M in ing Recursive Function s Ba sed on Gene Expression Programm ing

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1 39 5 ( ) Vol. 39 No JOURNAL OF SICHUAN UN IVERSITY ( ENGINEER ING SC IENCE ED ITION) Sep t : (2007) , 2, 1, η 1, 3, 1, 1, 1, 2 (1., ; 2., ; 3., ) : ( GEP), GEP, GEP2RecurM iner (DSCMS) GEP2RecurM iner GEP, GEP, GEP2RecurM iner, 20%, 10% : ; ; ; GEP2RecurM iner : TP : A M in ing Recursive Function s Ba sed on Gene Expression Programm ing WU J iang 1, 2, TANG Chang2jie 1, J IAN G Yue 1, 3, YE Shang2yu 1, DUAN L ei 1, L I Tai2yong 1, 2 (1. School of Computer Sci., Sichuan Univ., Chengdu , China; 2. School of Econom ic Info. Eng., Southwestern Univ. of Finance and Econom ics, Chengdu , China; 3. School of Computer Sci. and Technol., Southwest Univ. for Nationalities, Chengdu , China) Abstract: Traditional Gene Exp ression Programm ing ( GEP) is bare of discovering recursive functions. The lim ita2 tion of function m ining of the traditional GEP was analyzed. Revised algorithm GEP2RecurM iner based on recursive chromosomes and Dynam ic Selection, Crossover and M utation Strategy (DSCM S) based on best fitness were p ro2 posed. The theoretical p roof and experiments showed that GEP2RecurM iner extremely extends the domain of func2 tion m ining and can discover recursive functions. The experim ents also showed that the perform ance of GEP2Recur2 M iner is imp roved by the com bination of DSCM S. The number of average evolution generations decreases 10%, and the success rate increases 20%. Key words: Gene Exp ression Programm ing ( GEP) ; recursive function; function m ining; GEP2RecurM iner, ( Gene Ex2 p ression Programm ing, GEP) : : ( ) ; SRFDP ( ) : ( ),,. :., ( Genetic A lgorithm, GA) ( Genetic Programm ing, GP),,, GEP, GEP : 1) GEP ; 2)

2 128 ( ) 39 GEP2RecurM iner; 3) (Dynam ic Selection, Crossover and Mutation Strategy, DSCMS) ; 4) GEP2RecurM iner GEP : GEP, GEP2RecurM iner ;, GEP2 RecurM iner, 20%, 10% , Candida Ferreira [ 1 ], GEP, GEP [ 1-10 ] GEP,,,, Ferreira [ 2 ] GEP GA GP 2 4 Ferreira GEP,,,,,,, GEP 3 GEP2RecurM iner 3. 1 [ 3 ] GEP GEP,, ; ( ) ( ),, GEP :,, ( x 2 + y 2 ) / ( x + y), 1 1, / xyxxyy, 2 : ( ) ( ),, +,,, : : 1 ( ) f ( x 1,, x m ) g i ( x 1,, x n ), i = 1, 2,, m, h ( x 1,, x n ) = f ( g 1 ( x 1,, x n ),, g m ( x 1,, x n ) ) f g i (m, n) 2 ( ) g ( t 1,, t r ) h ( t 1,, t r, x, v), f : f ( t 1,, t r, 0) = g ( t 1,, t r ), f ( t 1,, t r, S (x) ) = h ( t 1,, t r, x, f ( t 1,, t r, x) ) f GEP, 1 F ig. 1 Expression tree, ( 1) t = h ( n - 1) + 1 (1), h, t, n GEP2RecurM iner, GEP2RecurM iner : 1),

3 5, : 129 ; 2), ( ) :, ( f ),,, :, m = t 1: { / }, { x}, h = 3, t = 4, m = 4, GEP2Recur2 M iner : + + x f3 xx 0121 (2), f,,, (3) A = { ( x - 1), ( x - 2), ( x - 3), ( x - 4) } (3), 0 A, ( x - 1), f ( x) = f ( x - 1) + x 2 + x 3. 2 GEP2RecurM iner,, (4) f i = ( 6 n j =1 1 (C i, j - T j ) 2 / n) + 1 (4), C i, j i j ; T j j ; n, C i, j = T j ( j = 1, 2,, n),, f i GEP2RecurM iner : : 1), P c ; 2) ; 3) ; 4), : 1), P m ; 2), ; 3), ;, ; 3. 4 GEP2RecurM iner GEP2RecurM iner : 1 GEP2RecurM iner : Cases( ), P s, P c, P m, N max : Best_Exp ( ) Begin 1. p retreat Cases / / 2. S = Initial Population; / / 3. Best_Exp = null; / / 4. m = MaxGeneration; / / 5. repeat 6. S = Selection ( S) by P s ; / / P s 7. S = CrossOver( S) by P c ; / / P c 8. S = Mutation ( S) by P m ; / / P m 9. keep (Best_Exp) ; / / 10. m = m - 1; 11. until ( (m = 0) ( Best_Exp doesn t im2 p rovement in N max iterations) ) 12. posttreat Best_Exp; / / 13. return (Best_Exp) ; End. 1 n, m, N max, k, GEP2Re2 curm iner O (m n k),, GEP2RecurM iner O (N max k) n :, k O ( k),, O ( k n) ;

