Multi-Innovation Gradient Iterative Locally Weighted Learning Identification for A Nonlinear Ship Maneuvering System

This paper explores a highly accurate identification modeling approach for the ship maneuvering motion with fullscale trial. A multi-innovation gradient iterative (MIGI) approach is proposed to optimize the distance metric of locally weighted learning (LWL), and a novel non-parametric modeling technique is developed for a nonlinear ship maneuvering system. This proposed method's advantages are as follows: first, it can avoid the unmodeled dynamics and multicollinearity inherent to the conventional parametric model; second, it eliminates the over-learning or underlearning and obtains the optimal distance metric; and third, the MIGI is not sensitive to the initial parameter value and requires less time during the training phase. These advantages result in a highly accurate mathematical modeling technique that can be conveniently implemented in applications. To verify the characteristics of this mathematical model, two examples are used as the model platforms to study the ship maneuvering.


Introduction
Generally, ship maneuvering modeling techniques can be divided into two main groups: parametric modeling and non-parametric modeling. Parametric modeling is a classical method in which the model structure is established first and then the parameters are calculated by various types of methods. Non-parametric modeling, however, ignores the working principles of the system and instead it studies the relationships between the system input and output directly, i.e., black-box modeling.
The Abkowitz model (Abkowitz, 1980) and the Mathematical Model Group (MMG) model (Ogawa and Kasai, 1978) are the most significantly classical parametric models.
The key idea underlying these models is that the multiple polynomial equations of the variables related to maneuvering motions are established first. Then, the hydrodynamic coefficients are obtained by various methods such as the captive model tests, empirical formulas, computational fluid dynamics (CFD), system identification, and so forth (Sutulo and Guedes Soares, 2014;Uzunoglu et al., 2016). The main task is to calculate the hydrodynamic derivatives. Hence, these models are called the parametric models. Un-fortunately, the structure of a hydrodynamic model typically does not accurately depict a real vessel, and this phenomenon is attributed to unmodeled dynamics and multicollinearity. These problems have attracted the attention of researchers for many years but have not yet been solved satisfactorily.
Non-parametric modeling mainly studies the mappings between the system inputs and outputs, thereby, eliminating the unmodeled dynamics and multicollinearity. Many AI techniques have been applied to the non-parametric modeling such as neural networks (NNs) (Moreira and Guedes Soares, 2012;Luo and Zhang, 2016) and the support vector machine (SVM) (Luo and Zou, 2010). NNs do not depend on a priori knowledge of the structure of the ship mathematical model, and they have significant nonlinear mapping capabilities. Unfortunately, NNs cannot be extended, modified or tuned for ship mathematical models. In addition, NNs have certain natural disadvantages. For example, the optimal number of the hidden layer neurons cannot be determined reliably and their generalization capability is restricted. The SVM has been proposed for small training points, and it has excellent generalization. However, the training data size is not easy to be determined for engineering systems.
In recent years, a kind of non-parametric learning technique called locally weighted learning (LWL) has attracted much attention (Kaneko and Funatsu, 2016;Jia and Song, 2017). LWL is a non-parametric statistical technique for learning complex engineering control systems (Atkeson and Schaal, 1995;Cleveland and Loader, 1995). The development of LWL can be divided into two different strategies. The first is memory-based LWL (Aha, 1997), in which all training points are stored in the memory and looked up when making a prediction for each query point. The second strategy is non-memory-based LWL, which can avoid storing training points in memory by using recursive techniques (Ljung and Sööderström, 1986). Non-memory LWL is an online algorithm; the lookup time for new data is shorter than that of memory-based LWL, although data reevaluation becomes impossible (Schaal et al., 2002). Nonmemory-based LWL is usually adopted to estimate system states and to design controllers, where as it is not suitable for modeling.
