Abstract

The journey to crime is a well-researched phenomenon in environmental criminology.  Various studies have measured the distance between criminals’ homes and their crimes, while examining the influences of offender and offense factors.  Other research has explored the nature and characteristics of criminal travel distributions, including changes in offending probability by distance.  This last requires an understanding of the shape of the journey-to-crime function, particularly its key components of distance decay (offending probability decreases with distance) and the buffer zone (a lower probability area surrounding an offender’s home).  However, it is difficult to study the shape of these distributions because of the data requirements.  This article offers an overview of the buffer zone, its definition and characteristics, and two proposed explanatory theories.  We then outline four critical research considerations and the ecological fallacy problem.  Finally, we propose a testing procedure for the presence of a buffer zone in a sample of individual-level crime trips.  Using simulation analysis, we conclude 50 or more observations are needed to reliably determine the shape of an offender’s travel distribution, a number much higher than found in previous studies.  We provide two case studies of prolific predatory offenders that show strong evidence of the buffer zone.

1. INTRODUCTION

The journey to crime (JTC) is an integral component of many criminal acts. It is also one of the most studied offender behaviors in criminology; for almost 100 years, researchers have measured the distance between criminals’ homes and their offenses, while examining the influences of crime type, offender age, sex, race, and other factors. An understanding of crime travel is important for both theoretical and practical reasons. Crime trips are of central interest in environmental criminology, particularly for crime pattern and routine activity theories. Travel possibilities influence offender decision-making, target choice, and spatial displacement (Rossmo & Summers, 2019, 2022). Crime scripts are informed by mobility, and findings from JTC research help focus criminal investigations (Beauregard et al., 2007; Rossmo, 2000). Such knowledge also guides crime prevention strategies and risk assessments.

While early research was primarily interested in crime journeys at the macrolevel, more recently scholars have begun to explore microlevel questions. The development of powerful computers, police record management systems, and geographic information system (GIS) software greatly facilitated this avenue of research. One of the better studied microlevel phenomena is the nature and shape of the JTC function (e.g., Ackerman & Rossmo, 2015; Davies & Dale, 1995; Phillips, 1980; Rossmo, 2000; Sapp et al., 1994; Turner, 1969; Warren et al., 1998). Most of these studies, however, have been limited to aggregate offender patterns, which do not reflect the probability morphology of individual crime trips. This is the level relevant for understanding discrete offender spatial behavior.

The two defining characteristics of individual JTC distributions are distance decay and the buffer zone (Brantingham & Brantingham, 1981). The former describes the decrease in offending probability with the increase in distance from the offender’s residence, while the latter is an area of lower probability immediately surrounding that home. In combination, they create a function that looks in cross section like a volcano with a caldera (Rossmo, 2000). Several studies have examined the first feature (Block & Bernasco, 2009; Emeno & Bennell, 2013; Kent, Leitner, & Curtis, 2006; Rengert, Piquero, & Jones, 1999; Rossmo, 2000; van Koppen & de Keijser, 1997). Little research, however, has been done on the second, hindering attempts to understand the shape of the JTC distribution.

As distance is a quantitative measure, offender travel can be mathematically modeled to provide predictive capabilities. But to do this correctly, the structure of the function must be known. It is surprisingly difficult to study the overall shape of JTC distributions because of the intensive data requirements. If JTC curve fitting only becomes a connect-the-dots exercise, projections cannot be made from the data; robust equations must inform the curves before they can serve as analytic tools (Arlinghaus, 1994). Here, we explore the buffer zone concept in detail, discuss measurement issues and methodological challenges, and outline important research considerations.

This article begins by discussing the definition of a buffer zone and its spatial characteristics. The two explanatory theories proposed for its existence are outlined next. Research criteria for studying the buffer zone – crime types, crime journeys, distance resolution, and number of crimes – are then presented, followed by an explanation of the ecological fallacy. Finally, a simulation analysis is used to illustrate requirements and techniques for detecting the buffer zone.

2. BUFFER ZONE

1. Definition

Journey-to-crime1 distributions typically exhibit distance decay, consistent with rational choice theory and the principle of least effort (Clarke & Felson, 1993; Zipf, 1949). Several researchers have also observed an opposite pattern in the region directly surrounding an offender’s residence. This buffer zone was first reported by Turner (1969) in his study of juvenile delinquency and crime trip distances in Philadelphia:

The offender tends to commit offenses nearer to his residence, and his tendency wanes as distance increases. However, very close to his residence, say a block or two, he is less likely to commit as many offenses as we would expect. (p. 17)

Figure 1 displays the standard conception of a criminal search pattern incorporating distance decay and a buffer zone (Brantingham & Brantingham, 1981, Figure 1.2, p. 33).

