Performance Evaluation of Linear and Nonlinear Models for Short-Term Forecasting of Tropical-Storm Winds

Abstract

Wind-sensitive structures usually suffer from violent vibrations or severe damages under the action of tropical storms. It is of great significance to forecast tropical-storm winds in advance for the sake of reducing or avoiding consequent losses. The model used for forecasting becomes a primary concern in engineering applications. This paper presents a performance evaluation of linear and nonlinear models for the short-term forecasting of tropical storms. Five extensively employed models are adopted to forecast wind speeds using measured samples from the tropical storm Rumbia, which facilitates a comparison of the predicting performances of different models. The analytical results indicate that the autoregressive integrated moving average (ARIMA) model outperforms the other models in the one-step ahead prediction and presents the least forecasting errors in both the mean and maximum wind speeds. However, the support vector regression (SVR) model has the worst performance on the selected dataset. When it comes to the multi-step ahead forecasting, the prediction error of each model increases as the number of steps expands. Although each model shows an insufficient ability to capture the variation of future wind speed, the ARIMA model still appears to have the least forecasting errors. Hence, the ARIMA model can offer effective short-term forecasting of tropical-storm winds in both one-step and multi-step scenarios.

1. Introduction

Tropical storms frequently occur over tropical oceans, and they are highly likely to evolve into a typhoon event when the wind speed is greater than 118 km/h [1]. Owing to the high wind speed, they usually cause extensive damage to engineering structures and infrastructure facilities. Specifically, wind-sensitive structures, e.g., long-span bridges, high-rise buildings, and large-expanse roofs, may suffer from high risks to structural safety [2,3]. Hence, tropical storms attract significant attention in the view of structural safety and serviceability in coastal regions. In light of the destructive power and inevitable occurrences of tropical storms each year, the prediction of wind speeds based on recorded data becomes an important topic in the area of structural wind engineering. If the wind speed can be accurately forecasted in advance, emergency measures can be taken to reduce the losses as far as possible. Effective approaches are thus essential for the short-term prediction of tropical storms.

In recent decades, various approaches have been proposed for the prediction of wind speeds with the purpose of wind energy forecasting or disaster prevention [4,5]. The approaches can be classified into two categories, including the physical method and the statistical method [6,7]. The physical method forecasts the wind speed based on formulas interpreting the laws of physics and considers the site conditions, including meteorology (temperature, pressure, humidity, and so on) and terrain effects [8,9,10]. Nevertheless, the calculations used in solving the associated governing equations are so complicated that they require substantial computational resources, which makes the simulation extremely time-consuming. The physical method is thus deemed to not be appropriate for the purposes of early warning. In contrast, the statistical method employs time-series models or black-box models, established based on historical data, to predict the future wind speed. This approach tacitly assumes that the future data have a linear or nonlinear relationship with the previous data. Compared with the physical method, the statistical method does not require many computational resources and is suitable for short-term forecasting.

The time-series models frequently used in the statistical method mainly include the autoregressive moving average (ARMA) model [11,12,13,14], auto-regressive integrated moving average (ARIMA) model [15,16,17], and other variants like the fractional ARIMA model [18] and vector autoregressive model [19,20]. These models have extensive applications in the forecasting of wind speeds, and effective predictions can be obtained when distinct linear features exist in the data. When it comes to the case with strong nonlinearities, nonlinear models are required to describe the connection between input and output data, and the prediction accuracy is usually improved compared with the results of linear models.

To characterize the nonlinear features, the artificial intelligence (AI) technique offers effective networks that can establish a black-box model for the input and output data. The function of the black-box model is similar to the time-series model, but the nonlinear patterns between future data and historical data can be represented. AI approaches mainly include back-propagation neural network (BPNN) [21,22], long short-term memory (LSTM) network [23,24], gated recurrent units (GRUs) network [25,26,27], support vector regression (SVR) [28,29,30], and other related networks [31,32,33]. Although, in some applications, the nonlinear model displays better performance compared with the time-series model, the training of the network is time-consuming and the trained network is sensitive to the preset parameters.

The aforementioned approaches have been widely used in the forecasting of conventional boundary-layer winds, but their applicability in tropical storms is rarely reported in the literature. Considering the significance of wind speed prediction of tropical storms for slender structures in the view of structural health monitoring, this paper presents a performance evaluation of linear and nonlinear models for short-term wind speed forecasting. Five commonly used models are taken into account, namely, ARIMA, BPNN, LSTM, GRU, and SVR models. The measured wind samples of the tropical storm Rumbia are employed for the analysis. One-step and multi-step ahead predictions are both conducted, and the performances are evaluated in light of the mean and maximum wind speeds in a given period. Finally, the advantages and disadvantages of these models in the short-term forecasting of tropical storms are summarized, and corresponding suggestions are made for the purposes of early warning.

