Файл: [D._Kim_Rossmo]_Geographic_Profiling(BookFi.org).pdf

© 2000 by CRC Press LLC

delineated by the Venn diagram in 

Figure 10.1

 are given in Equations 10.1

to 10.4:

p

(

C

i

) = 

p

d

(1 – 

p

d

)

2

(10.1)

p

(

C

i

∩

C

j

) = 

p

d

2

(1 – 

p

d

)

(10.2)

p

(

C

i

∩

C

j

∩

C

k

) = 

p

d

3

(10.3)

p

((

C

i

∩

C

j

∩

C

k

)

′

) = (1 – 

p

d

)

3

(10.4)

where:

p

(

A

) is the probability the offender’s residence lies within area 

A

;

C

x

is the area around crime site 

x

 circumscribed by radius 

d

; and

p

d

is the probability the offender’s crime journey is less than or

equal to 

d

.

Relative probabilities for the points within the various areas are obtained

by dividing the areal probabilities by area size (or number of “points”). This
process is a simple dichotomous function dependent only upon whether a
point lies within one of the circles or not. Points in the overlaps of two, or
all three of the circles, are given double or triple the value, respectively.

This process is conceptually similar to the function of the criminal geo-

graphic targeting algorithm, the primary tool used in geographic profiling,
but dichotomizing distance oversimplifies journey-to-crime patterns. The
Brantingham and Brantingham model suggests the criminal search process
is more correctly modeled by a distance-decay curve, incorporating a buffer
zone centred around the residence of the offender. A more sophisticated
method of predicting offender residence location results therefore by replac-
ing the circles in 

Figure 10.1

 with a Pareto function 

f

(

d

 ), in a fuzzy logic

approach that better describes journey-to-crime behaviour (Kosko & Isaka,
1993; Yager & Zadeh, 1994): the value assigned to point (

x

, 

y

), located at

distance 

d

 from crime site 

i

, equals 

f

 (

d

i

). The final value for point (

x

, 

y

) is

determined by adding together the 

n

 values derived for that point for the 

n

different crime sites.

Research conducted at Simon Fraser University and the Vancouver Police

Department following this approach led to the development of the criminal
geographic targeting (CGT) model which has been developed into a com-
puterized geographic profiling system. Crime site coordinates are analyzed
with a patented criminal hunting algorithm that produces a probability sur-
face showing likelihood of offender residence within the hunting area. A
three-dimensional depiction of this probability is referred to as a jeopardy

© 2000 by CRC Press LLC

surface. A two-dimensional perspective integrated with a street map is termed
a geoprofile. These are discussed further, and examples shown, below.

The hunting area is defined as the rectangular zone oriented along the

street grid containing all crime locations. These locations may be victim
encounter points, murder scenes, body dump sites, or some combination
thereof. The term hunting area is therefore used broadly in the sense of the
geographic region within which the offender chose — after some form of
search or hunting process — a series of places for criminal action. Locations
unknown to authorities, including those where the offender searched for
victims or dump sites but was unsuccessful or chose not to act, are obviously
not included.

While the primary purpose of determining offender hunting area is

calculation of search area size, there may be other value in such measures.
The FBI used convex hull polygons to analyze the point patterns formed by
serial rapists (Warren et al., 1995). For local serial rapists (travel under 20
miles), the mean CHP area was 7.14 square miles. The average CHP size was
larger for commuters than for marauders (11.38 vs. 7.62 mi

2

), for rapists who

burgled (15.24 vs. 2.49 mi

2

), and for offenders who lived outside of the CHP

area enclosing their crimes (23.53 vs. 3.22 mi

2

). If replicated, this last finding

could be useful in helping narrow offender residence area in cases of serial
rape.

Any geometric method of determining hunting area has strengths and

weaknesses, and the optimal approach depends ultimately upon the under-
lying purpose. Many predators exhibit a high hunting to offending ratio. 

A

priori

, we do not know where this hunting area is — we only know the

locations of the reported, and connected, crimes. Technically, a geoprofile
stretches to infinity; the hunting area is only a standardized method of
displaying results so that important information is shown, and unimportant
information is not. Special methods are used to deal with unusual patterns,
including elimination of outliers, division of crimes into separate analyses,
and geometric transformations of point patterns (e.g., rotations, “straight-
ening,” trimming, reflections, etc.). The decision on how to proceed is ulti-
mately based on the crime locations and their underlying landscape, guided
by theory and methodology. Criminal geographic targeting considers
offender hunting methods and mental maps within a framework informed
by routine activity, rational choice, and pattern theories.

The CGT analysis uses a Manhattan metric. This may appear to be a less

than optimal approach for crimes in cities characterized by concentric, as
opposed to grid, street layouts. Model testing and experiences with European
cases have demonstrated otherwise. The Manhattan metric slightly overesti-
mates travel in a concentric street layout, but crow-flight distance in turn
results in a small underestimate. Neither are far off; on average, the Manhat-

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tan distance is approximately 1.273 times the length of the crow-flight dis-
tance (Larson & Odoni, 1981). Wheel distance, or path routing, is the most
accurate estimate of shortest available travel distance — which may or may
not be the actual route taken. It is less the physical distance than its psycho-
logical perception that is important. Factors such as traffic congestion, travel
time, cost, and familiarity will influence “distance,” regardless of the metric
used.

