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Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion On using Different Distance Measures for Fuzzy Numbers in Fuzzy Linear Regression Models Duygu cen 1 Marco

  1. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion In the Monte Carlo method, several random crisp or fuzzy vec- tors are generated as regression coefficient vector. Then using these random vectors, the dependent variable is calculated. Two error measures are obtained by the difference of observed and estimated values of dependent variable to decide the best random vector for parameter estimation. One of these error measures depends on the error measure de- fined by Kim and Bishu (1998). In this error measure, distance of two fuzzy numbers has to be calculated. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 5 / 39

  2. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion In the Monte Carlo method, several random crisp or fuzzy vec- tors are generated as regression coefficient vector. Then using these random vectors, the dependent variable is calculated. Two error measures are obtained by the difference of observed and estimated values of dependent variable to decide the best random vector for parameter estimation. One of these error measures depends on the error measure de- fined by Kim and Bishu (1998). In this error measure, distance of two fuzzy numbers has to be calculated. Therefore, distance measure between two fuzzy numbers plays an important role in fuzzy regression with Monte Carlo method. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 5 / 39

  3. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion In the Monte Carlo method, several random crisp or fuzzy vec- tors are generated as regression coefficient vector. Then using these random vectors, the dependent variable is calculated. Two error measures are obtained by the difference of observed and estimated values of dependent variable to decide the best random vector for parameter estimation. One of these error measures depends on the error measure de- fined by Kim and Bishu (1998). In this error measure, distance of two fuzzy numbers has to be calculated. Therefore, distance measure between two fuzzy numbers plays an important role in fuzzy regression with Monte Carlo method. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 5 / 39

  4. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Aim of the study 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 6 / 39

  5. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Aim of the study Highlight the utility of distance measures 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 6 / 39

  6. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Aim of the study Highlight the utility of distance measures Calculate different distance measures in fuzzy linear regression with Monte Carlo method. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 6 / 39

  7. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Aim of the study Highlight the utility of distance measures Calculate different distance measures in fuzzy linear regression with Monte Carlo method. Estimate the parameters of fuzzy linear regression with Monte Carlo method according to the different distance measures 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 6 / 39

  8. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Aim of the study Highlight the utility of distance measures Calculate different distance measures in fuzzy linear regression with Monte Carlo method. Estimate the parameters of fuzzy linear regression with Monte Carlo method according to the different distance measures 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 6 / 39

  9. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Outline 1 Introduction 2 Preliminaries 3 Fuzzy Regression with Monte Carlo Method 4 Distance Measure for Fuzzy Numbers 5 Application Application for Second Category Application for Third Category Solutions 6 Conclusion 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 7 / 39

  10. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 8 / 39

  11. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.1. µ A ( x ) is the membership function of an element x belonging to a fuzzy set ˜ A , where 0 ≤ µ A ( x ) ≤ 1. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 8 / 39

  12. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.1. µ A ( x ) is the membership function of an element x belonging to a fuzzy set ˜ A , where 0 ≤ µ A ( x ) ≤ 1. Definition 2.2. A general fuzzy number ˜ A is a normal convex fuzzy set of ℜ with a piecewise continuous membership func- tion. The left and right sides of fuzzy numbers are L ( x ) = a 2 − x a 2 − a 1 and R ( x ) = x − a 3 a 4 − a 3 respectively. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 8 / 39

  13. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.1. µ A ( x ) is the membership function of an element x belonging to a fuzzy set ˜ A , where 0 ≤ µ A ( x ) ≤ 1. Definition 2.2. A general fuzzy number ˜ A is a normal convex fuzzy set of ℜ with a piecewise continuous membership func- tion. The left and right sides of fuzzy numbers are L ( x ) = a 2 − x a 2 − a 1 and R ( x ) = x − a 3 a 4 − a 3 respectively. Definition 2.3. The α -cut of a fuzzy number ˜ A is a non-fuzzy set defined as ˜ A ( α ) = { x ∈ ℜ , µ A ( α ) ≥ α } . 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 8 / 39

