A new tropospheric delay combination prediction model based on time series decomposition and deep learning

A new tropospheric delay combination prediction model based on time series decomposition and deep learning | 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 A new tropospheric delay combination prediction model based on time series decomposition and deep learning Xiao Xu, YingChun Yue, Ming ShangGuan, YiFan Liang, ShaoFeng Bian, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3933886/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 Zenith tropospheric delay (ZTD) prediction is of great significance for high-precision navigation. However, ZTD modeling has proved to be challenging due to the presence of linear and nonlinear characteristics. In this paper, we propose a combination ZTD prediction model (SLA), which considers the trend-based and seasonal variations respectively. It decomposes ZTD time series via seasonal-trend decomposition procedure based on loess (STL), individually predicting nonlinear components with long short-term memory network (LSTM) and linear components with autoregressive integrated moving average model (ARIMA). Finally, the individual predictions are recombined. The SLA model is compared with LSTM, extreme learning machine model (ELM), ARIMA, and the empirical global pressure and temperature (GPT3) model. The SLA model shows the best result in all models by analyzing the evaluation indicators including root mean square error (RMSE, 1.32 cm), the average normalized root mean square error (NRMSE, 0.56%), mean absolute error (MAE, 0.98 cm) and the mean coefficient of determination (R 2 , 0.83). In addition, the data of different months was tested separately, and the result showed that the SLA model has the best performance of ZTD prediction. Moreover, the SLA model has good results up to 12h, with RMSE < 1.60 cm, NRMSE < 0.7%, MAE = 0.75. This study provides a new model to predict the ZTD, which is helpful for the precise positioning of GNSS and can be further applied in the study of meteorology. tropospheric delay modeling time-series decomposition deep learning combined prediction models Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 1 Introduction As the Global Navigation Satellite System (GNSS) signal traverses the atmosphere, it encounters a delay attributed to the non-vacuum environment, simultaneously undergoing trajectory elongation due to the bending phenomenon (Wu et al. 2023 ). The part due to the troposphere is termed tropospheric delay, which is an important error source in GNSS high-precision positioning (Ma et al. 2022 ). Several studies have shown that high-precision tropospheric delay can effectively accelerate the convergence speed of precise point positioning (PPP) and improve the accuracy of positioning accuracy (Zhao et al. 2022 ; Bahadur 2022 ; Huang et al. 2023 ). In addition, ZTD is widely used in numerical weather forecasts and climate monitoring (Zhao et al. 2020 ; Li et al. 2021 ; Liu et al. 2022 ). Therefore, it is important to establish a high-precision tropospheric delay model for climate research and high-precision positioning. In recent years, many researchers have focused on the establishment of high-precision tropospheric delay models (Yao et al. 2016 ; Mao et al. 2021 ; Xu et al. 2023 ). Tropospheric delay models can be categorized into models based on measured meteorological parameters and empirical models without meteorological parameters. Tropospheric delay models based on measured meteorological parameters (e.g., Hopfield model (Hopfield 1969 ), Saastamoinen model (Saastamoinen 1972 ), Black model (Black 1978 ), Askne & Nordius model (Askne and Nordius 1987 )) require inputs such as temperature, water vapor pressure, and relative humidity. Since meteorological parameters are often unavailable, empirical meteorological parameters are generally used, which leads to additional errors in the models. The empirical models (e.g., UNB series models (Collins and Langley 1997 ; Leandro et al. 2006 ; 2008 ), EGNOS models (Penna et al. 2001 ), Tropgrid models (Krueger et al. 2004 ; Schüler 2014 ), GPT models (Böhm et al. 2015 ; Lagler et al. 2013 ; Landskron and Böhm 2018 ), and GZTD models (Yao et al. 2013 ; 2016 ), are mostly based on historical radiosonde data, GNSS ZTD products or reanalysis data. With the rapid development of computer technology, deep learning technology has become one of the main methods for time series modeling in the troposphere with its excellent nonlinear modeling ability (Ding 2022 ; Shangguan et al. 2023 ). Since tropospheric delays are correlated with multiple parameters such as meteorological conditions, geographic location, and time, which have complex linear and nonlinear characteristics, it makes deep learning-based tropospheric delay modeling one of the research hotspots (Xiao et al. 2018 ; Osah et al. 2021 ; Su et al. 2022 ). Zhang et al. ( 2020 ) through the LSTM network established a model to predict ZTD, in which the average RMSE for the next 6h is 7.2mm. Yang et al. ( 2021 ) adopted the artificial neural network (ANN) to construct the correlation between ZTD derived from GPT3 and GNSS observations which effectively improved the systematic deviation of the GPT3 model. Li et al. ( 2022 ) improved the GPT3 model in the Antarctic region by incorporating the LSTM network and radial basis function (RBF). Zhang et al. ( 2022 ) predict ZTD used the transformer model with an average RMSE of 1.8 cm at 505 VMF3 stations. Lu et al. ( 2023 ) utilized the convolutional LSTM network (ConvLSTM) to build a tropospheric delay network (TropNet) with an accuracy better than 11 mm for the GNSS-ZWD. Previous experiments have demonstrated that the learning model based on one or more sites has good accuracy in predicting ZTD. However, the variation of ZTD is complex in space and time, including trend, seasonal variation and etc. These models do not take into account the variations characteristics in ZTD data specifically, which cannot make full use of the advantage of the deep learning model. In this paper, a combination of STL algorithm, LSTM neural network, and ARIMA model is used to predict ZTD, hereinafter referred to as SLA model. Among them, the STL algorithm is used to decompose the original ZTD time series into trend, seasonal and, residual terms. The LSTM neural network and the ARIMA model are used to predict the nonlinear and linear parts of the ZTD components, respectively. The prediction results of different components are reconstructed to obtain highly accurate ZTD values. Finally, the performance of the model is verified through multi-dimensional experimental analyses (ZTD forecasting for different regions, months, and prediction steps). 2 Data and Methods 2.1 Data Source As shown in Fig. 1 , we utilized ZTD data from 505 stations (green points in Fig. 1 ), sampled at 6h intervals, provided by VMF3 from 2019 to 2022 for model training and testing. In addition, we used ZTD data from 405 IGS stations (red points in Fig. 1 ) offered by the IGS analysis center to validate the reliability of the data. 2.2 Data preprocessing. Due to missing values in the data of some IGS stations. The completeness rate of IGS station data was statistically analyzed first. As shown in Fig. 2 , 13 stations have completeness rates below 60%, while 299 stations have completeness rates exceeding 90%. Among them, 277 stations are common to both IGS and VMF3. Therefore, the comparison is made by these 277 stations. In order to further investigate the correlation and agreement between VMF3- and IGS-ZTD, data from January 2022 to December 2022 were selected for the comparison. We first resampled IGS-ZTD data in 6h intervals and utilized the Grubbs test to remove outliers and subsequently applied the K Nearest Neighbor algorithm for missing value interpolation. Finally, using IGS-ZTD as the real value, the accuracy of VMF3-ZTD was evaluated at 277 global stations using four indicators: RMSE, NRMSE, MAE, and R 2 . The formulas for these indicators are as follows: (1)-(4), where is the total number, \(Z_{i}^{{true}}\) is the original value, \(Z_{i}^{{pre}}\) is the predicted value, \(\overline {{Z_{i}^{{true}}}}\) is the mean of \(Z_{i}^{{true}}\) . $$RMSE=\sqrt {\frac{1}{N}\sum\nolimits_{{i=1}}^{N} {{{(Z_{i}^{{pre}} - Z_{i}^{{true}})}^2}} }$$ 1 $$NRMSE=\frac{{RMSE}}{{\overline {{Z_{i}^{{true}}}} }}$$ 2 $$MAE=\frac{1}{N}\sum\nolimits_{{i=1}}^{N} {\left| {Z_{i}^{{pre}} - Z_{i}^{{true}}} \right|}$$ 3 $${R^2}=1 - \frac{{{{(Z_{i}^{{pre}} - Z_{i}^{{true}})}^2}}}{{{{(Z_{i}^{{true}} - \overline {{Z_{i}^{{true}}}} )}^2}}}$$ 4 The RMSE, NRMSE, MAE, and R 2 between VMF3-ZTD and IGS-ZTD are 1.18 cm, 0.51%, 0.91 cm, and 0.92, which indicate a small difference between VMF3-ZTD and IGS-ZTD in general. Figure 3 (a-d) depicts the statistical distribution of the number of stations at different ranges for RMSE, NRMSE, MAE, and R 2 . The results show that 97.5% of the stations have an RMSE less than 2 cm, 99.6% of the stations have an MAE less than 2 cm, 82.3% of the stations have a NRMSE less than 0.6%, and only 0.7% of the stations exceed 1%. Additionally, 82.3% of the stations have an R 2 greater than 0.9, while only 4.6% of the stations have an R 2 less than 0.6. In summary, the majority of sites have a high level of accuracy between VMF-ZTD and IGS-ZTD. Furthermore, the global distribution of RMSE, NRMSE, MAE, and R 2 for VMF3-ZTD and IGS-ZTD is shown in Fig. 4 . The accuracy tends to be generally lower for sites located in maritime areas compared to mainland areas, which could be due to the drastic water vapor variations over the oceans. 2.3 Prediction models Traditional ZTD forecasting prediction models are generally modeled for a single ZTD time series, which makes it difficult to capture the trending and seasonal variations. In this paper, three single models are combined to predict ZTD. The used methods are described below: 2.3.1 STL The STL is a classical method in time series decomposition. Based on STL method, the sequence can be decomposed into the following three components: $${Z_t}={T_t}+{S_t}+{R_t}$$ 5 where \({Z_t}\) is the original, \({T_t}\) is the trend component, \({S_t}\) is the period component and \({R_t}\) is the residual component (Cleveland et al. 1990 ). The STL algorithm can be used to process the arbitrary time series data, and the STL decomposition implementation mainly consists of inner and outer loop recursive process, and the two loops belong to a nested relationship. The inner loop is used for the calculation of trend and period components, while the outer loop is used to calculate the robustness weights for adjusting the neighborhood weights of the next inner loop based on the results of the previous inner loop (Chen et al. 2023 ). 2.3.2 LSTM The LSTM is a network improved by the recurrent neural network (RNN) and is a powerful method for deep learning of time series (Ding 2022 ). LSTM introduces a memory cell to solve the problems of gradient disappearance and explosion which occur in RNN invariably. The memory cell contains a memory block, and each memory block has three gate structures including the forgetting gate, the input gate, and the output gate. These three gate structures can read, write, and reset data. Because the output value of the Sigmoid function is between 0 and 1 and it can let the information flow through the door or not, the activation functions of the three gates are all S-type functions ( Hochreiter et al. 1997 ). 2.3.3 ARIMA The ARIMA can identify complex patterns in data and generate predictions, which can be used to analyze and predict univariate time series data (Box et al. 1976). The function of ARIMA is represented by p, d, q (p represents the number of autoregression items, d represents the number of non-seasonal differences, and q represents the number of lag prediction errors in the prediction equation). The three steps of establishing ARIMA are identification, estimation, and prediction (Adamowski and Chan 2011 ). 