Evolutionary Polynomial Regression

evolutionary Polynomial RegressionEvolutionary polynomial regression (EPR) is a information mining technique based on evolutionary computing that was capture by Giustolisi and Savic (2006). It combines the power of genetic algorithm with numerical regression to beget symbolical puzzles. EPR is a two-step technique in which, at the primary step, exp starnts of symbolic structures are hunt clubed using a genetic algorithm (GA) that is the cardinal judgment behind the EPR, and in the second step, the parameters of the symbolic structures are ascertain by solving a linear least squares bother. The general symbolic expression used in EPR can be presented as followsWhere y is the estimated output of the process, m is the keep down number of the polynomial harm which excludes the diverge landmark a0, F is a function constructed by the process, X is the hyaloplasm of independent arousal variable quantitys, f is a function defined by the user, and aj is a constant value for jth term.The first step and key idea in identification of the model structure in EPR is to transfer equivalence 1 into the avocation vector formWhere is the least-squares estimate vector of the N target values is the vector of d=m+1 parameters aj and a0 ( is the transposed vector) and is a intercellular substance formed by (unitary vector) for bias a0, and m vectors of variables. For a fixed j, the variables are a product of the independent predictor vectors of inputs, .EPR starts from comparison 2 and searches for the best structure, i.e. a combination of vectors of independent variables (inputs) .The matrix of input X is given as 15Where the kth column of X represents the candidate variable for the j th term of Equation 2. Therefore the jthterm of Equation 2 can be written asWhere, Z jis the jthcolumn vector in which its elements are products of candidate independent inputs and ES is a matrix of exp starnts. Therefore, the problem is to prevail the matrix ESkmof exponent s whose bounds are specified by the user. For example, if a vector of candidate exponents for inputs, X , (chosen by user) is EX=0,1,2 and number of legal injury (m) (excluding bias) is 4, and the number of independent variables (k) is 3, then the polynomial regression problem is to find a matrix of exponents ES 4-3 15. An example of such a matrix is given hereEach exponent in ES corresponds to a value from the user-defined vector EX. Also, individually row of ES determines the exponents of the candidate variables of jth term in comparisons (2). By implementing the above values in equation (4), the following set of expressions is obtainedTherefore, based on the matrix given in equation (5), the expression of equation (2) is given asIn the next breaker point, the adaptable parameters, aj, can now be computed, by means of the linear least(prenominal) Squares (LS) method.The original EPR methodological analysis was based on Single-objective Genetic Algorithm (SOGA) for look the space of solutions while penalizing complex model structures using some punishment strategies.In this method, in the first stage, the maximum value for the number of terms (m) is assumed then a consecutive search for the formulas having 1 to m terms is undertaken. To accelerate convergence, the results obtained in each stage of search could be randomly entered into the population of the next stage search 15.However the single-objective EPR methodology showed some drawbacks, and therefore the multi-objective genetic algorithm (MOGA) strategy has been added to EPR.In 2006, Guistolisi and Savic (2006) change the EPR technique to overcome these shortcomings, using Multi-Objective Generic Algorithm (MOGA) instead of SOGA. The principal(prenominal) features of the developed method are as follows 221) Increasing the model accuracy,2) cut down the number of polynomial coefficients,3) Minimization of the number of inputs (e.g. the number of times each Xi appears in the model).In the devel oped version, a simultaneous search is conducted for polynomials having 1 to m coefficients consequently, it is faster than the previous version (i.e., SOGA).In aver to determine all models corresponding to the optimal trade-off between geomorphologic complexity and fitness level of the model, The EPR technique is Equipped with a enjoin of objective functions which help to optimize the result based on Pareto authority criterion.The objective functions used are (i) Maximization of the fitness (ii) Minimization of the total number of inputs selected by the modeling strategy (iii) Minimization of the length of the model expression.The objective functions mentioned above can be used in a two objective configuration or all together. In which one of them will limit the complexity of the models, while at least one objective function controls the fitness of the models.In this study the multi-objective EPR is used to develop the EPR-based models.The coefficient of determination (COD) whic h is used to evaluated the level of models accuracy at each stage isWhere Ya is the actual measured output value Yp is the EPR-predicted value, and N is the number of selective information points in which the COD is computed.