Development of Stacked Long Short-Term Memory Neural Networks with Numerical Solutions for Wind Velocity Predictions

Taiwan, being located on a path in the west Ocean where typhoons often strike, is often aﬀected by typhoons. The accompanying strong winds and torrential rains make typhoons particularly damaging in Taiwan. Therefore, we aimed to establish an accurate wind speed prediction model for future typhoons, allowing for better preparation to mitigate a typhoon’s toll on life and property. For more accurate wind speed predictions during a typhoon episode, we used cutting-edge machine learning techniques to construct a wind speed prediction model. To ensure model accuracy, we used, as variable input, simulated values from the Weather Research and Forecasting model of the numerical weather prediction system in addition to adopting deeper neural networks that can deepen neural network structures in the construction of estimation models. Our deeper neural networks comprise multilayer perceptron (MLP), deep recurrent neural networks (DRNNs), and stacked long short-term memory (LSTM). These three model-structure types diﬀer by their memory capacity: MLPs are model networks with no memory capacity, whereas DRNNs and stacked LSTM are model networks with memory capacity. A model structure with memory capacity can analyze time-series data and continue memorizing and learning along the time axis. The study area is northeastern Taiwan. Results showed that MLP, DRNN, and stacked LSTM prediction error rates increased with prediction time (1–6 hours). Comparing the three models revealed that model networks with memory capacity (DRNN and stacked LSTM) were more accurate than those without memory capacity. A further comparison of model networks with memory capacity revealed that stacked LSTM yielded slightly more accurate results than did DRNN. Additionally, we determined that in the construction of the wind speed prediction model, the use of numerically simulated values reduced the error rate approximately by 30%. These results indicate that the inclusion of numerically simulated values in wind speed prediction models enhanced their prediction accuracy.


