Prediction of 10,000-hour rate of civil aviation incidents based on grey Markov model

Abstract

The 10,000-hour rate of civil aviation incidents is an important index parameter to measure flight safety. Predicting the development trend of the 10,000-hour rate of civil aviation incidents plays an important role in aviation accident prevention and safety decision-making. Many complex factors influence the occurrence of civil aviation incidents, so the 10,000-hour rate of civil aviation incidents changes randomly and volatilely. This study proposed the idea of prediction by combining the grey GM (1, 1) model and the Markov model. Specifically, the grey GM (1, 1) prediction model was constructed using the statistical data on the 10,000-hour rate of civil aviation incidents in China during 2005–2020. On this basis, a grey Markov prediction model was established. The prediction of the 10,000-hour rate of incidents in 2021 based on the two models showed that the grey Markov model displayed higher prediction accuracy than the grey GM (1, 1) model and conformed to the change laws of the 10,000-hour rate data of civil aviation incidents better. Moreover, the grey Markov model could effectively improve the accuracy of the grey prediction model, compensate for its deficiencies, and facilitate the mastery of the change laws of civil aviation incidents, providing a reliable basis for aviation safety management and incident prevention.

Keywords

Civil aviation safety 10 000-hour rate of incidents grey prediction Markov prediction

1. Introduction

In recent years, COVID-19 has had a huge impact on the global civil aviation industry. Thanks to the precise and scientific epidemic prevention and control measures, China’s civil aviation industry has become the world’s fastest-recovering and best-running aviation market under the COVID-19 epidemic. According to the statistical bulletin on the development of the civil aviation industry, from 2020 to 2022, the transportation turnover of civil aviation decreased significantly compared with that in 2019, with an average annual growth rate of $-$ 20.37% [1], as shown in Fig. 1. However, the sudden decrease in air traffic volume has not positively impacted aviation safety. According to the changing trend of the global aviation accident rate from 2010 to 2022 by IATA’s Accident Classification Technical Group (ACTG) [2], the accident rate of the whole industry showed a downward trend from 2016 to 2019. Still, under the COVID-19 epidemic crisis since 2020, the accident rate of the whole industry has somehow risen in comparison with that in 2019 (Fig. 2). Aviation safety faces severe challenges.

Figure 1.

Total turnover of civil aviation transportation during 2018–2022.

Figure 2.

Global aviation accident rate during 2010–2022.

The statistical data from the Civil Aviation Administration of China revealed 440 transportation aviation incidents and four severe incidents in 2020, 559 transportation aviation incidents and six severe incidents in 2021. What is more noteworthy is that on March 21, 2022, flight MU5735 from Kunming to Guangzhou of China Eastern Airlines Yunnan Co., Ltd. crashed in Wuzhou, Guangxi, and disintegrated after hitting the ground, killing all 123 passengers and nine crew members on board, which also terminated the record of 100 million hours of continuous and safe flight time of China civil aviation transportation.

With the recovery of civil aviation transportation production capacity and the increase in flight number and flight hours, the subsequent flight operation safety problem is still the focus of civil aviation operation safety. During aircraft operation, flight accidents or flight incidents may occur due to defects in aircraft design and manufacture, human factors, and weather and environmental factors [3, 4]. The 10,000-hour rate of civil aviation incidents refers to the probability for serious unsafe behaviours or unsafe states of civil aircraft to cause aircraft damage, personal injury, etc., during the flight for 10,000 hours but not cause serious flight accidents or even disaster events. To realize the continuous safety of civil aviation and master the development trend of the 10,000-hour rate of civil aviation incidents [5], accurately predicting the 10,000-hour rate of civil aviation incidents plays a vital role in accident prevention and safety decision-making of relevant units.

