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Estimation  of  the  load  carrying  capacity  of  pile  foundation  is  one  of  the  most  sought  after  research  areas  in geotechnical   engineering.   Static   equilibrium   and   other dynamic equations are used to predict  the  axial  load  capacity  of  pile.  The  prediction  of lateral  load  capacity  of  piles,  used  in  tall  and  offshore structures is more complex and requires solution of non-linear differential equations. The elastic analysis adopting Winkler   soil   model is   not suitable for the non-linear soil behavior.  Estimating the load capacity of such piles using experimental methods is always associated with error and makes the modeling result far from reality. Today, intelligent methods have shown a high capability in predicting and estimating unknown variables and can replace experimental and analytical methods. In this research, we tried to accurately predict the lateral load capacity of piles in clay soils by creating an intelligent hybrid model called optimized relevant vector regression with the artificial bee colony algorithm. The relevant vector regression is a probabilistic method based on Bayesian approach. The relevant vector regression does not need to predict the error/margin tradeoff parameter C, which can decrease the time and the kernel function, does not need to satisfy the Mercer condition. For those relevant vector regression advantages compared with the support vector regression approach, relevant vector regression model is successfully applied in regression prediction problems. In this method, relevant vector regression is used as a predictive model and artificial bee colony algorithm is used to optimize the parameters of relevant vector regression method. The artificial bee colony algorithm is a swarm based meta-heuristic algorithm for optimizing numerical problems. It was inspired by the intelligent foraging behavior of honey bees. The algorithm is specifically based on the model for the foraging behavior of honey bee colonies. The model consists of three essential components: employed and unemployed foraging bees, and food sources. The first two components, employed and unemployed foraging bees, search for rich food sources, which is the third component, close to their hive. The model also defines two leading modes of behavior which are necessary for self-organizing and collective intelligence: recruitment of foragers to rich food sources resulting in positive feedback and abandonment of poor sources by foragers causing negative feedback.  In artificial bee colony, a colony of artificial forager bees (agents) search for rich artificial food sources (good solutions for a given problem). To apply artificial bee colony, the considered optimization problem is first converted to the problem of finding the best parameter vector which minimizes an objective function. Then, the artificial bees randomly discover a population of initial solution vectors and then iteratively improve them by employing the strategies: moving towards better solutions by means of a neighbor search mechanism while abandoning poor solutions.
In this modeling, the data used are related to a laboratory data set of small-scale pile load capacity. Various statistical indicators were used to evaluate the modeling accuracy. Finally, the results showed that the combined relevant vector regression with the artificial bee colony algorithm for test data with R² = 0.975 and RMSE = 0.001, has a high ability to predict the lateral load capacity of spark plugs. In addition, the sensitivity analysis performed in this study showed that the variables of eccentricity of load and the length of pile are more important and effective compared to other parameters.

Keywords: Relevant vector regression, Artificial bee colony algorithm, Lateral load capacity, Sensitivity analysis

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