PIECEWISE MULTIPLE LINEAR MODELS FOR ... - Chieh (Ross) Wang

PIECEWISE MULTIPLE LINEAR MODELS FOR PAVEMENT MARKING RETROREFLECTIVITY PREDICTION UNDER EFFECT OF WINTER WEATHER EVENTS

Chieh (Ross) Wang (Corresponding Author) Graduate Research Assistant School of Civil and Environmental Engineering, Georgia Institute of Technology 790 Atlantic Drive, Atlanta, GA 30332 Tel: 404-457-9967; Fax: 404-894-5418; Email: [email protected] Zhaohua Wang, Ph.D., P.E. Senior Research Engineer Center for GIS, Georgia Institute of Technology 760 Spring Street NW, Suite 217, Atlanta, GA 30308 Tel: 404-385-0904; Fax: 404-385-0450; Email: [email protected] Yi-Chang (James) Tsai, Ph.D., P.E. Professor School of Civil and Environmental Engineering, Georgia Institute of Technology 790 Atlantic Drive, Atlanta, GA 30332 Tel: 404-894-6950; Fax: 404-894-5418; Email: [email protected]

Word Count: 5470 words + 3 figure(s) + 5 table(s) = 7470 words Submission Date: August 17, 2015

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ABSTRACT Most existing pavement marking retroreflectivity prediction models had been developed using data collected in locations under similar weather conditions; therefore, the effect of different weather conditions, such as winter weather events, has not been extensively studied. The primary purpose of this paper is to develop degradation models that can predict retroreflectivity of durable pavement marking materials under different winter weather conditions. Piecewise multiple linear models (PMLMs) are developed to explicitly account for the effect of winter weather events. By applying and comparing the proposed models with conventional multiple linear models (MLMs) developed using the same set of data, the proposed models outperformed the MLMs with the overall root mean square error improved from 204.6 mcd/m2 /lux for MLMs to 106.5 mcd/m2 /lux for PMLMs. The proposed method also show robust and consistent results in predicting different materials’ retroreflectivity in different states. This indicates that the proposed method can be adopted by different states and regions to develop comprehensive retroreflectivity prediction models under different winter weather conditions.

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INTRODUCTION Pavement markings are important traffic control devices used to convey messages to road users. Markings can be used as the primary source of regulations, guidance, or warning to road users, they can also serve as secondary traffic control devices that are supplements to other devices such as signs and signals (1). For example, longitudinal pavement markings, or long lines, define the rightof-way of the road and provide guidance to vehicles in all lanes and directions. They also provide information such as the roadway conditions ahead so that drivers can be prepared to slow down for horizontal curves, or to stop at intersections. From traffic operations and safety standpoints, the visibility of pavement markings under all weather and lighting conditions is vitally important. Transportation agencies, such as the American Association of State Highway and Transportation Officials (AASHTO), the Federal Highway Administration (FHWA), and state Departments of Transportation (DOTs), have developed extensive standard specifications, manuals, and procedures to ensure the proper use and maintenance of pavement markings. Moreover, agencies and industries have been working on improving the visibility, durability, and cost-effectiveness of pavement marking materials that can function under various weather, lighting, and roadway conditions. Pavement Marking Retroreflectivity To quantitatively evaluate the visibility of pavement markings, specific measures such as retroreflectivity has been developed. Retroreflectivity is the ability of pavement markings (or other materials) to reflect light back in the direction from where it came. A pavement marking’s retroreflectivity allows it to be visible to drivers especially at nighttime as light reflects off the pavement marking. Retroreflectivity of pavement markings can be measured by a special apparatus called a retroreflectometer, which can be either handheld for manual data collection or mounted on a vehicle for mobile data collection. Retroreflectivity, or the coefficient of retroreflected luminance (RL ), is measured in millicandela (mcd) per square meter per luminous flux (lux), or mcd/m2 /lux. Commonly used test methods for pavement marking retroreflectivity include dry and wet testing methods. Dry retroreflectivity readings are collected according to the ASTM Standard E 1710 (2), which specifies requirements for conducting the dry retroreflectivity test. Wet retroreflectivity readings are commonly collected according to the ASTM Standard E 2177. Here, measuring retroreflectivity under the standard condition of wetness, as described in the ASTM standard, is to slowly and evenly pour a bucket of 2 to 5 liters of water to the test area, and measure the retroreflectivity 45 seconds after the completion of pouring the water (3). Both ASTM standards require the measurement geometry of the measuring instrument to be at a viewing distance of 30 meters, a headlight mounting height of 0.65 meter and an eye height of 1.2 meter, which is equivalent to an observation angle of 1.05±0.02◦ between the light source and receiver of the instrument. Effect of Factors on Performance Traffic Wear The effect of traffic on pavement marking retroreflectivity has been extensively studied in the literature. In existing models, traffic has been considered in the following forms: (1) average daily traffic (ADT) or annual average daily traffic (AADT) (4–11); and (2) commercial or truck traffic (7). Some studies considered the joint effect of both time and traffic and used independent variables such as the cumulative traffic passage (CTP) for degradation modeling (10, 12, 13). Another factor related to traffic is roadway characteristics, which include the functional class and the geometry of the road. For example, in some studies, analysis has been conducted

