An Analysis of the Factors Inﬂuencing the Retroreﬂectivity Performance of In-Service Road Trafﬁc Signs

: The road trafﬁc signs in Sweden have no inventory system and it is unknown when a sign has reached the end of its service life and needs to be replaced. As a result, the road authorities do not have a systematic maintenance program for road trafﬁc signs, and many signs which are not in compliance with the minimum retroreﬂectivity performance requirements are still found on the roads. Therefore, it is very important to ﬁnd an inexpensive, safe, easy, and highly accurate method to judge the retroreﬂectivity performance of road signs. This will enable maintenance staff to determine the retroreﬂectivity of road signs without requiring measuring instruments for retroreﬂectivity or colors performance. As a ﬁrst step toward the above goal, this paper aims to identify factors affecting the retroreﬂectivity of road signs. Two different datasets were used, namely, the VTI dataset from Sweden and NMF dataset from Denmark. After testing different models, two logarithmic regression models were found to be the best-ﬁtting models, with R 2 values of 0.50 and 0.95 for the VTI and NMF datasets, respectively. The ﬁrst model identiﬁed the age, direction, GPS positions, color, and class of road signs as signiﬁcant predictors, while the second model used age, color, and the class of road signs.


Introduction
Road traffic signs are fundamental tools to regulate traffic and provide clear and important information to road users [1,2].They are used on roads to inform road users about traffic rules and aim to avoid traffic accidents.Therefore, the biggest challenge for the road authorities is to make sure that these signs are readable and visible during their lifetime to provide consistent information to road users.This is particularly important at night when driving is more difficult, with more accidents per vehicle happening on roads during this time [3].
At night, road traffic signs reflect the light coming from vehicles to the road users so that they can see these signs [4,5].The property of returning the light to the source (vehicle, in this case) is called retroreflectivity [6].Retroreflectivity of the signs, therefore, plays an important role in increasing traffic safety, especially during the night, but also in daylight when the weather conditions may not always provide enough light.
Fading of the retroreflectivity or color of the road traffic signs makes these signs useless.The service life of road traffic signs is determined by several parameters, but primarily by retroreflection and color requirements according to the road authorities' regulation and based on European standards [7].As the retroreflective material ages, its retroreflectivity coefficient (R A ) constantly weakens and the sign legibility declines [8].The maintenance regulation specifies the minimum level of retroreflectivity that the road traffic signs should have.The road authorities require signs to be replaced if retroreflectivity reaches this minimum level.
European standards [7] regulate the color (chromaticity) of road traffic signs based on the CIE color space that was created by the International Commission on Illumination.Permissible colors are indicated by color regions or color boxes in the chromaticity diagram shown in Figure 1.The two chromaticity coordinates, X and Y, for retroreflective sheeting determine a location in the CIE chromaticity diagram which represents a certain color.Accepted colors must be within the color boxes to ensure that road traffic signs have the right color and are suitable for the intended purpose.The color is no longer accepted when it deteriorates beyond these chromaticity limits (color boxes).
Appl.Sci.2022, 12, x FOR PEER REVIEW 2 of 18 European standards [7] regulate the color (chromaticity) of road traffic signs based on the CIE color space that was created by the International Commission on Illumination.Permissible colors are indicated by color regions or color boxes in the chromaticity diagram shown in Figure 1.The two chromaticity coordinates, X and Y, for retroreflective sheeting determine a location in the CIE chromaticity diagram which represents a certain color.Accepted colors must be within the color boxes to ensure that road traffic signs have the right color and are suitable for the intended purpose.The color is no longer accepted when it deteriorates beyond these chromaticity limits (color boxes).The coefficient of retroreflection can be measured using retroreflectometers, while spectrophotometers are used for measuring colors.In the field, measurements can be recorded using either handheld or mobile equipment.Currently, handheld devices are most frequently used for measuring sign retroreflectivity and colors, although this approach needs to address safety challenges and the high cost of data collection.Handheld instruments are relatively expensive, must be in direct contact with the sign, and data collection must take place under special weather conditions.Maintenance workers who use these instruments must stand close to the road sign for a relatively long time and they may need to block the traffic, in particular, on motorways or when using a lift.Although mobile devices have several advantages over handheld devices, mobile equipment is much more expensive and has lower accuracy compared to handheld devices.
To summarize, the methods used to measure the retroreflection and colors of in-use road traffic signs are costly, time-consuming, need to be performed under special weather conditions, and involve an accident risk for maintenance staff.Therefore, many countries, including Sweden, do not have a systematic program for the inventory of the performance of road traffic signs.At the same time, it is very important to specify when a sign reaches the end of its service life and must be replaced.
This study aims to find a safe, easy, inexpensive, and accurate method that can be used by road authorities to analyze the performance and deterioration of retroreflective road traffic signs mounted on roads.Two datasets were used to fit models to predict the retroreflectivity of road traffic signs.The proposed method is to use linear and logarithmic regression models to predict retroreflectivity without needing to use measuring instruments.
Previous studies have suggested prediction models based on two features (sheeting type and color).Most of these studies focused on age as the only predictor for retroreflectivity performance, with Alkhulaifi [9], who considered color and observation angle, in The coefficient of retroreflection can be measured using retroreflectometers, while spectrophotometers are used for measuring colors.In the field, measurements can be recorded using either handheld or mobile equipment.Currently, handheld devices are most frequently used for measuring sign retroreflectivity and colors, although this approach needs to address safety challenges and the high cost of data collection.Handheld instruments are relatively expensive, must be in direct contact with the sign, and data collection must take place under special weather conditions.Maintenance workers who use these instruments must stand close to the road sign for a relatively long time and they may need to block the traffic, in particular, on motorways or when using a lift.Although mobile devices have several advantages over handheld devices, mobile equipment is much more expensive and has lower accuracy compared to handheld devices.
To summarize, the methods used to measure the retroreflection and colors of in-use road traffic signs are costly, time-consuming, need to be performed under special weather conditions, and involve an accident risk for maintenance staff.Therefore, many countries, including Sweden, do not have a systematic program for the inventory of the performance of road traffic signs.At the same time, it is very important to specify when a sign reaches the end of its service life and must be replaced.
This study aims to find a safe, easy, inexpensive, and accurate method that can be used by road authorities to analyze the performance and deterioration of retroreflective road traffic signs mounted on roads.Two datasets were used to fit models to predict the retroreflectivity of road traffic signs.The proposed method is to use linear and logarithmic regression models to predict retroreflectivity without needing to use measuring instruments.
Previous studies have suggested prediction models based on two features (sheeting type and color).Most of these studies focused on age as the only predictor for retroreflectivity performance, with Alkhulaifi [9], who considered color and observation angle, in addition to age and sheeting type, being an exception.Therefore, the accuracy of the models in these studies was low.
This study proposed regression models with multiple predictors to identify (statistically) significant factors that affect retroreflectivity.Furthermore, efforts have been made to find and evaluate models that include only the predictors that can be collected without the need for any expensive measuring instruments.For this reason, the predictors included were the age of the road traffic sign, its GPS position, and the direction of the sign, as well as the color and class of the retroreflective sheeting.
One other important issue in this paper was investigating the significance of the chromaticity, luminance factor, and GPS positions on the retroreflectivity of road traffic signs.These factors were not previously investigated and their effects on retroreflectivity were not included in the state of the art.
The main contributions of this study are: • Firstly, the relationship between retroreflectivity and other factors such as the CIE color coordinates, sign age, its class, GPS position, and direction are investigated.Linear and logarithmic regression models were conducted to identify the significant factors that affect retroreflectivity.

