AGENERAL FRAMEWORK FOR PROBABILISTIC SENSITIVITY ANALYSIS WITH RESPECT TO DISTRIBUTION PARAMETERS A P REPRINT

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AGENERAL FRAMEWORK FOR PROBABILISTIC SENSITIVITY

ANALYSIS WITH RESPECT TO DISTRIBUTION PARAMETERS

A PREPRINT

Jiannan Yang

Department of Engineering

University of Cambridge

Trumpington Street, Cambridge CB2 1PZ, UK

jy419@cam.ac.uk

February 1, 2023

ABSTRACT

Probabilistic sensitivity analysis identiﬁes the inﬂuential uncertain input to guide decision-making.

We propose a general sensitivity framework with respect to the input distribution parameters that

uniﬁes a wide range of sensitivity measures, including information theoretical metrics such as the

Fisher information. The framework is derived analytically via a constrained maximization and the

sensitivity analysis is reformulated into an eigenvalue problem. There are only two main steps to

implement the sensitivity framework utilising the likelihood ratio/score function method, a Monte

Carlo type sampling followed by solving an eigenvalue equation. The resulting eigenvectors then

provide the directions for simultaneous variations of the input parameters and guide the focus to

perturb uncertainty the most. Not only is it conceptually simple, but numerical examples demonstrate

that the proposed framework also provides new sensitivity insights, such as the combined sensitivity

of multiple correlated uncertainty metrics, robust sensitivity analysis with an entropic constraint,

and approximation of deterministic sensitivities. Three different examples, ranging from a simple

cantilever beam to an offshore marine riser, are used to demonstrate the potential applications of the

proposed sensitivity framework to applied mechanics problems.

Keywords

sensitivity matrix, parametric sensitivity, combined sensitivity, information theoretical sensitivity, decision

under uncertainty

1 Introduction

The use of mathematical models to simulate real-world phenomena is ﬁrmly established in many areas of science and

technology. The input data for the models are often uncertain, as they could be from multiple sources and of different

levels of relevance. The uncertain inputs of a mathematical model induce uncertainties in the output and sensitivity

analysis identiﬁes the inﬂuential inputs to guide decision-making. A broad range of approaches can be found in the

literature, but in practice, the input uncertainties are commonly quantiﬁed by a joint probability distribution. The

analysis of the input and output relationship in this probabilistic setting is called probabilistic sensitivity analysis [1].

A suitable measure can be used to summarise the induced output uncertainties. Commonly used metrics are the (central)

moment functions of the uncertain output, such as the mean and variance, and the probability of failure, i.e., the

probability that the random output would exceed a certain threshold. In addition, the average uncertainty or information

content can be measured using entropy that is based on the entire distribution function of the random output [

]. The

probabilistic sensitivity analysis then examines the relationship between the uncertain input and the induced uncertainty

of the output. In particular, we are interested in identifying which input parameters would impact the output metrics the

most, i.e., the largest output change for the same input variation, to guide decision-making.

In this setting, the sensitivity of the point estimates, such as the moment functions and the failure probabilities, can be

obtained using the partial derivatives of the metrics with respect to (w.r.t) the input distribution parameters. Although

arXiv:2210.01010v3 [stat.ME] 1 Feb 2023

Probabilistic Sensitivity Framework A PREPRINT

the general application of the derivative-based sensitivity analysis can be limited by the difﬁculty of computing the

derivatives, the derivatives w.r.t the input distribution parameters can be more easily evaluated by differentiation inside

the expectation operator (c.f. eqs. (1) to (4) in Section 2 ). This is possible because the individual samples of the random

output are not directly dependent on the input distribution parameters. As a result, the partial derivative operation is

only evaluated w.r.t the joint probability density function (PDF) of the input, and this approach is called the likelihood

ratio/score function method (LR/SF) [3, 4].

As described, the LR/SF method is merely a mathematical trick. Nevertheless, if used together with a sampling method,

it is efﬁcient as the uncertainty metric and its sensitivity can be evaluated in a single simulation run (c.f. Section 3.4).

