Influence of Regular Wave and Ship Characteristics on Mooring Force Prediction by Data-Driven Model

The study of mooring forces is an important issue in marine engineering and offshore structures. Although being widely applied in mooring system, numerical simulations suffer from difficulties in their multivariate and nonlinear modeling. Data-driven model is employed in this paper to predict the mooring forces in different lines, which is a new attempt to study the mooring forces. The height and period of regular wave, length of berth, ship load, draft and rolling period are considered as potential influencing factors. Input variables are determined using mutual information (MI) and principal component analysis (PCA), and imported to an artificial neural network (NN) model for prediction. With study case of 200 and 300 thousand tons ships experimental data obtained in Dalian University of Technology, MI is found to be more appropriate to provide effective input variables than PCA. Although the three factors regarding ship characteristics are highly correlated, it is recommended to input all of them to the NN model. The accuracy of predicting aft spring line force attains as high as 91.2%. The present paper demonstrates the feasibility of MI-NN model in mapping the mooring forces and their influencing factors.


Introduction
With the development of coastline and deep-water port, safe operation of ships and offshore platform is attracting more attention. The study of mooring forces with large open moored ships can improve the stability of mooring conditions, reduce the cable-breaking accidents and provide necessary data for wharf design. Since the ships are exposed to complex hydrology and meteorology conditions, the measurement of mooring forces is difficult and dependents on many variables. Research of mooring forces is mainly based on physical model tests (Chu et al., 2014;Cornett, 2014;Rosa-Santos et al., 2014;Sande et al., 2019) and numerical simulation. Physical model tests allow for the reproduction of the most significant physical phenomena involved in the moored ship behavior (Pinto et al., 2008), but they are costly and time consuming, and highly depend on the accuracy and feasibility of tests. Static methods , quasi-static approaches (Ahmed et al., 2015) and dynamic analysis (Wang and Wan, 2015) are carried out to numerically simulate the mooring forces. These studies are relatively less expensive and more flexible yet have certain limitations due to the difficulties in multivariate and highly nonlinear modeling.
In order to avoid details of the physical processes, datadriven models are good alternatives and capable of addressing the nonlinear nature. As one of the most widely-used data-driven models, artificial neural network (ANN) is powerful in real-time prediction since early 1990s (Cui, 2009;Karunanithi et al., 1994). Simoes et al. (2002) predicted mooring forces based on neural network and analyzed the dynamic behavior of a floating production storage and offloading system. The wavelet analysis has been combined with neural network for mooring load prediction (Yang and Zhang, 2016). A ship mooring force prediction of the open sea terminal was proposed based on genetic algorithm and neural network (Li and Qiu, 2017). Although the number of these studies is scanty, they provide the possibility of applying ANN in mooring force prediction.
The influencing factors on mooring force can be classified into three main groups: (1) environmental dynamic factors such as the tides, waves, flows and winds; (2) factors regarding the mooring system, such as the length and type of lines, and the mooring condition; (3) factors representing the characteristics of ship, such as the ship tonnage and draft. It is imperative to figure out the relationship and determine the most essential variables. Jiang et al. (2007) studied the influence of ship's tonnage, carrying quantity and type on mooring forces through model experiments.  drew a conclusion that the order of factors influencing on the cable tension was flow direction, wave height, flow speed, wave direction and wind speed from model experiment of a 50k DWG barge. Li and Qiu (2017) chose draft, wind direction, wind strength, tide level, wave height, wave direction and wave speed as the most important variables from many environmental factors. Mutual information (MI) and principal component analysis (PCA) have proven to be effective input determination techniques in practical applications (Bowden et al., 2005;Gao et al., 2016;Wang et al., 2012), and therefore, are employed in this paper to preprocess the potential input variables.
The purpose of this paper is to investigate the influences of factors on mooring forces prediction using MI and PCA, which is a new attempt to study the mooring force. The outline of this paper is as follows. Firstly, a description of the data-driven models including the ANN, MI and PCA methods is presented. A study case with oil tankers of 200 and 300 thousand tons is then provided. The computed results are discussed and compared between MI and PCA methods, and the conclusions are finally drawn.
2 Data-driven model for mooring force prediction

