CCLDBO-TCN-BiGRU-Attention: Research and Application of Hybrid Models for Temperature Prediction

CCLDBO-TCN-BiGRU-Attention: Research and Application of Hybrid Models for Temperature Prediction | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article CCLDBO-TCN-BiGRU-Attention: Research and Application of Hybrid Models for Temperature Prediction Rongchuan Yu, Linli Jiang, Wenxin Li, Jiansheng Wu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6230198/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Meteorological data present prominent spatio-temporal features and complex non-linear relationships, significantly challenging meteorological forecasting. Hence, to study the performance of hybrid models in weather prediction, this paper proposes a hybrid model named CCLDBO-TCN-BiGRU-Attention and experimentally compares it with current models. The proposed method improves the dung beetle optimization algorithm by introducing Circle mapping, the si-ne-cosine algorithm MSCA, and the dung beetle optimization algorithm using Levy flight (CCLDBO). The experimental results demonstrate that the developed method achieves outstanding results in a wide range of single-peak and multi-peak tests. Additionally, the evaluation index R2 reaches 0.9925 in the hybrid model for predicting the temperature in the Guizhong area, and the RMAE, RMSE, MSE, and MAPE metrics are the best compared with other models. The hybrid model is also closer to the real values than other models in predicting real value changes. Overall, the hybrid model performs well in temperature prediction and provides a feasible solution for weather prediction. Trial registration number : Not applicable. TCN-BiGRU-Attention CCLDBO hybrid model temperature prediction Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 1 Introduction With frequent human activities, various industrial activities affect the climate [ 1 ]. However, climate is crucial for human production, affecting agricultural production. The proverb, "There is no harvest in the water, more harvest, less harvest lies in the fertilizer", highlights that climate plays a determining factor in agricultural production. The Guizhong region is dominated by sugarcane cultivation as a cash crop and is the leading sugar production area in Guangxi [ 2 ]. Since temperature prediction is a good guide to agricultural production, it is vital to predict the pattern of tapping temperature changes [ 3 ]. Wang et al. used the Elman neural network model to predict the temperature data in Chongqing and concluded that the model has a good fit [ 4 ]. Wang et al. [ 5 ] proposed the SA-PSO-XGBoost prediction model and compared it with XGBoost, GRU, and LSTM neural net-works. They concluded that SA-PSO-XGBoost has high accuracy and robustness. Nevertheless, with the rapid development of computer technology and the massive accumulation of meteorological data, temperature prediction has new possibilities and challenges [ 6 ]. Traditional machine learning methods, such as Support Vector Machines (SVM), Back Propagation Neural Networks (BPNN), Relevance Vector Machines (RVM), and Radial Basis Functions (RBF), have some limitations in dealing with non-smooth, non-linear data [ 7 ]. In recent years, deep learning methods such as Recurrent Neural Networks (RNN), Long Short-Term Memory Neural Networks (LSTM), and Convolutional Neural Networks (CNN) have been rapidly applied for time series prediction [ 8 ]. However, as the requirements for prediction accuracy increase, a single model often fails to fully exploit the hidden features in the data. Thus, hybrid models have become increasingly important [ 9 ]. The benefit of hybrid models is combining the features and advantages of multiple models, thus im-proving the accuracy and robustness of prediction [ 10 ]. Specifically, hybrid models can better adapt to complex data characteristics and variations by integrating predictions from different models or utilizing the output of one model as input to another [ 11 ]. Combining multiple models helps to improve the effectiveness of temperature forecasts and achieve better results in practice. For example, Wang et al. [ 12 ] used three classical chaotic time series predictions in a TCN hybrid model, achieving an R2 value close to 1. Satish [ 13 ] developed an ANN model to predict the rainfall in Loktak Lift Irrigation in Manipur, India, from 1985 to 2018 and concluded that the model has high predictive accuracy. Mohammad et al. [ 14 ] developed the CONN-SVM-GPR for rainfall in the Terengganu River Basin, Malaysia, and derived a 5.9%-23% and 20%-60% reduction in daily and monthly scale MAE for the model. Anteneh et al. [ 15 ] used a multilayer artificial neural network model to predict salinity in Florida Bay using two climate change scenarios, where the Representative Concentration Pathway (RCP) 4.5 and 8.5 salinity may increase by 30% and 70%, respectively. Gao et al. [ 16 ] relied on deep learning to predict winter temperatures in North China, using six CMIP5 models with relatively small biases to train the CNN to derive the best performance for temperatures over 10 years. Temperature prediction in the Guizhong region is a challenging time series forecasting task because the temperature is affected by many factors, including seasonal changes, geographic location, and altitude. This paper combines the Temporal Convolutional Network (TCN), Bidirectional Gated Recurrent Unit (BiGRU), Attention mechanism, and Improved Dung Beetle Optimization Algorithm (Cir-cle- MSCA-Levy Dung Beetle Optimizer, CCLDBO) to propose a new hybrid model called CCLDBO-TCN-BiGRU-Attention. The improved dung beetle optimization algorithm optimizes the learning rate, the number of neurons in BiGRU, the key value of the attention mechanism, and the regularization parameter of the TCN-BiGRU-Attention model to enhance the prediction performance of the hybrid model. The hybrid model CCLDBO-TCN-BiGRU-Attention is used to predict the temperature in the Guizhong region. 2 Improved Dung Beetle Optimization Algorithm The Dung Beetle Optimizer (DBO) algorithm is inspired by the dung beetle's behaviors of rolling, dancing, foraging, breeding, and stealing. Each dung beetle cluster comprises four different kinds of agent dung beetles, namely, the ball-rolling dung beetle, the breeding dung beetle (breeding ball), the little dung beetle, and the stealing dung beetle. The diverse position update strategies of the dung beetle algorithm can explore the search space more comprehensively and effectively solve complex ground search and optimization problems in practical applications. 2.1 Motivation for Improvement The DBO algorithm is characterized by solid optimality-seeking ability and fast convergence speed. Still, the uneven distribution of the initial population and unbalanced global exploration and local exploitation ability cause the algorithm to fall into the local optimum easily. Therefore, to overcome these problems of DBO, this paper improves DBO using Circle mapping, the Modified Sine Cosine Algorithm (MSCA), and the Levy flight strategy. 2.2 Process Improvement Figure 1 illustrates the flowchart of CCLDBO. The CCLDBO algorithm contains three improvements: the Circle mapping strategy, fusing the Cauchy variation algorithm into the improved sine-cosine algorithm, and the Levy flight. To verify the independent validity of the different improvement strategies proposed in this paper, DBO is combined with each of these three strategies independently: 1) CDBO is obtained after initializing the population by adding a Circle mapping strategy in DBO. 2) SCDBO is obtained after embedding MSCA in CDBO. 3) The improved algorithm model, CCLDBO, is obtained after adding Levy flight at the end of SCDBO. 2.3 Introducing Population Initialization for Circle Mapping The essence of the DBO algorithm, which is a typical representative of intelligent optimization algorithms, is a stochastic search algorithm. The initial population is generated through random initialization. Still, the process is prone to problems such as uneven distribution of the positions of dung beetle individuals, weak global exploration ability, and easy falling into the local optimum due to low population diversity. To address these issues, a chaotic mapping characterized by stochasticity, ergodicity, and regularity is used in the initialization phase of the DBO population to generate a highly diverse initial population. Through comparison and analysis, this paper employs a trigonometric function-based Circle mapping, which promotes a more even distribution of the initial position of the population. Circle mapping is defined as follows: $$\:{x}_{i+1}={mod}({x}_{i}+0.2-\left(\frac{0.5}{2\pi\:}\right){sin}(2\pi\:{x}_{i}),1)$$ 1 where \(\:{x}_{i+1}\:\) denotes the i-th chaotic sequence number and \(\:{mod}(a,b)\) is the remainder operation of a on b . Figure 2 depicts the distribution of 1000 sequence values generated by each of the four chaotic mappings: common random numbers, Cubic mapping, Bernoulli mapping, and Circle mapping. Figure 2 reveals that the distribution of the chaotic sequence values generated by the Circle mapping be-tween 0 and 1 is more uniform than that of the sequences generated by the ordinary random numbers, the Cubic mapping, and the Bernoulli mapping. The uniform distribution is more helpful in expanding the search range of the dung beetle population in space, increasing the diversity of the population location, improving the defect that the algorithm easily falls into the local extremes to some extent, and further improving the convergence accuracy and optimization efficiency of the algorithm. 2.4 Improved Sine Cosine Algorithm Fused with Kersey Variation Algorithm During the iterative search for an optimal solution in the dung beetle algorithm and when iterating several times, current dung beetles will quickly gather near the optimal position and exhibit a near-optimal solution. However, the location updates of the dancing dung beetles are too random, increasing randomness in the distance and direction of the population search, i.e., the optimization method. Suppose the current optimal position is not the globally optimal point. In that case, the dung beetle population will be stuck in a local optimal solution and cannot find the genuinely optimal position. Thus, this paper introduces the improved sine-cosine algorithm and incorporates the Cauchy variation algorithm, promoting the rapid propagation of information in the population, controlling the optimization searching method, increasing the search space of the dung beetle, and increasing the freedom of each individual. This strategy prevents the population from falling into a locally optimal solution and makes it more likely to find a globally optimal solution. The improved sine algorithm (MSA) is inspired by various algorithms related to SCA, such as the Sine Cosine Algorithm (SCA), Sine Algorithm (SA), and Exponential Sine Cosine Algorithm (ESCA) functions, as well as improved Sine Cosine Algorithm (ISCA). The ESCA, ISCA, and other SCA-related algorithms involve an iterative optimization strategy using the sine function, which has a robust global exploration capability. Additionally, the Cauchy distribution function is relatively flat at the origin but shows a significant difference in distribution at both ends. The Cauchy variation can generate more significant perturbations in the vicinity of dung beetle individuals, broadening their distribution and helping to jump out of local optima. Hence, to better utilize this advantage, this paper improves the sine-cosine algorithm by incorporating the Cauchy operator, exploring the effect of the two-end variability of the Cauchy distribution function to optimally count the globally optimal individuals, thereby achieving the global optimum. The standard Cauchy distribution function is formulated as follows: $$\:f\left(x\right)=\frac{1}{\pi\:}\left(\frac{1}{{x}^{2}+1}\right)$$ 2 Accelerating algorithm convergence and improving problem-solving efficiency can be achieved by introducing the Cauchy distribution function on the improved sine-cosine algorithm and changing the square brackets in the sine function to absolute values in the original improved sine algorithm. In addition, introducing non-linear inertia weight coefficients during the position update process enables the algorithm to search the local region adequately, balancing global exploration and local exploitation capabilities. The position update formula for the improved sine-cosine algorithm incorporating the Cauchy operator is formulated as follows: $$\:{x}_{i}(t+1)={\omega\:}_{t}{x}_{i}\left(t\right)+{r}_{1}\times\:\text{cauchyrnd(}{r}_{2})\times\:|{r}_{3}{p}_{i}\left(t\right)-{x}_{i}\left(t\right)|$$ 3 where t denotes the number of current iterations, \(\:{\omega\:}_{t}\) is the inertia weight, \(\:{x}_{i}\left(t\right)\) represents the i-th position component of individual X in the t-th iteration, and \(\:{p}_{i}\left(t\right)\) denotes the i-th component of the best individual position variable in the t-th iteration. Additionally, \(\:{r}_{1}\) is a non-linear decreasing function with a random number \(\:{r}_{2}\) generated from the interval [0, 2π] and a random number \(\:{r}_{3}\) randomly generated in the interval [-2, 2]. Cauchyrnd() is the Cauchy distribution function. The Cauchy operator determines the change in the value of A between 0 and 1. The expression is: $$\:{r}_{1}=\frac{{{\omega\:}}_{\text{m}\text{a}\text{x}}-{{\omega\:}}_{\text{m}\text{i}\text{n}}}{2}\text{c}\text{a}\text{u}\text{c}\text{h}\:\text{y}\text{r}\text{n}\text{d}\left(\frac{\pi\:t}{M}\right)+\frac{{{\omega\:}}_{\text{m}\text{a}\text{x}}+{{\omega\:}}_{\text{m}\text{i}\text{n}}}{2}$$ 4 where \(\:{\omega\:}_{max}\) and \(\:{\omega\:}_{min}\) denote the maximum and minimum values, t represents the current number of iterations, and M indicates the maximum number of iterations. This paper employs a non-linear inertia weighting strategy, where the inertia weights are exponentially decreasing with the number of iterations: $$\:{\omega\:}_{t}=\frac{{e}^{\frac{t}{M}}-1}{{e}^{1}-1}$$ 5 To further improve the ability of the DBO algorithm to coordinate global exploration with local exploitation, the MSCA bootstrapping mechanism is introduced as an alternative to the dung beetle tangent-dancing strategy embedded in the DBO algorithm, i.e., a Cauchy variation operation is performed on the entire dung beetle individual to bootstrap the dung beetle position update in the ball-rolling phase. The improved formula is: $$\:{x}_{i}(t+1)=\left\{\begin{array}{cc}{x}_{i}\left(t\right)+\alpha\:\times\:k\times\:{x}_{i}(t-1)+b\times\:\varDelta\:x,&\:R2<ST\\\:{\omega\:}_{t}{x}_{i}\left(t\right)+{r}_{1}\times\:\text{cauchyrnd}\left({r}_{2}\right)\times\:\left|{r}_{3}{p}_{i}\right(t)-{x}_{i}(t\left)\right|,&\:R2\ge\:ST\end{array}\right.