**4. Results and discussion**

#### **4.1. Models MRA**

**Table 3** presents the results of MRA models for data set with complete bridge components condition rating data and data set after substituting missing data with the proposed methods. The results show that the performance of a network trained with the entire range scale of data sets (M1) seems better than the network trained with missing range scale ratings as in data sets M0, as indicated by the *R*<sup>2</sup> values. For data sets M1, the handling method with SMD, SMC, and SBR yields almost similar *R*<sup>2</sup> values that are better in comparison to SM and SMN methods. Furthermore, in term of RMSE value, SBR method shows better performance in comparison to other methods. Referring to RMSE and *R*<sup>2</sup> value of training, validation, and training data sets, the performance of SBR method appears to yield slightly better results than other methods.

The typical MRA models linking bridge condition rating to its explanatory components for data sets M1-SBR are presented in **Table 4**. The significance of each coefficient was determined by t-value and P-value. The F-ratio presented in **Table 3** measures the probability of chance departure from a straight line. On review of the output found in **Table 3** shows that the overall model was found to be statistically significant as (F = 53.6379, Sig.-F < .0000), (F = 138.6713, Sig.-F < .0000), (F = 99.1484, Sig.-F < .0000), (F = 101.3906, Sig.-F < .0000), (F = 104.8520, Sig.-F < .0000), and (F = 148.6167, Sig.-F < .0000) for data sets M0, M1-SM, M1-SMN, M1-SMD, M1-SMC, and M1-SBR, respectively.

#### **4.2. Models ANN**

The training process is evaluated based on plotting the training, validation, and Mean Squared Errors (MSEs) versus number of epochs. **Figure 4** shows the training progress of the selected ANN for the data sets M0. The network seems unstable, as indicated in **Figure 4**, where the characteristics of the validation and test error are not similar, which may indicate that there is a gap in the data set distribution or a poor division of the data set [17].


**Figure 5** shows the typical training progress of data sets where the missing value is substituted with the bridge condition rating value (SBR). **Figure 5** shows the decrease of the MSE versus the number of epochs during the training process of the ANN. Indeed, **Figure 5** shows that the validation and testing set errors show similar characteristics, which provide reasonable evidence of network training, and it does not appear that any significant overfitting has occurred. If the error in the test set reaches a minimum at a significantly different epoch num-

ber than the validation set error, it may indicate a poor division of the data set [17].

**Figure 4.** Typical training performance versus number epochs for data sets M0.

**Factors Coefficients t-value P-value** Intercept −0.0658 −0.8026 0.4239 Surfacing 0.0682 2.1352 0.0350\* Expansion joint 0.0402 1.2845 0.2017 Parapet 0.0980 3.2732 0.0014\* Drainage 0.0536 1.8002 0.0746 Slope protection 0.0542 2.0419 0.0436\* Abutment 0.2138 6.1275 0.0000\* Bearing 0.0100 0.2033 0.8392 Deck slab 0.2422 6.2569 0.0000\* Beam/Girder 0.2754 6.1422 0.0000\*

Developing a Bridge Condition Rating Model Based on Limited Number of Data Sets

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37

**Table 4.** Model coefficients estimated by MRA for data sets M1 substituting with SBR (M1-SBR).

\*

Significant at p < 0.05.

**Table 3.** The statistic summaries of MRA result.

Developing a Bridge Condition Rating Model Based on Limited Number of Data Sets http://dx.doi.org/10.5772/intechopen.71556 37


**Table 4.** Model coefficients estimated by MRA for data sets M1 substituting with SBR (M1-SBR).

hidden and output layer, *b*<sup>1</sup>

**4. Results and discussion**

**4.1. Models MRA**

**4.2. Models ANN**

M0, as indicated by the *R*<sup>2</sup>

SBR yields almost similar *R*<sup>2</sup>

other methods. Referring to RMSE and *R*<sup>2</sup>

for the neurons in the output layer [21].

