(College of Shipbuilding Engineering,Harbin Engineering University,Harbin 150001,China)

Extending the Scope of AR Model in Forecasting Non-stationary Ship Motion by Using AR-EMD Technique

HUANG Li-min,DUAN Wen-yang,HAN Yang,YU Dong-hua,Aladdin ELHANDAD

(College of Shipbuilding Engineering,Harbin Engineering University,Harbin 150001,China)

Accurate short-term prediction of ship motions allows better improvements in safety and control quality in ship motion sensitive maritime operations.Inspired by the high adaptive and effective nature of auto-regressive(AR)model,it was widely studied in substantial papers concerning shortterm prediction of ship motion.However,it suffers theoretical difficulty when the ship motion becomes non-stationary.In this paper,an extended AR model designated as EMD-AR for non-stationary ship motion forecast is developed by using AR-EMD technique.Where,AR-EMD technique refers to empirical mode decomposition(EMD)applying AR prediction method in boundary extension. EMD-AR model overcomes the non-stationarity in ship motion by decomposing the complex ship motion data into a couple of simple intrinsic mode functions(IMFs)and residual.Each sub-component is predicted individually,and predictions are then aggregated to attain the final results.Comparative study with linear AR model and nonlinear support vector regression(SVR)model employing model testing ship motion data was conducted.The results show that AR-EMD is effective in handling the negative effect on the prediction accuracy resulting from non-stationarity in ship motion and EMD-AR model produces better prediction compared to AR and SVR models.

non-stationary ship motion;short-term prediction;AR model; Empirical Mode Decomposition;EMD-AR model

0 Introduction

Six degrees of freedom ship swaying motions occur through lifetime to the ocean environmental disturbances including sea waves,wind and ocean current,etc.,which are dangerous in ship related maritime operations such as aircraft landing in carriers,ship-borne helicopter recovery,float over,launch and recovery of submarines,and cargo transfer between ships,and so on,especially in harsh conditions.

A short term prediction of the ship motions 5 to 10 seconds ahead of time may be very useful for offshore operations in both operational safety and efficiency aspects.For example,the prediction information can help to provide motion compensation which may prevent crash of cargo in cargo transfer,improving the fire accuracy of the ship-borne weapon systems and performance of the motion control systems.Besides,another important application of the motion prediction is to extend the operational limits by forecasting the quiescent periods where the ship motions are within acceptable limits to perform a desired maritime activity.Classical prediction approaches employ statistical data to assess whether a task can be executed,this may result in the outcome that an operation is never executed,whereas quiescent periods do exist[1].

Short term prediction of the ship motion is widely studied for its great engineering application value,so far,enormous number of forecast models have been studied,where some of them were already carried out in marine trial.According to the basis of different modeling principles,short-term prediction approaches are categorized into three types of models:hydrodynamic based prediction models,classical time series prediction models and nonlinear and artificial theories based on the short term prediction models[2].

The early research works on short-term prediction are related to hydrodynamics.Usually, hydrodynamic related prediction method contains the convolution based and state-space based Kalman filter forecast techniques.Kaplan(1965)[3]developed a predictor by using wave height measurements at the bow serving as input data which was then convoluted with the ship response kernel function to obtain the motion estimation in the coming seconds.Whereas,the ship response kernel function is derived out under the consumption of linear hydrodynamic theory.However,accurate response function and wave inputs are must,which always tend to be limited in the engineering application.Latter in(Kaplan,1969)[4],the Wiener filter was proposed for linear prediction of ship motion.It was successfully applied to a carrier,obtaining 5～6 seconds of prediction horizon.But the implementation was complicated and the time was consumed.Short term prediction using state-space approaches have been studied in a considerable number of papers.Triantafyllou et al(1981,1982,1983)[5-7]addressed Kalman filtering techniques for the prediction of six-degree-freedom motions using a precise state-space model.Numerical simulation results of DD-963 destroyer show that the prediction precision of Kalman filter greatly depends on the ocean wave frequency,and estimation results of 8-10 seconds advance obtained for the roll while 5 seconds with respect to pitch without noise condition.For the noise condition,prediction horizon can reach 6-8 seconds for roll and 2-3 seconds with respect to pitch.However,Kalman filter is still difficult to be applied to forecast the ship motion in the real world for its two shortcomings.First of all,accurate state-space equations and noise statistics are necessary in implementing Kalman filter which are hard to obtain in the real engineering problems.Besides,tremendous computational efforts required[8]to solve the ship hydrodynamic coefficients for the state-space equations,resulting in difficulty for real time implementation.