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separately for pavement markings on interstate highways and on other routes (12) because interstate highways have limited access and usually do not have intersections that require vehicles to accelerate, decelerate, or turn frequently. In addition, pavement markings installed along a vertical and/or horizontal curve may perform differently than those installed along a longitudinal road segment (14). Different lane and shoulder widths may also affect the performance of markings (10). Installation Properties Initial retroreflectivity is the initial value measured shortly after the installation, usually within 30 days of installation. It has been considered to be a significant variable in several existing models (5, 6, 8, 9, 12). Material properties can also have significant effect on the performance of markings. For instance, the composition of marking materials, such as binder, pigment, and beads, is essential to the performance of markings. Construction properties, such as material temperature at installation and the dispersion rate, density, and embedment of beads, as well as the applied thickness, are all crucial factors that affect the performance of pavement markings (6, 10, 15). Different line properties, including color, lateral location, and installation direction, can also affect pavement marking retroreflectivity (6, 9). For example, lane lines usually deteriorate faster than edge lines because the former tend to be more exposed to traffic wear and tear. Weather and Environmental Conditions Winter weather events and their associated activities, such as snowplows and pavement surface treatments (e.g. the use of salt), can significantly damage and degrade the durability and retroreflectivity of pavement markings, especially in northern states (8, 12, 16). Moreover, external environmental factors during installation, such as the ambient temperature, road surface temperature, as well as humidity, can also have important effect on the performance of marking materials (10, 17). Degradation Modeling Since the late 1990s, researchers have developed extensive statistical models to predict the degradation of pavement marking materials. As summarized in Table 1, majority of these models used retroreflectivity as the dependent variable, and used elapsed time, traffic volume, initial retroreflectivity, and other roadway and pavement marking properties mentioned previously as independent variables. Simple linear and non-linear regression models that predict retroreflectivity based on a single variable, such as time or traffic volume, were used among the first several studies in the late 1990s and early 2000s (4, 7, 13, 18, 19). However, due to the low goodness-of-fit (i.e., Rsquared values) of these models, they have been gradually replaced by multiple linear regression models in recent years, for the latter can better account for the effects of multiple variables on retroreflectivity (5, 6, 8–10, 20, 21). Higher goodness-of-fit was achieved because of the inclusion of these variables. For example, Fu and Wilmot (21) developed a multiple linear regression model based on time, initial retroreflectivity, AADT, and snowplow events to predict retroreflectivity of paint in North Carolina. Their final models were able to achieve a R-squared value of 0.76. Other variables, such bead types (e.g., standard beads and high reflective elements (6)), line types (e.g., center line or skipped line (6, 9)), and land and shoulder widths (10), have also been used.

CTP

Material(s) Multiple Polyester, Thermoplastic, Waterborne Paint, Tape Multiple

R2 0.85+ 0.14 to 0.18 N/A

CTP

Paint and Thermoplastic

Simple Linear Regression

Time

Thermoplastic and Epoxy

Inverse Polynomial Model

Time

Multiple

0.32 and 0.58 0.21 to 0.78 N/A

Smoothing Spline and Time Series ARIMA Model Logarithmic

Time

Multiple

N/A

Time

Thermoplastic, Epoxy, and Polyurea Epoxy and Waterborne

0.53 to 0.87 N/A

Thermoplastic and Paint

Paint Polyurea

0.60 and 0.75 N/A 0.64

NC NC

Paint

0.76

NC

Waterborne Paint and HighBuild Paint Thermoplastic, Tape, and Inverted Profile Thermoplastic Thermoplastic

0.24 to 0.34 0.18 to 0.89

SC

0.45 to 0.49

AB

Year 1997 1999

Author(s) Andrady Lee et al.

Model(s) Logarithmic Simple Linear Regression

Variable(s) Time, Initial Retroreflectivity Time

2001

Migletz et al.

2002

Abboud Bowman

Simple Linear Regression, Quadratic, and Exponential Models Logarithmic regression

2003 2006

Thamizharasan et al. Bahar et al.

2006

Zhang and Wu

2007

Fitch

2009

Multiple Linear Regression

2009

Sasidharan et al. Sitzabee et al.

2011 2012

Hummer et al. Sitzabee et al.

Linear Mixed-Effects Model Multiple Linear Regression

2012

Multiple Linear Regression

2012

Mull and Sitzabee Robertson et al.

