esa-sceva/satcom-chunk-collection huggingface dataset:
## 3 Results
### London Dispersive Surface Energy of Modified Copolymers
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Copolymer** & **Equation of \(\gamma_{s}^{d}(T)\)** & \(R^{2}\) & \(-\varepsilon_{s}^{d}\)=\(\gamma_{s}^{d}/dT\) & \(\gamma_{s}^{d}(T=0\) K) & \(\gamma_{s}^{d}(T=298.15\) K) & \(T_{Max}\) \\  & **(m/m\({}^{2}\))** & **(m) m\({}^{-2}\) K\({}^{-1}\))** & **(m/m\({}^{2}\))** & **(m/m\({}^{2}\))** & **(m/m\({}^{2}\))** & **(K)** \\ \hline S-DVB & \(\gamma_{s}^{d}(T)=-1.105\) T & & & & & \\  & + 587.13 & 0.9980 & 0.835 & 587.13 & 257.7 & 531.4 \\ \hline
1\% Melamine/S- & \(\gamma_{s}^{d}(T)=-0.907\) T & & & & & \\ DVB & + 496.21 & 0.9972 & 0.907 & 496.21 & 225.79 & 547.1 \\
2\% Melamine/S- & \(\gamma_{s}^{d}(T)=-0.656\) T & & & & & \\ DVB & + 341.59 & 0.9590 & 0.656 & 341.59 & 146.12 & 521.0 \\ \hline
3\% Melamine/S- & \(\gamma_{s}^{d}(T)=-0.827\) T & & & & & \\ DVB & + 439.96 & 0.9608 & 0.827 & 439.96 & 193.51 & 532.3 \\ \hline
4\% Melamine/S- & \(\gamma_{s}^{d}(T)=-0.876\) T & 0.975 & 0.876 & 473.64 & 212.40 & 540.6 \\ DVB & + 473.64 & & & & & \\ \hline
5\% Melamine/S- & \(\gamma_{s}^{d}(T)=-0.935\) T & 0.9825 & 0.935 & 501.23 & 222.46 & 536.07 \\ DVB & + 501.23 & & & & & \\ \hline
10\% Melamine/S- & \(\gamma_{s}^{d}(T)=-1.220\) T & 0.9915 & 1.220 & 584.42 & 220.68 & 479.03 \\ DVB & + 584.42 & & & & & \\ \hline Copolymer & Equation of \(\gamma_{s}^{d}(T)\) & & & & & \\  & (m/m\({}^{2}\)) & \(R^{2}\) & \(-\varepsilon_{s}^{d}=\gamma_{s}^{d}/dT\) & \(\gamma_{s}^{d}(T=0\) K) & \(\gamma_{s}^{d}(T=298.15\) K) & \(T_{Max}\) \\
1\% HMU/S-DVB & \(\gamma_{s}^{d}(T)=-0.874\) T & & & & & \\  & + 444.68 & 0.9837 & 0.874 & 444.68 & 184.16 & 508.9 \\ \hline
3.5\% & \(\gamma_{s}^{d}(T)=-1.096\) T & 0.9886 & 1.096 & 544.23 & 217.52 & 496.7 \\ HMU/S-DVB & + 544.23 & & & & & \\ \hline
5\% HMU/S-DVB & \(\gamma_{s}^{d}(T)=-1.120\) T & 0.9973 & 1.198 & 552.77 & 218.84 & 493.6 \\  & + 552.77 & & & & & \\ \hline
10\% & \(\gamma_{s}^{d}(T)=-1.198\) T & 0.9973 & 1.198 & 582.13 & 224.95 & 485.9 \\ HMU/S-DVB & + 582.13 & & & & & \\ \hline Copolymer & Equation of \(\gamma_{s}^{d}(T)\) & \(R^{2}\) & \(-\varepsilon_{s}^{d}=\gamma_{s}^{d}/dT\) & \(\gamma_{s}^{d}(T=0\) K) & \(\gamma_{s}^{d}(T=298.15\) K) & \(T_{Max}\) \\  & (m/m\({}^{2}\)) & & & & & \\ \hline
1\% 5-FU/S-DVB & \(\gamma_{s}^{d}(T)=-0.497\) T & & & & & \\  & + 290.95 & 0.8900 & 0.497 & 290.95 & 142.77 & 585.4 \\ \hline
5\% 5-FU/S-DVB & \(\gamma_{s}^{d}(T)=-0.780\) T & & & & & \\  & + 424.67 & & & & & \\ \hline
10\% 5-FU/S-DVB & \(\gamma_{s}^{d}(T)=-0.913\) T & & & & & \\  & + 490.08 & & & & & \\ \hline \end{tabular}
\end{table} 