Time series analysis is another possible solution accomplishing the short term prediction of ship motions,which only requires the time history of the ship motions or the ocean waves for modeling.Practical limitations of requiring accurate state-space and noise estimation in the Kalman filter and precise response kernel function in the convolution predictor are avoided.Research works involving determination of the model order and corresponding coefficients of AR model,identification techniques for nonlinear AR model,etc have been widely studied by Zhao et al[9-10].

To improve the prediction performance,Yumori(1981)[11]developed a novel auto-regressive moving average(ARMA)model based on leading indicator method using a statistical way that finds a time domain model which best fits an input wave sensor time history to the ship response time history.It showed good predictions of phase and amplitude for 2 to 4 seconds in advance and phase for 8 to 10 seconds in 8 second waves.But satisfactory prediction results are only obtained if it could sense waves at a distance from the ship which is not always available in the real situation.

Among time series forecast models,linear prediction theory is mostly focused for advantages like less computational complexity and memory demands,convenient for real-time realization.But prediction results are far from expected in harsh sea conditions.Furthermore,the real motions of the ship and ocean waves are always non-stationary that conflicts with the stationary assumptions in time series analysis models.

To overcome the nonlinearity and non-stationarity involved in the real-life ship motions, nonlinear theory and artificial intelligent identification methods are employed to short term prediction.Nonlinear linear autoregressive(NAR)model[10]determined by applying orthogonalization gave better prediction precise than AR model. Investigations into the application of artificial neural network methods for short term prediction of ship motion by Khan(2005)[12]suggest that the artificial neural network produces excellent predictions and is able to predict the ship motion satisfactorily for up to 7 seconds.To deal with the chaos characteristic in the ship motion,prediction model based on chaotic time series theory and radial basis function (RBF)artificial neural network(ANN)are implemented for short-term prediction(Gu Min et al,2012)[13].And simulation results show that it is able to predict ship motion acceptably up to 10 seconds.

Though the above nonlinear and intelligent models perform well in data fitting,their applications in real engineering problem are still constrained because of disadvantages such as high computational cost,demanding substantial samples,non-adaptive in model identification and so forth.

Hybrid estimation methods are possible solutions,and attempts are carried out.EMD based prediction models have been extensively researched in substantial papers,and proved to be an effective way to improve prediction performance of various models.Zhou et al[14]invented an empirical mode decomposition method(EMD)based least mean square support vector machines(LSSVM)for the prediction issue.In previous study,Hou et al[15]developed an empirical mode decomposition based radial basis function neural network(EMD-RBFNN)model to deal with the nonlinearity and non-stationarity of the ship swaying motions.However,these hybrid prediction models are still suffering difficulties inherited from the nonlinear and artificial intelligent models,such as requiring large size of samples,non-adaptive nature in determining model structure and parameters,unstable prediction accuracy,etc.,which strictly limit theirfeasible applications in on-line forecast.What is more,the influence of end effects on prediction results has not been focused.

In contrast,inspired by the high adaptive nature of AR model and EMD method,an extended AR model is developed for non-stationary ship motion forecasting by using AR-EMD technique in this paper.Where,AR prediction model is applied in boundary extension to reduce the effects on prediction results in EMD processing.Non-stationarity in ship motion is overcome by decomposing the complex data into a couple of simple IMFs and residual.Each sub-component is predicted individually,and predictions of each component are then reconstructed to attain the final results.Comparative study with linear AR model and nonlinear SVR model employing experimental ship motion data has been conducted.Results suggest that AREMD is effective in handling the negative effect on prediction accuracy resulting from non-stationarity and EMD-AR model produces better prediction compared to AR and SVR models.

The remainder of the paper is organized as follows:Chapter 1 presents methodology formulations of AR model,EMD processing,EMD-AR model and SVR model.Chapter 2 briefly describes the ship motion data and forecast evaluation method.Simulation results and comparative study of AR,EMD-AR and SVR models are given in Chapter 3.Finally in Chapter 4, some concluding remarks on this research work are presented.