2012

Fu and Wilmot

Multiple Linear Regression

Time Time, AADT, Bead Type, Initial Retroreflectivity, Line Lateral Location Time, Initial Retroreflectivity, AADT, and Plow Events Time, AADT, CTP, Lane Width, and Shoulder Width Time, AADT, CTP

2014

Ozelim Turochy

Multiple Linear Regression

Time, AADT, Initial Retroreflectivity

and

and

Multiple Linear Regression

Multiple Linear Regression

Time, Directional ADT, Line Type, Pavement Type Time, Initial Retroreflectivity, AADT, Line Lateral Location, Line Color

Location(s) Across the US MI 19 States in the US

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TABLE 1 Summary of Pavement Marking Degradation Models in the Literature (Adapted from (6))

AB

SC AB, CA, MN, MS, PA, TX, UT, WI MS VT PA NC

LA

4

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Although most variables included in existing models have been found significant, it should be noted that the inclusion of variables that are highly correlated could lead to insignificant modeling results. For example, CTP is highly correlated with ADT and time, therefore, if ADT and time are already included in the model, CTP should not be included (12). Among all material types that have been studied, paint and thermoplastic were two most popular materials (4, 7–10, 12, 13, 18, 22, 23). With the rising interests of other durable marking materials, such as epoxy and polyurea, researchers have also developed models to predict performance of these materials (6, 12, 24). Degradation models of other pavement marking materials, such as preformed tape and methyl methacrylate (MMA), however, have not yet been extensively studied. Objectives and Organization of Paper Objectives From the literature, it has been commonly accepted that multiple linear models can predict retroreflectivity better (i.e., with higher goodness-of-fit) than single or non-linear regression models. However, several challenges remain unaddressed. First, most multiple linear models in the literature were developed for the individual state in which the data were collected, thus these models and results cannot be directly applied by other states. Second, most existing multiple linear models did not include diverse datasets that consist of different weather conditions, hence the effect of different weather conditions, such as winter weather events, has not been fully studied. Last, the performance of a few durable pavement marking materials, such as preformed tape and MMA, has not been extensively studied. Therefore, the objectives of this paper are to 1. Observe the effect of winter weather events on retroreflectivity and incorporate the effect into degradation modeling; 2. Develop comprehensive pavement marking retroreflectivity prediction models that can be adopted by different states, whether or not winter weather events are a primary concern; 3. Develop retroreflectivity prediction models for durable pavement marking materials, including preformed tape and MMA, which have not yet been extensively studied; and 4. Provide suggestions for state DOTs to incorporate the effect of winter weather events into their pavement marking management procedures. Paper Organization Organization of this paper is as follows: the second section describes the data used in this study, followed by some preliminary observations of the winter weather effect. The third section describes the methodology, including a brief introduction of piecewise multiple linear regression, variable selection, and model development and cross-validation. The fourth section shows the results of our analysis and the final models for predicting retroreflectivity of preformed tape and MMA. The fifth section provides further analysis and discussion on the models and results. Finally the last section concludes this paper with conclusions and recommendations on future research. DATA DESCRIPTION AND OBSERVATIONS Raw Data Since 1994, the American Association of State Highway and Transportation Officials’ (AASHTO) National Transportation Product Evaluation Program (NTPEP) has tested numerous transportation

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products and provided results to participating state DOTs and manufacturers. The most recent test results are stored in the DataMine 2.0 database (DataMine hereafter), which is a publicly accessible online database (25). Available field test data in DataMine include (1) installation data, such as installation date, air, road, and material temperatures, applied thickness, material composition, and bead properties; and (2) inspection data, such as inspection date and interval, retroreflectivity, durability, and color measurements. Data used in this study include available field test data of preformed tape and MMA in DataMine, which consists of data from 7 test decks in 3 states (Pennsylvania, Florida, and Minnesota) between 2008 and 2015. On each test deck, yellow and white pavement marking products were installed on asphalt and concrete sub-decks and tests were conducted periodically (every month in first year and every three months after) throughout a three-year analysis period. For the two subject materials, there were 1,112 lines (640 tape and 472 MMA) installed on the 7 test decks and a total of 11,921 inspection data entries (7,028 for tape and 4,893 for MMA) used in this study. Note that all pavement marking lines were installed in transverse direction based on NTPEP’s test deck standard design, and accelerated degradation in retroreflectivity may be expected (22, 23). With that in mind, all retroreflectivity data used in this study were measurements collected in ‘skip areas’ (i.e., within 9 in. to the long skip line in the corresponding lane), which are “considered to represent long line retroreflectivity performance” (26). Interested readers can find more details on the relationships between the performance of transverse and long lines in the studies published by Zhang et al., Pike and Songchitruksa (27, 28). In addition to NTPEP data, other data including average daily traffic (ADT) and average truck traffic (ADTT) were retrieved from the respective state DOT’s traffic data websites (29–31) or provided by the corresponding DOTs. Preliminary Observation of the Effect of Winter Weather Events In order to better understand the effect of winter weather events and incorporate it into retroreflectivity modeling, the change of retroreflectivity must be examined. Figure 1 shows the degradation of MMA in Florida and Pennsylvania over time. Each dot in the figure represents one retroreflectivity measurement and each line in the figure connects all measurements made on an actual pavement marking line at different times. For each inspection interval, a boxplot is drawn to see the distribution of retroreflectivity measurements in this interval. The bounding box in each boxplot shows the range of the middle 50 percentile of the points. Three horizontal lines of a bounding box denote the 25, 50, and 75 percentiles of the measurements in one interval. For instance, the three percentiles of interval 0 in Florida are approximately 360, 640, and 800 mcd/m2 /lux respectively. Note that these graphs were summarized from multiple test decks within each state (2 decks in Florida and 3 decks in Pennsylvania), and some intervals were either not collected due to weather conditions (32) or have not yet been collected/reported. For example, measurements between intervals 6 and 8 in all three Pennsylvania test decks were not collected due to severe winter weather conditions. By comparing the changes in retroreflectivity in the two plots in Figure 1, interesting findings can be observed. In the first six intervals, the two states shared some similar degradation patterns. Retroreflectivity measurements in both states ranged from approximately 200 to 1,700 mcd/m2 /lux at interval 0 and these ranges gradually decreased in the first 6 intervals. Moreover, most median readings of these intervals were around 500 mcd/m2 /lux. Nevertheless, these similar patterns disappeared thereafter. While retroreflectivity measurements continued to gradually