Table 1: Equations of \(\gamma_{s}^{d}(T)\) of S-DVB-L-285 varying the percentage of 5-hydroxy-6-methyluracil, the linear regression coefficients R\({}^{2}\), the London dispersive surface entropy \(\varepsilon_{s}^{d}\), the values of London dispersive surface energy extrapolated at 0 K and 298.15 K, and the temperature maximum \(T_{Max}\), using the Hamieh thermal model.
## Fiber Beam Element Model of A CFST Member with the Strain Rate Effect
### Strain Rate Effect
#### 2.2.2 Strain Rate Effect of Concrete  
As mentioned above, as the strain rate increases, the uniaxial tensile strength, compressive strength, and elastic modulus of concrete all increase to various degrees. Therefore, the influences of strain rate on the material properties of concrete are considered in the following aspects.  
Figure 5: Considerations of the strain-rate effect in steel subroutine.  
The equations of both tension and compression skeleton curves are functions of the strength and elastic modulus. Therefore, the influence of the strain rate on concrete can be considered in real-time by updating only the tensile strength, compressive strength, and elastic modulus in real-time, according to the actual strain rate for the loading stress of the skeleton curve. The influence of the strain rate on the compressive and tensile strengths of concrete are manifested as the dynamic compressive strength, dynamic tensile strength, and dynamic elastic modulus, according to References [46; 47; 48], as shown in Formulas (16) through (18).  
\[\frac{f_{cd}}{f_{cs}}=1.0+0.0648\lg\frac{\dot{\epsilon}_{c}}{\dot{\epsilon}_{ cs}} \tag{16}\]  
\[\frac{f_{td}}{f_{ts}}=1.0+0.057\lg\frac{\dot{\epsilon}_{t}}{\dot{\epsilon}_{ ts}} \tag{17}\]  
\[\frac{E_{c}^{d}}{E_{c}^{s}}=1.0+0.0519\lg\frac{\dot{\epsilon}_{c}}{\dot{\epsilon}_ {cs}} \tag{18}\]  
where \(f_{cd}\) and \(f_{cs}\) are the dynamic and quasi-static compressive strength, respectively. \(f_{td}\) and \(f_{ts}\) denote the dynamic and quasi-static tensile strength, respectively. \(\dot{\epsilon}_{t}\) and \(\dot{\epsilon}_{c}\) are the tensile and compressive strain rate, respectively. \(\dot{\epsilon}_{ts}\) and \(\dot{\epsilon}_{cs}\) denote the tensile and compressive quasi-static strain rate, respectively. \(\dot{\epsilon}_{c}^{d}\) and \(E_{c}^{s}\) are dynamic and quasi-static [PERSON]'s modulus, respectively.  
For the stress in the reloading branch of concrete in tension and in compression, the strain rate does not affect the existing damage of the concrete. Therefore, regardless of the variation in the loading rate, the tensile and compressive reloading stiffness are all determined by the unloading stiffness of the last skeleton curve, that is, the reloading stress is determined by the last unloading stiffness. Currently, no consistent conclusions on the influence of the strain rate on concrete strain have been reached. Therefore, the influence of the strain rate on the concrete strain is ignored in this study.
