An analysis about the accuracy of geographic profiling in relation to the number of observations and the buffer zone

Geographic Profiling (GP) is a data analysis concept that aims to identify the origin of a series of events that can be represented on a map. It was firstly proposed in criminology as Rossmo´s Criminal Geographic targeting algorithm -CGT- (Verity et al. 2014; Gorski 2021) as a method to deal with serial killers (Butkovic et al. 2019) and later for the spreading of populations of invasive species, for targeting writers (Hauge et al. 2016) and also in epidemiology (Papini et al. 2013, 2017a; Santosuosso and Papini 2016).

The model requires that there is only one spreading centre and that this centre is surrounded by a buffer zone (Rossmo 2000), that is an area around the spreading centre itself where the probability of finding an event is extremely low. In criminology, but still more in biological invasions, the existence and the extension of the buffer zone should be checked case by case. B (the parameter used to represent the buffer zone around the offender’s home (see Fig. 1S in supplementary material), is a number that is between zero and the maximal distance of an event from the spreading point, otherwise the analysis would not produce any result. Moreover, in the model the probability of finding an event tends to decline proportionally to the increase in distance of the event from the offender’s home.

A GP uses coordinates of points (corresponding to the events of interest) to calculate a probability surface called a geoprofile (Rossmo 2000). The geoprofiles provide areas of different priority on the map with a varied probability density (Rossmo 1993, 2000). The geoprofile consists in an approximated localization of an area (often represented as a red area) containing the spreading centre, where the probability of finding it is higher than a given threshold (typically 95%). After its first use in criminology, GP was applied to biological problems such as the targeting of an infectious disease (Papini and Santosuosso 2016), the prediction of nest locations of bumble bees (Suzuki-Ohno et al. 2010), animal foraging (Le Comber et al. 2006; Raine et al. 2009) and shark hunting patterns (Martin et al. 2009). The obtained results can be compared to those of other analytical methods of mapping the higher or lower probability of crimes occurrence in a given area, as in Quick (2019).

More recently, GP was used to guess the source of an invasion by alien organisms using the positions of their current populations (Stevenson et al. 2012; Cini et al. 2014; Papini et al. 2017a). In this case, in the place of the offender’s home in criminology, the spreading centre will be the first place of introduction of an alien organism with invasive capability (Papini et al. 2013). Such data are increasing with respect to the past, also due to the diffuse monitoring activity on the territory, as with the Citizen science initiatives (Baker et al. 2019). This analysis is useful, since the knowledge of the source of events can suggest hints about the offender’s home or give ideas about the control methods of the invasion of alien plants and animals (Cini et al. 2014). Recently, further refinements of the method were proposed to a) improve the reliability of GP from the point of view of possible presence of multiple waves of invasion, rather than from a single starting point (Cini et al. 2014), b) to increase the robustness of the results with a jackknife procedure (Papini et al. 2017b), c) to give different weights to data on a quantitative basis (for instance on the basis of the population dimension in a given point in the case of an invasion) or several methods of data partitioning (Santosuosso and Papini 2018; Cini et al. 2019). GP may even be associated with other methods that could record the presence of events to be considered outliers of the distribution, as with the Isolation Forest method as defined by Liu et al. (2012). This method was implemented by Santosuosso et al. (2020) on biological invasions. In order to overcome some of the pitfalls of GP analysis, some authors proposed alternative approaches, such as Dirichlet process mixture -DPM- (Verity et al. 2014), the Topological Weighted Centroid -TWC- (Buscema et al. 2018a,b) or the O'Leary's simple Bayesian model (2009, 2010, 2012), even if GP remained the method of choice for many analyses aiming to find a centre of a series of events. Stevenson et al. (2012) showed also that GP gave better results compared to other techniques in 52 of the 53 data sets explored for invasive species spreading analysis in Great Britain.

One of the main concern with GP is that some parameters must be provided by the users, such as the buffer zone dimension (the above mentioned B parameter). This parameter represents an area around the spreading point where the probability of finding an event is low, is easy to be understood in criminology, but less in the case of a biological invasion. Nevertheless, evidence of buffer zone in many cases of spreading of invasive species have been recorded and calculated for many organisms, both animal, plants and algae (Stevenson 2012). In these cases the B value must be evaluated case by case on the basis of the biology of the involved organism. It is important to understand how the approximation in providing this parameter (that must be chosen ad hoc, depending on the type of investigated events) can influence the precision of the results. Moreover, it must be kept in mind that the use of the method always produces a result, independently of the correctness of the field data (observations/events) and the precision of the parameters used for the analysis.