FIGURE 1 Criminal search pattern with distance decay and buffer zone

Various names have been used for the buffer zone, including the “coal-sack effect” (Newton & Swoope, 1987) and the “safety space” (Canter & Larkin, 1993). However, the notion of “safety” is misleading and misinterprets the buffer zone as an area with no criminal activity. This is incorrect. A decrease in probability is not the same as zero probability; in fact, the likelihood of offending can still be relatively high within the region of the buffer zone, as can be seen from Figure 1. It is the direction of change in offending probability that is relevant here, not absolute magnitude.

2. Size

The buffer zone size is equal to the distance (radius) from the offender’s residence to the point where offending probability by distance changes from increasing to decreasing. More formally, the derivative of the crime-distance probability function is positive within the buffer zone, zero at the radius, and negative outside (after distance decay begins). The highest point of this curve is at the radius and represents an optimized balance between risk minimization and opportunity maximization.

Turner (1969, p. 24) suggests the buffer zone occurs within a block, perhaps two, of an offender’s residence. In Philadelphia, where his study was conducted, a block is approximately 500 feet (152 meters) in length. However, Bernasco and van Dijke (2020), noting a lack of standardization for the concept, point out that the size of a buffer zone depends on its causal mechanism:

If the presumed mechanism underlying the buffer zone is the offender’s fear of being recognized by local residents, it seems that any buffer zone detected should be within a range of up to 500 m (i.e. an area of .79 km²) in dense urban areas and up to 1000 m (i.e. an area of 3.13 km²) in more rural areas. At larger distances fear of recognition by local residents seems implausible. If, however, the presumed underlying mechanism is availability of suitable targets, deciding on an appropriate threshold value is more complicated, as it will depend on the type of crime under consideration and on the spatial density of potential targets. Because targets may not be available nearby the offender’s home, a minimum home crime distance of several kilometers might be a natural condition for some types of crime, such as a commercial robbery.... (p. 6)

A buffer zone may therefore vary substantially in radius and area depending on its underlying cause, the specific offense type, and neighborhood characteristics.

3. THEORIES

Two different theories have been proposed for the presence of a buffer zone in journey-to-crime distance distributions: (1) a desire for anonymity by criminals; and (2) the geometry of crime target opportunity. These causal explanations are not mutually exclusive, and both may operate on occasion.

3. Offender Anonymity

The offender anonymity hypothesis regards the buffer zone as an area centred on a criminal’s residence within which targets of instrumental crimes are viewed as less desirable because of the perceived risk associated with offending near home.

In their search behavior criminals are looking for “good” victims or targets. Part of what makes a victim or target “good” or “bad” is availability, potential payoff, and the risk of apprehension or confrontation associated with it.... Information about potential victims and targets is probably spatially biased toward the home base, but information is also spatially biased for the other people who live close to the criminal’s home base. While criminals know more of the area close to home and are more likely to locate a target easily, they are also more likely to be known and increase their risks close to home. One would expect that there would be an area right around the home base where offenses would become less likely (see Figure 1.2). (Brantingham & Brantingham, 1981, pp. 31-32)

It has been mistakenly argued under the anonymity hypothesis that a buffer zone would not apply to covert crimes such as break ins or theft. The reality is that most offending involves some risk of identification, and burglars and thieves are worried about being seen by witnesses or surveillance cameras before, during, and after the criminal act (Cromwell, Olson, & Avary, 1991; Rossmo & Summers, 2019; Wright & Decker, 1996). Offending near home is even riskier for criminals with a criminal record or who are known to police.

4. Target Opportunity Geometry

The crime opportunity/target availability hypothesis is derived from a simple geometric relationship (Rossmo, 2025). As distance from an offender’s residence increases, so does the number of potential targets. The combination of a linear increase in opportunity with a nonlinear decrease in travel inclination produces a distance-decay function with a buffer zone.

The manner in which an offender searches for criminal targets influences the shape of his or her crime trip distribution. According to crime pattern theory, the intersection of an offender’s awareness space with the target backcloth (the spatial-temporal pattern of suitable targets/victims) determines where and when crimes will most likely occur (Brantingham & Brantingham, 1981, 1984). Criminals search outward from their residence, work, or social locations, generally following some form of distance-decay (Rhodes & Conly, 1981). The probability of crime occurrence can be modeled by a Pareto function, starting from the nodes and routes that comprise an offender’s activity space and then decreasing with distance.

A journey-to-crime graph displays crime frequencies by distance from an offender’s residence, providing an estimate of spatial offending probability (see Figures 1 and 3). However, this so-called journey is often a composition of several search efforts, rather than a simple trip, occasionally integrated into non-criminal travel. The geometric pattern of crime occurrence is a function of street layout, target backcloth, offender travel, and criminal search techniques and hunting styles (Brantingham & Brantingham, 1981; Rossmo, 2025). The resulting two-dimensional pattern is then collapsed onto the single dimension of distance from offender residence to generate the typical journey-to-crime graph.

