Evolutionary Algorithms with Competing Heuristics in Computational ...

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Evolutionary Algorithms with Competing Heuristics in Computational Statistics Josef Tvrd´ık1 , Ivan Kˇriv´ y1 and Ladislav Miˇs´ık2 1 2

Department of Computer Science, University of Ostrava, 701 03 Ostrava, Czech Republic Department of Mathematics, University of Ostrava, 701 03 Ostrava, Czech Republic

Abstract. The paper presents a new class of evolutionary algorithms based on the competition of different heuristics. The algorithm was applied to solving some optimization problems of computational statistics, namely to estimating the parameters of non-linear regression models, constrained Mestimates and optimizing the smoothing constants in the Winters exponential smoothing. The results showed that the evolutionary algorithm with competing heuristics can be successfully used in solving some global optimization problems of computational statistics. Keywords. Evolutionary algorithms, heuristics, global optimization, nonlinear regression, robust estimates, time series analysis

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Algorithm and its convergence

We consider evolutionary algorithms (EAs) intended for finding the global minimum of complicated objective functions. A more detailed formulation of the optimization problem is given in Miˇs´ık et al. (2001). The algorithm can be described as follows: I: Generate an initial population P0 = {x1 , x2 , · · · , xN } chosen as an independent identically distributed sample according to the probability measure p0 and set k = 0. Ck : Copy a portion of M best points of Pk directly into the new population. Here the best points are those with the lowest values of the objective function f and M is an integer from {1, 2, · · · , N − 1}. Gk : Generate a new trial point according to the probability measure pk+1 = π(Pk ) and include it to the new population when fulfilling the condition C. Repeat this procedure until the new population is complete. Mk : With the probability mk+1 replace a randomly chosen point by its mutation. R: Set k = k + 1 and go back to Ck . According to the results of Solis and Wets (1981) and Miˇs´ık (2000), the EAs are convergent provided Q∞ that for every measurable subset S of the searching domain D we have k=1 (1−pk (S)) = 0, where pk (S) denotes the probability of producing a new trial point in the set S just in the k-th iteration. It is not necessary to use the only heuristic for producing a new trial point in the step Gk . The use of more different competing heuristics was proposed,

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see e.g. Tvrd´ık et al. (2001). Considering h heuristics at disposal, the new trial points can be generated by any of the heuristics with the probability qi , i = 1, 2, . . . , h, depending on the successfulness of the i-th heuristic during preceding part of the optimization process. Several strategies for evaluating the successfulness of the individual heuristics can be applied. One of them is based on simple counting the number of successful insertions of new trial points, ni , ni + n0 , (1) qi = Ph j=1 (nj + n0 ) where n0 > 0 is a constant. Setting n0 > 1 prevents a dramatic change in qi by one random successful use of the i-th heuristics. Another strategy takes into account the relative position wi of the trial points when they are inserted into the new population of points ordered in a nondecreasing sequence with respect to their objective function values, qi = Ph

Wi + w0

j=1 (Wj

+ w0 )

,

(2)

P where Wi = wi over a previous period of the process and w0 > 0 is an input parameter, which can be set to the mean value of wi . The EA with competing heuristics and without explicit mutation (i.e. the probability of mutation mk+1 = 0 in the step Mk ) is convergent if at least one of the heuristics in use is convergent when using alone. In practice this can be ensured by including the heuristics generating new trial points uniformly distributed over D. On the other hand, it was shown in Miˇs´ık et al. (2001) that in some practical tasks better results can be achieved using a nonconvergent heuristics instead of the theoretically convergent algorithm. That is why a natural question arises to estimate the probability that the random search produces a proper trial point. Therefore, we tried to estimate the probability p that just the uniform search produces a new trial point in a set S ⊂ D with its Lebesgue measure λ(S) > 0 during all the process. Starting from the sufficient condition for the convergence of the EA (see Solis and Wets, 1981), the following approximate relation between p and λ(S) was derived (Tvrd´ık et al., 2002) n λ(S) n λ(S) 0 subject to the constraint n

1 X ri ρ( ) ≤ ερ(∞), n i=1 σ where ε ∈ (0, 1). Using truncated Huber’s function for ρ(t)  2  t for |x| ≤ G, 2 ρ(t) =  G2 for |x| > G 2 with G = 2.795 and ε = 0.5, we could find the CM-estimates of both the location and scatter parameters for the following two tasks: • Hertzsprung-Russel diagram for the star cluster CYG OB1 (Rousseeuw and Leroy, 1987), • example from Robust Regression Tutorial of NCSS by Hintze (2001). The values of R, NE and the standard deviation of NE are given in Table 2. As regards the CYG OB1 task, the CM-solution is yˆi = −8.533 + 3.058xi .

