A study on ship collision conflict prediction in the Taiwan Strait using the EMD-based LSSVM method

Abstract

Ship collision accidents are the primary threat to traffic safety in the sea. Collision accidents can cause casualties and environmental pollution. The collision risk is a major indicator for navigators and surveillance operators to judge the collision danger between meeting ships. The number of collision accidents per unit time in a certain water area can be considered to describe the regional collision risk However, historical ship collision accidents have contingencies, small sample sizes and weak regularities; hence, ship collision conflicts can be used as a substitute for ship collision accidents in characterizing the maritime traffic safety situation and have become an important part of methods that quantitatively study the traffic safety problem and its countermeasures. In this work, an EMD-QPSO-LSSVM approach, which is a hybrid of empirical mode decomposition (EMD) and quantum-behaved particle swarm optimization (QPSO) optimized least squares support vector machine (LSSVM) model, is proposed to forecast ship collision conflicts. First, original ship collision conflict time series are decomposed into a collection of intrinsic mode functions (IMFs) and a residue with EMD. Second, both the IMF components and residue are applied to establish the corresponding LSSVM models, where the key parameters of the LSSVM are optimized by QPSO algorithm. Then, each subseries is predicted with the corresponding LSSVM. Finally, the prediction values of the original ship collision conflict datasets are calculated by the sum of the forecasting values of each subseries. The prediction results of the proposed method is compared with GM, Lasso regression method, EMD-ENN, and the predicted results indicate that the proposed method is efficient and can be used for the ship collision conflict prediction.

1. Introduction

The global shipping industry is witnessing a boom as economic globalization gains speed and the world economic integration trend intensifies in recent decades. According to the Review of Maritime Transport 2019, about 90 percent of global trade in terms of the weight of goods is undertaken by shipping, there is no doubt that shipping plays an irreplaceable role in the global economy [1]. However, shipping has long been regarded as a complex and high-risk activity, and maritime accidents often lead to serious damage, death, loss, injury or pollution, and may also have significant political, economic and environmental consequences [2]. The greater the role that shipping plays in international trade, the greater the impact on the world economy from the loss arising from maritime accidents. There are various international safety regulations to regulate the operation of ships and prevention of accidents, such as SOLAS 74/78/88 (International Convention for the Safety of Life at Sea), MARPOL 73/78 (Marine Pollution), STCW 78 (Standards of Training, Certification and Watch keeping for Seafarers) and COLREG 72 (International Regulations for Preventing Collisions at Sea), but the complex and high-risk environment at sea make it difficult to eliminate ship accidents [3]. Therefore, studies on maritime accidents will be helpful in guiding the management of maritime traffic safety and consequently reduce life and property loss [4].

The Taiwan Strait is a large channel between northern and southern China and is an important maritime passage connecting the Korean Peninsula, Japan, Southeast Asian countries, Hong Kong and Macao. With the steady increase in cargo throughput in Chinese ports, the number of ships sailing along the coast of China has also gradually increased. Taking the Taiwan Strait as an example, the number of 300 GT and above merchant ships passing through the Taiwan Strait every day during the three years from 2015 to 2017 is as high as 483 [5]. The increase in ship density and flow will inevitably lead to an increase in maritime traffic accident probability, among which ship collision accidents rank first among all kinds of accidents. The collision risk is a major indicator for navigators and surveillance operators to judge the collision danger between meeting ships [6], as well as the surveillance on shore plays an important role in preventing ship collision accidents [7]. Based on the historical statistical data, the number of collision accidents per unit time in a certain water area was considered to describe the regional collision risk by researchers, for example, the Formal Safety Assessment concept and Bayesian network method were used to evaluate the collision risk of ships in Yangtze River waters in China with real accident data [8]. Since historical ship collision accidents have the features of strong contingencies, small sample sizes and weak regularity, in general it is difficult to extract valuable information from historical data. So ship collision conflicts can be used as a substitute for ship collision accidents in characterizing the maritime traffic safety situation and have become an important part of methods that quantitatively study the traffic safety problem and its countermeasures. Therefore, it is of practical significance to carry out research on the analysis of collision conflicts and the prediction of future situations with the purpose of providing data support for early warning and future implementation of the maritime security strategy in China [9].