The buffer zone target availability hypothesis derives from a basic geometric relationship – there are more potential targets at greater distances from a given offender’s residence. The increase in criminal opportunity is linear;2 for example, there are twice as many potential targets at two miles from an offender’s residence than there are at one mile. Distance decay may be the product of the least-effort principle or simply the consequence of a first-choice mechanism. Travel probability is computed by a shifted geometric probability density function3 composed of multiple Bernoulli trial failures followed by one final success:

$D = dp{(1 - p)}^{s - 1}$ (Equation 1)

where:

D = total distance traveled

d = average distance of each search step

p = probability of search success

s = number of search steps.

Combining a linear increase in opportunity with a nonlinear decrease in the likelihood of travel (Figure 2) produces a distance-decay function with a buffer zone (Figure 3). The probability of target/victim selection (search success) influences the shape of this distribution and its journey-to-crime distances, with longer crime trips the product of smaller probabilities (p, in Equation 1) that result from low target densities or more discerning offenders.

FIGURE 2 Travel probability decay and opportunity increase by distance

FIGURE 3 Offending probability by distance

Changes in opportunity over distance depend upon the offender’s specific search process. For example, a burglar who simply travels linearly will encounter similar target frequencies in each distance interval (assuming an even distribution). “A constant number of offenses would only occur in the unlikely event of the criminal always traveling in a straight line in the same direction” (Rengert et al., 1999, pp. 436-437). However, straight-line travel is not an ideal least-effort strategy as crime journeys must also include return trips. More optimal would be a method of search wherein the sum of the distance to the next crime opportunity and the distance to return home is minimized. This results in a spiral-shaped hunting strategy, a two-dimensional pattern often found in animal foraging behavior (Smith, 1974a, 1974b; Smith & Sweatman, 1974).

As spiral searches have also been observed in criminal hunting, this pattern may provide a better description of how offenders find and select targets (Rossmo, 2025). A burglar’s decision of which house to break into is usually not spontaneous. Many offenders first become aware of potential targets by being attentive to their surroundings during routine daily activities; repeated observations then allow for risks and rewards to be assessed (Cromwell, Olson, & Avary, 1990, 1991; Rossmo & Summers, 2019; Wright & Decker, 1996). When a burglar makes the decision to offend, a number of possible options already exist for consideration. Criminal target selection is therefore a mix of both physical and mental processes involving many potential targets in a given neighborhood.

Rossmo et al. (2011) studied the spatial-temporal patterns of 14 reoffending parolees on the Florida Department of Corrections electronic monitoring and global positioning system (EM GPS) program. To obtain an accurate picture of their crime travels and other spatial activities, movement data were mapped and analyzed over a period of eight days, the seventh of which was the day of the crime. Amongst other quantities, distance traveled daily, number of sites visited, area covered, trip density, time spent at home, and time traveling were measured. The analysis suggests that the number of turns and the distances between turns were useful indicators of spiral search and criminal hunting behavior.

Raymond Lopez, known as the Chair Burglar, was a serial residential burglar in Southern California who employed a spiral search strategy (Rossmo & Velarde, 2008). After he was identified as a suspect from a geographic profile, police surveilled Lopez using a GPS tracker covertly attached to his vehicle. During one particular 32-minute period of this surveillance, Lopez traveled 6.7 miles at an average speed of 12.6 mph, but only covered 3.2 miles of unique roads4 – meaning he repeatedly drove over the same city blocks (52%). The target search pattern also involved numerous turns; during the half hour depicted in Figure 4, Lopez made approximately 50 turns, an average of one every 38 seconds.

FIGURE 4 Serial burglar search pattern

4. RESEARCH CONSIDERATIONS

Research on the JTC buffer zone must consider the following issues, the first two of which are related to appropriateness, the last two to detectability:

1. Crime types. Analyses need to be limited to only those crime types for which the buffer zone is predicted to occur.

2. Crime journeys. Offenses that do not involve travel should be excluded to avoid zero-distance crimes that distort the nature of the crime-distance probability function.

3. Distance resolution. The resolution of the distance measures must be sufficiently fine to detect the presence of a buffer zone, which may be as small as a city block.

4. Number of crimes. A large number of offenses committed by the same offender are necessary to reliably determine the shape of that individual’s crime trip distribution. The unit of analysis is the entire distribution, not single crime locations.

The first two issues are proscriptive, the last two prescriptive. Their rationale is discussed below; while all four are important, we focus on the last point as it is the most challenging for researchers.

