,,,,,,,, Stabilization of stock prediction by cross entropy optimization Kazuki Miura, Hideitsu Hino and Noboru Murata Prediction of series data is a long standing important problem Especially, prediction of stock is much in demand for forecasting the economic trend and guideline for asset maintenance Although there are growing number of studies on learning theory based series prediction, the prediction of stock s is still being very difficult task In this study, the stock s is predicted not only using one predictor, but using a set of predictors generated by the method of Genetic Programming (GP) Each element predictor is given non-negative weight, and the weight is optimized to minimize the cross entropy between the true learning stock s and the weighted sum of predicted values The proposed stock prediction method is evaluated using both an artificial data and real-world stock data,, ), 2),,, 3), 4), 5), 6),,,,,,,,,,,,, ( ),,, (Genetic Programming:GP 7) ) GP, ( : ),, ( ),,,,, 2 GP, Waseda University, Genetic Programming Polynomial Models 8) c 200 Information Processing Society of Japan
GP 7) 8) 2 GP,,, 2, τ x t = {x t τ,, x t }, x t τ ( ) 2 2 y i (u, v), ( ),, GP f i (x t ), a (i) j,, x t τ = 0 x t 0, x t 9,, x t x t y 6, x 5, x 3 x t 5, x t 3 t = T, x t 5, x t 3, x T 5, x T 3 y 6, x t 5 x t 5 x t 3 x 2 t 5 x t+ 5 x t+ 5x t+ 3 x t+ 5 x T 5 x T 5x T 3 x 2 T 5 a (6) 0 a (6) a (6) 3 = ( ),, y 4,, x t 2, x t x t+, y 2, 22 GP,, {f i } n i= (N ), x T (),, GP, Rational Average Error (RAE) Generalized Cross-Validation (GCV) 9) T T A RAE = (x t f(x t )) 2 / (x t x t ) 2 + λ a 2 j, (2) i= GCV = RAE/( A/T ) 2 t=, A GCV, GCV,,,,, GP, 8),, ( ), x t x t t xi l i=t l+, 8) l = 5, 3 GP f i (x t ), x t Bagging ), Bagging 2(a),, 0),, 2(b) 3 f(x), g(x), H(f, g) = E f [ log g(x)] = f(x) log g(x)dx (3), f(x) D x = {x i} n i=, f(x), g(x) 2 c 200 Information Processing Society of Japan
f i Table Transfer function f f2 N f f2 y y 2 y 4 x x 4 x 2 x 5 y 6 x 3 Fig Tree structure Set of transfer polynomials y (u, v) = a () 0 + a () u + a() 2 v + a() 3 y 2 (u, v) = a (2) 0 + a (2) u + a(2) 2 uv + a() 4 u2 + a () 5 v2 uv + a(2) 3 u2 + a (2) 4 v2 uv + a(3) 3 u2 + a (3) 4 u2 y 3 (u, v) = a (3) 0 + a (3) v + a(3) 2 y 4 (u, v) = a (4) 0 + a (4) u + a(4) 2 v + a(4) 3 u2 + a (4) 4 v2 y 5 (u, v) = a (5) 0 + a (5) y 6 (u, v) = a (6) 0 + a (6) y 7 (u, v) = a (7) 0 + a (7) y 8 (u, v) = a (8) 0 + a (8) uv + a(5) 2 u2 + a (5) 3 v2 u + a(6) 2 u + a(7) 2 v + a(8) 2 uv + a(6) 3 u2 uv + a(7) 3 v2 uv + a(8) 3 u2 y 9 (u, v) = a (9) 0 + a (9) v + a(9) 2 uv + a(9) 3 v2 y 0 (u, v) = a (0) 0 + a (0) u + a (0) 2 v 2 + a (0) 3 uv y (u, v) = a () 0 + a () u + a () 2 v + a () 3 u 2 y 2 (u, v) = a (2) 0 + a (2) u + a (2) 2 v + a (2) 3 v 2 y 3 (u, v) = a (3) 0 + a (3) u 2 + a (3) 2 v 2 y 4 (u, v) = a (4) 0 + a (4) u + a (4) 2 v 2 y 5 (u, v) = a (5) 0 + a (5) v + a (5) 2 u 2 x i,, D x g(x), f(x), g(y) 2 D x = {x i } n i=, D y = {y j } m, D y y j, D yv = {D y, V} = {(y j, v j )} m, v j =, v j 0 (4), y F (y) m vjf (yj) 0), Ĥ(D x, D yv ) n i= v j log y j x i (5) V, D yv,,, x t τ fn x t 2 x t x t ˆxt = N fi(xt ) N i= (a): Fig 2 x t τ x t 2 x t x t fn v i N ˆxt = vifi(xt ) i= (b): 2 Proposed stock prediction method V, V D y β = {β j} m Dirichlet h(v; β) m vβ j j, Ĥ(D x, D yv ) log h(v; β) = Ĥ(D x, D yv ) m (β j ) log v j, V, j β j = µ + 32 x t D (t) x, t τ x t = {x t τ,, x t } GP x t D (t) y = {ˆx t = f j (x t )} m, {D (t) x, D y (t) } T t=τ+ T τ, x t {ˆx t = f j (x t )} m m v j f j (x t ) x t µ m (βj ) log vj, J(V) = T τ T t=τ+ v j log f j (x t ) x t µ (β j ) log v j (6), J(V),, J(V) µ, µ = 0 0 6 3 c 200 Information Processing Society of Japan