With the development of time series analysis, artificial intelligence, fuzzy logic, chaos theory, artificial neural network and statistical learning theory, a large number of methods have been proposed for maritime traffic accident prediction [10–15]. The performance of some classic time series prediction models fail to satisfy expectations due to the ship motion process complexity with nonlinearity and uncertainty in harsh climates [16]. Support vector machine (SVM), a novel type of machine learning algorithm, has a strong capacity for processing nonlinear data. Based on SVM, the support vector regression model (SVR model) is an effective method in solving regression problems [17]. Compared to the neural network model, the SVR model needs less training data. Even though SVR is an effective prediction method, non-stationary time series have a great impact on its prediction accuracy [18]. As a new type of SVM, the LSSVM greatly improves the convergence speed by solving the function estimation problem with the quadratic programming method [19], and it can be used for ship collision prediction research [20]. However, due to the intrinsic complexity of ship collision conflicts, it is difficult to describe the variation trend in ship collision conflicts. In order to construct a suitable prediction model, the original dataset features of ship accidents need to be considered. Since a ship accident depends on the climate, which has specific cycles such as year, month, and week, the ship collision conflict time series can be considered as a combination of subseries characterized by different frequencies. Each subseries corresponds to a range of frequencies, shows much more regularities and is predicted more accurately than the original ship collision conflict series. EMD, proposed by Huang [21], exhibits a strong generality in dealing with non-stationary data. This method can reflect the physical characteristics of the original time series signal without pre-set basis function. As a special signal processing technique, EMD can decompose a complex signal into a collection of IMFs and a residue, which are relatively stationary subseries and can be readily modelled [22, 23]. Discrete wavelet transform (DWT) is also a powerful method in dealing with non-stationary and nonlinear signals [24, 25]. But the processing procedure of DWT is not autoregressive and the decomposition accuracy is affected by the band-pass filters which are chose to decompose target signals. Wavelet basics function and decomposed layer also affect the decomposition results. Therefore, the decomposition accuracy of DWT is relatively lower than EMD, and EMD is used in the decomposition of ship collision conflict time series.

According to the above comprehensive analysis, in this work, an EMD-QPSO-LSSVM approach, which is a hybrid of empirical mode decomposition and quantum-behaved particle swarm optimization optimized least squares support vector machine model, is proposed to forecast ship collision conflicts. In the approach, the original ship collision conflict time series are decomposed into a collection of IMFs and a residue with EMD. Then, both the IMF components and the residue are used to establish the corresponding LSSVM models, where the key parameters of each LSSVM models are optimized by quantum- behaved PSO algorithm. Finally, the prediction values of the original ship collision conflict datasets are calculated by summing the forecasting values of every subseries. The effectiveness of the proposed model is verified using the real data from ship collision conflicts in the Taiwan Strait in 2014. The prediction results can, to some extent, provide a theoretical basis for the maritime department to develop an effective maritime management countermeasure and will be helpful in guiding the management of maritime traffic safety.

2. Objectives and contributions

Maritime transport plays an extremely important role in international trade and makes great contributions to national economic development. Shipping has long been regarded as a complex and high-risk activity, and maritime accidents often lead to serious damage, death, loss, injury or pollution, and may also have significant political, economic and environmental consequences. The collision risk is a major indicator for navigators and surveillance operators to judge the collision danger between meeting ships. In order to measure the collision risk, ship collision conflicts are used as an important index for measuring maritime traffic safety and maritime management. The objective of this study is to propose an efficient method to predict the future state by analysing the historical data of ship collision conflicts in the Taiwan Strait. The contribution of the work is the reference value for the administrative department in developing a maritime management countermeasure to reduce ship collision accidents.

3. Methodology

A hybrid of empirical mode decomposition and a least squares support vector machine model, named EMD-QPSO-LSSVM method, is proposed to forecast ship collision conflicts. The flowchart is shown in Fig 1. In the approach, the original ship collision conflict time series are decomposed into a collection of IMFs and a residue by EMD. Then, both the IMF components and the residue are used to establish the corresponding LSSVM models, where the key parameters of each LSSVM models are optimized by quantum-behaved particle swarm optimization algorithm. Finally, the prediction values of the original ship collision conflict datasets are calculated by summing the forecasting values of every subseries.

