4.1. Obtaining the Optimal Fit Alignment
In this section, a comparison between the outcome of the alignment extraction methods and the existing inventory database is provided.
Table 1 presents the number of alignments according to the inventory database, those obtained by the proposed method, and the matching alignments. It contains the data broken down by type of alignment, considering the spirals within the same curve as the corresponding circular arc. As a first approximation, matching alignments are considered those that share more than 50% of their length in common in both databases. The column of matching alignments indicates the number of alignments that are found in both the database and the outcome of the method proposed under this criterion. According to the data shown in the table, important differences exist in terms of the number of alignments between the outcome of the alignment extraction methods and the existing inventory database. Although the inventory database contains more alignments than the number of alignments extracted with the procedure presented, some segments were missing in the inventory database. Moreover, the mismatched alignments correspond to either missing alignments or duplicated alignments in the inventory database, and to differences in the alignment sequences between the inventory and the extracted alignment. The matching tangents represent 79.4% of those in the inventory database, the matching curves 98.7% of those in the inventory database and the total matching alignments 91.0% of those in the inventory database.
Next, a comparison of the length, curve radii, and deflection angle of the matching alignments was conducted to verify the parameter values of the alignments obtained with the method presented.
Figure 4 shows histograms of the differences in length of the matching alignments, the differences in radii and the differences in deflection angles. T-tests for paired samples were performed to determine whether there are significant differences between the parameters of the alignments in the inventory database and their counterparts in the matching extracted alignments. The difference in length was not found to be significant at the 95% confidence level. The differences in the curve radii were also not found to be significant between the matching curves of two databases at the 95% confidence level. The differences in curve deflection angles were not found to be significant either at the 95% confidence level.
The root mean square error (RMSE) was computed for each of the three variables as an additional metric to compare the values obtained in the extracted alignment with the values of the counterpart matching alignments from the inventory database using the formula shown below:
where
n is the sample size,
is the value of the variable in the inventory database for the
i-th alignment and
is the counterpart value of the extracted alignment. The results are shown in
Table 2. The average percentage error (APE) has also been calculated for each of the three variables, and the results are also shown in
Table 2:
The results shown in
Table 2 indicate that the deflection angle values are the most accurate and the radius values are the least accurate. It should be noted that the error in the estimation of the alignment variables is in any case below 10%.
As seen previously, part of the sample of motorway sections in the inventory database contains information of high accuracy for the development of an SPF, while the rest contains information that differs significantly from the results obtained with the presented method.
The differences found between the inventory and the horizontal alignment are illustrated with an example section showing the difference between the alignment inventory and the reconstructed alignments.
Figure 5 shows a sequence of azimuth values obtained from the centerline sketched on the image for one roadway, and the subsequent extracted alignments on the A25 motorway. The former corresponds to the black line with steps. The latter includes horizontal blue lines representing tangents, inclined green straight lines representing circular arcs, and purple parabolas representing spirals. A curve with unsymmetrical spirals was adjusted from station 1+836 to 2+798, describing a turn of 41.6 gon. This sequence of alignments could be adjusted without difficulty using the best-fit alignment tool. Then, several small-angle deflection curves are observed. The first of these extends from stations 4+040 to 4+303, with no transition curves and with a deflection angle of only 4 gon. However, given that this alignment is connected to two tangents with a considerable length, the best-fit alignment tool was able to correctly identify the bend. Otherwise, the error in planimetry would be very large. Next, a reverse curve composed of two small-angle deflection curves is observed between stations 5+184 and 5+528. This is an alignment sequence to which the best-fit tool typically fails to adjust since the approach and departure tangents have approximately the same heading direction and, therefore, the planimetry error is low. As a result, these alignments required the azimuth graph and the circular adjustment to be extracted. It can also be noticed that the first curve of the sequence was adjusted with small transition curves at both ends, whereas the second one was adjusted without spirals. The completion of the sequence required a short interim tangent to connect both curves. Finally, a reverse curve is found between stations 5+800 and 6+112. In this case, only the first curve was adjusted with an approach transition curve. The other ends were adjusted without transition curves.
Figure 6 displays the curvature graphs as per (a) the inventory database and (b) the optimal-fit alignment of the same section. According to the alignment inventory, there is a curve between stations 1+476 and 2+250 with identical transition curves on both curve ends, whereas the analysis of the azimuth sequence shows that this curve is located 360 m ahead and the spirals are unsymmetrical. The rest of the section shows much more noticeable differences. According to the inventory database, there are three curves located between stations 2+400 and 3+700, while the analysis of the azimuth sequence reveals that none of these actually exist. Although the following curve is found in both the inventory database and the optimal-fit alignment, their parameters differ significantly. The radius, length and especially the location do not coincide. According to the inventory, from station 4+500, there are two curves with radii of 1100 m and 800 m. The analysis of the azimuth sequence reveals that there are actually two reverse curves, each consisting of two small-angle deflection curves. The curve radii do not match in any case.
In terms of the motorways analyzed, several sections can be found where the centerlines of each roadway are independent. Although this is not frequent, the different alignment features should be considered in the accident databases for each independent roadway, even though the differences may seem small, to ensure accurate crash prediction models. The case studies found both sections where the inventory was considered to match the existing alignments and sections where the clearly different characteristics of each roadway were not differentiated.
Figure 7 illustrates a section of the A25 motorway where the roadways diverge over a short distance and then rejoin, running completely parallel again. The parameters of horizontal alignment elements such as the lengths and radii of curves vary from one roadway to another in this section.
