# Sensitivity Analysis of an Implanted Antenna within Surrounding Biological Environment
## 3 Polynomial Chaos Expansion (PCE)
### Surrogate Model Based on PCE  
A truncation of the Polynomial Chaos Expansion provides a surrogate model at low cost avoids if the evaluation of the coefficients is performed using least-square regressions. Let us consider an approximate model \(\tilde{M}\) of the exact model \(M\). The corresponding random output is given by a truncated sum of \(P\) polynomials expressed as:  
\[\tilde{Y}=\sum_{k=0}^{P-1}a_{k}\Psi_{k}(X) \tag{7}\]  
The unknown coefficients of the truncation can be estimated through least square regression while minimizing a root mean square error. If we denote \(y\) the output vector collecting n values in the vector \(y=\left(y^{(1)},y^{(2)},\ldots,y^{(n)}\right)\) corresponding to the n inputs \(x^{(i)}\) (\(i=1,\ldots,n\)) given by \(x^{(i)}=\left(x_{1}^{(i)},x_{2}^{(i)},\ldots,x_{N}^{(n)}\right)\), then the estimated unknown coefficients derived from a regression approach are given by:  
\[\tilde{a}=\left\{\Psi^{T}\Psi\right\}^{-1}\Psi^{T}y \tag{8}\]  
where \(\Psi\) is the matrix whose coefficients are \(\Psi_{k}^{i}=\Psi_{k}\Big{(}x^{(i)}\Big{)}\).  
The Latin Hypercube Sampling approach, known as LHS, is often used for planning the design of experiments. The validation of the truncated model can be checked using Bootstrap or LOO (Leave-One-Out) validation error. This last approach is a natural definition. If we consider n input values \(x^{(i)}\) giving n output \(y^{(i)}=M\Big{(}x^{(i)}\Big{)}\), one sample point \(x^{(j)}\) can be removed and a new surrogate model \(M^{\text{PC}\,ij}\) can be built on the basis of \(n-1\) sample values. Then a comparison between the predictedoutput value with this surrogate model and the value \(y^{(j)}\) can reflects the accuracy of the approach. This leads to the LOO criterion defined as:  
\[LOO=\frac{1}{n}\sum_{i=1}^{n}\left(M(x_{i})\ -\ M^{\text{PC}\backslash i}\left(x^{(i )}\right)\right)^{2}/Var[M] \tag{9}\]  
Depending on the targeted accuracy, new samples can be added. From the computational efficacy point of view, it is important to select the most important polynomials. Then, a further reduction of the basis size can be performed using an adaptive technique, the so-called LARS algorithm (Least angle regression method) [24]. This algorithm identifies the polynomials having the most impact on the output response and on the sensibility indices.  
For a sensitivity analysis the \"Sobol\" decomposition [25] is well known. Quantitative estimates of the output sensitivity both to each input individually and to each of possible combinations as well, are straightforward; thanks to the orthonormality of the polynomial chaos basis, the global variance \(\bar{D}\) and the partial variances \(\bar{D}_{i_{1},\ldots,i_{k}}\) can be expressed with the coefficients of the expansion:  
\[D=Var\left(\tilde{\Upsilon}\right)=\sum_{k=1}^{P}{\color[rgb]{0,0,0}-1}a_{k}^ {2} \tag{10}\]  
\[\bar{D}_{i_{1},\ldots,i_{k}}=\sum_{\alpha\in\tau_{i_{1},\ldots,i_{k}}}a_{ \alpha}^{2} \tag{11}\]  
where \(\tau_{i_{1},\ldots,i_{k}}\) is the set of \(\alpha\) tuples such that only the indices \(i_{1},\ldots,\ i_{s}\) are nonzero:  
\[\tau_{i_{1},\ldots,i_{s}}=\left\{\alpha\,\ \alpha_{k}>0,\ \forall\ k=1, \ldots,N\ k\in(i_{1},\ldots,i_{s})\right\} \tag{12}\]  
The indices can be analytically extracted from the PCE [23] and are defined as:  
\[S_{i_{1},\ldots,i_{k}}=\frac{\bar{D}_{i_{1},\ldots,i_{s}}}{\bar{D}} \tag{13}\]