ASYMPTOTIC APPROXIMATIONS FOR MULTINORMAL INTEGRALS

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1 ASYMPTOTIC APPROXIMATIONS FOR MULTINORMAL INTEGRALS By Karl Breitung 1 Downloaded from ascelibrary.org by GEORGE MASON UNIVERSITY on 07/16/13. Copyright ASCE. For personal use only; all rights reserved. ABSTRACT: The problem of deriving approximations for multinormal integrals is examined using results of asymptotic analysis. The boundary of the integration domain given by g(x) = 0 is simplified by replacing g(x) by its Taylor expansion at the points on the boundary with minimal distance to the origin. Two approximations which are obtained by using a linear or quadratic Taylor expansion are compared. It is shown that, applying a quadratic Taylor expansion, an asymptotic approximation for multinormal integrals can be obtained, whereas using linear approximations large relative errors may occur. INTRODUCTION In many reliability problems it is necessary to calculate the probability content of part of an n-dimensional probability space. Especially, if the dimension of the space is large, the direct computation by numerical or Monte-Carlo integration is too time consuming. Therefore, approximation methods for this task were developed (5,7,9,12). The basic idea of the proposed methods is to transform the random variables into a probability space, where they are mutually independent, standardized, normal distributed random variables and then to approximate the integral by simplifying the boundary of the integration domain. For independent random variables with continuous densities the transformation is quite simple (5,7,9). In the case of dependent random variables, a numerical procedure is put forward in Ref. 9 which transforms pointwise into the standardized space. But in this case no theoretical results about the performance of this algorithm are known to the author (6). Nevertheless, it can be assumed that in many cases the problem of integration can be reduced to the problem of integrating the n-dimensional standard normal density function over a part of the n-dimensional space. In this space the so-called limit state function g(x) (x = {x x,..., x n )) is defined, it divides the space into the safe domain S = \x;g(x) > 0], the failure domain F = [x; g(x) < 0] and the limit state surface G = [%;g(x) - 0]. The failure occurs, if g(x) < 0. The quantity to be computed is probability of failure P, i.e. P = (2TT)-" /2 exp (-- \z\ 2 )dx. (1) in which \x\ = (2" =1 x 2 ) 1/2 the euclidean norm of the vector x. It is assumed in the following that the origin is not in F. First, all points x,..., x k on the limit surface G with minimal distance to the or- 'Diplom-Mathematiker; formerly, Research Asst, Institut fur Massivbau, Technische Universitat Miinchen, Arcisstr. 21, D-8000 Miinchen 2, FR Germany; Address: Schellingstr.21, D-8000 Miinchen 40, FR Germany. Note. Discussion open until August 1, To extend the closing date one month, a written request must be filed with the ASCE Manager of Technical and Professional Publications. The manuscript for this paper was submitted for review and possible publication on January 4, This paper is part of the Journal of Engineering Mechanics, Vol. 110, No. 3, March, ASCE, ISSN /84/ /$ Paper No

2 igin are computed, i.e. the points with y, = min gfe)=0 \x\. A globally convergent algorithm to minimize a function on hypersurfaces which can be used to find these points, is described in Ref. 11. The approximations are now obtained by replacing g(x) atx lf..., x by its Taylor expansion g*(x) to the first or second order. Then, P(g*(X) < 0) is taken as approximation for P(g(X) < 0). The usual method which is described in Refs. 5 and 9 is to take only the first derivatives of g(x), i.e. to replace the limit surface by a hyperplane through the point with the same gradient as g(x) at this point. In the cited papers it is stated that the quality of the obtained estimate is good and that it produces accurate approximations for the integral but no theoretical considerations or general error estimates to underline this proposition are given. For a fixed failure domain the error of the approximation can be found only by comparing it with the exact result obtained by numerical integration. Since reliability problems are considered the probability content of the failure region which is in general, small, and the distance of the limit surface tp the origin is usually large due to the standardization. It therefore appears reasonable to study the asymptotic behavior of the approximations for sequences of increasing safe domains to obtain information on their quality. Methods of asymptotic analysis are used to do this. Asymptotic Behavior of Approximations. In the case of n independent, standard normal random variables, the limit state function g(x) is given as a twice continuously differentiable function. Further, it is assumed that min gfe)=0 \x\ = 1, g(0) > 0 and that there is only one point x 0 on the surface G = [x; g(x) = 0] with x 0 = 1. The restriction that there is only one point on the limit surface is usually made since for the case of several minimal distance points no general procedure to deal with them was developed so far, it was only proposed that polyhedrical approximations should be used (5,9). Beginning with this one limit state function now a sequence g(x;fi) of limit state functions depending on the parameter p is g(z; P) = gft' 1 *) (2) It is obvious that g(x; 1) = g(x) and that the limit surface G(P) = {%; g{x; P) = 0} has only one point with minimal distance to the origin, namely px 0. For these limit surfaces two approximations are considered. The first approximation is obtained by replacing g(x; p) by its first order Taylor expansion g L (x; p) at $x 0, i.e. gdx; P) = (Vg(px 0 ; p)) T (x - p* 0 ) = (3) (Vg(x; p) denotes the gradient of the function g(x; p) atx). This is due to the definition of g(x; p) = (Vgfeo; l)) 7^" 1 * - aeo) (3«) Then, instead of the hypersurface G(P) the hyperplane defined by gi{x; P) = 0 is taken as approximating limit surface. The second approximation is given by replacing g(x; P) by its second order Taylor expansion g Q (x; P) at px 0, i.e.: 358

