esa-sceva/satcom-chunk-collection huggingface dataset:
## 3 RLBM Parallel Strategy
### Generate the Cartesian Grid  
According to the domain decomposition, each sub-domain can judge their lattice type based on the read surface mesh file of geometry, respectively, and there is no need to communicate with other MPI processes. The parallel algorithm of Cartesian grid generation is shown in Algorithm 1.  
```
1:Obtain the lattice count (noted as \(latticeCount\)) of the sub-domain based on Equations (21) and (22);
2:Each MPI process generates the initial Cartesian grid, repectively;
3:Judge the lattice type;
4:For (int \(i=0\); \(i<latticeCount\); \(i++\))
5:Product type (type[i]) of all lattice in each MPI process according to the relationship between point and geometry;
6:if(type[\(i\)] == 'boundary')
7: Compute the distance between curved boundary lattice and boundary wall [52] and obtain \(q\) by Equation (19);
```  
**Algorithm 1** Parallel algorithm of Cartesian grid generation.  
As is shown in Algorithm 1, each MPI process has to determine each lattice type in the buffer which has an effect on the parallel efficiency of Cartesian grid generation. The amount of Cartesian grid of three kinds of domain decomposition needed to be judged in each MPI process and can be described as follows:  
\[A_{1D}=\left(\frac{nLatX}{nDDX}+2\right)\times nLatY\times nLatZ, \tag{26}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**DDM** & \(nDDX\) & \(nDDY\) & \(nDDZ\) & \(\frac{nLatX}{nDDX}\) & \(\frac{nLatY}{nDDY}\) & \(\frac{nLatZ}{nDDZ}\) & **Communication Amount** \\ \hline
1D & (0, \(nLatX\)) & 1 & 1 & (0, \(nLatX\)) & 512 & 512 & 5,242,880 \\ \hline
2D & 256 & 8 & 1 & 2 & 64 & 512 & 679,936 \\
2D & 128 & 16 & 1 & 4 & 32 & 512 & 372,736 \\  & 64 & 32 & 1 & 8 & 16 & 512 & 249,856 \\ \hline  & 256 & 4 & 2 & 2 & 128 & 256 & 673,808 \\  & 128 & 8 & 2 & 4 & 64 & 256 & 355,872 \\  & 128 & 4 & 4 & 4 & 128 & 128 & 350,240 \\  & 64 & 16 & 2 & 8 & 32 & 256 & 212,288 \\
3D & 64 & 8 & 4 & 8 & 64 & 128 & 196,160 \\  & 32 & 32 & 2 & 16 & 16 & 256 & 171,264 \\  & 32 & 16 & 4 & 16 & 32 & 128 & 134,528 \\  & 32 & 8 & 8 & 16 & 64 & 64 & 124,032 \\  & 16 & 16 & 8 & 32 & 32 & 64 & 103,424 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Communication data amount among three kinds of DDMs.  
\[A_{2D}=\left(\frac{nLatX}{nDDX}+2\right)\times\left(\frac{nLatY}{nDDY}+2\right) \times nLatZ, \tag{27}\]  
\[A_{3D}=\left(\frac{nLatX}{nDDX}+2\right)\times\left(\frac{nLatY}{nDDY}+2\right) \times\left(\frac{nLatZ}{nDDZ}+2\right), \tag{28}\]  
where \(A_{1D}\), \(A_{2D}\), and \(A_{3D}\) denote the amount of lattice needed to judge the lattice type in 1D, 2D, and 3D DDM, respectively. It can be observed from Table 2 that \(A_{1D}\), \(A_{2D}\), and \(A_{3D}\) are compared based on the lattice scale \(2048\times 512\times 512\). Different DDMs have similar computational complexities; the difference depends on the buffers of different domain decomposition. When \(\frac{nLatX}{nDDX}\), \(\frac{nLatY}{nDDY}\), \(\frac{nLatZ}{nDDZ}\) gradually become equal, 3D DDM has least amount of lattice needed to be judged.
