COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Reliability Analysis using Saddlepoint Approximation
Jia Wang1*, Ka-Veng Yuen1, Siu-Kui Au2
1
2
Faculty of Science and Technology, University of Macau, Macao, China
Department of Building and Construction, City University of Hong Kong, Hong Kong, China
Email: ma46616@umac.mo
Abstract Saddlepoint approximation is a powerful tool for accurate estimation of the cumulative distribution function
(CDF) of a random variable, if its cumulant generating function (CGF) exists. In this paper, the saddlepoint
approximation is introduced for reliability analysis which is one of the most important tasks in engineering design. The
efficiency of the proposed method, is demonstrated by a 10-story shear building subjected to white noise excitation.
Key words: Failure probability, Monte Carlo simulation, reliability analysis, saddlepoint approximation
INTRODUCTION
The assurance of the performance of engineering structures within the design life is one of the principal responsibilities
of engineers. However, due to inherent uncertainties such as material properties and loading conditions to which they
are exposed to during their daily operation, the assurance of performance can seldom be perfect and risk generally
exists in engineered structures. Consequently, reliability analysis of the structures, concerned with the probability that
the structure will not reach some specified state of failure, arises as a probabilistic measure of the assurance of the
structural performance.
PROBLEM STATEMENT
The determination of the reliability of a structure usually requires calculating its complement, the probability of failure,
which is essentially the integration of the probability density function (PDF) in the failure domain. For an engineering
application, a random variable Y is used to specify the performance of the structure, and the failure event is defined as
F = {Y > b} where b is a corresponding bound. If the PDF f (y) of the performance variable Y can be obtained, the
failure probability will be given by the following integration:
In reliability analysis, all the uncertainties are parameterized and denoted by a random vector Z = [z1, …, zn], where n is
the number of random variables. The performance variable Y is a function of Z, but the relationship between them is
often complex such that it is not feasible to compute f (y) from the joint PDF of the uncertain parameters, q(Z) .
Consider an ns-story building under random excitation as an example. The time duration of interest T is discretized into
nt time instants {t1, …, tnt}, and the failure event is defined as the exceedance of a threshold b by any of the inter-story
drifts during T. Let Xj(tk; Z) denote the jth inter-story drift response at the kth time instant, then the failure event can be
defined as
where the performance variable Y is
The failure probability can be computed from the distribution of Z as:
⎯ 421 ⎯
where F is the failure domain within the parameter space; IF is an indicator function: IF(Z) = 1 if Z∈F and IF(Z) = 0
otherwise. Although the evaluation of this integral is straight-forward in theory, it is computationally prohibitive when
the integration dimension n is large and the failure domain is complex.
MONTE CARLO SIMULATION METHOD
Monte Carlo simulations (MCS) offer a robust methodology well suited for solving such high-dimensional reliability
problems [1- 2]. In this method, the failure probability PF = IF(Z)q(Z) dZ is viewed as the mathematical expectation of
IF(Z) with Z distributed according to q(Z). This implies that the failure probability can be estimated as a sample average
of the indicator function IF(.) over the sequence of independent and identically distributed (i.i.d.) samples {Z1, …, ZN}
generated according to their PDF q(Z) :
where N is the number of samples. The MCS estimator
in Eq. (5) converges to the failure probability PF with
probability 1 (Strong Law of Large Numbers), and is asymptotically normal distributed as N→∞ (Central Limit
Theorem). Therefore, it is unbiased, i.e., E[ ] = PF. The efficiency of the MCS method, and in general the efficiency
of a simulation-based reliability method, can be measured by the coefficient of variation (C.O.V.) of the estimator,
which is defined as the ratio of its standard deviation to its expectation, i.e.,
Note that the expression depends only on the failure probability PF and the number of samples N, regardless of the
number of uncertain parameters n. The applicability of MCS is independent of the specifications of the problem. The
specifications of the problem, such as the type of excitation models and the type of parameter PDFs as well as the
number of uncertain parameters, enter the MCS procedure through only the indicator function IF(.), whose value is
determined by structural analysis. The main drawback of MCS, however, is that it is not efficient to estimate small
failure probabilities. For small PF, Eq. (6) can be approximated as
In practice, the failure probability is expected to be small, because one does not anticipate to construct a fragile
engineering structure. In this case, Eq. (7) implies that the computational effort required by MCS to obtain an
acceptable accuracy is prohibitively large. For example, if PF = 10−6, one needs N = 108 samples (structural analyses) to
achieve the accuracy level of 10% C.O.V.
