Principal component analysis is a multivariate statistical technique that transforms a set of possible correlation variables into a set of linearly uncorrelated variables by orthogonal transformation. The converted set of variables is called the principal component. The basic idea is to reduce the dimensionality of the original variable data to obtain several principal component integrated variables that are not related to each other instead of a large number of original variables, and these integrated variables carry most of the information in the original variables [3]. The first comprehensive variable selected is denoted as F₁, and F₁ has the largest Var(F₁), which means that F₁ contains the largest amount of information, and F₁ is called the first principal component. If the first principal component is insufficient to represent the information carried by the original p variables, then the second principal component F₂ is selected, and F₂ is independent of F₁ linear, and the mathematical language expression requires Cov(F₁, F₂) = 0. By analogy, the third principal component and the fourth principal component can be constructed up to the No. p principal component.

This article uses related software to perform principal component analysis. The main steps are as follows:

1) Normalizing raw data.

In order to make the indicators comparable, the first thing is eliminating the different dimensions of each indicator and standardizing the indicators to obtain standardized data. This standardized process is actually doing the following transformation on the raw data X:

Z X i j = x i j − x j ¯ V a r ( x j )   ( i = 1 , 2 , 3 , ⋯ , n , j = 1 , 2 , ⋯ , p )

in which:

x j ¯ = ∑ i = 1 n x i j n ,     V a r ( x j ) = ∑ i = 1 n ( x i j − x j ¯ ) 2 n − 1

2) Calculating the correlation coefficient matrix of a normalized data matrix.

3) Typing the output result can directly obtain the eigenvalue and the corresponding eigenvector.

4) Calculating the variance contribution rate and the cumulative contribution rate of each principal component.

The contribution rate of the main component F_i:

α i = λ i ∑ k = 1 p λ k   ( i = 1 , 2 , ⋯ , p )

Cumulative contribution rate:

β = ∑ k = 1 m λ k ∑ k = 1 p λ k

In the practical application of the principal component analysis method, the corresponding first, second, …, m-th main components to the λ 1 , λ 2 , ⋯ , λ m ( m ≤ p ) are chosen, in which the, λ 1 , λ 2 , ⋯ , λ m ( m ≤ p ) of the eigenvalues must be the those that cumulative contribution rate is higher than 60%.

5) Calculating the principal component coefficients and principal component scores.

Let the load matrix be A, then the coefficient of the principal component F_i is the square root of each load matrix divided by the square root of the corresponding principal component variance, and propose the coefficient of F_i is the matrix C_i, then the score of the principal component F_i is:

Z i = C i × Z X i

where ZX_i represents the i-th column of the matrix ZX of data normalization.

6) Comprehensive score assessment.

The first m principal components with cumulative contribution rate of 60% are selected, and take the variance contribution rate α k ( k = 1 , 2 , ⋯ , m ) as the weight to construct a linear combination as a comprehensive evaluation function:

R = α 1 Z 1 + α 2 Z 2 + ⋯ + α m Z m

From the above formula, the comprehensive score R of the evaluation can be obtained, and then the magnitude of the R value is calculated according to the value of each data of each evaluation object and these R values are comprehensively sorted, thereby obtaining a comprehensive evaluation of each object to be evaluated [5].

According to the overseas railway macro-level risk assessment indicators established in Section 2, here are 15 indicators in total, the indicator data are from the World Bank (https://www.worldbank.org/) and the China National Bureau of Statistics. In order to increase the reliability of the data, this paper selects 63 countries (All the Belt and Road project routes and participating countries) as samples, because there is no direct relationship between the risks, and the difference between the dimensions Large, it is necessary to standardize the various risk data, and use the unified standard to judge, so the original data is first standardized, and the standardized data is shown in Table 2 [6].

The normalized data is used in the dimension reduction factor analysis, and the data is subjected to KMO and Bartley test. If the result of KMO value is greater than 0.6 and the significance of the Bartley test is less than 0.01, principal component analysis or factor analysis can be performed. Because the amount of samples are huge, so on the basis of principal component analysis, the rotation of the factors is actually rotating the factor load matrix, which can simplify the structure of the factor load matrix, so that the square of the element of each column or row in the load matrix is polarized to 0 and 1, through the factor rotation (actually coordinate rotation), it makes each original variable have a close relationship between as few factors as possible, so the actual meaning of the factor solution is easier to explain [7].