Fig 1. The flowchart of the EMD-QPSO-LSSVM method.

3.1 Empirical mode decomposition

Empirical mode decomposition method was first proposed by Huang [21]. In the prediction of non-stationary time series, EMD processing is very beneficial. This method can reflect the physical characteristics of the original time series signal without setting the basis function beforehand. The basic idea of empirical mode decomposition is that any set of signals consists of a limited number of intrinsic mode functions. According to the time scale characteristics of the data itself, the time series are decomposed step by step to extract IMF with different characteristic scales. Each IMF represents an intrinsic characteristic vibration form of the signal. The IMF needs to satisfy the following two basic conditions: i) The number of extrema and the number of zero-crossings should be equal or differ by one; ii) The average value of the upper envelope formed by the local maxima and the lower envelope formed by the local minima point should be zero.

Given an original ship collision conflict time series x(t), the EMD calculation can be described as follows:

$$\mathrm{x}(\mathrm{t})={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{imf}}_{\mathrm{k}}\left(\mathrm{t}\right)+}\mathrm{res}(\mathrm{t}),$$ (1)

where imf_k is the k_th IMF and res(t) is the residue after the IMFs are derived. The empirical mode decomposition steps are as following:

• Step 1. Find all the maximum and minimum points of original data sequence x(t), and fit all the maximum points with a cubic spline function. This curve is the upper envelope of data. All minimum points, similarly, are fitted with a cubic spline function to fit the lower envelope of data. Let m₁(t) be the mean of the upper and the lower envelopes. By subtracting the mean value m₁(t) from x(t), a new data sequence h₁(t) is achieved.

$${\mathrm{h}}_1{\left(\mathrm{t}\right)=\mathrm{x}\left(\mathrm{t}\right)\mathrm{‐}\mathrm{m}}_1\left(\mathrm{t}\right).$$ (2)

If h₁(t) does not satisfy the two basic requirements of IMF, the work above should be repeated with h₁(t) as the original data until h_k(t) meets the two requirements after k times. At this time h_k(t) is imf₁(t).

• Step 2. A new data sequence x₂(t) is achieved by subtracting IMF₁(t) from the original data x(t).

$${\mathrm{x}}_2{\left(\mathrm{t}\right)=\mathrm{x}\left(\mathrm{t}\right)\mathrm{‐}\mathrm{imf}}_1\left(\mathrm{t}\right).$$ (3)

• Step 3. Repeat the above steps n times until the last data sequence x_n+1(t) cannot be decomposed into IMF. This data sequence x_n+1(t) is named the residue res(t) of the original data.

3.2 Quantum-behaved PSO-LSSVM

Least-squares-SVM is a very active artificial intelligence method and is widely applied in modelling and control problems [19, 26]. To optimize the LSSVM parameters, different algorithms were used in literature [20, 27–31]. Quantum-behaved particle swarm optimization algorithm is a kind of intelligent optimization algorithm developed on particle swarm optimization, and can be used to solve the nonlinear and complex optimization problems with the features of less control parameters, easily to set up, strong search capability and good global search ability [32, 33].

In this work, a modified QPSO algorithm is adopted [20], where the swarm updates the individuals’ positions in the following way:

$$\begin{array}{l}{\mathbf{m}}_{best}[t]=\frac{1}{N}{\displaystyle \sum _{i=1}^N{\mathbf{p}}_{besti}[t]}=\left(\frac{1}{N}{\displaystyle \sum _{i=1}^N{\mathbf{p}}_{besti1}}[t],\cdots ,\frac{1}{N}{\displaystyle \sum _{i=1}^N{\mathbf{p}}_{bestiD}}[t]\right),\\ \mathbf{p}[t+1]=\phi [t]\cdot {\mathbf{p}}_{best}[t]+(1-\phi [t])\cdot {\mathbf{g}}_{best}[t],\\ \mathbf{x}[t+1]=\mathbf{p}[t]-\beta [t]\cdot |{\mathbf{m}}_{best}[t]-\mathbf{x}[t]|\ln (2u[t]),\end{array}$$ (4)

where φ[t],u[t] are random numbers in [0,1] at step t, N is the size of the swarm, D is the dimension of the particles, g_best(t) is the entire swarm’s best known position, p_besti[t] is the ith particle’s best known position, and p[t] is called a local attractor.

