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This manuscript has been published in Mechanical Systems and Signal Processing. The DOI of this manuscript
is: https://doi.org/10.1016/j.ymssp.2022.109775. Please cite as: C. Ding, C. Dang, M. A. Valdebenito, M. G. Faes,
M. Broggi, M. Beer, First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems
by a fractional moments-based mixture distribution approach, Mechanical Systems and Signal Processing 185 (2023)
109775.
First-passage probability estimation of high-dimensional nonlinear stochastic
dynamic systems by a fractional moments-based mixture distribution approach
Chen Dinga, Chao Danga,∗, Marcos A. Valdebenitob, Matthias G.R. Faesc, Matteo Broggia, Michael Beera,d,e
aInstitute for Risk and Reliability, Leibniz University Hannover, Callinstr. 34, Hannover 30167, Germany
bFaculty of Engineering and Sciences, Universidad Adolfo Ib´
a˜
nez, Av. Padre Hurtado 750, 2562340 Vi˜
na del Mar, Chile
cChair for Reliability Engineering, TU Dortmund University, Leonhard-Euler-Str. 5, Dortmund 44227, Germany
dInstitute for Risk and Uncertainty, University of Liverpool, Peach Street, Liverpool L69 7ZF, United Kingdom
e
International Joint Research Center for Resilient Infrastructure & International Joint Research Center for Engineering Reliability and Stochastic
Mechanics, Tongji University, Shanghai 200092, PR China
Abstract
First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems is a significant task to
be solved in many science and engineering fields, but remains still an open challenge. The present paper develops a
novel approach, termed ‘fractional moments-based mixture distribution’, to address such challenge. This approach is
implemented by capturing the extreme value distribution (EVD) of the system response with the concepts of fractional
moment and mixture distribution. In our context, the fractional moment itself is by definition a high-dimensional
integral with a complicated integrand. To efficiently compute the fractional moments, a parallel adaptive sampling
scheme that allows for sample size extension is developed using the refined Latinized stratified sampling (RLSS).
In this manner, both variance reduction and parallel computing are possible for evaluating the fractional moments.
From the knowledge of low-order fractional moments, the EVD of interest is then expected to be reconstructed. Based
on introducing an extended inverse Gaussian distribution and a log extended skew-normal distribution, one flexible
mixture distribution model is proposed, where its fractional moments are derived in analytic form. By fitting a set
of fractional moments, the EVD can be recovered via the proposed mixture model. Accordingly, the first-passage
probabilities under different thresholds can be obtained from the recovered EVD straightforwardly. The performance
of the proposed method is verified by three examples consisting of two test examples and one engineering problem.
Keywords:
First-passage probability, Stochastic dynamic system, Extreme value distribution, Fractional moment, Mixture
distribution
∗Corresponding author
Email address: chao.dang@irz.uni-hannover.de (Chao Dang)
Preprint submitted to Elsevier October 11, 2022
arXiv:2210.04715v1 [stat.ME] 10 Oct 2022
1. Introduction
Stochastic dynamic systems which involve the randomness in internal system properties and/or external dynamic
loads are widespread in various science and engineering fields, such as meteorology, quantum optics, circuit theory and
structural engineering [
1
]. To assess the effects of input randomness on the system performance, dynamic reliability
analysis has drawn increasing attention. Generally, dynamic reliability analysis for stochastic dynamic systems can be
classified as the first-passage probability evaluation and the fatigue failure probability estimation [
2
]. In the literature,
the first-passage probability evaluation has been extensively studied over the past several decades. However, finding
efficient and accurate solutions to the first-passage problem still remains challenging. The reason is twofold: (1) the
high-dimensional input randomness and strongly nonlinear behavior of stochastic dynamic systems may be encountered
simultaneously; (2) the first-passage probabilities of such systems under certain thresholds may be relatively small.
The existing approaches for first-passage probability estimation can be broadly divided into four kinds: the out-
crossing rate approaches, the diffusion process approaches, the stochastic simulation approaches and the extreme
value distribution (EVD) estimation approaches. For the out-crossing rate approaches, the first-passage probability
is evaluated considering the time of out-crossing within a time duration on the basis of Rice’s formula [
3
–
6
]. Such
approaches are based on the Poisson assumption that level-crossing events are mutually independent and each happens
at most once, or the Markovian assumption that the next crossing event only relates to the present event [
7
]. Although
these solutions can be accurate in some special cases, they may be not applicable for general cases. Besides, it is
hard to derive the joint probability density function (PDF) and its derivatives of the system response of interest when
complicated nonlinear stochastic dynamic systems are encountered. The diffusion process approaches evaluate the
first-passage probability by solving a partial differential equation, such as the Kolmogorov backward equation [
8
] or
the Fokker Planck equation [
9
]. Solutions to such equations could be derived via the path integration method [
10
–
12
],
stochastic average technique [
13
,
14
], ensemble-evolving-based generalized density evolution equation [
2
,
15
], etc.
Nevertheless, this kind of approach is mostly applicable for nonlinear stochastic dynamic systems enforced by white
noise. For the stochastic simulation approach, the extensively used Monte Carlo simulation (MCS) [
16
] is able to
address problems regardless of their dimensions and nonlinearities. However, MCS is inefficient and even infeasible
to assess a small probability for an expensive-to-evaluate model since a considerably large number of simulations
are required. Although some variants of MCS have been developed, such as important sampling [
17
–
20
] and subset
simulation [21–23], they still suffer their respective limitations concerning efficiency, accuracy and applicability, etc.
Recently, the EVD estimation approaches have attracted lots of attention. This is because once the EVD of system
response of interest is obtained, the first-passage probability can be straightforwardly and conveniently evaluated [
24
].
Nevertheless, the analytical solution to the EVD is difficult and even impossible to be obtained for a general nonlinear
stochastic dynamic system. Therefore, various approximation methods have been developed to estimate the EVD, which
can be roughly classified as probability conservation-based methods and moments-based methods. According to the
principle of probability conservation, the probability density evolution method (PDEM) [7,24] and direct probability
integral method (DPIM) [
25
] are derived, which can be used for the purpose of EVD estimation. However, since such
methods are typically dependent on the partition of random variable space, their application for high-dimensional
problems may be challenging. Moment-based methods, on the other hand, estimate the first-passage probability by
fitting an appropriate parametric distribution model to the EVD, and the free parameters of the distribution model are
obtained from the estimated moments of the EVD. The integer moments-based methods can be adopted to recover the
2
EVD [
26
,
27
], where high-order integer moments, i.e., skewness and kurtosis, need to be considered. Yet it is difficult
to evaluate such high-order integer moments using a small sample size, due to their large variability [
28
]. To alleviate
such difficulty, a series of methods based on non-integer moments, such as fractional moments and linear moments,
have been developed. The fractional moments-based maximum entropy methods [
29
–
32
] can estimate the first-passage
probabilities of nonlinear stochastic dynamic systems from low to high dimensions. However, it is difficult to solve the
non-convex optimization problem that is typically encountered, and the obtained results can be easily trapped into local
optimum. Besides, due to the polynomials involved in the maximum entropy density, the recovered EVD can have
unexpected oscillating distribution tail, which then leads to an inaccurate evaluation of the first-passage probability. Two
mixture parametric distribution methods in conjunction with fractional moments [33] or moment-generating function
[
34
] are developed. These methods enable to evaluate first-passage probabilities of high-dimensional and strongly
nonlinear stochastic dynamic systems from a small number of simulations. Furthermore, a fractional moments-based
shifted generalized lognormal distribution method [
35
] is utilized to assess seismic reliability of a practical bridge
subjected to spatial variate ground motions. Besides, the linear moments-based polynomial normal transformation
distribution method [
36
] is developed to analyze high-dimensional dynamic systems with deterministic structural
parameters subjected to stochastic excitations.
Overall, the fractional moments-based methods offer the possibility to deal with both high-dimensional and strongly
nonlinear stochastic dynamic systems from a reduced number of simulations, even with small first-passage probabilities.
In view of this, the present paper mainly focuses on such methods. Despite those attractive features, the fractional
moments-based methods still have two main problems to be solved. On one hand, the sample size for evaluating
fractional moments is usually empirically fixed. This is primarily because the sampling-based schemes adopted
by the existing methods do not allow for the sample size extension. However, the optimal sample size should be
problem-dependent. With a predetermined sample size, the adopted sampling methods may encounter over-sampling or
under-sampling, leading to a waste of over-all computational efforts or unsatisfactory accuracy of estimated fractional
moments. On the other hand, the success of fractional moments-based methods for first-passage probability evaluation
also depends on the selection of an appropriate distribution model. Although the existing distribution models are
capable of representing EVDs for some problems, their flexibility and applicability are limited. Hence, for a wide
range of problems, they may still lack the ability to accurately recover the EVDs over the entire distribution domain,
especially for the tails.
In this paper, we propose a fractional moments-based mixture distribution approach to estimate the first-passage
probabilities of high-dimensional and strongly nonlinear stochastic dynamic systems. It is worth mentioning that the
randomness from both internal system properties and external excitations is taken into account. The main contributions
of this study are summarized as follows. First, a parallel adaptive sampling scheme is proposed for estimating the
fractional moments, as opposed to the traditional fixed sample size scheme. Such a new scheme enables to extend
the sample size sequentially, i.e., one at a time or several at a time. The optimal sample size for fractional moment
estimation is determined by introducing a convergence criterion. In fact, a sequential sampling method with the ability
to effectively reduce variance in high-dimensional problems, named Refined Latinized stratified sampling (RLSS) [
37
],
is suitable for achieving our purposes and is employed within the proposed scheme. Second, one novel and versatile
mixture distribution model is proposed to reconstruct the EVD with the knowledge of its estimated fractional moments.