$$ 6 For this process, we define R2 = rand() and ST ∈ (0.5,1] is the basis for determining whether the dung beetle rolls purposefully. If R2 < ST, Dung Beetle possesses a clear rolling target and is in the normal global exploration phase. Conversely, if R2 ≥ ST, the dung beetle does not have a specific rolling target but will adopt Cauchy's variation function to search for movement, which is more random than the original sine function. Such an approach effectively improves the shortcomings of the DBO algorithm regarding excessive randomness in position updating. Each dung beetle individual can explore the search space better and exchange information with the current optimal individual. This process speeds up information propagation throughout the entire group and solves the original algorithm's lack of inter-individual information communication. The bootstrapping mechanism of MSCA, which allows individual dung beetles to freely explore and locally optimize within the region delineated by the algorithm, somehow expands the search space and allows the algorithm to converge to the optimal solution gradually. This enhances the algorithm's global search capability. Meanwhile, according to Eq. ( 3 ), the parameter \(\:{r}_{1}\) is the key to regulating the dung beetle search, which can be used to regulate the dung beetle's search range and direction and optimize the solution of the DBO algorithm. In addition, from Eq. ( 5 ), \(\:{\omega\:}_{t}\) will keep shrinking the search space, and the inertia weights will decrease as the number of iterations increases. In the early stages of an algorithm's iteration, relatively large inertia weights can give the algorithm a global solid exploration capability. In contrast, in the later stages, relatively small inertia weights are beneficial to improve its local exploitation capability. 2.5 Formula Levy Flight Strategy According to the updated formulas for the four individual behaviors of the dung beetle, the foraging, breeding, and stealing behaviors are caught in a local search, except for the ball-rolling behavior, which has a better global search performance. To solve the problem, this paper adopts the Levy flight strategy, which realizes random wandering by providing a random factor and generating a step size that conforms to the Levy flight to increase the dung beetle to improve the global optimization-seeking ability in different individual behaviors. The Levy flight is a probability distribution with long-tailed properties. By introducing the step size of Levy's flight generation to increase the algorithm's exploration of space, dung beetles can jump out of the local optimum and improve the ability to search globally during the search process of random wandering. 3 Optimization Algorithm Testing Experiments We select benchmark functions with different characteristics from CEC2005 to compare the search speed and quality of the optimal solutions of various algorithms. Additionally, we verify the convergence performance of the CCLDBO algorithm and whether it can jump out of the local optimum. 3.1 Multiple Benchmark Function Testing and Analysis In this section, we analyze CCLDBO based on the 23 benchmark functions reported in Tables 1 , 2 , and 3 , where dimension denotes the dimension of the set function. The range is the function's search space boundary, and minf is the function's best fitness value. The number of search agents is uniformly set to 30, the maximum number of iterations is 100, and each algorithm is run independently for 30 times. The following experiments compare the GWO, BWO, DBO, SCDBO, and CCLDBO algorithms. Seven single-peak benchmark functions (F1-F7) are selected to analyze each algorithm's single-objective solving ability, and 16 multi-peak benchmark functions (F8-F23) compare the models' convergence speed and accuracy. These trials aim to verify the effectiveness of each improvement strategy and analyze whether the algorithms can jump out of the local optimum where the dimension of the benchmark function is uniformly set to 30. 3.2 Single Peak Function Analysis Table 1 (F1 – F7) presents the seven single-peak benchmark test functions. Since the single-peak function has only one minimum, it can be used to check the performance of the algorithm development. Figure 3 illustrates the single-peak convergence process to compare the individual algorithmic strategies. Table 1 and Fig. 3 reveal that CCLDBO attains the highest convergence accuracy on the F1-F4 single-peak test functions and successfully converges to the theoretically optimal solution. For the F5 single-peak test function, although the convergence accuracy of BWO is higher than that of CCLDBO, the convergence speed of CCLDBO is significantly better than that of the other algorithms, reaching the desired accuracy in a short time. For the F6-F7 single-peak test function, the superiority of CCLDBO is even more apparent, dominating in convergence speed and reaching the theoretical optimum in both convergence accuracy and convergence theory. Table 1 Single-peak benchmark functions. Basis function Dimension Realm \({F_1}(x)=\sum\limits_{{i=1}}^{n} {x_{i}^{2}}\) 30, 50, 100 [-100, 100] \({F_2}(x)=\sum\limits_{{i=1}}^{n} | {x_i}|+\prod\limits_{{i=1}}^{n} | {x_i}|\) 30, 50, 100 [-10, 10] \({F_3}(x)=\sum\limits_{{i=1}}^{n} {{{\left( {\sum\limits_{{j=1}}^{i} {{x_i}} } \right)}^2}}\) 30, 50, 100 [-100, 100] \({F_{\text{4}}}(x){\text{=ma}}{{\text{x}}_i}\left\{ {\left| {{{\text{x}}_{\text{i}}}} \right|,{\text{1}} \leqslant i \leqslant n} \right\}\) 30, 50, 100 [-100, 100] \({F_{\text{5}}}(x){\text{=}}\sum\limits_{{{\text{i=1}}}}^{{\text{n}}} {\left[ {{\text{100}}{{\left( {{x_{{\text{i+1}}}}{\text{-x}}_{{\text{i}}}^{{\text{2}}}} \right)}^{\text{2}}}{\text{+}}{{\left( {{x_{\text{i}}}{\text{-1}}} \right)}^{\text{2}}}} \right]}\) 30, 50, 100 [-30, 30] \({F_{\text{6}}}(x){\text{=}}\sum\limits_{{{\text{i=1}}}}^{{\text{n}}} {{{\left( {[{x_i}{\text{+0}}.{\text{5}}]} \right)}^{\text{2}}}}\) 30, 50, 100 [-100, 100] \({F_{\text{7}}}(x){\text{=}}\sum\limits_{{{\text{i=1}}}}^{{\text{n}}} {x_{{\text{i}}}^{{\text{4}}}} {\text{+random}}[{\text{0}},{\text{1}})\) 30, 50, 100 [-128, 128] In conclusion, the CCLDBO algorithm shows a significant advantage in terms of average search accuracy and convergence speed over the test function of a single peak when compared to GWO, BWO, DBO, and SCDBO. This demonstrates that the improved CCLDBO algorithm is robust in balancing global exploration and jumping out of local optimal solutions. 3.3 Multi-peak Function Analysis Table 2 presents six multi-peak functions (F8-F13). Unlike the single-peak functions, each of the multi-peak functions in Table 2 has multiple extremes, and the dimensions of each function are 30, 50, and 100, respectively. The exploration capability of the proposed algorithm is evaluated with the help of a multi-peak function. The functional expressions for the multi-peak functions F14-F23 are reported in Table 3 . Besides, the dimension of each function is fixed, and the convergence process is presented in Fig. 4 . Table 2 Multi-peak benchmark functions for non-fixed dimensions. Basis function Dimension Realm \({F_8}(x)=\sum\limits_{{i=1}}^{n} - {x_i}\sin \left( {\sqrt {|{x_i}|} } \right)\) 30, 50, 100 [-500, 500] \({F_9}(x)=\sum\limits_{{i=1}}^{n} {\left[ {x_{i}^{2} - 10\cos \left( {2\pi {x_i}} \right)+10} \right]}\) 30, 50, 100 [-5.12, 5.12] \({F_{10}}(x)= - 20\exp \left( { - 0.2\sqrt {\frac{1}{n}\sum\limits_{{i=1}}^{n} {x_{i}^{2}} } } \right) - \exp \left( {\frac{1}{n}\sum\limits_{{i=1}}^{n} {\cos } \left( {2\pi {x_i}} \right)} \right)+20+e\) 30, 50, 100 [-32, 32] \({F_{11}}(x)=\frac{1}{{4000}}\sum\limits_{{i=1}}^{n} {x_{i}^{2}} - \prod\limits_{{i=1}}^{n} {\cos } \left( {\frac{{{x_i}}}{{\sqrt i }}} \right)+1\) 30, 50, 100 [-50, 50] \(\begin{gathered} {F_{12}}(x)=\frac{\pi }{n}\left\{ {10\sin (\pi {y_1})+\sum\limits_{{i=1}}^{n} {{{\left( {{y_i} - 1} \right)}^2}} {{\left[ {1+10{{\sin }^2}(\pi {y_{i+1}})} \right]}^2}} \right\} \hfill \\ +\sum\limits_{{i=1}}^{n} u \left( {{x_i},10,100,4} \right) \hfill \\ {y_i}=1+\frac{{{x_i}+1}}{4} \hfill \\ \end{gathered}\) 30, 50, 100 [-50, 50] \(\begin{gathered} {F_{13}}(x)=0.1\{ {\sin ^2}(3\pi {x_1})+\sum\limits_{{i=1}}^{n} {{{({x_i} - 1)}^2}} \left[ {1+{{\sin }^2}(3\pi {x_i}+1)} \right] \hfill \\ +{\left( {{x_n} - 1} \right)^2}\left[ {1+{{\sin }^2}\left( {2\pi {x_n}} \right)} \right]\} +\sum\limits_{{i=1}}^{n} u \left( {{x_i},5,100,4} \right) \hfill \\ \end{gathered}\) 30, 50, 100 [-50, 50] Table 3 Multi-peak benchmark functions for fixed dimensions. Basis function Dimension Realm \({F_{14}}(x)={\left( {\frac{1}{{500}}+\sum\limits_{{j=1}}^{{25}} {\frac{1}{{j+\sum\limits_{{i=1}}^{2} {{{({x_i} - {a_{ij}})}^6}} }}} } \right)^{ - 1}}\) 2 [-65, 65] \({F_{15}}(x)=\sum\limits_{{i=1}}^{{11}} {{{\left[ {{a_i} - \frac{{{x_1}\left( {b_{i}^{2}+{b_i}{x_2}} \right)}}{{b_{i}^{2}+{b_i}{x_3}+{x_4}}}} \right]}^2}}\) 4 [-5, 5] \({F_{16}}(x)=4x_{1}^{2} - 2.1x_{1}^{4}+\frac{1}{3}x_{1}^{6}+{x_1}{x_2} - 4x_{2}^{2}+4x_{2}^{4}\) 2 [-5, 5] \({F_{17}}(x)={\left( {{x_2} - \frac{{5.1}}{{4{\pi ^2}}}x_{1}^{2}+\frac{5}{\pi }{x_1} - 6} \right)^2}+10\left( {1 - \frac{1}{{8\pi }}} \right)\cos {x_1}+10\) 2 [-5, 5] \(\begin{gathered} {F_{18}}(x)=\left[ {1+{{({x_1}+{x_2}+1)}^2}\left( {19 - 14{x_1}+3x_{1}^{2} - 14{x_2}+6{x_1}{x_2}+3x_{2}^{2}} \right)} \right] \hfill \\ \times \left[ {30+{{\left( {2{x_1} - 3{x_2}} \right)}^2} \times (18 - 32{x_1}+12x_{1}^{2}+48{x_2} - 36{x_1}{x_2}+27x_{2}^{2})} \right. \hfill \\ \end{gathered}\) 2 [-2, 2] \({F_{19}}(x)=\sum\limits_{{i=1}}^{4} {{c_i}} \exp \left( { - \sum\limits_{{j=1}}^{3} {{a_{ij}}} {{\left( {{x_j} - {p_{ij}}} \right)}^2}} \right)\) 3 [0, 1] \({F_{20}}(x)=\sum\limits_{{i=1}}^{4} {{c_i}} \exp \left( { - \sum\limits_{{j=1}}^{6} {{a_{ij}}} {{\left( {{x_j} - {p_{ij}}} \right)}^2}} \right)\) 6 [0, 1] \({F_{21}}(x)= - \sum\limits_{{i=1}}^{5} {{{\left[ {\left( {X - {a_i}} \right){{\left( {X - {a_i}} \right)}^T}+{c_i}} \right]}^{ - 1}}}\) 4 [0, 10] \({F_{22}}(x)= - \sum\limits_{{i=1}}^{7} {{{\left[ {\left( {X - {a_i}} \right){{\left( {X - {a_i}} \right)}^T}+{c_i}} \right]}^{ - 1}}}\) 4 [0, 10] \({F_{23}}(x)= - \sum\limits_{{i=1}}^{{10}} {{{\left[ {\left( {X - {a_i}} \right){{\left( {X - {a_i}} \right)}^T}+{c_i}} \right]}^{ - 1}}}\) 4 [0, 10] Tables 2 and 3 , and Fig. 4 infer that CCLDBO excels in exploring and developing balanced algorithms compared to other algorithms. 3.4 Validation of the Effectiveness of Various Strategies in CCLDBO Table 4 compares the experimental results of CCLDBO with GWO, BWO, DBO, and SCDBO. Combined with the results in Table 4 , which compares CCLDBO with GWO, BWO, DBO, and SCDBO, and Figs. 4 and 5 , which present the convergence process of each algorithmic strategy, we conclude the following: Table 4 Experimental results for benchmark functions (comparison of individual strategies). DBO SCDBO CCLDBO BWO GWO F 1 8.8838e − 22 3.1582e − 34 2.9978e − 79 2.420267e − 3 4.2163e − 13 F 2 1.5335e − 16 2.5518e − 17 8.42e − 40 1.355398e − 2 1.11e − 7 F 3 8.6666e − 14 5.6797e − 24 4.7423e − 75 5.353404e − 2 0.00011079 F 4 1.4483e − 11 1.7699e − 14 2.6278e − 36 1.625666e − 2 6.9567e − 5 F 5 7.0049 6.6892 5.4402 1.269396e − 1 7.2053 F 6 5.155e − 6 2.8527e − 7 4.5594e − 7 6.564202e − 2 4.5235e − 5 F 7 0.00015061 0.00049843 3.9512e − 6 5.898160e − 5 9.0054e − 5 F 8 -3377.4633 -3821.8994 -6693.304 -4.146216e + 3 -2577.516 F 9 2.986 0 0 2.075062e − 3 2.2186e − 6 F 10 1.5099e − 14 8.8818e − 16 8.8818e − 16 1.286579e − 2 8.2078e − 7 F 11 0.0098713 0 0 6.054247e − 3 0.017069 F 12 6.7549e − 6 4.4018e − 7 3.0172e − 8 2.920900e − 4 0.041928 F 13 0.010988 9.7001e − 8 9.9088e − 8 1.502878e − 5 0.095086 F 14 0.998 0.998 0.998 6.903484e + 00 0.998 F 15 0.0014887 0.00072378 0.00032654 3.489317e − 4 0.00067064 F 16 -1.0316 -1.0316 -1.0316 -1.031404e + 00 -1.0316 F 17 0.39789 0.39789 0.39789 4.175370e − 1 0.39791 F 18 3 3 3 7.628276e + 00 3.0002 F 19 -3.8628 -3.8628 -3.8628 -3.767481e + 00 -3.8621 F 20 -3.2031 -3.1521 -3.2031 -2.797446e + 00 -3.1863 F 21 -10.1532 -5.0552 -10.1532 -8.334367e + 00 -10.1198 F 22 -10.3154 -9.7596 -10.4029 -1.003241e + 1 -10.3695 F 23 -10.3632 -8.007 -10.5364 -1.049222e + 1 -10.5091 For the single-peak benchmark functions (F1-F7), BWO outperforms CCLDBO in terms of convergence accuracy in the F5 function. Overall, the CCLDBO algorithm converges quickly and can balance exploration and exploitation, which is more conducive to jumping out of the local optimal solution. For the non-fixed dimensional multi-peak benchmark functions (F8-F13), CCLDBO performs better than some popular algorithms on most functions. It shows better exploration ability in finding the optimal solution of the multi-peak function. For the fixed-dimension multi-peak benchmark functions (F14-F23), CCLDBO finds the optimal value accurately. It strikes a better balance between global and local exploration and exploitation than the rest of the strategy algorithms. `Therefore, the CCLDBO algorithm's optimization performance on the 23 benchmark functions is significantly better than the other metaheuristic algorithms, such as the basic DBO. Hence, the CCLDBO algorithm's comprehensive performance is outstanding among many metaheuristic algorithms. 