The Offset Position Prediction of Light Buoy Based on Grey Optimization Multiplicative Seasonal Model

Wu, Zhizheng; Ren, Lüzhen; Zhou, Shibo; Wen, Jiandong; Xu, Wenpeng; Zhang, Xinliang

doi:https://doi.org/10.1155/2022/5198315

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5198315 | https://doi.org/10.1155/2022/5198315

The Offset Position Prediction of Light Buoy Based on Grey Optimization Multiplicative Seasonal Model

Zhizheng Wu,¹Lüzhen Ren,¹Shibo Zhou,¹Jiandong Wen,¹Wenpeng Xu,¹and Xinliang Zhang¹

Academic Editor: Pengwei Wang

Received28 Dec 2021

Revised14 Apr 2022

Accepted22 Apr 2022

Published06 May 2022

Abstract

Light buoy is a navigation aid sign to guide ship navigation, which plays an important role in ensuring the safety of ship navigation. To predict the offset distance of light buoy and provide accurate position information of light buoy for ship navigation safety, and to solve the problem of low prediction accuracy caused by the excessive amplitude of initial training data in the traditional multiplicative seasonal model, a grey optimization multiplicative seasonal model is proposed. By analyzing the characteristics of the time series of the light buoy offset data, as well as the regularity and seasonal characteristics of the light buoy offset distance, an optimization model is established. Based on the real offset distance data of the 1 # light buoy in Meizhou Bay, the model is simulated, and the prediction accuracy of the model is evaluated by MAPE, RMSE, and the Monte Carlo sampling method. The results show that, compared to the traditional multiplicative seasonal model, the average absolute percentage error of the grey optimization multiplicative seasonal model is reduced by 6.66%, and the root mean square error is reduced by 2.96. It shows that the model can effectively deal with the problem that the error of the prediction results caused by the instability of the initial time series of the traditional multiplicative seasonal model is too large, which provides a new idea for the prediction of the offset distance of the light buoy and the navigation safety of the ship.

1. Introduction

The main function of the light buoy is to mark the range of waterways, indicates shoals or obstructions, and is also a manual sign to guide ship navigation [1]. The design position, the throwing position, and the position observed by the light buoy navigator are not consistent. How to make the position observed by the navigator and the design position as consistent as possible and predict the offset law of the light buoy according to the historical position data of the light buoy are the goals and responsibilities of the beacon maintenance unit [2]. Meizhou Bay Port, located on the west coast of the Taiwan Strait, is a famous natural deep-water port in China. However, its traffic flow on the ship is large and the channel flow is complex, so the position of the accuracy of the light buoy is required when the ship is sailing. To provide accurate light buoy position information to the ship and study the law of light buoy offset, this paper uses the grey optimization multiplicative seasonal model to predict the offset position of the light buoy based on the offset distance data of the Meizhou Bay No. 1 light buoy and compares it with the actual position data to analyze the accuracy of the prediction results, to provide a scientific theoretical basis for future light buoy position prediction.

