Imputing Erroneous Data of Single-Station Loop

Imputing Erroneous Data of Single-Station Loop Detectors for Non-incident Conditions: Comparison between Temporal and Spatial Methods

Weihao Yin Graduate Research Assistant Department of Civil and Environmental Engineering Virginia Tech Tel: (571) – 329 2030 [email protected] Pamela Murray-Tuite* Assistant Professor Department of Civil and Environmental Engineering Virginia Tech 7054 Haycock Road Falls Church, VA, 22043 Tel: (703) 538-3764 Fax: (703) 538-8450 Email: [email protected] Hesham Rakha Professor, Civil & Environmental Engineering - Virginia Tech Director, Center for Sustainable Mobility - VTTI Virginia Tech Transportation Institute 3500 Transportation Research Plaza Blacksburg, VA 24061, USA. Tel: (540) 231-1505 Fax: (540) 231-1555 [email protected]

*Corresponding Author

1 1

ABSTRACT

2

Effective use of loop detector data for traffic management requires that errors be efficiently

3

detected, diagnosed, and corrected. We present two new spatial approaches and compare them to

4

state-of-the-art correction procedures for station flow estimation when detectors within that

5

station malfunction in non-incident conditions. One new method exploits the relationship

6

between individual detector flow and station flow using linear regression. The second

7

incorporates lane use percentages through kernel regression. To comprehensively compare the

8

procedures, systematic and random-error evaluations are conducted for two detector stations with

9

distinct lane configurations. Lane configuration is important for spatial correction methods,

10

which perform well under certain detector failure combinations. The random-error evaluation

11

indicates that temporal correction performs better at all error levels and spatial approaches are

12

inaccurate under light traffic conditions, especially when estimates are based on zero flow

13

readings. When choosing a correction procedure, one should consider facility configurations,

14

error types and magnitudes, and traffic conditions, and calibrate the method for location-specific

15

characteristics.

16 17 18

Keywords: detector, data correction, linear regression, kernel regression

2 1

1. INTRODUCTION

2

In 2010, congestion cost U.S. urban travelers 4.2 billion hours of delay and 2.8 billion gallons of

3

wasted fuel, which resulted in monetary costs of $101 billion (Schrank & Lomax, 2011).

4

Advanced Traveler Information Systems, supported by the Intelligent Transportation System

5

(ITS) infrastructure, help relieve congestion by providing route and decision guidance (FHWA,

6

1998). To provide useful advisory messages, data must be collected, synthesized, and translated

7

(FHWA, 1998). The data's accuracy and reliability impact the user-benefits (Maier et al., 2009;

8

Toppen & Wunderlich, 2003); travelers may encounter worse conditions if they follow advisory

9

messages based on erroneous data. The data often include volume and occupancy, which are

10

readily retrieved from loop detectors (Pignataro, 1973); however, loop detector data is error

11

prone (Peeta & Anastassopoulos, 2002; Vanajakshi & Rilett, 2004). Therefore, it is essential to

12

efficiently and effectively detect, diagnose, and impute erroneous and missing data for ITS

13

applications (Smith & Venkatanarayana, 2005), travel time estimation and prediction, origin-

14

destination demand estimation, and incident detection relying on loop detector data (Fernandez-

15

Moctezuma et al., 1997).

16

Loop detectors are usually installed on each lane and parallel detectors constitute a

17

station. Typical measurements are occupancy, volume, and speed at a pre-defined time resolution.

18

Occupancy is a surrogate for density and volume can be transformed into flow (May, 1990).

19

Therefore, loop detector data essentially provide microscopic (time headways and vehicle speeds)

20

and macroscopic characteristics of traffic flow (May, 1990).

21

Issues associated with loop detector data are erroneous recordings and missing data,

22

caused by improper installation, communication malfunction, and/or wire failures (Bikowitz &

23

Ross, 1985). Data collection is usually continuous, which makes equipment malfunctions more

3 1

likely than if occasional collection techniques are used (Vanajakshi & Rilett, 2004). Continuous

2

collection leads to large amounts of data, making manual detection and correction impractical.

3

Error detection and correction algorithms typically use historical data from the same

4

detector, data from neighboring detectors, or a combination. Depending on the data used, a

5

procedure can be classified as temporal or spatial. "Temporal" procedures rely on historical data

6

of the same detector, while "spatial" approaches use data from detectors within the same station.

7

Spatial correction procedures (e.g., Chen et al. (2003)) use historical data to derive the

8

relationships between in-station detectors for subsequent correction, but are categorized as

9

spatial since correction is based on in-station detectors. Some approaches (e.g., Smith and

10

Conklin (2002)) combine spatial and temporal information.

11

Selection of the best approach for imputing missing and erroneous data is important for

12

the archival of traffic data (Smith & Venkatanarayana, 2005). To aid this selection for non-

13

incident

14

comprehensively using field data in the context of a single station. Among the approaches are

15

two proposed spatial correction procedures. One uses kernel regression to derive station flows

16

based on lane use percentages, and the other modifies Chen et al.'s (2003) approach by using

17

lane flow to estimate station flow (instead of one lane’s flow to predict another lane’s flow).

18

These two new procedures are evaluated along with Smith and Conklin’s (2002) approach and

19

the time-of-day historical mean approach. The comparison is conducted through systematic and

20

random-error evaluations at two detector stations with different lane configurations. The

21

systematic evaluation uses deterministic error configurations while errors are stochastically

22

introduced for the random-error evaluation. The results are analyzed to identify the most accurate

23

method(s) to correct a large dataset and obtain accurate station flow estimates.

conditions,

the performance of four

correction procedures are compared

4 1

The remainder of this paper is divided into six sections. Section 2 reviews loop detector

2

data error detection and correction studies. Section 3 describes the study's dataset. Section 4

3

presents the methods, followed by experimental procedures in Section 5. The results are

4

provided in Section 6. The last section presents conclusions and future directions.

5

2. PREVIOUS STUDIES

6

Error identification is necessary before correction. Though this study does not deal with error

7

detection, these methods are discussed for completeness and some studies jointly addressed error

8

detection and correction.

9

While techniques for error detection and correction generally fall into temporal or spatial

10

categories, some methods detected errors by checking for impossible combinations of traffic

11

flow characteristics. Such combinations were identified by volume-to-occupancy or flow-to-

12

occupancy ratios (Cleghorn et al., 1991; Jacobson et al., 1990). Turner (2004) later incorporated

13

speed. Chen et al. (2003) checked for impossible combinations over an entire day and detected

14

implausible time series. Another widely-used technique compared volumes or flows,

15

occupancies, and speeds with specific thresholds (Payne & Thompson, 2007; Weijermars & Van

16

Berkum, 2006).

