Feature selection in a fuzzy student sectioning algorithm

Feature Selection in a Fuzzy Student Sectioning Algorithm Mahmood Amintoosi1 and Javad Haddadnia2 1 2 Mathematics Department, Sabzevar Teacher Training University, Iran, 397 Engineering Department, Sabzevar Teacher Training University, Iran, 397 {amintoosi, haddadnia}@sttu.ac.ir Abstract. In this paper a new student sectioning algorithm is proposed. In this method a fuzzy clustering, a fuzzy evaluator and a novel feature selection method is used. Each student has a feature vector, contains his taken courses as its feature elements. The best features are selected for sectioning based on removing those courses that the most or the fewest numbers of students have taken. The Fuzzy c-Means classifier classifies students. After that, a fuzzy function evaluates the produced clusters based on two criteria: balancing sections and students’ schedules similarity within each section. These are used as linguistic variables in a fuzzy inference engine. The selected features determine the best students’ sections. Simulation results show that improvement in sectioning performance is about 18% in comparison with considering all of the features, which not only reduces the feature vector elements but also increases the computing performance. 1 Introduction The course timetabling problem essentially involves the assignment of weekly lectures to time periods and lecture room in such a way that a set of constraints satisfy. Current methods for tackling timetabling problems include evolutionary algorithms [14], [15], [17], [29], [30], genetic algorithms [12], [13], [18], [19], [20], graph-based methods [10], [11], [35], simulated annealing [2], [3], tabu search [24], [33], [34], neural networks [26], hybrid and heuristic approaches [6], [7], [28], [38], constraint logic programming [1], [9], [21], [23], [32], ant colony optimization [36] and fuzzy expert systems [5]. A particular problem related to timetabling is student sectioning. This problem is due to courses which involve a large number of students. For a variety of reasons, splitting these students into a few smaller sections is desirable: for example, 1. Room capacity requirements: the when number of students in a course is greater than every room capacity. 2. The policies of the institution: some institutions have rules about maximum capacity of courses (e.g. 50 for specialized courses and 60 for public courses). 3. A good student sectioning may reduce the number of edges in the conflict matrix [16]. E. Burke and M. Trick (Eds.): PATAT 2004, LNCS 3616, pp. 147–160, 2005. c Springer-Verlag Berlin Heidelberg 2005
148 M. Amintoosi and J. Haddadnia Most previous works related to the course scheduling problem have concentrated on timetable construction, with little regard to student sectioning. Selim [35] introduced the idea of split vertices and made a start to determine those vertices, which should be split in order that the chromatic number may be reduced. Selim treated the problem as a conflict graph and showed how one could pick out certain vertices in the conflict graph to split, reducing the chromatic number of the graph to the desired value by increasing the number of sections in the timetable. With this idea Selim decreases the chromatic number of the conflict matrix, from 8 to 3. Thus the total number of periods needed is reduced. The main algorithm of Laporte and Desroches [25] has the following stages: 1. Constructing student schedules without taking into account section enrollments and room capacities. 2. Balancing section enrollments. 3. Respecting room capacities. One of its interesting features is the weight it gives to the overall quality of student schedules. Aubin and Ferland [6] generate an initial timetable with an assignment of the students to the course sections; then an iterative procedure is used which adjusts the timetable and the grouping successively until no more improvement of the objective function can be obtained. At each iteration, two procedures are used: 1. Given the grouping generation during the preceding iteration, the timetable is modified to reduce the number of conflicts. 