4 Temperature Projections 4.1 Establishment of the Data Set This paper selected temperature data from 12 meteorological stations in the Guizhong region from 1965 to 2020. A correlation analysis revealed that the correlation coefficients between the data of each observation point were above 0.9, reaching a statistically strong correlation for further study. As shown in Table 5 . Table 5 Correlation matrix of temperature data from various weather stations. A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A1 1 0.998 0.997 0.997 0.997 0.998 0.999 0.998 0.993 0.996 0.987 0.918 A2 0.998 1 0.999 0.999 0.998 0.998 0.998 0.996 0.997 0.999 0.982 0.918 A3 0.997 0.999 1 0.998 0.997 0.998 0.998 0.997 0.997 0.997 0.985 0.917 A4 0.997 0.999 0.998 1 0.999 0.998 0.998 0.997 0.995 0.998 0.981 0.92 A5 0.997 0.998 0.997 0.999 1 0.997 0.997 0.995 0.994 0.998 0.98 0.92 A6 0.998 0.998 0.998 0.998 0.997 1 0.999 0.999 0.994 0.996 0.988 0.916 A7 0.999 0.998 0.998 0.998 0.997 0.999 1 0.999 0.993 0.996 0.987 0.918 A8 0.998 0.996 0.997 0.997 0.995 0.999 0.999 1 0.991 0.993 0.988 0.916 A9 0.993 0.997 0.997 0.995 0.994 0.994 0.993 0.991 1 0.998 0.98 0.916 A10 0.996 0.999 0.997 0.998 0.998 0.996 0.996 0.993 0.998 1 0.979 0.919 A11 0.987 0.982 0.985 0.981 0.98 0.988 0.987 0.988 0.98 0.979 1 0.904 A12 0.918 0.918 0.917 0.92 0.92 0.916 0.918 0.916 0.916 0.919 0.904 1 Note: Table A1: Xincheng; A2: Liucheng; A3: Shatang; A4: Luzhai; A5: Liuzhou; A6: Xiangzhou; A7: Laibin; A8: Wuxuan; A9: Sanjiang; A10: Rongshui; A11: Jinxiu; A12: Liujiang. To better simulate the temperature change in the Guizhong area, which is affected by other weather factors, we added sunshine duration, monthly average water vapor pressure, and monthly average relative humidity to the dataset to enrich its diversity. 4.2 Predictive Modeling 4.2.1 Temporal Convolutional Network The Temporal Convolutional Network (TCN) can process time series data and has a primary structure of residual blocks containing dilated causal convolutions. It captures local patterns and long-range dependencies in sequences through its convolutional layers and is generally faster and easier to parallelize than traditional recurrent neural networks (RNNs). The causal convolution of TCN ensures that the output result depends only on the past input information, thus effectively avoiding the leakage of future information. The dilation convolution allows the input to be sampled at intervals during the convolution, which solves the problem of extracting the information from multivariate time series while enlarging the sensory field. This has enabled TCN to achieve good results in many time series pre-diction and analysis tasks. In the proposed model, the processed data series from the TCN layer is input to the BIGRU layer for the subsequent data analysis and processing step. 4.2.2 Bidirectional Gated Recurrent Unit A Bidirectional Gated Recurrent Unit (BiGRU) is a neural network architecture combining bidirectionality properties and a Gated Recurrent Unit (GRU). It consists of two GRU layers, one processing the input sequence in forward order and the other in reverse order. This allows the network to capture both forward and reverse dependencies in the input sequence, thus better capturing long-range dependencies in the sequence data. The output of BiGRU is usually a concatenation of the outputs of the GRU layers in both directions. It is used for various tasks, such as sequence prediction, classification, or sequence-to-sequence tasks. BiGRUs are widely used in natural language processing and time series analysis and can efficiently deal with complex relationships in sequence data. 4.2.3 Attention Mechanism Attention Mechanism is a technique used to enhance the performance of deep learning models, especially for processing sequential data. Its basic idea is to introduce learnable weights between different model parts to allocate different attention or importance between various parts. In the attention mechanism, given a sequence of inputs, the model can learn to dynamically assign different weights to each input element to capture the information more efficiently. This helps the model focus on the most relevant parts of the input sequence, improving its expressiveness and generalization. 4.2.4 CCLDBO-TCN-BiGRU-Attention Hybrid Model The TCN-BiGRU-Attention hybrid model stacks three layers of TCN residual modules to obtain a larger range of sensory fields of the input sequences and extract and downscale the features while avoiding problems such as gradient explosion and gradient vanishing. Each residual block has the same kernel size k, and its dilation factors D are 1, 2, and 4, respectively. Additionally, BiGRU ac-quires the TCN-processed data sequences and processes them in both directions using a time step from front to back (forward) and back to front (reverse). In this way, BiGRU can explore the dependencies of the timesteps more thoroughly and acquire contextual associations. The BiGRU-processed data sequences are then output to the attention layer to enhance the feature capture of the data sequences. Finally, the fully connected layer maps the high-dimensional features to the final prediction results. In this process, the learning rate, the number of neurons in BiGRU, the key value of the attention mechanism, and the regularization parameter of the TCN-BiGRU-Attention model are optimized using the improved dung beetle optimization algorithm (CCLDBO). The structure of the hybrid model is illustrated in Fig. 5 . 5 Results and Analysis 5.1 Evaluation Indicators The prediction effect of CCLDBO-TCN-BiGRU-Attention was evaluated based on all features of the first 10 samples to predict the temperature of the following sample. 5.2 Analysis of Results The reliability of various hybrid models, such as RVM-Adaboost, CNN-BiLSTM-Adaboost, CNN-BiLSTM-ATTENTION, CNN-BiGRU-ATTENTION, and TCN-BiGRU-ATTENTION was com-pared. The performance of each model and the vital role of each module were evaluated using the R2, MAE, RMSE, MSE, and MAPE metrics, with Fig. 6 presenting the corresponding results. Figure 6 highlights that the proposed CCLDBO-TCN-BiGRU-Attention model has an optimal performance in all indicators. Comparing the CNN-BiGRU-ATTENTION and TCN-BiGRU-ATTENTION reveals that the hybrid model of TCN is significantly better than the hybrid model of CNN in the pre-diction of this time series. This study also made predictions for data at different points in time and then compared the predictions of the six competitor models. The experimental results are presented in Fig. 7 , which reveals that the prediction results of each model are close to the real value, and all of them can predict the trend of the real power. However, the proposed hybrid model is closer to the real value. Combining the results in Figs. 6 and 7 , CCLDBO-TCN-BiGRU-Attention shows a good performance in the fitting effect of prediction. Table 6 Comparison of prediction results for different step sizes t + 1 t + 2 t + 3 t + 4 t + 5 t + 6 TCN-BiGRU-Attention 0.49532 0.96577 0.57397 0.59167 0.62841 1.237 MAE 0.029195 0.052057 0.032778 0.033228 0.034195 0.062077 MAPE 0.42893 1.4185 0.60733 0.7068 0.84823 2.9281 MSE 0.65493 1.191 0.77931 0.84071 0.921 1.7112 RMSE 0.989869 0.965691 0.9855 0.982986 0.955187 0.929656 R2 CCLDBO-TCN-BiGRU-Attention 0.42355 0.44561 0.3872 0.39941 0.4024 0.4167 MAE 0.023695 0.024529 0.023547 0.024052 0.023766 0.023854 MAPE 0.32842 0.34296 0.27683 0.30164 0.26521 0.30673 MSE 0.57308 0.58563 0.52615 0.54922 0.51498 0.55383 RMSE 0.9925 0.99208 0.99353 0.99296 0.99384 0.99288 R2 The prediction period was increased further to test the performance of the CCLDBO-TCN-BiGRU-Attention hybrid model. All features of the first 10 samples were used to predict the temperatures at step t + 1, step t + 2, step t + 3, step t + 4, step t + 5, and step t + 6. The corresponding results are reported in Table 6 , demonstrating that the prediction accuracy of TCN-BiGRU-Attention decreases as the step size of the predicted time series gradually increases. Additionally, the difficulty of capturing the data correlation in the previous moments improves. However, adding CCLDBO smooths the evaluation index, affecting the parameter optimization process of the hybrid model. 6 Conclusion This paper introduces the CCLDBO-TCN-BiGRU-Attention hybrid model for weather prediction. Specifically, this study proposes a multi-strategy improved algorithm, CCLDBO, which is based on the dung beetle optimization algorithm. Verified by 23 benchmark test functions, CCLDBO demonstrates higher efficiency and more reliable results. Secondly, the developed TCN-BiGRU-Attention hybrid model is proven better than the one based on CNN-BiGRU in prediction. After introducing the CCLDBO optimization algorithm to optimize the model parameters, its prediction accuracy is improved for single- and multi-step predictions across time. Finally, the experimental results of temperature prediction in the Guizhong region show that the proposed CCLDBO-TCN-BiGRU-Attention hybrid model performs well in weather prediction. Declarations Acknowledgment: Not applicable. Author Contributions: R.Y.: conceptualisation, methodology, modelling, writing original draft preparation. L.J.: editing, revision. W.L.: revision. J.W.: writing reviewing and editing. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (NSFC) Project: Deep Learning-Co-evolutionary Support Vector Machine Short-term Climate Hybrid Prediction Modeling of Monthly Precipitation [Project No. 42065004]; the Guangxi Innovation Drive Development Special Project (Science and Technology Major Special Project) "Human-Machine Intelli-gent Interaction Touch Terminal Manufacturing Key Technology and Industrial Cluster Applica-tion" [Project No.: Gui Ke AA21077018]; the Sub-project: Touch Display Integration Intelligent Touch System and Industrial Cluster Application [Project No.: Gui Ke AA21077018-2]; and the 2024 Guangxi University Young and Middle-aged Teachers' Scientific Research Basic Ability Enhancement Project: Research on the Application of Improved YOLOv5 in Detecting and Recognizing Sugar Cane Red Rot Disease [Project No.: 2024KY0868]. Data Availability Statement: The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author/s. Competing Interests: The authors declare that they have no conflict of interest. Ethical approval: Not applicable. Consent to participate: Not applicable. Consent for publication: Not applicable References Yin X, Zhu H, Gao J, Gao J, Guo L, Wang J: Effects of climate change and human activities on net primary productivity in the Northern Slope of Tianshan. J. Agric. Eng. 36, 195–202 (2020) Zhang S, Wang W: Discussion on high yield cultivation technology and pest control of sugarcane. Trop. Agric. Eng. 46, 88–90 (2022) Mou S. (2018). Bayesian modeling for temperature prediction and top ventilation regulation in solar greenhouse [Master thesis, Shenyang Agricultural University]. Shenyang. https://xueshu.baidu.com/usercenter/paper/show?paperid=1j210xn02p0w0cj 027240pk019154436&site=xueshu_se Wang F, Tu C, Gou Y: Temperature prediction based on Elman neural network. Agric. Sci. Technol. 12, 1680–1681 (2011) Wang Q, Qin H, Qi C, Wang Q: An XGBoost temperature prediction method based on adaptive SA-PSO improvement. Electron. Meas. Technol. 46, 67–72 (2023) Tran TTK, Bateni SM, Ki SJ, Vosoughifar H: A review of neural networks for air temperature forecasting. Water (Basel) 13, 1294 (2021) Lu X, Yang B, Zhang H, Zhang J, Wang Q, Jin Z: Inversion of leaf essential oil yield of cinnamomum camphora based on UAV multi-spectral remote sensing. Trans. Chin. Soc. Agric. Mach. 54, 191–197,213 (2023) Dai L, Shen J, Zhang F: An automatic energy and power demand forecasting model based on time series algorithm. Auto. Technol. Appl. 43, 49–51, 65 (2024) Yan L. (2023). Research on time series forecasting algorithm based on hybrid model [Master thesis, Statistics, Shandong University of Technology]. Shandong. https://d.wanfangdata.com.cn/thesis/D03215519 Mei z. (2018). Hybrid test method and application of RC structure based on material principal model parameter updating [Doctoral thesis, Harbin Institute of Technology Disaster Prevention and Mitigation Engineering and Protection Engineering]. Harbin. https://d.wanfangdata.com.cn/thesis/D01822984 Yang D, Wang H, He R, Cheng R, Zhang G, Liu D: Wind speed prediction based on improved empirical mode decomposition and hybrid deep learning model. Intell. Power 52, 1–7 (2024) Wang M, Qin F: A TCN-linear hybrid model for chaotic time series forecasting. Entropy 26, 467 (2024) Yumkhaibam S, Kusre BC: Application of artificial neural networks for time series rainfall forecasting in the loktak lift irrigation command area of manipur, india. Irrigat. Drain. 