The Bezier Curve as a Fuzzy Membership Function Shape∗

ISSN 2066-6594 Ann. Acad. Rom. Sci. Ser. Math. Appl. Vol. 10, No. 2/2018 THE BEZIER CURVE AS A FUZZY MEMBERSHIP FUNCTION SHAPE∗ Tania Yankova† Galina Ilieva‡ Stanislava Klisarova§ Abstract Membership functions are a key concept in fuzzy set theory and their correctness and precision are essential for the accuracy of obtained results. This article discusses the use of Bezier curve to construct a membership function. Based on the frequency distribution of data by minimization, the coordinate formulas of the control points that define the curve are derived. Use of the described membership function is illustrated by an example. These formulas are applied to bispectral index data sets in order to compare with other published method. MSC: 03E72, 65D05, 65D10, 65D15 keywords: Fuzzy set, Membership function, Bezier curve, Least-squares minimization ∗ Accepted for publication on March 5, 2018 [email protected] Faculty of Economics and Social Sciences, University of Plovdiv Paisii Hilendarski, 24 ”Tzar Assen” str., Plovdiv, Bulgaria ‡ [email protected] Faculty of Economics and Social Sciences, University of Plovdiv Paisii Hilendarski, 24 ”Tzar Assen” str., Plovdiv, Bulgaria § [email protected] Faculty of Economics and Social Sciences, University of Plovdiv Paisii Hilendarski, 24 ”Tzar Assen” str., Plovdiv, Bulgaria † 245 246 1 T. Yankova, G. Ilieva, S. Klisarova Introduction The fuzzy set theory was first introduced by Lotfi Zadeh in the 1960s as a way to capture uncertainty and vagueness often overlooked in complex systems. It can be considered as a generalization of classical set theory. Constructing fuzzy rules and building a proper membership function have been challenges for several decades by now. The fuzzy membership function is a key concept in designing fuzzy systems. Proper and precise use of the membership function is essential for the accuracy of obtained results. Therefore, construction a membership function and determining its parameters continues to be a current issue that many researchers are focused on and have been proposing new approaches and algorithms in recent years. For example, Wu and Chen in [16] created an algorithm for developing membership functions based on α-cuts of equivalence relations and induced the fuzzy rules from the numerical training data set. Yang and Bose [17] introduced automatic fuzzy membership generation with unsupervised learning where the proper cluster is generated and then the fuzzy membership function is generated according to this cluster. Feng, Li and Hu [3] suggested a training algorithm for Hierarchical Hybrid FuzzyNeural Networks, based on Gaussian membership function. Viattchenin, Tati and Damaratski [15] presented the problem of constructing Gaussian membership functions derived from the data by using heuristing algorithm of possibilistic clustering. Hasuike, Katagiri and Tsubaki [5] suggested that an appropriate membership function algorithm integrate the fuzzy Shannon entropy with a piecewise linear function into subjective intervals estimation by heuristic method. Jain and Khare [8] presented a mechanism for generating membership functions that exploits the properties of Bezier curves. For the construction of membership functions by numerical data set, Nasibov and Ulutagay in [10] used Gaussian function. Later Nasibov and Peker in [11] suggest using another exponential function as a better option for solving the same task and prove its advantage. The idea of this study was born by [11] and consists in