Maximum likelihood principle and its application in soil liquefaction assessment - Risk and Reliability in Geotechnical Engineering - page 201

4.3.4 Calibration of liquefaction models
In this calibration study (involving database shown in Table 4.5, and models shown in Table
4.4 ), θ = { b 0 , b 1 , b 2 , …, b r } are the uncertain parameters to be calibrated (in this case, r = 2).
To begin with, let I denote the indicator variable, with I = 1 denoting liquefaction and I = 0
denoting nonliquefaction, respectively. Let I i denote the indicator variable of the i th case. The
observed data can be denoted as d i = { I i }, and the calibration database is d = { d 1 , d 2 , …, d n }.
Suppose in a calibration database, there are n L liquefied cases and n NL nonliquefaction cases.
Based on a generalized linear model, the probability of observing a liquefied case is P L , and the
probability of observing a nonliquefied case is 1 − P L . The likelihood function, or the probabil-
ity to observe n L liquefied cases and n NL nonliquefied cases, can be calculated by multiplying the
probability to observe each case in the database. Using model M 1 (logistic regression model; see
Table 4.4 ) as an example, the weighted likelihood function can be written as follows:

n

L

1

∑

L ( θ| d = ) w ln

{ }

L

1 + exp[ − bbq + + b ln( CSR )]

0 1 tNcs 1 , 2

i = 1

n NL

1

∑

+ w ln 1 − (4.23)

{ }

NL

1 + exp[ − bbq + + b ln( CSR )]

0 1 tNcs 1 , 2

j = 1

The spreadsheet template for maximizing Equation 4.23 is shown in Figure 4.6. With
Excel Solver, the optimal values of θ are obtained as θ = { b 0 , b 1 , b 2 } = {21.932, −0.123, 6.182}.

4.3.5 ranking of liquefaction models
Using the spreadsheet template shown in Figure 4.6 , all four models are calibrated and the
optimal regression coefficients for these models are summarized in Table 4.4 along with the
model probabilities that are computed with Equation 4.19 . Note that since the weighted like-
lihood function is used for model calibration to remove the effect of sampling bias, BIC, and
model probability should be calculated accordingly (i.e., based on the weighted likelihood
function). Model ranking based on BIC is likely supported by Laplace's approximation of the
weighted log-likelihood function, although further research is needed. The results in Table
4.4 s how that M 4 (c-log-log regression) has the highest model probability, indicating that it
is the most suitable for the database shown in Table 4.5. The commonly used logistic regres-
sion model, however, has the least model probability with the database shown in Table 4.5.
It should be noted that the results obtained in the above study are not exactly the same as
those obtained by Zhang et al. (2013) based on an earlier database. In that similar study, they
determined that while the c-log-log model was indeed most suitable for constructing liquefac-
tion probability models, the logistic regression model was the next best for such construc-
tions. The model ranking results are thus shown as dependent on the adopted calibration
database, and the commonly used logistic regression model is far from being the best option.

4.4 ConVertIng a DeterMInIStIC lIqueFaCtIon
MoDel Into a ProbabIlIStIC MoDel

The Robertson and Wride method (Robertson and Wride 1998), which is later updated in
Robertson (2009), is one of the most widely used CPT-based simplified models for evaluat-
ing the potential of soil liquefaction. The Robertson and Wride method is a deterministic
model, which yields a factor of safety ( F S ) as the outcome of its evaluation of liquefaction