2. With this timetable, the grouping of students is modified to reduce the number of conflicts. Hertz [24] used a tabu search technique for both timetabling and sectioning problems. He assumed that the numbers of students in each section are fixed. The neighborhood N (s) of a solution s consists of all those grouping which can be obtained from s by exchanging the two students of two different sections of a course. The initial student sectioning of Muller and Rudova [27] is based on Carter’s [16] homogeneous sectioning and it is intended to minimize future student conflicts. They attempt to improve the solution with respect to the number of student conflicts. This is achieved via section changes during the search. Each student enrollment in a course with more than one section was processed. An attempt was made to switch it with a student enrollment from a different section. If this switch decreased the total number of student conflicts, it would be applied. Amintoosi et al. [4] introduced a fuzzy sectioning method which decreases the average number of conflicts in a genetic timetabling program. In this paper we concentrate on initial student sectioning prior to timetabling. A new student-sectioning algorithm is proposed. In the proposed method a fuzzy clustering, a fuzzy evaluator and a novel feature selection method are used. Feature Selection in a Fuzzy Student Sectioning Algorithm 149 A Fuzzy c-Means algorithm classifies students in a large class into smaller sections. Each student has a feature vector in fuzzy classifier. The courses taken by each student are its feature elements. The produced clustering is evaluated with a fuzzy function, according to some criteria: size of clusters and students’ schedules similarity of each section. The above parameters have been used in a fuzzy inference engine as linguistic variables for clustering evaluation. By removing those courses that have been taken by the most or by the fewest numbers of students, the best features (courses) are selected. Appropriate values for the most and the fewest values are determined with an iteration procedure. In each iteration, courses which contain each of the following properties are removed: – courses which have been taken by a high percentage of students, greater than a specified threshold, and – those that have been taken by a low percentage of students, lower than another threshold. A Fuzzy c-Means classifier classifies remaining courses. The produced clusters are evaluated by the mentioned fuzzy function. The best classification of students will be ready at the end of the above loop. Simulation results in the average case show that about 53% of courses are essential for clustering and with these selected courses the clustering performance would be about 18% more efficient. The reminder of this paper is organized as follows. Section 2 describes the fuzzy C-means clustering algorithm. In Section 3, the proposed method is explained in more detail and in Section 4 simulation results are considered. Section 5 concludes. 2 Fuzzy c-Means Clustering Fuzzy c-Means (FCM) is a data clustering algorithm in which each data point is associated with a cluster through a membership degree. Most analytical fuzzy clustering approaches are derived from Bezdeck’s FCM [8], [31]. This technique partitions a collection of NT data points into r fuzzy groups and finds a cluster center in each group, such that a cost function of a dissimilarity measure is minimized [22]. The algorithm employs fuzzy partitioning such that a given data point can belong to several groups with a degree specified by membership grades between 0 and 1. A fuzzy r-partition of input feature vector X = {x1 , x2 , . . . , xNT } ⊂ ℜn is represented by a matrix U = [µik ], where the entries satisfy the following constraints: µik ∈ [0, 1] , 1 ≤ i ≤ r , 1 ≤ k ≤ NT r
µik = 1 , 1 ≤ k ≤ NT (1) (2) i=1 0< N
T k=1 µik < NT , 1 ≤ i ≤ r. (3) 150 M. Amintoosi and J. Haddadnia U can be used to describe the cluster structure of X by interpreting µik as the degree of membership of Xk to cluster i. A proper partition U of X may be defined by the minimization of the following objective function: Jm (U, C) = NT  r  (µik )m d2ik (4) k=1 i=1 where m ∈ [1, +∞] is weighting exponent called the fuzzifier, C = {c1 , c2 , . . . , cr } is the vector of the cluster centres, and dik is the distance between Xk and the ith cluster. Bezdek proved that if m ≥ 1, d2ik > 0, 1 = i = r, then U and C minimize Jm (U, C) only if the entries of them are computed as follows: µ∗ik =
r 1 2 (5) (dik /djk ) m−1 j=1 c∗i = N
T (µik )m xk k=1 N
T . (µik (6) )m k=1 One of the major factors that influences the determination of appropriate clusters of points is the dissimilarity measure chosen for the problem. Indeed, the computation of the membership degrees µ∗ik depends on the definition of the distance measure dik , which is the inner product of norms (quadratic norms) on Rn . The squared quadratic norm (distance) between a pattern vector Xk and the center ci of the ith cluster is defined as follows: d2ik = ||xk − ci ||G = (xk − ci )T G(xk − ci ) (7) where G is any positive definite (n × n) matrix. The identity matrix is the simplest and most popular choice of G. The FCM algorithm consists of a series of iterations alternating between Equations (5) and (6). This algorithm converges to either a local minimum or