Uncertainty quantification using the Unscented Transformation (UT) 

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Random sampling and stratified sampling

Choosing the input values x1, . . . , xN as random samples from the PDF Wx (x), it is random sampling. Each individual is chosen randomly and entirely by chance. New sample points of random sampling are generated without taking into account the previously generated sample points. One does not necessarily need to know beforehand how many sample points are needed. The stratified sampling is that all areas of the sample space of x are represented by input values. Let the sample space S of x be partitioned into I disjoint strata Si . Obtain a random sample xi j , j = 1, . . . , ni from Si . The ni sum to N, and if I = 1, we have random sampling over the entire sample space. Comparing with random sampling, the advantage of stratified sampling is that if population density varies greatly within a region, stratified sampling will ensure that estimates can be made with equal accuracy in different parts of the region, and that comparisons of sub-regions can be made with equal statistical power. The disadvantage of stratified sampling is it is not useful when the population cannot be exhaustively partitioned into disjoint subgroups. It would be a misapplication of the technique to make subgroups’ sample sizes proportional to the amount of data available from the subgroups, rather than scaling sample sizes to subgroup sizes. Data to the subgroups’ sizes within the total population. For an efficient way to partition sampling resources among groups that vary in their means, their variances, and their costs, see « optimum allocation ».

Latin hypercube sampling (LHS)

LHS is proposed by McKay et al. [38], and then it was further elaborated in 1977 [40] and in 1981 [41]. In the context of statistical sampling, a square grid containing sample positions is a Latin square if (and only if) there is only one sample in each row and each column. A Latin hypercube is the generalisation of this concept to an arbitrary number of dimensions, whereby each sample is the only one in each axis-aligned hyperplane containing it. Let x1, . . . , xK are K uncertain inputs. The PDF of the kth uncertain input xk is split into N segments of equal probability. One random sample is taken from each segment, producing N samples for parameter k that span the whole of the PDF. This process is repeated for all K parameters. By then randomly combining the N samples
from the K parameters, N sets of input parameter values are produced. The maximum number of combinations for a Latin hypercube of N divisions and K variables (i.e., dimensions) can be computed with the following formula: ‚YN−1 i=0 (N − i) ŒK−1 (N!)K−1 (2.3).
For example, a Latin hypercube of N = 4 divisions with K = 2 variables will have 24 possible combinations. In LHS, the number of sample points and segments must be decided firstly. One advantage of the LHS is when the output y = f (x) is dominated by only a few of the components of x, and it ensures that each of those components is represented in a fully stratified manner, no matter which components might turn out to be important. Using LHS with the MCM has been shown to increase the rate of convergence compared to using random sampling or stratified sampling [38].

Unscented transformation (UT)

UT is proposed by Julier and Uhlmann in 1997 [27] as a method to propagate mean and covariance information through non-linear transformations. The authors’ motivation for proposing UT is that applying the Extended Kalman Filter (EKF) [42] to non-linear systems is difficult to implement, difficult to tune, and only reliable for systems that are 22 Uncertainty quantification using the Unscented Transformation (UT) almost linear on the time scale of the updates, so an alternative must be proposed as a practical estimator to non-linear systems. Then the use of UT to efficiently estimate uncertainties has been proposed in [23, 29].

The basic idea of UT

In [23], the author gives detailed description about UT. Considering a non-linear mapping f of a random variable x with mean μx and standard deviation σx , the output y of the non-linear transformation is simply expressed as: y = f (x) (2.4) and the objective is to obtain the statistics of y, such as the mean μy and variance σ2 y , from which the standard deviation is derived. For this objective, let ˆx be a random variable with zero mean and the same standard deviation as x, that is ˆx = 0 ± σx . Respective to this, x is expressed as: x = μx +0.

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Example 2: using second-order UT for UQ with two random variables in the calculation

In the case of two random variables, x1 and x2 in the calculation, the results of 900MHz and 1800MHz obtained by MCM and UT are presented in Fig. 2.5 and Fig. 2.6, respectively. At 900MHz, the result of UT corresponding to 36 runs is 0.896 ± 1.29%, and the result of MCM corresponding to 1000 runs is 0.895 ± 1.26%. At 1800MHz, the result of UT in the case of two random variables in the calculation is 1.719 = ±1.71%, and the result of MCM running 1000 times is 1.719 ± 1.98%. The results whether for 900MHz or 1800MHz obtained by MCM and second-order UT are similar.