This investigation is not intended as a validation of GP, that has already been object of several contributions, but it aims, rather, to answer an important question: is it possible to calculate how the choice of the B parameter influences the result? Or in other words: is it possible to evaluate if the inevitable inaccuracy of the B parameter can anyway produce feasible results? And after how many observations/events? In this article we faced the problem by simulating the data sets with a variable error to be added to the B parameter (known a priori, since it was chosen during the simulation itself) and calculating how the growing inaccuracy of the B parameters affected the results, together with an increase in the number of observations. The idea is to find an evaluation about the lowest possible number of observations/events that can still provide a reliable geoprofile, together with the highest allowed variation (inaccuracy) of the B parameter. The question is relevant, since the number of available events is sometimes relatively low, independently of the researcher’s will, and the evaluation of the B parameters not completely understood, depending on the biological properties of the organism from the point of view of propagation.

The research question is hence: how “wrong” (inaccurate) can be the parameters chosen for the analysis, in order to obtain still a reliable result? And how many cases/events do we need to obtain a reliable result?

In order to generalize the results to other fields outside criminology, we will use in the next lines the terms of cases/objects/events instead of crimes; spreading centre or centre of origin instead of “anchor points” as previously used by other authors, depending on the type of data we are talking about. The model requires that all the observed events derive from a single spreading centre and that around this spreading centre there is a belt (the buffer zone), where the probability of finding an event is extremely low. The buffer zone is inserted in the model as parameter B. Another requirement is that the probability of finding an event decreases with the increase in distance from the spreading centre (Rossmo 2000). To describe this function the model requires parameters f and g that control the shape of the decay function (Rossmo 2000; Stevenson et al. 2012). F and g are typically set to 1.2 as values originally used in criminology (Stevenson 2013) and later evaluated as giving reliable results also in biological invasions (Stevenson et al. 2012). The model was shown to be much more dependent on variation of the B parameter, rather than of f and g (Le Comber et al. 2006; Raine et al. 2009; Butkovic et al. 2019) and this is the reason why we focused on B. However, for testing if the choice of the parameters influence too much the approximation of the results, we performed also an analysis on a solved case famous in criminology to check possible strong deviation from the known position of the offender’s home (see later).

The analysis based on a single event will always produce a single ring around the event of maximum probability of finding the spreading centre, with B as the radius (Fig. 1). The map should be chosen wisely, large enough, but not too large, to avoid part of the results being drawn outside of it (as shown, as an example, in Fig. 1).

Rossmo’s formula as modified by Papini and Santosuosso (2016): where B is the radius of the buffer zone, \(i\) and \(j\) are variable indexes varying from one to n (numbers of events) and representing the coordinates on the map. The parameters f and g control the shape of the distance-decay function on either side of the buffer zone radius, that is how fast the probability of finding an event increases or, respectively, decreases, moving away from the radius. F and g hence represent the reduced probability of dispersal within the buffer zone and the fact that dispersal probability declines with distance and their user provided values are relevant mainly very close to circumference formed by the radius.

\(W_{i}\) is the weight associated to an event (possibly corresponding to the population dimension in biological invasions, to the number of infections occurring at the same address or the number of crimes occurred in given place): it is one if all the events have the same weight; f and g are parameters that control the shape of the distance-decay function on either side of the buffer zone radius as better explained by Rossmo (1995) and Le Comber et al. (2006).

\(D(X_{i},j,C_{n})\) is the distance between \(x_{i,j}\) (point of spread) and \(C_{n}\) (coordinates of the observed event), that may be calculated through several different metrics (Table 1: 2-dimensions case). The most used distance for criminology (with events most frequently occurring in urban environment), is the Manhattan distance that takes into account the geometry of the streets network calculating a 90 degrees angle in the distance, while for investigating the pattern of distribution of an invasive species, the Euclidean distance is normally preferred. More recently even a distance taking into account the google maps routing system calculation has been proposed as a way to provide more exact models for travelling in urban areas (Stamato et al. 2021). As a matter of fact, the model of distribution of crimes or other events around the centre of origin, is approximated to a circular distribution that is a simplification of reality. The use of the Manhattan distance offers a partial correction in urban areas.

The model postulates that the “object” coming out from the origin and reaching its final destination in the recorded event location will reduce its probability of being in a given point on the map when the distance from the origin increases. Correspondingly, the production of one of the event will have a “cost” (from energy or economic point of view) proportional to the extent of its distance from the origin. For crimes committed by an offender, it would simply mean that the farther the offender travels, the more he/it will spend in terms of energy/money. If the distance exceeds a given value, the probability of finding an event tends asymptotically to zero.

The two components of Rossmo’s formula can be then considered as a Pareto’s distribution, that is the best solution/tradeoff or group of solutions that tend to nearing as much as possible the maximization of two (in this case) or more different parameters, with the name of the distribution after the mathematician Vilfredo Pareto (Carapezza et al., 2013). Here the model tends to minimize on the one hand the distance from the origin and, on the other hand, the probability of finding an event very close to the origin (the offender’s home, in criminological terms). A Pareto optimality represents the solution(s) to a problem in which more variables need to be optimized and the optimization of one of them affect the optimization of the other(s). For instance, in biology an organism must perform several conflicting tasks trying to maximize the general fitness (Steuer 1986; Shoval et al. 2012). A simple example for an organism is that of maximizing visibility to enhance probability of mating but trying not to be too visible to avoid predation.

