Prediction of Offshore Wave at East Coast of Malaysia—A Comparative Study

Abstract

Exploration of oil and gas in the offshore regions is increasing due to global energy demand. The weather in offshore areas is truly unpredictable due to the sparsity and unreliability of metocean data. Offshore structures may be affected by critical marine environments (severe storms, cyclones, etc.) during oil and gas exploration. In the interest of public safety, fast decisions must be made about whether to proceed or cancel oil and gas exploration, based on offshore wave estimates and anticipated wind speed provided by the Meteorological Department. In this paper, using the metocean data, the offshore wave height and period are predicted from the wind speed by three state-of-the-art machine learning algorithms (Artificial Neural Network, Support Vector Machine, and Random Forest). Such data has been acquired from satellite altimetry and calibrated and corrected by Fugro OCEANOR. The performance of the considered algorithms is compared by various metrics such as mean squared error, root mean squared error, mean absolute error, and coefficient of determination. The experimental results show that the Random Forest algorithm performs best for the prediction of wave period and the Artificial Neural Network algorithm performs best for the prediction of wave height.

1. Introduction

Human activities in the offshore region are continually increasing due to oil and gas exploration. Adverse conditions can often occur due to environmental disasters during offshore operations. Predicting wave characteristics is a crucial prerequisite for offshore oil and gas development (Figure 1), considering the safety of lives and the avoidance of economic damage. Wave height and period are typically significantly increased by the wind associated with storms passing across the ocean’s surface. Weather forecasting departments usually predict wind forces rather than wave periods and heights. There are several empirical approaches for estimating wave height and wave period from wind force [1,2,3,4,5]. Estimating wave height and period is inherently inaccurate and random, making it difficult to simulate using deterministic equations [6]. The numerical approach for calculating wave height and period is a difficult and complex procedure that, despite substantial breakthroughs in computational tools, produces solutions that are neither dependable nor consistently applicable. Machine learning methods are perfect for modeling inputs with corresponding outputs since they do not necessitate an understanding of the underlying physical mechanism [7].

Several studies have been performed using artificial neural networks (ANN) to measure important wave heights and mean-zero-up-crossing wave period history for different locations in the seas. These parameters were predicted three, six, twelve, and twenty-four hours in advance using two different neural network methods [8,9,10]. The time series of these wave parameters has been investigated using simulations in Portugal’s western coast area. Time series with wave height have been disintegrated into multi-resolution time series using wavelet transformation hybridized with ANN and wavelet transformation [11,12]. As the input of the ANN, the multi-resolution time series has been used to predict the important wave height at an unlike multi-step lead time near Mangalore, India’s west coast. Mandal et al.  [13] expected wave heights from observed ocean waves off the west coast of India, Marmugao. To predict wave height, recurrent neural networks with a resilient propagation (Rprop) update algorithm have been implemented.

The time series of wave height and mean-zero-up crossing wave period were simulated using data from the local wind. The application of various types of machine learning models [14,15,16,17] was performed to improve the accuracy of the prediction. The authors [18] used ANN to estimate wave height from wind force at 10 chosen locations in the Baltic Sea. The WAM4 wave model was used to figure out the time series for the waves that had been forecasted in the past. There were two different techniques, Feed-forward Back-propagation (FFBP) and Radial Basis Function (RBF), used to develop a machine learning system for predicting wave heights at a particular coastal point in deeper offshore areas [19,20,21]. Data on wave height, average wave period, and wind speed were obtained from remotely sensed satellite data on India’s west coast. Tsai et al. [22] employed the ANN with a back-propagation algorithm to estimate wave height and cycle from wind input based on the wind-wave relationship [23]. Time records of waves at either station may be predicted based on data from the neighboring station. Several deterministic neural network models have been developed by Deo et al. [24] to predict wave height and wave periods from generated wind speed. However, the model can offer adequate results in deep water and open areas, and the prediction periods are extensive.

This research uses wind force/metocean data to correctly predict wave height and wave period. Critical comparisons are performed for the current investigation to provide more accurate findings in predicting the wave parameters using three machine learning algorithms: ANN (Artificial Neural Network), SVM (Support Vector Machine), and RF (Random Forest).

The main contributions of the paper are (i) performing the experiments on the data sets obtained on the east coast of Malaysia and (ii) understanding the best predictive models for future prediction.