5. Crime Types

The literature is clear that a buffer zone is only expected for certain crime types or subtypes. It is most likely to be present in instrumental and property offenses that are goal directed or planned, or for predatory crimes involving offender search behavior. It is less likely to occur with high-affect and violent crimes, risky offenses, or crimes lacking search behavior, such as domestic violence, date rape, argument homicides, drug use, Internet fraud, and the like (Brantingham & Brantingham, 1981, 1984; Turner, 1969). When examining juvenile arrests in Lexington, Kentucky, for example, Phillips (1980) found a peak in assaults at the offender’s residence but a buffer zone in the 1,000-foot range for petty larceny.

6. Crime Journeys

As the buffer zone is a spatial feature of the journey-to-crime distribution, any effort to study it must analyze actual crime journey data. It is particularly important to exclude offenses with no travel as these distort the distance function and obscure the buffer zone.5 While the potential risk of being observed by witnesses increases with proximity to an offender’s home, they dramatically decrease inside the home. Criminals who offend in their own residence have control over the environment, and by definition have a suitable target at hand; therefore, neither of the postulated reasons for the existence of a buffer zone – anonymity or decreased target availability – comes into play. There is an obvious connection between crime type and crime journeys.

7. Distance Resolution

The resolution of a study’s distance measurements (i.e., the precision of the data, or the width of crime trip distribution bins) must be sufficiently fine to detect the existence of a buffer zone. A buffer zone radius may be as small as 500 feet in urban areas. If the resolution is too coarse, the buffer zone will be subsumed into the next (usually higher-frequency) distance interval and disappear. For example, rounding data to half-mile intervals (2,640 feet) risks obscuring relevant detail. Distance resolutions that are too fine may also be problematic, however, as more cases will be required to reveal the shape of the crime trip distribution (as discussed below).

8. Number of Crimes

A small number of data points does not provide sufficient information to accurately establish the shape of a travel distribution. Andresen, Frank, and Felson caution, “most offenders are not prolific and, hence, may not have enough observations regarding their criminal behavior to confirm or deny a distance-decay propensity when it may, in fact, exist” (2014, p. 315). This is not to say an inherent travel probability function is absent, only that it cannot be reliably recreated from only a few crime locations.6 It is therefore only possible to observe a buffer zone in the distance distributions of prolific serial offenders who have been linked to multiple known crimes.

This challenge is often an underlying contributor to crime type/journey and distance resolution problems. In an effort to obtain larger numbers, researchers sometimes sample inappropriate crimes or group them in overly large bins. An unfortunate complication is the lack of guidance in the literature on the number of observations necessary to accurately estimate an individual’s JTC distribution.

Some researchers suggest at least 5 observations are needed for each continuous class interval, with 30 observations usually sufficient to fit a normal distribution.7 However, more information is required to determine the parameters of an unknown distribution. Sources recommend using 20 observations per degree of freedom (or per DF+1), with 60 to 80 observations for a robust fit (the number of categories in the distance distribution equate to its degrees of freedom). Simulation studies are advised for greater confidence. We follow this guidance in the next section.

Others have tried to circumvent this issue by aggregating single crimes from multiple different offenders. However, such mixtures can produce a variety of JTC distribution shapes in the aggregate because of natural variations in individual travel (see Andresen et al., 2014; Drawve, Walker, & Felson, 2015; Townsley & Sidebottom, 2010). Unless all the crime-distance probability distributions are similar, their groupings can distort the shape of the function and mask its characteristics, including the presence of a buffer zone. This problem, known as the ecological fallacy, is discussed below.

Ecological Fallacy

The buffer zone, as defined in the environmental criminology literature (Brantingham & Brantingham, 1981), contextualized in geographic profiling (Rossmo, 2025), and discussed by Bernasco and van Dijke (2020), is an individual-level phenomenon. It reflects a decreased probability of offending by an offender in the area near their residence. Merging different individual crime trip distances without first standardizing the data creates an ecological fallacy, changing the JTC distribution’s shape and obscuring the presence of a buffer zone.8

An ecological fallacy is the incorrect inference of an individual-level process from grouped data (Langbein & Lichtman, 1978). As most social science databases are aggregations, convenience has typically been the justification for their use in studies of individuals (Robinson, 1950). However, this expediency risks distorting relationships at the individual level. “Estimates of how individuals behave based on grouped data may be misleading if the process of aggregation alters parameter estimates or blurs individual or contextual effects” (Langbein & Lichtman, 1978, p. 61). Estimating risk from aggregated data is a well-known problem in criminology, as both crimes and offenders exhibit significant geographic clustering (Sherman, Gartin, & Buerger, 1989; Spelman & Eck, 1989).