Introduction
A typhoon is a severe natural disaster that affects tropical and subtropical coastal countries, and it occurs most frequently in the northwestern Pacific Ocean. Taiwan is to the east of the Eurasian Continent and at the western side of the Pacific Ocean; its climate is intermediate between tropical and subtropical climate. us, typhoons frequently occur in Taiwan, generally in summer and fall. Typhoons affecting Taiwan typically develop at the sea surface southeast of Taiwan, and most typhoons are accompanied by torrential rains and strong winds [1]. Such rain and wind add to the damage from typhoons, posing a great threat to the transportation, economic, agricultural, and fishery activities in Taiwan even if the typhoon itself does not hit Taiwan. erefore, typhoons constitute a serious natural disaster in Taiwan.
For example, the 2015 Typhoon Soudelor was the most destructive typhoon that occurred in Taiwan in recent history, with gust intensity exceeding 12 on the Beaufort wind force scale (32.7 m/s). Its strong gusts caused widespread damage to infrastructure, affecting gas supply, power and utilities, transportation and communication, and weather radar stations. Electricity was cut off in approximately 4.5 million households simultaneously during Typhoon Soudelor-the greatest recorded number in recent history. e economic loss from the typhoon was estimated to be as high as US$76 million [2]. erefore, we aimed to establish an accurate wind speed prediction model for future typhoons, allowing for better preparation to mitigate a typhoon's toll on life and property. In this study, cutting-edge machine learning (ML) techniques were used to improve predictive accuracy. In general, ML algorithms learn from a huge dataset, improving their ability to identify patterns in the data. Specifically, ML involves creating algorithms to make prediction from sets of unknown data. Given their ability to perform parameter adjustment and achieve optimization through self-learning, neural network algorithms in ML are particularly powerful [3]. Such algorithms have been extensively used in recent wind speed prediction models, and these models are increasingly data-driven due to developments in ML [4][5][6][7][8][9][10][11]. In the development of neural networks, multilayer perceptron (MLP) networks are a classic approach that are often used and compared with other neural network models. For example, Wei [12] compared the accuracy of MLP with that of adaptive network-based fuzzy-inference-system neural networks in the construction of typhoon wind speed prediction models.
Deep learning has become possible due to the exponential increase in computing power in recent years. is approach is the further derivation of multiple neural layers from the original neural layers of a model. Such derivation improves an algorithm's ability to learn, better approximating the complex neural network structure of a human being. For example, Hu et al. [13] formulated multilayer deep neural networks that were trained using data from data-rich wind plants. ese networks extracted wind speed patterns, and the mapping was finely tuned using data from newly constructed wind plants. Tiancheng et al. [14] proposed a sandstorm prediction method that considered both the effect of atmospheric movement and ground factors on sandstorm occurrence, called improved naive Bayesian convolutional neural network classification algorithm.
In the field of neural networks, recurrent neural networks (RNNs), which can analyze sequential (or time-series) data, have recently been developed [15][16][17][18][19][20]. RNNs are connectionist models with the ability to selectively pass information across sequence steps while processing sequential data one element at a time [21]. erefore, RNNs are important, especially in the analysis of sequential data. A particular type of RNNs is long short-term memory (LSTM), a class of artificial neural networks where connections between units form a directed cycle [22] introduced the LSTM primarily to overcome the problem of diminishing gradients. LSTM creates an internal network state that allows it to exhibit dynamic temporal behavior. Unlike feedforward neural networks, RNNs can use their internal memory to process arbitrary sequences of input [23]. To model the time series of wind speed data, Byeon et al. [24] developed the LSTM for the prediction of typhoon wind speeds. Hence, feature enhancement from RNNs has been explored in wind prediction. e widespread application of ensembles in numerical weather prediction (NWP) has helped researchers improve weather forecasts [25][26][27][28][29][30]. Numerical models can be used to calculate all climate parameters through atmospheric dynamics and numerical methods, allowing for simulated results to be explained in terms of physical relationships. NWP models simulate the atmosphere on user-defined grid scales as a moving fluid. rough several types of parameterizations, NWP models also account for the influence of subgrid physical processes on grid-resolved motions [31,32]. NWP models, such as the Weather Research and Forecasting model (WRF), have been increasingly popular as a low-cost alternative source of data for such assessments of climate parameters [33]. WRF offers a wide variety of physical and dynamical elements to choose from; these elements must be put together to form model configurations, with which the model can be run [34]. However, because of imperfect models and uncertain initial boundary atmospheric conditions, errors exist in the NWP output [35,36].
Recent studies have applied the NWP model to typhoons and tropical cyclones [37][38][39]. However, the prediction of severe meteorological phenomena (such as typhoons), which result from a multiplicity of mutually interacting multiscale processes, remains a major challenge for NWP systems [25,40,41]. ese models are typically unable to predict wind intensity with satisfactory accuracy, even at short forecast times and a high horizontal resolution [42,43]. Some studies have tried combining NWP with ML models to develop an integrated climate prediction model. For example, Zhao et al. [44] evaluated the performance and enhanced accuracy of a day-ahead wind-power forecasting system. e system comprised artificial neural networks and an NWP model. Furthermore, to enhance the predictive accuracy of constructed ML models, other researchers have used numerically simulated results as input data for the construction of ML models. For example, Zhao et al. [45] developed the ARIMAX model, where wind speed results from the WRF simulation were chosen as an exogenous input variable. However, to the best of our knowledge, in the literature on short-term wind speed prediction for typhoons (or tropical cyclones), numerically simulated values have seldom been used as an input variable for ML prediction models. erefore, in relation to the construction of a ML-based wind speed prediction model, we evaluated the improvements to predictive accuracy afforded by the use of numerically simulated values (by comparing between its use and nonuse). erefore, our study has two primary aims: (1) develop an ML-and neural network-based wind speed prediction model and compare the predictive accuracy of various neural network-based algorithms and (2) evaluate the improvements to predictive accuracy afforded by the use of numerical solutions obtained from NWP models (by comparing between its use and nonuse) in a typhoon-surface wind-speed prediction model. Figure 1 illustrates the flow of the construction, involving NWP numerical solutions, of our neural network-based typhoon wind-velocity prediction model. In the first stage, we collected data on the typhoon characteristics and ground wind speed of the research area's historical typhoon events.

