Comparative Analysis of Asphalt Pavement Condition Prediction Models

Abstract

There is a growing global interest in preserving transportation infrastructure. This necessitates routine evaluation and timely maintenance of road networks. The effectiveness of pavement management systems (PMSs) heavily relies on accurate pavement deterioration models. However, there are limited comparative studies on modeling approaches for rural roads in arid climatic conditions using the same datasets for training and testing. This study compares three approaches for developing a pavement condition index (PCI) model as a function of pavement age: classical regression, machine learning, and deep learning. The PCI is a pavement management index widely adopted by many road agencies. A dataset on pavement age and distress was collected over a twenty-year period to develop reliable predictive models. The results demonstrate that the regression model, machine learning model, and the deep learning model achieved a coefficient of determination ($R^2$) of 0.973, 0.975, and 0.978, respectively. While these values are technically equal, the average bias for the deep learning model (1.14) was significantly lower than that of the other two models, signaling its superiority. Additionally, the trend predicted by the deep learning model showed more distinct phases of PCI deterioration with age than the machine learning model. The latter exhibited a wider range of PCI deterioration rates over time compared to the regression model. The deep learning model outperforms a recently developed regression model for a similar region. These findings highlight the potential of using deep learning to estimate pavement surface conditions accurately and its efficacy in capturing the PCI-age relationship.

1. Introduction

Well-designed Pavement Maintenance Management Systems (PMMSs) are crucial for ensuring high-quality rideability (smoother roads) and cost-effectiveness. Early-stage maintenance can lead to significant cost savings, prompting governments to allocate sufficient funding for informed decision-making by pavement managers [1,2]. Collecting comprehensive road condition data is essential for understanding pavement deterioration patterns and making effective maintenance decisions at both project and network levels [3,4]. The network level provides a comprehensive overview of the condition of a road network, which typically spans many kilometers of roadway. Due to the prohibitively high costs of conducting in-depth surveys of every street or roadway segment, most agencies cannot perform detailed analyses. As a result, only the minimum necessary information is typically collected at the network level. This limited data, however, support an assessment of the overall condition of the pavement network and help to identify critical repairs. Specific roads and detailed scopes of work are developed at the project level. However, since the maintenance alternatives proposed by the network-level analysis are not derived from thorough surveys, additional detailed information is essential to finalize the list of roads and establish comprehensive scopes of work for each road. This information is crucial to ensure effective road design and construction and provide more accurate cost estimates [5]. Over the past five decades, the implementation of PMMSs has been critical for Departments of Transportation (DOTs) and other agencies, helping them manage pavement networks and control repair and rehabilitation expenditures within budgets [6,7,8].

The pavement condition index (PCI), introduced by the U.S. Army Corps of Engineers in 1970, is widely used to assess pavement conditions based on a range of distresses, roughness, and deflection [9,10,11,12]. Accurate PCI deterioration models are vital for predicting future pavement conditions and planning maintenance and rehabilitation activities [13,14]. Timely maintenance actions are necessary to maximize the benefits of repairs and minimize costs, as maintaining pavements in poor condition is significantly more expensive than preventive maintenance [15,16]. Performance models enable PMMS to optimize maintenance treatments and predict future conditions, facilitating effective decision-making and resource allocation at both the network and project levels [17,18,19,20]. Continuous development and updating of these models are essential due to the complexity of predicting pavement performance [21].

The budget allocated for maintaining the pavement network at an acceptable level of service necessitates the use of accurate pavement performance models [19]. The success of these models is influenced by factors such as the size and accuracy of measured data and the algorithms used [21]. Key elements that impact the loss of performance (serviceability) of a pavement include traffic loads, age, and climate factors [22,23]. Many studies in the literature employ deterministic and stochastic methods to forecast pavement performance [23]. In addition to these two methods, PMSs can also incorporate advanced models like neural networks and knowledge-based systems to forecast the future condition of pavement sections [24]. Extensive research has been conducted using diverse methodologies such as linear and non-linear regression, time-series analysis, multivariate adaptive regression splines, Bayesian techniques, data mining, machine learning, and deep learning. However, these models are often developed for specific regions or countries with distinct traffic and climatic conditions, limiting their direct applicability elsewhere [25].

Machine and deep learning approaches, including artificial neural networks (ANNs), have been adopted in many pavement performance modeling studies due to their ability to deal with massive amounts of data and provide high accuracy [26]. These approaches have shown themselves to be worthwhile for solving time-series problems and forecasting road pavement conditions. For instance, Badr et al. [27] utilized ANNs in MATLAB to forecast the pavement condition index of flexible pavement sections, achieving high $R^2$ values ranging from 83% to 91%. Random Forest (RF) is frequently cited as highly accurate for PCI prediction. It has been used to predict PCI for up to 11 years with high accuracy [28,29]. Support Vector Regression (SVR) is effective for specific applications like airfield asphalt pavement condition prediction [30]. Gradient-Boosting Regression (GBR) demonstrated highly accurate short-term predictions [31]. CatBoost is another algorithm that has shown some success, though it is often outperformed by RF [29].