17

Temporal detection and correction procedures were implemented by Chen et.al (1987)

18

and many European Intelligent Transportation Systems (Turner, 2004). Park (2003) constructed

19

a multi-variant screening methodology to identify outliers based on a variant of Mahalanobis

20

distance. Ishak (2003) developed an algorithm using fuzzy theory that clusters the input space of

21

the parameters speed, occupancy, and volume into regions based on normalized Euclidian

22

distance. Each observation’s uncertainty was then measured with one parameter, which was used

23

to identify errors. Later, this work was improved using probability theory and applied to real-

5 1

time situations (Ishak et al., 2007; Lee et al., 2010). Nihan (1997) modeled occupancy and flow

2

time-series as Autoregressive Moving Average processes and predicted values in the near future.

3

Other efforts relying on time-series techniques include Maier et al. (2009), who applied robust

4

estimation techniques to flow, occupancy, and speed time-series to estimate corrected

5

measurements. Using a spectral-domain time-series technique, Peeta and Anastassopoulos (2002)

6

developed a Fourier-Transform based algorithm that considered traffic characteristics as a time-

7

dependent function and distinguished abnormal data caused by incidents from detector

8

malfunctions and subsequently corrected erroneous data. One of the major concerns about

9

temporal detection and correction was that traffic flows could deviate from historical values due

10

to special circumstances, such as incidents, instead of malfunctioning detectors (Weijermars &

11

Van Berkum, 2006).

12

Spatial detection and correction procedures exploit the relationships between parallel

13

detectors and/or upstream and downstream detectors. Kwon et al. (2004) developed an algorithm

14

for detecting configuration errors based on the assumption that spatially close detectors should

15

show similar temporal patterns if no major disturbances, such as interchanges, exist in between.

16

Chen et al. (2003) identified strong correlation among neighboring loop detector measurements

17

and modeled this correlation with linear regression models. For a specific detector, a regression

18

model was constructed using the data from this detector and one of its neighbors. The final

19

detector flow estimate was the median of all pair-wise estimates. Chen et al. (2003) reported a

20

mean absolute error of 132 vehicles/hour with a mean hourly flow of 1220 vehicles/hour.

21

Detection and correction methods based on the vehicle conservation principle compare flow

22

measurements between upstream and downstream locations. For example, Vanajakshi et al.

23

(2004) compared cumulative volumes at each location and corrected discrepancies using

6 1

constrained non-linear optimization. Kikuchi et al. (1999) applied fuzzy-optimization to adjust

2

each link's flow to achieve consistency. One potential deficiency of spatial correction procedures

3

was the inability to capture temporal variation of traffic throughout a typical day (Maier et al.,

4

2009).

5

Smith and Conklin's (2002) correction procedure relied on both temporal and spatial

6

information, using lane distribution patterns to derive missing data values. They reported an

7

average error less than 10% for almost all test cases. This method is discussed in detail later.

8

From the previous approaches, Smith and Conklin’s (2002), the time-of-day historical

9

mean, and our modified version of Chen et al.’s (2003) approach were selected for comparison,

10

along with our kernel regression approach, based on implementation ease and reported correction

11

accuracy. The variant of Chen et al.'s (2003) approach constructs a linear relationship between

12

the detector and the station flow instead of between station-mates. The kernel regression

13

technique is inspired by its applications in traffic flow prediction (e.g., (Han et al., 2010; Sun &

14

Chen, 2008)). The main strength is its non-parametric nature, which requires no prior

15

assumptions regarding the model structure (Takezawa, 2006). It models the relationship between

16

lane use percentage and lane flow and subsequently corrects errors. The models' details are

17

presented in Section 4.

18

3. DATASET DESCRIPTION

19

The Virginia Department of Transportation (VDOT) has 91 detectors grouped into 34 stations on

20

eastbound Interstate 66 (I-66E) between Manassas and Falls Church. VDOT provided detector

21

data and incident records from 2008, which are used for this study. From this dataset, we extract

22

the data for two stations. Station 1 is near the Lee Jackson Memorial Highway interchange. This

23

station, which has five detectors, is selected because it relates to ongoing research efforts. Station

7 1

2, with four detectors, is near West Ox Rd. Figure 1 illustrates the detector configurations and

2

station locations. Station 1 has an exit lane while Station 2 has only general-purpose lanes.

3

Insert Figure 1 Here

4

Each detector should report speed, occupancy, and volume every minute; 1440 daily

5

station flow records should exist if no malfunctions occur. The real-time hourly flow is the

6

volume reading multiplied by 60. A detector's working status is recorded by an indicator, which

7

has three possible values. A "0" indicates the detector malfunctioned; a “1” indicates possible

8

erroneous readings. Only readings from a detector with a status of “2” are considered correct.

9

When the status is "0" or "1", the readings have known errors (type 1). In addition to

10

these errors, the status indicator may be incorrectly reported. Type 2 errors include unreasonably

11

high flows; since detectors are lane based, the hourly flow obtained from 1-minute data for a

12

freeway lane cannot exceed 5000 veh/hr (Payne & Thompson, 2007). Hence, the lane's detector

13

status is set to "0" in these situations. Type 3 errors, missing data, are common. Sometimes, an

14

individual detector's readings are missing, and, at others, no detectors report values for an

15

extended duration (more than 10 minutes). Station flow is the sum of the individual detector

16

flows and is correct only when all detectors report correctly. In this study, all error types are

17

converted to missing-data errors since data from detectors with a "0" or "1" status are unreliable.

18

Hence, correction is equivalent to estimating missing data points, and the two terms are used

19

interchangeably.

20

implementation.

Additionally,

this

conversion

simplifies

the

correction

procedures'

21

To ensure fair comparison, three datasets are carefully chosen and constructed for each

22

station. The first is the correction target. The data for Tuesday, April 22, 2008 from 4:00 AM to

23

10:00 PM were selected for technique comparison, similar to previous studies (Payne &

24

Thompson, 2007; Peeta & Anastassopoulos, 2002; Smith & Conklin, 2002). The second (Base 1)

8 1

is the correction base dataset, which contains the data for the three months prior to the target day

2

(01/23/2008 to 04/20/2008), excluding holidays. The third (Base 2) is a subset of Base 1 and

3

contains the data for Tuesdays in Base 1. Base 2 is the correction base dataset for temporal

4

correction, described in the next section. Erroneous data and those for incident durations are

5

eliminated from Base 1 and 2, as in Peeta and Anastassopoulos (2002). Though the data for the

6

incident duration is excluded, potential reporting inaccuracies prohibit a guarantee that the

7

incident impact is completely eliminated.

8

The reasons for selecting April 22, 2008 are:

9

•

If all detectors report correct readings, 1080 correct station flows (4:00 AM - 10:00

10

PM) would be obtained. The chosen day is among those with the highest number of

11

correct station flow readings - 937 for Station 1 and 930 for Station 2. The 143 and

12

150 incorrect station flows for Stations 1 and 2, respectively, are not used for the

13

accuracy assessment since they are inaccurate/invalid observations.

14

•

15 16

Base 1 does not involve a seasonal shift, which may cause traffic pattern changes due to school holidays.