Methods to input variable determination
This paper employs mutual information (MI) and principal component analysis (PCA) for input variable determination. The output variable is mooring force while the potential input variables are wave height, ship load, draft, berth length and so on.
(1) Mutual information Mutual information (MI) is a criterion to capture linear and/or nonlinear dependence between any two variables (Fraser and Swinney, 1986). Supposing that there are two variables of X and Y with N pairs of observations (x 1 , y 1 ), (x 2 , y 2 ), …, (x N , y N ), their mutual information is defined as: where p(x) and p(y) are the marginal probability density functions (PDFs) of X and Y respectively, and p(x, y) is the joint PDF of X and Y.
As can be seen, MI tends to measure the similarity between the joint distribution p(x, y) and the products of factored marginal distribution p(x) and p(y). If there is no dependence between the two variables, the joint probability of occurrence would be theoretically equal to the product of the individual probabilities. In that case, the MI scored in Eq. (1) would be equal to zero. A high value of MI indicates a strong relevance between variables. Thus, MI can be used as an input determination technique while the two variables are the target output and its potential factor respectively. The factors would be chosen as input variables when the corresponding MI scores are high enough to reveal strong dependence.
(2) Principal component analysis The essential idea of principal component analysis (PCA) is to reduce the dimensionality of a data set while retaining most of the information. PCA is a linear transformation method, attempting to find linear combination of the original variables with an objective of maximizing the variance of the data (Hotelling, 1933). These new linear combinations are known as principal components (PCs). They are a set of variables which are orthogonal and uncorrelated with each other. Given a set of original variables with P observations, a set of new variables can be written with a decreasing variance from the first to the last is the eigenvector with i=1, 2, …, P. The variable is the first order of PCs with the largest variance while gives the last most significant relation in the original variables. For the coefficients in the eigenvector, the following constraints are required to ensure the self-orthogonal transformation ∑ PCA is capable of converting the correlated factors into uncorrelated ones and finding the directions of the maximum variance in input data, and thus determining the most appropriate input variables from many factors.

Prediction model for mooring force
The input variables determined by the above data preprocessing methods can be introduced to artificial neural network (ANN), in order to predict the mooring forces. The most widely-used ANN model consists of input, hidden and output layers. The input layer introduces input variables to the network. The nodes in the hidden layer are computed by a predetermined activation function f(.) as follows: in which (i=1, 2, …, s) and (j=1, 2, …, k) represent nodes in the hidden and input layers, respectively. The weight parameter from the input layer to the hidden layer is denoted by , and is the bias value. Similarly, the nodes in the output layer are obtained using another activation function F(.): where (h=1, 2, …, r) represent nodes in the output layer, and and denote the weights from the hidden layer to the output layer and the bias, respectively. In the present study, mooring force is the only output node in the output layer. Details of the ANN model can be found in Thirumalaiah and Deo (1998).