LWL has been extended to numerous control systems, such as humanoid robots (Schaal and Atkeson, 1994), remote sensing imagery (Ma et al., 2014), soft sensor (Leung et al., 2004), and process monitoring (Jia and Song, 2017). These uses are the evidences that LWL has become a power tool in the engineering control area. However, few studies have considered LWL modeling in the field of ship maneuvering. Nevertheless, using LWL modeling for ship maneuvering may suffer from two major limitations. The first limitation is one-to-many mapping. Compared with robot motions, ship motions have a large inertial characteristic that represents the delay between a change in the input and a change in the ship's motion. Thus, the relationship between the system input and output is one-to-many mapping. As is well known, one-to-many mappings are not the functions which are difficult for most methods to approximate. The second limitation is the non-separable problem, in which one of the ship's motion states (such as the surge speed) changes not only through the input variation (in this paper, the input is the rudder angle) but also through variations in other motion states such as the sway speed or yaw rate. This is the so-called coupling problem. Moreover, LWL does not have information on the time sequence because the algorithm does not have information on the next or last point for a given query point. In this paper, the characteristics discussed above are called the general non-separable problem. These problems should be solved before using the LWL modeling for ship maneuvering. In this paper, to avoid these problems, the input dimension of the system is raised by following the basic concept underlying the SVM.
Practically, the key issue in memory-based LWL lies in calculating the distance metric. Predicting a query point is highly dependent on the neighborhood. Nevertheless, the size and shape of the neighborhood are determined by a distance metric. When an optimal distance metric is obtained, LWL cannot only enhance the learning of the ship motion characteristics but also improve the generalization. In Bai et al. (2017), the distance metric is obtained by setting a certain condition that can easily result in over-learning or under-learning (see Fig. 1). Many optimization algorithms, such as the gradient descent method (Schaal et al., 2002) and a genetic algorithm (Leung et al., 2004), have been applied to optimize the distance metric. However, few works have considered over-learning or under-learning.
To solve the problems of over-learning or under-learning, the novel multi-innovation gradient iterative (MIGI) algorithm is proposed for non-parametric learning modeling in this work. Additionally, because multi-innovation gradient algorithm (Ding and Chen, 2007;Xu and Ding, 2017) depends on the information vector of the system model, it is not suitable for non-parametric learning. However, the MIGI does not rely on the information vector of the system because the innovation gradient is adopted to update the distance metric in the MIGI algorithm. In contrast with other iterative algorithms, the MIGI uses not only the current data but also previous data of each iteration step. Hence, the MIGI has a faster convergence rate and is not sensitive to the initial parameter value and convergence factor.
The main features of the proposed scheme also include the following advantages: (1) Unmodeled dynamics and multicollinearity in the conventional parametric model can be eliminated. LWL is a type of non-parametric learning method that learns the mapping between the system input and output directly; therefore, the problems of unmodeled dynamics and multicollinearity are solved naturally using LWL; (2) The proposed method can avoid the over-learning or under-learning problem of the conventional LWL algorithm. In our algorithm, the innovation gradient is used in the update law. When the distance metric converges to the optimal value, the innovation gradient equals zero. Thus, the distance metric stops updating, and this algorithm is more accurate and effective. BAI Wei-wei et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 288-300 289 2.1 Abkowitz model The Abkowitz model of the Mariner Class Vessel is studied in this paper. However, to simplify the problem, the input state refers only to the rudder angle. The dynamic equations for the surge, sway and yaw are described as follows: where is the ship's mass, is the moment of the inertia of the ship about the -axis, and is the longitudinal coordinate of the ship's center of the gravity in the body-fixed coordinate system. , and are the force disturbing quantity at the -axis, -axis and -axis, respectively. Here, , and are the surge speed, the sway speed and the yaw rate, respectively, and , , , are the disturbing quantity, respectively. A superscript "′" indicates that the corresponding variable is normalized using the Primesystem (Jia and Yang, 1999). More details of non-dimensional hydrodynamic coefficients for the Mariner Class Vessel can be found in Fossen (2011).

Distance
The relevance of the training points and query point are determined by the distance. In this study, the Mahalanobis distance was adopted.
where is the query point, and it represents the system input of next time in this paper. is the j-th contribution of , and is the input of the training points which are stored in the memory. Therefore, represents the distance between the query point and training point.

Gaussian kernel function
The LWL algorithm calculates a relative weight between the training point and query point using the Gaussian Kernel function. The only open parameter remaining is the distance metric, which determines the size and shape of the local neighborhood.
h where is the distance metric.

Cost function
In this paper, the locally optimal LWL algorithm is proposed for the training points. This algorithm only needs to train on a single training point to implement the target. Moreover, the ship's motion states vary slowly. Therefore, the selected cost function is y q y q y q where is the system output corresponding to the query point, is the estimation of .