The CGT model follows a four-step process:

1. Map boundaries delineating the offender’s hunting area are first cal-

culated from the crime locations. In the case of a Manhattan grid
oriented along northerly and easterly axes, borders are determined by
adding edges equal to 1/2 the mean 

x

 and 

y

 interpoint distances to

the most eastern and western, and northern and southern points,
respectively (for a discussion of alternative techniques for dealing with
edge effects, see Boots & Getis, 1988):

y

high

 = 

y

max

 + (

y

max

 – 

y

min

)/2

(

C

 – 1)

(10.5)

y

low

 = 

y

min

 – (

y

max

 – 

y

min

)/2

(

C

 – 1)

(10.6)

x

high

 = 

x

max

 + (

x

max

– 

x

min

)/2

(

C

 – 1)

(10.7)

x

low

 = 

x

min

 – (

x

max

 – 

x

min

)/2

(

C

 – 1)

(10.8)

where:

y

high

 is the 

y

 value of the northernmost boundary;

y

low

 is the 

y

 value of the southernmost boundary;

y

max

  is the maximum 

y

 value for any crime site;

y

min

  is the minimum 

y

 value for any crime site;

x

high

 is the 

x

 value of the easternmost boundary;

x

low

 is the 

x

 value of the westernmost boundary;

x

max

  is the maximum 

x

 value for any crime site;

x

min

  is the minimum 

x

 value for any crime site; and

C

is the number of crime sites.

2. For every point on the map, Manhattan distances to each crime loca-

tion are determined. While there are an infinite number of mathemat-
ical points in an area, the model uses a finite number of pixels (40,000)
based on the measurement resolution of the 

x

 and 

y

 scales.

3. The distance is used as an independent variable in a distance decay

function; if the distance is less than the radius of the buffer zone,

© 2000 by CRC Press LLC

however, the function is reversed. Values are computed for each crime
location (e.g., 12 crime locations equates to 12 values for every map
point).

4. These values are summed

64

 to produce a final score for each map point.

The higher the resultant score, the greater the probability that point
contains the offender’s anchor point. The score function is presented
in Equation 10.9:

(10.9)

where:

| 

x

i

 - 

x

n

 | + | 

y

j

 - 

y

n

 | > 

B

⊃

φ

 = 1

(10.10)

| 

x

i

 - 

x

n

 | + | 

y

j

 - 

y

n

 | 

≤

B

⊃

φ

 = 0

(10.11)

and:

p

ij

is the resultant probability for point 

ij

;

φ

is a weighting factor;

k

is an empirically determined constant;

B

is the radius of the buffer zone;

C

is the number of crime sites;

f

is an empirically determined exponent;

g

is an empirically determined exponent;

x

i

, 

y

j

  are the coordinates of point 

ij

; and

x

n

, 

y

n

 are the coordinates of the 

n

th crime site.

A three-dimensional surface is produced when the probability for every

point on the map is calculated, which can be represented by an isopleth or
“fishnet” map with different scores on the 

z

-axis representing probability

density (Garson & Biggs, 1992). These maps, a form of virtual reality (in the
term’s original sense), are generated through computer-aided mathematical
visualization techniques. They are referred to as jeopardy surfaces.

The probability surface may be viewed from a top-down perspective and

shown two-dimensionally, similar to how a topographic map displays altitude
(Harries, 1990). When overlaid on a city map of the targeted region, specific.

64 

Alternatively, the logarithms of the values can be summed (a process equivalent to

generating the product).

p

k

x

x

y

y

B

B

x

x

y

y

ij

i

n

j

n

f

n

C

g f

i

n

j

n

g

=

−

+

−

(

)

+





−

(

)

( )

−

−

−

−

(

)





=

−

∑

φ

φ

/

1

1

2

© 2000 by CRC Press LLC

streets or blocks can be prioritized according to the CGT probability values.
The resulting map is termed a geoprofile. 

Figures 10.2

, and Chapter 10 Colour

Figures 1

 and 

2

 (following page 230) show, respectively, the crime sites,

jeopardy surface, and geoprofile for a series of armed robberies of insurance
agencies in Vancouver, British Columbia. A geoprofile can also be expressed
as a series of confidence intervals; 

Figure 10.3

 displays a hypothetical example

for the District of Columbia.

A geoprofile dictates less where an offender lives than it describes an

optimal search process. A search that starts in the highest (i.e., most probable)
area and works down is more likely to find the offender’s residence sooner
than a random process would. Search efficiency is therefore an indicator of
the performance of the CGT model, and can be measured by determining
the proportion of the total hunting area covered before the offender’s resi-
dence is encountered. This ratio is referred to as the hit score percentage,
and the actual size of the region it represents is called the search area. These
terms are discussed further later in the chapter. Parameter specification opti-
mizes predictive ability, but sophistication must be balanced with robustness;
complicated models may perform better under specific conditions, but at the
cost of losing their general applicability.

Figure 10.2

Vancouver robberies — crime sites.