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Dependence modelling with regular vine copula models: A case-study for car crash simulation data

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The analysis of car crash output parameters such as firewall intrusion points assist the overall engineering process. Such data are nowadays collected from many numerical simulations and it is not possible for the engineer to analyse this growing amount of data by hand. Therefore, data mining and statistical methods are needed. Here, we propose to use the flexible class of regular vine (R-vine) copulas for modelling the dependence between such output variables. R-vine copulas are multivariate copulas constructed hierarchically from bivariate copulas as building blocks. We introduce the concept of such constructions and their graphical tree representation. Applied to simulated frontal crash data of a Ford Taurus such graphs help us to illustrate the dependence structure among different firewall intrusion locations. The big advantage of R-vines compared with standard approaches such as the multivariate normal distribution or the multivariate Gaussian copula is the ability to model asymmetries and dependence in the tails. Our application demonstrates the strong potential of R-vines in the engineering context and opens further application areas.
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Dependence modeling with R-vine copula models: A
case study for car crash simulation data
Ulf Schepsmeier
Lehrstuhl f¨ur Mathematische Statistik, Technische Universit¨
at M¨unchen, Parkring 13,
85748 Garching-Hochbr¨uck, Germany.
Claudia Czado
Lehrstuhl f¨ur Mathematische Statistik, Technische Universit¨
at M¨unchen, Parkring 13,
85748 Garching-Hochbr¨uck, Germany.
Summary. The analysis of car crash output parameters such as fire wall intrusion
points assist the overall engineering process. Such data are nowadays collected
from many numerical simulations and it is not possible for the engineer to analyze the
growing amount of data by hand. Therefore, data mining and statistical methods are
needed. Here, we propose the flexible class of regular vine (R-vine) copula models
for modeling the dependence among such output variables.
R-vine copulas are multivariate copulas constructed hierarchically from bivariate cop-
ulas as building blocks. We introduce the concept of such constructions and their
graphical tree representation. Applied to simulated frontal crash data of a Ford Tau-
rus such graphs help us to assist in the engineering process showing the depen-
dence structure among different fire wall intrusion locations. The big advantage of
R-vines compared to standard approaches like the multivariate normal distribution or
the multivariate Gaussian copula is the ability to model different tail dependencies.
Our example demonstrates the strong potential of R-vines in the engineering context
and opens further application areas.
Keywords: car crash simulation, dependence modeling, FEM model, R-vine
copula
2Schepsmeier and Czado
1. Introduction
Preventing mortal injuries of car drivers in a crash and virtual product development
are probably the main research and innovation areas in the automotive industry.
Nowadays research is based on hundreds of computer simulation runs with (stochas-
tically) varied input parameters such as speed or the thickness of different plates.
Finite element models (FEM; see for example Chaskalovic, 2008) are essential tools
to analyze the influence of design parameters on the functional properties of the car
structure. Since CPU costs are falling exponentially and FEM-crash model sizes
are increasing at the same time the amount of data is growing fast (Blumhardt,
2001). Therefore, data mining and statistical methods are more and more in focus
of engineers (Mei and Thole, 2008). Here we will introduce and apply a new statis-
tical model class to analyze such high dimensional data in the engineering context.
Regular vine (R-vine) copula models are a flexible class of multivariate copulas and
are especially suited for dependence modeling in high dimensions.
One analysis aspect is the look at so called output parameters like the Head
Injury Criterion (HIC) Index or the fire wall intrusion. The fire wall is a group of
metal plates between the engine and the passenger space shielding the car passenger
from noise, smell, emissions and most importantly from fire and deformed metal
pieces in case of a frontal crash. The fire wall intrusion is measured at different
FEM-nodes and indicates the displacement of parts in a crash. Thus, the analysis
of the fire wall intrusion grants important information about the structural behavior
of the car during a crash and the safety of the passengers. Thereby it significantly
assist in the overall engineering process.
Such an evaluation is performed in this case study using R-vine copula models.
In general, copulas allow to model univariate effects and joint dependence structures
separately. By Sklar’s Theorem (Sklar, 1959) we can decompose a continuous mul-
tivariate cumulative distribution function (cdf) F(x1,...,xd) of random variables
X1,...,Xdwith marginal cdfs F1(x1), . . . , Fd(xd) into a unique copula distribution
Dependence modeling with R-vines 3
function Cand the marginals, i.e.
