esa-sceva/satcom-chunk-collection huggingface dataset:
# Incorporation of UV Filters into Oil-in-Water Emulsions--Release and Permeability Characteristics
## 2 Materials and Methods  
### Preparation of Emulsions Containing UV Filters  
Four oil-in-water emulsions with UV filters were obtained: two contained avobenzene in 0.5 and 2.0% \(w/w\) and two others consisted of octyl methoxycinnamate in 0.5% and 2.0% \(w/w\). Additionally, as a reference, a formulation without sunscreen agents was prepared. The ingredients of the sunscreen formulations are collected in Table 2.  
In the first step, the aqueous phase was prepared by dissolving the xanthan gum in glycerin. Next, an appropriate amount of water was added to this mixture. Concomitantly, the ingredients of the oil phase (glycerol stearate, polyglyceryl-3 methylglucose disciterate, cetyl alcohol, shea butter, and vegetable oil) were weighed and heated to 70 \({}^{\circ}\)C. Avobenzene or octyl methoxycinnamate was introduced to the oil phase. Then, two phases were mixed and homogenized by a homogenizer (SilentCrusher M, [PERSON], Schwabach, Germany). After cooling the formulations, phenoxyethanol, used as a preservative, was added to all the emulsions. The formulations obtained were denoted as E (emulsion without sunscreen agents), E\({}_{0.5\%\text{Avo}}\), E\({}_{2\%\text{Avo}}\), E\({}_{0.5\%\text{OMC}}\), and E\({}_{2\%\text{OMC}}\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**UV Filter** & **Structure** & **State of Matter** & **MW (g/mol)** & **Area (Å2)** & **Volume (Å3)** & **Topological Polar Surface Area (Å2)** & **Log K\({}_{\text{ow}}\)1** & **MP (°C)** \\ \hline Avo & solid & 310 & 360.12 & 342.12 & 43.4 & 4.51 & 83.5 \\ \hline OMC & liquid & 290 & 367.44 & 334.87 & 35.5 & 6.10 & \(-\)68.3 \\ \hline \multicolumn{7}{c}{\({}^{1}\) log partition coefficient octanol/water.} \\ \end{tabular}
\end{table}
Table 1: Physicochemical and geometric properties of avobenzene (Avo) and octyl methoxycinnamate (OMC) (Spartan ’18. Version 1.4.5. 29 June 2020).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Ingredients** & **E** & **E\({}_{0.5\%\text{Avo}}\)** & **E\({}_{2\%\text{Avo}}\)** & **E\({}_{0.5\%\text{OMC}}\)** & **E\({}_{2\%\text{OMC}}\)** \\  & **(\% w/tw)** & **(\% w/tw)** & **(\% w/tw)** & **(\% w/tw)** & **(\% w/tw)** \\ \hline Water & 69.3 & 68.8 & 67.3 & 68.8 & 67.3 \\ \hline Glycerin & 7.5 & 7.5 & 7.5 & 7.5 & 7.5 \\ \hline Xanthan gum & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 \\ \hline Glycerol stearate & 2.0 & 2.0 & 2.0 & 2.0 & 2.0 \\ \hline Polyglyceryl-3 metylglucose disciterate & 2.0 & 2.0 & 2.0 & 2.0 & 2.0 \\ \hline Cetyl alcohol & 3.0 & 3.0 & 3.0 & 3.0 & 3.0 \\ \hline Shea butter & 7.5 & 7.5 & 7.5 & 7.5 & 7.5 \\ \hline Vegetable oil & 7.5 & 7.5 & 7.5 & 7.5 & 7.5 \\ \hline Avo/OMC & - & 0.5 & 2.0 & 0.5 & 2.0 \\ \hline Phenoxyethanol & 0.9 & 0.9 & 0.9 & 0.9 & 0.9 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Components of oil-in-water emulsions.
