.1 Proposed Triangular Fuzzy Cognitive Maps (TrFCMs) Triangular Fuzzy Cognitive Maps (TrFCM) are more applicable when the data in the first place is an unsupervised one. The TrFCM works on the opinion of three experts. TrFCM is a models of the world which is a collection of classes and causal relations between classes. It is a different process when we compare to FCM. Usually, the FCM gives only the ON-OFF position. But this Triangular Fuzzy Cognitive Maps is more precise and it gives the ranking for the causes of the problem by using the weightage of the attributes. It is main advantage of the new Triangular Fuzzy Cognitive Maps. 2.2 BASIC DEFINITIONS OF TRIANGULAR FUZZY COGNITIVE MAPS Definition 2.1. When the nodes of the TrFCM are fuzzy sets then they are called as fuzzy triangular nodes. Definition 2.2. Triangular FCMs with edge weights or causalities from the set { 1, 0, 1} are called simple Triangular FCMs. Definition 2.3. An TrFCM is a directed graph with concepts like policies, events etc., as nodes and causalities as edges. It represents causal relationships between concepts. Definition 2.4. Consider the nodes/concepts T rc 1, T rc 2, T rc n of the Triangular FCM. Suppose the directed graph is drawn using edge weight T re ij { 1, 0, 1}. The triangular matrix M be defined by

3 254 P. Selvam / STrFCMs&CETDM T r(m) = (T re ij ) where T re ij is the triangular weight of the directed edge T rc i T rc j. Tr(M) is called the adjacency matrix of Triangular Fuzzy Cognitive Maps, also known as the connection matrix of the TrFCM. It is important to note that all matrices associated with a TrFCM which are always square matrices with diagonal entries as zero. Definition 2.5. Let T rc 1, T rc 2, T rc n be the nodes of an TrFCM. A = (a 1, a 2,, a n ) where T re ij { 1, 0, 1}. A is called the instantaneous state vector and it denotes the on-off position of the node at an instant. T ra i = 1 Instantaneous vector = T ra i = 0 Maximum(Weight) otherwise Definition 2.6. Let T rc 1, T rc 2, T rc n be the triangular nodes of an TrFCM. Let (T rc 1, T rc 2 ),(T rc 2 T rc 3 ),(T rc 3 T rc 4 )...(T rc i, T rc j ) be the edges of the TrFCM (i j). Then the edges form a directed cycle. AnTrFCM is said to be cyclic if it possesses a directed cycle. An TrFCM is said to be acyclic if it does not possess any directed cycle. Definition 2.7. An TrFCM is said to be cyclic is said to have a feedback. Definition 2.8. When there is a feedback in an TrFCM, i.e, when the causal relations flow through a cycle in a revolutionary way, the TrFCM is called a dynamical system. Definition 2.9. Let (T rc 1 T rc 2 ), (T rc 2 T rc 3 ), (T rc 3 T rc 4 )...(T rc i T rc j ) be a cycle. When T rc i is switched ON and if the causality flows through the triangular edges of a cycle and if it again causes C i, we say that the dynamical system goes round and round. This is true for any triangular node T rc i for i = 1, 2, n. The equilibrium state for this dynamical system is called the hidden pattern. Definition If the equilibrium state of a dynamical system is a unique state vector, then it is called a fixed point. Consider a TrFCM with T rc 1, T rc 2, T rc n as nodes. For example let us start the dynamical system by switching on T rc 1. Let us assume that the TrFCM settles down with T rc 1 and T rc n ON i.e., in the state vector remains as (1,0,...,0) is called fixed point. Definition If the TrFCM settles down with a state vector repeating in the form A 1 A 2 A i A 1 then this equilibrium is called a limit cycle. 3 METHOD OF DETERMINING THE HIDDEN PATTERN OF TRIANGULAR FUZZY COGNITIVE MAPS (TrFCMs) Step 1 :Let T rc 1, T rc 2 T rc n be the nodes of an TrFCM, with feedback, Let Tr(M) be the associated adjacency matrix.

