Adaptive method of fuzzy sets

In the course of the last 20 years the theory of fuzzy sets is being actively developed. The results of the theory are widely used in testing. The paper proposes a modification of the known adaptive method of fuzzy sets for evaluation of students’ knowledge level. By using Rasch’s testing model and Monte Carlo’s method the author verifies the accuracy of the modification proposed. The results have shown that the method modification of fuzzy sets has some higher accuracy than the classical method. Except for the classical method of fuzzy sets the two methods often used in United State Examination were also considered: a scaling The modification method of fuzzy sets is an adaptive algorithm. Thus, the test results in the form of primary points are used for construction of membership functions of fuzzy sets. Here are the steps of the algorithm making it possible to “tweak” the primary points. 1) The relative frequencies p_ij of appearance of points B = j by solving the i-th task are calculated, where i =1, ..., M ( M is the number of test’s tasks), j=0,..., max . 2) The membership functions are formed so that the middle lines of their curvilinear trapezoid were equal to p_ij . 3) The numbers of E_k, k = 0, ..., max, are calculated for each membership function. These numbers are the result of diffusivities for fuzzy sets by the method of severity’s center. 4) Adjustment of the primary point B for the i-th task is made by the formula: . 5) The sum of all “adjusted” points calculates: 6) Test points ТБ_кор is calculated by using scaling. The main results of the paper: 1) The accuracy of the method of fuzzy sets depends significantly on the distance “averages” E₀ and E_m of extreme points from 0 and 1, respectively. The highest accuracy of the method of fuzzy sets is achieved if we zoom the average values of E₀ and E_m , calculated by the method of center gravity, to 0 and 1 respectively by the formulas: E_0,кор = E ₀ / 5 , E_m,кор = 0,6+ 2. E_m/ 5. 2) The accuracy of the modified method of fuzzy sets is higher than the accuracy of the classical method in 0,2 %. 3) The accuracy of the modified method of fuzzy sets coincides with the accuracy of the method of the Rasch’s logarithm. 4) The method of the Rasch’s logarithm does not depend on the choice of the scaling function. Therefore, this method is the most convenient for use. The accuracy of this method is not worse than the accuracy of the other methods.

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