constructing a membership function through approximation of a frequency distribution by a Bezier curve. The rest of the paper is organized as follows: Section 2 consists of two parts. In 2.1, basic concepts of fuzzy sets theory are outlined briefly. In 2.2, the approach and results of previous research related to the present study are described. In Section 3, formulas for determining the coordinates of the Bezier curve control points are derived. Section 4 presents an algorithm for building a membership function via the Bezier curve base on the frequency distribution points and specifies an analytical expression of the proposed The Bezier curve as a fuzzy membership function shape 247 membership function. The frequency distributions published in [11] are used to illustrate the proposed method. In Section 5, the described function is compared to the exponential function of Nasibov and Peker [11]. Some summaries have been made and guidelines for future research have been identified. 2 2.1 A brief preliminary Some basic concepts of fuzzy sets theory The concepts and principles of fuzzy sets theory can be found in [1, 7, 12, 18]. Definitions of the some basic notions used in the following presentation will be briefly listed here. Let X be a collection of objects and x ∈ X. A fuzzy set A in X is the set of ordered pairs A = {(x, µA (x)) | x ∈ X}, where µA (x) : X → T ⊆ [0, 1] is called a membership function for the fuzzy set A. If sup µA (x) = 1, then A is called normal fuzzy set. If sup µA (x) < 1, then A is subnormal. The support of A is the subset of points of X at which µA (x) is positive, i.e. support(A) = {x ∈ X | µA (x) > 0}. For any α > 0, α ∈ T ⊆ [0, 1], an α-cut or α-level of the fuzzy set A in X is the set Aα = {x | x ∈ X, µA (x) ≥ α}. The fuzzy set A is convex if and only if for any x1 , x2 ∈ X and any λ ∈ [0, 1] is fulfilled: µA (λx1 + (1 − λ)x2 ) ≥ min{µA (x1 ), µA (x2 )}. (1) A fuzzy number A is a fuzzy set in the real line ℜ with the membership function µA (x) : ℜ → [0, 1] that satisfies the conditions for normality, convexity and piecewise continuity and support(A) is bounded. Remark 1 For a fuzzy number, the convexity defined by (1) means that the membership function is monotonic or that it is first monotonically increasing and then monotonically decreasing. 2.2 Publications related to this research The choice of membership function type is determined by the ability of the shape of its graph to approximate with sufficient accuracy the shape of the frequency distribution of x1 , x2 , . . . , xN data. 248 T. Yankova, G. Ilieva, S. Klisarova Nasibov and Ulutagay [10] recorded bispectral index data during sleep and analyzed it by using the Fuzzy c-Means and Fuzzy Neighborhood DBSCAN algorithms. As a result of these computational experiments, Nasibov and Ulutagay concluded that FN-DBSCAN method gives more realistic results in recognizing stable duration intervals and bispectral index stages in the measurement series. Remark 2 Bispectral index scale is a continuous processed electroencephalogram parameter that correlates to the level of brain activity. The numerical value of bispectral index varies from 0 (no cerebral activity) to 100 (fully awake patient) [10]. Sedation is the depression of the human’s awareness to environment and the reduction of responsiveness to external stimulation. Data from the formed sedation stages in [10] and used in [11] contain the frequencies of the class intervals of a bispectral index. With some changes to the parameter markings, the overall data type is presented in