4 P. Selvam / STrFCMs&CETDM 255 Step 2 :Let us find the hidden pattern when T rc 1 is switched ON. When an input is given as the vector A 1 = (1, 0, 0), the data should pass through the relation matrix M. This is done by multiplying A i by the triangular matrix M. Step 3 :Let A i T r(m) = (a 1, a 2, a n ) will get a triangular vector. Suppose A 1 T r(m) = (1, 0, 0) it gives a triangular weight of the attributes, we call it as A i T r(m) weight. Step 4 : Adding the corresponding node of the three experts opinion, we call it as A i T r(m) sum. Step 5 : The threshold operation is denoted by( ) ie., A 1 T r(m) Max(weight). That is by replacing a i by 1 if a i is the maximum weight of the triangular node (ie., a i = 1), otherwise a i by 0 (ie., a i = 0). Step 6 :Suppose A 1 T r(m) A 2 then consider A 2 T r(m) weight is nothing but addition of weightage of the ON attribute and A 1 T r(m) weight. Step 7 : Find A 2 T r(m) sum (ie., summing of the three experts opinion of each attributes). Step 8 : The threshold operation is denoted by( ) ie., A 2 T r(m) Max(weight). That is by replacing a i by 1 if a i is the maximum weight of the triangular node (ie., a i = 1), otherwise a i by 0 (ie., a i =0). Step 9 : If the A 1 T r(m) Max(weight) =A 2 T r(m) Max(weight). repeat the same procedure. Then dynamical system end otherwise Step 10 : This procedure is repeated till we get a limit cycle or a fixed point. 4 CONCEPT OF THE PROBLEM We have taken the following eleven concepts T rc 1, T rc 2, T rc 11 to analyze the various causes for dropout using linguistic questionnaire and the expert s opinion. The following concepts are taken as the main nodes of our problem. Tr C 1 - Language problem Tr C 2 - Friends Tr C 3 - Hostel facilities Tr C 4 - Economical Problem Tr C 5 - Sickness Tr C 6 - Transport Problem Tr C 7 - Lack of Basic Facilities Tr C 8 - Family Problem Tr C 9 - Not Interested Tr C 10 - Difficult to understanding

5 256 P. Selvam / STrFCMs&CETDM Tr C 11 - Minority Marriage, Pregnant etc., Now we give the connection matrix related with the TrFCM. LINGUISTIC VARIABLES FOR THE TRIANGULAR FUZZY NODES LINGUISTIC VALUES OF THE TRIANGULAR FUZZY NODES

6 P. Selvam / STrFCMs&CETDM 257 Attribute T rc 1 is ON: = ( ) T r(m) W eight = (0, (0, 0.15, 0.40) (0, 0, 0.15), (0, 0.15, 0.40), (0, 0.15, 0.40), (0, 0, 0.15), (0, 0, 0.15), (0, 0.15, 0.401), (0.40, 0.65, 0.90), (0.40, 0.65, 0.90), (0, 0, 0.15)) T r(m) Average = (0, (0.0500), (0.1833), (0.0500), (0.1833), (0.0500), (0.0500), (0.1833), (0.6500), (0.6500), (0.0500)) T r(m) Max(W eight) = ( ) = 1 1 T r(m) Average = (0.2968, (0.1517), (0.0650), (0.6500), (0.1517), (0.1517), (0.1517), (0.7952), (0.4225), (0.5308), (0.1417)) 1 T r(m) Max(W eight) = ( ) = 2 2 T r(m) Average = (0.6494, (0.1458), (0.0398), (0.1458), (0.3234), (0.0398), (0.1458), (0.0000), (0.6494), (0.6494), (0.0398)) 2 T r(m) Max(W eight) = ( ) = 3 3 T r(m) Average = (0.8422, (0.4871), (0.1515), (0.0325), (0.1191), (0.0325), (0.0325), (0.1191), (0.4221), (0.4221), (0.0325)) 3 T r(m) Max(W eight) = ( ) = 4 4 T r(m) Average = (0, (0.0422), (0.1548), (0.0422), (0.1548), (0.0422), (0.0422), (0.1548), (0.5487), (0.5487), (0.0422)) 4 T r(m) Max(W eight) = ( ) = 5 = 1