Commonly used accident prediction methods include BP neural network prediction method [6, 7], regression analysis prediction method [8, 9], grey prediction method [10, 11], and Markov prediction method [12, 13]. Therein, the BP neural network prediction model needs sufficient sample data, which has a large calculation workload and a slow convergence speed and easily falls into the local minimum. The accuracy of the regression analysis prediction method is highly correlated with the number of original data and samples, and it is not suitable for prediction in case of a small data size. The grey prediction method’s accuracy will decline if the data fluctuates greatly. The Markov prediction method applies to series with large data fluctuations, but more data should ensure accuracy. Given the differences in the applicable conditions of different prediction methods, combined prediction methods have also become the choice of many scholars to make up for the shortcomings of the methods. A combined method means integrating different prediction models and comprehensively using the information provided by various prediction methods to establish a combined prediction model, which can effectively improve the accuracy of the prediction model. According to the characteristics of sampled data, the aviation incident rate would be predicted using the grey Markov prediction model, expecting to provide a reference for aviation safety risk management and decision-making.

2. Research status

The 10,000-hour rate of civil aviation incidents is an important index parameter to measure flight safety. The greater the 10,000-hour rate of civil aviation incidents, the greater the possibility of civil aviation accidents. The existing literature mostly focuses on the research on civil aviation incidents, including incident influencing factors, incident prediction, and risk assessment. Aiming at the randomness and uncertainty of factors inducing flight accidents, Chen et al. [14] established an improved entropy and grey correlation algorithm for importance identification and identified key causes for flight incidents. Silagyi and Liu [15] evaluated and ranked the importance of accident factors to predict the severity of the consequences of aircraft approaching and landing accidents, established a support vector machine prediction model based on the RBF kernel function and verified the model’s prediction accuracy. Claros et al. [16] proposed a risk data modelling method based on a negative multinomial distribution. They integrated it into the FAA safety management system’s Safety Risk Management (SRM) plate, realizing the quantitative analysis of safety risks. To improve the ability of airlines to control flight operation risks, Zheli. [17] put forward an environmental risk assessment method of flight operation control based on support vector regression and verified its effectiveness. Fan [18] explored the change laws of China’s aviation incidents using the grey prediction theory. They established a grey prediction model, providing decision reference for civil aviation safety and disaster warning.

Grey prediction and Markov prediction have been widely investigated and applied by predecessors, involving various fields such as transportation, fault diagnosis, accident prediction, and aviation safety. To realize short-term traffic flow prediction, Comert et al. [19] proposed several new grey system models, expounded on the advantages of each model in reducing errors by comparing with the grey GM (1, 1) benchmark model, and verified the method’s feasibility. Liu et al. [20] proposed a new method to predict transformer faults by predicting the variation trend of dissolved gas content in transformer oil. They optimized the system state division method of the Markov model by using the Fibonacci sequence. Wang et al. [21] established three grey prediction models of hazardous chemicals considering the small size and information of accident data samples regarding hazardous chemicals. By comparison, it was concluded that the unbiased grey model achieved the highest prediction accuracy, which provided a new idea for the accident prediction of hazardous chemicals. Edem et al. [22] put forward a grey-fuzzy-Markov time series model for industrial incident prediction, which could predict accidents according to a small amount of or incomplete data information. Wang and Lv [23] established a grey Markov prediction model for incidents based on the grey prediction model, verified the feasibility and accuracy of the model with an example, and improved the prediction accuracy of the variables with large fluctuations. Ni et al. [24] integrated the deep belief network (DBN) and principal component analysis (PCA) in combination with the characteristics of big data to put forward a method to predict the serious flight accident rate of unsafe events based on deep learning. The results obtained by this combined prediction model are consistent with the reality.

To sum up, the current research on civil aviation incidents mainly focuses on analyzing influencing factors, predicting the number of incidents, and risk assessment. Although the key influencing factors of incidents can be identified as small-probability events, incidents are highly volatile, with little significance if used to predict the number of accidents quantitatively, making it difficult to reflect the safety performance level of civil aviation. Given the above problems, this study used the 10,000-hour rate of incidents as the prediction index. Then, a grey Markov prediction model was established to mitigate the influence of data fluctuations, which could help master the development trends of civil aviation safety and provide a reference for safety management and decision-making.

