esa-sceva/satcom-chunk-collection huggingface dataset:
## 2 Methodology
### DL Networks
#### 2.5.1 Networks and Performance Metrics  
Convolutional Neural Networks (CNN) introduced in [93] and popularized in [94], are a specific type of NN used for processing data that has a known grid-like topology [68]. These networks have also been utilized for time-series classification in conjunction with Convolutional Neural Networks (CNN) [95]. When applying a supervised algorithm the typical aim is to optimize the weights and biases \((w,b)\) and ultimately reduce the error between predicted (\(\hat{y}\)) and actual (\(y\)) values. Binary cross-entropy (BCE) loss has been utilized across many ML applications for assessing the difference between a predicted label and a true label of CNN models for classification [96; 97; 98]. This experiment aims to minimize error, ergo the cost (sometimes called loss) function utilized to optimize error of the CNN is [96]:  
\[L=\sum_{x}-(T_{x}\log P_{x}+(1-T_{x})\log(1-P_{x})) \tag{8}\]  
where, \(T\) is the true label, \(T_{x}\) is an individual element of that label, and \(P\) is the prediction of the output and \(P_{x}\) is a single element of that particular prediction.  
The confusion matrix \(C_{i,j}\) is defined in Table 2:  
Where \(C_{i,j}\) represents the \(i\) actual observations and \(j\) predictions. True Positive (TP) occurs when the sample is positive and predicted to be positive. True Negative (TN) takes place when a negative case occurs and is predicted to be negative. An occurrence of False Positive (FP) errors happens with an actual negative case predicted positive (Type I error) and False Negative (FN) error occurs when an actual case is positive and prediction is negative (Type II error). Some situations may prepare for Type I and Type II errors  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & & \multicolumn{2}{c}{**True**} \\  & & **Positive** & **Negative** \\ \hline \multirow{3}{*}{**Prediction**} & Positive & TP & FP \\  & Negative & FN & TN \\ \hline \hline \end{tabular}
\end{table}
Table 2: Confusion Matrix.  
differently. Prediction models can be 'adjusted' to reduce Type II errors which in turn could allow for more Type I errors and vice versa. In our case there is not a detrimental effect of either of these errors, so we do not penalize our prediction model to account for the errors.  
We define performance metrics from the binary confusion matrix as [99]:  
\[Accuracy=\frac{TP+TN}{TP+TN+FP+FN} \tag{9}\]  
\[Sensitivity=\frac{TP}{TP+FN} \tag{10}\]  
\[Specificity=\frac{TN}{TN+FP} \tag{11}\]
## 2 Mathematical Models of Pneumatic Flotation
### FSI Numerical Simulations
A structural analysis of the floating substrate was used to investigate the deformation between the inlet and outlet and the middle area due to the induced pressure distribution. The two sorts of deformation were depicted along the \(x\)-axis for the inlet and outlet and along the \(y\)-axis for the middle area (Figure 5). Table 1 summarizes the deformation and pneumatic pressure of the substrate. Among Types 0, A, and B, the increased air-hole density of the additional rows decreased the negative pressure, while the variation in the deformation increased. Therefore, the alternative design of Type C was employed with air-hole densities of 0.052 and 0.013 for the two rows, which represented an intermediate design between those of Types A and B. The inlet and outlet pressures for Type C were similar to those for Type A, but the difference in the deformation was decreased by more than five-fold. The following cost function (\(J\)) was proposed (each design is illustrated in Figure 6) [21]  
\[J=\int_{0}^{\infty}\Bigl{(}x^{T}Qx+u^{T}Ru\Bigr{)}dt \tag{6}\]  
where \(x\) is the minimum pressure at inlet/outlet, \(x^{T}\) is the transpose of \(x\), \(u\) is the difference in the deformation between the inlet and outlet and the middle area, \(u^{T}\) is the transpose of u, Q and R are weighting factors, and \(J\) is a cost function. Figure 4: Pressure distribution on the substrate for each air-bar design. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Geometry**} & \multicolumn{2}{c}{**Air-Hole Density**} & \multicolumn{2}{c}{**No. of**} & \multicolumn{2}{c}{**Inlet/Outlet**} & \multicolumn{2}{c}{**Deformation (mm)**} & \multirow{2}{*}{**Cost Function (\(\mathbf{l}\))**} \\ \cline{2-2} \cline{5-10}  & **Base Area** & **Additional Rows** & **Modeles** & **Maximum** & **Minimum** & **Minimum** & **Middle** & **Inlet/Outlet** & **Difference** & **(\(\mathbf{u}\) = R = 1)** \\ \hline Type 0 & 0.026 & none & 552 & 8.773 & \(-\)3.899 & 4.140 & \(-\)5.103 & 9.243 & 100.63 \\ \hline Type A & 0.026 & \begin{tabular}{c} 0.026 \\ (both rows) \\ \end{tabular} & 688 & 7.792 & \(-\)2.923 & \(-\)2.872 & 3.480 & 6.352 & 48.89 \\ \hline Type B & 0.026 & \
\begin{tabular}{c} 0.094 \\ (both row) \\ \end{tabular} & 742 & 3.899 & \(-\)1.957 & \(-\)3.660 & 4.474 & 8.134 & _69.99_ \\ \hline Type C & 0.026 &
\begin{tabular}{c} 0.052 + 0.013 \\ (two rows) \\ \end{tabular} & 622 & 8.799 & \(-\)2.912 & 0.518 & \(-\)0.659 & 1.177 & 9.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of the PSI simulations of the pneumatic pressure and deformation of the substrate in the side and middle areas as a function of the air-hole density.