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FIGURE 1 Methyl methacrylate retroreflectivity readings plotted against inspection intervals in two states (a) Florida (test decks FL09 and FL12); and (b) Pennsylvania (test decks PA08, PA11, and PA14).

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deteriorate in Florida, dramatic decrease in retroreflectivity was observed between intervals 5 and 9, i.e., the first winter, in Pennsylvania. The median measurements dropped from approximately 500 to 300 mcd/m2 /lux after the first winter and the distribution of measurements became slightly more skewed to the right (i.e., more low-retroreflectivity measurements). Using interval 10 for example, the range of measurements was approximately 150 to 950 mcd/m2 /lux in Florida and 100 to 700 mcd/m2 /lux in Pennsylvania, and the median readings were approximately 375 and 275 mcd/m2 /lux respectively. Similar but less dramatic patterns can also be observed in the second winter (between intervals 15 and 21) and in the third winter (between intervals 27 and 33). From these observations, it is noted that winter weather events, especially those in the first winter, can have significant non-gradual impact on the performance of pavement markings. This

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type of physical damage should be explicitly considered in degradation models. Some existing models, as stated in the review section, used the number of snowplows as an independent variable in their models (8, 16). However, the number of snowplows on a specific road segment is often not a routine record with which every state DOT keeps track. Moreover, even if the number of snowplows is available, retroreflectivity measurements during snow seasons are typically not collected, since pavement markings are constantly covered by snow, ice, and/or other surface treatments. Because of these limitations, actual effect of winter weather events on retroreflectivity cannot be adequately modeled. Therefore, how to account for the effect of winter weather events and accurately predict pavement marking retroreflectivity, especially when records of snowplow events and measurements of retroreflectivity are not available, is an important question to be answered. As described later in this study, piecewise multiple linear regression was proposed to address this issue. METHODOLOGY Piecewise Multiple Linear Regression Piecewise regression, also known as segmented regression, is a regression method that divides the sample/dataset into multiple segments, and for each segment, a separate regression models is fitted. Here, a segment is defined as a subset of the entire dataset. This regression method is often used when the regression coefficients are known to change over a certain range of some variables such that fitting the dataset with one regression model may not be ideal. To implement a piecewise regression, the dataset is first ordered according to the ordering variable, i.e., the variable in which change is observed. The dataset is then divided into multiple segments according to the change in this variable based on prior knowledge or statistical analysis results. Finally, a regression model is fitted to predict values of the dependent variable in each segment. Note that the ordering variable does not need to be an independent variable. For illustration purposes, an example of piecewise regression is shown in Figure 2, in which the sample is first ordered according to the value of the ordering variable, x, then the sample is divided into two segments with x = 10 as the breakpoint; and finally, for each segment, a simple linear model is fitted. 600

y

400

200

0 5

10

15

20

x

FIGURE 2 A piecewise simple linear regression model with two segments. 29 30 31 32

Similarly, a piecewise multiple linear model (PMLM) for a dataset that has n observations, m independent variables, and p segments can be formulated in the manner shown in Equation 1 (33). Let yˆi be the ith predicted dependent variable of the sample, and Xi j (i = 1, ..., n; j = 1, ..., m) be the jth independent variable for the ith observation. Then

Wang, Wang, and Tsai   α1 + β11 Xi1 + β12 Xi2 + ... + β1m Xim , 1 ≤ i ≤ n1    α2 + β21 Xi1 + β22 Xi2 + ... + β2m Xim , n1 ≤ i ≤ n2 yˆi = ..  .    α + β X + β X + ... + β X , n p pm im p1 i1 p2 i2 p−1 ≤ i ≤ n. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

9

(1)

