Hydraulic conductivity (k)
1D univariate random field
k1 k2 k3 k4 ...
...
...
...
...
...
...
...
kn-2 kn-1 kn
K∼f(µ
k,σ
k) ki
ki+∆
ρ(ki,k
i+∆)=ρ(|∆|,λ)
Figure 6.1: An illustration of one dimensional univariate random field of hydraulic conductivity (k) in vertical direction. µand σ are mean and standard deviation respectively, ∆ is the lags andλis the scale of fluctuation.
f(k1, k2, ..., kn) =
n
Y
x=1
f(kx)
n−1
Y
j=1 n−j
Y
i=1
Di,i+j|i+1,...,i+j−1[F(ki|ki+1, ...ki+j−1), F(ki+j|ki+1, ...ki+j−1)]
(6.4) where J denotes the index for Tree and i for edges in a particularTj. Similarly, using the C-vine formulation after Aas et al. (2009), Eq. (6.2) can be expressed as follows.
f(k1, k2, ..., kn) =
n
Y
x=1
f(kx)
n−1
Y
j=1 n−j
Y
i=1
Dj,j+i|1,...,j−1[F(kj|k1, ...kj−1), F(kj+i|k1, ...kj−1)] (6.5)
It can be noted in Eqs. 6.4 and 6.5 that evaluation of joint density involves n(n−1)/2 copula densities. However, it is tedious to keep track of conditional copula densities involved in the formulations. To organize the copula densities into easily tractable graphs, a tree-like structure called vine is often used, hence the name vine-copula. For illustration, consider the case of n= 5 in Eqs. 6.4 and 6.5. The resulting vine structures are known as Drawable (D) vine, and Canonical (C) vine and are depicted in Figs. 6.2a and 6.2b respectively. Each vine structure forn-dimensions is made up ofn−1 trees (Ti) consisting of nodes and edges. Each of the edges can be assigned a bivariate (pair) copula. C-vine resembles (Fig. 6.2b) a star-shaped structure wherein every tree (Ti) contains a base node connected to each of then−j edges. D-vine (Fig. 6.2a) resemble a parallel stack like structure and every node in any Ti can be connected to only two edges at maximum.
In Fig. 6.2b, for T1 it can be noted that ∆ = 1 : 1 :n−1. For example, ∆ = 1 for D12, ∆ = 2
1 D
12 2 D
23 3 D
34 4 D
45 5
D 45 34 35|4 D24|3 D 23
12 13|2
13|2 D14|23
24|3 D25|34 35|4
14|23 D15|234 25|34
(a)
1 2
3 4
D14
5
12 13
14
D24|1
15
23|1 24|1
25|1
35|12 34|12
D23|1
D34|12 D35|12
D25|1
D13
D45|123 D12 D15
(b)
Figure 6.2: An example illustration of (a) D-vine (b) C-vine inn= 5 dimensions withn−1 trees and n(n−1)/2 edges.
forD13, ∆ = 3 forD14 and so on. This implies that for C-vine the first pair trees can be uniquely determined from the spatial dependence model fitted to the auto-dependence data. Similarly, in Fig. 6.2a, it can be noted that ∆ = 1 is constant in T1 for all the pair copulas. For example,
∆ = 1 forD12,D23,D34 and so on. This implies that for D-vine all the parameters for a particular Ti will be same. This is a significant advantage since it reduces the number of parameters to be estimated fromn(n−1)/2 ton−1 only. However, from theT2onwards estimation of copula requires conditional variables and cannot be directly estimated from the data. However since parameters for tree Ti is determined fromTi−1, the pattern will we similar toT1. For example, the parameters of copula at eachTifor the D-vine will be same, and for C-vine the parameters at eachTiwill follow a decaying sequence similar to the spatial dependence model. This aspect will also be demonstrated in subsequent sections for real data.
Copula selection
Information criteria such as AIC is most popular (Aas et al., 2009) for estimating the best-fit dependence structure. The same can be expressed as follows
AIC =−2L(θn) + 2(k) (6.6)
whereL(θn) is the maximum log-likelihood, andkis the number of parameters. The copula families among the chosen candidates can be selected corresponding to the minimum AIC value. Since all the four copula utilized in this study (ref. Table DA2) have the same no of parameters, i.e. 1, minimum AIC is equivalent to maximumL(θn). The full likelihood estimation is popular for random variable modelling as the number of dimensions is usually low in geotechnical engineering e.g. 7 in
Ching et al. (2017), 9 in Ching, Phoon, Li and Weng (2018) and 10 in Ching and Phoon (2014).
However, for random fields, the dimension can be very large. Hence a full likelihood estimate will be computationally expensive and unfeasible for the random field. In such cases, an alternate pair- wise likelihood estimate is preferred (Padoan et al., 2010; R´ozs´as and Mogyor´osi, 2017; Wang and Li, 2019, e.g.). This is suitable for random fields since the pairwise likelihood can be constructed solely from the auto-dependence data. The pair wise likelihood can be expressed as follows.