1 Methodology formulations

1.1 AR prediction model

The advantages of AR model[16]are 1)Convenient in model identification,2)An AR spectrum can have much better frequency resolution,3)Convenient for realization,especially in the small embedded system where memory and computation complexity are strictly limited. Time series analysis theory considers that relations are existing among variables of the time sequence,therefore,present variable is able to be represented by the previous in time.For a given time seriesthe AR model can be formulated as:

Once the prediction model in Eq.(1)is determined,a k-step-ahead adaptive predictor can be developed as follows:

(1)Parameter estimation

Among various approaches,Least Mean Square(LMS)method,Recursive Least Square (RLS)and Levinson-Durbin(L-D)algorithms are mostly used in the identification of AR model.Compared with LMS algorithm,RLS and L-D algorithms perform faster convergence speeds and get rid of eigenvalue spread problem.As initial condition and forgetting factor affecting the identification results are required for RLS algorithm,therefore,L-D algorithm is adopted.

Levinson-Durbin algorithm is an efficient way to solve the Yule-Walker equations,which is derived from the recursive nature of AR model,is shown as follows:

Considering the general AR model:

Multiply both sides of the model by xt+kand take expectance,yields the Yule-walker equations:

where k=1～p,and rkare the auto-covariance function of xtas shown in Eq.(5).

The Eq.(4)can also be written as:

or

Therefore,

It can be seen in Eq.(8)that the model parameters Φ can be got by directly inverting the auto-correlation matrix R.However,difficulties,such as high complexity in algorithms and expensive computational cost,arise when the model order becomes large.Therefore,more efficient algorithm for Eq.(8)should be addressed.Levinson-Durbin algorithm is a faster,easier and better way to work out Yule-Walker equations.This efficient algorithm can be derived thanks to the Toeplitz structure of the auto-correlation matrix R.The algorithm is first invented by Levinson[17]and independently reformulated at a later date by Durbin[18].

The basic ideas of the Levinson-Durbin recursion are first that it is easy to solve the system for p=1,and second that it is also very simple to solve a p+1 coefficients sized problem when a p coefficients sized problem has already been solved.Summary of the Levinson-Durbin al-gorithm for solving equation system(8)is presented in Tab.1.

Tab.1 Summary of Levinson-Durbin algorithm

(2)Order Selection

One of the major problems in AR modeling is selecting an optimal order.In the past decades,a variety of criteria have been proposed to determine the AR order of specified time series. Although it has been long since being proposed,Aikaike information criterion(AIC)(Akaike, 1974)[19]and Bayesian information criterion(BIC)(Akaike,1979)[20]are still the most popular approaches.Simple and convenient as AIC principle is in order determination,difficulty is still suffered.For even with infinite sample size,the model order determined by AIC principle fails converging to the true order[20].Therefore,BIC criterion is applied for model order selection.BIC value of generalis defined as:

where σ2is the covariance.The model order p0leading to the minimum BIC value is chosen as the optimal order.

1.2 Empirical mode decomposition

Decomposition is a critical part in signal processing.We frequently decompose a complex into several components having simple forms and then analyze the information contained in each component to reduce the complexity and enhance interpretability.Among various decomposition methods,the Fourier transform and wavelet analysis are the mostly adopted.Fourier analysis decomposes a signal into a sum of sinusoid with different frequencies.However,it is well known that for the non-stationary signals,Fourier fails to extract the frequency information from the signals.Although wavelet analysis is able to deal with the non-stationarity when analyzing signals,it suffers from a non-adaptive nature as it applies the same type of basis functions to the entire range of data.Similarly,wavelet analysis also represents a signal by a linear combination of wavelet basis functions.Therefore,its decomposition results for nonlinear data can be misleading.Thus,a set of basic functions that reflects the time-varying property of a signal is required[21].

A data-driven method designated as Empirical mode decomposition(EMD)was proposed by Huang[22],which is powerful and adaptive in analyzing the nonlinear and the non-stationary data sets.It provides an effective approach to decompose a signal into a collection of so-calledintrinsic mode functions(IMFs),which can be treated as empirical basis functions driven by data.IMF results from the EMD procedure should satisfy two conditions:(1)the number of extrema and the number of zero-crossings should be equal or differ by one and(2)the local average should be zero,i.e.the mean of the upper envelope defined by the local maxima and the lower envelope defined by the local minima are zero.

Fig.1 Flow chart of EMD process

Fig.1 shows the framework of EMD processing procedure.For a given sequenceEMD algorithms can be summarized as:

(1)Identify the local extrema.

(2)Generate the upper envelopeand the lower envelopevia the spline interpolation among all the local maxima and the local minima,respectively.Then,then mean envelope is obtained:

(5)Get the n-th IMF component represented asafter n shifting processes and the corresponding residue

(6)Repeat the whole algorithmobtained in step(5)until residue is a monotonic function.

