Response Surface ModelsMetamodels

 

Response surface Method is an integration of statistical and mathematical techniques useful for developing, improving, and optimizing process. The most extensive applications of RSM are in the industrial world, particularly in situations where several variables potentially influence some performance measure or quality characteristics of the product or process. In general, a product or system response  depends on the controllable input variables . The relationship i

 

where the form of true response function is unknown and perhaps very complicated, and  is a term that represents other sources of variability not included in . In general, the statistical error  is a normal distribution. Suppose that the response surface models  may be represented by

 (Linear Model) and

 (Quadratic Model)

 

Where  are the unknown coefficients. If there is more than one data point under consideration, the linear model is extended to the matrix form

 

 

where is a vector of observations,  is a matrix of known constant,  is a vector of  parameters, and  is the vector of random errors. In order to obtain the unknown coefficients, we solve the sum of squares of residuals as . To minimize, we solve a set of equations. Hence the normal equation  is obtained. The matrix is a symmetric matrix with  rows and columns. Its rank is the same as the rank of , which is the number of linearly independent column of .

If the columns of  are linearly independent,  exists and the normal equations have a unique set of solutions. However, if  is less than full rank because the columns are not linearly independent, is singular and  does not exist. It will, in fact, be true of almost the experimental design models. Hence special numerical techniques are required to construct the general response surface modeling tools.