Soft Sensing - Advanced Control and Supervision of Mineral Processing Plants - page 170

The total soft sensor output was then obtained by a weighted combination of
the outputs of soft sensors for clusters C 1 and C 2 using the relative membership
functions as weights. Figure 4.7 shows the evolution of these relative membership
functions for the data in Figures 4.2 and 4.8. The clustering approach was also used
for this plant to design a Takagi and Sugeno model [31].
Clustering has also been used for a soft sensor designed to model an event and
the proximity to the occurrence of such event [26]. In this case the objective was
to give early warning for the event defined by the overload of a ball mill in an
industrial grinding plant. C-means clustering was used to group the plant's operating
points. Later developments led to retain two clusters representing two different types
of overload. The distances from the operating point to each of these two clusters,
computed on-line, give an indication of how far the operation was from reaching
one of the two overload conditions. Different corrective actions could then be taken
according to the distance to the nearest cluster.
4.2.4.3 Stepwise Regression
This is one of the methods used to determine which are the most significant com-
ponents (bases) associated with the soft sensor model primary measurement. One
version of stepwise regression begins by selecting that component from a set of
candidate components which is most closely correlated with the variable to be mod-
eled, i.e. , the primary variable. In this way a first partial model is determined. The
residue of this model is then correlated with the remaining candidate components.
In each stage that follows, the component to be included is the particular component
that gives the largest partial correlation with the previous step residue, calculated af-
ter a multilinear regression is performed with the previously selected components.
With this procedure, the model structure increases through the addition of one com-
ponent at a time, such that the added component is the one that contributes the
greatest improvement to the goodness of fit to the model. Before each new compo-
nent is included in the model, though, it is tested for its statistical significance. Each
model parameter estimate has an estimated standard deviation. The ratio between
this standard deviation and the coefficient value, is used to decide the inclusion of
every new component. If for any component this ratio exceeds a given threshold
then the corresponding component is not included, or excluded if it had been pre-
viously included. This procedure is repeated until no component from the list of
candidate components is either deleted or included in the model being determined.
Alternatively the procedure stops if inclusion of new candidates does not cause a
significant improvement according to an F-test [52].
MATLAB ® stepwise or stepwisefit functions may be used to perform stepwise
regressions. It should be mentioned that stepwise models are locally optimal, but
may not be globally optimal (Mathworks™ 1 , stepwisefit) [53]. Another MATLAB ®
function for model structure determination is lsselect [54] with which models with
1 Mathworks is a registered trademark of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA
01760-2098, USA, http://www.mathworks.com