•

No incidents occur on this day within five miles upstream or downstream of both

17

stations. The non-incident condition is explicitly required by Smith and Conklin's

18

(2002) procedure.

19 20

Figure 2 illustrates the station flow profiles for the two stations on the target day. Insert Figure 2 Here

21

The traffic pattern is characterized by light traffic in the early morning and late evening.

22

The morning peak traffic reaches around 8000 veh/hr for both stations. The volume for the

9 1

afternoon peak period is considerably lower since both stations are located on I-66E, on which

2

home-to-work trips concentrate in the morning peak.

3

4. DATA CORRECTION PROCEDURES

4

The four data correction procedures are examined within the context of the target dataset.

5

First, the basic techniques are presented and then their adaptations for this study are discussed.

6

4.1. Temporal Correction (TC)

7

The temporal correction (TC) applied for this study is the Time-of-Day (TOD) historical

8

mean (Daganzo, 1997; Nguyen & Scherer, 2003; Maier et al., 2009). Let d denote the number of

9

different days within Base 2. The corrected flow for detector i at time t, fˆt i is calculated by Eq.(1). d

10

fˆt i =

∑ y ( j) i t

j =1

d

(1)

11

where yti ( j ) denotes the flow reading for detector i at time t on day j. TC takes the arithmetic

12

average of the flows at time t of all days available in Base 2. If a detector is functional, no

13

correction is made to the target dataset and the corrected flow fˆt i is the reported flow ft i . The

14

station flow is the summation of all the estimated detector flows as in Eq.(2).

15

qˆtTC = ∑ fˆt i

(2)

i

16

where qˆtTC denotes the corrected station flow (veh/hr) for time t using TC.

17

4.2. Spatial Correction using Linear Regression (LR)

18

Unlike Chen et al. (2003) who use relationships between flows of individual detectors,

19

we construct a series of linear regression models to estimate the station flow from functional

20

detectors. The target dataset's time span includes the HOV-effective duration (5:30 AM to 9:30

21

AM). To handle traffic pattern shifts due to HOV restriction, we build two sets of regression

10 1

models - one for the HOV period and one for the regular period. For notational simplicity, the

2

two sets of models are written in a unified format. An estimate of the station flow qˆti at time t

3

based on the flow ft i of functional detector i at time t is obtained through the regression model

4

expressed in Eq.(3) qˆti= β1i ft i + β 2i + e, {detector i is functional (status =2)}

5

(3)

6

Only readings from functional detectors are used. The coefficients β1i and β 2i are estimated

7

using the ordinary least squares method. For the models when the HOV lane is activated, only

8

the data points during the HOV time in Base 1 are used to estimate the coefficients; data points

9

for the regular period in Base 1 are used to estimate the coefficients of the regular period's

10

models. The final estimated station flow at time t is the arithmetic average of available station

11

flow estimates using Eq.(4), where n denotes the number of functional detectors at time t.

12

LR t

qˆ

=

∑ qˆ

i t

i

n

(4)

13

For a given t, one must determine whether this time belongs to the HOV duration or not and

14

choose the set of models accordingly. If three detectors, for example, function at this time, an

15

estimate of station flow can be derived from each detector flow reading. The final estimated

16

station flow is the arithmetic mean of these three station flow estimates. The LR procedure

17

requires at least one functional detector to estimate station flow at a specific time. If no detectors

18

report correctly, the station flow is estimated using the temporal procedure qˆtTC, i.e., TC is the

19

default technique.

20

4.3. Spatial Correction using Kernel Regression (KR)

11 1

Under different congestion levels, drivers use different lanes, creating lane-to-lane flow

2

variability in multi-lane freeway sections (Carter et al, 1999;Amin & Banks, 2005). The lane use

3

pattern depends on the location and flow conditions (TRB, 2000).

4

We propose a new spatial correction procedure that exploits the relationship between lane

5

use patterns and flow conditions. A functional detector i at time t reports flow ft i . The

6

corresponding lane use percentage pti is defined as in Eq.(5). pti =

7

ft i qti

(5)

8

If the lane use percentage pti corresponding to the flow reading is known, the station flow can be

9

estimated by Eq.(6). qˆti =

10

ft i pti

(6)

11

However, no parametric function describes the relationship between lane use percentage and lane

12

flow. Therefore, kernel regression is used to obtain the empirical relationship between lane flow

13

and use percentage.

14

In parametric regression with a single regressor, the equation takes the form y = F(x)+e

15

like Eq.(3), where F(x) is a smooth function with known shape. The m observations (xj, yj) (j =

16

1,2,3,…,m) are used to estimate the unknown parameters and the fitted value yˆ j is given by F(xj).

17

For any xi, one can obtain yˆi using the constructed parametric relationship F. Similarly, the

18

output of kernel regression is a relationship that can be used to estimate yˆ i based on a provided xi.

19

The difference lies in the fact that kernel regression does not make assumptions about the shape

20

of F(x) (Bowman & Azzalini, 1997).

21

Kernel regression's underlying logic is that every observation other than the point of

22

interest (xi, yi) is used to derive the fitted value yˆi and the observations with the most information

12 1

about F(xi), or yˆ i , should be those close to this point of interest (xi, yi) (Bowman & Azzalini,

2

1997; Takezawa, 2006). The closeness of an observation (xj, yj) (j = 1,2,3,…,m and j ≠ i) to the

3

observation in question (xi, yi) is measured by the distance on the horizontal axis |xj – xi|. The

4

immediate neighboring observations, with smaller distance |xj – xi|, contribute more to deciding

5

F(xi) than distant ones. A weight wij measures the magnitude of observation j 's contribution to

6

the fitted value F(xi). Subsequently, F(xi) is a weighted average of all y’s of the observations (xj,

7

yj) (j = 1,2,3,…,m and j ≠ i). Since the observations far from the observation of interest (xi, yi)

8

should receive little or no weight, a decreasing function is desired to assign weights to each

9

observation (xj, yj) based on the distance |xj – xi| (i ≠ j). After the weights for the value yj are

10

determined, F(xi) is computed as the weighted average of all the yj’s. After this process is

11

repeated for each observation, the shape of the function F(x) can be described by a series of

12

derived points (xi, F(xi)) for i = 1,2,3,…,m where m is the total number of observations. The

13

choice of the weighting function should not influence the relationship captured by the kernel

14

regression. Therefore, an additional parameter bandwidth (h), rescales the horizontal distance to

15

minimize the effect of different weighting function choices. The ultimate measure of distance,

16

used to decide an observation's contribution to the calculation of fitted value, is the horizontal

17

distance scaled by bandwidth, i.e. |xj – xi|/h.

18

We treat individual detectors separately by constructing a kernel regression based

19

relationship between lane flow and use percentage for each of them. Suppose m pairs of lane

20

flow fj and use percentage pj (j = 1,2,3,…,m) for one particular detector are available in Base 1.

