esa-sceva/satcom-chunk-collection huggingface dataset:
# Path Loss Determination Using Linear and Cubic Regression Inside a Classic Tomato Greenhouse
## 2 Materials and Methods  
### The Wireless Sensor Network  
Wireless sensor networks (WSN) are one of the most important technologies of the 21 st century [22] being an optimized form of acquisition and transmission of information [23], overcoming deployment difficulties and high installation and maintenance costs [24]. Wirelessly, a network can be quickly built automatically, using hierarchical network communication protocols and distributed algorithms. On the other hand, given that the nodes of the sensors are small in size and have a good capacity for cooperation, their deployment has a small impact on the agricultural environment. They also have other advantages, such as low energy consumption, self-organization capacity and local processing, constituting a promising platform for the implementation of monitoring systems that are increasingly used in agriculture [25; 26; 27; 28; 29; 30].  
The WSN in our study is based on the IEEE 802.15.4 standard, operating in the 2.4 GHz band, has lower width of the Fresnel zones compared to the 868, 915 MHz bands and with a faster transmission speed [6]. It is an important tool for environmental monitoring [31; 32], and its use in rural areas and in the presence of vegetation is growing exponentially [33; 34]. It establishes communication and detection infrastructures in areas where, otherwise, it would be impractical or impossible to do so [35]. However, each sensor used in the WSN has a limited scope [36]. Therefore, to efficiently plan and deploy WSNs in the presence of vegetation, it is crucial to have knowledge of the position, the level of transmission power and the propagation of the radio signal in the deployment environment [37]. The attenuation of the propagated signal increases with distance [38], and in areas of dense vegetation, such as orchards and forests, where the line of sight (LOS) between the nodes is typically non-existent, the foliage of trees may cause additional attenuation [39] by diffraction, reflection and scattering if there is no line of sight (NLOS) [40], especially due to the presence of water within the leaves and stems [41].
## 3 Outage Performance Analysis  
### Outage Probability Analysis  
The outage event takes place when R and FU cannot properly decode the \(x_{F}\) signal. Therefore, the OP of FU can be determined as  
\[P_{FU}=\Pr\bigl{(}\min\bigl{(}\gamma_{R\to x_{F}},\gamma_{FU\to x_{F}} \bigr{)}<\gamma_{FU}\bigr{)} \tag{16}\]  
where \(\gamma_{FU}=2^{\frac{2R_{F}}{1-\beta}}-1\) denotes the threshold, and \(R_{F}\) denotes the target rate.  
**Theorem 1**.: _The exact expression OP of FU is expressed as_  
\[P_{FU}=1-\frac{4\theta_{FU}e^{-2\theta_{FU}\theta\left(1-\varepsilon^{2} \right)\rho_{PB}}}{\sqrt{\Omega_{SR}\Omega_{PS}\Omega_{FI}\Omega_{PR}}}K_{1} \left(2\sqrt{\frac{\theta_{FU}}{\Omega_{SR}\Omega_{PS}}}\right)K_{1}\left(2 \sqrt{\frac{\theta_{FU}}{\Omega_{FU}\Omega_{PR}}}\right) \tag{17}\]  
Proof of Theorem 1.: It can be seen in Appendix A.  