  14. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.1. µ A ( x ) is the membership function of an element x belonging to a fuzzy set ˜ A , where 0 ≤ µ A ( x ) ≤ 1. Definition 2.2. A general fuzzy number ˜ A is a normal convex fuzzy set of ℜ with a piecewise continuous membership func- tion. The left and right sides of fuzzy numbers are L ( x ) = a 2 − x a 2 − a 1 and R ( x ) = x − a 3 a 4 − a 3 respectively. Definition 2.3. The α -cut of a fuzzy number ˜ A is a non-fuzzy set defined as ˜ A ( α ) = { x ∈ ℜ , µ A ( α ) ≥ α } . { ˜ A ( α ) = [ A L ( α ) , A U ( α )] } 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 8 / 39

  15. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.1. µ A ( x ) is the membership function of an element x belonging to a fuzzy set ˜ A , where 0 ≤ µ A ( x ) ≤ 1. Definition 2.2. A general fuzzy number ˜ A is a normal convex fuzzy set of ℜ with a piecewise continuous membership func- tion. The left and right sides of fuzzy numbers are L ( x ) = a 2 − x a 2 − a 1 and R ( x ) = x − a 3 a 4 − a 3 respectively. Definition 2.3. The α -cut of a fuzzy number ˜ A is a non-fuzzy set defined as ˜ A ( α ) = { x ∈ ℜ , µ A ( α ) ≥ α } . { ˜ A ( α ) = [ A L ( α ) , A U ( α )] } 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 8 / 39

  16. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.4. v k = ( v 0 k , ..., v mk ) is called random crisp vector. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 9 / 39

  17. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.4. v k = ( v 0 k , ..., v mk ) is called random crisp vector. v ik are all real numbers in intervals I i , i = 0 , 1 , ..., m . 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 9 / 39

  18. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.4. v k = ( v 0 k , ..., v mk ) is called random crisp vector. v ik are all real numbers in intervals I i , i = 0 , 1 , ..., m . Firstly, random crisp vectors v k = ( x ok , ..., x mk ) with all x ik ∈ [0 , 1] are generated. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 9 / 39

  19. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.4. v k = ( v 0 k , ..., v mk ) is called random crisp vector. v ik are all real numbers in intervals I i , i = 0 , 1 , ..., m . Firstly, random crisp vectors v k = ( x ok , ..., x mk ) with all x ik ∈ [0 , 1] are generated. Then all x ik are put in the interval I i = [ c i , d i ] by v ik = c i + ( d i − c i ) x ik , i = 0 , 1 , ..., m . 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 9 / 39

  20. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.4. v k = ( v 0 k , ..., v mk ) is called random crisp vector. v ik are all real numbers in intervals I i , i = 0 , 1 , ..., m . Firstly, random crisp vectors v k = ( x ok , ..., x mk ) with all x ik ∈ [0 , 1] are generated. Then all x ik are put in the interval I i = [ c i , d i ] by v ik = c i + ( d i − c i ) x ik , i = 0 , 1 , ..., m . Definition 2.5. � V k = ( � V 0 k , ..., � V mk ) is called random fuzzy vector 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 9 / 39

  21. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.4. v k = ( v 0 k , ..., v mk ) is called random crisp vector. v ik are all real numbers in intervals I i , i = 0 , 1 , ..., m . Firstly, random crisp vectors v k = ( x ok , ..., x mk ) with all x ik ∈ [0 , 1] are generated. Then all x ik are put in the interval I i = [ c i , d i ] by v ik = c i + ( d i − c i ) x ik , i = 0 , 1 , ..., m . Definition 2.5. � V k = ( � V 0 k , ..., � V mk ) is called random fuzzy vector � V ik are all triangular fuzzy numbers. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 9 / 39

  22. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.4. v k = ( v 0 k , ..., v mk ) is called random crisp vector. v ik are all real numbers in intervals I i , i = 0 , 1 , ..., m . Firstly, random crisp vectors v k = ( x ok , ..., x mk ) with all x ik ∈ [0 , 1] are generated. Then all x ik are put in the interval I i = [ c i , d i ] by v ik = c i + ( d i − c i ) x ik , i = 0 , 1 , ..., m . Definition 2.5. � V k = ( � V 0 k , ..., � V mk ) is called random fuzzy vector � V ik are all triangular fuzzy numbers. First crisp vectors v k = ( v 1 k , . . . , v (3 m +3 , k ) ) with all the x ik in [0 , 1], k = 1 , ..., N are generated. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 9 / 39