•
Secondly, the study tries to find a connection between the performance and deterioration of retroreflectivity and the other factors mentioned above.

•
Finally, the study proposes a way to analyze and calculate the performance and deterioration of retroreflectivity using factors that can be collected without using expensive instruments.Such factors include the traffic sign's color, age, class, GPS position, and direction.
The rest of the paper is organized as follows.Section 2 discusses the prior literature, while Section 3 presents the description of the datasets used in this study and reviews the statistical methods applied.The results and discussion are provided in Section 4. Finally, the limitations, suggestions for further research, and conclusions are presented in Sections 5 and 6.

Systematic Literature Review
A systematic review was conducted in this paper, with Google Scholar used to identify peer-reviewed articles.The keywords used in the literature search were "retroreflectivity performance", "road traffic signs", "regression models", "retroreflective sheeting", and "aging".Additionally, the synonyms of these keywords were used in the search.A total of 192 articles were collected from 2000 to 2021 but after screening their titles and abstracts, only 17 articles were related.
The issue of a controlled study for sign deterioration was addressed in 1983 by Kenyon et al., who studied the deterioration of road signs in New York [10].Retroreflectivity deterioration is an interactive process between many factors [6].The main factors that affect retroreflectivity deterioration were identified by different researchers and were summarized in [6].Among the important factors were the geographic area, weather conditions, sheeting type, sheeting age, sheeting color, and sunlight exposure [6,11].Many researchers identified age as being the most significant factor that affects retroreflectivity deterioration, as shown in Table 1.The orientation of the sheeting in relation to the sun and the distance of the sign from the edge of the pavement were found to be insignificant [6,12,13].
Although many studies aiming to understand traffic sign retroreflection degradation have been conducted in the past three decades [1], only a few researchers have developed models that can accurately predict retroreflectivity performance and the durability of sign sheeting.In a 2017 study, developed regression models showed an average coefficient of determination (R-square) of 0.57 for linear models, 0.52 for logarithmic models, and 0.56 for exponential models [1].Regression models in earlier studies produced R-squared values that were relatively low, in the range of 0.10 to 0.30 [12].Swargam [6] compared the performance of multi-linear regression models with artificial neural network models (NN) used to predict the reflectivity of sign sheeting.The NN models predicted the retroreflectivity of signs closer to the in-field values for eight out of ten models, while two regression models estimated the reflectivity values closer to the in-field values than did the NN models.Even the other regression models produced nearly the same accuracy as the NN models.Generally, NN models are more competitively expensive than multi-linear regression models.
In a 2021 study, Alkhulaifi et al. [9] used regression models and neural networks based on deep learning models.The neural network with one-hot encoding achieved the best results with an R 2 value of 0.976, while polynomial regression models achieved an R 2 of 0.759.
Of all the sign types and colors studied in the past, attention has been paid to seven factors that are used as predictors for the retroreflectivity performance of road traffic signs.These factors are: sheeting type, sheeting age, sheeting color, the orientation of the sign concerning the sun, region, observation angle, and the distance from the edge of the pavement.Previously, only five factors-age, color, observation angle, sheeting type, and sign orientation-were included as predictors for retroreflectivity performance [1,14].
The developed regression models in previous studies showed relatively low R-squared values, with the suggested models needing improvement to achieve an accurate method of predicting the retroreflectivity performance of road traffic signs.For example, sheeting type, sheeting color, region, and distance from the edge of the pavement can be used as predictors, since including more predictors gives better prediction accuracy [6] and increases the goodness of fit of the models.
Although the state of the art found that the color of the retroreflective sheeting has a big effect on the deterioration of road traffic signs, no studies have evaluated the significance of chromaticity of the color (X and Y coordinates) on retroreflectivity.Despite the requirements according to the European standards 12899-1 [7] for the chromaticity and luminance factor of reflective sheeting, the effect of both the chromaticity and luminance factor on retroreflectivity was not investigated in any study.

Data Description and Data Pre-Processing
Two datasets were used in this study; the first one was collected in Sweden and the second one was collected in Denmark.Generally, the weather conditions and sheeting materials are almost identical in both countries.
The Road and Transport Research Institute (VTI) in Sweden collected the data from randomly selected signs mounted in the Swedish Transport Administration's West and Mid regions, with examples shown in Figure 2. The purpose of the collected data (VTI data) was to analyze the life cycle costs for road signs.The selected road signs were equally distributed between the northern and southern parts of Sweden.Only road signs on the right side of the road in urban and rural areas were investigated.Since road traffic signs can be in stock and the year of manufacture can thus be earlier than the year of installation, the inventory of four location signs was carried out.The selected location signs were manufactured to order and were thus set up soon after delivery.Stop signs were also investigated in VTI data to analyze the red retroreflective material.

Data Description and Data Pre-Processing
Two datasets were used in this study; the first one was collected in Sweden and the second one was collected in Denmark.Generally, the weather conditions and sheeting materials are almost identical in both countries.
The Road and Transport Research Institute (VTI) in Sweden collected the data from randomly selected signs mounted in the Swedish Transport Administration's West and Mid regions, with examples shown in Figure 2. The purpose of the collected data (VTI data) was to analyze the life cycle costs for road signs.The selected road signs were equally distributed between the northern and southern parts of Sweden.Only road signs on the right side of the road in urban and rural areas were investigated.Since road traffic signs can be in stock and the year of manufacture can thus be earlier than the year of installation, the inventory of four location signs was carried out.The selected location signs were manufactured to order and were thus set up soon after delivery.Stop signs were also investigated in VTI data to analyze the red retroreflective material.The NMF group, a voluntary Nordic research cooperation, measured 10 × 10 cm reflective sheeting samples (shown in Figure 3) which were mounted on a test stand and placed on the roof of the Danish Technical University, at its campus in Risø.
The datasets were cleaned of incomplete and unnecessary data, such as the manufacturer's name and description of the surrounding environment, because these data were not suitable for this study.The relevant features that remained after cleaning are listed in Table 2.The NMF group, a voluntary Nordic research cooperation, measured 10 × 10 cm reflective sheeting samples (shown in Figure 3) which were mounted on a test stand and placed on the roof of the Danish Technical University, at its campus in Risø.