3 / ox ru la our, a x, fe-pgo) r P(P)(g-px 0 ) g 0 (-i; P) = [Vg(p* 0 ; P)] (* - Pxo) + (4) with D(P) the matrix d 2 g(x; p)/3x,axy J=Pn of the second derivatives of g(x; P) at px 0. This is, due to the definition: ^0(x;P) = [V 5 (xo;p)] T (P~ 1 x-xo) + (p- 1 * - Xo) r D(l) (P *~* o) (4a) The hypersurface G(P) is replaced by the hypersurface given by g<}( x '' P) = 0- Also, this new surface has only $x 0 as point with minimal distance to the origin. This can be shown by the Lagrange multiplier theorem see Ref. 13, p. 19. Then there are three probability contents to be compared: P(P) = P[g(X; p) < 0] L(P) = P[g L (X; p) < 0] (56) Q(P) = P[g Q (X; p) < 0] The second and the third are approximations for the first one. To evaluate P(p) a substitution is made: P(P) = {l*)-"' 1 I exp (^f) dx = (6) Jg(x;fi)<0 \ Substituting (xj,..., x n ) -» (yi,..., y ) with y ; = p -1 *,-: = (2TT)-" /2 P" Jg(y;l)<0 Define J(p) = [ exp ( Z / (5a) (5c) -P 2 lyl 2 ' exp( ^ - j ^ (7) P lyl )dy (8) J(P) is an integral over a fixed domain whose integrand is an exponential function depending linearly on the parameter p 2. These integrals are called Laplace type integrals. An extensive study of their asymptotic behavior for p 2 can be found in Ref. 2, Chapt. 8. Using the results given there, the asymptotic form of I(P) is (details see Appendix I) I(P) ~ (2-ir) exp (^)p- ( " +1) /r 1/2 (P -> oo) (9) in which / (defined in Eq. 23a, Appendix I) is a quantity independent of p depending only on the first and second derivatives of g(x; 1) at x 0. This yields for P(P) P(P) ~ (2^)- I/2 p- 1 exp (^-j \J\- 1/2 (p-> oo) (10) Using 4>(-x) ~ (2IT)~ 1/2 exp (-x 2 /2)x~ 1 (x standard normal integral 359 -» oo) with $( ) denoting the

4 P(p)~$(-P) ;r/ 2 (p^oo) (ii) The asymptotic behavior of P(P) depends only on the value of p and the first and second derivatives of g(x; 1) at x 0. For the second probability Q(P) which is given by Q(P) = P[g Q (X; p) < 0], the same result is obtained analogously since Q(X; 1) and g(x; 1) have the same first and second derivatives at x 0 due to the definition of g Q (x; p) Q(p) ~ <D(-p) / - 1/2 (p-> oo) (12) This yields P(P) ~ Q(P) (P ^ oo) (12«) The result shows that Q(P) is an asymptotic approximation for P(p) in the sense of asymptotic analysis, i.e. the relative error tends to zero for p -> oo. For the linear approximation L(P) the estimate is L(P) = *(-p) (13) This gives, using Eqs. 12 and 12a: ^f~l/r 1/2 <P ) 03a) The relative error of the linear approximation given by L(P)/P(p) - 1 converges therefore to / ~ 1/2-1, a quantity depending on the first and second derivatives of g(x; 1) at x 0. Due to this, if an estimate for the failure probability P(p) is needed, the quadratic approximation should be used. In particular, if further arithmetic operations with this estimate are performed, as in the case of decision problems, in which the result is multiplied by the costs in case of a failure, at least asymptotically correct results can be obtained using the outlined procedure of quadratic approximation. Generalized Reliability Index. In Ref. 4 Ditlevsen defines the generalized reliability index which is a generalization of the reliability index of Hasofer/Lind (8). For the failure domain F(p) = {x;g(x; p) < 0} the generalized reliability index p G is given by P G = -$- 1 (P(p)) (14) The Hasofer/Lind index for F(P) is, due to the definition of F(P), simply p. This is also the estimate for the generalized index which is obtained by a linearization of the limit surface at x Q. For p -> oo, for P(P) the asymptotic approximation given in Eq. 11 is valid; this yields <K-PG)~<f>(-P) J - 1/2 (P->») (14a) By taking logarithms and retaining only the dominant terms i^b-'-'i'i-"'^ <15 > Rewriting the equation - ( p G -PKP G+ p) Mn( rl/2) _ in(p) + in(pg) (i5fl) which yields, divided by (P + p G ) 360