Table 1. The total number of classes in Table 1 is l. The midpoints mi , i = 1, l of the class intervals and the frequencies fi , i = 1, l are filled in the table. The l P fi fi . The class interval relative frequencies are pi = N , i = 1, l, where N = i=1 with a maximum frequency: pM = max{pi } i=1,l and its midpoint m = mM are determined. The normalized frequencies are pei = ppMi , i = 1, l, such as peM = 1. Class interval Midpoint mi Frequency fi Relative frequency pi [x0 , x1 ) [x1 , x2 ) ... [xl−1 , xl ) 1 m1 = x0 +x 2 x1 +x2 m2 = 2 ... x +x ml = l−12 l f1 f2 ... fl p1 = fN1 p2 = fN2 ... fl pl = N N 1 Total Table 1: Frequency table Normalized frequency pei pe1 = ppM1 pe2 = ppM2 ... pel = ppMl The Bezier curve as a fuzzy membership function shape 249 Tables 4-8 of Appendix section are filled in with numerical values and have the same appearance as Table 1. Each one corresponds to one of the five stages of sedation. For the bispectral index values at each stage, Nasibov and Ulutagay [10] construct a Gaussian fuzzy membership function of the type: 1 x−m 2 ) σ µA (x) = e− 2 ( (2) where m (in the article the original symbol is α) is the mean value of the data, and σ is their standard deviation. Later, Nasibov and Peker [11], solving a classification problem, suggested instead of (2) to use the following exponential membership function: ( x−m S e−( σ ) L , x ≤ m (3) µA (x) = )S R −( x−m β ,x > m e with unknown parameters sL , σ, sR , β. They output the formulas for the unknown parameters by least squares minimization. Then they verified the efficiency of the proposed exponential membership function with respect to the bispectral index data. Based on 21 sets of bispectral data, each of which containing 306 measurements, Nasibov and Peker [11] determined the classification accuracies based on exponential membership functions (3) and Gaussian membership functions (2). They calculated the classification accuracy as the ratio of the number of correctly detected points in the data set to the total number of points in the set. By the paired t-test (α = 0,10), they come to the conclusion that the mean of classification accuracy, based on exponential membership functions (3) is greater than the one based on Gaussian membership functions (2). 3 Approximation of a series of points by a Bezier curve The Bezier curve is a parametric curve that is determined by a set of control points C0 , C1 , . . . , Ck . The number of control points determines the order of the curve as at (k + 1) control points the curve is of order k. The curve is defined: B(t) = C0 (1 − t)2 + C1 2(1 − t)t + C2 t2 , at k = 2 (4) B(t) = C0 (1 − t)3 + C1 3(1 − t)2 t + C2 3(1 − t)t2 + C3 t3 , at k = 3 (5) 250 T. Yankova, G. Ilieva, S. Klisarova B(t) = C0 (1 − t)4 + C1 4 (1 − t)3 t + C2 6 (1 − t)2 t2 + + C3 (1 − t)t3 + C4 t4 , (6) at k = 4, where t ∈ [0, 1]. The beginning and end of the curve are coincident with the first and last control point, respectively, i.e. B(0) = C0 and B(1) = Ck . Let in the plane be given (n + 1) consecutive points Pi (xi , yi ), i = 0, n (Figure 1). Let us denote with ri the length of the line between two adjacent points Pi−1 and P i, i.e.: p ri = |Pi−1 Pi | = (xi − xi−1 )2 + (yi − yi−1 )2 , i = 1, n (7) and assume r0 = 0. Figure 1: Series of (n + 1) points in the plane and the marks associated with them The total length of the broken line that is obtained from all segments is i n P P ri . The length of the broken line between P0 and Pi is di = dn = rj . j=0 i=0 At each point Pi (xi , yi ), i = 0, n we match quantity: ti = di ∈ [0, 1], i = 0, n dn (8) If we treat ti as values of the parameter t ∈ [0, 1], then t = 0 corresponds to the first point P0 of the series of points and t = 1 corresponds to the last point Pn . For n ≥ 4 we will produce the formulas for determining the coordinates of the internal control points C1 (C1x , C1y ), C2 (C2x , C2y ), C3 (C3x , C3y ) of the fourth-order Bezier curve, and set requirement C0 (C0x , C0y ) ≡ P0 (x0 , y0 ) The Bezier curve as a fuzzy membership function shape 251 and C4 (C4x , C4y ) ≡ Pn (xn , yn ). For this purpose, we will minimize the squares of the deviations of the curve from the abscisses and the ordinates of the points P1 , P2 , . . . , Pn−1 : n−1 X E(Cx ) = (xi − Bx (ti ))2 → min (9) (yi − By (ti ))2 → min (10) i=1 n−1 X E(Cy ) = i=1 where Bx (ti ) and By (ti ) are obtained from (6) using respectively the abscisses C0x , C1x , C2x , C3x , C4x and the ordinates C0y , C1y , C2y , C3y , C4y of the control points. Theorem 1 For each set of points Pi (xi , yi ), (n ≥ 4) in the plane, for C0 (C0x , C0y ) ≡ P0 (x0 , y0 ) and C4 (C4x , C4y ) ≡ Pn (xn , yn ), there exist singular internal control points C1 (C1x , C1y ), C2 (C2x , C2y ), C3 (C3x , C3y ) of the fourth-order Bezier curve for which the sums E(Cx ) and E(Cy ) defined by (9) and (10) assume their minimum values. Proof: Let’s first look at only the abscisses of the points. From (9) and (6) there is obtained: n−1 X ∂Bx (ti ) ∂E(Cx ) =2 =0 (xi − Bx (ti )) ∂Cx ∂Cx i=1 Let we denote: Sp,q = n−1 X (1 − ti )p tqi ; (11) i=1 SXp,q = n−1 X xi (1 − ti )p tqi ; (12) yi (1 − ti )p tqi ; (13) i=1 SYp,q = n−1 X i=1 S(4)  4S6,2 6S5,3 4S4,4 =  4S5,3 6S4,4 4S3,5 . 4S4,4 6S3,5 4S2,6  252 T. Yankova, G. Ilieva, S. Klisarova Then the resulting system:   C1x 4S6,2 + C2x 6S5,3 + C3x 4S4,4 = SX3,1 − C0x S7,1 − C4x S3,5 C1x 4S5,3 + C2x 6S4,4 + C3x 4S3,5 = SX2,2 − C0x S6,2 − C4x S2,6  C1x 4S4,4 + C2x 6S3,5 + C3x 4S2,6 = SX1,3 − C0x S5,3 − C4x S1,7 has the solution:     SX3,1 − C0x S7,1 − C4x S3,5 C1x  C2x  = S −1  SX2,2 − C0x S6,2 − C4x S2,6  . (4) SX1,3 − C0x S5,3 − C4x S1,7 C3x (14) Similarly, from (10) and (6) for the ordinates of the three control points we obtain:     SY3,1 − C0y S7,1 − C4y S3,5 C1y  C2y  = S −1  SY2,2 − C0y S6,2 − C4y S2,6  . (15) (4) SY1,3 − C0y S5,3 − C4y S1,7 C3y The matrix S(4) is positive definite and therefore its determinant is positive [13]. This ensures the existence and the uniqueness of the solution. Theorem 2 For each set of points Pi (xi , yi ), (n ≥ 3) in the plane, for C0 (C0x , C0y ) ≡ P0 (x0 , y0 ) and C3 (C3x , C3y ) ≡ Pn (xn , yn ), there exist singular internal control points C1 (C1x , C1y ) and C2 (C2x , C2y ) of the third-order Bezier curve for which the sums E(Cx ) and E(Cy ) defined by (9) and (10) assume their minimum values. Proof: For the coordinates of the internal control points C1 (C1x , C1y ) and C2 (C2x , C2y ) of the third-order Bezier curve, similar to the proof of Theorem 1, we obtain:  C1x C2x  = −1 S(3)  SX2,1 − C0x S5,1 − C3x S2,4 SX1,2 − C0x S4,2 − C3x S1,5  (16)  C1y C2y  −1 = S(3)  SY2,1 − C0y S5,1 − C3y S2,4 SY1,2 − C0y S4,2 − C3y S1,5  (17)  ,  3S4,2 3S3,3 where S(3) = is a positive definite matrix. Therefore, its 3S3,3 3S2,4 determinant is positive, which guarantees the existence of the solution. The Bezier curve as a fuzzy membership function shape 253 Theorem 3 For each set of points Pi (xi , yi ), (n ≥ 2) in the plane, for C0 (C0x , C0y ) ≡ P0 (x0 , y0 ) and C2 (C2x , C2y ) ≡ Pn (xn , yn ), there exist singular internal control point C1 (C1x , C1y ) of the second-order Bezier curve