By implementing the presented algorithm,the signal can be decomposed according to the equation:

Inspired by its powerful capability in handing the nonlinear and non-stationary signals, EMD has been popularly applied to analyze all kinds of nonlinear and non-stationary signals in disciplines of science and engineering[23].

1.3 EMD-AR prediction model

(1)Hybridization process of EMD-AR model

Time series of ship motion is a kind of complicated nonlinear and non-stationary signal which consists of some different characteristic sub-components.Therefore,direct prediction results on original signal are always unsatisfactory because of the linear and stationary theoretical limitations of AR model.Another,a single AR model is not reasonable for all different subcomponents.If we decompose original signal into some more stationary IMFs,better results can be obtained when we predict each IMF other than the whole original signal.

Fig.2 The framework of EMD-AR prediction model

The empirical mode decomposition based AR model is designated as EMD-AR model. As shown in Fig.2,the framework of EMD-AR model is presented as follows:1)Decomposing the ship motion time series into a couple of IMFs and a residual by EMD processing.2)Identifying AR model corresponds to each sub-component,and implementing short term prediction on all components.3)Attaining the final prediction results by aggregating the predictions of all the decomposed components.

(2)Boundary processing using AR prediction approach

EMD based prediction models have been extensively researched in substantial papers[24-26]and proved to be an effective way to improve prediction performance of various models.How-ever,the influence of end effects on prediction results is not focused.In this section,an analytical signal is shown in Fig. 3 which consists of three sine functions is deployed as an example to compare the performance of EMD-AR model with conventional symmetric extension method and AR based extension method.

Fig.4 presents the extension results of AR prediction model and conventional symmetric extension method,from which it is seen that AR prediction model provides more reasonable and satisfactory extension results.And exterma produced by conventional symmetric extension method fail to match the true ones. Results in Fig.5 further highlight that considerable reduction of end effect on prediction accuracy has been reached by employing AR prediction method in boundary processing instead of conventional symmetric extension method.

Fig.3 Analytical signal obtained through the superposition of three sine function

Fig.4 Boundary extension results using AR model and symmetric extension approach

Fig.5 Prediction results of EMD-AR models with different extension methods

1.4 SVR prediction model

Support Vector Machine(SVM)is a statistical learning theory based method with strong capacity to handle nonlinear problems.Its basic idea is to map the nonlinear data into highdimension feature space using a nonlinear mapping function,where linear techniques are available[27].

Given a training data set of N pointswith input dataand output data yi∈R.According to the SVM theory,the input space RNis mapped into a feature space Z withbeing the corresponding mapping function.In the feature space,regression function form(11)designated as the support vector regression(SVR)model is applied to estimate the unknown nonlinear function,and w and b are the parameters to be selected.

To minimize the empirical risk and overall fitting errors,the regression problem should be equal to the optimization problem as follows:

where C is a regularization constant.

To solve the optimization problem,Lagrangian function L is introduced:

Then,according to the Karush-Kuhn-Tucker(KKT)conditions,equations are obtained,

Solving linear equation system(13)obtains the parameters of b and a,then the regression is rewritten as

2 Brief descriptions of ship motion data and forecast evaluation

2.1 Ship motion data and analysis

(1)Ship motion data by model testing

Model testing based on motion time series of a large container ship moving at a speed of 24 knots in heading waves with the condition of sea state 5 is used in prediction simulation, which are shown in Fig.6.Ship motion status(heave,pitch)are sampled at 50 Hz rate in model testing and down-sampled to 2 points per second when simulation implemented.

Fig.6 Time history of a large container ship in model testing

(2)Analysis of the non-stationarity in ship motion data

According to conventional definition,a time series,is stationary in all wide sense, if,for all t,

It is known from the stationary definition in Eq.(16)that the covariance and the mean of a stationary process do not vary with time.In other words,they are identical to time variable t.Therefore,quantitative methods of consecutive statistics including the covariance and mean are used to analyze the non-stationary nature of the ship motions shown in Fig.6.Fig.7 presents the covariance and mean of pitch and heave time series,respectively.From which the time-varying nature of the covariance and mean are noticed,indicating that the ship motion is non-stationary.

Fig.7 Covariance and mean of ship motion time sequence

To further confirm the non-stationarity in ship motion data,recurrence plots of pitch and heave motion are studied.For stationary process,the corresponding recurrence plot latticesshould be subjected to uniform distribution,otherwise,the recurrence plot must be non-uniform distributed.Recurrence plots respecting to pitch and heave motion have been shown in Fig.8,where the non-uniform distribution nature is easily to be found,further demonstrating the existence of non-stationarity in ship motion.Therefore,it may be concluded that the data given in the model testing ship motion for prediction simulation study is non-stationary.