73, 741–756 (2024) Ehteram M, Ahmed AN, Sheikh Khozani Z, El-Shafie A: Convolutional neural network -support vector machine model-gaussian process regression: A new machine model for predicting monthly and daily rainfall. Water Resourc. Manage. 37, 3631–3655 (2023) Abiy AZ, Wiederholt RP, Lagerwall GL, Melesse AM, Davis SE: Multilayer feedforward artificial neural network model to forecast florida bay salinity with climate change. Water (Basel) 14, 3495 (2022) Gao L, Yang YM, Li Q, Ham YG, Kim JH: Deep learning for predicting winter temperature in north China. Atmosphere 13, 702 (2022) Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Wu","email":"","orcid":"","institution":"Guangxi Science and Technology Normal University","correspondingAuthor":false,"prefix":"","firstName":"Jiansheng","middleName":"","lastName":"Wu","suffix":""}],"badges":[],"createdAt":"2025-03-15 03:53:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6230198/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6230198/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":81030597,"identity":"ef08bb39-6cb4-4161-a50e-0c37ff7dbdfc","added_by":"auto","created_at":"2025-04-21 11:17:25","extension":"jpeg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":125385,"visible":true,"origin":"","legend":"\u003cp\u003eFlowchart of the improved algorithm.\u003c/p\u003e","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6230198/v1/087b3f5fa500c44682d9eb2e.jpeg"},{"id":81031870,"identity":"0dae834c-a1db-426a-afd8-3fdd37fa769b","added_by":"auto","created_at":"2025-04-21 11:25:25","extension":"jpeg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":486829,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of initialized populations for chaos mapping.\u003c/p\u003e","description":"","filename":"floatimage2.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6230198/v1/155322ad9a1390137f45fed0.jpeg"},{"id":81030603,"identity":"6af0f614-d16b-4233-8e7a-aa1f2de12c18","added_by":"auto","created_at":"2025-04-21 11:17:26","extension":"jpeg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":352521,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of convergence process of different algorithms for single peak function.\u003c/p\u003e","description":"","filename":"floatimage3.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6230198/v1/32a99cd3140b67c601b53c81.jpeg"},{"id":81032430,"identity":"b8e26314-13e3-4a5e-aaf7-0005aa0ee3a8","added_by":"auto","created_at":"2025-04-21 11:33:25","extension":"jpeg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":580118,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of convergence process of different algorithms for multi-peak function.\u003c/p\u003e","description":"","filename":"floatimage4.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6230198/v1/35a60c0f3d3d08e7356f1ff9.jpeg"},{"id":81030606,"identity":"e5e75496-22e5-45ab-bd83-11c92c9e2a4d","added_by":"auto","created_at":"2025-04-21 11:17:26","extension":"jpeg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":265700,"visible":true,"origin":"","legend":"\u003cp\u003eStructure of the hybrid model.\u003c/p\u003e","description":"","filename":"floatimage5.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6230198/v1/d0a070ddef56137d6de5c0b5.jpeg"},{"id":81030607,"identity":"6db32417-5e88-483a-926d-f4f92b01fc4f","added_by":"auto","created_at":"2025-04-21 11:17:26","extension":"jpeg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":144644,"visible":true,"origin":"","legend":"\u003cp\u003eGraph of evaluation results of different models.\u003c/p\u003e","description":"","filename":"floatimage7.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6230198/v1/3c7738b3c918bf78cd6e29ca.jpeg"},{"id":81030601,"identity":"ab3e7967-b47c-420c-9626-96eab2bdff2c","added_by":"auto","created_at":"2025-04-21 11:17:25","extension":"jpeg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":123932,"visible":true,"origin":"","legend":"\u003cp\u003eComparison between predicted values and True values of different models.\u003c/p\u003e","description":"","filename":"floatimage6.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6230198/v1/4943f3590d316de55a06cfdf.jpeg"},{"id":82471661,"identity":"e0df038b-ebb7-447c-8678-265cf5cce48e","added_by":"auto","created_at":"2025-05-11 21:16:31","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3252410,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6230198/v1/03147399-a442-41a1-bcc9-6bd53022dfc5.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eCCLDBO-TCN-BiGRU-Attention: Research and Application of Hybrid Models for Temperature Prediction\u003c/p\u003e","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eWith frequent human activities, various industrial activities affect the climate [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. However, climate is crucial for human production, affecting agricultural production. The proverb, \"There is no harvest in the water, more harvest, less harvest lies in the fertilizer\", highlights that climate plays a determining factor in agricultural production. The Guizhong region is dominated by sugarcane cultivation as a cash crop and is the leading sugar production area in Guangxi [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Since temperature prediction is a good guide to agricultural production, it is vital to predict the pattern of tapping temperature changes [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eWang et al. used the Elman neural network model to predict the temperature data in Chongqing and concluded that the model has a good fit [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Wang et al. [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] proposed the SA-PSO-XGBoost prediction model and compared it with XGBoost, GRU, and LSTM neural net-works. They concluded that SA-PSO-XGBoost has high accuracy and robustness. Nevertheless, with the rapid development of computer technology and the massive accumulation of meteorological data, temperature prediction has new possibilities and challenges [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. Traditional machine learning methods, such as Support Vector Machines (SVM), Back Propagation Neural Networks (BPNN), Relevance Vector Machines (RVM), and Radial Basis Functions (RBF), have some limitations in dealing with non-smooth, non-linear data [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. In recent years, deep learning methods such as Recurrent Neural Networks (RNN), Long Short-Term Memory Neural Networks (LSTM), and Convolutional Neural Networks (CNN) have been rapidly applied for time series prediction [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. However, as the requirements for prediction accuracy increase, a single model often fails to fully exploit the hidden features in the data. Thus, hybrid models have become increasingly important [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe benefit of hybrid models is combining the features and advantages of multiple models, thus im-proving the accuracy and robustness of prediction [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Specifically, hybrid models can better adapt to complex data characteristics and variations by integrating predictions from different models or utilizing the output of one model as input to another [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. Combining multiple models helps to improve the effectiveness of temperature forecasts and achieve better results in practice. For example, Wang et al. [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] used three classical chaotic time series predictions in a TCN hybrid model, achieving an R2 value close to 1. Satish [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] developed an ANN model to predict the rainfall in Loktak Lift Irrigation in Manipur, India, from 1985 to 2018 and concluded that the model has high predictive accuracy. Mohammad et al. [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] developed the CONN-SVM-GPR for rainfall in the Terengganu River Basin, Malaysia, and derived a 5.9%-23% and 20%-60% reduction in daily and monthly scale MAE for the model. Anteneh et al. [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] used a multilayer artificial neural network model to predict salinity in Florida Bay using two climate change scenarios, where the Representative Concentration Pathway (RCP) 4.5 and 8.5 salinity may increase by 30% and 70%, respectively. Gao et al. [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] relied on deep learning to predict winter temperatures in North China, using six CMIP5 models with relatively small biases to train the CNN to derive the best performance for temperatures over 10 years.\u003c/p\u003e \u003cp\u003eTemperature prediction in the Guizhong region is a challenging time series forecasting task because the temperature is affected by many factors, including seasonal changes, geographic location, and altitude. This paper combines the Temporal Convolutional Network (TCN), Bidirectional Gated Recurrent Unit (BiGRU), Attention mechanism, and Improved Dung Beetle Optimization Algorithm (Cir-cle- MSCA-Levy Dung Beetle Optimizer, CCLDBO) to propose a new hybrid model called CCLDBO-TCN-BiGRU-Attention. The improved dung beetle optimization algorithm optimizes the learning rate, the number of neurons in BiGRU, the key value of the attention mechanism, and the regularization parameter of the TCN-BiGRU-Attention model to enhance the prediction performance of the hybrid model. The hybrid model CCLDBO-TCN-BiGRU-Attention is used to predict the temperature in the Guizhong region.\u003c/p\u003e"},{"header":"2 Improved Dung Beetle Optimization Algorithm","content":"\u003cp\u003eThe Dung Beetle Optimizer (DBO) algorithm is inspired by the dung beetle\u0026apos;s behaviors of rolling, dancing, foraging, breeding, and stealing. Each dung beetle cluster comprises four different kinds of agent dung beetles, namely, the ball-rolling dung beetle, the breeding dung beetle (breeding ball), the little dung beetle, and the stealing dung beetle. The diverse position update strategies of the dung beetle algorithm can explore the search space more comprehensively and effectively solve complex ground search and optimization problems in practical applications.\u003c/p\u003e\n\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n \u003ch2\u003e2.1 Motivation for Improvement\u003c/h2\u003e\n \u003cp\u003eThe DBO algorithm is characterized by solid optimality-seeking ability and fast convergence speed. Still, the uneven distribution of the initial population and unbalanced global exploration and local exploitation ability cause the algorithm to fall into the local optimum easily. Therefore, to overcome these problems of DBO, this paper improves DBO using Circle mapping, the Modified Sine Cosine Algorithm (MSCA), and the Levy flight strategy.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n \u003ch2\u003e2.2 Process Improvement\u003c/h2\u003e\n\u003cp\u003eFigure 1 illustrates the flowchart of CCLDBO.\u003c/p\u003e\n \u003cp\u003eThe CCLDBO algorithm contains three improvements: the Circle mapping strategy, fusing the Cauchy variation algorithm into the improved sine-cosine algorithm, and the Levy flight. To verify the independent validity of the different improvement strategies proposed in this paper, DBO is combined with each of these three strategies independently:\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003e1) CDBO is obtained after initializing the population by adding a Circle mapping strategy in DBO.\u003c/p\u003e\n\u003cp\u003e2) SCDBO is obtained after embedding MSCA in CDBO.\u003c/p\u003e\n\u003cp\u003e3) The improved algorithm model, CCLDBO, is obtained after adding Levy flight at the end of SCDBO.\u003c/p\u003e\n\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\n \u003ch2\u003e2.3 Introducing Population Initialization for Circle Mapping\u003c/h2\u003e\n \u003cp\u003eThe essence of the DBO algorithm, which is a typical representative of intelligent optimization algorithms, is a stochastic search algorithm. The initial population is generated through random initialization. Still, the process is prone to problems such as uneven distribution of the positions of dung beetle individuals, weak global exploration ability, and easy falling into the local optimum due to low population diversity. To address these issues, a chaotic mapping characterized by stochasticity, ergodicity, and regularity is used in the initialization phase of the DBO population to generate a highly diverse initial population. Through comparison and analysis, this paper employs a trigonometric function-based Circle mapping, which promotes a more even distribution of the initial position of the population.\u003c/p\u003e\n \u003cp\u003eCircle mapping is defined as follows:\u003c/p\u003e\n \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$$\\:{x}_{i+1}={mod}({x}_{i}+0.2-\\left(\\frac{0.5}{2\\pi\\:}\\right){sin}(2\\pi\\:{x}_{i}),1)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{x}_{i+1}\\:\\)\u003c/span\u003e\u003c/span\u003edenotes the i-th chaotic sequence number and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{mod}(a,b)\\)\u003c/span\u003e\u003c/span\u003e is the remainder operation of \u003cem\u003ea\u003c/em\u003e on \u003cem\u003eb\u003c/em\u003e.