# Weak Fault Feature Extraction and Enhancement of Wind Turbine Bearing Based on OCYCBD and SVDD
## 5 Experiment Verification  
### Introduction of Experimental Platform  
The vibration signal collected from experimental platform is utilized for feasibility verification of proposed method. Figure 7 displays the experimental test site. Two SKF6025 bearings were used to support the rotating shaft, which was driven by chain wheel. The rotating speed of main shaft was 1470 rpm, viz. the rotating frequency \(fr\) = 24.5 Hz. It can be found the bearing on the right side is normal while the left side bearing is defective in Figure 7a. The injury was inserted on the roller of experimental bearing, as displayed in Figure 7b. The width and the depth of local defect were respective 0.2 mm and 1.53 mm. The bearing structure parameters are listed in Table 1. The installed PCB acceleration sensors on the bearing seat are shown in Figure 7c and the signal acquisition process is displayed in Figure 7d. During the process of bearing running, the sample frequency was set as \(fs\) = 12,800 Hz and the collected signal with length of 8192 points is intercepted for analysis. The defect feature frequencies of inner ring \(fi\), outer ring \(fo\), roller \(fe\), and cage \(fc\) of experimental bearing can be respectively calculated as \(fi\) = 132.67 Hz, \(fo\) = 87.83 Hz, \(fe\) = 115.48 Hz and \(fc\) = 9.76 Hz based on the structure parameters and the theoretical equations in literature [32].  
Figure 6: Diagnosis procedures of proposed method.
# Estimation of River Velocity and Discharge Based on Video Images and Deep Learning
## 2 Theory and Methods
### Improvement of ShuffleNetV2  
We take ShuffleNetV2_2.0 as the backbone network in this research and improve it. The overall structure of the improved network is shown in Figure 5. Firstly, the second \(1\times 1\) convolution in the main branch of both the downsampling unit and basic unit is deleted; secondly, the kernel size of all depthwise separable convolutions is increased from \(3\times 3\) to \(5\times 5\); additionally, the Bottleneck Attention Module (BAM) [21] is introduced, and it is added after the channel concatenation operation of the downsampling unit.
## 2 Materials and Methods
### Data Processing
#### 2.5.2 Signal Purity  
The signal purity \((P)\) assesses how much power of the spectral density of the measured signals is concentrated close to the \(f_{\text{in}}\). This is important to ensure that \(T\) calculations are feasible. In particular, it is considered that the \(P\) is the signal proportion of the PSD located between \(\pm 1\) Hz of the excitation frequency \(f_{\text{in}}\)[22]. Thus, \(P\) (1) can be computed as follows:  
\[P=\frac{\sum_{i=i_{-}}^{i=i_{+}}\left[S_{xi}+S_{yi}+S_{zi}\right]}{\sum_{i=1}^ {N}\left[S_{xi}+S_{yi}+S_{zi}\right]} \tag{1}\]  
where \(S_{x}\), \(S_{y}\), and \(S_{z}\) are the power spectral densities of the three components of vibration of a triaxial accelerometer; \(i\) is the index that moves along the frequency sampling; \(N\) is the block size of 2000; and \(i_{-}\) and \(i_{+}\) are defined as \(f_{i_{-}}\approx f_{\text{in}}-1\) and \(f_{i_{+}}\approx f_{\text{in}}+1\). Low \(P\) values indicate large distortion due to non-linear effects, indicating an exposure at the output position mainly governed by an excitation frequency multiple of \(f_{\text{in}}\) instead of itself \(f_{\text{in}}\). In this work, it is considered that \(P\) values below 80% imply that the result at that accelerometer is discarded; satisfactory values were considered above 80% [22; 24; 25].