2.3.4 SLA The SLA model combines the STL, LSTM, and ARIMA, where the STL algorithm is employed to decompose the original ZTD sequences and extract characteristic information from different dimensions. These dimensions exhibit distinct linear and nonlinear features. Previous studies (Li et al. 2022 ; Zhang et al 2022 ; Shangguan et al. 2023 ) have indicated that LSTM has advantages in predicting nonlinear features, while the ARIMA model excels in forecasting linear features and stationary sequences. Therefore, this study utilizes the LSTM and ARIMA model to predict the ZTD characteristic information in different dimensions separately. Finally, the predictions are reconstructed to achieve high-precision ZTD forecasting results. The flow diagram of the SLA model is shown in Fig. 5 . In Fig. 5 , the prediction model based on STL decomposition and deep learning can be divided into the following four steps: Step 1: STL decomposes the original ZTD data into trend terms, seasonal terms, and residual terms; Step 2: ARIMA model is used to predict the trend term, and then the LSTM model is used to predict the seasonal term and residual term to get the predicted values; Step 3: Reconstruct the predicted values of the three parts of step 2 to get the final prediction results; Step 4: Model Evaluation: Evaluate the prediction accuracy of the model using four evaluation indexes: MAE, RMSE, R 2 , and NRMSE. 3 Results and Discussion In this section, we first present the result of the STL method in the SLA model. Subsequently, we analyze the optimal parameter settings for LSTM in predicting ZTD. Finally, we conduct multidimensional experimental analyses of the SLA composite model, including comparisons of different regions, different months, and different prediction steps. 3.1 STL results Figure 6 shows the STL decomposed ZTD time series of station ADIS (38.77°E, 9.04°N, 2439.2m). It can be concluded that the trend component retains the variable characteristics of the original time series, the volatility is smaller, and the sequence is relatively stable. Therefore, it is suitable for the ARIMA model considering the time relationship to predict. The residual component and the seasonal component have greater data volatility, retaining the nonlinear variation characteristics of the original time series, so the nonlinear LSTM model is suitable for predicting. It is feasible and meaningful to construct a combined model based on STL decomposition combined with the LSTM and ARIMA model for prediction research in the following. 3.2 LSTM results In order to analyze the influence of different training lengths on the predicted result of the model, this paper uses different training lengths to train the model and uses the ZTD data from January to March 2022 as the prediction data. Using the data of 12 steps, we predict the data of the next step (6h). The length of training is set to 1, 2, 3, and 5 years. Table 1 shows evaluation indicators in ZTD prediction results of different training lengths at 505 VMF3 sites. It can be concluded that when the length of training data is only 1 year, all evaluation indexes are poorest, but when the training data increases to 3 years or more, all evaluation indexes do not improve. The reason is that when the length of training data is less than 3 years the results are unstable. As the length of training data increases to 3, 4, and 5 years, the prediction results remain stable. Therefore, the 3 years data is selected as the training set data for the following experiments. Table 1 Average accuracy indicators of LSTM ZTD prediction experiments with different training lengths in 505 VMF3 stations from January to March 2022. Metric 1 year 2 years 3 years 4 years 5 years RMSE (cm) 1.99 1.68 1.56 1.57 1.55 NRMSE (%) 0.84 0.71 0.66 0.66 0.66 MAE (cm) 1.52 1.36 1.15 1.15 1.14 R 2 0.60 0.71 0.76 0.76 0.76 In order to analyze the influence of different input steps on the predicted result, this study uses different input steps (4, 8, 12, 16, and 20) to train the model with 1 to 4 prediction steps, respectively. The three years data from 2019 to 2022 are used as the training set. The test data is from January to March 2022. The experimental results are shown in Table 2 . The accuracy of the prediction results is the worst with the 4 input steps. With the increase of the length of the input steps, all the evaluation indexes improve. However, when the length of the input data is more than 16 steps, the evaluation indexes do not increase significantly. When the input steps are 16, the prediction has the best result. Therefore, the ZTD sequence with an input step size of 16 is determined hereafter. 3.3 SLA results 3.3.1 ZTD results for different regions Under the optimal LSTM strategy for predicting ZTD, we compared the accuracy of the SLA model with models such as LSTM, ARIMA, ELM, and GPT3. Table 3 shows that the SLA model has better accuracy compared to other models. Specifically, the average RMSE of the SLA model is 1.32 cm, which is 14.8% lower than that of LSTM, 16.5% lower than that of ARIMA, 21.9% lower than that of ELM and 61.5% lower than that of ARIMA. The average NRMSE is 0.56%, which is 15.2%, 16.4%, 25.3%, and 61.6% lower than that of LSTM, ARIMA, ELM, and GPT3 models respectively. Similarly, MAE improved by 14%, 16.2%, 21%, and 64.3%, respectively. The SLA model had the largest average R 2 value, up to 0.83, followed by LSTM(0.76), while the GPT3 model had the lowest R 2 only − 0.11. This shows that the combined model SLA can significantly improve the accuracy of prediction compared with other models. In order to verify the advantage of the combined model over the single decomposition model, the STL algorithm combined with the LSTM model (namely STL-LSTMA) and the STL algorithm combined with the ARIMA model (namely STL-ARIMA), are also included in Table 3 . The results show that the STL-LSTM model improves the average RMSE, NRMSE, MAE, and R 2 by 9%, 9.1%, 7.9%, and 3.9% compared to LSTM, respectively. Similarly the STL-ARIMA model improved by 3.8%, 3.0%, 3.4%, and 2.7% compared to ARIMA. The reason is that the LSTM and ARIMA models were able to capture more features such as seasonality and trend of the original sequences from the STL decomposed sequences. In addition, the forecasting results of STL-LSTM are better than those of STL-ARIMA, which is mainly due to large variations of the residual and seasonal terms with the ARIMA model. The SLA model used in this paper combined the advantages of ARIMA and LSTM models, and the average RMSE, NRMSE, MAE, and R 2 of 505 VMF3 stations are 1.32 cm, 0.56%, 0.98 cm, and 0.83, respectively, which are improved by 6.4%, 6.7%, 6.7%, and 5.1% relative to the STL-LSTM, respectively. It concludes that the combination model STA can utilize the advantages of different models to better learning more features in the original data, which has a better prediction effect and can reduce the error of ZTD prediction. Table 3 Average accuracy of SLA ZTD prediction experiments in 505 VMF3 stations from January to March 2022. Model RMSE (cm) NRMSE (%) MAE (cm) R 2 LSTM 1.55 0.66 1.14 0.76 ARIMA 1.58 0.67 1.17 0.75 ELM 1.69 0.75 1.24 0.70 GPT3 3.43 1.46 2.75 -0.11 STL-LSTM 1.41 0.60 1.05 0.79 STL-ARIMA 1.52 0.65 1.13 0.77 SLA 1.32 0.56 0.98 0.83 In order to further investigate the predictive ability of the models globally, the paper analyzes the global distribution of the indicators of the different models. Figure 7 (a-d), shows the distribution of RMSE, NRMSE, MAE, and R 2 of the SLA model in global regions, respectively. The results of six regions 90°N-60°N, 60°N-30°N, 30°N-0°, 0°-30°S, 30°S-60°S, and 60°S-90°S are shown in Table 4 . The accuracy of the SLA model is better in the high latitude than in the low latitude region, the accuracy on land is higher than that on the sea level. The overall mean values of RMSE, NRSEM, MAE, and R 2 of the SLA model increased with increasing latitude. The model has the best accuracy in the 60°S-90°S region, with RMSE, NRMSE, MAE, and R 2 of 0.69 cm, 0.31%, 0.49 cm, and 0.92, respectively, which is an improvement of 47.7%, 44.6%, 50%, and 9.7% compared to the global average accuracy. Table 4 Average accuracy of SLA ZTD prediction results for January-March 2022 at different latitudinal bands for the 505 VMF3 sites of the SLA model. Latitude Number of sites RMSE (cm) NRMSE (%) MAE (cm) R 2 90°N-60°N 38 0.77 0.34 0.57 0.93 60°N-30°N 247 1.17 0.51 0.84 0.81 30°N-0° 82 1.46 0.61 1.12 0.83 0°-30°S 77 1.59 0.66 1.24 0.84 30°S-60°S 48 2.05 0.87 1.52 0.79 60°S-90°S 13 0.69 0.31 0.49 0.92 3.3.2 ZTD forecasts for different months Due to the obvious seasonal variation of ZTD, the performance of ZTD prediction models in different months in 2022 is tested. Table 5 shows evaluation indexes of five prediction models in 12 months. All models have better results in winter, but with the arrival of summer, the model performance is gradually decreasing. However, the RMSE and MAE of the SLA model adopted in all months are smaller than other models, and R 2 is larger than other models. Figures 8 and 9 show the predicted value and error of the ZTD prediction by five models and three combined models at the BJCO (6.38°E, 2.45°N, 30.7m) site in January and July. As shown in Figs. 8 and 9 , the SLA model has the smallest error in predicting ZTD, which is most suitable for ZTD prediction. Table 5 Average accuracy of SLA ZTD prediction results of 505 VMF3 stations in different months in 2022. Month Model RMSE (cm) NRMSE (%) MAE (cm) R 2 January LSTM 1.53 0.65 1.15 0.73 ARIMA 1.55 0.66 1.16 0.73 ELM 1.69 0.72 1.28 0.67 GPT3 3.33 1.42 2.70 -0.21 STL-LSTM 1.41 0.61 1.06 0.76 STL-ARIMA 1.49 0.64 1.12 0.75 SLA 1.29 0.55 0.97 0.81 February LSTM 1.53 0.65 1.15 0.67 ARIMA 1.55 0.66 1.17 0.67 ELM 1.69 0.72 1.28 0.61 GPT3 3.36 1.43 2.74 -0.58 STL-LSTM 1.42 0.61 1.07 0.70 STL-ARIMA 1.50 0.64 1.13 0.69 SLA 1.31 0.56 0.99 0.76 March LSTM 1.53 0.65 1.13 0.73 ARIMA 1.59 0.68 1.18 0.70 ELM 1.68 0.72 1.26 0.67 GPT3 3.44 1.46 2.80 -0.30 STL-LSTM 1.41 0.61 1.05 0.76 STL-ARIMA 1.53 0.65 1.14 0.72 SLA 1.32 0.56 0.98 0.80 April LSTM 1.56 0.67 1.17 0.73 ARIMA 1.64 0.70 1.22 0.71 ELM 1.76 0.75 1.33 0.67 GPT3 3.41 1.45 2.77 -0.17 STL-LSTM 1.44 0.62 1.10 0.75 STL-ARIMA 1.57 0.67 1.18 0.73 SLA 1.36 0.58 1.02 0.80 May LSTM 1.66 0.70 1.22 0.71 ARIMA 1.76 0.75 1.31 0.67 ELM 1.85 0.78 1.40 0.64 GPT3 3.56 1.51 2.93 -0.27 STL-LSTM 1.55 0.66 1.15 0.73 STL-ARIMA 1.69 0.72 1.26 0.70 SLA 1.46 0.62 1.06 0.77 June LSTM 1.82 0.77 1.35 0.68 ARIMA 1.96 0.83 1.46 0.64 ELM 2.0 0.85 1.52 0.63 GPT3 3.67 1.55 3.0 -0.27 STL-LSTM 1.70 0.73 1.27 0.71 STL-ARIMA 1.88 0.79 1.41 0.67 SLA 1.63 0.68 1.22 0.75 July LSTM 1.84 0.77 1.39 0.67 ARIMA 1.96 0.82 1.49 0.63 ELM 2.05 0.86 1.57 0.60 GPT3 3.86 1.62 3.16 -0.34 STL-LSTM 1.72 0.73 1.31 0.70 STL-ARIMA 1.89 0.79 1.44 0.65 SLA 1.64 0.69 1.25 0.74 August LSTM 1.86 0.77 1.38 0.70 ARIMA 1.99 0.83 1.48 0.66 ELM 2.05 0.86 1.55 0.64 GPT3 4.11 1.73 3.32 -0.41 STL-LSTM 1.75 0.73 1.30 0.73 STL-ARIMA 1.91 0.80 1.44 0.68 SLA 1.64 0.69 1.24 0.76 September LSTM 1.77 0.74 1.33 0.72 ARIMA 1.89 0.79 1.42 0.69 ELM 2.01 0.85 1.53 0.66 GPT3 4.01 1.69 3.29 -0.29 STL-LSTM 1.65 0.69 1.25 0.75 STL-ARIMA 1.83 0.77 1.38 0.71 SLA 1.56 0.66 1.19 0.79 October LSTM 1.74 0.74 1.29 0.71 ARIMA 1.85 0.78 1.38 0.67 ELM 1.93 0.82 1.46 0.64 GPT3 3.88 1.64 3.17 -0.34 STL-LSTM 1.63 0.69 1.22 0.74 STL-ARIMA 1.78 0.75 1.33 0.69 SLA 1.54 0.65 0.15 0.77 November LSTM 1.67 0.71 1.24 0.71 ARIMA 1.76 0.75 1.31 0.68 ELM 1.85 0.79 1.39 0.65 GPT3 3.74 1.59 3.06 -0.39 STL-LSTM 1.55 0.67 1.16 0.74 STL-ARIMA 1.69 0.72 1.26 0.71 SLA 1.46 0.62 1.09 0.78 December LSTM 1.58 0.68 1.16 0.76 ARIMA 1.63 0.69 1.19 0.74 ELM 1.77 0.76 1.32 0.69 GPT3 3.73 1.59 3.03 -0.26 STL-LSTM 1.56 0.64 1.09 0.78 STL-ARIMA 1.58 0.67 1.17 0.76 SLA 1.37 0.59 1.02 0.81 3. 