Bearing trains with heavy weight axle, the heavy haul railway line subjects to a larger load, which affects its regularity [13] . The direct results are the changes of the geometric dimensions [14] , such as deviated track height and gauge by different degrees, leading to stress concentration on local track structure. Consequently, rail surface fatigue defects would be aggravated [15] , such as spalling and scaling [16] . As the connecting part of rails, the fasteners [17] would break, lose or shift under the lateral force of the wheel [18] , aggravating the geometric state degradation of the track and causing safety issues; Placed under rails, the sleeper [19] is to support and maintain the geometric shape and position between the rails. Under the impact of the train and the open air environment, the wooden sleepers are prone to aging, corrosion, breakage, etc. As the under-rail structure, the ballast-subgrade [20] is influenced by train operation [21] , its geographical location and natural environment [22] , therefore prone to damages such as smudging, ballast compaction, and settlements [23] [24] . This will cause the rails to change geometry, leading to a loss of regularity. It can even result in derailment accidents [25] . Furthermore, with an increase in the total mass of the train [26] , the quantity and growth rate of rail damages ascend nonlinearly [27] . As a consequence, the rail service life will be reduced. Both the rail status and train operation quality will be affected by those damages. Hence, the status of heavy haul railway line is not the state of a single structure, but the combined condition of various foundation structures.

According to the above discussion, the heavy haul line state is mainly composed of three kinds of status: rail geometry status, track structure status and operating status. The rail geometry includes elevation, gauge, alignment, level, warp, curvature, TQI, etc.; the track structure status involves rail system, subgrade ballast system, connecting part system, sleeper system, etc. The main defects of them affecting the rail status concerns the rail surface defects of the rail system, fastener missing of the connecting part system, broken sleepers of the sleeper system, deviated ballast bed thickness and ballast bed settlement of the subgrade ballast system For the operating status, the main indicator is the total mass of passing trains.

Considering the requirements of TG/GW102-2019 “Rules for repair of general speed railway lines” of China, “Comprehensive inspection vehicle of Shuo-Huang heavy haul railway line” books and so on, a set of variables is constructed for the status evaluation of heavy haul railway line. It takes 3 kinds of status into account, i.e. the rail geometry, the track structure and the operating system. And it has 3 layers, i.e. status, system, variable, as shown in Table 1.

At present, the deduction score system stipulated by industry standards is used to evaluate all variables in China. The basic evaluation variables for heavy haul railway line are mostly derived from inspections. Not only there is no unified standard for such inspection, but also some variables have little impact on the status of heavy haul railway line. Thus, evaluating all variables is not necessary and will reduce the efficiency. In this paper, key variables are selected out for the status evaluation of heavy haul railway line. In order to evaluate the line more reasonably, the association rule is applied. According to the definition of support and confidence in the association rule, the four levels of state evaluation of heavy haul railway line over the years, namely Level I, Level II, Level III and Level IV, are quantified into a set of 4-dimensional arrays to build a quantification matrix.

The concept of association rule is to find an interrelated relationship between things and factors, reflecting the interdependence and relevance between them. It is defined that I = { i 1 , i 2 , i 3 , ⋯ , i m } is a finite item set composed of M items to be studied, and T = { T 1 , T 2 , T 3 , ⋯ , T N } is the transaction data table, where T i = { i 1 , i 2 , i 3 , ⋯ , i k } ⊂ I , called k-item set. X and Y are subsets that appear in finite itemset and k itemset.

The association rule has two basic measures: support and confidence. Support S is defined as the probability of X and Y appearing simultaneously in a transaction. It is calculated by ratio of transactions with X and Y item appearing simultaneously in the sample data set I to total transactions. It reflects the probability of X and Y appearing simultaneously. The equation is:

S ( X → Y ) = | T ( X ∨ Y ) | | T | (1)

where: | T ( X ∨ Y ) | indicates transactions contain X and Y at the same time. | T | represents the total number of transactions.