Despite the extensive research conducted on pavement performance modeling, no previous study has directly compared ANNs, SVR, and polynomial regression specifically for modeling the pavement performance of rural roads in arid climatic conditions. Mahmoud et al. [32] conducted a quadratic regression analysis to predict the pavement condition index (PCI) of desert roads, utilizing polynomial regression but without considering advanced machine learning methods like ANNs or SVR. Mahmood, et al. [33] developed an ANN model incorporating multiple factors such as pavement age, cracking length and area, and cumulative traffic loading across four climatic zones (dry freeze, dry non-freeze, wet freeze, and wet non-freeze) in the USA. Their ANN model achieved high precision in predicting PCI; however, the study did not focus on arid climates, nor did it compare ANNs with SVR or polynomial regression.

Recent studies have demonstrated the effectiveness of ANN models for predicting PCI; for example, Ali et al. [34] compared ANN and multiple linear regression (MLR) models in different climate regions, concluding that ANN models outperform traditional regression techniques, especially in wet and no-freeze climates. Similarly, Sirhan et al. [35] implemented deep neural networks (DNNs) to predict PCIs using a large dataset covering multiple distress types and severities. The study found that DNN models significantly outperformed linear and non-linear regression models in predicting PCIs. However, neither study focused on arid climates nor compared ANNs with SVR or polynomial regression specifically for rural roads.

In Iran, Shahnazari, Tutunchian, Mashayekhi and Amini [9] employed optimization procedures using ANN and genetic programming (GP) to estimate PCI based on visual survey results. The ANN-based model yielded more accurate results than the GP-based model, yet the study did not include SVR or polynomial regression in the comparison, nor was it specific to rural roads in arid conditions. Kırbaş and Karaşahin [23] developed models using multivariate adaptive regression splines (MARSs), deterministic regression analysis, and ANNs to predict PCI based on pavement age. While all models showed comparable prediction accuracies, the ANN model was more precise. However, their study did not include SVR, and the focus was not on arid climates. Therefore, the current research has compared the performance of ANNs to SVR and polynomial regression in modeling the deterioration of asphalt concrete pavement.

The accuracy and performance of pavement condition models are significantly influenced by the choice of predictive variables [26]. For instance, Georgeet et al. [17] developed models for forecasting pavement deterioration, utilizing historical data comprising structural information and information about the traffic volume and load and pavement age. They categorized the regions under study into flexible pavements with no overlay, flexible pavements with one or more overlays, and composite pavements. Pavement age was found to be the most significant variable in determining road serviceability, surpassing traffic volume, load, and structural configuration.

The literature survey review reveals the significance of comprehensive data collection to understand pavement deterioration and inform maintenance decisions. Various models, including neural networks and regression techniques, are used to predict pavement conditions, but they are often region-specific, limiting their general applicability, as they exhibit a differential performance. Advanced models like ANNs outperform traditional methods in predicting pavement conditions but lack comparative studies, particularly in arid climates and rural roads. This insufficient presence of comparative studies was identified as a critical gap in knowledge. The current study presents a comparative analysis of deterioration-related predictive models for rural roads in arid climates, namely ANNs, SVR, and polynomial regression. The rural roads in the agriculture zone of Fayoum City, Egypt, were used as a case study. The PCI models developed for these roads did not consider traffic conditions due to the restricted access to heavy vehicles in urban areas, where traffic levels tend to remain steady with minimal growth rates [23]. Similarly, climatic conditions were not considered, as rural roads often incorporate drainage systems that minimize the influence of weather, such as rainfall, on pavement surface conditions. Notably, Fayoum City experiences a predominantly arid climate with high temperatures, significant daily temperature fluctuations, and low rainfall [36]. However, air temperature exhibits minimal long-term changes, further reducing the impact of climate variability on pavement deterioration. Pavement thickness is also uniform in the study area. As such, this study employed pavement age as the sole variable in building the models proposed to predict the PCI. In doing so, a dataset collected over a twenty-year period covering eighty kilometers of the road network was utilized.

2. Methodology

Four main steps were undertaken to achieve the study’s objective. The first step involved collecting and preparing the actual database from the Fayoum region. The second step entailed data preparation, ensuring the removal of outliers, illogical data, and any unsuitable information. The third step involved developing and optimizing a PCI model using non-linear regression, machine learning, and deep learning. Finally, the validation process was performed to compare the proposed models against each other and published models.

3. Data Collection and Analysis

3.1. Study Area and Field Data

The dataset utilized in this study consists of observations from pavement sections, with a focus on two key variables: the Input Variable (Predictor), which is pavement age (in years), and the output variable (response), which is the PCI. A total road length of 80.3 km was selected in the Fayoum region, which included the ring road around Fayoum City and three connecting roads, for data collection and analysis (Figure 1). Each road was divided into sections, as shown in Table 1; each section was further divided into samples to ensure homogeneity. The road structure data were provided by the General Authority for Roads, Bridges, and Land Transport (GARBLT) in Egypt, as shown in Table 2. Handheld computers and global positioning system (GPS) receivers were utilized to assess pavement conditions and accurately mark specific distress locations on the roads. The pavement samples were inspected to determine the distress type, severity, and quantity. Identified distresses ranged from structural failures such as fatigue cracking, potholes, and depressions to functional failures like bleeding and patching. The distresses were classified according to the 19 types listed in the PAVER system. Homogeneous sections were then identified based on the collected distress data. All collected distress data underwent meticulous preparation and pre-evaluation. An example of the collected distress data for the Fayoum-Itsa Road is presented in Table 3. The subsequent step involved entering and recording the data using Micro PAVER software, version 7.0.9, a well-established PMS tool developed by the Army Corps of Engineers, Washington, DC, USA [37]. This allowed the estimation of PCI values for the surveyed road sections and samples. Samples of age and PCI samples are shown in Table 4.