## 3 The Proposed DSEC Method
### AdaBoost Algorithm  
As a widely used machine learning method, AdaBoost is employed as the classifier in this paper. With the advantages of a sound theoretical foundation and simple implementation [21], it has been applied to many classification problems in practice [22; 23].  
Since AdaBoost itself is also an ensemble learning method, the classification and regression tree (CART) is adopted as its base learner and the basic processes of the AdaBoost algorithm are described as follows:  
Step 1: Given the training set \(D=\{(\mathbf{x}_{1},y_{1})\), \((\mathbf{x}_{2},y_{2})\),..., \((\mathbf{x}_{N},y_{N})\}\), \(y_{i}\in\{-1,+1\}\), set the iteration round to \(R\).  
Step 2: Set \(r=1\), and initialize the weight of sample \(i\), \(i=1\),..., \(N\)  
\[\omega_{r}(i)=\frac{1}{N}\,. \tag{11}\]  
Step 3: Create \(D_{r}\) by randomly selecting \(N\) samples from \(D\) with the probability \(\omega_{r}(i)\).  
Step 4: Train the CART \(h_{r}\) by using \(D_{r}\).  
Step 5: Calculate the classification error on \(D\).  
\[\varepsilon_{r}=\frac{1}{N}{\sum_{i=1}^{N}I(h_{r}(\mathbf{x}_{i})\
eq y_{i})} \tag{12}\]  
Step 6: Calculate the weight \(\alpha_{r}\) of CART \(h_{r}\).  
\[\alpha_{r}=\frac{1}{2}\ln\left(\frac{1-\varepsilon_{r}}{\varepsilon_{r}}\right) \tag{13}\]  
Step 7: Update the weight distribution.  
\[\omega_{r+1}(i)=\left\{\begin{array}{l}\omega_{r}(i)\times\exp(-\alpha_{r}) \,\,if\,h_{r}(\mathbf{x}_{i})==y_{i}\\ \omega_{r}(i)\times\exp(\alpha_{r})\,\,\,\,\,\,\,\,\,\,\,Step 8: Output decision value.  
\[H(\mathbf{x}_{i})=\sum_{r=1}^{R}\alpha_{r}h_{r}(\mathbf{x}_{i}). \tag{16}\]
# Compact Ultra-Wideband Series-Feed Microstrip Antenna Arrays for IoT Communications
## 3 Results
### Five Elements Array  
The five-element microstrip array with log-periodic configuration, shown in Figure 4a has global dimensions of 14 \(\times\) 25.5 mm\({}^{2}\), and was characterized in regarding to its main operating parameters, reflection coefficient and radiation pattern.  