  23. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.4. v k = ( v 0 k , ..., v mk ) is called random crisp vector. v ik are all real numbers in intervals I i , i = 0 , 1 , ..., m . Firstly, random crisp vectors v k = ( x ok , ..., x mk ) with all x ik ∈ [0 , 1] are generated. Then all x ik are put in the interval I i = [ c i , d i ] by v ik = c i + ( d i − c i ) x ik , i = 0 , 1 , ..., m . Definition 2.5. � V k = ( � V 0 k , ..., � V mk ) is called random fuzzy vector � V ik are all triangular fuzzy numbers. First crisp vectors v k = ( v 1 k , . . . , v (3 m +3 , k ) ) with all the x ik in [0 , 1], k = 1 , ..., N are generated. Then the first three numbers in v k are chosen and ordered from smallest to largest. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 9 / 39

  24. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.4. v k = ( v 0 k , ..., v mk ) is called random crisp vector. v ik are all real numbers in intervals I i , i = 0 , 1 , ..., m . Firstly, random crisp vectors v k = ( x ok , ..., x mk ) with all x ik ∈ [0 , 1] are generated. Then all x ik are put in the interval I i = [ c i , d i ] by v ik = c i + ( d i − c i ) x ik , i = 0 , 1 , ..., m . Definition 2.5. � V k = ( � V 0 k , ..., � V mk ) is called random fuzzy vector � V ik are all triangular fuzzy numbers. First crisp vectors v k = ( v 1 k , . . . , v (3 m +3 , k ) ) with all the x ik in [0 , 1], k = 1 , ..., N are generated. Then the first three numbers in v k are chosen and ordered from smallest to largest. Let us assume that x 3 k < x 1 k < x 2 k , 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 9 / 39

  25. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Definition 2.4. v k = ( v 0 k , ..., v mk ) is called random crisp vector. v ik are all real numbers in intervals I i , i = 0 , 1 , ..., m . Firstly, random crisp vectors v k = ( x ok , ..., x mk ) with all x ik ∈ [0 , 1] are generated. Then all x ik are put in the interval I i = [ c i , d i ] by v ik = c i + ( d i − c i ) x ik , i = 0 , 1 , ..., m . Definition 2.5. � V k = ( � V 0 k , ..., � V mk ) is called random fuzzy vector � V ik are all triangular fuzzy numbers. First crisp vectors v k = ( v 1 k , . . . , v (3 m +3 , k ) ) with all the x ik in [0 , 1], k = 1 , ..., N are generated. Then the first three numbers in v k are chosen and ordered from smallest to largest. Let us assume that x 3 k < x 1 k < x 2 k , then the first triangular fuzzy numbers is � V 0 k = ( x 3 k / x 1 k / x 2 k ). 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 9 / 39

  26. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Outline 1 Introduction 2 Preliminaries 3 Fuzzy Regression with Monte Carlo Method 4 Distance Measure for Fuzzy Numbers 5 Application Application for Second Category Application for Third Category Solutions 6 Conclusion 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 10 / 39

  27. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Choi and Buckley (2008) classified fuzzy regression models in three categories: Input and output data are both crisp (First Category) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 11 / 39

  28. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Choi and Buckley (2008) classified fuzzy regression models in three categories: Input and output data are both crisp (First Category) Input data is crisp and output data is fuzzy (Second Category) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 11 / 39

  29. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Choi and Buckley (2008) classified fuzzy regression models in three categories: Input and output data are both crisp (First Category) Input data is crisp and output data is fuzzy (Second Category) Input and output data are both fuzzy (Third Category) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 11 / 39

  30. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Fuzzy linear regression model (Second Category) Y l = � � A 0 + � A 1 x 1 l + � A 2 x 2 l + ... + � l = 1 , 2 , .., n (1) A m x ml Fuzzy linear regression model (Third Category) Y l = a 0 + a 1 � � X 1 l + a 2 � X 2 l + ... + a m � l = 1 , 2 , .., n (2) X ml 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 12 / 39

  31. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Predicted values 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 13 / 39