The inertia weight β[t] takes the following form

$$\beta [t]={\beta }_0-{\beta }_1\chi [t]+{\beta }_2\lambda [t],$$ (5)

where χ[1] = λ[1] = 0 and

$$\chi [t]\triangleq \frac{\mathrm{FIT}({\mathbf{g}}_{best}[t])}{\mathrm{FIT}({\mathbf{g}}_{best}[t-1])},\lambda [t]\triangleq \frac{\mathrm{FIT}({\mathbf{g}}_{best}[t-1])}{\frac{1}{N}{\displaystyle \sum _{i=1}^N\mathrm{FIT}({\mathbf{p}}_{best}{}_i[t-1])}},2\le t\le t_{\max },$$

and β₀,β₁,β₂ satisfy the constraints β₁<β₀ and β₀+β₂<1.78 as it was proved in [33] that as long as β[t]<1.78, the convergence of QPSO can be guaranteed.

For given a dataset $\mathrm{S}=\{({\mathbf{x}}_{\mathrm{i}}{,\mathrm{y}}_{\mathrm{i}}{)\}}_{\mathrm{i}=1}^{\mathrm{N}}$, where ${\mathbf{x}}_{\mathrm{i}}\in {ℝ}^m$ is input data in input space and $y_i\in ℝ$ is output value for given value of specific input variable, the LSSVM-based prediction model for the nonlinear function is

$$y(\mathbf{x})={\displaystyle \sum _{l=1}^N{\alpha }_l}\cdot \mathrm{K}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{l}(\mathbf{x},{\mathbf{x}}_l)+b.$$ (6)

The parameters α = [α₁,α₂,⋯,α_N]^T and b can be determined by

$$\left[\begin{array}{c}\mathrm{b}\\ \mathbf{\alpha }\end{array}\right]={\left[\begin{array}{cc}0& {\mathbf{L}}^T\\ \mathbf{L}& \mathrm{\Phi }+{\mathrm{\Gamma }}^{\mathrm{‐}1}\end{array}\right]}^{-1}\left[\begin{array}{c}0\\ \mathbf{Y}\end{array}\right],$$ (7)

where Y = [y₁,y₂,⋯,y_N]^T, L = [1,1,⋯,1]^T, Φ = (Φ_ij)_N×N with general element Φ_ij = φ(x_i)^Tφ(x_j) = Kernal(x_i,x_j) and Γ = (Γ_ij)_N×N with general element

$${\mathrm{\Gamma }}_{\mathrm{ij}}=\{\begin{array}{cc}{\mathrm{\gamma }}_{\mathrm{i}}\triangleq {\mathrm{\gamma }}_0\exp \left(\frac{\mathrm{i}}{\mathrm{N}}\rho \right),& \mathrm{j}=\mathrm{i},\\ 0,& \mathrm{j}\ne \mathrm{i}.\end{array}$$ (8)

The kernel function Kernal(⋅) is chosen as the RBF kernel function, and the parameters γ₀,ρ and σ² are determined by QPSO algorithm. The flow chart of parameters adjustment QPSO-based is depicted in Fig 2. The optimization procedure has been repeated several times as attempts to reach the most probable global optimum of the fitness function.

Fig 2. Flow chart of the parameters of the LSSVM model optimization by QPSO algorithm.