*t*1

36 Bridge Engineering

is the bias in the hidden layer, *b*<sup>2</sup>

**Table 3** presents the results of MRA models for data set with complete bridge components condition rating data and data set after substituting missing data with the proposed methods. The results show that the performance of a network trained with the entire range scale of data sets (M1) seems better than the network trained with missing range scale ratings as in data sets

Furthermore, in term of RMSE value, SBR method shows better performance in comparison to

The typical MRA models linking bridge condition rating to its explanatory components for data sets M1-SBR are presented in **Table 4**. The significance of each coefficient was determined by t-value and P-value. The F-ratio presented in **Table 3** measures the probability of chance departure from a straight line. On review of the output found in **Table 3** shows that the overall model was found to be statistically significant as (F = 53.6379, Sig.-F < .0000), (F = 138.6713, Sig.-F < .0000), (F = 99.1484, Sig.-F < .0000), (F = 101.3906, Sig.-F < .0000), (F = 104.8520, Sig.-F < .0000), and (F = 148.6167, Sig.-F < .0000) for data sets M0, M1-SM, M1-SMN, M1-SMD, M1-SMC, and M1-SBR, respectively.

The training process is evaluated based on plotting the training, validation, and Mean Squared Errors (MSEs) versus number of epochs. **Figure 4** shows the training progress of the selected ANN for the data sets M0. The network seems unstable, as indicated in **Figure 4**, where the characteristics of the validation and test error are not similar, which may indicate that there is

M0 0.2740 53.6379 4.31E−31 0.8518 0.8060 0.8700 M1-SM 0.2741 138.6713 1.83E−55 0.9197 0.8770 0.9140 M1-SMN 0.2816 99.1484 1.36E−48 0.9158 0.8610 0.9160 M1-SMD 0.2723 101.3906 4.58E−49 0.9213 0.8920 0.9140 M1-SMC 0.2692 104.8520 4.64E−50 0.9272 0.8830 0.8900 M1-SBR 0.2655 148.6167 5.72E−57 0.9246 0.8840 0.9140

a gap in the data set distribution or a poor division of the data set [17].

**Data sets RMSE F Significance-F** *R<sup>2</sup>*

**Table 3.** The statistic summaries of MRA result.

the performance of SBR method appears to yield slightly better results than other methods.

values. For data sets M1, the handling method with SMD, SMC, and

values that are better in comparison to SM and SMN methods.

value of training, validation, and training data sets,

**-Train** *R<sup>2</sup>*

**-Val** *R<sup>2</sup>*

**-Test**

is the transfer function for the neurons in the hidden layer, and *t*<sup>2</sup>

is the bias in the output layer,

is the transfer function

**Figure 4.** Typical training performance versus number epochs for data sets M0.

**Figure 5** shows the typical training progress of data sets where the missing value is substituted with the bridge condition rating value (SBR). **Figure 5** shows the decrease of the MSE versus the number of epochs during the training process of the ANN. Indeed, **Figure 5** shows that the validation and testing set errors show similar characteristics, which provide reasonable evidence of network training, and it does not appear that any significant overfitting has occurred. If the error in the test set reaches a minimum at a significantly different epoch number than the validation set error, it may indicate a poor division of the data set [17].

0.9057 for the training, validation, and testing sets, respectively, with the substitution of the missing value with the bridge condition rating value. The network of the incomplete data sets does not seem to perform well in fitting the entire range of the rating scale. This problem may be due to the effect of a gap between the available bridge condition rating scale (where rating 4 is unavailable) and the bridge component ratings, which have a rating scale from 1 to 5.