2. Linear and Nonlinear Methods

The statistical method conducts the prediction using models established based on historical data. Thus, this method is more straightforward and convenient to implement compared with the physical method. In this section, an overview of the commonly used linear and nonlinear models is presented.

2.1. Time-Series Models

Time-series models build a linear relationship between future values and previously observed values, and mainly refer to the ARMA model and its variants [34]. These models can be generally expressed as a sum of the autoregression (AR) part and the moving average (MA) part. The AR part presents a regression on its own lagged values, which shows the memory effect of p previously predicted values in the time series. The MA part describes the regression error as a linear combination of error terms that occurred in past periods. As the model depends on the orders of AR and MA parts, the model is often expressed as the ARMA(p, q) model. Furthermore, if the data show evidence of non-stationarity in the sense of mean, an initial differencing step can be applied to eliminate the time-varying trend. Then, the difference between consecutive observations can be treated as wide-sense stationary. The series of differences can be characterized using the ARMA model, and the future value can be calculated from the predicted difference. These two consecutive steps form the ARIMA model. The expression of the ARIMA model is given as

$$Y_t={(1-L)}^d{y}_t,$$

(1)

$$\left(1-{\displaystyle \sum _{i=1}^p{\phi }_i{L}^i}\right)Y_{\mathrm{t}}=\left(1+{\displaystyle \sum _{i=1}^q{\theta }_i{L}^i}\right){\epsilon }_t,$$

(2)

where $y_t$ represents the future value to be predicted at period t; $Y_t$ is the difference with the order $d$; $L$ is the lag operator; p and q are non-negative integers, referring to the orders of AR and MA parts, respectively; ${\phi }_i$ is the ith AR coefficient; ${\theta }_i$ is the ith MA coefficient; and ${\epsilon }_t$ is the error term that follows a normal distribution with a zero mean. As the prediction depends on the order of differencing $d$ apart from the AR order p and MA order q, the ARIMA model is also named as the ARIMA(p, d, q) model. If the parameter $d$ equals zero, the ARIMA model is reduced to the ARMA model.

Generally, there are three indispensable steps to establish an ARIMA model, which can be summarized as follows [35]:

• Model order identification: an initial differencing step is applied to make the data stationary in the mean sense. Then, the autocorrelation function (ACF) and partial autocorrelation function (PACF) of differenced time series will be calculated for the order determination of AR and MA parts.
• Estimation of parameters: the parameters to be determined are estimated using the maximum likelihood method or the least squares method.
• Residual diagnostics: the goodness of fit of the given data is estimated by checking the prediction errors. The prediction errors of a good model correspond to white noise, and its ACF will be at a low level.

2.2. Artificial Neural Network Approach

The artificial neural network-based approach is a typical nonlinear method for wind speed forecasting. Nonlinear artificial neural networks consist of an input layer, one or more hidden layers, and an output layer. Each layer has a number of artificial neurons, and the activation functions between neurons of different layers make the network capable of non-linear mapping. This approach offers a black-box model to simulate the complicated nonlinear relationship between input and output layers and requires no explicit mathematical expressions as used in the physical and time-series approaches. The networks of BPNN, LSTM, and GRU are summarized as follows.

2.2.1. BPNN

The back-propagation neural network (BPNN), which is a type of multilayer feed-forward neural network, has been widely applied in various fields of forecast and recognition [36]. As shown in Figure 1, BPNN includes three types of basic network layers, namely, the input layer, the hidden layer(s), and the output layer, and each layer has various numbers of nodes that aim to obtain a scalar result by computing the inner product of the input vector and weight vector through an activation function. The calculating process of each layer can be given as

$${\mathit{a}}^l=\sigma \left(w^l{\mathit{a}}^{l-1}+{\mathit{b}}^l\right),$$

(3)

where ${\mathit{a}}^l$ represents the output vector of the l^th layer; ${\mathrm{w}}^l$ and ${\mathit{b}}^l$ are the connection weight vector from the (l − 1)^th layer to the l^th layer and the bias vector of the l^th layer, respectively; and $\sigma$ is the nonlinear activation function, which is usually a rectified linear unit (ReLU) activation function or a sigmoid activation function.

This artifical neural network can be trained via the gradient descent method that minimizes the loss function of the network to tune the weight vectors and threshold vectors of the joints between two layers. The most commonly used loss function is the squared error loss, which can be expressed as

$$\mathrm{\Gamma }({\mathit{y}}^{\mathit{o}} ,\hat{\mathit{y}})=k{‖{\mathit{y}}^{\mathit{o}}-\hat{\mathit{y}}‖}_2^2,$$

(4)

where ${\mathit{y}}^{\mathit{o}}$ and $\hat{\mathit{y}}$ represent the desired output and forecasting output, respectively; $k$ is a constant weight and can be set as 1; and ${‖\cdot ‖}_2$ denotes an operator of L² norm.

