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In-Depth Information
The analysis phase of the algorithm possesses computational complexity O(mN log
N) and can be efficiently implemented using data stream operations similar to FORM.
Precision of the method is estimated using standard deviation formula above.
Remarkably, the precision depends neither on dimension of parameter space n, nor on
the length of simulation result m, but only on sample size N=Nexp and user-specified
confidence level F=C. For instance, CL determination at the level 68% (F=0.16) with
4% precision requires Nexp=84, while for 68%±1% one needs Nexp=1344.
3.4
Monte Carlo Combined with RBF Metamodel (MC-RBF)
Large sample size is required for precise determination of CL with Monte Carlo
method. To reduce the number of required simulations, RBF metamodel can represent
the mapping f(x) during analysis phase of CL-MC. While a metamodel can be
constructed using a moderate number of simulations, e.g. Nexp~100, determination of
CL can be done with N>>Nexp. Application of RBF metamodel for CL computation
proceeds similarly to CL-MC. The only difference is that Nexp parameter points
generated at phase (CL1) are used as input for the metamodel. They should not
necessarily possess user specified distribution, but one providing better precision of
metamodel, i.e. better covering "the corners" of parameter space. It is especially
important for populating tails of distribution, corresponding to high confidence e.g.
99.7% CL. Uniform distribution is suitable for this purpose. Then, after numerical
simulations at phase (CL2), and after filtering out failed experiments, the actual
distribution
(x) is used to generate N parameter points, and construct RBF weight
matrix w ij =w i (x j ), i=1..Nexp, j=1..N. This matrix is used in phase (CL3) for
multiplication with simulation results y ik , k=1..m, comprising O(m N Nexp)
operations, which usually prevails over O(m N log N) operations needed for sorting
of interpolated samples.
ρ
4
Causal Analysis
Causal analysis is determination of cause-effect relationships between events. In
context of crash test analysis, this usually means identification of events or properties
causing the scatter of the results. This allows to find sources of physical or numerical
instabilities of the system and helps to reduce or completely eliminate them.
Causal analysis is generally performed by means of statistical methods,
particularly, by estimation of correlation of events. It is commonly known that
correlation does not imply causation (this logical error is often referred as "cum hoc
ergo propter hoc": "with this, therefore because of this"). Instead, strong correlation of
two events does mean that they belong to the same causal chain. Two strongly
correlated events either have direct causal relation or they have a common cause, i.e. a
third event in the past, triggering these two ones. This common cause will be
revealed, if the whole causal chain i.e. a complete sequence of causally related events
will be reconstructed. Practical application of causal analysis requires formal methods
for reconstruction of causal chains.
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