Prediction of significant wave height using machine learning and its application to extreme wave analysis

Abstract

Waves of large size can damage offshore infrastructures and affect marine facilities. In coastal engineering studies, it is essential to have the probability estimates of the most extreme wave height expected during the lifetime of the structure. This study predicts significant wave height using the machine learning (ML) technique with generalized extreme value (GEV) theory and its application to extreme wave analysis. The wind speed, wind direction, sea temperature, and swell height data consisting of wave characteristics for 60 years has been obtained from the European Centre for Medium-Range Weather Forecasts (ECMWF). While analyzing extreme waves, the block maxima approach in GEV was used to incorporate the seasonal variations present in the data. The estimated scale parameter, shape parameter, and location parameter of GEV are used in the ML model to predict the significant wave heights along with their return periods. The ML algorithms such as linear regression (LR), artificial neural networks (ANN), and support vector machines (SVM) are evaluated in terms of R² performance. The model comparison results suggested that the SVM model outperforms the LR and ANN models with an accuracy of 99.80%. Finally, the GEV analysis gives the extreme wave height results of 2.348, 3.470, and 4.713 m with a return period of 5, 20, and 100 yrs, respectively. Hence, the model developed is capable of predicting both significant wave height and extreme waves for the design of coastal structures.

Appendix A

1.1 A.1 Machine learning models

1.1.1 A.1.1 Linear regression (LR)

The linear regression method generally formulates the output as a linear function of the input variables in such a way that the values obtained using the formulated function are close to the actual output. This formulated linear function (hypothesis) can be further used to predict the output values for any given input. The method of LR involves creating a cost function which is a function of these parameters (constants). The lesser the value of the cost function the lower the difference between the predicted values and the actual values. Hence, the values of these parameters are determined by minimizing the value of the cost function. This is achieved using optimization techniques such as gradient descent.

1.1.2 A.1.2 Artificial neural network (ANN)

An artificial neural network (ANN) consists of an input layer, a series of hidden layers and an output layer, each having a various number of nodes. The input layer contains an equal number of nodes as input features. Similarly, the output layer is the same as the number of outputs desired by the model, which is generally one for regression models. The hidden layers contain a number of nodes that act as a parameter for the model.

In this neural network technique, the input layer uses feature value as input. The input layer contains an activation function (such as relu, tanh, or sigmoidal), which allow input to pass through it and provides the output from the input layer. The input layer is connected with the hidden layer with some weight factor. The weight factor determines the relative contribution of different nodes in the hidden layer. The output obtained from the input layer act as an input to the hidden layer. Further, the activation function determines the output of the hidden layer. Now, the hidden layer output is multiplied with the subsequent weight factor to obtain input to the output layer. Therefore, the final output is obtained by adding all the input of the output layer. After the first iteration, the weight is updated, and the processes are repeated until the minimum difference is obtained from the actual and predicted output, this process is also called back propagation algorithm.

The multilayer perceptron neural network is used for this study, which comprises three hidden layers with 10, 20 and 30 nodes, respectively. The number of nodes in each layer is decided by performing iterations for a different number of nodes in each layer. The relu (rectified linear unit) activation function was used for training the model. The other two widely used activation functions, namely tanh and logistic activation functions have output values in the range of [−1, 1] and [0, 1], respectively, and hence are more suitable for classification problems. As this is not the case with relu function, it is used as the activation function for training the model.

1.1.3 A.1.3 Support vector machine (SVM)

Support vector machine (SVM) algorithms employ a number of mathematical formulations, which are defined as the kernel. The kernel’s function uses input data and converts them into the required form. Basically, it returns the internal product between two points in an appropriate feature space. The linear, non-linear, polynomial, and radial basis function is used as kernel function in the SVM algorithm (Gunn et al. 1998). The kernel function gained considerable attention in the last few years, especially as the SVM algorithm gained popularity. In many applications, kernel functions provide a simple relationship between linearity and non-linearity for the algorithms, which can be expressed as dot products (Fadel et al. 2016). In this paper, the Gaussian radial basis function kernel (popular kernel function used in various kernelized learning algorithms) has been used for the prediction of significant wave height using the SVM model as it outperformed other RBF kernels by significant difference (comparison is shown in the results section).

Gamma (γ) is an important parameter of the Gaussian RBF kernel. The gamma parameter depicts the extent to which an individual training example influences, with low values that mean ‘far’ and high values that mean ‘close’. The gamma parameters can be viewed as the inverse of the sample’s influence radius selected as support vectors by the model. Analytically, the narrower the Gaussian RBF kernels get (larger gammas) the ‘spikier’ the hypersurface is going to get. On the other hand, if the Gaussian RBF kernels are too wide (small gammas), it would end up with a hypersurface that is almost flat. Hence, choosing an optimum value of gamma is very important.

The advantages of the RBF are ease of construction, good generalization, high input noise tolerance, and the capacity to learn. The radial basis function can be used to develop the SVM model as well as ANN Model. When dealing with a large number of high-dimensional datasets, the ANN with RBF techniques is ineffective due to their sensitivity to the Hughes phenomenon. SVM with radial basis function has been effectively applied for high dimensional datasets in recent years (Huang et al. 2002; Camps-Valls et al. 2004; Melgani and Bruzzone 2004). The properties of SVM with RBF kernel allow them to deal with a large set of high-dimensional datasets. The SVM with RBF kernel can handle large input spaces, cope with noisy samples robustly and create sparse solutions effectively.