Soft Sensing - Advanced Control and Supervision of Mineral Processing Plants - page 169

dergoes changes. It may also happen that models with different structures may be
the best choice when the operating point moves to other operating regions, so the
model must be changed. For a smooth transition between two regions clustering and
membership functions may be used, as described below under Clustering. There are
various indexes defined for the training stage of structure determination.
In order to reduce the complexity of the model to be determined the number
of parameters or the number of bases is penalized in relation to the length of the
training record. Examples are the FPE (final prediction error) and AIC (Akaike's
information criterion) indexes:

log V

1 + m / T

2 m FPE = AIC = ( 1 +

T ),

(4.74)

1 − m / T

where T is the length of the data sets, m is the number of model parameters and
V is the estimation of the mean square prediction error or the modeling error (4.2)
estimated by (4.4) [44]. Another index that has been used in connection with re-
ducing complexity of neural network models is the Lipschitz number to select the
inputs, and a weighted sum of the modeling error and a term penalizing the sum of
the squares of the parameters [14]. Other indexes used for model structure determi-
nation are given by Fortuna et al. [1].
All these indexes tend to avoid overfitting, which may cause the mean square
output error to be very small for the training data set but very large for validation
(generalization) data sets. To ensure that this situation does not occur, in addition
the structure determination should be done with one set of data called the training
set and tested with a different set, the test or validation set.

4.2.4.2 Clustering

The advantages of dealing with a simple, parsimonious model may sometimes be
achieved by means of separating the operation region into clusters, each of which
has a good simple model. Then if the operating point is identified as belonging to a
given cluster, the corresponding model is used. Such a procedure has been used in
building a soft sensor for concentrate grade measurement in an industrial rougher
flotation plant [26]. Much better results were obtained with clustering than with the
case of a single model determined for the complete operating range.
In the design of a soft sensor for the rich copper concentrate grade g cc of an
industrial flotation rougher flotation plant two clusters, C 1 and C 2 were employed
using fuzzy C-means clustering [31, 51]. For each cluster a soft sensor model was
determined. From the relevant membership functions μ 1 ( g cc ) and μ 2 ( g cc ) relative

membership functions were defined as

μ 1 ( g cc ) μ 2 ( g cc )

ψ 1 ( g cc )= ψ 2 ( g cc )=

) .

(4.75)

μ 1 ( g cc )+ μ 2 ( g cc ) μ 1 ( g cc )+ μ 2 ( g cc