## 5 Experimental Results
### Influence of Model Parameters on Classification Performance
#### 5.5.2 Influence of the Resampling Rate
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Max-Margin** & **Unsupervised Max-Margin** & **Sum-Margin** & **Unsupervised Sum-Margin** \\ \hline Car & **87.8** & 86.4 & 86.8 & 86.9 \\ Cleveland & 31.4 & 30.7 & 30.7 & **32.2** \\ Covtype.data & 52.0 & 51.7 & **52.5** & 52.0 \\ Glass & **93.4** & **93.4** & 93.2 & 93.2 \\ Hayes-roth & 84.2 & 83.5 & **85.2** & 84.5 \\ Newthyroid & **95.7** & **95.7** & 95.5 & 95.1 \\ Optdigit & 88.5 & **88.9** & 88.5 & 88.2 \\ Page-blocks & 75.5 & **76.0** & **76.0** & 75.9 \\ Penbased & 92.8 & 93.0 & 93.2 & **93.4** \\ Pendigit & 86.9 & 86.7 & **87.5** & 87.2 \\ Segment & 94.4 & 94.5 & **94.7** & 94.5 \\ Statlog & **83.0** & 82.9 & **83.0** & 82.9 \\ Urbanlandcover & 74.8 & 75.1 & 74.6 & **76.1** \\ Vehicle & 75.8 & 76.0 & **76.3** & **76.3** \\ Wilt & 86.2 & 86.3 & **86.4** & 86.3 \\ Wine & 99.1 & 99.1 & 99.1 & **99.2** \\ Winequality-red & 30.9 & 30.8 & 30.6 & **31.9** \\ Winequality-white & **29.5** & 29.3 & **29.5** & 29.1 \\ \hline Mean & 75.7 & 75.6 & 75.7 & **75.8** \\ \hline \hline \end{tabular}
\end{table} 
Table 8: F-measure of margin-based bagging involving four margins with optimal resampling range. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Max-Margin** & **Unsupervised Max-Margin** & **Sum-Margin** & **Unsupervised Sum-Margin** \\ \hline Car & 89.1 & 89.1 & 89.3 & **89.8** \\ Cleveland & **11.2** & 10.1 & 10.3 & 8.3 \\ Covtype.data & 40.1 & 39.9 & 39.9 & **40.8** \\ Glass & 80.0 & 80.0 & 80.0 & 80.0 \\ Hayes-roth & 72.8 & **73.1** & 72.6 & 70.7 \\ Newthyroid & 90.9 & 90.9 & 92.0 & **92.2** \\ Optdigit & 80.4 & **81.1** & 80.4 & **81.1** \\ Page-blocks & 90.3 & 90.4 & **90.8** & **90.8** \\ Penbased & 79.6 & 79.6 & 79.7 & **80.1** \\ Pendigit & **75.8** & 73.3 & 72.7 & 73.7 \\ Segment & 84.4 & 84.7 & **85.1** & 84.9 \\ Statlog & 70.2 & 71.0 & **71.8** & 69.8 \\ Urbanlandcover & 55.6 & 55.6 & **56.8** & **56.8** \\ Vehicle & **51.2** & 48.6 & 50.4 & 47.7 \\ Wilt & 95.5 & 95.5 & 95.5 & 95.5 \\ Wine & 98.0 & 98.0 & 98.1 & **99.0** \\ Winequality-red & 17.5 & 20.3 & **27.3** & 19.9 \\ Winequality-white & **16.0** & 14.4 & 15.2 & 13.9 \\ \hline Mean & 66.6 & 66.4 & **67.1** & 66.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Minimum accuracy per class of margin-based bagging involving four margins with optimal resampling range. Figure 6: Average accuracy with the optimal resampling rate \(a\) in margin based bagging involving four different margins for all the data sets. Figure 8: Minimum accuracy per class with the optimal resampling rate \(a\) in margin based bagging involving four different margins for all the data sets. Figure 7: F-measure with the optimal resampling rate \(a\) in margin based bagging involving four different margins for all the data sets.