# Non-Destructive Study of the Microstructural Effects of Sodium and Magnesium Sulphate Attack on
## 3 Results  
### 0% Silica Fume Mortars  
The impedance spectroscopy parameter results for 0% silica fume mortars are depicted in Figure 5. In general, these results were very similar in all of the samples, regardless of the aggressive sulphate solution. A general increase in \(R_{1}\) may be observed (Figure 5a), while \(R_{2}\) remained relatively constant after an initial increase (Figure 5b) and \(C_{1}\) slightly decreased (Figure 5c). For the samples immersed in the mixed sulphate solution, \(R_{2}\) increased and \(C_{1}\) decreased after approximately 60 days. In the case of the capacitance \(C_{2}\) (Figure 5d), this parameter showed a slight decreasing tendency in all samples. On the other hand, the resistivity results (presented in Figure 6) increased with the exposure period for the majority of the specimens.  
Figure 4: Electrical resistivity measurement of a 10% silica fume sample made using the four-electrode “Wenner” method.  
Figure 5: (**a**) Variation in impedance spectroscopy resistance \(R_{1}\) in mortars with 0% silica fume; (**b**) Variation in impedance spectroscopy \(R_{2}\) in mortars with 0% silica fume; (**c**) Variation in impedance spectroscopy capacitance \(C_{1}\) in mortars with 0% silica fume; (**d**) Variation in impedance spectroscopy capacitance \(C_{2}\) in mortars with 0% silica fume.  
Figure 6: Variation in resistivity in mortars containing 0% silica fume.  
Regarding the mercury intrusion porosimetry results (see Figure 7), these suggest that the porosity was greater in the samples exposed to the sodium sulphate solution, and increased slightly with exposure time, whereas in the other solutions it remained relatively constant. The Hg retained at the end of the experiment was very similar for all the 0% silica fume mortars, and it remained practically constant during the study.  
The loss in compressive strength was greatest in the samples exposed to the 15% Na\({}_{2}\)SO\({}_{4}\) solution, as can be observed in Figure 8, and the samples exposed to the 15% MgSO\({}_{4}\) solution showed the greatest values of this parameter.
# Real-Time Localization of Epileptogenic Foci EEG Signals: An FPGA-Based Implementation
## 2 Methodology, Materials and Methods
### Dataset  
In this work, the publicly available Bern-Barcelona database [7] was used. This is an open source EEG dataset that has been used for a large number of epilepsy studies [16; 17; 18; 19; 20]. This dataset collects intracranial EEG recordings from five pharmacoresistant epilepsy patients, including two classes of EEG signals: focals and non-focals. _Focal_ signals (F) are those captured from an epileptogenic area (acquired from those channels that detected ical EEG signal changes first, as decided by at least two neurologists via visual inspection) and _non-focal_ signals (NF) are captured in channels out of this area. Each class contains 3750 pairs of simultaneously acquired signals \"x\" and \"y,\" all of them randomly selected, and consisting of 20 s windows of simultaneous recording, sampled at 512 Hz. Each focal pair consists of one of the focal EEG channels for the \(x\) signal, and one of this channel's neighboring focal channels for the \(y\) signal, both simultaneously acquired from the same patient. The non-focal pairs were selected from nonfocal EEG channels in the same way [7; 29]. All EEG signal were band-pass filtered by an fourth order Butterworth (0.5 Hz-150 Hz). In addition, before being included into the database, signal pairs were visually inspected to discard prominent measurement artifacts.  
Note that all recordings of seizure activity, and three hours after the last seizure, were excluded. Thus, this database contains neither ictal nor postictal stage activity.
