A quick semantic artificial bee colony programming (qsABCP) for symbolic regression

Abstract

Artificial bee colony programming (ABCP) is a novel evolutionary computation based automatic programming method, which uses the basic structure of artificial bee colony (ABC) algorithm. In this paper, some studies were conducted to improve the performance of ABCP and three new versions of ABCP are introduced. One of these improvements is related to the convergence performance of ABCP. In order to increase the local search ability and achieve higher quality solutions in early cycles, quick ABCP algorithm was developed. Experimental studies validate the enhancement of the convergence performance when the quick ABC approach is used in ABCP. The second improvement introduced in this paper is about providing high locality. Using semantic similarity based operators in the information sharing mechanism of ABCP, semantic ABCP was developed and experiment results show that semantic based information sharing improves solution quality. Finally, combining these two methods, quick semantic ABCP is introduced. Performance of these novel methods was compared with some well known automatic programming algorithms on literature test problems. Additionally, ABCP based methods were used to find approximations of the Colebrook equation for flow friction. Simulation results show that, the proposed methods can be used to solve symbolic regression problems effectively.

Introduction

Automatic programming is a method, which is used to automatically generate computer programs. Symbolic regression problems can be solved by automatic programming techniques because, a system model or a computer program that produces the desired outputs according to the input values is searched in this problem. In literature, many evolutionary computation based automatic programming algorithms like genetic programming (GP) [32], ant programming [43], dynamic ant programming [45], artificial immune system programming [23], immune programming [37], clone selection programming [11], and biogeography based programming [14] are used for symbolic regression. Including symbolic regression problems, these algorithms have a wide range of application fields such as circuit design [32], [43], [49], data mining [5], [38], [39], classification [5], [10], [12], [39], path planning [24], [32], [47], neural network design [7], fuzzy system design [6], time series prediction [6], [7], prediction [15], [17], [18], [35], [46], etc.

Artificial bee colony programming (ABCP) is one of the evolutionary computation based automatic programming algorithms and it was introduced into literature in 2012 [31] by Karaboga et al. Using this algorithm, various types of problems like symbolic regression can be successfully solved. Golafshani and Ashour used ABCP for the prediction of self-compacting concrete elastic modulus [16]. Golafshani and Behnood also employed ABCP to predict the elastic modulus of recycled aggregate concrete (RAC)[17]. Boudardara and Gorkemli applied ABCP to the Santa Fe and Los Altos Hills trails of artificial ant problem, which is a kind of robotic path planning problem [3]. ABCP is based on artificial bee colony (ABC) algorithm [25], which is a very popular, simple, and flexible optimization algorithm that simulates the foraging behavior of honeybees and this algorithm has been widely used successfully in different fields [1], [27], [30]. However, some researchers argue that the convergence performance of the ABC algorithm for local minimum needs to be improved [30]. In order to cope with this problem, many studies have been carried out [9], [22], [30], [36], [50]. Since having the same algorithmic structure as ABC, disadvantages of ABC and the provided improvements on this point also are related to ABCP. However, ABC and ABCP employ different solution representations, so we cannot use the same formulations in some basic mechanisms of these methods. For example, the formulation of the candidate solution production of ABC cannot be used directly in ABCP. This situation causes some problems when some new ideas which work for ABC algorithm are applied to ABCP, especially ones based on string representation. Considering this kind of problem, we use the quick ABC (qABC) idea introduced in [28], [29] for improving the convergence performance of the ABCP because, the qABC algorithm does not employ a distinct modification of a neighborhood searching formulation or other representation dependent distinct modifications. It is a new version of ABC, which models the behavior of onlooker bees more accurately and improves the performance of the standard ABC in terms of local search ability [28], [29]. More over being a modified version of the standard ABC, qABC uses a new idea for modelling the foraging behavior of honeybees. Therefore, in this paper, as one of the improvements in the ABCP algorithm, this qABC idea was applied to standard ABCP.

In evolutionary computation, locality is a highly important property for the performance of the search [20], [41], [42]. Actually, the locality is about a balance as a small change in genotype also causes a small change in phenotype. Generally, it is hard to design searching operators that provide locality using existing GP representations. In order to achieve locality, some researchers consider syntax [21] while others consider semantics [33], [48] of the representation of individuals. In automatic programming methods like GP, semantics of the program (or the individual) is usually understood as the behavior of this program when it uses the given input values [33], [48]. Uy et al. showed that semantic locality is more efficient than syntactic locality on the GP crossover operator [48]. In our study, in order to improve the performance of the standard ABCP, we also try to achieve semantic locality in the neighborhood searching mechanism, which is used for producing a candidate solution. Using this modified version of ABCP, we expect to get more controlled fluctuations in the employed and onlooker bees’ phases.

As mentioned above, standard ABCP has some drawbacks like poor convergence speed and low locality. The main contribution of this study is improving the performance of ABCP in terms of two different ways. One of them is the algorithmic structure of the swarm intelligence algorithm that ABCP is based on. In this subject, we consider the qABC idea to achieve a better convergence performance with ABCP and propose a new automatic programming method called quick ABCP (qABCP). We can say that, this improvement is about what the algorithm does. The other improvement in standard ABCP provides a higher locality while using the same representation (tree structure) for individuals. This time, considering the semantic locality we modified the information sharing mechanism of the standard ABCP and called this algorithm semantic ABCP (sABCP). By applying this, we modified how the algorithm does something. Then, we applied these two modifications together and called this new version of ABCP quick semantic ABCP (qsABCP). In the experimental studies, ABCP, qABCP, sABCP, and qsABCP algorithms were compared with each other, and with some GP and Cartesian ant programming (CAP) versions on literature test problems. Then, the proposed approaches and ABCP were applied to a real world problem for finding the approximations of the Colebrook equation for flow friction, and their results were compared. The results show that, these new modifications obtain new successful automatic programming methods comparing to competitive ones in the literature and improve the performance of the standard ABCP.

The rest of the paper is organized as follows: The standard ABCP method is explained in Section 2. The proposed new versions of ABCP (qABCP, sABCP, and qsABCP) are introduced in Section 3. Then, the experimental studies and the simulation results are given in Section 4 and finally, Section 5 concludes the paper.