The LR/SF method has been applied to general objective functions in stochastic optimization [

], the failure probability

in reliability engineering [5] and some distribution-free properties of the LR/SF method are discussed in [6].

The sensitivity of entropy, on the other hand, cannot be directly evaluated using the LR/SF method. Instead, sensitivity

related to entropy is often analysed using the Kullback–Leibler (K-L) divergence (aka relative entropy), by measuring

the divergence between two PDFs (probability density functions) corresponding to two different cases. This approach

is studied in [

] for safety assessment to explore the impact on risk proﬁle due to input uncertainties and in [

] for

engineering design before and after uncertainty reduction of the random variables of interest. A similar approach using

the mutual information between the input and the output has also been studied for sensitivity analysis [

]. The mutual

information can be regarded as a special form of the K-L divergence except that it requires the use of the joint PDF. As

the K-L divergence is not a metric, alternative distance measures such as the Hellinger distance has been proposed to

quantify the difference between two PDFs and the corresponding sensitivities [10].

It should be noted although the relative entropy is not a metric, its inﬁnitesimal form is directly linked to the Fisher

information [

] which is a metric tensor and this link has been explored in [

] for probabilistic sensitivity analysis

using the Fisher information matrix (FIM). The LR/SF method can then be used to compute the FIM efﬁciently for

sensitivity analysis of the output entropy [12].

In this paper, we propose a new sensitivity matrix

that uniﬁes the sensitivity of a wide range of commonly used

uncertainty metrics, from moments of the uncertain output to the entropy of the entire distributions, in a single

framework. This is made possible by the likelihood ratio/score function method (LR/SF) where the sensitivity to the

input distribution parameters of different metrics can be expressed in the same form (c.f. Eq 4). The 2nd moment of the

sensitivity matrix,

ErrT

, arises naturally when the impact of input perturbation on the output is examined. Moreover,

the maximization of the perturbation of the output uncertainty metrics leads to an eigenvalue problem of the matrix

ErrT

. The eigenvalues represent the magnitudes of the sensitivities with respect to (w.r.t) simultaneous variations

of the input distribution parameters

, and the relative magnitudes and directions of the variations are given by the

corresponding eigenvectors. Therefore, the eigenvectors corresponding to the largest eigenvalues are the most sensitive

directions to guide decision-making.

The sensitivity matrix

can be seen as a counterpart of the deterministic sensitivity matrix (Jacobian matrix) as the

elements of

are the normalised partial derivatives of the output uncertainty metrics w.r.t to the distribution parameters

of the uncertain input (c.f. Eq 6). The resulting eigenvectors, therefore, have direct sensitivity interpretation. It should

be noted that although the sensitivity matrix

is formulated and estimated using the LR/SF method, the use of the 2nd

moment matrix and its eigenvectors for sensitivity analysis additionally captures the interactions of the sensitivities of

different metrics.

In addition, the current work is motivated by a recent study [

] where a special case of the proposed sensitivity matrix

has been applied successfully to the combined sensitivity analysis of multiple failure modes. We are going to show

that, not only does

ErrT

capture the combined perturbation effect of multiple metrics, e.g., multiple failure modes

or multiple moment functions, but also include the Fisher information matrix (FIM) as a special case. Application

of the FIM for sensitivity analysis can be found in many areas of science and engineering. For example, the Fisher

Information Matrix (FIM) has been applied to the parametric sensitivity study of stochastic biological systems [

], to

assess the most sensitive directions for climate change given a model for the present climate [

] and as one of the

process-tailored sensitivity metrics for engineering design [12].