This model is based on the extension of the conventional inverse Gaussian distribution and the log transformation of the
extended skew-normal distribution. The analytical expression of the fractional moments for such mixture distribution is
3
derived, and a fractional moments-based parameter estimation technique is developed.
The remainder of this paper is organized as follows. Section 2outlines the first-passage probability estimation of a
stochastic dynamic system from the perspective of EVD. In section 3, the proposed fractional moments-based mixture
distribution approach is described in detail, including a parallel adaptive scheme for fractional moments evaluation and
a flexible mixture distribution model for EVD reconstruction. Three examples are given in section 4to demonstrate the
performance of the proposed method. The paper is closed with some concluding remarks in section 5.
2. First-passage probability estimation of stochastic dynamic systems
2.1. Stochastic dynamic systems
Consider a stochastic dynamic system that is governed by the following state-space equation:
˙
Y(t) = Q(Y(t),U, t),(1)
with an initial condition
Y(0) = y0,(2)
where
Y= (Y1, Y2, ..., Ynd)
is a
nd
-dimensional state vector;
Q= (Q1, Q2, ..., Qnd)
is a dynamics operator vector;
U= (U1, U2, ..., Uns)
is a
ns
-dimensional random parameter vector with a known joint probability density function
(PDF)
pU(u)
;
u= (u1, u2, ..., uns)
denotes a realization of
U
;
t
denotes the time. Note that Eq. (1) can be strongly
nonlinear, which may be caused by material, geometrical, or contact nonlinearities inherent in the stochastic dynamic
system. In addition, hundreds or thousands of random variables can be included in the vector
U
due to the randomness
from system properties and external excitations.
For a well-posed stochastic dynamic system, the solution to Eq. (1) is unique and depends on the vector
U
, which
can be assumed to be:
hY(t),˙
Y(t)i=HY(U, t),∂HY(U, t)
∂t ,(3)
where HYand ∂HY
∂t are the deterministic operators.
If we consider the system responses of interest for reliability analysis, say
W(t) = {W1(t),W2(t), ..., Wnd(t)}
,
they can be evaluated from their relations to the state vectors:
W(t) = ΨhY(t),˙
Y(t)i=H(U, t),(4)
where
Ψ
is the transfer operator; and
H
denotes the mapping relation from
U
and
t
to
W(t)
. Accordingly, the
q
-th
component of
W(t)
is denoted by
Wq(t) = Hq(U, t), q = 1, ..., nd
. For notational simplicity, the subscript
q
is
omitted hereafter, and only a component W(t)is considered in the following.
2.2. First-passage probability estimation by EVD
For a stochastic dynamic system, the first-passage probability is the probability that the system response of interest
exceeds a certain safe domain for the first time within a given time range. Accordingly, assuming
T
is the time duration,
we have
Pf= Pr {W (t)/∈Ωsafe,∃t∈[0, T ]},(5)
4
where
Pf
is first-passage probability;
Pr
is probability operator;
Ωsafe
denotes the safe domain. According to different
application backgrounds, the boundary of
Ωsafe
can be different, such as one boundary, double boundary, and circle
boundary [
7
]. In the case of symmetric double boundary problem, the first-passage probability can be further written
as:
Pf= Pr {|W (t)|> blim,∃t∈[0, T ]},(6)
where
blim
is the given threshold that limits the symmetric bounds of
Ωsafe
, and
|·|
is the absolute value operator. In the
present study, the first-passage probability defined by Eq. (6) is of concern.
Note that if the system response in the time period
[0, T ]
remains below the boundary of
Ωsafe
, the first-passage
probability will be equal to zero. From this perspective, once the extreme value of system response exceeds the
boundary, the system fails. Accordingly, Eq. (6) can be rewritten as
Pf= Pr {max {|W (t)|} > blim,∀t∈[0, T ]}= Pr {Z > blim},(7)
where
Z= max
t∈[0, T ]{|W (t)|}
. Note that
Z
is always positive, and depends on the random parameter vector
U
. If we de-
note the functional relationship between
Z
and
U
as
G
, then we have
Z=G(U)
and
Pf= Pr {Z =G(U)> blim}
.
According to classical probability theory, once the probability distribution of
Z
, which is also referred to as extreme
value distribution (EVD), is obtained, Eq. (7) can be straightforwardly calculated from the EVD. Let
fZ(z)
and
FZ(z)
be the PDF and cumulative distribution function (CDF) of Z. Then the first-passage probability reads
Pf=Z+∞
blim
fZ(z)dz= 1 −FZ(blim ).(8)
It should be pointed out that the first-passage probability is easy to be obtained from Eq. (8) once the PDF or CDF
of
Z
is known. However, how to estimate the EVD of
Z
is quite challenging. This is because deriving an analytical
expression for the EVD is intractable even for some simple stochastic responses, not to mention the stochastic responses
of high-dimensional and strong-nonlinear stochastic dynamic systems. Therefore, to tackle such challenge, an EVD
estimation method is proposed in the following section.
Remark 1.
For system failure probability evaluation, the above-mentioned EVD estimation method can also be applied
by using the theory of equivalent extreme-value events [
38
]. Briefly speaking, the system failure can be regarded as a
compound event of multiple random events, where a single random event can be described by an inequality associated
with a single response and its threshold. Based on the inequality relationship between the involved random events,
the compound event can be equated to an equivalent extreme-value event whose threshold can be obtained by a linear
combination of the thresholds of the involved random events. In this manner, the system failure probability can be
assessed by the Eq. (8), where
Z
is the equivalent extreme-value event. The interested readers can refer to Ref. [
38
] for
more details.
3. A fractional moments-based mixture distribution approach
In this section, we propose a novel fractional moments-based mixture distribution approach to approximate the
EVD in an efficient and accurate way. The proposed method consists of two main parts. First, a parallel adaptive
scheme is proposed for fractional moments estimation, which allows sequential sample size extension until a prescribed
convergence criterion is satisfied. Second, from the knowledge of estimated fractional moments, an eight-parameter
5
mixture distribution model with increased flexibility is developed to capture the main body and distribution tail of the
EVD.
3.1. Characterizing EVD by fractional moments
The analytical expression of EVD can not be directly obtained for a general high-dimensional and nonlinear
stochastic dynamic system, as discussed earlier. To this end, we have to resort to some indirect methods that can
approximate the EVD from a limited number of sample data. The fractional moment, as a generalization of the
traditional integer moment, has received a growing interest to characterize a positive random variable in many fields.
More recently, it has also been introduced to the area of EVD characterization [31–33,35].
3.1.1. Concept and properties of fractional moments
The r-th fractional moment of the positive random variable Zis defined as [33]
Mr
Z=E[Zr] = Z+∞
0
zrfZ(z) dz, (9)
where
r
can be any real number and
E[·]
denotes the expectation operator. Note that when
r
takes an integer value,
Eq. (9) yields the
r
-th integer moment of
Z
. Therefore, for any positive random variable, the integer moment of the
variable is a special case of its fractional moment.
If one expands Zraround its mean value µZ=M1
Zusing the Taylor series expansion, we have
Zr=∞
X
k=0 r
kµr−k
Z(z−µZ)k,(10)
where the fractional binomial coefficient
r
k
can be computed as
r
k=r(r−1)···(r−k+1)
k(k−1)···1
, and
k
can be any non-negative
integer. Taking the expectation of both sides of Eq. (10) yields:
E[Zr] = ∞
X
k=0 r
kµr−k
ZEh(z−µZ)ki.(11)
It can be seen that the right-hand side of Eq. (11) contains an infinite number of integer moments, i.e.,
Eh(z−µZ)ki
,
and the left-hand side of Eq. (11) is exactly the
r
-th fractional moment. Hence, Eq. (11) implies that a single
r
-order
fractional moment can embody statistical information of numerous integer moments. Further, as observed from Eq.
(11), when
r
is fixed, the value of coefficient
r
kµr−k
Z
decreases as
k
increases; when
k
is fixed,
r
kµr−k
Z
increases
as
r
increases. This indicates that the higher the fractional order, the greater the contribution of higher-order integer
moments. Since higher-order integer moments can provide more information about the shape of EVD, higher-order
fractional moments reflect more statistical features of EVD than lower-order fractional moments. In addition, it
should be mentioned that higher-order fractional moments have higher variability and are more difficult to obtain than
lower-order fractional moments [
28
,
33
]. Note that one is able to generate any number of fractional moments given the
range of fractional orders. However, one can only generate a fixed number of integer moments if the maximum integer
order is given. As a compromise, a set of fractional moments up to second order, as adopted in Ref. [
33
], is used in this
work.
6
3.1.2. Parallel adaptive estimation of fractional moments
According to the principle of probability conservation, Eq. (9) can be rewritten in the random variable space of
U
:
Mr
Z=ZΩU
Gr(u)pU(u) du,(12)
where
ΩU
denotes the random variable space of
U
. For a general stochastic dynamic system, a considerably large
number of random variables are collected in
U
, and strong nonlinearity exists in
G(U)
. In addition, the expression of
G(U)
cannot be explicitly given. Hence, a high-dimensional integral with a complex and implicit integrand is involved
in Eq. (12), which is impossible to solve analytically.
Alternatively, we can resort to the sampling methods to approximate the high-dimensional integral involved in Eq.
