Inmathematics,statistics, andcomputational modelling, agrey box model[1][2][3][4] combines a partial theoretical structure with data to complete the model. The theoretical structure may vary from information on the smoothness of results, to models that need only parameter values from data or existing literature.[5] Thus, almost all models are grey box models as opposed toblack box where no model form is assumed orwhite box models that are purely theoretical. Some models assume a special form such as alinear regression[6][7] orneural network.[8][9] These have special analysis methods. In particularlinear regression techniques[10] are much more efficient than most non-linear techniques.[11][12] The model can bedeterministic orstochastic (i.e. containing random components) depending on its planned use.
The general case is anon-linear model with a partial theoretical structure and some unknown parts derived from data. Models with unlike theoretical structures need to be evaluated individually,[1][13][14] possibly usingsimulated annealing orgenetic algorithms.
Within a particular model structure,parameters[14][15] or variable parameter relations[5][16] may need to be found. For a particular structure it is arbitrarily assumed that the data consists of sets of feed vectorsf, product vectorsp, and operating condition vectorsc.[5] Typicallyc will contain values extracted fromf, as well as other values. In many cases a model can be converted to a function of the form:[5][17][18]
where the vector functionm gives the errors between the datap, and the model predictions. The vectorq gives some variable parameters that are the model's unknown parts.
The parametersq vary with the operating conditionsc in a manner to be determined.[5][17] This relation can be specified asq =Ac whereA is a matrix of unknown coefficients, andc as inlinear regression[6][7] includes a constant term and possibly transformed values of the original operating conditions to obtain non-linear relations[19][20] between the original operating conditions andq. It is then a matter of selecting which terms inA are non-zero and assigning their values. The model completion becomes anoptimization problem to determine the non-zero values inA that minimizes the error termsm(f,p,Ac) over the data.[1][16][21][22][23]
Once a selection of non-zero values is made, the remaining coefficients inA can be determined by minimizingm(f,p,Ac) over the data with respect to the nonzero values inA, typically bynon-linear least squares. Selection of the nonzero terms can be done by optimization methods such assimulated annealing andevolutionary algorithms. Also thenon-linear least squares can provide accuracy estimates[11][15] for the elements ofA that can be used to determine if they are significantly different from zero, thus providing a method ofterm selection.[24][25]
It is sometimes possible to calculate values ofq for each data set, directly or bynon-linear least squares. Then the more efficientlinear regression can be used to predictq usingc thus selecting the non-zero values inA and estimating their values. Once the non-zero values are locatednon-linear least squares can be used on the original modelm(f,p,Ac) to refine these values .[16][21][22]
A third method ismodel inversion,[5][17][18] which converts the non-linearm(f,p,Ac) into an approximate linear form in the elements ofA, that can be examined using efficient term selection[24][25] and evaluation of the linear regression.[10] For the simple case of a singleq value (q =aTc) and an estimateq* ofq. Putting dq = aTc − q* gives
so thataT is now in a linear position with all other terms known, and thus can be analyzed bylinear regression techniques. For more than one parameter the method extends in a direct manner.[5][18][17] After checking that the model has been improved this process can be repeated until convergence. This approach has the advantages that it does not need the parametersq to be able to be determined from an individual data set and the linear regression is on the original error terms[5]
Where sufficient data is available, division of the data into a separate model construction set and one or twoevaluation sets is recommended. This can be repeated using multiple selections of the construction set and theresulting models averaged or used to evaluate prediction differences.
A statistical test such aschi-squared on the residuals is not particularly useful.[26] The chi squared test requires known standard deviations which are seldom available, and failed tests give no indication of how to improve the model.[11] There are a range of methods to compare both nested and non nested models. These include comparison of model predictions with repeated data.
An attempt to predict the residualsm(, ) with the operating conditionsc using linear regression will show if the residuals can be predicted.[21][22] Residuals that cannot be predicted offer little prospect of improving the model using the current operating conditions.[5] Terms that do predict the residuals are prospective terms to incorporate into the model to improve its performance.[21]
The model inversion technique above can be used as a method of determining whether a model can be improved. In this case selection of nonzero terms is not so important and linear prediction can be done using the significanteigenvectors of theregression matrix. The values inA determined in this manner need to be substituted into the nonlinear model to assess improvements in the model errors. The absence of a significant improvement indicates the available data is not able to improve the current model form using the defined parameters.[5] Extra parameters can be inserted into the model to make this test more comprehensive.
