# Reliability Investigation of Pavement Performance Evaluation Based on Blind-Number Theory: A Confidence Model

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Testing Methods for Pavement Performances

#### 2.1. Evaluation Indexes and Data Collection Methods

_{PQI}, w

_{RQI}, w

_{RDI}, and w

_{SRI}are the calculated weights of PCI, RQI, RDI, and SRI, respectively.

#### 2.2. Probability Distribution Analyses of Evaluation Indexes

_{0}represents the assumption that the observed data conform to the specified distribution, while H

_{1}suggests that the data deviate from the specified distribution.

_{1}, …, x

_{n}denote a series of random samples extracted from the theoretical probability density function F(x), with n representing the number of sample sequences. Based on the predetermined significance level (typically 0.05) in the hypothesis test, the critical value is determined by referencing the appropriate table. If the test statistic D exceeds the critical value, the null hypothesis H

_{0}is rejected; otherwise, it is accepted.

_{n}(x) represents the empirical cumulative probability density function. By comparing the test statistic A

^{2}with the critical value size of each distribution cluster, it is determined at the given significance level α (e.g., 0.01, 0.05, etc.) whether to accept or reject the null hypothesis H

_{0}.

_{i}is the observed frequency of the sample falling in the i-th interval, whereas E

_{i}signifies the expected frequency of the sample falling within the same interval. F(x) denotes the probability density function of the calculated sample, and x

_{1}and x

_{2}specify the range of interval i.

#### 2.3. Analysis Framework Pavement Performance Reliability

#### 2.3.1. Principle of Blind-Number Theory

_{i}, where each a

_{i}belongs to H(I). If α

_{i}∊ [0, 1] for i = 1, 2, …, n, the gray function in H(I) can be defined as f(x), as shown in Equation (2). If i ≠ j, a

_{i}≠ a

_{j}, and ∑

^{n}α

_{i}= α ≤ 1, the f(x) should be called a blind number, which can be expressed by Equation (3) [30].

_{i}is the confidence of the value a

_{i}of f(x); α is the total confidence of f(x). The greater n, the greater the accuracy of the blind number f(x).

_{1}, x

_{2}, …, x

_{k}and y

_{1}, y

_{2}, …, y

_{n}as lists of real numbers in descending order, referred to as the sequence of possible values for A^ and B^. The vertical side of a matrix is denoted by x

_{1}, x

_{2}, …, x

_{k}, while the horizontal edge is represented by y

_{1}, y

_{2}, …, y

_{m}. The bounded matrix is perpendicular horizontal and vertical axes, and the possible values of A^ and B^ are associated with the edges and matrices, as depicted in Table 2.

_{1}), f(x

_{2}), …, f(x

_{k}) and g(y

_{1}), g(y

_{2}), …, g(y

_{k}) represent the vertical and horizontal sides of the matrix with an edge, referred to as the confidence sequence of A^ and B^. The horizontal and vertical axes of the matrix with an edge are perpendicular lines. The resulting confidence matrix obtained by performing an edge product on A^ and B^ is presented in Table 3.

#### 2.3.2. Confidence Modeling of Pavement Quality Index

## 3. Analysis of Examples

#### 3.1. Statistical Result Analysis

#### 3.2. Confidence Analysis Based on Blind-Number Theory

^{^}results do not align with those obtained using the current standard. The PQI

^{^}interval determined by the current standard is [83.64, 92.94], whereas the PQI interval computed is [73.32, 91.01]. Three of the judgment values for the optimal pavement performance evaluation grade exceed 1, indicating that the confidence of judgment values greater than 1 in the sample group is 0.075. There are 36 judgment values for the good pavement performance evaluation grade that surpass 1, signifying a confidence of 0.9 or 90% for judgment values greater than 1 in the sample group. Moreover, confidence is considered as 1 for values exceeding 0.9. According to the current standard, the evaluation outcome indicates that 30 samples are classified as good, while 10 samples are deemed excellent. For instance, considering sample 14, the current standard assigns a PQI evaluation result of 83.64, corresponding to a good evaluation grade. However, when employing the information entropy weight determination calculation for PQI

^{^}, the result is 73.32, indicating a medium evaluation grade. These evaluation results differ significantly.