4 130 ( ) 39 m, O (m n k), N max,, N max, O (N max n k), 2 U GEP, V GEP2RecurM iner, U Α V : GEP2RecurM iner f,,, GEP GEP2RecurM iner, GEP2RecurM iner, GEP,, 2 GEP2RecurM iner GEP,, GEP2RecurM iner,,,, ;,,, 4, GEP2 RecurM iner, (Dy2 nam ic Selection, Crossover and Mutation Strategy), (DSCMS), ;,, :,,,,,,, P c, P m, P s (5) P s = 1 - P c - P m, 0 < P c + P m < 1 (5) (1 - P c - P m )M,, M n max ( n max < N max ),,, (6) P c P m = (1 + rand 1 ) P c = (1 + rand 2 ) P m P s = 1 - (1 - rand 1 ) P c - (1 - rand 2 ) P m (6), rand 1 rand 2, (7) 0 < rand 1 < 1 / P c < rand 2 < 1 / P m - 1 (7),, n max,, P s, P c P m 5 GEP GEP2RecurM iner GEP2RecurM iner(dscms) 1, (8) : f ( x, y) = f ( x - 1, y) + f ( x, y - 1) + xy (8) , 100, GEP, 1 1 Tab. 1 Results of exper im en t GEP 0 - GEP2RecurM iner , GEP2RecurM iner GEP 2 2 F ig. 2 Com par ison of best f itness and average f itness 1 :,

5, : 131 GEP2RecurM iner GEP 2, (9) : f ( x, y) = xy + y (9) 20 0 19, 100, GEP, 2 2 Tab.

+ x + 1, = xy + y 3 (DSCMS) GEP2 RecurM iner ( n max = 60, N max = 100), (10) (11) : f 1 ( x) = x 3-3x 2 + x + 1 (10) f 2 ( x, y) = sin (2xy + x 2 ) (11) 20, [ - 10, 10 ],, 100, 3 4 4 F ig.

5 5, : 131 GEP2RecurM iner GEP 2, (9) : f ( x, y) = xy + y (9) , 100, GEP, 2 2 Tab. 2 Results of exper im en t GEP f ( x, y) = xy + y GEP2 RecurM iner f ( x, y) = xy + y f ( x, y) = f ( x, y - 1) + x + 1 2, GEP2RecurM iner GEP, f ( x, y) f ( x, y) 2 2 = f ( x, y - 1) + x + 1, = xy + y 3 (DSCMS) GEP2 RecurM iner ( n max = 60, N max = 100), (10) (11) : f 1 ( x) = x 3-3x 2 + x + 1 (10) f 2 ( x, y) = sin (2xy + x 2 ) (11) 20, [ - 10, 10 ],, 100, F ig. 4 Com par ison of average evolution genera tion s DSCMS GEP2RecurM iner, 10%, DSCMS GEP2RecurM iner 6 GEP,, : GEP2RecurM iner GEP, (DSCMS), GEP2Recur2 M iner, 20%, 10%,, : [ 1 ] Ferreira C. Gene Exp ression Programm ing: a new adap tive algorithm for solving p roblem s[ J ]. Comp lex System s, 2001, 13 (2) : [ 2 ] Ferreira C. Gene Exp ression Programm ing [M ]. 1 st Ed. Portugal: Angra do Heroismo, [ 3 ] Huang Xiaodong, Tang Changjie, L i Zhi, et al. M ining 3 F ig. 3 Com par ison of success ra te 3, DSCMS GEP2RecurM iner, DSCMS GEP2RecurM iner, 20% ; 4 functions relationship based on Gene Exp ression Program2 m ing[ J ]. Journal of Software, 2004, 15 ( Supp l) : [,,,. [ J ]., 2004, 15 ( ) : ] [ 4 ] Peng J ing, Tang Changjie, L i Chuan, et al. M 2GEP: a new evolution algorithm based on multi2layer chromosomes Gene

6 132 ( ) 39 Exp ression Programm ing [ J ]. Journal of Computer, 2005, 28 (9) : [,,,. M 2GEP: [ J ]., 2005, 28 (9) : ] [ 5 ] Yuan Changan, Tang Changjie, Zuo J ie, et al. Function m ining based on Gene Exp ression Programm ing convergen2 cy analysis and remnant2guided evolution algorithm [ J ]. Journal of Sichuan University: Engineering Science Edition, 2004, 36 (6) : [,,,. cial immune system and Gene Exp ression Programm ing[ J ]. Journal of Sichuan University: Engineering Science Edition, 2006, 38 (5) : [,,,. [ J ]. :, 2006, 38 ( 5) : ] [ 8 ] Ferreira C. Gene Exp ression Programm ing in p roblem solving [ C ] / / invited tutorial of the 6 th Online World Conference on Soft Computing in Industrial App lications. 2001: [ 9 ] Ferreira C. Mutation, transposition, and recombination: an [ J ]. :, 2004, 36 analysis of the evolutionary dynam ics[ C ] / /4 th International (6) : ] [ 6 ] Yuan Changan, Tang Changjie, W en Yuanguang, et al. Imp lementation of intelligent model base system based on Genetic Exp ression Programm ing[ J ]. Journal of Sichuan U2 Workshop on Frontiers in Evolutionary A lgorithm s. North Carolina, USA, 2002: [ 10 ] Ferreira C. D iscovery of the Boolean Functions to the best density2classification rules using Gene Exp ression Program2 niversity: Engineering Science Edition, 2005, 37 ( 3) : 99 - m ing[ C ] / /Proceedings of the 4 th European Conference on 104. [,,,. [ J ]. :, 2005, 37 (3) : ] [ 7 ] Zeng Tao, Tang Changjie, Zhu M ingfang, et al. M ining Genetic Programm ing, EuroGP 2002, volume 2278 of Lec2 ture Notes in Computer Science. Berlin: Sp ringer2verlag, Germany, 2002: ( ) multi2dimensional comp lex association rule based on artifi2

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