2.2.2. LSTM

The long short-term memory (LSTM) is an artificial recurrent neural network structure [37]. Different from feedforward neural networks, it has feedback connections that can consider the memory effect of time series. Cyclic connections are used to acquire short-term memory and capture information from sequences of data. Figure 2 shows the structure of a memory cell of an LSTM network.

Apart from the cell, the LSTM network has three gates, namely, an input gate, an output gate, and a forget gate. The cell remembers values over abitrary intervals and the three gates regulate the flow of information into and out of the cell. More specifically, the transfer of input and output activations is controlled by the input gate and output gate, respectively. The forget gate enables the LSTM to update the interior state adaptively. The processes of updating the cell state and calculating the output of LSTM can be given as

$${\mathit{f}}_{\mathit{t}}={\mathsf{\sigma }}_g\left({\mathit{W}}_{\mathit{f}}{\mathit{x}}_{\mathit{t}}+{\mathit{U}}_{\mathit{f}}{\mathit{h}}_{\mathit{t}-\mathbf{1}}+{\mathit{b}}_{\mathit{f}}\right),$$

(5)

$${\mathit{i}}_{\mathit{t}}={\mathsf{\sigma }}_{\mathit{g}}\left({\mathit{W}}_{\mathit{i}}{\mathit{x}}_{\mathit{t}}+{\mathit{U}}_{\mathit{i}}{\mathit{h}}_{\mathit{t}-\mathbf{1}}+{\mathit{b}}_{\mathit{i}}\right),$$

(6)

$${\mathit{o}}_{\mathit{t}}={\mathsf{\sigma }}_g\left({\mathit{W}}_{\mathit{o}}{\mathit{x}}_{\mathit{t}}+{\mathit{U}}_{\mathit{o}}{\mathit{h}}_{\mathit{t}-\mathbf{1}}+{\mathit{b}}_{\mathit{o}}\right),$$

(7)

$${\mathit{c}}_{\mathit{t}}^{\prime }={\mathsf{\sigma }}_h\left({\mathit{W}}_{\mathit{c}}{\mathit{x}}_{\mathit{t}}+{\mathit{U}}_{\mathit{c}}{\mathit{h}}_{\mathit{t}-\mathbf{1}}+{\mathit{b}}_{\mathit{c}}\right),$$

(8)

$${\mathit{c}}_{\mathit{t}}={\mathit{f}}_{\mathit{t}}\odot {\mathit{c}}_{\mathit{t}-\mathbf{1}}+{\mathit{i}}_{\mathit{t}}\odot {\mathit{c}}_{\mathit{t}}^{\prime },$$

(9)

$${\mathit{h}}_{\mathit{t}}={\mathit{o}}_{\mathit{t}}\odot {\mathsf{\sigma }}_h\left({\mathit{c}}_{\mathit{t}}\right),$$

(10)

where ${\mathit{x}}_{\mathit{t}}$ refers to input vector to the LSTM cell; ${\mathit{i}}_{\mathit{t}}$, ${\mathit{f}}_{\mathit{t}}$, and ${\mathit{o}}_{\mathit{t}}$ represent activation vectors of the input gate, forget gate, and output gate, respectively; ${\mathit{h}}_{\mathit{t}}$ denotes the hidden state vector also known as the output vector of the LSTM unit; and ${\mathit{c}}_{\mathit{t}}^{\prime }$ and ${\mathit{c}}_{\mathit{t}}$ represent the cell input activation vector and cell state vector, respectively. Additionally, $\mathit{W}$ and $\mathit{U}$ stand for the weight matrices; $\mathit{b}$ is the corresponding bias vector parameter,${\mathsf{\sigma }}_g$ is the logistic sigmoid function, ${\mathsf{\sigma }}_h$ is the hyperbolic tangent function, and $\odot$ denotes the Hadamard product (element-wise product).

2.2.3. GRU

Similar to the LSTM neural network, the gated recurrent unit (GRU) neural network is also a variant of recurrent neural networks [38], and it has fewer parameters because there is no output gate. The cell structure of GRU is shown in Figure 3, and the procedure on updating the cell state and calculating the output of GRU can be expressed as

$${\mathit{z}}_{\mathit{t}}={\mathsf{\sigma }}_g\left({\mathit{W}}_{\mathit{z}}{\mathit{x}}_{\mathit{t}}+{\mathit{U}}_{\mathit{z}}{\mathit{h}}_{\mathit{t}-\mathbf{1}}+{\mathit{b}}_{\mathit{z}}\right),$$

(11)

$${\mathit{r}}_{\mathit{t}}={\mathbf{\sigma }}_g\left({\mathit{W}}_{\mathit{r}}{\mathit{x}}_{\mathit{t}}+{\mathit{U}}_{\mathit{r}}{\mathit{h}}_{\mathit{t}-\mathbf{1}}+{\mathit{b}}_{\mathit{r}}\right),$$