Methodology and Algorithms
In the second step, wind speed solutions were obtained from a wind field simulation of typhoons, involving an NWP numerical model. In our study, we also employed a WRF numerical model to simulate circulation distribution, thus obtaining the wind speed values in the research area. Two datasets could be built, one comprising typhoon data and the other comprising wind simulation results from the NWP model. e datasets were split into testing and trainingvalidation subsets. e training-validation set was used for the learning of several ML-based wind velocity prediction models, and the testing set was used for the identification of the optimal prediction model among these models. e best model among a set of ML neural network-based models, involving MLP, DRNN, and stacked LSTM, was determined. According to [46], real-time dynamics constitute the most challenging aspect of wind speed forecasting. We determined the stacked LSTM model to be the best for wind speed forecasting due to its appropriate handling of long-and short-term time dependency. In the final stage, the forecast accuracy of the stacked LSTM model was evaluated against other neural networks. We also evaluated whether forecast accuracy in ML-based models improved when NWP numerical solutions were used as input.

Frameworks Underlying the Proposed Neural Networks.
In this section, we describe the neural network-based architectures, which used the MLP, DRNN, and stacked LSTM algorithms, that were adopted for model construction. As illustrated in Figure 2(a), the MLP is a typical type of feedforward backpropagation neural network that uses processing units placed in the input, hidden, and output layers [47][48][49]. Each unit (with an associated weight) in a layer is connected to the units in adjacent layers [50,51]. In the study, to enhance learning efficacy-and by implication, approximation, and prediction accuracy-we added hidden layers to a simple MLP neural network; the MLP was trained through backpropagation. Generally, the weight updates Verify the optimal neural networks-based wind velocity prediction models Advances in Meteorology between layers are calculated in terms of the stochastic gradient descent [52]. Specifically, where w ij (t) is the weight set connecting layers i and j at time t, Δw is the weight correction, η is the learning rate, β is a momentum coefficient, and E is a cost function that indicates the difference between the target and predicted values. In particular, η and β are hyperparameters for adjusting the spacing of weight correction. As illustrated in Figure 2(b), the multilayers of the RNN structure comprise an input layer, multiple recurrent layers, and an output layer. When the length of recurrent layer is 1, the framework is a simple RNN. Here, the recurrent network is based on the networks developed by [53]. In the RNN, the hidden units are connected to context units; in the successive time step, the units feed back into the hidden units. e hidden state at any time step can contain information from an (almost) arbitrarily long context window [21]. e DRNN model framework has multiple recurrent layers before the forwarding to a dropout layer and output layer at the final output. In this paper, the dropout layer excludes 10% of the neurons to avoid overfitting.
A stacked LSTM architecture is defined as an LSTM model that comprises multiple LSTM layers (Figure 2(c)). e stacked LSTM, also known as deep LSTM, was first formulated by [54] and was applied to speech recognition problems. Similar to the framework underpinning the DRNN model, the stacked LSTM model uses multiple LSTM layers that are stacked before the forwarding to a dropout layer and output layer at the final output. In a stacked LSTM, the first LSTM layer produces sequence vectors used as the input of the subsequent LSTM layer. Moreover, the LSTM layer receives feedback from its previous time step, thus allowing for the capturing of data patterns. e dropout layer also excludes 10% of the neurons to avoid overfitting. e basic structure of LSTM, as illustrated in the LSTM layer in Figure 2(c), comprises an input gate i t , output gate o t , forget gate f t , and memory cell c t . A single LSTM layer has a second-order RNN architecture that excels at storing sequential short-term memories and retrieving them at many time steps later [55]. An LSTM network is identical to a standard RNN, with the exception of the summation units in the hidden layer being replaced by memory blocks [56]. Equations (2)-(6) describe how output values are updated at each step [22,46,57]. Specifically, where x t is the input vector; σ is the activation function; W f ,  ReLU activation function in the middle layers (the hidden, recurrent, and LSTM layers) of the aforementioned three models. e ReLU is defined as the positive part of its argument; that is, f(x) � max (0, x), where x is the input to a neuron. e ReLU function has been recently used to replace the sigmoid function in neural networks, resulting in good performance and fast training times [58]. Moreover, in these proposed models, the input layers receive the observed and simulated meteorological values as their input.

Study Area and Data
Taiwan is located at the western Pacific Ocean, a region frequently in the path of typhoons. Typhoons in Taiwan are often accompanied by torrential rains, which severely endanger life and property. us, more accurate regional wind speed prediction is needed to improve typhoon protection.
In this study, we chose northeastern Taiwan as the study area ( Figure 3), wherein Taipei City and Yilan City are the two major cities.