We arrange the remaining parts of the manuscript as follows. First, we explain the data collection procedures, the three regressor models, and performance metrics used for the comparison among the models in Section 2. Then, we show the experimental results and discuss the findings in Section 3. Finally, Section 4 contains a concise conclusion.

2. Materials and Methods

In this section, first, the data collection procedure is described, followed by the regression methods, and finally the evaluation metrics are described.

2.1. Data Collection

On a 2° × 2° grid in South China, environmental data are collected along Malaysia’s east coast, including the basins of Sabah (longitude 114.39° E, latitude 5.83° N) and Sarawak (longitude 111.82° E, latitude 5.15° N) (Figure 2). The Malaysian Meteorological Service and Fugro OCEANOR provided the data that was acquired from satellite altimetry [25,26] using oceanographic SEAWATCH meteorological (metocean) buoys and sensors, calibrated and corrected by Fugro OCEANOR [27].

The summary of common statistical values of the collected data is given in

Table 1. Summary of basic statistics of the collected data.

Statistics	Wind Speed	Wave Height	Wave Period
Number of Samples	1460	1460	1460
Minimum	0.00	0.17	4.05
Average	4.90	1.29	6.67
Maximum	15.41	5.13	13.68
Standard Deviation	3.34	0.75	1.24

.

There are 1460 data samples and three features: wind speed (m/s), wave height (m), and wave period (s).

2.2. Methods

There are a lot of algorithms [28,29,30,31,32,33] that can be used for regression analysis. In this study, three well-known and commonly used regression algorithms (ANN, SVM, and RF) were chosen to predict wave height and period from wind force. The wind force is employed as an input for training the network, while the wave height and period are used as outputs. Before discussing the details of each method,

Table 2. Advantages and disadvantages of different regressors.

Name	Advantages	Disadvantages
ANN	state-of-the-art algorithm; complex relations can be modeled	blackbox and hard to understand; many hyperparameters need to be tuned
SVM	provide non-linear solutions minimum generalization error	require knowledge of kernels weak regressor for large dataset
RF	most commonly used with good performance can handle missing values	interpretability of ensembles are questionable can lead to overfitting

shows the advantages and disadvantages of each method [34]:

The ANN algorithm is the most frequently utilized algorithm in such problems. The difference between the state-of-the-art ANN algorithm and two commonly used algorithms, SVM and RF, is then determined. Note that ANN is a parametric regression algorithm while SVM and RF are nonparametric regression algorithms. However, because the data set is small enough, it is not possible to employ advanced deep learning techniques (such as convolutional neural networks or recurrent neural networks).

2.2.1. Artificial Neural Network (ANN)

Inspiring by biological neurons, the researchers created artificial neurons [35]. It is possible to create a network of artificial neurons to predict the desired functions (i.e., the target variables).

The basic idea behind a single artificial neuron is that it takes an input function I, which is a product of weight vectors with the sample input vector, and feeds it into an activation function g to produce an output (Figure 3). Generally, the usual practice is to create a network of neurons based on many layers: input layers, hidden layers, and output layers. The input layers basically consist of one or more neurons based on the supplied inputs (for the present study, only one input is used), then there can be one or more hidden layers (only one hidden layer is used), and finally, there can be one output layer consisting of the expected outputs (Figure 4). The general trend is to use a back-propagation algorithm to update the weights involved in each connection between neurons so that it is possible to minimize the errors of regression between true output and the predicted output.

The hyperbolic tangent function, $tanh\left(x\right)=\frac{e^x-e^{-x}}{e^x+e^{-x}}$ is used as the activation function.

The steps of the Python implementation are described in Algorithm 1.

Algorithm 1 The Python implementation of the ANN algorithm

Input: Data set containing features and target Output: Prediction of ANN for the given data set 1. Read the data set using the Python read_excel function; 2. Extract features and target variables; 3. Scale the features and target using the MinMaxScaler function; 4. Split the data set into training and test parts using the train_test_split function; 5. Choose the best hyperparameter using the GridSearchCV function; 6. Derive the MLPRegressor using the above results; 7. Predict using the obtained regressor and calculate the performance metrics on the test data set;

2.2.2. Support Vector Machine (SVM)

The basic idea behind regression using SVM (i.e., also popularly known as Support Vector Regression) is that given input examples $\{(i_1,o_1),\dots ,(i_n,o_n)\}$$\subset X\times R$, where X denotes the example space (e.g., $X=R$ for our problem), a function $f\left(x\right)$ is obtained that has at most $ϵ$ deviation from the actual outputs oi for all the input examples (Figure 5) [36].

