Chapter 28: 

Crime Trip Distribution 

Ned Levine 

Ned Levine & Associates 

Houston, TX 

Richard Block 

Loyola University 

 Chicago, IL

Dan Helms 

Scytale Consulting

Reston, VA 

Phil Canter 

Towson University 

Towson, MD 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table of Contents 

Theoretical Background 28.1

 Logic of the model 28.1 

 Observed and Predicted Distributions 28.3 

The Gravity Model 28.4

 Social Applications of the Gravity Concept 28.5 

 Trips as Interactions 28.6 

 Negative Exponential Distance Function 28.7 

Travel Impedance 28.8

 Distance v. Travel Time 28.8 

 Travel Cost 28.11 

 Travel Utility 28.12 

 Impedance Function 28.13 

Alternative Model: Intervening Opportunities 28.14 

Method of Estimation 28.15 

CrimeStat IV Trip Distribution Module 28.16

 Describe Origin-Destination Trips 28.17 

Example of Observed Trip Distribution from Baltimore County 28.22 

Calibrate Impedance Function 28.24 

Example of Empirical Impedance from Baltimore County 28.27 

Setup of Origin-Destination Model 28.28

 Fitting the Impedance Function 28.41 

Running the Origin-Destination Model 28.42

 Calibrate Origin-Destination Model 28.42 

 Apply Predicted Origin-Destination Model 28.42 

Example of the Predicted Trip Distribution from Baltimore County 28.45 

Comparing Observed & Predicted Trips 28.49 

Estimating Impedance Parameters and Exponents of the Gravity Model 28.51 

 Comparing Intra-zonal Trips 28.52 

Illustration 28.52 

 Comparing Trip Length Distributions 28.53 

 Graphical fit 28.54 

 Coincidence ratio 28.54 

 Komolgorov-Smirnov two-sample test 28.55 

Illustration 28.56 
 

 

 

 

 

 

 

 

 

 

Table of Contents (continued) 

Comparing the Trips of the Top Links 28.56 

 Number of links to test 28.61 

Illustration 28.61 

 Optimizing the Three Evaluation Criteria 28.63 

 One solution for optimizing decisions 28.64 

Illustration 28.65 

Implementing the Comparisons in CrimeStat 28.69 

 Observed trip file 28.69 

 Predicted trip file 28.70 

 Select bins 28.71 

 Compare top links 28.71 

 Save comparison 28.72 

 Table output 28.72 

 File output 28.72 

Graph 28.73 

Uses of Trip Distribution Analysis 28.73 

Utility of Observed Trip Distribution Analysis 28.73 

 Crime prevention efforts 28.73 

 Improved journey-to-crime analysis 28.73 

Utility of Predicted Trip Distribution Analysis 28.74 

References 28.76 

Attachments 28.79 

A. Modeling DWI Trips That End in Crashes 

in Baltimore County, MD 

By Ned Levine & Phil Canter 28.79 

B. Targeting Crime on Public Transport: An Example 

from Greater Manchester, England 

By Daisy Smith & Steph Winstanley 28.80 
 

Chapter 28: 

Crime Trip Distribution 

In this chapter, the mechanics of the second crime travel demand modeling stage -trip 

distribution, is explained. Trip distribution is a model of the number of trips that occur between 

each origin zone and each destination zone. It uses the predicted number of trips originating in 

each origin zone (trip production model) and the predicted number of trips ending in each 

destination zone (trip attraction model). Thus, trip distribution is a model of travel between zones 

- trips or links. The modeled trip distribution can then be compared to the actual distribution to 

see whether the model produced a reasonable approximation. 

Theoretical Background 

The theoretical background behind the trip distribution module is presented first. Next, 

the specific procedures and tests are discussed with the model being illustrated with data from 

Baltimore County. 

Logic of the Model 

Trip distribution usually occurs through an allocation model that splits trips from each 

origin zone into distinct destinations. That is, there is a matrix which relates the number of trips 

originating in each zone to the number of trips ending in each zone. Figure 28.1 illustrates a 

typical arrangement. In this matrix, there are a number of origin zones, M, and a number of 

destination zones, N. The origin zones include all the destination zones but may also include 

additional ones. The reasons that there would be different numbers of zones for the origin and 

destination models are that crime data for other jurisdictions are not available but that many 

crimes that occurred in the study jurisdiction were committed by individuals who lived in other 

jurisdictions. 

For example, with crimes that occurred in Baltimore County, approximately 35% were 

committed by offenders who lived in the City of Baltimore. Thus, it is important to include the 

City of Baltimore as an originating area for Baltimore County crimes. Hence, there are 325 

destinations zones for Baltimore County while the origin zones include both the 325 in Baltimore 

County and 207 more from the adjacent City of Baltimore. If it were possible to obtain crime 

data for the City of Baltimore, then it would be possible to have the same number of zones for 

both the origin file and the destination file. As Chapter 26 pointed out, the study area should 

extend beyond the modeling area until the origins of at least 95% of all trips ending in the study 

area are counted. 

28.1 
Figure 28.1: 

Example Crime Origin-Destination Matrix 
 

Each cell in the matrix indicates the number of trips that go from each origin zone to each 

destination zone. To use the example in Figure 28.1, there were 15 trips from zone 1 to zone 2, 

21 trips from zone 1 to zone 3, and so forth. Note that the trips are asymmetrical; that is, trips 

in one direction are different than trips in the opposite direction. To use the table, there were 15 

trips from zone 1 to zone 2, but only 7 trips from zone 2 to zone 1. 

The trips on the diagonal are intra-zonal trips, trips that originate and end in the same 

zone. Again, to use the example above, there were 37 trips that both originated and ended in 

zone 1, 53 trips that both originated and ended in zone 2. 

In such a model, constancy is maintained in that the number of trips originating from all 

origins zones must equal the number of trips ending in all destination zones. This is the 

fundamental balancing equation for a trip distribution. In equation form, it is expressed as: 

(28.1) 

where the origins, Oi, are summed over M origin zones while the destinations, Dj, are summed 

over N destination zones. To use the example in Figure 28.1, the total number of origins is equal 

to the total number of destinations, and is equal to 43,240. 

The balancing equation is implemented in a series of steps that include modeling the 

number of crimes originating in each zone, adding in trips originating from outside the study area 

(external trips), and statistically balancing the origins and destinations so that equation 28.1 

holds. This was done in the trip generation stage. But, it is essential that the step should have 

been completed for the trip distribution to be implemented. 

Observed and Predicted Distributions 

There are two trip distribution matrices that need to be distinguished. The first is the 

observed (or empirical) distribution. This is the actual number of trips that are observed 

traveling between each origin zone and each destination zone. In general, with crime data, such 

an empirical distribution would be obtained from an arrest record where the residence (or arrest) 

location of each offender is listed for each crime that the offender was charged with. In this 

case, the residence/arrest location would be considered the origin while the crime location would 

be considered the destination. 

In Chapter 26, it was mentioned that there is always uncertainty as to the true origin 

location of a crime incident, whether the offender actually traveled from the residence location to 

the crime location or even whether the offender was actually living at the residence location. 

28.3 
 

 

 

 

But absent any alternative evidence, a meaningful distribution can still be obtained by simply 

treating the residence location as an approximate origin. 

The observed distribution is calculated by simply enumerating the number of trips by 

each origin-destination combination. This is sometimes called a trip link (or trip pair). The 

second distribution, however, is a model of the trip distribution matrix. This is usually called 

the predicted distribution. In this case, a simple model is used to approximate the actual 

empirical distribution. The trips originating in each origin zone are allocated to destination zones 

usually on the basis of being directly proportional to attractions and inversely proportional to 

costs (or impedance). 

Thus, a model of the trip distribution is produced that approximates the actual, empirical 

distribution. There are a number of reasons why this would be useful - to be able to apply the 

model to a different data set from which it was calibrated, to use the model for evaluating a 

policy intervention, or to use the model for forecasting future crime trip distribution. But, 

whatever the reason, it has to be realized that the model is not the observed distribution. There 

will always be a difference between the observed distribution from which a model is constructed 

and the resulting predicted distribution of the model. It is useful to compare the observed and 

predicted model because this allows a test of the validity of the impedance function. But, 

rarely, if ever, will the predicted distribution be identical to the empirical distribution. 

Another way to think of this is that the actual distribution of crime trips is complex, 

representing a large number of different decisions on the part of offenders who do not 

necessarily use the same decision logic. The model, on the other hand, is a simple allocation on 

the basis of three or, sometimes, four variables. Almost by definition, it will be much simpler 

than the real distribution. Still, the simple model can often capture the most important 

characteristics of the actual distribution. Hence, modeling can be an extremely useful analytical 

exercise that allows other types of questions to be asked that are not possible with just the 

observed distribution. 

The Gravity Model 

A model that is usually used for trip distribution is that of the gravity function, an 

application of Newton=s fundamental law of attraction (Oppenheim, 1980; Field & MacGregor, 

1987; Ortuzar & Willumsen, 2001). Much of the discussion below is also repeated in Chapter 13 

on journey-to-crime modeling since there is a common theoretical basis. In the original 

Newtonian formulation, the attraction, F, between two bodies of respective masses M1 and M2, 

separated by a distance D, will be equal to 

28.4 
 

(28.2)

where g is a constant or scaling factor which ensures that the equation is balanced in terms of the 

measurement units (Oppenheim, 1980). As we all know, of course, g is the gravitational 

constant in the Newtonian formulation. The numerator of the function is the attraction term (or, 

alternatively, the attraction of M2 for M1) while the denominator of the equation, d2, indicates 

that the attraction between the two bodies falls off as a function of their squared distance. It is an 

impedance (or resistance) term. 

Social Applications of the Gravity Concept 

The gravity model has been the basis of many applications to human societies and has 

been applied to social interactions since the 19th century. Ravenstein (1895) and Andersson 

(1897) applied the concept to the analysis of migration by arguing that the tendency to migrate 

between regions is inversely proportional to the squared distance between the regions. Reilly=s 

>law of retail gravitation= (1929) applied the Newtonian gravity model directly and suggested that

retail travel between two centers would be proportional to the product of their populations and

inversely proportional to the square of the distance separating them:

(28.3)

where Iij is the interaction between centers I and j, Pi and Pj are the respective populations, dij is 

the distance between them raised to the second power and α is a balancing constant. In the 

model, the initial population, Pi, is called a production while the second population, Pj, is called 

an attraction. 

Stewart (1950) and Zipf (1949) applied the concept to a variety of phenomena (migration, 

freight traffic, information) using a simplified form of the gravity equation: 

(28.4)

where the terms are as in equation 28.3 but the exponent of distance is only 1. Given a 

particular pattern of interaction for any type of goods, service or human activity, an optimal 

location of facilities should be solvable. 

In the Stewart/Zipf framework, the two P=s were both population sizes. However, in 

modern use, it is not necessary for the productions and attractions to be identical units (e.g., Pi 

could be population while Pj could be employment). 

28.5 
.

ഊ

 

Trips as Interactions 

It should be obvious that this interaction equation can be applied to trips from one area 

(zone) to another. Changing the symbols slightly, the total volume of trips from a particular 

origin zone, i, to a single location, j, is directly proportional to the product of the productions at i 

and the attractions at j, and inversely proportion to the impedance (or cost) of travel between the 

two zones: 

(28.5) 

where Pi are the productions for zone I, Aj are the attractions zone j, α is a production constant, β

is an attraction constant, and dij is the impedance (cost) of travel between zone ii and zone j. 

Over time, the concept has been generalized and applied to many different types of travel 

behavior. For example, Huff (1963) applied the concept to retail trade between zones in an 

urban area using the general form of: 

(28.6)

where Aij is the number of purchases in location j by residents of location݅, Sj is the 

attractiveness of zone ݆ (e.g., square footage of retail space), dij is the distance between zones ݅

and ݆, α is a constant, λ is the exponent of Sj, and ρ is the exponent of distance (Bossard, 1993). 

Dij

-ρ

 is sometimes called an inverse distance function. This differs from the traditional gravity 

function by allowing the exponents of the production from location ݅, the attraction from 

location ݆, and the distance between zones, , to vary. 

Equation 28.6 is a single constraint model in that only the attractiveness of a commercial 

zone is constrained, that is the sum of all attractions for j must equal the total attraction in the 

region. Again, it can be generalized to all zones by, first, estimating the total trips generated 

from one zone, i, to another zone, j, 

(28.7)

where Tij is the interaction between two locations (or zones), Pi is productions of trips from zone 

݅, Aj is the attractiveness of zone ݆, dij is the distance between zones ݅ andj݅, λ is the exponent 

of Pi, τ is the exponent of Aj, ρ is the exponent of distance, and α is a constant. 