Case study
A study case is employed to predict the mooring forces of moored ships at a large open terminal. The data are collected from experiments in the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The ships in the experiment are oil tankers of 200 and 300 thousand tons. The experiment was conducted in a wave tank, which is 50 m in length, 24 m in effective width and 1.0 m in depth, as shown in Fig. 1. The geometric scale of this experiment is 1:65, considering the impact of regular wave. Measured by cable force dynamometers of MJ800 system, the mooring forces were studied in terms of different wave conditions, mooring layouts and loads. More information about the physical model tests can be referred to Liu et al. (2008) and Lu et al. (2009). The layout of mooring lines studied in this paper is shown in Fig. 2, in which the lines are made of polyamides cables and the diameter is 75 mm. Eight lines for the moored ship, that is, two head lines (#1), two breast lines (#2, #5), two spring lines (#3, #4) and two stern lines (#6) are tied to mooring bitts. The mooring forces corresponding to each line are named head line force (F HL ), fore breast line force (F FB ), fore spring line force (F FS ), aft spring line force (F AS ), aft breast line force (F AB ) and stern line force (F SL ).
The layout of these mooring lines is well designed to resistant the movement of the ship, and the angles between cables and wharf front line and horizontal plane meet the requirement of "Code of Practice for Design of Fendering and Mooring Systems (BS 6349-4-1994)" (BSI, 1994) and "Design and Construction Technical Code of Open Sea Terminal (JTJ 295-2000)" (MOT, 2000).
There are 101 samples for each test, of which the first 86 samples are used in the training period for calibration and the left 15 samples used in the testing period. The corresponding statistic parameters of the training and testing data, in terms of different mooring forces, are summarized in Table 1, in which X max , X min , X mean , S x, C v and C s denote the maximum, minimum, mean, standard deviations, coefficient of variation and skewness coefficient, respectively. The training set can fully include the testing data, and each of them has similar statistical values, assuring the availability of the selected data for the purpose of generalization and prediction. In addition, datasets in this paper are scaled linearly to the range of 0.1−0.9, as a method of normalization to avoid larger data dominating smaller data.
Six influencing factors are considered as potential input variables, namely wave height (H), wave period (T), length of berth (L B ), ship load (M), draft (D) and rolling period (T 0 ), respectively. Herein, H and T are wave characteristic factors; L B can reflect the length of mooring lines; M, D and T 0 are factors representing the characteristics of ship. The values of the six variables are summerized in Table 2. The heights of the waves are within the range of 0.5−2.5 m, in which all the waves are regular and perpendicular to the longitudinal axis of the ships.

Evaluation criterion
Three statistical indices, including the root mean squared error (RMSE), mean relative error (MRE) and Pearson correlation coefficient (R) were used to evaluate the performance of prediction models. They are defined as follows: RMSE in which and are the observed and the predicted forces respectively, and N is the number of data. The parameter denotes the covariance between and ; and are the standard deviation of and , respectively. RMSE scales the mean squared error, therefore, can reflect the performance on high values. A smaller value of RMSE indicates a better prediction performance. MRE measures the relative error, and a zero value of MRE stands for a perfect fit. Index R detects the linear correlation between the observed and predicted forces. A higher positive linear correlation between two variables would result in a larger R, gradually approaching to 1.

Input selection using MI and PCA
The challenge of input determination is to select input variables that have maximum dependency with output variable and minimum redundancy for calibration. Mutual information (MI) is firstly employed, where the MI scores between each mooring line force and factor are summarized in Table 3. It is interesting to note that the MI scores in terms of the ship load, draft and rolling period are the same for all the six mooring forces. Thus the three factors representing ship characteristics have equal dependence with the output forces. Further, the MI scores regarding ship characteristics are smaller as compared with the other factors. For example, the MI scores for H, T and L B are respectively 0.433, 0.414 and 0.448 in the case of head line force, and those for the three ship characteristic factors are 0.360. It can be concluded that the head line force has greater dependences on the length of berth, and is less relevant to the ship load, draft and rolling period. Therefore, the wave height, period and berth's length are firstly chosen as the three most influencing input variables. The ship load, draft and rolling period can also be included in the input combination, since their corresponding MI scores are comparable. In brief, H, T, L B , M, D and T 0 are regarded as effective influencing factors for all the six line forces using the MI method.
As indicated by the analysis of mutual information, the three factors representing the characteristics of ship are relevant. Pearson's correlation coefficient (R) is further employed as a measure of the linear correlation between two influencing factors. As can be seen in Table 4, the wave height is totally uncorrelated with the other five factors since R values regarding H are zero. Similarly, there are minor correlations between the wave period/length of berth and the other factors. Thus, H, T and L B would be three independent input variables. On the contrary, the values of R obtained between the ship load and draft is 0.932, indicating strong linear correlation between these two factors. The same phenomenon can also be observed between the rolling period and ship load/draft. Therefore, the three factors in terms of ship characteristics are interdependent and correlated.
In this perspective, there is a need for PCA method to extract the principal components from the three correlated factors. It is found that the eigenvalue of y 1 is 1.2×10 10 , while that of y 2 is 0.6408. That is, the first order of PCs obtains much larger variance than that of the second order of PCs. Thus, the first PCs is selected as giving the only and most significant relation in the original three variables. The new variable PC is attained as PC = −0.6163M+0.7875T 0 , as a linear combination of the variables. The influence of draft is totally ignored. As the correlation between draft and the other two ship factors are 0.932 and 0.872, the influencing of draft can be adequately represented by the ship load and rolling period. The variable PC is selected as an input variable in addition to the wave height, period and berth's length.
After input variables are determined, they can be imported to neural network for mooring forces prediction. The input combination for MI-NN mode is {H, T, L B , M, D, T 0 } and that for PCA-NN model is {H, T, L B , PC}, respectively. The difference lies in the consideration of the ship characteristic factors, where the PC variable attempts to move correlation and give the largest variance of the three factors.