Generally, it is easy to describe the case of linear functions [37] f,

$f\left(x\right)=<w,x>+b$, with $w\in X,b\in R$

where $< , >$ denotes the dot product in X. Furthermore, it is possible to rewrite this problem as a convex optimization problem:

minimize $\frac{1}{2}\left|\right|w^2\left|\right|$ subject to

$$y_i-<w,x>-b\le ϵ$$

$$<w,x>+b-y_i\le ϵ$$

The steps of the Python implementation are described in Algorithm 2.

Algorithm 2 The python implementation of the SVR algorithm

Input: Data set containing features and target Output: Prediction of SVR for the given data set 1. Read the data set using the Python read_excel function; 2. Extract features and target variables; 3. Scale the features and target using the StandardScaler function; 4. Split the data set into training and test parts using the train_test_split function; 5. Choose the best hyperparameter using the GridSearchCV function; 6. Derive the SVR using the above results; 7. Predict using the obtained regressor and calculate the performance metrics on the test data set;

2.2.3. Random Forest (RF)

A decision tree has been widely used from the very beginning of machine learning research to find the best hypothesis as the classifier and regressor. A decision tree is a tree-like structure where a sequence of actions is taken from the root to the leaf nodes based on the values of each decision node in the concerned path. A sample decision tree is depicted in Figure 6, where the course of action of playing outside is taken based on the value of whether it is training outside or not.

There are a number of different variants of decision tree algorithms, from ID3 [38], C4.5 [39], and CART [40] to ensembles of decision trees (e.g., Random Forest [28])) that have been proposed to improve the accuracy of the classifiers and regressors. For a single decision tree construction, the most widely used splitting criteria, “Gini index” or “Entropy”, can be mathematically stated as follows [32,38,39,40,41]:

$$p_t=\frac{N_t\left(T\right)}{N\left(T\right)}$$

• Entropy $ent\left(T\right)=-{\sum }_{t\in D\left(T\right)}{p}_t{log}_2\left(p_t\right)$;
• Gini index $gini\left(T\right)=1-{\sum }_{t\in D\left(T\right)}{p}_t^2$.

  where T is the data set and $D\left(T\right)$ is the set of labels in the data set T. In addition, $N\left(T\right)$ represents the number of samples and $N_t\left(T\right)$ represents the number of samples with the label t.

A Random Forest (RF) is an ensemble of decision tree models such that each tree in the ensemble is built based on the bootstrap samples of the training data set. Furthermore, during the construction of the decision trees, the random forest selects only a subset of the features at each split point. In this way, the constructed decision trees are more different from each other to reduce the correlation among them and to have a better prediction. In contrast to the single decision tree (CART), random forests do not use any pruning of the tree.

Like any ensemble algorithm, the random forest also takes the average among all decision tree regressors to produce the final predictions (Figure 7).

The steps of the Python implementation are described in Algorithm 3.

Algorithm 3 The Python implementation of the RF algorithm

Input: Data set containing features and target Output: Prediction of RF for the given data set 1. Read the data set using the Python read_excel function; 2. Extract features and target variables; 3. Scale the features and target using the MinMaxScaler function; 4. Split the data set into training and test parts using the train_test_split function; 5. Choose the best hyperparameter using the GridSearchCV function; 6. Derive the RandomForestRegressor using the above results; 7. Predict using the obtained regressor and calculate the performance metrics on the test data set;

2.3. Performance Metrics

Four performance metrics [42] have been used for the comparison of the results.

