Prediction of Ship Heaving Motion Based on Chaos Theory and Improved Extreme Learning Machine

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(1.School of Mechanical Engineering,Xiangtan University,Xiangtan 411105,China;2.Department of Mechanical and Aerospace Engineering,Hong Kong University of Science and Technology,Hong Kong 999077,China)

Abstract:Ship heaving motion forecasting is an important aspect of the active heave compensation system.To satisfy the real-time characteristics of ship heave motion prediction with high accuracy,a hybrid method that combines chaos theory and extreme learning machine with enhancing search model parameters(CES-ELM)was proposed in this paper.The error-minimization-based method was used to grow ELM’s hidden nodes and unceasingly update weights based on phase space reconstruction of the chaotic dynamical system.Optimized model parameters were used to establish the ship motion forecasting model.The simulation results in different sea conditions indicate that the prediction mean-absolute-percentage error of the proposed method is less than 10%,and that this model can effectively improve the forecast accuracy and robustness in comparison with the traditional ELM and LSSVM.

Key words:heave motion prediction;phase space reconstruction;extreme learning machine

0 Introduction

Ship heaving motion forecasting plays a significant role in the active heave compensation system[1].Forecast results will directly influence the accuracy of compensation of the active heave compensation system,how to develop more accurate and robustness motion forecasting methods becomes an important research topic.

Recently,many methods have been applied extensively for ship motion forecasting,such as the time series analysis method[2-4],artificial neural network method[5-6],support vector machine(SVM)[7],and so on.The time series analysis method does not require a priori information of the waves or state equation of ship’s navigation state,merely using historical data of ship movement to seek the law of historical data and predict future value.However,this method only works with the smooth changes of heave values.In support vector machine regression algorithm,penalty coefficientCand normal coefficientσgreatly influence the precision of the SVM model,bringing intelligent optimization algorithms in the SVM algorithm to look for the optimum parameters[8].Nevertheless,intelligent optimization algorithms need plenty of iteration,which requires much time and makes it difficult to ensure real-time prediction.Recently,artificial neural networks(ANN)have been extensively applied for forecasting.The ANN is a universal function approximator which is capable of mapping any linear or nonlinear functions.However,most ANN-based forecasting methods use gradientbased learning algorithms,which cannot avoid many difficulties such as learning epochs,learning rate and local minima.

Extreme learning machine(ELM)is a novel machine-learning algorithm based on single-hidden-layer feedforward neural networks(SLFN)proposed in Ref.[9].In the learning process,the input weights and hidden biases are randomly chosen,and the output weights are analytically determined by using the Moore-Penrose generalized inverse.ELM can learn much faster with a higher generalization performance than the ANN and SVM,and can solve the problem of learning epochs,learning rate and local minima.However,because of random generation of hidden nodes,the model of the ELM is poorly stabilized.To avoid this problem,an enhancing search ELM(ES-ELM)method is proposed in this paper.ES-ELM randomly grows hidden nodes and updates the network’s weights according to the error-minimization-based method.Ship heaving motion is usually highly complex due to the internal and external environment’s influence.It is concluded in many studies that ship motion is a nonlinear chaotic system on the sea.Therefore,ES-ELM should be combined with chaos theory[10]to develop more accurate and robust motion forecasting methods.

Based on the above analysis,a hybrid method that combines chaos theory and extreme learning machine with enhancing search model parameters(CES-ELM)for heaving motion forecasting is proposed in this paper.The rest of the paper is organized as follows.Chaos characteristic identification and phase space reconstruction are first presented in Chapter 1,followed by introduction of the proposed method’s fundamental principle in Chapter 2.Prediction steps of CES-ELM are listed in Chapter 3,and the experimental results are shown in Chapter 4.The last chapter concludes this study.

1 Phase space reconstruction

1.1 Data

Time series data of heaving displacement come from the Marine Systems Simulator(MSS)developed by the Norwegian University of Science and Technology.Data A is generated at a significant wave height of 5 m,a mean wave direction of 10°,and a sample period of 0.5 s.

1.2 Phase space reconstruction

Phase space reconstruction(PSR)is the basic theory in the analyzing chaotic dynamic systems,founded by Taken[11].The theorem proves that a phase space from a one dimension chaotic time series can be reconstructed with an equal phase space with a dynamic system in topology.The discrimination,analysis and prediction are employed in the reconstructed phase space.Thus PSR is the key point in the chaotic time-series study.The embedding dimensionmand time delayτare the most important variables in PSR.The selections ofmandτhave been well studied in the relevant literatures.In this paper,the mutual information method[12]is used to calculate the time delayτ.The mutual information method’s computing results as shown in Fig.1 show that the first minimum point of mutual information functionG(tau)corresponding to thetauis 6,so the time delayτis 6.Cao’s method[13]is used to determine the embedding dimensionm.The computing results of two judgment indicatorsE1(m)=E(m+1)/E(m)andE2(m)=E*(m+1)/E*(m)in Cao’s method are shown in Fig.2.It is indicated thatE1(m)increases with the embedding dimension between 0 and 13,then presents saturation phenomenon inm=14.However,with the growth of the embedding dimensionm,E2(m)is changing all the way,which has not been fixed in the vicinity of 1.So the embedding dimensionmis 14.