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Table 2. Reasults of making CM-estimates in linear regression models Competing1 Task CYG OB1 NCSS

d 3 7

R 90 85

NE 3040 6420

sd(NE) 200 678

Competing2 R 80 95

NE 2893 6275

sd(NE) 166 591

3.3 Optimization of smoothing constants This section deals with an application of evolutionary algorithms to solving the problem of time series smoothing. The time series of some Czech labour market descriptors were treated using the Winters exponential smoothing algorithm with multiplicative seasonal adjustment. The Winters multiplicative seasonality algorithm is based on the following formulas: Xt a(t) = α Szt−L (t−L) + (1 − α)[a(t − 1) + b(t − 1)],

b(t) = β[a(t) − a(t − 1)] + (1 − β)b(t − 1), Xt + (1 − γ)Szt−L (t − L), Szt (t) = γ a(t)

where a(t), b(t) and Szt (t) are estimates of the time series level, its slope and its seasonal factor at time t, respectively, α, β, γ are smoothing constants and L represents the number of periods per year. The smoothing constants α, β and γ were optimized with respect to the criterion SSE, the searching space being constrained to 0 < α, β, γ ≤ 1. The empirical formulas proposed by Cipra (1986) were used for estimating the initial values a(0), b(0) and Szt (t) for t = −(L − 1), −(L − 2), . . . , 0. The original data (extended by the Czech Ministry of Labour and Social Affairs) covered the period from January 1995 to January 2001. The following four labour market descriptors were studied: labour force, number of job applicants, unemployment rate, and number of job vacancies. For each individual time series at least ten independent runs were performed. In all the cases the global minimum was found with the reliability R = 100. The resulted optimum values of smoothing constants showed that some of them reach their upper limit value, which indicates that the mechanism generating them has recently gone through some fundamental changes. Using the optimum values of smoothing constants, all the time series were smoothed and the forecasts for the year 2001 were calculated and compared with the real data as well as with the forecasts resulted from the Box-Jenkins methodology when applicable. It was found that in some cases the forecasts resulted from the Winters smoothing algorithm are even better than those based on the Box-Jenkins approach.

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Conclusions

The evolutionary algorithm with competing heuristics was applied to several problems of the searching for the global minimum in computational statistics. As regards nonlinear regression models, the algorithm was highly reliable but time-consuming in two tasks. The CM-estimates were a bit less reliable. In

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smoothing constants optimization the algorithm proved to be sufficiently fast and reliable in all runs. The self-adaptation feature of the algorithm with competing heuristics makes the setting of input parameters easier comparing with other evolutionary algorithms. Thus, the algorithm can be used with its default parameter settings in statistical programs. References Arslan, O., Edlund, O. and Ekblom, H. (2000). Algorithms for Constrained M-Estimates. In: COMPSTAT 2000, Short Communications and Posters, 143–144, Utrecht: Statistics Netherlands. B¨ ack, T. (1996). Evolutionary Algorithms in Theory and Practice. New York: Oxford University Press. Cipra, T. (1986). Time Series Analysis with Applications in Economics (in Czech). Prague: SNTL Press. Hintze, J. (2001). NCSS and PASS, Number Cruncher Statistical Systems. Kaysville, Utah: WWW.NCSS.COM. Kˇriv´ y, I. and Tvrd´ık, J. (1995). The Controlled Random Search Algorithm in Optimizing Regression Models. Comput. Statist. and Data Anal., 20, 229–234. Kˇriv´ y, I. and Tvrd´ık, J. (2001). Stochastic optimization in smoothing time series of labour market descriptors. In: Bulletin of ISI, Tome LIX, Book 1 - Contributed Papers, 253-254. Seoul: ISI Press. Kvasniˇcka, V., Posp´ıchal, J. and Tiˇ no, P. (2000). Evolutionary algorithms (in Slovak). Bratislava: Slovak Technical University Press. McCullough, B.D., Wilson, B. (1999). On the accuracy of statistical procedures in Microsoft Excel 97, Comput. Statist. and Data Anal., 31, 27–37. McCullough, B. D. (2000). The Accuracy of Mathematica 4 as a Statistical Package. Comp. Statistics, 15, 279–299. Miˇs´ık, L. (2000). On Convergence of a Class of Evolutionary Algorithms. In: Proceedings of MENDEL 2000, 6th International Conference on Soft Computing, 97–100. Brno: Technical University Press. Miˇs´ık, L., Tvrd´ık J. and Kˇriv´ y, I. (2001). On Convergence of a Class of Stochastic Algorithms. In: Proceedings of ROBUST 2000 (J. Antoch and ˇ G. Dohnal eds), 198 - 209, Prague: JCMF Press. Solis, F. J. and Wets, R. J-B. (1981). Minimization by Random Search Techniques. Mathematics of Operations Research, 6, 19–30. Rousseeuw, P. J. and Leroy, A. M. (1987). Robust Regression and Outlier Detection. New York: John Wiley & Sons. Storn, R. and Price, K. (1997). Differential Evolution – a Simple and Efficient Heuristic for Global Optimization. J. Global Optimization, 11, 341–359. Tvrd´ık, J., Kˇriv´ y, I. and Miˇs´ık, L. (2001). Evolutionary Algorithm with Competing Heuristics. In:Proceedings of MENDEL 2001, 7-th Int. Conference on Soft Computing, 58-64. Brno: Technical University Press. Tvrd´ık, J., Miˇs´ık, L. and Kˇriv´ y, I. (2002). Competing Heuristics in Evolutionary Algorithms. Accepted to the 2nd Euro-ISCI (Koˇsice, July 16-19, 2002) to appear in Studies in Computational Intelligence series. Berlin: Springer-Verlag. Winters, P. R. (1960). Forecasting Sales by Exponentially Weighted Moving Averages. Management Science, 6, 324–342. Acknowledgement. This research was supported by the grant 402/00/1165 of the Czech Grant Agency.