28.6 
ഊ

Second, the total number of trips generated by a single location, ݅, to all destinations is 

obtained by summing over all destination locations, ݆: 

(28.8)

and generalizing this to all zones, we get: 

(28.9)

where α is a constant for the productions, Pi 

λ

 and β is a constant for the attractions, Aj 

τ

. This 

type of function is called a double constraint model because the equation has to be constrained 

by the number of units in both the origin and destination locations; that is, the sum of Pi over all 

locations must be equal to the total number of productions while the sum of Aj over all locations 

must be equal to the total number of attractions. Adjustments are usually required to have the 

sum of individual productions and attractions equal the totals (usually estimated independently). 

Negative Exponential Distance Function 

One of the problems with the traditional gravity formulation is in the measurement of 

travel impedance (or cost). For locations separated by sizeable distances in space, the gravity 

formulation can work properly. However, as the distance between locations decreases, the 

denominator approaches infinity. Consequently, an alternative expression for the interaction 

uses the negative exponential function (Hägerstrand, 1957; Wilson, 1970). 

 (28.10) 

where Tji is the attraction of location j for residents of location I, Aj is the attractiveness of 

location j, dij is the distance between locations i and j, β is the exponent of Aj, α is a coefficient 

of dij (and, also, an exponent) and e is the base of the natural logarithm (i.e., 2.7183...). Derived 

from principles of entropy maximization, the latter part of the equation is a negative exponential 

function that has a maximum value of 1 (i.e., e-0 = 1; Wilson, 1970). This has the advantage of 

making the equation more stable for interactions between locations that are close together. For 

example, Cliff and Haggett (1988) used a negative exponential gravity-type model to describe 

the diffusion of measles into the United States from Canada and Mexico. It has also been 

argued that the negative exponential function generally gives a better fit to urban travel patterns, 

particularly those by automobile (Bossard, 1993; Foot, 1981). Figure 28.2 shows a typical 

negative exponential function and one recommended for home-based work trips by the 

Transportation Research Board as a default value (NCHRP, 1995). 

28.7 
Figure 28.2: 
 

Note that by moving the distance term to the numerator, strictly speaking it no longer is 

an impedance term since impedance increases with distance. Rather it is a discount factor (or 

disincentive); the interaction is discounted with distance. Nevertheless, the term >impedance= is 

still used primarily for historical reasons. 

There are other distance functions, as well. Chapter 13 explored some of these. For 

example, we are finding that, for crime trips, the lognormal function may produce better results 

than the negative exponential primarily because many crimes are committed at short-to-moderate 

distances. Chapter 17 discusses the MCMC Poisson-lognormal regression model which is 

useful with a low mean (e.g., very short distance traveled) and small sample sizes. It is possible 

that the lognormal function is more useful for very localized crime trips than the negative 

exponential. 

Travel Impedance 

One of the biggest advances in the negative exponential model of equation 28.10 has 

been to increase the flexibility of the denominator. In the traditional gravity model, the 

denominator is distance. This is a proxy for a discount factor (or cost); the farther two zones 

are from each other, the less likely there is to be interaction between them, all other things being 

equal. Conversely, the closer two zones are, the more likely there is to be interaction, all other 

things being equal. 

Distance v. Travel Time 

It has been realized, however, that distance is only an approximation for impedance. In 

real travel, travel time is a much better indicator of the cost of travel in that time varies by the 

time of day, day of week, direction of travel, type of road used, and other factors. For example, 

travel across town in any metropolitan area is generally a lot easier at 3 in the morning, say, than 

at the peak afternoon rush period. The difference in travel time can vary as much as 

two-to-three times between peak and off-peak hours. Using only distance, however, these 

variations are never picked up because the distance between locations is invariant. 

This realization has led to the concept of travel impedance which, in turn, has led to the 

concept of travel cost. >Impedance= is the resistance (or discounting) in travel between two 

zones. Using travel time as an impedance variable, the longer it takes to travel between two 

zones, the less likely there will be interaction between them, all other things being equal. 

Conversely, a shorter travel time leads to greater interaction between zones, again, all other 

things being equal. Similarly, a travel route that shortens travel time will generally be selected 

over one that takes longer even if the first one is longer in distance. For example, it has been 

28.9 
 

 

 

 

 

documented that people will change work locations that are farther from their home if traveling 

to the new work location takes less time (e.g., traveling in the >opposite= direction to the bulk of 

traffic; Wachs, Taylor, Levine & Ong, 1993). 

If travel time is a critical component of travel, why then don=t offenders commit more 

crimes at, say, 3 in the morning than at the peak afternoon travel times? Since the impedance is 

less at 3 in the morning than at, say, 5 in the afternoon, would not the model predict more trips 

occurring in the early morning hours than actually occur in those hours? The answer has to do 

with the numerator of the gravity equation and not just the denominator. At 3 in the morning, 

yes, it is easier to travel between two locations, at least by personal automobile (not by bus or 

train when those services are less frequent). But the attraction side of the equation is also less 

strong at 3 in the morning. For a street robber, there are fewer potential >victims= on the street at 

3 in the morning than in the late afternoon. For a residential burglar, there is more likely to be 

someone at home at night than in the afternoon. The travel time component is only one 

dimension of the likelihood of travel between two locations. The distribution of opportunities 

and other costs can alter the likelihood considerably. 

Nevertheless, shifting to an impedance function allows a travel model to better replicate 

actual travel conditions. Most travel demand models used by transportation planners use an 

impedance function, rather than a distance function.1 Distance would only be meaningful if the 

standards were invariant with respect to time (e.g., a model calculated over an entire year, 24 

hours a day). As will be demonstrated in Chapter 30 on network assignment, a travel time 

calculation leads to a very different network allocation than a distance calculation. For 

example, if distance is used as an impedance variable, then the shortest trips will rarely take the 

freeways because travel to and from a freeway usually makes a trip longer than a direct route 

between an origin and a destination. But as most people understand, taking a freeway to travel 

a sizeable distance is usually a lot quicker than traversing an urban arterial system with many 

traffic lights, stop signs, crossing pedestrians, cross traffic from parking lots and shopping malls, 

and other urban >obstacles=. Today, the use of distance in travel demand modeling has virtually 

been dropped by most transportation planners. 

Distance can be used as a rough approximation for impedance, but is rarely a good predictor of actual travel 

behavior. For example, in the mode split mode that will be discussed in Chapter 29, the distance between 

a location and the nearest bus or rail route can be used to quickly select trip pairs that might travel by 

transit. However, the actual prediction must be based on a network calculation of travel time or travel 

cost in traversing the system. 

28.10 

1 
 

 

Travel Cost 

An even better concept of impedance is that of travel cost (sometimes called generalized 

cost) which incorporates real and perceived costs of travel between two locations. Travel time 

is one component of travel cost in that there is an implicit cost to the trip (e.g., an hourly wage or 

price assigned). In this case, two different individuals will value the time for a trip differently 

depending on their hourly >wage=. For example, for an individual who prices his/her travel at 

$100 an hour, the per minute cost is $1.67. For another individual who prices his/her travel at 

$12 an hour, the per minute cost is 204. These relative prices assigned to travel will 

substantially affect individual choices in travel modes and routes. For instance, these two 

hypothetical individuals will probably use a different travel mode in getting from an airport to a 

hotel on a trip; the former will probably take a taxi whereas the latter will probably take a bus or 

train (if available). 

But cost involves other dimensions that need to be considered. There are real operating 

costs in the use of a vehicle - fuel, oil, maintenance, and insurance. Many travel studies have 

suggested that drivers incorporate these costs as part of their implicit hourly travel price (Ortuzar 

& Willumsen, 2001; 323-327). But, there are also real, >out-of-pocket= costs such as parking or 

toll costs. Parking is particularly a major expense for intra-urban driving behavior. In many 

built-up business districts, parking costs can be considerable, for example as much as $90 a day 

in major metropolitan centers. In most busy commercial areas, there are some parking costs, if 

only at on-street parking meters. Thus, a travel cost model needs to incorporate these real costs 

as the out-of-pocket costs may overwhelm the implicit value of the travel time. For example, 

an offender who lives 10 minutes from the downtown area by car would probably not drive into 

the downtown to commit a robbery since that individual will have to bear the price of parking. 

There are lots of well known stories that circulate about bank robbers who are caught because 

they incur parking tickets while committing their crime. How often this has occurred is not 

known from any study that we are aware, but the story line is cognizant of the actual costs of 

travel that must be incurred as part of travel. 

In addition to real costs are perceived costs. For transit users, particularly, these 

perceived costs affect the ease and time of travel. One of the standard questions in travel 

surveys for transit users is the time it takes to walk from their home to the nearest bus stop or 

intra-urban rail system (if available) and from the last transit stop to their final destination; the 

longer it takes to access the transit system, the less likely an individual will use it. Similarly, 

transfers between buses or trains decrease the likelihood of travel by that mode, almost in 

proportion to the number of transfers. The reason is the difficulty in getting out of one bus or 

train and into another. But, the time between trains adds an implicit travel cost; the longer the 

wait between buses, the less likely that mode will be used by travelers. In short, ease of access 

and convenience are positive incentives in using a mode or a route while difficulty in accessing 

28.11 
 

it, lack of convenience, and even fear of being vulnerable to crime will decrease the likelihood of 

using that mode or route.2 

If the concept is expanded to that of an offender, there are other perceived costs that 

might affect travel. One obvious one is the likelihood of being caught. It may be easy for one 

offender to travel to an upscale, high visibility shopping area, but if there are many police and 

security guards around, the individual is more likely to be caught. Hence, that likelihood (or, 

more accurately, an assumption that the offender makes about that likelihood since he/she does 

not really know what is the real likelihood) is liable to affect the choice of a destination and, 

possibly, even a route. 

Another perceived cost is the likelihood of retaliation from other gangs. Bernasco and 

Block (2009) showed that robbers in Chicago will usually not commit robberies in the territories 

of rival gangs even if those areas are closer to where the robbers live. 

Another perceptual component affecting a likely choice is the reliability of the 

transportation mode. Many offenders are poor and do not have expensive, well maintained 

vehicles. If the vehicle is not capable of higher speeds or is even likely to break down while an 

offence is being committed that vehicle is not liable to be used in making a trip or the choice of 

destination may be altered. It is well known that many offenders steal vehicles for use in a 

crime. Fears about not being identified are clearly a major factor in those decisions, but the 

reliability of their own vehicles may also be a factor. 

Thus, in short, a more realistic model of the incentives or disincentives to make a trip 

between two locations requires a complex function that weights a number of factors affecting the 

cost of travel - the travel time, implicit operating costs, out-of-pocket costs, and perceived costs. 

Many travel demand models used by Metropolitan Planning Organizations use such a function, 

usually under the label of >generalized cost=. The more complex the pricing structure for 

parking and travel within a metropolitan area, the more likely a generalized cost function will 

provide a realistic model of trip distribution. 

Travel Utility 

The final concept that is introduced in defining impedance is that of travel utility. 

>Utility= is an individual concept, rather than a zonal one. Also, it is the flip side of cost (i.e.,

higher cost is associated with less utility). A generalized cost function calculates the objective

Most of the research on factors affecting use of transit were conducted in the 1960s and 1970s. These 

assumptions are more or less assumed by travel demand modelers, rather than documented per se. See 

Schnell, Smith, Dimsdale, & Thrasher, 1973; Roemer & Sinha, 1974; WASHCOG, 1974; Carnegie-Mellon 

University, 1975; Johnson, 1978; Levine & Wachs, 1986 for some examples. 

28.12 

2 
 

and average perceived costs of travel between two zones. But the utility of travel for an 

individual is a function of both those real costs and a number of individual characteristics that 

affect the value placed on that travel. Thus, two individuals living in the same zone (perhaps 

even living next door to each other) who travel to the same destination location may >price= their 

trip very differently. Aside from income differences which effect the average hourly >wage=, 

there may be differences due to convenience, attractiveness, or a host of other factors. Other 

factors are more idiosyncratic. For example, a trip by a gang member into another gang=s >turf=

might be expected to increase the perceived costs to the individual of traveling to that location, 

above and beyond any objective cost factors. Alternatively, a trip to a location where a close 

friend or relative is located might decrease the perceived cost of travel to that zone. In other 

words, there are both objective costs as well as subjective costs in travel between two zones. 

The concept of utility may be less useful for crime analysis than for general travel 

behavior. For one thing, since the concept is individual, it can only be identified by individual 

surveys (Domencich & McFadden, 1975). For crime analysis, this makes it virtually 

impossible to use since it is very difficult to interview offenders, at least in the United States. 

But, for completeness sake, we need to understand that the likelihood or disincentive to travel 

between two locations is a function of individual characteristics as well as objective travel cost 

components. A mixture of aggregate and individual variables can be used to produce a 

synthetic utility model for modeling locations where individuals commit crimes (Block & 

Bernasco, 2009). 

The modeling of individual utility can be done with either a multinomial logit model for a 

limited number of discrete choices or a more general conditional logit model for many choices. 

Chapter 21 discusses these models while Chapter 22 presents the CrimeStat discrete choice 

module. At this point, it is impractical to utilize either model for predicting trip distribution 

links since the number of origin-destination pairs would require an enormous data set. So, we 

are left for the time being with the gravity function being the only practical approach to trip 

distribution. 