4. Numerical simulations

4.1 Error measures

To assess the performance of the prediction models, three error measures are used for model comparison, i.e., the mean absolute error (MAE), the mean relative error (MRE), the mean square error (MSE) and the mean absolute percentage error (MAPE).

$$\begin{array}{l}{\mathrm{e}}_{\mathrm{MAE}}=\frac{1}{N_{\mathrm{Pr}ed}}{\displaystyle \sum _{j=1}^{N_{\mathrm{pred}}}\left|z^P(j)-z(j)\right|}\\ {\mathrm{e}}_{\mathrm{MAPE}}=\frac{1}{N_{\mathrm{Pr}ed}}{\displaystyle \sum _{j=1}^{N_{\mathrm{pred}}}\left|\frac{z^P(j)-z(j)}{z(j)}\right|}\\ {\mathrm{e}}_{\mathrm{MSE}}=\frac{1}{N_{\mathrm{Pr}ed}}{\displaystyle \sum _{j=1}^{N_{\mathrm{Pr}ed}}\sqrt{{\left(z^p(j)-z(j)\right)}^2}}\\ {\mathrm{e}}_{\mathrm{MAPE}}=\frac{1}{N_{\mathrm{Pr}ed}}{\displaystyle \sum _{i=1}^{N_{\mathrm{Pr}ed}}\left|\frac{z^p(j)-z(j)}{z(j)}\right|}\end{array}$$ (9)

where N_Pred is the prediction sample size and z(j) and z^p(j) are the actual and forecast values during a time period, respectively.

4.2 Ship collision conflict datasets

To verify the validity of the proposed hybrid approach, ship collision conflict data from the Taiwan Strait are employed. The data consist of actual daily ship collision conflicts from 1999 to 2014 [34], and the verification is processed on the data in 2014, as presented in Table 1.

Table 1. Ship collision conflicts in the Taiwan Strait in 2014.

No.	Count	No.	Count	No.	Count	No.	Count	No.	Count	No.	Count	No.	Count
1	198	54	149	107	241	160	193	213	250	266	240	319	211
2	211	55	139	108	151	161	201	214	169	267	303	320	323
3	246	56	156	109	199	162	193	215	158	268	291	321	241
4	182	57	117	110	220	163	195	216	157	269	262	322	301
5	217	58	115	111	219	164	196	217	226	270	283	323	423
6	206	59	131	112	222	165	198	218	266	271	343	324	376
7	255	60	134	113	172	166	160	219	429	272	271	325	377
8	201	61	177	114	261	167	370	220	386	273	217	326	288
9	233	62	195	115	268	168	212	221	300	274	304	327	286
10	275	63	186	116	270	169	250	222	261	275	276	328	321
11	224	64	108	117	219	170	282	223	394	276	223	329	301
12	222	65	139	118	149	171	238	224	265	277	320	330	302
13	170	66	176	119	168	172	175	225	200	278	201	331	264
14	188	67	186	120	180	173	202	226	193	279	142	332	284
15	258	68	162	121	290	174	248	227	297	280	296	333	264
16	217	69	149	122	246	175	225	228	253	281	295	334	275
17	229	70	163	123	260	176	162	229	230	282	303	335	346
18	185	71	130	124	180	177	227	230	252	283	181	336	250
19	231	72	131	125	176	178	196	231	294	284	384	337	275
20	211	73	149	126	308	179	178	232	221	285	401	338	264
21	152	74	131	127	242	180	152	233	293	286	196	339	249
22	201	75	154	128	285	181	210	234	269	287	301	340	291
23	187	76	135	129	184	182	223	235	271	288	200	341	260
24	163	77	130	130	185	183	225	236	194	289	244	342	320
25	151	78	128	131	181	184	223	237	262	290	317	343	319
26	128	79	146	132	210	185	233	238	299	291	250	344	332
27	163	80	117	133	138	186	240	239	254	292	304	345	276
28	140	81	180	134	143	187	276	240	230	293	266	346	267
29	167	82	191	135	218	188	185	241	261	294	302	347	295
30	159	83	177	136	201	189	228	242	324	295	223	348	278
31	178	84	158	137	111	190	147	243	223	296	260	349	290
32	102	85	140	138	174	191	170	244	215	297	226	350	248
33	128	86	122	139	196	192	283	245	297	298	269	351	309
34	127	87	152	140	175	193	276	246	300	299	268	352	294
35	110	88	131	141	178	194	210	247	264	300	179	353	254
36	135	89	136	142	220	195	233	248	277	301	282	354	230
37	121	90	120	143	143	196	240	249	270	302	222	355	320
38	102	91	128	144	174	197	227	250	267	303	265	356	314
39	111	92	176	145	161	198	163	251	240	304	203	357	348
40	100	93	169	146	156	199	207	252	383	305	299	358	294
41	105	94	241	147	229	200	231	253	302	306	201	359	290
42	101	95	253	148	178	201	292	254	277	307	287	360	298
43	103	96	176	149	186	202	191	255	188	308	444	361	278
44	100	97	255	150	171	203	151	256	230	309	392	362	245
45	100	98	158	151	180	204	130	257	180	310	318	363	257
46	150	99	187	152	201	205	133	258	139	311	242	364	210
47	140	100	212	153	171	206	274	259	282	312	274	365	186
48	127	101	203	154	192	207	287	260	256	313	291
49	121	102	210	155	196	208	258	261	349	314	359
50	152	103	115	156	150	209	294	262	193	315	352
51	138	104	207	157	250	210	231	263	250	316	234
52	133	105	227	158	229	211	207	264	190	317	239
53	131	106	300	159	208	212	198	265	259	318	243