(12)

$${\hat{\mathit{h}}}_{\mathit{t}}={\varphi }_h\left[{\mathit{W}}_{\mathit{h}}{\mathit{x}}_{\mathit{t}}+{\mathit{U}}_{\mathit{h}}\left({\mathit{r}}_{\mathit{t}}\odot {\mathit{h}}_{\mathit{t}-\mathbf{1}}\right)+{\mathit{b}}_{\mathit{h}}\right],$$

(13)

$${\mathit{h}}_{\mathit{t}}=\left(1-{\mathit{z}}_{\mathit{t}}\right)\odot {\mathit{h}}_{\mathit{t}-\mathbf{1}}+{\mathit{z}}_{\mathit{t}}\odot {\hat{\mathit{h}}}_{\mathit{t}},$$

(14)

where ${\mathit{z}}_{\mathit{t}}$ and ${\mathit{r}}_{\mathit{t}}$ are activation vectors of the update gate and reset gate, respectively; and ${\hat{\mathit{h}}}_{\mathit{t}}$ and ${\mathit{h}}_{\mathit{t}}$ represent the candidate activation vector and output vector, respectively.

2.3. SVR

The support vector regression (SVR) is a typical extension of the support vector machine in regression [39]. The basic principle is to map the data into a high dimensional feature space via nonlinear mapping, after which the linear regression is performed in the feature space. It avoids overfittings in the traditional machine learning algorithms. Similar to artificial neural networks, the performance of SVR heavily depends on the input parameters and associated coefficients. The regression formula of SVR can be expressed as

$$f(\mathit{x})=w\phi (\mathit{x})+b,$$

(15)

where $\phi (\cdot )$ denotes a nonlinear mapping function and $W$ and $b$ represent the weight vector and bias, respectively. To find an optimal SVR model, it needs to minimize the optimization function subject to tolerance and slacks. The governing target function is given as

$$\begin{gathered} \underset{w,b}{\min }\frac{1}{2}\parallel w{\parallel }^2+C{\displaystyle \sum _{i=1}^N\left({\xi }_i+{\xi }_i^{\ast }\right)}, \\ \mathrm{s}.\mathrm{t}. \left\{\begin{array}{c}{y}_i-f(\mathit{x})\le ϵ+{\xi }_i\\ f(\mathit{x})-y_i\le ϵ+{\xi }_i^{\ast }\\ {\xi }_i\ge 0,{\xi }_i^{(\ast )}\ge 0\end{array}\right., \end{gathered}$$

(16)

where C is a factor for tradeoff between over-fitting and underfitting; $ϵ$ is the tolerance that determines the width of the decision boundary; ${\xi }_i$ and ${\xi }_i^{\ast }$ are slack variables; and N stands for the number of samples.

Using the Lagrange multiplier method with Karush–Kuhn–Tucker conditions, the optimization function can be converted to a quadratic optimization problem. Then, the regression function is simplified as

$$\begin{array}{c}\hfill f\left(\mathit{x}\right)=\sum _{i=1}^N\left({\alpha }_i-{\alpha }_i^{\ast }\right)k(x_i,x)+b,\end{array}$$

(17)

where ${\alpha }_i$ and ${\alpha }_i^{\ast }$ represent positive and negative Lagrange multipliers, respectively; and $k(x_i,x)$ is the kernel function, including linear, polynomial, Gaussian kernels, and so on. In this study, the Gaussian kernel, also known as the radial basis function (RBF) kernel, is employed because of its popularity [40].

3. Measured Tropical-Storm Winds

The tropical storm Rumbia, which occurred in 2018, is selected as an example for short-term wind-speed forecasting. It was the third tropical storm that hit the east coast of China and caused widespread damage to properties with concomitant economic losses and casualties. Rumbia originally developed as a tropical depression in the southeast of the Ryukyu Islands and then moved northward on 13 August. It went towards the west on 16 August and made landfall at Shanghai in the early morning of 17 August. During its landfall, the intensity of the wind power achieved its maximum. After it moved into the continent, the wind power of Rumbia began to decrease and finally disappeared.

The Sutong Bridge, which was the longest cable-stayed bridge in the world, is very close to Shanghai, so the wind data of Rumbia were well recorded by the anemometers in the structural health monitoring system. Details on the anemometers can be found in [41]. The data from the anemometer, which is located in the middle of the main span, with a height of 76.9 m, are employed for the forecasting. The data span from 20:48 on 16 August to 08:48 on 17 August. In order to obtain sufficient reaction time to take emergency measures, the time interval during the wind speed forecasting is taken as 1 min. Figure 4 depicts the 1 min peak wind speeds of the measured wind samples, in which the zero time instant corresponds to 20:48 on 16 August.

The wind speed in Figure 4 shows a typical wind sample measured in the outer region of the tropical storm Rumbia. The wind speed generally increases before 07:45 on 17 August and then decreases after the peak wind speed, which equals to 33 m/s, although a dramatical plunge occurs between 03:56 and 04:03. The observation above indicates that a distinct time-varying trend exists in the wind sample. Hence, the wind sample is visibly judged as a non-stationary process, which will be quantitively analyzed in the following section.