SADDLEPOINT APPROXIMATION METHOD
The saddlepoint approximation technique was introduced to statistics for approximating the PDF by Daniels (1954) [3].
Since then, the research and applications of saddlepoint approximations have vastly increased [4-6]. A detailed
account of saddlepoint approximations and related techniques can be found in [7].
Let Y be a random variable with PDF f (y). The moment generating function of Y is defined as
and the cumulant generating function (CGF) of Y is defined as the natural logarithm of M(ξ):
To restore f (y) from K(ξ), one can apply the inverse Fourier transformation:
⎯ 422 ⎯
Using the exponential power series and Hermite polynomials, Daniels (1954) [3] developed the saddlepoint
approximation formula for f (y):
(.) is the second derivative of the CGF, and ξs is called the saddlepoint, which satisfies the following
where
relationship for a given value of y:
Based on the saddlepoint approximation for f (y) in Eq. (11), Lugannani and Rice (1980) [8] derived the following
formula to approximate the CDF of Y:
where Φ(.) and φ(.) are the CDF and PDF of the standard normal random variable, with w and v given by
where sgn(.) is the signum function.
To obtain the CDF value of Y at a point y, it requires only to find its corresponding saddlepoint ξs instead of integration.
Consequently, the existence and properties of the real roots for Eq. (12) is the key to the application of the saddlepoint
approximation. According to Eq. (9), Eq. (12) can also be rewritten as
The slope of the ξ-y curve is
By applying the Cauchy-Schwarz inequality, one can show that
. Thus,
which ensures that the curve y = (ξ) is monotonically increasing and so the root ξs is unique if it exists. The property
(ξ) > 0 also guarantees that the square root in Eq. (14) gives real values. Define a function g(ξs) ≡ ξs y −K(ξs) =
ξs
(ξs) −K(ξs), then (ξs) = ξs (ξs). Since
(ξs) > 0, the function g(ξs) is monotonically decreasing when ξs < 0,
and monotonically increasing when ξs > 0. Thus, we can obtain
to give a real value for w.
CONSTRUCTION OF THE CGF OF Y
As discussed in the previous section, the key to the application of the saddlepoint method is to estimate the CGF of the
system performance Y = g(Z). In engineering applications, various sampling techniques such as MCS are available for
generating the sample values of Y provided that the distribution of Z and the analysis model Y = g(Z) are given.
According to the construction way of the CGF based on the sample values {Y1, …, YN}, the saddlepoint method is
divided into three classes in this paper.
1. Cumulant method The cumulant generating function (CGF) K(ξ) can be expanded in Taylor series at the point
ξ = 0:
⎯ 423 ⎯
Since K(0) = 0, the above equation can be simplified as
where κj ≡ K( j)(0) is called the jth cumulant of the random variable Y. For example, κ1 = E[Y] and κ2 = var[Y]. In practice,
the first four cumulants are used in the series expansion [9], i.e.,
The unbiased estimators for the cumulants of a population are generally provided by k statistics [10]. Based on the
sample values of the random variable Y, the unbiased estimators for the first four cumulants are given by
. For the application of the cumulant method, we
where Sr (r = 1, …, 4) is the rth moment estimate and Sr =
(ξ) > 0 and thus the curve y = (ξ) is monotonically increasing in the domain of ξ concerned.
have to ensure that
Noting that
(0) = κ2 > 0, according to the continuity of the function
(ξ), we can find a region of ξ in the
neighborhood of ξ = 0 such that the Lugannani-Rice formula can be used to approximate the CDF P(Y ≤ y). By
examining the curve of
(ξ), this region can be found in four different cases:
Case 1 κ4 < 0
> 0. Typical curves of y = (ξ) and its slope function
Then, Δ =
can be given by (ξl ;ξu) where ξl and ξu can be computed from
(ξ) are shown in Fig. 1, and the region
(23)
Case 2 κ4 ≥ 0 and Δ =
≤0
(ξ) ≥ 0 and y = (ξ) is monotonically increasing for all ξ, as shown in Fig. 2. A special subcase occurs
Then =
≤ 0, κ3 = 0 is required. Thus
(ξ) = κ2 is a positive
when κ4 = 0. In order to satisfy the condition Δ =
constant for all ξ.