## 2 Results
### Sequence Alignment and Phylogenetic Analysis  
Partial or complete _CpFIL_, _CpCRC_, _CpYABBY2_, _CpYABBY5-1_, and _CpYABBY5-2_ cDNAs contain open reading frame of 636, 519, 546, 555, and 552 bp, respectively. The predicted CpFIL, CpCRC, CpYABBY2, CpYABBY5-1, and CpYABBY5-2 proteins of 212, 173, 182, 185, and 184 amino acid residues contains a zinc-finger domain in the N-terminus and a YABBY domain in the C-terminus (Figure 3A). The putative CpFIL protein shares 89% and 82% similarity with MgFIL from _Magnolia grandiflora_ and Ny.coFIL from _Nymphaea colorata_. CpCRC protein shows 67% and 59% identity with the products of NnCRC-1/2 in _Nelumbo nucifera_ and AtCRC in _Arabidopsis_. CpYABBY2 protein shares 72% and 57% similarity with NnYAB2 from _N. nucifera_ and AtYAB2 from _Arabidopsis_, respectively. The putative CpYABBY5-1 protein shared high similarity (81%) with CpYABBY5-2, and they both showed 89% and 93% identity with the products of CsYAB5 in _Chloranthus serratus_, and 72% and 69% with that of AtYABBY5 in _Arabidopsis_ (Figure 4A and Figure S1; Table S1).  
The sequences of the five YABBYs protein homologues were aligned with the respective FIL, CRC, YABBY2, YABBY5-1, and YABBY5-2 proteins of the multiple angiosperm taxa for phylogenetic analysis. The CpYAB2 homologue clustered with non-core NnYAB2 of _Nelumbo nucifera_, DIYAB2 of _Dimocarpus longan_, Am.trYAB2 from _Amborella trichropoda_ and core AtYAB2 from _A. thaliana_. The two YABBY5 sequences from _C. praecox_, CpYAB5-1 and CpYAB5-2, formed a sister group to the basal eudicot, CsYAB5 of _Chloranthus serratus_, formed a clade YABBY5 with AtYAB5 of _A. thaliana_. The CpFIL sequence aligned near to none-core basal eudicots, MgFIL of _Magnolia grandiflora_, NjFIL of _Nubar japonica_, _Nymphaea coloration_ Ny.coFIL and _A. trichropoda_ Am.trFIL, and formed FIL clade. Together with _A. thaliana_ AtCRC, the CpCRC, _A. trichropoda_ Am.trCRC, _Oryza sativa_ OsCRC, EsCRC of _Eppelium sagittatum_, EcCRC of _Eschscholzia californica_, _N. nucifera_ NnCRC-1, and NnCRC-2 formed a clade of CRC (Figure 4B).  
Figure 3: Distribution of the nectaries in FB and OF in the _Chimonantlus praecox_ by SEM. Little white dot shown the nectaries concentrated near the axis of petals. Red line represented the axis of the petal. Red asterisks show the bottom of the petal.
## 3 Results
### Comparison of Estimation Capabilities with Different Variable Combinations  
VIF analysis was conducted to evaluate multicollinearity among the environmental variables (Table 1). The analysis revealed varying degrees of multicollinearity among the environmental variables. PAR and SR exhibited severe multicollinearity with extremely high VIF values (>100), suggesting they contain highly redundant information. Several vegetation indices and environmental parameters (NR, EVI, VPD, Temp, and NDVI) showed moderate multicollinearity with VIF values ranging from 5.17 to 8.46, slightly exceeding the  
Figure 7: Seasonal comparison of model performance in GPP estimation. The subpanels show (**a**) \(R^{2}\), (**b**) \(RMSE\) (g Cm\({}^{-2}\) d\({}^{-1}\)), and (**c**) \(MAE\) (g Cm\({}^{-2}\) d\({}^{-1}\)) metrics across four seasons: spring (March–May), summer (June–August), autumn (September–November), and winter (December–February). \(RMSE\) and \(MAE\) are expressed in g Cm\({}^{-2}\) d\({}^{-1}\).  
commonly used threshold of 5. Only LAI demonstrated low collinearity with a VIF value of 2.84. Based on these findings, we implemented strategic variable selection in subsequent modeling to minimize redundancy while preserving essential predictive information.  