To analyze the statistical property of the wind sample, the PDF of the wind speeds is calculated, as shown in Figure 4b. It is observed that the wind speeds are mostly larger than 10 m/s and are not well satisfied with the normal distribution. In such a case, the models based on a Gaussian assumption may present a relatively poor prediction. Considering the non-stationarity, non-Gaussianity, and even nonlinearity in tropical-storm winds, it is necessary to discuss the applicability of different forecasting approaches.

To facilitate the wind speed forecasting, the measured data in Figure 4 are classified as three parts with a length ratio of 3:1:1. The first part is the training set to build the basic forecasting model. The second part is the validation set that provides an unbiased evaluation of the forecasting model when tuning hyperparameters, and this part can eliminate the overfitting problem. The last part is the test set, which is utilized to verify the forecasting performance. The stationarity and statistical properties of the three parts are different to some extent. This is the big challenge for short-term forecasting of tropical storms.

The linear and nonlinear models are both established based on in situ measured data, so the wind characteristics and topography effects are inherently involved in the prediction. As the wind direction of the tropical storm will not change significantly in a short amount of time, the trained model by the previously collected data in a determined period can represent the upstream topography. However, for the wind speed near the eye wall, the wind direction changes rapidly, thus the model trained by the previously collected data would not work for the future data on this condition, as the upstream terrain may be different.

It should be noted that only artificial neural network-based models with parameters optimized using the stochastic gradient descent algorithm usually have the problem of overfitting, while this problem is rarely observed in the ARIMA and SVR models because the fitting criterion is based on the maximum likelihood method. Accordingly, the validation can be ignored in the model training of the ARIMA and SVR models.

4. Performance Comparison of Different Models

4.1. Stationary Test and Model Building

For tropical-storm winds that often have prominent non-stationary properties [42], it is necessary to perform a stationary test for the samples before short-term forecasting, which facilitates the selection of stationary and non-stationary models. In statistics, the well-known Dickey–Fuller test is frequently utilized to test the stationarity of a sequence. Once the sequence is non-stationary, its autoregressive model would have a unit root. A general form of the conventional Dickey–Fuller test is given as [43,44]

$$\Delta u_t=\delta u_{t-1}+\alpha +\beta \lambda +{\epsilon }_t,$$

(18)

where $\Delta u_t=u_t-u_{t-1}$ and $u_t$ represents the sample; $\alpha$, $\beta$, and $\delta$ are coefficients; and $\lambda$ represents the deterministic trend. The null hypothesis H₀: $\delta =0$, which means the sample is non-stationary, can be tested against the stationary alternative H₁: $\delta <0$ based on the t-statistic [45]. A smaller t-statistic indicates a strong evidence to reject the null hypothesis, so a stationary sample has a small t-statistic.

The results of the Dickey–Fuller test for the wind samples in Figure 4a are presented in

Table 1. Results of the Dickey–Fuller test.

Items	Test Value
	Original Data	Differenced Data
t-statistic	−1.067	−11.894
Critical Value (95%) 1	−2.868	−2.868

¹ Note: the value in the bracket indicates the confidence to accept the null hypothesis.

with a comparison with the critical value that the null hypothesis can be accepted with a 95% confidence level. It can be seen from Table 1 that the t-statistic of the original wind records is −1.067, which is greater than the critical value, which means the null hypothesis can be accepted. Thus, the wind data are regarded as a non-stationary process.

In the ARIMA model, the differenced data will be employed for forecasting. Based on this consideration, a first-order differencing step is applied to the measured wind data, and the results are depicted in Figure 5a. It is found that the differenced data show no distinct time-varying trend in the mean sense. The Dickey–Fuller test is further conducted to the differenced data, and the results are presented in Table 1 as well. It is found that the corresponding t-statistic is −11.894, which is much smaller than the critical value. Hence, the null hypothesis can be rejected and the differenced data are judged as a stationary process. The PDF is also analyzed for the differenced data and then compared to the normal distribution, as shown in Figure 5b. Different from the observation in Figure 4b, the PDF of the differenced data follows the normal distribution very well. On this condition, the order of the differencing step of the ARIMA model can be taken as one.

Figure 6 depicts the ACF and PACF of the differenced data. A 95%-confidence interval, in which the wind speeds are thought to be uncorrelated, is also plotted. It is observed that the ACF has a cutoff after the second lag, and the PACF cuts off after the sixth lag, which implies that the values of q and p in the time-series model can be taken as 2 and 6, respectively. Hence, an ARIMA(6,1,2) model can be utilized to characterize the selected dataset.