3.3 SLA model with different prediction steps Tropospheric delay prediction for a longer period of time is more useful. Therefore, in this section, the prediction step increased from 1 to 2, 3, and 4 steps. Table 6 shows the results of the prediction experiments the SLA model. Figure 12 shows the ZTD predicted results of AREG(16.47° S, 71.49 °W, 2489.3m) stations with 1–4 prediction steps. The prediction result of 6h is the best. The average RMSE, NRMSE, and MAE are 1.32 cm, 0.56%, and 0.98 cm respectively, and R 2 is 0.83. The prediction curve has a high degree of fitting with the reference value, and the prediction result can correctly reflect the change of ZTD (Fig. 10 ). With the increase of the predicted time to 12h, the average RMSE, NRMSE, and MAE increased to 1.59 cm, 0.67%, and 1.21 cm, and R 2 decreased to 0.75, and the experimental results were slightly worse than 6h. Although in some time periods the predicted results of the model have some deviation, the overall change of ZTD can be predicted well (Fig. 12b). When the forecasting time is 24h, the average RMSE, NRMSE, and MAE of all stations are 1.86 cm, 0.79%, and 1.44 cm, respectively, and R 2 is only 0.68. To sum up, the predicted results of SLA model in 6h and 12h have a good performance, but when the prediction time is longer, the prediction accuracy drops obviously, and the prediction value of the model has a big error with the original value. In the case of low accuracy (R 2 < 0.7), the ZTD prediction result of the SLA model for 24h can be used as a reference. Table 6 Average accuracy of SLA ZTD prediction results of 505 VMF3 stations from January to March 2022 with different step sizes. Prediction step size RMSE (cm) NRMSE (%) MAE (cm) R 2 6h 1.32 0.56 0.98 0.83 12h 1.59 0.67 1.21 0.75 18h 1.72 0.73 1.31 0.72 24h 1.86 0.79 1.44 0.68 3.4 Discussion In this study, we utilized the STL, LSTM, and ARIMA constructed combined model SLA to predict ZTD. The results show that the SLA model has a better result compared to other model due to its better capture of seasonal and trend features. Our findings are in accord with recent study, such as Zhang et al. ( 2022 ) established ZTD prediction model based on transformer indicated that the model performs better in high-latitude regions than in low-latitude regions and Zhang et al. ( 2020 ) established ZTD prediction model based on different blind source separation methods and LSTM shows that as the prediction duration increases, the predicted results also deteriorate. We use a combination model to predict ZTD, which only requires ZTD products to quickly construct the model. However, it should be pointed out that our proposed short-term prediction model has the advantage of quickly obtaining high-precision ZTD, but as the prediction steps increase to 4, the predicted results can only be used as a reference. To address the limitations of this question, it is possible to consider adopting a cyclic prediction approach to improve the accuracy of long-term predictions. Conclusion and outlook In this study, the SLA model which combined STL, LSTM, and ARIMA was constructed to predict ZTD. 505 VMF3 station data are used by comparing the SLA and other models including LSTM, ARIMA, ELM, and GPT3, and it is demonstrated that SLA has the highest accuracy. The average RMSE, NRMSE, and MAE of the SLA model is around 1.32 cm, 0.56%, and 0.98 cm, which are about 15% lower than other models. The R 2 is about 0.83, which is about 9.2% higher than the other models. The 505 VMF3 stations are located in different regions of high, low, and middle latitudes in the world, which indicates that the SLA model is feasible to predict ZTD in global regions. The specific work and results of this paper are as follows: Firstly, we make a comparative analysis between VMF3-ZTD and IGS-ZTD. The results show that RMSE, NRMSE, MAE, and R 2 between VMF3-ZTD and IGS-ZTD are 1.18 cm, 0.51%, and 0.05, respectively. This suggests a strong consistency between VMF-ZTD and IGS-ZTD, ensuring the reliability of the data source used in this study. Secondly, the combined forecasting model of tropospheric delay based on STL algorithm and deep learning is explained and analyzed. The influence of different training data and different input steps of the LSTM model on the accuracy of the combined model is discussed. Experiments show that the results are the best when the training data is 3 years and the input step size is 16. Finally, the accuracy of the combined model SLA is analyzed in a multi-dimensional experiment. It is mainly divided into three parts: The first part is the comparison experiment of different models, and the results show that the combined model SLA has better accuracy in all evaluation indicators; The second part is the ZTD prediction experiment in different months, and the results show that the prediction of the SLA model is the best in each month; The third part is the different prediction length of the prediction experiments, the results show that the SLA model has better prediction results at 6h and 12h, with RMSE < 1.60 cm, NRMSE < 0.7%, MAE = 0.75. However, when the prediction time is longer than 18h, the prediction accuracy decreases significantly, and the predicted value of the model has a large error with the original value. Overall, this study demonstrates that the SLA model has high accuracy in the prediction of ZTD. In future work, it can be explored how to extend the prediction duration while ensuring prediction accuracy. Additionally, it is possible to integrate reanalysis data such as ERA5 to improve prediction accuracy. Declarations Availability of data and materials The VMF-ZTD analyzed in this study is available from GGOS Atmosphere (https://vmf.geo.tuwien.ac.at/trop_products/GNSS/VMF3/VMF3_OP/, accessed on 5 March 2023). The IGS-ZTD is available from the IGS repository (ftp://cddis.gsfc.nasa.gov/pub/gps/data, accessed on 5 April 2023). Funding This work was supported by the National Natural Science Foundation of China (Grant NO. 42374050). Author contributions Xiao Xu: Methodology, Software, Validation, Writing - Original Draft, Visualization; YingChun Yue: Project Administration, Supervision, Writing - Review & Editing; Ming ShangGuan: Conceptualization, Resources, Supervision, Writing - Review & Editing; YiFan Liang: Formal Analysis, Investigation, Writing - Original Draft; ShaoFeng Bian: Supervision, Resources, Writing - Review & Editing; GuoJun Zhai: Funding Acquisition, Resources, Writing - Review & Editing. Acknowledgments The authors would like to thank the IGS, and Global Geodetic Observing System (GGOS) for providing the ZTD data for us to obtain and predict ZTD. Competing interests All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. 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12:46:20","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3933886/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3933886/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":51025670,"identity":"be6b2084-0c04-4aa1-ba9c-99048bf7cbfc","added_by":"auto","created_at":"2024-02-12 21:48:03","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":2584650,"visible":true,"origin":"","legend":"\u003cp\u003eGlobal distribution map of 505 VMF3 sites (green points) and 405 IGS sites (red points).\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-3933886/v1/8339968e3582a515bf853794.png"},{"id":51025668,"identity":"ecc493bc-9451-45c5-9a30-752a87c9c154","added_by":"auto","created_at":"2024-02-12 21:48:03","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":1071637,"visible":true,"origin":"","legend":"\u003cp\u003eData completeness rates for 405 IGS stations ZTD data.\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-3933886/v1/3c9f2317cd2cd247a70686de.png"},{"id":51025669,"identity":"76e68754-762e-4872-af21-58540ee22048","added_by":"auto","created_at":"2024-02-12 21:48:03","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":428167,"visible":true,"origin":"","legend":"\u003cp\u003eHistograms of accuracy indicators ((a)RMSE, (b) NRMSE, (c)MAE, (d)R\u003csup\u003e2\u003c/sup\u003e) for VMF3-ZTD and IGS-ZTD at 277 stations.\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-3933886/v1/2c3b7868e17a54deec757d02.png"},{"id":51025671,"identity":"d3a59d81-9dc9-455a-91e0-ab3ff6a82c97","added_by":"auto","created_at":"2024-02-12 21:48:04","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":4786613,"visible":true,"origin":"","legend":"\u003cp\u003eGlobal distribution of accuracy indicators ((a)RMSE, (b) NRMSE, (c)MAE, (d)R\u003csup\u003e2\u003c/sup\u003e) for VMF3-ZTD and IGS-ZTD at 277 VMF3 sites.\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-3933886/v1/e26d5b08ec84c241b2385e80.png"},{"id":51026088,"identity":"0db90bf1-6a2f-4ac8-bb1b-315f41d1a6cc","added_by":"auto","created_at":"2024-02-12 21:56:04","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":289198,"visible":true,"origin":"","legend":"\u003cp\u003eThe flowchart of SLA model.\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-3933886/v1/822e4d0aafaffa3b507a6a57.png"},{"id":51025672,"identity":"0bc0bbe9-8c7f-4afb-a139-2d19fef30dc3","added_by":"auto","created_at":"2024-02-12 21:48:04","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":1280973,"visible":true,"origin":"","legend":"\u003cp\u003eThe STL algorithm decomposition results ((a) original observation, (b) trend terms, (c) seasonal terms, and (d) residual terms) of ZTD time series in 2022 of VMF3 ADIS site.\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-3933886/v1/3a5e89db552a1749e30120e2.png"},{"id":51025674,"identity":"288d63bc-705a-4c78-b513-4ea4e7195298","added_by":"auto","created_at":"2024-02-12 21:48:04","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":5017063,"visible":true,"origin":"","legend":"\u003cp\u003eGlobal distribution of assessment indicators ((a) RMSE, (b) MAE, (c) NRMSE, and (d) R\u003csup\u003e2\u003c/sup\u003e) for the January-March 2022 ZTD forecast results for the 505 VMF3 sites of the SLA model.\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-3933886/v1/adb004abed9ff50156c04691.png"},{"id":51025677,"identity":"f2382d33-450e-4e24-8338-856c4b2d7b59","added_by":"auto","created_at":"2024-02-12 21:48:04","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":2593400,"visible":true,"origin":"","legend":"\u003cp\u003ePredicted ZTD and dZTD (difference between predictions and true value) by five models at BJCO in January and July. (a) and (b) are ZTD predicted results by different models in January and July. (c) and (d) are their corresponding differences. Orange line is the original ZTD in (a) and (b). Grey lines are the zero line in (c) and (d). Predicted ZTD of LSTM, ARIMA, ELM, GPT3, and SLA are represented by green, purple, blue, black, and red lines, respectively.\u003c/p\u003e","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-3933886/v1/d423ee340afe1740f4715f30.png"},{"id":51025675,"identity":"03380b49-ba99-4237-acdf-07cd248ef2bd","added_by":"auto","created_at":"2024-02-12 21:48:04","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":2245776,"visible":true,"origin":"","legend":"\u003cp\u003ePredicted ZTD and dZTD (difference between predictions and true value) by three combined models at BJCO in January and July. SL represents STL-LSTM, SA represents STL-ARIMA. (a) and (b) are ZTD predicted results by different models in January and July. (c) and (d) are their corresponding differences. Orange line is the original ZTD in (a) and (b). Grey lines are the zero line in (c) and (d). Predicted ZTD and dZTD of LSTM, ARIMA, ELM, GPT3, and SLA are represented by green, purple, blue, black, and red lines, respectively.\u003c/p\u003e","description":"","filename":"floatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-3933886/v1/abb527adc4aa5ed837c86884.png"},{"id":51025676,"identity":"51a9d756-5a79-483c-8d77-569b1fdf5a68","added_by":"auto","created_at":"2024-02-12 21:48:04","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":1763537,"visible":true,"origin":"","legend":"\u003cp\u003eZTD prediction results (orange line) and true values (blue lines) of AREG station in SLA model from January to March. ((a) -(d) are the results of 1, 2, 3, and 4 steps respectively.\u003c/p\u003e","description":"","filename":"floatimage10.png","url":"https://assets-eu.researchsquare.com/files/rs-3933886/v1/6b107fb78188fff8bacc7af1.png"},{"id":66476884,"identity":"e732a0cb-cef5-4d31-be90-b500af283ccf","added_by":"auto","created_at":"2024-10-12 20:01:55","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":20827286,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3933886/v1/e6dec1d3-72a2-4738-a7eb-9068aa32ed62.