Position prediction refers to the establishment of a complete model by obtaining the position data of the mobile terminal and then using the relevant mathematical theory and conversion formula to analyze and experiment with the existing data, to predict the future position data. Position prediction is widely used in intelligent transportation, physical science, resident travel, natural disasters, and other fields [3–5]. For example, Tian [6] proposed a new prediction method for short-term wind speed based on local mean decomposition (LMD) and combined kernel function least squares support vector machine (LSSVM); the results show that the proposed prediction method has higher prediction accuracy and can reflect the laws of wind speed correctly. Qiao et al. [7] used the Gaussian mixture model to predict the trajectory of uncertainty in the position of moving objects more accurately. The method has been successfully applied in the field of intelligent traffic control, auxiliary driving, and military digitization. Li et al. [8] used an artificial neural network to predict the position of people's travel and analyzed the gathering place of people. The research results show that the model can effectively predict the position of people's travel. Ma et al. [9] used the Gaussian Markov mobility model to predict the position of the vehicle at the next moment given the problem that the vehicle terminal with variable speed mobility will face more frequent switching, resulting in poor service quality of users. From the results, the model effectively improves the service quality of users. Hou et al. [10] applied the dynamic Bayesian network to the prediction of the position of the user based on whether the mobile user will leave the living area and where it will go, and the prediction result is better. Jierula et al. [11] used an artificial neural network to predict the position of damage and used seven different accuracy indicators to evaluate the prediction accuracy of the six algorithms. This algorithm can be widely used in engineering prediction. To solve the problems of large loss of trajectory information and low prediction accuracy of existing prediction methods in vehicle position prediction of the intelligent transportation systems, Xiao et al. [12] proposed a vehicle position prediction algorithm based on spatial feature transformation method and hybrid long short-range memory (LSTM) neural network model. The algorithm is better than other models in vehicle position prediction. To predict the mobility of people, Huang et al. [13] proposed an LP-HMM model by combining user position preference with a hidden Markov model. The prediction accuracy of this model is 6.4% and 7% higher than that of the Gaussian mixture model and the traditional HMM model. Tian et al. [14] proposed a prediction approach for short-term wind speed using ensemble empirical mode decomposition-permutation entropy and regularized extreme learning machine; the results show that the prediction approach in this paper has higher reliability under the same confidence level. Wu et al. [15] proposed the MC-SARIMA prediction model; the experimental results show that the prediction error of this model is low.

The above-related studies have achieved good application results by using different methods to predict the trajectory and position of mobile terminals or personnel, but there are also some shortcomings. For example, the Gaussian mixture model is the fastest in the mixed model, but if the data node is insufficient, it will make the covariance difficult to estimate and require high data integrity. The Markov model can well predict the process state, but if the prediction period is long, it is not suitable to apply this model. The artificial neural network can build a nonlinear model with complex relationships, which can be very close to various complex relationships in daily life. However, there are also some shortcomings, such as the nature of the black box, the time required for development being too long, and the cost of calculation being too high. The Bayesian model has relatively stable classification efficiency and is not very sensitive to missing data. However, the Bayesian model needs to know the prior probability in advance, and in most cases, the prior probability is taken from the assumption. Therefore, in the case of the assumed prior model, it is easy to lead to poor prediction accuracy and has high requirements for the form of input data. Based on the above factors, dealing with the initial data accurately, dealing with various external factors, and accurately predicting the target over a long-time span are an important research direction for position prediction.

The multiplicative seasonal model refers to the use of a specific mathematical model to describe the trend between a set of random variables related to time. It can comprehensively consider components such as season, trend, and random interference and can predict the future value of time series. However, the prediction of time series data often appears high and low or even seriously deviates from the actual situation and requires a large amount of training data for mathematical fitting to be able to effectively predict the future value and cannot be predicted for a long time [16]. The training data used in the grey (1, 1) model are not the original time series data, but the training data generated by a series of mathematical analyses. The model does not need a large number of training data to fit, which can reduce the dependence on the training data, and the operation is simple. It has the advantage of not considering the distribution law and the trend of change [17]. As the navigation aid sign of ships, the light buoy plays an important role in the safe navigation of ships. Whether the offset position of the light buoy can be accurately predicted has a great influence on the safe navigation of the ship and the working efficiency of the light buoy offset reset department. In some sudden conditions, such as typhoons, ship collisions, and other special factors, it is easy to cause the offset distance of the light buoy to be abnormal.

Therefore, by analyzing the historical data of the offset distance of the light buoy, this paper proposes a grey optimization multiplicative seasonal model, to deal with the problem that the deviation of the historical data of the offset distance of the light buoy is too large and reduce the prediction error of the offset distance of the light buoy so that the position of the light buoy can be adjusted more accurately in the process of adjusting the position of the light buoy, and the navigation aid efficiency of the light buoy can be improved, to reduce the risk of collision between the ship and the light buoy in navigation and improve the safety of ship navigation.