  32. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Predicted values Fuzzy linear regression model (Second Category) � lk = � V 0 k + � V 1 k x 1 l + � V 2 k x 2 l + ... + � Y ∗ V mk x ml l = 1 , 2 , .., n (3) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 13 / 39

  33. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Predicted values Fuzzy linear regression model (Second Category) � lk = � V 0 k + � V 1 k x 1 l + � V 2 k x 2 l + ... + � Y ∗ V mk x ml l = 1 , 2 , .., n (3) Fuzzy linear regression model (Third Category) Y ∗ � lk = v 0 k + v 1 k � X 1 l + v 2 k � X 2 l + ... + v mk � X ml ; l = 1 , 2 , .., n (4) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 13 / 39

  34. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Since the dependent variable has a membership function, the estimated fuzzy output, which is also represented by a mem- bership function, should be close to the membership function of the given data. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 14 / 39

  35. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Since the dependent variable has a membership function, the estimated fuzzy output, which is also represented by a mem- bership function, should be close to the membership function of the given data. The sum of the differences is calculated as 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 14 / 39

  36. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Since the dependent variable has a membership function, the estimated fuzzy output, which is also represented by a mem- bership function, should be close to the membership function of the given data. The sum of the differences is calculated as � D = | µ ˜ Y ( x ) − µ ˜ lk ( x ) | dx Y ∗ 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 14 / 39

  37. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Since the dependent variable has a membership function, the estimated fuzzy output, which is also represented by a mem- bership function, should be close to the membership function of the given data. The sum of the differences is calculated as � D = | µ ˜ Y ( x ) − µ ˜ lk ( x ) | dx Y ∗ � | µ � Y ( x ) − µ � lk ( x ) | dx Y ∗ S � Y ∪ S � Y ∗ E = lk � µ � Y ( x ) dx S � Y 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 14 / 39

  38. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 15 / 39

  39. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Error Measure (Abdalla & Buckley (2007)) �� ∞ � � n −∞ | � Y l ( x ) − � Y ∗ lk ( x ) | dx l =1 E 1 = �� ∞ � (5) −∞ � Y l ( x ) dx 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 15 / 39

  40. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Error Measure (Abdalla & Buckley (2007)) �� ∞ � � n −∞ | � Y l ( x ) − � Y ∗ lk ( x ) | dx l =1 E 1 = �� ∞ � (5) −∞ � Y l ( x ) dx Y l = ( y l 1 / y l 2 / y y 3 ) and � � Y ∗ lk = ( y lk 1 / y lk 2 / y lk 3 ) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 15 / 39

  41. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Error Measure (Abdalla & Buckley (2007)) �� ∞ � � n −∞ | � Y l ( x ) − � Y ∗ lk ( x ) | dx l =1 E 1 = �� ∞ � (5) −∞ � Y l ( x ) dx Y l = ( y l 1 / y l 2 / y y 3 ) and � � Y ∗ lk = ( y lk 1 / y lk 2 / y lk 3 ) V k ∈ { � � V 1 , ..., � V N } and v k ∈ { v 1 , ..., v N } 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 15 / 39

  42. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Outline 1 Introduction 2 Preliminaries 3 Fuzzy Regression with Monte Carlo Method 4 Distance Measure for Fuzzy Numbers 5 Application Application for Second Category Application for Third Category Solutions 6 Conclusion 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 16 / 39

  43. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion The methods of measuring the distance between fuzzy numbers have become important due to the significant applications in diverse fields like data mining, pattern recognition, multivariate data analysis and so on. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 17 / 39

  44. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion The methods of measuring the distance between fuzzy numbers have become important due to the significant applications in diverse fields like data mining, pattern recognition, multivariate data analysis and so on. Kaufmann (1991) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 17 / 39

  45. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion The methods of measuring the distance between fuzzy numbers have become important due to the significant applications in diverse fields like data mining, pattern recognition, multivariate data analysis and so on. Kaufmann (1991) Heilpern (1997) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 17 / 39

  46. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion The methods of measuring the distance between fuzzy numbers have become important due to the significant applications in diverse fields like data mining, pattern recognition, multivariate data analysis and so on. Kaufmann (1991) Heilpern (1997) Heilpern-1 (1997) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 17 / 39