Fig.1 Calculated mutual information function

Fig.2 Calculated results of two judgment indicators

Given a chaotic time series{X(n),n=1,2,…,N},the reconstructed phase space can be represented by

wheret=1,2,…,N-( )m-1τ，τis the time delay,mis the embedding dimension,provided thatm≥2D+1 andDis the fractal dimension of the attractor.

Lyapunov exponent is a useful tool to characterize a chaotic attractor quantitatively,which effectively measures the chaotic orbit’s sensitivity to its initial conditions and quantifies the attractor’s dynamics of a complex system.When the Lyapunov exponentλof a time series is positive,then the time series will become chaotic.The calculation of Lyapunov exponent in this work is conducted using the classical Wolf-method[14].Lyapunov exponentλ=0.239 2＞0 proves that time-series data of heaving displacement has chaos characteristics.

2 ELM and ES-ELM

2.1 ELM

The ELM is a flexible computing framework for a broad range of nonlinear problems[15].A single hidden-layer feedforward network(SLFN)is the most widely used model for forecasting modeling[16].As shown in Fig.3,the model is characterized by a network of three layers of simple processing units connected by acyclic links.The hidden layers can capture the nonlinear relationship among variables.Each layer consists of multiple neurons connected to neurons in adjacent layers.Suppose there are distinct samples ofN,D={(Xi,Yi),i=1,2,…,N} where

Fig.3 Structure of the ELM model

The SLFN withLhidden neurons and an activation function vectorg(X)are described as

whereαi=[αi1,αi2,…,αiL]is the weight vector connecting theith input neurons and the hidden neuron,βi=[βi1,βi2,…,βim]Tis the weight vector connecting theith hidden neuron and the output neurons,biis the threshold of theith hidden neuron andTi=[ti1,ti2,…,tim]T∈Rmis the output value of the network.If the SLFNs can approximate theseNsamples with a zero error viz.0,thus,there also exists parametersαi，biandβi,such that

Thus,the above equations can be compactly described as

whereHis the hidden layer output matrix,

Training an SLFN is simply equivalent to finding a least-squares solutionβ^of the linear functionHβ=T:

The smallest norm least-squares solution of the above linear system is

whereH+is the Moore-Penrose generalized inverse of matrixH.Owing to the Moore-Penrose generalized inverse,the learning speed is dramatically increased for the network.

2.2 ES-ELM

In the learning process of ELM,the input weights and hidden biases are randomly generated,and the output weights are analytically determined by using the Moore-Penrose generalized inverse.Thus,non-optimal input weights and hidden biases may be generated,which would produce unwanted consequences.To avoid this problem,this paper proposes an enhanced search ELM(ESELM)capable of growing the randomly-generated hidden nodes and updating the weights of the network according to the error-minimization-based method.

ES-ELM Algorithm:Given a training setthe hidden node output functionG( )X,α,b,the maximum numberLmaxof hidden nodes,εis the expected learning accuracy andLis a small positive integer given by the user.

Step1 Initialization:LetL=0 andε=0.01

Step2 Learning step:whileL＜Lmax,E(HL)＞ε

(a)Increase by 1 the number of hidden nodesL:L=L+1

(b)Fori=1:k

(i)Assign random parameters(αi,bi)for the hidden node，Calculate the hidden layer output matrix

(ii)Calculate the corresponding output error

End for

(d)Update the output weightβ^according to formula(7).

End while

3 Prediction step of CES-ELM

Excitation function is selected as sigmoid functiong(x)=sigmoid(x)=The prediction algorithm of CES-ELM is summarized as follows:

(1)Normalize processing of data,use Cao’s method to determine the embedding dimensionm.Use the mutual information method to calculate the time delayτ.

(2)Reconstruct the time seriesX(i)to get set.Meanwhile,separate them into a training set and testing set.

(3)Train neural networks of the ES-ELM algorithm by using the set1,2,…,Mbefore obtaining the optimum parametersL,αandb,and updating the output weightβ.