2. Multiplicative Seasonal Model

The multiplicative seasonal model refers to the multiple of the differential autoregressive moving average model and the random seasonal model. It is an important method to deal with time series with seasonality and trend. The general form is SARIMA (p, d, q) × (P, D, Q, S). Among them, the parameters p, d, q, P, D, Q, and S represent the nonseasonal regression order, the moving average order, the nonseasonal moving average order, the seasonal autoregressive order, the seasonal difference order, the seasonal average moving order, and the unit cycle, respectively. The general form is shown in the following formula:where .

2.1. ARIMA Model

The ARIMA [18] model is a time series prediction method proposed by Jenkins and Box. It is used mainly to study time series with periodicity, trend, and seasonality. The ARIMA model (p, d, q) is made up of the autoregressive model AR (p) and the moving average model MA (q). The derivation formula is shown in the following equation:where is the delay operator; is the zero-mean white noise sequence; is the time series; is a polynomial of self-regression coefficients; is a moving average coefficient polynomial; , are the corresponding coefficients, respectively.

After the integration of AR (p) and MA (q) and the d-order difference of their trends, the ARIMA (p, d, q) model is obtained. See the following formula:where is the difference operator; is the trend difference; .

2.2. Random Seasonal Model

The random seasonal model [19] ARIMA (P, D, Q, S) is the time series with only periodicity and seasonality obtained by the integration of the seasonal autoregressive model AR (P) and the seasonal moving average model MA (Q) through seasonal periodic difference. The general forms of AR (P) and MA (Q) are shown in where is a polynomial of seasonal autoregressive coefficients; is the seasonal moving average coefficient polynomial; and are the corresponding coefficients, respectively.

The SARIMA (P, D, Q S) model is obtained by integrating AR (P) and MA (Q) and performing S and D order difference processing on their seasonal and periodic parts, respectively. The general expression sees the following formula:where is the periodic difference; is the seasonal difference.l

3. GM (1, 1) Model

The grey model (GM) [20] refers to turning a system with hierarchical and fuzzy institutional relations and uncertainty of target data into a grey system and establishing a grey prediction model to describe the fuzziness of the development of target things for a long time, to achieve the prediction of the development of things within the system. The grey model has the properties of differential, differential, and exponential compatibility. The differential and differential equations are established according to the time series of the target so that the time series with exponential properties are mathematically fitted to predict future data. In the grey model, the grey model of the first order and one variable is abbreviated as GM (1, 1), and the steps are as follows. Step 1: Let the variable be the initial training time series, be the first-order cumulative sequence, and be the adjacent mean equal weight sequence, as shown in the following equation: where ; . Step 2: Let be a parameter vector, and according to the least square criterion method, is derived. The derivation formulas of B and y are shown in According to equation (8), the whitening formula of the grey differential equation is derived, where a is the development coefficient and b is the grey action. The specific formula is shown in equation (9). Step 3: After the whitening formula of grey differential equation is obtained, parameter b is calculated, and the time corresponding function of whitening differential equation is obtained by bringing parameter a and parameter b into the following formula: Cumulative reduction prediction formula is

4. Grey Optimization Multiplicative Seasonal Model

Due to uncertainty factors such as the instability of the initial time series data in the single multiplicative seasonal model, the deviation between the predicted and the actual data is too large in the model fitting process. Therefore, this paper proposes a grey optimization multiplicative seasonal model to reduce the excessive error caused by the instability of the data in the multiplicative seasonal model fitting process. The multiplicative seasonal model of grey optimization is based on the average relative error of the theoretical value obtained by the grey model and the multiplicative seasonal model, and the theoretical value is mathematically fitted according to the weight, to improve the accuracy of the target prediction value. Firstly, to meet the requirements of the model for the initial time series data, it is necessary to test and process the time series data first, and then whether the optimal parameters can be selected is also an important condition for reducing the prediction error. After obtaining the relevant theoretical values needed by the model and calculating the average relative error, the appropriate weight of each theoretical value is given, and finally, the target prediction value of the grey optimization multiplicative seasonal model is obtained. The steps are as follows.

Step 1. Missing data supplement. Due to the influence of natural factors, human factors, and the stability of the light buoy telemetry remote control system, the light buoy telemetry data often appears missing and redundant. Therefore, the collected light buoy telemetry data need to be interpolated to make the data continuous.

