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

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6.3.1 Anchor Points

Within an activity space are anchor points or bases, the most important
places in one’s spatial life (Coucelis, Golledge, Gale, & Tobler, 1987). The
main anchor point for the vast majority of people is their residence, but other
bases may exist such as a work site or close friend’s home. Some street
criminals do not have a fixed address, basing their activities out of a bar, pool
hall, or some other social activity location (Rengert, 1990). They can also be
transient, homeless, living on the street, or mobile to such a degree that their
anchor point constantly shifts. Offender anchor points are important in the
understanding of crime patterns.

Especially significant are the “anchor points” which focus the routine activ-
ities of criminals on specific sites in our urban environments .... If a criminal
routinely visits the same location nearly every day, this location may serve
as an “anchor point” about which other activities may cluster. This propo-
sition is implicitly recognized when we document that most crime occurs
near the home of the criminal (Brantingham and Brantingham, 1984). The
home is the dominant anchor point in the lives of most individuals. How-
ever, other anchor points also are important influences on the spatial behav-
ior of criminals. (Rengert, 1990, pp. 4–5)

Criminals, to the extent that they live in everyday society, are bound by

the normal limitations on human activity, shaped by the dictates of work,
families, sleep, food, finances, transportation, and so forth. Canter (1994)
suggests that environmental psychology and an understanding of offenders’
mental maps (“criminal maps”) can assist in the investigation of violent
crime. Offenders operate within the confines of their experience, habits,
awareness, and knowledge. “Like a person going shopping, a criminal will
also go to locations that are convenient” (p. 187).

Criminal predators may be stable or nomadic. Stable offenders possess

a permanent anchor point during their period of criminal activity. Nomadic
offenders are transient, lacking a fixed address or anchor point. Albert
DeSalvo, for example, resided in the same dwelling throughout his killing
period, while Ottis Toole lived on the road, travelling from city to city, and
state to state. Other offenders fall somewhere between these two positions.
David Berkowitz resided in two different locations in New York City during
his crimes, while Ted Bundy, though not nomadic, moved several times
during his murder spree (Terry, 1987; U.S. Department of Justice, 1992).

6.4 Centrography

The spatial mean (sometimes referred to as the centroid or mean centre) is
a univariate measure of the central tendency of a point pattern (Taylor, 1977),

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and has been used to analyze crime site patterns. This geographic “centre of
gravity” minimizes the sum of the squared distances to the various points in
a pattern. It provides a single summary location for a series of points and
has a variety of geostatistical uses collectively referred to as centrography.

The spatial mean is defined as:

(

SM

x

 , 

SM

y

)

(6.1)

where:

(6.2)

(6.3)

and:

SM

x

is the 

x

 coordinate of the spatial mean;

SM

y

is the 

y

 coordinate of the spatial mean;

C

is the total number of crime sites; and

x

n

, 

y

n

are the coordinates of the 

n

th crime site.

It is possible to determine a weighted mean centre if certain points are

more important in a centrality analysis than others. The median centre –
also known as the centre of minimum travel – is another measure of central
tendency in point patterns and is found by locating the position from which
travel to all points in a spatial distribution (i.e., the sum of the distances) is
minimized. There is no general method for its calculation and the median
centre must be calculated through an iterative process.

Changes over time in the location of the spatial mean allow for the

calculation of the geographic equivalents of concepts of velocity (rate of
spatial change), acceleration (rate of change in velocity), and momentum
(velocity multiplied by number of points) (LeBeau, 1987b). The spatial mean
is the basis for calculating the standard distance of a point pattern, a measure
of spatial dispersion analogous to the standard deviation (Taylor, 1977).
When used with the mean centre it can help describe two-dimensional dis-
tributions, and through the concept of relative dispersion (the ratio of two
standard distances), allow for comparisons of spread between different sets
of points. Similarly, the median distance is the radius which encompasses
one half of the points in a spatial distribution.

The standard distance is defined as:

SM

x

C

x

n

n

C

=













=

∑

1

SM

y

C

y

n

n

C

=













=

∑

1

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(6.4)

where:

Sd

  is the standard distance;

C

is the total number of crime sites; and

r

ns

is the distance between the spatial centre and the 

n

th crime site.

Centrography has been used in a variety of criminological studies and

investigative contexts. The spatial mean and changes in its location over time
were calculated for rape incidents in San Diego (LeBeau, 1987b). An inves-
tigative review team helped locate the hometown of the Yorkshire Ripper
from the geographic centre of the murder sites (Kind, 1987a). A similar
approach in a blackmailing case used cash withdrawal points from automated
teller machines (ATMs) to determine the offender’s residence area east of
London (Britton, 1997). Such techniques were also employed in a retrospec-
tive analysis of the Hillside Stranglers (Newton & Swoope, 1987). The FBI
and ATF analyze serial arson cases by determining the spatial mean of fire
sites (Icove & Crisman, 1975). Traditionally, centrography has been the pri-
mary form of geographic analysis used to support criminal investigations.