Figure 1: The typical curves of
(ξ) and its slope function
⎯ 424 ⎯
(ξ) in the first case with the cumulant method
Figure 2: The typical curves of
Case 3 κ4 ≥ 0 and Δ =
(ξ) and its slope function
(ξ) in the second case with the cumulant method
> 0 as well as κ3 > 0
Then the region of ξ which is applicable for the cumulant method is (ξl ,∞) with ξl given by
(24)
The curves of y = (ξ) and its slope function
the expression of ξl given by
(ξ) are shown in Fig. 3. A special subcase occurs when κ4 = 0, with
(25)
Case 4 κ4 ≥ 0 and Δ =
> 0 as well as κ3 < 0
Then the region of ξ which is applicable for the cumulant method is (−∞, ξu) with ξu given by
(26)
Figure 3: The typical curves of
(ξ) and its slope function
(ξ) in the third case with the cumulant method
Figure 4: The typical curves of
(ξ) and its slope function
(ξ) in the fourth case with the cumulant method
⎯ 425 ⎯
The curves of y = (ξ) and its slope function
the expression of ξu given by
(ξ) are shown in Fig. 4. A special subcase occurs when κ4 = 0, with
(27)
From the bounds of ξ, we can compute the bounds yu (corresponding to ξu) and yl (corresponding to ξl ). If the specified
y ∈ (yl ; yu), then the corresponding CDF value can be computed using the Lugannani-Rice formula.
2. Expectation method The main idea of the expectation method is to use the sample average to substitute for the
expectation,
where E0(ξ) ≡ (1/N)
where E1(ξ) ≡
. Then the functions
= (1/N)
and E2(ξ) ≡
(ξ) and
(ξ) can be obtained through differentiation:
= (1/N)
. According to the Cauchy-Schwarz
2
inequality, we can get
, which ensures
(ξ) > 0 and
monotonically increasing for every ξ as shown in Fig. 5. Another important feature of the curve y =
Figure 5: The typical curves of
(ξ) and its slope function
(ξ) with the expectation method
Figure 6: The typical curves of (ξ) and its slope function
(ξ) with a combination of
the expectation method and the cumulant method
⎯ 426 ⎯
(ξ) is
(ξ) is
and
which implies that the expectation method is applicable for estimating P(Y ≤ y) if y ∈ [Ymin, Ymax].
3. Proposed method In practical applications, the performance variable Y is always distributed over [0,∞). A
combination of the cumulant method and the expectation method is proposed to estimate the CGF as follows:
Then,
(ξ) and
(ξ) are given by
Table 1 Selected threshold levels
Table 2 Results for the sample mean of the estimated P(Y > b)
and
with the typical curves shown in Fig. 6. The saddlepoint corresponding to a specified threshold y is obtained from the
relationship y = (ξ).
The continuity of the function
(ξ) is examined at the point ξ = 0, i.e.,
⎯ 427 ⎯
From the discussion of the cumulant method and the expectation method, the proposed method is applicable in the
interval [Ymin, yu).
EXAMPLE
A 10-story building model is used to examine the proposed method. The inter-story stiffness and floor mass is taken to
be identical for all stories. The stiffness-to-mass ratio is taken to be 1762.2s−2 so that the building has fundamental
frequency 1.0Hz. Furthermore, Rayleigh damping is used such that the damping ratios of the first two modes are 2.0%.
The building is subjected to Gaussian white noise ground accelerations with spectral intensity 0.001m2s−3. The
response of the system is computed at the discrete time instants, where the sampling interval is assumed to be Δt =
0.01s and the duration of study is T = 10s, so that the number of the time instants is nt = T/Δt +1 = 1001. The failure
event is defined as the exceedance of a threshold b by any of the inter-story drifts during the time duration. Let Xj(tk)
denote the inter-story drift response for the jth story at time tk, j = 1, …, 10, k = 1, …, 1001, then the failure event of
.
interest can also be defined as F = {Y > b}, where Y =
Table 3 Results for the sample C.O.V. (×10−3) of the estimated P(Y > b)
Table 1 shows some reference threshold levels b corresponding to typical failure probabilities (for example 10−3) using
the MCS method. In order to ensure the accuracy of the results, 10 million samples are generated so that the C.O.V. of
the estimated P(Y > b) are very small. Note that the largest C.O.V. which occurs at the failure probability of 10−4 is
316×10−4, i.e., 3.16%.