F(x1,...,xd) = C(F1(x1),...,Fd(xd)).(1)
The problem of standard multivariate copulas such as the multivariate Gaussian
copula is that they are less flexible in higher dimensions. In the bivariate case
parametric copulas are very flexible and are able to model most of the data’s fea-
tures, e.g. tail dependence and asymmetries. Pair-copula constructions (PCC)
make use of the bivariate copulas as building blocks to decompose a multivari-
ate distribution function. Joe (1996) and later Bedford and Cooke (2001, 2002)
independently constructed multivariate densities using d(d1)/2 bivariate copu-
las. Bedford and Cooke (2001) introduced a set of nested trees to help to orga-
nize the decomposition and called the resulting graph structure a vine. Aas et al.
(2009) developed the statistical inference to this construction and their work form
the basis for subsequent work. In particular Czado et al. (2012) developed model
selection algorithms for a sub-class, while Dißmann et al. (2013) consider the gen-
eral class. Parsimony can be achieved through truncation (Brechmann et al., 2012)
and analytic expressions for standard errors of the parametric estimates are derived
(St¨ober and Schepsmeier, 2013). Surveys for estimation and model selection are also
available (Czado, 2010; Czado et al., 2013). The books by Kurowicka and Cooke
(2006) and Kurowicka and Joe (2010) provide details and further properties. Be-
cause of their very flexible and easy construction R-vine copula models became quite
popular in many fields. For example Hanea et al. (2006) use them for Bayesian
Belief Nets. But the main application area so far for the vine copulas are fi-
nance (Brechmann and Czado, 2013) or agricultural science and electricity loads
(Smith et al., 2010). Applications in the engineering context are not known to
the authors. We will show that vine copula models are very useful to analyze car
crash simulations gaining additional knowledge about model behavior with regard
to safety aspects. They improve the engineer’s understanding of the car structure
and behavior in a frontal crash.
In Section 2 we describe the fire wall intrusion data set extracted from a car crash
4Schepsmeier and Czado
simulation of a Ford Taurus, while our proposed model, the R-vine, is introduced
in Section 3. The analysis applying R-vines and our results are given in Section
4. The final Section 5 is summing up and suggests further research areas for the
R-vine copula models.
2. Fire wall intrusion data of a Ford Taurus
We consider a frontal crash simulation of a Ford Taurus, a model from the National
Crash Analysis Center, from 2001 with around 875.000 FE nodes and 300 time
steps. With LS-DYNA (http://www.lstc.com/products/ls-dyna) wecomputed 289
simulations of the whole crash varying selected parameters. In Figure 1 the Taurus
is illustrated for time step t= 300, i.e. after the crash.
With n1,...,nMNwe denote the finite element nodes, whose positions in the
three dimensional space are rxti
1,...,rxti
MR3, where r= 1,...,R = 289 denotes
simulation index and ti∈ {1,...,T = 300}the current time step. The fire wall
intrusion is now defined as the displacements of the 23 finite element nodes listed
in Table A of the Appendix, minus the displacement of a fixed point in the back of
the car, i.e.
rdti,tj
l=rxti
lrxtj
lrxti
fix rxtj
fixl= 1,...,23.(2)
The fire wall intrusion points are illustrated in Figure 1, right panel. For each
intrusion point the displacement vector (2) can be calculated for each time step ti
to tj, stored in a data matrix
Xti,tj=
k1dti,tj
1k2k2dti,tj
1k2... kRdti,tj
1k2
k1dti,tj
2k2k2dti,tj
2k2... kRdti,tj
2k2
.
.
..
.
..
.
.
k1dti,tj
Mk2k2dti,tj
Mk2... kRdti,tj
Mk2
R23×289 (3)
The data is provided by the Fraunhofer Institute (SCAI), Schloss Birlinghoven, 53754
Sankt Augustin, Germany, in a joint project supported by the German Minister for Educa-
tion and Research (BMBF).
Dependence modeling with R-vines 5
Fig. 1. Ford Taurus from the right (left panel) for time step t= 300 and the fire wall with 23
marked intrusion points (right panel).
with M= 23 and R= 289. Here kxk2denotes the Euclidean norm of a vector x.