LWL algorithm
We should determine the training point in advance, where and are the system input and output of ith training point, respectively. In addition, , and is one of the ship motion states, e.g. , , , , , .
The basic goal of LWL is to approximate nonlinear functions using linear models in the neighborhood. The standard linear regression model is presented as follows: where is a query point; is the extended input vector, is an m-dimensional input vector; and is the regression parameter vector. The relative weight between the training point and query point is calculated according to Eq. (2) and the Gaussian Kernel Eq. (3). In this step, one of the training points is left out as a query point, which is called the leave-one-out cross validation approach.
(6) X y The training points input matrix and output vector are multiplied by the weight: where , , y = , and n is the number of the training points. And the regression parameter vector can be obtained by the least square method: Then, LWL can predict the query point as follows: Usually, the magnitude of the ship's acceleration is small, which may cause a singularity of . To overcome this problem, a smaller design parameter metric, , is applied (Bai et al., 2017).
h Next, the distance metric will be updated according to a certain scheme in a leave-one-out cross validation approach.

Multi-innovation gradient iterative algorithm
We introduce several notations. represents the maximum value of , represents the maximum eigenvalue of the symmetric matrix , and The MIGI algorithm was developed to avoid the problems of over-learning or under-learning. The quantity in Eq. (4) (where is the iterative step) is called the single innovation (Ding and Chen, 2007), and a multi-innovation algorithm is studied. The basic idea of MIGI is that it calculates the derivation of every innovation in the multi-innovation vector. Let where grad(·) is the partial differentials with respect to h, this p-dimensional vector derivative is called the gradient, and denotes the innovation gradient length. The MIGIL-WL algorithm uses the innovation gradient length as: The convergence factor will be given later. Here, , whose elements are 1. To prove the convergence of the MIGI algorithm, we provide the following description, lemma and theorem.
Consider the following dynamical discrete time system: x ∈ R n A ∈ R n×n g where is the iterative vector, is the parameter matrix, and is an arbitrarily uncertain term.
Let be the metric and be the eigenvalues of . The spectral radius is the maximum absolute value of the eigenvalues, such that ρ(A)= max{|λ i |}.
(15) Lemma 1. The null solution of the system Eq. (14) is uniformly stable if and only if there exists a Lyapunov function (Daafouz and Bernussou, 2001) (17) and whose difference along the solution of Eq. (14) is a negative definite decrescent that is κ ∞ for all and where are the functions.

Remark 1:
The function is a function if it is continuous, strictly increasing zero at zero and unbounded ( as ).
Lemma 2. System Eq. (14) converges to the exact solution (i.e., ) for any initial values if and only if the spectral radius is smaller than 1, such that g Finally, the convergence is independent of .

Convergence of the MIGI algorithm
We next establish the MIGI algorithm's convergence. The basic idea is to extend the single innovation gradient to a multi-innovation gradient: that is, the single innovation gradient iterative is proved first, and then, the multi-innovation gradient iterative is proved.
The single innovation gradient iterative algorithm is presented as follows: (20) Without the loss of generality, one can obtain a unique parameterization after abstracting the virtual black-box model: is the unknown information function consisting of the training points, is the training point input matrix, is the training point output vector, and is the distance metric corresponding to the query point.
µ Theorem 1. For the iterative algorithm Eq. (20), assume that the system parameter Eq. (21) has a unique solution and that is finite. Algorithm Eq. (20) takes the innovation length . Then, the iterative solution converges to the exact solution (i.e., ) for any finite initial values if a convergence factor exists such that 0 < µ < 2 ψ 2 max (q,X,Y) .
Theorem 2. For the iterative algorithm Eq. (13), assume that Theorem 1 holds. Then, take the innovation length ( is the iterative step) which is a positive integer. The error converges to zero for any finite initial value if a convergence factor exists such that In summary, we can obtain when from Theorem 2, however, this situation is not allowed. In addition, Theorem 1 is a special case to guarantee the integrity of Theorem 2. Therefore, by combining Theorem 1 with Theorem 2, we can conclude that the estimation error converges to zero when . (The proofs of Lemma 2 and Theorems 1-2 in the sequel are described in the Appendix.)