Data from Gauge and Typhoon Weather Stations.
According to typhoon data from the Central Weather Bureau (CWB), between 2000 and 2018, 29 typhoons have directly struck Taiwan (Table 1;  In these collected typhoons, the numbers for the category 1, 2, and 3 typhoons were 9, 5, and 8, respectively, and the number for the tropical storm was 7. e data included two types of datasets: one on typhoon dynamic characteristics and another (comprising 2621 hourly records) on surface wind speed observations. Surface wind speed data were released by the CWB and measured from eight gauge stations: Anbu, Banqiao, Keelung, Pengjiayu, Su-ao, Taipei, Tamsui, and Yilan ( Figure 3). In addition, data on typhoon dynamics were released by the CWB. e data comprised six variables: pressure at the typhoon center, latitude of the typhoon center, longitude of the typhoon center, typhoon radius (i.e., the distance from typhoon center for points with winds greater than 15.5 m/s), moving speed of the typhoon, and maximum wind speed at the typhoon center.

Data from Numerical
Solutions. WRF model simulations using typhoon data were used to generate wind speed values at each meteorological station. To set the initial field and boundary conditions, we used data from the Final Operational Global Analysis dataset. e dataset is a part of the Global Data Assimilation System of the US Government's National Center for Environmental Protection. e grid was set to be a two-domain nested grid. e horizontal grid spacing for the coarser domain (118°E-126°E, 21°N-29°N) was 15 km, and that of the finer domain (121°E-123°E, 24°N-26°N) was 3 km. Both domains had 32 vertical levels. is study used the WRF physical parameters recommended by [59,60]. ey studied the tracks and rainfall of typhoons that have invaded Taiwan in addition to other physical parameters, and they conducted wind forecasts using the WRF model. ey determined the following physical parameters to be suitable for wind speed forecasts: with regard to the planetary layer, those from the Yonsei University (YSU) scheme; with regard to microphysics, those from the WRF Single-Moment 5-Class (WSM5) scheme; for cumulus parameterization, those from the Kain-Fritsch scheme; and with regard to longwave radiation, those from the Rapid Radiative Transfer Model scheme.
After our simulation of a large number of typhoon events, the WRF model generated wind velocity simulations at eight ground gauge stations. In the verification of subsequent WRF wind outcomes, the output objective for the WRF model was the simulated wind value at an altitude of 10 m, because the ground gauge-station wind speed reached the observation value at an altitude of 10 m. Figure 5 displays the scatter plot of the WRF model simulation values and observed gauge-station values. According to Figure 5, several observation stations (Keelung, Anbu, Yilan, and Su-ao) have consistently higher wind speeds. ese higher wind speeds are caused by their surrounding topographies-which cause them to face windwards-in conjunction with the anticlockwise circulatory flow characteristic of typhoons. Pengjiayu Station is similar to the four stations. However, being located on an island on the sea around northeastern Taiwan, the station is directly affected only by a typhoon's circulatory flow and not the station's surrounding topography; this station also records higher wind speed values. e Taipei, Banqiao, and Tamsui Stations are located within the Tamsui Basin and are thus fully enclosed by mountains. Because the circulatory flow of typhoons can be easily disrupted by the surrounding topography, the wind speed values observed at these stations are slightly lower.
For the four windward observation stations, wind speed simulation values were lower and approximately equal to actual values for high and low wind speeds, respectively. For the three observation stations in the Tamsui Basin, wind speed simulation values were approximately equal to the actual values, for all wind speeds.
To evaluate the simulation outcomes, this study used the mean absolute error (MAE) and root mean squared error (RMSE) that are defined as follows: where Y i is the estimated value of record i, O i is the observation of record i, and n is the total number of records.
Advances in Meteorology e performance of the estimation indicators (MAE and RMSE) in terms of errors is presented in Figure 6.