  47. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion The methods of measuring the distance between fuzzy numbers have become important due to the significant applications in diverse fields like data mining, pattern recognition, multivariate data analysis and so on. Kaufmann (1991) Heilpern (1997) Heilpern-1 (1997) Heilpern-2 (1997) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 17 / 39

  48. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion The methods of measuring the distance between fuzzy numbers have become important due to the significant applications in diverse fields like data mining, pattern recognition, multivariate data analysis and so on. Kaufmann (1991) Heilpern (1997) Heilpern-1 (1997) Heilpern-2 (1997) Heilpern-3 (1997) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 17 / 39

  49. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion The methods of measuring the distance between fuzzy numbers have become important due to the significant applications in diverse fields like data mining, pattern recognition, multivariate data analysis and so on. Kaufmann (1991) Heilpern (1997) Heilpern-1 (1997) Heilpern-2 (1997) Heilpern-3 (1997) Chen & Hsieh (1998) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 17 / 39

  50. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion The methods of measuring the distance between fuzzy numbers have become important due to the significant applications in diverse fields like data mining, pattern recognition, multivariate data analysis and so on. Kaufmann (1991) Heilpern (1997) Heilpern-1 (1997) Heilpern-2 (1997) Heilpern-3 (1997) Chen & Hsieh (1998) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 17 / 39

  51. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Kaufmann (1991) � 1 � � d ( � A , � | A L ( α ) − B L ( α ) | + | A U ( α ) − B U ( α ) | B ) = d α 0 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 18 / 39

  52. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Kaufmann (1991) � 1 � � d ( � A , � | A L ( α ) − B L ( α ) | + | A U ( α ) − B U ( α ) | B ) = d α 0 � � � � A L ( α ) , A U ( α ) B L ( α ) , B U ( α ) and are the closed intervals of α -cuts 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 18 / 39

  53. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Heilpern-1 (1997) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 19 / 39

  54. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Heilpern-1 (1997) � A = ( a 1 , a 2 , a 3 , a 4 ) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 19 / 39

  55. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Heilpern-1 (1997) � A = ( a 1 , a 2 , a 3 , a 4 ) � ∞ E ∗ ( � A ) = a 2 − ( a 2 − a 1 ) 0 L ( x ) dx � ∞ E ∗ ( � A ) = a 3 + ( a 4 − a 3 ) 0 R ( x ) dx 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 19 / 39

  56. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Heilpern-1 (1997) � A = ( a 1 , a 2 , a 3 , a 4 ) � ∞ E ∗ ( � A ) = a 2 − ( a 2 − a 1 ) 0 L ( x ) dx � ∞ E ∗ ( � A ) = a 3 + ( a 4 − a 3 ) 0 R ( x ) dx � � EV ( � E ∗ ( � A ) − E ∗ ( � A ) = 1 A ) 2 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 19 / 39

  57. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Heilpern-1 (1997) � A = ( a 1 , a 2 , a 3 , a 4 ) � ∞ E ∗ ( � A ) = a 2 − ( a 2 − a 1 ) 0 L ( x ) dx � ∞ E ∗ ( � A ) = a 3 + ( a 4 − a 3 ) 0 R ( x ) dx � � EV ( � E ∗ ( � A ) − E ∗ ( � A ) = 1 A ) 2 σ ( � A , � B ) = | EV ( � A ) − EV ( � B ) | (6) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 19 / 39

  58. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Heilpern-2 (1997) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 20 / 39

  59. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Heilpern-2 (1997) � 1 d p ( � A , � d p ( � A ( α ) , � B ) = B ( α ) d α ) (7) 0 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 20 / 39

  60. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Heilpern-2 (1997) � 1 d p ( � A , � d p ( � A ( α ) , � B ) = B ( α ) d α ) (7) 0 � A ( α ) = [ A L ( α ) , A U ( α )] and � B ( α ) = [ B L ( α ) , B U ( α )] 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 20 / 39

  61. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Heilpern-2 (1997) � 1 d p ( � A , � d p ( � A ( α ) , � B ) = B ( α ) d α ) (7) 0 � A ( α ) = [ A L ( α ) , A U ( α )] and � B ( α ) = [ B L ( α ) , B U ( α )] � � A ( α ) , � � d p B ( α ) = � (0 . 5)( | A L ( α ) − B L ( α ) | p + | A U ( α ) − B U ( α ) | p ) 1 / p , 1 ≤ p ≤ ∞ ; max | A L ( α ) − B L ( α ) | , | A U ( α ) − B U ( α ) | , p = ∞ . (8) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 20 / 39