# Transport-Layer Limitations for NFV Orchestration in Resource-Constrained Aerial Networks
## 4 Practical Evaluation of Transport-Layer Options
### Orchestration Traffic Performance Using TCP
These problems could be potentially solved using UDP instead, since the connection will not be interrupted (there is no connection per-se, but rather a source sending traffic to the destination) while also dealing with reliable transmission at the application level in order to avoid resending outdated/useless information since UDP does not guarantee reliability against packet losses. Figure 4: Received throughput on SUAV (4,4) using TCP as transport-layer protocol.
# Integration and Field Evaluation of an IoV System for Enhancing Road Safety
## 7 Conclusions  
In the present work, an integrated IoV system is presented for enhancing road safety by monitoring various traffic and road behavior parameters and by sending specialized warning messages in case the traffic conditions or a certain behavior of the driver is prone to lead to an accident. The system comprises dedicated hardware and software developed for the project and employs versatile technologies and communication protocols that make the system scalable for use in large road and highway/motorway infrastructures, while it can be used by virtually all types of vehicles, despite their age and connectivity capabilities. The system's technical performance and user acceptance were evaluated during integration and field tests that were carried out in an actual motorway environment with two different types of vehicles. The system's performance in all aspects was very good and showed its potential in accomplishing its goal. The authors intend to further improve the system's usability by providing more warnings for more real use cases, expand the system's connectivity features by incorporating next generation connectivity options, improve the system's autonomy and reduce its installation and maintenance cost.  
The authors declare no conflict of interest.
## 3 Results  
### Effect on Boiler Temperature Profile during Experiments  
Detailed measuring of local temperatures in different parts of an industrial boiler is a complex issue. However, also a relatively simple analysis can provide quantitative insights. Figure 3 describes the temperature in four different measuring points when the boiler was operated at a low air-to-fuel ratio (1.07), while Figure 4 describes the same temperatures when operating the boiler at a higher air to fuel ratio (1.17). These temperatures were measured with conventional thermocouples of type-K.  
Figure 3: Temperature profile over the boiler when operated on lower air-to-fuel-ratio (\(\approx\)1.07). \(T_{bottom}\) was in the range 866–877 \({}^{\circ}\)C and have been used as reference.  
In Figures 3 and 4, the following phenomena are evident:  
* The cyclone is water-cooled, so there should always be and is a temperature drop here. However, it is much more pronounced for experiments involving manganese ore than when sand is used. This suggests that fewer combustibles enter the cyclone and are burnt here.
* From the temperature difference between the bottom of the boiler and the top of the dense bed it can be seen that more heat is generated in the dense bed when manganese ore is present. Especially with 100% manganese ore with sulphur addition, there is considerable temperature increase for both high and low air-to-fuel ratio, which should be compared to slight temperature decrease when sand is used. The cases with partial substitution is somewhere in between.
* The effect on the temperature profile of adding manganese ore is larger at the lower air-to-fuel ratio than at the higher.  
Even though it should be acknowledged that the measured temperature data are only good for a qualitative analysis, Figures 3 and 4 are consistent with what could be expected. Addition of an oxygen carrying bed material is supposed to facilitate combustion in the dense part of the bed and to some extent also in the freeboard above it. This should result in a larger temperature increase in this part of the boiler compared to when sand is used. Another effect should be that fewer combustibles remain to be burned in the cyclone, leading to a sharper drop in temperature here. The effect should be more evident at lower air-to-fuel ratio than at higher.  
Figures 3 and 4 indicate that 100% manganese ore has superior ability to facilitate combustion in the boiler compared to partial substitution. This is noteworthy since the effects on CO emissions are not as clear cut, as will be explained in Section 3.2 below. These results underlines that the ultimate level of emissions are dependent not only on what happens inside the boiler, but also on final burnout which normally takes place in the vortex finder, i.e., the exit tube from the cyclone.