Modeling
We used ML-based models to construct our wind speed prediction model for our study area. Observation stations at Taipei and Yilan were chosen as the test location. When constructing the model, the attribute data entered into the model included data on typhoon dynamics, data from ground meteorological observation stations, and data obtained from the aforementioned simulation. In this study, we performed data splitting for all typhoon episodes. Our training-validation set comprised data on 23 typhoon episodes between 2000 and 2013 (2093 records in total); our testing set comprised data on six typhoon episodes between 2014 and 2017 (528 records in total). Model training and validation were performed through 10-fold cross-validation, in which the training set was divided into 10 subsamples, one of which was retained for model verification, and the other nine were used for model   Figure 4: Tracks of historical typhoon incidents that affected Taiwan. 6 Advances in Meteorology training. In the verification process, each subsample must be verified. As mentioned in Section 2.1, multilayer neural networks were used to construct our ML neural network-based models. We used the adaptive moment estimation optimization algorithm (also known as the Adam optimizer) to optimize the momentum and learning rate [61]. Generally, the Adam optimizer is more broadly applied in neural networks [62][63][64]. e Adam optimizer can be used instead of the classical stochastic gradient descent procedure to iteratively update network weights based on training data. We also calibrated the hyperparameters, specifically, the number of neurons in a middle layer and the length of the middle layers in the MLP, DRNN, and stacked LSTM models. ese hyperparameters were evaluated through trial and error. First, the number of neurons in a middle layer was adjusted from 10 to 100 until the curves of the RMSE of the RMSE errors were approximated. e middle layers of the MLP, DRNN, and stacked LSTM models are the hidden, recurrent, and LSTM layers, respectively. Subsequently, the prediction errors corresponding to the minimum RMSE were obtained in the form of the optimal solution for the number of neurons in the middle layer. When the lead time was 1 h, for the Taipei Station, the optimal numbers of neurons were 30, 50, and 50 for the MLP, DRNN, and LSTM models, respectively; for the Yilan Station, those numbers for the three models were 40, 30, and 30, respectively (Figures 7(a) and 8(a)). Subsequently, we calibrated the length of the middle layers in   Advances in Meteorology the networks, adjusting them to be between 1 and 10 layers. For the Taipei Station, the optimal lengths of the middle layers were 7, 5, and 4 for the MLP, DRNN, and LSTM models, respectively; for the Yilan Station, those for the three models were 7, 6, and 5, respectively (Figures 7(b) and 8(b)).
Using the aforementioned method, we conducted parameter testing for the prediction models. Tests were       Advances in Meteorology conducted using the training-validation set at six hourly lead times between 1 and 6 h. e RMSE performance values (for all lead times) for the three models are presented in Figure 9. e stacked LSTM model outperformed the MLP and DRNN models for all lead times and for both the Taipei and Yilan Stations. We further tested and fine-tuned these models using the testing set to confirm the accuracy and feasibility of each model.

Predictions of Forecasting Horizons.
We tested and evaluated the MLP, DRNN, and stacked LSTM models using the testing set. e testing set comprised data (measured at the Yilin and Taipei Stations) on six typhoon episodes that occurred between 2014 and 2017.  Figure 10(a), the maximum observed wind speeds for the Taipei Station were between 7.8 m/s (Fung-Wong) and 14.9 m/s (Soudelor). According to Figure 11(a), those for the Yilan Station were between 9.9 m/s (Fung-Wong) and 26.8 m/s (Soudelor). As mentioned, the windward Yilan Station is more susceptible to the circular flow of the typhoon, unlike the Taipei Station, which is shielded by the mountains of the Tamsui Basin. erefore, wind speed was consistently higher at the Yilan station. Figures 10(a)-10(c) illustrate the prediction results for the Taipei Station at the lead times of 1, 3, and 6 h, respectively; for all prediction models, prediction accuracy was inversely related to the lead time. e data for the Yilan Station exhibited a similar trend (Figures 11(a)−11(c)). Such a situation is not atypical for prediction models. Intuitively, the further a model predicts into the future, the harder it is to obtain useful, real-time features for prediction. erefore, to evaluate predictive accuracy, we used the error evaluation indicators to quantify each model's rate of error in predicting the subsequent hour's wind speed.
To compute term-by-term comparisons of the relative error in the prediction with respect to the actual value of the variable, the mean absolute percentage error (MAPE) and root mean square percentage error (RMSPE) were calculated. MAPE and RMSPE can be calculated using the following formulae: Generally, MAPE is used to express the MAE as a percentage of the observations. MAPE is an unbiased statistic for measuring the predictive capability of a model [65]. RMSPE has the same properties as the RMSE but is expressed as a percentage [66].   Table 2 presents the average measurement values of the four error indicators for all lead times. In terms of the two unbiased statistical indicators, MAPE and RMSPE, the stacked LSTM and MLP models exhibited the most and least favorable performance, respectively. Stacked LSTM's superiority is attributable to its model structure. Specifically, LSTM is a neural network that contains LSTM blocks. LSTM blocks can be described as a type of smart network unit that can memorize numerical values of different time lengths (to determine the quantity of useful information). Additionally, a gate in the LSTM blocks can help determine if the input is important enough to be remembered and if the input can be exported as outputs. Because the data involved in wind speed prediction are sequential, the decision on how far back the retention of memory data should go becomes consequential for predictive accuracy. By contrast, because the MLP model has no memory capacity, its predictions are based on limited and currently available information, thus decreasing its predictive accuracy. For the DRNN model, although it can receive memory information from long ago, the lack of a gate filter system can cause the receiving of too much (i.e., overly long) memory information, potentially undermining predictive accuracy.    Figures 14 and 15, respectively. We determined that the use of WRF simulation values as input increased the model's predictive accuracy. erefore, the use of numerically simulated values as a part of the input data aids in the reduction of predictive error.