Data Description and Data Pre-Processing
Two datasets were used in this study; the first one was collected in Sweden and the second one was collected in Denmark.Generally, the weather conditions and sheeting materials are almost identical in both countries.
The Road and Transport Research Institute (VTI) in Sweden collected the data from randomly selected signs mounted in the Swedish Transport Administration's West and Mid regions, with examples shown in Figure 2. The purpose of the collected data (VTI data) was to analyze the life cycle costs for road signs.The selected road signs were equally distributed between the northern and southern parts of Sweden.Only road signs on the right side of the road in urban and rural areas were investigated.Since road traffic signs can be in stock and the year of manufacture can thus be earlier than the year of installation, the inventory of four location signs was carried out.The selected location signs were manufactured to order and were thus set up soon after delivery.Stop signs were also investigated in VTI data to analyze the red retroreflective material.The NMF group, a voluntary Nordic research cooperation, measured 10 × 10 cm reflective sheeting samples (shown in Figure 3) which were mounted on a test stand and placed on the roof of the Danish Technical University, at its campus in Risø.
The datasets were cleaned of incomplete and unnecessary data, such as the manufacturer's name and description of the surrounding environment, because these data were not suitable for this study.The relevant features that remained after cleaning are listed in Table 2.The datasets were cleaned of incomplete and unnecessary data, such as the manufacturer's name and description of the surrounding environment, because these data were not suitable for this study.The relevant features that remained after cleaning are listed in Table 2.The coefficient of retroreflection (RA) was measured in cd/lx/m 2 at an observation angle of 0.33 • and measuring angle of 5 • .In this study, the coefficient of retroreflection is named retroreflection, which is the coefficient of luminous intensity of a plane retroreflection surface to its area, or a ratio of the returned intensity to incident illumination divided by the area of the retroreflection [1].
Daylight chromaticity (X, Y) was measured according to the CIE color space system.The luminance factor (β) is the ratio of the flux reflected from a specimen relative to the flux reflected from a perfect reflecting diffuser under the same geometric and spectral conditions of measurement.B is used to measure the brightness of the material and ranges from 0 for perfect black to 1 for perfect white.
This work investigated five colors: blue, green, red, white, and yellow At the same time, three different types of retroreflective sheeting were included in both datasets (Class 1, 2, and 3).The main difference between the classes is the amount of reflected light by each sheeting, with Class 3 reflecting the most, as described below: • Class 1: Three different sheetings belong to this class.The first one is Engineer Grade (EG) sheeting that has glass beads as a reflective material and it represents the least reflective type of foil.The second one is the Engineer Grade Prismatic (EGP) that has prisms instead of glass beads as reflective material.The last one is the Super Engineer Grade (SEG), which is similar to the Engineer Grade but with less variance in the size of the glass beads.

•
Class 2: High Reflectivity (HR) or High Intensity (HI) has glass beads as reflective material, but the encapsulation of the beads is of a different type, which gives the reflected light a narrower beam.

•
Class 3: Diamond Grade (DG) is a highly reflective prismatic material.Finally, the direction in degrees gives the azimuth angle to which the sign is facing, where north, east, south, and west are 0, 90, 180, and 270 degrees, respectively.

Statistical Analysis
Results from previous studies indicated that the best-fitting relationships between retroreflectivity and age were generally linear and that the linear models were significant despite their low R 2 values [1,6,12,13,[15][16][17].Therefore, linear regression models were conducted in this study.However, possible nonlinearity in the regression model was checked using residual plots, with their remedies attempted through Box-Cox transformations [18] of the response factor.
The Box-Cox transformation was used to transform the dataset into a more normally distributed one.This transformation procedure is used to modify the distributional shape of the response variable so that the residuals are more normally distributed, which is important for ensuring a stable estimate and valid inference [19,20], especially with a small sample size.The Box-Cox transformation is a family of power transformation and has the following mathematical form: where Y represents the data and λ is the "power" to which each data value is raised.The Box-Cox plot is a plot of the correlation from the data transformation for given values of λ.The Box-Cox plot, shown in Figure 4, is considered as a guide to choosing a value of the parameter of power transformation (λ) that is as close as possible to the one that maximizes the log-likelihood function and is easy to interpret [18].Figure 4 indicated the optimal value of λ was close to zero for the two datasets, therefore logarithmic regression models were used to transform the data according to Equation (1).The plot of the profile likelihood in Figure 4 is not very flat, indicating that the Box-Cox transformation is likely to be successful with the data [21].