## 3 Results and Discussion
### Effects of SGS Location on Glottal Flow Dynamics, Vocal Fold Vibration and Acoustics  
In this paragraph, we investigate the effects of SGS location on voice production. Since the above results show that SGS only shows noticeable effects on voice production when the area reduction is above 90%, the effects of SGS location are only studied on the cases with SGS area reduction of 90% and 96%. Two SGS distances of 3.13 and 2.1 cm were employed in the simulations with the varied SGS severity. Figure 12 compares the glottal flow rate waveform, glottal area waveform, \(f_{0}\), sound intensity, and the similarity of the vibration modes with those of the baseline case. The overall observation is that the SGS location does not change its effects. The changes in the peak flow rate, maximum glottal area, and other quantities in Figure 12, due to the change of the SGS location, are all within 2%. We have also computed other quantities including the skewness and open quotients, MFDR, mode energies, SNR, and vocal efficiency. Overall, we do not observe noticeable effects of SGS location on them. This clearly indicates that the voice outcome is insensitive to the location of SGS.
## 5 Experimental Results and Analysis
### Experimental Results and Discussion on Algorithms Comparison
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Best** & **Worst** & **Median** & **Average** & **StdDev** \\ \hline NSGAII & \(2.15\times 10^{2}\) & \(6.19\times 10^{3}\) & \(6.86\times 10^{2}\) & \(1.51\times 10^{3}\) & \(1.85\times 10^{3}\) \\ MOPSO & \(1.39\times 10^{3}\) & \(7.35\times 10^{3}\) & \(2.76\times 10^{3}\) & \(3.22\times 10^{3}\) & \(1.83\times 10^{3}\) \\ MOAO & \(1.22\times 10^{3}\) & \(1.06\times 10^{4}\) & \(2.93\times 10^{3}\) & \(3.78\times 10^{3}\) & \(2.47\times 10^{3}\) \\ MOHBA & \(6.23\times 10^{2}\) & \(1.17\times 10^{4}\) & \(2.99\times 10^{3}\) & \(4.38\times 10^{3}\) & \(3.71\times 10^{3}\) \\ MOWOA & \(1.39\times 10^{3}\) & \(9.61\times 10^{3}\) & \(3.94\times 10^{3}\) & \(4.54\times 10^{3}\) & \(2.63\times 10^{3}\) \\ MOGWO & \(1.75\times 10^{3}\) & \(9.40\times 10^{3}\) & \(2.96\times 10^{3}\) & \(3.80\times 10^{3}\) & \(2.10\times 10^{3}\) \\ MOFECO & \(1.62\times 10^{2}\) & \(3.90\times 10^{2}\) & \(3.15\times 10^{2}\) & \(2.90\times 10^{2}\) & \(5.98\times 10^{1}\) \\ LSMOFECO & \(9.59\times 10^{1}\) & \(2.76\times 10^{2}\) & \(1.71\times 10^{2}\) & \(1.89\times 10^{2}\) & \(5.92\times 10^{1}\) \\ MOFECO-MS & \(4.91\times 10^{1}\) & \(2.11\times 10^{2}\) & \(1.11\times 10^{2}\) & \(1.20\times 10^{2}\) & \(4.68\times 10^{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 27: Statistical results of SP indicator of the eight comparison algorithms for \(ch150\_n745\). Figure 11: Plots of the non-dominated solutions in the objective space of \(ch150\_n149\) obtained by nine algorithms in 15 independent runs (Pareto front obtained in each run are denoted by each color). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Best** & **Worst** & **Median** & **Average** & **StdDev** \\ \hline NSGAII & \(2.29\times 10^{7}\) & \(1.17\times 10^{7}\) & \(1.70\times 10^{7}\) & \(1.70\times 10^{7}\) & \(3.06\times 10^{6}\) \\ MOPSO & \(2.51\times 10^{7}\) & \(1.08\times 10^{7}\) & \(1.83\times 10^{7}\) & \(1.81\times 10^{7}\) & \(4.42\times 10^{6}\) \\ MOAO & \(2.59\times 10^{7}\) & \(1.48\times 10^{7}\) & \(1.82\times 10^{7}\) & \(1.91\times 10^{7}\) & \(3.44\times 10^{6}\) \\ MOHBA & \(3.09\times 10^{7}\) & \(1.06\times 10^{7}\) & \(2.03\times 10^{7}\) & \(1.95\times 10^{7}\) & \(5.22\times 10^{6}\) \\ MOWOA & \(2.24\times 10^{7}\) & \(1.41\times 10^{7}\) & \(1.70\times 10^{7}\) & \(1.72\times 10^{7}\) & \(2.37\times 10^{6}\) \\ MOGWO & \(3.23\times 10^{7}\) & \(1.19\times 10^{7}\) & \(1.79\times 10^{7}\) & \(1.92\times 10^{7}\) & \(6.13\times 10^{6}\) \\ MOFECO & \(7.34\times 10^{7}\) & \(3.39\times 10^{7}\) & \(4.73\times 10^{7}\) & \(4.88\times 10^{7}\) & \(9.50\times 10^{6}\) \\ LSMOFECO & \(1.24\times 10^{7}\) & \(7.59\times 10^{6}\) & \(9.84\times 10^{6}\) & \(1.00\times 10^{7}\) & \(1.49\times 10^{6}\) \\ MOFECO-MS & \(1.25\times 10^{7}\) & \(5.86\times 10^{6}\) & \(8.66\times 10^{6}\) & \(8.67\times 10^{6}\) & \(1.95\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 28: Statistical results of PD indicator of the eight comparison algorithms for \(ch150\_n745\).