It should be noted that there are two main differences between the proposed framework and the commonly used

variance-based sensitivity analysis [

]. First, variance-based approaches study how the variance of the output can

be decomposed into contributions from uncertain inputs. It ranks the factors based on the assumption that the factor

can be ﬁxed to its true value, i.e., complete reduction of the uncertainties, which is rarely possible in practice [

]. In

contrast, the proposed framework uses partial derivatives to examine the perturbation of the output metrics due to a

variation of the input distribution parameters. As the distribution parameters are often based on data, it is equivalent

to asking which uncertain dataset the decision-makers should focus on to change the output the most. And this is

particularly pertinent to data-driven applications like digital twins [

]. Second, the output sensitivity measure from

Probabilistic Sensitivity Framework A PREPRINT

the variance-based methods is the percentage contribution, of each factor or the interactions between factors, to the

output variance. The proposed framework, on the other hand, outputs the eigenvectors of the sensitivity moment matrix

ErrT

as the principal sensitivity directions for simultaneous variations of the input distribution parameters. This is

based on a more pragmatic view that given a ﬁnite budget to change the parameters, maximizing the impact on the

output follows the principal sensitivity directions, which tend to be a simultaneous variation of the parameters because

their effects on the output are likely to be correlated. More discussions on the budget constraint can be found in Section

3.2 with a generalization to the generalised eigenvalue problem.

It should be noted that despite the differences, for some cases, the aggregated index for individual parameters given in

Eq 25 can be used to compare the Fisher sensitivity results against the variance-based main and total sensitivity indices.

This has been done in [

] for a 15-dimensional problem, and in that case, the dominant ﬁrst eigenvector of the FIM

seems to correspond to the main effects from the variance-based sensitivity analysis.

In what follows, the general sensitivity framework is introduced in Section 2 where the sensitivity analysis is refor-

mulated as a standard eigenvalue problem. In Section 3, we discuss various properties of the proposed framework,

including the link to the Fisher information matrix and the possible extension to a generalised eigenvalue problem

for robust sensitivity analysis. A benchmark study, using two commonly used functions for sensitivity analysis, is

conducted in Section 4 for the Fisher sensitivity. Three different examples are considered in Section 5, ranging from a

simple cantilever beam to an offshore marine riser, to demonstrate the potential applications of the proposed sensitivity

framework. Concluding remarks are given in Section 6.

2 Sensitivity framework

Consider a general function

y=h(x)

, the probabilistic sensitivity analysis characterises the uncertainties of the outputs

that are induced by the random inputs

. It is assumed that the uncertainties of

can be described by parametric

probability distributions, i.e., x∼p(x|b), where bare the distribution parameters.

One commonly used summary statistic is the (central) moment function of the uncertain output, such as the mean and

variance. More generally, the moment function is taken with respect to a function of the uncertain output

g(y)

. This

might arise when there is a stochastic process present, such as the random forces considered in some of the examples in

Section 5, and the

g(·)

function could represent max, min or root mean square (r.m.s). In this setting, the

qth

moment

function and its partial derivative w.r.t the input distribution parameters can be expressed as:

mq=EX[gq(h(x))] = Zgq(h(x))p(x|b)dx(1a)

∂mq

∂b=Zgq(h(x))∂p(x|b)

∂bdx(1b)

where it has been assumed that the differential and integral operators are commutative, i.e. the order of the two

operations can be exchanged under regularity conditions of continuous and bounded functions.

Another metric is the probability of failure and its gradient:

Pf=EX[H [g(h(x)) −z]] = ZH [g(h(x)) −z]p(x|b)dx(2a)

∂Pf

∂b=ZH [g(h(x)) −z]∂p(x|b)

∂bdx(2b)

where

H(·)

is the Heaviside step function and

represents the failure threshold. It is noted in passing that the application

of failure probability is not limited to reliability engineering. For example, the probability of cost-effectiveness in health

economics [18] and the probability of acceptability in design [19] can both be formulated in the same way as Eq 2a.

When the quantity of interest is the underlying distribution function of the uncertain outputs, the density function and

its gradient w.r.t the distribution parameters can be expressed as [12]:

p(y) = EX"Y

δ[yn−hn(x)]#=ZY

δ[yn−hn(x)] p(x|b)dx(3a)

∂p(y)

∂b=ZY

δ[yn−hn(x)] ∂p(x|b)

∂bdx(3b)

where δ(·)is the Dirac delta function.