## 4. Conclusions and Outlook

- (1)
- The Pavement Condition Index (PCI), Riding Quality Index (RQI), Rutting Depth Index (RDI), and Skidding Resistance Index (SRI) of pavement facilities do not exhibit complete adherence to the normal distribution. Furthermore, the probability distribution of the pavement performance evaluation index differs.
- (2)
- A blind-number expression for the pavement performance evaluation index is developed in this study. Additionally, a confidence model for analyzing the confidence of pavement performance is constructed using the method of determining the weight information entropy weight of the pavement performance evaluation index. The model effectively integrates blind information into the pavement performance evaluation system, making it independent of the probability distribution function.
- (3)
- Compared to the traditional method, the proposed confidence model for pavement performance confidence analysis has several advantages. Firstly, it provides a clear evaluation level for pavement performance, allowing for a more precise assessment. Secondly, it also assigns corresponding credibility to the evaluation level, which enhances the scientific and rational nature of the evaluation process. This improvement ensures that the evaluation takes into account the confidence of the data and the assessment results.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Comparison of probability density of pavement performance indexes and fitting analysis results: (

**a**) PCI; (

**b**) RQI; (

**c**) RDI; (

**d**) SRI.

**Figure 2.**Cumulative probability density function of pavement performance evaluation indexes: (

**a**) PCI; (

**b**) RQI; (

**c**) RDI; (

**d**) SRI.

**Figure 3.**Comparison of measured values and fitting values of pavement performance indexes: (

**a**) PCI; (

**b**) RQI; (

**c**) RDI; (

**d**) SRI.

Type | Test Method |
---|---|

Kolmogorov–Smirnov Test | $D=\underset{1\le i\le n}{\mathrm{max}}(F({x}_{i})-\frac{i-1}{n},\frac{i}{n}-F({x}_{i}))$ |

Anderson–Darling Goodness Test | ${A}^{2}=-n-\frac{1}{n}{\displaystyle \sum _{i=1}^{n}(2i-1)\cdot \left[\mathrm{ln}F({X}_{i})+\mathrm{ln}(1-F({X}_{n-i+1}))\right]}$ |

Chi-Squared Goodness Test | ${\chi}^{2}={\displaystyle \sum _{i=1}^{k}\frac{{({O}_{i}-{E}_{i})}^{2}}{{E}_{i}}}$, ${E}_{i}=F({x}_{2})-F({x}_{1})$ |

${x}_{1}$ | ${x}_{1}+{y}_{1}$ | ${x}_{1}+{y}_{2}$ | $\cdots $ | ${x}_{1}+{y}_{j}$ | $\cdots $ | ${x}_{1}+{y}_{m}$ |

${x}_{2}$ | ${x}_{2}+{y}_{1}$ | ${x}_{2}+{y}_{2}$ | $\cdots $ | ${x}_{2}+{y}_{j}$ | $\cdots $ | ${x}_{2}+{y}_{m}$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

${x}_{i}$ | ${x}_{i}+{y}_{1}$ | ${x}_{i}+{y}_{2}$ | $\cdots $ | ${x}_{i}+{y}_{j}$ | $\cdots $ | ${x}_{i}+{y}_{m}$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

${x}_{k}$ | ${x}_{k}+{y}_{1}$ | ${x}_{k}+{y}_{2}$ | $\cdots $ | ${x}_{k}+{y}_{j}$ | $\cdots $ | ${x}_{k}+{y}_{m}$ |

${y}_{1}$ | ${y}_{2}$ | $\cdots $ | ${y}_{j}$ | $\cdots $ | ${y}_{m}$ |

$f({x}_{1})$ | $f({x}_{1})g({y}_{1})$ | $f({x}_{1})g({y}_{2})$ | $\cdots $ | $f({x}_{1})g({y}_{i})$ | $\cdots $ | $f({x}_{1})g({y}_{m})$ |