21

For a specific observation (fi, pi), a series of weights wij (j = 1,2,3,…,m and j ≠ i) are assigned to

22

each lane use percentage pj selected for this detector to determine fitted lane use percentage pˆ i

23

based on the distance measure | f j − fi |. This step is repeated for every pair of lane flow and use

13 1

percentage (fi,pi) for i = 1,2,3,…,m. The weighting function in this study is the one proposed by

2

Nadaraya (1964) and shown in Eq.(7). wij =

3

(f

K h ( fi − f j ) / h 

∑

j =m j =1

  ( f i − f j ) / h 

(7)

− f j ) / h. Then K h ( u ) is a function of u, and h is a positive-

4

To simplify the notation, let= u

5

value bandwidth. The kernel function K h ( u ) used here is the standard Gaussian kernel (Eq.(8)).

Kh (u ) =

6 7 8

i

1 −u2 e 2π

(8)

The equation (Eq.(9)) proposed by Bowman and Azzalini (2005) is used to determine the optimal bandwidth for the standard Gaussian kernel. 1

9

hopt

 4 5 = ,  3m 

(9)

10

where m is the sample size of Base 1 for this particular detector. Using Eqs.(7-9), the fitted data

11

point ( fi , pˆ i ) can be found and the relationship between lane use percentage and flow F can be

12

constructed for this detector. By applying this procedure to all the detectors of a particular station,

13

the corresponding lane use percentage can be estimated for any lane flow through interpolation,

14

based on the relationship F.

15

Once the lane use percentage is determined, station flow is estimated using Eq.(6).

16

Similar to the LR-based spatial correction, the final estimate qˆtKRof station flow at time t, is the

17

arithmetic mean of all the estimates from individual functional detectors. Also, the relationship

18

between the lane flow and use percentage is treated separately for the HOV and regular periods.

19

4.4.Lane Distribution Correction (LD)

14 1

Smith and Conklin's (2002) method exploits both temporal and spatial information. A historical

2

lane use percentage pti is derived from Base 1, computed as the arithmetic mean of all available

3

lane use percentages for lane i at time t as in Eq.(10). d

pti =

4

∑ p ( j) j =1

i t

,

d

(10)

5

where d denotes the number of different days within Base 1 and pti ( j ) is the lane use percentage

6

of lane i at time t of day j, calculated using Eq. (5). This historical lane use percentage is used to

7

estimate the station flow as in Eq.(11).

qˆti =

8 9 10

fti pti

(11)

The final estimate of the station flow for time t is the arithmetic mean of all available estimates based on individual detectors (Eq.(12)).

qˆtLD =

11

∑i qˆti n

,

(12)

12

where n is the number of functional detectors at time t.

13

5. EXPERIMENTAL PROCEDURES

14

The experiment consists of systematic and random-error evaluations. The former assesses the

15

performance of each method under all possible error combinations and serves two purposes. First,

16

it guides the selection of individual detectors for aggregation to derive the final station flow

17

estimate for the kernel regression and lane distribution approaches. Second, it reveals the

18

methods' accuracies under different error configurations and aids the selection of the best method

19

when a specific error configuration is dominant. For n detectors in a station, the number of

20

possible error combinations is (2n – 2). The two excluded cases are (1) all detectors malfunction

21

and (2) all detectors function. Figure 3 visualizes the error combinations.

15 1

Insert Figure 3 Here

2

The systematic evaluation targets two periods (peak and off-peak). The selected peak

3

period is 7:00 to 9:00 AM while the off-peak period is 1:00 to 3:00 PM. For each error

4

combination, the flow readings of the desired detectors are treated as missing for the two-hour

5

duration.

6

The systematic evaluation consists of two stages corresponding to the two purposes

7

aforementioned. In the first (SysEval-I), the KR and LD approaches adopt all available station

8

flow estimates to derive the final estimate for a given time. The results are analyzed by

9

comparing the procedures' performances under different error scenarios to provide insights into

10

each detector's relative importance to the method's accuracy. These insights are used to modify

11

the LD and KR spatial methods by changing the detectors selected for aggregation. The second

12

systematic evaluation (SysEval-II) incorporates these modifications and compares the methods'

13

performances under different error configurations to identify the one that provides the best

14

performance. Detectors often malfunction for some time, resulting in missing readings for this

15

duration. The observations derived from SysEval-II provide practical guidance about the choice

16

of method when prolonged malfunctions occur.

17

Random error evaluation follows the systematic evaluation. The target day contains 5400

18

individual detector flow records (for Station 1) or 1080 station flows. Errors are randomly

19

introduced into the target dataset at different levels: 10, 20, 30, 40, and 50%. The 10% error

20

level means that 10% of the individual flow readings (i.e., 10%×5400 = 540 records) are

21

artificially set to missing values based on random numbers. Since the target dataset contains pre-

22

existing incorrect detector readings, the actual error percentage could be less than the specified

23

error level when the existing incorrect records are flagged as introduced error. For each error

16 1

level, 10 replications are conducted by revising the random number seed. Each replication hosts

2

a particular combination of error configurations defined in the systematic evaluation.

3

The correction methods' performances are compared using two widely-accepted measures

4

of effectiveness (MOEs): Mean Absolute Error (MAE) (Eq. (13)) and Mean Absolute Percentage

5

Error (MAPE) (Eq. (14)), which converts the absolute quantity to a relative one.

6

MAPE measure the average correction accuracy for the target dataset. = MAE

7

1 T ∑ | qt − qˆt |, T t =1

MAE and

(13)

8

where T represents the total number of records in the target dataset and qt is real station flow and

9

qˆt is the estimated station flow. 1 T | qt − qˆt | MAPE = ∑ T t =1 qt

10

(14)

11

Results of the random-error evaluation are analyzed and interpreted from several angles.

12

Each procedure's overall performance is contrasted at different error levels. The corrected value

13

is supplied by TC if the other three procedures (LR, KR, and LD) fail to produce an estimate of

14

the station flow for a given time. Such correction failures are assessed to provide insights to each

15

procedure's robustness. Finally, since the traffic's inherent random variation may not be captured

16

by aggregated MOEs, the overall performance measures (MAPEs) are disaggregated by time of

17

day to uncover temporal performance variations.

18

6. EXPERIMENTAL RESULTS

19

We constructed two sets of linear regression models using Base 1 before conducting the LR

20

correction. Since Base 1 includes readings from functional detectors for the three months prior to

21

the target day, it includes normal traffic fluctuations. The regression models for the HOV and

22

regular periods are shown in Table 1. At Station 1, the linear relationship between detector 609's

17 1

lane flow and the station flow for the HOV period is weak, as shown by the relatively low

2

adjusted R2 metric 0.01. The same findings apply for detectors 607 and 609 for the regular period,

3

which may be explained by detector 609's installation on the exit lane and detector 607 on the

4

adjacent lane. For the HOV period, the final estimate of Station 1's flow is the arithmetic mean of

5

available estimates from detectors 601, 603, 605 and 607. For the regular period, only the

6

available flow estimates based on detectors 601, 603, and 605 are used. All the linear regression

7

models for Station 2 enjoy reasonably good fit and are used in the final estimate.