## 4 Results
### Pearson Correlation
#### 4.1.4 Confusion Matrix  
A confusion matrix is a fundamental tool for evaluating the performance of classification models. It provides a comprehensive view of how well a model predicts different classes. The confusion matrix is based on a binary classification problem with two classes: 0 and 1. These classes could represent various outcomes, such as pass/fail, yes/no, or true/false. The matrix is organised into four quadrants:  
* True Negatives (TN): Instances correctly predicted as class 0. These are desirable predictions--correctly identifying instances as negative (class 0);
* False Positives (FP): Instances incorrectly predicted as class 1 when they are actually class 0. These are errors--instances wrongly classified as positive (class 1) when they are not;
* False Negatives (FN): Instances incorrectly predicted as class 0 when they are actually class 1. These are also errors--instances wrongly classified as negative when they are positive;  
Figure 7: Kendall correlation.  
* True Positives (TP): Instances correctly predicted as class 1. These are desirable predictions--correctly identifying instances as positive.  
The confusion matrix shown in Figure 9 summarises the classification model's performance. It provides insights into how well the model's predictions aligned with the actual outcomes.  
1. True Negative (TN): The model correctly predicted negative instances (the actual outcome was negative). There are 997 true negatives;
2. False Positive (FP): The model incorrectly predicted positive instances (the actual negative outcome). Here, there are nine false positives;
3. False Negative (FN): The model incorrectly predicted negative instances (the actual positive outcome). The matrix shows four false negatives;
4. True Positive (TP): The model correctly predicted positive instances (the actual outcome was positive). In this matrix, there are four true positives.  
Figure 8: The training and test loss function.  
Figure 9: The confusion matrix.
# Application of Machine Learning Models to Multi-Parameter Maximum Magnitude Prediction
## 4 Results
### Future Maximum Magnitude Prediction  
In this study, the last time window ends in December 2018. We attempted to supplement the seismic data from January to December 2019 to predict the maximum magnitude of an earthquake that may occur before December 2020. The results are shown in Figure 8: The region will likely experience earthquakes with a maximum magnitude of about \(M_{\rm L}6.0\) in the future. As time progresses, moderate to strong earthquakes indeed occur, including the \(M_{\rm L}6.3\) earthquake in Changing, Sichuan, on 17 June 2019, the \(M_{\rm L}5.8\) earthquake in Gongxian, Sichuan, on 4 July 2019, and the \(M_{\rm L}5.7\) earthquake in Weiyuan, Sichuan, on 8 September 2019.
# Power Sources for Unmanned Aerial Vehicles: A State-of-the Art
## 2 Unmanned Aerial Vehicle (UAV)
This may comprise actuators, cameras, radar systems, or other specialized tools and instruments [63]. The propulsion system of a UAV plays a critical role as it consumes a significant portion of the onboard power. Its main purpose is to convert stored electrical energy into mechanical power, which propels the UAV through the motor propeller system called a propulsion unit. In fact, the propulsion system can constitute more than half of the total weight of the UAV. Figure 3 provides a typical schematic diagram of a UAV propulsion system. The power from the onboard sources is supplied to the DC bus through unidirectional and bidirectional converters, allowing for battery charging and discharging. These converters enable control over the power flow and receive control signals from the energy management system (EMS), which manages power distribution. Among the various motor types utilized in small UAVs, the brushless DC (BLDC) motor is widely preferred. This motor type offers significant advantages, including high efficiency and power density [65], as well as features such as high speed, favorable torque characteristics, reliability, ease of control via electronic speed controllers, ESCs, and long lifespan [66]. While induction m  
Figure 2: A typical UAV block diagram [64]. and robust, they have limitations such as relatively low efficiency, cooling challenges, and lower torque output.