$f({x}_{2})$ | $f({x}_{2})g({y}_{1})$ | $f({x}_{2})g({y}_{2})$ | $\cdots $ | $f({x}_{2})g({y}_{i})$ | $\cdots $ | $f({x}_{2})g({y}_{m})$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

$f({x}_{i})$ | $f({x}_{i})g({y}_{1})$ | $f({x}_{i})g({y}_{2})$ | $\cdots $ | $f({x}_{i})g({y}_{i})$ | $\cdots $ | $f({x}_{i})g({y}_{m})$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

$f({x}_{k})$ | $f({x}_{k})g({y}_{1})$ | $f({x}_{k})g({y}_{2})$ | $\cdots $ | $f({x}_{k})g({y}_{i})$ | $\cdots $ | $f({x}_{k})g({y}_{m})$ |

$g({y}_{1})$ | $g({y}_{2})$ | $\cdots $ | $g({y}_{i})$ | $\cdots $ | $g({y}_{m})$ |

Serial Number | Distribution Function | Kolmogorov–Smirnov | Anderson–Darling | Chi-Squared | |||
---|---|---|---|---|---|---|---|

Statistic | Sort | Statistic | Sort | Statistic | Sort | ||

PCI | Log-Logistic | 0.10416 | 5 | 2.5466 | 5 | 18.017 | 6 |

Log-Logistic (3P) | 0.05657 | 1 | 1.3167 | 1 | 8.0729 | 2 | |

Logistic | 0.07701 | 2 | 1.6292 | 2 | 6.5873 | 1 | |

Lognormal | 0.10857 | 6 | 3.0129 | 6 | 17.641 | 5 | |

Lognormal (3P) | 0.10026 | 4 | 2.5372 | 4 | 13.059 | 4 | |

Normal | 0.09952 | 3 | 2.3536 | 3 | 13.057 | 3 | |

RQI | Log-Logistic | 0.08961 | 5 | 2.83 | 6 | 14.394 | 3 |

Log-Logistic (3P) | 0.06286 | 1 | 1.6038 | 1 | 10.768 | 1 | |

Logistic | 0.09492 | 6 | 2.3974 | 2 | 11.416 | 2 | |

Lognormal | 0.08381 | 4 | 2.8015 | 5 | 19.881 | 6 | |

Lognormal (3P) | 0.07966 | 3 | 2.6147 | 4 | 18.457 | 5 | |

Normal | 0.07928 | 2 | 2.4687 | 3 | 15.721 | 4 | |

RDI | Log-Logistic | 0.169 | 6 | 9.2652 | 6 | 79.025 | 6 |

Log-Logistic (3P) | 0.10865 | 1 | 4.6751 | 1 | 41.345 | 1 | |

Logistic | 0.16246 | 4 | 6.3528 | 2 | 60.395 | 2 | |

Lognormal | 0.16794 | 5 | 8.9946 | 5 | 77.588 | 5 | |

Lognormal (3P) | 0.15262 | 2 | 6.606 | 4 | 63.712 | 4 | |

Normal | 0.15298 | 3 | 6.4046 | 3 | 62.267 | 3 | |

SRI | Log-Logistic | 0.09108 | 6 | 2.1105 | 6 | 5.8546 | 1 |

Log-Logistic (3P) | 0.0554 | 1 | 1.2219 | 1 | 8.7292 | 5 | |

Logistic | 0.07973 | 2 | 1.8442 | 4 | 10.447 | 6 | |

Lognormal | 0.08903 | 5 | 2.031 | 5 | 7.0076 | 4 | |

Lognormal (3P) | 0.08081 | 3 | 1.7982 | 3 | 6.5312 | 3 | |

Normal | 0.08172 | 4 | 1.6595 | 2 | 6.3648 | 2 |

No. | PQI (Grade) | PQI^{^} (Grade) | K_{PQI}^{^} (90) | K_{PQI}^{^} (80) |
---|---|---|---|---|