8 9

Insert Table 1 Here

6.1. Systematic Evaluation

10

The results of the first systematic evaluation (SysEval-I) are shown in Figures 4 and 5 for Station

11

1 and Figures 6 and 7 for Station 2. SysEval-I helps identify the relative importance of lane

12

detectors to LD's and KR's accuracies.

13

Insert Figure 4 Here

14

Insert Figure 5 Here

15

Insert Figure 6 Here

16

Insert Figure 7 Here

17

6.1.1. SysEval-I Results for Station 1

18

The MOEs are not uniform within a scenario group of the same number of malfunctioning

19

detectors, suggesting that the location of malfunctioning detectors impacts the accuracy. (To

20

simplify the narration, detector 601 malfunctioning is represented by “d601-X”.) For the off-

21

peak period, TC (Figure 4's upper left panel) achieves lowest MAPE and MAE for combination

22

5 (d609-X, see Figure 3). For TC, the MAPE for combination 6 (d601-X&d603-X) is

23

significantly higher than that for combination 5. Information from detectors 601 and 603 play

18 1

important roles in TC and for all methods considered in this study. This observation is further

2

substantiated by combinations 26 and 27 exhibiting high MAPE for all methods; for these

3

combinations, detectors 601, 603, and 605 malfunction.

4

Another interesting comparison is among error combinations 4 (d607-X), 5 (d609-X), 15

5

(d607-X & d609-X), and 16 (d601-X, d603-X, & d605-X) in the off-peak period. For KR,

6

combination 15's MAPE is lower than combinations 4 and 5, indicating that including more

7

station flow estimates from individual detectors does not necessarily lead to more accurate final

8

station flow estimates. The addition of information from either detector 607 or 609, in the

9

absence of another detector, decreases KR's effectiveness. Additionally, combination 16's MAPE

10

is higher than that of combinations 4, 5, and 15, any of which has at least one functional detector

11

of the group (d601, d603, and d605). A similar pattern is observed for the peak period. The slight

12

difference lies in KR achieving the lowest MAPE (8.40%) for combination 25 (d605-X, d607-X,

13

& d609-X) for the off-peak period while the lowest MAPE 7.12% is achieved for combination

14

15 (d607-X & d609-X) during the peak period. The practical significance of this finding is that

15

an adaptation to the KR approach is needed. During the off-peak period, only available estimates

16

of station flow based on detectors 601 and 603 are used. During the peak period, detectors 601,

17

603, and 605 are used.

18

A similar pattern exists for LD (lower right panel in Figures 4 and 5). Combination 15's

19

MAPE is much lower than that for combination 16. The estimated station flows based on the two

20

right-most lanes are relatively inaccurate because of the exit lane, whose flow pattern is different

21

from those of its in-station counterparts. Based on this finding, the LD correction method is

22

revised to use only available estimates from detectors 601, 603, and 605 for both peak and off-

23

peak periods.

19 1

6.1.2. SysEval-I Results for Station 2

2

For Station 2, the comparison of interest involves error configurations 4, (d547-X), 10 (d545-X,

3

d547-X), 12 (d541-X, d543-X, d547-X), 13 (d541-X, d545-X, d547-X), and 14 (d543-X, d545-X

4

and d547-X). For the off-peak period (Figure 6's lower left panel), combination 4's MAPE is the

5

lowest and configuration 10's is slightly higher. The common feature of these two combinations

6

is the functional detectors 541 and 543, which are vital to KR's accuracy. The comparison among

7

error configurations 12, 13, and 14 lends additional credence. When both detectors 541 and 543

8

malfunction, the MAPE is higher than when at least one of them works properly (configurations

9

13 and 14). However, the peak period lacks this pattern. Hence, the aggregation step for the KR

10

approach uses station flow estimates from detectors 541, 543, and 545 for off-peak period and all

11

available estimates for peak-period estimation.

12

For LD, in the off-peak period, error configuration 7 (d541-X, d547-X) has the lowest

13

MAPE, suggesting the importance of detectors 543 and 545 for station flow estimation.

14

However, the peak period lacks this pattern. Therefore, the LD approach adopts only the

15

estimates from detectors 543 and 545 for the off-peak period but takes all available estimates for

16

the peak period.

17

The findings of the systematic evaluation for both stations for KR and LD suggest that

18

certain station flow estimates based on individual lane distribution can be inaccurate. The

19

fundamental reason is that the lane distribution varies according to multiple factors, such as

20

traffic composition and the number and location of access points (TRB, 2000). The systematic

21

evaluation highlights the impact of access/egress points and the facility characteristics (number

22

of lanes). Therefore, spatial correction approaches based on lane distribution need calibration and

23

adaptation before application.

20 1

6.1.3. SysEval-II Results for Station 1

2

The modified KR and LD approaches are evaluated in SysEval-II. The intent of SysEval-II is to

3

identify the best method for every error configuration. Tables 2 and 3 present the results for

4

Stations 1 and 2, respectively.

5

The improvement achieved by the modifications is confirmed by the drop of LD's

6

MAPEs for both the peak and off-peak periods and the decrease of KR's MAPEs in the off-peak

7

period.

8

The number of malfunctioning detectors influences TC's and LR's performance relatively

9

more deterministically than it does the other two approaches during the off-peak period.

10

Generally, the MOEs increase when the number of malfunctioning detectors increases from one

11

to four. However, this trend is not present for KR and LD, thus these two approaches are less

12

sensitive to the number of malfunctioning detectors during the off-peak period.

13

Insert Table 2 Here

14

Table 2 highlights the lowest MAPE for each error configuration and shows the MAPE

15

differences for the other three procedures. For example, TC has the lowest MAPE (5.42%) for

16

error combination 1 during the peak period. The MAPE difference is “+1.79%” for LR, or a total

17

MAPE of 7.21% (5.42%+1.79%). For the off-peak period, TC presents dominant performance

18

with a few exceptions. One is error combination 21, in which LD provides the lowest MAPE

19

(8.85%). LR performs nearly as well when TC is the best, with an average MAPE difference of

20

0.76% for the off-peak period. For the peak period, a similar pattern exists for scenarios in which

21

two or fewer detectors malfunction (before error combination 15). TC provides the most

22

accuracy with only one exception (combination 8). When the number of malfunctioning

23

detectors exceeds 2 (beyond error combination 15), KR is competitive for several error

21 1

configurations for the peak period. In contrast to the off-peak period, KR is the second best with

2

an average MAPE difference of 1.26% while the LR approach is third with an average MAPE

3

surplus of 1.59%. Hence, LR exhibits robustness across different error configurations due to its

4

competitive performance for both periods (less than 2% margin).

