A natural question which arises is that how the proposed Bayesian approach in this study com- pares against other alternate approaches for probabilistic analysis of SWCC. Therefore this section, compares the parameters estimatesW = [λα, ζα, λn, ζn, θα,n] (ref. Eq. 4.13) obtained using the pro- posed approach in this study and the approaches in literature. Popular approach in literature is to fit a parametric joint distribution to the observed data ofα, neither using translational approaches (Carsel and Parrish, 1988; Phoon et al., 2010) or copula approach (Prakash et al., 2018b). Partic- ularly the approach (to be referred as conventional approach hereafter) followed in Prakash et al.
(2018b) will be used for comparison since the copula approach was utilized in these studies and a one to one comparison between estimated parameters is possible.
A numerical experiment, similar to the previous two subsections was conduced. For each
t
Lag
0 500 1000 1500 2000
Autocorrelation
-0.2 0 0.2 0.4 0.6 0.8
1 (a)
Parameter n Parameter α
Lag
0 500 1000 1500 2000
Autocorrelation
-0.2 0 0.2 0.4 0.6 0.8
1 (b)
Lag
0 500 1000 1500 2000
Autocorrelation
-0.2 0 0.2 0.4 0.6 0.8
1 (c)
Figure 4.10: Markov chain Monte Carlo (MCMC) chains autocorrelation plot for the (a) laom (b) fly ash and (c) bentonite database.
database of loam, fly ash and bentonite, random trials were made. Number of data points randomly picked from the databases areN = 2,5,10,20,30. For eachN, 10 random trials were made. The same priors as in Table 4.2 were utilized. In the conventional approach, (Prakash et al., 2018b) only the available data will be used to estimate the parameters of the joint distribution i.e. the vector W = [λα, ζα, λn, ζn, θα,n]. In the proposed approach, these random trials will be updated to obtain MCMC samples corresponding to the target PDFf(α, n|data) using the M-H algorithm (ref. 4.13) and the parameters will be estimated over these MCMC samples.
It is worthwhile to note that the conventional approach is straight forward to use as the marginals and copula parameters can be fitted simply to the available data. However, this ap- proach works satisfactorily only for data sufficient cases. For a real case scenario, the number of SWCC will be most likely limited in numbers, and hence the statistical uncertainty will be high for the estimated joint distribution parameters ( to be also shown later). The proposed approach, although comparatively computationally intensive, supplements the available data, i.e. α, n with prior constructed from a statistical analysis conducted in the literature to obtain the posterior distributionf(α, n|data). Not only it reduces uncertainty in the estimates at a low number of data points, but it also allows for subjective and expert judgement (in the form of prior) to be incorpo- rated. In the conventional approach, even at abundant data cases, the parameters estimate is solely governed by the available data, and there is no scope for the incorporation of expert judgement or experience.
For the comparison, a total of 150 MCMC simulations were performed i.e. 50 (length(N) = 5×T rials = 10) each for the loam, fly ash and bentonite dataset. Before making any statisti- cal inference, it is necessary to inspect for the convergence of the MCMC chain. Although the convergence can be judged by visually observing trace plots (e.g. Fig. 4.3), it is worthwhile to sta- tistically confirm the convergence. Statistical confirmation becomes important in this section also, since 150 MCMC simulations were performed, and it is not possible to visually present and judge every chain for inspection. Popular approaches for checking convergence are autocorrelation plots and Gelman-Rubin statistics (Gelman et al., 2013). In this study, autocorrelation plots are used
λ
-20 0 20
N=2
-3 -2 -1 0 1
N=5 N=10 N=20 N=30
λ
-15 -10 -5 0 5
-4.5 -4 -3.5 -3 -2.5
λ
-40 -20 0 20 40
-6 -4 -2 0 2
(a) Comparison of lognormal marginal parameter
λαestimate using conventional (red) and proposed
(black) approach.
ζ
0 20 40 60
N=2
0 1 2 3
N=5 N=10 N=20 N=30
ζ
0 10 20 30
0 1 2
ζ
0 50 100 150
0 2 4 6 8
(b) Comparison of lognormal marginal parameter
ζαestimate using conventional (red) and proposed
(black) approach.
λ
-2 0 2 4
N=2
-2 0 2
N=5 N=10 N=20 N=30
λ
-2 0 2 4
-1 0 1 2
λ
-2 0 2 4
-1 0 1 2 3
(c) Comparison of lognormal marginal parameter
λnestimate using conventional (red) and proposed
(black) approach.
ζ
0 0.5 1 1.5
N=2
0 0.2 0.4 0.6 0.8
N=5 N=10 N=20 N=30
ζ
0 0.5 1 1.5
0 0.2 0.4 0.6 0.8
ζ
0 0.5 1 1.5
0 0.2 0.4 0.6 0.8
(d) Comparison of lognormal marginal parameter
ζn estimate using conventional (red) and proposed
(black) approach.
θ
-1 0 1
N=2
-1 0 1
N=5 N=10 N=20 N=30
Conventional Proposed
θ
-1 0 1
-1 0 1
θ
-1 0 1
-1 0 1
(e) Comparison of copula parameterθestimate us-
ing conventional (red) and proposed (black) ap- proach.
Figure 4.11: Comparison of joint distribution parameters W = [λα, ζα, λn, ζn, θα,n] obtained using conventional (red) and proposed (black) approach. First, second and third row in each subplot denotes the statistics obtained for loam, fly ash and bentonite database respectively.
for convergence diagnostics since calculating Gelman-Rubin statistic requires multiples chains with different starting points for the MCMC simulation. For example, even if only 10 chains were used for each MCMC simulations, the total number of MCMC simulations will reach 1500. This will result in a lot of wastage of MCMC chains and is computationally expensive and time-consuming.
Hence only autocorrelation plots will be used for demonstrating the convergence. Fig. 4.10 presents the MCMC simulations performed for loam, fly ash and bentonite dataset. It can be noted that that the autocorrelation dies down quickly for all the simulations to zero hence demonstrates a proper mixing and thus the convergence of the MCMC chain.
Given that the MCMC chains have converged the parameter estimated over MCMC samples (referred to as proposed approach) and the approach mentioned in Prakash et al. (2018b) (re- ferred to as conventional approach) can be compared. Fig. 4.11 presents the comparison of joint distribution parametersW = [λα, ζα, λn, ζn, θα,n] estimated from the proposed approach and a con- ventional approach. The parameter estimates are accompanied by their 95 % confidence interval corresponding to the estimate. In Fig. 4.11, it can be easily observed that at a lower number of data points (e.g. N = 2, N = 5) the scatter in the parameter estimate using the proposed approach is significantly lower than the conventional approaches. The proposed approach not only utilizes the available data points but also supplements it with the prior information and therefore, statistical uncertainty associated with the estimates are comparatively lower. In Fig. 4.11 it can also be noted that with an increase in the number of data points, uncertainty in estimates of both conventional and proposed approach decrease and at a higher number of data points (e.g. N = 30) both the conventional and proposed estimate converge. These observations not only demonstrates the superiority of proposed approach in terms of lower uncertainty in the parameter’s estimate but also demonstrate the robustness of the proposed approach since with the increase in the number of data points both the Bayesian and conventional estimate converge towards the same results.