A comprehensive database was curated for pavement age values (floating-point numbers) and the corresponding PCI values (integers) to facilitate model development and validation. Statistical synthesis of 1034 data points is displayed in Table 5, highlighting mean, minimum, and maximum values. Figure 2 depicts the dispersion or variability in PCI values, represented by buffers, around the line of central tendency with age. The line of central tendency was determined using linear regression, while buffers (confidence intervals) were plotted to reflect the dispersion of the data points around the line of central tendency. The figure confirms an inverse relationship between PCI and age and demonstrates that the PCIs are clustered around an average value for all pavement ages. PCI values exceed 90% for up to 2 years, whereas the PCI approaches zero after approximately 20 years. The database was divided into two categories: one for model development, representing 75% of the total data, and the other for the validation process, representing 25% of the data. The validation dataset was carefully chosen to encompass a significant amount of the original data and ensure variety (different sites) for sound statistical representation.

3.2. Missing Value Analysis

The analysis indicates no missing values for the pavement age and PCI variables (zero values for both). This completeness of the dataset eliminates the need for imputation strategies or the exclusion of data points due to missing entries, thereby ensuring the integrity and robustness of subsequent analyses. This thorough validation of data quality is essential for maintaining the scientific accuracy and reliability of the research findings.

3.3. Assessment of Data Normality

The Shapiro–Wilk test was used to assess the normality of the pavement age and PCI variables. This test effectively detects deviations from normality, particularly in small- to medium-sized samples (n < 2000). The Shapiro–Wilk test evaluates the null hypothesis (H0) that a given sample x₁, x₂, …, x_n is drawn from a normally distributed population. The test statistic W is calculated as follows:

(1)$W={\displaystyle \frac{{\left({\sum }_{i=1}^n{\alpha }_i{x}_{\left(i\right)}\right)}^2}{{\sum }_{i=1}^n{(x_i-\overline{x})}^2}}$

where

• $n$ is the sample size.

• $x_{\left(i\right)}$ is the i-th order statistic (the i-th smallest value in the sample).

• $\overline{x}$ is the sample mean.

• ${\alpha }_i$ is a coefficient constantly calculated from the expected values and variances of the order statistics of a standard normal distribution.

A $W$ value close to 1 indicates that the sample data are likely normally distributed, while a value significantly less than 1 suggests a deviation from normality. The results of the Shapiro–Wilk test conducted on pavement age and PCI data are presented in Table 6.

For pavement age, the W-Statistic is 0.8471 with a p-value of less than 0.0011. Since the p-value is significantly lower than the significance level of 0.02, we reject the null hypothesis, concluding that Pavement Age is not normally distributed. Similarly, for the Pavement Condition Index (PCI), the W-Statistic is 0.8963 with a p-value of less than 0.0021. Again, the p-value is significantly below 0.02, indicating that the PCI is also not normally distributed. Consequently, non-parametric algorithms such as Support Vector Regression (SVR) and Artificial Neural Networks (ANNs) were employed. These models do not require the predictors or response variables to follow a normal distribution, making them well suited for the observed data characteristics.

4. Developing PCI Deterioration Models

Three PCI deterioration models were developed in this study: deterministic regression analysis, machine learning, and deep learning.

4.1. Regression Analysis

This study attempted to use numerous mathematical forms to capture the relationship between PCI and pavement age. Table 7 illustrates the results of the regression analysis models. The polynomic model best fits the data. Polynomial regression, a parametric machine learning model, was used in regression analysis to model the relationship between the independent variable X (age) and the dependent variable Y (PCI) as an nth-degree polynomial. Statistical regression analysis of the developed model resulted in $R^2$ value of 0.974, as shown in Table 7, indicating a moderate-to-strong fit. The first term of Equation (2) represents the y-intercept, indicating a PCI value of 94% when the pavement age is zero. This value is comparable to the y-intercept of 91% from the model developed by Mahmoud et al. [32], which is a regression model that was recently developed for a similar region. The second term represents the linear component of the forecasting model, showing that as the pavement age increases, the PCI value decreases by a coefficient of approximately 5.057. The quadratic term captures the non-linear aspects of the data. The cubic term also addresses non-linearity and data scatter. The positive coefficient of the cubic term could indicate periods where the cumulative effect of maintenance led to temporary improvements before the overall decline continued.

(2)$PCI=94.081-5.057\times Age-0.0126\times {Age}^2+0.00242\times {Age}^3$

Figure 3 depicts the suggested correlation between PCI and age. The rate of PCI deterioration with age remains nearly constant until the pavement reaches 14 years old, followed by a milder rate until the pavement reaches the age of 20 years. The milder rate of PCI deterioration at the age of 14 can be justified by a combination of maintenance and rehabilitation efforts, reduced traffic loads, environmental adaptation, and load redistribution. These factors collectively contribute to a slower rate of decline in a pavement’s condition near the end of its service life [38].