5

Error combinations exist for which all methods exhibit higher error rates. For the off-

6

peak period, combinations 17 and 18 result in MAPEs exceeding 13% for all procedures because

7

detectors 601 and 603, considered the most important, malfunction. For the peak period, the

8

similar combination is 22 (only detectors 601 and 609 function).

9

6.1.4. SysEval-II Results for Station 2

10

Distinct from the findings for Station 1, TC is not obviously dominant for Station 2. In the off-

11

peak period, when more than two detectors malfunction (after error configuration 4), the spatial

12

correction methods show obvious advantages. LR enjoys the lowest MAPE values for six

13

combinations. For the peak period, KR shows the best performance for six error combinations.

14

Insert Table 3 Here

15

The performance difference for Stations 1 and 2 can be attributed to the lane

16

configurations. Since Station 2 has no exit lane, the traffic is more consistent than Station 1,

17

which has one exit lane. This lane configuration distinction impacts the methods' performances,

18

which will be confirmed by the random-error evaluation.

19

6.2. Random-Error Evaluation

20

KR and LD refer to the modified versions. Figure 8 presents the MAE and MAPE averaged over

21

10 realizations for the four methods. The TC replacements when spatial correction fails due to no

22

functional detectors are not included in the MAPE calculations to guarantee a fair comparison

23

among the spatial correction methods.

22 1

Insert Figure 4 Here

2

6.2.1. Analysis of the Results of Random-Error Evaluation for Station 1

3

TC performs the best at all error levels for Station 1. At error level 10%, TC's MAPE is only

4

6.76% compared to LD's 15.17%, LR's 13.06%, and KR's 11.58%. Even at 50% error for Station

5

1, TC achieves a MAPE of 11.08%. Considering that the error percentage within a dataset

6

usually does not surpass 50%, TC provides reliable correction performance for non-incident

7

conditions. (In 2008, only 33 out of 314 days had error levels exceeding 50%).

8

LR maintains a relatively steady performance at all error levels. The MAPEs range from

9

13.06% to 15.09% while the ranges for the other three approaches each exceed 5%. This steady

10

trend is determined by LR's inherent characteristics. The linear relationship predicts the expected

11

station flow conditioned on an individual lane flow. The station flow can be estimated even if

12

only one lane flow is valid, so, even if the error level is high, LR's performance is stable.

13

However, the standard deviation of LR's MAPE is larger than those of KR and TC because LR

14

performs relatively poorly when traffic is light in early morning and late evening as

15

demonstrated in Section 6.3. The correction accuracy, in descending order, ranks TC, LR, KR,

16

and LD.

17

LD's performance is not as good as anticipated for Station 1. The lowest MAPE for error

18

level 10% is higher than that reported (less than 8%) in Smith and Conklin (2002). The

19

difference between the results can be ascribed to several reasons. The original study used 10-

20

minute aggregate data, which had less noise while our dataset has a higher resolution of one

21

minute. Station 1 has five lanes, whereas the original study had three (without an exit lane).

22

Though only information from the three left-most lanes is used, the traffic pattern can lead to

23

different results. Additionally, incident impacts cannot be ruled out completely due to possible

23 1

documentation inaccuracy. Since being free of incidents is required for the LD approach,

2

potential incident impacts may undermine its performance.

3

6.2.2. Random-Error Evaluation for Station 2

4

Similar to the findings for Station 1, TC outperforms the other three. However, the most obvious

5

distinction lies in LD's competitive performance, which is better than LR and KR. LD's MAPE at

6

the 10% error level is 9.43% and 11.60% for the 50% error level. LR gives MAPEs ranging from

7

10.38% for error level 10% to 13.34% for error level 50%. The rankings for correction accuracy

8

based on MAPE are TC, LD, KR, and LR. This ranking contradicts the one for Station 1 in

9

which LD has the largest MAPEs for all error levels. Traffic patterns due to different numbers of

10

lanes for the two stations plays an important role in the accuracy for the lane distribution

11

methods. LD performs better when the lanes are all general-purpose lanes, which exhibit more

12

homogenous traffic patterns than those with exit lanes. The three spatial correction methods

13

uniformly present better performance for Station 2 than they do for Station 1, which again

14

underscores the significance of lane configuration. This also accounts for the lower standard

15

deviation of LD's MAPE for Station 2 than for Station 1.

16

6.2.3. Correction Failure Rate for Random-Error Evaluation

17

As mentioned in Section 4, LR, KR, and LD require at least one functional detector. Therefore,

18

they are compared in terms of their correction failure rate. Correction failure indicates the

19

procedure fails to produce an estimate and defaults to the TC estimate. Figure 9 shows the 10-

20

replication average for the percentage of correction failure. For error levels 10% to 30%, all three

21

correction procedures enjoy failure rates less than 10% for both stations and approximately 6%

22

for LR and KR for both stations at the 40% error level, suggesting that they are robust under

23

most circumstances, considering that most datasets have less than 50% error.

24 1

LD's relatively high failure rate for Station 2 at the error levels 40% and 50% is due to

2

the station having only four detectors. Therefore, Station 2 is more likely to have a station

3

malfunction than Station 1, which has five detectors.

4

Insert Figure 9 Here

5

6.3. Method Performance by Time-of-Day

6

To examine time-of-day effects, the MAPEs are calculated for each hour of the target day. The

7

hourly MAPEs are averaged across the 10 random realizations. The results are visualized

8

through Figures 10 and 11. One or more detector malfunctions leads to unreliable station flow

9

for a given time, which calls for correction. The maximum possible station corrections made for

10

an hour is 60. (Random errors are distributed among the possible individual detector readings). A

11

general pattern shared by all random error levels for both stations is the relatively high early

12

morning and late evening MAPEs. Station 2's MAPEs fluctuate at a lower level due to overall

13

better performance (see Figure 8 and Section 6.2). Figures for Station 1 at two error levels (10%

14

and 40%) are shown and analyzed. The relationships for the 20 and 30% error levels closely

15

resemble the one for 10% and the 50% error level is similar to the 40% level.

16

Insert Figure 10 Here

17

Insert Figure 11 Here

18

The gap for 7:01 to 8:00 exists because the original dataset does not have correctly

19

reported flows and the performance measurements could not be calculated. The number of

20

corrections made for each hour is relatively steady. Hence, the temporal variation of performance

21

is due to detector malfunction combinations and possibly traffic flow variations instead of error

22

quantity.