for which the sums E(Cx ) and E(Cy ) defined by (9) and (10) assume their minimum values. Proof: For the coordinates of the internal control point C1 (C1x , C1y ) of the second-order Bezier curve, analogously to the previous theorems, we obtain: C1x = SX1,1 − C0x S3,1 − C2x S1,3 2S2,2 (18) C1y = SY1,1 − C0y S3,1 − C2y S1,3 2S2,2 (19) and with that S2,2 6= 0 according to (8) and (11). Remark 3 Generally, for formulas (14)-(19) it is not required that the sequence of abscisses or ordinates of the points Pi (xi , yi ), i = 0, n to be monotone. The plane curve B(t) is set parametrically and the correspondence between the abscisses and the ordinates of its points is implicit. Therefore, we will take notice of defining the ordinate of a point of the curve by a given abscissa. Task 1. Let the curve B(t) be determined by the set of points Pi (xi , yi ), i = 0, n, for which x0 < x1 < . . . < xn and Bx (t) is a strictly monotonic function of t ∈ [0, 1]. Determine the ordinate yB at a point of the curve by the given abscissa xB ∈ [x0 , xn ]. Solution: We first localize the numerical interval that contains xB . Suppose that xB ∈ [xi−1 , xi ]. By linear interpolation according to the values ti−1 and ti defined by (8), for tB (Figure 2) there is obtained: tB = ti−1 + x − xi−1 (ti − ti−1 ), tB ∈ [0, 1], xi − xi−1 (20) where we calculate yB = By (tB ). Determining of a α-cut of the membership function proposed in the next section is the reverse of Task 1: Task 2. Let the curve B(t) be determined by set points Pi (xi , yi ), i = 0, n for which x0 < x1 < . . . < xn ; By (t) ∈ [0, 1] and Bx (t), By (t) are strictly monotonic functions of t ∈ [0, 1]. Determine the abscissa xB ∈ [x0 , xn ] at a 254 T. Yankova, G. Ilieva, S. Klisarova Figure 2: Linear interpolation to determine the value of tB point of the curve by the specified ordinate yB ∈ [0, 1]. Solution: Similarly to the solution of the previous task, we define: tB = ti−1 + y − yi−1 (ti − ti−1 ), tB ∈ [0, 1], yi − yi−1 (21) where: • yB ∈ [yi−1 , yi ] if By (t) is a monotonic increasing function of t; • yB ∈ [yi , yi−1 ] if By (t) is a monotonic decreasing function of t and xB = Bx (tB ). 4 Constructing a membership function by approximating a frequency distribution with a Bezier curve We will use the frequency distributions for the five stages of sedation, which are published in [11] and are listed in Tables 4-8 of Appendix section. From the corresponding table for each stage, a series of points are formed which have evenly distributed abscissas. The points coordinates Pi (mi , pei ), i = 1, l are input data for the following algorithm: 1. Points P0 (left) and Pl+1 (right) with zero ordinates added to the left and right of the series of points, so that the even distribution of the points abscissas is preserved (Figure 3). 2. We determine the midpoint (mM , peM ) of the class interval with a maximum frequency, i.e. peM = 1. Let us denote m = mM . 3. The series of points Pi (mi , pei ), i = 0, l + 1 is divided into two groups: The Bezier curve as a fuzzy membership function shape 255 Figure 3: A histogram of the frequency distribution and used parameter markers • left Pi (mi , pei ), i = 0, M , containing (M + 1) points, i.e. n = M ; • right Pi (mi , pei ), i = M, l + 1, containing (l − M + 2) points, i.e. n = l − M + 1. 4. Using the formulas given in Section 3 for the left and right group of points, we successively define left B L (t) and right B R (t) Bezier curve under the following conditions: (a) The number (k + 1) of control points of B(t) is determined by:  n, n = 2; 3 k= . 