\u003c/p\u003e\n \u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e depicts the distribution of 1000 sequence values generated by each of the four chaotic mappings: common random numbers, Cubic mapping, Bernoulli mapping, and Circle mapping. Figure \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e reveals that the distribution of the chaotic sequence values generated by the Circle mapping be-tween 0 and 1 is more uniform than that of the sequences generated by the ordinary random numbers, the Cubic mapping, and the Bernoulli mapping. The uniform distribution is more helpful in expanding the search range of the dung beetle population in space, increasing the diversity of the population location, improving the defect that the algorithm easily falls into the local extremes to some extent, and further improving the convergence accuracy and optimization efficiency of the algorithm.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n \u003ch2\u003e2.4 Improved Sine Cosine Algorithm Fused with Kersey Variation Algorithm\u003c/h2\u003e\n \u003cp\u003eDuring the iterative search for an optimal solution in the dung beetle algorithm and when iterating several times, current dung beetles will quickly gather near the optimal position and exhibit a near-optimal solution. However, the location updates of the dancing dung beetles are too random, increasing randomness in the distance and direction of the population search, i.e., the optimization method. Suppose the current optimal position is not the globally optimal point. In that case, the dung beetle population will be stuck in a local optimal solution and cannot find the genuinely optimal position. Thus, this paper introduces the improved sine-cosine algorithm and incorporates the Cauchy variation algorithm, promoting the rapid propagation of information in the population, controlling the optimization searching method, increasing the search space of the dung beetle, and increasing the freedom of each individual. This strategy prevents the population from falling into a locally optimal solution and makes it more likely to find a globally optimal solution.\u003c/p\u003e\n \u003cp\u003eThe improved sine algorithm (MSA) is inspired by various algorithms related to SCA, such as the Sine Cosine Algorithm (SCA), Sine Algorithm (SA), and Exponential Sine Cosine Algorithm (ESCA) functions, as well as improved Sine Cosine Algorithm (ISCA). The ESCA, ISCA, and other SCA-related algorithms involve an iterative optimization strategy using the sine function, which has a robust global exploration capability. Additionally, the Cauchy distribution function is relatively flat at the origin but shows a significant difference in distribution at both ends. The Cauchy variation can generate more significant perturbations in the vicinity of dung beetle individuals, broadening their distribution and helping to jump out of local optima. Hence, to better utilize this advantage, this paper improves the sine-cosine algorithm by incorporating the Cauchy operator, exploring the effect of the two-end variability of the Cauchy distribution function to optimally count the globally optimal individuals, thereby achieving the global optimum. The standard Cauchy distribution function is formulated as follows:\u003c/p\u003e\n \u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e$$\\:f\\left(x\\right)=\\frac{1}{\\pi\\:}\\left(\\frac{1}{{x}^{2}+1}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cp\u003eAccelerating algorithm convergence and improving problem-solving efficiency can be achieved by introducing the Cauchy distribution function on the improved sine-cosine algorithm and changing the square brackets in the sine function to absolute values in the original improved sine algorithm. In addition, introducing non-linear inertia weight coefficients during the position update process enables the algorithm to search the local region adequately, balancing global exploration and local exploitation capabilities. The position update formula for the improved sine-cosine algorithm incorporating the Cauchy operator is formulated as follows:\u003c/p\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e$$\\:{x}_{i}(t+1)={\\omega\\:}_{t}{x}_{i}\\left(t\\right)+{r}_{1}\\times\\:\\text{cauchyrnd(}{r}_{2})\\times\\:|{r}_{3}{p}_{i}\\left(t\\right)-{x}_{i}\\left(t\\right)|$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cp\u003ewhere t denotes the number of current iterations, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\omega\\:}_{t}\\)\u003c/span\u003e\u003c/span\u003e is the inertia weight, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{x}_{i}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e represents the i-th position component of individual X in the t-th iteration, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{p}_{i}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e denotes the i-th component of the best individual position variable in the t-th iteration. Additionally, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{r}_{1}\\)\u003c/span\u003e\u003c/span\u003e is a non-linear decreasing function with a random number \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{r}_{2}\\)\u003c/span\u003e\u003c/span\u003e generated from the interval [0, 2\u0026pi;] and a random number \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{r}_{3}\\)\u003c/span\u003e\u003c/span\u003e randomly generated in the interval [-2, 2]. Cauchyrnd() is the Cauchy distribution function.\u003c/p\u003e\u003cp\u003eThe Cauchy operator determines the change in the value of A between 0 and 1. The expression is:\u003c/p\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e$$\\:{r}_{1}=\\frac{{{\\omega\\:}}_{\\text{m}\\text{a}\\text{x}}-{{\\omega\\:}}_{\\text{m}\\text{i}\\text{n}}}{2}\\text{c}\\text{a}\\text{u}\\text{c}\\text{h}\\:\\text{y}\\text{r}\\text{n}\\text{d}\\left(\\frac{\\pi\\:t}{M}\\right)+\\frac{{{\\omega\\:}}_{\\text{m}\\text{a}\\text{x}}+{{\\omega\\:}}_{\\text{m}\\text{i}\\text{n}}}{2}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\omega\\:}_{max}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\omega\\:}_{min}\\)\u003c/span\u003e\u003c/span\u003e denote the maximum and minimum values, t represents the current number of iterations, and M indicates the maximum number of iterations.\u003c/p\u003e\n \u003cp\u003eThis paper employs a non-linear inertia weighting strategy, where the inertia weights are exponentially decreasing with the number of iterations:\u003c/p\u003e\n \u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e$$\\:{\\omega\\:}_{t}=\\frac{{e}^{\\frac{t}{M}}-1}{{e}^{1}-1}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003cp\u003eTo further improve the ability of the DBO algorithm to coordinate global exploration with local exploitation, the MSCA bootstrapping mechanism is introduced as an alternative to the dung beetle tangent-dancing strategy embedded in the DBO algorithm, i.e., a Cauchy variation operation is performed on the entire dung beetle individual to bootstrap the dung beetle position update in the ball-rolling phase. The improved formula is:\u003c/p\u003e\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e$$\\:{x}_{i}(t+1)=\\left\\{\\begin{array}{cc}{x}_{i}\\left(t\\right)+\\alpha\\:\\times\\:k\\times\\:{x}_{i}(t-1)+b\\times\\:\\varDelta\\:x,\u0026amp;\\:R2\u0026lt;ST\\\\\\:{\\omega\\:}_{t}{x}_{i}\\left(t\\right)+{r}_{1}\\times\\:\\text{cauchyrnd}\\left({r}_{2}\\right)\\times\\:\\left|{r}_{3}{p}_{i}\\right(t)-{x}_{i}(t\\left)\\right|,\u0026amp;\\:R2\\ge\\:ST\\end{array}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003cp\u003eFor this process, we define R2\u0026thinsp;=\u0026thinsp;rand() and ST \u0026isin; (0.5,1] is the basis for determining whether the dung beetle rolls purposefully. If R2\u0026thinsp;\u0026lt;\u0026thinsp;ST, Dung Beetle possesses a clear rolling target and is in the normal global exploration phase. Conversely, if R2\u0026thinsp;\u0026ge;\u0026thinsp;ST, the dung beetle does not have a specific rolling target but will adopt Cauchy\u0026apos;s variation function to search for movement, which is more random than the original sine function. Such an approach effectively improves the shortcomings of the DBO algorithm regarding excessive randomness in position updating. Each dung beetle individual can explore the search space better and exchange information with the current optimal individual. This process speeds up information propagation throughout the entire group and solves the original algorithm\u0026apos;s lack of inter-individual information communication.\u003c/p\u003e\u003cp\u003eThe bootstrapping mechanism of MSCA, which allows individual dung beetles to freely explore and locally optimize within the region delineated by the algorithm, somehow expands the search space and allows the algorithm to converge to the optimal solution gradually. This enhances the algorithm\u0026apos;s global search capability. Meanwhile, according to Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e), the parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{r}_{1}\\)\u003c/span\u003e\u003c/span\u003e is the key to regulating the dung beetle search, which can be used to regulate the dung beetle\u0026apos;s search range and direction and optimize the solution of the DBO algorithm. In addition, from Eq. (\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\omega\\:}_{t}\\)\u003c/span\u003e\u003c/span\u003e will keep shrinking the search space, and the inertia weights will decrease as the number of iterations increases. In the early stages of an algorithm\u0026apos;s iteration, relatively large inertia weights can give the algorithm a global solid exploration capability. In contrast, in the later stages, relatively small inertia weights are beneficial to improve its local exploitation capability.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\u003ch2\u003e2.5 Formula Levy Flight Strategy\u003c/h2\u003e\u003cp\u003eAccording to the updated formulas for the four individual behaviors of the dung beetle, the foraging, breeding, and stealing behaviors are caught in a local search, except for the ball-rolling behavior, which has a better global search performance. To solve the problem, this paper adopts the Levy flight strategy, which realizes random wandering by providing a random factor and generating a step size that conforms to the Levy flight to increase the dung beetle to improve the global optimization-seeking ability in different individual behaviors.\u003c/p\u003e\u003cp\u003eThe Levy flight is a probability distribution with long-tailed properties. By introducing the step size of Levy\u0026apos;s flight generation to increase the algorithm\u0026apos;s exploration of space, dung beetles can jump out of the local optimum and improve the ability to search globally during the search process of random wandering.\u003c/p\u003e\u003c/div\u003e"},{"header":"3 Optimization Algorithm Testing Experiments","content":"\u003cp\u003eWe select benchmark functions with different characteristics from CEC2005 to compare the search speed and quality of the optimal solutions of various algorithms. Additionally, we verify the convergence performance of the CCLDBO algorithm and whether it can jump out of the local optimum.\u003c/p\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Multiple Benchmark Function Testing and Analysis\u003c/h2\u003e \u003cp\u003eIn this section, we analyze CCLDBO based on the 23 benchmark functions reported in Tables\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, and \u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, where dimension denotes the dimension of the set function. The range is the function's search space boundary, and minf is the function's best fitness value. The number of search agents is uniformly set to 30, the maximum number of iterations is 100, and each algorithm is run independently for 30 times.\u003c/p\u003e \u003cp\u003eThe following experiments compare the GWO, BWO, DBO, SCDBO, and CCLDBO algorithms. Seven single-peak benchmark functions (F1-F7) are selected to analyze each algorithm's single-objective solving ability, and 16 multi-peak benchmark functions (F8-F23) compare the models' convergence speed and accuracy. These trials aim to verify the effectiveness of each improvement strategy and analyze whether the algorithms can jump out of the local optimum where the dimension of the benchmark function is uniformly set to 30.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Single Peak Function Analysis\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e (F1 \u0026ndash; F7) presents the seven single-peak benchmark test functions. Since the single-peak function has only one minimum, it can be used to check the performance of the algorithm development. Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e illustrates the single-peak convergence process to compare the individual algorithmic strategies. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e reveal that CCLDBO attains the highest convergence accuracy on the F1-F4 single-peak test functions and successfully converges to the theoretically optimal solution. For the F5 single-peak test function, although the convergence accuracy of BWO is higher than that of CCLDBO, the convergence speed of CCLDBO is significantly better than that of the other algorithms, reaching the desired accuracy in a short time. For the F6-F7 single-peak test function, the superiority of CCLDBO is even more apparent, dominating in convergence speed and reaching the theoretical optimum in both convergence accuracy and convergence theory.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSingle-peak benchmark functions.