Table 2 shows the sample mean of the estimates of PF over 1000 independent simulation runs with 10000 samples for
each run at the selected b using the MCS method ( ), the cumulant method ( ), the expectation method ( ) and the
are actually the estimated PF using 10 million samples.
proposed method ( ). Note that the sample mean
are considered as ‘exact’ PF and used to measure the accuracy
Considering their small C.O.V. as shown in Table 1,
are very close to
with the largest relative error of
of other results. It is clearly seen that the sample mean and
−3
at PF = 0.90,0.80
4% and 8% (both for PF = 1.0×10 ). Note that although the deviation of the sample mean from
and , it can be seen that the
is large, its relative error is still small (1.8% and 2.1%). By comparing the results of
(than ) when PF is small, and the results
results of the sample mean using the cumulant method ( ) are closer to
of the sample mean using the expectation method ( ) are preferable when PF is large. Combining the advantages of
the cumulant method and the expectation method, the results of the sample mean using the proposed method ( ) are
shown in the last column.
Table 3 shows the sample C.O.V. of the estimates of PF over the same 1000 independent simulation runs as those for
Table 2 using the MCS method (α0), the cumulant method (α1), the expectation method (α2) and the proposed method
(α3). In order to judge the efficiency of the cumulant method, the expectation method and the proposed method, the
number of required samples using MCS to obtain the same accuracy as α1, α2 and α3 (i.e., N1, N2 and N3), is compared
with the present sample number N0. According to Eq. (6), Ni can be measured by (1−PF)=(PF ), i=0, 1, 2, 3. Thus the
ratio Ni=N0 = (α0=αi)2, i=1, 2, 3, is used to compare the required samples to obtain the same accuracy between MCS
and the saddlepoint methods, as shown in the fourth, sixth and eighth columns. It is clearly seen that the cumulant
⎯ 428 ⎯
method is efficient to estimate PF when PF is not too large (≤ 0.60), with the observation that more samples are required
for MCS to achieve the same accuracy as α1 (N1/N0=1.2 to 20); whereas the required samples using MCS to obtain the
accuracy α1 are less for most of large PF, with N1/N0 = 0.21, 0.37, 0.62 for PF = 0.99, 0.8, 0.7, respectively. It is also
noted that the efficiency of the expectation method is obvious for large PF (≥ 0.40), with more required samples for
MCS to obtain the same accuracy as α2 (N2/N0=1.2 to 1.5); while the efficiency is not always maintained for small PF,
with the ratio N2/N0 = 0.93, 0.84, 0.91 for PF = 0.2, 0.1, 1.0×10−4, respectively. The seventh and eighth columns show
the results using the proposed method, a combination of the cumulant method and the expectation method for the
estimation of small PF and large PF respectively. Note that the results using the proposed method are consistent with
those using the cumulant method for PF ≤ 0.40, and with those using the expectation method for PF ≥ 0.50, i.e., the
efficiency of the proposed method can be maintained for all PF (N3/ N0 = 1.2 to 20).
CONCLUSIONS
The saddlepoint approximation technique is presented for the reliability assessment. In practice, the key of the
application of the saddlepoint method is to estimate the cumulant generating function (CGF) of the performance
variable based on its samples. As shown in the illustrative example, the saddlepoint method, with the CGF constructed
by a combination of the cumulant method and the expectation method, can provide good estimations of the failure
probabilities, and is significantly more efficient than MCS.
Acknowledgement
The financial support from the research committee of the University of Macau under grant RG068/04-05S/YKV/FST
UMAC is gratefully acknowledged.
REFERENCES
1. Rubinstein RY. Simulation and the Monte-Carlo method. Wiley, NewYork, 1981.
2. Fishman GS. Monte Carlo: Concepts, Algorithms, and Applications. Springer-Verlag, NewYork, 1996.
3. Daniels HE. Saddlepoint approximations in statistics. Annals of Mathematical Statistics, 1954; 25: 631-650.
4. Reid N. Saddlepoint methods and statistical inference. Statistical Science, 1988; 3: 213-227.
5. Huzurbazar S. Practical saddlepoint approximations. American Statistician, 1999; 53(3): 225-232.
6. Goutis C, Casella G. Explaining the saddlepoint approximation. American Statistician, 1999; 53(3): 216-224.
7. Jensen JL. Saddlepoint Approximations. Clarendon Press, Oxford, England, 1995.
8. Lugannani R, Rice SO. Saddlpoint approximation for the distribution of the sum of independent random
variables. Advances in Applied Probability, 1980; 12: 475-490.
9. Sudjianto A, Du X, Chen W. Probabilistic sensitivity analysis in engineering design using uniform sampling and
saddlepoint approximation. SAE 2005 Transactions Journal of Passenger Cars: Mechanical Systems, Paper No.
2005-01-0344.
10. Fisher RA. Moments and product moments of sampling distributions. Proceedings of the London Mathematical
Society, 1928; 2(30): 199-238.
⎯ 429 ⎯