The value of nk is determined by counting the number of zi where zi < z∗k . Here, zi denotes the ith ordering variable and z∗k is the value of breakpoint for segment k; αk and βk j are the coefficients for the regression model of segment k, where k = 1, ..., p. Variable Selection Ordering Variable In this study, a PMLM was developed to account for the effect of winter weather events. Although it may be intuitive to assume that the elapsed time can be a good ordering variable, it may not be an ideal one. The reason is that agencies may install markings at different times of the year, thus the change of retroreflectivity due to winter weather events would also be observed at different times of the year, which then make it difficult to identify a clear changing point of retroreflectivity over time. Since the first winter typically has the largest impact on retroreflectivity, as observed in Figure 1, we proposed to use a binary variable – first winter passed – as the ordering variable. The value of this variable is 0 if the first winter has not passed or if there is no snow in the specific state; and is 1 if the first winter has passed. The dataset was divided into two segments based on this variable. Variable Selection As stated previously, the dependent variable in this study is retroreflectivity. Independent variables, on the other hand, are selected through the following process. First, a list of potential variables is summarized below based on literature and the availability of data in this study. Note that we added the maximum retroreflectivity of each line (MaxRetro) as a potential independent variable. Our assumption was that, when compared with the InitialRetro, this variable can better improve the accuracy of the model by accounting for the ‘polish effect’, of which the retroreflectivity ‘picks up’ in the first couple of months then starts to deteriorate afterwards. Among these variables, the ADT was pre-selected (starred) as a final variable for implementation purposes – so that the proposed model can easily be implemented by state DOTs. • ADT? : average daily traffic per lane (veh/day/ln) • Days: elapsed days from installation • MaxRetro: maximum retroreflectivity from installation • InitialRetro: initial retroreflectivity from installation • Thickness: the average applied thickness • MultipleBeads: a binary variable, 1 if multiple types of beads were applied, 0 otherwise • ADTT: average daily truck traffic per lane • RoadTemp: the average of road temperatures during installation Second, analysis of variance (ANOVA) and t-test were conducted to test the prediction power of these potential independent variables. In Table 2, the larger the absolute t-value is, the higher prediction power the corresponding variable has. Statistically significant t-values at the 95% level

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were highlighted in bold in this table. From the results, Days and MaxRetro were the two variables with t-values that are high and significant. MaxRetro turned out to be a better predictor than InitialRetro for both materials, which verifies our previous assumption. Consequently, the final selected independent variables were ADT, Days, and MaxRetro. Note that two additional variables, Days2 and MaxRetro2, representing the number of days and the maximum retroreflectivity after the first winter, were created and used for the second segment modeling (see Equation 2). Ten-fold cross-validation analyses were conducted to ensure that the proposed models were not over-fitting. TABLE 2 Prediction Power (Absolute t Value) of Potential Independent Variables

Days? Max Retro? Initial Retro Thickness Multiple Beads ADTT Road Temp ?

8 9 10 11 12 13 14 15 16

17 18 19 20 21 22 23 24 25 26 27 28

Preformed Tape Asphalt Concrete White Yellow White Yellow 35.78 31.60 20.17 17.85 13.97 10.03 14.19 8.30 1.97 1.62 7.32 6.63 6.38 1.36 5.09 1.32 2.35 0.97 0.80 0.10 1.68 4.49 1.33 1.45 4.91 2.05 4.58 4.09

Methyl Methacrylate Asphalt Concrete White Yellow White Yellow 20.11 7.42 15.87 5.14 15.51 27.64 13.19 20.72 0.28 9.54 4.22 0.15 7.80 4.43 0.59 0.35 12.29 10.60 3.33 4.42 0.37 1.44 2.15 1.52 4.48 1.10 1.50 1.09

Final variables

RESULTS Model Coefficients and Goodness-of-Fit The final models were formulated as in Equation 2, where n1 denotes the number of measurements before the first snow winter or in non-snow state. A total of eight PMLM models were developed, one for each combination of material type, pavement surface type, and line color, and the results are shown in Table 3. Note that the results, in terms of model coefficients and R-squared values are reported separately for the two segments in each model. For example, for yellow preformed tape markings installed on concrete pavements, the R-squared values for the first and second segments were 0.793 and 0.828 respectively. ( α1 + β11 ADTi + β12 Daysi + β13 MaxRetroi , i = 1, ..., n1 (2) RLi = α2 + β21 ADTi + β22 Days2i + β23 MaxRetro2i , i = n1 + 1, ..., n Several findings can be summarized from Table 3. First, all coefficients in all developed models were significant, indicating that the variable selection procedure using t-test was robust. Second, the signs of coefficients were mostly consistent with the literature that retroreflectivity gets lower when markings get older, expose to more traffic, and have lower maximum (or initial) retroreflectivity. Third, R-squared values for both segments, ranging from 0.640 to 0.941, showed promising goodness-of-fit of the final models. For states that snow is not a primary concern or for snow states before the first winter, the coefficients of the first segment can be used to predict retroreflectivity. Using white MMA on asphalt pavement as an example, with ADT = 5,000 veh/day/ln and MaxRetro = 800 mcd/m2 /lux, the predicted retroreflectivity in 730 days (i.e., 2 years from installation) is (137.060 − 0.004 × 5000 − 0.470 × 730 + 0.771 × 800) = 391 mcd/m2 /lux. Similarly, to predict retroreflectivity in regions