(12). In the literature, various variance reduction sampling methods with fixed sample sizes are employed to facilitate
the estimation of fractional moments. Under this setting, Mr
Zcan be approximated as:
ˆ
Mr
Z=
N
X
k=1
$k·Gr(uk),(13)
where
N
denotes the sample size;
$k
represents the
k
-th sample weight,
k= 1, ..., N
;
uk
is the
k
-th sample of random
variables
U
. Note that most variance reduction sampling methods do not allow sample size extension, and thus require
N
to be specified in advance from experience. However, for estimating fractional moments, an “optimal sample size”
is desired, which is problem-dependent, and cannot be known in advance for a specified first-passage problem. The
optimal sample size enables the estimation to strike a balance between accuracy and computational efficiency. However,
with a predefined sample size, the fractional moment estimation may lose such balance, and may be trapped into
over-sampling or under-sampling situations. Specifically, if an overly conservative sample size is pre-specified, i.e., too
many samples are taken, oversampling occurs and leads to unnecessary computational waste. On the other hand, if the
predefined sample size is too small, under-sampling takes place, resulting in inaccurate evaluation of the fractional
moments.
To tackle with such dilemma, an adaptive sampling scheme should be developed for estimating fractional moments.
One feasible strategy is to generate samples one at a time or several at a time, and enrich the sample size progressively
until a specified convergence criterion is satisfied. In this manner, sample size extension is allowed, and the sample size
can be obtained adapted to different problems, which enables the estimated fractional moments to achieve both the
desired accuracy and computational efficiency. In addition, parallel computing technique can be equipped to further
accelerate the computational speed of such process. As such, we shall name this sampling scheme as parallel adaptive
sampling scheme. To illustrate the advantages of proposed scheme, Fig. 1shows the comparison between traditional
sampling scheme and proposed parallel adaptive sampling scheme. In this figure,
l
denotes the
l
-th time of sample
size extension, and
l∈Z+
. As seen, by the proposed sampling scheme, the sample size for a given first-passage
problem can be determined in an adaptive way, where fractional moments can be approximated with a desired accuracy.
In addition, it is quite time-saving to evaluate additional samples of
Z
only when it is required. In the process of
estimating the additional samples of
Z
, the analysis time can be further decreased by adopting parallel computing
technique.
By employing the proposed parallel adaptive sampling scheme,
ˆ
Mr
Z
after the
l
-th sample size extension can be
7
Define a sample design
Obtain samples of
Traditional
Define an initial sample design
Obtain new samples of
Proposed parallel adaptive
Estimate the
fractional moments
Converged?
Perform sample size extension
Yes
No
vs
Fixed
sample size
Problem-dependent sample size
Estimate the
fractional moments
Figure 1: Comparison of traditional sampling scheme and proposed parallel adaptive scheme
computed as follows:
ˆ
Mr
Z=
(l−1)~
X
k=1
$(k)·Gru(k)+
l~
X
k=(l−1)~+1
$(k)·Gru(k),(14)
where the number of samples added in each time of sample size extension is denoted as
~
and
~∈Z+
; the cur-
rent sample size is
l~
; the weight is reallocated in the
l
-th sample size extension and satisfies
Pl~
k=1 $(k)= 1
;
u((l−1)~+1), ..., u(l~)
are the newly added samples in the
l
-th sample size extension, while
u(1), ..., u((l−1)~)
are
samples generated in the previous
(l−1)
sample size extensions. Note that when
l= 1
, initial samples of
Z
, i.e.,
Gu(k)~
k=1
are evaluated. Since
Gu(k)(l−1)~
k=1
have been already obtained in the previous
(l−1)
sample size
extensions, one only needs to evaluate Gu(k)l~
k=(l−1)~+1 in the l-th sample size extension.
In order to achieve the proposed parallel adaptive sampling scheme, the key is to employ a sampling strategy that
allows sequential sample size extension. Simple random sampling method, i.e., Monte Carlo simulation (MCS), can
naturally meet such aim. To obtain a better precision of fractional moments with fewer computational efforts, one
can apply a variance reduction sampling method to the proposed sampling scheme. In addition, sampling methods
that are applicable to high-dimensional problems are also desired. In fact, one recently developed sequential stratified
sampling technique, termed refined Latinized stratified sampling (RLSS) [
37
], is suitable for our purposes. On one
hand, RLSS is advantageous as it owns the ability to achieve effective variance reduction in terms of both main/additive
effects and variable interaction that appear in
G(U)
. On the other hand, RLSS is applicable to problems involving low-
and high-dimensional input random variables. By using the RLSS technique, we can evaluate
ˆ
Mr
Z
according to Eq.
(14). Since the samples of RLSS are generated in the
[0,1]ns
hyper-rectangular space, we need to transform the RLSS
sample points to the original distribution domain of random variables
U
. Denote
ˆϕ(k)
and
$(k)
to be the
k
-th sample
point and corresponding weight obtained by RLSS and
Γ
to be the transformation operator,
ˆ
Mr
Z
by RLSS at the
l
-th
8
sample size extension can be evaluated as:
ˆ
Mr
Z=
(l−1)~
X
k=1
$(k)·GrΓˆϕ(k)+
l~
X
k=(l−1)~+1
$(k)·GrΓˆϕ(k).(15)
A brief illustration of the RLSS technique is discussed in the following. For more details, the interested readership
can refer to Appendix A or Ref. [37].
The first step of RLSS is generating
N ≥ 1
samples that follow a so-called Latinized stratifed sampling (LSS)
scheme [
39
], which implies that these samples fulfill both the properties of Latin hypercube sampling (LHS) and
stratified sampling (SS). An schematic diagram of a LSS design is shown in Fig. 2(a), considering
N= 4
and
ns= 2
.
In this figure, the strata associated with LHS are shown with dashed black line, the strata associated with SS are marked
with solid green line, the samples per each dimension of analysis are marked with blue cross marks and the actual
samples are marked with blue dots. It is readily observed that the strata associated with SS possess the same area, and
boundaries of the strata associated with LHS match those associated with SS, which are the key properties of LSS.
The second step of RLSS consists of applying a Hierarchical Latin hypercube sampling (HLHS) design [
37
] over
the existing LHS design. This implies applying a refinement of each LHS strata by subdividing it
δ
times, which is
illustrated schematically in Fig. 2(b), where
δ= 1
. The new strata associated with LHS are shown with red dashed line
and the new candidate samples per each dimension on those strata are marked with orange cross marks. Note that up to
this point, no new actual samples have been generated. In addition, one identifies candidate strata for refining the SS
design by dividing the existing strata, which is shown schematically in Fig. 2(b) with blue solid lines.
The third step involves generating new candidate samples for RLSS. In this sense, candidate samples are those that
may include the already existing
N
samples. These candidate samples must be identified following a special procedure
such that the properties of LSS continue being fulfilled. For materializing this third step, one must identify the strata
which must contain candidate samples in order to enforce the LSS condition, and the strata where candidate samples
can be generated randomly. This is illustrated schematically in Fig. 2(c). The pink color indicates those strata that must
contain candidate samples, while the yellow color shows those strata where a candidate sample may be generated at
random. With all these considerations, one can generate
Nδ
candidate designs, as shown schematically in Fig. 2(c)
with 4 orange dots.
The fourth step of RLSS is to incorporate a batch of
~
samples to the existing set of
N
samples. This is performed
by selecting at random from the existing
Nδ
candidate samples. Note that in this process, it is necessary to update the
strata associated with SS taking into account the candidate strata defined in the second step. Clearly, in such update,
one must also update the weights (areas) of the selected strata. Fig. 2(d) illustrates the case where
~= 4
and also
shows the updated strata with green solid line.
It should be mentioned that the fourth step described above can be repeated as many times as necessary to select
many batches of
~
samples as long as there are candidate samples left. In case one runs out of candidate samples,
it is necessary to return to the second step and perform a new run of HLHS, which implies subdividing the strata
associated with LHS. Furthermore, after each sample size extension, generated RLSS samples contain not only batches
of additional samples, but also samples from the initial LSS design. In this work, we take
~≥ N
in order to include
the initial LSS design in the initial RLSS samples when l= 1 in Eq. (15).
In the proposed sampling scheme, a proper convergence criterion should be developed to determine the desired
number of sample size extensions. It is found that higher-order fractional moment always exhibits larger variability
9
(a) (b)
(c) (d)
Figure 2: Schematic description of the RLSS technique for generating 8 samples in two dimensions
than its lower-order counterpart. Accordingly, if the variability of maximum order fractional moment is controlled,
the variability of the lower-order ones will be automatically below a desired level. Note that the maximum order of
fractional moments is set to be 2 in this work, as mentioned in Section 3.1.1. Therefore, a convergence criterion is
proposed by judging the variability of the second-order fractional moment
ˆ
M2
Z
evaluated by RLSS. Specifically, the
coefficient of variation (COV) of the
ˆ
M2
Z
is compared with a user-defined small value
E
(e.g.,
E= 0.02
) to determine
when to stop the sample size extension. The stopping criterion is defined as:
COV nˆ
M2
Zo<E.(16)
Although the expression of
COV nˆ
M2
Zo
is not available for RLSS, the bootstrap resampling technique [
40
] can be
alternatively implemented here to estimate it. Note that traditional bootstrap method generates samples with equal
probability of occurrence, which is not the case for RLSS samples. To consider the unequal weight property of
RLSS samples, the approach proposed in Ref. [
41
] is adopted here, such that samples with higher weights have more
probability of being chosen for bootstrap. For more details on this approach, it is referred to Ref. [41].