Keeping ti= 1 fixed we get an array of dimension 23 ×289 ×300 as our fire wall
intrusion data set. Preliminary analysis shows no significant displacement in time
up to tj= 149, therefore we concentrate on the time horizon tj= 150,...,300.
3. Regular vine copula model
Considering X1,...,Xdto be random continuous variables with marginal distribu-
tions F1(x1),...,Fd(xd) it can be a quite challenging problem to state the joint
cumulative distribution function (cdf) F(x1,...,xd) of them. Copulas are a pop-
ular statistical tool to solve this problem. The famous Theorem of Sklar (Sklar,
1959) given in (1) decomposes the problem into modeling of the margins and the
joint dependence structure modeled by the copula. In the bivariate case we have
F(x1, x2) = C(F1(x1), F2(x2)). If Cis two-times differentiable the bivariate copula
density is c(u1, u2) = 2C(u1,u2)
∂u1 u2, where (u1=F1(x1), u2=F2(x2)) are so called cop-
ula data on [0,1]2. This allows to write f(x1, x2) = c(F1(x1), F2(x2))f1(x1)f2(x2).
Classical one or two-parametric copulas are the Gaussian or Student’s t copula
arising from the elliptical copula family, or the Archimedean copulas like Clayton,
Gumbel or Frank copula (Joe, 1997).
A commonly well known dependence measure for copulas is the rank correlation
6Schepsmeier and Czado
coefficient Kendall’s τ. Let U1and U2be uniform distributed random variables on
[0,1], then Kendall’s τis defined as
τ= 4 Z Z C(u1, u2)dC(u1, u2)1,
where C(·,·) is the copula distribution function. For the Archimedean copulas a
closed form expression of Kendall’s τis available based on the copula specific gener-
ator function (Embrechts et al., 2003), while for the Gauss and Student’s t copula
the calculation is more involved (Frahm et al., 2003). An overview of Kendall’s τ
for different parametric copula families is given in Table A in the Appendix.
Beside Kendall’s τthere are several other dependence measures studied in the
literature, e.g. Spearman’s ρ. But for dependence among extreme values the concept
of tail dependence is more adequate. The concept of bivariate tail dependence
involves the amount of dependence in the upper-quadrant tail or lower-quadrant
tail of a bivariate distribution (Joe, 1997). The upper tail dependence coefficient
λUand lower tail dependence coefficient λLare defined as
λU= lim
uր1
12u+C(u, u)
1u(0,1] and λL= lim
uց0
C(u, u)
u(0,1]
respectively. Again, Table A in the Appendix lists the tail behavior of some bivariate
copula families.
Multivariate copulas, extended from bivariate elliptical and Archimedean cop-
ulas, lack flexibility in higher dimensions. In particular in modeling different tail
dependencies for different pairs of variables. Vine copula models overcome this
problem. Using only bivariate (conditional) copulas as building blocks vine copula
models are very flexible in catching the underlying dependence and tail dependence
structure. First discussed by Joe (1996) and later by Bedford and Cooke (2001,
2002) a multivariate density is constructed by d(d1)/2 bivariate (conditional)
copulas. This process is called a pair-copula construction (PCC)(Aas et al., 2009),
which we will illustrate in a 3-dimensional example. In the following we denote
by fx|v(Fx|v) the conditional density (distribution) function of Xgiven V=v.
Further let cxy;vbe the copula density associated with the bivariate conditional
Dependence modeling with R-vines 7
distribution of (X, Y ) given V=v, while cxy denotes the bivariate copula density
corresponding to (X, Y ). In general cxy;vcan depend on the conditioning values v.
To allow for tractable statistical inference we assume that cxy;vis independent of
v. This is called the simplifying assumption (see for example St¨ober et al. (2013)
for details on this restriction).
Example 1 (3-dim pair-copula construction)
Let X1F1, X2F2and X3F3. We can decompose the joint density as
f(x1, x2, x3) = f3|12(x3|x1, x2)f2|1(x2, x1)f1(x1).
By Sklar’s Theorem applied to bivariate densities it is obvious that
f2|1(x2|x1) = c12(F1(x1), F2(x2))f2(x2)
f3|12(x3|x1, x2) = c13;2(F1|2(x1|x2), F3|2(x3|x2))f3|2(x3|x2)
f3|2(x3|x2) = c23(F2(x2), F3(x3))f3(x3).