Performances of prediction models
The performances of MI-NN models for mooring forces prediction are firstly discussed. The evaluation criteria RMSE, MRE and R during the training and testing periods are provided in Table 5. It can be observed that the head line force (F HL ) is accurately predicted. The obtained MRE val-ues in training and testing periods are smaller than 20% and the values of R are much larger than 0.80. The same phenomenon can be noted for the fore breast line force (F FB ) prediction, which demonstrates the ability of MI-NN models in generalization and prediction for these two forces. The fore spring line force (F FS ) also obtains an accurate prediction since the values of MRE are smaller than 10%. With regards of the aft spring line force (F AS ), the values of RMSE in the training and testing periods are 30.9818 and 29.4356 kN 2 respectively, which are quite close to each other. This indicates that the MI variables are effective for the NN modeling, since the excellent performances in the training period are well extrapolated to the testing stage. Besides, the accuracy of predicting F AS attains as high as 91.2%. The value of R for the aft breast line force (F AB ) case is 0.9241 in the testing period, revealing a strong correlation between the predicted and observed forces. The MI-NN model yields sufficiently accurate results for predicting the stern line force (F SL ), with the minor errors and high correlation between the observed and predicted values.
To further illustrate the above results, Fig. 3 exhibits the observed and predicted forces during the training and testing period. It can be perceived that the predicted values match most of the observations for all the six cases. The results of the testing data are compatible with the training data, showing the generalization ability of MI-NN models. As for F HL , the predicted values below 500 kN are mostly close to the observed ones, and those larger than 1000 kN are overpredicted. It is acceptable for the mooring line prediction. While the maximum forces are over-ratted in advance, measures can be taken before the mooring lines break. Similarly, the model for F FB can almost fit the observations larger than 1000 kN. F FS and F AS predictions are comparably better than F HL and F FB , as indicated by the large number of predicted dots located around the observations. This can ex-  Table 2, the ship loads acting on the ships are quite different, which can be described as M 1 =115112 t, M 2 =135700 t, M 3 =213243 t, M 4 =223757.5 t and M 5 =342586 t, respectively. The prediction of maximum value of mooring lines is important in practice, and can present dissimilar performances according to the ship loads. As can be seen in Fig. 4, the maximum values of F HL , F FB , F AB and F SL are over-predicted, while those of spring line forces are underpredicted with M 1 . These relatively small errors indicate that the MI-NN model can yield appropriate prediction for the mooring lines with small ship loads. As for the cases with M 5 , F FB , F AS , F AB and F SL attain large relative errors when computing the maximum values. This can be explained by that the tail part of the ship might move outwards with large ship load, thus the breast and stern lines forces are important and difficult to be computed. The performances of F FB and F FS are relatively stable with different ship loads as compared with the other four cases. It may be due to the fact that the movement of the forward part of the ships is less relevant with the ship load. Overall, most of the maximun values of mooring lines are over-predicted, indicating the ability of MI-NN models to prevent cable-breaking accidents.
The comparisons between the MI and PCA combined with NN model in terms of RMSE, MRE and R values are provided in Table 6. As for F HL case, there is decrement of 14.78% in RMSE value, indicating the superiority of MI-NN  model. The R values attained by the two models are comparable, revealing the effectiveness of PCA for the HL prediction. With regard to F FB and F SL prediction, the PCA-NN cannot yield accurate results as the MRE values are approaching 20%. The results for F FS also reveal superiority of MI-NN over PCA-NN model in predicting mooring forces. As for the F AS and F AB cases, the PCA-NN model attains larger correlation between the predicted and observed values when compared with the MI-NN model. However, MI is found to be superior to PCA as the result of smaller RMSE and MRE values. This may be due to excellent performances in high values prediction.
