1. Local sensors Round 1: 30 km/h Round 2: 20 km/h Round 3: 10 km/h
At (local) detector:
utms= 20 km/u !
(18% > space mean)
TT = 3 min/km!
(17% < real TT)
Round 1: 30 km/h
Round 2: 20 km/h
Round 3: 10 km/h
Average travel time
TT = 3.6 min/km
So space mean speed:
usms= km/u
a) If we observe each vehicle only once:
Average travel time / km:
(1/30 + 1/20 + 1/10)/3 = 11/(60*3) =11/180 u/km = 3.6 min/km
Average speed
180/11 = km/u
at detector: u = 20 km/u (+18%); TT = 3 min (-17%)
b) If we observe for one hour
Average travel time / km
(30/30 +20/20 +10/10)/( ) = 3/60 u/km = 3 min/km
60/3 = 20 km/u
at detector: u = (30*30 +20*20 +10*10)/( ) = 23.3 km/u (+17%)
TT = 2.6 min (-14%)
Detector
December 28, 2018, CT5804 Lecture 5: traffic monitoring & state estimation

2. Local sensors Red car: 30 km/u Yellow car: 20 km/u Blue car: 10 km/u
At detector after 1 hour:
utms= 23 km/u !
(17% > space mean)
TT = 2.6 min/km! (14% < real TT)
Red car: 30 km/u
Yellow car: 20 km/u
Blue car: 10 km/u
After 1 hour:
TT = 3 min/km
thus
usms=20 km/u !
a) If we observe each vehicle only once:
Average travel time / km:
(1/30 + 1/20 + 1/10)/3 = 11/(60*3) =11/180 u/km = 3.6 min/km
Average speed
180/11 = km/u
at detector: u = 20 km/u (+18%); TT = 3 min (-17%)
b) If we observe for one hour
Average travel time / km
(30/30 +20/20 +10/10)/( ) = 3/60 u/km = 3 min/km
60/3 = 20 km/u
at detector: u = (30*30 +20*20 +10*10)/( ) = 23.3 km/u (+17%)
TT = 2.6 min (-14%)
Detector
December 28, 2018, CT5804 Lecture 5: traffic monitoring & state estimation

3. Does it matter ?
December 28, 2018, CT5804 Lecture 5: traffic monitoring & state estimation

4. Yes it matters: time mean induces serious bias (speeds)
Difference instantaneous space mean, time and harmonic mean
Data from measured vehicle trajectories (over a section)
Differences in speeds > 100%!
Strongly related to measurement location
Proportional to variance in individual speeds
December 28, 2018, CT5804 Lecture 5: traffic monitoring & state estimation

5. Yes it matters: time mean induces serious bias (also in estimated travel times)
Measured travel time
Estimated travel time
(harmonic mean speed)
Estimated travel time
(Time mean speed)
December 28, 2018, CT5804 Lecture 5: traffic monitoring & state estimation

6. Solution: harmonic averaging local quantities
Average of quantity z over space:
Fill in z = v (speed) and you get
December 28, 2018, CT5804 Lecture 5: traffic monitoring & state estimation

7. Relevance harmonic averaging
Most ITS need the space mean of a quantity: unbiased proxy with harmonic average (premise: stationarity and homogeneity)
Continuity relation: q = ku; density from flow and speed: k = q/u
Only valid with space mean speed (thus harmonic local mean!)
Differences (estimated and real density) up to 100% and more
December 28, 2018, CT5804 Lecture 5: traffic monitoring & state estimation