Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 71

1.8 Future ChallengeS

This chapter discusses how to construct and how to identify a virtual site from limited geo-
technical data within the framework of a multivariate non-normal probability model. The
practical usefulness of a virtual site in reducing the uncertainties in the estimation of design
parameters is illustrated. The effect of reducing these parametric uncertainties on RBD is
discussed elsewhere (Ching et al. 2014a).
Ideally, a virtual site should emulate every aspect of a real site. The realism of the virtual
site presented in this chapter can be improved in at least six ways. First, spatial variability
should be included as it is a fundamental feature of geotechnical data. The mathematical
framework in the form of a random vector field exists, but very few studies have been con-
ducted so far on how to identify this field statistically from limited data. Hence, the chal-
lenge here is statistical, not theoretical.
Second, different tests are coupled using the standard concept of product-moment corre-
lation. This concept is strictly applicable only to linear relationships. Currently, one attempts
to linearize nonlinear relationships using a CDF transform, but there is no guarantee that
this method will work for all real cases. Copulas can accommodate more general dependen-
cies, but they appear to be of limited use beyond modeling bivariate data.
Third, the proposed multivariate non-normal probability model is constructed from an
underlying multivariate normal probability model. The K-S test is routinely used to verify
goodness of it to a normal distribution. It cannot be extended to verify multivariate normal-
ity. Goodness-of-fit tests for multivariate normality do exist, but they are more difficult to
apply and no “gold standard” has arisen in the statistics literature.
Fourth, genuine multivariate data are rare in geotechnical engineering. It is not trivial to
construct a valid positive-definite correlation matrix from bivariate data, which are commonly
available in a site investigation program. A partial expedient solution is offered in this chapter,
but it could be possible to develop a more rigorous and directed approach in the future.
Fifth, it is important to realize that fitting each parameter to a marginal distribution is
only an intermediate goal and cannot be conducted without regard to the more stringent
constraints imposed by the multivariate model, which is the final goal in the characteriza-
tion of geotechnical engineering data. If the multivariate nonnormal probability model is
constructed from an underlying multivariate normal probability model as proposed in this
chapter, then the non-normal marginal distribution should preferably be transformed from
a normal distribution using the closed-form equation. The Johnson system of distributions is
elegant in this sense. However, distribution identification using percentiles is not robust for
a small sample size, say <30 data points. In addition, its model parameters are not related to
physical bounds in a direct way. Physical lower/upper bounds are important to avoid absurd
realizations such as negative shear strengths or parameters such as SPT-N, OCR, sensitivity,
and so on taking values <1.
Sixth, the basic framework of constructing a multivariate non-normal probability
model by transforming each component in a multivariate normal probability model indi-
vidually is not general and may not work for actual data. In addition, this framework
cannot accommodate more information beyond marginal distributions and a correlation
matrix.

lISt oF SYMbolS

Φ(⋅) Cumulative distribution function for standard normal

φ(⋅) Probability density function for standard normal