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Table 15.7 Point 1 2 3 4 5 6 7 8
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Results from Hong s method with four variables for Culmann slope c(kPa) 2.445 5.863 4.333 4.333 4.333 4.333 4.333 4.333 (deg) 15.667 15.667 12.556 19.167 15.667 15.667 15.667 15.667 (kN/m3 ) 19.333 19.333 19.333 19.333 17.445 20.863 19.333 19.333 (deg) 20.000 20.000 20.000 20.000 20.000 20.000 18.367 21.663 M (kPa) 0.636 4.054 1.274 3.978 2.701 2.381 2.900 2.521 Mest = Sum of M p = p 0.112 0.138 0.132 0.118 0.112 0.138 0.125 0.125 2.576 1.202
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Variance = Sum of M2 p (Mest)2 = = 1.097, = 2.350 H = 10 m, = 26 deg
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Points far from Mean for Large n
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Lind s, Haar s, and Hong s methods give results superior to those from FOSM methods with little or no increase in computational effort. Because they use fewer points than the original point-estimate methods, they may be somewhat less accurate for some functions, but limited testing indicates that they are usually quite satisfactory. All the methods locate the points at or near the surface of a hypersphere or hyperellipsoid that encloses the points from the original Rosenblueth method. The radius of the hypersphere is proportional to
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Figure 15.13 Distribution of evaluation points for three uncertain variables. The variables have been normalized by subtracting the mean and dividing by the standard deviation. (a) Coordinate system, (b) black squares are the eight points in the original Rosenblueth procedure, (c) ellipses are circular arcs de ning a sphere circumscribed around the Rosenblueth points, (d) black dots are the intersections of the circumscribed sphere with the coordinate axes, which are the six points in the Harr and Hong procedures. (Christian, J. T. and Baecher, G. B., 2002, The Point-Estimate Method with Large Numbers of Variables, International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 126, No. 15, pp. 1515 1529, reproduced by permission of John Wiley & Sons.)
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Figure 15.13 (continued )
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THE PROBLEM OF THE NUMBER OF COMPUTATION POINTS
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1E +07 1E +06 1E +05 2n 1E +04 1E +03 1E +02 1E +01 1E +00 0 5 10 15 Number of Uncertain Quantities 20 (n 3+ 3n = 2)/2
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Number of Calculations
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2n + 1
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Figure 15.14 Numbers of calculations by various algorithms. (Christian, J. T. and Baecher, G. B., 2002, The Point-Estimate Method with Large Numbers of Variables, International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 126, No. 15, pp. 1515 1529, reproduced by permission of John Wiley & Sons.)
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n in Harr s method and nearly so in Lind s and Hong s. It is possible that, when n is large, the values of the xi s at which the evaluation is to take place may be so many standard deviations away from the means that they are outside the range of meaningful de nition of the variables. Figure 15.15 shows what happens in a case of normalized uncorrelated and unskewed variables. The four black points are the points in the conventional Rosenblueth procedure for two variables. The four open points on the circle drawn through the black points are the evaluation points that would be used in the Harr, Hong, or Lind procedures. There is obviously no advantage in doing this for two variables as four points are needed in either case. However, if there were nine variables, a reduction from 512 to 18 evaluations would be well worth the effort. The outer circle in Figure 15.15 represents a two-dimensional section through the nine-dimensional hypersphere, and the evaluation points now lie three standard deviations form the mean. Thus, critical values of the variables will be located far from the region where the distributions are known best. In Table 15.8, which gives the coordinates of the points to be used in the analysis with four uncertain variables triangularly distributed, it is clear that the values of some of the variables are quite close to the limits of their triangular distributions. If there were seven variables, Table 15.9 shows that the values of all the variables would fall outside the bounds of their distributions. Figure 15.16 is the same as Figure 15.4, except that the locations of the evaluation points for cases with 5 and 10 variables are shown. The evaluations fall far out along the tails of the distribution, where the values of the distribution are known poorly.
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