Example 3: using fourth-order UT for UQ with one random variable in the calculation

For fourth-order UT, the sigma points are calculated by the method proposed in [23], and the sigma points are given by equation (2.44) using this method. When there is only one random variable x1 in the calculation, the mean and standard variance of the output for 900MHz and 1800MHz calculated by MCM, second-order UT and fourthorder UT are shown in Fig. 2.7 and Fig. 2.8, respectively. The result of fourth-order UT is 0.895 ± 1.35% at 900MHz and 1.719 ± 3.57% at 1800MHz. The results displayed show that the mean of the output calculated by second-order UT and fourth-order UT are similar for both 900MHz and 1800MHz, however, the standard variance of the output calculated by second-order UT is closer to the results of MCM than the standard deviation calculated by fourth-order UT. Theoretically, the fourth-order UT should be more accurate than the second-order UT since higher-order of Taylor series are considered. However, since the equations of fourth-order UT cannot be calculated directly, the sigma points of fourth-order UT are approximated by the method proposed in [23], and some error is existing. The error results in the poor performance of fourthorder UT. In contrast, the sigma points of the second-order UT can be obtained directly by solving the equations without approximation or error.

Table of contents :

Résumé
References
List of figures
List of tables
1 Introduction 
1.1 Uncertainty in SAR calculation
1.1.1 Specific absorption rate (SAR)
1.1.2 Finite Integration Technique (FIT) and SAR calculation in CST .
1.1.3 Uncertainty in modelling
1.2 Probability approach for uncertain quantification
1.2.1 Random variables
1.2.2 Random vectors
1.3 Overview of the methods related to uncertainty quantification
1.3.1 Method of moments
1.3.2 Quadrature method
1.4 Aims and objectives
1.5 Outline of thesis
2 Uncertainty quantification using the Unscented Transformation (UT) 
2.1 Introduction
2.2 The Monte Carlo method
2.2.1 Random sampling and stratified sampling
2.2.2 Latin hypercube sampling (LHS)
2.3 Unscented transformation (UT)
2.3.1 The basic idea of UT
2.3.2 The formulation of two-dimensional UT
2.3.3 Higher-order UT
2.4 Results and discussions
2.4.1 Example 1: using second-order UT for UQ with one random variable in the calculation
2.4.2 Example 2: using second-order UT for UQ with two random variables in the calculation
2.4.3 Example 3: using fourth-order UT for UQ with one random variable in the calculation
3 Uncertainty quantification using non-intrusive methods 
3.1 Introduction
3.2 Stochastic collocation (SC) method
3.3 Polynomial chaos (PC) expansion
3.3.1 Homogeneous chaos
3.3.2 The generalized polynomial chaos (gPC) method
3.3.3 Using non-intrusive polynomial chaos for UQ
3.3.4 Sensitivity analysis (SA) using PC expansion
3.4 Combined uncertainties
3.5 Results and discussions
3.5.1 Uncertainty quantification using a simple mobile phone model .
3.5.2 Numerical model for SAR calculation
3.5.3 Uncertainty in SAR calculation
3.5.4 Combined uncertainties
3.5.5 Sensitivity analysis in SAR calculation
3.6 Conclusion
4 Intrusive polynomial chaos method for uncertainty quantification 
4.1 Introduction
4.2 Stochastic system formulation
4.3 The Finite Difference Time Domain method
4.4 Generalized polynomial chaos in finite difference time domain simulation
4.5 Uncertainty quantification applied to a one dimensional example
4.6 Results and discussions
4.7 Conclusion
5 Conclusions and Future Work 
5.1 Conclusions
5.2 Future Work
Appendix A Intrusive gPC expansion in 3D-FDTD 103
A.1 Finite difference time domain in three dimensions
A.2 Generalized polynomial chaos in three-dimension finite-difference timedomain simulation
A.3 The inner products of intrusive polynomial chaos expansion
Appendix B Glossary 
References 

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