Step 2. Data detection. The stationarity of the data is an important prerequisite for the use of the multiplicative seasonal model. The augmented dickey-fuller test (ADF) [21] can be used to test the stationarity of the data and white noise. If the data is not stable, it needs to be differentially processed.

Step 3. Data process. The difference is an important method to deal with the stability of time series. After difference processing, it is necessary to calculate the first-order cumulative sequence and adjacent mean equal weight sequence.

Step 4. Parameter acquisition. The Akaike information criterion (AIC) [22] is used to select the optimal parameters p, d, q, P, D, Q and then according to formula (7)–(9) to calculate the parameters B, y, a, b.

Step 5. Acquisition of theoretical value. The parameters are entered into the multiplicative seasonal model and the GM (1, 1) model, and the theoretical values of the multiplicative seasonal model and GM (1, 1) are obtained.

Step 6. Average relative error of calculation. Relative error [23] refers to the absolute error multiplied by 100% of the actual value, which can accurately reflect the credibility of the measurement; see the following formula:where x is the theoretical value; is the actual value; e is the average relative error.

Step 7. Weight assignment. Weight refers to the degree of importance of a factor relative to something; its meaning is different from proportion; the meaning expressed is not only the percentage of a factor to something; it is more focused on the importance of a factor to something; see weight formula:where and are expressed as weights of GM (1, 1) model and multiplicative seasonal model, respectively; e₁ and e₂ are the average relative errors of predicted values of the GM (1, 1) model and multiplicative seasonal model, respectively. is the predict value; is the sequence of theoretical values.

5. Case Analysis

In this paper, the data of Meizhou Bay 1 # light buoy from 1:00 on April 2, 2019, to 12:00 on April 6, 2019, for 108 consecutive hours were used as training data, and the data from 13:00 to 24:00 on April 6, 2019, for 12 consecutive hours were used as test data. Data are merged into the multiplicative seasonal model and the GM (1, 1) model, respectively. The theoretical value data of 1:00 to 12:00 on April 6, 2019, are obtained and the average relative error between the theoretical value of GM (1, 1) and the actual value of multiplicative seasonal model is calculated. According to equations (13) and (14), the offset distance of the light buoy from 13:00 to 24:00 on April 6, 2019, was predicted.

5.1. Missing Data Supplement

When we only know the position of the observation data in some nodes but do not know the specific expression of the data, we can use the algebraic interpolation method to give the approximate form of the function. As a commonly used numerical fitting method, Newton interpolation is widely used in the experimental analysis because it is convenient to calculate a large number of interpolation points. The time interval of light buoy telemetry data points is about 1 h discrete data points, which is suitable for using the Newton interpolation method to supplement missing data; the processed data are shown in Table 1.

5.2. Offset Distance

Due to seasonal cycles, tidal fluctuations, water waves, and other factors, the light buoy will have a certain offset with the position of the sunken stone of the light buoy. Therefore, the distance between the actual position of the light buoy and the position of the sunken stone is called the offset distance. The caisson position of the buoy is (119.046°E, 24.915°N). The calculation formula is shown in equation (15), and some calculation results are shown in Table 1. (J₁, W₁) are the light buoy position coordinates (latitude and longitude); (J₀, W₀) are the coordinates of the sunken rock position of the light buoy (longitude and latitude); L is offset distance; 6371 is the average radius of the Earth.

5.3. Visual and Statistical Feature Analysis of Data

The grey optimization multiplicative seasonal model can deal with the problem of large deviation of initial time series data. Therefore, this paper selects the data from April 2, 2019, to April 5, 2019, for analysis. The data trend figure is shown in Figure 1.

It can be seen from Figure 1 that, from 13:00 on April 4 to 24:00 on April 5, the peak value of the initial training data showed a continuous downward trend, and the average value on April 2 to April 4 was 56.357115, and the average value on April 5 was 43.90706918. The difference between the average value on April 5 and the average value on April 2 to April 5 was greater than 20%, which can prove that the data had the problem of large deviation.

