Condition Rating Prediction for Highway Bridge Based on Elman Neural Networks and Markov Chains

: Bridges are a critical component of transportation infrastructure, playing a vital role in connectivity. The safe operation of bridges demands significant resource and capital investment, particularly as the operation phase is the most extended period in a bridge’s life cycle. Therefore, the efficient allocation of resources and funds is crucial for the maintenance and repair of bridges. This study addresses the need to predict changes in bridge condition over time. The commonly used state-based Markov chain method for bridge condition rating prediction is straightforward but limited by its assumptions of homogeneity and memorylessness. To improve upon this, we propose a novel method that integrates an Elman neural network with a Markov chain to predict the bridge condition rating. Initially, the ReliefF algorithm conducts a sensitivity analysis on bridge features to obtain the importance ranking of these features that affect the bridge condition. Next, six significant features are selected for data classification: bridge age, average daily truck traffic volume, material type, skew angle between bridges and roads, bridge deck structure type, and bridge type. The Elman neural network is then trained to train a prediction model for bridge condition ratings using the classified data, which can predict the condition levels of bridges. The Markov chain’s transition probability matrix is derived using a genetic algorithm to match the deterioration curve predicted by the Elman neural network. This proposed method, when applied to actual bridge data, demonstrates its effectiveness as evidenced by the condition rating of an actual bridge.


Introduction
Bridges are a key link in the road network, serving as a critical component to ensure road safety and control traffic.According to the 2022 Statistical Bulletin on the Development of the Transport Industry in China [1], there were 1,033,200 highway bridges across the country at the end of 2022, representing an increase of 72,000 bridges from the previous year.Consequently, China has been transitioning from a major country in bridge construction to a major country in bridge maintenance.During the operation and maintenance phase, the bridge condition rating serves as the basis for bridge rehabilitation and maintenance planning.At present, bridge condition levels are evaluated through existing inspection data from regular inspections conducted every two years.However, current inspection data only provide the current condition of the bridge; they do not enable predictions of the performance deterioration trend during the bridge maintenance stage.Since the degradation trend of bridge performance is crucial for decision-making in bridge maintenance and repair, predicting the future condition rating of bridges is necessary to allocate resources reasonably during the maintenance stage.condition.Then, the ANN model was used to predict future bridge conditions; the results indicate that the established neural network's prediction accuracy has reached 73.6%, and the accuracy of the model can reach 98.5% if the allowable error in the bridge condition rating is ±1.Artificial intelligence models aim to explain bridge deterioration by capturing the connections between bridge features from inspection reports.Artificial neural networks are the most commonly used approach in artificial intelligence.Ariza et al. [12] proposed an artificial neural network model to predict bridge deterioration performance and, when comparing it with the Markov model and the semi-Markov model, found that the artificial neural network prediction accuracy was higher.Srikanth et al. [13] summarized the features of ANNs in bridge deterioration prediction, highlighting that they can generate missing inspection data to fill data gaps resulting from unscheduled inspections.Because of the neural networks' generalization ability, they can handle large amounts of data.
In summary, bridge deterioration models based on the Markov chain or ANN have been widely used in bridge condition prediction, with significant results achieved.However, the process of selecting bridge features for training neural networks in previous methods was based on the experience of bridge engineers and literature.In this study, a sensitivity analysis of bridge feature parameters is firstly carried out using the ReliefF algorithm to effectively select bridge features.Then, based on the importance and influence of the feature parameters on the bridge condition, a combination of feature parameters is selected for neural network training, reducing the data dimensionality of training.Finally, a novel prediction model is proposed that combines the Elman neural network (ENN) with the state-based Markov chain method to predict the long-term performance of bridges.To verify the accuracy of the model, the prediction results from the proposed model are compared with actual bridge condition levels.The bridge condition prediction process based on the ENN and Markov chain is illustrated in Figure 1.
tions; the results indicate that the established neural network's prediction acc reached 73.6%, and the accuracy of the model can reach 98.5% if the allowable e bridge condition rating is ±1.Artificial intelligence models aim to explain bridg ration by capturing the connections between bridge features from inspection re tificial neural networks are the most commonly used approach in artificial in Ariza et al. [12] proposed an artificial neural network model to predict bridge tion performance and, when comparing it with the Markov model and the sem model, found that the artificial neural network prediction accuracy was higher et al. [13] summarized the features of ANNs in bridge deterioration prediction, ing that they can generate missing inspection data to fill data gaps resulting from uled inspections.Because of the neural networks' generalization ability, they c large amounts of data.
In summary, bridge deterioration models based on the Markov chain or A been widely used in bridge condition prediction, with significant results achiev ever, the process of selecting bridge features for training neural networks in methods was based on the experience of bridge engineers and literature.In th sensitivity analysis of bridge feature parameters is firstly carried out using the gorithm to effectively select bridge features.Then, based on the importance and of the feature parameters on the bridge condition, a combination of feature par selected for neural network training, reducing the data dimensionality of trainin a novel prediction model is proposed that combines the Elman neural netwo with the state-based Markov chain method to predict the long-term perfor bridges.To verify the accuracy of the model, the prediction results from the model are compared with actual bridge condition levels.The bridge condition p process based on the ENN and Markov chain is illustrated in Figure 1.