3. Theory and method

3.1 Grey theory

Chinese professor Deng put forward the grey prediction method in 1982, suitable for series prediction with a small data size and insufficient information. The most common grey prediction method is the GM (1. 1) model [25]. The model regards random variables as grey variables and random processes as grey processes, thus not needing a lot of historical data. Basic grey models filter out the possible random quantities in the original sequence through the accumulation of time series data, look for some hidden regularity from the fluctuating time series, and then get new series with weakened randomness and enhanced regularity, thus mining the inherent characteristics of the original sequence. The method of accumulation or subtraction in grey prediction is easy to produce errors, and the fluctuation of the number of civil aviation incidents and flight time data leads to the partially low accuracy of prediction results.

3.2 Markov theory

Markov is a random process with discrete time and state, which is characterized by no aftereffect; that is, the state at time $t$ is only related to the state at the previous time but has nothing to do with other time states [26, 27, 28]. The occurrence and rate of civil aviation incidents are dynamic systems that change with time and have nothing to do with the past state, and their occurrence is a random, non-stationary process. The Markov theory can describe this dynamic system with random fluctuation characteristics. This method predicts the future development and change of the system according to the transition probability between states. The Markov prediction method effectively predicts data series with large fluctuations, but the premise for the Markov model-based prediction is a definite system state. When the 10,000-hour rate of civil aviation incidents is predicted, the data series fluctuates greatly, making it difficult to meet this premise.

3.3 Grey Markov theory

The grey and Markov models can be used to predict time series. The grey prediction curve shows a monotonic increasing or decreasing trend, which can reflect the overall changing trend. The Markov model predicts the future development direction of the system according to the transition probability of the system state, and the transition probability reflects the influence of various random factors, so it is suitable for the prediction of time series with large random fluctuations. The grey Markov combined prediction model combines these two methods, which can learn from each other’s strong points. The development trend of the 10,000-hour rate of civil aviation incidents can be more accurately predicted by taking the respective advantages of the two models; namely, the grey model can reflect the development of trends with small data sizes, while the Markov model can process data fluctuations.

The grey Markov model was constructed using the following method. Firstly, the GM (1, 1) prediction model was established according to historical civil aviation incidents data in the first $n$ years, and the first step prediction was implemented. Then, a Markov multi-step transition probability matrix model was established by using the relative change rate between the predicted value of GM (1, 1) and the actual historical data, and the state of the next year’ s time series was inferred by using the initial state information of the first $n$ years and the corresponding multi-step transition probability sum. Finally, the predicted value of the GM (1, 1) model was corrected according to the predicted state of the Markov model, followed by the accuracy calculation and applicability verification, as shown in Fig. 3.

Figure 3.

Grey Markov model.

3.3.1 Grey GM (1, 1) model

The calculation process of the GM (1, 1) model does not involve other related variables of the predicted variables but only needs to test or process the original data series, so the GM (1, 1) model has been widely used. The specific modeling steps are as follows [29, 30, 31].

(1) The original sequence $X^{(0)}$ of predicted variables is established, namely:

$\displaystyle X^{(0)}=\{{X^{(0)}(1),X^{(0)}(2),\ldots X^{(0)}(n)}\}$

Where $X^{(0)}(k)\geqslant 0,k=1,2,\ldots n$ .

(2) The original sequence is accumulated once to obtain a new sequence.

$\displaystyle X^{(1)}=\{{X^{(1)}(1),X^{(1)}(2),\ldots X^{(1)}(n)}\}$

Where $X^{(1)}(k)=\mathop{\sum}\limits_{i=1}^{k}X^{(0)}(i),k=1,2,\ldots n$ .