Impedance Function 

For a zonal type model, we can think of the gravity function as a generalized impedance 

function. For travel between any one zone and all other zones, we have: 

(28.11) 

where the number of trips from zone ݅ to all other zones is a function of the productions at zone 

݅ and the relative attraction of any one zone, ݆, to the impedance of that zone for ݅, Iij. The 

28.13 
ഊ

impedance function, Iij , is some declining function of cost for travel between two zones. It 

does not have to be any particular form and can be (and usually is) a non-linear function. The 

costs can be in terms of distance, travel time, speed (which is converted into travel time) or 

general costs. The greater the separation between two zones (i.e., the higher the impedance), 

the less likely there will be a trip between them. Generalizing this to all zones, we get: 

(28.12)

where Pi is the production capacity of zone ݅, Aj is the attraction of zone ݆, Iij, is a generalized 

function that discounts the interaction with increasing separation in distance, time, or cost, α and 

β are constants that are applied to the productions and attractions respectively, and λ and τ are 

>fine tuning= exponents of the productions and attractions respectively. This is the gravity

function that we will estimate in the CrimeStat trip distribution model.

Alternative Model: Intervening Opportunities 

There are alternative allocations procedures to the gravity model. One well known one 

is that of intervening opportunities. Stouffer (1940) modified the simple gravity function by 

arguing that the attraction between two locations was a function not only of the characteristics of 

the relative attractions of two locations, but of intervening opportunities between the locations. 

His hypothesis A..assumes that there is no necessary relationship between mobility and distance... 

that the number of persons going a given distance is directly proportional to the number of 

opportunities at that distance and inversely proportional to the number of intervening 

opportunities@ (Stouffer, 1940, p. 846). This model was used in the 1940s to explain interstate 

and inter-county migration (Isard, 1979; Isbell, 1944; Bright & Thomas, 1941). Using the 

gravity type formulation, this can be written as: 

(28.13)

where Aji is the attraction of location j by residents of location ݅, Sj is the attractiveness of zone

݆, Sk is the attractiveness of all other locations, 

, that are intermediate in distance between 

locations ݅ and ݆ (with there being O such locations), dij is the distance between zones ݅ and

݆, β is the exponent of Sj, ξ is the exponent of Sk, and λ is the exponent of distance. While the 

intervening opportunities are implicit in equation 28.7 in the exponents, β and λ, and coefficient, 

α, equation 28.13 makes the intervening opportunities explicit. The importance of the concept is 

that travel between two locations becomes a complex function of the spatial environment of 

nearby areas and not just of the two locations. 

28.14 
 

 

 

 28.15 

In practice, in spite of its more intuitive theoretical model, the intervening opportunities 

model does not improve prediction much beyond that of the gravity model since it includes the 

attractions associated with the destination zones. Also, it is a more difficult model to estimate 

since the attractions of all other zones must be considered for each zone pair (origin-destination 

combination). Consequently, it is rarely used in actual practice (Ortuzar & Willumsen, 2001). 

Another alternative method was conducted by Porojan (2000) in applying the gravity 

model to international trade flow. He added a spatial autocorrelation component in addition to 

impedance and obtained a slightly better fit than the pure gravity function. However, whether 

this approach would improve the fitting of intra-regional crime travel patterns is still unknown. 

Nevertheless, this and other approaches might improve the predictability of a gravity function for 

intra-urban crime travel. 

Method of Estimation 

The CrimeStat trip distribution model implements equation 28.12. The specific details are 

discussed below, but the model is iterative. The steps are as follows: 

1. Depending on whether a singly constrained or doubly constrained model is to be

estimated, it starts with aN initial guess of the values for α or β (or both for a

doubly constrained model). Table 28.1 illustrates the three models.

Table 28.1: 

Three Methods of Constraining the Gravity Model 

Single constraint 

Constrain origins: 

Constrain destinations: 

(28.14) 

(28.15)

Double constraint 

Constrain both origins and destinations: 

(28.16) 
2. The routine proceeds to estimate the value for each cell in the origin-destination

matrix (see Figure 28.1 above) using the existing estimates for α and β.

3. The routine then sums the rows and columns in the matrix. Then, depending on

whether a single- or double-constraint model is to be estimated and, if a

single-constraint, whether origins or destinations are to be held constant, it then

calculates the ratio of the summed values (row totals or column totals or both) to

the initial row or column sum. The inverse of that ratio is the subsequent estimate

for α or β (or both for a double-constrained model).

4. The routine repeats steps 2 and 3 until the changes from one iteration to the next

are very small.

5. The last estimate of α or β (or both for a double-constrained model) is taken as the

final values of these parameters.

6. Once the parameters have been estimated, the model can be applied to the

calibration data set or to another data set. Note that the parameters are row or

column specific (or both). That is, in the >constrain origins= model, there is a

separate coefficient for each row. In the >constrain destinations= model, there is a

separate coefficient for each column. In the >constrain both origins and

destinations=, there is a separate coefficient for each cell (row-column

combination).

A comparison can be made between the observed distribution and the predicted 

(modeled) distribution. Because most origin-destination matrices are very large, the vast 

majority of cells will have zero in them. Thus, a chi-square test would be inappropriate. 

Instead, a comparison of the trip length distribution is made using two different statistics - a 

coincidence ratio and the Komologorov-Smirnov Two-sample statistic. Details are provided 

below. 

CrimeStat IV Trip Distribution Module 

Next, we examine the actual tools that are available in the CrimeStat trip distribution 

module. The tools are illustrated with examples from Baltimore County. The CrimeStat trip 

distribution module includes one setup screen and five routines that implement the model: 

1. Calculate observed origin-destination distribution. If there is a file available

with the coordinates for individual origins and destinations (e.g., an arrest record),

this routine will calculate the empirical trip distribution matrix;

28.16 
 

2. Calibrate impedance function. If there is a file available with the coordinates

for individual origins and destinations, this routine will calibrate an empirical

impedance function.

3. Setup origin-destination model. This screen allows the user to define the

parameters of a trip distribution (origin-destination) model with either a

mathematical or empirical impedance function.

4. Calibrate origin-destination model. This routine calibrates the parameters of

the trip distribution model (equation 28.12) using the parameters defined on the

setup page.

5. Apply predicted origin-destination model. This routine applies the estimated

parameters to a data set. The data set can be either the calibration file or another

file.

6. Compare observed and predicted origin-destination trip lengths. This

routine compares the trip lengths from the observed (empirical) trip distribution

with that predicted by the model. Comparisons are made graphically by a

coincidence ratio, the Komologorov-Smirnov Two-Sample test, and a Chi square

test on the most frequent trip links.

Each of these routines is described in detail below. Figure 28.3 shows a screen shot of the 

trip distribution module. 

Describe Origin-Destination Trips 

An empirical description of the actual trip distribution matrix can be made if there is a 

data set that includes individual origin and destination locations. The user defines the origin 

location and the destination location for each record and a set of zones from which to compare 

the individual origins and destinations. The routine matches up each origin location with the 

nearest zone, each destination location with the nearest zone, and calculates the number of trips 

from each origin zone to each destination zone. This is an observed distribution of trips by zone. 

The steps in running the model are as follows: 

1. Calculate observed origin-destination trips. Check if an empirical

origin-destination trip distribution is to be calculated.

28.17 
Figure 28.3: 

Trip Distribution Module 
 

 

 

2. Origin file. The origin file is a list of origin zones with a single point

representing the zone (e.g., the centroid). It must be input as either the primary

or secondary file. Specify whether the data file is the primary or secondary file.

A. Origin ID. Specify the origin ID variable in the data file (e.g.,

CensusTract, Block, TAZ).

3. Destination file. The destination file is a list of destination zones with a single

point representing the zone (e.g., the centroid). It must be input as either the

primary or secondary file. Specify whether the data file is the primary or

secondary file.Specify the destination ID variable in the data file (e.g.,

CensusTract, Block, TAZ).

A. Note: all destination ID=s should be in the origin zone file and must have

the same names and both should be character (string) variables.

4. Select data file. The data set must have individual origin and destination

locations. Each record must have the X/Y coordinates of an origin location and

the X/Y coordinates of a destination location. For example, an arrest file might

list individual incidents with each incident having a crime location (the

destination) and a residence or arrest location (the origin).

A. Select the file that has the X and Y coordinates for the origin and

destination locations. CrimeStat can read ASCII, dbase '.dbf', ArcGIS

'.shp' and MapInfo 'dat' files.

B. Select the tab and specify the type of file to be selected. Use the browse

button to search for the file. If the file type is ASCII, select the type of

data separator (comma, semicolon, space, tab) and the number of columns.

C. Variables. Define the file which contains the X and Y coordinates for both

the origin (residence) and destination (crime) locations.

D. Column. Select the variables for the X and Y coordinates respectively for

both the origin and destination locations (e.g., Lon, Lat, HomeX, HomeY,

IncidentX, IncidentY.) Both locations must be defined for the routine to

work.

E. Missing values. Identify whether there are any missing values for these

four fields (X and Y coordinates for both origin and destination locations).

28.19 
By default, CrimeStat will ignore records with blank values in any of the 

eligible fields or records with non-numeric values (e.g.,alphanumeric 

characters, #, *). Blanks will always be excluded unless the user selects 

<none>. There are 8 possible options: 

a. <blank> fields are automatically excluded. This is the default 

b. <none> indicates that no records will be excluded. If there is a 

blank field, CrimeStat will treat it as a 0 

c. 0 is excluded 

d. B1 is excluded 

e. 0 and B1 indicates that both 0 and -1 will be excluded 

f. 0, -1 and 9999 indicates that all three values (0, -1, 9999) will be 

excluded. 

g. Any other numerical value can be treated as a missing value by 

typing it (e.g., 99)Multiple numerical values can be treated as 

missing values by typing them, separating each by commas (e.g., 

0, -1, 99, 9999, -99). 

F. Type of coordinate system and data units. The coordinate system and 

data units are listed for information. If the coordinates are in longitudes 

and latitudes, then a spherical system is being used and data units will 

automatically be decimal degrees. If the coordinate system is projected 

(e.g., State Plane, Universal Transverse Mercator B UTM), then data units 

could be either in feet (e.g., State Plane) or meters (e.g., UTM.). 

5. Table output. The full origin-destination matrix is output as a table to the screen 

including summary file information and: 

a. The origin zone (ORIGIN) 

b. The destination zone (DEST)=

c. The number of observed trips (FREQ) 

6. Save observed origin-destination trips. If specified, the full origin-destination 

matrix output is saved as a >dbf= file named by the user. The file output includes: 

a. The origin zone (ORIGIN) 

b. The destination zone (DEST) 

c. The X coordinate for the origin zone (ORIGINX) 

d. The Y coordinate for the origin zone (ORIGINY) 

e. The X coordinate for the destination zone (DESTX) 

28.20 
 

 

 

f. The Y coordinate for the destination zone (DESTY)

g. The number of trips (FREQ)

Note: each record is a unique origin-destination combination. There are M x N 

records where M is the number of origin zones (including the external zone) 

and N is the number of destination zones. 

7. Save links. The top observed origin-destination trip links can be saved as

separate line objects for use in a GIS. Specify the output file format (ArcGIS

'.shp', MapInfo '.mif' or Atlas*GIS '.bna') and the file name.

8. Save top links. Because the output file is very large (number of origin zones x

number of destination zones), the user can select a sub-set of zone combinations

with the most observed trips. Indicating the top K links will narrow the number

down to the most important ones. The default is the top 100 origin-destination

combinations. Each output object is a line from the origin zone to the

destination zone with an ODT prefix. The prefix is placed before the output file

name. The line graphical output for each object includes:

a. An ID number from 1 to K, where K is the number of links output

(ID)

b. The feature prefix (ODT)

c. The origin zone (ORIGIN)

d. The destination zone (DEST)

e. The X coordinate for the origin zone (ORIGINX)

f. The Y coordinate for the origin zone (ORIGINY)

g. The X coordinate for the destination zone (DESTX)

h. The Y coordinate for the destination zone (DESTY)

i. The number of observed trips for that combination (FREQ)

j. The distance between the origin zone and the destination zone.

9. Save points. Intra-zonal trips (trips in which the origin and destination are the

same zone) can be output as separate point objects as an ArcGIS '.shp', MapInfo

'.mif' or Atlas*GIS '.bna' file. Again, the top K points are output (default=100).