Confidence C of the association rule is used to indicate the dependence of the consequent item Y to the preceding item X. It is calculated as the ratio of Y appearing in the transactions with X item, as follows:

C ( X → Y ) = | T ( X ∨ Y ) | | T ( X ) | (2)

where: | T ( X ∨ Y ) | means the quantity of transactions with both X and Y. And | T ( X ) | indicates number of transactions with X. Confidence of association rule is a relative variable that measures the accuracy of association rule. The higher this value, the higher the dependent possibility of Y to X.

Confidence can be used to quantify the correlation between each basic variable and the operation status of heavy haul railway line. The higher the confidence, the stronger the correlation between the variable and the actual operation. Taking the quantification of the initial variables of the rail according to the state statistics over the years as an example, there is:

1) Transaction database I = {Operation and maintenance status of heavy haul railway line}

2) Itemset X_i,j = {damage occurrence on the jth variable of the ith system}

3) Itemset Y_i = {damage occurrence in the ith system}

By Equation (1), the support of each variable can be calculated first:

S ( X i , j → Y i ) = P ( X i , j ∪ Y i ) = σ ( X i , j ∪ Y i ) | I | × 100 % (3)

where, X i , j and Y i are elements of itemset X and Y. S ( X i , j → Y i ) is the support of X i , j and Y i appearing simultaneously. P ( X i , j ∪ Y i ) is the conditional possibility of I containing both X i , j and Y i . σ ( X i , j ∪ Y i ) is the support count of X i , j and Y i . Based on support, confidence C of the variable is obtained according to Equations (2) and (3). The calculation formula is as follows:

C ( X i , j → Y i ) = P ( X i , j ∪ Y i ) P ( X i , j ) = σ ( X i , j ∪ Y i ) / | I | σ ( X i , j ) / | I | = σ ( X i , j ∪ Y i ) σ ( X i , j ) × 100 % (4)

where, C ( X i , j → Y i ) indicates the confidence of X i , j and Y i appearing at the same time. It is an element of the matrix C. P ( X i , j ) gives the possibility of I containing X i , j . σ ( X i , j ) is the support count of X i , j . Its meaning is the same as defined by Equation (3). Taking the first three variables of the rail (spalling and chipping, top surface scratches, squat damage) as an example, the overall operation and maintenance status of the rail is expressed as I, the 3 variables are given as X 2 , 1 , X 2 , 2 , X 2 , 3 , and the occurrence of rail damage is recorded as Y 2 .

In this paper, the operation and maintenance status data of a certain line from 2015 to 2019 is used.

According to its statistics, there 483 times of damage recordings in total for its Level III state, including 107, 208 and 338 times of spalling, corrugation wear and rail side wear, respectively. While for the rail system state, the recorded damages 234 times, 15, 36 and 103 times for spalling, corrugation wear and rail side wear, respectively.

According to Equation (3), there is | Y 2 | = 234 , σ X 2 , 1 = 107 , σ X 2 ， 2 = 208 , σ X 2 , 3 = 338 , σ X 2 , 1 ∪ Y 2 = 15 , σ X 2 , 2 ∪ Y 2 = 36 , and σ X 2 , 3 ∪ Y 2 = 103 .

According to Equation (4), confidence of the variable spalling can be calculated as C 2 , 1 .

C 2 , 1 = P ( X 2 , 1 ∪ Y 2 ) P ( X 2 , 1 ) × 100 % = 15 / 483 107 / 483 = 14 %

The confidence values of corrugation wear C 2 , 2 and rail side wear C 2 , 3 can be obtained in the same way. And the results were 17.31% and 30.18% respectively. This calculation method was then applied also to confidence of other variables of the rail system. Combining the statistic databases of Level I, Level II, Level III, and Level IV, confidence values of rail systems were calculated, as shown in Figure 1.

The variables appeared in Figure 1 are listed out in the following Table 2 with their sequence number.