The five models exhibited varying prediction accuracies on the test set when using different combinations of explanatory variables (Figure 5), as illustrated in Figure 8. Variable combination 2, which includes NDVI, NR, Temp, VPD, and LAI, yielded the optimal predictive performance, explaining an average of 73.7% of GPP variance (\(R^{2}\) = 0.737) with an \(RMSE\) of 1.52 g Cm\({}^{-2}\) d\({}^{-1}\) and an \(MAE\) of 1.16 g Cm\({}^{-2}\) d\({}^{-1}\). Compared to using all variables (combination 7), combination 2 improved the average \(R^{2}\) by 3.96% and reduced \(RMSE\) by 0.77 g Cm\({}^{-2}\) d\({}^{-1}\). Combination 5, which incorporated EVI, NR, Temp, VPD, and LAI, demonstrated the second-best performance, explaining an average of 73% of GPP variance with an \(RMSE\) of 1.54 g Cm\({}^{-2}\) d\({}^{-1}\) and an \(MAE\) of 1.17 g Cm\({}^{-2}\) d\({}^{-1}\). Variable combinations 1, 3, and 7 exhibited comparable performance, with \(R^{2}\) values ranging from 0.71 to 0.72, \(RMSE\) from 1.56 to 1.60 g Cm\({}^{-2}\) d\({}^{-1}\), and \(MAE\) from 1.19 to 1.22 g Cm\({}^{-2}\) d\({}^{-1}\). Combinations 4 and 6 showed similar average performance to combinations 1, 3, and 7; however, most individual models demonstrated inferior stability with these two variable sets. Overall, variable combination 2 demonstrated superior predictive performance and model stability compared to all other variable combinations.
# Input Shaping Control of a Flexible Structure for Rest-to-Rest and Non-Rest-to-Rest Maneuvers
## 6 Non-Rest-to-Rest Problem with Conventional ZVDD Input Shaping  
When the ZVDD shaped input is applied to a system with non-resting initial conditions, the system states at the final time \(t_{f}+3\delta t\) are shown below:  
\[\theta_{1}(t_{f}+3\delta t) = (2c_{3}^{3}\theta_{0}+6\pi c_{1}\dot{\varrho}_{2}+2c_{1}c_{3}e_{2} +2c_{1}\dot{\varrho}_{2}\sin(c_{3}(2t_{21}+f_{1}))+6\pi c_{3}^{2}\dot{\varrho} _{1}+4t_{21}c_{3}^{3}\dot{\varrho}_{1} \tag{42}\] \[+2c_{3}^{3}\dot{\varrho}_{1}f_{1}+2c_{1}c_{3}e_{2}\cos(c_{3}(2t_{ 21}+f_{1}))+2t_{21}^{2}b_{2}c_{3}^{3}u_{2}-b_{2}c_{3}^{3}f_{1}^{2}u_{2}+4t_{21} c_{1}c_{3}\dot{\varrho}_{2}\] \[+2c_{1}c_{3}\dot{\varrho}_{2}f_{1}-3\pi b_{4}c_{1}f_{1}u_{2}-3\pi b _{2}c_{3}^{2}f_{1}u_{2}+2t_{21}^{2}b_{4}c_{1}c_{3}u_{2}-b_{4}c_{1}c_{3}f_{1}^{2 }u_{2})/(2c_{3}^{3})\] \[\dot{\theta}_{1}(t_{f}+3\delta t) = (c_{1}\dot{\varrho}_{2}+c_{3}^{2}\dot{\varrho}_{1}+c_{1}\dot{ \varrho}_{2}\cos(c_{3}(2t_{21}+f_{1}))-c_{1}c_{3}e_{2}\sin(c_{3}(2t_{21}+f_{1} ))-b_{2}c_{3}^{2}f_{1}u_{2}\] (43) \[-b_{4}c_{1}f_{1}u_{2})/c_{3}^{2}\] \[\theta_{2}(t_{f}+3\delta t) = -(\dot{\varrho}_{2}\sin(c_{3}(2t_{21}+f_{1}))+c_{3}e_{2}\cos(c_{3 }(2t_{21}+f_{1})))/c_{3}\] (44) \[\dot{\theta}_{2}(t_{f}+3\delta t) = c_{3}e_{2}\sin(c_{3}(2t_{21}+f_{1}))-\dot{\varrho}_{2}\cos(c_{3 }(2t_{21}+f_{1})) \tag{45}\]  
Note that all four elements are non-zero even after using the ZVDD shaper. The impact of using the ZVDD shaper can be measured by subtracting Equations (42)-(45) from Equations (38)-(41). Clearly, if \(\dot{\varrho}_{1}\), \(e_{2}\), \(\dot{\varrho}_{2}\), and \(f_{1}\) are zero, Equations (42)-(45) will be exactly the same as Equations (29)-(33).  