4.2. Machine Learning

The support vector machine (SVM) algorithm is a non-parametric machine learning algorithm that is widely used for regression and classification problems [39,40,41]. SVMs are often referred to as kernel techniques, as they combine a group of algorithms with high prediction accuracy. The most important characteristics of SVM are its minimal number of modifying factors and its strong generalization capacity [42]. Support Vector Regression (SVR) is an extension of SVM for regression tasks, adapting the principles of SVM to predict continuous outcomes rather than class labels [43]. The SVR algorithm is particularly well suited for complex but small- or medium-sized dataset regression [44]. In this study, various trial models were examined, including decision tree models. However, these models tended to overfit small datasets, failing to capture the underlying patterns effectively. In contrast, SVR with the radial basis function (RBF) kernel exhibited superior performance in reducing the risk of overfitting, improving predictive accuracy on unseen data, and the ability to handle complex patterns within the dataset. This is essential for accurately predicting PCI based on age [45]. During training, SVR learns the coefficients that define the best-fit line. For prediction, these coefficients are used to calculate the output for a given input vector based on the best-fit line. To address overfitting during training, the hyperparameter associated with the RBF kernel was set at a “C” value of 4. The “C” value plays a crucial role in balancing the trade-off between achieving a low error on the training data and minimizing model complexity, which ultimately enhances generalization. A “C” value of 4 was found to effectively balance bias and variance, resulting in better performance on unseen data. For this model, the RBF kernel was selected, as indicated by the parameter kernel = ‘rbf’. The RBF kernel is particularly suitable for this task because it can manage non-linear relationships between features by transforming the data into a higher-dimensional space where linear separation becomes possible. This kernel can be mathematically represented as follows:

(3)$K \left(X1, X2\right)=\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{{‖X1-X2‖}^2}{2{\sigma }^2}})$

where σ is the variance of the hyperparameter and $‖X1-X2‖$ is the Euclidean distance between two points X1 and X2.

Regarding Epsilon (ε), the default epsilon value in SVR (from the library used, epsilon = 0.1) controls the width of the epsilon-insensitive tube, within which no penalty is associated with predictions. Adjusting the epsilon was part of the hyperparameter tuning process, and the default was found to provide a good balance between sensitivity to the training data and the robustness of the model. For further optimization, additional parameters such as gamma (γ) in the RBF kernel, which defines the influence of a single training example, could be tuned. However, gamma was set to ‘auto’ (defaulting to 1/n features) in our model, providing a balance between computational efficiency and model performance. These hyperparameters were selected based on a series of numerical experiments that included a grid search with cross-validation to determine the combination that delivers the best predictive performance while avoiding overfitting.

Table 8 summarizes the statistical evaluation of the SVM model. The model yielded an $R^2$ value of 0.977, which was slightly better than the value achieved by the regression model. The SVM model also demonstrates lower mean absolute and squared errors. This suggests that the chosen machine learning model outperforms the classical polynomial regression model. Figure 4 depicts the relationship between PCI and age versus the SVM model. Upon comparing Figure 3 and Figure 4, it is evident that the SVM model exhibits a superior ability to capture and represent the relationship between PCI and age. The overall trend predicted by the SVM model shows a non-linear relationship between age and PCI, with distinct phases of rapid, moderate, and intensified deterioration. The PCI drops sharply from 93% to 75% within the first five years. During the initial years, pavements experience significant changes due to the settling of road materials, initial traffic loads, and environmental factors such as weather conditions. Rapid deterioration often occurs during this period as the pavement adjusts to these stresses. Between 5 and 13 years, the rate of PCI decline becomes more gradual, with PCI values decreasing from around 65% to approximately 50%.

After the initial adjustment period, the pavement enters a more stable phase. The pavement cross-sections have undergone initial deformations, and the pavement has adapted to regular traffic loads and environmental conditions. During this period, the rate of deterioration slows down and becomes more consistent. The SVM model captures this phase by depicting a more linear relationship between age and PCI, indicating a steady but moderate rate of deterioration. After 13 years, the rate of PCI decline accelerates again, with PCI values dropping from around 50% to merely 12% in the 17th year. As the pavement ages, the cumulative effects of long-term distress caused by traffic loads, weathering, and material fatigue become more pronounced. These factors lead to an accelerated decline in pavement conditions. After 17 years, the rate of PCI deterioration with age slows down until the age of 20 years. This could be attributed to a combination of maintenance and rehabilitation efforts, reduced traffic loads, environmental adaptation, and load redistribution [38]. With its ability to handle non-linear relationships, the SVM model effectively captures the variability in the rate of PCI deterioration across different age ranges, reflecting the real-world behavior of pavement deterioration.

4.3. Deep Learning

In the current era of deep learning, PCI can be predicted using a deep learning model. Deep learning involves the incorporation of hidden layers into a neural network structure using algorithms. Several deep learning models exist, such as Artificial Neural Networks (ANNs), Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs) and their extensions, Gated Recurrent Units, and Attention Mechanisms. CNNs and RNNs excel in specific domains—CNNs are more suited for image-related tasks, while RNNs are more suitable for processing sequential data. These models are often less optimal for general regression problems. CNNs rely on spatial hierarchies and are specifically designed for processing grid-like data, such as images, where local pixel dependencies are crucial. However, they may not fully leverage the power of regression tasks that do not involve spatial patterns. Similarly, RNNs are well suited for temporal data with strong sequential dependencies, but their recurrent nature can introduce complexity and unnecessary overhead for regression problems without inherent time-series or sequence structure [46].