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\"\u0026minus;\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBasis function\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDimension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRealm\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_1}(x)=\\sum\\limits_{{i=1}}^{n} {x_{i}^{2}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-100, 100]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_2}(x)=\\sum\\limits_{{i=1}}^{n} | {x_i}|+\\prod\\limits_{{i=1}}^{n} | {x_i}|\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-10, 10]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_3}(x)=\\sum\\limits_{{i=1}}^{n} {{{\\left( {\\sum\\limits_{{j=1}}^{i} {{x_i}} } \\right)}^2}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-100, 100]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{\\text{4}}}(x){\\text{=ma}}{{\\text{x}}_i}\\left\\{ {\\left| {{{\\text{x}}_{\\text{i}}}} \\right|,{\\text{1}} \\leqslant i \\leqslant n} \\right\\}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-100, 100]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{\\text{5}}}(x){\\text{=}}\\sum\\limits_{{{\\text{i=1}}}}^{{\\text{n}}} {\\left[ {{\\text{100}}{{\\left( {{x_{{\\text{i+1}}}}{\\text{-x}}_{{\\text{i}}}^{{\\text{2}}}} \\right)}^{\\text{2}}}{\\text{+}}{{\\left( {{x_{\\text{i}}}{\\text{-1}}} \\right)}^{\\text{2}}}} \\right]}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-30, 30]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{\\text{6}}}(x){\\text{=}}\\sum\\limits_{{{\\text{i=1}}}}^{{\\text{n}}} {{{\\left( {[{x_i}{\\text{+0}}.{\\text{5}}]} \\right)}^{\\text{2}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-100, 100]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{\\text{7}}}(x){\\text{=}}\\sum\\limits_{{{\\text{i=1}}}}^{{\\text{n}}} {x_{{\\text{i}}}^{{\\text{4}}}} {\\text{+random}}[{\\text{0}},{\\text{1}})\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-128, 128]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn conclusion, the CCLDBO algorithm shows a significant advantage in terms of average search accuracy and convergence speed over the test function of a single peak when compared to GWO, BWO, DBO, and SCDBO. This demonstrates that the improved CCLDBO algorithm is robust in balancing global exploration and jumping out of local optimal solutions.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Multi-peak Function Analysis\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e presents six multi-peak functions (F8-F13). Unlike the single-peak functions, each of the multi-peak functions in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e has multiple extremes, and the dimensions of each function are 30, 50, and 100, respectively. The exploration capability of the proposed algorithm is evaluated with the help of a multi-peak function. The functional expressions for the multi-peak functions F14-F23 are reported in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Besides, the dimension of each function is fixed, and the convergence process is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMulti-peak benchmark functions for non-fixed dimensions.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\"\u0026minus;\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBasis function\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDimension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRealm\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_8}(x)=\\sum\\limits_{{i=1}}^{n} - {x_i}\\sin \\left( {\\sqrt {|{x_i}|} } \\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-500, 500]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_9}(x)=\\sum\\limits_{{i=1}}^{n} {\\left[ {x_{i}^{2} - 10\\cos \\left( {2\\pi {x_i}} \\right)+10} \\right]}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-5.12, 5.12]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{10}}(x)= - 20\\exp \\left( { - 0.2\\sqrt {\\frac{1}{n}\\sum\\limits_{{i=1}}^{n} {x_{i}^{2}} } } \\right) - \\exp \\left( {\\frac{1}{n}\\sum\\limits_{{i=1}}^{n} {\\cos } \\left( {2\\pi {x_i}} \\right)} \\right)+20+e\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-32, 32]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{11}}(x)=\\frac{1}{{4000}}\\sum\\limits_{{i=1}}^{n} {x_{i}^{2}} - \\prod\\limits_{{i=1}}^{n} {\\cos } \\left( {\\frac{{{x_i}}}{{\\sqrt i }}} \\right)+1\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-50, 50]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\begin{gathered} {F_{12}}(x)=\\frac{\\pi }{n}\\left\\{ {10\\sin (\\pi {y_1})+\\sum\\limits_{{i=1}}^{n} {{{\\left( {{y_i} - 1} \\right)}^2}} {{\\left[ {1+10{{\\sin }^2}(\\pi {y_{i+1}})} \\right]}^2}} \\right\\} \\hfill \\\\ +\\sum\\limits_{{i=1}}^{n} u \\left( {{x_i},10,100,4} \\right) \\hfill \\\\ {y_i}=1+\\frac{{{x_i}+1}}{4} \\hfill \\\\ \\end{gathered}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-50, 50]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\begin{gathered} {F_{13}}(x)=0.1\\{ {\\sin ^2}(3\\pi {x_1})+\\sum\\limits_{{i=1}}^{n} {{{({x_i} - 1)}^2}} \\left[ {1+{{\\sin }^2}(3\\pi {x_i}+1)} \\right] \\hfill \\\\ +{\\left( {{x_n} - 1} \\right)^2}\\left[ {1+{{\\sin }^2}\\left( {2\\pi {x_n}} \\right)} \\right]\\} +\\sum\\limits_{{i=1}}^{n} u \\left( {{x_i},5,100,4} \\right) \\hfill \\\\ \\end{gathered}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e30, 50, 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026minus;\" colname=\"c3\"\u003e \u003cp\u003e[-50, 50]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMulti-peak benchmark functions for fixed dimensions.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBasis function\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDimension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRealm\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{14}}(x)={\\left( {\\frac{1}{{500}}+\\sum\\limits_{{j=1}}^{{25}} {\\frac{1}{{j+\\sum\\limits_{{i=1}}^{2} {{{({x_i} - {a_{ij}})}^6}} }}} } \\right)^{ - 1}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[-65, 65]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{15}}(x)=\\sum\\limits_{{i=1}}^{{11}} {{{\\left[ {{a_i} - \\frac{{{x_1}\\left( {b_{i}^{2}+{b_i}{x_2}} \\right)}}{{b_{i}^{2}+{b_i}{x_3}+{x_4}}}} \\right]}^2}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[-5, 5]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{16}}(x)=4x_{1}^{2} - 2.1x_{1}^{4}+\\frac{1}{3}x_{1}^{6}+{x_1}{x_2} - 4x_{2}^{2}+4x_{2}^{4}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[-5, 5]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{17}}(x)={\\left( {{x_2} - \\frac{{5.1}}{{4{\\pi ^2}}}x_{1}^{2}+\\frac{5}{\\pi }{x_1} - 6} \\right)^2}+10\\left( {1 - \\frac{1}{{8\\pi }}} \\right)\\cos {x_1}+10\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[-5, 5]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\begin{gathered} {F_{18}}(x)=\\left[ {1+{{({x_1}+{x_2}+1)}^2}\\left( {19 - 14{x_1}+3x_{1}^{2} - 14{x_2}+6{x_1}{x_2}+3x_{2}^{2}} \\right)} \\right] \\hfill \\\\ \\times \\left[ {30+{{\\left( {2{x_1} - 3{x_2}} \\right)}^2} \\times (18 - 32{x_1}+12x_{1}^{2}+48{x_2} - 36{x_1}{x_2}+27x_{2}^{2})} \\right. \\hfill \\\\ \\end{gathered}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[-2, 2]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{19}}(x)=\\sum\\limits_{{i=1}}^{4} {{c_i}} \\exp \\left( { - \\sum\\limits_{{j=1}}^{3} {{a_{ij}}} {{\\left( {{x_j} - {p_{ij}}} \\right)}^2}} \\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[0, 1]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{20}}(x)=\\sum\\limits_{{i=1}}^{4} {{c_i}} \\exp \\left( { - \\sum\\limits_{{j=1}}^{6} {{a_{ij}}} {{\\left( {{x_j} - {p_{ij}}} \\right)}^2}} \\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[0, 1]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{21}}(x)= - \\sum\\limits_{{i=1}}^{5} {{{\\left[ {\\left( {X - {a_i}} \\right){{\\left( {X - {a_i}} \\right)}^T}+{c_i}} \\right]}^{ - 1}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[0, 10]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{22}}(x)= - \\sum\\limits_{{i=1}}^{7} {{{\\left[ {\\left( {X - {a_i}} \\right){{\\left( {X - {a_i}} \\right)}^T}+{c_i}} \\right]}^{ - 1}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[0, 10]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_{23}}(x)= - \\sum\\limits_{{i=1}}^{{10}} {{{\\left[ {\\left( {X - {a_i}} \\right){{\\left( {X - {a_i}} \\right)}^T}+{c_i}} \\right]}^{ - 1}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[0, 10]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTables\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e and \u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, and Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e infer that CCLDBO excels in exploring and developing balanced algorithms compared to other algorithms.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Validation of the Effectiveness of Various Strategies in CCLDBO\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e compares the experimental results of CCLDBO with GWO, BWO, DBO, and SCDBO. Combined with the results in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, which compares CCLDBO with GWO, BWO, DBO, and SCDBO, and Figs.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e and \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, which present the convergence process of each algorithmic strategy, we conclude the following:\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eExperimental results for benchmark functions (comparison of individual strategies).\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDBO\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSCDBO\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eCCLDBO\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eBWO\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eGWO\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e8.8838e\u003csup\u003e\u0026minus;\u0026thinsp;22\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.1582e\u003csup\u003e\u0026minus;\u0026thinsp;34\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2.9978e\u003csup\u003e\u0026minus;\u0026thinsp;79\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.420267e\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e4.2163e\u003csup\u003e\u0026minus;\u0026thinsp;13\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.5335e\u003csup\u003e\u0026minus;\u0026thinsp;16\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.5518e\u003csup\u003e\u0026minus;\u0026thinsp;17\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e8.42e\u003csup\u003e\u0026minus;\u0026thinsp;40\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.355398e\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.11e\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e8.6666e\u003csup\u003e\u0026minus;\u0026thinsp;14\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5.6797e\u003csup\u003e\u0026minus;\u0026thinsp;24\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4.7423e\u003csup\u003e\u0026minus;\u0026thinsp;75\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5.353404e\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.00011079\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e4\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.4483e\u003csup\u003e\u0026minus;\u0026thinsp;11\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.7699e\u003csup\u003e\u0026minus;\u0026thinsp;14\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2.6278e\u003csup\u003e\u0026minus;\u0026thinsp;36\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.625666e\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e6.9567e\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e5\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e7.0049\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e6.6892\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5.4402\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.269396e\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e7.2053\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e6\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e5.155e\u003csup\u003e\u0026minus;\u0026thinsp;6\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.8527e\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4.5594e\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e6.564202e\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e4.5235e\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e7\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.00015061\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.00049843\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3.9512e\u003csup\u003e\u0026minus;\u0026thinsp;6\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5.898160e\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e9.0054e\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e8\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-3377.4633\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-3821.8994\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-6693.304\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-4.146216e\u003csup\u003e+\u0026thinsp;3\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-2577.516\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e9\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2.986\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.075062e\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.2186e\u003csup\u003e\u0026minus;\u0026thinsp;6\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e10\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.5099e\u003csup\u003e\u0026minus;\u0026thinsp;14\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e8.8818e\u003csup\u003e\u0026minus;\u0026thinsp;16\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e8.8818e\u003csup\u003e\u0026minus;\u0026thinsp;16\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.286579e\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e8.2078e\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e11\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0098713\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e6.054247e\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.017069\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e6.7549e\u003csup\u003e\u0026minus;\u0026thinsp;6\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4.4018e\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3.0172e\u003csup\u003e\u0026minus;\u0026thinsp;8\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.920900e\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.041928\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e13\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.010988\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e9.7001e\u003csup\u003e\u0026minus;\u0026thinsp;8\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e9.9088e\u003csup\u003e\u0026minus;\u0026thinsp;8\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.502878e\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.095086\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e14\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e6.903484e\u003csup\u003e+\u0026thinsp;00\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e15\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0014887\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e 