The results obtained by executing the GP analysis on the simulated data sets with the different \(B\) values shown in Table 2 produced the following results: if the number of cases/events increases with time, as it happens normally by waiting (think about the examples of criminal events or invasion sites of an invasive species), the average distance of the barycentre of the (95%) red zone from the real spreading centre tends to decrease asymptotically (Fig. 4), as expected if the spreading model is correct. In Fig. 2S (supplementary material) the same calculation is represented, but taking into consideration the barycentre of the red + the yellow zone, that is the area containing the 90% of probability to find the spreading centre (containing the red zone).

After seven points (cases/events), the average distance of the barycentre from the spreading centre is less than 10% of the \(B\) value chosen for the reconstruction (Fig. 4).

By increasing or decreasing the B value used for the reconstruction from the real \(B\), the curves behave in a similar way. In other words: if we use \(B\) values that are +/-20% of the real \(B\) value (B = 120% of the real B or B = 80% of the real B), the average curves overlap with an error decreasing proportionally to the number of points (cases/events), see Fig. 4. These results confirm that increasing the number of events we are allowed to reduce the precision of the B value.

The concern about the B value (related to the buffer zone) was well clear since the beginning of the use of the geographic profiling method, with most authors declaring the need for an empirical evaluation of this parameter (Rossmo, 2000; Beauregard et al. 2005; Kent et al. 2006; Stevenson et al. 2012; Papini et al. 2013). These results provide evidence about how the error in this estimation may influence the results of the following geoprofiling analysis, and what an approximation we can use without significantly affecting the results.

The total number of simulations is relatively low (1311 simulated data sets) with respect to more recent analyses, and this limitations to our results is due to computational requirements for the calculations of the geographic profiling for each data set (with 6 different values of B). Lerche and Mudford (2005) suggested to evaluate the mean and standard error of an output of interest. The process should be repeated until one determines that the change in the mean value with the increase in number of runs is less than a pre-specified degree of accuracy. Marriott (1979) also considered the estimation of the error a critical evaluation to assess the number of replicates to obtain a reliable result. We observed the convergence of the results already with 15 simulations each with 23 data sets of events, and hence, we consider our results reliable for the parameters here used. A higher number of replicates may lead to a better generalization of our results.

To check the correctness of the previously tested algorithm with simulated data, we applied the method to a real case, that is the crimes committed by the so called Boston strangler (Albert De Salvo, https://​en.​wikipedia.​org/​wiki/​Albert_​DeSalvo) of whom the identity and home address was known after the trial, together with the locations of the crimes (Kanchan et al., 2015), listed in Table 1S and visualized on a map in Fig. 5 and with a zoomable view in https://​umap.​openstreetmap.​fr/​it/​map/​de-salvo_​237784. We considered B as a median value of the distance calculated in pixels on the map between the crimes location and the offender’s home. A list of the cases is reported in Table 1S (supplementary material), while Table 2S (supplementary material) shows the translation of the position of the crimes to the (x,y) positions on our map. Table 2S indicates also the distance, in pixels and Km, of the crimes from the offender’s home.

The results are reported in Fig. 6, expressing B as a percentage of the real B value. It corresponds to the analysis done with the simulated data.

Figure 6 reports the B values used for the sequence of reconstructions, normalized with respect to the supposedly real B value. The more that the used B value gets closer to the real B value, the more the distance between the red zone and the spread centre decreases.

Passing the real B value, the distance of the barycentre of the red zone from the offender’s home tends to rapidly increase again. Analogous results can be obtained for the dimension of the 90% zone. Hence, values close enough to the real B value or lower are able to approximate well the results. Increasing the number of cases, the distance between the curves tend to reduce and to overlap with the real value curve (Fig. 6). It means that the approximation of B can be compensated by increasing the number of events. We see also that passing 6–7 cases, the distance between the real values curves and +/- 7% appear to be low (Fig. 6).

This distance does not tend to zero as in the simulations, since the cases are not spread homogeneously around the spread centre, but are rather grouped in three different clusters. The results of the geoprofiling are shown in Fig. 7. The calculated red zone included the offender´s home (as from the trial). This result confirms that the choice of the f and g parameters (that are empirically chosen) did not affect too much the results of the analysis that is always to be considered as an approximation of the offender’s position.

The GP method is sensitive to variations of the B parameter. Hence, it is necessary to find criteria allowing the use of a B value, with an approximation of 5–20% in order to obtain feasible results with a number of observations lower than ten.