The only way to detect the presence of a buffer zone is to examine the crime-distance function for a particular single offender (Brantingham & Brantingham, 1993). It cannot be derived by simply combining data because aggregating individual crime trips produces a collective distribution very distinct from individual distributions. Different offenders engage in different spatial behavior. One walks to his crime sites, another rides a bicycle, while a third drives a vehicle. One offender, inexperienced, impatient, and careless, tends to offend in his own neighborhood, while another is experienced, careful, and risk averse. Yet a third might specialize in a particular type of crime, which requires greater search efforts and travel to more distant locations. Rengert, Piquero, and Jones warn, “researchers cannot and should not make inferences about individual behavior with [spatial] data collected at the aggregate level” (1999, p. 432; see also van Koppen & de Keijser, 1997).

The journey to crime is particularly susceptible to aggregation problems and the ecological fallacy (Smith, Bond, & Townsley, 2009, p. 217). Aggregate-level models are only surrogates for models of individual-level behavior and their equivalency should not be assumed. “Investigators should always ask ... what changes does the process of grouping introduce in the proper specification of individual level relations?” (Langbein & Lichtman, 1978, p. 61). The onus is on the researcher to establish that data aggregation does not undermine integrity (Robinson, 1950).

Theory provides some guidance for accurate ecological inference. The appropriateness of potential solutions (e.g., homogenous groups, methods of bounds, additional variables) cannot be determined without first understanding the nature of the aggregation bias (see Langbein & Lichtman, 1999, for a review of solutions to this problem). The radius of an offender’s buffer zone – which may be anywhere from 150 meters to several kilometers – depends on its underlying causal mechanism, the relevant crime type, and such area characteristics as population density, land use, and target backcloth. Compare Turner’s finding of one block to that of one mile for serial arsonists observed by Sapp et al. (1994).9 Combining the spatial patterns of disparate offenders can distort their individual configurations and wash out important detail, including the presence of a buffer zone.

The column graph in Figure 5 was derived from aggregating the simulated journey-to-crime distributions of 20 offenders, all of whom followed the same basic travel function (shown in insert), but with varying buffer zone radii (2 to 40 units). The bin size was set to 4 units (different resolutions produce similar results). Even though each individual distribution had a clear and sharply demarcated buffer zone, this feature disappears from the grouped distribution pattern.

FIGURE 5 Aggregated journey-to-crime distributions

To resolve these aggregation problems, data collected on serial offenders’ journeys to crime should be standardized before comparison. One method for doing so involves converting the distance data for a given offender into a probability function, and then dividing each individual crime trip by their median travel distance (see Rossmo, 2025). While imperfect, this approach helps address the major issue of varying spatial scale in crime journeys. For example, Warren and her colleagues (1998) failed to find a buffer zone in their first analysis of aggregated serial rape journey-to-crime data; when they normalized the individual-level journey-to-crime distributions, however, a distinct buffer zone emerged (Raine, Rossmo, & Le Comber, 2009).

It has been argued that requiring the use of individual measures to study individual phenomena is too strict and could disqualify a substantial portion of empirical work in criminology and other social sciences. However, our primary concern must be with scientific integrity, not the difficulty of the research or the proportion of the literature that is flawed. When he first published his work on the ecological fallacy, Robinson recognized its serious consequences because of the doubt it threw upon the validity of some important studies (1950, p. 357).

5. SIMULATION ANALYSIS

Samples in most JTC studies include only a single case per offender (Rengert et al., 1999). As such, it is not possible to verify whether individual crime trip distributions have a buffer zone. This is the motivation for our simulation analysis – how many observations are necessary to accurately estimate an individual’s JTC distribution?

We show using simulation analysis that to identify a buffer zone in JTC data typically requires a minimum of 50 observations – a threshold rarely met when examining individual JTC distributions. We then analyze two case studies of serial predatory offenders, each responsible for a large number of crimes (79 and 129); in both cases, their JTC distributions closely conform to the buffer zone hypothesis.

9. Analysis

We begin the analysis by simulating random variates from a gamma distribution. We simulate from a gamma distribution because it is a fairly standard, two parameter distribution, that can reasonably approximate the buffer zone hypothesis. It offers a simple test with smaller sample size requirements and standard errors than more complicated distributions.

The probability density function (PDF) for the gamma distribution is defined by its shape, $\alpha$, and its rate, $\beta$:

$f(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha - 1}e^{- \beta x}$ (Equation 2)

Some formulations of the gamma distribution parameterize the distribution in terms of scale instead of rate; however, these are simply reciprocals.

The gamma distribution has a positive mode when its shape ($\alpha$) is above 1, consistent with the buffer zone hypothesis. When $\alpha > 0$, the mode for the distribution is at $(\alpha - 1)\text{/}\beta$. When $\alpha$ is below 1, the PDF monotonically decreases, consistent with a distance-decay hypothesis. The rate parameter is somewhat arbitrary; varying the distance units (e.g., from miles to kilometers) will change the rate but not the shape estimate. Figure 6 illustrates five example gamma distributions, with shapes varying from 0.8 to 1.6 ($\beta$ = 1). The gamma distributions shapes of 0.8 and 1 monotonically decrease; shapes greater than 1 show a buffer, with larger shapes corresponding with more flattened distributions.