23

At all error levels, TC generally outperforms the other procedures for both stations,

24

except for 15:00 to 16:00 for Station 1 at 40% error levels. In both figures for Station 1, LR,

25 1

followed by KR, achieves the lowest MAPE, indicating these approaches outperform TC under

2

certain circumstances. One example is the time 15:35:00 at which the real station flow is 2280

3

veh/hr and the working detectors for this time under one realization of the 40% error level are

4

detectors 605, 607, and 609. The TC estimate is 2979 veh/hr, derived from the individual lane

5

flows for detectors 601 and 603, which are 675 veh/hr and 1224 veh/hr while the observed

6

values are 420 veh/hr and 780 veh/hr, respectively. The predicted values overestimate the

7

readings, possibly because the flow for this particular time deviates from the past average at the

8

same time. The LR estimate is 2411veh/hr and KR's estimate is 2255 veh/hr in this realization,

9

which are good estimates. As the general trend suggests, TC performs the best in terms of

10

estimation accuracy though cases of lower accuracy occur when the flow deviates from the

11

historical value.

12

LR, KR, and LD's performances exhibit relatively high MAPE for 4:00 to 5:00 and

13

deteriorate for the duration after 19:00 for both stations. Selected individual data points between

14

these time durations at the 10% error level are more closely examined. Traffic conditions are

15

light; for example, the real flows at 4:10 AM are 0(d601), 0(d603), 300(d605), 0(d607), and

16

0(d609). In one realization, detector 603 malfunctions and the final station flow is based on at

17

least one zero flow reading (detector 601). LR's estimated station flow is 1221 veh/hr, which is

18

four times the real station flow. Similar cases are found for KR and LD after 20:00, suggesting

19

LR, KR, and LD may be inaccurate under light conditions especially when the estimates are

20

based on zero flow readings.

21

7. CONCLUSIONS

22

In this study, four correction procedures were examined for non-incident conditions. Temporal

23

correction exploited the inherent temporal trend of historical data. The spatial correction based

26 1

on linear regression, a modification of a previous approach, used the relationship between

2

individual detector flows and station flow. The proposed kernel regression spatial method had

3

the unique feature of incorporating lane use percentage into the correction process. Smith and

4

Conklin's (2002) method based on lane distribution was included as a benchmark.

5

To comprehensively compare the procedures, systematic and random-error evaluations

6

were conducted for two stations. After the results of systematic evaluation were analyzed, the

7

KR and LD approaches were modified. Individual lane flows provided by the detectors on

8

particular general purpose lanes produced more accurate estimates. The two procedures were

9

revised and their station flow estimates were compared to those of the TC and LR approaches at

10

five random error levels.

11

The two detector stations had distinct lane configurations. One station had five lanes,

12

including an exit lane and the other station had four general-purpose lanes. The lane

13

configuration had a significant impact on the spatial correction procedures' performances.

14

7.1. Summary of Findings and Practical Recommendations

15

Lane configurations must be considered when choosing a correction procedure. Traffic patterns

16

associated with the number and types of lanes influence the spatial methods' accuracies.

17

Homogenous lane configurations are more suitable for spatial methods, evidenced by improved

18

accuracy under random errors, compared to heterogeneous lane configurations. The spatial

19

methods are also more accurate when more than one detector malfunctions, based on the

20

systematic evaluation. However, when an exit lane is present, TC is generally more accurate.

21

Correction methods should be calibrated according to location-specific characteristics

22

since each detector has a different impact on accuracy. This is especially true for approaches

23

involving lane distribution. Careful selection of aggregation rule is warranted as shown by the

27 1

systematic evaluations. In this study, including the estimates from the exit and adjacent lanes

2

undermine KR's and LD's accuracies.

3

The distribution of error configurations can guide correction procedure selection.

4

Systematic evaluation indicates that TC is the most accurate for most configurations. However,

5

as the number of malfunctioning detectors increases, KR provides better results under certain

6

error combinations for the station with an exit lane. LR is robust under both lane configurations

7

and achieves accuracy levels consistent with the best approaches under various error

8

configurations; thus, it could be a viable choice for a dataset with diverse error configurations. If

9

one error configuration is dominant, the best approach corresponding to this configuration should

10

be chosen.

11

Time-of-day performance assessment confirms TC's superiority. Overall, this approach

12

outperforms the others at all random error levels, although sporadic cases exist in which other

13

methods are better. Associated with time-of-day are traffic conditions, which are important to the

14

spatial correction procedures, especially when valid observations of zero flow are incorporated

15

into an average. Thus, time-of-day and traffic conditions should be considered when selecting a

16

procedure.

17

In addition to performance-based suggestions, consideration should be given to practical

18

implementation. Though TC is generally the most accurate, it requires data archiving. One query

19

is necessary for each lane correction and may be time-consuming. The storage space and

20

processing time might become prohibitively expensive. When the size of the target dataset is

21

small or speed is not the top priority, TC would be ideal out of the procedures examined,

22

provided the dataset is incident free. When accuracy is not strictly demanded, correction speed is

28 1

an issue, or diverse error configurations are present, the linear regression-based method can be

2

used.

3

7.2. Future Directions

4

This study focuses on incident-free data. However, incidents occur frequently and evaluation and

5

adaptation of the methods for incident conditions remains an important research avenue. The

6

temporal correction procedure’s ability to handle flow deviations from the past average is

7

relatively insufficient and should be improved. This might be achieved by accommodating

8

information from temporally adjacent readings. Additionally, the correction of occupancy and

9

speed readings needs to be evaluated. The use of occupancy might improve spatial methods'

10

performances. Finally, the methods' performances will be assessed for datasets with longer

11

aggregation intervals.

12

ACKNOWLEDGEMENT

13

The authors thank MAUTC for funding the project of which this study is a part. The material in

14

this paper is not necessarily MAUTC's view; the authors remain solely responsible for the

15

content.

16

29 1

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33 1

Figure Caption List

2

Figure 5 Station Location and Detector Configuration

3

Figure 6 Station Flow Profiles for 4:00 AM to 22:00 PM, 04/22/2008

4

Figure 7 Error Configurations for Systematic Evaluation

5

Figure 8 Results of SysEval-I for the Off-peak Period for Station 1

6

Figure 9 Results of SysEval-I for the Peak Period for Station 1

7

Figure 10 Results of SysEval-I for the Off-peak Period for Station 2

8

Figure 11 Results of SysEval-I for the Peak Period for Station 2

9

Figure 12 Comparison between Correction Methods

10

Figure 13 Percentage of Correction Failures (Averaged over 10 Realizations)

11

Figure 14 MAPE by Time-of-Day for Station 1 at Error Level 10%

12

Figure 15 MAPE by Time-of-Day for Station 1 at Error Level 40%

13

34 1

Tables

2

Table 4 Linear Regression Models for Station 1 and Station 2 HOV Period (5:30 AM to 9:30 AM) Station 1

Station 2 2

Detector

Model

R

Detector

Model

R2

601 (HOV)

HOV HOV qˆ601 =1.96 × f 601 + 3670.97

0.45

541 (HOV)

HOV HOV qˆ541 =1.89 × f541 + 3486.23

0.40

603 (GP)

HOV HOV qˆ603 =1.97 × f 601 + 2212.81

0.63

543 (GP)

HOV HOV qˆ543 =1.95 × f543 + 2358.62

0.63

605 (GP)