Figure 5 shows the comparison between the simulated and measured curves of the reflection coefficient (parameter S\({}_{11}\)). Assuming the common rule of S\({}_{11}\) \(<-10\) dB for a good impedance matching, and consequently for considering the operation bandwidth, it is possible to observe that the antenna is well adapted in a 6.91 GHz band, from 24.52 GHz to 31.43 GHz, a little higher than the simulated value (6.1 GHz).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\mathbf{D_{n}}\) (mm) & \(\mathbf{D_{1}}\) & \(\mathbf{D_{2}}\) & \(\mathbf{D_{3}}\) & \(\mathbf{D_{4}}\) & \(\mathbf{D_{5}}\) & \(\mathbf{D_{6}}\) & \(\mathbf{D_{7}}\) & \(\mathbf{D_{8}}\) & \(\mathbf{D_{9}}\) \\ \hline Array 5 & 2.44 & 1.95 & 2.59 & 2.27 & 2.63 & & & & \\ Array 7 & 2.14 & 1.94 & 1.86 & 2.40 & 2.14 & 1.86 & 1.52 & & \\ Array 9 & 3.08 & 2.66 & 2.23 & 1.98 & 2.99 & 2.52 & 2.20 & 2.05 & 1.98 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Final optimized distances \(\mathbf{D_{n}}\) between elements in the array with 5, 7, and 9 elements.  
Figure 4: Fabricated prototypes of the log-periodic microstrip arrays: (**a**) 5 elements, (**b**) 7 elements, and (**c**) 9 elements.  
Figure 5: Simulated and measured reflection coefficient of the 5 elements antenna array.  
Figure 6 shows the different radiation patterns of the antenna array, simulated and measured, for the two main and orthogonal planes, at three different frequencies within the band, 27 GHz, 28 GHz, and 29 GHz. Despite the reduced size of the antenna, which makes its connector relevant in this type of measurements (and impossible to be removed from them), it is possible to verify that both simulated and measured curves present a similar behavior, practically coincident, allowing to prove the correct functioning of the array in terms of radiation.  
The evolution of the simulated gain of the antenna as well as its total efficiency, along the frequency are shown in Figure 7. Regarding the gain, it is possible to verify that it is always higher than 8 dBi in the entire operating band of the antenna, although there is a band from 25.5 GHz to 29.5 GHz where the gain is always greater than 10 dBi, reaching its maximum point around 11 dBi. In terms of efficiency, the antenna has a total efficiency always above 70% in its operating band, mostly between 80% and 90% in the band, which demonstrates that the structure is well designed to operate at these frequencies.
## 2 Materials and Methods
### Measurement of Dielectric Properties
This aligns with the selective heating observed during system testing. To accurately measure the reflection coefficient, the VNA system is calibrated using standard materials such as air, distilled water, and a dielectric block, ensuring precise computation of \(\Gamma\). Once the reflection coefficient is measured, the complex  
Figure 2: Experimental setup for dielectric property measurements of _Cirrhinus microlepis_ tissues. The system includes a computer with Keysight N1500A software, a vector network analyzer (VNA), and an open-ended coaxial probe and tissue samples of fish scales, skin, and muscle. permittivity is derived using numerical inversion of the reflection coefficient model for the coaxial probe. Dielectric constant and dielectric loss are obtained as the real and imaginary components of \(\varepsilon\), respectively. These values are critical for understanding the material's response to electromagnetic fields, particularly in applications involving dielectric heating. The penetration depth (\(PD\)) is another key parameter in dielectric heating, representing the depth at which the wave intensity decreases to approximately 36.8% of its original value. This is calculated using the following:  
\[PD=\frac{\lambda_{0}}{2\pi e^{\mu}\sqrt{\varepsilon^{\prime}}} \tag{2}\]  
where \(\lambda_{0}\) is the wavelength in a vacuum. In this study, the \(PD\) values indicated that electromagnetic energy predominantly penetrated shallow layers (e.g., scales and skin), with minimal diffusion into muscle tissue. This supported the design choice of using a 2.45 GHz horn antenna and a 9 cm antenna-to-sample distance to maximize energy targeting of parasite-prone tissues while protecting edible muscle layers.
## 4 Results
\begin{table}
\begin{tabular}{c c c c c} \hline
**Classifier** & **ACC (\%)** & **P (\%)** & **R (\%)** & **F1 (\%)** \\ \hline SVM & 93.98 & 95.66 & 93.98 & 94.81 \\ LSTM & 95.18 & 96.05 & 95.18 & 95.61 \\ BiLSTM & 93.98 & 95.66 & 93.98 & 94.81 \\ ConvLSTM & **96.39** & **96.71** & **96.39** & **96.55** \\ \hline \end{tabular}
\end{table} 
Table 2: Weighted average classification results. \begin{table}
\begin{tabular}{c c} \hline
**Classifier** & **Classification Error (Zero-One Loss)** \\ \hline SVM & **0.0180 \(\pm\) 0.0178** \\ LSTM & 0.0241 \(\pm\) 0.0197 \\ BiLSTM & 0.0218 \(\pm\) 0.0188 \\ ConvLSTM & 0.0250 \(\pm\) 0.0210 \\ \hline \end{tabular}
\end{table} 
Table 3: Average classification error results. Figure 9: Error curves of the classifiers. features and features related to the type of mechanical ventilators rank first in both SVM- and LSTM-based classifiers.
