## 2 Globalized Miniaturization Using Variable-Fidelity EM Models and PCA
### Globalized Size Reduction: Constructing a Surrogate Model  
The designs \(\{\mathbf{x}_{c}^{(i)}\}_{j=1,\,\ldots,\,N_{3}}\) provide an approximate allocation of a part of the feasible region boundary. These vectors will be used to establish a metamodel domain to be assessed for the purpose of globalized optimization of antenna size. In order to facilitate and lower the surrogate model setup cost, the domain will be of reduced dimensionality using the procedure proposed in [43], which is explained in Figure 3.  
We define a covariance matrix: \(\mathbf{S}_{z}=\dfrac{1}{N_{3}-1}\sum_{r=1}^{N_{3}}(\mathbf{x}_{z}^{(i)}-\mathbf{x}_{n}) (\mathbf{x}_{z}^{(i)}-\mathbf{x}_{n})^{T}\) where:  
\(\mathbf{x}_{n}=[\sum_{z}-1,\,N_{3}\,\mathbf{x}_{z}^{(i)}]/N_{3}\) - the set's centre \(\{\mathbf{x}_{i}^{(i)}\}\)  
\(v_{n}\), \(k=1,\,\ldots,\,n\) - the eigenvectors of \(\mathbf{S}_{g}\)  
\(\lambda_{n}\), \(k=1,\,\ldots,\,n\) - the eigenvalues, arranged in a descending order, \(\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{n}\geq 0\)  
Consider the expansion: \(\mathbf{x}_{z}^{(i)}=\mathbf{x}_{n}+\sum_{k=1}^{n}b_{k}\mathbf{v}_{k}:\) where \(b_{k}\) are unique expansion coefficients. We also define the center point: \(\mathbf{x}_{i}=\mathbf{x}_{n}+\{\mathbf{v}_{1}\,v_{2}\,\ldots\,v_{n}\}|\mathbf{b}_{0}\) with:  
\(\mathbf{b}_{0}=\left[\begin{array}{c}b_{1,0}\\ \vdots\\ b_{n,0}\end{array}\right]=\left[\begin{array}{c}b_{1,0}\\ \vdots\\ b_{n,0}\end{array}\right]=\left[\begin{array}{c}b_{1,0}\\ \vdots\\ b_{n,0}\end{array}\right]\)  
We also have: \(\lambda_{n}=\left[\lambda_{11}\ldots\lambda_{n}\right]^{T}=\left[\dfrac{b_{1, \,\min}+b_{1,\,\max}}{2}\ \ \...\ \ \dfrac{b_{n,\,\min}+b_{n,\,\max}}{2}\right]^{T}\)  
Model domain defined using \(p\) most dominant eigenvectors: \(\mathbf{X}_{p}=\left[\begin{array}{c}\mathbf{x}=\mathbf{x}_{c}+\sum_{k=1}^{p}(2\lambda_ {k}-1)\lambda_{n}\mathbf{v}_{k}\\ 0\leq\lambda_{k}\leq 1,\ k=1,...,p\end{array}\right]\)  
Figure 3: Procedure for surrogate domain identification with dimensionality reduction using spectral analysis of the observable set [85] (for details see [43]). Notation: \(\mathbf{S}_{g}\)—covariance matrix, \(N_{3}\)—the number of the pre-optimized designs with the lowest constraint violation; \(\{\mathbf{x}_{g}^{(i)}\}\)—the observable set; \(\mathbf{x}_{m}\)—the center of the observable set; \(\mathbf{x}_{c}\)—the center point; \(\mathbf{v}_{k}\)—the eigenvectors of \(\mathbf{S}_{g}\); \(\lambda_{k}\)—the eigenvalues of \(\mathbf{S}_{g}\); \(b_{k}\)—expansion coefficients.  
In our approach, we assume that \(N_{3}>n\) (dimensionality of the design space; see also Figure 3). The model domain \(X_{p}\) refers to the set encompassing every design \(x_{c}\)\({}^{(0)}\), \(j=1\),..., \(N_{3}\) alongside the selected p directions. In most cases, the correlation between designs \(x_{c}\)\({}^{(l)}\) is rather strong, which means that restricting the domain to the first few dominant eigenvectors is sufficient. Dimensionality reduction allows for rendering reliable models with the use of relatively small training datasets. In our experiments in Section 3, we use \(p=3\) for all verification cases.  
The surrogate model itself is set up as a kriging interpolation model [87]. The training set \(x_{B}\)\({}^{(0)}\in X_{p}\), \(j=1,\ldots,N_{4}\) is allocated with the use of Latin hypercube sampling (LHS) [88]. In Section 3, we use \(N_{4}=200\) high-fidelity training data samples, which allows us to maintain the relative RMS error in the metamodel of around a few percent.  
However, the alignment of the domain with the coordinate system axes is not preserved; therefore, the design of the experiment technique has to be slightly more involved. More specifically, the dataset is first allocated using LHS in the unity interval \(0\leq z_{j}\leq 1\), \(j=1,\ldots,k\). Subsequently, the normalized data points \(z=[z_{1}\ldots z_{n}]^{T}\) are mapped into the domain \(X_{p}\) as \(\mathbf{y}=h(z)=\mathbf{x}_{c}+\sum\limits_{k=1}^{p}{(2z_{k}-1)\lambda_{b_{k}}a_{k}}\).