Feature Selection Based on ReliefF Algorithm
Feature selection plays a critical role in data-mining techniques.When processing a vast amount of data, datasets with irrelevant and redundant features may hinder subsequent analysis, such as key information extraction, classification, and prediction.Irrelevant and redundant features can weaken model performance.Hence, it is necessary to perform feature selection on the data to remove these irrelevant and redundant features.Feature selection algorithms help to build simpler and more understandable models, reduce training time, and improve model performance by identifying the most distinctive subset of features from a large number that are relevant to the problem under study [14].Feature selection algorithms combined with classification models fall into three categories: filter, wrapper, and embedded [15].
Filter feature selection algorithms are independent of any learning algorithm; they evaluate and select features based on the data's characteristics to determine the relevance of each feature [16].The Relief algorithm is a typical filter feature selection method.It is a sample-based learning algorithm for calculating feature weights, measuring the distinctiveness power of features by examining their differences between similar nearneighbor samples and dissimilar samples.If a feature shows a small difference between similar samples and a large difference between dissimilar samples, it is considered to have a strong distinctiveness ability.The Relief algorithm is regarded as one of the most straightforward and effective filter feature selection methods available [17].The ReliefF algorithm, an extension of Relief, is a multi-category feature selection algorithm that extends the study object from a two-category problem to a multi-category problem by transforming the multi-category problem into multiple one-to-many two-category problems [18].
Due to its high efficiency and good performance, the ReliefF algorithm has been one of the most important algorithms implemented successfully in many feature selection applications [19].It is also used for feature selection in this study.
The ReliefF algorithm finds k nearest neighbor samples of R for each instance by randomly selecting one sample from the same class and k nearest neighbors from different class samples for each instance when dealing with multi-class problems.It then updates the weights of each feature as follows: where H i (i = 1,2,. ..,k) is selected from the sample set of the class as the sample R. M i (C) denotes the i-th nearest neighbor sample in class C. W j (A) is the weight of the feature A that is updated according to the following formula until the sampling number m is reached.P(C) is the prior probability of class C. diffj(A, R 1 , R 2 ) denotes the difference between samples R 1 and R 2 on feature A at the j-th sampling, and it can be calculated by the following equation: In the ReliefF algorithm, different weights are assigned according to their correlation between features and classes.The weights are then sorted in descending order, and the features corresponding to the highest weights are selected as a subset of features.The principle of the ReliefF algorithm shows that the selection of the feature set mainly depends on the correlation between features and classes and the ability of features to distinguish between similar samples.
According to the ReliefF algorithm, bridge inspection data can be analyzed, and feature parameters with a high impact on bridge condition ratings can be classified and selected.The purpose of this classification is to group bridge features with similar causes of deterioration to enhance the identification of deterioration patterns in bridges.