(3) A new sequence GM (1, 1) model is constructed to obtain its differential equation:

$\displaystyle\frac{dx^{(1)}}{dt}+ax^{(1)}=b$ (1)

Where a and b stand for undetermined coefficients.

(4) A data matrix $B$ and a data vector $Y$ are constructed

$\displaystyle B=\left({{\begin{array}[]{*{20}c}{-\frac{1}{2}({x^{(1)}(1)+x^{1}% (2)})}&1\\ {-\frac{1}{2}({x^{(1)}(2)+x^{1}(3)})}&1\\ \vdots&\vdots\\ {-\frac{1}{2}({x^{(1)}({n-1})+x^{1}(n)})}&1\\ \end{array}}}\right)$ (2) $\displaystyle Y=[{x^{(0)}(2),x^{(0)}(3),\ldots,x^{(0)}(n)}]^{T}$ (3)

(5) The undetermined coefficients a and b are solved by the least square method.

$\displaystyle\hat{a}=(B^{T}B)^{-1}B^{T}Y$ (4)

(6) The differential equation is solved and the time response equation of the prediction model is determined.

$\displaystyle\hat{x}_{k+1}^{(1)}=\left(x^{1}(1)-\frac{b}{a}\right)e^{-ak}+% \frac{b}{a}$ (5)

(7) The original sequence reduction model is established.

Finally, the predicted sequence $\hat{X}^{(0)}$ of the original sequence ${X}^{(0)}$ is finally acquired:

$\displaystyle\left\{{{\begin{array}[]{l}\hat{x}^{(0)}(1)=x^{(0)}(1)\\ \hat{x}^{(0)}(k)=\hat{x}^{(1)}(k)-\hat{x}^{(1)}(k-1)=\left[x^{(0)}(1)-\frac{b}% {a}\right](1-e^{a})e^{-a(k-1)},k>1\end{array}}}\right.$ (6)

(8) Residual test

The residual refers to the difference between the observed value and the predicted value, and the residual test is one of the effective methods for data reliability tests. The specific inspection process is as follows:

The residual is recorded as $e^{(0)}(k)$ and the relative error as $\delta^{(0)}(k)$ , and then:

$\displaystyle e^{(0)}(k)=x^{(0)}(k)-\hat{x}^{(0)}k$ (7) $\displaystyle\delta^{(0)}(k)=\left|{\frac{e^{(0)}(k)}{x^{(0)}(k)}}\right|% \times 100\%$ (8)

According to the relative error, the average relative error of the samples can be obtained as follows:

$\displaystyle\bar{\delta}=\frac{1}{n}\sum\limits_{k=1}^{n}{\delta^{(0)}(k)}$ (9)

The feasibility of the prediction result can be judged by the result of the average relative error. If the average relative error is $\bar{\delta}<$ 20%, the prediction result passes the test. If the average relative error satisfies $\bar{\delta}>$ 20%, the prediction can be done only after the model is corrected.

3.3.2 Markov model

(1) System state analysis

According to the relative change rate between the actual data and the predicted value of the grey model, the system is divided into $n$ states, and any state interval can be recorded as: $E_{i}=[{Q_{li},Q_{ui}}]$ , $i=1,2,3,\ldots n$ , where: $Q_{ui}$ is the upper limit of the system state, and $Q_{li}$ is the lower limit of the system state. In general, the number and interval size of system states should be considered comprehensively according to the actual situation of the predicted object.

(2) Construction of a transition matrix

After the state interval is divided, the state transition probability is calculated by counting the frequency of state transition, and finally the state transition matrix is obtained.