When the ZVDD-shaped input is applied to the non-rest-to-rest example problem defined in Section 2, the resulting state trajectories are shown in Figure 7. In this case, unlike the rest-to-rest problem, none of the states come to rest at the final time.  
Figure 6: Non-rest-to-rest case with single-switch, bang-bang torque. The plot units are degree (\(\theta_{1}(t)\) and \(\theta_{2}(t)\)), degree/second (\(\dot{\theta}_{1}(t)\) and \(\dot{\theta}_{2}(t)\)) and N.m (\(u(t)\)).
# Internal Flow Prediction in Arbitrary Shaped Channel Using Stream-Wise Bidirectional LSTM
## 3 Experiments  
### Dataset and Pre-Processing  
A total of 5000 channel shapes were created through the process described in Section 2.1. The 5000 samples were then flipped upside down to double the quantity to 10,000. For all shapes, the samples were used as input into the NN, where pixels inside the binary array indicated the wall and fluid regions by 0 and 1, respectively. The data values obtained using the CFD tool in the domain were converted into target data by assigning the simulation results at the center position of each pixel set in the fluid region to an empty array with dimensions of 192 \(\times\) 192. As a result, we constructed a dataset with 10,000 sets of an input channel shape and corresponding target fields. For the training, validation, and test datasets, we randomly selected and divided the generated samples into 8000, 1000, and 1000 cases, respectively.  
During pre-processing, channel normalization was applied to the target field. Each pixel was divided by the standard deviation obtained from corresponding fluid regions to generate non-dimensional values. The pre-processing task is expressed as follows:  
\[u^{*}=\frac{u}{u_{std}},\quad v^{*}=\frac{v}{v_{std}},\quad P^{*}=\frac{P}{P_ {std}}. \tag{5}\]
# A Review of Research on Diagnosability of Control Systems Based on Structural Analysis
## 2 Diagnosability Architecture for Control Systems Based on Structural Analysis
### A Structural Model for Diagnosability Based on Structural Analysis
#### 2.1.4 Analytical Models and Patterns of System Behavior  
The equations \(\{E_{c_{1}},E_{c_{2}},\ldots,E_{c_{i}},\ldots\}\) of each module \(\{c_{1},c_{2},\ldots,c_{i},\ldots\}\), i.e., \(\mathbf{E}=\underset{c_{i}\in\mathbf{C}}{\cup}E_{c_{i}}\), jointly describe the control system analytical model. The variables of \(E\) can be categorized into the known variables \(\mathbf{Z}\) and the unknown variables \(\mathbf{X}\), and the domains of \(\mathbf{x}\in\mathbf{X}\) and \(\mathbf{z}\in\mathbf{Z}\) are \(\mathbb{X}\) and \(\mathbb{Z}\).  