Developing a Bridge Condition Rating Model Based on Limited Number of Data Sets

In terms of RMSE values, the missing bridge component data handled using SBR method yield lowest RMSE value in comparison to other substitution methods, which indicates that the SBR method improves the performance of the ANN model. The RMSE values of the data sets where the missing data are handled by SM, SMN, SMD, SMC, and SBR are 0.2349, 0.2386,

The linear regression analysis of all data sets is then also performed to evaluate the developed models' response on all the data sets. Here, all the data sets, namely training, validation, and test sets, are introduced through the models, and a linear regression between the model outputs and the corre-

The result shows that the MRA model and ANN model trained with the missing data substituted by the bridge condition rating value (SBR) has a higher accuracy of prediction in com-

and 0.9328 for all data sets by MRA and ANN methods, respectively. This indicates that, for the bridge condition rating model with these data conditions, the SBR method is a more

**-Train** *R<sup>2</sup>*

parison to other methods, as shown in **Table 6**. The SBR method yields *R*<sup>2</sup>

M0 0.2079 0.9131 0.7515 0.8115 M1-SM 0.2349 0.9375 0.8789 0.9045 M1-SMN 0.2386 0.9363 0.8688 0.9059 M1-SMD 0.2257 0.9429 0.8884 0.9063 M1-SMC 0.2404 0.9372 0.8849 0.8792 M1-SBR 0.2066 0.9553 0.8922 0.9057

**Conditions of data sets MRA models ANN models** M0 0.8380 0.8605 M1-SM 0.9050 0.9192 M1-SMN 0.8980 0.9167 M1-SMD 0.9090 0.9247 M1-SMC 0.9040 0.9151 M1-SBR 0.9100 0.9328

value of entire data sets for all substituting methods.

values of all data sets for all handling methods.

http://dx.doi.org/10.5772/intechopen.71556

39

**-Val** *R<sup>2</sup>*

values of 0.9100

**-Test**

0.2257, 0.2404, and 0.2066, respectively.

sponding targets is performed. **Table 6** shows the *R*<sup>2</sup>

**Data sets RMSE** *R<sup>2</sup>*

**Table 5.** The ANN models performance for data sets M0 and M1.

**Table 6.** The *R*<sup>2</sup>

**Figure 5.** Typical training performance versus number epochs for data sets M1-SBR.

Regression analysis between the bridge condition rating predicted by the ANN model and the corresponding bridge condition rating target is performed using the routine postreg using MATLAB software. The format of this routine is [m,b,r] = postreg (a,t), where m and b correspond to the slope and the intercept, respectively, of the best linear regression that relates the targets to the ANN outputs. If the fit is perfect, the ANN outputs are exactly equal to the bridge condition rating targets, and the slope is 1 and the intercept with the Y-axis is 0. The third variable, r, is the correlation coefficient between the ANN outputs and targets. It is a measure of how well the variation in the predicted bridge condition rating is explained by the target. If r is equal to 1, then there is perfect correlation between the targets and ANN outputs [22].

The performance of ANN models for complete data sets (M0) and data sets (M1) after the missing data are handled by the above-mentioned methods are presented in **Table 5**. The regression analysis for data sets with the missing data substituted with the SM method yields *R*<sup>2</sup> values of 0.9375, 0.8789, and 0.9045 for the training, validation, and testing sets, respectively. The training of data sets where the missing data are substituted with the SMN method yields *R*<sup>2</sup> values of 0.9363, 0.8688, and 0.9059 for the training, validation, and testing sets, respectively. The training of data sets where the missing data are substituted with SMD method yields *R*<sup>2</sup> values of 0.9429, 0.8884, and 0.9063 for the training, validation, and testing sets, respectively. Meanwhile, the training of data sets where the missing data are substituted with the SMC method yields *R*2 values of 0.9372, 0.8849, and 0.8792 for the training, validation, and testing sets, respectively. The training of data sets where the missing data are substituted with the SBR method yields *R*<sup>2</sup> values of 0.9553, 0.8922, and 0.9057 for the training, validation, and testing sets, respectively.

The results also show that the predictions of a network trained with the data sets M0 are less accurate than those of the handled data sets, as indicated by the *R*<sup>2</sup> values in **Table 5**. The *R*<sup>2</sup> values for the data sets M0 are 0.9131, 0.7515, and 0.8115 for the training, validation, and testing sets, respectively. The treated data sets yield the highest *R*<sup>2</sup> values of 0.9553, 0.8922, and 0.9057 for the training, validation, and testing sets, respectively, with the substitution of the missing value with the bridge condition rating value. The network of the incomplete data sets does not seem to perform well in fitting the entire range of the rating scale. This problem may be due to the effect of a gap between the available bridge condition rating scale (where rating 4 is unavailable) and the bridge component ratings, which have a rating scale from 1 to 5.