Too many variables will increase the difficulty and complexity of status evaluation to a certain extent. In order to reasonably extract key variables, principal component analysis is used to extract variables and remove redundant variables, so as to establish key variables for state evaluation. The extracting of key variables by principal component analysis is carried out as follows:

Step 1: Convert confidence values in Figure 1 into a matrix for standard orthogonalization. A correlation coefficient matrix R = X X T is constructed and the singular value decomposition is carried out, and then the eigenvalues and eigenvectors of the correlation coefficient matrix are obtained. The eigenvalues are ranked according to their size and expressed λ 1 ≥ λ 2 ≥ ⋯ ≥ λ p ≥ 0 . α 1 , α 2 , ⋯ , α p is the eigenvector corresponding to the eigenvalue.

Step 2: Determine the number of principal components. The cumulative contribution rate C R i of the eigenvalue λ i is defined as:

C R i = ∑ k = 1 i λ k ∑ k = 1 p λ k ( i = 1 , 2 , ⋯ , n ) (5)

The eigenvalues with the cumulative contribution rate in 85% - 95% are selected. Provided that the number of them is m, it is defined that m eigenvalues can represent the information of the original p basic variables, i.e. F = ( F 1 , F 2 , ⋯ , F m ) T

F = A X = ( λ 1 α 1 , λ 2 α 2 , ⋯ , λ m α m ) T ⋅ ( X 1 , X 2 , ⋯ , X m ) T (6)

where: ( λ 1 α 1 , λ 2 α 2 , ⋯ λ m α m ) T is the factor loading matrix, α 1 , α 2 , ⋯ , α m , and λ 1 , λ 2 , ⋯ , λ m is the eigenvector corresponds to eigenvalue.

Step 3: Calculate the comprehensive score of principal components. By weighted summation of principal components whose cumulative contribution rate meets the requirements, the comprehensive score is calculated as follows:

F ^ = ω F = ω A X = ( λ 1 α 1 , λ 2 α 2 , ⋯ , λ m α m ) T ⋅ ( X 1 X 2 , ⋯ , X m ) ​ ​ T = ( λ 1 α 1 + λ 2 α 2 + ⋯ + λ m α m ) T ( X 1 , X 2 , ⋯ , X p ) T (7)

ω = ( ω 1 , ω 2 , ⋯ , ω m ) is the weight of the principal component in the comprehensible score. The weight vector H of basic variables is:

H = ( h 1 , h 2 , ⋯ , h p ) = ( λ 1 α 1 + λ 2 α 2 + ⋯ + λ m α m ) T (8)

According to the above steps, the confidence data of the rail system was input into MATLAB operation, and the eigenvalues obtained after standardization, orthogonalization and singular value decomposition of the matrix were as follows:

λ 1 = 3.02 , λ 2 = 0.498 , λ 3 = 0.349 , λ 4 = 0.133 , λ 5 = 0 , λ 6 = 0 , ⋯ , λ 24 = 0

According to Equation (6), the contribution rates of the 1st - 4th principal components were 75.495%, 12.449%, 8.733% and 3.324% respectively. The cumulative contribution rate of the 1st and 2nd principal components was 87.944%, which is in the 85% to 95% confidence range required by the algorithm. Therefore, the eigenvector corresponding eigenvalue λ 1 , λ 2 can be used to calculate the weight of each variable in the comprehensive score. Nine variables with a weight greater than 0.5 were extracted as the key evaluation variables of rail condition, such as spalling and chipping, flow of rail head or working edge, rail joint malocclusion on the top or inner side, rail vertical wear, rail side wear, scaling, rail head transverse damage, abnormal light band and screw hole transverse crack, as shown in Figure 2. The principal component analysis method was then applied to the whole initial variable system, and key variables of each sub-system were extracted. A total of 30 key variables were obtained. Finally, the evaluation system with key variables for heavy haul railway line was established, and shown in Table 3.