4.3 Data processing

The data processing follows the following steps:

• Step 1: EMD of the ship collision conflict time series

Due to the intrinsic complexity of the original ship accident time series, the variation tendency is difficult to predict. To improve the prediction accuracy, EMD is used to decompose the original ship collision conflict time series z = (z₁,z₂,⋯,z_T) with T = 365, which yields seven IMF components imf_k = (z_k1,z_k2,⋯,z_kT)(k = 1,2,⋯,7) and a residue res = (r₁,r₂,⋯,r_T), as illustrated in Fig 3.

Fig 3. Schematic diagram of the EMD components.

• Step 2: Data normalization

For the sake of expression, denote imf_k by z_k (k = 1,2,⋯,6) and res by z₇ = {z₇₁,z₇₂,⋯,z_7T}. Then normalize the sequence ${\mathbf{z}}_k={\{z_{k1},z_{k2},\cdots ,z_{kT}\}}_{k=1}^7$ by Min–Max Normalization method [35] in the following form:

$${\overline{z}}_{ki}=\frac{z_{ki}-z_{k\min }}{z_{k\max }-z_{k\min }},i=1,2,\cdots ,T,k=1,\cdots ,7.$$

• Step 3: Data phase space reconstruction

To sufficiently extract the useful information from time series ${\overline{\mathbf{z}}}_k=\left({\overline{z}}_{k1},{\overline{z}}_{k2},\cdots ,{\overline{z}}_{kT}\right)$, the commonly used method is the phase space reconstruction (PSR) method in delay coordinates proposed by Packard et al. [36]. Theoretically speaking, a time series can sufficiently reconstruct an original dynamic system according to Takens [37]. From this procedure, time series ${\overline{\mathbf{z}}}_k=\left({\overline{z}}_{k1},{\overline{z}}_{k2},\cdots ,{\overline{z}}_{kT}\right)$ can be reconstructed in a multidimensional phase space as follows:

$${\mathbf{x}}_{ki}=\left({\overline{z}}_{ki},{\overline{z}}_{k(i+\tau )},\cdots ,{\overline{z}}_{k(i+(m-1)\tau )}\right),y_{ki}={\overline{z}}_{k(i+m\tau )},i=1,\cdots ,T-m\tau ;k=1,\cdots ,7$$ (10)

where τ is the delay parameter and m is the embedding dimension. It is very important to select a suitable pair of embedding dimensions m and time delay τ when performing PSR [38–40]. There is no exact way to determine the values of τ and m, the result in [41] indicates that a larger value for τ than necessary should be selected to prevent system information from being ignored. Besides, according to the result in Brock et al [42], the appropriate values for embedded dimension m should be between 2 and 5. In the following simulations, the embedded dimension m is set equal to 4 and the time delay is assumed to be day to day.

