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A practical problem of causal analysis in crash-test simulations is often not a
removal of a prime cause of scatter, which is the crash event itself. It is more an
observation of propagation paths of the scatter, with a purpose to prevent this
propagation, by finding regions where scatter is amplified (e.g. break of a welding
point, pillar buckling, slipping of two contact surfaces etc). Since a small cause can
have large effect, formally earliest events in the causal chain can have a microscopic
amplitude ("butterfly effect"). Therefore it is reasonable to search for amplifying
factors and try to eliminate them, not the microscopic sources.
As input for causal analysis the centered data matrix dy
ij
, i=1..m, j=1..Nexp is used.
Here every column forms one experiment, every row forms a data item varied in
experiments, and the mean value <y> is row-wise subtracted from the matrix. Then
every data item is transformed to a z-score vector [8]:
z
ij
=dy
ij
/|dy
i
|, |dy
i
|=sqrt(
j
dy
ij
2
),
(18)
or by means of the equivalent alternative formula
z
ij
=dy
ij
/(s(y
i
)(N
exp
)
1/2
), s(y
i
)=(
j
dy
ij
2
/N
exp
)
1/2
.
(19)
Here s(y
i
) is the root mean square deviation of the i-th data item, which can serve as a
measure of scatter. In this way the data items are transformed to m vectors in Nexp-
dimensional space. All these z-vectors belong to an (Nexp-2)-dimensional unit-norm
sphere, formed by intersection of a sphere |z|=1 with a hyperplane
j
z
ij
=0. The scalar
product of two z-vectors is equal to Pearson's correlator of data items:
(z
1
,z
2
) =
j
z
1j
z
2j
= corr(y
1
,y
2
).
(20)
An important role of this representation is the following. Strongly correlated data
items correspond either to coincident (z
1
=z
2
) or opposite (z
1
=-z
2
) z-vectors. If not the
sign but only the fact of dependence is of interest, one can glue opposite points
together formally considering a sphere of z-vectors as projective space. Using this
representation, one can apply [13,14] general purpose clustering methods such as k-
means to group data items distributed on this sphere to a few strongly correlated
components.
In spite of their numerical efficiency, these clustering methods neglect temporal
ordering of events, while in causal analysis the task is to find an earliest physically
significant event in the causal chain. In crash test simulation such events correspond
to bifurcation points, where the scatter appears "ex nihilo". Such points are clearly
visible as spikes in dynamical scatter plots s(y), the problem is that there are too many
of them. Although decision between potential candidates by a formal algorithm can be
difficult, an engineering knowledge allows narrowing the search to significant parts
where scatter propagation can be really initiated by physical effects, such as buckling
of longitudinal, break of welding point etc. The other problem is that in bifurcation
points new scatter is just appeared and it is generally hidden under the consequences
of previous effects. At first one needs to separate scatter contributions.
Considering two data items dy(a) and dy(b), one can define contribution relevant to
the data item (a) in (b) as follows:
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