# An Effective Surrogate Ensemble Modeling Method for Satellite Coverage Traffic Volume Prediction
## 2 Preliminary
### Kriging  
The Kriging model assumes that the true deterministic response \(y(\mathbf{x})\) is realized with a trend function and a stochastic process \(z(\mathbf{x})\). The formulation can be written as [24]  
\[y(\mathbf{x})=\mathbf{f}(\mathbf{x})^{T}\mathbf{\beta}+z(\mathbf{x}), \tag{5}\]  
where \(\mathbf{f}(\mathbf{x})\) and \(\mathbf{\beta}\) are the basis functions and regression coefficients of the trend function respectively. \(z(\mathbf{x})\) is assumed to have mean zero and covariance \(\sigma^{2}R\left(\mathbf{x}^{(i)},\mathbf{x}^{(j)}\right)\) between \(\mathbf{x}^{(i)}\) and \(\mathbf{x}^{(j)}\), where \(\sigma^{2}\) is the process variance and \(R\left(\mathbf{x}^{(i)},\mathbf{x}^{(j)}\right)\) is the correlation model. Given a set of training points, the Kriging predictor can be obtained as  
\[\hat{y}_{Kriging}=\mathbf{f}(\mathbf{x})^{T}\mathbf{\hat{\beta}}+\mathbf{r}( \mathbf{x})^{T}\mathbf{R}^{-1}\left(\mathbf{y}-\mathbf{F}\mathbf{\hat{\beta}}\right), \tag{6}\]  
where \(\mathbf{\hat{\beta}}\) can be solved by the generalized least-squares estimation  
\[\mathbf{\hat{\beta}}=\left(\mathbf{F}^{T}\mathbf{R}^{-1}\mathbf{F}^{T}\right)^{-1 }\mathbf{F}^{T}\mathbf{R}^{-1}\mathbf{y}. \tag{7}\]  
The matrix \(\mathbf{F}=\left[\mathbf{f}\left(\mathbf{x}^{(1)}\right),\mathbf{f}\left( \mathbf{x}^{(2)}\right),\ldots,\mathbf{f}\left(\mathbf{x}^{(n)}\right)\right]^ {T}\) is constructed by evaluating the basis functions \(\mathbf{f}(\mathbf{x})=\left[f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),\ldots,f_{N_{b }}(\mathbf{x})\right]\) at the training points. The correlation matrix \(\mathbf{R}\) can be constructed as  
\[\mathbf{R}=\left[\begin{array}{cccc}R\left(\mathbf{x}^{(1)},\mathbf{x}^{(1 )}\right)&\cdots&R\left(\mathbf{x}^{(1)},\mathbf{x}^{(n)}\right)\\ \vdots&\ddots&\vdots\\ R\left(\mathbf{x}^{(n)},\mathbf{x}^{(1)}\right)&\cdots&R\left(\mathbf{x}^{(n)},\mathbf{x}^{(n)}\right)\end{array}\right], \tag{8}\]  
and the stationary Gaussian correlation model \(R\left(\mathbf{x}^{(i)},\mathbf{x}^{(j)}\right)=\exp\left[-\sum\limits_{k=1}^ {m}\theta_{k}\left(\mathbf{x}_{k}^{(i)}-\mathbf{x}_{k}^{(j)}\right)^{2}\right],1\leq i,j\leq n\) is often used, where \(\mathbf{\theta}=\left[\theta_{1},\theta_{2},\ldots,\theta_{m}\right]\) are the correlation parameters. The vector \(\mathbf{r}\left(\mathbf{x}_{p}\right)\) of correlation between the training points and an unsampled point \(\mathbf{x}_{p}\) is defined as  
\[\mathbf{r}\left(\mathbf{x}_{p}\right)=\left[R\left(\mathbf{x}^{(1)},\mathbf{x }_{p}\right),\cdots,R\left(\mathbf{x}^{(n)},\mathbf{x}_{p}\right)\right]^{T}. \tag{9}\]