4.4 Prediction by QPSO-LSSVM and representation

The data pair ${\{({\mathbf{x}}_{ki},y_{ki})\}}_{i=1}^{T_1}$ obtained in Eq (10) is used to train the QPSO-LSSVM and obtain an optimal parameter pair $({\gamma }_{k0},{\varrho }_k,{\sigma }_k^2)$, where T₁ is the number of sample data in the training set. Then, the trained LSSVM is used to make a prediction

$${\overline{y}}_{kj}=\mathrm{L}\mathrm{S}\mathrm{S}\mathrm{V}\mathrm{M}({\mathbf{x}}_{kj}),j=T_1+1,\cdots ,T-m\tau .$$ (11)

The final step is to carry out the reverse normalization on

$${\overline{\mathbf{z}}}_k=\left({\overline{z}}_{k1},{\overline{z}}_{k2},\cdots ,{\overline{z}}_{k{T}_1},{\overline{y}}_{k(T_1+1)},\cdots ,{\overline{y}}_{k(T-m\tau )}\right),$$

which yields the sequence ${\mathbf{z}}_k^{\prime }=\left(z_{k1}^{\prime },z_{k2}^{\prime },\cdots ,z_{kT}^{\prime }\right)$ and the prediction result is

$$z_j^p={\displaystyle \sum _{k=1}^7{z}_{kj}^{\prime }},j=T_1+1,\cdots ,T.$$ (12)

4.5 Analysis of prediction results

To evaluate the prediction accuracy, the dataset is partitioned into a training dataset (90%) and a validation dataset (10%). The training dataset can be applied to establish the prediction model, and the validation dataset can be applied to validate the effectiveness of the model.

Grey model is easily set up, and the prediction result is presented in Fig 4. It can be observed that the prediction of GM is unsatisfied, and most of predictions are higher than the actual data. When LSSVM with key parameters γ₀ = 10,ρ= 0,σ = 2 is applied, the prediction results for training dataset and testing dataset are shown in Fig 4. It is obvious that the performance of the LSSVM is better than that of the GM. The maximum error is about 24% and the mean square error is about 5, it is still not suitable for real applications.

Fig 4. The prediction results of the GM and LSSVM for the ship collision conflicts dataset.

In order to improve the prediction accuracy, QPSO algorithm is applied to search an optimal key parameters (γ₀,ϱ,σ²). Here, the K-fold cross-validation is adopted to prevent the overfitting issue, and the training dataset is divided randomly into 9 folds, one of which was selected as the validation set each time for model selection, and the rest was used for model training. Table 2 illustrates the performance of LSSVM with 9-fold cross-validation.

Table 2. The result of 9-fold cross-validation.

Models	MSE on fold k	MSE on validation dataset
LSSVM1	4.68	4.62
LSSVM2	4.79
LSSVM3	4.46
LSSVM4	4.32
LSSVM5	4.58
LSSVM6	4.63
LSSVM7	4.72
LSSVM8	4.83
LSSVM9	4.57

Besides, due to the intrinsic complexity of ship collision, the regularity of the conflict time series is unobvious, and the prediction results directly from the original dataset is unsatisfied. Since a ship accident depends on the climate, which has specific cycles such as year, month, and week, the ship collision conflict time series can be considered as a combination of subseries characterized by different frequencies. Each subseries corresponds to a range of frequencies, shows much more regularities and is predicted more accurately than the original ship collision conflict series. The IMF components and residue by EMD is shown in Fig 3. The regularity of the latter five IMFs and residue is obviously stronger than the first two IMFs. By establishing different LSSVMs to the IMF components and residue, it can obtain a satisfied prediction results. The parameters of each LSSVM can be achieved by the flow chart of Fig 2. The prediction of the quantum-behaved PSO-LSSVM for each IMF component and residue are shown in Fig 5.

Fig 5. The prediction of the LSSVM for each IMF component.

The final prediction of the original ship collision conflict numbers are calculated by the sum of the prediction of each subseries, as shown in Fig 6. It can be seen that the prediction accuracy has been greatly improved. This indicates that the proposed method can be used for the prediction of ship collision conflicts as a substitute for ship collision accidents in characterizing the maritime traffic safety situation.