a saddle point of Jm (U, C). FCM is used to determine the cluster centers ci and the membership matrix U for a given r value as follows: Step1 : Initially the membership matrix is constructed using random values between 0 and 1, such that constraints (1)–(3) are satisfied. Step2 : For each cluster i (i = 1, 2, . . . , r) the fuzzy cluster centres ci are calculated using Equation (6). Step3 : For each cluster i, the distance measures dik are computed using Equation (7). Step4 : The cost function in Equation (4) is computed and if either it is found to be below a certain tolerance value, or its improvement over the previous iteration (dJm ) is below a certain threshold, then it is stopped and the clustering procedure is terminated. Feature Selection in a Fuzzy Student Sectioning Algorithm 151 Step5 : A new U using Equation (5) is computed and steps 2–5 are repeated. By using the above fuzzy clustering procedure, the students in large classes are divided into r clusters. Clustering evaluation is done with a fuzzy function. This fuzzy evaluation is based on a fuzzy inference engine. In the next section the proposed method is explained. 3 The Proposed Method The aim is to allocate students of a course into smaller sections, satisfying the following criteria: 1. Student course selections must be respected. 2. Section enrollments should be balanced, i.e. all sections of the same course should have roughly the same number of students; 3. Section capacities and policies of institute should not be exceeded. 4. Student schedules in each section would be the same as each other (as much as possible). A fuzzy c-Means algorithm is used for student sectioning [4]. This algorithm satisfies criterion 1. Other criteria are evaluated at the evaluation phase. A fuzzy inference engine evaluates the produced clustering. With a well-defined set of rules [4], the criteria 2–4 are considered. Removing those courses that have been taken by the most or the fewest numbers of students achieved the best feature elements. The simulation results show that about 53% of features are important for classification; in addition, the clustering performance with selected features is better than the performance in the case that all features were considered. The proposed algorithm contains three basic parts as follows: – method of data representation, – fuzzy clustering and fuzzy evaluation, – feature selection method. The following sections explain the basic parts of the proposed method. 3.1 Data Representation In the proposed method each student has a feature vector. The courses taken by each student are its feature elements, represented by a bit array. Suppose that P is the number of all courses and Vi is the list of taken courses by student i. As shown in Equation (8), Vi is the feature vector of the ith object (student):  1 if student i has taken lesson j Vij = (8) Vi = (Vi1 , . . . , ViP ) , 0 otherwise . Table 1 shows an example for five students. Column 2 contains the selected courses list from three total courses, which are taken by each student. Column 3 shows the corresponding feature vector for each student. If sectioning of A is desirable, students 1 and 2 will be in Section 1, and others remain in the second section. Column 4 displays the sectioning results. 152 M. Amintoosi and J. Haddadnia Table 1. List of courses taken by five students and their corresponding feature vectors Student 1 2 3 4 5 3.2 Courses taken A, A, A, A, A, B B C C C Feature vector Section no. 110 110 101 101 101 1 1 2 2 2 Fuzzy Clustering and Fuzzy Evaluation In our algorithm Fuzzy C-Mean has been used as classifier. The input of the classifier is the students’ feature vectors, as explained in the previous section. For simplicity, the number of clusters is assumed to be 2. Clustering evaluation has been done with a fuzzy function. Rates of section balancing and similarity of students’ schedules in each section (criteria 1 and 2) are its inputs and its output is the clustering performance. Two lingual variables “Density” and “N1PerN2” (N1/N2) are defined as inputs of a fuzzy inference engine. Density of clusters is the sum of the common courses of all student pairs (a, b) such that students a, b lie in the same section. By dividing the mentioned summation with its maximum value, the value of “Density” will be normalized.