4, n ≥ 4 (b) By formula (7) we define the lengths ri , i = 1, n of the segments between each two adjacent points. We set zi = 1, i = 1, n. (c) ri = zi ri , i = 1, n. (d) We calculate di and ti , i = 0, n. Then we find the coordinates of the internal control points (Theorems 1-3) of curves B(t). (e) We check the monotony of Bx (t) and By (t) and the non-negativity of By (t): • If Bx (t) and By (t) are monotonic and By (t) ≥ 0, t ∈ [0, 1], we move to step 5 of the algorithm; 256 T. Yankova, G. Ilieva, S. Klisarova • For each interval [ti−1 , ti ] in which Bx (t) or By (t) is not a monotonic function or By (t) < 0, the value of zi is reduced by half:  zi , if Bx (t) and By (t) ≥ 0 are monotonic in [ti−1 , ti ] zi = 1 2 zi , otherwise. We return to step 4 (c) of the algorithm. 5. We construct the membership function from the two parametrically defined curves:   x = BxL (t)   , t ∈ [0, 1], where x ≤ m  L  y = ByR(t) (22) µA (x(t), y(t)) = x = Bx (t)    , t ∈ [0, 1], where x > m y = ByR (t) and build it. Following the algorithm, for each of the five sedation stages the coordinates of the control points of Bezier’s curves are calculated (Table 9 in the Appendix section) and the membership functions are built (Figures 4-8, in black). The convexity of the constructed fuzzy numbers is guaranteed in step 4 (e) of the algorithm by the requirement for monotonicity of the obtained Bezier curves. Adding points P0 and Pl+1 with zero ordinates in step 1 of the algorithm provides support(A) to be bounded. 5 Comparing the results and conclusion Nasibov and Peker [11] compared the classification accuracy based on exponential membership functions (3) and Gaussian membership functions (2). Figure 4: A histogram and membership functions of first sedation stage The Bezier curve as a fuzzy membership function shape 257 They conclude that the average degree of classification accuracy of (3) is higher than that of (2). Since we do not have the necessary data to compare the classification accuracy of the proposed functions (22) and the functions (3), we will only compare them by the sum of the squares of the deviations from the points. The curves that we will compare are shown on Figures 4-8, where: • The functions in black correspond to (22) and are built according to the algorithm in Section 4. The control points are listed in Table 9 in the Appendix section. • The functions in red (or gray in the gray scale) correspond to (3) and are constructed according to the analytical type and the parameter values specified in [11]. The parameter values are listed in Table 10 in the Appendix section. Figure 5: A histogram and membership functions of second sedation stage Figure 6: A histogram and membership functions of third sedation stage 258 T. Yankova, G. Ilieva, S. Klisarova Figure 7: A histogram and membership functions of fourth sedation stage Figure 8: A histogram and membership functions of fifth sedation stage Functions (22) are defined parametrically, but functions (3) are clearly defined. This determines the difference in the measurement of the deviations of the curves from the frequency distribution points. The deviations of curves (22) must be reported on both coordinate axes, while the deviations of the curves (3) are defined as deviations only on the ordinate axis. The formulas for sum of squares deviations on which the values of Table 2 are calculated are: • for (22): E(22) (e pi , µ A ) = • for (3): E(3) (e pi , µA ) = l+1 P [(mi − Bx (ti ))2 + (e pi − By (ti ))2 ]; i=0 l+1 P (e pi − µA (mi ))2 . i=0 The result of