Fig.8 Recurrence plots of the model testing ship motions

2.2 Forecast evaluation approaches

For the purpose of forecasting performance evaluation,prediction results are studied by (I)Comparing time histories of above models’forecasts with actual ship motions,(II)Computing the correlation coefficientand the root mean square error(RMSE)as shown in Eqs. (17)and(18),and(III)Drawing scatter diagrams of prediction results and corresponding measured ship motions.

3 Simulation results and discussion

To implement the above prediction models,a sliding data window with the size of 500 points is used for modeling and 1 000 points are employed to test.In order to notice how AR, SVR and EMD-AR prediction model perform in various lead times forecasts,short-term prediction of heave and pitch motion with lead times varying from 1 second to 10 seconds areimplemented using the three prediction models.

Figs.9 and 10 present 5 second ahead predicted time histories using the above three models.Tab.2 shows the values of all error measures in pitch prediction by AR,SVR and EMDAR models with respects to various lead times.And the same of error measures in heave prediction are given in Tab.3.To directly conduct comparison between the three prediction models and better understanding of the evolutionary process of error measures respect to lead times,RMSE and r are plotted as shown in Figs.11 and 12.

Fig.9 Predicted time history of pitch motion 5 seconds ahead

Fig.10 Predicted time history of heave motion of 5 seconds ahead

Observations can be noticed from the results given in Figs.9 and 10.First of all,from the comparisons among the time histories,it is proved that the proposed EMD-AR model matches best with the true ship motion in both amplitude and phase tracking.Second,results predicted by AR and EMD-AR model suggest that EMD technique is an effective way to improve the prediction performance of AR model,especially for non-stationary ship motion.Third,SVR model suffers non-stationary difficulty,which leads to failure of prediction results to match the true ship motion.

Fig.11 Error measures comparison regarding to various lead times of pitch prediction

Fig.12 Error measures comparison regarding to various lead times of heave prediction

The observations have been further confirmed by the error measures with a lead time of 5 seconds.As shown in Tab.2,the RMSE values of AR,SVR and EMD-AR models are 0.09, 0.1 and 0.08 degrees,while the same of correlation coefficients r are 0.9,0.91 and 0.92,respectively.The lowest RMSE value and highest r value suggest EMD-AR’s being the superior model in predicting non-stationary pitch motion.Larger improvements in heave prediction by using EMD-AR model show also the superiority of EMD-AR model in a further sense. From Tab.3,it may be found that the RMSE values of EMD-AR model is 0.1 m,80%smaller than RMSE values of AR and SVR models.And the correlation coefficient r value of EMD-AR model is 0.93,while the same of AR and SVR models are only 0.75 and 0.8.

Tab.2 Error measures comparison of pitch prediction

Continue Tab.2

Tab.3 Error measures comparison of heave prediction

Results shown in Figs.11 and 12 reflect the trend of the error measures regarding forecast lead times.Where,it can be found that the increase of forecast lead time leads to growth in the RMSE values and decrease in r values.Furthermore,the results also conduct confirmation that EMD-AR model’s performing better in non-stationary short-term prediction comparing to AR and SVR models,showing high agreement with discussions on the predicted time histories given in Figs.9 and 10.Under the forecast error condition that correlation coefficient value r is larger than 85%,the valid forecast lead times can reach to 10 seconds for pitch and 9 seconds for heave,while the same of AR model is 8 seconds for pitch and 3 seconds for heave, and for SVR model they are 7 seconds in pitch prediction and 4 seconds in heave prediction.

However,it should be also noticed that SVR model produces higher accuracy than EMDAR model when forecast lead times less than 4 seconds.Once the lead time exceeds 4 seconds, the prediction accuracy of SVR model decreases sharply,which leads to poor predictions compared to EMD-AR model.Therefore,two useful conclusions can be drawn from the results of SVR model.First of all,although nonlinear SVR model can provide satisfied forecast while the lead time is small,the prediction accuracy becomes unstable if lead time grows.Second, the degree of non-stationarity affecting the prediction results depends on forecast lead times, the larger lead time is,the worse influence on prediction results will be.