Probabilistic Sensitivity Framework A PREPRINT

Although the aforementioned diverse metrics measure different aspects of the uncertain output, it is clear that all of

them can be more compactly described using a general utility function:

U=EX[u(x)] = Zu(x)p(x|b)dx(4a)

∂U

∂b=Zu(x)∂p(x|b)

∂bdx(4b)

where the utility function

u(x)

represents the

gq(·)

in Eq 1,

H(·)

in Eq 2 and

δ(·)

in Eq 3. It should be noted that the

utility function could also depend on other variables, such as the failure threshold

for the case of failure probability.

However,

u(x)

is not directly dependent on the parameters

, as

b→x→u(x)

forms a Markov chain. As a result, it

is possible to differentiate the joint PDF

p(x|b)

within the integral in Eq 4b. And that is the same for eqs. (1) to (3). As

mentioned in the introduction, this approach is sometimes called the likelihood ratio/score function method (LR/SF).

An advantage of this approach is that, if used together with a sampling method such as the Monte Carlo method, the

uncertainty quantiﬁcation and sensitivity analysis can be conducted in a single simulation run and more details are

given in Section 3.4.

The purpose of our sensitivity analysis is to identify the most important uncertain parameters, i.e., which set of

parameters would perturb the output of interest the most. This perturbation can be quantiﬁed as

Ω = (∆U/U)2

, where

the normalisation leads to percentage perturbation and the square operation quantiﬁes the absolute value. If a ﬁrst-order

perturbation is assumed, a general form of the normalised perturbation is:

Ω = E"X

k∆Uk

Uk2#

=E"X

k1

∂Uk

∂b∆b2#

∆bi∆bjE"X

rikrjk #

= ∆bTErrT∆b

(5)

where the jkth entry of the matrix ris deﬁned accordingly as:

rjk =1

∂Uk

∂bj

(6)

The matrix

can be seen as a counterpart of the deterministic sensitivity matrix (Jacobian matrix) and therefore

called sensitivity matrix in this paper. It is interesting to note that the 2nd moment of the sensitivity matrix,

ErrT

arises naturally from the perturbation analysis. As it is in the form of a Gram matrix,

ErrT

is symmetric positive

semi-deﬁnite (also evident from the quadratic form of Eq 5).

The general form of the perturbation in Eq 5 considers the combined effect of multiple utilities. For example, there

could be multiple failure modes where

Uk=P(k)

denotes the

kth

failure mode; it is also often of interest to consider

the combined sensitivity of multiple responses or moments of the same uncertain output where

Uk=mk

denotes the

kth

moment. It is noted in passing that a weighting could be added to each

and that would result in a weighting of

rjk

in Eq 6. The weighted scenario is not considered further in this paper as the weighting is strongly case-dependent

but will not alter the general form of Eq 5. The expectation operation

E[·]

in Eq 5 takes account of any additional

uncertainties that might arise in different cases. For example, the failure threshold

could be uncertain in Eq 2; for the

case of the joint density function, where

U=p(y)

, the gradient of the log utility described in Eq 6 is uncertain due to

randomness of the output y.

Using the general perturbation function described in Eq 5, the sensitivity analysis can be formulated as a constrained

optimization problem:

max 1

2Ω = 1

2∆bTErrT∆b

s.t. ∆bT∆b=

(7)

where the method of Lagrange Multiplier can be used:

L= ∆bTErrT∆b−λ(∆bT∆b−)(8a)

∂L

∂∆b=ErrT∆b−λ∆b(8b)

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AGENERALFRAMEWORKFORPROBABILISTICSENSITIVITYANALYSISWITHRESPECTTODISTRIBUTIONPARAMETERSAPREPRINTJiannanYangDepartmentofEngineeringUniversityofCambridgeTrumpingtonStreet,CambridgeCB21PZ,UKjy419@cam.ac.ukFebruary1,2023ABSTRACTProbabilisticsensitivityanalysisidentiestheinuentialuncertaininputtoguided...

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