# Predictive Maintenance for Remanufacturing Based on Hybrid-Driven Remaining Useful Life Prediction
## 3 Experiment and Result
### Results and Analysis
#### 3.3.3 Dynamic Maintenance Scheduling  
The maintenance schedule is the outcome of the proposed predictive maintenance framework. Each bearing has its own reliability function, the manager could decide the maintenance time period based on the separate reliability functions. We assume that there are four bearings on a machine, then we could obtain all overall reliability as shown in Figure 11. Considering the suitable cost of remanufacturing, the threshold period of reliability is set to \([0.25,0.4]\). Then the separate maintenance period can be calculated as shown in Table 4. Based on the separate maintenance period, we could obtain the optimized period which could reduce the planned downtime frequency from four times to three times. Compared with the optimized results, the second strategy is better because the \(Bearing1\_1\) could work more time.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \(Bearing1\_3\) & & & \(Bearing2\_6\) & \\ \hline \multirow{2}{*}{\begin{tabular}{c} **Time** \\ **Proportion** \\ \end{tabular} } & \multirow{2}{*}{**Actual RUL (s)**} & \multirow{2}{*}{\begin{tabular}{c} **Predicted RUL** \\ **(s)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Er (\%)** \\ \end{tabular} } & \multirow{2}{*}{**Actual RUL (s)**} & \multirow{2}{*}{
\begin{tabular}{c} **Predicted RUL** \\ **(s)** \\ \end{tabular} } \\  & & & & & **(s)** & \\ \hline
30\% & 16,590 & 483,400 & \(-\)2813.80 & 4870 & 472,840 & \(-\)9609.24 \\
40\% & 14,210 & 100,030 & \(-\)603.94 & 4170 & 3570 & 14.38 \\
50\% & 11,840 & 23,430 & \(-\)97.88 & 3470 & 2490 & 28.24 \\
60\% & 9460 & 9960 & \(-\)5.28 & 2770 & 2720 & 1.80 \\
70\% & 7090 & 5780 & 18.47 & 2070 & 2710 & \(-\)30.91 \\
80\% & 4710 & 3190 & 32.27 & 1370 & 3140 & \(-\)129.19 \\
90\% & 2340 & 1210 & 48.29 & 670 & 3510 & \(-\)423.88 \\ \hline \hline \end{tabular}
\end{table}
Table 2: RUL prediction results under different historical data percentages.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \
\begin{tabular}{c} **Current Time** \\ **(s)** \\ \end{tabular} & **Actual RUL (s)** &
\begin{tabular}{c} **Predicted RUL** \\ **(s)** \\ \end{tabular} & **Er (\%)** & **Er of [36] (\%)** & **Er of [38] (\%)** \\ \hline \(Bearing1\_3\) & 18,010 & 5730 & 4100 & 28.44 & \(-\)31.76 & 43.28 \\ \(Bearing2\_6\) & 5710 & 1290 & 3170 & \(-\)145.73 & \(-\)51.94 & \(-\)13.95 \\ \hline \hline \end{tabular}
\end{table}
Table 3: RUL prediction results compared with other methods.