R PEER REVIEW 7 of 18
of the response variable so that the residuals are more normally distributed, which is important for ensuring a stable estimate and valid inference [19,20], especially with a small sample size.The Box-Cox transformation is a family of power transformation and has the following mathematical form: where Y represents the data and λ is the "power" to which each data value is raised.
The Box-Cox plot is a plot of the correlation from the data transformation for given values of λ.The Box-Cox plot, shown in Figure 4, is considered as a guide to choosing a value of the parameter of power transformation (λ) that is as close as possible to the one that maximizes the log-likelihood function and is easy to interpret [18].Figure 4 indicated the optimal value of λ was close to zero for the two datasets, therefore logarithmic regression models were used to transform the data according to Equation (1).The plot of the profile likelihood in Figure 4 is not very flat, indicating that the Box-Cox transformation is likely to be successful with the data [21].Linear and logarithmic regression models with multiple predictors were applied to analyze the retroreflectivity, as shown in Figure 5. Firstly, the models were used to find the relationship between retroreflectivity and all the factors in the datasets.A backward stepwise selection [18] was then used to find the best subset selection for the significant factors, as described in Section 4.1.Secondly, logarithmic regression models were used to calculate the performance and deterioration of retroreflective road traffic signs mounted on the roads, as shown in Sections 4.2 and 4.3.
The analysis performed in this work is two-fold.Firstly, all the factors in the datasets (X, Y, age, color, class, GPS position, direction, and β) were included in the analyses and models (Sections 4.1 and 4.2).Secondly, only the factors that can be collected without the need for expensive measuring instruments were used in the models to calculate the retroreflectivity (Section 4.3).The predictors included the age of the road traffic sign, its GPS position, and the direction of the sign, as well as its color and the class of its retrore- Linear and logarithmic regression models with multiple predictors were applied to analyze the retroreflectivity, as shown in Figure 5. Firstly, the models were used to find the relationship between retroreflectivity and all the factors in the datasets.A backward stepwise selection [18] was then used to find the best subset selection for the significant factors, as described in Section 4.1.Secondly, logarithmic regression models were used to calculate the performance and deterioration of retroreflective road traffic signs mounted on the roads, as shown in Sections 4.2 and 4.3.Linear and logarithmic regression were used to determine equations that produce the minimum distance between the fitted values and all data points, with R 2 used to measure, statistically, how close the data are to the fitted values.R 2 was selected to assess the goodness of fit (model performance) because it had been used in previous studies investigating retroreflectivity deterioration, thereby leaving scope for comparability between studies.
Because the predictive performance of the model is not the prime concern of this work, we did not conduct any cross-validation or other type of out-of-sample performance assessment.

The Relationship between Retroreflectivity and Other Factors
The results from linear regression models, two logarithmic models (Model 1 and Model 2), and backward selection on Model 1 for the VTI and NMF data are shown in Tables 3 and 4, respectively.Model 1 is a logarithmic regression model that uses all factors in the data as predictors without considering interaction between any two predictors.Model 2 is a logarithmic regression model that investigates the effects of interaction between age and three other predictors (direction, colors, and classes).As color and classes are categorical factors, the blue color was used as a reference factor for colors and Class 1 was used as a reference factor for classes in the regression models (Model 2).The analysis performed in this work is two-fold.Firstly, all the factors in the datasets (X, Y, age, color, class, GPS position, direction, and β) were included in the analyses and models (Sections 4.1 and 4.2).Secondly, only the factors that can be collected without the need for expensive measuring instruments were used in the models to calculate the retroreflectivity (Section 4.3).The predictors included the age of the road traffic sign, its GPS position, and the direction of the sign, as well as its color and the class of its retroreflective sheeting.
Linear and logarithmic regression were used to determine equations that produce the minimum distance between the fitted values and all data points, with R 2 used to measure, statistically, how close the data are to the fitted values.R 2 was selected to assess the goodness of fit (model performance) because it had been used in previous studies investigating retroreflectivity deterioration, thereby leaving scope for comparability between studies.
Because the predictive performance of the model is not the prime concern of this work, we did not conduct any cross-validation or other type of out-of-sample performance assessment.