The outage event of NU takes place when R and NU cannot properly decode the \(x_{N}\) signal. Thus, the OP of NU can be given by  
\[P_{NU}=\Pr\bigl{(}\min\bigl{(}\gamma_{R\to x_{N}},\gamma_{NU\to x_{N}} \bigr{)}<\gamma_{NU}\bigr{)} \tag{18}\]  
**Theorem 2**.: _The exact expression OP of NU is expressed as_  
\[P_{NU}=1-\frac{4\theta_{NU}\varepsilon^{-2\theta_{NU}\theta\left(1- \varepsilon^{2}\right)\rho_{PB}}}{\sqrt{\Omega_{SR}\Omega_{PS}\Omega_{NU} \Omega_{PR}}}K_{1}\left(2\sqrt{\frac{\theta_{NU}}{\Omega_{SR}\Omega_{PS}}} \right)K_{1}\left(2\sqrt{\frac{\theta_{NU}}{\Omega_{NU}\Omega_{PR}}}\right) \tag{19}\]  
**Proof of Theorem 2.** With help from (10) and (15), the OP of NU is expressed as  
\[P_{NU} =1-\Pr\left(\left|\hat{\beta}_{SR}\right|^{2}>\left(\theta\Big{(}1- \varepsilon^{2}\Big{)}\Omega_{SR}\theta_{NU}\rho_{PB}+\frac{\theta_{NU}}{\left| g_{PS}\right|^{2}}\right)\right)\] \[\times\Pr\left(\left|\hat{\beta}_{NU}\right|^{2}>\left(\theta \Big{(}1-\varepsilon^{2}\Big{)}\Omega_{NU}\theta_{NU}\rho_{PB}+\frac{\theta_{ NU}}{\left|g_{PR}\right|^{2}}\right)\right) \tag{20}\]  
where \(\theta_{NU}=\frac{\gamma_{NU}}{\theta\varepsilon^{2}\rho_{PB}\delta_{N}}\). Then, the \(P_{NU}\) in (20) can be calculated as  
\[P_{NU} =1-\int\limits_{0}^{\infty}f_{\left|g_{PS}\right|^{2}}(x)\int \limits_{0}^{\infty}f_{\left|g_{PR}\right|^{2}}(y)\] \[\times F_{\left|g_{SR}\right|^{2}}\left(\theta\Big{(}1- \varepsilon^{2}\Big{)}\Omega_{SR}\theta_{NU}\rho_{PB}+\frac{\theta_{NU}}{x} \right)dz\] \[\times F_{\left|g_{NU}\right|^{2}}\left(\theta\Big{(}1- \varepsilon^{2}\Big{)}\Omega_{NU}\theta_{NU}\rho_{PB}+\frac{\theta_{NU}}{y} \right)dy \tag{21}\]  
Similarly, in Appendix A, the exact expression OP of NU is given in (19). The proof is completed. \(\square\)
## Appendix A
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Penetration Resistance (N)**} \\ \hline  & \multicolumn{6}{c}{**Medium-Grained Sand**} & \multicolumn{6}{c}{**Fine-Grained Silica Sand**} \\ \hline  & \multicolumn{2}{c}{**1 wt\%**} & \multicolumn{2}{c}{**2 wt\%**} & \multicolumn{2}{c}{**1 wt\%**} & \multicolumn{2}{c}{**2 wt\%**} \\ \cline{2-9}  & \multicolumn{1}{c}{**M**} & \multicolumn{1}{c}{**SD**} & \multicolumn{1}{c}{**M**} & \multicolumn{1}{c}{**SD**} & \multicolumn{1}{c}{**M**} & \multicolumn{1}{c}{**SD**} & \multicolumn{1}{c}{**M**} & \multicolumn{1}{c}{**SD**} \\ \hline Polysaccharides & & & & & & & & \\ \hline Carboxymethyl cellulose & 11.5 & 2.0 & 21.1 & 3.6 & 14.9 & 3.2 & 16.4 & 2.9 \\ Corn starch & 14.2 & 2.0 & 27.4 & 4.4 & 18.0 & 4.8 & 25.4 & 8.6 \\ Sodium lignocellulosic & 1.7 & 0.2 & 4.3 & 1.3 & 14.2 & 3.1 & 21.1 & 3.1 \\ Peak starch & 8.0 & 1.2 & 20.1 & 4.3 & 10.8 & 2.2 & 14.4 & 1.1 \\ Potato starch & 4.4 & 0.6 & 16.8 & 1.3 & 10.1 & 1.7 & 19.0 & 1.6 \\ Wheat starch & 13.3 & 0.9 & 14.4 & 2.1 & 10.0 & 1.3 & 8.8 & 0.4 \\ Xantham gum & 12.4 & 5.5 & 26.5 & 3.9 & 20.7 & 5.3 & 16.0 & 2.8 \\ \hline Average & 9.4 & 4.5 & 18.7 & 7.3 & 14.1 & 3.8 & 17.3 & 4.9 \\ \hline Proteins & & & & & & & & \\ \hline Fava bean protein concentrate & 8.8 & 2.9 & 15.9 & 1.7 & 15.0 & 4.1 & 30.3 & 3.9 \\ Hen egg albumen & 2.5 & 0.7 & 5.6 & 3.0 & 10.8 & 2.4 & 13.9 & 0.6 \\ Haemoglobin protein & 2.1 & 0.5 & 10.2 & 1.7 & 6.7 & 1.9 & 11.2 & 2.7 \\ Plasma protein & 3.5 & 1.6 & 15.4 & 2.8 & 15.8 & 2.6 & 21.0 & 4.8 \\ Technical gelatin & 8.0 & 0.2 & 34.0 & 6.0 & 20.0 & 4.5 & 37.9 & 2.1 \\ Wheat protein & 3.5 & 1.2 & 13.5 & 3.8 & 9.1 & 0.9 & 18.5 & 2.9 \\ Whey protein concentrate & 2.4 & 0.5 & 2.5 & 0.7 & 10.0 & 1.0 & 15.6 & 3.2 \\ \hline Average & 4.4 & 2.6 & 13.9 & 9.5 & 12.5 & 4.3 & 21.2 & 8.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Moisture retention of medium-grained sand and fine-grained silica sand samples treated at biopolymer concentrations of 1 and 2 wt% (XG = 0.25 and 0.50 wt%). Tests were performed in triplicates (\(n=3\)), with M = mean and SD = standard deviation. On medium-grained sand the water-treated control group achieved a mean moisture retention of 6.9 wt% (SD = 1.2) and on fine-grained silica sand 2.5 wt% (SD = 0.7).