Advantages of the MIGI algorithm
Motivated by the superiority of the MIGI algorithm proposed in this paper, several comments are made in this subsection. After choosing a training point, the relationship between the value of the cost function and the iterative step during the training phase is shown in Fig. 1.

Remark 2:
At the beginning of the training phase in Fig. 1, the non-smooth points are caused by over-learning or under-learning. Essentially, over-learning reduces the generalization of the system, whereas under-learning fails to learn the system characteristics effectively. This non-smooth curve phenomenon may be caused by over-learning and under-learning. However, the optimal point strikes a good compromise between the generalization and learning performance.
p ⩾ 20 p p Comment 1. For comparison purposes, Fig. 1 also plots the iterative curve of the locally optimal LWL algorithm. At the beginning of the training phase, the locally optimal LWL algorithm has no ability to avoid the local minimum point, which is caused by the over-learning or under-learning problem, however, by choosing the proper initial value we can solve this problem. Unfortunately, the proper initial value is difficult to obtain. In the training phase for the same training point, different innovation lengths were tested. Fig.  1 shows that the MIGI algorithm has higher accuracy when . MIGI can converge to a fixed value with no additional conditions. Moreover, the MIGI can effectively avoid the over-learning or the under-learning problem. In Fig. 1, the optimal value of the cost function becomes increasingly smaller as the innovation length increases. Additionally, as the innovation length increases, the MIGI estimations approach the optimal LWL estimation. The optimal cost function values for different innovation lengths are listed in Table 1, and the optimal value obtained by LWL is selected by manually searching.
Comment 2. An important criterion for evaluating an iterative algorithm is the sensitivity of the initial parameter value. Although the initial parameter values vary, the MIGI algorithm still converges to the same optimal value as shown in Fig. 2, indicating that the MIGI is not sensitive to the initial parameter. p = 50 Comment 3. Compared with the locally optimal LWL (Bai et al., 2017), the MIGI algorithm achieves a fast convergence rate and reduces the training time. The innovation gradient length is . As shown in Table 2, the MIGI LWL requires 3723 s for 4835 training points, whereas the locally optimal LWL requires 78613 s. Comment 4. Fig. 3 and Table 3 show the change in the cost function value with different values of p and μ. When p=50, the cost function value change is approximately 26.9% with different μ values. When p=30, the change rises to 38.3%. However, the change can reach 82.3% when p=10. Therefore, for p=50, 30, 10, a larger p corresponds to a smaller sensitive convergence factor.

Illustrative examples and discussion
In this section, the illustrative learning examples will be presented to verify the performance of the proposed scheme. To achieve this goal, the Abkowitz model of the Mariner Class Vessel and full-scale trial of YUKUN of Dalian Maritime University are studied. All these simulations are conducted in a MATLAB environment on a computer with a 3.2GHz CPU and 4GB of RAM.

Abkowitz model of the Mariner Class Vessel
In this paper, the propeller rotation change is not con-