Thus, we can represent the joint density as a product of pair-copulas and marginal
densities, i.e.
f(x1, x2, x3) = f3(x3)f2(x2)f1(x1)c12(F1(x1), F2(x2))c23(F2(x2), F3(x3))
·c13;2(F1|2(x1|x2), F3|2(x3|x2)),
where the arguments of the conditional pair-copula are of the form F(x|v). For
every vjin the vector vwe can express F(x|v)as
F(x|v) = ∂Cx,vj;vj{F(x|vj), F (vj|vj)}
∂F (vj|vj)
with vj=v\{vj}in the general case.
Note that the construction is not unique. There is a huge number of possible
constructions (Morales-N´apoles, 2010). A set of nested trees Ti= (Vi, Ei) is used
to illustrate and order all these possible constructions. Each edge Eiin a tree
corresponds to a pair-copula in the PCC, while the nodes Viidentify the pair-copula
arguments. This set of trees is called a vine (Bedford and Cooke, 2001). The vine
tree structure for the 3-dimensional example is illustrated in Figure 2.
8Schepsmeier and Czado
1 2 3
1,2 2,3 T1
1,2 2,3
1, 3|2 T2
Fig. 2. Tree structure of the 3-dimensional example.
In general, a nested set of trees is a regular vine if and only if the trees fulfill the
following conditions (Bedford and Cooke, 2001):
(a) T1is a tree with nodes V1={1,...,d}and edges E1.
(b) For i2, Tiis a tree with nodes Vi=Ei1and edges Ei.
(c) If two nodes in Ti+1 are joint by an edge, the corresponding edge in Timust
share a common node (proximity condition).
A PCC is called a regular vine (R-vine) copula if all marginal densities are
uniform. We denote the R-vine structure given by the nested set of trees with V.
Since every pair-copula in the composition can be selected independently from a set
of bivariate copula families we get a set of pair-copulas B(V) with corresponding
parameter set θ(B(V)). Thus the R-vine is fully specified by RV (V,B(V),θ(B(V))).
For further details on vine copulas, their construction, estimation and properties we
refer to the work of Aas et al. (2009), Czado (2010), Chapter 5 of Mai and Scherer
(2012) and Dißmann et al. (2013).
To select the “best” fitting R-vine tree structure we follow the approach of
Dißmann et al. (2013) maximizing the sum of absolute Kendall’s τtree-wise by
a Maximum Spanning Tree (MST) algorithm. Here the main idea is to capture
most of the dependence in the first trees. Due to its sequential way of construction
Dependence modeling with R-vines 9
by nested trees the vine can be selected tree by tree. The copula family of the corre-
sponding pair-copulas illustrated by the edges is chosen by the Akaike Information
Criterion (AIC). The associated copula parameters can be estimated either sequen-
tially or in a joint maximum likelihood (ML) approach (Aas et al., 2009). Alter-
natively a Bayesian approach can be followed (Smith et al., 2010; Min and Czado,
2010; Gruber and Czado, 2014).
4. Analysis and Results
Given the proposed fire wall intrusion data three modeling scenarios are possible.
First, each simulation run can be modeled separately given the data from time
t= 151,...,300. Further, each time point can be modeled separately given all
simulations, and third all time points and simulations can be modeled in a joint
approach. We will concentrate on the second one to characterize the dynamic
behavior. Further, in an additional analysis we will first reduce the dimensionality
of the data set using Principal Component Analysis (PCA) and apply our R-vine
copula models to this processed data.
4.1. R-vine models for separate time steps
Keeping the time fixed one data set to be analyzed is of dimension 289 ×23, given
289 simulation runs for the 23 fire wall intrusion points. Assuming independently
generated simulation runs we utilize the empirical distribution function as an esti-
mate of the marginal distribution function and define copula data u1,...u23 using
the corresponding probability integral transform for each component. Fitting an
R-vine copula model (Vt,Bt(Vt),θt(Bt(Vt))), t = 151,...,300,to each of the 150
data sets, denoted as ( ˆ
Vt,ˆ
Bt(ˆ
Vt),ˆ
θt(ˆ
Bt(ˆ
Vt))), we get a dynamic picture of the un-
derlying dependence structure by extracting the joint distribution of the fire wall
intrusion points during the frontal car crash. For the selection of the pair-copula
families Bt(Vt) we allowed the elliptical bivariate Gaussian and t-copula, as well
as the Archimedean bivariate Clayton, Gumbel, Frank and Joe copula and their
10 Schepsmeier and Czado
rotated versions to fit negative dependencies. The selection of Vtand Bt(Vt) and
the estimation of θt(Bt(Vt)) were performed with the R-package VineCopula of
Schepsmeier et al. (2012).