Choosing a suitable model for non-linear regression tasks is crucial [47]. For most non-linear regression tasks, a well-designed ANN is the most suitable choice due to its flexibility, theoretical foundation, and ability to approximate complex non-linear functions [48]. Thus, it was selected for this research’s deep learning model. ANNs consist of interconnected artificial neurons or nodes, which mimic the behavior of biological neurons. These neurons transmit signals to other connected neurons, combining and transforming the received signals. ANNs are mathematical systems composed of a series of interconnected processing units (neurons) with associated weights, forming a neural structure. The neurons process data and are commonly referred to as processing elements [32]. ANNs have proven to be invaluable in pavement deterioration modeling due to their data-processing neurons [23]. These neurons are typically organized into different layers: the input layer receives the external data, the output layer produces the result, and there may be zero or more hidden layers in between.

Multilayer Feed-forward Neural Network (MFNN) is the type of ANN that is most widely used for non-linear regression, as it is capable of approximating any continuous function given a sufficient number of neurons and layers [49]. Multiple hidden layers and various neuron alternatives at each layer were considered to achieve the highest predictive accuracy, resulting in the development of the most appropriate neural network architecture. The training was conducted using the Levenberg–Marquardt algorithm, with performance evaluated based on $R^2$, MAE, and MSE. The optimum MFNN developed in this research consisted of multiple layers: one input layer for age, two hidden layers with 8 to 16 neurons, and one output layer for PCI, as depicted in Figure 5.

The rectified linear unit (ReLU) activation function was used for the layers, along with a learning rate of 0.001 for the optimizer, as indicated in Table 9. Hence, there are two main parameters ahead. The first is the activation function, which is divided into a ReLU (rectified linear unit). The ReLU activation function was chosen for the hidden layers because it mitigates the vanishing gradient problem, allowing models to learn faster and perform better. ReLU is defined as f(x) = max(0,x), which introduces non-linearity into the model without affecting the output scale for positive values. Secondly, linear activation (the output layer) uses a linear activation function to perform regression, where we predict a continuous value. The linear activation allows the output to assume any value in the range of real numbers, matching the regression needs. The second parameter is Adam’s optimizer. This optimizer was selected for its efficient computation and lower memory requirement. It combines the benefits of two other extensions of stochastic gradient descent: adaptive gradient algorithm (AdaGrad) and root mean square propagation (RMSProp). Adam’s optimizer is well suited for problems involving large amounts of data and/or a number of parameters. It is also adaptive in nature, adjusting the learning rate throughout training, which improves the convergence speed. The model architecture and these parameter choices were iteratively refined through a series of experiments involving validation sets. This allows the optimization of performance metrics, specifically the mean squared error (MSE), which is used as the loss function in this model setup.

Seventy-five percent of the data were randomly selected to train the model, while the remaining twenty-five percent were reserved for testing. Figure 6 provides a graphical representation of the pavement deterioration model created using ANN. As shown in Table 10, the obtained $R^2$ value, approximately equal to 0.980, indicated a satisfactory correlation between the predicted and actual values and was the highest among the three developed models. The low values of MAE (3.11) and MSE (15.25) further support this observation. The overall trend predicted by the ANN model shows a non-linear relationship between age and PCI, with more distinct phases of rapid, moderate, and intensified deterioration than those predicted by the SVM model. The PCI drops sharply from nearly 100% to approximately 85% within the first 1.5 years. Between 1.5 and 10 years, the rate of PCI decline becomes gradual, with PCI values decreasing from around 85% to approximately 45%. Between 10 and 14 years, the PCI declines more gradually, with PCI values decreasing from around 45% to approximately 30%. After 14 years, the PCI declines sharply, with the PCI dropping from around 30% to 10% by the 17th year, and slows down thereafter, with PCI values dropping from 10% to around 2% by the 20th year.

5. Model Performance

Model validation is crucial for judging its statistical performance and accuracy of prediction [9]. When constructing regression models, it is important to assess their reliability by validating predictions using an independent dataset separate from the dataset used for model development [37,50]. To accomplish this, 25% of the data in this study were randomly selected from the main dataset to validate all three models. Figure 7 displays the prediction of the proposed models versus measurements. The figure demonstrates that the three models effectively fit most of the validation dataset, indicating their validity.

To further evaluate the authenticity of the models, other error norms were considered [47,51]: the average bias (B) and index of agreement (IG). The average bias (B) is expressed as the ratio between the average estimated and measured PCI values (Equation (4)). The index of agreement (IG) is defined as the ratio between MSE and potential error (PE) (Equation (5)). Table 11 summarizes the results of the coefficient of determination, index of agreement, and bias for the three developed models. The ANN model consistently achieves the best performance considering all three metrics. It has the highest R² value, indicating the best fit to the data; the highest index of agreement, suggesting the closest match to the actual values; and the lowest average bias, indicating the highest overall accuracy. The SVM model also performs well, with a slightly lower $R^2$ and index of agreement values compared to the ANN model, but still better than the regression model. Its average bias is also lower than that of the regression model. While performing reasonably well, the regression model lags behind the SVM and ANN models in all three metrics. It has the lowest $R^2$ and index of agreement and the highest average bias.

