City of Calgary

1
Logit Gravity Trip Distribution Model for a
Personal Travel External Trip Table
Authors Dr. John Douglas HUNT Zoran CARKIC
Presentation prepared by Zoran CARKIC Nina
GANCHEV September 2005
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City of Calgary

• 2003 Population 922 315
• (data from 2003 civic census)

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RTM Study Area Zone System

• Current RTM was updated in 2001 and expanded to
  include the surrounding region
• 1187 Zones in City
• 235 Zones in Region
• 25 External Entry/Exit Points
• Total of 1447 Zones

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Regional Transportation Model

• Nested Logit Structure
• Personal travel model with choice behaviour for
  trip generation, mode choice, time-of-day and
  distribution
• 25 travel segments (person/purpose)
• 9 possible modes choices
• 3 times of day choices
• Crown/Shoulder choice for auto 1, 2, 3 persons
• 378 practical combinations of person/purpose/time
  of day/mode

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Model Structure
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Full Trip Matrix for Total Trips
D e s t i n a t i o n T r a n s p
. Z o n e s
I-E
I-I
O r i g i n T r a n s p. Z o n e s
E-I
E-E
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Internal Trips - Calgary 2001 RTM
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Personal Travel External Trip Table -
Distribution Model
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Data Sources

• Traffic Volume Counts
• 2001 Household Activity Survey
• Population and Employment Data
• 1981 External Cordon Survey

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Traffic Volume Counts

• Traffic volume counts were conducted on roads and
  highways bordering with the Region
• Provided information about the total number of
  journeys Te to/from External Zones

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2001 Household Activity Survey

• Completed a total of 9769 surveys
• City of Calgary 8537
• Regional Households 950
• Banff/Canmore 282
• Travel Activity Questions Included
• Start and end times
• How did you travel?
• Where was the activity? (address)
• Where you picked up or dropped off by someone?
• Was a vehicle available for the trip?

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2001 Household Activity Survey
Trip Purpose TZ O-H H-Sc H-W H-O N
H Sc-H W-H 101 0 31 345 1535 1696 0 0
102 1357 0 2040 2149 2300 362 719
103 195 0 1520 483 1092 0 291
104 1713 137 4380 1734 4700 0 1187
105 1614 252 4230 641 3737 0 686
106 946 40 400 3427 3275 0 370 107 59
0 2415 918 2753 62 351 108 280 0 6487
1678 6619 0 205 109 74 133 6774 1216
7698 0 151 110 0 0 5997 1525 4846 0 0
111 0 0 604 1115 2749 0 0 112 0 0
2011 855 3478 0 0 113 158 0 3617 1345
4051 0 0 114 0 0 1639 44 1908 0 0
115 0 69 1749 2917 6157 0 0 . . . .
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1981 External Cordon Survey

• Summary statistics from the 1981 External Cordon
  Survey provided ratios for all external trip
  types
• In 1981 it was found that 5.9 of external trips
  were through trips with no stops in Calgary or
  the region
• The impact of through traffic on regional traffic
  is relatively small

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Model Selection

• A simple growth factor trip distribution model
  was considered but rejected due to its
  disadvantages.

Advantages - Simple - Direct use of observed
values Disadvantages - Requires extensive data
(therefore expensive) - Accuracy of results are
heavily influenced by accuracy of input trip
matrix - Components of input trip matrix with
zero in cells continue to have zero in solution -
Preserving patterns in observed behaviour - only
applicable for short term planning horizon -
Changes in transport cost ignored - therefore of
limited use when analysing policies involving new
modes, new links, pricing changes and/or new
regulations
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Model Selection

• A gravity model is appropriate if the assumptions
  made are acceptable for the situation modelled
• The general expression for the Gravity Model
  comes from
• Analogy approach is based on Newton's
  gravitational law estimates trips without using
  observed trip pattern directly (synthetic).
  Entropy-maximization and Intervening
  Opportunities two other approaches to form a
  gravity model
• The generalized cost function must reflect the
  influences of relevant factors.

G m1 m2 F ----------------------- r2
F
Calibration of a gravity model involves -Selecti
ng a functional form for the function -Selecting
the variables to include in the
function -Estimating values for the associated
coefficients
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Transportation Zone Variables

• Distance
• Population
• Employment
• Decisions regarding which variables to combine
  for the utility functions were made independently
  of the t-statistic analysis.