Then, the factor analysis tool is used for dimensionality reduction. Based on the principal component analysis, the maximum variance method is used to perform the factor rotation, and the result Table 3 is obtained.

It can be seen from the above test results that the KMO value is greater than 0.6 and the significance is less than 0.01, so it is suitable for principal component analysis or factor analysis.

As can be seen from Table 4, the first five principal components contain nearly 66% of the information, and it can be considered that these five principal components contain most of the information of the original elements.

The load matrix after the rotation of these five principal components is shown in Table 5, the coefficient indicating the risk of each component, generally greater than 0.5 - 0.6, is attributed to the component.

The above data was processed to obtain Table 6, where the gray shading marks were the portions with coefficients greater than 0.58.

Name each principal component according to the data marked in the below table.

The fifth item (power coverage rate) and the 12th item of urbanization rate in F₁ have large coefficients. These two indicators are related to the level of urban development. Therefore, F₁ is called “the main component of urban modernization level”;

The 14th (political stability) and 15th (government efficiency) factors in F₂ are relatively large. Both of these indicators are related to the political situation, so F₂ is called “the main component of the political environment”;

The third item in F₃ (the turnover of China’s foreign contracted projects) and the 13th (the establishment of cooperative relations with China) have a large coefficient. These two indicators are related to bilateral cooperation, so F₃ is called “bilateral with China”. The main component of the partnership;

The coefficient of item 10 (population growth rate) in F₄ is relatively large, and F₄ is called “the main component of population development trend”;

The coefficient of item 7 (traffic accident rate) in F₅ is relatively large, and F₅ is called “main component of traffic safety index”.

In summary, the five main components of national line selection risk are shown in Table 7.

Next, each column of the load matrix is divided by the square root of the variance of the corresponding principal component, and the coefficient of each principal component is obtained, and the matrix is denoted as A; then the variance matrix is normalized, which are the Weights of each principal component, this is regarded as matrix B.

Matrix B:

B = ( 0.27929 0.21273 0.18081 0.17293 0.15424 )

Matrix A:

A = ( − 0.47136 0.03028 0.10537 0.13193 − 0.04228 0.02501 − 0.49417 0.15823 − 0.11215 0.17072 0.07624 0.08342 0.62446 0.02401 − 0.04938 − 0.25111 0.36531 0.11270 0.13971 0.23499 0.42956 0.08841 0.01350 − 0.02911 0.00320 − 0.29322 − 0.01949 0.19150 0.15453 0.13921 − 0.01022 − 0.09820 0.27012 0.25967 0.55058 − 0.00918 − 0.26565 0.14840 − 0.50938 0.06616 0.04578 0.05147 0.14674 0.16255 − 0.66155 − 0.14158 − 0.12123 0.04950 0.62279 0.04328 − 0.22008 − 0.09239 0.40058 − 0.19332 − 0.12596 0.43733 0.11563 0.13456 0.36634 − 0.11936 − 0.12579 − 0.23893 0.44631 0.03156 0.19908 0.20211 0.50417 − 0.01946 − 0.10390 − 0.00303 0.34499 0.41885 0.16485 − 0.02726 − 0.27457 )

Let the data after standardization be matrix X, then the composite score of the sample countrie: S = X × A × B .

The results are generally irregular, for example, South Korea is 1.6243, and there is a negative number in the score, which only indicates that the score is lower than the average. Since this score does not intuitively judge the country’s risk, it is converted into a familiar percentage system score [8].

Suppose:

M = f ( S ) = 1 1 + exp ( − S )

Then control the score between (0,1), then suppose:

R = M × 60 + 40

The score can be converted to a percentile, as shown in Table 8:

In response to the planned plan for the passage of the Asia-Europe Railway, experts proposed several planning options, of which the two most important programs in the Middle East [9] (Figure 1 from China Map Network [4] ) are:

1) Channel plan one: China-Kyrgyzstan-Tajikistan-Afghanistan-Iran, finally arrived in Germany;

2) Channel plan two: China-Kazakhstan-Kyrgyzstan-Uzbekistan-Turkmenistan-Iran, finally arrived in Germany.

According to the risk score rankings in Table 8, it can be seen that in the plan 1, Tajikistan and Afghanistan rank at the bottom, while in the plan 2, although Uzbekistan and Kyrgyzstan are ranked lower. However, compared with the plan 1, the advantages are obvious; the total score of the country of plan 2 is much higher than that of plan 1, so the plan 2 was chosen as the Middle East connectivity scheme.