(4)${\mathrm{P}\mathrm{C}\mathrm{I}}_{\mathrm{s}}\mathrm{b}\mathrm{i}\mathrm{a}\mathrm{s}(\mathrm{B})={\displaystyle \frac{{\mathrm{P}\mathrm{C}\mathrm{I}}_{\mathrm{s}-\mathrm{p}}}{{\mathrm{P}\mathrm{C}\mathrm{I}}_{\mathrm{s}-\mathrm{m}}}}$

(5)$\mathrm{Index} \mathrm{of} \mathrm{Agreement} (\mathrm{IG})=1-{\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{[{\left({\mathrm{P}\mathrm{C}\mathrm{I}}_{\mathrm{s}-\mathrm{m}}\right)}_{\mathrm{i}}-{\left({\mathrm{P}\mathrm{C}\mathrm{I}}_{\mathrm{s}-\mathrm{p}}\right)}_{\mathrm{i}}]}^2}{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{[\left|{\left({\mathrm{P}\mathrm{C}\mathrm{I}}_{\mathrm{s}-\mathrm{p}}\right)}_{\mathrm{i}}-\overline{{\mathrm{P}\mathrm{C}\mathrm{I}}_{\mathrm{s}-\mathrm{m}}}\right|+\left|{\left({\mathrm{P}\mathrm{C}\mathrm{I}}_{\mathrm{s}-\mathrm{m}}\right)}_{\mathrm{i}}-\overline{{\mathrm{P}\mathrm{C}\mathrm{I}}_{\mathrm{s}-\mathrm{m}}}\right|]}^2}}=1-\mathrm{n}{\displaystyle \frac{\mathrm{M}\mathrm{S}\mathrm{E}}{\mathrm{P}\mathrm{E}}}$

Figure 8 demonstrates the performance of three regression models—polynomial regression (third power), SVR, and ANN—in predicting the PCI over pavement age. Figure 8a presents the models’ predictions of pavement age, while Figure 8b provides the corresponding residuals for each model. The ANN model (green dashed-dotted line) exhibits the best predictions across all age ranges. It captures subtle variations in the PCI values while maintaining an overall smooth trend. This demonstrates its ability to generalize complex patterns effectively. The polynomial regression model (a black solid line) follows the overall trend of the data, capturing the non-linear relationship between age and PCI. However, it overestimates PCI values for mid-range pavement age (around 8–12 years). The SVR model (red dashed line) performs well, particularly at the extremes of the data range, but exhibits slightly less precision than the ANN model when it comes to capturing the detailed variations.

The ANN residuals (green dots) demonstrate the least scatter around zero over age, indicating minimal error. This trend confirms the ANN’s ability to provide accurate predictions with localized variations. The residuals for polynomial regression show larger variability, particularly for mid-range age values. SVR residuals are the smallest at extreme pavement ages but show more variation in the middle range, indicating slightly larger errors. The ANN model consistently outperforms the polynomial regression and SVR models, maintaining minimal residual errors. The residual plot confirms that the ANN model achieves the best balance between accuracy and generalization, with the smallest deviations across the entire data range. To this end, the ANN emerges as the most effective model for predicting PCI based on pavement age.

6. Evaluation of Predictive Models

A comprehensive evaluation of the predictive models proposed in the study is necessary. The three models were assessed using 10-fold cross-validation, and their performances were statistically compared using the Friedman Test and the Nemenyi Post Hoc Test. The respective R² values were calculated to complement the statistical evaluation.

6.1. Cross-Validation

Ten-fold cross-validation was employed to ensure the robustness of the predictive models. This method involves partitioning the dataset into ten equal subsets (folds). Each model is trained and validated ten times: nine folds are used for training and the remaining fold is used for validation. This approach reduces the variance associated with random sampling and provides a more reliable estimate of the model’s performance on unseen data. The use of 10-fold cross-validation is particularly important for statistical testing, as it generates multiple performance estimates for each model, allowing for deep comparisons using non-parametric tests like the Friedman test.

6.2. Statistical Analysis

The Friedman test was employed to statistically compare the performances of the three models, followed by the Nemenyi Post Hoc Test. The Friedman test (Equation (6)) is a non-parametric statistical test used to detect treatment differences across multiple test attempts. It compares more than two related samples, making it ideal for the cross-validation setup where each model’s performance is measured across the same folds. The test ranks the models within each fold based on their RMSE values and evaluates whether the observed rank differences are statistically significant. The null hypothesis reveals no differences in the median performances of the models.

Upon determining the significance based on the results of the Friedman Test, the Nemenyi Test (Equation (7)) is conducted for pairwise comparisons between models to identify where the significant differences lie. The test determines whether the difference in average ranks between any two models exceeds the critical difference at the specified confidence level.

(6)$X_F^2={\displaystyle \frac{12N}{k\left(k+1\right)}}\left(\sum _{j=1}^k{\overline{R}}_j^2\right)-3N(k+1)$

where

• $N$ is the number of blocks (folds).

• $k$ is the number of groups (models).

• ${\overline{R}}_j$ is the average rank of the j-th model across all folds.