The frequency matrix for the system transition from state $E_{i}$ to state $E_{j}$ through $k$ steps is as below:

$\displaystyle N=\left[{{\begin{array}[]{*{20}c}{n_{11}}&{n_{12}}&\ldots&{n_{1j% }}\\ {n_{21}}&{n_{22}}&\ldots&{n_{2j}}\\ \vdots&\vdots&\ddots&\vdots\\ {n_{i1}}&{n_{i2}}&\ldots&{n_{ij}}\\ \end{array}}}\right]$ (10)

$n_{ij}$ – the frequency of transition from state $E_{i}$ to state $E_{j}$ through $k$ steps.

The system state transition probability is the ratio of each element in the transition frequency matrix to the sum of the elements in the row, denoted as $P_{ij}$ , and the system state transition matrix can be obtained as follows:

$\displaystyle P=\left[{{\begin{array}[]{*{20}c}{P_{11}}&{P_{12}}&\ldots&{P_{1j% }}\\ {P_{21}}&{P_{22}}&\ldots&{P_{2j}}\\ \vdots&\vdots&\ddots&\vdots\\ {P_{i1}}&{P_{i2}}&\ldots&{P_{ij}}\\ \end{array}}}\right]$ (11)

Where $0\leqslant P_{ij}\leqslant 1$ and $\mathop{\sum}\limits_{j=1}^{n}P_{ij}=1,i=1,2\ldots,n$ .

Generally, the matrix $P$ is referred to as the one-step transition probability matrix, which can also be written as $\mbox{P}_{ij}^{(1)}=P$ , and the transition probability of each state after the system experiences $n$ -step transition is $\mbox{P}_{ij}^{(n)}=P^{n}$ .

(3) System state prediction

Knowing the initial state ${\pi}(0)$ and transition probability matrix of the system, the probability ${\pi}(n)$ of each state of the system after arbitrary transition can be obtained.

$\displaystyle{\pi}(n)={\pi}(0)\cdot P^{n}$ (12)

Where $\pi(0)=[{\pi_{1}(0),\pi_{2}(0),\ldots,\pi_{n}(0)}]$ , $\pi_{i}(0)$ is the probability that the system is in the state $i$ at the initial time $t=0$ ; $\pi(n)=[{\pi_{1}(n),\pi_{2}(n),\ldots,\pi_{n}(n)}]$ , $\pi_{i}(n)$ stands for the probability that the system is in the state $i$ after $n$ -step transition.

(4) Calculation of the predicted value

The $k$ (the number of states) years closest to the prediction year is selected. According to the closeness to the prediction year, the number of transition steps are respectively defined as $1,2,\ldots,k$ . In the transition matrix corresponding to the transition steps, the row vector corresponding to the initial state is taken, i.e., the occurrence probability of each state, and all probabilities are summed to obtain the system state corresponding to the maximum number of transition steps. Furthermore, the predicted value of the grey GM (1, 1) model is corrected as follows:

$\displaystyle\hat{x}(k)=\hat{x}^{(0)}(k)\left[1+\left(1-\frac{Q_{ui}+Q_{li}}{2% }\right)\right]$ (13)

Where $\hat{x}(k)$ is the corrected value of the grey Markov model.

The test method of grey Markov prediction results is consistent with that of grey prediction, so it will not be hereby repeated.

4. Results analysis and discussion

Taking the statistical data of civil aviation incidents during 2005–2020 released by the official website of the Civil Aviation Administration of China as an example (Table 1) [32] (CAAC, 2005–2020), the civil aviation incidents in 2021 were predicted to verify the effectiveness of the model.