The Relationship between Retroreflectivity and Other Factors
The results from linear regression models, two logarithmic models (Model 1 and Model 2), and backward selection on Model 1 for the VTI and NMF data are shown in Tables 3 and 4, respectively.Model 1 is a logarithmic regression model that uses all factors in the data as predictors without considering interaction between any two predictors.Model 2 is a logarithmic regression model that investigates the effects of interaction between age and three other predictors (direction, colors, and classes).As color and classes are categorical factors, the blue color was used as a reference factor for colors and Class 1 was used as a reference factor for classes in the regression models (Model 2).In this study, GPS position is used to locate the region where the signs are mounted and included as predictors in the models to find the relationship between the location of the traffic sign and its deterioration.
Model 2 for the VTI data shows that the interaction between age and the other three predictors (direction, white color, and classes) was significant, but the estimate of the interaction effect with age was very small and had no visible impact on the prediction models (almost identical R 2 values for Model 1 and Model 2).Model 2 for the NMF data shows that age has no significant interaction with colors or classes.Because of the small or marginal effect of interaction between age and other predictors, the models used in the additional analysis do not consider interaction between any factors in the data.
As depicted in Tables 3 and 4 for Model 1, the age of the road traffic sign was significant in both the VTI and NMF data.In addition, the X coordinate of the CIE color space and the class of road traffic signs were also significant in both datasets.
The backward stepwise selection begins with the full least squares model containing all predictors and then iteratively removes the least useful predictor, one at a time [22].The reduced model that best explains the VTI and NMF data is shown in Tables 3 and 4, respectively.It is worth mentioning that logarithmic models were used for backward stepwise selection because they satisfied all the model diagnostics reasonably well and gave higher R 2 values compared to the linear models (no transformation of the response factor).GPS longitude was insignificant in the backward stepwise models for the VTI data, while the Y coordinate of the CIE color space and β were insignificant in the model for the NMF data.
An important finding is that the effect of GPS position on retroreflectivity is significant for the latitude but not longitude (shown in Table 3).It is clear from the results that road signs mounted in the southern regions of Sweden fade faster compared with those in the northern parts.There are many reasons why road signs fade faster in the southern regions.The high intensity of traffic and high humidity in the south may be the main factors that distinguish it from the north.The population size in the north is considerably smaller than that in the southern and central regions, while the summer period lasts longer in the south.In addition, Sweden's four largest lakes lie within the southern regions and they may increase the humidity in these regions.It was also noticed that logarithmic regression models of the yellow, red, and white colors in NMF data produced relatively higher R 2 values than other colors.These three colors have the highest β among the other colors.
As shown in Tables 3 and 4, the logarithmic regression models applied on the datasets that include all factors were more accurate than the linear regression models.The best regression models came from using backward stepwise selection on logarithmic regression models, as shown in Equations ( 2) and (3).
Based on VTI data (R 2 = 0.57): Based on NMF data (R 2 = 0.95): where: R A = The calculated coefficient of retroreflectivity.X, Y = Daylight chromaticity.Age = The age of the road sign in years.G = A constant must be set to 1 if the color is Green, otherwise set to 0. R = A constant must be set to 1 if the color is Red, otherwise set to 0. W = A constant must be set to 1 if the color is White, otherwise set to 0. Ye = A constant must be set to 1 if the color is Yellow, otherwise set to 0. Cl2 = A constant must be set to 1 if the class is 2, otherwise set to 0. Cl3 = A constant must be set to 1 if the class is 3, otherwise set to 0. Dir: The compass direction (Direction) in degrees gives the direction that the sign is facing (0, 90, 180, and 270 represent north, east, south, and west, respectively).lat, long: The coordinates (GPSlat, GPSlong) where the sign is mounted at the geographic coordinate system.
The performance of the prediction models in Equations ( 2) and (3) differs for each dataset because of the differences in the size, age, and placing of the retroreflective sheeting.
The logarithmic regression models on the NMF data showed a higher R 2 value than for the VTI data because the NMF data were collected from samples that have the same size and are exposed to the same conditions.
Age has a negative coefficient in Equations ( 2) and ( 3), confirming that the retroreflectivity decreases (by 4-5%) for a unit increase in age of the road sign, given all other factors in the model are held constant.