“N1PerN2” represents the section’s balancing rate. It is supposed that N1 is the size of the smaller section and hence, the range of this variable is between 0 and 1. Since the number of students in each section should be as equal as possible, the suitable values of N1PerN2 are close to 1. Figure 1 shows the membership functions of the mentioned variables. The output of the fuzzy inference engine is named “Performance” and has the following values: Bad, NotBad, Medium, Good and Excellent. Our Fuzzy rules were defined as follows: Fuzzy Rules Rule 1: if (Density Rule 2: if (Density Rule 3: if (Density Rule 4: if (Density Rule 5: if (Density Rule 6: if (Density is High) and (N1PerN2 is Suitable) then (Performance is Excellent) is High) and (N1PerN2 is Middle) then (Performance is Good) is High) and (N1PerN2 is UnSuitable) then (Performance is Bad) is Med ) and (N1PerN2 is Suitable) then (Performance is Good) is Med ) and (N1PerN2 is Middle) then (Performance is Medium) is Med ) and (N1PerN2 is UnSuitable) then (Performance is Bad) Feature Selection in a Fuzzy Student Sectioning Algorithm 153 Fig. 1. Membership functions of our linguistic variables: “Density” (upper), “N1PerN2” (lower) Rule 7: if (Density is Low ) and (N1PerN2 is Suitable) then (Performance is NotBad) Rule 8: if (Density is Low ) and (N1PerN2 is Middle) then (Performance is Bad) Rule 9: if (Density is Low ) and (N1PerN2 is UnSuitable) then (Performance is Bad) The rules are defined such that the influence of unsuitable sections sizes is more than the effect of students’ schedules similarity. Rules 3, 6 and 9 reflect this. Hence the decision surface of our rules’ database shown in Figure 2 has an asymmetric face. Fig. 2. Decision surface of our rules’ database 154 3.3 M. Amintoosi and J. Haddadnia Feature Selection Method Feature selection plays an important role in classification problems. Advances in feature selection not only reduce the dimension of feature vector but also reduce the complexity of classifier. The most important rule in feature selection is reducing the feature elements as far as possible such that their class discrimination remains [37]. As you can see in Table 1, the first feature (course A) is common between all vectors (students). Removing it should not influence the clustering results. In our problem it seems that removing the following courses is a good idea: 1. courses that the most number of students have been taken; 2. courses that the fewest number of students have been taken. Removing those courses that none of the students or all of them have taken is done in a pre-processing stage. One problem is to specify the appropriate threshold values for the most and the fewest parameters. An exhaustive search procedure finds them as follows: the percentage of students that have taken each course is determined and these values are finite. The number of such courses is equal to or less than the number of all courses. If P is the number of all courses taken by the students of the class, a procedure with worst time complexity O(P 2 ) will find the appropriate values for the most and the fewest. Before entering a loop, the percentage of students that have taken each course, and the percentage that have not taken each course, are determined. Each value of these two lists can be a threshold for the most (T1 ) and the fewest (T2 ) parameters, respectively. In each iteration, those courses for which the percentage of students which have taken them is greater than T1 , or for which the percentage of students that have not taken them is greater than T2 , will be removed. The fuzzy C-Means algorithm classifies students based on these selected courses (features). After that, the fuzzy function evaluates the produced clusters based on two criteria: balancing sections and students’ schedules similarity of each section. After the last iteration the best classification of students will be arrived at. The following pseudo-code illustrates the overall procedure. In this algorithm Clustering() is a classifier function and ClusteringEvaluation() is an evaluator function. AllCourses = Set of all courses that students of this course have taken; //(All Features) List1 = Percentage of students that have taken each course, sorted in non increasing order; Thresholds1 = Distinct elements of List1; List2 = Percentage of students that have not taken each course, sorted in non increasing order; Thresholds2 = Distinct elements of List2; for i: = 1 to length(Thresholds1 ) do begin Feature Selection in a Fuzzy Student Sectioning Algorithm 155 MaxSet = Set of Courses that percentage of students which have taken them > Thresholds1[i]; for j: = 1 to length(Thresholds2 ) do begin MinSet = Set of Courses that percentage of students which have not taken them > Thresholds2[j]; SelectedFeatures = AllCourses – (MaxSet ∪ MinSet ); Clusters = Clustering( SelectedFeatures); if ClusteringEvaluation(Clusters) is better than previous clusters’ performance then begin BestClusters = Clusters; T1 = Thresholds1[i]; T2 = Thresholds2[j]; end; // end if end; // end for j end; // end for i 4 Simulation Results The information used for simulation is taken from the Mathematics department at Sabzevar