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A new tropospheric delay combination prediction model based on time series decomposition and deep learning","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eAs the Global Navigation Satellite System (GNSS) signal traverses the atmosphere, it encounters a delay attributed to the non-vacuum environment, simultaneously undergoing trajectory elongation due to the bending phenomenon (Wu et al. \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). The part due to the troposphere is termed tropospheric delay, which is an important error source in GNSS high-precision positioning (Ma et al. \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Several studies have shown that high-precision tropospheric delay can effectively accelerate the convergence speed of precise point positioning (PPP) and improve the accuracy of positioning accuracy (Zhao et al. \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Bahadur \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Huang et al. \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). In addition, ZTD is widely used in numerical weather forecasts and climate monitoring (Zhao et al. \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Li et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Therefore, it is important to establish a high-precision tropospheric delay model for climate research and high-precision positioning.\u003c/p\u003e \u003cp\u003eIn recent years, many researchers have focused on the establishment of high-precision tropospheric delay models (Yao et al. \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Mao et al. \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Xu et al. \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Tropospheric delay models can be categorized into models based on measured meteorological parameters and empirical models without meteorological parameters. Tropospheric delay models based on measured meteorological parameters (e.g., Hopfield model (Hopfield \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e1969\u003c/span\u003e), Saastamoinen model (Saastamoinen \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e1972\u003c/span\u003e), Black model (Black \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e1978\u003c/span\u003e), Askne \u0026amp; Nordius model (Askne and Nordius \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e1987\u003c/span\u003e)) require inputs such as temperature, water vapor pressure, and relative humidity. Since meteorological parameters are often unavailable, empirical meteorological parameters are generally used, which leads to additional errors in the models. The empirical models (e.g., UNB series models (Collins and Langley \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e1997\u003c/span\u003e; Leandro et al. \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2006\u003c/span\u003e; \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2008\u003c/span\u003e), EGNOS models (Penna et al. \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2001\u003c/span\u003e), Tropgrid models (Krueger et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Sch\u0026uuml;ler \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2014\u003c/span\u003e), GPT models (B\u0026ouml;hm et al. \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Lagler et al. \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Landskron and B\u0026ouml;hm \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), and GZTD models (Yao et al. \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2016\u003c/span\u003e), are mostly based on historical radiosonde data, GNSS ZTD products or reanalysis data.\u003c/p\u003e \u003cp\u003eWith the rapid development of computer technology, deep learning technology has become one of the main methods for time series modeling in the troposphere with its excellent nonlinear modeling ability (Ding \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Shangguan et al. \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Since tropospheric delays are correlated with multiple parameters such as meteorological conditions, geographic location, and time, which have complex linear and nonlinear characteristics, it makes deep learning-based tropospheric delay modeling one of the research hotspots (Xiao et al. \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Osah et al. \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Su et al. \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Zhang et al. (\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) through the LSTM network established a model to predict ZTD, in which the average RMSE for the next 6h is 7.2mm. Yang et al. (\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) adopted the artificial neural network (ANN) to construct the correlation between ZTD derived from GPT3 and GNSS observations which effectively improved the systematic deviation of the GPT3 model. Li et al. (\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) improved the GPT3 model in the Antarctic region by incorporating the LSTM network and radial basis function (RBF). Zhang et al. (\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) predict ZTD used the transformer model with an average RMSE of 1.8 cm at 505 VMF3 stations. Lu et al. (\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) utilized the convolutional LSTM network (ConvLSTM) to build a tropospheric delay network (TropNet) with an accuracy better than 11 mm for the GNSS-ZWD. Previous experiments have demonstrated that the learning model based on one or more sites has good accuracy in predicting ZTD. However, the variation of ZTD is complex in space and time, including trend, seasonal variation and etc. These models do not take into account the variations characteristics in ZTD data specifically, which cannot make full use of the advantage of the deep learning model.\u003c/p\u003e \u003cp\u003eIn this paper, a combination of STL algorithm, LSTM neural network, and ARIMA model is used to predict ZTD, hereinafter referred to as SLA model. Among them, the STL algorithm is used to decompose the original ZTD time series into trend, seasonal and, residual terms. The LSTM neural network and the ARIMA model are used to predict the nonlinear and linear parts of the ZTD components, respectively. The prediction results of different components are reconstructed to obtain highly accurate ZTD values. Finally, the performance of the model is verified through multi-dimensional experimental analyses (ZTD forecasting for different regions, months, and prediction steps).\u003c/p\u003e"},{"header":"2 Data and Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Data Source\u003c/h2\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, we utilized ZTD data from 505 stations (green points in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e), sampled at 6h intervals, provided by VMF3 from 2019 to 2022 for model training and testing. In addition, we used ZTD data from 405 IGS stations (red points in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) offered by the IGS analysis center to validate the reliability of the data.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Data preprocessing.\u003c/h2\u003e \u003cp\u003eDue to missing values in the data of some IGS stations. The completeness rate of IGS station data was statistically analyzed first. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, 13 stations have completeness rates below 60%, while 299 stations have completeness rates exceeding 90%. Among them, 277 stations are common to both IGS and VMF3. Therefore, the comparison is made by these 277 stations.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn order to further investigate the correlation and agreement between VMF3- and IGS-ZTD, data from January 2022 to December 2022 were selected for the comparison. We first resampled IGS-ZTD data in 6h intervals and utilized the Grubbs test to remove outliers and subsequently applied the K Nearest Neighbor algorithm for missing value interpolation. Finally, using IGS-ZTD as the real value, the accuracy of VMF3-ZTD was evaluated at 277 global stations using four indicators: RMSE, NRMSE, MAE, and R\u003csup\u003e2\u003c/sup\u003e. The formulas for these indicators are as follows: (1)-(4), where \u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003e is the total number, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Z_{i}^{{true}}\\)\u003c/span\u003e\u003c/span\u003e is the original value, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Z_{i}^{{pre}}\\)\u003c/span\u003e\u003c/span\u003e is the predicted value, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\overline {{Z_{i}^{{true}}}}\\)\u003c/span\u003e\u003c/span\u003e is the mean of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Z_{i}^{{true}}\\)\u003c/span\u003e\u003c/span\u003e.\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$RMSE=\\sqrt {\\frac{1}{N}\\sum\\nolimits_{{i=1}}^{N} {{{(Z_{i}^{{pre}} - Z_{i}^{{true}})}^2}} }$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$NRMSE=\\frac{{RMSE}}{{\\overline {{Z_{i}^{{true}}}} }}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$MAE=\\frac{1}{N}\\sum\\nolimits_{{i=1}}^{N} {\\left| {Z_{i}^{{pre}} - Z_{i}^{{true}}} \\right|}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${R^2}=1 - \\frac{{{{(Z_{i}^{{pre}} - Z_{i}^{{true}})}^2}}}{{{{(Z_{i}^{{true}} - \\overline {{Z_{i}^{{true}}}} )}^2}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe RMSE, NRMSE, MAE, and R\u003csup\u003e2\u003c/sup\u003e between VMF3-ZTD and IGS-ZTD are 1.18 cm, 0.51%, 0.91 cm, and 0.92, which indicate a small difference between VMF3-ZTD and IGS-ZTD in general. Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(a-d) depicts the statistical distribution of the number of stations at different ranges for RMSE, NRMSE, MAE, and R\u003csup\u003e2\u003c/sup\u003e. The results show that 97.5% of the stations have an RMSE less than 2 cm, 99.6% of the stations have an MAE less than 2 cm, 82.3% of the stations have a NRMSE less than 0.6%, and only 0.7% of the stations exceed 1%. Additionally, 82.3% of the stations have an R\u003csup\u003e2\u003c/sup\u003e greater than 0.9, while only 4.6% of the stations have an R\u003csup\u003e2\u003c/sup\u003e less than 0.6. In summary, the majority of sites have a high level of accuracy between VMF-ZTD and IGS-ZTD.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFurthermore, the global distribution of RMSE, NRMSE, MAE, and R\u003csup\u003e2\u003c/sup\u003e for VMF3-ZTD and IGS-ZTD is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. The accuracy tends to be generally lower for sites located in maritime areas compared to mainland areas, which could be due to the drastic water vapor variations over the oceans.