Application of Multi-level Recursive Method for Ship-Sway prediction

The ship-sway is a non-stationary time series while ship sailing in the sea , when using the traditional orthogonal polynomial fitting,index filtering or Kalman filtering to predict ship-sway, the prediction error may be large, because the model parameters are fixed and cannot be adjusted in real time according to the measured data. Multi-level recursive method treats the dynamic system as a time-varying parameters of the system, and can be more in line with the objective reality of rocking the ship. After analyzing the characteristics of the ship-sway, the predict model established by multi-level recursive has been built and test results showed that this model can improve the prediction accuracy of the ship-sway data, and has some practical value in the prediction of the ship-sway.


Introduction
In the tasks of maritime spacecraft tracking and control, the prediction of the measure base directly influences the data accuracy of radar while tracking the target in the way of following number-leading calculated by computer.At present, the ship-sway is the most important factor in the measure base, and the common ship-sway prediction methods include statistics-predict, Kalman filter-predict, orthogonal polynomial fitting-predict and maximum likelihood estimation-predict [1] .In reality application, some weakness of these methods is found, just like needing too much prior condition, too difficult to build the accurate ship-sway model, and too big predict deviation etc.Using multi-level recursive method, we can treat the ship-sway data as a random dynamic time varying system, further, separate the system predict to two steps, predict the dynamic time varying parameters and system status based on ahead parameters.By this way, the dynamic parameters will update real-time while the ship-sway make a sudden change, and the system prediction can have a good adaptability.

Principle of Multi-level Recursive
The Multi-level Recursive treats a dynamic system as a stationary time series composed by one or more dimensional.By analysis the external characteristic of the system, establish the connection between inputs and outputs.Under the general circumstance, all systems can be simplified to one model includes one or more one-dimensional model [2] .Here, we suppose the input of a stationary time series is: Inside the above expression, ( ) y k is the observed value in the time of k , so the one-step prediction is: is the one-step prediction. Suppose: ( ) Inside the above expression, 0 1 r c c c are dynamic time varying parameters If the parameters are time varying, we can establish third, fourth or more further expression [3] , else the whole expressions make up the predict progress.

Parameters Update
According to the expression (1) (2) and reference documentation [4], the common expression of a predict progress is: In the real-time prediction of the ship-sway by the multi-level recursive method, the factor k U can be ignored, only analyses the ship-sway data series.So, the estimate of parameters θ is: values δ is adjust factor.
According to reference documentation [5], generally speaking, the different initial values of the parameters will not influence  4), the l steps prediction of y can be calculated.

Points of Multi-level Recursive Method
According to reference documentations [6][7] [8], whatever the model points selected, the predict precision will satisfy requirement by parameters stochastic makeup, But just like the model parameters initial values, the most suitable model points can shorten the predict time.So, Final Prediction Error criterion is used to define the most suitable model points: Inside the above expression, k is number of the model independent parameters, n is the number of the samples, 2 e δ is the error mean square deviation of the model.

Multi-level Recursive Model for Ship-sway
Using the method from reference documentation [9], we can simulate 1000 points ship-sway samples, after difference of first order, the new series is a stationary time series, and multi-level recursive model is built for predicting it.Here Eviews soft is used to get the model points and parameters initial values.Select Final Prediction Error criterion to have a unit-root test for the series and set the biggest lag order is 21 [10] , the result shows figure 1.
Though the orders D(-1) D(-14) D(-15) have no conspicuousness, considering the randomness, the model points is defined 16.Using least squares method to estimate initial values of the parameters, the result shows figure 2. And meanwhile, the predict model expression is: According to expression (5), estimate and update parameters i a expressions are: ( 1) ( ) ( ) Predict Steps: Step1: According to the expressions (8) and the initial values of (0) ~(16) a a , using the before 30 points samples, each parameter will form a series; Step2: For each parameter series, establish a AR(3) model, and the one step prediction of the parameter can be get;  Compute the maximum,mean and variance of the deviation,and compare with the results from reference documentation [10], the result shows table1.

Table1 Forecast accuracy deviation comparison
For verify when varying parameters dynamically adjusting benefits, when the ship rolling simulation data, select the data points between 400-500 increases a trend item (endpoint regression), using the same method as described above and spreadsheet modeling, comparative prediction deviation maximum, mean and variance, the results shown in table 2.

Table2 Forecast accuracy deviation transition sequence comparison
From the spreadsheet results, when the status of the observed series change suddenly,the multi-level recursive method advantage is more obvious, with a faster adaptability and higher prediction accuracy

Conclusion
Compared with traditional fixed parameter prediction method, multi-level recursive method forecast by layer by layer when varying parameters, dynamic adjustment of the estimated parameters in the prediction process, the study can be closer to the actual situation, a more sensitive and state recognition ability to adjust the prediction accuracy has also been greatly improved, with high practical value and popularization meaning.

Fig1.
Fig1.Unit-Root Test Result the predicted result of the parameters, the progress for predicting the ship-sway has been shown by the expression (7);Step4: Compute the deviation between the samples and the prediction with the same time,figure them and the result shows figure3.

Fig3.
Fig3.Dynamic curve fitting step prediction and deviation results