## 3 Results
### Compatibility
#### 3.5.2 Compatibility and Issues  
Across all tested environments and configurations, no compatibility issues were encountered. Deployments on Windows 11 and Ubuntu 24.04 with pre-installed containerization tools required no additional modifications, ensuring smooth and efficient setups. The Ubuntu 18.04 setup, while necessitating the installation of the Container Toolkit and Docker, was successfully completed without any compatibility concerns. The presence or absence of the NVIDIA Container Toolkit and Docker was the sole determinant of the setup time variations observed, with no adverse effects on system compatibility. The specific versions of GPU drivers and CUDA did not materially affect the setup times, underscoring that the primary factor was the presence of necessary containerization tools.
# 3D Ear Normalization and Recognition Based on Local Surface Variation
## 4 Experiments
### Comparison and Discussion
Finally, the groups based on modified ICP-LSV algorithms obtained better performances than the groups based on traditional ICPIF algorithm did. This proves the superior performance of the proposed ICP-LSV algorithm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Matching Time (s)** & **Average Iterations** & **Recognition Rate (\%)** \\ \hline ICP & 0.08 & 18.5 & 94.46 \\ ICPIF + Curvature & 0.22 & 18.4 & 96.39 \\ ICP-LSV & 0.10 & 17.2 & 98.55 \\ \hline \hline \end{tabular}  
* ICP, Iterative Closet Point; ICPIF, a modified ICP algorithm named ICP using Invariant Features; LSV, Local Surface Variation.  
\end{table} 
Table 4: Comparison of different algorithms on UND-J2 database 1. Figure 14: Comparing recognition performance of different algorithms: (**a**) The Cumulative Matching Characteristics (CMC) curves; (**b**) The Receiver Operating Characteristic (ROC) curves. A comparison of the proposed approach with similar approaches is given in Table 5. The ear recognition system of this paper shows higher identification accuracy than other recognition systems in the table. Notice that although [PERSON]'s approach achieved a lower EER, without an initial guess of the full 3D (translation and rotation) transformation, its computational time of 5-8 s cannot be ignored for a recognition system. Although a higher EER is also obtained in [PERSON] and [PERSON]'s approach, the scale of the database used by our proposed system is different from theirs. All of the 415 subjects were tested in our proposed ear recognition system but only 404 subjects were selected in [PERSON] and [PERSON]'s experiment. Moreover, the time consumption of the feature extraction and the two-step matching was not mentioned. As such, the conclusion can be drawn that the proposed 3D ear recognition system outperforms other well-known 3D ear recognition systems. The computational time cost of the proposed system is displayed in Table 6. The time cost in the table is for reference only, as the algorithms were implemented on different hardware environments. Nevertheless, the computational complexity can be used to estimate the efficiency of these algorithms. The computational complexity of each ICP registration is denoted as \(O(Iter\times N_{\text{p}}\times\log_{2}(N_{\text{g}}))\), where _Iter_ is the iteration times, and \(N_{\text{p}}\) and \(N_{\text{g}}\) are the number of probe and gallery ear data points, respectively. Compared with other algorithms, the proposed recognition system reduces the ear data to one third the size of coarsely extracted ear region data, and the modified ICPIF algorithm converges to the minimum distance with less iterations than the traditional ICP. Moreover, there is only one step ICP matching in the recognition process. As such, this ear recognition system with its reduced computational complexity is more efficient.