Distance between internal TZ and external
entry/exit points was calculated directly from
the model network Population and Employment
values for internal zones were used to develop
general attraction ratios in the Logit Gravity
Model calculations.
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Utility Function

• A linear function that assigns utility values to
  the trip maker alternatives is called a utility
  function

U i V ( a, i ) E ( a, i ) or U i F1
X1i F2 X2i .. Fn Xni
where V ( a, i ) the measurable conditioning
component of the utility individual i associates
with alternative a, E ( a, i ) the error
component of the utility individual i associates
with alternative a
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Utility Function
General formula for the utility function chosen
is
U i a Ln(Attr i ) b Distance i
Where a, b - parameters to be estimated Attr i
- weighted attraction factor Distance i -
distance between TZ
Using different mathematical combinations for
selected variables, alternative utility functions
were examined.
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General Logit Expression

• The logit expression could be presented in the
  following form
• Regardless of the magnitude of the coefficients
  of the variable values, this model will always
  produce values in the range 0 to 1.

exp( l V(a,i )) Pi S exp( l V(a ,i
)) a ? A
Pi is a PROBABILITY that trip maker alternative
a is selected out of full set of alternatives A
being considered
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Logit Gravity Trip Distribution Model

• By combining the logit expression with a utility
  function that assigns utility values to the trip
  maker alternatives, we created a logit gravity
  model formula

exp (a Ln(Attr i ) b Distance i ) Tei
Te --------------------------------------------
------- S i exp(a Ln(Attr i ) b
Distance i )
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Constants Determination

• Maximum likelihood technique was used for
  constant determination
• The A-logit computer program is used for the
  calibration of utility functions
• Some of the outputs are
• 1) Initial likelihood
• 2) Final value of likelihood
• 3) r2(0) - "Rho-Squared" with respect to zero
• 4) r2(C) - "Rho -Squared" with respect to
  constants
• 5) Standard error
• 6) T - Ratio

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Model Quality of Fit

• The quality of fit of the logit model could be
  considered by using a goodness-of-fit index
  labeled as r2. The following is the mathematical
  definition of r2

r2(0) 1 - L (K) - N
L (0)
Where L (K) log-likelihood for model with
the full vector of parameters K L (0) -
log-likelihood for model with no parameters,
where all parameters are set to 0 N number of
coefficients in estimated models The r2(0) index
is used in the same way that the R2 is used with
linear regression models. Larger values
indicate a better fit.
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The Zero Hypotheses Test

• The mathematical definition for the t-ratio is
• If the absolute value of the t-ratio is greater
  than 1.96 than there is a less than 5 chance
  that the associated difference is due to random
  effects only

t - ratio K(1) _ ___K(1)_______ VAR
K (1) 0.5
The t-ratio is used to test the zero hypotheses
that there is no difference between a given
parameter K (1) and 0. If a given parameter is
not significantly different from 0 it is not
going to be used in the utility function because
it has no influence on the choice behavior of a
trip maker.
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Results
In absence of the field data it was hard to find
attributes and/or variables that influence trip
maker decisions. Even though decisions regarding
choice of variables combined for the utility
functions were made independently of the
t-statistic analysis, the logic behind the
gravity model gave us a lot of comfort in
choosing the distance and the attraction between
transportation zones as variables. Results
obtained from statistical analyses in this
assignment suggest that the model was reasonable.
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External - Internal Trips

• Journeys originating outside the region and
  destined to a TZ within the region
• Final logit formula for the logit gravity model

Ui 0.6589 Ln(Attr i) 0.07737 Distancei
1 (2.1) (-5.7)
exp (0.6589 Ln(Attr i) 0.07737
Distancei) Tei Te ----------------------------
--------------------------------- 2 S i
exp (0.6589 Ln(Attr i) 0.07737 Distancei)
The coefficient estimates signs were consistent
with what would be expected in case corresponding
variable values were changed. For example, if
distance is increased then that alternative would
be less attractive, so the coefficient for
distance is negative.
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Internal - External Trips

• Journeys originating from a TZ within the region
  destined outside the region
• Final logit formula for the logit gravity model

Ui 0.4744 Ln(Attr i) 0.06328 Distancei
3 (1.3) (-3.7)
exp (0.4744 Ln(Attr i) 0.06328
Distancei) Tie Te ----------------------------
--------------------------------- 4 S i
exp (0.4744 Ln(Attr i) 0.06328 Distancei)
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External - External Trips

• Represents through traffic in the model
• 1981 External Cordon Survey was used to obtain
  the percentage of the regional through traffic

Tee S Te 0.059
It was found that 5.9 of the trips were through
trips with no stop in Calgary or the Region. A
simple adjustment of the Logit Gravity Model is
appropriate to reflect the overall influence of
the through traffic.
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Questions?
Authors Dr. John Douglas HUNT Zoran CARKIC
Presentation prepared by Nina GANCHEV Zoran
CARKIC
With limited resources to do an external travel
survey, the logit gravity model was found
appropriate for the situation modelled.