5 PG - Pi -» 0 (P -» ) (15b) The relative and the absolute error for estimating p G by the usual Hasofer/Lind reliability index converges to zero for p > <*>. Case of Several Minimal Distance Points. If there are k points 3fi,... x k on the surface G with \x t \ =... = \x k \ = 1, the quadratic approximation Q(P) is derived (as demonstrated in Ref. 2) by calculating the contribution given by Eq. 11 for each point x t separately and then adding up the results, yielding (2(0) ~ 4>(-P) E \J'\' in (P-* ) ( 16 ) >=i with /, denoting the value given in Eq. 23 computed for the point x, instead of x 0. From the mathematical considerations given in Appendix I it follows that it would be sufficient to postulate that g(x) is twice continuously differentiable near the points on the surface G which have minimal distance to the origin and continuous elsewhere. The results are valid also in this case. Calculation of the Quadratic Approximation for Given Limit State. Let a limit state function g(x) be given in the n-dimensional space of standard normal independent variables with g(0) > 0. This function defines a limit surface G = [x; g(x) = 0], Define g(x) by g(x) = g{$ox) with Po = min 2eG \z\. Then, g(x) = g(x; p 0 ) with g(x p) defined as before. If p 0 is large the asymptotic approximation Q(p 0 ) can be used to approximate P(g(X) < 0) = P(g(X; Po) < 0). The approximation can be found according to the following procedure: (1) Calculate all points z x,..., z K on G with si =... = sit = Po (=min2 E G s ); (2) for each point Po 2; calculate J, as defined in Eq. 23; and (3) the approximation for P(g(X) < 0) is P(g(X) < 0) ~ $(-p 0 )(X \h\~ 1/2 ) (17) Using Eq. 29 in Appendix I, /, = 11";/ (1 - K, /; ) with the K,-, ; - (/' = 1,..., n - 1) denoting the ordered main curvatures of the surface G = [x;g(x; 1) = 0] at Po" 1?,. But due to the definition of g(x) the following relation is valid (see also Ref. 3) K,,; = PoK,-, ; - (17a) with the K,, ; being the ordered main curvatures of the surface G = [x; g(x) = 0] at z,. This yields: P(g(X) < 0) ~ *(-Po) 2 (f[(i-pok, ; r 1/2 ) (17b) This formula gives the estimate in terms of the distance of the limit surface from the origin and the curvatures of the surface at the minimal distance points. Examples 1. Approximation of Ellipse: In a two-dimensional space the limit state 361

6 a ((3) Downloaded from ascelibrary.org by GEORGE MASON UNIVERSITY on 07/16/13. Copyright ASCE. For personal use only; all rights reserved FIG. 1. Q(P)/P(P) versus (3 for a = 1.25, 2, 4 functions g(x; 0) = (xi/a) 2 + x - p 2 (a > 1) are given. There are two points on the curve g(x; 1) = 0 with minimal distance to the origin Xi = (0,1) and x 2 = (0,-1). This yields, using Eq. 16: Q(P) = 2<i>(-p)(l-«- 2 )- 1 / 2 (18) for the quadratic approximation Q(P). This approximation is compared with the exact probability P(P) (which is given in Ref. 10, p. 164, Eq. 78) in Fig. 1 for a = 1.25, 2, 4. The relative error converges to zero for p» oo. 10' _ ^^^^^. s \ \ \ ;. ^ exact quadratic approximation linear approximation X. X X. \ N \ X FIG. 2. Sum of 10 Exponentially Distributed Random Variables \ a. 362