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\u003cp\u003e-3.767481e\u003csup\u003e+\u0026thinsp;00\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-3.8621\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e20\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-3.2031\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-3.1521\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-3.2031\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-2.797446e\u003csup\u003e+\u0026thinsp;00\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-3.1863\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e21\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-10.1532\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-5.0552\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-10.1532\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-8.334367e\u003csup\u003e+\u0026thinsp;00\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-10.1198\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e22\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-10.3154\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-9.7596\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-10.4029\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-1.003241e\u003csup\u003e+\u0026thinsp;1\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-10.3695\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eF\u003csub\u003e23\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-10.3632\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-8.007\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-10.5364\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-1.049222e\u003csup\u003e+\u0026thinsp;1\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-10.5091\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFor the single-peak benchmark functions (F1-F7), BWO outperforms CCLDBO in terms of convergence accuracy in the F5 function. Overall, the CCLDBO algorithm converges quickly and can balance exploration and exploitation, which is more conducive to jumping out of the local optimal solution.\u003c/p\u003e \u003cp\u003eFor the non-fixed dimensional multi-peak benchmark functions (F8-F13), CCLDBO performs better than some popular algorithms on most functions. It shows better exploration ability in finding the optimal solution of the multi-peak function.\u003c/p\u003e \u003cp\u003eFor the fixed-dimension multi-peak benchmark functions (F14-F23), CCLDBO finds the optimal value accurately. It strikes a better balance between global and local exploration and exploitation than the rest of the strategy algorithms.\u003c/p\u003e \u003cp\u003e`Therefore, the CCLDBO algorithm's optimization performance on the 23 benchmark functions is significantly better than the other metaheuristic algorithms, such as the basic DBO. Hence, the CCLDBO algorithm's comprehensive performance is outstanding among many metaheuristic algorithms.\u003c/p\u003e \u003c/div\u003e"},{"header":"4 Temperature Projections","content":"\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Establishment of the Data Set\u003c/h2\u003e \u003cp\u003eThis paper selected temperature data from 12 meteorological stations in the Guizhong region from 1965 to 2020. A correlation analysis revealed that the correlation coefficients between the data of each observation point were above 0.9, reaching a statistically strong correlation for further study. As shown in Table \u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eCorrelation matrix of temperature data from various weather stations.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"13\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv 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colname=\"c11\"\u003e \u003cp\u003e0.996\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.987\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e0.918\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eA2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e 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align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e0.995\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.981\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e 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align=\"left\" colname=\"c8\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e0.994\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e0.996\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.988\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e0.916\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eA7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e0.993\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e0.996\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.987\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e0.918\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eA8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.996\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.995\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e0.991\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e0.993\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.988\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e0.916\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eA9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.993\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.995\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.994\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.994\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e0.993\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e0.991\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.98\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e0.916\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eA10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.996\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.996\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e0.996\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e0.993\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.979\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e0.919\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eA11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.987\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.982\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.985\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.981\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.98\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.988\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e0.987\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e0.988\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e0.98\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e0.979\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e0.904\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eA12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.918\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.918\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.917\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.92\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.92\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.916\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e0.918\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e0.916\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e0.916\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c11\"\u003e \u003cp\u003e0.919\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c12\"\u003e \u003cp\u003e0.904\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c13\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"13\"\u003eNote: Table A1: Xincheng; A2: Liucheng; A3: Shatang; A4: Luzhai; A5: Liuzhou; A6: Xiangzhou; A7: Laibin; A8: Wuxuan; A9: Sanjiang; A10: Rongshui; A11: Jinxiu; A12: Liujiang.\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTo better simulate the temperature change in the Guizhong area, which is affected by other weather factors, we added sunshine duration, monthly average water vapor pressure, and monthly average relative humidity to the dataset to enrich its diversity.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Predictive Modeling\u003c/h2\u003e \u003cdiv id=\"Sec18\" class=\"Section3\"\u003e \u003ch2\u003e4.2.1 Temporal Convolutional Network\u003c/h2\u003e \u003cp\u003eThe Temporal Convolutional Network (TCN) can process time series data and has a primary structure of residual blocks containing dilated causal convolutions. It captures local patterns and long-range dependencies in sequences through its convolutional layers and is generally faster and easier to parallelize than traditional recurrent neural networks (RNNs). The causal convolution of TCN ensures that the output result depends only on the past input information, thus effectively avoiding the leakage of future information. The dilation convolution allows the input to be sampled at intervals during the convolution, which solves the problem of extracting the information from multivariate time series while enlarging the sensory field. This has enabled TCN to achieve good results in many time series pre-diction and analysis tasks. In the proposed model, the processed data series from the TCN layer is input to the BIGRU layer for the subsequent data analysis and processing step.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section3\"\u003e \u003ch2\u003e4.2.2 Bidirectional Gated Recurrent Unit\u003c/h2\u003e \u003cp\u003eA Bidirectional Gated Recurrent Unit (BiGRU) is a neural network architecture combining bidirectionality properties and a Gated Recurrent Unit (GRU). It consists of two GRU layers, one processing the input sequence in forward order and the other in reverse order. This allows the network to capture both forward and reverse dependencies in the input sequence, thus better capturing long-range dependencies in the sequence data.\u003c/p\u003e \u003cp\u003eThe output of BiGRU is usually a concatenation of the outputs of the GRU layers in both directions. It is used for various tasks, such as sequence prediction, classification, or sequence-to-sequence tasks. BiGRUs are widely used in natural language processing and time series analysis and can efficiently deal with complex relationships in sequence data.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section3\"\u003e \u003ch2\u003e4.2.3 Attention Mechanism\u003c/h2\u003e \u003cp\u003eAttention Mechanism is a technique used to enhance the performance of deep learning models, especially for processing sequential data. Its basic idea is to introduce learnable weights between different model parts to allocate different attention or importance between various parts. In the attention mechanism, given a sequence of inputs, the model can learn to dynamically assign different weights to each input element to capture the information more efficiently. This helps the model focus on the most relevant parts of the input sequence, improving its expressiveness and generalization.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec21\" class=\"Section3\"\u003e \u003ch2\u003e4.2.4 CCLDBO-TCN-BiGRU-Attention Hybrid Model\u003c/h2\u003e \u003cp\u003eThe TCN-BiGRU-Attention hybrid model stacks three layers of TCN residual modules to obtain a larger range of sensory fields of the input sequences and extract and downscale the features while avoiding problems such as gradient explosion and gradient vanishing. Each residual block has the same kernel size k, and its dilation factors D are 1, 2, and 4, respectively. Additionally, BiGRU ac-quires the TCN-processed data sequences and processes them in both directions using a time step from front to back (forward) and back to front (reverse). In this way, BiGRU can explore the dependencies of the timesteps more thoroughly and acquire contextual associations. The BiGRU-processed data sequences are then output to the attention layer to enhance the feature capture of the data sequences. Finally, the fully connected layer maps the high-dimensional features to the final prediction results. In this process, the learning rate, the number of neurons in BiGRU, the key value of the attention mechanism, and the regularization parameter of the TCN-BiGRU-Attention model are optimized using the improved dung beetle optimization algorithm (CCLDBO). The structure of the hybrid model is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"5 Results and Analysis","content":"\u003cdiv id=\"Sec23\" class=\"Section2\"\u003e \u003ch2\u003e5.1 Evaluation Indicators\u003c/h2\u003e \u003cp\u003eThe prediction effect of CCLDBO-TCN-BiGRU-Attention was evaluated based on all features of the first 10 samples to predict the temperature of the following sample.