The behavioral patterns of the modules \(\wedge_{c_{i}\in\mathbf{C}}(c_{i}=b_{c_{i}}),b_{c_{i}}\in\mathbf{B}_{c_{i}}\) jointly characterize the behavioral patterns. The system's behavioral patterns can be classified into single-fault and multi-fault behavioral patterns depending on each module's behavioral pattern. Therefore, \(e\in E\) has a corresponding behavioral pattern \(\text{assump}(e)\). \(\text{assump}:\mathbf{E}\mapsto\mathcal{P}(\mathbf{B})\) is the mapping from the equations to the behavioral patterns. The behavioral pattern determined by \(e\) can be defined as \(\text{\it sys}\in\text{assump}(e)\to e\), \(\text{assump}(e_{1})=\mu(c_{1}=NF)\).
## 2 Materials and Methods
### Image Calibration and Roi Identification  
In order to reduce the influence of illumination and dark current of the CCD detectors as well as the differences in camera and physical configuration in the imaging system, raw acquired images (\(R_{0}\)) were calibrated by two reference images, dark (\(R_{d}\)) and white (\(R_{w}\)). The dark image (with 0% reflectance) was collected by recording a spectral image after turning off the light source and fully covering the camera lens with its black cap. The white image was acquired from a standard Teflon white reference board (with 99% reflectance). Therefore, the calibrated image (\(R\)) was calculated according to the following formula:  
\[R=\frac{R_{0}-R_{d}}{R_{w}-R_{d}} \tag{1}\]  
After image acquisition and calibration, regions of interest (ROIs) in radial samples were identified. In order to extract spectra of each radial, the sub-image at 688 nm, which represents the best contrast between sample and background, was selected from the spectral space, and segregated using global thresholding as a mask to produce an image containing only the radish, avoiding any interference from the background. All images were processed and analyzed individually. The pixels in the mask were used as the region of interest (ROI) to extract spectral data from the calibrated hyperspectral image, and the mean reflectance spectrum of the hyperspectral image was calculated as an average of the spectral values of all pixels in the ROIs. All of the calibrated images were processed and analyzed by accessory image acquisition software and Environment for Visualizing Images software program (ENVI 4.7, Exelis Visual Information Solutions Inc., Redlands, CA, USA, 2010).
## 2 Materials and Methods
### Analysis of Color  
The color of infusions was evaluated by instrumental analysis as well as by a sensory panel. Instrumental color analysis was performed by using a Konica Minolta 3600d spectrophotometer (Konica-Minolta, Osaka, Japan). The color parameter of the CIE L*a*b* system was assessed using D65 illuminant and a 2\({}^{\circ}\) observation angle. Light was transmitted through a sample held in a 10 mm thick glass cuvette. L* (lightness), a* (green-red component) and b* (yellow-blue component) were recorded. Then, the total color differences (\(\Delta\)E) between infusions with and without the addition of chokeberry pomace were calculated according to the following equation [29]:  
\[\Delta E=\sqrt{\left(L_{0}^{*}-L_{CP}^{*}\right)^{2}+\left(a_{0}^{*}-a_{CP}^{* }\right)^{2}+\left(b_{0}^{*}-b_{CP}^{*}\right)^{2}} \tag{1}\]  
where \(L_{0}^{*}\), \(a_{0}^{*}\), \(b_{0}^{*}\) are the color parameters of infusions without pomace addition, and \(L_{CP}^{*}\), \(a_{CP}^{*}\), \(b_{CP}^{*}\) are the color parameters of infusions with pomace addition.  
Sensory color assessment was performed by ten panelists, in the range of 20 to 45 years old, the same number of males and females, with formal classroom training in sensory evaluation. Samples were evaluated on a 5-point scale: 1--brown, cloudy, non-accepted; 2--weakly brown, detectable clouds, weakly desirable; 3--pink, brown shadow, clear, average desirable; 4--pink towards red, clear, intensive, desirable; 5--intensive red, clear, very desirable. About 200 mL of samples were presented in transparent plastic cups (top diameter 70 mm, bottom diameter 44 mm, height 95 mm). Coded samples, about 1 h after preparation, were tested by panelist in the room intended for sensory analysis under white light [30].
