PoLogCem 4.2. How to build the nonlinear regression cement plant?

4.2. How to build the nonlinear regression
models for the pollution generated by a
cement plant?
The images presented in the next figures illustrate the steps to follow in designing the
data models. Firstly, the user will select the output measurement and then push the
DISPLAY TABLE button and the results will be displayed as correspondence table
for the selected output measurement with all the values of the input measurements for
the corresponding output. Both classical and piecewise modeling is possible. In the
Fig. 6, is shown a classical modeling. Considering interval data, the piecewise
modeling is possible as shown in the Fig. 7.
Press here to see the table of data
The selection of the
output model
Here, there is the table containing the
data related to dependent (Y) and
independent (X) variables.
Fig. 6. The automatic loading of values for the output dependent variable
The table of intervals to be considered for
the selected parameter
The data can be exported to XLD / Dbase files
Fig. 7. Piecewise modelling
Functions defined
by user. After
solving the model,
the coefficients
are shown in the
third column.
Press the button to
solve the model.
Fig. 8. Defining the structure of the regression model
In order to determine the mathematical models the following information is valuable:
the output measurement must be selected and then in the left-down table will be filled
the linear/nonlinear functions in a manner enabling us to obtain a model with the
following expression:
y = A0+ A1*F1(x1, ..., xm) + A2*F2(x1, ..., xm) + AnFn(x1, x2, …, xm),
where m is the number of independent variables, and y is the dependent variable. The
functions F1, F2,…, Fm must be defined by the user in the editing window (the right
down corner in the Fig. 8). Let us denote by Yk si Yk the measured respective the
computed output variable, and by Y , the mean value of the variable Y given by the
values Yk , k = 1, 2, … . Let Sy be the deviance to the mean of the measured data, and
S ^ be the deviance from the mean of the data obtained after the solving of the
regression model. Therefore, S y = ∑ (Yk − Y ) 2 and S ^ = ∑ (Yk − Y ) 2 . To compare the
k =1
k =1
measured model against de regression model, the software uses the following indicator
(adequacy level): R 2 = Y .
The adequacy
Press button for
more Knowledge
to be added in the
Database Rules.
It is shown the measured versus modeled
data in the case of the dependent variable.
Fig. 9. Updating the Knowledge Database
Note: The software PoLogCem uses the notation R instead of R2 (Fig. 8 and Fig. 9).
After the model was created (the structure was established and the coefficients were
found) it will be added (using the button APPEND TO DATABASE RULES) to the
existing general set of rules (Fig. 9). The name (label) of the new rule is automatically
generated. The rules are needed to the optimization module.