There is a big difference in the age of traffic signs included in the two datasets.While the age of traffic signs in the NMF data is just five years, the age of the traffic signs in the VTI dataset has a maximum of 36 years.Referring to Equations ( 2) and (3), the VTI dataset gives a negative X coordinate (−1.74), while the coefficient of the NMF dataset is positive (+1.47).One possible explanation for this difference is that at the beginning of the life cycle of the traffic sign, the color deteriorates much faster than the retroreflectivity which makes the traffic sign reflect more light and hence increases retroreflectivity.This happens in the case of the NMF dataset.By the end of the traffic sign life cycle, both the color and retroreflectivity deteriorate, which explains the negative sign in the equation.This was seen in the case of the VTI dataset.

Prediction Models Using Factors That Can Be Collected without Instruments
Thus far, regression models were applied to all factors in the data to investigate the relationship between retroreflectivity and the other predictors, as well as find the best prediction models.
To achieve the aim of building a model to calculate retroreflectivity using predictors that can be collected easily, quickly, safely, and cheaply, further regression models were generated.Predictors that need expensive measuring instruments and complex methods to be collected should not be included in the predicting models.Therefore, X, Y, and β were removed from the linear and logarithmic regression models, with the results shown in Tables 7-9.Firstly, linear and logarithmic regression were conducted on the datasets (see Table 7), using age, color, class, direction, and GPS position as predictors for models on the VTI data, while age, color, and class were predictors for models on the NMF data.Secondly, the regression models were further applied on the datasets that were divided according to classes (see Table 8).Finally, regression models were applied on the data divided according to class and color (see Table 9).The number of observations in the divided data is shown in Table 10.
The divided data, according to colors and classes, have few observations and give less information (see Table 10).Therefore, a decrease in the performance of models, expressed by lower R 2 values, was expected in models including only one color or one class (see Tables 8 and 9).The most accurate regression models producing the highest R 2 values (0.50 and 0.95) are logarithmic regression models, which include the class and color as predictors together with age (see Table 7).
As shown in Table 7, using logarithmic regression models produces R 2 values of 0.50 and 0.95 for VTI and NMF data, respectively, when omitting X, Y, and β.Removing these significant predictors (X, Y, and β) from the predicting models has no impact on the performance of models, with the R 2 values unchanged when compared with those for Equations ( 2) and (3).
The best regression models using only the factors that can be collected without instruments were the logarithmic regression models in Equations ( 4) and (5), as shown in Table 7: Based on VTI data (R  Six predictors can be used in Equation ( 4) to predict retroreflectivity: the class of the retroreflective sheeting, the color of the sign, the direction of the sign, the GPS latitude of the sign, the GPS longitude of the sign, and the age of the sign.These predictors can be easily collected without the need for expensive measuring instruments.Three predictors can be used in Equation (5) to predict retroreflectivity: the class of the retroreflective sheeting, the color of the sign, and the age of the sign.
The two models described in Equations ( 4) and (5) give a different predictive performance in terms of R 2 values, which are 0.50 and 0.95 for prediction models based on the VTI and NMF datasets, respectively.The two datasets differ in size and the age, size, and placing of the tested sheeting in each dataset are different, explaining why the performance of the prediction models differs for each dataset.
Age has a negative coefficient in Equations ( 4) and (5), confirming that retroreflectivity decreases (by 4-5%) for a unit increase in age of the road sign, given all other factors in the model are held constant.
The negative coefficient (−0.26) in Equation ( 4) for the direction of the road indicates that direction is negatively correlated with RA and that retroreflectivity decreases (by 26%) when the azimuth angle of the road traffic sign increases (by one unit).In particular, the road signs that faces south and west (directions of 180 and 270 degrees from north) fade faster than signs facing north and east (directions of 0 and 90 degrees) and need to be replaced more often.This conclusion is supported by the fact that solar radiation can affect road signs negatively.
Four assumptions are necessary so that the regression model can be applied for hypothesis testing, confidence intervals, and prediction [20].The assumptions of a linearity relationship, homogeneity of variance, uncorrelated errors, and normally distributed errors can be checked using diagnostic plots [19].The diagnostic plots of the logarithmic models (Figures 6 and 7) showed that:

•
The Residuals vs. Fitted values plot shows an almost horizontal line around the zero value.This indicates a linearity relationship between retroreflectivity and the predictors for the VTI and NMF data.

•
The residuals are more normally distributed in the NMF data because the Normal Q-Q plot shows that the residual points almost follow the straight dashed line.However, the VTI data residuals follow the dashed line only in the middle part of the plot.

•
The Scale-Location plot that was used to check the homogeneity of variance of the residuals (homoscedasticity) gives no indications of heteroscedasticity problems.A horizontal line with equally spread points is expected to indicate good homoscedasticity, and this is the case with both the VTI and NMF data.

•
The Residuals vs. Leverage plots indicate the presence of some outliers in both datasets.
fade faster than signs facing north and east (directions of 0 and 90 degrees) and need to be replaced more often.This conclusion is supported by the fact that solar radiation can affect road signs negatively.Four assumptions are necessary so that the regression model can be applied for hypothesis testing, confidence intervals, and prediction [20].The assumptions of a linearity relationship, homogeneity of variance, uncorrelated errors, and normally distributed errors can be checked using diagnostic plots [19].The diagnostic plots of the logarithmic models (Figures 6 and 7) showed that:

•
The Residuals vs. Fitted values plot shows an almost horizontal line around the zero value.This indicates a linearity relationship between retroreflectivity and the predictors for the VTI and NMF data.

•
The residuals are more normally distributed in the NMF data because the Normal Q-Q plot shows that the residual points almost follow the straight dashed line.However, the VTI data residuals follow the dashed line only in the middle part of the plot.

•
The Scale-Location plot that was used to check the homogeneity of variance of the residuals (homoscedasticity) gives no indications of heteroscedasticity problems.A horizontal line with equally spread points is expected to indicate good homoscedasticity, and this is the case with both the VTI and NMF data.

•
The Residuals vs. Leverage plots indicate the presence of some outliers in both datasets.The performance in terms of the R 2 values of the two models that were suggested in Equations ( 4) and (5) are almost equal to those of the models in Equations ( 2) and ( 3).This means that predictors requiring instruments to measure them can be excluded from the predicting models without affecting the models' performance.In summary, the models described in Equations ( 4) and (5) can be used to predict retroreflectivity and provide an accurate, safe, and unexpressive method.The regression models from studies as early as 2011 produced R 2 values ranging from 0.10 to 0.30 [12], while a 2017 study had R 2 values ranging from 0.52 to 0.57 [1].However, the regression models in Equations ( 4) and (5) of this study showed a better fit with the data than previous models, with R 2 values of 0.50 and 0.95.The performance in terms of the R 2 values of the two models that were suggested in Equations ( 4) and (5) are almost equal to those of the models in Equations ( 2) and (3).This means that predictors requiring instruments to measure them can be excluded from the predicting models without affecting the models' performance.In summary, the models described in Equations ( 4) and (5) can be used to predict retroreflectivity and provide an accurate, safe, and unexpressive method.The regression models from studies as early as 2011 produced R 2 values ranging from 0.10 to 0.30 [12], while a 2017 study had R 2 values ranging from 0.52 to 0.57 [1].However, the regression models in Equations ( 4) and (5) of this study showed a better fit with the data than previous models, with R 2 values of 0.50 and 0.95.