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Prediction models\u003c/h2\u003e \u003cp\u003eTraditional ZTD forecasting prediction models are generally modeled for a single ZTD time series, which makes it difficult to capture the trending and seasonal variations. In this paper, three single models are combined to predict ZTD. The used methods are described below:\u003c/p\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e2.3.1 STL\u003c/h2\u003e \u003cp\u003eThe STL is a classical method in time series decomposition. Based on STL method, the sequence can be decomposed into the following three components:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${Z_t}={T_t}+{S_t}+{R_t}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Z_t}\\)\u003c/span\u003e\u003c/span\u003e is the original, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T_t}\\)\u003c/span\u003e\u003c/span\u003e is the trend component, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({S_t}\\)\u003c/span\u003e\u003c/span\u003e is the period component and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({R_t}\\)\u003c/span\u003e\u003c/span\u003e is the residual component (Cleveland et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1990\u003c/span\u003e). The STL algorithm can be used to process the arbitrary time series data, and the STL decomposition implementation mainly consists of inner and outer loop recursive process, and the two loops belong to a nested relationship. The inner loop is used for the calculation of trend and period components, while the outer loop is used to calculate the robustness weights for adjusting the neighborhood weights of the next inner loop based on the results of the previous inner loop (Chen et al. \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section3\"\u003e \u003ch2\u003e2.3.2 LSTM\u003c/h2\u003e \u003cp\u003eThe LSTM is a network improved by the recurrent neural network (RNN) and is a powerful method for deep learning of time series (Ding \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). LSTM introduces a memory cell to solve the problems of gradient disappearance and explosion which occur in RNN invariably. The memory cell contains a memory block, and each memory block has three gate structures including the forgetting gate, the input gate, and the output gate. These three gate structures can read, write, and reset data. Because the output value of the Sigmoid function is between 0 and 1 and it can let the information flow through the door or not, the activation functions of the three gates are all S-type functions (\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eHochreiter et al. 1997\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section3\"\u003e \u003ch2\u003e2.3.3 ARIMA\u003c/h2\u003e \u003cp\u003eThe ARIMA can identify complex patterns in data and generate predictions, which can be used to analyze and predict univariate time series data (Box et al. 1976). The function of ARIMA is represented by p, d, q (p represents the number of autoregression items, d represents the number of non-seasonal differences, and q represents the number of lag prediction errors in the prediction equation). The three steps of establishing ARIMA are identification, estimation, and prediction (Adamowski and Chan \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2011\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section3\"\u003e \u003ch2\u003e2.3.4 SLA\u003c/h2\u003e \u003cp\u003eThe SLA model combines the STL, LSTM, and ARIMA, where the STL algorithm is employed to decompose the original ZTD sequences and extract characteristic information from different dimensions. These dimensions exhibit distinct linear and nonlinear features. Previous studies (Li et al. \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Zhang et al \u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Shangguan et al. \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) have indicated that LSTM has advantages in predicting nonlinear features, while the ARIMA model excels in forecasting linear features and stationary sequences. Therefore, this study utilizes the LSTM and ARIMA model to predict the ZTD characteristic information in different dimensions separately. Finally, the predictions are reconstructed to achieve high-precision ZTD forecasting results. The flow diagram of the SLA model is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, the prediction model based on STL decomposition and deep learning can be divided into the following four steps:\u003c/p\u003e \u003cp\u003eStep 1: STL decomposes the original ZTD data into trend terms, seasonal terms, and residual terms;\u003c/p\u003e \u003cp\u003eStep 2: ARIMA model is used to predict the trend term, and then the LSTM model is used to predict the seasonal term and residual term to get the predicted values;\u003c/p\u003e \u003cp\u003eStep 3: Reconstruct the predicted values of the three parts of step 2 to get the final prediction results;\u003c/p\u003e \u003cp\u003eStep 4: Model Evaluation: Evaluate the prediction accuracy of the model using four evaluation indexes: MAE, RMSE, R\u003csup\u003e2\u003c/sup\u003e, and NRMSE.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"3 Results and Discussion","content":"\u003cp\u003eIn this section, we first present the result of the STL method in the SLA model. Subsequently, we analyze the optimal parameter settings for LSTM in predicting ZTD. Finally, we conduct multidimensional experimental analyses of the SLA composite model, including comparisons of different regions, different months, and different prediction steps.\u003c/p\u003e\n\u003cdiv id=\"Sec11\"\u003e\n \u003ch2\u003e3.1 STL results\u003c/h2\u003e\n \u003cp\u003eFigure\u0026nbsp;\u003cspan\u003e6\u003c/span\u003e shows the STL decomposed ZTD time series of station ADIS (38.77°E, 9.04°N, 2439.2m). It can be concluded that the trend component retains the variable characteristics of the original time series, the volatility is smaller, and the sequence is relatively stable. Therefore, it is suitable for the ARIMA model considering the time relationship to predict. The residual component and the seasonal component have greater data volatility, retaining the nonlinear variation characteristics of the original time series, so the nonlinear LSTM model is suitable for predicting. It is feasible and meaningful to construct a combined model based on STL decomposition combined with the LSTM and ARIMA model for prediction research in the following.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec12\"\u003e\n \u003ch2\u003e3.2 LSTM results\u003c/h2\u003e\n \u003cp\u003eIn order to analyze the influence of different training lengths on the predicted result of the model, this paper uses different training lengths to train the model and uses the ZTD data from January to March 2022 as the prediction data. Using the data of 12 steps, we predict the data of the next step (6h). The length of training is set to 1, 2, 3, and 5 years. Table \u003cspan\u003e1\u003c/span\u003e shows evaluation indicators in ZTD prediction results of different training lengths at 505 VMF3 sites. It can be concluded that when the length of training data is only 1 year, all evaluation indexes are poorest, but when the training data increases to 3 years or more, all evaluation indexes do not improve. The reason is that when the length of training data is less than 3 years the results are unstable. As the length of training data increases to 3, 4, and 5 years, the prediction results remain stable. Therefore, the 3 years data is selected as the training set data for the following experiments.\u0026nbsp;\u003c/p\u003e\n \u003ctable id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 1\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eAverage accuracy indicators of LSTM ZTD prediction experiments with different training lengths in 505 VMF3 stations from January to March 2022.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eMetric\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003e1 year\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003e2 years\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003e3 years\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003e4 years\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003e5 years\u003c/p\u003e\n \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eRMSE (cm)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.99\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.68\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.56\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.57\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.55\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eNRMSE (%)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.84\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.71\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eMAE (cm)\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.52\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.36\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.15\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.15\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.14\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.60\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.71\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n \u003cp\u003eIn order to analyze the influence of different input steps on the predicted result, this study uses different input steps (4, 8, 12, 16, and 20) to train the model with 1 to 4 prediction steps, respectively. The three years data from 2019 to 2022 are used as the training set. The test data is from January to March 2022. The experimental results are shown in Table \u003cspan\u003e2\u003c/span\u003e. The accuracy of the prediction results is the worst with the 4 input steps. With the increase of the length of the input steps, all the evaluation indexes improve. However, when the length of the input data is more than 16 steps, the evaluation indexes do not increase significantly. When the input steps are 16, the prediction has the best result. Therefore, the ZTD sequence with an input step size of 16 is determined hereafter.\u003c/p\u003e\n \u003cdiv align=\"char\"\u003e\u003cbr\u003e\u003cimg src=\"https://myfiles.space/user_files/122228_c8a1650c59388082/122228_custom_files/img1707736591.png\"\u003e\u003c/div\u003e\n \u003cp\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec13\"\u003e\n \u003ch2\u003e3.3 SLA results\u003c/h2\u003e\n \u003cdiv id=\"Sec14\"\u003e\n \u003ch2\u003e3.3.1 ZTD results for different regions\u003c/h2\u003e\n \u003cp\u003eUnder the optimal LSTM strategy for predicting ZTD, we compared the accuracy of the SLA model with models such as LSTM, ARIMA, ELM, and GPT3. Table \u003cspan\u003e3\u003c/span\u003e shows that the SLA model has better accuracy compared to other models. Specifically, the average RMSE of the SLA model is 1.32 cm, which is 14.8% lower than that of LSTM, 16.5% lower than that of ARIMA, 21.9% lower than that of ELM and 61.5% lower than that of ARIMA. The average NRMSE is 0.56%, which is 15.2%, 16.4%, 25.3%, and 61.6% lower than that of LSTM, ARIMA, ELM, and GPT3 models respectively. Similarly, MAE improved by 14%, 16.2%, 21%, and 64.3%, respectively. The SLA model had the largest average R\u003csup\u003e2\u003c/sup\u003e value, up to 0.83, followed by LSTM(0.76), while the GPT3 model had the lowest R\u003csup\u003e2\u003c/sup\u003e only − 0.11. This shows that the combined model SLA can significantly improve the accuracy of prediction compared with other models.\u003c/p\u003e\n \u003cp\u003eIn order to verify the advantage of the combined model over the single decomposition model, the STL algorithm combined with the LSTM model (namely STL-LSTMA) and the STL algorithm combined with the ARIMA model (namely STL-ARIMA), are also included in Table \u003cspan\u003e3\u003c/span\u003e. The results show that the STL-LSTM model improves the average RMSE, NRMSE, MAE, and R\u003csup\u003e2\u003c/sup\u003e by 9%, 9.1%, 7.9%, and 3.9% compared to LSTM, respectively. Similarly the STL-ARIMA model improved by 3.8%, 3.0%, 3.4%, and 2.7% compared to ARIMA. The reason is that the LSTM and ARIMA models were able to capture more features such as seasonality and trend of the original sequences from the STL decomposed sequences. In addition, the forecasting results of STL-LSTM are better than those of STL-ARIMA, which is mainly due to large variations of the residual and seasonal terms with the ARIMA model. The SLA model used in this paper combined the advantages of ARIMA and LSTM models, and the average RMSE, NRMSE, MAE, and R\u003csup\u003e2\u003c/sup\u003e of 505 VMF3 stations are 1.32 cm, 0.56%, 0.98 cm, and 0.83, respectively, which are improved by 6.4%, 6.7%, 6.7%, and 5.1% relative to the STL-LSTM, respectively. It concludes that the combination model STA can utilize the advantages of different models to better learning more features in the original data, which has a better prediction effect and can reduce the error of ZTD prediction.\u0026nbsp;\u003c/p\u003e\n \u003ctable id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 3\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eAverage accuracy of SLA ZTD prediction experiments in 505 VMF3 stations from January to March 2022.