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Electronic Companion-“Short-term traffic state prediction based on temporal-spatial correlation” by T.L. Pan, A. Sumalee and R.X. Zhong T. L. Pana , A. Sumaleea , R. X. Zhongb,a a

 Department of Civil and Structural Engineering, The Hong Kong Polytechnic Polytechnic Unive University rsity,, Hong Kong SAR, China.  Research Center of Intelligent Transportation Systems, School of Engineering, Sun Yat-Sen University, Guangzhou, China.

b

1. Test site description and model parameters parameters calibration A segment of Interstate 210 West bound, approximately two miles in length, is chosen in this case study as depicted in Fig.   1. This short segment, located in Monrovia, Los Angeles, stretches from S Myrtle Ave (A) through W. Huntington Dr(B) to N Santa Anita Ave (C) with two on-ramps and two o ff -ramps. -ramps. This segment of freeway is chosen here for the following reasons: 1. The high level of recurrent congestion within the section can be observ observed ed in the early morning period (6 am-10 am). 2. The segment possesses possesses necessary necessary infrastructure infrastructure and traffic detectors embedded in the on-ramps and mainline lanes for data collection. This section This section is instru instrument mented ed with with singlesingle-loop loop induct inductanc ancee detect detectors ors,, which which are embedde embedded d in the paveme pavement nt along the mainline, HOV lane, on-ramps, and off --ramps. ramps. Each Each loop detecto detectorr pro provide videss ra raw w dat dataa such such as traffic volume (veh / time-step) time-step) and occupancy (%) for the corresponding lane for 30 seconds, where in this empirical study the first item is conversed to general(for all lanes) tra ffic flow rate (veh / hour)and hour)and the second one become general densities (veh / mile) mile) via dividing the occupancy by   g-factor (feet / veh) veh) which is the eff ective ective vehicle length. For the convenie convenience nce of analyzing analyzing the prediction prediction of boundary boundary var variables iables and supply functions, functi ons, we present present an schematic representation representation of the freeway freeway segment segment and its detector detector configuration configuration in Fig.   2. Boundary Boundary variables variables of this freeway freeway segment, i.e.   qu ,   r 1 ,   f 1 ,  q o ,   r 2 , and   f 2 , are predicted using the best linear predictor predictor.. Here, we pretend the detector in the middle, i.e.   qm , was missing (i.e. we do not use it in the simulation) simulation) for validation validation of the performance performance of the proposed proposed method. Tr Traaffic flow data of seven hours (4:00 am-11:00 am) collected on Tuesday, Wednesday and Thursday of March and April of 2008 and 2009 provided by the Performance Measurement System (PeMS) is used in this test to calculate spatial and temporal correlations of the related traffic characteristics. 2. Spatial-temporal Spatial-temporal correlation phenomena phenomena and measurement: measurement: data preparation Spatial-temporal correlations can be utilized to predict short-term tra ffic state. Rather than introducing the mathematical definitions of these correlations, we would like to give an intuitive example by analyzing the detected traffic flow data for this freeway freeway segment segment to support this empirical empirical study. Figures Figures   3-6  depict

 Email addresses:   [email protected] [email protected] (T. L. Pan),   [email protected] [email protected] .hk  (A. Sumalee), [email protected] (R. X. Zhong) Prep Preprrint int submi ubmitt tted ed to IEEE IEEE Transac nsacti tion onss on Int Intel elli lige gent nt Trans ansport portat atio ion n Syst System emss

Febr ebruary uary 1, 2013 2013

 

 

Figure 1: Map of the test site (Source: Google map)

 

qo

0.45 mile Cell 4

 f  4 

 

0.45 mile  

qm

 

0.5 mile

Cell 3

Cell 2

 f  2



3



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0.5 mile  

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qu

Cell 1

r 1 

Detector 

Figure 2: A section of I210-W divided into 4 cells and its detector configuration