4 2 GP Table 2 Setup of GP 00 3 0 40 0 (size:2) 0 (2):λ 00000, GP 2 4,, (6),,,, 4 3, 20 3, (42) 7000 f,t (x t 4, x t 7 ) = a () + a () 2 y 3(x t 4, x t 7 ) + a () 3 y 3(x t 4, x t 7 )y 5 (x t 4, x t 7 ) +a () 4 y 3(x t 4, x t 7 ) 2 + a () 5 y 5(x t 4, x t 7 ) 2, f 2,t(x t 5, x t ) = a (2) + a (2) 2 y6(xt 5, xt ) + a(2) 3 y6(xt 5, xt )y2(xt 5, xt ) +a (2) 4 y 6(x t 5, x t ) 2 + a (2) 5 y 2(x t 5, x t ) 2, f 3,t (x t 6, x t 3 ) = a (3) + a (3) 2 y (x t 6, x t 3 ) + a (3) 3 y (x t 6, x t 3 )y 5 (x t 6, x t 3 ) +a (3) 4 y (x t 6, x t 3 ) 2 + a (3) 5 y 5(x t 6, x t 3 ) 2 a (i) j,,, 3, 3,, f v = 0, f 2 v 2 = 03, f 3 v 3 = 06,, D f = {f 3,t, f 3,t+, f 3,t+, f 3,t+2, f 2,t+3, f 2,t+4, f 3,t+5, f 3,t+6, f 2,t+7, f,t+8, }, 0, 03, 06, (6) 3 3, 3 3, 42, 00 (42) 7000,, 00, σ, [ σ/3, σ/3], ±, (6), 45, 705, 384%, ±, 3, ± 42, (2009/08/ 2009/2/3), 20 8000, (GP) 3000, 5000, 0000, (Mean Prediction Error:MPE), 4 4,,,, 42 3 ( ) (Best Model) 4 c 200 Information Processing Society of Japan
3 Table 3 Defined weights for element models and optimized weights v v2 v3 f f 2 f 3 02 03 05 02 029 050 05 02 03 048 020 032 03 05 02 029 050 02 0 2 3 4 0220 0225 0230 0235 0240 0245 0250 0255 True Value Orignal Weighted 3 Fig 3 Cut down dispersion ( 2 ) (Orignal Model) ( 3 ) (Weighted Model) 3, (MPE), [%](Hit Rate:HR) 4 4, 5 (sd), 3 5 4 5,,,,,, t 5% 422 ±, ( ) (Orignal Model) ( 2 ) (Weighted Model) 90 95 200 205 0006 0008 000 002 004 MPE weight 4 Fig 4 Correspondence between MPE and weight 40 4 42 43 44 0230 0240 0250 0260 0270 0280 0290 0300 True Value Best Orignal Weighted 5 Fig 5 Result of prediction 2, (Variance), ± [%](Including) 5, 5, 5 (sd), 6 5,, ±,,, 5% t 5,,,,,, GP, 5 c 200 Information Processing Society of Japan
4 Table 4 Predictive accuracy MPE (sd) HR (sd) Best 888 5682 Model (0078) (0072) Orignal 545 58 Model (0089) (050) Weighted 532 583 Model (0088) (083) 5 Table 5 Predictive distribution Variance Including (sd) (sd) Orignal 7489 637 Model (230) (0073) Weighted 7044 6338 Model (295) (07), 2, GP 0, GP,,,,,,, 6,,,, 22800067 0 40 4 42 43 44 0226 6 Fig 6 Result of prediction 0298 ) :, (2003) 2) :, (2000) 3) YYoon and GSwales: Predicting stock performance: A neural network approach, In Proc Twenty-Fourth Annual Hawaii International Conference on System Sciences, (99) 4) WHuang and YNakamori and SWang: Forecasting stock market movement direction with support vector machine, Computers and Operations Research, Vol32, No0, pp253 2522 (2005) 5) SMahfoud and GMani: Financial forecasting using genetic algorithms, Applied Artificial Intelligence, Vol0, pp543 565, (996) 6) HIba and TSasaki: Using Genetic Programming to Predict Financial Data, In Proc 999 Congress on Evolutionary Computation, (999) 7) (200) 8) HIba and NNikolaev: Genetic programming polynomial models of financial data series, In Proc the 2000 Congress on Evolutionary Computation (2000) 9) THastie and RTibshirani and JFriedman: The Elements of Statistical Learning Data Mining, Inference, and Prediction (2nd edition), Springer, (2009) 0), IBISML (200) ) L Brieman: Bagging Predictors, Machine Learning, Vol 24, No 2, pp 23 40 (996) 6 c 200 Information Processing Society of Japan