Evaluation of Forecast Efficiency with and without
We define the improvement rates for the MAE and RMSE (denoted IR MAE and IR RMSE , respectively) as follows: where MAE with and MAE without are the MAE results on the use and nonuse of WRF simulation values, respectively, and RMSE with and RMSE without are the RMSE results on the use and nonuse of the WRF simulation values, respectively. Figure 16

Conclusions
To accurately predict the wind speed of future typhoons, we constructed a typhoon wind speed prediction model using cutting-edge ML techniques. RNNs have been recently developed as a type of neural network that can analyze sequential data. e structure of such networks facilitates the effective processing of wind speed-relevant climate data over an extended period.
at is, the structure imbues RNN models with long-term memory capacity. erefore, such networks are suitable for predicting typhoon wind speeds. LSTM is a type of RNN that allows the user to decide the memory time's length. Additionally, LSTM gives users the option to filter output results, increasing LSTM's predictive accuracy. According to current developments in deep learning, learning performance is enhanced when the layers of neural networks are deepened. erefore, we used deep learning neural networks in this study. Additionally, we compared the performance of three types of RNNs-MLP, DRNN, and stacked LSTM-in predicting wind speed values. ese three types of model structure differ by their memory capacity: MLPs are model networks with no memory capacity, whereas DRNNs and stacked LSTM are model networks with memory capacity.
We chose northeastern Taiwan as the study area, and the observation stations at Taipei and Yilan were selected as the study subjects. e results indicated that for both the Taipei and Yilan Stations and for the MLP, DRNN, and stacked LSTM models, prediction error is positively related to the prediction lead time (of which there were six, one for each hour between 1 and 6 h). In other words, the lead time is inversely related to predictive accuracy. Model networks with memory capacity (DRNN and stacked LSTM) are more accurate than those without memory capacity (MLP). Stacked LSTM was also discovered to be more accurate than DRNN. Stacked LSTM is better than the traditional RNN method. Because stacking the LSTM's hidden layers deepens the model, model accuracy is increased; LSTM is closer to being a deep learning technique than its counterparts. us, stacked LSTMs is a stable technique for overcoming problems in wind velocity prediction. Finally, we examined if model accuracy is increased by the use of WRF simulation values as an input variable in the wind speed prediction model. e stacked LSTM model was used in the evaluation, given its best performance in most evaluation tests of the current study. Evaluating predictive accuracy in terms of the four error indicators, namely, MAE, RMSE, MAPE, and RMSPE, verified that the inclusion of the numerically simulated input greatly increased model accuracy. Specifically, for all lead times, the use of numerically simulated values reduced the MAE and RMSE error rates of the Taipei Station by 27.3% and 28.7%, respectively. Similarly, the MAE and RMSE rates for the Yilan Station were reduced by 30.3% and 31.1%, respectively. erefore, the inclusion of numerically simulated values in wind speed prediction models enhanced their prediction accuracy.

Data Availability
e Final Operational Global Analysis data were obtained from the Global Data Assimilation System of the US Government's National Center for Environmental Protection, which is available at https://rda.ucar.edu/datasets/ ds083.2/. e typhoon data were obtained from the Central Weather Bureau of Taiwan, which are available at https://rdc28.cwb.gov.tw/and https://e-service.cwb.gov.tw/ wdps/.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.