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This paper presents a comparative study of sensitivity analysis (SA) and simplification on artificial neural network (ANN) based model used for ship motion prediction. Considering traditional structural complexity of ANN usually results in slow convergence, SA, as an efficient tool for correlation analysis, can help to reconstruct the ANN model for ship motion prediction. An ANN-Garson method and an ANN-EFAST method are proposed, both of which utilize the ANN for modeling but select the input parameters in a local and a global fashion, respectively. Through the benchmark tests, ANN-EFAST exhibits superior performance in both linear and nonlinear systems. Further test on ANN-EFAST via a case study of ship heading prediction shows its cost-effective and timely in compacting the ANN based prediction model. INTRODUCTION With the development and prosperity of the world’s shipping industry, the maritime transportation has become more and more busy. In order to ensure the safety of navigation, great concern has been put toward the ship motion prediction. Furthermore, some special operations, such as submarine cable laying, marine survey, etc., need more accurate ship motion prediction and control precisely. Therefore, how to establish an efficient ship motion prediction model has great theoretical and practical value in the maritime applications. However, mathematical model based ship motion prediction is challenging due to the nonlinear and ∗Corresponding author, Email: guoyuan.li@ntnu.no. Xu Cheng and Guoyuan Li have equal contribution to this paper. time-varying dynamical model of ship, as well as complex dynamic nature of sea [1, 2]. Our partner in Norway therefore started to collect on-board ship sensor data long time ago and intended to create robust predictive models for ship maneuvering technologies. There would be a possibility to combine those ship sensor data with modeling methods to design and implement ship motion prediction model. To date, a variety of novel intelligent approximation-based techniques and algorithms like fuzzy logic, Kalman filtering, Bayesian network, regression analysis and ANN have been applied to create predictive models [3–6]. Those methods have their own pros and cons at specific aspects. For example, regression analysis is not suitable for complex, high dimensional and non-linear system; Fuzzy logic relies more on mathematical model; Kalman filtering works only for Gaussian noise process; The performance of Bayesian network in high dimensional data set is poor. None of them except ANN are suitable for modeling the ship motion, as situations in which lack precise mathematical model and only input-output sample data are available. Indeed, an ANN is a “black box” and has the ability to explicitly identify possible causal relationships from the inputoutput sample data. However, there is no standard to construct a compact ANN for prediction purposes. Input parameters and hidden units are the main factors to obtain an optimized model [7]. If there are too few inputs, the network cannot represent the input-output mapping of system with sufficient accuracy. If there are too many inputs, the network dimension will increase, which in turn aggravates computational complexity. Both cases will deteriorate the generalization capability of the network. Therefore, selection of input parameters is a key issue when applying ANN 1 Copyright c © 2017 by ASME to ship motion prediction. SA investigates how the variation in the output of a numerical model can be attributed to variations of its input factors, and it plays an important role in prediction model construction and simplification, and thus the generalization ability of prediction model. The main purpose of SA is to estimate the contribution of each model input, either main or interaction contribution, on the model output and to identify the main contributors to the output. SA has been widely used in areas such as engineering, economics, and sociology [8]. Taking advantages of SA’s characteristics, it is possible to use it to select the input parameters of an ANN based model used for ship motion prediction. The rest of the paper is organized as follows. The related work section is a brief recall of some of the existing methods in ANN and SA. In the next section, we describe the input selection procedure and the case ship, then the methods we used in this paper is introduced and the calculation of local sensitivity analysis (LSA) and global sensitivity analysis (GSA) are explained. After that, the proposed algorithm is tested using two analytical models and a case study of SA on heading of ship motion prediction model is described in detail. The results are shown and the calculated first order sensitivity index are compared with analytical results. A comparison of the performance of the LSA and GSA is also presented in this section. Finally conclusions are given. RELATED WORK Artificial Neural Network Inspired by biological neural network, ANN could build up the mathematical relationship between the input parameters and the output parameters, with the advantage that it can be modeled without prior knowledge. An ANN facilitates the ability to learn complex nonlinear relationships between input and output parameters. Thanks to the powerful potential (massive parallelism, generalization capacity and fault-tolerance), ANN has been widely used in fields like pattern recognition, reliability analysis, classification, ship motion control and prediction. The basic architecture of ANN consists of single input, hidden and output layer, with each layer containing one or more neurons, in addition to bias neurons connected to the hidden and output layers. The back-propagation (BP) algorithm is the most widely used learning algorithm for ANN, which is a self-adapted learning procedure that minimizes the error between the desired and the predicted outputs. The learning process consists of two parts: feed-forward and backward pass. The output of ANN is calculated in the process of feed-forward pass, with the output error propagated backward to adjust the weights and bias of the ANN. The number of hidden layer nodes and the maximum iteration number should be carefully chosen to overcome the over-fitting and under-fitting problems. Over-fitting means that a trained ANN has weak capability of generalization. An over-fitted ANN usually has a good prediction capability over train samples, but has a bad prediction capability over test samples. Under-fitting means that a trained ANN is too simple to be capable of representing the relationship between input parameters and output targets. An under-fitted ANN usually has bad prediction capabilities over both training and testing samples. Sensitivity Analysis SA could be implemented in either local or global manner. The LSA explore the response of the model output to a small change of the parameter from its nominal value. Garson algorithm is one of the popular LSA algorithms [9]. This method has shown to be computational efficient and conceptually simple when quantifying the relative importance of input parameters. It has been used in some ship motion prediction applications, such as the work in [5, 6, 10]. Local sensitivity index is calculated at the nominal point or a fixed point, which is not representative for all inputs in the whole parameter space. In addition, the LSA do not explore the interactions between input parameters. In contrast, a GSA estimates the effect of input parameters across the whole input parameter space. GSA is generally divided into four categories: Traditional methods, Analysis of Variance methods (ANOVA) methods, Derivative-Based Methods and Surrogate-Based Methods. ANOVA methods are also called variance-based methods, which makes ANOVA decomposition of model response variances into the contributions from individual parameters and their interactions. Cukier, et al. presented Fourier Amplitude Sensitivity Test (FAST) [11]. Later, Salteli et al. introduces a global, quantitative, model independent SA method for calculating both main effect and total effect indices based on the FAST — extended FAST (EFAST) [12]. EFAST is model independent, which can be used in ANN based prediction. Currently, most of the study only focuses on studying either LSA or GSA in ANN based ship prediction model. There is not a systematic comparison between them. In this study, efforts are made to combine the ANN with the Garson algorithm and the EFAST algorithm respectively, aiming to find out which one is preferable for nonlinear ship motion prediction. SIMPLIFICATION OF ANN MODEL VIA SENSITIVITY ANALYSIS System Structure This paper aims to construct a compact ANN model for ship motion prediction using the SA approach. The main idea is to use the SA method to evaluate the importance of each input and select the inputs according to their importance. The input selection procedure consists of four components: data cleaning, surrogate model, SA and result visualization. Data cleaning is to minimize the affection of noisy, redundant information of sensor data on further analysis and modeling. In general, it is difficult to estimate the contribution of each input parameter and the interaction 2 Copyright c © 2017 by ASME

Simplifying Neural Network Based Model for Ship Motion Prediction: A Comparative Study of Sensitivity Analysis