Prediction Model of Bridge Degradation Based on Elman Neural Network
The Elman neural network (ENN) was proposed by Elman in 1990 for speech signal processing problems and is a typical local recurrent network suitable for time series issues [20].The ENN has been applied in various fields related to time and is one of the well-known recurrent neural networks.While recurrent neural networks have a feedback architecture, the Elman neural network is a dynamic feedback network that includes a special context unit in addition to the input layer, hidden layer, and output layer units [21][22][23].The context unit is used to retain the hidden layer's output values from previous time steps, effectively functioning as a short-term delay operator, which endows the network with dynamic memory [24].
In predicting bridge condition levels, the ENN is also employed, and its generalized structure is depicted in Figure 2. In the Elman neural network model, the condition ratings at time t − 1 are utilized as inputs to predict the condition ratings at time t.Local feedback loops provide the mechanism for incorporating the predicted ratings as inputs.Compared to other typical feedforward neural network models, the ENN has advantages in timedependent prediction, making it an effective neural network for predicting the condition of bridges with irregular deterioration over time.
guish between similar samples.
According to the ReliefF algorithm, bridge inspection data can be analyzed, an ture parameters with a high impact on bridge condition ratings can be classified a lected.The purpose of this classification is to group bridge features with similar cau deterioration to enhance the identification of deterioration patterns in bridges.

Prediction Model of Bridge Degradation based on Elman Neural Network
The Elman neural network (ENN) was proposed by Elman in 1990 for speech processing problems and is a typical local recurrent network suitable for time series [20].The ENN has been applied in various fields related to time and is one of the known recurrent neural networks.While recurrent neural networks have a feedba chitecture, the Elman neural network is a dynamic feedback network that includes cial context unit in addition to the input layer, hidden layer, and output layer unit 23].The context unit is used to retain the hidden layer's output values from previou steps, effectively functioning as a short-term delay operator, which endows the ne with dynamic memory [24].
In predicting bridge condition levels, the ENN is also employed, and its gener structure is depicted in Figure 2. In the Elman neural network model, the condition r at time t-1 are utilized as inputs to predict the condition ratings at time t.Local fee loops provide the mechanism for incorporating the predicted ratings as inputs.Com to other typical feedforward neural network models, the ENN has advantages in dependent prediction, making it an effective neural network for predicting the con of bridges with irregular deterioration over time.

Bridge Deterioration Curve Fitting based on Markov Chain
State-based models use a transition probability matrix obtained from the diffe between two condition states at a given discrete time interval to predict the long performance of bridges [25].Jiang et al. [26] proposed a relationship between bridg dition and bridge age, concluding that the deterioration performance of bridges re constant throughout the prediction period; however, this method does not accurate flect the true deterioration process of bridges.Conversely, Butt et al. [27] suggested multiple transition probability matrices to predict bridge conditions by dividing b age into six-year age groups, assuming that the deterioration performance is cons