5.4. Data Detection and Process

5.4.1. First-Order Cumulative Sequence and Adjacent Mean Equal Weight Sequence

The first-order cumulative sequence is gradually accumulated by the initial time series, and the adjacent mean equal weight sequence is expressed as the sum of the corresponding values of each value starting from the second value and the previous value. The calculation formula is shown in equation (6). Because all the data is too large, only part of the data is shown in Table 2.

5.4.2. Stationarity and White Noise Test

To meet the requirements of the multiplicative seasonal model for the stability of the initial time series and determine whether it is a white noise sequence, it is necessary to carry out differential processing, and the time series is tested to detect whether the time series can meet the stability requirements after the differential processing. Establish the trend graph and the first-order difference graph of the initial data of the offset distance; see Figures 2 and 3.

It can be seen from Figure 2 that the data are not stable, so it is necessary to perform differential processing. It can be seen from Figure 3 that, after the first-order difference treatment, the stability of the model has been achieved, but for the accuracy of the data, ADF needs to be carried out. If the stability requirement cannot be met, the second-order difference treatment is carried out. ADF is one of the important methods to detect whether the time series is a stationary sequence and a white noise sequence. The test results are shown in Table 3.

Table 3 shows that the offset distance ADF test statistics are less than the corresponding test level of 1%, 5%, 10% of the critical value and the probability , to meet the requirements of the test observations, so the offset distance time series is a stationary sequence and a nonwhite noise sequence.

5.5. Parameter Selection

After the first-order difference of the migration distance time series, the parameter d = 1 can be determined when the trend of the time series disappears. After the first-order difference of the migration distance time series, its seasonal disappearance can determine the parameter D = 1. The values of the parameter p and the parameter q can be determined by combining the Autocorrelation Function (ACF) [24] and the Partial Autocorrelation Function (PACF), and the general range of parameters p and q is [0, 2]. The ACF and PACF analysis is shown in Figures 4 and 5.

After analyzing ACF and PACF, it can be seen from the figure that the offset distance time series has no obvious tailing or truncation. Therefore, it is necessary to separate the time series into seasonal trend, random fluctuation trend, and growth trend. See Figure 6.

It can be seen from Figure 6 that there are two peaks every 24 hours, so the time series of offset distance is obviously affected by periodicity and seasonality. Therefore, it can be determined that the period of the time series is S = 24, and there is no obvious rule between the random fluctuation trend and the growth trend. After getting the parameters d = 1, D = 1, S = 24, to optimize the combination of various parameters, the Akaike information criterion (AIC) is used to select the optimal parameters. The smaller the AIC value, the better the model. The AIC calculation results of the time series are shown in Table 4.

Table 4 shows that the minimum AIC value is 297.2971083, so we can determine the parameters of the model p = 0, q = 1, P = 0, Q = 1. Therefore, the combination of model should select SARIMA (0, 1, 1) × (0, 1, 1, 24) as the optimal multiplicative seasonal model.

After selecting the optimal parameters p, d, q, P, D, Q, and S, the values of B and y can be calculated according to formula (7). The results of the calculation are shown below.

After calculating the values of B and y, the development coefficient a = 0.00271603 and the grey action b = 61.958295-52 are calculated according to and formulas (8) and (9).

5.6. Acquisition of Theoretical Value

According to SARIMA (0, 1, 1) × (0, 1, 1, 24), the time series from 1:00 on 2 April 2019 to 24:00 on 5 April 2019 are fitted mathematically, and the theoretical value of the multiplicative seasonal model from 1:00 to 12:00 on 6 April 2019 is obtained. Bring the parameters a and b into (5), and then the time corresponding function of the whitening differential equation is obtained. Then according to equation (11), the theoretical value of GM (1, 1) is obtained. Some specific values are shown in Table 5.

5.7. Prediction Optimization and Error Test

After calculating the theoretical values of the multiplicative seasonal model and GM (1, 1) model from 1:00 to 12:00 on April 6, 2019, the average relative error between the theoretical values of the multiplicative seasonal model and GM (1, 1) model is calculated according to formula (13), where = 15.7% and = 6.07%. According to equation (14), the two theoretical values are weighted to obtain = 28.49% and = 71.54%. According to equation (15), the offset distance of the light buoy from 13:00 to 24:00 on April 6, 2019, is predicted. The prediction results are shown in Figure 7.