With such parallel adaptive scheme above, once the samples of
Z
that meet the convergence condition are obtained,
a set of lower-order (only up to 2) fractional moments can be estimated according to Eq. (15), which are then used to
represent the EVD.
10
3.2. Representing EVD by a mixture distribution with fractional moments
After obtaining the fractional moments of
Z
, an adequate probability distribution model should be employed for the
EVD estimation. Generally, the state-of-art distribution models represent the EVD by adopting either maximum entropy
density [
31
,
32
] or positively skewed distributions such as shifted generalized lognormal distribution [
35
] and a mixture
of lognormal distribution and inverse Gaussian distribution [
33
]. However, their flexibility is still limited for the EVDs
with heavy tails, leading to the inaccuracy of EVD reconstruction for some first-passage problems. To increase the
flexibility and enlarge the application scope, we first extend the traditional inverse Gaussian distributions by introducing
an exponential transformation with an additional shape parameter. Then, we introduce the log transformation to
the extended skew-normal distribution, to enhance its ability to accommodate fat tails. Further, these two improved
distributions are mixed together to produce a more flexible mixture distribution model, whose involved parameters can
be estimated from the estimated fractional moments.
3.2.1. Proposed extended inverse Gaussian distribution
The inverse Gaussian distribution (IGD) is a two-parameter skewed unimodal distribution and applies for positive
real values [
42
]. It is a first-passage time distribution for the Brownian motion with positive drift [
43
]. The PDF of the
IGD is:
fIGD (z;a, b) = rb
2πz3exp "−b(z−a)2
2za2#,with z > 0,(17)
where a > 0is the location parameter; b > 0is the shape parameter.
Denote the random variable which follows an IGD as ZIGD. The r-th fractional moment of ZIGD is given as:
Mr
ZIGD =E[Zr
IGD] = Z+∞
0
zrfIGD (z)dz= exp b
ar2b
πar−1/2K1/2−rb
a,(18)
where Kα(β)is the modified Bessel function of the second kind.
In fact, the IGD can be extended to obtain higher flexibility in its shape. Here, we introduce a transformation
X=Z1/η
to extend the original distribution, where
η > 0
is a shape parameter. The resulting distribution is called
extended inverse Gaussian distribution (EIGD). To obtain the PDF and fractional moments of the EIGD, the following
theorem is first given:
Theorem 1.
Assume
X
and
Z
are two continuous and positive real-valued random variables, and
fZ(z)
is already
available. Let
X=Z1/η
where
η > 0
, then we have
fX(x) = fZ(xη)·η·xη−1
. Additionally, the
r
-th fractional
moment of Xis E[Xr] = EZr/η .
Proof.
Since
X=Z1/η
, according to the principle of conservation of probability, it is straightforward to derive
fZ(z) dz=fX(x)dx
. Thus, the PDF of
X
can be derived as
fX(x) = fZ(z)dz
dx =fZ(xη)·η·xη−1
. We may also
derive the relationship between the
r
-th fractional moment of
X
and that of
Z
as
E[Xr] = EhZ1/ηri=EZr/η
.
Therefore, the PDF of EIGD reads:
fEIGD (x;η, a, b) = ηrb
2πx−η/2−1exp "−b(xη−a)2
2xηa2#,with x > 0.(19)
Denote the random variable which follows the EIGD as
XEIGD
. According to Eq. (18) and
Theorem 1
, the
r
-th
fractional moment of XEIGD can be derived in analytic form:
Mr
XEIGD = exp b
ar2b
πar/η−1/2K1/2−r/η b
a.(20)
11
Note that when
η= 1
, the EIGD reduces to the IGD according to Eq. (19). The limit or special cases of IGD also
belong to the EIGD, such as the chi-square distribution with single degree of freedom, normal distribution and L
´
evy
distribution. Besides, the shape flexibility of the EIGD is illustrated by Fig. 3under four different sets of parameters.
In this figure, we make a comparison between the original IGD and the proposed EIGD by changing parameter
η
and
fixing
a= 1, b = 1
of the EIGD. It can be observed that, the proposed EIGD possesses much more flexibility in shape
of PDF than the original IGD.
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
Figure 3: PDFs of extended inverse Gaussian distribution under four different sets of parameters
3.2.2. Proposed log extended skew-normal distribution
The extended skew-normal distribution (ESND) was first introduced by Azzalini [
44
]. This distribution is a
four-parameter unimodal asymmetric distribution with support on
(−∞,+∞)
, which generalizes the traditional
skew-normal distribution and normal distribution. The statistical properties of the ESND are discussed in detail in Ref.
[45]. The PDF of the ESND of a real random variable ˜
X∈Ris:
fESND (˜x;c, d, θ, τ ) = 1
dφ˜x−c
dΦτ√1 + θ2+θ˜x−c
d
Φ (τ),with ˜x∈R,(21)
where
c∈R
is the location parameter;
d > 0
is the scale parameter;
θ∈R
is the shape parameter;
τ∈R
is the
truncation parameter; φ(·)and Φ (·)denote the PDF and CDF of the standard normal distribution.
The moment-generating function (MGF) of the ESND is:
M˜
X˜
t=Ehexp ˜
t˜
Xi= exp c˜
t+1
2d2˜
t2Φτ+θd˜
t
√1+θ2
Φ (τ),with ˜
t∈R.(22)
Although the ESND enables to accommodate asymmetry characteristics, its ability to fit heavier tails can be further
improved by introducing a log transformation to the ESND. We shall refer the newly generated distribution as log
extended skew-normal distribution (LESND). Denote the random variable which follows a LESND as
XLESND
. Then,
we have the relationship between
XLESND
and
˜
X
as
XLESND = exp ˜
X
. That is, the logarithm of
XLESND
follows
12
the original ESND. Hence, we can get the PDF of the LESND as:
fLESND (x;c, d, θ, τ ) = 1
dxφlog (x)−c
dΦτ√1 + θ2+θlog(x)−c
d
Φ (τ),with x > 0.(23)
From the relationship between the fractional moment of the LESND and the MGF of the ESND, it is easy to derive
Mr
XLESND =E[Xr
LESND] = Ehexp ˜
Xri=M˜
X(r)
. Hence, the
r
-th fractional moment of
XLESND
can be
given in analytic form as:
Mr
XLESND = exp cr +1
2d2r2Φτ+θdr
√1+θ2
Φ (τ).(24)
Note that according to Eq. (23), when
τ= 0
, the LESND reduces to the log skew-normal distribution [
46
]; and
when
θ= 0
, the LESND reduces to the traditional lognormal distribution. It should be mentioned that if
θ= 0
, the
shape of LESND will not be affected by changing the value of parameter τ. Besides, to illustrate the flexibility of the
LESND, Fig. 4depicts the LESND with four sets of parameters. In this figure, the log skew-normal distribution is
given for comparison by setting the parameters of LESND as
c= 0, d = 1, θ = 3, τ = 0
. As can be seen, the LESND
provides richer distribution shapes compared to the log skew-normal distribution, showing the increased flexibility of
LESND.
0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 4: PDFs of log extended skew-normal distribution under four different sets of parameters
3.2.3. Proposed mixture distribution
It is worth mentioning that the first-passage probability estimation is closely associated to the distribution tail
of EVD. Besides, the EVD is usually asymmetric and possesses heavy tail in many cases. Hence, a highly flexible
distribution model is needed, which is suitable for fitting distributions with various tail properties, especially the
heavy-tailed distributions. For accurate EVD estimation, two single-component skewed distributions proposed above,
i.e., the EIGD and LESND, may still not be flexible enough and their applicability to various first-passage problems is
limited. To further improve the flexibility, one potential way is to mix the proposed single-component distributions
together by introducing a weight parameter. Such distribution model enables to incorporate both characteristics of two
single-component distributions, and can accommodate asymmetry in a more flexible way so as to properly estimate
13
the EVD. Therefore, motivated by the above, a novel mixture of the extended inverse Gaussian and log extended
skew-normal distributions (M-EIGD-LESND) is developed herein.
The PDF of M-EIGD-LESND is given as:
fM−EIGD−LESND (x;Υ) = wfEIGD (x;η, a, b) + (1 −w)fLESND (x;c, d, θ , τ)
=wηqb
2πx−η/2−1exp h−b(xη−a)2
2xηa2i+ (1 −w)1
dx φlog(x)−c
dΦ(τ√1+θ2+θlog(x)−c
d)
Φ(τ),with x > 0,(25)
where
Υ= [w, η, a, b, c, d, θ, τ ]
is the set of eight unknown parameters and
w∈[0,1]
is the weight parameter of
M-EIGD-LESND.
According to Eqs. (20) and (24), the r-th fractional moment of M-EIGD-LESND can be given in analytic form:
Mr
XM−EIGD−LESND =EXr
M−EIGD−LESND;Υ=wE [Xr
EIGD] + (1 −w)E[Xr
LESND]
=wexp b
aq2b
πar/η−1/2K1/2−r/η b
a+ (1 −w) exp cr +1
2d2r2Φτ+θdr
√1+θ2
Φ(τ).
(26)
Note that the proposed M-EIGD-LESND can reduce to the mixture of lognormal and inverse Gaussian distributions
[
33
] if set
η= 1
and
θ= 0
. To illustrate the flexibility of the proposed mixture distribution model, Fig. 5shows the
plot of the PDFs associated with M-EIGD-LESND with different parameters. It can be seen that the proposed mixture
distribution model is highly flexible with rich shapes and enables to accommodate various heavy tails. In addition, the
M-EIGD-LESND is able to represent not only unimodal distributions but also bimodal distributions.