Bridge Deterioration Curve Fitting Based on Markov Chain
State-based models use a transition probability matrix obtained from the difference between two condition states at a given discrete time interval to predict the long-term performance of bridges [25].Jiang et al. [26] proposed a relationship between bridge condition and bridge age, concluding that the deterioration performance of bridges remains constant throughout the prediction period; however, this method does not accurately reflect the true deterioration process of bridges.Conversely, Butt et al. [27] suggested using multiple transition probability matrices to predict bridge conditions by dividing bridge age into six-year age groups, assuming that the deterioration performance is consistent within each age group, and the transition probability matrix for each group can be calculated by optimizing the objective function.
where N = the number of years in an age group; U = the number of unknown probabilities; A(t) = the conditional rating of the performance curve at time t, estimated from the deterioration curve predicted by the neural network; and E(t) = the estimated value of the conditional rating at time t, predicted by the Markov chain.The bridge conditional ratings to be predicted by the Markov chain are generated by the following equation: where P(0) = the initial state vector, e.g., if the initial state of the bridge is Level 9, then P(0) is denoted by [1 0 0 0 0 0 0]; and P is the transition probability matrix, R = [9 8 7 6 5 4 3].
The reliability of state-based Markov stochastic models depends on the accuracy of the transition probability matrix.This matrix is established in an n by n format, where n represents the number of possible condition ratings.It is used to calculate the distribution of bridge condition ratings over time.According to the National Bridge Inventory (NBI) bridge data, the lowest condition rating typically observed for a bridge component is 3. Thus, the minimum rating for bridge condition transitions is set at 3, below which no further deterioration is calculated.
Transition probability matrices can be divided into two types: one-step and multi-step.In a one-step transition probability matrix, the condition state can only transition to the next lower state.In a multi-step transition probability matrix, the condition state can transition to multiple lower states.Although it is theoretically possible for a condition state to deteriorate by two or more levels within a year, such occurrences are rare in bridge condition data [28].Therefore, the one-step transition probability matrix is described using the following equation: where P(i) = probability of staying at condition rating i, i = 1, 2, . .., 6; 1 − P(i) = probability of switching to the next condition rating.In the NBI condition rating scale, the lowest condition rating for maintenance is 3, and it is not possible to continue deteriorating to the next rating, so the corresponding probability P(7) = 1.

Genetic Algorithms
Genetic algorithms were originated from computer simulation studies of biological systems and were first proposed by John Holland.The algorithm is a search heuristic used to solve optimization problems.A genetic algorithm is a stochastic global search and optimization method developed by simulating the mechanism of natural selection and genetics.Genetic algorithms generate a near-optimal solution from a population of potential solutions based on the principle of survival of the fittest.In each generation, a new set of solutions is created by selecting individuals according to their fitness values and applying genetic operators such as crossover and mutation.This process results in a population of individuals that are better adapted to their environment than the previous generation [29].
Unlike most classical optimization algorithms, which are based on the gradient or higher-order derivatives of a single objective function to produce a deterministic sequence of trial solutions, genetic algorithms do not rely on gradient information.Instead, they search for the optimal solution by emulating the process of natural evolution.The steps to implement a genetic algorithm include the generation of the initial population, calculation of individual fitness values, proportional selection operation, crossover operation, mutation operation and generating a new populations, and finally, a convergence upon an optimal solution when a stopping criterion is met [30].The flowchart of the genetic algorithm is illustrated in Figure 3.
Sci. 2024, 14, x FOR PEER REVIEW search for the optimal solution by emulating the process of natural evolu to implement a genetic algorithm include the generation of the initial popu tion of individual fitness values, proportional selection operation, crosso mutation operation and generating a new populations, and finally, a conv an optimal solution when a stopping criterion is met [30].The flowchart algorithm is illustrated in Figure 3.In this study, the genetic algorithm is primarily applied to solve the M tion probability matrix.Since calculating the transition probability matrix is problem requiring the identification of an optimal solution, the genetic algo ibly and adaptively search for and determine the best solution in the search the genetic algorithm can effectively utilize its strengths in this study.

Case Study
In the ENN prediction model for bridge deterioration, it is first neces the bridge data to predict the deterioration of a specific bridge type.Howev selection for evaluating bridge conditions relies on the engineering judgme rigor, and no scientific method has been established.In this study, the mod bridge inspection data from the NBI Texas Bridge Inventory from 1992 to tures influencing bridge deterioration are identified using the ReliefF algor by the classification of data according to the extracted feature values.Thes used to train the ENN prediction model.The transition probability matrix i the trained ENN model, and the bridge deterioration curve is fitted using a In this study, the genetic algorithm is primarily applied to solve the Markov transition probability matrix.Since calculating the transition probability matrix is a multivariate problem requiring the identification of an optimal solution, the genetic algorithm can flexibly and adaptively search for and determine the best solution in the search process.Thus, the genetic algorithm can effectively utilize its strengths in this study.

