Maximum likelihood principle and its application in soil liquefaction assessment - Risk and Reliability in Geotechnical Engineering - page 183

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Spreadsheet Template for Calibrating a Bivarate Normal Distribution

Parameters to
be estimated

µ 1 σ 1 µ 2 σ 2 ρ

6 156.44 122.229 18.052 5.01425-0.4068

Log-likelihood

Eq. (4.7)

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Oberved data

Test No. c (kPa) φ (°) Α B C D f ( d i ln[f f ( d i |θ)] L (θ| D )

9 1 165 11.86 0.0049 .52492 -0.0704 0.87448 .000119 -9.04006 -229.14

10 2 127 14.04 .05801 .64019 0.15681 .5123 .00017 8.67789

11 3 253 13.5 0.62409 .82412 -0.5836 0.51808 .000169 -8.68366

12 4 427 10.2 4.89981 .45212 -2.8205 2.71516 .88E-05 10.8807

13 5 106 11.31 .17029 .80785 0.45149 .45579 .63E-05 9.62137

14 6 242 12.95 0.49 1.0353 -0.5796 0.56667 .000161 -8.73225

15 7 209 12.41 .18491 .26605 -0.3937 0.63349 .000151 -8.79907

16 8 328 13.5 1.97008 .82412 -1.0368 1.05299 .92E-05 9.21857

17 9 98 12.95 0.2286 .0353 0.39585 .99449 .000105 -9.16007

18 10 10 15.64 1.4354 .23139 0.46894 .27968 .91E-05 9.44526

19 11 213 16.17 .21413 .14087 -0.1413 0.12803 .00025 8.29361

20 12 365 17.22 .91148 .02753 -0.2304 1.62296 .61E-05 9.78854

21 13 324 20.81 .87929 .30254 0.61355 .67493 .33E-05 9.84051

22 14 85 20.81 .34161 .30254 -0.2616 0.22922 .000226 -8.3948

23 15 18 25.64 .28285 .29004 -1.3947 1.30515 .71E-05 9.47073

24 16 15 22.29 .33905 .71435 -0.7958 0.75352 .000134 -8.9191

25 17 78 24.7 0.41184 .75781 -0.6923 0.88518 .000117 -9.05076

26 18 12 26.1 1.39646 .57611 -1.5433 1.45555 .63E-05 9.62113

27 19 34 22.78 .00346 .88909 -0.7686 0.67346 .000145 -8.83904

28 20 70 19.8 0.50013 .12153 -0.2006 0.25229 .000221 -8.41787

29 21 20 17.74 .24605 .00387 0.05651 .78279 .00013 8.94837

30 22 20 20.81 .24605 .30254 -0.4996 0.62853 .000152 -8.79411

31 23 217 20.3 0.24548 .201 0.18075 .37582 .000195 -8.5414

32 24 221 20.3 0.27898 .201 0.19268 .40305 .00019 8.56863

33 25 254 27.47 .63708 .52782 1.21987 .22644 .13E-05 11.392

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Notes:
(1) To faciliate spreadsheet implementation, four intermedidate variables are defined as follows:

2 2

d − µ d − µ d − µ d − µ ABC ++

A = , B = , C =− 2 ρ , and D =

i 1 1 i 2 2 i 1 1 i 2 2

( )

σ σ σ σ 21 − ρ

2

1 2 1 2

(2) The setting in Solver is "Maximize Cell L9 by changing the values in Cells E6, F6, G6, H6, and I6."

Figure 4.2 Spreadsheet template for calibrating the bivariate normal distribution.

4.2.2 Correlated observations
The maximum likelihood method is also applicable to correlated observations, as illustrated
in the following example.

eXaMPLe 4.3
Due to the existence of spatial variability, the properties of soil samples taken from
the same soil layer are not necessarily the same but are usually correlated. Consider a
hypothetical saturated clay soil as shown in Figure 4.3 . Suppose the undrained shear
strength of the clay, c u , is a realization of a stationary lognormal random field with a
mean μ, a standard deviation σ, and an exponential correlation function defined as fol-
lows (Vanmarcke 1983):