Sensitivity Analysis for Epistemic Uncertain System Based on Sobol’ Method

SUO Bin ( )， GUO Huixin ()

1 Institute of Electronic Engineering， China Academy of Engineering Physics， Mianyang 621999， China2 Department of Mechanical and Electrical Engineering， Changsha University， Changsha 410003， China

Abstract： Considering the characteristics of sensitivity analysis in epistemic uncertain system， a new sensitivity analysis index based on Sobol’ method was proposed. Then the index calculation method with single cycle Monte-Carlo sampling was studied， and an application was developed. Numerical examples show that， the new method could solve the system sensitivity analysis problem with mixed aleotory and epistemic uncertainties. In this method， aleotory uncertain parameters could be sampled randomly according their distributions， and need not be translated to evidence bodies. Epistemic uncertain parameters were sampled randomly based on evidence theory. Therefore the loss of information in the sampling process was small， and the shortcomings of the existing methods were overcome.

Key words： sensitivity analysis; aleotory uncertainty; epistemic uncertainty; Sobol’ method

Introduction

Sensitivity analysis was a scientific study of how the variance of model output varies with the variance of each input parameter[1-3]. In other word， sensitivity analysis studies how the variation of input parameters， affects the variability of model output. With sensitivity analysis， the importance of quantifying the influence of input parameters on model output uncertainty could be quantified， and the key factors that affect the model response variability could be found， which provides the direction and approach for improving the robustness and reliability of the system output.

At present， sensitivity analysis methods could be broadly divided into two categories， which were local sensitivity analysis[4-6]and global sensitivity analysis[7-9]. In sensitivity analysis， analytic method， graphical method or the Monte Carlo stochastic simulation method could be used to calculate the one order or the total sensitivity for input parameters to the system response， sorting the importance of the input parameters.

Existing sensitivity analysis methods usually studied the sensitivity of stochastic uncertain systems. In the model of the system， the input factors of the system are random variables. It is worth pointing out that not all the uncertainty of the factors could be characterized by randomness. In practical engineering problems， in addition to thealeotory uncertainty， epistemic uncertainty was widespread. Using fuzzy variables， interval variables and evidence to characterize the epistemic uncertainty may be in line with the actual situation[10-12]. Therefore， the study of the sensitivity analysis of uncertain cognitive systems becomes a practical problem to be solved. In recent years， the theory of system sensitivity analysis with cognitive uncertainty has made great progress， but there are still many problems that need further study. This paper attempts to make a useful attempt in this area and has made some progress.

1 Sensitivity Index of Aleotory and Epistemic Uncertainty Coexisting Systems

A mixed uncertainty system with aleotory uncertain variablesV=(V1，V2， …，VKV) and epistemic uncertain variablesU=(U1，U2， …，UKU)， which mathematical model isY=h(x)=h(x1， …，xKV，xKV+1， …，xKV+KU)=h(V，U)， wherex=(V，U)， andn=KV+KU.

Now the sensitivity index calculation method forUi(i=1， 2， …，k) will be studied. According to the definition of Sobol’ sensitivity index， the first-order sensitivity ofUican be expressed as

(1)

where |Uis in the condition of epistemic，Var(·) is the variance， andE(·) is the mean. Do mathematical transformations to the upper formula， then

(2)

And

(3)

whereCis defined as

(4)

Under epistemic uncertainty， according to the definition of Sobol’ sensitivity index， the total sensitivity ofUican be expressed as

(5)

(6)

(7)

That is，

(8)

3 Calculation and Sorting of SensitivityIndex

3.1 Calculation method of sensitivity index

When the epistemic uncertainty.But if U is handled by a random variable，some of its cognitive uncertainty is neglected． So in this situation，the sensitivity of the epistemic uncertainty variables needs to be corrected，which can be calculated by Eqs． ( 2 ) and ( 6 ) ． For aleotory uncertain variables V， it is not necessary to correct the uncertainties because the sampling process does not change the nature and distribution of their uncertainties，that is

SVi|U=SVi,

(9)

(10)

Step1A subsample ofU，ui， is generated randomly bydu(u). The epistemic uncertainty ofUis neglected in part， and a random sampling method (i.e.， Pignistic probability distribution density function sampling method based on evidence theory) is adopted to generateui.