0 5 10 15 20 25 30 35 40
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Figure 5: PDFs of the proposed mixture distribution under four different sets of parameters
3.2.4. Parameter estimation
The proposed mixture distribution model has the potential to characterize the EVD. Hence, in order to recover the
EVD of
Z
, we assume that the EVD follows the proposed mixture distribution model, and determine the free parameters
of this model in an appropriate way. Note that the proposed distribution contains a set of eight free parameters. To
estimate these unknown distribution parameters, a natural way is to match the fractional moments of the proposed
14
mixture distribution model with the estimated fractional moments of the corresponding orders (hereafter referred to as
the fractional moment matching technique). Accordingly, the following nonlinear system of equations requires to be
solved:
ˆ
Mr1
Z=Mr1
XM−EIGD−LESND
ˆ
Mr2
Z=Mr2
XM−EIGD−LESND
···
ˆ
Mr8
Z=Mr8
XM−EIGD−LESND ,
(27)
where
ˆ
Mri
Z, i = 1,2, ..., 8
are the
ri
-th fractional moments estimated by RLSS;
Mri
XM−EIGD−LESND
can be obtained by
Eq. (26); and the fractional order
ri
takes
[r1, r2, ..., r8] = 2
8×[1,2, ..., 8]
. Here, the equally spaced fractional orders
are introduced for convenience, since it is straightforward to take such value without any prior knowledge of fractional
orders. Besides, as adopted in Ref. [
33
], the maximum fractional order is set to be 2, since the second-order fractional
moment can be estimated efficiently from a small number of samples, and it reflects more shape information of EVD
than lower-order fractional moments, as discussed in Section 3.1.1.
Solution to Eq. (27) can be obtained in seconds by any appropriate nonlinear solver, such as lsqnonlin in Matlab. To
facilitate the solving process, initial values for the free parameters are required. Denote the initial values of Eq. (27) as
w0,ˆη0,ˆa0,ˆ
b0,ˆc0,ˆ
d0,ˆ
θ0,ˆτ0
.
w0
is set to be 0.5 to assign an equal initial weights to the two single-component functions.
The other initial values, i.e.,
ˆη0,ˆa0,ˆ
b0,ˆc0,ˆ
d0,ˆ
θ0,ˆτ0
, can be obtained by another low-order fractional moment matching
technique, where a nonlinear system of equations is involved:
ˆ
M1/2
Z=M1/2
XEIGD
ˆ
M1
Z=M1
XEIGD
ˆ
M3/2
Z=M3/2
XEIGD
,(28)
and
ˆ
M1/2
Z=M1/2
XLESND
ˆ
M1
Z=M1
XLESND
ˆ
M3/2
Z=M3/2
XLESND
ˆ
M2
Z=M2
XLESND
,(29)
where
ˆη0>0,ˆa0>0,ˆ
b0>0,ˆc0∈R,ˆ
d0>0,ˆ
θ0∈R,ˆτ0∈R
. Note that the M-EIGD-LESND can reduce to
the inverse Gaussian distribution (if set
w= 0, η = 1
) or the lognormal distribution (if set
w= 1, θ = 0
), and the
relationships between the parameters and the first two central moments of each reduced distribution are easy to be
obtained. Besides, as discussed earlier, the value of parameter
τ
will be irrelevant if
θ= 0
. Hence, the initial values
for Eqs (28) and (29) can be determined as:
a0= ˆµZ, b0= ˆµ3
Z/ˆσ2
Z, η0= 1
,
c0= log ˆµ2
Z/pˆσ2
Z+ ˆµ2
Z, d0=
plog (ˆσ2
Z/ˆµ2
Z+ 1), θ0= 0, τ0= 0
, where
ˆµZ=ˆ
M1
Z
and
ˆσZ=rˆ
M2
Z−ˆ
M1
Z2
. The parameter estimation
process of proposed M-EIGD-LESND is briefly summarized in Algorithm 1.
3.3. Procedure of the proposed method
Once the EVD is reconstructed by the proposed probability distribution model, the first-passage probability can
be evaluated by Eq. (8) for a given threshold. A flowchart of the proposed method is shown in Fig. 6, and a brief
procedure is summarized as follows:
15
Algorithm 1 Parameter estimation for M-EIGD-LESND using the fractional moment matching technique
Input: central moments ˆµZ,ˆσZ, and fractional moments ˆ
Mr
Z(r=1
4,1
2,3
4,1,5
4,3
2,7
4,2).
Output: estimated distribution parameters Υ= [w, η, a, b, c, d, θ, τ ].
1: Use ˆµZand ˆσZto evaluate η0, a0, b0, c0, d0, θ0, τ0as the initial values of Eqs. (28) and (29);
2:
Solve Eqs. (28) and (29) with
η0, a0, b0, c0, d0, θ0, τ0
to estimate the initial values
ˆη0,ˆa0,ˆ
b0,ˆc0,ˆ
d0,ˆ
θ0,ˆτ0
of Eq.
(27).
3:
Solve the fractional moment matching equations (Eq. (27)) by means of any appropriate nonlinear solver
with
ˆη0,ˆa0,ˆ
b0,ˆc0,ˆ
d0,ˆ
θ0,ˆτ0
and
w0= 0.5
, and then obtain the estimated distribution parameters
Υ=
[w, η, a, b, c, d, θ, τ ]of M-EIGD-LESND.
Step 1
: Initialization. Set the initial sample size
N
of LSS, the refinement factor
δ
of HLHS, the number of samples
~added in each sample size extension and the value of tolerance E. Determine the threshold blim.
Step 2
: Generate
~
new samples by RLSS. Produce
~
new samples and update the weights by RLSS method
according to Algorithm 2in Appendix A, and then compute the new samples of Z.
Step 3
: Judge the convergence criterion. Evaluate the COV of
ˆ
M2
Z
by using bootstrap technique. If Eq. (16) is
satisfied, then turn to step 4; otherwise, return to step 2.
Step 4
: Moment evaluation. Calculate a set of fractional moments
ˆ
Mr
Z
(
r=1
4,1
2,3
4,1,5
4,3
2,7
4,2
) according to
Eq. (15), and then compute the first-two central moments ˆµZand ˆσZby ˆµZ=ˆ
M1
Zand ˆσZ=rˆ
M2
Z−ˆ
M1
Z2
.
Step 5
: EVD representation. Represent the EVD by using the proposed distribution model, i.e., M-EIGD-LESND,
where the involved free distribution parameters are estimated by the low-order fractional moment matching technique
described in Algorithm 1.
Step 6
: First-passage probability estimation. Evaluate the first-passage probability
Pf= Pr {Z > blim }
via
obtained EVD and Eq. (8).
4. Numerical examples
In this section, three examples, including two test examples and one practical engineering example, will be
investigated to verify the efficacy of the proposed method. In all examples, the parameters of the proposed method
are set as
N= 1
,
δ= 1
,
~= 8
and
E= 0.015
. The computational efficiency and accuracy of proposed methods
for first-passage probability estimation are compared with MCS, Subset simulation (SS) [
21
,
23
] and two state-of-art
mixture distribution methods presented in Ref. [
33
] and [
34
]. Note that in SS, the number of samples per layer is
1000 and the conditional probability is 0.1. Both the existing mixture distribution methods for comparison employ the
Latinized partially stratified sampling (LPSS) to evaluate fractional moments of
Z
. The mixture distribution method
in Ref. [
33
] develops a mixture distribution combining conventional inverse Gaussian and lognormal distributions
(MIGLD), and thus this method is referred as LPSS+MIGLD in the following examples. Another existing mixture
distribution method [
34
] develops a mixture of two generalized inverse Gaussian distributions (MTGIG), and this
method is denoted as LPSS+MTGIG in the following examples.
16
Start
Set N,δ,~,E,blim
Generate ~new RLSS samples and update the weights
Calculate ~new samples of Z
Compute COV nˆ
M2
Zoby bootstrap sampling
COV nˆ
M2
Zo<E
Calculate ˆ
Mr
Z(r=1
4,1
2,3
4,1,5
4,3
2,7
4,2) based on Eq. (15) and then compute ˆµZand ˆσZ
Estimate distribution parameters and represent the EVD by the proposed distribution model
Calculate Pfvia Eq. (8) with blim
Stop
Yes
No
Figure 6: Flowchart of the proposed method
4.1. Example 1: a Duffing oscillator under Gaussian white noise
The first example considers a Duffing oscillator with uncertain parameters under Gaussian white noise, which is
governed by
¨
Y(t) + γ˙
Y(t) + Y(t) + εY 3(t) = G(t),(30)
where
¨
Y
,
˙
Y
and
Y
are the acceleration, velocity and displacement at time
t
;
γ
denotes the damping control coefficient;
ε
is the parameter controlling the degree of nonlinearity in the restoring force; and
G(t)
is the Gaussian white noise.
Differential equation solver Ode45 in Matlab is utilized to solve Eq. (30). Both
γ
and
ε
follow the lognormal
distributions with mean values as 0.5 and 0.3, and standard deviation values as 0.2 and 0.1, respectively. The Gaussian
white noise is expressed as
G(tk) = θ(tk)p2πS/∆t, (31)
where
S= 1
is the spectral intensity;
∆t = 0.01 s
is the time interval;
T= 30 s
is the time period;
tk=k∆t, k =
0,1, ..., nt
is the discrete time; and here we consider
nt=T/∆t + 1 = 3001
random variables
θ(tk)
in the Gaussian
white noise following the standard normal distributions. Therefore, a total number of
2 + nt= 3003
random variables
are involved in the present example.