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec24\" class=\"Section2\"\u003e \u003ch2\u003e5.2 Analysis of Results\u003c/h2\u003e \u003cp\u003eThe reliability of various hybrid models, such as RVM-Adaboost, CNN-BiLSTM-Adaboost, CNN-BiLSTM-ATTENTION, CNN-BiGRU-ATTENTION, and TCN-BiGRU-ATTENTION was com-pared. The performance of each model and the vital role of each module were evaluated using the R2, MAE, RMSE, MSE, and MAPE metrics, with Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e presenting the corresponding results.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e highlights that the proposed CCLDBO-TCN-BiGRU-Attention model has an optimal performance in all indicators. Comparing the CNN-BiGRU-ATTENTION and TCN-BiGRU-ATTENTION reveals that the hybrid model of TCN is significantly better than the hybrid model of CNN in the pre-diction of this time series.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThis study also made predictions for data at different points in time and then compared the predictions of the six competitor models. The experimental results are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, which reveals that the prediction results of each model are close to the real value, and all of them can predict the trend of the real power. However, the proposed hybrid model is closer to the real value. Combining the results in Figs.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e and \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, CCLDBO-TCN-BiGRU-Attention shows a good performance in the fitting effect of prediction.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eComparison of prediction results for different step sizes\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"8\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003et\u0026thinsp;+\u0026thinsp;1\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003et\u0026thinsp;+\u0026thinsp;2\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003et\u0026thinsp;+\u0026thinsp;3\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003et\u0026thinsp;+\u0026thinsp;4\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003et\u0026thinsp;+\u0026thinsp;5\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003et\u0026thinsp;+\u0026thinsp;6\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"4\" rowspan=\"5\"\u003e \u003cp\u003eTCN-BiGRU-Attention\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.49532\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.96577\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.57397\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.59167\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.62841\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.237\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eMAE\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.029195\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.052057\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.032778\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.033228\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.034195\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.062077\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eMAPE\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.42893\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.4185\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.60733\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.7068\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.84823\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2.9281\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eMSE\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.65493\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.191\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.77931\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.84071\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.921\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.7112\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.989869\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.965691\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.9855\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.982986\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.955187\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.929656\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eR2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"4\" rowspan=\"5\"\u003e \u003cp\u003eCCLDBO-TCN-BiGRU-Attention\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.42355\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.44561\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.3872\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.39941\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.4024\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.4167\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eMAE\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.023695\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.024529\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.023547\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.024052\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.023766\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.023854\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eMAPE\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.32842\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.34296\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.27683\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.30164\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.26521\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.30673\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eMSE\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.57308\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.58563\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.52615\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.54922\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.51498\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.55383\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.9925\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.99208\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.99353\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.99296\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.99384\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.99288\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eR2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe prediction period was increased further to test the performance of the CCLDBO-TCN-BiGRU-Attention hybrid model. All features of the first 10 samples were used to predict the temperatures at step t\u0026thinsp;+\u0026thinsp;1, step t\u0026thinsp;+\u0026thinsp;2, step t\u0026thinsp;+\u0026thinsp;3, step t\u0026thinsp;+\u0026thinsp;4, step t\u0026thinsp;+\u0026thinsp;5, and step t\u0026thinsp;+\u0026thinsp;6. The corresponding results are reported in Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, demonstrating that the prediction accuracy of TCN-BiGRU-Attention decreases as the step size of the predicted time series gradually increases. Additionally, the difficulty of capturing the data correlation in the previous moments improves. However, adding CCLDBO smooths the evaluation index, affecting the parameter optimization process of the hybrid model.\u003c/p\u003e \u003c/div\u003e"},{"header":"6 Conclusion","content":"\u003cp\u003eThis paper introduces the CCLDBO-TCN-BiGRU-Attention hybrid model for weather prediction. Specifically, this study proposes a multi-strategy improved algorithm, CCLDBO, which is based on the dung beetle optimization algorithm. Verified by 23 benchmark test functions, CCLDBO demonstrates higher efficiency and more reliable results. Secondly, the developed TCN-BiGRU-Attention hybrid model is proven better than the one based on CNN-BiGRU in prediction. After introducing the CCLDBO optimization algorithm to optimize the model parameters, its prediction accuracy is improved for single- and multi-step predictions across time. Finally, the experimental results of temperature prediction in the Guizhong region show that the proposed CCLDBO-TCN-BiGRU-Attention hybrid model performs well in weather prediction.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgment:\u003c/strong\u003e Not applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions:\u003c/strong\u003e R.Y.: conceptualisation, methodology, modelling, writing original draft preparation. L.J.: editing, revision. W.L.: revision. J.W.: writing reviewing and editing. All authors have read and agreed to the published version of the manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding:\u003c/strong\u003e This research was funded by the National Natural Science Foundation of China (NSFC) Project: Deep Learning-Co-evolutionary Support Vector Machine Short-term Climate Hybrid Prediction Modeling of Monthly Precipitation [Project No. 42065004]; the Guangxi Innovation Drive Development Special Project (Science and Technology Major Special Project) \u0026quot;Human-Machine Intelli-gent Interaction Touch Terminal Manufacturing Key Technology and Industrial Cluster Applica-tion\u0026quot; [Project No.: Gui Ke AA21077018]; the Sub-project: Touch Display Integration Intelligent Touch System and Industrial Cluster Application [Project No.: Gui Ke AA21077018-2]; and the 2024 Guangxi University Young and Middle-aged Teachers\u0026apos; Scientific Research Basic Ability Enhancement Project: Research on the Application of Improved YOLOv5 in Detecting and Recognizing Sugar Cane Red Rot Disease [Project No.: 2024KY0868].\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability Statement:\u003c/strong\u003e The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author/s.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting Interests:\u003c/strong\u003e The authors declare that they have no conflict of interest.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthical approval:\u003c/strong\u003e Not applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent to participate:\u003c/strong\u003e Not applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent for publication:\u003c/strong\u003e Not applicable\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eYin X, Zhu H, Gao J, Gao J, Guo L, Wang J: Effects of climate change and human activities on net primary productivity in the Northern Slope of Tianshan. 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Atmosphere 13, 702 (2022)\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"TCN-BiGRU-Attention, CCLDBO, hybrid model, temperature prediction","lastPublishedDoi":"10.21203/rs.3.rs-6230198/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6230198/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eMeteorological data present prominent spatio-temporal features and complex non-linear relationships, significantly challenging meteorological forecasting. Hence, to study the performance of hybrid models in weather prediction, this paper proposes a hybrid model named CCLDBO-TCN-BiGRU-Attention and experimentally compares it with current models. The proposed method improves the dung beetle optimization algorithm by introducing Circle mapping, the si-ne-cosine algorithm MSCA, and the dung beetle optimization algorithm using Levy flight (CCLDBO). The experimental results demonstrate that the developed method achieves outstanding results in a wide range of single-peak and multi-peak tests. Additionally, the evaluation index R2 reaches 0.9925 in the hybrid model for predicting the temperature in the Guizhong area, and the RMAE, RMSE, MSE, and MAPE metrics are the best compared with other models. The hybrid model is also closer to the real values than other models in predicting real value changes. Overall, the hybrid model performs well in temperature prediction and provides a feasible solution for weather prediction.\u003c/p\u003e \u003cp\u003e \u003cb\u003eTrial registration number\u003c/b\u003e: Not applicable.\u003c/p\u003e","manuscriptTitle":"CCLDBO-TCN-BiGRU-Attention: Research and Application of Hybrid Models for Temperature Prediction","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-04-21 11:17:21","doi":"10.21203/rs.3.rs-6230198/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"fc50b08e-f706-4e9e-9896-1094e5997764","owner":[],"postedDate":"April 21st, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-05-11T21:08:21+00:00","versionOfRecord":[],"versionCreatedAt":"2025-04-21 11:17:21","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6230198","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6230198","identity":"rs-6230198","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}