4 Conclusions

In this paper,a extended AR model for non-stationary ship motions’short-term forecastis developed by using AR-EMD technique.The model designated as EMD-AR overcomes the non-stationarity in ship motion by decomposing the complex data into a couple of simple IMFs and residual.Each sub-component is predicted individually,and predictions are then aggregated to attain the final results.Comparative analysis with linear AR model and nonlinear SVR model using experimental ship motion data highlights the superiority of EMD-AR model.Results suggest the non-stationarity’s negative effects on prediction performance of both linear AR and nonlinear SVR prediction models,and AR-EMD technique is an effective way to handle the difficulty as it provides better forecasting than the two comparative models.

Acknowledgement

This work was financially supported by the project of Natural Nature Science Foundation of China(No.51079032).

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Aladdin ELHANDAD（1980-），男，哈爾濱工程大學(xué)船舶工程學(xué)院博士研究生。

基于AR-EMD方法的擴(kuò)展非平穩(wěn)船舶運(yùn)動(dòng)極短期預(yù)報(bào)AR模型

黃禮敏，段文洋，韓 陽(yáng)，余冬華，Aladdin ELHANDAD

（哈爾濱工程大學(xué) 船舶工程學(xué)院，哈爾濱 150001）

準(zhǔn)確的極短期預(yù)報(bào)技術(shù)能夠提高對(duì)船舶搖蕩運(yùn)動(dòng)敏感的海洋特種作業(yè)安全性和效率。自回歸（auto-regressive，AR）預(yù)報(bào)模型由于其自適應(yīng)性強(qiáng)、計(jì)算效率高而被廣泛應(yīng)用于船舶運(yùn)動(dòng)的極短期預(yù)報(bào)研究。但該模型基于平穩(wěn)隨機(jī)假設(shè)，因而在非平穩(wěn)船舶運(yùn)動(dòng)的極短期預(yù)報(bào)中存在困難。針對(duì)非平穩(wěn)船舶運(yùn)動(dòng)極短期預(yù)報(bào)，文章提出一種基于AR-EMD方法的擴(kuò)展AR模型，稱為EMD-AR預(yù)報(bào)模型。其中，AR-EMD方法是指在經(jīng)驗(yàn)?zāi)B(tài)分解（empirical mode decomposition，EMD）的過(guò)程中，采用AR預(yù)報(bào)的方法處理端點(diǎn)效應(yīng)問(wèn)題。EMD-AR預(yù)報(bào)模型將非平穩(wěn)信號(hào)分解成若干平穩(wěn)的固有模態(tài)函數(shù)分量及余項(xiàng)，然后對(duì)各個(gè)分量分別用AR模型預(yù)報(bào)，得到最終的預(yù)報(bào)結(jié)果，以此克服非平穩(wěn)性對(duì)AR預(yù)報(bào)模型的影響。研究基于船舶試驗(yàn)數(shù)據(jù)將EMD-AR模型與線性AR模型、非線性支持向量機(jī)回歸（support vector regression，SVR）預(yù)報(bào)模型進(jìn)行對(duì)比分析，結(jié)果表明，AR-EMD方法能夠有效處理船舶運(yùn)動(dòng)非平穩(wěn)性對(duì)AR預(yù)報(bào)模型的影響，提高該模型的預(yù)報(bào)精度，且EMD-AR模型預(yù)報(bào)性能較線性AR模型和非線性SVR模型更優(yōu)。

非平穩(wěn)船舶運(yùn)動(dòng)；極短期預(yù)報(bào)；AR模型；經(jīng)驗(yàn)?zāi)B(tài)分解；EMD-AR模型

U661.32

:A

黃禮敏（1988-），男，哈爾濱工程大學(xué)船舶工程學(xué)院博士研究生；

U661.32

A

10.3969/j.issn.1007-7294.2015.09.002

1007-7294（2015）09-1033-17

段文洋（1967-），男，哈爾濱工程大學(xué)船舶工程學(xué)院長(zhǎng)江學(xué)者特聘教授，博士生導(dǎo)師；

韓 陽(yáng)（1988-），女，哈爾濱工程大學(xué)船舶工程學(xué)院博士研究生；

余冬華（1988-），男，哈爾濱工程大學(xué)船舶工程學(xué)院碩士研究生；

Received date：2015-05-05

Foundation item:Supported by the National Nature Science Foundation of China(No.51079032)

Biography：HUANG Li-min(1988-),male,Ph.D.candidate of Harbin Engineering University,E-mail: huanglimin@hrbeu.edu.cn;

DUAN Wen-yang(1967-),male,professor/tutor of Harbin Engineering University,E-mail: duanwenyang@hrbeu.edu.cn.