## 2 Algorithm for Optimal Deployment
### Estimate of the Communication Performance  
Real-time communication with a high data transfer rate and low bit error rate (BER) is required for a robust UWASN system. There is a limit to using a single sensor node in the UWASN because underwater communication is severely affected by the time-varying multipaths in the ocean environment. Recently, in a number of studies on underwater acoustic communication, sensor array systems have been used to achieve the high data rate with low BER [15; 16; 17]. In general, as the element spacing and the number of sensors increase, communication performance becomes enhanced. However, the spatial diversity aggravates the spatial efficiency of the UWASN system. Here, it is assumed that each sensor node in the UWASN is a vertical line array composed of three hydrophone receivers with an element spacing of 1.5 m, which corresponds to 10 \(\lambda\) (where \(\lambda\) is the acoustic wavelength) based on a frequency of 10 kHz.  
The communication sequence used for the communication performance simulation is a binary phase-shift keying (BPSK) sequence with a center frequency of 10 kHz and a bit rate of 1 kbps. To simulate the communication signals received after propagating through the multipath ocean channel, the BPSK communication sequence is convolved with the channel impulse responses simulated using the method described in Section 2.2, and then isotropic white Gaussian noise was added.  
Figure 3: Eigenray tracing results for the azimuthal angles of (**a**) 90\({}^{\circ}\) and (**b**) 270\({}^{\circ}\). Channel impulse responses as a function of arrival time for the azimuthal angles of (**c**) 90\({}^{\circ}\) and (**d**) 270\({}^{\circ}\).  
Figure 4 shows a signal processing block diagram of the SIMO system used to decode the communication data. The communication data received by each sensor component are multiplied by \(\mathrm{e}^{-i\omega t}\) to recover the baseband waveform, where \(\omega\) is the angular frequency. After that, the data were low-pass filtered and passed through an adaptive Decision feedback equalizer (DFE) [29] to compensate for the channel distortion by ISI. Recursive Least-Squares (RLS) algorithm was used to adaptively update the filter weights of the equalizer, and a forgetting factor was 0.995. The number of feedforward and feedback filter taps in the DFE is related to the delay time spread due to the multipaths in the underwater communication channel [10; 17]. In this paper, the tap numbers covering the delay time spread sufficiently were used in the equalizer process. Then, they are summed to eliminate residual ISI and increase SNR (Signal-to-Noise Ratio) [30; 31]. Finally, the communication performance is evaluated with a BER estimate.
# Correction Compensation and Adaptive Cost Aggregation for Deep Laparoscopic Stereo Matching
## 3 Materials and Methods
### Loss  
In addition to the correction loss we designed, we retained all the losses from the baseline model [21], including the photometric loss \(L_{\text{P}}\), disparity smoothness loss \(L_{\text{S}}\), and parallax attention mechanism loss \(L_{\text{PAM}}\). The difference lies in the computation of the photometric loss and disparity smoothness loss.  