  (7)$CD=q_{\alpha }\sqrt{{\displaystyle \frac{k(k+1)}{6N}}}$

6.3. Results

The RMSE values obtained for the ten folds are presented in Table 12. The Friedman test yielded a Friedman’s statistic (X²_F) of 9.6000 and a p-value of 0.00823. Since the p-value was less than the significance level of 0.02 (corresponding to a 98% confidence level), the null hypothesis was rejected. This is indicative of statistically significant differences in the performance of the folds. The p-values for the pairwise comparisons are presented in Table 13. Polynomial regression versus SVM showed a p-value of 0.019945, which is less than 0.02, indicating a statistically significant difference at the 98% confidence level. This was also observed for polynomial regression versus ANN. In contrast, the comparison between SVM and ANN yielded a p-value of 1.000000, suggesting that there was no significant difference between these two models. These results demonstrate that the SVM and ANN models outperform the polynomial regression model in terms of predicting PCI based on age.

7. Comparisons with Existing Models

The ANN model proposed in this study is compared to a model developed for Al-Minea city in Upper Egypt, as shown in Equation (8). This model estimates the PCI as a function of time using ten sections of desert roads (approximately 500 m each) located in the east area of Al-Minea [32], which is situated close to our study area. The selected segments are nearly identical in terms of their age and characteristics, including traffic loads, materials, and dimensions. Data from these segments were collected every 3 months over a period of 30 months. A deterioration model was then developed using regression analysis of these data. Mahmoud et al. [32] model was employed for training and validation using our data.

(8)$PCI=-0.0244t^2-0.031 t+91.045$

As shown in Table 14, the $R^2$ value of Mahmoud et al.’s [32] model is notably lower than this study’s ANN model. This deficiency is further confirmed by the IG, which is significantly lower than that achieved using the ANN model. Additionally, the average bias value for this model is significantly greater than that of this study’s ANN model. This model appears unsuitable to predict the PCI values for this study’s area beyond three years. As such, the developed ANN model demonstrates broader applicability and generality.

8. Conclusions

This study presented a comparative investigation of three distinct approaches—regression analysis, machine learning, and deep learning—developed to model the deterioration of asphalt concrete pavements for rural roadways in arid climates. The models attempted to predict PCI as a function of pavement age, thereby aiding in effective pavement management and rehabilitation planning. The regression model achieved an $R^2$ value of 0.974, indicating a strong fit to the data. The machine learning models using the SVM algorithm and deep learning model (utilizing ANN) both attained slightly higher $R^2$ values of 0.977 and 0.980, respectively. The ANN model demonstrated the highest accuracy, with the lowest MAE of 3.11 and MSE of 15.25. The SVM model showed high accuracy, with an MAE of 3.35 and MSE of 17.93, also outperforming the regression model. The ANN model provided the most distinct phases of PCI deterioration with age. The SVM model also effectively captured non-linear relationships, showing distinct PCI deterioration phases. While capturing the general trend, the regression model was less precise than ANN and SVM models at representing the variability in deterioration rates. This was also echoed in the high MAE and MSE values (compared to ANN and SVM models).

All three models were validated using an independent dataset, confirming their reliability and accuracy. The ANN model consistently performed the best—as indicated by $R^2$, IG, and B. The model also outperformed an existing model developed for a similar region, with notably higher $R^2$ and IG values, and lower B. This study underscores the potential of advanced modeling techniques in improving the accuracy and reliability of pavement deterioration predictions, particularly deep learning. The current outcomes demonstrate that the ANN model performs well in predicting the PCI using pavement age as the sole predictor. The high-quality predictions from the regression and machine learning models suggest a degree of linearity in the relationship, which the ANN effectively captures.

To generalize the proposed models to different conditions, it is necessary to consider additional predictors such as rainfall intensity, temperature, traffic load, the California bearing ratio of pavement layers, pavement thickness, and the percentage of voids in mineral aggregate. Incorporating these factors would require additional data collection to enrich the models’ learning capacity in capturing potential nonlinearity. As such, the proposed models are currently limited to dry and non-freezing regions. It should also be noted that the PCI can be influenced by several factors beyond pavement age, including pavement distresses, asphalt layer thickness, and climatic conditions. To this end, PCI could be combined with other pavement performance indexes to develop a new index that can more holistically consider pavement conditions.

Author Contributions

Conceptualization, M.M.R. and E.M.M.Z.; Supervision, E.M.M.Z. and A.M.; Methodology, M.M.R.; Validation, E.M.M.Z.; Resources, M.M.R.; Software, M.M.R.; Formal analysis, M.M.R.; Data curation, M.M.R.; Visualization, E.M.M.Z. and A.M.; Writing—original draft, M.M.R.; Writing—review & editing, E.M.M.Z., O.D., Z.A. and A.M.; Project Administration, O.D., Z.A. and A.M. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Dataset available on request from the authors.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

AdaGrad	Adaptive Gradient Algorithm.
ANN	Artificial Neural Network.
B	Average Bias.
CNNs	Convolutional Neural Networks.
DOT	Department of Transportation.
GARBLT	General Authority for Roads, Bridges and Land Transport.
GP	Genetic Programming.
GPS	Global Positioning System.
HDM	Highway Development and Management.
IG	Index of Agreement.
LTPP	Long-Term Pavement Performance.
MAE	Mean Absolute Error.
M&R	Maintenance and Rehabilitation.
MAPE	Mean Absolute Percentage Error.
MARS	Multivariate Adaptive Regression Spline.
ME	Mechanistic–Empirical.
MFNN	Multilayer Feed-forward Neural Network.
MSE	Mean Square Error.
PCI	Pavement Condition Index.
PE	Potential Error.
PMMS	Pavement Maintenance Management System.
PMS	Pavement Management System.
PR	Polynomial regression.
$R^2$	Coefficient of Determination.
RBF	Radial Basis Function.
RELU	Rectified Linear Unit.
RMSE	Root Mean Square Error.
RMSProp	Root Mean Square Propagation.
RNN	Recurrent Neural Networks.
SVM	Support Vector Machine.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. PCI distribution of the study roads’ network.