2

 

the temporal and spatial correlated traffic flow phenomena via the historical tra ffic data of the freeway segment, e.g. the density and inflow profiles detected at locations A, B and C on Tuesday, Wednesday, and Thursday Thursd ay of March and April 2008 and 2009. Figure  Figure  3(a) shows the temporal similarity (or correlation) of the flow profile detected at location A for the three weekdays from 5:00 to 6:00 am with a detection frequency freque ncy (i.e. resolution, resolution, the sample time interval) interval) of 5 minutes. It is observed observed that these flow profiles profiles are positively correlated among adjacent time steps, which imply that the higher value a certain flow profile at a certain time step, the flow profile intends intends to have a larger value value in the near future. Temporal emporal covariance covariance and correlation coefficient of  q  q (k ) between two adjacent time steps  k  and  and  k-z  are defined as: Covtm (q(k ), q(k  −  z))  = γ ttm m (q(k ), q(k  −  z))  =

∑ N   (q (k ) −  q¯ (k )) (q (k  − z) −  q¯ (k  − z)) i

i=1

i

 N   − 1

  Covtm (q(k ), q(k  −  z)) sq(k ) sq(k − z)

 

(1)

,   z  =  1 , 2 · · · , le ,   le   ≤  k  − 1

(2)

where  q i (k ) is the value of the detected flow during interval  k  on   on the  i th day,  z  is a positive integer less than or equal to  l s  which is a predefined bound (also an integer) for the calculation, and  s q(k ) denotes the standard (b) presents the correlation coe fficient deviation of the detected flow  q (k ) for the  N   sample days. days. Figure 3 Figure 3(b) γ ttm erent values of  z  z  (e.g. distances of 5 minutes, 15 m  of flow profiles detected at location A with three di ff erent minutes and 30 minutes) from 4:00 am to 12:00 am over N =54 days. The result illustrates that the temporal correlations of the flow profiles will generally decrease as  z   increases ,  which can be interpreted interpreted as the flow pattern of a given time is more similar to a neighboring flow pattern than a distant one in temporal domain, e.g.   γ ttm m (q(5 : 30), q(5 : 25))   > γ tm tm (q(5 : 30), q(5 : 00))   > γ ttm m (q(5 : 30), q(4 : 35)).  Temporal correlation of the parameters parameters of a fundamental fundamental diagram (or supply functions) for a given given location also can be analyzed similarly:

    ∑ N i=1 δi (k  p p ) −  ¯δ(k   pp ) δi (k   pp   − l p ) −  ¯δ(k   pp  − l p )



Covtm δ(k  p  p ), δ(k  p  p   − l p )  =





γ tm tm δ(k   p p ), δ(k  p  p  − l p )  =

 N   − 1

Covtm (δ(k  p  p ), δ(k  p  p  − l p )) sδ(k   pp ) sδ(k   pp −l p )

 

,   l p   =  1 , 2 · · · , l p,ε ,   l p,ε   <  k   pp

(3)  

(4)

where δi (k   pp ) is a certain parameter calibrated with data detected during the time period  k   pp  (e.g. half an hour) on the   ith day.   δ¯ (k   pp ) is the mean of   δ k  p  p , and   sδ (k  p  p ) is the corresponding standard deviation over the   N 

 

days. As shown in Figure Figure 4,  4, temporal correlations of free-flow speed  v f  during the non-rush hours are much more evident than the correlations of congestion wave speed   w  duri  during ng the rus rush h hours. Compare Compared d with with c the correlation analysis of detected flow patterns, the supply functions are more complicated because the flow-density relationship might not constitute a complete triangular fundamental diagram for particular time periods, e.g.   wc  cannot be observed during 4:00-6:00 am while  v f  cannot be evaluated during rush hours 7:00-9:00 am. This is reflected in the figure as zero entries. Because the temporal correlation coe fficients of  Qm ,  ρ c  and  ρ J  are very small, the results are omitted for brevity.  5 (upper  (upper side) illustrates the spatial correlated phenomenon of the free-flow and congestion wave Figure 5 Figure speeds calibrated calibrated at three adjacent adjacent sites based on the flow-density flow-density data collected. collected. In this exam example, ple, the freefreeflow speeds are found to be positively correlated in spatial domain during the 4:00-6:00 am. To interpret this positive correlation, we depicted tra ffic flow-density data during 4:00-4:30 am for two days (April 24, 2008 and April 21, 2009) under diff erent erent weather conditions with the corresponding free-flow speeds calibrated in Figure 5 Figure  5  (low  (lower er side). The positive positive correlation correlation implies that if the free-flow free-flow speed at site A (S. Myrtle) increase / decrease, decrease, then the free-flow free-flow speed at site B (W. (W. Huntington Dr) and site C (N. Santa Anita) would increase / decrease decrease accordingly (as illustrated by Figure 5 Figure  5). ). 3