Limitations and Suggestions for Further Research
This study has some limitations with implications for future research.The first limitation is due to the used datasets (the variables measured in each dataset, including the age of retroreflected sheeting, the region included, and the type of retroreflective sheeting).The present study had a focus on linear and logarithmic models, which limits the performance of the models in terms of R 2 values of 0.50.Additionally, the study had a national focus on road signs mounted in Sweden.
The NMF data has some limitations because the age of signs was only up to five years old and the fading rate of retroreflectivity may deteriorate quicker after these first five years.The Nordic group plans to continue collecting data by measuring the retroreflectivity, daylight chromaticity, and luminance factor once a year.A follow-up is suggested on the data that will be collected by NMF to obtain a more accurate prediction.Future research must consider data collected from other regions and with higher ages to accomplish more accurate predictions.Nonlinear models such as neural networks, support vec-

Limitations and Suggestions for Further Research
This study has some limitations with implications for future research.The first limitation is due to the used datasets (the variables measured in each dataset, including the age of retroreflected sheeting, the region included, and the type of retroreflective sheeting).The present study had a focus on linear and logarithmic models, which limits the performance of the models in terms of R 2 values of 0.50.Additionally, the study had a national focus on road signs mounted in Sweden.
The NMF data has some limitations because the age of signs was only up to five years old and the fading rate of retroreflectivity may deteriorate quicker after these first five years.The Nordic group plans to continue collecting data by measuring the retroreflectivity, daylight chromaticity, and luminance factor once a year.A follow-up is suggested on the data that will be collected by NMF to obtain a more accurate prediction.Future research must consider data collected from other regions and with higher ages to accomplish more accurate predictions.Nonlinear models such as neural networks, support vector machines, regression trees, generalized additive models, etc., need to be tested in future work for better prediction.Given the short time series on the retroreflectivity of road traffic signs, it is not currently feasible to develop a more complex prediction, and we leave this issue for future work when we acquire more data.Finally, the significance of other factors needs to be investigated, including the distance from the sea, weather zones, surrounding environment, and weather.The road traffic signs mounted in high-speed or crowded roads need to be studied separately to other road signs to obtain more accurate models for prediction.

Conclusions
The main goal of this study was to build an accurate, easy, fast, inexpensive, and safe method to predict the retroreflectivity performance for road traffic signs mounted on roads.The logarithmic models suggested in this study provide an accurate and easy method to predict the retroreflectivity performance of road signs, to help decide if these signs should be replaced.

Figure 1 .
Figure 1.CIE color space with color boxes according to SS-EN 12899-1.

Figure 1 .
Figure 1.CIE color space with color boxes according to SS-EN 12899-1.

Figure 2 .
Figure 2. Randomly selected road signs in VTI data.

Figure 3 .
Figure 3. Reflective sheeting samples in the NMF data.

Figure 2 .
Figure 2. Randomly selected road signs in VTI data.

Figure 2 .
Figure 2. Randomly selected road signs in VTI data.

Figure 3 .
Figure 3. Reflective sheeting samples in the NMF data.Figure 3. Reflective sheeting samples in the NMF data.

Figure 3 .
Figure 3. Reflective sheeting samples in the NMF data.Figure 3. Reflective sheeting samples in the NMF data.

Figure 4 .
Figure 4. Profile-likelihood plot for the Box-Cox transformation of the VTI data (left) and NMF data (right).

Figure 4 .
Figure 4. Profile-likelihood plot for the Box-Cox transformation of the VTI data (left) and NMF data (right).

Figure 5 .
Figure 5. Statistical analysis design to determine those factors/predictors that significantly affect retroreflectivity and also predict retroreflectivity.

Figure 5 .
Figure 5. Statistical analysis design to determine those factors/predictors that significantly affect retroreflectivity and also predict retroreflectivity.

Figure 6 .
Figure 6.Checking the assumptions for logarithmic regression of the VTI data.

Figure 6 .
Figure 6.Checking the assumptions for logarithmic regression of the VTI data.

Figure 7 .
Figure 7. Checking the assumptions for logarithmic regression of the NMF data

Figure 7 .
Figure 7. Checking the assumptions for logarithmic regression of the NMF data.

Table 1 .
Summary of the previous literature discussing retroreflectivity deterioration.

Table 2 .
Description of the datasets used in this study.

Table 3 .
The relationship between retroreflectivity and other factors in the VTI data.

Table 4 .
The relationship between retroreflectivity and other factors in the NMF data.

Table 6 .
R 2 values for each color and class for VTI and NMF data (using logarithmic regression).

Table 7 .
Regression models for estimating retroreflectivity performance for VTI and NMF data.

Table 8 .
R 2 values for VTI and NMF regression models for all colors.

Table 9 .
R 2 values for VTI and NMF regression models for each color and class.

Table 10 .
The number of observations for divided data.