During the training process, the left and right images are alternately compensated. When compensating the left image, the input for the photometric loss consists of the compensated left image and predicted left image obtained by warping the right image. On the other hand, when compensating the right image, the input for the photometric loss consists of the left image and the predicted left image obtained by warping the compensated right image. This can be formulated as follows:  
Figure 4: An illustration of the IGCS. Interleaved group convolution, compared to conventional group convolution, introduces two additional stages: feature interleaving and feature rollback. These stages are located before and after the conventional group convolution, respectively.  
Figure 5: An illustration of the DCS. All features learned by APABs are propagated to subsequent modules using either concatenation or summation.  
\[L_{\text{P}}=\frac{1}{N}\sum_{p}\alpha\frac{1-\mathcal{S}(\mathbf{I}(p),\mathbf{I}( p))}{2}+(1-\alpha)\|\mathbf{I}(p)-\mathbf{I}(p)\|_{1}, \tag{8}\]  
where \(\mathbf{I}=\mathcal{W}(\mathbf{I}_{left},\mathbf{O}_{l})\), \(\mathbf{I}=\mathcal{W}(\mathbf{I}_{right},\mathbf{D}_{refined})\) or \(\mathbf{I}=\mathbf{I}_{left}\), \(\mathbf{I}=\mathcal{W}(\mathcal{W}(\mathbf{I}_{right},\mathbf{O}_{r}), \mathbf{D}_{refined})\), depending on which side of the image has been compensated. \(\mathbf{I}_{left}\) and \(\mathbf{I}_{right}\) denote the left and right input images, respectively. \(\mathcal{W}\) is a warping operation and \(\mathcal{S}\) is the structural similarity metric (SSIM) function. \(p\) denotes a pixel and \(N\) denotes the number of pixels. \(\alpha\) is a regulatory factor, set to 0.85.  
Similarly, if the left image is compensated, the input image for computing the disparity smoothness loss should be the compensated image. Otherwise, it is the original image. The smoothness loss can be formulated as follows:  
\[L_{\text{S}}= \frac{1}{N}\sum_{p}(\|\
abla_{x}\mathbf{D}_{refined}(p)\|_{1}e^ {-\|\
abla_{x}\mathbf{I}(p)\|_{1}}+ \tag{9}\] \[\|\
abla_{y}\mathbf{D}_{refined}(p)\|_{1}e^{-\|\
abla_{y} \mathbf{I}(p)\|_{1}}),\]  
where \(\mathbf{I}=\mathbf{I}_{left}\) or \(\mathbf{I}=\mathcal{W}(\mathbf{I}_{left},\mathbf{O}_{l})\). \(\
abla_{x}\) and \(\
abla_{y}\) denote the gradients along the \(x\) and \(y\) axes, respectively.  
Finally, the total loss of our network can be expressed as  
\[L=L_{\text{P}}+\eta_{\text{S}}L_{\text{S}}+\eta_{\text{PAM}}L_{\text{PAM}}+ \eta_{\text{C}}L_{\text{C}}. \tag{10}\]  
where \(\eta\) is a weight coefficient.
# A Survey on Hardware Accelerators for Large Language Models
## 5 ASIC Accelerators
### H3D Transformer  
In 2024, a new 3D heterogeneous accelerator design was proposed for Transformer models [31]. It combines compute-in-memory (CIM) and digital tensor processing units (TPUs) to address chip area and energy consumption issues. The design uses 22 nm FeFET digital CIM chips for the high-density on-chip memory and processes MatMul tasks efficiently. It achieves 10 TOPS/W (10,000 GOPs/W) for BERT and GPT-2 models, which is 2.6\(\times\) to 3.1\(\times\) better than 7 nm TPU and FeFET memory baselines.
# Development of a Predictive Tool for the Parametric Analysis of a Turbofan Engine
## 2 Materials and Methods  
The Pratt & Whitney F100-PW229, a twin-spool, low BPR, afterburning turbofan engine that has powered combat planes, including the F-15 and F-16, was selected as the reference engine. A commercially available gas turbine simulation software, GasTurb 14, was used to create a dataset of engine performance response variables and input design parameters. After the validation of key performance parameters, like thrust and specific fuel consumption for afterburning and non-afterburning conditions, parametric analysis was carried out. Two parameters were simultaneously varied to generate a range of output variables. A total count of 24,636 data points was exported from the GasTurb 14 Parametric Studies module into Excel. The influence of every input parameter on the corresponding result parameter was visualized in graph form. The effects of the Mach number, fan pressure ratio, altitude, turbine entry temperature, and bypass ratio on the specific thrust, propulsive efficiency, specific fuel consumption, and total fuel flow were investigated. The regression learning models and deep neural networks were then programmed on this dataset to predict responses for new input data.
## 6 Summary and Conclusions  
In this document, we summarize the basic relations of free-space APD receivers with an emphasis on optical LEO data downlinks. Such data reception is prone to fast and strong signal- and background-light variations due to atmospheric and mechanical effects. Furthermore, changes in atmospheric attenuation and the strength of the background light with elevation add to this dynamic parametrization. We derive a model for the optimum multiplication factor and evaluate its relation to signal power, background light, and other parameters.  
We find that the dependence of the optimum multiplication factor from the background light and data signal strength suggests that both parameters need to be measured. Accordingly, the control of the APD biasing-voltage, \(U_{R}\), is recommended to ensure optimum receiver performance in all FSO situations, in addition to its dependency on temperature.  
Figure 14: BER and \(M_{\text{opt}}\) over \(P_{BCL}\) at a \(P_{Rx}\) of 100/200/300 Ppb, for BGL from 0 to 100 nW (no \(I_{dm}\) to visualize the effect of BGL only).  
Figure 15: Measured BER performance with and without BGL for \(M_{opt}\) (\(i_{n}\) = 2.5 pA/Hz\({}^{0.5}\), \(I_{dm}\) = 9 nW, \(P_{BCL}\) = 50 nW, \(B\) = 500 MHz, \(r\) = 300 Mbps).  
With a high amount of background, the optimization of the multiplication becomes less important, and rather a fixed value can be used.  
The RFE performance also suffers from other effects, like the non-matching bandwidths of TIA, reception filters and limiters, and noise effects from the limited stability of the biasing voltage.  
One of the parameters that can be practically controlled is the limitation of the background light through denser chromatic filtering; however, the data signal's channel width (spectral broadness of data signal) sets a limit to this reduction. Furthermore, the receiver telescope's FoV can be reduced by opto-mechanic measures, with this then requiring more precise pointing and tracking regarding the signal source during communication. Other parameters lie in the sophistication of the APD itself, namely the tapering of dark current and excess noise factors.  
This research received no external funding.  
The author wants to thank [PERSON] and [PERSON] for their discussions and their assistance with the measurements. This research was partly supported by the DLR Responsive Space Cluster Competence Center (RSC3) on behalf of the German Federal Ministry of Defense (BMVg).  
The authors declare no conflict of interest.
# Evaluating the Effects of Deep Excavation on Nearby Structures Through Numerical Simulation
## 4 Scenarios Simulation Under Adjacent Building Conditions
### Effect of Changing the Distance Factor of Adjacent Buildings
#### 4.1.3. HSS Model  
This section uses the HSS model to analyze the effect of varying distances between the adjacent building and the diaphragm wall, comparing 1 He, 2 He, and 3 He distances. The fixed loads were 5 F (91 kPa) and 12 F (195 kPa), and basement depths were 0 He, 0.25 He, 0.5 He, 0.75 He, and 1.0 He. Figure 12 shows that at 1 He, the maximum wall deformation was 15.5 mm with a 5 F load and 16.2 mm with 12 F load. At 2 He, it was 14.29 mm with a 5 F load and 14.99 mm with 12 F load. At 3 He, the deformation was 14 mm with a 5 F load and 14.47 mm with a 12 F load. This indicates that distances of 1 He have a greater impact on deformation, while distances beyond 2 He show reduced impact. These findings suggest that if an adjacent building is within 2 He of the site, the excavation's impact on it can be estimated.  
Figure 11. Comparison of maximum deformation under different loads (HS model): (**a**) 5F; (**b**) 12F.  
Figure 12. _Cont._
# A Terminal-Oriented Distributed Traffic Flow Splitting Strategy for Multi-Service of V2X Networks
## 2 Related Works
### Analytic Hierarchy Process Based on Service Distribution Weight  
The analytical hierarchy process (AHP) is a multi-attribute decision making method that combines qualitative analysis with quantitative analysis. Its basic idea is to decompose complex problems into several levels and elements, make simple comparisons and judgments between those elements at the same level, and finally, to derive the importance of the relationship between parameters [16].  