In the following we will analyze the estimated R-vine copula models in detail
and justify why they are preferable against much simpler but limited models such
as the multivariate Gauss copula. The multivariate normal distribution or the
multivariate Gauss copula are often used models for multivariate data assuming
a linear dependence structure and no tail dependencies. The multivariate Gauss
copula can be represented as any R-vine with Gaussian pair-copulas, where the
parameters are determined by the associated partial correlations.
Log-likelihood, AIC and BIC: The log-likelihood of the estimated R-vine
copula models for each time point illustrated in Figure 3 is one commonly used
measure of goodness of fit. The AIC or BIC are classical model comparison mea-
sures, taking the model complexity into account. The log-likelihood of the R-vine
models are higher than the one achieved by the Gaussian copula by up to 150 points
over time. Further, due to the significant smaller number of model parameters in
the R-vine, on average 150 model parameters against 23*24/2=253 in the multi-
variate Gauss copula, the corresponding AIC and BIC of the R-vine copula models
are much better. On average the R-vine copula outperforms the multivariate Gauss
copula by 260 AIC points. The maximum difference in the AIC is 520. Similar huge
differences can be seen in a BIC based comparison.
Dependence strength modeling: Another aspect in our analysis is the depen-
dence behavior over time. A single measure of dependence at each time point could
be the relative sum of absolute pair-wise Kendall’s τs, aggregating the estimated
bivariate dependence, i.e.
¯τt:= 1
d(d1)/2
d1
X
i=1
d
X
j=i+1 |ˆτt
i,j |,
where ˆτt
i,j is the empirical Kendall’s τvalue based on the observed data (ut
i,ut
j)
R2·150 at time point t. We plotted ¯τtin Figure 3 (top row), too, and find that
Dependence modeling with R-vines 11
the log-likelihood follows ¯τt, thus catching the time dependent overall dependence.
Thus the proposed R-vines are able to model the mean dependence. But much
simpler models such as the multivariate Gauss model achieve this too (see dotted
line in Figure 3 (top row)).
Tail dependence: R-vines are particularly well suited to account for tail de-
pendence of different pairs of variables. This can be seen from the percentage of
Non-Gaussian pair-copulas (Figure 4) over time. About 20% of the selected pair
copulas have either only lower or upper tail dependence such as Clayton, Gumbel
or Joe copula (tail asymmetric copula). Further 3-5% have upper and lower tail
dependence modeled with a Student’s t-copula.
Furthermore, the tail dependence is significant since the tail dependence coeffi-
cient calculated form the estimated parameters is in mean always greater than 0.15.
The maximum tail dependence per time step is between 0.7 and 0.95, and espe-
cially high for the last time steps (t > 225). An analysis of the degrees-of-freedom
parameters of the chosen Student’s t copula reveal a similar picture. The estimated
degrees-of-freedom parameter νis always smaller than 20. A high νindicates a
weak tail dependence and a convergence towards the Gauss copula, which is the
limiting case for ν→ ∞.
In Figure 4 (bottom line) we have a closer look at the very important first R-vine
tree. The percentage of Gaussian to Non-Gaussian pair-copulas remains the same
as in Figure 4 (top line) considering all trees. However a somewhat higher volatility
is observed in the first tree compared to all trees overall. No independence copulas
are chosen in the first R-vine trees.
Dependence structure: The vine tree structure visualizes the dependencies
between the 23 intrusion points. In Figure 5 we can clearly recognize how the
connections change in the progressing crash. Some of the edges of the first trees do
not vary at all, while others are only chosen at one time point. It is quite interesting
that between time step t= 200 and t= 250 only one edge, i.e. pair-copula, differ.