University. Students and courses are randomly selected with the following characterization: – total number of students: 210; – number of courses: 38. The simulator program has been written with Matlab on a Windows platform. Figure 3(a) shows the feature vectors for a course with 67 students and total 31 features. Figure 3(b) highlights the selected features. As can be seen in the last row of Table 2 (related to this course) the number of selected features (NSF column) is 16 out of 31. With these selected features the performance is better by about 35% (P1/P2 column) than with consideration of all features. At each iteration of the feature selection algorithm, performance is evaluated with the selected features. The values of the fuzzy function’s inputs are varying with every set of selected courses (features). Figure 4 demonstrates the inputs and output values of fuzzy evaluator function in a total of 179 iterations for the mentioned course with 67 students. The best solution is shown with a circle on iteration no 87 in Figures 4 and 5. The values of the “Density”, “N1PerN2”, “Performance” and the number of selected features are 0.73, 0.97, 0.75 and 16 respectively. The first iteration (values: 0.78, 0.60, 0.56 and 31) corresponds with the case in which all features have been considered in clustering. As can be seen, performance increases by about 35% (0.75/0.56 = 1.35). 156 M. Amintoosi and J. Haddadnia The simulation results on 12 courses are shown in Table 2. As can be seen, the performance with the proposed method (P1) is always better than or equal to the performance when all features are considered (P2) in clustering. All of the features are required only in three examples (Examples 3, 10 and 11). In the average case, 53% of features are required and performance is increased by about 18%. (a) (b) Fig. 3. (a) Feature vectors for a course with 67 students and 31 features (courses). (b) The selected features (columns in the figure) are highlighted. Fig. 4. The values of inputs and output of fuzzy evaluator function in the Feature Selection Algorithm. Dashed, dotted and solid lines represent the value of “Density”, “N1PerN2” and “Performance”, respectively. The best solution is shown with a circle on iteration no. 87. Feature Selection in a Fuzzy Student Sectioning Algorithm 157 Fig. 5. The best number of selected features is 16 and demonstrated with a circle on iteration no. 87. The point formation depicts the outer and the inner loops in the proposed feature selection algorithm. Table 2. The simulation results for 12 courses. P1 is the best performance achieved (with selected features), P2 is the performance taking into account all the courses. NSF is the best number of selected features, NTF is the number of total features, T1 and T2 are the values of the most and the fewest parameters. LessonNo N1PerN2 1 2 3 4 5 6 7 8 9 10 11 12 Average 5 0.8 0.95 0.8 0.76 0.75 0.56 0.7 0.86 0.76 0.89 0.79 0.97 0.80 P1 0.76 0.5 0.75 0.54 0.65 0.39 0.63 0.65 0.69 0.66 0.58 0.75 P2 P1/P2 NSF NTF NSF/NTF 0.49 0.49 0.75 0.52 0.58 0.24 0.57 0.51 0.64 0.66 0.58 0.56 1.56 1.03 1 1.03 1.13 1.61 1.1 1.28 1.08 1 1 1.35 1.18 5 27 29 26 3 2 7 10 6 33 31 16 18 34 29 34 22 11 30 31 34 33 31 31 T1 T2 0.28 1 0.44 0.79 0.33 0.95 1 1 0.97 0.76 1 0.93 0.14 1 0.64 0.18 0.5 0.57 0.23 1 0.71 0.32 0.33 0.77 0.18 1 0.71 1 1 0.97 1 1 0.98 0.52 0.25 0.9 0.53 Concluding Remarks In this paper a novel course selection method and a fuzzy evaluation function for the student sectioning problem is proposed. Our aim is to allocate students of a course to smaller sections—prior to timetabling—so as to satisfy the following criteria: – student course selections must be respected, – section enrollments should be balanced, 158 M. Amintoosi and J. Haddadnia – section capacities and rules of institution should not be exceeded, – student schedules in each section should be the same as each other (as far as possible). Firstly, with a Fuzzy c-Means algorithm, students in a large class have been classified. Each student has a feature vector in the fuzzy classifier. The courses taken by each student are his/her features. In contrast to the usual graph-based sectioning, not only is the number of courses common to two students important, but also the number of courses that is not taken by both of them is significant. Secondly, the produced clusters are evaluated with a fuzzy function, according to two criteria: balancing sections and students’ schedules similarity of each section. By removing those courses that the most students or the fewest students have taken, the best features (courses) are selected. Simulation results in the average case show that about 53% of courses are essential for clustering. 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