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eModel\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eRMSE (cm)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eNRMSE (%)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eMAE (cm)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\n \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.55\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.14\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.58\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.17\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.24\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.70\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.43\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.46\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e2.75\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.11\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.41\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.60\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.05\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.79\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.52\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.13\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.77\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.32\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.56\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.98\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.83\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n \u003cp\u003eIn order to further investigate the predictive ability of the models globally, the paper analyzes the global distribution of the indicators of the different models. Figure \u003cspan\u003e7\u003c/span\u003e(a-d), shows the distribution of RMSE, NRMSE, MAE, and R\u003csup\u003e2\u003c/sup\u003e of the SLA model in global regions, respectively. The results of six regions 90°N-60°N, 60°N-30°N, 30°N-0°, 0°-30°S, 30°S-60°S, and 60°S-90°S are shown in Table \u003cspan\u003e4\u003c/span\u003e. The accuracy of the SLA model is better in the high latitude than in the low latitude region, the accuracy on land is higher than that on the sea level. The overall mean values of RMSE, NRSEM, MAE, and R\u003csup\u003e2\u003c/sup\u003e of the SLA model increased with increasing latitude. The model has the best accuracy in the 60°S-90°S region, with RMSE, NRMSE, MAE, and R\u003csup\u003e2\u003c/sup\u003e of 0.69 cm, 0.31%, 0.49 cm, and 0.92, respectively, which is an improvement of 47.7%, 44.6%, 50%, and 9.7% compared to the global average accuracy.\u003c/p\u003e\n \u003cdiv align=\"char\"\u003e\u003cbr\u003e\u003c/div\u003e\u0026nbsp;\u003ctable id=\"Tab4\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 4\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eAverage accuracy of SLA ZTD prediction results for January-March 2022 at different latitudinal bands for the 505 VMF3 sites of the SLA model.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eLatitude\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eNumber of sites\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eRMSE (cm)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eNRMSE (%)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eMAE (cm)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\n \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e90°N-60°N\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e38\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.77\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.34\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.57\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.93\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e60°N-30°N\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e247\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.17\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.51\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.84\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.81\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e30°N-0°\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e82\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.46\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.61\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.12\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.83\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e0°-30°S\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e77\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.59\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.24\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.84\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e30°S-60°S\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e48\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e2.05\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.87\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.52\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.79\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003e60°S-90°S\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.49\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.92\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Sec15\"\u003e\n \u003ch2\u003e3.3.2 ZTD forecasts for different months\u003c/h2\u003e\n \u003cp\u003eDue to the obvious seasonal variation of ZTD, the performance of ZTD prediction models in different months in 2022 is tested. Table \u003cspan\u003e5\u003c/span\u003e shows evaluation indexes of five prediction models in 12 months. All models have better results in winter, but with the arrival of summer, the model performance is gradually decreasing. However, the RMSE and MAE of the SLA model adopted in all months are smaller than other models, and R\u003csup\u003e2\u003c/sup\u003e is larger than other models. Figures \u003cspan\u003e8\u003c/span\u003e and \u003cspan\u003e9\u003c/span\u003e show the predicted value and error of the ZTD prediction by five models and three combined models at the BJCO (6.38°E, 2.45°N, 30.7m) site in January and July. As shown in Figs. \u003cspan\u003e8\u003c/span\u003e and \u003cspan\u003e9\u003c/span\u003e, the SLA model has the smallest error in predicting ZTD, which is most suitable for ZTD prediction.\u003c/p\u003e\u0026nbsp;\u003ctable id=\"Tab5\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 5\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eAverage accuracy of SLA ZTD prediction results of 505 VMF3 stations in different months in 2022.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eMonth\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eModel\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eRMSE (cm)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eNRMSE (%)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eMAE (cm)\u003c/p\u003e\n \u003c/th\u003e\u003cth align=\"left\"\u003e\n \u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\n \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eJanuary\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.53\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.15\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.73\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.55\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.16\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.73\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.72\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.28\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.33\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.42\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e2.70\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.21\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.41\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.61\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.06\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.49\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.64\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.12\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.29\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.55\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.97\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.81\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eFebruary\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.53\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.15\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.55\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.17\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.72\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.28\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.61\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.36\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.43\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e2.74\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.58\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.42\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.61\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.07\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.70\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.50\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.64\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.13\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.31\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.56\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.99\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eMarch\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.53\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.13\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.73\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.59\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.68\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.18\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.70\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.68\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.72\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.26\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.44\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.46\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e2.80\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.30\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.41\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.61\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.05\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.53\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.14\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.72\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.32\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.56\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.98\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.80\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eApril\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.56\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.17\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.73\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.64\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.70\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.22\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.71\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.76\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.33\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.41\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.45\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e2.77\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.17\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.44\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.62\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.10\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.57\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.18\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.73\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.36\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.58\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.02\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.80\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eMay\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.66\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.70\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.22\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.71\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