When we analyze all R-vine copula models and in particular their first tree structure
12 Schepsmeier and Czado
150 200 250 300
0.30 0.32 0.34 0.36 0.38 0.40
time step
rel. sum of Kendall's tau
7500 8000 8500 9000
loglikelihood
150 200 250 300
17500 16500 15500 14500
time step
AIC
140 150 160 170
# parameter
Fig. 3. top row: Average absolute empirical Kendall’s τand the log-likelihood; bottom row:
Akaike Information Criterion (AIC) and number of parameters (R-vine) of the fitted R-vine
copula and multivariate Gauss copula models over time.
Dependence modeling with R-vines 13
150 200 250 300
0 20 40 60 80 100
Gaussian vs. NonGaussian
time step
percentage of used paircopulas
Gaussian
NonGaussian
150 200 250 300
0 20 40 60 80 100
NonGaussian
time step
percentage of used paircopulas
independence copula
Tail asymmetric copulas
tcopula
150 200 250 300
0 20 40 60 80 100
Gaussian vs. NonGaussian
time step
percentage of used paircopulas
Gaussian
NonGaussian
150 200 250 300
0 10 20 30 40
NonGaussian
time step
percentage of used paircopulas
Tail asymmetric copulas
tcopula
Fig. 4. top row: all trees; bottom row: only the first tree; left column: percentage of
Gaussian and Non-Gaussian pair-copulas selected; right column: composition of the Non-
Gaussian case.
we can recognize great similarities in the selected edges. The estimated probability
of appearance of a specific edge is illustrated in Figure 6. Here the thickness of an
edge corresponds to its probability of appearance. Only 42 of 253 possible edges are
selected at all. Nine of them appear in all 150 R-vine tree structures, while 17 still
appear with a probability greater than 75%. More than half of all used edges (22
of 42) appear in more than 75 of 150 cases. As interpretation we can conclude that
the 23 intrusion points more or less deform parallel over time, in particular between
14 Schepsmeier and Czado
Table 1. Percentage of Gaussian, Students’t and tail asymmetric copulas of the 9 most selected edges in the
copula Gauss Student’s t asym. copula
(22,23) 78 3 20
(8,22) 84 10 6
(1,9) 58 20 22
(10,17) 70 3 27
(2,3) 87 11 3
(3,4) 94 4 1
(11,12) 78 16 5
(5,13) 88 12 0
(7,15) 97 3 0
time step t= 200 and t= 250.
Furthermore, we can recognize that almost every intrusion location has its strongest
dependencies with locations in its closest neighborhood. The locations 21 and 23
are not connected since they are not on the same metal plate, thus differ in their
z-coordinates, which can not be seen in the two-dimensional plots. But their phys-
ical Euclidean distance would be the smallest among the neighbors of location 23.
The same is true for the locations 19 and 22.
Table 4.1 lists the percentage of the selected copula families on the nine most
chosen edges. In particular, the edges (22,23), (1,9) and (10,17) have a high per-
centage of tail asymmetric copulas. Thus, these points and their correlation may
be of higher interest to the engineer due to their clearly non-Gaussian behavior.
The time points t= 211 and t= 264 are chosen due to their relative sum of
empirical Kendall’s τ’s (¯τt) values. We highlighted two local maxima and two local
minima in Figure 3. The first peak is at t= 211, the second peak is at t= 224, while
the local minimum between them is a t= 219. Interestingly the R-vine structure of
the first tree is the same for all three time points (see Figure 5 top right panel). The
local minimum of ¯τtbetween t= 200,...,300 is the last chosen point (t= 264).
Dependence modeling with R-vines 15
Fig. 5. Fire wall intrusion points modeled by an R-vine. First R-vine tree plot of t= 150 (top
left), t= 211 ˆ=”first peak” (top right), t= 264 ˆ=”local minimum” (bottom left) and t= 300
(bottom right).
Fig. 6. Selected edges in the first R-vine tree for all 150 R-vines. The size of an edge
indicates its appearance probability.