## 2 Materials and Methods
### Description of the Datasets Used for the Analysis  
All the networks used in this work for object detection were trained using a set of real photographs depicting navigational signs of inland navigation. The photographs show signs on various backgrounds, including bridges, buildings, elements of coastal infrastructure, and vegetation. The collection includes intakes taken both from the water and whaves, all of which were taken during the day in various meteorological conditions. The following situations can be specified: sunny days (normal and against the sun photos), slightly cloudy days, or foggy and heavily cloudy days. A total of 1923 photos with resolutions from 2304 \(\times\) 3072 to 6000 \(\times\) 4000 were collected, on which a total of 4153 objects were marked. The image dataset used in the study includes high-resolution images in the JPEG format. The dimensions of the images range from 2304 \(\times\) 3072 to 6000 \(\times\) 4000 pixels, with an average size of 4543.7 \(\times\) 3100.4 pixels and an average aspect ratio of 1.45. File sizes range from 746.5 KB to 22 MB, with an average of 6.9 MB, reflecting the detailed visual content of the dataset. The images were taken with different cameras: Nikon D5000 (558 images), Canon PowerShot A710 IS (544), Nikon D7200 (556), DSC-RX100M3 (206), Sony Xperia Z3 Compact (23), and uTough-8010 (36). The time distribution covers the years 2008-2022, with the highest number of images recorded in 2022 (913) and 2010 (502). The images were acquired on the Oder River basin, covering the range from Szczecin to the tributary of the Nysa Luzycka River. The pictures were taken from boat and on land, often using changes in zoom to differentiate the scale of the pictures. Due to image acquisition within urbanized and non-urbanized areas, signs had various backgrounds: high and low vegetation, buildings, bridges, and often a mix of them.  
Labeled marks are in various states, from looking new to old and faded. Some have been spray-painted, and some show signs of rust. The signs are visible from different angles and in different positions, and some of them are partially covered by vegetation. Examples of photos of signs are shown in Figure 3.  
The collection of images intended for working with classifiers was created on the basis of the previous collection of photos. Labeled navigation marks were automatically cut out and then manually classified. The resulting set was subjected to argumentation using rotation, vertical and horizontal cropping in the range, the modification of contrast and brightness, and the addition of random noise. A total of 1000 images were generated for  
Figure 2: Workflow of the research process for GPU usability evaluation.  
each of the 50 mark classes. An example of the signs detected by the YOLO 11 [49] neural network is presented in Figure 4.
# A PMSG Wind Energy System Featuring Low-Voltage Ride-through via Mode-Shift Control
## 3 System under Consideration
### Currents' Reference Generation
#### 4.2.2 During LVRT Mode  
In this mode, the MPPT is seized to limit rises in grid current and the boost chopper overtakes DC-link voltage regulation. Thus, the reference for the boost converter current \(I_{b}^{*}\) is produced from the DC-link voltage regulation loop. On the other hand, according to grid regulations, reactive power should be injected during sag conditions. The amount of reactive power that should be injected is dependent on the voltage sag depth and the inverter rated current, noting zero reactive grid current before the voltage dip. Thus, the reference for the grid current \(q\)-axis \(I_{q}^{*}\) component varies, as shown in (36), in accordance with Eon-Netz grid codes [25; 48].  
\[I_{q}^{*}=\left\{\begin{array}{cc}0&0.9\leq V_{d}(p.u.)<1.1\\ -kI_{N}[1-V_{d}(p.u.)]&0.5\leq V_{d}(p.u.)<0.9\\ -I_{N}&V_{d}(p.u.)<0.5\end{array}\right. \tag{36}\]  
where \(k\) is a constant given by (37):  
\[k=\frac{\Delta I_{q}(p.u.)}{\Delta V_{d}(p.u.)}\geq 2 \tag{37}\]  
where \(I_{N}\) is the nominal grid current limited by the inverter maximum rating current, \(\Delta I_{q}\) is the change in grid current \(q\)-axis component and \(\Delta V_{d}\) is the change in grid voltage \(d\)-axis component. Thus, based on voltage sag depth, \(I_{q}^{*}\) is determined and, accordingly, the reference for the grid current \(d\)-axis component \(I_{d}^{*}\) is calculated from (38).  
\[I_{d}^{*}=\sqrt{I_{N}^{2}-I_{q}^{*2}} \tag{38}\]  
It is clear that the proposed scheme features grid current control decoupled from the DC-link voltage control during LV conditions. This can be concluded from Figure 6, where \(I_{d}^{*}\) and \(I_{q}^{*}\) are computed independently of \(V_{dc}\) loop. This improves the robustness of the applied control system during faults and prevents the transfer of noise and power ripples at the DC-link into the grid, thus preserving the grid current's high quality.