Case Study
In the ENN prediction model for bridge deterioration, it is first necessary to classify the bridge data to predict the deterioration of a specific bridge type.However, the feature selection for evaluating bridge conditions relies on the engineering judgment, which lacks rigor, and no scientific method has been established.In this study, the model is based on bridge inspection data from the NBI Texas Bridge Inventory from 1992 to 2022.The features influencing bridge deterioration are identified using the ReliefF algorithm, followed by the classification of data according to the extracted feature values.These data are then used to train the ENN prediction model.The transition probability matrix is calculated by the trained ENN model, and the bridge deterioration curve is fitted using a Markov chain to predict the condition rating of an actual bridge.This predicted rating is then compared with the actual detected value to verify the model's effectiveness.

Data Pre-Processing
The Texas NBI data, published by the Federal Highway Administration, is utilized in this study.The NBI database encompasses over 130 types of bridge characterization information, coded and stored according to the uniform American bridge inspection standards.Each bridge component is rated on a scale from 0 to 9, and the final rating of the entire bridge is derived from the lowest condition rating among its components.Studies have indicated that predicting the condition of individual bridge components has a minimal impact on the overall bridge condition.Consequently, the Elman neural network will be employed to directly predict the bridge's overall condition.
Before predicting future bridge deterioration, it is necessary to process the bridge features.Initially, records with missing indicators are excluded from the NBI data, as well as data coded with 'N', which applies to culverts and other structures without deck slabs, and bridges with a rating of less than 3.The latter are considered failed in the bridge inventory description.Subsequently, the characteristics of each bridge are extracted and organized into a time series format that correlates bridge age with its condition.
To accommodate computer performance and ensure the completeness of the time series for each bridge, this study focuses on data for Texas bridges from 1992 to 2022.

Feature Selection
In the NBI data, numerous features correlate with bridge conditions, but not all data are necessary for the feature selection database.In this study, bridge feature data are selected based on prior research, which includes the basic structural information of the bridge, traffic volume, and bridge age, as shown in Table 1.The detailed descriptions of these features can be found in Ref. [19].Since the study addresses bridge deterioration over time and does not account for maintenance and reconstruction measures during the operational phase, the features for bridge reconstruction and structural rehabilitation lengths are set to 0, indicating that no maintenance or reconstruction measures have been taken for these bridges.Existing studies have concluded that these feature sets are closely related to the future condition rating of the bridge, and a modified ReliefF algorithm was used to determine the appropriate feature sets.A Pareto chart illustrating feature importance is shown in Figure 4, which indicates the degree of influence of features on bridge condition.It can be seen that bridge age, average daily truck traffic, material type, and skew have a significant effect on bridge condition.In contrast, the number of approach spans does not have a significant effect.When training an Elman neural network, it is first necessary to classify the brid features.Thus, it is essential to select bridge features that have a significant impact bridge condition.If too few features are selected, they cannot represent the entire datas however, if too many features are chosen, the amount of neural network training da cannot be guaranteed.Therefore, this study will apply the following four different typ of feature parameter combinations for Elman neural network training and prediction.T accuracy of the prediction results is shown in Table 2.It can be seen that classifying t data through six features yields the highest prediction accuracy.Consequently, this stu will select six features for data classification, as shown in Table 2, including bridge ag average daily truck traffic, type of material, skew, deck structure type, and type of bridg

Elman Neural Network for Bridge Condition Prediction
The data are classified based on the features mentioned above.Out of the datase from Texas, 5600 are eligible; 70% of the data are used for training, 20% for validation, a 10% for testing.The data with selected features and completed classification are used ENN inputs to predict the long-term performance of a given bridge.Assuming the p diction for Year 60 results, the Elman neural network predicts these results by adding t predicted values of t59 to the original ENN inputs from t1 -t58.In this study, this iterati of the one-step prediction process lasts for 60 cycles, as shown in the figure, and the to number of long-term predictions (LTP) for the ENN of the bridge is 10, as shown in Figu 5.  When training an Elman neural network, it is first necessary to classify the bridge features.Thus, it is essential to select bridge features that have a significant impact on bridge condition.If too few features are selected, they cannot represent the entire dataset; however, if too many features are chosen, the amount of neural network training data cannot be guaranteed.Therefore, this study will apply the following four different types of feature parameter combinations for Elman neural network training and prediction.The accuracy of the prediction results is shown in Table 2.It can be seen that classifying the data through six features yields the highest prediction accuracy.Consequently, this study will select six features for data classification, as shown in Table 2, including bridge age, average daily truck traffic, type of material, skew, deck structure type, and type of bridge.