• Mean squared error ($MSE$)
• Root mean squared error ($RMSE$)
• Mean absolute error ($MAE$)
• Coefficient of determination ($R^2$)

For each sample, the error ($e_i$) is the difference between actual output ($o_i$) and predicted output ($\widehat{o_i}$). The average of the squares of the error ($e_i$) is the $MSE$. The average is taken by dividing the summation of the square of all the errors by the total number of samples (n).

$$MSE=\frac{{\sum }_{i=1}^n{(o_i-\widehat{o_i})}^2}{n}$$

(1)

Root mean squared error ($RMSE$) is the square root of the average difference in squared error between actual output ($o_i$) and predicted output ($\widehat{o_i}$). It is the most commonly used metric.

$$RMSE=\sqrt{\frac{{\sum }_{i=1}^n{(o_i-\widehat{o_i})}^2}{n}}$$

(2)

Mean absolute error ($MAE$) is the average absolute difference of error between actual output ($o_i$) and predicted output ($\widehat{o_i}$). It is the most commonly used metric. The advantage of $MAE$ is that it is softer to outliers. The reason is that it does not have any square term associated with the equation, therefore the penalty for the outlier points is not that much compared to $MSE$ and $RMSE$ where there are heavy penalties imposed by the square term.

$$MAE=\frac{{\sum }_{i=1}^n\left|(o_i-\widehat{o_i})\right|}{n}$$

(3)

The coefficient of determination or $R^2$ explains the degree to which the independent variables explain the variation of the output variable. It is a measure of how well new samples can be predicted by the model through the proportion of explained variance. If $o_i$ is the true value of the i-th sample and $\widehat{o_i}$ is the corresponding output or predicted value, the estimated $R^2$ can be calculated as:

$$R^2=1-\frac{{\sum }_{i=1}^n{(o_i-\widehat{o_i})}^2}{{\sum }_{i=1}^n{(o_i-\overline{o})}^2}$$

(4)

where $\overline{o}=\frac{{\sum }_{i=1}^n{o}_i}{n}$.

3. Results and Discussion

The experiments have been executed based on the available field data from the east coast of Malaysia, which consists of 1460 samples, one input variable, wind speed (m/s), and one target variable, either wave height (m) or wave period (s). The regression results are compared among the three regressors (ANN, SVM, and RF). The Python programming environment (version $3.6$) with scikit-learn (version $0.24.1$) is used for the implementation.

Initially, the ANN regressor is trained using the training data and then the accuracy is validated using the testing data. The network is chosen with one hidden layer of 4 neurons, and each neuron uses the hyperbolic tangent function. The weight of the ANN is optimized using “lbfgs” in the family of quasi-Newton methods. Furthermore, the SVM regressor is trained using the same training data and validated using the same testing data as mentioned above. For SVM, the standard radial basis function (RBF) is used as the kernel function. Finally, the RF regressor is trained and validated in a similar fashion. For RF, 400 trees are used for the number of trees, 3 is used for the maximum depth of the tree, and $1460\times 0.05=73$ samples are used for the training of each decision tree in the ensembles. Each regressor is compared by the state-of-the-art performance metrics.

3.1. Cross Validation Results

It is a common practice to keep a part of the available data for testing and the remaining part for training. However, the model evaluation is particularly dependent on specific pairs of (training and testing) fractions, and the result can be overfitting. To overcome this problem, cross validation [43] is used to test the model’s fitness to predict new data that was not used to train the model and how it can generalize to unknown data.

There are many variants of cross-validation. The standard method is to use k-fold cross-validation. In general, the procedure is to use k rounds in cross-validation. It splits the data into k subsets. In a single round, it completes the analysis on one subset (the training set), and then validates the performance on the remaining $k-1$ subsets (the validation set). In the next round, another subset is chosen for training and the remaining is chosen for validation. To reduce variability, these steps are repeated k times for k subsets so that each subset is selected for training. Finally, an average among the k validation results is reported as the fitness of the model’s predictive performance.

Nevertheless, it is common practice to repeat the k-fold cross-validation process multiple times and report the average performance among all folds and all repeats. This approach is called repeated k-fold cross-validation.

The average and standard deviation of the regression performance of wave height were determined using 10-fold cross validation that was done for three methods as presented in

Table 3. Regression results for 10-fold cross validation with three times repeated for wave height (m).

Regressor	MAE		MSE		RMSE		${\mathit{R}}^2$
	Avg	Std	Avg	Std	Avg	Std	Avg	Std
SVM	0.4920	0.0385	0.4881	0.0920	0.6956	0.0651	0.5015	0.0871
ANN	0.0765	0.0049	0.0107	0.0016	0.1034	0.0079	0.5164	0.0872
RF	0.0775	0.0042	0.0110	0.0016	0.1046	0.0075	0.5104	0.0725

. The four-performance metrics are reported in the same way as in the preceding section.