Table 1
Relevant data of civil aviation incidents in China during 2005–2020

Year	Number/occurrence of civil aviation incidents	Total flight time/ 10,000 hours	10,000-hour rate of civil aviation incidents (number of civil aviation incidents/ total flight time)
2005	106	273.3	0.3879
2006	109	321	0.3396
2007	111	368.9	0.3009
2008	103	393.5	0.2618
2009	142	446.8	0.3178
2010	196	509.8	0.3845
2011	207	556.1	0.3722
2012	295	618.9	0.4767
2013	302	691.3	0.4369
2014	324	764.1	0.4240
2015	394	851.6	0.4627
2016	519	949.4	0.5467
2017	587	1059.8	0.5539
2018	568	1153.52	0.4924
2019	570	1231.13	0.4630
2020	440	876.22	0.5022
2021	559	932.16	0.5997

(1) GM (1, 1) prediction model

From the data during 2005–2020 in Table 3, the original time series regarding the 10,000-hour rate of civil aviation incidents can be known as follows:

$\displaystyle X^{(0)}=[0.3879,0.3396,0.3009,0.2618,0.3178,0.3845,0.3722,0.4767% ,0.4369,0.4240,0.4627,0.5467,0.5539,0.4924,0.4630,0.5022]$

A one-time accumulated data sequence is generated to obtain:

$\displaystyle X^{(1)}=[0.3879,0.7275,1.0284,1.2902,1.6080,1.9925,2.3647,2.8414% ,3.2783,3.7023,4.1650,4.7117,5.2656,5.7580,6.2210,6.7232]$

The following matrix is established:

$\displaystyle B=\left[\begin{array}[]{cc}-\frac{1}{2}(x^{(1)}(1)+x^{1}(2))&1\\ \\ -\frac{1}{2}(x^{(1)}(2)+x^{1}(3))&1\\ \\ -\frac{1}{2}(x^{(1)}(3)+x^{1}(4))&1\\ \vdots&\vdots\\ -\frac{1}{2}(x^{(1)}(14)+x^{1}(15))&1\\ \\ -\frac{1}{2}(x^{(1)}(15)+x^{1}(16))&1\end{array}\right]=\left[\begin{array}[]{% cc}-0.5577&1\\ -0.8780&1\\ -1.1593&1\\ -1.4491&1\\ -1.8003&1\\ -2.1786&1\\ -2.6031&1\\ -3.0599&1\\ -3.4903&1\\ -3.9337&1\\ -4.4384&1\\ -4.9887&1\\ -5.5118&1\\ -5.9895&1\\ -6.4721&1\end{array}\right]$ $\displaystyle Y=[{x^{(0)}(2),x^{(0)}(3),x^{(0)}(4),x^{(0)}(5),\ldots,x^{(0)}({% 15}),x^{(0)}({16})}]^{T}=[0.3396,0.3009,0.2618,0.3178,0.3845,0.3722,0.4767,0.4% 369,0.4240,0.4627,0.5467,0.5539,0.4924,0.4630,0.5022]$

The following can be solved:

$\displaystyle\hat{a}=\left[\begin{array}[]{l}a\\ b\end{array}\right]=(B^{T}B)^{-1}B^{T}Y=\left[\begin{array}[]{l}0.039\\ 0.296\end{array}\right]$

Namely, $a=-$ 0.039 and $b=$ 0.296.

Therefore, the time response sequence is acquired as below:

$\displaystyle\hat{x}^{(1)}(k+1)=\left[x^{(0)}(1)-\frac{b}{a}\right]e^{-ak}+% \frac{b}{a}=7.9776e^{0.039k}-7.5897$

The calculation results are listed in Table 2.

Table 2

GM (1, 1) predicted value of 10,000-hour rate of civil aviation incidents during 2005–2020

