Model Validation and Reasonableness Checking Manual
4.0 Trip Distribution
Trip distribution links the trip productions in the region with the trip attractions to create matrices of interzonal and intrazonal travel, called trip tables. The critical outputs of trip distribution are trip length and travel orientation (suburb to CBD, CBD to suburb, etc. ), and the resulting magnitude of traffic and passenger volumes. The results of trip distribution are assigned (after mode split has been determined) to the highway and/or public transportation systems to determine the travel demand as related to the carrying capacity of the facilities in question.
The most common form of model used for trip distribution is the gravity model. Gravity models are implemented as mathematical procedures designed to preserve the observed frequency distribution of trip lengths for each modeled trip purpose. The gravity model theory states that the number of trip interchanges between two traffic analysis zones will be directly proportional to the number of productions and attractions in the zones, and inversely proportional to the spatial separation between the zones. The inputs for gravity model-based trip distribution models are productions and attractions for each zone and a matrix of interzonal and intrazonal travel impedances.
4.1 Determination of Travel Impedances
One of the major inputs to gravity model-based trip distribution models are the travel impedance matrices. Travel impedances reflect the spatial separation of the zones based on shortest travel time paths for each zone-to-zone interchange.
Some models use a generalized cost approach which converts highway travel time to cost and combines the time cost with other highway costs including operating expenses (i.e. gas, wear-and-tear), parking, and tolls.
In areas with minimal transit service, travel impedances for trip distribution are typically based only on highway times. For regions with extensive transit service, a "composite impedance" approach allows for the inclusion of multiple modes serving the trip interchange. One consequence of this approach is that overall predicted travel patterns will change when a transit improvement is made - this would not occur if only highway time is used. Transit travel times are separated into each component of the trip - walking or driving to a stop, waiting, in-vehicle travel, and transferring. Transit costs are the fares paid by the passenger.
The creation of highway impedances (also called skimming the network) involves determining the path of least resistance (impedance) between each pair of zones; summing the various components of highway impedance along that path (time, distance, toll, or a combination of these); adding the travel time for intrazonal trips and the terminal times at the trip ends; and then storing these components in travel time matrices (skims).
The use of feedback loops has been highlighted in a number of recent national publications and conferences as "best practice" for travel modeling. In past modeling practice, distribution used highway speeds that were estimated from static look-up tables for specified conditions (see Table 4-1). Best practice takes the congested speeds from the assignment step back to the distribution step through the use of a feedback loop.
Table 4-1
Example Look-Up
Table
Average Speeds for Trip Distribution (mph)
| Area Type | Facility Type | |||||
|---|---|---|---|---|---|---|
| Freeway | Class 1 Arterial | Class 2 Arterial | Class 3 Arterial | Collector | Centroid Connector | |
| Urban | 50 | 35 | 25 | 20 | 15 | 10 |
| Suburban | 55 | 40 | 35 | 25 | 20 | 15 |
| Rural | 60 | 45 | 40 | 35 | 25 | 20 |
Regardless of the procedure used to estimate travel speeds, several types of reasonableness checks can be performed to ensure that the highway skims contain realistic values. The first is a simple determination of implied speeds for each interchange. These can be estimated by simply dividing the skimmed highway distance by the highway travel time and converting for units:
Sij = Dij / Tij * 60
where:
- Sij = speed from zone i to zone j in miles per hour
- Dij = shortest path distance from zone i to zone j in miles
- Tij = shortest path time from zone i to zone j in minutes
- 60 = conversion of minutes to hours
Once the above calculations are made, several items can be checked. The first might be the minimum and maximum speed by interchange or from a group of zones (e.g. area type). The second might be a simple frequency distribution of speeds on all interchanges. This can be done by creating a matrix of "1's" and performing a trip length frequency distribution using the speeds as the impedance matrix and the "1's" as the trip table. In some software packages, this matrix histogram can be summarized directly as an unweighted matrix histogram (skipping the step of creating the matrix of 1's). The key items to review in this distribution are the extrema - any very slow or very fast interchange speeds.
Another aggregate network-level check is of terminal times. These represent the time spent traveling to/from a vehicle to/from the final origin or destination within the TAZ. Terminal times are generally determined using the area type of the TAZ. The terminal times may be adjusted as part of the trip distribution model calibration process in order to make the average trip lengths produced by the model more closely match the observed average trip lengths. If terminal times are used to adjust impedances, these will tend to shift the friction factor curve to the right making the distribution of trips from that zone less sensitive to impedance. Terminal times might also affect mode choice.
Two sets of terminal times are determined--one to be used at the home end of the trip and one to be used at the attraction end. An example of initial terminal times is shown in Table 4-2. These classifications of terminal times should be checked for reasonabableness by measuring actual terminal times for specific combinations of area types and trip end types.
