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Figure 7. Cross correlations resulting from the verification algorithm
The test for exogenous third variables inducing a spurious correlation between two elements of the
causal system is performed by the principles as proposed in this chapter. Since the variations of both, the
variable average salary and price of raw material are not only reflected in the time series of their direct
successor but also in those of the remaining elments it can be concluded that the potentially causal rela-
tions SR → CG as well as CG → SP are genuinely causal. As the latter proves that the direction of this
relation is unambiguous, the potentially causal relation SP → CG is marked as spurious association.
Approximating Causal f unctions
The next step according to the proposed approach is to approximate the unknown causal functions for
each causal function kernel which consists of a positive number of proven causal relations. Therefore
this approach derives three multi layer perceptrons (MLP) from the identified causal strategy model. For
the K SR this neural network consists of the output node sr t as well as of an input layer consisting of as t-1 ,
as t-2 and rm t-1 and a four-element hidden layer being dimensioned using a heuristic proposed by Baum
and Haussler (1988). Another MLP - approximating the causal function of K SP - is established with the
time series cg t-1 , cg t-2 and cg t-3 forming the input nodes and sp t as an output node. The causal input and
output nodes of the causal function approximator for K CG are derived analogously: The time series sr t-1 ,
cg t-2 , rm t-2 and rm t-3 representing the input layer and cg t as output node. However, since this CFK contains
the influence relation RM → SR which causes multicollinearity, it is necessary to distinguish between
first and second level input nodes and connect them with correcting function approximators. Therefore
the temporal disaggregation algorithm introduces two correcting MLPs: One of them contains sr t-1 as
output and rm t-2 as input node and the other receives input values from rm t-3 and has sr t-2 as output ele-
ment. The resulting MLP is shown in Figure 8, where dashed arcs represent auxiliary correcting MLPs
accounting for the relation RM → SR and solid links are part of the main function approximator.
After deriving the three neural function approximators as shown above, the latter have to be trained
in order to inductively learn the unknown causal function from the empirical data: Therefore the time
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