S/N	Year	Actual value	GM (1, 1) predicted value	Residual ( $e^{(0)}(k)$ )		Relative change rate (%)		Relative error
1	2005	0.3879	0.3879	0.	0000	100.	00	–
2	2006	0.3396	0.3173	0.	0223	107.	03	6.	5538%
3	2007	0.3009	0.3300	$-$ 0.	0291	91.	18	9.	6656%
4	2008	0.2618	0.3431	$-$ 0.	0813	76.	30	31.	0648%
5	2009	0.3178	0.3568	$-$ 0.	0390	89.	07	12.	2704%
6	2010	0.3845	0.3710	0.	0135	103.	64	3.	5092%
7	2011	0.3722	0.3858	$-$ 0.	0136	96.	47	3.	6500%
8	2012	0.4767	0.4012	0.	0755	118.	82	15.	8481%
9	2013	0.4369	0.4171	0.	0198	104.	75	4.	5249%
10	2014	0.4240	0.4337	$-$ 0.	0097	97.	76	2.	2986%
11	2015	0.4627	0.4510	0.	0117	102.	59	2.	5236%
12	2016	0.5467	0.4690	0.	0777	116.	57	14.	2147%
13	2017	0.5539	0.4877	0.	0662	113.	57	11.	9572%
14	2018	0.4924	0.5071	$-$ 0.	0147	97.	10	2.	9842%
15	2019	0.4630	0.5273	$-$ 0.	0643	87.	81	13.	8862%
16	2020	0.5022	0.5483	$-$ 0.	0461	91.	59	9.	1789%

According to Eq. (9), the average relative error of the grey GM (1, 1) model can be calculated as $\bar{\delta}=\frac{1}{n}\sum\limits_{k=1}^{n}{\delta^{(0)}(k)}=$ 9.6087%, and the prediction result has passed the test.

(2) System state interval division

According to the basic idea of the grey Markov model, after the grey prediction model is completed, the system state should be divided according to the relative change rate of data. According to the above data of relative change rate, the system states are divided as shown in Table 3.

Table 3

Division of state intervals

State	State no.	State interval (%)
Strong decline of predicted value	D₁	[76, 86]
Decline of predicted value	D₂	[87, 99]
Rise of predicted value	D₃	[100, 109]
Strong rise of predicted value	D₄	[110, 119]

(3) Construction of a transition matrix

According to the content in Table 3, the states of each year during 2005–2020 are divided, and a line chart of the system state transition is drawn, as shown in Fig. 4.

Figure 4.

Line chart of state transition during 2005–2020.

According to the system state transition diagram, the transition frequency matrix of one-step state transition during 2005–2020 can be obtained as follows:

$\displaystyle N=\left[{{\begin{array}[]{*{20}c}0&1&0&0\\ 1&2&2&1\\ 0&3&1&1\\ 0&1&1&1\\ \end{array}}}\right]$

Further, the system state transition probability matrix can be calculated as follows:

$\displaystyle P^{(1)}=\left[{{\begin{array}[]{*{20}c}0&1&0&0\\ {1/6}&{1/3}&{1/3}&{1/6}\\ 0&{3/5}&{1/5}&{1/5}\\ 0&{1/3}&{1/3}&{1/3}\\ \end{array}}}\right]$

Similarly, the state transition probability matrix of transition for 2, 3, and 4 steps can be calculated as follows:

$\displaystyle P^{(2)}=\left[{{\begin{array}[]{*{20}c}{0.1667}&{0.3333}&{0.3333% }&{0.1667}\\ {0.0556}&{0.5333}&{0.2333}&{0.1778}\\ {0.1000}&{0.3867}&{0.3067}&{0.2067}\\ {0.0556}&{0.4222}&{0.2889}&{0.2333}\\ \end{array}}}\right]$ $\displaystyle P^{(3)}=\left[{{\begin{array}[]{*{20}c}{0.0556}&{0.5333}&{0.2333% }&{0.1778}\\ {0.0889}&{0.4326}&{0.2837}&{0.1948}\\ {0.0644}&{0.4818}&{0.2591}&{0.1947}\\ {0.0704}&{0.4474}&{0.2763}&{0.2059}\\ \end{array}}}\right]$ $\displaystyle P^{(4)}=\left[{{\begin{array}[]{*{20}c}{0.0889}&{0.4326}&{0.2837% }&{0.1948}\\ {0.0721}&{0.4682}&{0.2659}&{0.1938}\\ {0.0803}&{0.4454}&{0.2773}&{0.1970}\\ {0.0746}&{0.4539}&{0.2730}&{0.1985}\\ \end{array}}}\right]$

(4) Results

Because the state was divided into 4 state intervals, the 4 years closest to 2021, namely 2020, 2019, 2018, and 2017, were selected to prepare the prediction table, as shown in Table 4.