The above conclusions can be strengthened by Fig. 5, where the scatter plots of the observed and predicted values are demonstrated using different models. As seen from Fig.  5a, the MI-NN model predictions are less scattered than the PCA-NN. The latter model under-predicts most of the val-ues. Similarly, the high values obtained by the PCA-NN are mostly smaller than the observations for F FB case. The MI-NN exhibits better matches since there are intensively distributed dots along the ideal line when predicting F FS . Although the maximum value is under-predicted, it is quite close to the observation. As presented in Fig. 5d, the values predicted by the PCA-NN model are apparently scattered and away from the line. When employing PCA input variables, most of the F AB predicted dots are below the ideal lines but the values of F SL are mostly over-predicted. This conforms to the inferior prediction results of PCA-NN presented in Table 6. A conclusion can be drawn that the MI provides better input variables to predict the mooring forces when compared with the PCA method. Although the PC variable can reduce the correlation of influencing factors, the absence of draft in the PC variable leads to insufficient mapping between the factors and mooring forces.
As seen from the above results, mooring forces can be accurately predicted although the data size is not huge. Generally, a small number of data renders an unreliable prediction since it might miss some necessary information. Table  2 demonstrates the values of input variables, showing the possible range of influencing factors on the mooring forces. That is, the provided data contains enough information of the mooring forces with regards to the experiment model test. The data are effectively classified as well to ensure the availability of training data. Besides, over-fitting might occur due to small data size, in which the training error is slight but the testing error is relatively large. However, the comparable training and testing errors in Table 5 show that the data in this paper are properly fitted, since they have been normalized before the generalization. By taking the LIU Bi-jin et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 589-596 595 above measures, the influence of small data size can be minimized and the efficiency of the data in predicting mooring forces is validated.

Conclusion
This study is concerned with the influencing factors on mooring forces prediction using data-driven models. Artificial neural network is employed as prediction model, in which MI and PCA are combined as data-preprocessing techniques to determine input variables. With a case study of oil tankers, the prediction of forces in the head line, breast lines, spring lines and stern line are considered. Six variables including height and period of regular wave, length of berth, ship load, draft and rolling period are imported to the MI-NN model, while the last three factors regarding ship characteristics are replaced with PC variable in the PCA-NN model. Results show that the input variables yielded by MI are more feasible as compared with PCA since the three ship characteristic factors are independent and individually influencing the mooring force. The MI-NN model demonstrates ability and stability in generalization and prediction of different mooring forces. In spite of the limitation of only 200 and 300 thousand tons ships experiment and small number of data, the present work is the first attempt to consider influencing factors such as ship load and draft in neural network for modeling mooring forces. It can be referred or extended for other marine engineering models. Besides, the availability of data-driven model in predicting mooring forces can be verified as compared with numerical or experiment results in future work.