.76\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.31\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.85\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.78\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.40\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.64\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.56\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.51\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e2.93\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.27\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.55\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.15\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.73\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.72\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.26\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.70\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.46\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.62\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.06\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.77\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eJune\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.82\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.77\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.35\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.68\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.96\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.83\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.46\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.64\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e2.0\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.85\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.52\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.63\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.67\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.55\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.0\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.27\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.70\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.73\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.27\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.71\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.88\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.79\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.41\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.63\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.68\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.22\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eJuly\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.84\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.77\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.39\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.96\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.82\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.49\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.63\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e2.05\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.86\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.57\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.60\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.86\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.62\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.16\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.34\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.72\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.73\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.31\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.70\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.89\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.79\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.44\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.64\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.25\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.74\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eAugust\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.86\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.77\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.38\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.70\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.99\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.83\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.48\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e2.05\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.86\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.55\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.64\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e4.11\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.73\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.32\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.41\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.75\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.73\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.30\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.73\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.91\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.80\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.44\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.68\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.64\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.24\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eSeptember\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.77\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.74\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.33\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.72\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.89\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.79\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.42\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e2.01\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.85\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.53\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e4.01\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.29\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.29\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.65\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.25\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.83\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.77\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.38\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.71\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.56\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.66\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.19\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.79\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eOctober\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.74\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.74\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.29\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.71\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.85\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.78\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.38\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.93\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.82\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.46\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.64\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.88\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.64\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.17\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.34\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.63\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.22\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.74\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.78\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.33\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.54\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.15\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.77\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eNovember\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.67\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.71\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.24\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.71\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.76\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.31\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.68\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.85\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.79\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.39\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.74\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.59\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.06\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.39\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.55\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.16\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.74\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.72\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.26\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.71\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.46\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.62\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.09\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.78\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"7\"\u003e\n \u003cp\u003eDecember\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eLSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.58\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.68\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.16\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.63\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.19\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.74\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eELM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.77\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.32\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eGPT3\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.73\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.59\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e3.03\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.26\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-LSTM\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.56\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.64\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.09\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.78\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSTL-ARIMA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.58\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.17\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\"\u003e\n \u003cp\u003eSLA\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.37\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.59\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e1.02\u003c/p\u003e\n \u003c/td\u003e\u003ctd align=\"char\"\u003e\n \u003cp\u003e0.81\u003c/p\u003e\n \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003ch3\u003e3. 3.3 SLA model with different prediction steps\u003c/h3\u003e\n\u003cp\u003eTropospheric delay prediction for a longer period of time is more useful. Therefore, in this section, the prediction step increased from 1 to 2, 3, and 4 steps. Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e shows the results of the prediction experiments the SLA model. Figure\u0026nbsp;12 shows the ZTD predicted results of AREG(16.47° S, 71.49 °W, 2489.3m) stations with 1–4 prediction steps. The prediction result of 6h is the best. The average RMSE, NRMSE, and MAE are 1.32 cm, 0.56%, and 0.98 cm respectively, and R\u003csup\u003e2\u003c/sup\u003e is 0.83. The prediction curve has a high degree of fitting with the reference value, and the prediction result can correctly reflect the change of ZTD (Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e). With the increase of the predicted time to 12h, the average RMSE, NRMSE, and MAE increased to 1.59 cm, 0.67%, and 1.21 cm, and R\u003csup\u003e2\u003c/sup\u003e decreased to 0.75, and the experimental results were slightly worse than 6h. Although in some time periods the predicted results of the model have some deviation, the overall change of ZTD can be predicted well (Fig.\u0026nbsp;12b). When the forecasting time is 24h, the average RMSE, NRMSE, and MAE of all stations are 1.86 cm, 0.79%, and 1.44 cm, respectively, and R\u003csup\u003e2\u003c/sup\u003e is only 0.68. To sum up, the predicted results of SLA model in 6h and 12h have a good performance, but when the prediction time is longer, the prediction accuracy drops obviously, and the prediction value of the model has a big error with the original value. In the case of low accuracy (R\u003csup\u003e2\u003c/sup\u003e \u0026lt; 0.7), the ZTD prediction result of the SLA model for 24h can be used as a reference.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cdiv class=\"gridtable\"\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\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\u003eAverage accuracy of SLA ZTD prediction results of 505 VMF3 stations from January to March 2022 with different step sizes.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"5\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePrediction step size\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRMSE (cm)\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNRMSE (%)\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMAE (cm)\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6h\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.32\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.56\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.98\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.83\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e12h\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.59\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.67\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.21\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.75\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e18h\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.72\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.73\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.31\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.72\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e24h\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.86\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.79\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.44\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.68\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e \u003cp\u003e\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \n\u003ch3\u003e3.4 Discussion\u003c/h3\u003e\n\u003cp\u003eIn this study, we utilized the STL, LSTM, and ARIMA constructed combined model SLA to predict ZTD. The results show that the SLA model has a better result compared to other model due to its better capture of seasonal and trend features. Our findings are in accord with recent study, such as Zhang et al. (\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) established ZTD prediction model based on transformer indicated that the model performs better in high-latitude regions than in low-latitude regions and Zhang et al. (\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) established ZTD prediction model based on different blind source separation methods and LSTM shows that as the prediction duration increases, the predicted results also deteriorate. We use a combination model to predict ZTD, which only requires ZTD products to quickly construct the model. However, it should be pointed out that our proposed short-term prediction model has the advantage of quickly obtaining high-precision ZTD, but as the prediction steps increase to 4, the predicted results can only be used as a reference. To address the limitations of this question, it is possible to consider adopting a cyclic prediction approach to improve the accuracy of long-term predictions.\u003c/p\u003e"},{"header":"Conclusion and outlook","content":"\u003cp\u003eIn this study, the SLA model which combined STL, LSTM, and ARIMA was constructed to predict ZTD. 505 VMF3 station data are used by comparing the SLA and other models including LSTM, ARIMA, ELM, and GPT3, and it is demonstrated that SLA has the highest accuracy. The average RMSE, NRMSE, and MAE of the SLA model is around 1.32 cm, 0.56%, and 0.98 cm, which are about 15% lower than other models. The R\u003csup\u003e2\u003c/sup\u003e is about 0.83, which is about 9.2% higher than the other models. The 505 VMF3 stations are located in different regions of high, low, and middle latitudes in the world, which indicates that the SLA model is feasible to predict ZTD in global regions. The specific work and results of this paper are as follows:\u003c/p\u003e \u003cp\u003eFirstly, we make a comparative analysis between VMF3-ZTD and IGS-ZTD. The results show that RMSE, NRMSE, MAE, and R\u003csup\u003e2\u003c/sup\u003e between VMF3-ZTD and IGS-ZTD are 1.18 cm, 0.51%, and 0.05, respectively. This suggests a strong consistency between VMF-ZTD and IGS-ZTD, ensuring the reliability of the data source used in this study.\u003c/p\u003e \u003cp\u003eSecondly, the combined forecasting model of tropospheric delay based on STL algorithm and deep learning is explained and analyzed. The influence of different training data and different input steps of the LSTM model on the accuracy of the combined model is discussed. Experiments show that the results are the best when the training data is 3 years and the input step size is 16.\u003c/p\u003e \u003cp\u003eFinally, the accuracy of the combined model SLA is analyzed in a multi-dimensional experiment. It is mainly divided into three parts: The first part is the comparison experiment of different models, and the results show that the combined model SLA has better accuracy in all evaluation indicators; The second part is the ZTD prediction experiment in different months, and the results show that the prediction of the SLA model is the best in each month; The third part is the different prediction length of the prediction experiments, the results show that the SLA model has better prediction results at 6h and 12h, with RMSE\u0026thinsp;\u0026lt;\u0026thinsp;1.60 cm, NRMSE\u0026thinsp;\u0026lt;\u0026thinsp;0.7%, MAE\u0026thinsp;\u0026lt;\u0026thinsp;1.25 cm, and R\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;\u0026gt;\u0026thinsp;=\u0026thinsp;0.75. However, when the prediction time is longer than 18h, the prediction accuracy decreases significantly, and the predicted value of the model has a large error with the original value.\u003c/p\u003e \u003cp\u003eOverall, this study demonstrates that the SLA model has high accuracy in the prediction of ZTD. In future work, it can be explored how to extend the prediction duration while ensuring prediction accuracy. Additionally, it is possible to integrate reanalysis data such as ERA5 to improve prediction accuracy.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAvailability of data and materials\u0026nbsp;\u003c/strong\u003eThe VMF-ZTD analyzed in this study is available from GGOS Atmosphere (https://vmf.geo.tuwien.ac.at/trop_products/GNSS/VMF3/VMF3_OP/, accessed on 5 March 2023). The IGS-ZTD is available from the IGS repository (ftp://cddis.gsfc.nasa.gov/pub/gps/data, accessed on 5 April 2023).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u0026nbsp;\u003c/strong\u003eThis work was supported by the National Natural Science Foundation of China (Grant NO. 42374050).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions\u0026nbsp;\u003c/strong\u003eXiao Xu: Methodology, Software, Validation, Writing - Original Draft, Visualization; YingChun Yue: Project Administration, Supervision, Writing - Review \u0026amp; Editing; Ming ShangGuan: Conceptualization, Resources, Supervision, Writing - Review \u0026amp; Editing; YiFan Liang: Formal Analysis, Investigation, Writing - Original Draft; ShaoFeng Bian: Supervision, Resources, Writing - Review \u0026amp; Editing; GuoJun Zhai: Funding Acquisition, Resources, Writing - Review \u0026amp; Editing.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003eThe authors would like to thank the IGS, and Global Geodetic Observing System (GGOS) for providing the ZTD data for us to obtain and predict ZTD.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthics approval and consent to participate\u0026nbsp;\u003c/strong\u003eThis study did not involve human or animal participants, and therefore, ethics approval and consent to participate are not applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent for publication\u0026nbsp;\u003c/strong\u003eAll authors have provided their explicit consent for the publication of this research work.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAdamowski J, Chan HF (2011) A wavelet neural network conjunction model for groundwater level forecasting. 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Sensors, 20 (8), 2343. https://doi.org/10.3390/s20082343\u003c/li\u003e\n\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":"tropospheric delay modeling, time-series decomposition, deep learning, combined prediction models","lastPublishedDoi":"10.21203/rs.3.rs-3933886/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3933886/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eZenith tropospheric delay (ZTD) prediction is of great significance for high-precision navigation. However, ZTD modeling has proved to be challenging due to the presence of linear and nonlinear characteristics. In this paper, we propose a combination ZTD prediction model (SLA), which considers the trend-based and seasonal variations respectively. It decomposes ZTD time series via seasonal-trend decomposition procedure based on loess (STL), individually predicting nonlinear components with long short-term memory network (LSTM) and linear components with autoregressive integrated moving average model (ARIMA). Finally, the individual predictions are recombined. The SLA model is compared with LSTM, extreme learning machine model (ELM), ARIMA, and the empirical global pressure and temperature (GPT3) model. The SLA model shows the best result in all models by analyzing the evaluation indicators including root mean square error (RMSE, 1.32 cm), the average normalized root mean square error (NRMSE, 0.56%), mean absolute error (MAE, 0.98 cm) and the mean coefficient of determination (R\u003csup\u003e2\u003c/sup\u003e, 0.83). In addition, the data of different months was tested separately, and the result showed that the SLA model has the best performance of ZTD prediction. Moreover, the SLA model has good results up to 12h, with RMSE\u0026thinsp;\u0026lt;\u0026thinsp;1.60 cm, NRMSE\u0026thinsp;\u0026lt;\u0026thinsp;0.7%, MAE\u0026thinsp;\u0026lt;\u0026thinsp;1.25 cm, and R\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;\u0026gt;\u0026thinsp;=\u0026thinsp;0.75. This study provides a new model to predict the ZTD, which is helpful for the precise positioning of GNSS and can be further applied in the study of meteorology.\u003c/p\u003e","manuscriptTitle":"A new tropospheric delay combination prediction model based on time series decomposition and deep learning","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-02-12 21:47:59","doi":"10.21203/rs.3.rs-3933886/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":"8d5e3b29-fbe0-4f64-a118-069b3b74626a","owner":[],"postedDate":"February 12th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-10-12T19:53:17+00:00","versionOfRecord":[],"versionCreatedAt":"2024-02-12 21:47:59","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3933886","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3933886","identity":"rs-3933886","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}