Figure 2. Measured PCI versus pavement age for the study area. The thick dashed line represents the trend of PCI values over time. The shaded region around the dashed line indicates the confidence interval, representing the variability in the PCI values.

Figure 3. Polynomial regression (age vs. PCI).

Figure 4. Support Vector Regression (age vs. PCI).

Figure 5. Artificial Neural Network architecture.

Figure 6. Deep regression network (age vs. PCI).

Figure 7. Predicted PCI vs. observed PCI.

Figure 8. Predicted PCI over pavement age: (a) versus measured PCI; (b) model residual.

Table 1

Roads investigated in this study.

Road Name *	Road Length (km)	Road Width (m)
Fayoum Ring Road (clockwise)	22	10
Fayoum Ring Road (anticlockwise)	22	10
Fayoum-Itsa Road	8.2	6
Fayoum-Abshaway Road	16.4	6
Senores-Tamia Road	11.7	7.5

* The typical flexible pavement section has a 30 cm base layer and an 11 cm asphalt layer.

Table 2

A sample of road structure data provided by GARBLT.

Road Name	Fayoum-Itsa
Road Code	205
Last Maintenance Date	1995
Road Length (km)	8.2
Pavement Width (m)	6
Lane Width	-
Base Layer (cm)	30
Surface Layer (cm)	11

Table 3

Samples of distress data for Fayoum-Itsa road.

Distress Type	Quantity	Severity Level
Corrugation (m2)	5.28	Medium
Depression (m2)	2.625	Medium
Bleeding (m2)	70	Medium
Shoving (m2)	0.5	Low
Pothole (No.)	2	Low
Bumps and sag (m)	9	Medium
Edge crack (m)	15	Medium
Patching (m2)	2.8	Medium

Table 4

Samples of age and PCI data used in the study.

Age	PCI
14.7	23
14.7	24
15	29
10	48
9.9	47
10	45
9.6	42
10	48
9.6	41
9.8	47
9.5	38

Table 5

Statistics of the dataset (1034 points).

	Age	PCI
Mean	7.19	60.45
Standard Deviation	6.46	27.93
Min.	0.00	0
First Quartile	2.48	40
Second Quartile	4.40	70
Third Quartile	11.18	81
Max.	20	100

Table 6

Shapiro–Wilk test results for normality assessment.

Variable	W-Statistic	p-Value	Normality Conclusion (α = 0.02)
Age	0.8471	<0.0011	Not Normally Distributed
PCI	0.8963	<0.0021	Not Normally Distributed

Table 7

Comparison of attempted regression analysis models.

Model	${\mathit{R}}^2$	Mean Absolute Error (MAE)	Mean Squared Error (MSE)
Logarithmic	0.791	9.84	165.7
Exponential	0.842	12.8	241.8
Linear	0.956	4.52	34.8
Polynomial	0.974	3.70	20.5

Table 8

Evaluation of Support Vector Regression model.

Norm	Value
$R^2$	0.977
MAE	3.35
MSE	17.93
MAPE	14.70
RMSE	4.23

Table 9

Specifications of the Artificial Neural Network Model.

Neural Network Specification	Value
Number of Layers	3
Number of Hidden Layers	2
Hidden Layer #1 Activation Function	ReLU
Number of Units in Hidden Layer #1	16
Hidden Layer #2 activation function	ReLU
Number of Units in Hidden Layer #2	8
Output Layer Activation Function	Linear
Number of Units in Output Layer	1

Table 10

Evaluation of deep regression network model.

Norm	Value
$R^2$	0.980
MAE	3.11
MSE	15.25
MAPE	10.82
RMSE	3.86

Table 11

Validation metrics of the three proposed models in this study.

Norm	Polynomial Regression	SVM	ANN
$R^2$	0.973	0.975	0.978
IG	0.993	0.993	0.994
B	1.33	1.20	1.14

Table 12

RMSE values across folds for each model.

Fold	Polynomial Regression	SVM	ANN
1	4.309	4.025	3.845
2	4.547	4.334	4.813
3	4.981	4.639	4.511
4	4.706	4.222	4.539
5	4.695	4.567	4.680
6	4.642	4.257	4.019
7	4.660	4.448	4.980
8	4.112	3.646	3.525
9	4.048	3.778	3.759
10	4.898	4.645	4.301

Table 13

Nemenyi Post Hoc Test p-values.

	Polynomial Regression	SVM	ANN
Polynomial regression	1.000000	0.019945	0.019945
SVM	0.019945	1.000000	1.000000
ANN	0.019945	1.000000	1.000000

Table 14

Statistical performance of the ANN model developed against Mahmoud et al. [32].

Norm	ANN	Mahmoud et al. [32]
$R^2$	0.978	0.829
IG	0.994	0.233
B	1.14	23.85

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