292
BAI Wei-wei et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 288-300 (u 0 , v 0 , r 0 ,u 0 ,v 0 ,ṙ 0 ) = (7.72, sidered. The rudder angle is the system input regarded as the system excitation. Therefore, the rudder angle is a known variable. The initial states are 0, 0, 0, 0, 0). To adequately excite the characteristics of the ship motion, a series of training experiments are designed. The first is a figure-of-eight experiment, and the second is a zigzag test. The trajectory of the figure-of-eight experiment is indicated in Fig. 4. In consideration of the computational burden, the sampling interval is set up with 2 s for the training data, but 1 s for test data, which are shown in Table 6. Table 4 lists the data from the figure-of-eight experiments. Moreover, the proposed scheme manages to identify the characteristics of the ship maneuvering, and the noise is not considered.
To further excite the ship motion characteristics, we will add the zigzag test data to the training dataset. Table 5 shows the data from zigzag tests.
The test experiment is composed of the 35°/35° zigzag test (No. 1 in Table 6) and the 22° turning test (No. 2 in Table 6).
In this subsection, the data in Table 4 and Table 5 are applied to train, and the data in Table 6 are applied to valid- ate. The main working principle is that the distance metric is trained by using the training points, the rudder angle from the test data is regarded as the query point, and then the ship motion states corresponding to the query points can be estimated. The superiority of the proposed algorithm is demonstrated by comparing it with the global optimal LWL and locally optimal LWL. The design parameters of the global optimal LWL algorithm are , , Λ = and . The parameters of the locally optimal LWL are given by Bai et al. (2017) and the parameters setting of the proposed scheme are , , diag[0.00001, ··· , 0.00001] 4835×4835 and h(0) = 0.005 for training the surge speed, yaw rate, surge acceleration, sway acceleration and yaw acceleration, respectively, and for training the sway speed. Normally, the generalization verification is necessary for identification modeling (Haddara and Wang, 1999), and it requires that the algorithm can well blindly predict the untrained data. To verify the generalization performance of the proposed scheme, the simulation results of the speed states and acceleration states with the measured data from the 35°/35° zigzag test and 22° turning test are compared with the simulation results of the proposed scheme. The simulated results are not included in the training data.      BAI Wei-wei et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 288-300 293 factory generalization capability of the proposed scheme, which is compared with global optimal LWL and locally optimal LWL. Figs. 5-11 show that the zigzag test motion states are highly nonlinear. The red curve denotes the reference signal from the zigzag test. However, the proposed scheme can learn the mapping between the rudder angle and ship motion states precisely and has the highest accuracy among all the algorithms applied in this paper. This result shows that the over-learning or under-learning problem can be effectively solved by using the MIGI algorithm. Table 7 shows that the mean error of the proposed scheme has been reduced by almost 50% compared with that of the locally optimal LWL, which shows that MIGI has a higher performance than the other schemes.
Global optimal LWL uses the same distance metric for the whole domain. Hence, its accuracy is of the lowest quality in the highly nonlinear part. Compared with the global optimal LWL, the simulation accuracy of the locally optimal LWL is improved which may be related to the adoption of different distance metrics with different training points. However, during the training phase of the locally optimal LWL, over-learning or under-learning persistently occurs. Typically, the MIGI algorithm can solve this problem and improve the ability of LWL of nonlinear mapping. The nonlinearity of the turning test is weaker than that of the zigzag test. The 22° turning test simulation results are depicted in Figs. 12-18. Note that the proposed scheme simulation results, which have only minor errors, are in accordance with the reference results. In particular, the nonlinear mapping learning results of MIGI are better than those of other algorithms.
It can be noted from Table 8 that the MIGI algorithm has produced a considerable reduction in the mean square

294
BAI Wei-wei et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 288-300 u error (reduced by approximately 20% compared with the locally optimal LWL), which means that MIGI LWL algorithm can accurately learn the ship motion characteristics. However, the mean error percentage of the global optimal LWL is smaller than that of the proposed scheme because the global optimal LWL has a smaller error over the whole domain and the turning test has weaker nonlinearity. According to the above simulations, the proposed scheme has obviously learned the characteristics of the Mariner Class Vessel maneuvering motions better than the other schemes. The mean error of the proposed scheme is smaller than that of the other schemes, which shows that it avoids over-learning or under-learning when compared with the locally optimal LWL. The 35°/35° zigzag test and 22°t urning test indicate that the generalization of the proposed scheme is satisfactory.
In the following discussion, the surge speed's training       BAI Wei-wei et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 288-300 295 results will be used as an example. Different learning principles have been applied to update the distance metric, and a comparison of the training results is illustrated in Fig. 19. The x-axis represents the sample index, which corresponds to the sequence of training points. The y-axis represents the distance metric corresponding to the training points. Obviously, the global optimal LWL obtains a global optimal distance metric, which means that a constant distance metric is applied over the whole domain. In contrast, different distance metrics are assigned to different training points in the locally optimal LWL, which improves the nonlinearity mapping of the algorithm. Thus, the mean error percentage of the locally optimal LWL is smaller than that of the global optimal LWL. However, the distance metric has larger fluctuations in the locally optimal LWL. In this paper, the result of the MIGI algorithm not only obtains different distance metrics for different training points but also has smaller fluctuations. Therefore, the MIGI LWL has the highest accuracy.
In order to obtain the range of the convergence factor, we need to calculate the range of ψ. According to the simulation results shown in Fig. 19 and Eq. (21), we can obtain ψ. As shown in Fig. 20, the x-axis represents the sample index, which is the same as the distance metric index, and the y-axis represents the value of ψ corresponding to the distance metric trained by the MIGI. The main target is to obtain the maximum value of ψ according to Eqs. (22) and (23). As shown in Fig. 20, the maximum value of ψ is 1.1874. With Theorem 1 and Theorem 2, we can obtain the range of the convergence factor.
µ=0.005 Obviously, the selected convergence factor of the surge speed is within the range, and the simulation result of the mean square percentage astringes to a smaller error based on Table 7 and Table 8. All the above factors show that the simulation results correspond with the theoretic analysis, which indirectly proves the proposed scheme.