In terms of RMSE values, the missing bridge component data handled using SBR method yield lowest RMSE value in comparison to other substitution methods, which indicates that the SBR method improves the performance of the ANN model. The RMSE values of the data sets where the missing data are handled by SM, SMN, SMD, SMC, and SBR are 0.2349, 0.2386, 0.2257, 0.2404, and 0.2066, respectively.

The linear regression analysis of all data sets is then also performed to evaluate the developed models' response on all the data sets. Here, all the data sets, namely training, validation, and test sets, are introduced through the models, and a linear regression between the model outputs and the corresponding targets is performed. **Table 6** shows the *R*<sup>2</sup> values of all data sets for all handling methods.

The result shows that the MRA model and ANN model trained with the missing data substituted by the bridge condition rating value (SBR) has a higher accuracy of prediction in comparison to other methods, as shown in **Table 6**. The SBR method yields *R*<sup>2</sup> values of 0.9100 and 0.9328 for all data sets by MRA and ANN methods, respectively. This indicates that, for the bridge condition rating model with these data conditions, the SBR method is a more


**Table 5.** The ANN models performance for data sets M0 and M1.

**Figure 5.** Typical training performance versus number epochs for data sets M1-SBR.

Regression analysis between the bridge condition rating predicted by the ANN model and the corresponding bridge condition rating target is performed using the routine postreg using MATLAB software. The format of this routine is [m,b,r] = postreg (a,t), where m and b correspond to the slope and the intercept, respectively, of the best linear regression that relates the targets to the ANN outputs. If the fit is perfect, the ANN outputs are exactly equal to the bridge condition rating targets, and the slope is 1 and the intercept with the Y-axis is 0. The third variable, r, is the correlation coefficient between the ANN outputs and targets. It is a measure of how well the variation in the predicted bridge condition rating is explained by the target. If r is

equal to 1, then there is perfect correlation between the targets and ANN outputs [22].

analysis for data sets with the missing data substituted with the SM method yields *R*<sup>2</sup>

ing of data sets where the missing data are substituted with SMD method yields *R*<sup>2</sup>

accurate than those of the handled data sets, as indicated by the *R*<sup>2</sup>

ing sets, respectively. The treated data sets yield the highest *R*<sup>2</sup>

*R*2

38 Bridge Engineering

The performance of ANN models for complete data sets (M0) and data sets (M1) after the missing data are handled by the above-mentioned methods are presented in **Table 5**. The regression

0.9375, 0.8789, and 0.9045 for the training, validation, and testing sets, respectively. The training of data sets where the missing data are substituted with the SMN method yields *R*<sup>2</sup>

of 0.9363, 0.8688, and 0.9059 for the training, validation, and testing sets, respectively. The train-

0.9429, 0.8884, and 0.9063 for the training, validation, and testing sets, respectively. Meanwhile, the training of data sets where the missing data are substituted with the SMC method yields

 values of 0.9372, 0.8849, and 0.8792 for the training, validation, and testing sets, respectively. The training of data sets where the missing data are substituted with the SBR method yields *R*<sup>2</sup> values of 0.9553, 0.8922, and 0.9057 for the training, validation, and testing sets, respectively.

The results also show that the predictions of a network trained with the data sets M0 are less

values for the data sets M0 are 0.9131, 0.7515, and 0.8115 for the training, validation, and test-

values of

values

values of

values in **Table 5**. The *R*<sup>2</sup>

values of 0.9553, 0.8922, and


**Table 6.** The *R*<sup>2</sup> value of entire data sets for all substituting methods. reasonable method for handling the missing value prior to training the network. The performance comparison of ANN models and MRA models is also made in terms of *R*<sup>2</sup> values of predicted value versus target value as shown in **Table 6**. It can be seen that all ANN models provide better agreement with the target of bridge condition rating.

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