Regression

Regression attempts to find the Base and Delta values which give the least-squares best fit for all the Data Sets included in the Model Generator Case.

where:

• aj is the value of the jth input

• âjbase is the base value of the jth input - this value is a constant offset taken directly from the base dataset

• βi is the base value for the ithoutput in the Base + Delta model - in general the βi are not identical to the outputs for the base dataset

• Δij is the full matrix of Δ values, each column being a single Δ vector

• σj is the user-defined scaling for the jthinput

• Ai is the adjuster value for the ithoutput - this is a user-defined value which is not used in the regression i.e. the regression takes all adjustor values to be zero

The primary inputs to the regression are the datasets which may be calculated from a core process model or entered manually. Each dataset contains a complete set of input and output variables. The regression problem consists of calculating the set of βiand Δij which minimise the squared deviation between the outputs taken from each of the datasets and the corresponding modelled values, summed over all datasets:

where:

• ûi is the ithoutput taken from a particular dataset

• ui is the corresponding modelled value calculated from the inputs for the same dataset.

This is a standard least-squares problem. If the outputs are not inter-related in any way, then the base and Δ vectors can be determined independently for each output.

It may be necessary to force the Base + Δ model through the base dataset in which case the base output values in which case output values βiare no longer regressed but are simply fixed to the outputs given for the base dataset:

It may also be necessary to manually override specific base or Δ values in which case the values are fixed and not regressed.

Δ values for a specific set of outputs U may be constrained to sum to zero, or the sum of the Δ values for a different set of outputs V, leading to a constraint of the following form:

For example the yields of a particular process unit may have to be constrained in order to balance the mass.