Table 4-2
Terminal Times (minutes)
| Area Type | Production End | Attraction End |
|---|---|---|
| Urban | 2 | 4 |
| Suburban | 1 | 2 |
| Rural | 1 | 1 |
The terminal times shown in Table 4-2 are used to augment the estimated congested and uncongested travel time matrices (including intrazonal times). The production end terminal times are added at the origins and the attraction end terminal times are added at the destinations.
4.2 Gravity Model
Model Description
The gravity model trip distribution technique is an adaptation of the
basic theory of gravitational force. This method is the most common
technique for distributing trips. Other approaches include Intervening
Opportunities and Destination Choice models. Types of aggregate validation
checks remain basically the same regardless of which method is used.
As applied in transportation planning, the gravity model theory states that the number of trips between two traffic analysis zones will be directly proportional to the number of productions in the production zone and attractions in the attraction zone. In addition, the number of interchanges will be inversely proportional to the spatial separation between the zones.
The gravity model for trip distribution is defined as follows:
where:
- Tij is the number of trips from zone i to zone j
- Pi is the number of trip productions in zone i
- Aj is the number of trip attractions in zone j
- Fij is the "friction factor" relating the spatial separation between zone i and zone j
- Kij is an optional trip distribution adjustment factor for interchanges between zone i and zone j
The friction factors are inversely related to spatial separation of the zones--as the travel time increases, the friction factor decreases. A number of different functional forms have been used for friction factors. In fact, early gravity models used "hand fitted" friction factor tables. More recently, however, it has been discovered that mathematical functions such as the "gamma" function produce a realistic trip distribution and can be easily calibrated. Other friction factor calibrations are based on the power or exponential functions.
It is important to note that the trip length frequency distributions, not the observed trip tables from an origin-destination survey, form the basis for model calibration. There was typically little statistical significance to zonal interchange data collected as part of a home interview survey. In fact, even the 1%, 4%, and 10% sample surveys performed throughout the U.S. in the 1950s and 1960s were not sufficiently large to produce statistically significant trip tables at the zonal interchange level. There is, however, a reasonable degree of statistical significance to the average trip lengths and trip length frequency distribution data collected in household travel surveys.
Validation Tests
Observed and estimated trip lengths are both calculated using
network-based impedance. A summary of some modeled trip lengths from
different regions is shown in Table 4-3. Most
packages automatically calculate average trip length for all trip
interchanges. In effect, it is finding the average travel time from the
skims matrix weighted by the trip matrix.
Most household travel surveys, and secondary sources such as the CTPP and NPTS, do ask respondents to report travel times to work. However, these times are not considered as reliable as the origin and destination information obtained from the survey. Reported times do serve a purpose in model validation by providing a "ballpark" estimate of trip length. Examples of Census Journey-to-Work reported trip lengths are listed in Appendix A.
The 1990 NPTS found the reported average commute travel time to be 19.7 minutes and 10.6 miles (see Table 4-4). Work trip lengths are typically in the 20 to 25 minute range, although these can be longer for large metropolitan areas and shorter for small metropolitan areas. Non-work trip lengths are typically less than those for work trips.
- Compare average trip lengths by purpose. The most standard
validation checks of trip distribution models used as part of the
calibration process are comparisons of observed and estimated trip
lengths. Modeled average trip lengths should generally be within five
percent of observed average trip lengths.
If a generalized cost is used as the measure of impedance, average trip lengths and trip length frequency distributions should be checked using the individual components of generalized cost (e.g., time and distance).
- Compare trip lengths for trips produced versus trips attracted by purpose by area type. An example of a summary showing trip lengths produced and attracted by area type is shown in Appendix B. Average trip lengths sent and received by district could be mapped using GIS.
- Plot trip length frequency distributions by purpose. The trip length frequency distribution shows how well the model can replicate observed trip lengths over the range of times (see Figure 4-1). Visual comparison of distributions is an effective method for validation. A quantitative measure which can be used to evaluate distribution validation is the coincidence ratio.
Table 4-3
Comparison of Trip Lengths Among
Cities
| City | Year of Survey | Average Trip Length in Minutes | |||||
|---|---|---|---|---|---|---|---|
| HBWork | HBShop | HBSchool | HBOther | HBNon-Work | NHB | ||
| San Juan | 1991 | 35.4 | 14.2 | 15.5 | 16.1 | -- | 16.2 |
| Denver | 1985 | 22.7 | -- | -- | -- | 12.9 | 13.8 |
| Northern N.J. | 1986 | 23.2 | 14.4 | -- | -- | 15.3 | 17.1 |
| Phoenix | 1988 | 19.3 | 10.6 | -- | -- | 13.0 | 13.6 |
| Charles-ton, WV | 1993 | 20.7 | 18.7 | 15.9 | 17.3 | -- | 15.7 |
| Reno | 1990 | 11.2 | 8.6 | 9.34 | 10.4 | -- | 8.1 |
| Houston | 1985 | 20.9 | 9.4 | 8.9 | 11.7 | 10.6 | 12.7 |
Table 4-4
Commuting Patterns of
Home-to-Work Trip by Mode
| Mode | 1969 | 1977 | 1983 | 1990 | Percent Change (69-90) |
|---|---|---|---|---|---|
| Trip Distance (Miles) | 9.9 | 9.2 | 9.9 | 10.6 | 7% |
| Travel Time (Minutes) | 22.0 | 20.4 | 20.4 | 19.7 | -10% |
| Source: NPTS | |||||
Coincidence Ratio
The coincidence ratio is used to compare two distributions. In using the
coincidence ratio, the ratio in common between two distributions is
measured as a percentage of the total area of those distributions.