1 | 83.64 | 73.31757 | 0.81464 | 0.91647 |

2 | 83.79 | 76.23695 | 0.847077 | 0.952962 |

3 | 84.89 | 76.85353 | 0.853928 | 0.960669 |

4 | 85.98 | 78.84341 | 0.876038 | 0.985543 |

5 | 84.90 | 80.23577 | 0.891509 | 1.002947 |

6 | 87.01 | 81.69958 | 0.907773 | 1.021245 |

7 | 84.92 | 82.01217 | 0.911246 | 1.025152 |

8 | 85.45 | 82.64352 | 0.918261 | 1.033044 |

9 | 84.96 | 82.85526 | 0.920614 | 1.035691 |

10 | 87.14 | 82.93161 | 0.921462 | 1.036645 |

11 | 84.72 | 83.26604 | 0.925178 | 1.040826 |

12 | 87.61 | 83.40383 | 0.926709 | 1.042548 |

13 | 87.67 | 83.48033 | 0.927559 | 1.043504 |

14 | 86.60 | 84.10160 | 0.934462 | 1.051270 |

15 | 86.36 | 84.60134 | 0.940015 | 1.057517 |

16 | 87.76 | 84.83564 | 0.942618 | 1.060446 |

17 | 87.91 | 84.84281 | 0.942698 | 1.060535 |

18 | 87.86 | 85.12389 | 0.945821 | 1.064049 |

19 | 88.06 | 85.74850 | 0.952761 | 1.071856 |

20 | 89.21 | 86.82397 | 0.964711 | 1.085300 |

21 | 89.70 | 87.03685 | 0.967076 | 1.087961 |

22 | 89.23 | 87.35512 | 0.970612 | 1.091939 |

23 | 89.22 | 87.47873 | 0.971986 | 1.093484 |

24 | 88.56 | 87.48089 | 0.972010 | 1.093511 |

25 | 89.42 | 87.75188 | 0.975021 | 1.096899 |

26 | 88.19 | 88.27680 | 0.980853 | 1.103460 |

27 | 89.83 | 88.34652 | 0.981628 | 1.104331 |

28 | 89.83 | 88.36494 | 0.981833 | 1.104562 |

29 | 90.35 | 88.42565 | 0.982507 | 1.105321 |

30 | 90.16 | 88.50354 | 0.983373 | 1.106294 |

31 | 89.74 | 88.56722 | 0.984080 | 1.107090 |

32 | 90.57 | 88.84937 | 0.987215 | 1.110617 |

33 | 89.18 | 88.98633 | 0.988737 | 1.112329 |

34 | 90.39 | 89.02159 | 0.989129 | 1.112770 |

35 | 90.99 | 89.17456 | 0.990828 | 1.114682 |

36 | 91.41 | 89.56480 | 0.995164 | 1.11956 |

37 | 90.84 | 89.72866 | 0.996985 | 1.121608 |

38 | 91.94 | 90.05702 | 1.000634 | 1.125713 |

39 | 92.94 | 90.97604 | 1.010845 | 1.137200 |

40 | 92.00 | 91.01112 | 1.011235 | 1.137639 |

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## Share and Cite

**MDPI and ACS Style**

Wei, H.; Liu, Y.; Li, J.; Liu, L.; Liu, H.
Reliability Investigation of Pavement Performance Evaluation Based on Blind-Number Theory: A Confidence Model. *Appl. Sci.* **2023**, *13*, 8794.
https://doi.org/10.3390/app13158794

**AMA Style**

Wei H, Liu Y, Li J, Liu L, Liu H.
Reliability Investigation of Pavement Performance Evaluation Based on Blind-Number Theory: A Confidence Model. *Applied Sciences*. 2023; 13(15):8794.
https://doi.org/10.3390/app13158794

**Chicago/Turabian Style**

Wei, Hui, Yunyao Liu, Jue Li, Lihao Liu, and Honglin Liu.
2023. "Reliability Investigation of Pavement Performance Evaluation Based on Blind-Number Theory: A Confidence Model" *Applied Sciences* 13, no. 15: 8794.
https://doi.org/10.3390/app13158794