Each output object is a point representing an intra-zonal trip with an

ODTPOINTS prefix. The prefix is placed before the output file name. The

point graphical output for each object includes:

28.21 
a. An ID number from 1 to K, where K is the number of links output

(ID)

b. The feature prefix (POINTSODT)

c. The origin zone (ORIGIN)

d. The destination zone (DEST)

e. The X coordinate for the origin zone (ORIGINX)

f. The Y coordinate for the origin zone (ORIGINY)

g. The X coordinate for the destination zone (DESTX)

h. The Y coordinate for the destination zone (DESTY)

i. The number of observed trips for that combination (FREQ)

Example of Observed Trip Distribution from Baltimore County 

Figure 28.4 shows the output of the top 1000 links for the observed trip distribution from 

a sample of 41,974 records for incidents committed between 1993 and 1997. The zonal model 

used was that of traffic analysis zones (TAZ). These were discussed in Chapter 26. Because 

there are a large number of links (532 origin zones by 325 destination zones), the top 1000 were 

taken. These accounted for 19,615 crime trips (or 46.7% of all trips). A larger number of 

links could have been selected, but the map would have become more cluttered. Of the 19,615 

trips that are displayed in the map, 7,913 or 40.3% are intra-zonal trips. These were output by 

the routine as points and have been displayed as circles with the size proportional to the number 

of trips. The remaining 11,702 trip links were output by the routine as lines and are displayed 

with the thickness and strength of color of the line being proportional to the number of trips. 

There are several characteristics of the trip pattern that should be noted. First, the 

intra-zonal trips tend to concentrate on the eastern part of Baltimore County. This is an area 

that is relatively poor with a high number of public housing projects. This suggests that there 

are a lot of intra-community crimes being committed in these locations. Second, the 

zone-to-zone pattern, on the other hand, tends to concentrate at five different locations relatively 

close to border with the City of Baltimore. These five locations are all major shopping malls. 

Third, the origins for those trips to the shopping mall tend to come from within the City of 

Baltimore. Fourth, in general, the locations with high intra-zonal trips do not have a large 

number of zone-to-zone trips. However, there is one exception in the southwest corner of the 

county. 

In other words, the observed distribution of crime trips is complex, but with several 

patterns being shown. A lot of crime trips occur over very short distances. But there is also a 

convergence of many crime trips on major shopping malls in the County. 

28.22 
Figure 28.4: 
 

 

Calibrate Impedance Function 

This routine allows the calibration of an approximate travel impedance function based on 

actual trip distributions. It is used to describe the travel impedance in distance only of an actual 

sample (the calibration sample). Unlike the remaining routines in this section, the ACalibrate 

impedance function cannot use travel time, or cost. A file is input which has a set of incidents 

(records) that include both the X and Y coordinates for the location of the offender's residence 

(origin) and the X and Y coordinates for the location of the incident that the offender committed 

(destination.) 

The routine estimates a travel distance function using a one-dimensional kernel density 

method. See the details in Chapter 13. Essentially, for each record, the separation between 

the origin location and the destination location is calculated and is represented on a distance 

scale. The maximum impedance is calculated and divided into a number of intervals; the 

default is 100 equal sized intervals, but the user can modify this. For each impedance point 

calculated, a one-dimensional kernel is overlaid. For each interval, the values of all kernels are 

summed to produce a smooth function of travel impedance. The results are saved to a file that 

can be used for the origin-destination model. 

Note, however, that this is an empirical distribution and represents the combination of 

origins, destinations, and costs. It is not necessarily a good description of the impedance (cost) 

function by itself. Some of the mathematical functions produce a better fit than the empirical 

impedance function. 

The steps in calculating an empirical impedance function are as follows: 

1. Select data file for calibration. Select the file that has the X and Y coordinates

for the origin and destination locations. CrimeStat can read ASCII, dbase '.dbf',

ArcGIS '.shp' and MapInfo 'dat' files. Select the tab and select the type of file

to be selected. Use the browse button to search for the file. If the file type is

ASCII, select the type of data separator (comma, semicolon, space, tab) and the

number of columns.

A. Variables. Define the file which contains the X and Y coordinates for

both the origin (residence) and destination (crime) locations

B. Columns. Select the variables for the X and Y coordinates respectively for

both the origin and destination locations (e.g., Lon, Lat, HomeX, HomeY,

IncidentX, IncidentY.) Both locations must be defined for the routine to

work.

28.24 
C. Missing values. Identify whether there are any missing values for these

four fields (X and Y coordinates for both origin and destination locations).

By default, CrimeStat will ignore records with blank values in any of the

eligible fields or records with non-numeric values (e.g.,alphanumeric

characters, #, *). Blanks will always be excluded unless the user selects

<none>. There are 8 possible options:

a. <blank> fields are automatically excluded. This is the default

b. <none> indicates that no records will be excluded. If there is a

blank field, CrimeStat will treat it as a 0

c. 0 is excluded

d. B1 is excluded

e. 0 and B1 indicates that both 0 and -1 will be excluded 

f. 0, -1 and 9999 indicates that all three values (0, -1, 9999) will be

excluded.

g. Any other numerical value can be treated as a missing value by

typing it (e.g., 99)Multiple numerical values can be treated as

missing values by typing them, separating each by commas (e.g.,

0, -1, 99, 9999, -99).

D. Type of coordinate system and data units. Select the type of coordinate

system. If the coordinates are in longitudes and latitudes, then a

spherical system is being used and data units will automatically be

decimal degrees. If the coordinate system is projected (e.g., State Plane,

Universal Transverse Mercator B UTM), then data units could be either in

feet (e.g., State Plane) or meters (e.g., UTM.) Directional coordinates are

not allowed for this routine.

2. Select Kernel Parameters. There are five parameters that must be defined.

A. Method of interpolation. There are fives types of kernel distributions

that can be used to estimate point density:

a. The normal kernel overlays a three-dimensional normal

distribution over each point that then extends over the area defined

by the reference file. This is the default kernel function.

b. The uniform kernel overlays a uniform function (disk) over each

point that only extends for a limited distance.

c. The quartic kernel overlays a quartic function (inverse sphere)

over each point that only extends for a limited distance.

28.25 
 

d. The triangular kernel overlays a three-dimensional triangle (cone)

over each point that only extends for a limited distance.

e. The negative exponential kernel overlays a three dimensional

negative exponential function ('salt shaker') over each point that

only extends for a limited distance

B. The methods produce similar results though the normal is generally

smoother for any given bandwidth.

3. Choice of bandwidth. The kernels are applied to a limited search distance,

called 'bandwidth'. For the normal kernel, bandwidth is the standard deviation of

the normal distribution. For the uniform, quartic, triangular and negative

exponential kernels, bandwidth is the radius of a circle defined by the surface.

For all types, larger bandwidth will produce smoother density estimates and both

adaptive and fixed bandwidth intervals can be selected.

A. Fixed bandwidth. A fixed bandwidth distance is a fixed interval for each

point. The user must define the interval, the interval size, and the

distance units by which it is calculated (miles, nautical miles, feet,

kilometers, meters.) The default bandwidth setting is fixed with intervals

of 0.25 miles each. The interval size can be changed.

B. Adaptive bandwidth. An adaptive bandwidth distance is identified by

the minimum number of other points found within a symmetrical band

drawn around a single point. A symmetrical band is placed over each

distance point, in turn, and the width is increased until the minimum

sample size is reached. Thus, each point has a different bandwidth size.

The user can modify the minimum sample size. The default for the

adaptive bandwidth is 100 points.

4. Specify Interpolation Bins. The interpolation bins are defined in one of two

ways:

A. By the number of bins. The maximum distance calculated is divided by

the number of specified bins. This is the default with 100 bins. The user

can change the number of bins.

B. By the distance between bins. The user can specify a bin width in miles,

nautical miles, feet, kilometers, and meters.

28.26 
 

 

5. Output (Areal) Units. Specify the density units as points per mile, nautical

mile, foot, kilometer, or meter. The default is points per mile.

6. Calculate Densities or Probabilities. The density estimate for each cell can be

calculated in one of three ways:

A. Absolute densities. This is the number of points per grid cell and is scaled

so that the sum of all grid cells equals the sample size.

B. Relative densities. For each grid cell, this is the absolute density divided

by the grid cell area and is expressed in the output units (e.g., points per

square mile)

C. Probabilities. This is the proportion of all incidents that occur in the

grid cell. The sum of all grid cells equals a probability of 1. Unlike the

Jtc calibration routine, this is the default. In most cases, a user would

want a proportional (probability) distribution as the relative differences in

impedance for different costs are what is of interest.

Select whether absolute densities, relative densities, or probabilities are to

be output for each cell. The default is probabilities.

7. Select Output File. The output must be saved to a file. CrimeStat can save the

calibration output to either a dbase 'dbf' or ASCII text 'txt' file.

8. Calibrate! Click on 'Calibrate!' to run the routine. The output is saved to the

specified file upon clicking on 'Close'.

9. Graphing the travel impedance function. Click on 'View graph' to see the

travel impedance function. The screen view can be printed by clicking on 'Print'.

For a better quality graph, however, the output should be imported into a graphics

or spreadsheet program.

Example of Empirical Impedance from Baltimore County 

An example of an empirical impedance function from Baltimore County is seen in Figure 

28.5. This was derived from the 41,974 incidents in which both the crime location and the 

offender=s origin location were known. As seen, the function looks similar to a negative 

exponential function. But there is a little >hitch= around 3 miles where the travel likelihood 

28.27 
Figure 28.5: 
 

increases, rather than decrease. This could possibly be due to the City of Baltimore border 

which abuts much of the southern part of the County. 

Whatever the reason, the empirical impedance function can be used as a proxy for travel 

>cost= by offenders. As we shall see, however, it may not produce as good a fit in the gravity

model as some of the mathematical functions, particularly the lognormal. The reason is that it

is a behavioral description. Consequently, the pattern reflects both the existence of crime

opportunities (attractions) as well as costs. While an empirical description is useful for

guessing where a serial offender might live, for a trip distribution model it apparently does not

cleanly estimate the real costs to an offender. Nevertheless, it is a tool that can be used.

Setup of Origin-Destination Model 

The page is for the setup of the origin-destination model. All the relevant files, models 

and exponents are input on the page and it allows the trip distribution model to be calibrated and 

allocated. Figure 28.6 shows the setup screen. There are a number of parameters that have to be 

defined: 

1. Predicted origin file. The predicted origin file is a file that lists the origin zones

with a single point representing the zone (e.g., the centroid) and also includes the

predicted number of crimes by origin zone. The file must be input as either the

primary or secondary file. Specify whether the data file is the primary or

secondary file.

A. Origin variable. Specify the name of the variable for the predicted

origins (e.g., PREDICTED, ADJORIGINS).

B. Origin ID. Specify the origin ID variable in the data file (e.g.,

CensusTract, Block, TAZ).

2. Predicted destination file. The predicted destination file is a list of destination

zones with a single point representing the zone (e.g., the centroid) and also

includes the predicted number of crimes by destination zone. It must be input as

either the primary or secondary file. Specify whether the data file is the primary

or secondary file.

A. Destination variable. Specify the name of the variable for the predicted

destination (e.g., PREDICTED, ADJDEST).

28.29 
Figure 28.6: 

Trip Distribution Model Setup 
Note: with a 32 bit operating system (e.g., Windows XP, 32 bit Windows 7), there is maximum 

allowable of 4 Gb. If M is the number of rows and N is the number of columns, then the total

number of grid cells (M x N) cannot be greater than where RAM is the available RAM. 

With a 64 bit operating system, on the other hand, 32 Gb are addressable. 

B. Destination ID. Specify the destination ID variable in the data file (e.g.,

CensusTract, Block, TAZ).

3. Exponents. The exponents are power terms for the predicted origins and

destinations. They indicate the relative strength of those variables. For

example, compared to an exponent of 1.0 (the default), an exponent greater than

1.0 will strengthen that variable (origins or destinations) while an exponent less

than 1.0 will weaken that variable. They can be considered >fine tuning=

adjustments.

A. Origins. Specify the exponent for the predicted origins. The default is

1.0.

B. Destinations. Specify the exponent for the predicted origins. The

default is 1.0.

4. Impedance function. The trip distribution routine can use two different travel

distance functions:

A. Use an already-calibrated distance function. If a travel distance

function has already been calibrated (see 'Calibrate impedance function'

above), the file can be directly input into the routine. The user selects the

name of the already-calibrated travel distance function. CrimeStat reads

dbase 'dbf', ASCII text 'txt', and ASCII data 'dat' files.

B. Use a mathematical formula. A mathematical formula can be used

instead of a calibrated distance function. Similar to the Journey-to-crime

module (see chapter 13), there are five mathematical functions. They

measure a separation between two zones and estimate a likelihood value.

>Separation= can be in terms of distance, travel time, speed (which is

converted into travel time), or travel costs.

28.31 
 

 

5. Mathematical functions. Briefly, the five functions are:

A. Linear. The simplest type of distance model is a linear function. This

model postulates that the likelihood of traveling to a zone from another by

an offender declines by a constant amount with distance from the

offender=s home. It is highest near the offender=s home but drops off by a

constant amount for each unit of distance until it falls to zero. The form of

the linear equation is;

݂൫݀௜௝൯ ൌ ߙ ൅ ߚܵ) ௝௜28.17(

where f(dij) is the likelihood that the offender will travel from an origin

zone, ݅, to a destination zone, ݆, Sij is the separation in distance, time or

cost between the offender=s residence, ݅, and location ݆, α is a slope

coefficient which defines the fall off in distance, and β is a constant. It

would be expected that the coefficient β would have a negative sign since

the likelihood should decline with separation. The user must provide

values for A and β. The default for A is 10 and for β is -1. When the

function reaches 0 (the X axis), the routine automatically substitutes a 0

for the function. Figure 28.7 illustrates this function.