17
The maximum absolute extreme value of displacement over time
t∈[0, T ]
, i.e.,
Z= maxt∈[0, T ]{|Y(t)|}
, is of
interest in this example. First, the proposed parallel adaptive sampling scheme is implemented for fractional moment
estimation. The proposed scheme performs sample size extension successively until the convergence criterion in
Eq. (16) is satisfied. In each sample size extension,
~= 8
new RLSS samples are firstly generated for deterministic
dynamic analysis. Then, 8 new samples of
Z
are produced at a time using parallel computing technique with 8 CPU
processors. After that, the RLSS weights are redistributed so that the weights produced by all performed sample size
extensions sum to 1. Subsequently, Eq. (16) is checked to determine whether to perform a new round of sample size
extension. Accordingly, a total of
ˆ
N= 520
samples of
Z
are produced that satisfy the convergence criterion, where the
corresponding
ˆ
Mr
Z
(
r=1
4,1
2,3
4,1,5
4,3
2,7
4,2
) can be obtained by Eq. (15). Table 1compares the first-two central
moments (
ˆµZ=ˆ
M1
Z
and
ˆσZ=rˆ
M2
Z−ˆ
M1
Z2
) with the benchmark results given by MCS with
106
runs. In this
table, relative errors of the first-two moments between proposed method and MCS are also given, i.e., 0.5656% and
0.7195%, which indicate that proposed parallel adaptive scheme using RLSS enables to obtain accurate low-order
central moments.
Table 1: Comparison of first-two central moments by the proposed method and MCS (Example 1)
Method( ˆ
N)ˆµZˆσZ
Proposed(520) 3.6570 0.6623
MCS(106) 3.6778 0.6671
R.E. 0.5656% 0.7195%
Note: R.E. = Relative error with reference to MCS.
Once the required fractional moments are obtained, eight unknown free parameters involved in the proposed mixture
distribution (i.e., M-EIGD-LESND) can be determined by the fractional moment matching technique. Specifically, the
nonlinear system of equations in Eq. (27) is solved according to Algorithm 1, where initial values of free parameters are
given to speed up the solving process. Afterwards, the EVD could be approximated by the proposed mixture distribution
model, where the PDF, CDF and probability of exceedance (POE) curves are all plotted in Fig. 7. For comparison,
the benchmark results by MCS and the results from LPSS+MIGLD and LPSS+MTGIG are also depicted in Fig. 7.
It can be found that both the PDF and POE curves obtained from the proposed method accord well with the MCS
results. Although there is almost no difference between the CDF curves obtained by proposed method and those by
existing mixture distribution methods, larger deviations appear in the POE curves obtained by the LPSS+MIGLD and
LPSS+MTGIG. Moreover, both of the LPSS+MIGLD and LPSS+MTGIG require 625 LPSS samples to estimate the
fractional moments used for distribution parameter evaluation, where the number of samples is empirically determined
in advance and is larger than that required by the proposed method. In this regard, the proposed method shows a
considerable improvement in both efficiency and accuracy to recover the EVD in this example.
After obtaining the reconstructed EVD, the first-passage probability can be evaluated by Eq. (8), where the
safe threshold of this example is set to be
blim = 7
. Table 2lists the first-passage probabilities estimated by the
proposed method, LPSS+MIGLD, LPSS+MTGIG, SS and MCS. In this table, the estimated first-passage probabilities
are denoted as
ˆ
Pf
. With reference to
ˆ
Pf
obtained by the MCS, i.e.,
1.2200 ×10−4
, the first-passage probability
evaluated by the proposed method has acceptable accuracy, which reads
1.2245×10−4
. Unfortunately, the first-passage
probabilities by SS, LPSS+MIGLD and LPSS+MTGIG largely deviate from the reference ˆ
Pfby the MCS.
18
0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
MCS
LPSS+MIGLD
LPSS+MTGIG
Proposed method
(a) PDF
0 2 4 6 8 10
10-8
10-6
10-4
10-2
100
MCS
LPSS+MIGLD
LPSS+MTGIG
Proposed method
(b) CDF
0 2 4 6 8 10
10-8
10-6
10-4
10-2
100
MCS
LPSS+MIGLD
LPSS+MTGIG
Proposed method
(c) POE
Figure 7: PDF, CDF and POE of Zin Example 1
Table 2: Comparison of first-passage probabilities by different methods (Example 1)
Method MCS SS LPSS+MIGLD LPSS+MTGIG Proposed
ˆ
N1064600 625 625 520
ˆ
Pf1.2200 ×10−48.3100 ×10−54.7154 ×10−54.5286 ×10−51.2245 ×10−4
4.2. Example 2: a 15-storey shear frame structure under fully nonstationary stochastic ground motion
A 15-storey nonlinear shear frame structure with uncertain structural properties under fully nonstationary stochastic
ground motion is investigated in this example, shown in Fig. 8. The equation of motion of this structure reads:
M(Ustr)¨
Y+C(Ustr)˙
Y+K(Ustr) [˜aY+ (1 −˜a)V] = −M(Ustr)I¨xg(Uexl, t),(32)
where
¨
Y
,
˙
Y
and
Y
are the lateral acceleration, velocity and displacement matrices of the structure with respect to
the ground;
M
,
C
and
K
denote the mass, damping and stiffness matrices, respectively; Term
I
denotes the unit
matrix. All of the lumped masses and the corresponding stiffnesses from bottom to top of the structure are assumed to
be independent random variables, following the lognormal distributions with same coefficients of variation
0.1
and
19
different mean values
6×104
kg and
7×107
N/m, respectively. Hence,
ns1= 30
random variables are involved
in the system properties, which are denoted as
Ustr
. The floor slabs are assumed to be rigid. Rayleigh damping is
implemented as
C= ˆαM+ˆ
βK
, where
ˆα
and
ˆ
β
are obtained by taking both the damping ratios of the first and second
modes as 0.05. The Bouc-Wen resilience model [
47
] is adopted to describe the nonlinear behavior of the structure,
where the hysteretic displacement Vsatisfies:
˙
V=A∆˙
Y− B ∆˙
Y|V|ρ−1V−ξ∆˙
Y|V|ρ,(33)
in which
∆˙
Y
is the relative velocity between two neighboring floors,
˜a= 0.1
,
A= 1,B=ξ= 50
and
ρ= 1
are
the dimensionless parameters controlling the hysteretic performance of Bouc-Wen model. The fully nonstationary
stochastic ground motion
¨xg(Uexl , t)
is modeled by the second family of spectral representation method (SRM) [
48
]:
¨xg(Uexl , t) = √2
ns2−1
X
j=0 q2S¨xg(ωj, t)∆ω cos (ωjt+Uexl,j ),(34)
where
Uexl =Uexl,1, Uexl,2, ..., Uexl,ns2
denotes the random vector with
ns2= 1600
independent random variables
uniformly distributed in
[0,2π]ns2
;
ωj=j∆ω , j = 1,2, ..., ns2
is the discrete frequency and
∆ω =ωup/ns2
denotes
the frequency interval with upper cut frequency
ωup = 240 rad/s
;
S¨xg(ωj, t)
is the double-sided evolutionary power
spectrum density (EPSD) function:
S¨xg(ω, t) = |A(ω, t)|2S(ω),(35)
in which
A(ω, t)
is the time-frequency modulation function and
S(ω)
is the power spectrum density represented by
Clough-Penzien spectrum [49], which are given as
A(ω, t) = e−χ0ωt
ωgT·t
C0·e1−t
C0κ
,(36)
S(ω) = ω4
g+ 4ζ2
gω2
gω2ω4
hω2
g−ω22+ 4ζ2
gω2
gω2iω2
f−ω22+ 4ζ2
fω2
fω2
¯a2
max
γ2
0hπωg2ζg+1
2ζgi,(37)
where
χ0
is the frequency modulation factor;
C0
is the approximate arrive time of peak ground acceleration (PGA);
κ
is the shape control coefficient;
ωg
and
ζg
are the parameters describing the dominant frequency and damping ratio of
site soil;
ωf
and
ζf
are similar parameters for the second filter that hinders the low-frequency component;
γ0
is the
peak factor;
T
is the time duration; and
¯amax
denotes the PGA. Values of these involved parameters in EPSD take
χ0= 0.15
,
C0= 9 s
,
κ= 2
,
ωf= 0.1ωg=4
7π
,
ζf=ζg= 0.64
,
γ0= 2.85
,
T= 20 s
,
¯amax = 400 cm/s2
. Note
that a total number of ns1+ns2= 1630 random variables are involved in this example.
The maximum absolute extreme value of inter-storey drift on each storey over the time duration is considered as
the response of interest in this example, which is denoted as
Zi, i = 1,2, ..., 15
. Function solver Ode45 in Matlab is
employed to perform deterministic dynamic analysis. Here, the second-order fractional moment of the maximum value
of
Z
of all layers, i.e.,
ˆ
M2
Zmax ,Zmax = max
16i615 {Zi}
, is considered in the convergence criterion (Eq. (16)). Accordingly,
a total of
ˆ
N= 520
samples of
Zi, i = 1,2, ..., 15
are generated, and the required fractional moments are obtained
according to Eq. (15). Besides, the speed up factor between the total computing time by using one CPU processor
T(1)
and that by using 8 CPU processors
T(8)
is computed, which is
Sp=T(1) /T (8) = 661 s/246 s = 2.7
. This
shows the benefit of using the parallel computing technique in the proposed parallel adaptive scheme.