comparing the mean values of E(e pi , µA ) for both methods with a level of significance α = 0, 05 are shown in Table 3. The tStat value falls into the critical area (−2, 82849 < −2, 13185) of the null hypotheses H0 : E(22) ≥ E(3) against its alternative H1 : E(22) < E(3) , which means that the E(22) values are significantly smaller than those of E(3) . The Bezier curve as a fuzzy membership function shape Stages of sedation 1 2 3 4 5 E(22) (e pi , µ A ) 0,0013931 0,0063041 0,0163010 0,0102372 0,0024799 259 E(3) (e pi , µA ) 0,0121480 0,0903728 0,1154965 0,0880298 0,2297327 Table 2: Sum of squares deviations Mean Variance Observations df t Stat P(T ≤ t) one-tail t Critical one-tail P(T ≤ t) two-tail t Critical two-tail E(22) 0,00734306 3,71881E-05 5 4 -2,828491142 0,023708788 2,131846782 0,047417576 2,776445105 E(3) 0,10715596 0,006192155 5 Table 3: t-Test: Paired Two Sample for Means By the t-criterion we have established that the membership functions (22) are closer to the frequency distribution points compared to the membership functions (3). This gives us a reason to consider (22) a good alternative to (3) and with that each of these two methods has advantages and disadvantages. In some cases, a disadvantage of the proposed membership function (22) may be its ”sensitivity” to fluctuations in frequency distribution. The key to overcoming this ”sensitivity” is step 4 (e) of the algorithm. Refining the values that are assigned to zi would improve the algorithm and may be subject to future research. Studies based on the values of the bispectral index are used in various sociological and marketing studies [4, 14]. The derived formulas for defining Bezier curves, which best describe a series of points, could be used as an analogue of exponential smoothing, and the proposed algorithm for designing membership functions can be used to solve a wide range of problems, including financial, marketing, macro- and microeconomic problems [6, 2, 8, 9], etc. 260 T. Yankova, G. Ilieva, S. Klisarova References [1] L. C. de Barros, R. C. 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Klisarova Appendix Class interval [0, 15; 0, 2) [0, 2; 0, 25) [0, 25; 0, 3) [0, 3; 0, 35) [0, 35; 0, 4) [0, 4; 0, 45) [0, 45; 0, 5) [0, 5; 0, 5)5 [0, 55; 0, 6) [0, 6; 0, 65) Midpoint mi 0,175 0,225 0,275 0,325 0,375 0,425 0,475 0,525 0,575 0,625 Frequency fi 8 65 322 266 130 28 7 3 3 1 Relative frequency pi 0,00960384 0,07803121 0,38655462 0,31932773 0,15606242 0,03361345 0,00840336 0,00360144 0,00360144 0,00120048 Normalized frequency pei 0,02484472 0,201863354 1 0,826086957 0,403726708 0,086956522 0,02173913 0,00931677 0,00931677 0,00310559 Table 4: Frequency table of first sedation stage Class interval [0, 2; 0, 25) [0, 25; 0, 3) [0, 3; 0, 35) [0, 35; 0, 4) [0, 4; 0, 45) [0, 45; 0, 5) [0, 5; 0, 55) [0, 55; 0, 6) [0, 6; 0, 65) [0, 65; 0, 7) [0, 7; 0, 75) [0, 75; 0, 8) [0, 8; 0, 85) [0, 85; 0, 9) [0, 9; 0, 95) Midpoint mi 0,225 0,275 0,325 0,375 0,425 0,475 0,525 0,575 0,625 0,675 0,725 0,775 0,825 0,875 0,925 Frequency fi 4 33 124 280 355 216 199 143 157 117 67 38 6 1 2 Relative frequency pi 0,00229621 0,01894374 0,07118255 0,16073479 0,20378875 0,12399541 0,11423651 0,08208955 0,09012629 0,06716418 0,03846154 0,02181401 0,00344432 0,00057405 0,00114811 Normalized frequency pei 0,011267606 0,092957746 0,349295775 0,788732394 1 0,608450704 0,56056338 0,402816901 0,442253521 0,329577465 0,188732394 0,107042254 0,016901408 0,002816901 0,005633803 Table 5: Frequency table of second sedation stage 263 The Bezier curve as a fuzzy membership