## 3 Multi-Parametric Formulations
### Ambiguity Set  
In reality, the vector \(w\) is uncertain due to the inherent natures of renewable power. Combining the scenario-based stochastic optimization and robust optimization, distributionally robust optimization is employed here to optimize the expected renewable energy curtailment, considering the worst probability distribution in a neighborhood of the empirical one built by the representative scenarios.  
Given the set \(\mathcal{W}^{0}=\{w_{s}\}_{s=1}^{S}\) composed of \(S\) representative scenario and their corresponding probabilities \(\{\rho_{s}\}_{s=1}^{S}\), we first build the empirical probability distribution, \(\mathcal{P}^{0}\), by using the following:  
\[\mathcal{P}^{0}=\frac{1}{S}\sum\limits_{s=1}^{S}\delta_{\rho_{s}^{0}} \tag{12}\]  
where \(\delta_{\rho_{s}^{0}}\) is the Dirac measure on \(\rho_{s}^{0}\); thus, \(\mathcal{P}^{0}\) is discrete and supported on \(\mathcal{W}^{0}\).  
To construct a neighborhood of \(\mathcal{P}^{0}\), i.e., the ambiguity set, we need to define the distance between two probability distributions. Given an \(S\)-dimensional distribution \(\mathcal{P}\) supported on \(\mathcal{W}^{0}\), whose \(s\)-th probability is \(\rho_{s}\), the conditions \(\rho_{s}\in[0,1]\) and \(\sum\limits_{s=1}^{S}\rho_{s}=1\) are satisfied. The infinite norm-based distance between \(\mathcal{P}\) and \(\mathcal{P}^{0}\) is as follows [31]:  
\[d_{I}\Big{(}\mathcal{P},\mathcal{P}^{0}\Big{)}=\|\mathcal{P}-\mathcal{P}^{0}\| _{\infty} \tag{13}\]  
Thus, the ambiguity set based on (13) is developed as follows:  
\[\mathcal{M}_{I}^{sc}\Big{(}\mathcal{P}^{0}\Big{)} =\Big{\{}\mathcal{P}|d_{I}\Big{(}\mathcal{P},\mathcal{P}^{0}\Big{)} \leq\varepsilon,\rho\in[0,1]^{S},1^{T}\rho=1\Big{\}} \tag{14}\] \[=\left\{\begin{array}{c}\rho_{s}-\rho_{s}^{0}\leq\varepsilon, \rho_{s}^{0}-\rho_{s}\leq\varepsilon,s=1,\cdots,S\\ 0\leq\rho_{s}\leq 1,s=1,\cdots,S;\ \sum\limits_{s=1}^{S}\rho_{s}=1\end{array}\right\}\]where \(\rho=\left[\rho_{1},\cdots,\rho_{S}\right]^{T}\) and \(\mathbf{1}\) is an all-one column vector with compatible dimensions; \(\varepsilon\) is an arbitrarily given radius of the ambiguity set, indicating that the infinite norm-based distance between \(\mathcal{P}\) and \(\mathcal{P}^{0}\) does not exceed \(\varepsilon\); meanwhile, a larger \(\varepsilon\) reflects a higher model conservativeness because some worse distributions are considered. The value of \(\varepsilon\) can be chosen according to the following [31]:  
\[\varepsilon=\frac{1}{2S}\ln\frac{2S}{1-\beta} \tag{15}\]  
where \(\beta\) is the confidence level; such a choice of \(\varepsilon\) guarantees the following:  
\[\Pr\Bigl{\{}\left\|\mathcal{P}-\mathcal{P}^{0}\right\|_{\infty}\leq \varepsilon\Bigr{\}}\geq\beta \tag{16}\]  
Furthermore, \(\mathcal{M}_{I}^{\varepsilon}(\mathcal{P}^{0})\) is a polyhedron and can be expressed as \(\mathcal{M}\left(\mathcal{P}^{0}\right)=\left\{\rho|Q\rho\leq U\right\}\), where \(Q\) and \(U\) are derived from (14).