16 Schepsmeier and Czado
4.2. R-vine models for PCA reduced data
The Principal Component Analysis (PCA) is the standard dimensionality reduction
method in the car engineering context to identify the intrinsic geometry of car crash
data or to reveal bifurcations in the time-dependent behavior of beams (Bohn et al.,
2013). Here, we apply this easy but powerful method to each of the 23 matrices of
the firewall intrusion point array defined at the end of Section 2. Each matrix repre-
sents the information for one intrusion point given the 289 simulation runs and the
last 150 time steps (t= 151,...,300). The first three principal components explain
96% of the data’s variability (in mean over the 23 locations: first component: 79%,
first two components: 92%, first three components: 96%) and are thus sufficient
to identfy the underlying structure. Thereby, we can reduce the dimension from
150 ×289 (or considering all 300 time steps, 300 ×289) to 3 ×289 for each of the
23 intrusion points. Thus, the analysis effort reduces from 300 (or in the subsection
before only the last 150 time steps) models to only three models.
Before we come to the analysis of the PCA reduced data some needed notation
and background on PCA. Let XRn×mbe our (standardized) data matrix and
R:= XTXRm×mits empirical correlation matrix. Further, let λ1λ2...
λmthe ordered eigenvalues of Rwith their corresponding orthonormal eigenvectors
T:= (v1,...,vm)Rm×m. Then, R·T= Λ ·Twith Λ := diag(λ1,...,λm) and
H:= Z·Tlead to the principal component decomposition
X=H T T=HΛ1
2Λ1
2TT=F LT,
where F:= HΛ1
2Rn×mare the principal components and L:= TΛ1
2Rm×m
the loading matrix (Fahrmeir et al., 1996).
As before we assume independent PCA entries of each PCA component vec-
tor fP CAj
m= (1fP CAj
m,...,RfP C Aj
m)T,j∈ {1,...,150},m∈ {1,...,M = 23},
R= 289 (principal component matrix Fm= (fP CA1
m,...,fP C A150
m) inherited from
independently generated simulation runs. We select the first three components.
Again we use the empirical distribution function for the marginal modeling result-
ing in copula data uP C Aj
1,...,uP C Aj
23 ,j∈ {1,2,3}and fit an R-vine copula model
Dependence modeling with R-vines 17
Table 2. Log-likelihood (), AIC, BIC, number of parameters (#par), percentage of
Non-Gaussian (%NG), t (%t), independence (%I) and tail-dependence (%Tail) pair-
copulas selected of the three R-vine copula models applied to the PCA reduced data
sets (top) and the min, mean and max of their analogs given the full data set (bottom).
PCA AIC BIC #par %NG %t %I %Tail
1 8830 -17342 -16760 159 75 4 41 19
2 11030 -21671 -20957 195 73 8 31 20
3 8704 -17072 -16456 168 79 5 38 22
R-vine models for t= 151,...,300
min 7367 -17610 -17020 132 68 1 37 11
mean 8259 -16210 -15660 151 73 3 43 16
max 8966 -14450 -13930 168 80 5 50 22
(Vj,Bj(Vj),θj(Bj(Vj))) to each of the three new data sets.
Results: The quantitative results including the log-likelihood (), AIC, BIC,
number of parameters (#par) and the composition of the selected bivariate copula
families are given in Table 4.2. All these results are comparable to the results of
the full data set analyzed before due to the same dimensionality of the (processed)
data sets. In particular, the log-likelihoods as well as the relative sums of Kendall’s
τvalues (¯τP C A1= 0.31, ¯τP CA2= 0.56, ¯τP C A3= 0.47) fit in the range of their
analogs given in Figure 3, except for P CA2having a higher mean sum of Kendall’s
τ. Considering the log-likelihood, AIC and BIC, again P C A2with = 11030 lies
beyond the maximum of the log-likelihoods given the full data set (= 8966).
Consequently the AIC and BIC are lower than their matching parts. The tail
asymmetry is comparable pronounced as in the full data set. Thus, R-vine copula
models are here preferable to multivariate Gaussian copulas, too. Although there are
some minor departures from the original data results the three PCA based models
can represent the underlying structure and properties of the fire wall intrusion
points.
Furthermore, the most interesting dependence structure among the fire wall in-
18 Schepsmeier and Czado
trusion points is still recognizable in the R-vine tree structures of the three PCA
based R-vine copula models. As analyzed before only 42 of 253 possible edges in
the first tree are selected at all in the full data set. The three PCA based models
need only 33.
5. Summary and future work
The analysis of the fire wall intrusion points revealed several interesting points
which are very helpful for the understanding of the car structure and its behavior in
a frontal crash. The engineer can draw important conclusions for the improvement
of passenger security or car design.