After the orders of different terms are determined, the remaining coefficients of the ARIMA model can be estimated via the maximum likelihood method. Then, the wind speeds of the training set are reconstructed via the established ARIMA(6,1,2) model and the residual errors with respect to the measured data are calculated. Figure 7a presents the PDF of the residual component, and the ACF of residual errors is depicted in Figure 7b. A comparison between the PDF of the residual component and the normal distribution indicates that the residual errors are Gaussian with a zero mean and standard deviation of 0.0141. The ACF suggests that there is no significant correlation between the errors and its own lagged values. Based on these observations, the established ARIMA(6,1,2) model provides a satisfied estimation of the training set.

For the training of BPNN, there are two main hyperparameters determining the topology, which are the number of layers and the number of neurons in each layer. The first and last layer are generally identified as the input layer and the output layer, whose functions are accessing data and providing results, respectively. The nodes of the input layer equal to the number of focused elements. In this study, the wind speed is the only dependent variable to conduct forecasting. The result of PACF presented in Figure 6 suggests that the wind speed is related with the previous 6min historical data. The number of nodes in the input layer is set as 6 in BPNN. When it comes to nodes in the output layer, the number depends on the steps ahead for prediction. For example, six nodes will be employed for a six-step ahead forecasting model. Although there is no unified standard to determine the number of hidden layers and the quantity of nodes in each hidden layer, studies suggest that a single hidden hidden layer is generally competent in the large majority of engineering problems, and this configuration is utilized for BPNN in this study [22,46,47,48]. Moreover, these studies also suggest that the optimal size of the hidden layer is usually taken between those of the input and output layers. Based on several training and associated comparisons, six nodes are employed for the single hidden layer of BPNN in this study.

With respect to the training of the LSTM network, a more accurate prediction can be obtained with more memory cells. To facilitate a comparison with the BPNN model, the LSTM network employs one hidden layer and six memory cells within the hidden layer as well. As the memory cell of LSTM can automatically access data and reserve the previous information, there is no need for an additional input layer, and the configuration of the output layer is similar to that of BPNN. For the network with GRU cells, the structure is the same as that with the LSTM cell. The only difference is that each neuron is represented via the GRU cell instead of the LSTM cell.

On the SVR model based on the radial basis function kernel, the dimension of the input feature vector is equal to 6 according to the PACF of observed data. Then, the undetermined parameters can be estimated using the least-square method. When it comes to the multi-step ahead forecasting, different SVR models need to be developed as the number of output variates varies.

Given the discussions above, there are five models included for the subsequent short-term wind speed forecasting. The models and their corresponding descriptions are given in

Table 2. Statistical models employed for short-term forecasting of tropical storms.

No.	Model	Description
1	ARIMA	The autoregressive, differential, and moving average terms have the order of 6, 1, and 2, respectively.
2	BPNN	A relatively simple model is applied, which contains an input layer with six neurons, a hidden layer with six neurons, and an output layer with six neurons.
3	LSTM	The memory cell takes place of the neurons in the hidden layer of BPNN.
4	GRU	The same structure as LSTM is adopted for this model.
5	SVR	The numbers of input and output features equal 6 and 1, respectively.

. Note that the assigned parameters of each model vary for different tropical storms.

As the wind speeds present typical time-varying features, a dynamic training strategy that updates the model coefficients when new observations are obtained is more appealing. However, this strategy is only suitable for ARIMA and SVR models because the training of artificial neural network-based models is time-consuming, which is contradictory to the purpose of early warning. Moreover, the optimization in training artificial neural networks has salient uncertainties in the trained results. Specifically, the error of the training set decreases with training epochs, while the error of the validation set firstly decreases with the training epochs before the optimum epoch and increases afterwards. Normally, the optimum epoch can be hardly determined automatically, and overfitting may be encountered. In such a case, the parameters of artificial neural network-based models are fixed after the training step is completed. However, for the ARIMA and SVR models, the parameters are updated in each prediction step.

4.2. Performance Evaluation Criteria

The linear and nonlinear models would present inevitable forecasting errors due to the statistical approximations. As each model employs the optimal set of parameters that lead to the smallest error for the training dataset, the forecasting errors can be viewed just from the performances of different models.

To evaluate the forecasting performance of different models, four error indexes including the mean absolute error (MAE), the mean relative percentage error (MRPE), the root mean squared error (RMSE), and the coefficient of determination (R²) are employed. The four indexes are defined as

$$\mathrm{MAE}=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|U_i-{\widehat{U}}_i\right|},$$

(19)

$$\mathrm{MRPE}=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|\frac{U_i-{\widehat{U}}_i}{U_i}\right|}\times 100\%,$$

(20)

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^N{\left(U_i-{\widehat{U}}_i\right)}^2}},$$

(21)

$${\mathrm{R}}^2=1-\frac{{\displaystyle \sum _{i=1}^N{(U_i-{\widehat{U}}_i)}^2}}{{\displaystyle \sum _{i=1}^N{(U_i-\frac{1}{N}{\displaystyle \sum _{i=1}^N{U}_i})}^2}},$$

(22)

where ${\mathrm{U}}_{\mathrm{i}}$ and ${\widehat{U}}_i$ represent the measured and predicted wind speeds at period i, respectively; and N is the number of data points involved in the performance evaluation.