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TABLE 3 Final PMLM Coefficients and R-Squared Values

α1 β11 β12 β13 R21 α2 β21 β22 β23 R22

Tape Asphalt Concrete White Yellow White Yellow 96.720 77.289 171.705 86.402 -0.010 -0.003 -0.007 -0.003 -0.656 -0.374 -0.618 -0.319 0.917 0.828 0.821 0.824 0.878 0.868 0.752 0.793 124.032 60.985 22.129 28.050 -0.007 -0.001 0.004 0.002 -0.223 -0.145 -0.316 -0.194 0.643 0.554 0.716 0.666 0.768 0.701 0.831 0.828

MMA Asphalt Concrete White Yellow White Yellow 137.060 39.100 126.905 74.688 -0.004 -0.003 -0.005 -0.006 -0.470 -0.124 -0.449 -0.138 0.771 0.883 0.796 0.887 0.804 0.941 0.784 0.919 153.497 114.484 87.460 75.205 -0.009 -0.007 -0.002 -0.003 -0.295 -0.168 -0.262 -0.155 0.693 0.643 0.713 0.718 0.640 0.747 0.744 0.800

Note: all independent variables were significant variables in these models

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17 18 19 20 21 22 23 24 25

that snow is a primary concern, the second segment results can be used. Using the same example above, with the maximum retroreflectivity after first winter being 250 mcd/m2 /lux, the predicted retroreflectivity for white MMA on asphalt pavement is (153.497 − 0.009 × 5000 − 0.295 × 730 + 0.693 × 250) = 66 mcd/m2 /lux. Note that in order to implement the proposed method for more accurate retroreflectivity prediction, the collection of retroreflectivity data after the first winter should be incorporated as a standard procedure in these states. Applying Proposed Models to Derive Service Life – An Example The proposed models can be used to further derive the service life of pavement markings under different weather, traffic, and surface conditions. By plugging the minimum acceptable retroreflectivity, MinRetro, into Equation 2, the expected service life (Li ) can be derived using Equation 3. For example, assuming the minimum acceptable retroreflectivity is 100 mcd/m2 /lux (the current practice of the Georgia DOT), with a maximum retroreflectivity of 773 mcd/m2 /lux and an ADT of 10,000 veh/day/ln, the expected life of white MMA on asphalt pavement with no winter weather events is (100 − 137.060 + 0.004 × 10000 − 0.771 × 773)/(−0.470) = 1, 261 days, or approximately 3.4 years.  MinRetro−α1 −β11 ADTi −β13 MaxRetroi , i = 1, ..., n1   β12 (3) Li =   MinRetro−α2 −β21 ADTi −β23 MaxRetro2i , i = n1 + 1, ..., n β 22

Table 4 shows an example of expected service life of preformed tape and MMA under different weather, traffic, and surface conditions. It also shows the MaxRetro and MaxRetro2 values used to calculate service life in Equation 3. The values of MaxRetro were derived using the mean maximum retroreflectivity of the corresponding material, line color, and surface type in the first segment of original dataset, and the values of MaxRetro2 were derived in a similar manner using data in the second segment. Note that the service life derived in the second segment only accounts for the life after the first winter, therefore, the actual life of pavement markings would be the sum of the derived

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life and the life before the first winter finished. Furthermore, because of the nature of regression methods, the predicted retroreflectivity at the service life is in fact a normal distribution with a mean of 100 mcd/m2 /lux. Hence, for each expected service life in Table 4, we added the precision of retroreflectivity prediction at the 95% confidence interval for uncertainty considerations. TABLE 4 Pavement Marking Service Life and Retroreflectivity Precision Tape MMA ADT Asphalt Concrete Asphalt Concrete (veh/day/ln) White Yellow White Yellow White Yellow White Yellow ‡ 5,000 3.8 (53) 3.7 (33) 4.6 (84) 5.3 (59) 3.6 (63) 7.8 (69) 4.0 (71) 8.2 (77) 10,000 3.5 (43) 3.6 (28) 4.4 (74) 5.2 (53) 3.4 (53) 7.4 (62) 3.8 (61) 7.6 (67) No Snow 20,000 3.1 (29) 3.4 (21) 4.1 (60) 4.9 (43) 3.2 (39) 6.7 (50) 3.5 (47) 6.4 (50) 5,000 2.5 (22) 1.4 (11) 2.0 (20) 1.3 (8) 3.1 (37) 2.2 (14) 2.7 (32) 1.9 (14) After Snow 10,000 2.0 (14) 1.3 (7) 2.2 (17) 1.5 (5) 2.7 (30) 1.6 (9) 2.6 (25) 1.6 (9) 20,000 1.1 (42) 1.1 (28) 2.6 (20) 1.8 (8) 1.9 (42) 0.6 (17) 2.4 (24) 1.0 (9) MaxRetro (mcd/m2 /lux) 1039 655 1212 784 773 483 883 532 331 209 407 229 473 237 392 211 MaxRetro2 (mcd/m2 /lux) ‡ All values in this table are presented in format a(b), where a denotes the expected service life (in years), and b denotes the precision of retroreflectivity prediction (in ± mcd/m2 /lux) at 95% confidence level Life