It is clearly evident that the method of ANN has a minimum average value of mean absolute error, mean square error, and root mean square error. Besides, it has the maximum average value of the coefficient of determination. As a result, this approach performs best for predicting wave height.

Similar findings for wave period have been presented in

Table 4. Regression results for 10-fold cross validation with three times repeated for wave period (s).

Regressor	MAE		MSE		RMSE		${\mathit{R}}^2$
	Avg	Std	Avg	Std	Avg	Std	Avg	Std
SVM	0.7136	0.0495	0.8317	0.1406	0.9089	0.0753	0.1629	0.0743
ANN	0.0935	0.0056	0.0139	0.0019	0.1176	0.0079	0.1552	0.0614
RF	0.0930	0.0054	0.0136	0.0018	0.1165	0.0076	0.1717	0.0652

using 10-fold cross validation that has been done for the consecutive three methods.

It is apparent that the RF approach has the lowest mean absolute error, mean square error, and root mean square error. Furthermore, it has the maximum value of the coefficient of correlation. As a result, this approach is the most accurate for predicting wave periods.

$R^2$ is close to 1, which indicates that regression predictions perfectly fit the data. However, in our case, the experimental results show the lower value of $R^2$ due to the inconsistencies in the collected data set.

3.2. Graphical Representation of a Sample Training and Testing Results

To illustrate the model’s performance graphically, one sample is chosen randomly as the pair of (training, testing), where the training is 80% and testing is 20% of the data. In

Table 5. A sample regression results for wave height (m).

Regressor	Training				Testing
	$\mathit{MAE}$	$\mathit{MSE}$	$\mathit{RMSE}$	${\mathit{R}}^{\mathit{2}}$	$\mathit{MAE}$	$\mathit{MSE}$	$\mathit{RMSE}$	${\mathit{R}}^{\mathbf{2}}$
SVM	0.4843	0.4739	0.6884	0.5196	0.5025	0.5009	0.7077	0.5249
ANN	0.0752	0.0105	0.1026	0.5367	0.0792	0.0111	0.1052	0.5449
RF	0.0751	0.0105	0.1024	0.5386	0.0814	0.0115	0.1074	0.5258

, the results for both training and testing data are shown in the case of the wave height regression problem. It is evident that the mean squared error, mean absolute error, and root mean squared errors are the smallest for RF compared to others in the training data. However, these metrics are the smallest for ANN compared to others in the testing data. Furthermore, the coefficient of correlation ($R^2$) is largest for RF compared to others in the training data, but it is largest for ANN compared to others in the testing data. As a result, the ANN approach is superior at predicting wave height in the future.

The regression results of ANN, SVM, and RF for wave height are depicted in Figure 8. Figure 8a shows the ANN regression findings for wave height. The black dots are actual testing data, and the green dots are the predicted values. It is clear that the predicted values follow the pattern of the actual testing data. Nevertheless, the regression results for the SVM regressor are shown in Figure 8b. The black dots are actual testing data, and the blue dots are the predicted values. It is clear that the predicted values follow the pattern of the actual testing data.

The regression results of wave height based on the RF regressor are presented in Figure 8c. The black dots are actual testing data, and the orange dots are the predicted values. It is clear that the predicted values follow the pattern of the actual testing data. The pattern is not as smooth as SVM or ANN because RF is not a single decision tree method; rather it is an ensemble method that works by taking the average of different decision trees’ predictions.

In addition, in the case of wave period,

Table 6. A sample regression results for wave period (s).

Regressor	Training				Testing
	$\mathit{MAE}$	$\mathit{MSE}$	$\mathit{RMSE}$	${\mathit{R}}^{\mathbf{2}}$	$\mathit{MAE}$	$\mathit{MSE}$	$\mathit{RMSE}$	${\mathit{R}}^{\mathbf{2}}$
SVM	0.694	0.7980	0.8933	0.1846	0.7654	0.9366	0.9678	0.1353
ANN	0.0926	0.0137	0.1169	0.1574	0.1006	0.0157	0.1251	0.1284
RF	0.0899	0.0128	0.1129	0.2140	0.099	0.0153	0.1235	0.1507

shows the findings for both training and testing data. It is evident that the mean squared error, mean absolute error, and root mean squared error is the smallest for RF compared to others in both training and testing data. Nevertheless, the coefficient of determination ($R^2$) is the largest for RF compared to others in both training and testing data. Therefore, for future prediction of wave period, the RF method is best.