Identification modeling for YUKUN with full-scale trial
In this subsection, the effectiveness of the proposed algorithm will be illustrated by using the data from full-scale trial at sea. The data to be analyzed were obtained from the YUKUN scientific research vessel of Dalian Maritime University (see Fig. 21). The main particulars of YUKUN are given in Table 9.
Full-scale trials were conducted under slight sea conditions (approximately level 2) with sufficient depth in the Yellow Sea of China. Visibility was excellent. The maneuvering trials were conducted by an experienced captain in  Dalian Maritime University in the daytime. The main trials include the moderate turning tests and zigzag tests. The ship's motion states, including the speed of the surge and sway during the sea trials, were measured by Doppler Log (type SKIPPER DL850, and the speed and distance error is 0.1 kts or 2%), and the ship's heading and yaw rate were measured by Fiber-Optic Gyrocompass (type NAVIGAT 2100, the error is smaller than 0.7° secant latitude, and the yaw rate error is smaller than 0.4°/min). Disturbances and data dropout are the major problems that must be considered. The main mission of this identification process is to acquire the natural ship characteristics; consequently, the disturbances should be filtered. Data dropouts seriously interfere with the identification process. The global optimal LWL is employed to smooth and pad the data points in this study.
In this study, the sampling interval is 1.0s, and the experimental data are listed in Tables 10-12. These data, including the 30° turning test and +20°/20° zigzag test, are used to verify the proposed scheme.
Remark 3: "-" indicates that in the zigzag experiment, the port rudder was moved first, while"+" indicates that the starboard rudder was moved first.
The data listed in Table 10 and Table 11 are applied to learn the ship's characteristics, and the data in Table 12 illustrate the performance and generalization of the proposed scheme in a practical application. During the training phase, the parameter settings of the proposed scheme for training h(0) = 0.0001 µ = 5 p = 50 diag{0.00001, · · · , 0.00001} 535×535 the turning test are , , and Λ = . Because of the weaker nonlinearity of the turning test, the convergence factor value is large.
Figs. 22-24 illustrate the simulation results of the 30°t urning test (No. 1 in Table 12). The simulation results of the speed states and acceleration states depicted in Fig. 22 and Fig. 23 clearly show the changes in the individual variables.
Compared with the wind and waves disturbance, the influence of the current is inevitable. Worse still, the current disturbances cannot be eliminated. In particular, the speed of the current is faster than the ship's speed. In Fig. 22, the sway speed of the proposed scheme is of low quality. One reason may be the influence of the current.
The data listed in Table 11 are applied to train while No.      BAI Wei-wei et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 288-300 297 2 test data in Table 12 are applied to validate the proposed scheme. The parameters of the proposed scheme are listed in Table 13. Compared with the turning test, the zigzag test has higher nonlinearity. Hence, a smaller convergence factor is selected. Figs. 25-27 show the maneuvering results of the +20°/ 20° zigzag test. The speed and acceleration states are predicted with high accuracy. However, the surge speed presents a relatively large error. This phenomenon may be caused by a lack of high quality training data.

Conclusions
This paper presents a novel black-box modeling technique by combining the LWL and MIGI algorithms for a nonlinear ship maneuvering system. The unmodeled dynamics and multicollinearity inherent in the conventional parametric model are avoided. Additionally, this paper shows that the MIGI iterative error can converge to zero via the convergence analysis and obtain the optimal distance metric, thereby eliminating the over-learning or under-learning problems in the locally optimal LWL. Several examples show that the proposed scheme is a powerful modeling tool for predicting the motions of ship maneuvering, and it can easily be implemented in applications. Although the proposed scheme includes many advantages, the LWL algorithm still needs a large computational cost. Therefore, our future research will focus on decreasing the computational burden for big data.    298 BAI Wei-wei et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 288-300