4.3. One-Step Ahead Forecasting

Based on the five models given in Table 2, one-step ahead forecasting is carried out for the wind data shown in Figure 4. The comparison of measured and forecasted wind speeds is presented in Figure 8. As observed in the figure, the forecasting results by the ARIMA, BPNN, LSTM, and GRU models have slight differences and can generally resemble the measured wind speeds. However, the SVR model presents noticeable deviations from the measured wind data at each instant, because the tacit assumption of Gaussian distribution to be required by SVR is not well satisfied for the measured winds. In addition, the peak wind speed of this extreme wind event is underestimated by all five models. This is probably attributed to the fact that these statistical methods based on the accumulated historical data are able to capture the primary trend of the time-series, but the abrupt changes around the peak wind speed of tropical-storm winds that present completely different patterns cannot be characterized. This point is thus proved to be a critical and difficult issue for the short-term forecasting of tropical storms using time-series models.

In order to further quantify the forecasting errors, the MAE, MRPE, RMSE, and R² of the five models are calculated and listed in

Table 3. Performance comparison of different methods.

Model	MAE	MRPE/%	RMSE	R2
ARIMA	1.26	5.24	1.64	0.68
BPNN	1.29	5.35	1.69	0.66
LSTM	1.27	5.32	1.64	0.68
GRU	1.29	5.27	1.68	0.67
SVR	1.67	6.76	2.24	0.40

. It can be seen from the table that the errors of GRU and LSTM models present negligible differences to the BPNN model, despite the consideration of the memory effect. The MAE, MRPE, RMSE, of the SVR model are 1.67, 6.67%, and 2.24, respectively. The values of these indexes are obviously larger than the other models. Moreover, the R² value of the SVR is only 0.40, which indicates relatively bad forecasting compared with the measured wind speeds. Hence, the SVR provides the worst forecasting among the five models. With respect to the ARIMA model, it has the best performance in each index, which means it provides the most accurate prediction from a global point of view. The better performance of the ARIMA model is attributed to the differencing procedure that considers the non-stationarity of wind speeds in the mean sense. Based on the existing observations, the short-term forecasting of tropical-storm winds should emphatically take account of the non-stationary patterns.

Figure 9 depicts the density heatmap of the forecasting errors with a comparison to the fitted normal distribution. It is observed that the errors of the ARIMA, BPNN, LSTM, and GRU models are generally satisfied with the normal distribution with small shifts from zero in the mean, especially for the results of the ARIMA and LSTM models. The ARIMA model has the smallest mean of 0.014, whereas the mean of the GRU model has reached 0.373. The error distribution of the SVR model is the flattest, which indicates the largest standard deviation. This observation is consistent with the aforementioned phenomenon that the SVR model provides the worst forecasting with large errors.

4.4. Multi-Step Ahead Forecasting

The emergency management hopes the forecasting can be as far ahead of time as possible. Hence, multi-step ahead forecasting is also appealing for tropical storms, although the error increases as the number of forecasting steps increases [49]. The difference between one-step and multi-step ahead forecasting is that more outputs are required in the model building of a multi-step case. Figure 10 shows a comparison of multi-step forecasted wind speeds by the ARIMA model with comparison with the measured data. The results support the fact that the deviations between forecasted and measured wind speeds become more and more prominent as the time horizon expands, as the correlation between the wind speeds in the more distant future and the historical data becomes weaker.

Figure 11 shows the prediction errors of different models with respect to the forecasting steps. The step varies from one to six, which means the maximum reaction time can be expanded to 6 min. It is found in Figure 11 that the forecasting errors of the SVR model are generally the largest for each forecasting step. Although a relatively large error is observed in the GRU model when the forecasting step is three, the differences in the errors of the LSTM, BPNN, and GRU models in other steps are not large. The ARIMA model forecasts the wind speeds with the least error even when the forecasting step is six. The error increment of the ARIMA model is relatively small. For example, from one to six steps ahead, the MAE, MRPE, and RMSE of the ARIMA model increase by 0.06, 0.27%, and 0.05, respectively, while the R² value decreases from 0.68 to 0.67. Hence, the ARIMA model provides a stable forecasting at varying forecasting steps, and the errors have slight changes.

Given the discussions above, it can be concluded that the performances of all the BPNN, LSTM, GRU, and SVR models are not satisfied as they cannot well consider the non-stationary and non-Gaussian properties. However, the ARIMA model can provide better forecasting, especially in the multi-step cases, as the involved differencing step can make the predicted variable to be stationary and Gaussian.

5. Forecasting of Mean Wind Speeds

The forecasting of the 1 min maximum wind speed is relatively challenging owing to the existence of some high-frequency components. The mean wind speed of tropical-storm winds is another important focus. Thus, multi-step forecasting of the mean wind speed is also conducted to investigate the performances of linear and nonlinear models. Figure 12 shows the mean wind speed of Rumbia, which is averaged by a time interval of 1 min. Meanwhile, the original wind speed is also plotted in Figure 13, and the comparison shows that mean wind speed has a lower rangeability and changing rate. In the forecasting, the data are divided following a scheme with the same maximum wind speed as that shown in Figure 4.