7 258 P. Selvam / STrFCMs&CETDM Do the process for the remaining attributes Table 1: Weightage of the attributes Now, we give the connection matrix related with the Combined Effect Time Dependent Data Matrix(CETD matrix). 5 Introduction CETD matrix are the techniques used to find the causes for the maximum number of dropouts. The data are transformed into four types of matrix called Initial Raw Data Matrix, Average Time Dependent Data matrix (ATD matrix), Refined Time Dependent Data matrix (RTD matrix), and Combined Effect Time Dependent Data matrix (CETD matrix). 5.1 The Concept of RTD Matrix Average Time Dependent (ATD) Matrix Raw data is transformed into a raw time dependent data matrix by taking various causes along the rows and level of education [standard wise] along the columns.we make it into the Average Time Dependent Data (ATD) matrix (a ij ) by dividing each entry of the raw data matrix by the number of

8 P. Selvam / STrFCMs&CETDM 259 years i.e., the time period. This matrix represents a data, which is totally uniform. At the third stage we find the average and Standard Deviation (S.D) of every column in the ATD matrix Refined Time Dependent (RTD) Matrix Using the average µ j of each jth column and σ j the S.D of the each jth column, choose a parameter α from the interval [0, 1] and form the Refined time dependent Matrix[RTD Matrix]. formula Using the a ij (µ j α σ j ) then e ij = 1 else if a ij (µ j α σ j, µ j + α σ j ) then e ij = 0 else if a ij (µ j + α σ j ) then e ij = 1 We redefine the ATD matrix into the Refined time dependent fuzzy matrix for here the entries are -1, 0 or 1. The row sum of this matrix gives the maximum age group Combined Effective Time Dependent Data (CETD) matrix Combine the above RTD matrices by varying the α [0, 1] so that we get the Combined Effective Time Dependent Data (CETD) matrix. The row sum is obtained for CETD matrix and conclusions are derived based on the row sums. These are represented by graphs also. Now use this technique to find which reason is mainly for the dropouts. 5.2 REASONS FOR THE MAIN CAUSES FOR THE STUDENT DROPOUTS 1. Language problem [R 1 ] 2. Not Interested [R 9 ] 3. Difficult to understand [R 10 ] 4. Friends [R 2 ] 5. Others [Minority Marriage, Pregnant etc.] [R 11 ] 6. Hostel facilities [R 3 ] 7. Transportation Problem [R 6 ] 8. Lack of Basic Facilities [R 7 ] 9. Economical Problem [R 4 ] 10. Sickness [R 5 ] 11. Family Problem [R 8 ] The level of education are taken as the column and the above attributes are classified as four categories as 1-3 [Education], 4-5[Society], 6-8[Facilities] and 9-11[Family] are taken as the row of

9 260 P. Selvam / STrFCMs&CETDM the initial raw data matrix. Based on the attributes, the initial raw data matrix has been obtained and it has been clearly stated from 500 students and experts opinion. Initial raw data matrix of dropouts of order 5 x 11 is given below. Case (i): Table 2: Initial Raw data matrix Table 3: ATD matrix Table 4: Average and SD of ATD matrix

10 P. Selvam / STrFCMs&CETDM 261 Case (ii): We have taken the value α =0.1,0.25,0.45,0.5 and 0.7 to find the CETD matrix

11 262 P. Selvam / STrFCMs&CETDM 6 CORRELATION COEFFICIENTS Assessing the correlation between different ranking patterns obtained by Triangular method and CETD Matrix or different decision makers and /or different scenarios for a given set of alternative forms a major part of comparative study. The range of the correlation coefficient is between -1 and +1. If the correlation coefficient is negative, then the variable are inversely proportional and it is maximum when it is -1; if the coefficient is 0, there is no association between the variable. If

12 P. Selvam / STrFCMs&CETDM 263 the coefficient is positive, then the variable are associated directly and it is maximum when it is +1. A non parametric rank correlation method study namely, Spearman which is used to compute correlation coefficient values is explained as follows. 6.1 CHARLES SPEARMAN S COEFFICIENTS OF RANK CORRELATION METHOD Charles Spearman s coefficients of rank correlation method are the technique of determining the degree of correlation between ranks achieved by TrFCM method and CETD matrix method or different decision-makers and /or different scenarios for a set of alternatives. This coefficient is determined as under: [ ] 6 d 2 r s = 1 i n(n 2 1) Where d i =difference between ranks of ith pair of the two variable. N = number of pairs of observations. Various critical values for Charles Spearman s coefficients of rank correlation for various significance levels is provided. Numerous case studies have used the Charles Spearman s coefficients of rank correlation method for computation of correlation coefficients values. Table 5: Characteristics of r s