20
O
Figure 8: A 15-storey nonlinear shear frame structure
Once the fractional moments are available, the EVDs of
Zi, i = 1,2, ..., 15
are then reconstructed by the proposed
M-EIGD-LESND. Figs. 9-11 depict the PDFs and POEs of
Z1
on the 1st storey,
Z7
on the 7th storey and
Z15
on
the 15th storey, respectively. As seen, the proposed mixture distribution model well captures the main parts and tail
information of the EVDs for selected storeys. Specifically, for all the selected storeys, the proposed method gives
almost same accurate results of PDF and POE compared to the reference results from MCS. Besides, to further illustrate
the advantages of the proposed method, a comparison of the PDF and POE curves of
Z1
is depicted in Fig. 12, where
results by LPSS+MIGLD and LPSS+MTGIG and those by the proposed method are given. As observed, with smaller
sample size, the proposed method is able to capture the tail information more accurately than LPSS+MIGLD and
LPSS+MTGIG, both of which require 625 samples.
0 20 40 60 80 100 120
0
0.01
0.02
0.03
0.04
0.05
MCS
Proposed method
(a) PDF
0 20 40 60 80 100 120
10-5
10-4
10-3
10-2
10-1
100
MCS
Proposed method
(b) POE
Figure 9: PDF and POE of Z1in Example 2
Further, we estimate the first-passage probabilities of the 1st, 7th and 15th storey of this example by Eq. (8), by
setting three different thresholds as
blim,1st = 95 mm
,
blim,7th = 80 mm
and
blim,15th = 67 mm
. Table 3gives the
21
0 20 40 60 80 100
0
0.01
0.02
0.03
0.04
0.05
0.06
MCS
Proposed method
(a) PDF
0 20 40 60 80 100
10-5
10-4
10-3
10-2
10-1
100
MCS
Proposed method
(b) POE
Figure 10: PDF and POE of Z7in Example 2
0 10 20 30 40 50 60 70 80
0
0.02
0.04
0.06
0.08
0.1
MCS
Proposed method
(a) PDF
0 10 20 30 40 50 60 70 80
10-5
10-4
10-3
10-2
10-1
100
MCS
Proposed method
(b) POE
Figure 11: PDF and POE of Z15 in Example 2
comparison results of proposed method, SS and MCS. As seen, with only 520 samples involved, all three first-passage
probabilities by the proposed method have better accuracy than probabilities by SS.
Table 3: Comparison of first-passage probabilities by the proposed method, SS and MCS (Example 2)
Method( ˆ
N)1st storey 7th storey 15th storey
blim(mm) ˆ
Pfblim(mm) ˆ
Pfblim(mm) ˆ
Pf
M-EIGD-LESND(520) 95 1.5075 ×10−480 2.1708 ×10−467 4.2208 ×10−4
SS(3700) 95 1.9300 ×10−480 4.3600 ×10−467 4.5300 ×10−4
MCS(106) 95 1.6300 ×10−480 2.3300 ×10−467 3.0000 ×10−4
22
0 20 40 60 80 100 120
0
0.01
0.02
0.03
0.04
0.05
MCS
LPSS+MIGLD
LPSS+MTGIG
Proposed method
(a) PDF
0 20 40 60 80 100 120
10-5
10-4
10-3
10-2
10-1
100
MCS
LPSS+MIGLD
LPSS+MTGIG
Proposed method
(b) POE
Figure 12: A comparison between the PDF and POE of Z1in Example 2
4.3.
Example 3: a spatial steel frame structure with viscous dampers under fully nonstationary stochastic ground
motion
To illustrate the practical applicability of the proposed method, a two-bay four-storey nonlinear spatial steel frame
structure with three viscous dampers under fully nonstationary ground motion is considered in this example, as shown
in Fig. 13. The whole structure is modeled and analyzed by the OpenSees software, where the bilinear constitutive
model shown in Fig. 14 is used to model the nonlinear stress–strain relationship of steel materials. The slab of each
floor is supposed to be rigid. The IPE270 beam and IPB300 column are adopted, where the column mass takes its self
weight, while the beam mass is defined by “self weight of beam + dead loads
DL
+ 0.2
×
live loads
LL
”. The viscous
dampers are all represented by the Maxwell model which includes a linear spring and nonlinear dashpot in series. Three
coefficients are involved in these viscous dampers, i.e., axial elastic stiffness of linear spring
Kd
, damping coefficient
Cd
, and velocity exponent
αd
. The Rayleigh damping is also employed here, where the damping ratios for both the
first and second modes are taken as 0.03. The fully nonstationary stochastic ground motion takes the same form and
parameters as employed in Example 2. It should be mentioned that the randomness of this structure comes from its
external loads (i.e., dead loads, live loads and ground motion) and its structural properties. The statistical information
of uncertain structural properties is collected in Table 4. In total, 1608 random variables are involved in this example.
We consider the maximum absolute inter-storey drift of the whole structure as the quantity of interest, denoted
by
Z
. By adopting the proposed parallel adaptive scheme,
ˆ
N= 1032
samples of
Z
are generated, where a set of up
to second order fractional moments can be estimated by Eq. (15). From the knowledge of the estimated fractional
moments, the EVD is represented by the proposed mixture distribution model, where the corresponding PDF and POE
curves are depicted in Fig. 15. For comparison, the results by LPSS+MIGLD and LPSS+MTGIG are also provided,
together with the benchmark results from MCS. Good accordance between results by proposed method and MCS
is readily observed. Admittedly, LPSS+MIGLD and LPSS+MTGIG are more computationally efficient since only
625 LPSS samples are employed. However, the tail distributions captured by the LPSS+MIGLD and LPSS+MTGIG
unfortunately deviate from the benchmark results to a large extent. Moreover, we calculate the first-passage probability
of this example by setting the threshold of
Z
as 38 mm. The first-passage probabilities by the MCS, SS, LPSS+MIGLD,
LPSS+MTGIG and proposed method are listed in Table 5. Remarkably, the proposed method yields a probability
23
O
Figure 13: A two-bay four-storey nonlinear spatial steel frame structure with viscous dampers
y
F
y
F
b
b
1
1
s
E
0
strain of deformation
stress of force
Figure 14: Bilinear constitutive model
Table 4: Statistical information of the uncertain structural properties in Example 3
Parameter Description Distribution Mean Standard variation
DLDead load Lognormal 10 N/m20.5 N/m2
LLLive load Lognormal 10 N/m21N/m2
FyYield strength of the steel Normal 250 ×106Pa 375 ×105Pa
EsYoung’s modulus of the steel Normal 2×1011 Pa 3×1010 Pa
bStrain-hardening ratio Normal 10−35×10−5
KdAxial stiffness of linear spring Normal 25 Pa 2.5 Pa
CdDamping coefficient Normal 20.7452 2.07452
αdVelocity exponent Normal 0.35 0.0175
24
that is quite close to what MCS gives, i.e.,
2.2439 ×10−4
by the proposed method, and
2.3600 ×10−4
by MCS.
The probability by LPSS+MIGLD and LPSS+MTGIG notably deviate from the probability by the MCS, reading
5.0859 ×10−5
and
5.0677 ×10−5
, respectively. In addition, the first-passage probability by SS is also less accurate,
reading 2.0400 ×10−4, but requires much more model evaluations.
0 10 20 30 40 50
0
0.02
0.04
0.06
0.08
0.1
0.12
MCS
LPSS+MIGLD
LPSS+MTGIG
Proposed method
(a) PDF
0 10 20 30 40 50
10-5
10-4
10-3
10-2
10-1
100
MCS
LPSS+MIGLD
LPSS+MTGIG
Proposed method
(b) POE
Figure 15: PDF and POE of Zin Example 3
Table 5: Comparison of first-passage probabilities by different methods (Example 3)
Method MCS SS LPSS+MIGLD LPSS+MTGIG Proposed
ˆ
N1063700 625 625 1032
ˆ
Pf2.3600 ×10−42.0400 ×10−45.0859 ×10−55.0677 ×10−52.2439 ×10−4
5. Concluding remarks
This paper proposes a novel fractional moments-based mixture distribution method to estimate the EVD and the
first-passage probabilities of high-dimensional nonlinear stochastic dynamic systems. Unlike the existing methods,
a parallel adaptive sampling scheme that allows for sample size extension is first proposed for estimating fractional
moments. By doing so, the sample size can be determined problem-dependently in conjunction with a proposed
convergence criterion. Such scheme is realized by a sequential sampling method, i.e., refined latinized stratified
sampling (RLSS), which also enables to achieve variance reduction in high dimensions. One versatile mixture
distribution model, namely, M-EIGD-LESND, is proposed to represent the EVD with enhanced flexibility, whose
free parameters are evaluated from obtained fractional moments. Three examples involving high-dimensional and
strong-nonlinear stochastic dynamic systems are investigated to demonstrate the efficacy of the proposed method. The
main conclusions are summarized as follows:
(1) The studied examples indicate that the proposed method is able to tackle with high-dimensional and strongly
nonlinear stochastic dynamic systems, where the uncertainties in both internal structural properties and external
excitations are considered. In addition, the proposed method is capable of accurately estimating small first-passage
probabilities in the order of 10−4.
25
(2) Several byproducts can be obtained by adopting the proposed method, i.e., fractional moments (including
integer moments such as mean and standard deviation) and EVD. Furthermore, for a general stochastic dynamic system,
multiple EVDs and first-passage probabilities under different thresholds can be estimated from only a single run of the
proposed method.