HOV HOV qˆ605 =2.73 × f 605 + 1878.34

0.69

545 (GP)

HOV HOV qˆ545 =1.95 × f545 + 2358.62

0.71

607 (GP)

HOV HOV qˆ607 =2.67 × f 607 + 3248.15

0.52

547 (GP)

HOV HOV qˆ547 =2.60 × f547 + 2151.15

0.54

609 (GP)

HOV HOV qˆ609 =0.01 × f 609 + 5840.27

0.01

Regular Period (4:00 AM to 5:30 & 9:30 AM to 10:00 PM) Station 1

3 4

Station 2 2

Detector

Model

R

Detector

Model

R2

601 (GP)

REG REG qˆ601 = 2.38 × f 601 + 1543.21

0.75

541 (GP)

REG REG qˆ541 = 2.39 × f541 + 1966.15

0.70

603 (GP)

REG REG qˆ603 = 2.84 × f 603 − 65.14

0.85

543 (GP)

REG REG qˆ543 = 2.67 × f543 + 214.45

0.85

605 (GP)

REG REG =3.15 × f 605 − 47.15 qˆ605

0.75

545 (GP)

REG REG qˆ545 =3.12 × f545 + 190.70

0.73

607 (GP)

REG REG =3.46 × f 607 + 2353.18 qˆ607

0.23

547 (GP)

REG REG qˆ547 =3.40 × f547 + 1320.98

0.53

609 (GP)

REG REG qˆ601 = 4.19 × f 601 + 2006.42

0.38

35 1

Table 5 The Results for SysEval-II for Station 1 Error Config.

2 3

MAPE (%) Peak (7:00 AM - 9:00 AM)

MAPE (%) Off-Peak (1:00 PM - 3:00 PM)

TC LR KR LD TC LR 1 5.42 +1.79 +1.61 +9.52 7.79 +1.19 2 5.63 +2.82 +2.85 +7.75 6.38 +2.17 3 3.12 +5.45 +4.66 +6.87 5.24 +3.16 4 5.62 +2.38 +1.31 +2.19 4.1 +2.88 5 3.9 +3.86 +3.03 +3.91 3.87 +3.11 6 7.2 +0.21 +0.95 +10.86 12.54 +0.68 7 7.01 +1.09 +1.02 +6.29 10.48 +1.43 8 +1.21 +0.34 7.03 +7.91 +0.75 +0.13 9 6.53 +0.68 +0.50 +8.41 +0.37 +0.13 10 7.47 +2.80 +5.59 +20.43 9.76 +2.75 11 5.79 +3.86 +2.69 +7.59 7.62 +0.93 12 6.62 +1.83 +1.86 +6.76 7.86 +0.69 13 5.58 +3.37 +2.20 +4.41 6.41 +1.99 14 4.89 +3.68 +2.89 +5.10 7.01 +1.39 15 +0.65 +1.07 6.93 +0.88 5.64 +1.34 16 9.20 +1.02 9.20(T) 9.20(T) 15.13 15.13(T) 17 8.15 +0.21 8.15(T) +9.91 0.73 13.22 18 +0.9 7.41 +0.74 +10.65 0.37 13.22 19 +0.78 +0.02 8.03 +5.27 11.71 +0.2 20 +0.01 +0.07 8.03 +5.27 11.72 +0.19 21 +2.45 +0.34 7.03 +7.91 +1.77 +0.13 22 7.3 +6.78 +5.76 +20.60 10.24 +2.27 23 8.13 +2.14 +4.93 +19.77 10.9 +1.61 24 7.95 +1.70 +0.53 +5.43 +0.28 8.55 25 7.42 +1.53 +0.36 +2.57 8.00 +0.4 26 9.45(T) 9.45(T) 9.45(T) 9.45(T) 16.19 16.19(T) 27 10.15 +0.07 10.15(T) 10.15(T) 16.31 16.31(T) 28 +2.01 +0.02 8.03 +5.27 +1.03 11.91 29 +1.9 +0.21 8.15 +9.91 +1.6 13.22 30 8.86 +5.22 +4.20 +19.04 11.6 +0.91 (T) indicates the temporal correction was used due to the correction failure

KR +4.42 +9.29 +5.01 +6.15 +6.38 12.54 +1.73 +3.36 +3.36 +5.91 +8.05 +7.81 +3.84 +3.24 +4.61 15.13(T) +0.73(T) +0.37(T) +0.5 +0.49 +3.36 +5.43 +4.77 +7.12 +2.25 16.19(T) 16.31(T) +0.3 +1.6 +4.07

LD +1.06 +4.55 +7.53 +2.55 +2.78 +1.06 +1.71 8.85 8.85 +16.6 +3.31 +3.07 +6.36 +5.76 +1.01 15.13(T) +0.38 +0.38 +0.48 +0.47 8.85 +16.12 +15.46 +2.38 +4.77 16.19(T) 16.31(T) +0.28 +0.38 +14.76

36 1

Table 6 The Results of SysEval-II for Station 2 Error Config.

2 3

MAPE (%) Peak (7:00 AM - 9:00 AM)

MAPE (%) Off-Peak (1:00 PM - 3:00 PM)

TC LR KR LD TC LR 1 +0.22 +2.68 +2.63 4.38 6.55 +0.34 2 +0.11 +1.17 +1.12 5.12 6.75 +0.28 3 4.06 +2.96 +2.76 +0.50 5.28 +0.73 4 4.26 +3.69 +3.48 +0.41 4.77 +0.77 5 +0.56 +0.03 6.55 +0.27 +2.19 9.37 6 7.08 +0.93 +0.88 +0.13 +1.8 8.45 7 +0.06 +2.62 +2.57 7.10 +1.98 +0.23 8 +0.24 +0.14 7.62 +1.28 +2.22 8.28 9 +0.29 +0.08 7.69 +0.79 +1.78 7.81 10 6.71 +2.07 +1.97 +1.02 +1.44 6.55 11 +0.6 +0.06 9.18 +3.45 +1.09 14.38 12 +0.31 +0.05 9.85 +1.75 +2.05 12.1 13 9.31 +3.58 +3.89 +3.13 +3.21 0.06 14 +0.22 +0.44 9.96 +6.98 +2.03 10.92 (T) indicates the temporal correction was used due to the correction failure

KR +0.95 +2.05 +1.67 +1.38 +3.00 +0.58 +0.64 +4.64 +0.99 +0.4 +1.09(T) +0.27 9.03 +2.00

LD +0.31 +5.58 +4.49 +2.09 +2.96 +1.32 6.86 +2.22(T) +4.52 +3.22 +1.09(T) +0.23 +0.74 +2.03(T)

37 1 2 3

4 5 6

Figures Figure 1

Map Data Source: ESRI.

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Figure 2

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Figure 3

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Figure 4

41 1

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Figure 5

42 1

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Figure 6

43 1

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Figure 7

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Figure 8

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Figure 9

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Figure 10

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Figure 11