FIGURE 6 Density of gamma distributions ($\beta$ = 1; $\alpha$ varying from 0.8 to 1.6). Shape parameters greater than 1 are consistent with the buffer zone hypothesis

This graph illustrates why we use the gamma distribution for our simulations – it offers a simple test ($\alpha > 1$) and has smaller standard errors around estimates compared to distributions with more parameters. Hence sample size requirements for more complicated distributions are likely to be larger.

Imagine we fit a gamma distribution to a simple set of 6 JTC distances: {1.5, 0.4, 0.8, 1.9, 0.3, 2.5}. We estimate the parameters (standard error) as shape of 2.2 (1.2) and rate of 1.3 (0.8). Here we can see that the point estimate for the shape parameter is above 1, but the standard error is quite large. A null hypothesis significance test of $\alpha = 1$ would fail to reject the null hypothesis. This test, however, does not confirm the buffer zone hypothesis. One could switch the null to a non-inferiority type test, common in drug testing (Walker, 2019), where the null is a shape parameter sufficiently above 1 to confidently confirm the buffer zone hypothesis. This is typically done by testing whether the lower bound of the confidence interval is some pre-specified value above 1. Here the lower bound of a 95% confidence interval is well below 1, so we would reject the null hypothesis in that particular inferiority testing scenario.

With a larger sample of JTC distances, say 50 observations, we can estimate the gamma distribution’s shape as 0.6 (0.1), clearly more discriminatory. This particular (hypothetical) offender’s JTC distribution is not consistent with a buffer zone hypothesis. The point estimate for the shape is below 1, with a standard error small enough for us to be reasonably confident in our inference.

Most published samples of JTC distributions have nowhere near 50 observations per offender. Townsley and Sidebottom (2010), for example, used an average of 10 burglaries per offender in their sample – a relatively large number in comparison with similar studies. Here we provide simulation evidence, given a range of different shape parameters, to determine how large a sample is necessary to confidently estimate a shape parameter in order to confirm or falsify the buffer zone hypothesis for any particular offender’s JTC distribution.

Table 1 shows simulated gamma random variates, fixing the rate parameter to $\beta$ = 1 but varying the shape parameter to either be monotonically decreasing or to have a buffer zone. We provide the mean standard error estimates over these 10,000 replications. In addition, as these estimates are fit via maximum likelihood, there is a potential small sample error. As such, we additionally estimate the bias, defined as “estimate – true.”

If the distribution’s shape is 0.6, sample sizes of 10 have both very large standard errors (on average 0.31) and a large positive bias. The latter is not surprising as a negative shape parameter is not possible. Sample sizes of 10 are therefore much too imprecise to either confirm or falsify the buffer zone hypothesis.

With a shape of 0.6, and 20 observations, the average standard error in the simulations is 0.18, with a positive bias of 0.08. Combined, these make the majority of the confidence intervals in the simulations cover 1. A sample size of 50 is needed to achieve reasonable precision (and lack of small sample bias) in falsifying the buffer zone hypothesis. But with shape parameters near 1, even larger samples are required. If we examine parameters between 0.8 and 1.2, we find sample sizes as high as 50 are still too imprecise to confidently discriminate between data consistent with the buffer zone hypothesis and strictly monotonically decreasing distances.

TABLE 1 Simulations for different gamma shapes, showing the mean standard error and bias for the fitted maximum likelihood estimate for the gamma distribution over 10,000 replications

10. Example Fitting Non-Parametric Density

Specifying a particular parametric distribution for an individual’s JTC distribution is arbitrary. While the gamma distribution can qualitatively capture distance decay or a buffer zone, it is not directly derived from our prior explicated theory of the supposition of the backcloth opportunities and particular offender travel patterns.

To entirely avoid relying on parametric distributions, it is common to employ non-parametric kernel density estimators (KDE) (Townsley & Sidebottom, 2010). Prior analyses of KDE estimators often use kernels that spread the density below zero distances, which is not possible (traveled distance must be a positive number). Here, we use the technique of calculating the density on the log scale and retransforming the density back to the original scale (Cox, 2004). This tends to provide better estimates of the density around 0 (Wand, Marron, & Ruppert, 1991), which is what is wanted in this situation. We then employ bootstrap resampling to estimate pointwise confidence intervals around the density estimate. We use a normal kernel, again on the log scale, for each of the graphs.

Case Study 1: Paperbag Rapist

The Paperbag Rapist series in Vancouver, British Columbia, involved 79 reported crimes (Alston, 1994; Eastham, 1989). Table 2 displays the gamma estimates for this case study, with a shape parameter of 2.1 and a standard error of 0.3. This is strong confirmatory evidence for the presence of a buffer zone for this particular offender.