Mathematically, the sum of the lower value of the two distributions at
each increment of X, is divided by the sum of the higher value of the two
distributions at each increment of X. Generally, the coincidence ratio
measures the percent of area that "coincides" for the two
curves.
The procedure to calculate the coincidence of distributions is as follows:
Coincidence = sum {min ( count+T/count+, count-T/count-
) }
Total = sum {max ( count+T/count+, count-T/count-
) }
Calculate for T = 1, maxT
Coincidence Ratio = coincidence /
total
where
- count+T = value of estimated distribution at Time T
- count+ = total count of estimated distribution
- count-T = value of observed distribution at time T
- count- = total count of observed distribution
The coincidence ratio lies between zero and one, where zero indicates two disjoint distributions and one indicates identical distributions. Thus, in the upper portion of Figure 4-2, the area in common is shaded. In the lower portion of the figure, the common area, also shaded is greater as the distributions are closer. Thus, the coincidence ratio will be higher for the second example.
Figure 4-1
Home-Based Work -
Trip Length Frequency Distribution
Figure 4-2
Coincidence Ratio
for Trip Distribution
- Plot normalized friction factors. If a gravity model is used for trip distribution, it is also worthwhile to plot the calibrated friction factors (scaled to a common value at the lowest impedance value). Such a plot provides a picture of the average traveler's sensitivity to impedance by trip purpose and can be compared to friction factors from other regions. For example, travelers might be expected to be less sensitive to travel time for work trips since these trips must be made every day and can usually not be shifted to off-peak conditions or to different locations. This is shown in Figure 4-3 where the friction factors for work trips show gradual change as travel time increases.
If there are significant differences between observed and estimated trip lengths, this may be due to a number of factors:
- Inadequate closure on production/attraction balancing.
- Travel impedances may be too high or too low.
After validating the trip distribution model at a regional level, the model results should be checked for subgroups of trips and segments of the region. Appendix C shows an example of a validation summary used in New Orleans.
- Calculate percent of intrazonal trips by purpose. The percent of intrazonal trips by purpose should be checked for the region and by zone size (e.g., ranges in area such as 0 to 0.5 square miles, 0.5 to 1 square mile, etc.). Typically intrazonal trips account for less than 5% of total person trips. However, this percentage is highly dependent on zone size and the ideal amount will depend on whether the travel model is used for regional or local-level analysis. Systemwide link volumes can be modified by varying the number of intrazonal trips through changes to intrazonal times.
- Compare observed and estimated district-to-district trip Interchanges and major trip movements. Although comparing trip lengths provides a good regional check of trip distribution, the model can match trip lengths without distributing trips between the correct locations. In order to permit easier review of the person trip tables, zonal interchanges can be summarized into districts, or groups of zones. Trips to the major employment area in the region (i.e. CBD) should be reviewed. Major trip movements across rivers or other physical barriers should be summarized as well.
- Stratify trip lengths and/or trip interchanges by income class. Often different income classes exhibit different travel characteristics.
Figure 4-3
Normalized Friction
Factors
K- Factors
K-factors are sector to sector factors which correct for major
discrepancies in trip interchanges. These factors are computed as the
ratio between observed and estimated trip interchanges. K-factors are
typically justified as representing socioeconomic characteristics that
affect trip making but are not otherwise represented in the gravity model.
Physical barriers, such as a river crossing, may also result in
differences between observed and modeled trip patterns. For example, trip
movements between zones separated by a bridge may not be as great as would
be expected using only quantifiable measures. In that case, the planner
can use either k-factors or artificial times on the bridge links to match
the actual interchange of travel.
A specific problem with trip distribution occurs when low income households are matched with high income jobs in the central business district, particularly for large metropolitan areas. Although there are certainly trips between low income residences and downtown business districts, trip distribution models can have a tendency to overstate these trips. This error can have an even greater impact on transit projections since low income riders tend to be more transit dependent and transit is usually more competitive with the automobile downtown.
The use of K-factors is generally discouraged and are seen as a major weakness with traditional gravity models when used to correct for socioeconomic factors. Since K-factors represent characteristics of the population which change over time, the assumption that K-factors stay constant in the future can introduce a significant amount of error in predictions of future trip distributions.
A preferred approach is to stratify trip productions and attractions by income class (or auto ownership) and perform separate distributions of trips by class. Each model can reflect the different distributions of employment types throughout the region, as well as the unique sensitivities of different classes of travelers to travel time.
