Analysis of Bulky Crash Simulation Results: Deterministic and Stochastic Aspects - Simulation and Modeling Methodologies, Technologies and Application - page 222

dy| a (b) = corr(a,b)s(b)z(a).

(21)
After subtraction of this contribution a residual dy(b)-dy| a (b) does not correlate with
dy(a), and the scatter
s 2 | a (b)=<(dy(b)-dy| a (b)) 2 >=s 2 (b)(1-corr 2 (a,b)) (22)

does not increase.
Armed with this subtraction procedure, we propose the following algorithm for
causal analysis.
Temporal clustering:
(T1) visualize scatter state-by-state;
(T2) isolate bifurcation point;
(T3) subtract its contribution from the scatter in consequent states;
(T4) if scatter is still remaining, goto (T1).
Here subtraction of scatter from previous bifurcations reveals new bifurcations hidden
under the consequences of previous ones. The remaining scatter monotonously falls
during the iterations, and the iterations can be stopped when the scatter becomes small
everywhere or in the regions of interest.
The geometrical meaning of subtraction procedure: b-b| a =b-(a,b)/(a,a)a is an
orthogonal projection in the space of data items and the whole sequence is Gram-
Schmidt (GS) orthonormalization procedure applied in the order of appearance of
bifurcation points a i . The obtained orthonormal basis g i =GS(a i ) can be used for
reconstruction of all data by the formula:

dy= i Ψ

i g i +res,

Ψ

i =(dyg i ).

(23)

The norm of residual is controlled by remaining scatter, which is small according to
our stop criterion:

|res| 2 /Nexp=s ⊥

2 (y)=s 2 (y)- i Ψ i 2 /Nexp.

(24)

Algorithmically every i-th iteration one computes a scalar field Ψ i describing

contribution of i-th bifurcation point to scatter of the model and a scalar field s i⊥

2 (y)
used for determination of the next bifurcation point a i+1 , or for stop criterion
s i⊥

2

(y)<threshold. This requires O(mNexp) floating point operations per iteration.
Matrix decomposition of the form dy= Ψ g is similar to PCA described above, with

the other meaning of the modes are scalar fields distributed over

dynamical model which are common for all experiments. They have the other
temporal profile than PCA modes, reflecting causal structure of scatter: they start at
corresponding bifurcation points and propagate forward in time. Differently from
PCA modes, they are not orthogonal columnwise, i.e. with respect to scalar product
over the model. g-coefficients form Nexp*Nmod columnwise orthonormal matrix.
Like corresponding matrix in PCA, they define an orthonormal basis in the space of
experiments, with respect to the scalar product coincident with Pearson's correlator.

Ψ . Like in PCA, Ψ