Elman Neural Network for Bridge Condition Prediction
The data are classified based on the features mentioned above.Out of the datasets from Texas, 5600 are eligible; 70% of the data are used for training, 20% for validation, and 10% for testing.The data with selected features and completed classification are used as ENN inputs to predict the long-term performance of a given bridge.Assuming the prediction for Year 60 results, the Elman neural network predicts these results by adding the predicted values of t 59 to the original ENN inputs from t 1 -t 58 .In this study, this iteration of the one-step prediction process lasts for 60 cycles, as shown in the figure, and the total number of long-term predictions (LTP) for the ENN of the bridge is 10, as shown in Figure 5. Before analyzing these prediction results, a filtering process of the predicted tion is required to further improve the quality of the long-term predictions.The fi process adheres to the guideline that the condition of the previous year should be than or equal to the condition of the following year.That is, if the bridge does not un maintenance, the condition rating should not improve, as bridge degradation is a uous, gradual, and slow progression.Therefore, if the predicted bridge condition is 5.45 in Year 53 and 5.47 in Year 54 for LTP10 in Figure 5, this does not correspond actual rule of bridge deterioration, so this prediction dataset for LTP10 should carded.The results of the bridge condition prediction are removed from the deterio curves with the highest and lowest predicted conditions, because these two curves d significantly, as shown in Figure 6.Before analyzing these prediction results, a filtering process of the predicted condition is required to further improve the quality of the long-term predictions.The filtering process adheres to the guideline that the condition of the previous year should be greater than or equal to the condition of the following year.That is, if the bridge does not undergo maintenance, the condition rating should not improve, as bridge degradation is a continuous, gradual, and slow progression.Therefore, if the predicted bridge condition rating is 5.45 in Year 53 and 5.47 in Year 54 for LTP10 in Figure 5, this does not correspond to the actual rule of bridge deterioration, so this prediction dataset for LTP10 should be discarded.The results of the bridge condition prediction are removed from the deterioration curves with the highest and lowest predicted conditions, because these two curves deviate significantly, as shown in Figure 6.Before analyzing these prediction results, a filtering process of the predicted tion is required to further improve the quality of the long-term predictions.The fi process adheres to the guideline that the condition of the previous year should be than or equal to the condition of the following year.That is, if the bridge does not u maintenance, the condition rating should not improve, as bridge degradation is a uous, gradual, and slow progression.Therefore, if the predicted bridge condition is 5.45 in Year 53 and 5.47 in Year 54 for LTP10 in Figure 5, this does not correspon actual rule of bridge deterioration, so this prediction dataset for LTP10 should carded.The results of the bridge condition prediction are removed from the deteri curves with the highest and lowest predicted conditions, because these two curves significantly, as shown in Figure 6.After completing network training and optimization analysis, the selected pre curve for the condition rating of the bridge is shown in Figure 7.The transition prob matrix of the Markov chain is calculated using the selected bridge deterioration Owing to the straightforward process of the genetic algorithm, the optimal solutio After completing network training and optimization analysis, the selected prediction curve for the condition rating of the bridge is shown in Figure 7.The transition probability matrix of the Markov chain is calculated using the selected bridge deterioration curve.Owing to the straightforward process of the genetic algorithm, the optimal solution of the objective function can be found through the continuous iteration of the function.Therefore, a genetic algorithm will be used.
Appl.Sci.2024, 14, x FOR PEER REVIEW objective function can be found through the continuous iteration of the function.fore, a genetic algorithm will be used.The genetic algorithm is an optimization search algorithm inspired by the pri of natural selection and genetics.It is a stochastic search technique commonly u solve various optimization problems.Specifically, the process of the genetic algori volves initializing the population, evaluating fitness, selecting, crossing over, mu and repeating the process.The advantage of the genetic algorithm is that it can con global search in the solution space and is well-suited for optimization problems wi complexity and a large search space.
To calculate the transition probability matrix of the bridge condition over a 60 years, the following steps are taken: first, the 60-year range is divided into 10 in of 6 years each, and the transition probability matrix for each interval is calculated the genetic algorithm code is written in MATLAB software 2022, and Equations (2 expressed as Equations ( 2)-( 6) with the function fun1, which is then inserted into netic algorithm to perform the iterative operation.Since each unknown quantity function represents a transition probability matrix-for example, P(1) denotes the tion probability matrix from Condition 9 to Condition 8-the range of each unkn between 0 and 1.
The genetic algorithm determines that the results of the transition probability for the first 6 years are: P(1) = 0.844, P(2) = 0.905, P(3) = 1, P(4) = 1, P(5) = 1, and P( 6) Following the same methodology, the transition probability matrix is solved for the of 1-60 years, and the transition probability matrix for the 60-year period is shown i 3.
Table 3. Markov chain transition probability matrix from 1 to 60 years.The genetic algorithm is an optimization search algorithm inspired by the principles of natural selection and genetics.It is a stochastic search technique commonly used to solve various optimization problems.Specifically, the process of the genetic algorithm involves initializing the population, evaluating fitness, selecting, crossing over, mutating, and repeating the process.The advantage of the genetic algorithm is that it can conduct a global search in the solution space and is well-suited for optimization problems with high complexity and a large search space.
To calculate the transition probability matrix of the bridge condition over a span of 60 years, the following steps are taken: first, the 60-year range is divided into 10 intervals of 6 years each, and the transition probability matrix for each interval is calculated.Then, the genetic algorithm code is written in MATLAB software 2022, and Equations ( 2) and ( 3) is expressed as Equations ( 2)-( 6) with the function fun1, which is then inserted into the genetic algorithm to perform the iterative operation.Since each unknown quantity in the function represents a transition probability matrix-for example, P(1) denotes the transition probability matrix from Condition 9 to Condition 8-the range of each unknown is between 0 and 1.
The genetic algorithm determines that the results of the transition probability matrix for the first 6 years are: P(1) = 0.844, P(2) = 0.905, P(3) = 1, P(4) = 1, P(5) = 1, and P(6) = 0.998.Following the same methodology, the transition probability matrix is solved for the period of 1-60 years, and the transition probability matrix for the 60-year period is shown in Table 3.The accuracy of the transition probability depends on the closeness of A(t) to E(t).The chi-square fit test can be used to verify the accuracy of the transition probability matrix.The chi-square test is conducted as follows: where χ 2 denotes the chi-square distribution with k − 1 degrees of freedom; E i represents the condition rating value in year i predicted by the state-based stochastic model; A i is the condition rating value in year i predicted by the ENN bridge deterioration curves; and k is the number of years predicted.The calculated χ 2 for the Markov chain and Elman neural network is 0.0006, which is significantly smaller than the chi-square value corresponding to the significance level α = 0.05.This indicates that the generated transition probability matrices are acceptable for predicting the long-term performance of bridges.The fitted values from the Elman neural network and Markov chain are shown in Figure 8.The accuracy of the transition probability depends on the closeness of A(t) The chi-square fit test can be used to verify the accuracy of the transition probabil trix.The chi-square test is conducted as follows:

Appl
 where  2 denotes the chi-square distribution with k-1 degrees of freedom; Ei rep the condition rating value in year i predicted by the state-based stochastic model; A condition rating value in year i predicted by the ENN bridge deterioration curves is the number of years predicted.The calculated  2 for the Markov chain and Elman network is 0.0006, which is significantly smaller than the chi-square value corresp to the significance level α = 0.05.This indicates that the generated transition prob matrices are acceptable for predicting the long-term performance of bridges.Th values from the Elman neural network and Markov chain are shown in Figure 8.After obtaining the transition probability matrix, the condition rating of the can be predicted using Equations ( 2)-( 4).Taking an actual bridge as an example, lected for the long-term deterioration performance prediction of the bridge.The lon performance prediction is based on the latest inspection records as the initial state which is used to predict the bridge state level over the next 20 years.The bridge d After obtaining the transition probability matrix, the condition rating of the bridge can be predicted using Equations ( 2)- (4).Taking an actual bridge as an example, it is selected for the long-term deterioration performance prediction of the bridge.The longterm performance prediction is based on the latest inspection records as the initial state vector, which is used to predict the bridge state level over the next 20 years.The bridge deterioration prediction model is validated by comparing the predicted values with the actual values, as shown in Table 4.In the real-world scenario, the bridge condition ratings are whole numbers.Therefore, the predicted data for the bridge condition levels from 1 to 20 years are rounded.The predicted values are then compared with the actual values, as depicted in Figure 9, which demonstrates that the prediction results align well with the actual values.Thus, a 60-year bridge condition rating for the case study is predicted.As illustrated in Figure 10, the 1-20 year condition levels from the bridge inspection reports are compared with the prediction results of the bridge condition level by the Markov chain model from 1 to 60 years, where the bridge condition prediction curve closely matches the actual bridge condition.This reflects the real deterioration performance of the bridge accurately.The bridge condition worsens over time, and the deterioration rate becomes increasingly slower, consistent with the actual deterioration performance of the bridge.
Appl.Sci.2024, 14, x FOR PEER REVIEW 13 In the real-world scenario, the bridge condition ratings are whole numbers.T fore, the predicted data for the bridge condition levels from 1 to 20 years are rounded predicted values are then compared with the actual values, as depicted in Figure 9, w demonstrates that the prediction results align well with the actual values.Thus, a 60 bridge condition rating for the case study is predicted.As illustrated in Figure 10, th 20 year condition levels from the bridge inspection reports are compared with the pr tion results of the bridge condition level by the Markov chain model from 1 to 60 y where the bridge condition prediction curve closely matches the actual bridge cond This reflects the real deterioration performance of the bridge accurately.The bridge dition worsens over time, and the deterioration rate becomes increasingly slower, sistent with the actual deterioration performance of the bridge.