Step2According to the distribution of random variablesdV(v|u=ui) randomly generates a sample ofV，vi. A sample ofx=(V，U)， (vi，ui)， is obtained from the above two steps. Repeat the first and second stepsNtimes， and get a sample matrixA. Repeat the above two steps N times， and get a sample matrixB.

It is assumed that all uncertain variables are independent of each other， so

(11)

(12)

whereduj(uj) is the pignistic probability distribution density function ofUj， anddvj(vj|ui) is the probability distribution density function ofVj. Since the two uncertain variables are independent of each other， thendvj(vj|ui)=dvj(vj).

3.2 Sensitivity correction factor C

(2) WhenY=h(V，U) does not containV， the system only contain epistemic uncertainty， andY=h(V，U) degenerates intoY=h(U). At this point，E(Y|U)=E(Y)， then

(13)

At this point，using Sobol's method to find the sensitivity index of the system，Stot Ui and are equal respectively，that is to say，at this time do not have to amend．

(3) WhenY=h(V，U) includesUandVsimultaneously， asVar(E(Y|U))=Var(Y)-E[Var(Y|U)]， then

(14)

From above，due to the existence of cognitive uncertainty in the system，the uncertainty of the system response to E ( Y) increases，and the sensitivity index will be scaled up to 1 /C． So the value of and Stot may be greater than 1．

If there are 2 mathematical models of uncertain systems，Y=f(V，U) andY=f(V′，U)， and it is assumed thatVandV′ are identically distributed random uncertainty vectors. Obviously there isVar(Y)=Var(Y′)，E(Y)=E(Y′)， andE(E(Y|U))=E(Y). So

(15)

(16)

If the number of samples isN， the single cycle Monte Carlo sampling ofCis designed as follows.

Step1Pressdu(u) to generate a sub sampleuiofUrandomly. Considering the uncertainty ofU， two random sampling methods (i.e.， sampling based on evidence theory， lower probability model) are adopted to generateui.

Step2According to the distribution lawdV′(v′|u=ui) of the random variableV， a sampleviis generated randomly.

Repeat Step 1 to Step 3 withNtimes， and get 2 sample matricesAcandBc， where

Ac=(vi，ui)，(i=1， 2， …，N),

(17)

(18)

The sample matrixAcis used for the statistical calculation ofY， and the sample matrixBcis used for the statistical calculation ofY′. The total evaluation times of modelY=h(V，U) is 2N.

3.3 Sensitivity index sorting

By Eqs． ( 2) ，( 6) and ( 7) ，the sensitivity ranking of all epistemic uncertainty factors is not affected by the sensitivity correction factor C，because the ordering is unchanged before and after the correction． If we only need to determine the order of and Stot ，we can first find SUi and Stot，and then the first order and the total sensitivity of the order can be obtained．

4 Numerical Examples

As shown in Fig. 1， in the crank slider mechanism， the crank length isa， the length of the connecting rod isb， and the external force isF. Considering the strength of the connecting rod， the cross section is hollow， and the size is shown Table 1. The elastic modulus of the connecting rod material isE， and the yield limit isS， which is a random variable that subjects to normal distribution. Set the bias distance ofe， and the friction coefficient between the slider and the frame isμ. The dimension of the connecting rod isd2=50 mm，d1=25 mm.

Fig.1 Offset slider-crank mechanism

Under the action of the external loadF， the condition that the connecting rod does not yield the damage is

The distribution law of variables is shown in Tables 1-2.

Tabel 1 Distribution law of variables

Table 2 Distribution law of epistemic uncertainty variables

The sensitivity of each variable to theG1is solved by MATLAB programming， and the number of random simulation timesNis 5 000. The sensitivity analysis reports generated by the simulation program are shown in Fig.2.

4 Conclusions

Considering the characteristics of sensitivity analysis in epistemic uncertain system， a new sensitivity analysis index based on Sobol’ method was proposed. Then the index calculation method with single cycle Monte-Carlo sampling was studied， and an application was developed. Numerical examples show that， the new method could solve the system sensitivity analysis problem with mixed aleotory and epistemic uncertainties. In this method， aleotory uncertain parameters could be sampled randomly according their distributions， and need not be translated to evidence bodies. Epistemic uncertain parameters were sampled randomly based on evidence theory. Therefore, the loss of information in the sampling process was small， and the shortcomings of the existing methods were overcome.