Table 4

Prediction table of 10,000-hour rate of civil aviation incidents

Initial year	Initial state	Number of transition steps	10,000-hour rate probability of civil aviation incidents				Probability source
			State 1	State 2	State 3	State 4
2017	D₄	4	0.0746	0.4539	0.2730	0.1985	$P^{(4)}$
2018	D₂	3	0.0889	0.4326	0.2837	0.1948	$P^{(3)}$
2019	D₂	2	0.0556	0.5333	0.2333	0.1778	$P^{(2)}$
2020	D₂	1	0.1667	0.3333	0.3333	0.1667	$P^{(1)}$
2021			0.3858	1.7531	1.1233	0.7378	–

It could be seen from the above table that the system was most probably at state 2 in 2021, i.e., “decline of predicted value”.

Given that the predicted value of GM (1, 1) in 2021 was 0.570, the corrected value of grey Markov is:

$\displaystyle\hat{x}(k)=\hat{x}^{(0)}(k)*\left[1+\left(1-\frac{Q_{ui}+Q_{li}}{% 2}\right)\right]=0.570*\left[1+\left(1-\frac{0.99+0.87}{2}\right)\right]=0.6099$

By comparatively analyzing the relative errors produced by the two prediction methods (Table 5), it could be seen that the results predicted by the grey Markov model were more accurate and could provide more accurate decision-making reference for civil aviation safety management and accident prevention.

Table 5

Comparison between GM (1, 1) and grey Markov model in prediction results

Year	Actual value	Prediction results of GM (1, 1) model		Prediction results of grey Markov model
		Predicted value	Relative error	Predicted value	Relative error
2021	0.5997	0.5700	4.95%	0.6099	1.70%

5. Conclusion

This study mainly aims to establish a prediction model for the 10,000-hour rate of civil aviation incidents in China. Based on the statistical data of civil aviation incidents during 2005–2020, a grey GM (1, 1) prediction model was established, and on this basis, a grey Markov prediction model was established, and the two models were combined to predict the 10,000-hour rate of civil aviation incidents in 2021. The results showed that the predicted value of the grey GM (1, 1) model was 0.5700 with a relative error of 4.95% and that of the grey Markov model was 0.6099 with a relative error of 1.70%, so the prediction accuracy of the grey Markov model was higher than that of the GM (1, 1) model, and the former conformed to data change laws more. To sum up, despite the suitability for not high requirements for the data sample size, the grey prediction model reaches relatively low prediction accuracy when used to cope with arrays of strong fluctuations. The Markov model can better solve the prediction problem in case of great fluctuations. If combined, the two methods can enhance the model accuracy and be more helpful for mastering the change laws of civil aviation incidents, providing a reliable basis for aviation safety management and accident prevention.

Footnotes

Acknowledgments

This work was supported by Science and Technology Planning Project of Henan Province (232102320049, 232102320011, 232102240016) Training Program for Young Core Teachers in University of Henan Province (2020GGJS174).

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Prediction of 10,000-hour rate of civil aviation incidents based on grey Markov model

Abstract

Keywords

1. Introduction

3. Theory and method

3.1 Grey theory

3.2 Markov theory

3.3 Grey Markov theory

(1) System state analysis

(2) Construction of a transition matrix

(3) System state prediction

(4) Calculation of the predicted value

Table 1 Relevant data of civil aviation incidents in China during 2005–2020

(1) GM (1, 1) prediction model

(2) System state interval division

(3) Construction of a transition matrix

(4) Results

Footnotes

Acknowledgments

References

Table 1
Relevant data of civil aviation incidents in China during 2005–2020