function shape Class interval [0, 25; 0, 3) [0, 3; 0, 35) [0, 35; 0, 4) [0, 4; 0, 45) [0, 45; 0, 5) [0, 5; 0, 55) [0, 55; 0, 6) [0, 6; 0, 65) [0, 65; 0, 7) [0, 7; 0, 75) [0, 75; 0, 8) [0, 8; 0, 85) [0, 85; 0, 9) [0, 9; 0, 95) [0, 95; 1) Midpoint mi 0,275 0,325 0,375 0,425 0,475 0,525 0,575 0,625 0,675 0,725 0,775 0,825 0,875 0,925 0,975 Frequency fi 1 3 25 119 112 199 189 344 304 347 379 499 129 30 14 Relative frequency pi 0,0003712 0,00111359 0,00927988 0,04417223 0,04157387 0,07386785 0,0701559 0,12769117 0,11284336 0,12880475 0,140683 0,18522643 0,04788419 0,01113586 0,00519673 Normalized frequency pei 0,002004008 0,006012024 0,0501002 0,238476954 0,224448898 0,398797595 0,378757515 0,689378758 0,609218437 0,695390782 0,759519038 1 0,258517034 0,06012024 0,028056112 Table 6: Frequency table of third sedation stage Class interval [0, 4; 0, 45) [0, 45; 0, 5) [0, 5; 0, 55) [0, 55; 0, 6) [0, 6; 0, 65) [0, 65; 0, 7) [0, 7; 0, 75) [0, 75; 0, 8) [0, 8; 0, 85) [0, 85; 0, 9) [0, 9; 0, 95) [0, 95; 1) Midpoint mi 0,425 0,475 0,525 0,575 0,625 0,675 0,725 0,775 0,825 0,875 0,925 0,975 Frequency fi 1 4 8 8 10 23 65 51 219 215 92 41 Relative frequency pi 0,00135685 0,00542741 0,01085482 0,01085482 0,01356852 0,0312076 0,08819539 0,06919946 0,29715061 0,2917232 0,12483039 0,05563094 Normalized frequency pei 0,00456621 0,01826484 0,03652968 0,03652968 0,0456621 0,105022831 0,296803653 0,232876712 1 0,98173516 0,420091324 0,187214612 Table 7: Frequency table of fourth sedation stage 264 T. Yankova, G. Ilieva, S. Klisarova Class interval [0, 35; 0, 4) [0, 4; 0, 45) [0, 45; 0, 5) [0, 5; 0, 55) [0, 55; 0, 6) [0, 6; 0, 65) [0, 65; 0, 7) [0, 7; 0, 75) [0, 75; 0, 8) [0, 8; 0, 85) [0, 85; 0, 9) [0, 9; 0, 95) [0, 95; 1) Midpoint mi 0,375 0,425 0,475 0,525 0,575 0,625 0,675 0,725 0,775 0,825 0,875 0,925 0,975 Frequency fi 3 4 3 11 2 4 2 4 7 46 260 316 64 Relative frequency pi 0,00413223 0,00550964 0,00413223 0,01515152 0,00275482 0,00550964 0,00275482 0,00550964 0,00964187 0,06336088 0,35812672 0,43526171 0,08815427 Normalized frequency pei 0,009493671 0,012658228 0,009493671 0,034810127 0,006329114 0,012658228 0,006329114 0,012658228 0,022151899 0,14556962 0,82278481 1 0,202531646 Table 8: Frequency table of fifth sedation stage Stages 1 2 3 4 5 Left σ SL 0,03366153 1,19058453 0,098304385 2,12921635 0,23903161 1,16559615 0,04244326 0,67457267 0,08407821 1,03766617 Right β SR 0,10954908 1,75612535 0,140004284 0,97277506 0,03494202 0,91429043 0,12082079 4,33694334 0,075 0,55743699 Table 10: Parameters of the membership functions (3) used as a benchmark The Bezier curve as a fuzzy membership function shape Stages 1 2 3 4 5 Ci C0 C1 C2 C3 C4 C0 C1 C2 C3 C4 C0 C1 C2 C3 C4 C0 C1 C2 C3 C4 C0 C1 C2 C3 C4 Left x y 0,125 0 0,247991904 -0,021264839 0,242565841 0,430096798 0,275 1 0,175 0,446856004 0,228966718 0,378416281 0,425 0,225 0,624874518 0,4019694 0,784104357 0,825 0,375 0,543533011 0,709677477 0,797493323 0,825 0,325 0,548083099 1,080709167 0,768476768 0,925 0 0,081682191 0,606975012 0,700014982 1 0 -0,002210513 0,85309558 0,444195245 1 0 -0,006419047 0,133690305 0,004591904 1 0 0,217777331 -0,688340048 1,035569291 1 Right x y 0,275 1 0,385177558 0,698654314 0,401145303 0,52556588 0,243719602 -0,07293049 0,675 0 0,425 1 0,432350881 0,162518 0,782069858 0,768679 0,585874127 -0,02454 0,975 0 0,825 1 0,919899079 0,681197 0,793213422 -0,11791 0,945898462 0,087821 1,025 0 0,825 1 1,000155204 1,038217807 0,816868712 0,414115227 0,966205507 0,293372071 1,025 0 0,925 1 0,940176064 0,188264274 1,025 0 Table 9: Coordinates of control points 265