## 4 Proposed Method for Sensitivity Analysis of SORM-Based Estimates
### Complex-Step Derivative Approximation
Substituting Equations (21) and (23) into Equations (13) and (15) provides the sensitivity of the second-order probability approximation with respect to a vector of \(\mathbf{d}\) as:  
\[\
abla_{\mathbf{d}}P_{f\mathrm{SORM}}=\overline{\mathbf{A}}(\beta,\kappa_{i })+\overline{\mathbf{B}}(\beta,\kappa_{i})\cdot\frac{\Im[g(\mathbf{d}+\mathrm{ i}h\mathbf{e}_{1},\mathbf{x}*),\ldots,g(\mathbf{d}+\mathrm{i}h\mathbf{e}_{2}, \mathbf{x}*)]^{\mathrm{T}}}{h\cdot\|\
abla g(\mathbf{d},\mathbf{x}*)\mathbf{ J}_{\mathbf{u},\mathbf{x}}^{-1}(\mathbf{x}*)\|} \tag{24}\]  
\[\
abla_{\mathbf{d}}P_{f\mathrm{SORM}\,-\,\mathrm{i}}=\overline{\mathbf{C}}( \beta,\kappa_{i})+\overline{\mathbf{D}}(\beta,\kappa_{i})\cdot\frac{\Im[g( \mathbf{d}+\mathrm{i}h\mathbf{e}_{1},\mathbf{x}*),\ldots,g(\mathbf{d}+ \mathrm{i}h\mathbf{e}_{2},\mathbf{x}*)]^{\mathrm{T}}}{h\cdot\|\
abla g( \mathbf{d},\mathbf{x}*)\mathbf{J}_{\mathbf{u},\mathbf{x}}^{-1}(\mathbf{x}*)\|} \tag{25}\]  
where,\[\overline{\mathrm{A}}(\beta,\kappa_{i}) =-\varphi(\beta).\frac{\Im[g(\mathbf{d}+\mathrm{i}\mathrm{e}_{1}, \mathbf{x}^{\mathrm{e}}),\ldots,g(\mathbf{d}+\mathrm{i}\mathrm{e}_{n},\mathbf{x }^{\mathrm{e}})]^{\mathrm{T}}}{h\cdot\|\
abla g(\mathbf{d},\mathbf{x}*)\|_{ \mathbf{x}^{\mathrm{e}}}^{-1}(\mathbf{x}*)\|}.\left(\prod\limits_{i=1}^{n-1} \frac{1}{\sqrt{1+\beta\kappa_{i}}}\right) \tag{26}\] \[\overline{\mathrm{B}}(\beta,\kappa_{i}) =\Phi(-\beta).\left(\sum\limits_{j=1}^{n-1}\left(-\frac{1}{2} \bigg{(}\frac{1}{\sqrt{1+\beta\kappa_{j}}}\bigg{)}\kappa_{j}\cdot\prod\limits _{m\in\{1,\cdots,n-1\}}\frac{1}{\sqrt{1+\beta\kappa_{m}}}\right)\right)\] \[\overline{\mathrm{C}}(\beta,\kappa_{i}) =-\varphi(\beta).\frac{\Im[g(\mathbf{d}+\mathrm{i}\mathrm{e}_{1}, \mathbf{x}^{\mathrm{e}}),\ldots,g(\mathbf{d}+\mathrm{i}\mathrm{e}_{n},\mathbf{ x}^{\mathrm{e}})]^{\mathrm{T}}}{h\cdot\|\
abla g(\mathbf{d},\mathbf{x}*)\|_{ \mathbf{x}^{\mathrm{e}}}^{-1}(\mathbf{x}*)\|}.\left(\prod\limits_{i=1}^{n-1} \frac{1}{\sqrt{1+\varphi(\beta)\kappa_{i}}}\right)\] \[\overline{\mathrm{D}}(\beta,\kappa_{i}) =\Phi(-\beta).\left(\sum\limits_{j=1}^{n-1}\left(-\frac{1}{2} \bigg{(}\frac{1}{\sqrt{1+\varphi(\beta)\kappa_{j}}}\bigg{)}(-\beta+\psi(\beta) )\cdot\psi(\beta)\kappa_{j}\cdot\prod\limits_{m\in\{1,\cdots,n-1\}}\frac{1}{ \sqrt{1+\psi(\beta)\kappa_{m}}}\right)\right)\]  
The proposed method integrates the RBDO with the CS derivative approximation (RBDO/CS) to perform the sensitivity analysis of the failure probabilities estimated using [PERSON]'s asymptotic approximation (SORM), and [PERSON]'s improved correction approach (SORM-i). Figure 2 illustrates the process flow of the proposed RBDO/CS\(-\)SORM/SORM-i algorithms.