5 6 7 8 9 10

ANALYSIS AND DISCUSSIONS In this section, proposed models were compared with commonly used conventional multiple linear models to predict retroreflectivity. Conventional MLMs, formulated as in Equation 4, were developed using the same set of data and independent variables. Table 5 summarizes the coefficients, R-squared values, and RMSE values of the multiple linear models. In addition, RMSE values of the proposed models were also summarized for comparison purposes. RLi = α + β1 ADTi + β2 Daysi + β3 MaxRetroi

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(4)

As shown in Table 5, the range of R-squared values for MLM models was between 0.566 and 0.676, a range that was consistent with the literature, but was not as good as that of the proposed PMLM models. Similar findings could be found in RMSE values. For each combination of material type, line color, pavement surface type, and state, the RMSE of PMLM was smaller than the RMSE of MLM, indicating that the proposed method improved the accuracy of retroreflectivity prediction under various conditions. Moreover, by examining the overall RMSEs summarized by state (the extra column on the right of Table 5), the proposed method improved retroreflectivity predictions in all three states – the RMSE changed from 170.6 to 144.7 mcd/m2 /lux in Florida, 252.1 to 95.9 mcd/m2 /lux in Minnesota, and 182.4 to 84.7 mcd/m2 /lux in Pennsylvania. This result implies that the proposed method is robust and applicable to different states with various winter weather conditions, and can especially improve the prediction in snow states. Consequently, the overall RMSE of PMLM, also shown in the extra column on the right of Table 5, outperformed the overall RMSE of MLM by almost 100 mcd/m2 /lux (106.5 versus 204.6 mcd/m2 /lux), indicating that the proposed method can generally reduce the prediction error by 100 mcd/m2 /lux – a significant improvement. For each model, we further plotted its predicted retroreflectivity against the observed retroreflectivity in Figure 3. A predicted-observed (P-O) plot shows how well the predicted values match

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TABLE 5 Coefficients, R-Squared Values, and Root Mean Square Errors of MLM Tape Asphalt Concrete White Yellow White Yellow α 155.830 96.357 103.345 134.178 β1 0.013 0.013 0.009 0.007 β2 -0.744 -0.500 -0.757 -0.516 0.523 0.450 0.631 0.526 β3 R2 0.676 0.640 0.660 0.600 RMSEFLmlm 175.1 94.8 264.6 191.4 RMSEMNmlm 307.9 235.6 324.6 246.2 RMSEPAmlm 207.8 130.8 248.1 156.3 RMSEmlm 235.9 158.3 277.5 193.9 RMSEFL pmlm 150.0 85.8 243.2 132.3 RMSEMNpmlm 108.3 90.0 114.4 62.9 RMSEPA pmlm 93.2 58.0 115.2 61.9 RMSE pmlm 114.0 75.9 156.3 87.7 † Not statistically significant at 95% level

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MMA Asphalt Concrete White Yellow White Yellow 234.968 123.548 188.004 215.013 0.003 0.000† 0.003 -0.007 -0.560 -0.340 -0.553 -0.369 0.496 0.585 0.538 0.570 0.566 0.599 0.625 0.570 119.2 107.2 158.0 144.8 216.2 191.7 222.8 201.8 178.5 152.8 170.4 141.0 180.2 160.2 187.4 167.9 119.0 77.6 147.5 87.7 118.9 52.3 127.5 62.3 111.9 58.4 89.5 49.7 116.2 61.4 119.6 65.2