# QRLIT: Quantum Reinforcement Learning for Database Index Tuning
## 4 QRLIT: Quantum-Classical Implementation of Classical SMARTIX Algorithm
\[\mathrm{2^{n}}=\mathrm{N}_{\mathrm{a}}\equiv\mathrm{n}=\mathrm{log}_{2}( \mathrm{N}_{\mathrm{a}})\equiv\mathrm{n}=\mathrm{log}_{2}(90)\equiv\mathrm{n }\approx 6.491\approx 7 \tag{21}\]  
\[\mathrm{N}=2^{\mathrm{number\ of\ qubits}}\equiv\mathrm{N}=2^{7}=128\]  
\[\mathrm{t}=\mathrm{int}(\tfrac{\pi}{4}\sqrt{\mathrm{N}}-\tfrac{1}{2})\equiv \mathrm{t}=\mathrm{int}(\tfrac{\sqrt{128}\pi}{4}-\tfrac{1}{2})\equiv \mathrm{t}=\mathrm{int}(8.386)=8 \tag{22}\]  
As identified before, to create the QRLIT, we replace the Epsilon-greedy strategy in the classical algorithm with [PERSON]'s search (line 6 in Algorithm 1). Figure 2 illustrates the interactions between the principal components of QRLIT. The agent component initiates the first interaction through the execution of [PERSON]'s algorithm, which returns the action a. In binary code, the action is converted to a decimal value and executed in the environment (line 7 in Algorithm 1). The environment then processes the value and enters a new state s/. In this new state, the benchmark is prepared by creating the necessary query instances using QGen to run the power and throughput tests. Once the benchmark has been executed, the reward r and the new state s/ of the environment are returned to the agent. With these two values, the agent calculates the number of Grover iterations L, selects the action of the new state that contains the highest Q-value, and sends these values to the operation that builds the [PERSON] algorithm circuit (line 8 in Algorithm 1). Then, the quantum circuit is constructed with all seven qubits initialized in the register in the state \(\ket{0}\). Our QRLIT proposal provides a natural balance between exploration and exploitation, allowing for more effective learning; as the agent learns and adjusts its policy, the exploration rate decreases (Equation (20)). Furthermore, given the properties of quantum parallelism and superposition states in [PERSON]'s algorithm, this proposal offers another advantage, namely that it is able to find an action faster (complexity of O(\(\sqrt{\text{N}}\))) (line 6 in Algorithm 1) than its classical counterpart (complexity of O(N)). ```
1: Random initialization of parameters \(\Theta\)
2: Empty initialization of replay memory \(D\)
3:for each episode do
4:\(s\gets DB\) initial index configuration mapped as initial state
5:for each step of episode do
6:\(a\leftarrow\) Run [PERSON] algorithm on \(s\)
7:\(st\), \(r\gets execute(a)\)
8: Build [PERSON]\({}^{\prime}\)s circuit with \(L\) and \(argmax_{a\in A}\)
9:for\(\theta_{i}\in\Theta\)do
10: Update \(\theta_{i}\) according to Equation (19)
11:endfor
12: Store experience \(e=\langle s,\ a,r,s\prime\rangle\) in \(D\)
13: Sample random mini batch of experiences \(e\sim\)\(D\)
14: Performance experience replay using sampled data
15:\(s\gets s^{\prime}\)
16:endfor
```  
**Algorithm 1** QRLIT algorithm with [PERSON]'s search, function approximation, and experience replay. Adapted from [3,34]. Figure 2: Flow diagram of QRLIT algorithm.
## 3 Validation of Analytical Solution
### Solution of Tri-Layer Thick-Walled Cylinders  
For tri-layer thick-walled cylinders, the contact radius are 33.5 mm and 70 mm, respectively; the interference of both contact pairs are 28.71 \(\upmu\)m; the outer cylinder is made of gray iron with the maximum radius of 115 mm; the medium cylinder is made of aluminum alloy; and the inner cylinder is made of steel with the minimum radius of 31 mm.  