Figure 7 shows the estimated gamma density (black line) and the non-parametric KDE (blue line with lighter blue areas signifying the 95% confidence interval of the density). The KDE uses a bandwidth equal to ln (3). The plot also shows a rug of markers, illustrating the exact observed distances in the sample, at the bottom of the plot. The KDE suggests a much more peaked and declining buffer, relative to the estimated gamma distribution. The KDE estimator shows the mode of the distribution at 1.8 kilometers, whereas the gamma estimate for the buffer mode is much larger at 9 kilometers.

TABLE 2 Gamma estimates for the two case studies (distances in kilometers)

FIGURE 7 Estimated gamma density (black line) and non-parametric KDE (blue line with lighter blue areas signifying 95% confidence interval of density)

Case Study 2: Covered Face Bandit

The Covered Face Bandit series in Orange County, California, involved 129 indecent exposings. As shown in Table 2, the gamma distribution estimate for this offender is 2.0 with a standard error of 0.2, again confirmatory evidence of the buffer zone hypothesis. The non-parametric KDE estimate, using a bandwidth equal to ln (2), is again shifted left (same as for Case Study 1), but comparatively is not as peaked (see Figure 8). The gamma distribution estimated mode is at 4 kilometers, whereas the KDE estimated mode is at 1.7 kilometers.

Both case studies offer strong confirmatory evidence of a buffer zone. They also suggest the gamma distribution may not be the best fit as the KDE distribution has a more peaked buffer and a longer tail. The inverse-gamma or Lévy based distributions might be better default options for offender crime journeys than the gamma (Brantingham & Tita, 2008). We leave such analysis to future researchers – our focus on the gamma distribution was intentional (as these alternate distributions cannot fit a monotonically decreasing PDF they cannot be used to confirm a simpler distance-decay hypothesis). KDE results, however, show strong promise to qualitatively estimate the JTC distribution in these examples given reasonable sample sizes. We suggest a sample size of 50 observations is likely sufficient, based on a combination of our simulations and case studies.

FIGURE 8 Estimated gamma density (black line) and non-parametric KDE (blue line with lighter blue areas signifying 95% confidence interval of density)

11. Simulation Contributions

The simulation analysis provides two contributions. First, it gives us practical advice about the sample sizes necessary to effectively test the buffer zone hypothesis using individual-level JTC data. Approximately 50 observations are needed to estimate the shape parameter of the gamma distribution with sufficient precision to confirm or reject the buffer zone hypothesis. Smaller sample sizes will likely have standard errors too large to usefully confirm (or falsify) the buffer zone hypothesis. While more observations will increase the estimation precision of a specific offender’s JTC distribution, even the non-parametric KDE showed reasonable discrimination in identifying either a monotonically decreasing distance-decay like function, or a positive mode consistent with the buffer zone hypothesis (see Appendix B).

Our case studies, given their large sample sizes, show strong evidence confirming the buffer zone hypothesis for both the gamma distribution estimates and the non-parametric KDE. Given our initial premise that it is necessary to examine individual crime trips to test the buffer zone hypothesis, it should also be clear we will never be able to confirm whether the hypothesis holds for every predatory offender. What we can do is show whether it tends to hold in the particular circumstances we expect it to. We also provide advice and replicable code for those who wish to conduct similar analyses with other samples of JTC distance data.

6. DISCUSSION AND CONCLUSION

Research on offender mobility has informed police investigations, crime prevention strategies, and environmental criminology theory. GPS and cell phone data provide novel opportunities for investigating the details of individual crime trips and for analyzing the morphology of the journey-to-crime. The buffer zone is a critical component of realistic crime-distance probability functions. Knowledge of the nature of the distribution and its underlying causes is relevant for many theoretical and practical purposes.

We obviously cannot claim a buffer zone exists in all JTC distributions as the necessary evidence is lacking. Reliable measurements are not possible in the absence of a surfeit of data. Differences invariably exist between individual offenders in their distance distributions, distance-decay rate, buffer zone presence and size, and travel range. Criminals vary in their propensity for travel as demonstrated in multiple prior studies (Andresen et al., 2014; Drawve et al., 2015; Rossmo, 2025; Smith et al., 2009; Townsley & Sidebottom, 2010). However, if the buffer zone was absent or rare, we would expect to see more predatory and instrumental crimes occurring near offender residences and much shorter median journey-to-crime distances.