13 264 P. Selvam / STrFCMs&CETDM [ ] 6 d 2 r s =1 i n(n 2 1) [ ] 6 32 = = Conclusions and suggestions In the course of this study, fuzzy data was converted to raw data applying the Triangular Fuzzy Cognitive maps and the CETD Matrix, and the causes of school dropout were correlated by using two methods and it was found that the nature of correlation was identified as Very High bearing a numerical value of 0.9 indicating a marked relationship between the two - both methods are interlinked. From the above mentioned data, it is found that the mentioned causes are the main reason for dropout. School, Society and Government can create awareness programme about the importance of education and make each individual to realize the value of education. References [1] W. B. Vasantha Kandasamy,Florentin Smarandache,K. Amal, K. Kandasamy (2013), Fuzzy Analysis of School Dropouts and their Life After, Education Publisher Inc., Ohio. [2] T.Pathinathan, K.Thirusangu and M.Mary John, (2005), On School dropouts of School Children- A Fuzzy Approach, Acta Ciencia Indica, Vol. XXXIM, No.4, pp [3] T.Pathinathan, K.Thirusangu and M.Mary John (2006), A Mathematical Approach to Issues which Increase Dropouts in School Education, Ind. Journal of Millennium Development studies - An International Journal, Vol.1, No.2, pp [4] T.Pathinathan, K.Thirusangu and M.Mary John (2007), School dropouts; breeding ground for exploited child labourers -A mathematical approach, Far East Journal of Mathematical Sciences, Vol.25, No.1, pp [5] M.Clement Joe Anand and A.Victor Devadoss (2013), Using New Triangular Fuzzy Cognitive Maps (TRFCM) to Analyze Causes of Divorce in Family, International Journal of Communications Networking System, Vol.02, pp [6] A.Rajkumar and A.Victor Devadoss (2014), A Study on Miracles through the Holy Bible using Triangular Fuzzy Cognitive Maps(TrFcms), International Journal of Computer Application, Vol.03, pp

14 P. Selvam / STrFCMs&CETDM 265 [7] Amiya K.Shyamal and Madhumangal Pal, Two new operators on fuzzy matrices, J. Appl. Math. And Computing, 15(2004) [8] W.B.Vasantha Kandasamy and A.Victor Devadoss (2004), Some New Fuzzy Techniques, Jour. Of Inst. Of. Math. and Comp. Sci. (Math.Ser.), Vol. 17, No.2, [9] D.Dubois and H.Prade (1978), Operations on fuzzy numbers, The International Journal of Systems Science, Vol. 9, No. 6, pp [10] Lotfi. A. Zadeh (1965), Fuzzy Set, Information and control, 8, pp [11] Michael Glykas, Fuzzy Cognitive Maps: Advances in Theory, Methodologies, Tools and Applications, Springer. [12] Victor Devadoss.A, Clement Joe Anand.M (2013), Analysis of Women Computer users affected by Computer Vision Syndrome (CVS) using CETD Matrix, International Journal of Scientific and Engineering Research,, Vol. 4, Issue3. [13] Narayanamoorthy.S,(2012) Application Of Fuzzy Cetd Matrix Technique To Estimate The Maximum Age Group Of Silk Weavers As Bonded Laborers, Int. J. of Appl. Math and Mech., 8 (2):pp [14] Sunita Chugh, Dropout in Secondary Education: A Study of Children Living in Slums of Delhi, National University of Educational Planning and Administration, February [15] G.Woodbury (2002), Introduction to Statistics, Thomson Learning, USA. [16] K.S.Raju and D.Nagesh Kumar (1999), Multi-criterion Decision-Making in Irrigation Planning, Agricultural Systems, vol. 62, no.2, pp [17] C.R Kothari, Research Methodology Methods and Techniques, 2nd edition, New Age International Publishers Pvt Ltd, India. Received: May 13, 2015; Accepted: June 19, 2015 UNIVERSITY PRESS Website: