esa-sceva/satcom-chunk-collection huggingface dataset:
## 5 Finite Element Approach  
As mentioned before, for a conical frustum, which is a straight finite element, as shown in Figure 3, with two nodes and three degrees of freedom per node, the displacements and rotation \(\mathrm{u}_{\mathrm{r}}\), \(\mathrm{u}_{\mathrm{r}_{1}}\) and \(\mathrm{\beta}_{\mathrm{i}}\) were used in this work.  
The simplest interpolations were given by [PERSON] et al. [8]: \(\mathrm{N}_{\mathrm{i}}=(1+\xi_{\mathrm{i}}\xi)/2\), with \(\xi_{\mathrm{i}}=\mp 1\) and \(\xi=2\,\mathrm{s}/\mathrm{L}\). Thus, \(\mathrm{N}_{1}=(1-\xi)/2\) and \(\mathrm{N}_{2}=(1+\xi)/2\). For numerical integration, reduced integration with one Gauss point was used at the coordinate \(\xi=0\), leading to accurate results for thin and thick structures, as validated elsewhere. From previous expressions, \(\mathrm{N}_{1}=\mathrm{N}_{2}=1/2\), \(\mathrm{dN}_{1}/\mathrm{d}\xi=-1/2\), \(\mathrm{dN}_{2}/\mathrm{d}\xi=1/2\) and \(\mathrm{d}\xi/\mathrm{d}\mathrm{s}=2/\mathrm{L}\).  
The displacement field can be represented in matrix form as  
\[\mathbf{u}=\mathbf{Z}\,\left(\sum_{\mathrm{i}=1}^{2}\mathrm{N}_{\mathrm{i}} \;\mathrm{d}_{\mathrm{i}}\right)=\mathbf{Z}\;\mathbf{N}\;\mathbf{a}^{\mathrm{ e}};\;\mathbf{d}=\sum_{\mathrm{i}=1}^{2}\mathrm{N}_{\mathrm{i}}\;\mathrm{d}_{ \mathrm{i}}=\mathbf{N}\;\mathbf{a}^{\mathrm{e}} \tag{25}\]  
where \(\mathbf{a}^{\mathrm{e}}\) is the element displacement vector, \(\mathrm{d}_{\mathrm{i}}=\left\{\mathrm{u}_{\mathrm{r}_{1}}\quad\mathrm{u}_{ \mathrm{z}_{\mathrm{i}}}\quad\mathrm{\beta}_{\mathrm{i}}\right\}\) is the displacement vector of node \(\mathrm{i}\) and \(\mathrm{N}_{\mathrm{i}}\) are the shape functions. Thus, we have  
\[\left\{\begin{matrix}\mathrm{u}_{\mathrm{r}}\\ \mathrm{u}_{\mathrm{z}}\\ \mathrm{\beta}\end{matrix}\right\}=\left[\begin{matrix}\mathrm{N}_{\mathrm{i} }&0&0\\ 0&\mathrm{N}_{\mathrm{i}}&0\\ 0&0&\mathrm{N}_{\mathrm{i}}\end{matrix}\right]\left\{\begin{matrix}\mathrm{u} _{\mathrm{r}i}\\ \mathrm{u}_{\mathrm{z}_{\mathrm{i}}}\\ \mathrm{\beta}_{\mathrm{i}}\end{matrix}\right\} \tag{26}\]  
From this displacement field, the linear strains are as given as follows:  
\[\varepsilon=\sum_{\mathrm{i}=1}^{2}\mathrm{B}_{\mathrm{i}}\;\mathrm{d}_{ \mathrm{i}} \tag{27}\]  
\[\mathrm{B}_{\mathrm{i}}=\left[\begin{matrix}\frac{\mathrm{dN}_{\mathrm{i}}}{ \mathrm{d}\xi}\cos\phi&\frac{\mathrm{dN}_{\mathrm{i}}}{\mathrm{d}\xi}\sin\phi &0\\ \frac{\mathrm{N}_{\mathrm{i}}}{\mathrm{r}}&0&0\\ 0&0&-\frac{\mathrm{dN}_{\mathrm{i}}}{\mathrm{d}\xi}\\ 0&0&-\mathrm{N}_{\mathrm{i}}\frac{\cos\phi}{\mathrm{r}}\\ -\frac{\mathrm{dN}_{\mathrm{i}}}{\mathrm{d}\xi}\sin\phi&\frac{\mathrm{dN}_{ \mathrm{i}}}{\mathrm{d}\xi}\cos\phi&-\mathrm{N}_{\mathrm{i}}\end{matrix}\right]; \;\frac{\mathrm{dN}_{\mathrm{i}}}{\mathrm{d}\xi}=\frac{\mathrm{dN}_{\mathrm{i}} }{\mathrm{d}\xi}\frac{\mathrm{d}\xi}{\mathrm{d}\xi}=\mp\frac{1}{\mathrm{L}} \tag{28}\]  
where \(\phi\) is the angle represented in Figure 3.  
Figure 3: Conical frustum finite element.
## 5 Energy-Efficient Resource Matching Algorithm for D2D Relay Communications
### Three-Dimensional Matching Algorithm
#### 5.2.1 Preference Establishment  
In the three-dimensional matching, D2D TRs on one side propose to establish pairs with RC units on the other side based on their preference lists. As the ultimate optimization goal is to maximize the EE of D2D links, the preference values of D2D TR on RC units can be obtained by solving a power control problem with the optimization of transmit power \(P_{m_{t}}\) and \(P_{r}\). When calculating the preference of the same D2D TR towards another RC unit, the formulated power control problem is different from the previous one, and the derived optimum power control strategy would not be the same. Hence, the determination of optimal power control strategy cannot be handled separately from matching problems because it depends on the specific matching assumption. To establish the preference lists, each D2D TR is temporarily paired with each RC unit to obtain the optimum EE and corresponding transmit power, which is also restricted to meet the QoS requirements of both D2D and cellular links. When TR \(m\) has been matched with RB \(c_{s}\) for the second hop, the EE of TR \(m\) can be rewritten as \(E\bar{E}_{m,r,c_{f},c_{s}}\). The preference value of TR \(m\) on each RC unit in a descending order is denoted as the set \(\bar{\mathcal{E}}_{m}=\{\bar{E}_{m,1},\bar{E}_{m,2},...,\bar{E}_{m,R\times C}\}\), which is firstly calculated as the achievable EE of the two-hop link, and the corresponding RC unit is \(\bar{\mathcal{O}}_{m}=\{\bar{O}_{m,1},\bar{O}_{m,2},...,\bar{O}_{m,R\times C }\}\), \(\forall\bar{O}_{m,i}\in\mathcal{R}C\), \(i\in[1,R\times C]\), which is the preference list of TR \(m\). For a specific matching \  
\[P_{min,r}=max\left(\frac{\gamma_{min}^{D}\left(1+\gamma_{m_{r}}\right)}{\left( \gamma_{m_{r}}-\gamma_{min}^{D}\right)B},\frac{\gamma_{min}^{D}\left(P_{c_{s}}h _{c_{s},m_{r}}^{2}+N_{0}\right)}{h_{r,m_{r}}^{2}}\right). \tag{15}\]  
The upper bounds are derived based on the QoS requirements of CUEs, and the lower bounds are derived based on the QoS requirements of D2D links. \(P^{max}\) denotes the maximum transmit power allowed for any mobile equipment. \(A\) and \(B\) in (14) and (15) are given by, respectively,  
\[A=h_{m_{r},r}^{2}/\left(P_{c_{s}}h_{c_{s},m_{r}}^{2}+N_{0}\right), \tag{16}\]  
\[B=h_{r,m_{r}}^{2}/\left(P_{c_{s}}h_{c_{s},m_{r}}^{2}+N_{0}\right). \tag{17}\]  
Finally, the optimum power of D2D TX \(m_{t}\) and RS \(r\) can be expressed as:  
\[P_{m_{t}}^{*}=max\left(min\left(\bar{P}_{m_{t}},P_{max,m_{t}}\right),P_{min,m_ {t}}\right), \tag{18}\]  
\[P_{r}^{*}=max\left(min\left(\bar{P}_{r},P_{max,r}\right),P_{min,r}\right), \tag{19}\]  
respectively. Note that if the calculated upper bound is lower than the lower bound, i.e., \(P_{max,m_{t}}<P_{m_{t},m_{t}}\) or \(P_{max,r}<P_{min,r}\), then TR \(m\) cannot be matched with unit \(RC_{r,c_{f}}\).
# Quantitative Study on the Impact of Energy Consumption Based Dynamic Selfishness in MANETs +
## 6 Experimental Results
### Impact of Selfish Node Densities
The average throughput of both networks with dynamic selfish and altruistic nodes decreases with increasing node densities in the MANET (Figure 6(c)). The reason is that the number of packets successfully received by the receiver is reduced because of increasing packet loss rates. In sum, the performance of the network is significantly reduced by the increased number of selfish nodes involved in the network. The increasing node densities have negative implications on the performance of MANETs.
## 4 Conclusions  
In this paper, a novel polymer concrete composite obtained by utilizing GFRP waste is investigated by 3D X-ray computed tomography and scanning electron microscopy analysis. This multi-material was used to manufacture ornamental synthetic panels for building decoration. The methodology used for the manufacturing allows for the development of a new class of materials from composite material waste. The manufacturing process was found to be stable.  
The following conclusions are drawn:  
* The density of the new material proposed is 33.46% lower than that of the marble used as a datum model. The panel mass for the datum model (marble) was 15 kg, whereas the measured mass of the proposed material was 9.3 kg;
* The ornamental panels used for building cladding must have a pleasant appearance, yet also have the lowest possible mass and best possible resistance to external factors. A further study will investigate the mechanical characteristics of the proposed material;
* The proposed polymer concrete composite includes a large amount of GFRP waste (30%), and thus helps to protect the environment with regards to sustainable development;  
Figure 19: SEM images of the interface between composite 1 and the composite 2: (**a**) 1:1 \(\times\) 10\({}^{-4}\) m scale; (**b**) 1:1 \(\times\) 10\({}^{-5}\) m scale.  
Figure 18: SEM images of composite 2: (**a**) 1:5 \(\times\) 10\({}^{-4}\) m scale (**b**); 1:5 \(\times\) 10\({}^{-5}\) m scale.  
* The new methodology proposed for CT analysis indicated results with a mean value of 0.146% for the porosity of composite 1 and 31.3% for composite 2. The porosity of composite 2 of the proposed multi-material offers advantages for building construction, such as being a lightweight material and providing good adhesive connection abilities. The porosity of composite 1 was observed as being lower than the porosity of the marble;
* The mean porosity of a large panel can be obtained as an arithmetic mean of the determined porosities across all five samples cut from the panel, based on a sampling method;
* The homogeneous distribution of the pores within the multi-material composite for all samples was shown. Most pores detected by CT analysis were shown to be of an irregular shape, which was later confirmed by microstructure analysis;
* The CT analysis only focused on the porosity investigation. The constituents of the composite material were investigated based only on SEM analysis. The surface morphology indicates constituents in a homogenous material, where a good connection between the surface layer material and structural composite material was detected.
## 5 Concluding Remarks and Further Studies  
An M&S software design method was proposed for multi-radio integrated navigation systems as a GNSS backup under navigation warfare. The M&S software in the proposed design method consisted of a message broker and function modules. A message broker-based M&S software was designed for a multi-radio integrated navigation system. In order to show the feasibility of the proposed design method, the M&S software was implemented for GPS, KNSS, eLoran, Loran-C, and DME/VOR and the performance evaluation was carried out. The results showed that the multi-radio integrated navigation system provides navigation results continuously, even when the GPS only navigation system cannot provide navigation results, due to the jamming under a navigation warfare environment.  
In further studies, additional ground stations and fixed/moving pseudolites will be included, which are newly assigned by the coverage analysis module in the M&S software. Performance evaluations of the M&S software based on integrity and continuity will be performed.  
Conceptualization, S.H.O, H.S. and D.-H.H.; software, H.K. and J.L.; supervision, D.-H.H.; validation, H.S.; writing--original draft, H.K., J.L., S.H.O. and D.-H.H.; writing--review & editing, D.-H.H. This research was funded by the National GNSS Research Center Program of Defense Acquisition Program Administration and Agency for Defense Development. This work has been supported by the National GNSS Research Center Program of Defense Acquisition Program Administration and Agency for Defense Development. The authors declare no conflict of interest.
###### Abstract  
Detecting the Region of Interest (ROI) for video clips is a significant and useful technique both in video codecs and surveillance/monitor systems. In this paper, a new model-based detection method is designed which suits video compression codecs by proposing two models: an \"inter\" and \"intra\" model. The \"inter\" model exploits the motion information represented as blocks by global motion compensation approaches while the \"intra\" model extracts the objects details through objects filtering and image segmentation procedures. Finally, the detection results are formed through a new clustering with fine-tune approach from the \"intra\" model assisted with the \"inter\" model. Experimental results show that the proposed method fits well for real-time video codecs and it achieves a good performance both on detection precision and on computing time. In addition, the proposed method is versatile for a wide range of surveillance videos with different characteristics.  
ROI detection; global motion detection & compensation; image segmentation; decision tree; camera status classification Article
# Optimal Energy Efficiency Fairness of Nodes in Wireless Powered Communication Networks
## Appendix A  
Proof of Lemma 1.: From Theorem 1, we know that the first order derivative of \(U_{EE}\) with respect to \(\theta_{k}\) is equal to zero when \(A_{k}\) is fixed. Similarly, we first write the energy efficiency of UE in \((7)\) function with respect to \(\theta_{k}\) as follows  
\[U_{EE}\left(\theta_{k}\right)=\sum_{k=1}^{K}A_{k}\{\log\frac{W}{A_{k}N}+\log \left(\log\left(1+\theta_{k}\right)\right)+\log[\mathbb{E}\left(C_{k}\right) ]-\log\left(\bar{P}+\beta\mathbb{E}[C_{k}]\right)\}, \tag{1}\]  
Figure 7: The time cost of brute-force two-dimensional search algorithm and two-step sub-optimal algorithm with respect to the accuracy of system parameters.  
where \(\bar{P}=P_{t,k}+P_{s,k}-\eta\sum_{j=1}^{K}\frac{2\pi^{2}\lambda_{j}p_{j}^{\frac{2}{ 2}}}{\alpha\sin\left(\frac{\pi\alpha}{\alpha}\right)}\) which is independent of \(A_{k}\), \(\beta=\eta\sum_{j=1}^{K}\frac{2\pi^{2}\lambda_{j}p_{j}^{\frac{2}{2}}}{\alpha\sin \left(\frac{\pi\alpha}{\alpha}\right)}\), and \(C_{k}=\exp[-\frac{2\pi l^{2}}{\alpha-2}\sum\limits_{j=1}^{K}\lambda_{j}(\hat{p }_{jk})^{\frac{2}{2}}(\hat{B}_{jk})^{\frac{2}{2}-1}]\). For the sake of simplicity,we make \(h\left(A_{k}\right)\) a function of \(A_{k}\) as shown below  
\[h\left(A_{k}\right)=\frac{2\pi l^{2}}{\alpha-2}\sum\limits_{j=1}^{K}\lambda_{ j}(\hat{p}_{jk})^{\frac{2}{2}}(\hat{B}_{jk})^{\frac{2}{2}-1},h\left(A_{k} \right)>0. \tag{10}\]  
Then we can have that \(C_{k}=\exp\left[-h\left(A_{k}\right)\theta_{k}\right]\) and \(\log\left[\mathbb{E}\left(C_{k}\right)\right]=-h\left(A_{k}\right)\theta_{k}\). Moreover, we can also get the further simplified expression of \(U_{EE}\left(\theta_{k}\right)\) as follows  
\[U_{EE}\left(\theta_{k}\right)\] \[=\sum\limits_{k=1}^{K}A_{k}\{\log\frac{W}{A_{k}N}+\log(\log(1+ \theta_{k}))-h\left(A_{k}\right)\theta_{k}-\log[\bar{P}+\beta\exp\left(-h \left(A_{k}\right)\theta_{k}\right)]\}. \tag{11}\]  
Hence, we define the function with respect to \(\theta_{k}\) given below  
\[U_{EE,k}(\theta_{k})=A_{k}\{\log\frac{W}{A_{k}N}+\log(\log(1+ \theta_{k}))-h\left(A_{k}\right)\theta_{k}-\log[\bar{P}+\beta\exp\left(-h \left(A_{k}\right)\theta_{k}\right)]. \tag{12}\]  
The target implicit function can be obtained via taking the first order derivative with respect to \(\theta_{k}\) as  
\[\bar{P}+\beta\exp\left(-h\left(A_{k}\right)\theta_{k}\right)=\bar{P}h\left(A_ {k}\right)(1+\theta_{k})\log(1+\theta_{k}), \tag{13}\]  
where \(\theta_{k}\) is the function of \(A_{k}\). According to the implicit function theorem, we can get the following expression  
\[\frac{d\theta_{k}}{dA_{k}}=-\frac{L_{A_{k}}}{L_{\theta_{k}}}\] \[=-\frac{h(A_{k})^{{}^{\prime}}[\beta h\left(A_{k}\right)\exp \left(-h\left(A_{k}\right)\theta_{k}\right)\theta_{k}+\bar{P}\log\left(1+ \theta_{k}\right)+\bar{P}\theta_{k}\log\left(1+\theta_{k}\right)]}{h\left(A_{ k}\right)[\beta\exp\left(-h\left(A_{k}\right)\theta_{k}\right)+\bar{P}\log \left(1+\theta_{k}\right)+\bar{P}]} \tag{14}\] \[<0.\]  
Hence, \(\theta_{k}\) decreases with \(A_{k}\).
## 3 Results and Discussion
### Microcyst  
The CyanoSTUN UV-C treatment demonstrated a sustained, dose-dependent effect on microcyst concentration over the 9-day regrowth period. Figure 2A shows that samples collected from the rear of the vessel had a significantly reduced microcystin concentration compared to samples collected from the front of the vessel. Figure 2B shows that the average reduction in microcystin concentration for 15, 30, and 45 mJ/cm\({}^{2}\) treatment doses was 31.6% (\(n=10\), SD = 20.1%; 1.3 \(\mu\)g/L reduced), 45.7% (\(n=10\), SD = 10.8%; 1.9 \(\mu\)g/L reduced), and 49.9% (\(n=10\), SD = 8.2%; 1.7 \(\mu\)g/L reduced), respectively, over the 9-day regrowth period. These findings suggest that the UV\({}_{254}\) emitted from the reactor may have induced direct photolysis of microccyst. It is also possible that the low-pressure mercury lamps within the Trojan UV3000 PTP reactor emitted radiation at other wavelengths than 254 nm, which had an impact on microcystin degradation. For example, low-pressure mercury lamps typically emit 185 nm radiation at approximately 8% of the intensity of 254 nm radiation [24]. If this 185 nm radiation is not completely eliminated by the quartz sleeve, then the 185 nm radiation would be absorbed by the ambient water, leading to photochemical ionization and the production of hydroxyl radicals capable of oxidizing microccyst. It is unclear why the vessel had an effect on microccystin concentration at the control dose (0 mJ/cm\({}^{2}\); 17.8% reduction (\(n=10\), SD = 13.3%; 0.8 \(\mu\)g/L reduced), however it is possible that some microccystin adsorption occurred on the polytetrafluorethylene diffuse reflector sheeting that lined the interior stainless-steel walls of the reactor to increase UV\({}_{254}\) reflectance.  
Figure 2: Changes in composite microccystin samples are shown by (**A**) the concentration of microccystin (\(\mu\)g/L) over a 9-day period post-field trial and (**B**) the percent reduction in microccystin concentration over a 9-day period post-field trial.
## 4 Evolution Knowledge MOABC Algorithm (EKMOABC) for WNN-Optimized Construction of PIs
### Probability Choice Equation Combining the Dominance and Distribution Relationships  
Probability choice equation is used to choose the solutions to evolve deeply and balance the capacity between exploit and explore. Different from the single-objective optimization, there may be a large number of solutions for multi-objective optimization which cannot dominate other solutions at the same time. The quality of the Pareto optimal set is related to both the dominance relationship and distribution relationship. If only one of them is considered in the probability choice equation, it will hardly obtain the high-quality of PIs. Therefore, a typical probability choice equation combining the dominance and distribution relationships is proposed as follows:  
\[P_{i}=\left\{\begin{array}{ll}\frac{1}{1+(1-s_{i})e^{-\eta_{1}t_{i}}},&s_{i }=0\\ \frac{1}{1+s_{i}e^{-\eta_{2}t_{i}}},&s_{i}=1\end{array}\right. \tag{18}\]  
where \(P_{i}\) is the probability, \(d_{i}\) is the crowding distance. \(\eta_{1}\) and \(\eta_{2}\) are the probability choice parameters and \(l_{i}\) is defined as:  
\[l_{i}=\underset{j=1}{\overset{s}{\min}}\sqrt{\underset{k=1}{\overset{m}{\sum }}\left(f_{ik}-f_{jk}\right)^{2}} \tag{19}\]  
where \(f_{i}\) is the dominant solution and \(f_{j}\) is the non-dominant solution. \(s\) is the number of solutions in the archive.  
From Equations (18) and (19), we can see that the proposed equation of probability choice combines the dominance relationship and the distribution relationship. When the solution is dominated by other solutions, it will be punished by the distance from the Pareto optimal set. From Equation (18), when the solution is dominated, the probability will be limited to \([0,0.5]\) by the parameter \(\eta_{1}\). However, when the solution is non-dominated, the probability will be chosen in the range of \([0.5,1]\) due to the superiority of non-dominant solutions. Thus, even though the dominant solution is closer to the Pareto optimal set, its probability will be never be greater than that of the non-dominant solution, which will be of great benefit for population evolution both in convergence and distribution.
## 3 Experimental Rock Characterization Methods
### Porosity and Pore Size Distribution with Porosimetry  
Small cuboidal pieces, with dimensions of \(\sim\)3 mm 3 mm 3 mm, were cut from core fragments F1, F3 and F4 for porosity measurements. Porosity of these pieces was determined using a Poremaster automated mercury intrusion porosimeter (Quantachrome Corporation, Boynton Beach, FL, USA) [33]. This apparatus progressively applies pressure to force mercury into pore-space in a sample [33; 34], using theory for capillary flow into cylindrical pores [35] to determine the cumulative size-distribution of pores in the sample.
# Performance Modeling Framework for IoT-over-Satellite Using Shared Radio Spectrum
## 3 IoT-over-Satellite Performance Modeling
### Satellite Channel Model  
As the wireless electromagnetic signal propagates from the IoT device toward the satellite, it suffers from the natural wave-front expansion. This expansion is captured in the free-space path loss \(L_{\text{FSPL}}\) and computed as  
\[L_{\text{FSPL}}=20\log_{10}(4\pi d(t)f), \tag{4}\]  
in which \(d\) is the slant range in meters and \(f\) is the carrier frequency in Hertz.  
The second important factor in the wireless satellite channel is the attenuation caused by the atmosphere air absorption \(L_{\text{Air}}\) as elaborated in the following subsection.
## 4 Design of the DRFTM
### Trust Evaluation Based on Random Forest
#### 4.2.2 Generation of Decision Trees  
A decision tree is a basic method for classification and regression tasks. It forms a tree-like structure, where data points in a specified feature space are progressively allocated from the root node to child nodes and ultimately reach a leaf node, which represents the data point's classification. Depending on the choice of the best attribute selection method, decision trees are generally categorized into three main methods, including ID3, C4.5, and CART [17]. In this paper, C4.5 is chosen as the decision tree generation algorithm. For the five types of trust evidence obtained through the method in Section 4.1, the trust evidence obtained within a time window can be expressed as sample \(D\), i.e., \(\left\{\left(T_{1}^{d},T_{1}^{en},T_{1}^{en},T_{1}^{rec}\right),T_{1}^{rec} \right),\left(T_{2}^{c},T_{2}^{d},T_{2}^{en},T_{2}^{env},T_{2}^{rec}\right), \ldots,\left(T_{n}^{c},T_{n}^{d},T_{n}^{en},T_{n}^{rec},T_{n}^{rec}\right)\right\}\). The primary focus in decision tree learning is discovering the best attribute for grouping data. The C4.5 algorithm selects attributes by calculating the information gain ratio and prioritizes the attributes with higher information gain: it is assumed that there are \(V(1,2,\ldots v,\ldots,V)\) possible values of \(\left\{a_{1},a_{2},\ldots,a_{v},\ldots,a_{V}\right\}\) for discrete attribute \(a\), where all the samples of attribute \(a\) in sample \(D\) with value \(a_{v}\) are denoted as \(D_{v}\). \(k\) denotes the classification result of the sample, and \(D_{vk}\) denotes the sample classified as \(k\). The corresponding information entropy is as follows:  
\[H(D_{v})=-\sum_{k\in N}p(k|v)\log p(k|v)=-\sum_{k\in N}\frac{|D_{vk}|}{|D_{v}|} \log\frac{|D_{vk}|}{|D_{v}|}, \tag{17}\]  
Each value contains a different number of samples, and we assign weights \(|D_{v}|/|D|\) to the likelihood of each value. Consequently, we can calculate the information gain obtained when attribute \(a\) is used to partition the sample set \(D\):  
\[Gain(D,a)=H(D)-\sum_{v=1}^{v}\frac{|D_{v}|}{|D|}H(|D_{v}|)=H(D)-H(D,a)=I(D,a). \tag{18}\]  
The information gain criterion tends to favor attributes with a larger number of values. To mitigate this preference, the C4.5 decision tree algorithm uses the information gain ratio to select the optimal splitting attribute. The information gain rate is defined as follows:  
\[Gain_{ratio}(D,a)=\frac{Gain(D,a)}{IV(a)}, \tag{19}\]  
where \(IV(a)=-\sum\limits_{v=1}^{V}\frac{|D_{v}|}{|D|}\log\frac{|D_{v}|}{|D|}=H(a)\).