Individual variation notwithstanding, general patterns exist. Distance decay, for example, is a long-established concept used in migration research, transportation modeling, and analyses of shopping behavior (Haynes & Fotheringham, 1984; Johnston, 1973). The buffer zone is an important concept in geographic profiling, an investigative methodology that outlines the probable area of offender residence by analyzing the location pattern of a crime series. It uses a suspect prioritization algorithm based on a crime probability distance-decay function with a buffer zone (Bernasco & van Dijke, 2020). Prior research has demonstrated that an effective geoprofiling algorithm requires an optimal balance between buffer zone radius and distance-decay exponent for the structure of its crime probability function (Rossmo, 2005).

Studies in biology and zoology using geographic profiling have found the buffer zone parameter to be the single most important variable in describing and differentiating animal foraging patterns (Le Comber, Nicholls, Rossmo, & Racey, 2006; Raine, Rossmo, & Le Comber, 2009; Suzuki-Ohno, Inoue, & Ohno, 2010).

The nature of the journey-to-crime distribution and the characteristics of the buffer zone require further study. Here, we discussed some of the measurement and methodological challenges that must be managed in such research, including the need to analyze only appropriate offense types involving actual crime trips, measure distances precisely, study individual offenders to avoid ecological fallacies, and collect a sufficiently large sample of data.

The last challenge is the most problematic. The number of samples required to differentiate between JTC distribution shapes is much larger than noted in previous research. Both the gamma distribution (with a subsequent shape parameter test) and a non-parametric KDE suggest at least 50 data points are necessary to test the buffer zone hypothesis. Smaller sample sizes in a range of scenarios are too variable for effective discrimination.

Finally, we want to suggest the potential value of employing mixed methods in JTC studies. Offender interviews, for example, can provide a useful perspective on the dynamics of criminal search and travel (see Rossmo & Summers, 2019). Triangulation increases the reliability and validity of any research, and including a qualitative perspective on these questions may help address some of the limitations in the quantitative data.

APPENDIX A

Deriving Closed Form Estimates of the Gamma Distribution Standard Errors

Here, we derive the exact standard error of the rate parameter for a gamma distribution, based on the Fisher Information matrix. The Fisher information matrix for the gamma distribution with shape $\alpha$ and rate $\beta$ is:

where $n$ is the sample size, and $\psi^{1}$is the trigamma function. The variance-covariance of the parameters of the gamma distribution can be estimated via the inverse of the Fisher information matrix. For a 2 by 2 matrix, the inverse is:

The standard error of the shape parameter, under the inverse matrix, is the upper left corner, $d$, multiplied by the determinant of the Fisher information matrix:

We can then collect $n^{2} \cdot \beta^{- 2}$ terms in the denominator. This allows the $nB^{- 2}$ term in the numerator to be cancelled out, eliminating all rate terms. What remains is:

This is the variance of the shape parameter. The standard error is the square root of this value. While the standard error of the shape parameter is a function of the square root of the sample size (as is typical for most statistics), it is also tied to the size of the shape parameter itself. Larger shapes have larger standard errors.

Figure 9 displays the standard error behavior for shapes of {0.5, 1.0, 1.5, 2.0} and sample sizes of 10 to 100. It can be seen that higher shape parameters result in larger standard errors given equal sample sizes.

FIGURE 9 Standard error behavior

APPENDIX B

Non-Parametric Bootstrap KDE Estimators

Figure 10 illustrates the validity of the non-parametric bootstrap KDE estimators. The graph on the left is a simulation of a gamma distribution with shape 0.7 (N = 50 observations). There is a close agreement between the true distribution (black line) and the estimated distribution (blue line, with associated 95% confidence interval). The graph on the right, with shape parameter 1.5 (N = 50 observations), shows a slight bias in the kernel density estimator, and is slightly shifted to the left. The KDE estimate shows the correct buffer. Each graph uses a bandwidth of ln (2).

This establishes that the KDE can uncover a monotonically decreasing function and our case study results are not the product of the KDE procedure itself.

FIGURE 10 Non-parametric bootstrap KDE estimators validity

Acknowledgements

Funding: No outside funding was used to support this work.

Availability of data and materials: See https://github.com/apwheele/buffer for replication code.

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AUTHOR BIOS

Kim Rossmo is a professor in the School of Criminal Justice and Criminology and the director of the Center for Geospatial Intelligence and Investigation at Texas State University.  He has researched and published in the areas of the geography of crime, environmental criminology, and criminal investigative failures.  Dr. Rossmo sits on several scholarly journal editorial boards and is a member of the IACP’s Police Investigative Operations Committee.  He was formerly a detective inspector with the Vancouver Police Department and then the director of research for the Police Foundation.  He received his PhD in criminology from Simon Fraser University.

Andrew Wheeler has a PhD in criminal justice from SUNY Albany (2015).  His research focuses on the applications of predictive policing and operations research within the criminal justice field.  He is currently a data scientist and consults with police departments and criminal justice agencies across the country.  See his portfolio of work at crimede-coder.com.