By sequence analysis method, the contact pressure of the outer contact pair can be calculated in the analysis order of inside to outside, while the contact pressure of the inner contact pair can be calculated in the analysis order of outside to inside. Thus, two-way sequential coupling calculations are needed to get the contact pressure of the two contact surfaces. It can be seen in Section 2.5.1 and Section 2.5.2 that these calculated contact pressure are the lower limits of the actual values. The example in this section shows that the contact pressure calculated by sequence analysis is 10%-20%  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Contact Pressure** & \multicolumn{2}{c}{**FEM Solution (MPa)**} & \multicolumn{2}{c}{**Theoretical**} & \multirow{2}{*}{**Relative Error**} \\ \cline{3-4} \cline{6-6}  & & **9.8894** & & & **9.8498** \\ \hline \multirow{4}{*}{Stress \({}^{\star}\)} & Inner cylinder & \(\sigma_{\mathrm{r}}\) & \(-10.00\)\(\pm\) 0.51 & \(-9.85\) & \(1.52\%\) \\ \cline{2-5}  & \multirow{2}{*}{\(\sigma_{\mathrm{\Theta}}\)} & \(-127.55\)\(\pm\) 0.22 & \(-127.25\) & \(0.24\%\) \\ \cline{2-5}  & & \(\sigma_{\mathrm{r}}\) & \(-9.805\)\(\pm\) 0.032 & \(-9.85\) & \(-0.46\%\) \\ \cline{2-5}  & & \(\sigma_{\mathrm{\Theta}}\) & \(15.715\)\(\pm\) 0.013 & \(15.70\) & \(0.10\%\) \\ \hline \hline \end{tabular}  
* Statistical analysis of 20 nodes selected randomly on the contact surface was conducted, with the expression of mean \(\pm\)STD; only the stress component on contact surface was concerned.  
\end{table}
Table 1: Contact pressure and stress of bi-layer thick-walled cylinders by FEM and analytical method.  
lower than the actual value, shown in Table 2. In contrast, the holistic coupling analysis is proved to be a reliable method with high accuracy, the relative errors of which is less than \(\pm 0.6\%\), shown in Table 2.  
The analytical method educes results consistent with FEM for both the examples of bi-layer and tri-layer thick-walled cylinders with high accuracy, as shown in Table 1 and Table 2, and which verifies the correctness of the analytical solutions. Besides, the stress on the interference contact surface derived from FEM tends to perform circumferential fluctuation, especially for the inner cylinder. In our simple example, the stress component of the nodes selected randomly on the contact surface of the inner cylinder disperses distinctly. Further numerical analysis indicates that such dispersion depends on the mesh scale, and it is caused by the numerical error in finite element discretization, which is difficult to avoid. There are no such problems with the analytical method.
# Special Issue on Future Intelligent Transportation System (ITS) for Tomorrow and Beyond
###### Abstract
Edge computing, which is a distributed computing service that splits complicated computing tasks into multiple small components and allocates the separate tasks to local devices, is utilized for vehicular applications in ITS. Edge computing, an extension of cloud computing, has several characteristics including dynamics, mobility, low latency, and location awareness. The nodes in edge computing can implement allocation and cooperation for energy-efficient operation. Recently, various machine learning models and technologies have been applied to the transportation domain because a large amount of traffic information collected from GPS, road devices, and front board cameras is available. Deep learning, which is one of the machine learning techniques, has a significant role in computer vision-based applications. A study relevant to deep learning technique was implemented to improve the performance and efficiency of transportation safety, control, and management [10]. With computer vision-based applications, the vehicle obtains a decision-making capability depending on its surrounding objects. Convolutional neural network (CNN), which is especially effective at image classification and detection, can be applied to advanced driver assistance systems including object detection, lane departure, collision detection, traffic sign detection, and pedestrian detection. This special issue of \"Future Intelligent Transportation System for Tomorrow and Beyond\" consists of recent research achievements from the experts in the field of ITS, bringing together contributions to the enhanced transportation systems. The goal of this special issue is to encourage research and development in future ITS while addressing their validation by using experimental data as possible.
# A Multi-Input Fusion Model for Privacy and Semantic Preservation in Facial Image Datasets
## 2 Related Work
### Gan  
Currently, generative models that synthesize synthetic but natural-looking images provide novel ideas for de-identification. Generative Adversarial Networks [34] (GANs) are the most relatively well-known and commonly used model among all generative models and have paved a novel way for research on face de-identification [35; 36; 37; 38]. The components of it are two competing deep models: a generative model and a second discriminator network. [PERSON] et al. [39] put forward a novel Loss-Sensitive Generative Adversarial Network (LS-GAN). It combines the set of mathematical principles and computational techniques, which involve constraining the Lipschitz constant and measuring the rate of change of the function or model with the utilization of the idea of game theory to promote the generator to generate the most realistic samples. The authors of reference [40] introduced a novel model based on the architecture of a Variational Generative Adversarial Network (VGAN), combining the Variational Autoencoder (VAE) [41] and CGAN [42]. This approach is capable of extracting image representations that are specifically disentangled from identity information. CGAN-based PPGAN was proposed by [PERSON] et al. [43], which utilized the discriminator in the pre-trained model to output a structurally similar image, using the extracted feature space related to identity privacy. To boost the effectiveness of concealing personal information, [PERSON] et al. [44] proposed a novel GAN called SF-GAN. This method combines both a geometry synthesizer and an appearance synthesizer to construct various external mechanisms, achieving the goal of facial-related feature concealing. An au-to-GAN-based identity protection method, which lowers the data dimensionality for the database utilized to facilitate machine learning operations, was proposed in [45]. It generates deidentified data through confrontation. Reference [46] put forward FPGA, an end-to-end method. It uses an improved convolutional neural network that involves both an encoder and a decoder pathway and two discriminators, and its loss function is devised depending on the requirement of the specific scenario of service. GAN-based methods lack the powerful grasping ability for facial features; thus, each of these methods has a common drawback in that the details of the generated facial images are not perfect.