B. Negative Exponential. A slightly more complex function is the negative

exponential. In this type of model, the likelihood of travel also drops off

with distance. However, the decline is at a constant rate of decline, thus

dropping quickly near the offender=s home until is approaches zero

likelihood. The mathematical form of the negative exponential is:

(28.18) 

where f(dij) is the likelihood that the offender will travel from an origin 

zone, ݅, to a destination zone, ݆, Sij is the separation in distance, time or 

cost between the offender=s residence, ݅, and location ݆, e is the base of 

the natural logarithm, α is the coefficient and β is an exponent of e. The 

user inputs values for α - the coefficient, and β - the exponent. The 

default for α is 10 and for β is 1. 

28.32 
Figure 28.7: 
 

a. This function is the one most used by travel demand modelers. It

has been recommended for use by the Federal Highway

Administration (NCHRP, 1995). Figure 28.8 illustrates a typical

negative exponential impedance function.

C. Normal. A normal distribution assumes the peak likelihood is at some

optimal distance from the offender=s home base. Thus, the function rises

to that distance and then declines. The rate of increase prior to the

optimal distance and the rate of decrease from that distance is symmetrical

in both directions. The mathematical form is:

28.34 

(28.19)

(28.20)

where f(dij) is the likelihood that the offender will travel from an origin zone, ݅, to 

a destination zone, ݆, Sij is the separation in distance, time or cost between the 

offender=s residence, ݅, and location ݆, ݀̅is the mean distance input by the user, σd 

is the standard deviation of distances, e is the base of the natural logarithm, and α

is a coefficient. The user inputs values for ݀̅, σd, and α. The default values are 1 

for each of these parameters.

a. By carefully scaling the parameters of the model, the normal

distribution can be adapted to a distance decay function with an

increasing likelihood for near distances and a decreasing likelihood

for far distances. For example, by choosing a standard deviation

greater than the mean (e.g., ݀̅= 1,Sd = 2), the distribution will be

skewed to the left because the left tail of the normal distribution is

not evaluated. Figure 28.9 illustrates a possible normal impedance 

function. 
Figure 28.8: 
Figure 28.9: 
 

 

 

 

D. Lognormal. The lognormal function is similar to the normal except it is

more skewed, either to the left or to the right. It has the potential of

showing a very rapid increase near the origin with a more gradual decline

from a location of peak likelihood. The mathematical form of the

function is:

(28.21)

where f(dij) is the likelihood that the offender will travel from an origin 

zone, ݅, to a destination zone, ݆, Sij is the separation in distance, time or 

cost between the offender=s residence, ݅, and location ݆, ݀̅ is the mean 

distance input by the user, σd is the standard deviation of distances, e is the 

base of the natural logarithm, and α is a coefficient. The user inputs 

values for ݀̅, σd, and α. The default values are 1 for each of these

parameters. Figure 28.10 illustrates a log-normal impedance function 

that had wide utility in several studies that are discussed below. 

E. Truncated Negative Exponential. Finally, the truncated negative

exponential is a joined function made up of two distinct mathematical

functions - the linear and the negative exponential. For the near distance,

a positive linear function is defined, starting at zero likelihood for distance

0 and increasing to dp, a location of peak likelihood. Thereupon, the 

function follows a negative exponential, declining quickly with distance.

The two mathematical functions making up this spline function are:

Linear: (28.22) 

(28.23) 

where f(dij) is the likelihood that the offender will travel from an origin 

zone, ݅, to a destination zone, ݆, Sij is the separation in distance, time or 

cost between the offender=s residence, ݅, and location ݆, β is the slope of 

the linear function (default=+1) and α is a coefficient and ξ is an exponent 

28.37 

Negative

Exponential: for
Figure 28.10: 
 

for the negative exponential function. Since the negative exponential 

only starts at a particular distance, Sp, α, is assumed to be the intercept if 

the Y-axis were transposed to that distance. Figure 28.11 illustrates a 

truncated negative exponential impedance function. 

F. Model parameters. For each mathematical model, two or three different 

parameters must be defined: 

1. For the negative exponential, the coefficient and exponent 

2. For the normal distribution, the mean distance, standard deviation 

and coefficient 

3. For lognormal distribution, the mean distance, standard deviation 

and coefficient 

4. For the linear distribution, an intercept and slope 

5. For the truncated negative exponential, a peak distance, peak 

likelihood, intercept, and exponent. 

The parameters will be obtained either from a previous analysis or from an 

iterative process of experimentation. See the example below under 

ACompare observed and predicted trips@. 

G. >Fine Tuning= Exponents. In addition, for each function, exponents for 

the attraction and production terms can adjusted. This allows a >fine 

tuning= of the impedance function to better fit the empirical distribution. 

5. Distance Units. The routine can calculate impedance in four ways, by: 

A. Distance (miles, nautical miles, feet, kilometers, and meters) 

B. Travel time (minutes, hours) 

C. Speed (miles per hour, kilometers per hour). Speed is then converted 

into travel time, in minutes. 

D. General travel costs (unspecified units). 

These must be setup under >Network Distance= on the Measurement Parameters 

page. In the Network Parameters dialogue, specify the measurement units. 

The default is distance in miles. 

6. Assumed Impedance for External Zones. For trips originating outside the study 

area (external trips), specify the amount and the units that will be assumed for 

these trips. The default is 25 miles. 

28.39 
Figure 28.11: 
 

 

 

 

7. Assumed Impedance for Intra-zonal Trips. For trips originating and ending in 

the same zone (intra-zonal trips), specify the amount and the units that will be 

assumed for these trips. The default is 0.25 miles. 

8. Model Constraints. In calibrating a model, the routine must constrain either the 

origins or the destinations (single constraint) or constrain both the origins and the 

destinations (double constraint). In the latter case, it is an iterative solution. 

The default is to constrain destinations as it is assumed that the destination totals 

(the number of crimes occurring in each zone) are probably more accurate than 

the number of crimes originating in each zone. Specify the type of constraint for 

the model. 

A. Constrain origins. If >constrain origins= is selected, the total number of 

trips from each origin zone will be held constant. 

B. Constrain destinations. If >constrain destinations= is selected, the total 

number of trips from each destination zone will be held constant. 

C. Constrain both origins and destinations. If >constrain both origins and 

destinations= is selected, the routine works out a balance between the 

number of origins and the number of destinations. 

Fitting the Impedance Function 

The impedance function is fit in an iterative manner. First, either an empirical impedance 

or a mathematical impedance is chosen. Second, the particular mathematical function is selected. 

For example, with the lognormal function, which has been found to produce the best fit for three 

different data sets, there are three parameters: 1) the mean distance; 2) the standard deviation of 

distance; and 3) the coefficient. 

Third, initial values of the parameters are chosen; one suggestion is to use the defaults 

available in the CrimeStat routines. The ACompare observed and predicted trips@ routine can be 

used to evaluate the fit of the model. Fourth, the parameters are adjusted in small increments, one 

at a time, on both side of the initial guess in order to improve the fit. For example, with the 

lognormal function, the mean distance is fit first because it has the greatest impact on the overall 

fit. Then, after a Abest@ mean distance has been found, the standard deviation of distance is 

adjusted until it produces a Abest@ fit. Then, the coefficient is adjusted until it produces a Abest@

fit. Fifth, and finally, the >fine tuning= exponents of the production and attraction functions are 

adjusted. Typically, these change the final fit only slightly. Hence, they represent a final 

adjustment. 

28.41 
 

 

This process is illustrated below in the discussion on the comparison of the observed and 

predicted trips. Essentially, the empirical (observed) distribution is being used as a calibration 

sample in order to find that impedance model and parameters that best approximate it. 

Running the Origin-Destination Model 

The trip distribution (origin-destination) model is implemented in two steps. First, the 

coefficients are calculated according to the exponents and impedance functions specified on the 

setup page. Second, the coefficients and exponents are applied to the predicted origins and 

destinations resulting in a predicted trip distribution. Because these two steps are sequential, 

they cannot be run simultaneously. 

Calibrate Origin-Destination Model. 

In this routine, the row or column parameters (or both if double constraint is used) are 

estimated using a calibration file. The steps are as follows: 

1. Check the >Calibrate origin-destination model= box to run the calibration model. 

2. Save Modeled Coefficients (parameters). The modeled coefficients are saved as 

a >dbf= file. Specify a file name. 

Apply Predicted Origin-Destination Model 

In this routine, the coefficients that were calibrated in the above routine can be applied to 

a data set. The data set can be the same as the calibration file or a different one. The reason 

for separating the calibration from application steps is that the coefficients can be used for many 

different data sets. The steps are as follows: 

1. Check the >Apply predicted origin-destination model= box to run the trip 

distribution prediction. 

2. Modeled Coefficients File. Load the modeled coefficients file saved in the 

>Calibrate origin-destination model= stage. 

3. Assumed Coordinates for External Zone. In order to model trips from the 

>external zone= (trips from outside the study area), specify coordinates for this 

zone. These coordinates will be used in drawing lines from the predicted origins 

to the predicted destinations. There are four choices: 

28.42 
 

 

A. Mean center (the mean X and mean Y of all origin file points are taken). 

This is the default. 

B. Lower-left corner (the minimum X and minimum Y values of all origin 

file points are taken). 

C. Upper-right corner (the maximum X and maximum Y values of all origin 

file points are taken). 

D. User coordinates (user-defined coordinates). Indicate the X and Y 

coordinates that are to be used. 

Because an arbitrary location is taken to represent the >external zone=, any lines that are 

shown from that zone will not necessarily represent any real travel behavior. However, if a 

very high proportion of all crime trips fall within the modeled origin zones (i.e., 95% or more), 

then it is very unlikely that any of the top trip links will come from the >external zone=. 

4. Table Output. The table output includes summary file information and: 

A. The origin zone (ORIGIN) 

B. The destination zone (DEST) 

C. The number of predicted trips (PREDTRIPS) 

5. Save Predicted Origin-destination Trips. Define the output file. The output is 

saved as a >dbf= file specified by the user. 

6. File Output. The file output includes: 

A. The origin zone (ORIGIN) 

B. The destination zone (DEST) 

C. The X coordinate for the origin zone (ORIGINX) 

D. The Y coordinate for the origin zone (ORIGINY) 

E. The X coordinate for the destination zone (DESTX) 

F. The Y coordinate for the destination zone (DESTY) 

G. The number of predicted trips (PREDTRIPS) 

Note: each record is a unique origin-destination combination and there are M x N 

records where M is the number of origin zones (including the external zone) and 

N is the number of destination zones. 

7. Save Links. The top predicted origin-destination trip links can be saved as 

separate line objects for use in a GIS. Specify the output file format (ArcGIS 

'.shp', MapInfo '.mif' or Atlas*GIS '.bna') and the file name. 

28.43 
 

 

 

Save Top Links 

Because the output file is very large (number of origin zones x number of destination 

zones), the user can select a sub-set of zone combinations with the most predicted trips. 

Indicating the top K links will narrow the number down to the most important ones. The 

default is the top 100 origin-destination combinations. Each output object is a line from the 

origin zone to the destination zone with an ODT prefix. The prefix is placed before the output 

file name. 

The graphical output includes: 

A. An ID number from 1 to K, where K is the number of links output (ID) 

B. The feature prefix (ODT) 

C. The origin zone (ORIGIN) 

D. The destination zone (DEST) 

E. The X coordinate for the origin zone (ORIGINX) 

F. The Y coordinate for the origin zone (ORIGINY) 

G. The X coordinate for the destination zone (DESTX) 

H. The Y coordinate for the destination zone (DESTY) 

I. The number of predicted trips for that combination (PREDTRIPS) 

J. The distance between the origin zone and the destination zone. 

8. Save Points 

Intra-zonal trips (trips in which the origin and destination are the same zone) can be 

output as separate point objects as an ArcGIS '.shp', MapInfo '.mif' or Atlas*GIS '.bna' file. 

Again, the top K points are output (default=100). Each output object is a point representing an 

intra-zonal trip with an ODTPOINTS prefix. The prefix is placed before the output file name. 

The graphical output for each includes: 

A. An ID number from 1 to K, where K is the number of links output (ID) 

B. The feature prefix (POINTSODT) 

C. The origin zone (ORIGIN) 

D. The destination zone (DEST) 

E. The X coordinate for the origin zone (ORIGINX) 

F. The Y coordinate for the origin zone (ORIGINY) 

G. The X coordinate for the destination zone (DESTX) 

H. The Y coordinate for the destination zone (DESTY) 

I. The number of predicted trips for that combination (PREDTRIPS) 

28.44 
 

 

 

Example of the Predicted Trip Distribution from Baltimore County 

The predicted origins and predicted destinations from Baltimore County were input into a 

trip distribution model and a predicted trip distribution was output. The impedance function was 

a lognormal distribution, which produced a good fit to the observed (empirical) distribution (see 

discussion below). 