4.2. Impact of Variable Accuracy on Crash Prediction Models
Crash prediction models have been adjusted from the final alignment database in the case studies using the geometric design features extracted. The objective is to estimate the effect of the error in estimating the geometric design parameters on the accident prediction.
Generalized linear models have been used to link the independent variables with the crash count in the period 2015–2018. The number of accidents is modeled as a negative binomial variable since the data distribution exhibits over-dispersion [
50]. The link function predicts a number of accidents equal to or greater than zero according to the exposure variables, and it is linearizable in order to estimate the coefficients, with a mathematical expression as follows [
51]:
where
is the estimated number of crashes in four years;
is the segment length;
stands for the annual average daily traffic;
is the geometric design independent variable; and
are the model estimates. The 548 motorway segments extracted were used to adjust the models. Three geometric design variables have been used to create variables to be incorporated into the models: segment length, curve radius and deflection angle (
Table 3). According to Equation (8), the natural logarithm of the segment length is incorporated into the model as an exposure measure, along with the natural logarithm of the AADT. To include tangents and curves likewise in the model, the variable curvature was calculated as the inverse value of the curve radius. The deflection angle (Def_angle) was taken directly from the final alignment database, and the curvature change rate (CCR) was computed as the ratio of the deflection angle to the length.
In order to achieve a balance between simplicity and explanatory power, three parsimonious generalized linear models (GLMs) have been adjusted with R-Studio software (version 1.3.959). The first model, denoted as Model 1, incorporated the length, the segment and the curvature of the segment. Substituting the curvature with the inverse value of the curve radius, the equation that predicts the number of accidents in four years is expressed below:
The estimates
and the level of significance of the fitted model are presented in
Table 4. All the variables included were found to be significantly different from zero at the 95% confidence level.
The second model, denoted as Model 2, incorporated the length of the segment and the deflection angle of the segment (
. Equation (12) predicts the number of accidents in four years from the AADT, the length and the deflection angle of the segment:
The estimates
and their level of significance are presented in
Table 5. All the variables included in this model were found to be significantly different from zero at the 95% confidence level.
The third model, denoted as Model 3, included the segment length and the CCR. The number of accidents in four years is predicted by Equation (13), where the CCR is replaced by the ratio of the deflection angle divided by the segment length.
The estimates
and their level of significance are presented in
Table 6. In this model, all the variables included were found to be significantly different from zero at the 95% confidence level.
It should be noted that the parameters obtained for the models fitted are consistent with similar models in the literature. First, all the exposure variables have positive values of the estimates. In all cases, the models indicate that the accident frequency increases as the traffic volume increases and as the segment length increases. Also, the positive correlation between the curvature, deflection angle, and CCR and the accident frequency is consistent with similar models and logical expectations.
Based on the adjusted crash prediction models, an error propagation analysis was conducted to determine how the error in determining the geometric design variables impacts the crash prediction output. Evaluating error propagation in a model equation involves determining how the uncertainty in the input variables affects the uncertainty of the output. As an initial approach, uncorrelated and independent errors were assumed. Equation (14) provides the expression to obtain the relative value of the propagated uncertainty
of the output of a SPF as a function of the partial derivative of the model function with respect to one of the geometric design input variables
, the model function itself
and the value of the error for that input variable
[
52].
Applying the expression of Equation (14) to the model that includes the curvature variable (
Table 4) to obtain the output error produced by an error of 10% in the estimation of the curve radius, as measured in km, the following expression is obtained:
The result is a relationship that is inversely proportional to the radius of the curve and is independent of the segment length and the AADT. Substituting the result of Equation (15) with different values for the curve radius and taking into account the 10% reference error in the radius value mentioned above, the error in estimating the accident frequency is −5.6% for curves with 400 m radius and −1.1% for curves with 2000 m radius. It can therefore be said that the error propagation in the radius estimation in this SPF is limited.
If Equation (14) is developed for the model that includes the curvature variable to obtain the output error produced by an error of 10% in the estimation of the segment length, the following result is obtained:
The result is a constant value of 7.8%. Therefore, in this case, the propagated error is smaller than the estimation error of the segment length.
In the case of the Model 2, using Equation (14) and considering an error of 10% in the estimation of the deflection angle
, the error propagation is:
The result is linearly related to the deflection angle and is independent of the segment length and the AADT. The error value is 0.4% for curves with a 5 gon deflection angle and 7.3% for curves with a 100 gon deflection angle.
Considering Equation (14) to deduce the error propagation produced by a 10% error in the estimation of the segment length, results in:
The result is a constant value of 6.5%. Consequently, the propagated error is less than the estimation error of the segment length.
In the third model, the result of applying Equation (14) with an estimation error of 10% in the deflection angle
is:
The result of the accident frequency prediction error in this case depends on two variables. It is directly proportional to the value of the deflection angle and inversely proportional to the segment length. The contour plot in
Figure 8a represents the relative error propagated to the accident prediction output as a function of both variables. Each line is annotated with the error associated with the accident prediction according to the values of the variables on the axes. The vertical axis represents the segment length, and the horizontal axis displays the deflection angle, each of them for the range of values in the sample of motorway alignments. It can be observed that the value of the maximum error in accident prediction is 0.05%, and it is obtained for the segments with a length of about 100 m long and curves with a deflection angle of almost 120 gon.
Also, in the third model, applying Equation (14) with an estimation error of 10% in the section length
results in:
The contour plot in
Figure 8b illustrates the relative error propagated to the accident prediction according to the result of Equation (22) for the ranges of the segment length and deflection angle in the motorway alignment sample. This shows that the relative error propagated is almost constant at 7.7% and increases slightly with an increasing segment length and a decreasing deflection angle.