7 2. Sum of Exponentially Distributed Random Variables: Given are n independent, identically distributed random variables, Y lt... Y. Each has a standardized exponential distribution with cumulative distribution function F(y) = 1 - exp (-y). Let the limit state function be given = by Siy) n + <*V" - 2?=i yt (in which a is constant). The transformation in the space of standard normally distributed variables X lf..., X n is given by x { = -<1>~ 1 (exp (-y,)) (/ = 1,..., n). The limit state function in this space is g(%) = 2" =1 In [4>(-x,)] + n + avn. Using the method of Lagrange multipliers (see Ref. 13, p. 19), the only point Z\ on the limit surface with minimal distance to the origin is found 2l = (z,...,z) with z = -$-'lexpl I 1 (19) The distance to the origin is Vn z. For / x using Eq. 23 and denoting the standard normal density by <j>(z) this yields h^1- Z M-z)-z] <t>(2) The approximation is then Ptf(X)<Q)-*(-Vn«)-{l-«[^r^]} -<B-l)/2 (20a) (20b) In Fig. 2 the exact probability (given by the incomplete gamma function), the approximation by Eq. 20(b) and the approximation <I>(-Vn z) obtained by linearizing the limit surface are compared for 0 < a < 3 and n = 10. SUMMARY AND CONCLUSIONS The problem of obtaining approximations for multinormal integrals is reviewed. Two approximations, the linear and the quadratic, are examined. The result shows that only the quadratic approximation gives an asymptotic approximation for the integral in the sense of asymptotic analysis, whereas the linear approximation produces an uncontrollable relative error. Only in the case in which the generalized reliability index has to be computed does the linear approach produce an asymptotic approximation. This implies that, if the failure probability has to be estimated, only the quadratic approximation should be used. ACKNOWLEDGMENT The author would like to thank I. Hordan-Klaucke, M. Hohenbichler and especially N. Zorn, for their help. The study was part of a project supported by the Deutsche Forschungsgemeinschaft. APPENDIX I. ASYMPTOTIC EXPANSION OF MULTINORMAL INTEGRAL In Ref. 2, chapt. 8, asymptotic expansions are given for integrals of the form 363

8 J(\) = exp (Xf(x))f -JL 0 (x)dx (X^ oo) (21) JD D is a fixed domain in the n-dimensional space, f(x) and f Q (x) are at least two times continuously differentiable functions. Further, it is assumed that the boundary or D is given by the points x with h(x) = 0, where h(x) is also at least twice continuously differentiable. It is shown that, if f(x) has no global maximum with respect to D at an interior point of D, the asymptotic behavior of I(X) depends on the points on the boundary where f(x) attains its global maximum on the boundary. Due to the Lagrange multiplier theorem, a necessary condition for these points is that V/(x) = K- Vh(x) (K is a constant). The contribution of one of these points to the asymptotic expansion of 1(X) is given in Eqs. 8, 3, 64, p. 340, Ref. 2. Defining D = \x;g{x; 1) < 0], X = p 2, f(x) = -\x\ 2 /2, h(x) - g(x; 1), the formula can be applied to obtain an asymptotic expansion for I((3). Due to the assumption made at the beginning, there is only one point x 0 on the surface g(x; 1) = 0 with minimal distance to the origin, i.e. only one point x 0 in which -\x\ 2 /2 achieves its maximum. Eqs. 8, 3, 64 yields then J(B) ~ (2ir) < "- 1)/2 exp f~ p2-2 ) p-c + i) / -i/2 (p _ oo) (22) with J = 2 2 xucot(-b -Kg,,) (23) i=l ;=1 x'o = «'th component of x 0 (23a) 8, 7 = delta Kronecker Symbol (23b) S 2 (x; 1) gti = ' (23c) dxjdxj K = Vgfe 0 ; l)! -1 (23d) cof (-8!; - Kgij) denotes the cofactor (see, e.g., Ref. 1, p. 372) of the element (-8, ; - - Kg tj ) in the matrix (-8 i; - - Kgij) it j=i,...,. Since \x 0 \ = 1, the formula simplifies I(p) ~ (2TT) <B - 1)/2 exp ^V (n+1) / - 1/2 (24) Due to the rotational symmetry it can be assumed for further considerations, that x 0 = (0,..., 0, 1) (i.e. the unit vector in the direction of the x -axis). Then, since x 0 is parallel to the gradient of g(x; 1) at x 0 due to the Lagrange multiplier theorem, the tangential space of the hypersurface G(l) at x 0 is spanned by the unit vectors in the direction of the first n 1 axes. Then / is given (using the definition of the cofactor): / = cof[-8 -^ ] = det(b), with B = (8 lra + ^lm ) lim=1 _! (25) Defining: D = (-Kg lm ) 1/tn=1 _j (25a) 364