Fig 6. The final prediction of the EMD-LSSVM for the original ship collision conflict numbers.

To evaluate the performance of the proposed method, the statistical test is carried out on the real data and the prediction result of EMD-QPSO-LSSVM, as shown in Table 3. The sig. is 0.212, which is greater than 0.05. Thus, the proposed method is suitable in predicting the ship collision conflict numbers.

Table 3. Statistical analysis on the performance of the proposed method.

Paired Samples Test
				95% Confidence Interval of the Difference
	Mean	Std. Deviation	Std. Error Mean	Low	Upper	t	df	Sig. (2-tailed)
Real-Prediction	-1.512	23.128	1.211	-3.893	0.868	-1.249	364	0.212

To verify the efficiency of the proposed method, it is compared with GM, Lasso Regression, Bayes Regression, LSSVR and EMD-ENN. The comparison results is shown in Fig 7 and Table 4, where ENN contains 15 neurons. It can be seen that the performance of EMD-QPSO-LSSVM is better than other methods. But it should also be pointed that EMD-ENN is also a suitable method for ship collision conflicts predication.

Fig 7. Comparison between EMD-QPSO -LSSVM and EMD-ENN for the original ship collision conflict numbers.

Table 4. Comparison between different methods.

error method	eMAE	eMAPE	eMSE	eMSPE	eMax
GM(1,1)	39.8732	19.9898	8.3172	131.6	74.642
Lasso Regression	31.0324	11.7076	6.4043	91.9915	48.8811
Bayes Regression	30.5444	11.4899	6.346	90.4829	47.9449
LSSVR	24.0912	8.6409	5.1562	66.2941	26.794
EMD-ENN	13.6143	5.0842	2.8862	39.1503	13.7046
EMD-QPSO-LSSVM	12.2919	4.6141	2.4832	34.1591	11.5604

Since there is no exact way to determine the choice of the embedded dimension, according to Brock et al [38], different simulations are carried out to show the influence of embedded dimension m, as shown in Table 5. For the ship collision conflicts, the embedded dimension can be set equal to 4 or 5 when the time delay is one.

Table 5. Influence of the embedded dimension on the error measures.

error m	eMAE	eMAPE	eMSE	eMSPE	eMax
3	16.7773	6.1433	3.5215	46.1189	15.6572
4	12.2919	4.6141	2.4832	34.1591	11.5604
5	12.5429	4.7489	2.4701	34.521	11.7386

5 Conclusion

The Taiwan Strait is a large channel between northern and southern China and is an important maritime passage connecting the Korean Peninsula, Japan, Southeast Asian countries, Hong Kong and Macao. The ship traffic flow is large, the navigation risk is high, and the daily average number of ship collision conflicts is approximately 220. The number of collision accidents per unit time in a certain water area can be used to describe the regional collision risk, which is the main index for evaluating maritime traffic safety and measuring maritime management. It is of great significance for maritime administrative authorities to formulate strategies to reduce ship collision accidents by predicting the occurrence of ship collision conflicts in the Taiwan Strait in a short period of time through historical collision conflicts. By considering the advantages of the empirical mode decomposition method, quantum-behaved PSO optimized least squares support vector machine, a hybrid of EMD and QPSO-LSSVM model, is proposed to forecast the ship collision conflicts. The original ship collision conflict time series are first decomposed into a collection of IMFs and a residue by EMD method. And then, both the IMF components and residue are applied to establish the corresponding LSSVM models, where the key parameters of the LSSVM are optimized by quantum-behaved PSO algorithm. Each subseries is predicted using the corresponding LSSVM. Finally, the prediction values of the original ship collision conflict datasets are calculated by the sum of the forecasting values of every subseries. The prediction results show that the EMD-QPSO-LSSVM is an efficient method and can be used in the forecasting of ship accidents.

Supporting information

S1 File

(DOCX)

S1 Data

(RAR)

Data Availability

All relevant data are within the manuscript and its Supporting Information files.

Funding Statement

This study is supported by the Natural Science Foundation of Fujian Province (Grant No. 2019J01326) and the Scientific Research Foundation of Jimei University, China. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.