L(θn) =ln
n
Y
i<j
f(ki, kj) (6.7)
Breaking the joint densityf(ki, kj) into copula and marginals, the above equation can be written as follows.
L(θn) =
n−1
X
i=1 n
X
j=i+1
ln{Dij[F(ki),(kj)]f(ki)f(kj)} (6.8)
L(θn) =
n−1
X
i=1 n−i
X
j=1
lnDj,j+i[F(kj), F(kj+i)] +
n
X
k=1
ln[f(xk)](n−1) (6.9)
The above equation can further be simplified as follows:
L(θn) =
n−1
X
i=1
Li+
n
X
k=1
ln[f(xk)](n−1) (6.10)
where Li =Pn−i
j=1lnDj,j+i[F(kj), F(kj+i)]. Since the second term in Eq. (6.10) is independent of copula, minimum AIC copula can be selected corresponding to the maximum value ofPn−1
i=1 Li as follows.
min(AIC)≡max(L(θn))≡max(
n−1
X
i=1
Li) (6.11)
Simulation
For the simulation of correlated random numbers (corresponding to a random field) from vine cop- ula, parameters of the copula associated with each edge in a vine (e.g. Fig. 5.3) are required along with conditional and inverse conditional copula functions. The general algorithm for simulating random fields are as follow.
For i= 1 : Number of realizations
• Generate a random sampler = (ri1, ri2, ri3, .., rin) from independent uniform distribution
• SetUi1 =ri1
• SetUi2 =C2|1−1(ri2|Ui1)
• SetUi3 =C3|12−1 (ri3|Ui, Ui2)
• ....
• ....
• SetUin=Cn|12...n−1−1 (rin|Ui, Ui2, ...Uin−1) End
The obtained random vector U = (U1, U2, ...U n) is a random field with specified dependence structure and uniform marginals. As already discussed in previous sections, only the first tree parameters of the D and C vine can be directly estimated from the data. For example, in C-vine structure, e.g. Fig. 5.3θ12 can be uniquely determined fromτ12which is equal to auto-dependence (not data but fitted) at ∆ = 1. Similarlyθ13corresponding to ∆ = 2 and in generalθ1ncorresponds to ∆ = n−1. Therefore the parameter estimates will decay in the same manner as the spatial dependence model. However, from the 2nd tree onwards, the parameters cannot be determined uniquely from the available data. It requires transformed data which in turn are obtained from the previous tree nodes. Therefore would follow the similar decaying trend as spatial auto dependence model. Since most of the cases, only a single realization is available, this approach is not possible for most practical purposes. In such a case, the simulation-based approach (Wang and Li, 2017, e.g.) is attractive and is utilized in this study. This approach is basically a moment matching method wherein the parameters corresponding to all edges are initialized and continued until the dependence produced are matched to the assigned one. In the simulation approach, although the transformed data is not required, conditional distributions and its inverses are still required. The conditional distributions Cj|j−1,..,1 required for simulating the random numbers can be obtained recursively using the following formula (Aas et al., 2009).
FX|Y(x|y) = ∂CX,Yj|Y−j[FX|Yj(x|y−j), FYj|Yj(y|y−j)]
∂FYj|y−j(y|y−j) (6.12)
whereX is a random variable, Yj is a element of random vectorY andY−j is vector consisting of all components ofY butYj. For the case where Y is univariate, the above reduces to
FX|Y(x|y) = ∂CX,Y[FX(x), FY(y)]
∂FY(y) (6.13)
For the case of X ∼U(0,1) and Y ∼U(0,1) i.e. FX(x) =uand FY(y) =v, the above reduces to
h(u, v, θ) =FX|Y(x|y) = ∂CX,Y[u, v]
∂v (6.14)
The h and h−1 functions for the four copulas used in this study along with copula function C and density D are summarized in Table DA2. Detailed algorithms for efficient implementation of
the above procedure can be found in Aas et al. (2009).
6.2.2 Multivariate random fields: Modelling cross dependence
After modelling the spatial dependence in univariate random fields, cross dependence can be im- parted in the multivariate random fields as follows.
1. Generatekunivariate random fieldsU1, U2, ...UK using the procedure mentioned in previous section.
2. Fori= 1 : Number of realizations
• Vi1 =Ui1
• Vi2 =C2|1−1(Ui2|Vi1)
• Vi3 =C3|12−1 (Ui3|Vi1, Vi2)
• ....
• ....
• Vik=Ck|12...k−1−1 (Uik|Vi1, Vi2, ..., Vik−1) End
This step essentially establishes cross-dependence among nuniform random variable of the dis- cretized random fields to the target cross-dependence correlation . Cross-correlation simply refers the correlation coefficients between the different random variables corresponding to the multivari- ate random field. Similar to the simulation-based approach followed in the previous section, the parameters are initialized and repeated until the target cross-dependence matrix is achieved. Fi- nally, the marginals among the spatially and cross-correlated random fields (till now uniform) can be incorporated after applying inverse CDF transform as follows:
Xik=Fk−1(Vik) (6.15)
whereXik is a stationary random field andFk−1 is the inverse CDF of the marginal forkth random field.