The regression results of ANN, SVM, and RF for wave period are depicted in Figure 9. For ANN, in Figure 9a, the actual testing data is shown in black dots and the predicted values are shown in green dots. It is clear that the predicted values follow the pattern of the actual testing data for ANN. For SVM, in Figure 9b, the actual testing data are represented by black dots, while the actual predicted values are represented by blue dots. It is clear that the predicted values follow the pattern of the actual testing data for SVM.

Finally, for RF, the real testing data is represented in black dots, whereas the actual predicted values are shown in orange dots in Figure 9c. It is clear that the predicted values follow the pattern of the actual testing data for RF. The pattern is not as smooth as SVM or ANN because RF is not a single decision tree method; rather it is an ensemble method that works by taking the average of different decision trees’ predictions.

3.3. Comparison with Standard Non-Parametric Kernel Regression

The non-parametric method of kernel regression (KR) in statistics is used to calculate the conditional expectation of a random variable. The goal is to find a non-linear relationship between two random variables, I and O [44]. The problem under consideration can be modeled using a standard non-parametric regression problem. As an example, I can be wind speed and O can be wave height. In

Table 7. A sample comparison with kernel regression for predicting wave height.

Methods	${\mathit{R}}^2$ for Predicting Wave Height
SVM	0.5249
ANN	0.5449
RF	0.5258
KR	0.5389

, the previous results are compared with the results of KR for the same sample.

It is clear that KR results are not far from those of the SVM, ANN, or RF. In fact, ANN produces the best results in the case of wave height prediction.

3.4. Overall Discussion

The goal of this study is to understand the best predictive model among ANN, SVM, and RF for the prediction of wave height and period from the wind speed. A detailed experiments are performed in terms of cross-validation and sample training and testing results. The summary of this result is given in the

Table 8. Summary of prediction results.

Methods	Prediction of Wave Height	Prediction of Wave Period	Final Comment
SVM	worst in MAE, MSE, RMSE and $R^2$	worst in MAE, MSE, RMSE, but second best in $R^2$	The results are not the best, however, not very far from the best one
ANN	best in MAE, MSE, RMSE, and $R^2$	second best in MAE, MSE, RMSE but worst in $R^2$	The results are promising for wave height and reasonable for wave period
RF	second best in MAE, MSE, RMSE, and $R^2$	best in MAE, MSE, RMSE, and $R^2$	The results are really promising for wave period and also for wave height

.

It is evident from Table 8 that the Random Forest (RF) method is truly performing well across two prediction problems. It is the second best for the prediction of wave height and the best for predicting wave period. Nevertheless, it has the advantage of being a non-parametric method. Moreover, the underlying tree structure has the advantages of interpretability and usage. However, to be specific, RF performs best for the prediction of wave period while ANN performs best for the prediction of wave height.

4. Conclusions

This study carried out three different and well-known machine learning algorithms for the prediction of offshore waves. These approaches are used to predict the wave height and wave period from the given wind forces. Multiple accuracy ranges are obtained in terms of the mean absolute error, mean square error, root mean square error, and coefficient of determination. Overall, these performance measures show average behavior. However, it is possible to compare the three employed methods and analyze the results. The regression analysis in the random forest performs best for the prediction of wave period, and the artificial neural network performs best for the prediction of wave height. Furthermore, it was compared with the standard non-parametric kernel regression and found to have a similar result.

In situations when a traditional analysis would be challenging, these machine learning techniques can produce very quick and reasonable predictions. These studies can benefit the community as a measurement of safety and precaution in the critical marine environment.

In this regard, there are numerous potential future research directions. One disadvantage of using such metocean data is the existence of discrepancies. Future work should incorporate advanced algorithms in addition to the aforementioned models to address such inconsistencies. Additionally, more machine learning techniques should be used to find the ideal answer for this specific prediction problem. The time dependence of the metocean data, which can be examined using time series analysis tools, is another interesting subject of research.