 

(a) Historical segment inflow profiles 8000 7000    )   r   u   o    h    / 6000   e    l   c    i    h 5000   e   v    (     w4000   o    l    F 3000

March 4,2008 March 12,2008 March 25,2008

2000 5:00

6:00

Time (clock) (b) Temporal correlation coefficient of inflow profiles

1

0.5

0

5 minutes 15 minutes 30 minutes

−0.5   4

5

6

7

8

9

10

Time (clock)

Figure 3: Samples of inflow profiles and their temporal correlation

Temporal correlation of waveback speed w

Temporal correlation of free flow speed v



1

c

1

Location A Location B Location C

0.5

0.5

0

0

−0.5  

0.5 5

6

7

8

9

5

10

6

7

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Time (clock)

Time (clock)

Figure 4: Temporal correlation coefficients of the supply functions

4

10

 

Spatial correlation coefficient of free flow speed v

Spatial correlation coefficient of congestion wave speed w



0.6

c

0.6

Location A and B 0.5

0.5

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Location A and C Location B and C

−0.2   4

10

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4000

3500

3500

     ) 3000      h      /     s     e 2500      l     c      i      h 2000     e     v      (     w1500     o      l      F

3000

3000

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April 24,2008 April 21, 2009

0 10

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Density(vehicles/mile)

Density(vehicles/mile)

Density(vehicles/mile)

10

Location C

4000

0  0

9

Location B

Location A 4000 3500

8

Time (clock)

Time (clock)

Figure 5: Spatial correlated phenomenon of supply functions

The spatial covariance and correlation coe fficient of the free-flow speeds during time interval  k   pp  are thus defined: Covsp (δm (k  p  p ), δn (k  p  p ))  =

∑ N  δ i=1





 ¯  ¯ p ) m,i (k  p  p ) − δm (k   p p ) δn,i (k   p p ) − δn (k   p

 

 N  −  − 1

(5)

Covsp (δm (k   pp ), δn (k   pp )) γ sspp (δm (k  p ), δn (k  p ))  =

(6)

sδm (k  p p ) sδn (k  p p )

(6) are similar with the ones in (3 (3) and (4 (4) but for the diff erentiation erentiation of  where the notations in (5 (5) and (6  presents the spatial adjacent locations. Figure 6 Figure 6 presents δm (k   pp ) and  δ n (k   pp ) denoted by  m  and  n  which represent the adjacent correlation of densities: Covsp ( ρm (k ), ρn (k ))))  =

∑ N  ( ρ i= 1

γ sspp ( ρm (k ), ρn (k ))))  =

  ¯ m (k ) m,i (k ) −  ρ

) ( ρ

  ¯ n (k ) n,i (k ) −  ρ

 N   − 1

Covsp ( ρm (k ), ρn (k )) )) s ρm (k ) s ρn (k )

)

 

(7)

(8)

 ρm,i (k ) is the tra ffic density detected at location  m  during time interval  k  on the  i th day,  ρ¯ m (k ) is the mean density over the  N  days, and  s ρm (k )  is the corresponding standard deviation.

5

 

Spatial correlation coefficients of density

1

0.8

0.6

0.4

0.2

0

Location A and B

−0.2

Location A and C Location B and C

−0.4 4

5

6

7

8

9

Time (clock)

Figure 6: Spatial correlated phenomenon of traffic densities

6

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