In the IoT (Internet of Things)-based healthcare system, the multiple deployments of medical devices in a narrow area results in signal interference, and a hierarchical interface analysis (AHP)-basednetwork interface and channel selection algorithm have been proposed. The multi-channel MAC (Media Access Control) protocol in the networked ecosystem ultimately achieves low latency and high reliability goals [17]. A priority of different QoS requirements of heterogeneous IoT applications in a fog network has been proposed by enforcing the analytical framework using an analytic hierarchy process (AHP), then, a two-sided matching game to initiate stable association between the network infrastructure (i.e., fog devices) and IoT devices is formulated. Finally, the externalities in the matching game that occur due to job delay, along with the solution of the network resource allocation problem by applying a \"best fit\" resource allocation strategy during matching, are verified [18].  
The AHP-based traffic flow offloading method needs to clarify the weight of each service, so that each service has a focus on the choice of network access. However, this scheme has its shortcomings. The AHP algorithm primarily allocates the weight of each service, but does not allocate the priority of each service. Therefore, it cannot meet the adaptive capability of service offloading in some scenarios, i.e., automatic network access and shunt control cannot be achieved.
## 1 Introduction and Literature Review
These domains include both controlled features influenced by government responses and decisions (such as school closures, restrictions on community mobility and transportation, workplace closures, and facial covering) and exogenous uncontrolled features (such as geographical indicators, population demographics, or citizens behavior). This paper aims to measure the correlation between a large number of features monitored daily for significant countries to analyze pre-vaccine government strategies and decisions and their ranked effectiveness against new confirmed COVID-19 cases (1) and deaths (2). We refer to (1) and (2) as responses and endpoints of this study. A comparative study to measure the effects of demographic, geographic, and healthcare system factors, with different anti-pandemic restrictions to citizens' mobility and social places for different countries before the spread of vaccines has not been conducted yet and is hence mandatory to prepare for new COVID-19-like pandemics in the future. The subsequent sections of this paper are structured as follows: Section 2 introduces the proposed methodology for data collection, database construction, and correlation analysis comparing different periods and countries. Section 3 outlines the selected data sources for the database construction, presenting the features involved in the correlation analyses with confirmed cases and new death cases. Section 4 presents the results of the correlation analyses and discusses significant findings from a comparative study of correlation measures for different countries and periods. Section 5 demonstrates a feature ranking analysis to identify the most relevant features. Finally, Section 6 delivers a conclusive discussion encompassing conclusions and areas of interest for further research.
# A Hybrid Deep Learning Model Based on FFT-STL Decomposition for Ocean Wave Height Prediction
## 3 Experimental Results and Analysis
### Evaluation Metrics  
To assess model performance, various quantitative evaluation metrics are used. The mean absolute error (MAE) and root mean square error (RMSE) quantify the average magnitude of the prediction error in the same units as the target variable, intuitively measuring the deviation between the predicted value and the actual value. The coefficient of determination \(R^{2}\) represents the proportion of the variance of the observed data explained by the model, reflecting the goodness of fit of the model.  
The \(R^{2}\) is 0\(\sim\)1. The closer it is to 1, the stronger its explanatory capacity. In the prediction of wave height, \(R^{2}\) is used to evaluate the degree to which the model captures the actual changes in wave height, reflecting the degree of fit and prediction accuracy of the model. The mean absolute percentage error (MAPE) measures the relative error between the predicted value and the actual value. The lower the MAPE value, the smaller the difference between the predicted value and the observed value. The specific calculation formula is as follows:  
\[MAPE=\frac{1}{N}\sum_{i=1}^{N}\left|\frac{\hat{x}_{i}-\hat{x}_{i}}{x_{i}}\right| \tag{14}\]  
\[R^{2}=1-\frac{\sum_{i=1}^{N}(x_{i}-\hat{x}_{i})^{2}}{\sum_{i=1}^{N}(x_{i}- \hat{x}_{i})^{2}} \tag{15}\]  
where \(x_{t}\) refers to the true value (actual observation) at time \(t\). \(\hat{x}_{t}\) is the value predicted at time \(t\). \(\overline{x}\) refers to the mean of the true values.