NNs variable selection method.
4.2.3 Interpreting Model Weights
Relatively few studies are carried out with the aim of
developing methods for variable contribution
analysis in NNs models – perhaps at least in part due
to seeming complexity of the task.
Variable contribution analysis methods have
been examined and compared by Gevrey,
Dimopoulos and Lek (2003). One of the seven
methods they surveyed included a computation that
used connection weights to provide explanatory
dimension to a NNs model using ecological data.
First proposed by Garson (1991) and later further
investigated by Goh (1995), the procedure is set to
determine the relative importance of the inputs by
partitioning the connection weights. Essentially,
hidden-output connection weight of hidden neurons
is partitioned into components associated with the
input neurons (please see more in Appendix A of the
(Gevrey et al., 2003)). Authors concluded that
method that uses connection weights was able to
provide a good classification of input parameters
even though it was found to lack stability.
One of the concerns conveyed regarding the
otherwise extensive investigation of different
methods was that the dataset originally employed in
2003 study (Gevrey et al., 2003) was empirical, and
therefore did not allow to ascertain the factual
precision and accuracy of each method as the true
relations between the variables are not known
(Olden et al., 2004). Instead, the artificial dataset
was created using the Monte Carlo simulation and
employed to assess true accuracy of each method
using the dataset with defined and therefore knows
relations. Results showed that weights method that
uses input-hidden and hidden-output connection
weights showed consistently best results out of all
methods assessed, contrary to Gevrey et al., (2003)
findings. Additionally, the weights method was able
to accurately identify the predictive importance
ranking, whereas other methods were only able to
identify the first few if any at all (Olden et al.,
2004).
Olden and Jackson (2002) also used ecological
data to demonstrate the predictive and explanatory
power of NNs. A number of methods surveyed,
including Neural Interpretation Diagram, Garson’s
algorithm and sensitivity analysis, aid in
understanding the mechanics of NNs and improve
the explanatory power of the models. Interpretation
of statistical models is imperative for acquiring
knowledge about the causal relationships behind the
phenomena studied. They also propose a
randomization approach for statistical evaluation of
the importance of connection weights and the
contribution of input variables in the neural network
(already discussed in details in the sections above).
Nord and Jacobsson (1998) have also addressed
the issue of explaining and interpreting NNs
structure and developed algorithms for variable
contribution analysis. The study compared the
proposed novel algorithmic approach for NNs model
interpretation with the analogous variable
contribution method of partial least squares
regression. Sensitivity analysis is also performed
through setting each input to zero in a sequential
manner. Linear regression coefficients for each of
the input variables have also been generated for the
purposes of examining the variable contribution
direction. The results of the two approaches are then
reviewed and compared to the results of the partial
least squares regression. What the study is able to
reveal is that in the linear dataset both the partial
least squares regression and NNs models show
similar performance in the variable contribution
task, whereas with the nonlinear data the differences
become obvious (Nord and Jacobsson, 1998).
Andersson et al., (2000) present two methods to
study variable contribution in NNs models: (1) a
variable sensitivity analysis and (2) method of
systematic variation of variables. Variable
sensitivity analysis is based on setting the
connection weights between the input and hidden
layer to a zero in a sequential manner, whereas the
systematic variation of variables method is based on
keeping the other variables constant or manipulated
simultaneously. In the course of their study, it is
shown that there is a high similarity between the
method proposed by the authors for the variable
contribution analysis in NNs models and the nature
of the processes used to develop the synthetic
datasets used. Thus, it is shown that the NNs models
are suitable not only for the function approximation
in nonlinear datasets, but are also able to accurately
reflect the characteristic qualities of the input data.
The transparency of highly interconnected NNs
models could be demonstrated in response to the
‘black box’ argument. Presented method is then able
to generate information about the variables that
could be useful in examination and interpretation of
variable contribution and relations.
The discussed earlier method of Nord and
Jacobsson (1998) is based on the saliency estimation
principles (such as Optimal Brain Surgeon, Optimal
Brain Damage, etc.) as it estimates the consequence
of weight deletion on prediction error. The
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