9 and denoting the unity matrix by I: det (B) = det (I - D) (26) Det (B) is given by the product of the eigenvalues of B, which are the roots of det (B - KI) = det ((1 K)I D). But these roots are given by 1 - Kj(i_ = 1,..., n - 1), in which the K,'S are the eigenvalues of the matrix D. This gives: 1/1 = 2 a - *>) These eigenvalues are the main curvatures of the surface G(l) at x 0 (see Ref. 3, chap. 12). The curvature is defined positive, if the surface is curved towards the origin, as in Ref. 13. The last formula shows, in which cases the approximation is not applicable. Since x 0 is a point on the surface with minimal distance to the origin, the main curvatures at x 0 must be not larger than unity. Elsewhere, consider a point 2 on the surface near x 0 in the direction of a principal axis of curvature at x 0 with curvature K, larger than unity. Due to the definition of the curvature, the curve on the surface connecting x 0 and x is approximated by a part of a circle in the same direction through x 0 with radius 1/K, and center (0,.,0,1 1/K,). Using elementary trigonometric relations, for small distances \x x 0 \ the squared distance of x to the origin is approximately,., (1 - K,) \x - X 0 \ 2 \x\ 2 «1 + - < 1 (28) This contradicts the assumption, that x 0 is a point on the surface with minimal distance to the origin with respect to the surface and therefore K ( < 1. Due to this n-l l/l = EI (1 - K ;) (29) In the case that one curvature is exactly equal to unity, the approximation cannot be used. Then it becomes necessary to study higher derivatives of g(x) and the global behavior of the function, but no general results are available for this problem. APPENDIX II. -REFERENCES 1. Bellman, R., Introduction to Matrix Analysis, 2nd Ed., McGraw Hill, New York, N.Y., Bleistein, N., and Handelsman, R. A., Asymptotic Expansions of Integrals, Holt, Rinehart and Winston, New York, N.Y., Breitung, K., "An Asymptotic Formula for the Failure Probability," DIALOG 82-6, Department of Civil Engineering, Danmarks Ingeniarakademi, Lyngby, Denmark, 1982, (to appear). 4. Ditlevsen, O., Uncertainty Modelling, McGraw-Hill, New York, N.Y., Ditlevsen, O., "Principle of Normal Tail Approximation," Journal of the Engineering Mechanics Division, ASCE, Vol. 107, No. EM6, Dec, 1981, pp (27)

10 6. Dolinski, K., "First-Order Second Moment Approximation in Reliability of Structural Systems; Critical Review and an Alternative Approach," Structural Safety, (to appear). 7. Fiessler, B., Neumann, H.-J., and Rackwitz, R., "Quadratic Limit States in Structural Reliability," Journal of the Engineering Mechanics Division, ASCE, Vol. 105, No. EM4, Aug., 1979, pp Hasofer, A. M., and Lind, N. C, "An Exact and Invariant First-Order Reliability Format," Journal of the Engineering Mechanics Division, ASCE, Vol. 100, No. EMI, Feb., 1974, pp Hohenbichler, M., and Rackwitz, R., "Non-Normal Dependent Vectors in Structural Safety," Journal of the Engineering Mechanics Division, ASCE, Vol. 107, No. EM6, Dec, 1981, pp Johnson, N. I., and Kotz, S., Distributions in Statistics; Continuous Unvariate Distributions 2, Wiley, New York, N.Y., Psenicnyj, B. N., "Algorithms for General Mathematical Programming Problems," Cybernetics, Vol. 6, No. 5, 1970, pp Rackwitz, R., and Fiessler, B., "Structural Reliability under Combined Random Sequences," Computers and Structures, Vol. 9, 1978, pp Thorpe, J. A., Elementary Topics in Differential Geometry, Springer, New York, N.Y., APPENDIX III. NOTATION The following symbols are used in this paper: B = matrix defined in Eq. 25; D = matrix defined in Eq. 25a; D(P) = matrix of second derivatives; F,F(P) = failure domains; G, G(p), G(p) = limit surfaces; g(x),g*(x),g(x),g(x; P) = limit state functions; gi(x; P) = linear approximation for the limit state function g(x; P); Q(X; P) = quadratic approximation for the limit state function g(x; P); I = unity matrix; 7(P) = integral defined by Eq. 8; / = quantity defined in Eq. 23; L(P) = approximation for P(P) using linear Taylor expansion; P,P(P) = failure probability; Q(P) = approximation for P(P) using quadratic Taylor expansion; S = safe domain; PG = generalized reliability index; Vg(2o) = gradient of g(x) at x 0 ; Kj/Kf, ; -,K(^. = main curvatures of surfaces; <E>(x) = standard normal integral; and <$>(x) = standard normal density. 366