  62. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Heilpern-3 (1997) � A = ( a 1 , a 2 , a 3 , a 4 ) � B = ( b 1 , b 2 , b 3 , b 4 ) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 21 / 39

  63. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Heilpern-3 (1997) � A = ( a 1 , a 2 , a 3 , a 4 ) � B = ( b 1 , b 2 , b 3 , b 4 ) � �� 4 i =1 | a i − b i | p � 1 / p 0 . 25 , 1 ≤ p < ∞ ; δ p ( � A , � B ) = (9) max ( | a i − b i | ) , p = ∞ . 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 21 / 39

  64. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Chen & Hsieh (1998) � L − 1( α )+ R − 1( α ) � � w 0 α d α 2 P ( A ) = � w 0 α d α 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 22 / 39

  65. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Chen & Hsieh (1998) � L − 1( α )+ R − 1( α ) � � w 0 α d α 2 P ( A ) = � w 0 α d α � A = ( a 1 , a 2 , a 3 , a 4 ) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 22 / 39

  66. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Chen & Hsieh (1998) � L − 1( α )+ R − 1( α ) � � w 0 α d α 2 P ( A ) = � w 0 α d α � A = ( a 1 , a 2 , a 3 , a 4 ) P ( A ) = a 1 +2 a 2 +2 a 3 + a 4 6 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 22 / 39

  67. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Chen & Hsieh (1998) � L − 1( α )+ R − 1( α ) � � w 0 α d α 2 P ( A ) = � w 0 α d α � A = ( a 1 , a 2 , a 3 , a 4 ) P ( A ) = a 1 +2 a 2 +2 a 3 + a 4 6 P ( A ) = a 1 +4 a 2 + a 4 6 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 22 / 39

  68. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Chen & Hsieh (1998) � L − 1( α )+ R − 1( α ) � � w 0 α d α 2 P ( A ) = � w 0 α d α � A = ( a 1 , a 2 , a 3 , a 4 ) P ( A ) = a 1 +2 a 2 +2 a 3 + a 4 6 P ( A ) = a 1 +4 a 2 + a 4 6 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 22 / 39

  69. Introduction Preliminaries Fuzzy Regression with Monte Carlo Method Distance Measure for Fuzzy Numbers Application Conclusion Chen & Hsieh (1998) � L − 1( α )+ R − 1( α ) � � w 0 α d α 2 P ( A ) = � w 0 α d α � A = ( a 1 , a 2 , a 3 , a 4 ) P ( A ) = a 1 +2 a 2 +2 a 3 + a 4 6 P ( A ) = a 1 +4 a 2 + a 4 6 | P ( A ) − P ( B ) | (10) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 22 / 39

  70. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion Outline 1 Introduction 2 Preliminaries 3 Fuzzy Regression with Monte Carlo Method 4 Distance Measure for Fuzzy Numbers 5 Application Application for Second Category Application for Third Category Solutions 6 Conclusion 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 23 / 39

  71. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion In this section, there are two different applications. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 24 / 39

  72. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion In this section, there are two different applications. First application is for the second fuzzy regression model category and the other one is for the third fuzzy regression model category. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 24 / 39

  73. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion In this section, there are two different applications. First application is for the second fuzzy regression model category and the other one is for the third fuzzy regression model category. We consider different distance measures for fuzzy numbers given in Section 4 in the error measure ( E 1 ) for fuzzy linear regression models with Monte Carlo approach. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 24 / 39

  74. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion In this section, there are two different applications. First application is for the second fuzzy regression model category and the other one is for the third fuzzy regression model category. We consider different distance measures for fuzzy numbers given in Section 4 in the error measure ( E 1 ) for fuzzy linear regression models with Monte Carlo approach. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 24 / 39