Error test is one of the important conditions to evaluate whether the prediction results meet the standard. In this paper, Mean Absolute Error (MAE), the Sum of Squares due to Error (SSE), and the R-Square (R²), the mean absolute percentage error (MAPE), the root mean square error (RMSE), and the Relative Root Mean-Squared Error (RRMSE) [25, 26] are used to test the error of grey optimization multiplicative seasonal model and multiplicative seasonal model, grey model, and ARIMA model. The calculation formula is shown in formulas (16)–(21).where x is actual value; is prediction value; is the average of actual values.

It can be seen from Figure 8 that the prediction MAPE of the offset distance of the grey model is 6.26%, and the RMSE is 3.02, MAE is 1.089320658, SSE is 27.50957492, R² is 0.898578, nd RRMSE is 0.130145678. MAPE and RMSE, MAE, SSE, R², and RRMSE of the SARIMA model are 9.27% and 4.47, 3.901110212, 240.298901, 0.2804, and 0.21758, respectively. The MAPE and RMSE of the GM-SARIMA model were decreased by 3.65% and 1.50, respectively, compared to the grey prediction model. Compared to the SARIMA prediction model, the MAPE and RMSE, MAE, SSE, and RRMSE of the GM-SARIMA prediction model are reduced by 6.66% and 2.96, 2.811789554, 212.7893261, and 0.249110681, respectively, R² is increased by 0.61818.

(a)

(b)

(c)

(d)

(e)

(f)

To further verify the advantages of the prediction model, the Monte Carlo sampling method [27] is used to evaluate the prediction model. The size of each data block in the iteration is set to 75, the number of iterations is set to 12, the fval is set to 0, and the figs is set to 0. The simulation results are shown in Table 6.

It can be seen from Table 6 that the average error performance of the prediction model of the Monte Carlo model is evaluated. The comparison shows that, compared to the traditional grey prediction model and SARIMA model, the GM-SARIMA model can better predict the problem that the prediction results are not ideal due to the large deviation of the initial training data.

6. Conclusion

In this paper, to effectively predict the offset distance of light buoy and deal with the problem of the excessive amplitude of initial training data, a combined optimization prediction model integrating multiplicative seasonal model and grey model is proposed. The historical offset data of Meizhou Bay 1 # light buoy are used as experimental data. Firstly, the offset distance data of the light buoy are processed completely. Secondly, the difference method is used to process the stability of the data, to meet the requirements of the model for the initial data. Then, the optimal model parameter combination is selected by the AIC minimum information criterion method, to obtain the theoretical values of the multiplicative seasonal model and the grey model, respectively. Then, the average relative error and the weight are calculated to optimize the predicted value. Finally, the prediction accuracy of the model is evaluated by MAPE, RMSE, and Monte Carlo sampling method. The results show that the MAPE and RMSE of the grey optimized multiplicative seasonal model are 2.61% and 1.51, respectively, which are 6.66% and 2.96 lower than those of the traditional product seasonal model, indicating that the model can effectively deal with the problem that the prediction error of the traditional multiplicative seasonal model is too large due to the instability of the amplitude the initial time series and effectively reduce the prediction error, which provides a new idea for the prediction of maritime traffic position and the early warning of ship navigation safety.

Data Availability

The light buoy position data used to support the findings of this study were supplied by Jinxing Shao under license and so cannot be made freely available. Requests for access to these data should be made to Jinxing Shao, affiliation: Aids to Navigation Department, Xiamen; e-mail: 13606931987@139.com.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of Fujian Province (Grant nos. 2020J01658 and 2019J01325), Open Project Fund of National Local Joint Engineering Research Center for Ship Assisted Navigation Technology (Grant no. HHXY2020002), and Doctoral Start-Up Fund of Jimei University (Grant no. ZQ2019012).

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Copyright © 2022 Zhizheng Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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