Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 51

1.5.5.2 Parameter estimation under prescribed lower and/or upper bound
There are soil parameters with clear lower and/or upper bounds. Many parameters can-
not be negative. It is very rare for the undrained shear strength of a clay to be <10 kPa. By
definition, the OCR and sensitivity ( S t ) are bounded from below by 1. The Poisson ratio is
bounded between 0 and 0.5. The percentile procedure described in Section 1.5.3 cannot
accommodate these bounds directly. This fact is demonstrated by the S t data points in the
Clay/10/7490 database (Ching and Phoon 2014a). The number of data points n = 1591.
The histogram of this Y = ln( S t ) dataset is shown in Figure 1.24a . There is a physical lower
bound of zero, because S t > 1. However, the percentile procedure gives the SU type, which is
unbounded from below. This fitted SU distribution is plotted in Figure 1.24a . It can be seen
that the PDF is not zero for y < 0. To reinforce the physical lower bound at y = 0, one method
is to bypass the percentile procedure and choose the SL distribution (SL is bounded from
below). The lower bound of Y (namely b Y ) is prescribed to be −0.01 (not 0). This is to avoid
the numerical problem that may occur when we have data points with S t = 1 (if b Y = 0, S t = 1
implies that Y = 0 and that ln(Y + b Y ) = ln(0) = negative infinity). The method of maximum
likelihood is used to estimate a X and b * With b y = −0.01 being prescribed, the PDF for the

SL distribution is (see Equation 1.90 )

( )

2

( ) =⋅ ( ) ( )

fy a exp. − 05 b

*

+ a ln y + 001 . 2 π ⋅ + y 001 . (1.98)

X X X

The maximum likelihood estimates for a X and b * can be found by maximizing the loga-
rithm of the likelihood function:

( )

ln fY Y

() 1

,

() 2

, ..., Y

() n

| ab

X *

,

( )

n

∑

( ) − ( )

2

( ) −

(1.99)

= ln a ln Y

() k

+ 0010 . . 5 ba Y

*

+ ln

( k

+ 0 01 .

X X X

k = 1

(a) (b)

500 500

SU distribution SL distribution

400 400

300 300

200 200

100 100

0 0

0 2 4

ln( S t )

6 8 0 2 4

ln( S t )

6 8

Percentile method Prescribed lower bound

Figure 1.24 Histograms of ln( S t ) and the identified Johnson distributions using (a) the percentile method and
(b) the maximum likelihood with a prescribed lower bound = − 0.01.