Figure 28.12 outputs the top 1000 links from the model. The top 1000 links account for 

14,271.9 trips, or 34.0% of the total number of trips. Compared to the observed distribution, 

the top 1000 links account for a smaller proportion of the total trips (14,272 v. 19,615). This 

suggests that the actual distribution is slightly more concentrated than the model suggests. Like 

the observed distribution, however, a sizeable number of the top links are intra-zonal trips (5,428 

or 12.9%). The intra-zonal trips have been displayed as circles in the figure. 

Comparing the predicted trip distribution to the observed trip distribution, some 

similarities and differences are seen. Figure 28.13 compares the top 1000 zone-to-zone links 

for the predicted and observed distributions. The model has captured many of the major links. 

For the five shopping malls that received a sizeable number of actual crime trips, the model has 

captured the majority of trips for three of them and some trips for a fourth. For the mall in the 

southeast corner of the county, on the other hand, the model has not allocated a large number of 

trips. Similarly, for a zone near the western edge of the county, the model has allocated more 

trips than actually occurred. 

There are, of course, only 325 intra-zonal trip links (one for each destination zone). 

Looking at a comparison of the intra-zonal trips (Figure 28.14), some similarities and differences 

are seen. Generally, the model captured the location of many intra-zonal trips, but it did not 

capture the quantity very accurately. Zones that had many intra-zonal trips are shown as having 

only some by the model and, conversely, the model predicts many intra-zonal trips for two zones 

which had only some. 

In other words, the fit between the actual distribution and the model is not perfect. 

Considering that only 1000 of the 172,900 trip links (532 origin zones x 325 destination zones) 

are shown, the model has still done a reasonable job of capturing the major links. 

It is not surprising that the model is not perfect. The model is a simple analogue using 

only three variables (productions, attractions, impedance) whereas the actual distribution 

represents a very complex set of individual decisions made by offenders. What is perhaps 

remarkable is that the model has done a decent job of capturing some of these relationships at all. 

28.45 
Figure 28.12: 
Figure 28.13: 
Figure 28.14: 
This brings up an important point, namely that a model is not reality. It is only a 

simplified set of relationships that approximates reality (in this case, the observed distribution). 

It is important in developing any model to evaluate it relative to an observed set of facts, and this 

applies no less to the trip distribution model. One has to understand, however, that a good 

model will not capture all the relationships. Hopefully, it captures enough of them to make the 

model useful for prediction and evaluating policy options. 

Comparing Observed & Predicted Trips 

It is important to conduct a number of tests on the predicted model to ensure that it is 

capturing the most important elements of the observed distribution. These are conducted by 

comparing the predicted distribution with the observed (empirical) distribution. Figure 28.15 

shows the setup page for comparing the observed with the prediction distribution 

There are a number of tests that can be used to evaluate a model by comparing the 

predicted distribution with the observed one. CrimeStat includes three of these and the steps 

are as follows: 

1. Estimate the parameters of the model and apply them to the calibration data set

2. Examine the intra-zonal trips to be sure that the predicted number corresponds to

the observed number

3. Compare the trip lengths of the observed and predicted distributions using two

tests:

A. The Coincidence Ratio

B. The Komolgorov-Smirnov Two-sample Test

4. Compare the number of trips for the top links using a pseudo-Chi square test.

That is, the number of trips for the most frequent links in the observed distribution

is compared to the number predicted by the model for the same links.

Unfortunately, not one of these tests is sufficient to validate a model. Further, 

minimizing the discrepancy for only one of them may distort the others. It is very unlikely that 

there will be a model that minimizes the errors for all three tests. Consequently, the user will 

have to choose a model that balances these factors in a desirable way (an optimum model). 

28.49 
Figure 28.15: 

Comparing Observed and Predicted Trip Length
 

Estimating Impedance Parameters and Exponents of the Gravity Model 

While this is not strictly an evaluation test, this step is essential in estimating the 

particular impedance parameters that are used in the first place. Typically, an analyst will 

approximate an impedance function. Using a comparison between the observed and predicted 

models, the parameters can be adjusted to produce a better fit. The steps are as follows: 

1. The model is estimated with a calibration data set. There is a file of predicted

origins and another file of predicted destinations; typically, these are defined as

the primary and secondary files respectively, though the order could be reversed

or the same file used for both origins and destinations (if the number of origins

zones was identical to the number of destination zones).

2. On the trip distribution setup page, select the type of impedance function that is to

be used, already-calibrated (empirical) or mathematical. For the

journey-to-crime routine, generally the empirical function led to better results

than the mathematical. However, with a trip distribution function, a

mathematical function may be as good, if not better. This was tested with three

data sets for Baltimore County, Las Vegas, and Chicago and, in all cases, a

mathematical function (the lognormal) gave a much better fit than an

empirically-derived function (see Chapters 31 and 32).

3. If a mathematical function is to be used, select the type of distribution. The

default value is a lognormal, but the user can choose a negative exponential, a

normal, a linear, or a truncated negative exponential function.

4. For the particular mathematical function, select initial guesses for the parameters.

For each mathematical model, two or three different parameters must be defined:

A. For the negative exponential, the coefficient and exponent

B. For the normal distribution, the mean distance, standard deviation

and coefficient

C. For lognormal distribution, the mean distance, standard deviation

and coefficient

D. For the linear distribution, an intercept and slope

E. For the truncated negative exponential, a peak distance, peak

likelihood, intercept, and exponent.

5. In addition, there are exponents of the production and attraction side that can be

made to >fine tune= the model. In general, these exponents will only affect the

28.51 
 

 

results slightly, compared to the basic choices of the type of model and the 

selection of values for the main parameters. 

6. Calibrate and apply the model to the calibration data set. Examine the three

criteria discussed below to minimize the error between the actual distribution and

that predicted by the model.

7. Modify the parameter values slightly.

8. Repeat steps 4 through 7 until a good fit is found between the actual and predicted

distribution and in which the errors are minimized and optimized. The process

by which this is done is discussed below.

Comparing Intra-zonal Trips 

The first evaluation test is to compare the percentage of trips that occur within the same 

zone - intra-zonal trips. The Travel Model Improvement Program manual indicates that 

intra-zonal trips should represent typically no more than 5% of all trips for home-to-work trips; 

that is, commuting trips (FHWA, 1997, chapter 4). However, given that most crime trips are 

quite short, the proportion of trips that are intra-zonal is liable to be much higher. In Baltimore 

County, for example, 19.7% of all crime trips were intra-zonal. Ideally, the predicted model 

should also have 19.7% of all crime trips being intra-zonal. 

The ACompare observed and predicted trip lengths@ routine is discussed below. The 

routine outputs the number of trips that are intra-zonal in both the observed and predicted 

distributions. A good model should produce approximately the same number of intra-zonal 

trips in the predicted distribution as what actually occurred. 

Illustration 

For example, in the Baltimore County model displayed in Figure 28.12 above, there were 

8,272 intra-zonal trips in the actual distribution (out of 41,979). On the other hand, there were 

only 5,428 predicted intra-zonal trips in the model. In other words, the predicted model 

assigned fewer intra-zonal trips than actually occurred. 

It may be necessary to modify the model to produce a closer fit for the intra-zonal trips. 

A simple way to do this to increase or decrease the relative impedance parameter in the model. 

So, to use the example, if the predicted model is assigning too few intra-zonal trips, then the cost 

function can be strengthened (i.e., making travel more expensive). In this case, in the original 

model the lognormal function was used with a mean distance of 6.18 miles. If the mean 

28.52 
 

 

distance of the impedance function is reduced to 3.5, then the number of predicted intra-zonal 

trips increases to 8,275, almost the same number as occurred in the observed distribution. 

In other words, by decreasing the mean distance for the lognormal function, the 

impedance function was strengthened (i.e., made more expensive) and a better fit was created 

between the observed and predicted distributions. 

In and of itself, a mismatch for intra-zonal trips between the predicted model and what 

actually occurred does not necessarily require a modification of the gravity function. Other 

criteria must be considered, namely how well the predicted model fits the trip length distribution 

and how well the predicted models captures the most frequent inter-zonal (zone-to-zone) trip 

links. Later in the discussion, the issue of optimizing a model by balancing these different 

criteria will be described. 

Comparing Trip Length Distributions 

The second evaluation test in comparing the observed with the predicted distribution is a 

calculation of the trip length distribution (see steps below). Because the trip distribution matrix 

will typically be very large, most cell values will be zero. Rarely will there be enough data to 

cover all the cells and, even if there was, the skewness in a crime distribution will leave most 

cells with no data. For example, for the Baltimore County model, with 532 origin zones and 

325 destination zones, there will be 172,900 cells (325 x 532). The calibration data set had only 

41,974 cases. Thus, the number of cells is more than four times the sample size and it is not 

possible to fill all cells with a number. 

Consequently, because of the large number of cells with zero counts, one cannot use the 

Chi square test to compare the observed and predicted distributions. The Chi square test 

assumes that, first, the distribution is relatively normal (which it is not since the data are highly 

skewed) and, second, that there are at least 5 cases per cell. The latter condition is impossible 

given the large number of cells. 

Therefore, what is usually done is to compare the trip length distribution of the observed 

and predicted models. >Trip length= is the length in distance, travel time, or cost of each trip. 

It is measured by the actual length (or separation) between two zones times the number of cases 

for that zone pair. For example, in Figure 28.1, there were 15 trips from zone 1 to zone 2 and 7 

trips in the opposite direction (from zone 2 to zone 1). Let=s assume that the distance between 

zone 1 and zone 2 is 1.5 miles. Thus, there are 22 trips that fall into a trip length of 1.5 miles (15 

in the direction of zone 1 to zone 2 and 7 in the direction of zone 2 to zone 1). 

28.53 
 

If travel time is used, the calculation uses time rather than distance. For example, if a 

vehicle was traveling 30 miles per hour, then it would take 3 minutes to cover 1.5 miles (1.5 

miles ) 30 miles per hour = 0.05 hours x 60 minutes per hour = 3 minutes). Thus, there are 22 

trips that fall into a trip >length= of 3 minutes. A similar logic would apply to travel cost 

categories. 

This process is repeated for all cells and the distribution of trips is allocated to the 

distribution of trip lengths (in distance, travel time, or travel cost). In general, one uses many 

intervals (or bins) for trip length (25 or more). In CrimeStat, the default number of trip lengths 

is 25, but it is not unknown to use up to 100. The problem in using too many is that the 

distributions become unreliable and differences that appear may not be real. 

Graphical fit 

Once the trip length distribution is calculated for both the observed and predicted 

distributions, it is possible to compare them. CrimeStat outputs a graph showing the fit of the 

two distributions. In general, they should be very close. An examination of differences 

between the distributions can indicate at what trip lengths the model is failing. This might 

allow the parameters to be adjusted in order to improve the fit on the next iteration. Examples 

will be given below of the graphing of the two distributions. But, it is important to come up 

with a model in which the two distributions >look= similar. 

Coincidence ratio 

The coincidence ratio compares the two trip length distributions by examining the ratio 

of the total area of those distributions that coincide, that are in common (FHWA, 1997, chapter 

4). It is defined as: 

(28.24) 

(28.25) 

(28.26) 

The steps are as follows: 

1. Essentially, the two distributions are broken into K bins (or intervals). That is, the

number of trips in each bin is enumerated (see example above).

28.54 

ൌ ݅݊ܿ݅݀݁݊ܿ݁݋ܥ

ൌ ݈ܽݐ݋ܶ

ൌ ݋݅ݐܽݎ ݅݊ܿ݅݀݁݊ܿ݁݋ܥ
 

 

2. Each of the two distributions is converted into a proportional distribution by

dividing the bin count by the total number of trips in the distribution. This step 

is not absolutely essential as the test can be conducted of the raw counts.

However, by converting into proportions, the two distributions are standardized.

3. A cumulative count is conducted of the minimum proportion in each interval.

That is, starting at the lowest interval, the smaller of the two proportions is taken.

At the next interval, the smaller of the two proportions is added to the count.

This is repeated for all K bins. This is called the coincidence and measure the

overlapping proportions over all intervals.

4. A similar cumulative count is conducted of the maximum proportion in each

interval. That is, starting at the lowest interval, the larger of the two proportions is

taken. At the next interval, the larger of the two proportions is added to the count.

This is repeated for all K bins. This is called the total and measures the unique

proportion over all intervals.

5. Finally, the coincidence ratio is defined as the ratio of the minimum count to the

total count.

The coincidence ratio is a proportion from 0 to 1. It is analogous to the R2

 statistic in 

regression analysis in that it measures the >explained= (or overlapping) variance. According to 

the Travel Model Improvement Program manual (FHWA, 1997, chapter 4), the higher the 

coincidence ratio, the better. A value of 0.9 would generally be considered good. 