(3) The proposed method is computational efficient since the proposed parallel adaptive scheme allows to determine
an optimal sample size for a particular problem at hand. In addition, only additional samples of extreme value need to
be evaluated in each sample size extension, where parallel computing technique can be adopted to further improve the
efficiency.
(4) The proposed eight-parameter mixture distribution model is highly flexible and can adapt to different levels
of distribution asymmetry. This model generalizes several single-component distributions, such as the lognormal,
skew-normal, log skew-normal, and inverse Gaussian distribution. In addition, the mixture of lognormal and inverse
Gaussian distributions is a special case of the proposed model. As a result, this model enables the proposed method to
accurately recover a wide variety of EVDs.
CRediT authorship contribution statement
Chen Ding:
Methodology, Software, Validation, Investigation, Writing - Original Draft, Writing - Revised draft;
Chao Dang:
Conceptualization, Methodology, Investigation, Visualization, Writing - Original Draft, Writing - Revised
draft, Funding acquisition;
Marcos Valdebenito:
Validation, Writing- Reviewing and Editing, Funding acquisition;
Matthias Faes:
Validation, Writing- Reviewing and Editing;
Matteo Broggi:
Validation, Supervision, Writing-
Reviewing and Editing; Michael Beer: Supervision, Project administration.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgement
Chen Ding is grateful for the support by the European Union’s Horizon 2020 research and innovation programme
under Marie Sklodowska-Curie project GREYDIENT – Grant Agreement n
°
955393. Chao Dang is mainly supported
by the China Scholarship Council (CSC). Marcos Valdebenito acknowledges the support of ANID (National Agency
for Research and Development, Chile) under its program FONDECYT, grant number 1180271.
Appendix A. Refined Latinized stratified sampling
To generate samples and weights by the Refined Latinized stratified sampling (RLSS) [
37
], first we need to generate
candidate samples and candidate strata by the combination of hierarchical Latin hypercube sampling (HLHS) [
37
] and
Latinized stratified sampling (LSS) [39].
Begin with a LSS design with
N
samples in
ns
dimensions. First, generate a
ns
-dimensional Latin hypercube
sampling (LHS) design with
N
one-dimensional LHS strata
Ωij
and samples in each stratum
ϕij
,
i= 1, ..., ns;j=
26
1, ..., N
. Denote
S
as the
[0,1]ns
space. Divide
S
equally into
N
mutually exclusive and collectively exhaustive strata
Ω(k), k = 1, ..., N
, where
Ω(k)TΩ(q)=∅, k 6=q
and
SN
k=1 Ω(k)=S
. Note that each
Ω(k)
is an equal-weighted
hyper-rectangle and its boundary coincides with the boundary of
Ωij
. Each
Ω(k)
can be described by its starting
coordinate near the origin
Λ(k)=nΛ(k)
1, ..., Λ(k)
nso
and its side length
λ(k)=nλ(k)
1, ..., λ(k)
nso
. The weight of each
Ω(k)can be calculated as [37]:
$(k)=
ns
Y
i=1
λ(k)
i,(A.1)
where
PN
k=1 $(k)= 1
. For each
Ω(k)
, randomly pair each
ϕij
without replacement to produce the
k
-th LSS sample
ϕ(k)=hϕ(k)
1, ..., ϕ(k)
nsi, k = 1, ..., N.
Afterwards, apply a
δ
-level refinement of each
Ωij
based on the idea of HLHS, where
δ∈Z+
is the refinement
factor. Specifically, along each dimension, divide
Ωij δ
times equally to obtain a total of
˜
N=N(δ+ 1)
strata
Ωijh , h = 1, ..., ˜
N
. Produce new candidate samples per each dimension by uniform sampling inside every empty newly
produced stratum
Ωijh
. Subsequently, generate the candidate strata of RLSS, denoted as
˜
Ω(k?), k?= 1, ..., ˜
N
, by
dividing all the
Ω(k)δ
times along the LHS stratum boundaries in the dimension of largest side length
λ?= max
inλ(k)
io
.
Then, identify the candidate stratum
Ξ(k?)
i=nΩij ∈hΛ(k?)
i, Λ(k?)
i+λ(k?)
iio
which intersects with
˜
Ω(k?)
in each
i
-th dimension. Count the number of
Ξ(k?)
i
as
ε(k?)
i, i = 1, ..., ns
, and then determine the minimum number of
Ξi,(k?)
as
?
i= min
k?nε(k?)
io
. The candidate samples of RLSS, denoted as
˜
ϕ(k?), k?= 1, ..., ˜
N
, are generated by drawing
samples to the stratum
Ω(k?)
satisfying
ε(k?)
i=?
i
: if
?
i= 1
,
Ω(k?)
contains only one single candidate LHS stratum,
one must draw a sample from it; if
?
i>1
, one can draw samples from
Ξ(k?)
i
at random without replacement. Repeat
the sample adding process until all the dimensions of Ω(k?)have one related sample.
Once the candidate samples
˜
ϕ(k?)
and strata
˜
Ω(k?)
of RLSS are obtained, we can generate
~
RLSS samples at
a time. First, randomly select
~
RLSS strata
ˆ
Ω(l), l = 1, ..., ~
from the candidate strata
˜
Ω(k?)
. Then form RLSS
samples
ˆ
ϕ(l), l = 1, ..., ~
by drawing corresponding samples from
˜
ϕ(k?)
to
ˆ
Ω(l)
. Update the stratum weight according
to Eq. (A.1) by specifying the side length of
ˆ
Ω(l)
. Repeat several times to add
~
RLSS samples continuously until a
user-defined convergence criterion is met or the number of remaining candidate samples
˜
ϕ(k?)
of RLSS is less than
~
.
Note that if the number of candidate samples is insufficient, a new extension of the sample candidate pool is required.
If
ς > 1
extensions of the candidate sample pool can finally produce enough samples and weights of RLSS that meet
the convergence criterion, then the total number of
˜
ϕ(k?)
and
˜
Ω(k?)
at this time will be
˜
N=N(δ+ 1)ς
. Briefly, the
procedure of RLSS scheme is summarized in Algorithm 2, where ˆ
Ndenotes the obtained optimal sample size.
References
[1]
H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, 2nd Edition, Springer Series in Synergetics 18, Springer-Verlag
Berlin Heidelberg, 1989.
[2]
M.-Z. Lyu, J.-B. Chen, First-passage reliability of high-dimensional nonlinear systems under additive excitation by the ensemble-evolving-based
generalized density evolution equation, Probabilistic Engineering Mechanics 63 (2021) 103119.
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30 (2) (2009) 255–262.
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27
Algorithm 2 Refined Latinized stratified sampling approach [37]
Input:
Dimension
ns
of the random parameter vector
U
, LSS size
N
, refinement factor
δ
and number of samples for
each new sample size extension ~.
Output: RLSS samples ˆ
ϕ=nˆ
ϕ(1), ..., ˆ
ϕ(ˆ
N)oand corresponding weights $=n$(1), ..., $(ˆ
N)o.
1:
Initialize with
ς= 1
. Define a LHS design with
N
ungrouped LHS sample components
ϕij
and corresponding
one dimension LHS strata Ωij , i = 1, ..., ns;j= 1, ..., N.
2:
Establish a
ns
-dimensional stratification
Ω(k), k = 1, ..., N
to form LSS strata such that each stratum is an
equal-weighted hyper-rectangle and its boundary coincides with the boundary of
Ωij
. Calculate the stratum weight
of Ω(k)according to Eq. (A.1).
3:
Generate LSS samples
ϕ(k)=hϕ(k)
1, ..., ϕ(k)
nsi, k = 1, ..., N
by randomly drawing
ϕij
to its related LSS stratum
without replacement.
4:
Produce candidate samples per each dimension by applying a
δ
-level refinement of each
ϕij
inherent in
Ω(k)
according to HLHS design.
5:
Generate candidate strata of RLSS
˜
Ω(k?), k?= 1, ..., N(δ+ 1)ς
by dividing all the strata
Ω(k)
equally
δ
times
along every dimension with largest side length λ?
i.
6:
Identify the strata
Ξ(k?)
i=nΩij ∈hΛ(k?)
i, Λ(k?)
i+λ(k?)
iio, k?= 1, ..., N(δ+ 1)ς
which intersect with
˜
Ω(k?)
in
each
i
-th dimension. Count the number of
Ξ(k?)
i
in the
i
-th dimension as
ε(k)
i
, and then calculate
?
i= min
k?nε(k?)
io
.
7:
Generate candidate samples of RLSS
˜
ϕ(k?), k?= 1, ..., N(δ+ 1)ς
inside the stratum
˜
Ω(k?)
satisfying
ε(k?)
i=?
i
:
if
?
i= 1
, draw samples from
Ωij
; if
?
i>1
, draw samples from
Ξ(k?)
i
at random; repeat sample selection until all
the dimensions are filled.
8:
Select
~
RLSS strata
ˆ
Ω(k), k = 1, ..., ~
randomly from candidate
˜
Ω(k?)
and generate
~
RLSS samples
ˆ
ϕ(k), k =
1, ..., ~
by drawing corresponding samples from candidate
˜
ϕ(k?)
to
ˆ
Ω(k)
. Calculate the stratum weight according
to Eq. (A.1) by specifying the side length of ˆ
Ω(k).
9:
Repeat step 8 to add samples continuously until Eq. (16) is satisfied or an enlargement of the pool of candidate
samples ˜
ϕ(k?)is required. Then return to step 4 with ς=ς+ 1 and Ω(k)=˜
Ω(k?).
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H. Vanvinckenroye, I. Kougioumtzoglou, V. Deno
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