# A Review of the Evolution Characteristics and Argillization of Clay Interbeds in Rockslides
## 4 Observation Technique and Analysis Method of Clay Interbed
### Observation Technique of Clay Interbed
#### 4.1.4 X-ray Diffraction (XRD)  
X-ray diffraction (XRD) is mainly used to study the crystal structure of metals, and it is widely used to obtain the composition and content of various substances in clay interbeds. However, this is only a preliminary application of qualitative and quantitative analysis of XRD. The final data can only be used for statistical analysis and comparative analysis. Further development is needed for dynamic changes, analysis of sample structure defects and other applications under special circumstances, such as low temperature, high pressure, instantaneous, etc.
# A Hybrid Wind Power Forecasting Model with XGBoost, Data Preprocessing Considering Different NWPs
## 4 Methodology
### Autoencoders  
An autoencoder is a type of unsupervised artificial neural network (ANN), which compresses the data into a lower dimension and then reconstructs the input back. The autoencoder is trained to learn the representation of the input data in a lower dimension by focusing on the important features, getting rid of noise and redundancy. It is based on Encoder-Decoder architecture, as shown in Figure 3, where an encoder encodes the high-dimensional data and reconstructs the original high-dimensional data. Moreover, the autoencoders are feed forward deep learning models, which represents the learning flexibility [24; 25; 26].  
The structure of an autoencoder is similar to that of a feedforward neural network (Multi-Layer Perceptron). In terms of its simplest form, an autoencoder uses hidden layers to recreate the inputs. The algorithm of an autoencoder is described in two parts:  
Part 1: Encoder function (\(Z=f(X)\)) that converts \(X\) to \(Z\) codings.  
Part 2: Decoder function (\(X^{\prime}=g(Z)\)) that produces a reconstruction of the inputs (\(X^{\prime}\)).  
The encoder and the decoder can be defined as transitions \(\phi\) and \(\psi\), such that:  
\[\phi\,:\mathcal{X}\longrightarrow\mathcal{F} \tag{11}\]  
\[\psi\,:\mathcal{F}\longrightarrow\mathcal{X} \tag{12}\]  
\[\phi,\psi=argmin_{\phi,\psi}||X-(\phi\circ\psi)X||^{2}. \tag{13}\]  
In the simplest case, given one hidden layer, the encoder stage of an autoencoder takes the input \(x\in\mathbb{R}^{d}=\mathcal{X}\) and maps it to \(z\in\mathbb{R}^{p}=\mathcal{F}\):  
\[z=\sigma(Wx+b). \tag{14}\]  
This image \(z\) is usually referred to as code. Hence, an element-wise activation function \(\sigma\), for example, sigmoid or rectified linear unit, a weight matrix \(W\), and a bias vector \(b\). Weight and bias are normally initialized randomly and then updated iteratively during training through back-propagation. Then, the decoder stage of the autoencoder maps \(z\) to the reconstruction \(X^{\prime}\) of the same shape as \(x\).  
To reduce the dimension of the learning model, this study has reduced the size of the code that adequately represents \(X\). An undercomplete autoencoder whose internal representation has a smaller dimensionality than the input data is represented in Figure 3.  
To learn the neuron weights, the autoencoder seeks to minimize the loss function, such as RMSE, which penalizes \(X^{\prime}\) for being dissimilar from \(X\):  
\[\text{minimize }L=f\big{(}X,X^{\prime}\big{)}. \tag{15}\]