(4)Apply the neural network to obtain the forecasting result ofYt(t),t=1,2,…,Min the testing set,the input data of the neural network areXt(t),t=1,2,…,Min the testing set.Keep predicting forwardstimes,then we can obtainspredicting heave series

(5)Predicting heave series is treated with anti-normalization,then calculate the prediction error.We further introduce Step(4)in detail.According to the training process in Step(3),we can get the predicted neural network model

The 1-step predicted value calculation:When the input datacome,the output of the predicted neural network modelis calculated by Eq.(3)with the optimum parametersL,α,b,and the output weightcan be updated.is the 1-step predicted value and

The 2-step predicted value calculation:We add the 1-step predicted value into the input data and remove the oldestXt(1),therefore the new input data areThrough the trained neural network model,we can get the 2-step predicted value

The 3-step predicted value calculation:Then we remove the oldest input data pointXt(2) in the previous step and add the 2-step predicted valueinto the input data to get a new input dataThe 3-step predicted valueorcan be calculated by the trained neural network model.The rest may be deduced by analogy.

The n-step predicted value calculation:Then-step predicted valuecan be cal-culated by the trained neural network model with the input datapresent 1 to(n-1)-step predicted values.

4 Numerical simulation studies

4.1 Forecast error measure method and the best forecast time determination

To compare the CES-ELM method with other methods,an adequate error measure method must be selected.The well-known statisticsRMSE(Root Mean Square Error)andMAPE(Mean Absolute Percentage Error)are used to measure the heaving displacement prediction models in this work.Their representation can be expressed as

Delay timeτ=6 and embedding dimensionm=14 of heaving displacement data are determined according to Chapter 2,and the related parameters ofL=1,k=10,ε=0.01 are set.We reconstruct the heave series,and then select 1 100 points in heaving sequence for prediction,in which the first 1 000 points are used for ES-ELM neural network learning and the remaining 100 points as a test set.

Prediction and actual curves of the 20 s-ahead forecast using the CES-ELM method are shown in Fig.4.The prediction starts atTpred=0 s and runs untilTpred=20 s.The parameters used for the prediction are the ones estimated atTpred=0 s.

Fig.4 20 s-ahead forecasting and actual curves of CES-ELM method

From the prediction result shown in Fig.4,the CES-ELM method can obtain a good result in 0-7 s.However,with the increase of predicted time,the prediction error is increased significantly.The literature[18]shows that the delay in the ship’s active heave compensation system is below 2 s.The 4 sahead forecast will give enough executive time to make an appropriate response.The 4 s-ahead forecast can not only save computing resources but also help the system to update sample sequence in complex sea conditions,constantly updating sample and doing 4 s-ahead forecast will ensure real-time of the forecasting model.The mean absolute percentage error of the 4 s-ahead forecast in the testing set is 6.2%.

4.2 Comparison among CES-ELM,LSSVM and ELM in the different marine environment

The other two groups of data are introduced to verify the stability of CES-ELM，Data B is generated at a significant wave height of 5 m,a mean wave direction of 20°,and a sample period of 0.5 s.Data C is generated at a significant wave height of 5 m,a mean wave direction of 30°,and a sample period of 0.5 s.Fig.5 displays the prediction and actual curves of the 4 s-ahead forecast of the testing set in three different marine environments.Tab.1 shows the performance of CES-ELM,LSSVM and ELM in variable marine environments.

Fig.5 Zoom of 4 s-ahead forecasting results of the testing set(50 s)using the CES-ELM model in different marine environments

Tab.1 Error statistics of three prediction models

As seen from Tab.1,the testing root mean square error(RMSE)of CEI-ELM is generally much smaller than those of the ELM and LSSVM when they are in the same marine environment.In different marine environments,the performance of ELM and LSSVM varies greatly.The maximumMAPEvariation of ELM has reached 14%.Nevertheless,the maximumMAPEvariation of CES-ELM is 4.7%,the maximumMAPEof CES-ELM is only 9.9%.Furthermore,as observed in Tab.1,CESELM obtains much smaller testingMAPEthan two other methods,which means that the performance of CES-ELM is more stable than those of ELM and LSSVM.

5 Concluding remarks

In this paper,the prediction method of ship heaving motion in an active heave compensation system is studied based on chaos theory and ELM with enhancing search model parameters.The obtained online time series are mapped intomdimension characteristics space by phase space reconstruction.Then ELM with enhancing search model parameters(ES-ELM)is used to build up the ES-ELM prediction model.An error-minimization-based method is used to automatically determine the hidden nodes and update weights in generalized SLFNs that need not be neuron alike.The experimental results generated by a set of consistent performance measures with different metrics(MAPE,RMSE)demonstrate that the proposed model produces smaller predicting errors than some other heaving motion forecasting methods,and can be employed in heaving motion forecasting.Thus,the CES-ELM is a better choice for the practical forecasting of ship heaving motion in an active heave compensation system.