First of all we recognized that the overall dependence of the fire wall intrusion
points vary over time. Furthermore, the relative sum of absolute Kendall’s τesti-
mates reveals a time dependent structure. The special R-vine construction allowed
us to detect more details. As expected many of the pair-wise dependencies are of an
elliptical shape. But plenty of them show tail dependence, violating the multivariate
normality assumption. Simpler models such as the multivariate Gauss copula can
not model such behaviors. The vine copula is much more flexible, since it allows
for different bivariate dependence structures.
Additionally, the R-vine tree structure allows us to investigate the whole de-
pendence structure over time. It turned out that the dependence structure varies
during the frontal crash but not much since only few edges of the first R-vine tree
structure change. Furthermore, the tree structure identifies which pairs of intrusion
points are most strongly related.
If the engineer is only interested in a rough picture of a crash or want to identify
the intrinsic geometry of a car crash the combination of PCA and R-vine copula
models is very helpful. Here, the R-vine tree structure can reveal the dependence
structure as well. Furthermore, all the major dependency properties characterized
in the full analysis of the fire wall intrusion data set can be found in the reduced
PCA data, too.
Dependence modeling with R-vines 19
This shows that the vine copula framework is useful to analyze the dynamic
behavior of crash simulation data. It is expected that for side crashes the vine
methodology will be even more successful than a Gaussian dependence model. The
R-vine tree structures show clear spatial features and approaches as proposed by
Erhardt et al. (2014) will be good starting points to capture this behavior. Since
these approaches were formulated in a 2D context, the extension to 3D will be nec-
essary in addition to incorporating the plate structures of the car. These extensions
will be part of future research.
Acknowledgments
The used model has been developed by The National Crash Analysis Center (NCAC)
of The George Washington University under a contract with the FHWA and NHTSA
of the US DOT (http://www.ncac.gwu.edu/).
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Dependence modeling with R-vines 23
Table 3. Coordinates of the 23 fire wall intrusion points.
nr. node-id x y z nr. node-id x y z
1 3184699 -1544.4 578.9 239.7 13 3188401 -1329.7 -191.2 400.5
2 3183912 -1520.7 489.5 223.1 14 3188442 -1330.5 -423.6 401.2
3 3183367 -1526.0 312.1 222.7 15 3179751 -1426.6 -511.1 372.7
4 3182514 -1522.8 146.5 227.6 16 3263455 -1649.4 -735.5 371.2
5 3180860 -1530.4 -185.4 239.0 17 3186460 -1311.8 475.1 537.5
6 3180230 -1543.1 -339.0 229.1 18 3186733 -1312.3 345.8 501.2
7 3179978 -1537.0 -479.6 224.6 19 3186555 -1310.8 469.5 688.9
8 3156178 -1649.7 737.0 366.8 20 3187573 -1297.5 223.7 613.5
9 3184686 -1486.9 571.2 344.5 21 3190789 -1280.1 -330.6 610.0
10 3186626 -1330.6 426.4 389.2 22 3405632 -1656.5 336.3 828.9
11 3187198 -1329.9 161.9 401.0 23 3405257 -1698.2 -298.4 740.0
12 3187318 -1331.3 0.9 386.2
A. Appendix
24 Schepsmeier and Czado
Table 4. Properties of parametric bivariate copula families.
Copula Parameter range Kendall’s τTail dependence
Gaussian ρ(1,1) 2
πarcsin(ρ) 0
Student-t ρ(1,1), ν > 12
πarcsin(ρ) 2tν+1 ν+ 1q1ρ
1+ρ
Clayton θ > 0θ
θ+2 (21,0)
Gumbel θ1 1 1
θ(0,221)
Frank aθR\ {0}14
θ+ 4 D1(θ)
θ(0,0)
Joe bθ > 12θ4+2γ+2 log 2+Ψ( 1
θ)+Ψ( 2+θ
2θ)
θ2(0,221)
aD1(θ) = Rθ
0
c/θ
exp(x)1dx (Debye function)
bγ= limn→∞(Pn
i=1 1
ilog n)0.57721 (Euler’s constant), Ψ(x) = d
dx log(Γ(x))
(Digamma function)
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