Figure 4 .
Figure 4. Pareto chart of importance of bridge features.

Figure 5 .
Figure 5.Ten prediction results using Elman neural network.

Figure 6 .
Figure 6.Worst and best bridge deterioration curves predicted by Elman neural network.

Figure 5 .
Figure 5.Ten prediction results using Elman neural network.

Figure 5 .
Figure 5.Ten prediction results using Elman neural network.

Figure 6 .
Figure 6.Worst and best bridge deterioration curves predicted by Elman neural network.

Figure 6 .
Figure 6.Worst and best bridge deterioration curves predicted by Elman neural network.

Figure 7 .
Figure 7. Selected prediction curve from Elman neural network.

Figure 7 .
Figure 7. Selected prediction curve from Elman neural network.

Figure 8 .
Figure 8.Comparison of Elman neural network and Markov chain predicted values.

Figure 8 .
Figure 8.Comparison of Elman neural network and Markov chain predicted values.

Figure 9 .
Figure 9.Comparison with predicted and detection values of bridge condition of the case from 20 years.

Figure 9 .
Figure 9.Comparison with predicted and detection values of bridge condition of the case from 1 to 20 years.

Table 1 .
Characteristic values of bridges in the United States.

Table 2 .
Different ways of combining bridge features.

Table 2 .
Different ways of combining bridge features.

Table 3 .
Markov chain transition probability matrix from 1 to 60 years.

Table 4 .
Measured condition level of this case from 1 to 20 years.

Table 4 .
Measured condition level of this case from 1 to 20 years.