To characterize the errors in forecasting the mean wind speed, the MAE, MRPE, RMSE, and R² values of different models at different forecasting steps are calculated, as shown in Figure 13. It is found that the errors of the BPNN, LSTM, GRU, and SVR models still increase as the forecasting step increases. However, the differences in the errors of BPNN, LSTM, and GRU decrease compared with the results of maximum wind speed shown in Figure 11. The ARIMA model offers the best forecasting of the mean wind speed among all five models and its errors change little when increasing the forecasting steps, which means the ARIMA model can achieve high accuracy in multi-step forecasting of wind speeds in the mean sense.

Moreover, the forecasting accuracy of these models improved compared with those in the 1 min maximum wind speed forecasting. As a typical example, herein, the accuracy improvement of the ARIMA model is presented in

Table 4. Accuracy improvement of the ARIMA model.

Error Index	Forecasting Step
	1	2	3	4	5	6
MAE (%)	19.8	18.9	18.6	18.2	18.7	20.4
MRPE (%)	11.0	9.8	9.3	8.8	9.3	11.2
RMSE (%)	22.7	20.9	20.8	20.6	20.7	21.9
R2 (%)	18.7	17.5	18.6	18.8	18.2	18.6

because it outperforms other models. It can be seen from the table that the MAE, MRPE, and RMSE of the forecasting results are reduced by more than 18%, 8%, and 20%, respectively, and the reduction proportions are not related to the forecasting step. As for the R² of the forecasting results of maximum wind speed and mean wind speed, the latter increases by at least 17.5% compared with the former. In summary, these statistical models can achieve preferable performance in terms of forecasting mean wind speed.

Based on the aforementioned discussions, the ARIMA model is found to be suitable for short-term forecasting of tropical storms for both the maximum and mean wind speeds. Especially for the mean wind speed, the forecasting error with six steps ahead is almost the same as that of the one with one step ahead. For the BPNN, LSTM, GRU, and SVR models, they present relatively large prediction errors for either the maximum or the mean wind speed, as the non-stationarity is not well represented by these models. Hence, for the purpose of early warning, the ARIMA model is suggested as an effective approach to forecast the wind speeds of tropical storms.

6. Application to Other Datasets

To evaluate the performance of the suggested model from the view of universality, the measured data of three other severe tropical storms, namely, Lekima (2019), Maria (2018), and Ampil (2018), are employed for further analyses. Among them, Lekima and Maria evolved as strong typhoons. Based on the datasets, the prediction of wind speeds is conducted using the five models, respectively. As an example, the MAE values of different models in forecasting the 1 min maximum wind speed are presented in

Table 5. MAE of different models in forecasting wind speeds.

Case	Forecasting Steps	ARIMA	BPNN	LSTM	GRU	SVR
Lekima	1	1.05	1.09	1.07	1.11	1.27
	3	1.09	1.26	1.15	1.38	1.33
	6	1.13	1.35	1.20	1.45	1.42
Maria	1	0.52	0.52	0.52	0.55	0.57
	3	0.54	0.57	0.59	0.63	0.62
	6	0.54	0.62	0.64	0.64	0.71
Ampil	1	1.10	1.10	1.11	1.12	1.14
	3	1.16	1.22	1.24	1.41	1.27
	6	1.14	1.20	1.25	1.35	1.32

.

It can be seen from Table 5 that, regardless of one-step or multi-step forecasting, the error of the ARIMA model is still the smallest among the five models in each case. This finding is the same as that obtained in Section 4 and Section 5, which validates the universality of the ARIMA model in better forecasting tropical-storm wind speeds than the other four models.

7. Conclusions

This paper presents a performance evaluation of linear and nonlinear models in short-term forecasting of tropical-storm winds. The following conclusions can be obtained based on existing analyses and discussions:

• The artificial neural network-based models provide similar prediction errors in the one-step ahead forecasting of maximum wind speeds, while the SVR model offers the worst forecasting as the trained model cannot well represent the inherent general features of measured winds.
• The forecasting errors of different models all increase with the augment of forecasting steps, but the error increment of the ARIMA model is the smallest. The SVR model still provides the worst prediction at different forecasting steps, and the difference in the errors of the BPNN, LSTM, and GRU models is small.
• The errors of all the models in forecasting the mean wind speed are smaller than those in forecasting the maximum wind speed, which means high accuracy can be achieved when forecasting the wind speeds in the mean sense.
• Among the investigated models, the ARIMA model provides the least forecasting error, and the error changes little when increasing the forecasting steps. Hence, the ARIMA model is suggested as an effective approach to forecast tropical-storm winds for the purpose of early warning.