Komolgorov-Smirnov two-sample test 

The Komolgorov-Smirnov Two-Sample Test is similar to the coincidence ratio, but it 

examines the maximum difference across all bins (Kanji, 1993). For each distribution, a 

cumulative sum is created. At each interval, the difference between the two cumulative sums is 

calculated. The maximum difference between the two distributions is taken as the test statistic: 

(28.27) 

where D is the maximum difference found, Oi is the cumulative sum of the actual (observed) trip 

lengths, and Pi is the cumulative sum of the predicted trip lengths. There are tables of critical 

values for the Komolgorov-Smirnov Two-Sample Test which are a function of the number of 

intervals, K (Smirnov, 1948; Massey, 1951; Siegel, 1956; Kanji, 1993). 

28.55 
 

 

Illustration 

To illustrate the trip length comparison, figures 28.16 through 28.19 show the results for 

four different impedance models - an empirical impedance function, a negative exponential 

impedance function, a truncated negative exponential impedance function, and a lognormal 

impedance function. As seen, the fit of the empirical impedance function is not particularly 

good, but gets progressively better with the three different mathematical functions. 

The best fit is clearly was with the lognormal function. With these parameters (mean 

center = 6.0 miles, standard deviation = 4.7 miles, coefficient = 1, origin exponent = 1, and 

destination exponent = 1.06), the Coincidence Ratio was 0.93. 

But, again, this is just one criterion, though it fits most of the distribution matrix. As 

with the number of intra-zonal trips, minimizing the error for a trip length distribution will not 

necessarily minimize the error for the other two criteria (intra-zonal trips and the top links). 

But, it is important that the trip length comparison be reasonably close. 

Comparing the Trips of the Top Links 

The third evaluation test focuses on the top links. That is, it evaluates how well the 

predicted model captures the major trip links, both intra-zonal and inter-zonal. Since crime trip 

distributions are skewed (i.e., a handful of zones contribute to most crime origins and a handful 

of zones attract many crimes), capturing the most important links is essential for a good crime 

distribution model. This is particularly true since a model that produces the best fit for the 

overall trip length distribution may not capture the top links very well. 

Therefore, simply comparing the trip length distribution may not adequately capture the 

top links. That is, on average a particular model may produce a good fit between the predicted 

and observed distributions, but may do this by minimizing error across the entire matrix of trip 

pairs without necessarily minimizing the error for the top links. 

Consequently, it is important to also compare the fit of the model for the top links. One of 

the lines in the dialogue for the ACompare observed and predicted trip lengths@ is ACompare top 

links@. The user should specify the number of top links to be compared; the default is 100. 

The top links are the trip pairs that have the most number of actual trips, starting from the pair 

with the most trips and sorting in descending order. The routine calculates a pseudo-Chi square 

test on just those links. Since the top links will all have a sufficient number of trips, it is 

possible to calculate a Chi square statistic. However, since not all links are being considered in 

this test, a significance test of this statistic cannot be calculated since the sampling error is not 

known. 

28.56 
Figure 28.16: 
Figure 28.17: 
Figure 28.18: 
Figure 28.19: 
 

Using the observed (actual) links as the reference, the test calculates: 

(28.28) 

where Oi is the observed (actual) number of trips for trip pair, ݅, Pi is the predicted number of 

trips for trip pair, ݅, and ݅ is the number of trip pairs that are compared up to K comparisons, 

where K is selected by the user. 

Number of links to test 

The number of top links that are to be compared depends on how skewed is the 

distribution. One good way to look at this is to plot the rank size distribution of the observed 

trips. Using the output >dbf= file for the observed trip distribution (see ACalculate observed 

origin-destination trips@ above), import the file into a spreadsheet. Sort the file in descending 

order of the trip frequency and create a new variable called ARank order@, which is simply the 

descending order of the trip frequencies. Then, plot the frequency of trips (FREQ) on the Y 

axis against the rank order of the trip pairs on the X axis. 

Figure 28.20 below shows the rank size distribution of the Baltimore County crime trips. 

Notice how the distribution is very skewed for the top crime trip pairs, but declines substantially 

after that. That is, the top trip link (which was an intra-zonal trip pair - zone 654 to itself) had 

278 trips. The second top link (also an intra-zonal pair - zone 714 to itself) had 226 trips. The 

third had 223; the fourth had 205; and so forth. As mentioned above, the top 1000 trip links 

account for about 47% of all the trips in the matrix, but the first 176 account pairs account for 

half of that. In other words, if the top 150 to 200 trip pairs are examined, the highest volume 

links will be included and most of the skewness in the distribution will be accounted for. The 

remaining distribution, which is not fitted, will be less skewed. 

Illustration 

An illustration of how comparing the top links can modify a trip distribution model can 

be given. The same model as shown in Figure 28.12 was run. The pseudo-Chi square test for 

the first 176 pairs was 5,832 (rounding-off to the nearest integer). However, by modifying the 

mean distance of the lognormal function a lower Chi square value was obtained. After several 

iterations, the lowest Chi square value was obtained for a mean distance of 5.2 miles (χ

2

 = 

5,448). Again, the top links represents only one criterion out of the three mentioned. A good 

model should balance all three of these. 

28.61 
Figure 28.20: 
 

 

 

Optimizing the Three Evaluation Criteria 

The ideal solution would be to have all three evaluation criteria minimized. That is, with 

an ideal model, there should be very little error between the predicted model and the observed 

distribution for the number of intra-zonal trips, the trip length distribution, and the top links. 

In practice, it is unlikely that any one model will minimize all three types of errors. 

Thus, a balance (a compromise) must be obtained in order to produce an optimal solution. 

Since a balance can be obtained in different ways, there are multiple solutions possible. 

Hint: In CrimeStat, it is very easy to run through different models. The parameters 

are input on the ASetup origin-destination model page@. The coefficients are 

calibrated in the ACalibrate origin-destination model@ routine on the 

AOrigin-Destination Model@ page. The coefficient file which is output is then input 

into the AApply predicted origin-destination model@ routine on the same page. The 

comparison between the observed and predicted values is found in the ACompare 

observed and predicted origin-destination trip lengths@ routine. Once set up, 

iterations of the models can be run very easily. A change is made on the setup page. 

The model is calibrated. It is then applied to the calibration data set. Finally, a 

comparison is made. Since the file names remain constant, an entire iteration will 

take less than a minute on a fast computer. 

To illustrate the multiple criteria, Table 28.2 shows the best models for each of the three 

tests with variations on the mean distance in the model shown in Figure 28.12. All other 

parameters were held constant. Many models were run to produce this table including testing 

other functions. These are the three best. 

As seen, different models produce the lowest error for each of the criteria. For 

obtaining the closest fit to the number of intra-zonal trips, the mean distance of the lognormal 

function was 3.5 miles. For producing the best fit to the top 176 links, the mean distance for the 

best model was 5.2 models. For producing the best fit for the entire trip length distribution, the 

mean distance of the best model was 6.0 miles. The question is which one to use? 

28.63 
 

 

 

 

 

Table 28.2: 

Multiple Criteria in Selecting a Distribution Function

Lognormal function 

Standard deviation = 4.7 miles 

Coefficient = 1 

Origin exponent = 1.0 

Destination exponent = 1.06 

Number of Chi square 

Mean Intra-zonal for top Coincidence 

distance Trips 176 Links Ratio 

Observed 8272 - -

6.0 5463 5814 0.93 

5.2 6296 5777 0.87 

3.5 8275 5986 0.74 

One solution for optimizing decisions 

One possible solution is to optimize in the following way: 

1. If the trip distribution matrix is highly skewed (which will occur with most crime 

data sets), then it is essential that the top links be replicated closely. This would 

take priority over the second criterion which is minimizing the error for the trip 

length distribution, and the third criterion which is minimizing the error in 

predicting intra-zonal trips. 

2. Next fit the model to minimize the Chi square value for the top links. In the 

example above, this would be the top 176 pairs. Typically, the mean distance has 

the biggest impact for a lognormal or normal function and this would be adjusted 

first. For a negative exponential function, the exponent has the strongest impact. 

For a linear function, the slope has the strongest impact and for a truncated 

negative exponential, both the peak distance, for the near distance, and the 

exponent, for the far distance, has the biggest impacts (see Chapter 13). Again, 

the aim is to produce the Chi square for the top links with the lowest value. 

4. Then, while trying to maintain a Chi square value as close to this minimal value 

as possible, adjust the model to minimize the error in the trip length comparison. 

In this case, the model with the highest Coincidence Ratio is that which 

minimizes the error. For lognormal and normal functions, the standard deviation 

28.64 
 

 

 

 

 

 

 

is the next parameter to adjust. For a negative exponential function, the 

coefficient should be adjusted next. For a linear function, the intercept would be 

adjusted next and for a truncated negative exponential the slope would be adjusted 

next. Again, the aim should be to obtain the highest Coincidence Ratio without 

losing the fit for the top links. 

5. Finally, if it is possible, adjust the exponents of the origins and destinations and 

the other parameters (e.g., the coefficient in the lognormal and normal 

distributions) to reduce the error in the total number of intra-zonal trips. 

Typically, however, these do not alter the results very much. They can be 

thought of as Afine tuning@ adjustments. 

Notice that this hierarchy fits the highest volume trip links first, then fits the overall trip 

length distribution, and finally fits the number of intra-zonal trips. 

Illustration 

To illustrate, we first start with the model that produced the lowest Chi square. That 

model used a lognormal function with a mean distance of 5.2 miles, a standard deviation of 4.7 

miles, a coefficient of 1, an origin exponent of 1.0 and a destination exponent of 1.06. Varying 

the standard deviation of the lognormal function produced the following results (Table 28.3). 

Table 28.3: 

Minimizing the Second Criteria in Selecting a Distribution Function 

Lognormal function 

Mean distance = 5.2 miles 

Standard Deviation = 4.6 miles 

Coefficient = 1 

Origin exponent = 1.0 

Destination exponent = 1.06 

Number of Chi square 

Standard Intra-zonal for top Coincidence 

deviation Trips 176 Links Ratio 

4.5 5809 5789 0.90 

4.6 6057 5779 0.88 

4.7 (baseline) 6296 5777 0.87 

4.8 6526 5780 0.86 

4.9 6746 5788 0.84 

28.65 
 

 

 

As the standard deviation was increased, the Coincidence Ratio decreased while the 

number of intra-zonal trips increased. Of these five different standard deviations, 4.5 produced 

the highest Coincidence Ratio, but also increased the Chi square statistic for the 176 top links. 

Since that criterion was set first, we do not want to loosen it substantially during the second 

adjustment. Consequently, a standard deviation of 4.6 was selected because this increased the 

Coincidence Ratio slightly while not substantially worsening the Chi square test. 

Subsequent tests varying the coefficient of the lognormal function and the exponents of 

the origin and destination terms did not alter these values. Consequently, the final model that 

was selected is listed in Table 28.4. 

Table 28.14: 

Baltimore County Crime Trips: 1993-1997 

Optimal Model Selected 

Lognormal function 

Mean distance = 5.2 miles 

Standard deviation = 4.6 

Coefficient = 1 

Origin exponent = 1.0 

Destination exponent = 1.06 

The model was re-run with the new parameters used. The top 176 predicted trip links 

were output and were compared to the top 179 observed trip links (which exceeded 176 because 

of tied values). The top predicted 176 links accounted for 7,241 trips, or 17.3% of the total 

number of trips. The top observed 179 links accounted for 9,900 trip, or 23.6% of the total. 

Compared to the observed distribution, the top 176 predicted links accounted for a smaller 

proportion of the total trips. 

However, the fit was generally better. Figure 28.21 shows the top predicted inter-zonal 

trip links and compares them to the top observed links while Figure 28.22 shows the top 

predicted intra-zonal (local) trip links and compares them to the top observed intra-zonal links. 

Comparing these maps to Figure 28.12 and 28.13 (which mapped the top 1000 links, not the top 

176), the fit is a bit better for the major links, which is what we optimized. The fit is not 

perfect; it probably will never be. But, it is reasonably close. 

Of course, this is not the only way to optimize and different users might approach it 

differently (e.g., minimizing the intra-zonal trips first, then the overall trip length distribution, 

and finally the top links). It has to be realized that optimizing in a different order will probably 

produce varying results; there is not, unfortunately, a single optimum solution to these three 

28.66 
Figure 28.21: 
Figure 28.22: 
 

 

criteria. That is why it is important to explicitly define how an optimal solution will be 

obtained. In that way, users of the model can be cognizant of where the model is most accurate 

and where it is probably less accurate. 

Implementing the Comparisons in CrimeStat 

The mechanics of conducting the tests is fairly straightforward. The three tests are 

implemented in the ACompare Observed and Predicted Trip Lengths@ routine on the last page of 

the Trip distribution module. 

Observed trip file 

Select the observed trip distribution file by clicking on the Browse button and choosing 

the appropriate file. 

Observed number of origin-destination trips 

Specify the variable for the observed number of trips. The default name is FREQ. 

Orig_ID 

Specify the ID name for the origin zone. The default name is ORIGIN. Note that the 

ID=s used for the origin zones must be the same as in the destination file and the same as in the 

predicted trip file if the top links are to be compared. 