  75. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion Table: Data for the application (Second category) Fuzzy Output x 1 l x 2 l x 3 l (2 . 27 / 5 . 83 / 9 . 39) 2 . 00 0 . 00 15 . 25 (0 . 33 / 0 . 85 / 1 . 37) 0 . 00 5 . 00 14 . 13 (5 . 43 / 13 . 93 / 22 . 43) 1 . 13 1 . 50 14 . 13 (1 . 56 / 4 . 00 / 6 . 44) 2 . 00 1 . 25 13 . 63 (0 . 64 / 1 . 65 / 2 . 66) 2 . 19 3 . 75 14 . 75 (0 . 62 / 1 . 58 / 2 . 54) 0 . 25 3 . 50 13 . 75 (3 . 19 / 8 . 18 / 13 . 17) 0 . 75 5 . 25 15 . 25 (0 . 72 / 1 . 85 / 2 . 98) 4 . 25 2 . 00 13 . 50 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 25 / 39

  76. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion Before the application we have to decide the intervals for I i , i = 0 , 1 , 2 , 3 to obtain the model coefficients as explained in Defi- nition 2.5. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 26 / 39

  77. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion Before the application we have to decide the intervals for I i , i = 0 , 1 , 2 , 3 to obtain the model coefficients as explained in Defi- nition 2.5. We use same intervals in order to compare the results we have with the results from Abdalla and Buckley (2007) in the liter- ature. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 26 / 39

  78. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion Before the application we have to decide the intervals for I i , i = 0 , 1 , 2 , 3 to obtain the model coefficients as explained in Defi- nition 2.5. We use same intervals in order to compare the results we have with the results from Abdalla and Buckley (2007) in the liter- ature. Four separate intervals ( MCI , MCII , MCIII , MCIV ) that they studied are given with Table 2. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 26 / 39

  79. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion Before the application we have to decide the intervals for I i , i = 0 , 1 , 2 , 3 to obtain the model coefficients as explained in Defi- nition 2.5. We use same intervals in order to compare the results we have with the results from Abdalla and Buckley (2007) in the liter- ature. Four separate intervals ( MCI , MCII , MCIII , MCIV ) that they studied are given with Table 2. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 26 / 39

  80. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion Table: Intervals for I i , i = 0 , 1 , 2 , 3 for second category Interval MCI MCII MCIII MCIV I 0 [-1,0] [0,1] [-18.174,-18.174] [28.000,47.916] [-1,0] [-1,0] [-1.083,-1.083] [-2.542,-2.542] I 1 I 2 [-1.5,-0.5] [-1.5,-0.5] [-1.150,-1.150] [-2.323,-2.323] [0,1] [0,1] [1.733,2.149] [-1.354,-1.354] I 3 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 27 / 39

  81. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion Results for using different definitions of distance measures in fuzzy linear regression with MC method for minimizing E 1 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 28 / 39

  82. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion Table: Data for the application (Third category) Fuzzy output X 1 l X 2 l (55.4/61.6/64.7) (5.7/6.0/6.9) (5.4/6.3/7.1) (50.5/53.2/58.5) (4.0/4.4/5.1) (4.7/5.5/5.8) (55.7/65.5/75.3) (8.6/9.1/9.8) (3.4/3.6/4.0) (61.7/64.9/74.7) (6.9/8.1/9.3) (5.0/5.8/6.7) (69.1/71.7/80.0) (8.7/9.4/11.2) (6.5/6.8/7.1) (49.6/52.2/57.4) (4.6/4.8/5.5) (6.7/7.9/8.7) (47.7/50.2/55.2) (7.2/7.6/8.7) (4.0/4.2/4.8) (41.8/44.0/48.4) (4.2/4.4/4.8) (5.4/6.0/6.3) (45.7/53.8/61.9) (8.2/9.1/10.0) (2.7/2.8/3.2) (45.4/53.5/58.9) (6.0/6.7/7.4) (5.7/6.7/7.7) 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 29 / 39

  83. Introduction Preliminaries Application for Second Category Fuzzy Regression with Monte Carlo Method Application for Third Category Distance Measure for Fuzzy Numbers Solutions Application Conclusion Before the application we have to decide the intervals for I i , i = 0 , 1 , 2 to obtain the model coefficients as explained in Definition 2.4. 31 th March, 2014 D. ˙ I¸ cen, M. Cattaneo On using Different Distance Measures in FLR 30 / 39

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