Overall 170.6 252.1 182.4 204.6 144.7 95.9 84.7 106.5

the actual measurements, the closer all P-O points are to the 45-degree dotted line, the smaller prediction error is expected. Moreover, the model is said to better meet the basic assumptions of linear regression methods (e.g., linearity, normality, and homogeneity) when the P-O points are (1) aligned with the 45-degree dotted line; (2) distributed randomly on both sides of the dotted line; and (3) deviated consistently from the dotted line no matter how much the value of the observation is. Several important findings can be summarized from these plots. First, PMLM models showed more precise and accurate predictions than MLM models, for the former had P-O points that were distributed tighter with each other and closer to the 45-degree line. Second, PMLM predictions showed good alignment with the 45-degree line. Predictions of MLM, on the other hand, followed a nonlinear-like trend that tended to overestimate retroreflectivity when the actual observation was low and underestimate when observation was high. Third, PMLM models showed more homogeneous predictions than MLM models, for the deviations of the former’s predictions remained mostly consistent and random on both sides of the dotted line no matter how high or low the observed retroreflectivity was; whereas MLM models tended to have larger deviations when the observed retroreflectivity were either lower or higher. This finding also show that PMLM models significantly improved the accuracy of prediction especially when the observed retroreflectivity was low, which indicates that using piecewise regression can better account for factors (e.g., winter weather events) that cause low retroreflectivity conditions. In fact, improving prediction accuracy at low retroreflectivity can also potentially correct the underestimation issues shown in MLM models at high retroreflectivity observations, which was likely caused by overcompensating low retroreflectivity readings. The second and third findings above also validate the basic assumptions of linear regression models, which assume that the errors of a model should be homogeneously and normally distributed (i.e., homogeneity and normality), and should align well with the 45-degree line (i.e., linearity). The predictions were also compared based on results in different states, as denoted in different colors in Figure 3. For example, both MLM and PMLM performed fairly similar on observations made in Florida, for the black points of both types of models show similar distributions.

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FIGURE 3 Predicted versus observed retroreflectivity of preformed tape and methyl methacrylate (MMA) with different line colors and pavement surface types. Comparison between multiple linear models (MLM) and piecewise multiple linear models (PMLM).

1 This finding is consistent with the RMSE results previously since there were no ‘after winter’ data 2 in Florida, and all data were fitted using the first segment model of PMLM, which is essentially

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a MLM. This piece of information may also help explain why MLM was chosen by most literature in recent years – almost all of those models were developed for states in which snow is not a primary concern. Moreover, MLM models predicted poorly in Pennsylvania (blue dots) and Minnesota (yellow dots) especially in low retroreflectivity conditions. This indicates that for snow states, fitting a MLM may not be ideal, because the effect of winter weather events, as shown in our preliminary observations, may not be linear. Interestingly, this information may also explain why most studies conducted in northern or multiple states, as shown previously in Table 1, used nonlinear models such as logarithmic and inverse polynomial regression, instead of linear regression. CONCLUSIONS In this study, robust piecewise multiple linear models were developed to predict retroreflectivity of durable pavement markings under different weather, traffic, and surface conditions. Key contributions of this study include: 1. A robust linear regression method – piecewise multiple linear regression – was first introduced and implemented to predict pavement marking retroreflectivity under effect of winter weather events; 2. Results of this study show that significant improvements in accuracy and precision were achieved. Moreover, the proposed method was able to perform consistently under various weather conditions in different locations, hence can be easily be adopted by different state DOTs; 3. Although PMLM shared similar limitations (e.g., precision issues) as most linear regression models, results of this study indicate that PMLM significantly improved the prediction of retroreflectivity, and more precise service life prediction can be achieved; 4. In-depth observation and discussions were made to better understand the effect of winter weather events. Useful insights on the characteristics of retroreflectivity degradation and potential models for different winter conditions were summarized (e.g., MLM for non-snow states, nonlinear regression for snow states, and PMLM for all states); and 5. Models were developed to predict retroreflectivity of preformed tape and methyl methacrylate materials that had not yet been extensively studied and expected service life of these materials were derived. FUTURE RESEARCH The proposed method may still have several limitations. For instance, the variables selected in this study were based on the prediction power of variables on all combinations of material type, color, and pavement surface, which may not be an ideal selection process if the goal was to predict retroreflectivity of a specific material. Second, the models were developed using 3-year data collected on 7 test decks in 3 states, which may not be representative for predicting retroreflectivity under conditions that were not considered in the study. For future research, more material-specific modeling can be carried out to account for the unique effect of other independent variables on specific materials. For example, as shown in Table 2, road temperature could be another important variable for predicting preformed tape retroreflectivity, and the use of multiple beads could be another useful variable for predicting MMA retroreflectivity. Additional data can be collected and added to improve the current models. For example, the longest data collection period was 3 years in the current NTPEP database, by adding

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data collected after 3 years of installation, more detailed observations can be made to take into account the effect of winter conditions in the following winters. Data collected under more diverse traffic conditions could also be added. Finally, the proposed method can be applied to analyze other types of pavement marking materials and a comprehensive life-cycle cost analysis can be conducted to support more cost-effective pavement marking management. ACKNOWLEDGMENTS This study was partly funded by the Georgia Department of Transportation. The authors would like to thank the support and valuable input from Mr. Richard Douds and Mr. Binh Bui of GDOT. We would also like to thank the National Transportation Product Evaluation Program for the data and input it provided, especially the tremendous help from Mr. David Kuniega, the Chair of NTPEP pavement marking materials committee, and Ms. Katheryn Malusky, Program Manager of NTPEP. Assistance from Pennsylvania, Florida, and Minnesota DOTs, is also appreciated.

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