Orig_X 

Specify the name for the X coordinate of the origin zone. The default name is 

ORIGINX. 

Orig_Y 

Specify the name for the Y coordinate of the origin zone. The default name is 

ORIGINY. 

28.69 
Dest_ID 

Specify the ID name for the destination zone. The default name is DEST. Note that all 

destination ID=s should be in the origin zone file and must have the same names and the same as 

in the predicted trip file if the top links are to be compared. 

Dest_X 

Specify the name for the X coordinate of the destination zone. The default name is 

DESTX. 

Dest_Y 

Specify the name for the Y coordinate of the destination zone. The default name is 

DESTY. 

Predicted trip file 

Select the predicted trip distribution file by clicking on the Browse button and choosing 

the appropriate file. 

Predicted number of origin-destination trips 

Specify the variable for the observed number of trips. The default name is PREDTRIPS. 

Orig_ID 

Specify the ID name for the origin zone. The default name is ORIGIN. Note that the ID=s 

used for the origin zones must be the same as in the destination file and the same as in the 

observed trip file if the top links are to be compared. 

Orig_X 

Specify the name for the X coordinate of the origin zone. The default name is ORIGINX. 

Orig_Y 

Specify the name for the Y coordinate of the origin zone. The default name is 

ORIGINY. 

28.70 
Dest_ID 

Specify the ID name for the destination zone. The default name is DEST. Note that all 

destination ID=s should be in the origin zone file and must have the same names and the same as 

in the observed trip file if the top links are to be compared. 

Dest_X 

Specify the name for the X coordinate of the destination zone. The default name is 

DESTX. 

Dest_Y 

Specify the name for the Y coordinate of the destination zone. The default name is 

DESTY. 

Select bins 

Specify how the bins (intervals) will be defined. There are two choices. One is to select 

a fixed number of bins. The other is to select a constant interval. 

Fixed number 

This sets a fixed number of bins. An interval is defined by the maximum distance 

between zone divided by the number of bins. The default number of bins is 25. Specify the 

number of bins. 

Constant interval 

This defines an interval of a specific size. If selected, the units must also be chosen. 

The default is 0.25 miles. Other distance units are nautical miles, feet, kilometers, and meters. 

Specify the interval size. 

Compare top links 

The ACompare top <value> links@ dialogue implements a comparison of the top links. 

The user specifies the number of links to be compared. The default is 100. The routine 

28.71 
calculates a Chi square statistic for these links. Note that in order to make the comparison, the 

origin and destination ID's must be the same for both the observed and predicted trip files. 

Save comparison 

The output is saved as a >dbf= file specified by the user. 

Table output 

The table output includes summary information and: 

1. The number of trips in the observed origin-destination file 

2. The number of trips in the predicted origin-destination file 

3. The number of intra-zonal trips in the observed origin-destination file 

4. The number of intra-zonal trips in the predicted origin-destination file 

5. The number of inter-zonal trips in the observed origin-destination file 

6. The number of inter-zonal trips in the predicted origin-destination file 

7. The average observed trip length 

8. The average predicted trip length 

9. The median observed trip length 

10. The median predicted trip length 

11. The Coincidence Ratio (an indicator of congruence varying from 0 to 1) 

12. The D value for the Komolgorov-Smirnov two-sample test 

13. The critical D value for the Komolgorov-Smirnov two-sample test 

14. The p-value associated with the D value of Komolgorov-Smirnov two-sample test 

relative to the critical D value. 

15. The pseudo-Chi square test for the top links 

and for each bin: 

16. The bin number 

17. The bin distance 

18. The observed proportion 

19. The predicted proportion 

File output 

The saved file includes: 

1. The bin number (BIN) 

28.72 
2. The bin distance (BINDIST) 

3. The observed proportion (OBSERVPROP) 

4. The predicted proportion (PREDPROP) 

Graph 

While the output page is open, clicking on the graph button will display a graph of the 

observed and predicted trip length proportions on the Y-axis by the trip length distance on the 

X-axis. This would produce a similar graph to that seen in Figures 28.16 through 28.19 above. 

Uses of Trip Distribution Analysis 

There are a number of uses for the trip distribution analysis. First, for policing, an 

analysis of the actual (observed) trip distribution can be valuable. Second, the predicted model 

has value, above-and-beyond the analysis of the actual distribution. 

Utility of Observed Trip Distribution Analysis 

This information by itself can be very useful for police. Two applications will be 

discussed. 

Crime prevention efforts 

A major application is using the data shown in a trip distribution map to guide 

enforcement efforts. For example, in Baltimore County, with the crimes occurring at the five 

shopping malls, the origin locations can be more easily seen. This has utility for police. First, 

the police can intervene more effectively on the routes leading from likely origin locations. 

They can patrol those routes more heavily and, perhaps, intervene more frequently. By using 

the information from the trip distribution analysis, they make their enforcement efforts smarter. 

Second, they can conduct crime prevention efforts more effectively. By knowing the likely 

origin of offenders, intervention efforts in the origin zones may head off some of these incidents. 

Programs such as weed-and-seed and after-school programs depend on providing alternative 

facilities for youth, hoping to redirect them to more constructive activities. These facilities can be 

placed in locations where many crimes originate. 

Improved Journey-to-crime analysis 

A second application is in guessing the likely origin of a serial offender. In Chapter 13, 

theories of travel behavior by a serial offender was discussed. The resulting analysis 

(geographic profiling, Journey-to-crime analysis) utilized information on the distribution of 

28.73 
 

incidents committed by the offender. On the other hand, the trip distribution pattern seen in 

Figure 28.4 provides a probability map of offender locations and gives more information than 

was evident in the Journey-to-crime model. That model assigned a likelihood of the offender 

living at a location (the origin) on the basis of the distribution of the incidents. There was no 

additional information used about likely origin locations. This trip distribution map, on the 

other hand, points to certain zones as being the likely origin for offenses committed at the major 

destination locations. There is more >structure= in this analysis than in the Journey-to-crime 

logic. This is the basis for the Bayesian Journey-to-crime approach discussed in Chapter 14. 

One can think of this in terms of a quasi-Bayesian approach to guessing the likely origin 

of an offender. The geographic profiling/Journey-to-crime logic assumes no prior probabilities. 

The only information that is used is the distribution of crimes committed by a serial offender and 

a model of crime travel distance (essentially, an impedance function). The trip distribution map, 

on the other hand, points to certain locations as being the likely origin for incidents. 

Admittedly, this is based on a large sample of cases rather than one particular serial offender. 

But, the map points to certain prior probabilities for an origin location. The Bayesian 

Journey-to-crime routine combines those two pieces of information. As mentioned in Chapter 14, 

tests on more than 1000 serial offenders in four cities (in three countries) showed that the method 

was 10-15% more accurate than the traditional journey-to-crime approach and as precise. 

In other words, the empirical description of crime travel patterns is useful for policing, 

above-and-beyond any modeling that is developed. 

Utility of Predicted Trip Distribution Analysis 

The model also has a lot of utility for both policing and crime analysis. A number of 

examples will be given. First, it can be used for forecasting. By calibrating the model on one 

data set, it be applied to a future data set. As mentioned in Chapter 26, much of the population 

and employment data that form the basis of a trip generation model comes from a Metropolitan 

Planning Organization (MPO). Most MPOs in the United States also make forecasts of future 

population and employment. Those forecasts can be, in turn, converted into forecasts of future 

crime origins and crime destinations. Thus, on the assumption that the distribution trends will 

remain the same over time, the trip distribution model can be applied to the forecast set of origins 

and destinations. This could allow an examination of possible changes in the crime distribution 

(assuming that the future forecasts are correct and that the trip distribution coefficients remain 

constant). 

Second, a model of crime trip distribution can be useful for modeling changes in land 

uses. For example, if a new shopping mall is being planned, one can take the existing trip 

generation model and adjust it to fit the planned situation (e.g., adding 500 retail jobs to the zone 

28.74 
 

 

in which the mall is being developed). Then, the trip generation model is re-run with the new 

expected data, and the trip distribution model is applied to the predicted crime origins and crime 

destinations. The result would be a model of likely crime trips to the new shopping mall. This 

can be useful to the mall developers, to future businesses, and to the police. If it turns out that 

the model forecasts there will be a sizeable number of crime trips to that mall, then preventive 

actions can be developed before the mall is built (e.g., improving security design in the mall; 

improving the parking lot arrangement). 

Third, a model of crime trip distribution can help in analyzing future interventions. For 

example, increasing police patrols in a high crime attraction area can be examined as to possible 

effectiveness before taking the trouble to reorganize deployment. Or, adding a new drug 

treatment center or a new youth center can be modeled as to its possible effectiveness in 

changing the nature of crime trips. Again, the input is at the data level, which affects the trip 

generation model. But the trip distribution model is applied to the new outputs from the trip 

generation model. The advantage of a model is that it explores a set of interventions without 

having to actually having to implement them; it is a >thinking= tool for planning change. 

Fourth, and finally, a crime trip distribution model is helpful in developing crime theory. 

As indicated in Chapter 25, the theory of crime travel has been very elementary up to now. The 

primary focus of analysis has been only on the destinations and on the trip lengths as measured 

by distance traveled. A trip distribution model, on the other hand, analyzes both trip 

destinations and trip origins, and can include a more sophisticated measure of impedance than 

simple distance. Because analysis is conducted over a larger area (a jurisdiction or a 

metropolitan area), the hierarchy of crime trips can be analyzed as an interaction between origins 

and destinations. In short, a crime trip distribution model is a >quantum leap= in sophistication 

and complexity compared to the usual Journey-to-crime types of models. Hopefully, it will 

generate even more sophisticated types of models. The attachment illustrates how the crime 

travel demand model was used to examine possible interventions to reduce DWI trips ending in 

crashes in Baltimore County. 

The next chapter continues the travel demand model by examining how crime trip links 

are split into different travel modes. That is, the trip distribution model estimates the number of 

trips flowing from each origin zone to each destination zone. The mode split model then breaks 

these trips into distinct travel modes. 

28.75 
 

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28.78 
 

 

Modeling DWI Trips That End in Crashes 

in Baltimore County, MD 

 Ned Levine

Ned Levine & Associates 

Houston, TX 

Phil Canter 

Towson University 

Towson, MD 

A crime travel demand study was conducted on 862 Driving While Intoxicated (DWI) motor 

vehicle crash trips that occurred in Baltimore County, Maryland between 1999 and 2001. 

Factors associated with both the residence location of the drivers and the crash location were 

identified. The crime travel demand model was used to simulate the likely outcome of 

concentrating on a few zones with targeted interventions. It was estimated that a 7.5% reduction 

in DWI crashes could be obtained by targeting 3% of the origin zones and 6% of the destination 

zones with anti-DWI efforts. The full study can be found in Levine, N. & Canter, P. (2011), 

Linking origins with destinations for DWI Motor Vehicle Crashes: An application of crime 

travel demand modeling”. Crime Mapping, 3, 7-41. 
 

 

 

 

 

 

 

 

Ex

Targeting Crime on Public Transport: 

An Example from Greater Manchester, England 

Daisy Smith & Steph Winstanley 

Strategic Analytical Partnership Co-ordinators 

Greater Manchester Against Crime Central Team 

The aim of the Greater Manchester Against Crime Central Team was to provide GMPTE 

(Greater Manchester Passenger Transport Executive) with an evidence base for their 

resources to address incidents of crime and anti-social behaviour on public transport during a 

Greater Manchester Partnership Day of Action. The analysis made use of the Crimestat Crime 

Travel Demand module to map the ‘journey to crime’ (home address to offence location) 

taken by personal robbery offenders within Greater Manchester. As a result GMPTE were 

able to identify their role in the partnership operation as they could easily visualise the bus 

and Metrolink tram system routes that ran coterminous with the most frequent journeys taken 

by offenders. 

Personal Robbery: Internal Offending and Predicted Cross Border Trips 

The map shows a high number of predicted trips (10 to 

30) between Stockport and central Manchester. This 

mirrors the route of the 192 bus service, which travels 

along the A6 between Stockport and Manchester that was 

then identified as a main target for the Partnership Day of 

Action. 

During the Day of Action, Gateway checks were conducted on the key public transport routes 

(bus routes and the Metrolink tram system) that were identified through Crimestat analysis. 

The Gateway checks consisted of staff from a range of agencies deployed on static and 

mobile patrols in order to identify fare evasion/ fraud and conduct intelligence checks. The 

agencies involved included Greater Manchester Police, GMPTE, Carlisle Security 

(independent enforcement agency) and the UK Border Agency. The public transport routes 

identified through Crimestat were targeted with much success and resulted in 7058 

passengers being checked, 496 buses boarded, 76 people identified without valid tickets, 28 

intelligence checks, 22 Bus Operator penalties issued and 22 arrests (including possession of 

illegal substances, robbery, fraud, outstanding warrant). The total fraud prevented through the 

Gateway Checks was estimated to be £3784.50 and extremely positive feedback was received 

from all agencies involved.