As helpful as centrographic analysis may be in certain cases, the spatial

mean suffers from three serious methodological difficulties: (1) it generally
provides only a single piece of information; (2) it is distorted by spatial
outliers; and (3) theory suggests the intersection between offender activity
space and target backcloth (the distribution of crime targets across the phys-
ical landscape) may produce crime locations unrelated to measures of central
tendency. If the activity space of an offender is not centred around his or her
home, or if the target backcloth is highly variable, then the spatial mean of
the crime sites and offender residence are not correlated.

A study of the spatial patterns in a sample of British serial rapists revealed

the limitations in centrographic analysis (Canter & Larkin, 1993). The study
plotted maximum distance from residence to crime site against maximum
distance between crime sites, producing the following regression equation:

y

 = 0.84 

x

 + 0.61

(6.5)

where:

y

  is the maximum distance in miles from residence to crime site; and

x

  is the maximum distance in miles between crime sites.

The gradient of 0.84 in Equation 6.5 indicates an eccentric placement of

the residence 

vis-à-vis

 the crime sites (a perfect centric placement would yield

a gradient of 0.5). Similar regressions for U.S. and British serial murder crime
location data yield values of 0.81 and 0.79, respectively (Canter & Hodge,

Sd

r

C

ns

=

( )

∑

2

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1997), and an Australian study found gradients of 0.77 for rape, 0.60 for
arson, and 0.65 for burglary (Kocsis & Irwin, 1997). This eccentricity suggests
the spatial mean is limited in its ability to predict offender residence location.

Additionally, the spatial mean lacks real world significance. The geo-

graphic centre of Canada is in the Northwest Territories which tells one little
about the demographic, economic, or political patterns of that country.
LeBeau (1987b) notes “an important property about the mean center to
remember is that it is a synthetic point or location representing the average
location of a phenomenon, and not the average of the characteristics of the
phenomenon at that location” (pp. 126–127; see also Taylor, 1977).

Studies of journey-to-crime trips, particularly those that are offence

specific, help determine the most likely radius within which offenders search
for victims. For example, research has consistently shown targets are typically
located within one or two miles of offender residence (see McIver, 1981).
When utilized in conjunction with the spatial mean, such information may
be of investigative value.

6.5 Nearest Neighbour Analysis

While the spatial mean provides a way to measure central tendency in a point
pattern, nearest neighbour analysis, first developed by plant ecologists, sup-
plies a way to quantify spacing between points (Taylor, 1977; see Boots &
Getis, 1988; Garson & Biggs, 1992). Distances between points and their
closest neighbours provides important information concerning a pattern’s
degree of randomness and underlying evolution. It is also possible to calculate
other proximity pattern measures including centroid, 

k

-nearest neighbour,

mean interpoint, and furthest neighbour distances (Garson & Biggs).

The random allocation of points to a map can be described by the Poisson

process (Taylor, 1977). The Poisson probability function is defined as:

p

(

x

) = 

e

–

λ

 λ

x

/

x

!

(6.6)

where:

p

(

x

)  is the probability that a given small area will contain 

x

 points;

and

λ

is the expected probability of finding a point within that area.

Connecting nearest neighbour analysis to the Poisson probability func-

tion allows the degree of clustering, dispersion, or randomness in a given
independent point pattern to be calculated. The 

R

 scale, the ratio between

the actual average nearest neighbour distance and that expected under an

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assumption of randomness, provides a simple index for measuring diver-
gence from randomness. It is calculated as follows:

R

 = 

r

a

/

r

e

(6.7)

where:

(6.8)

and:

R

is the 

R

 scale value;

r

a

is the actual average nearest neighbour distance;

r

e

is the expected average nearest neighbour distance;

n

is the number of points; and

A

is the area size.

Theoretically the 

R

 scale can fall between the limits of 0 to 2.149, though

real world patterns tend to range between 0.33 and 1.67 (Taylor, 1977). A
value of 1 (meaning that 

r

a

 = 

r

e

) indicates a random pattern, values smaller

than 1, a clustered pattern, and values larger than 1, a dispersed pattern.
Problems result in the interpretation of the 

R

 scale if boundary placement is

distorted (Garson & Biggs, 1992).

It is possible by chance for a randomly produced pattern to appear

aggregated or dispersed, therefore it is necessary to determine the signifi-
cance of the 

R

 scale value (Taylor, 1977). This can be accomplished through

the 

Z

-score calculated from the standard error of the expected average

nearest neighbour distance. The associated two-tailed probability may then
be determined from a table of normal distribution values (e.g., Blalock,
1972). The standard error (

SE

) is estimated as follows:

(6.9)

where:

SE r

e

is the standard error of the expected average nearest neighbour
distance  (

r

e

);

n

is the number of points; and

A

is the area size.

While the 

R

 scale provides a measure of spatial randomness, it says

nothing about the actual evolution of the point pattern. More than one
distinct process can be operating as might be found with a series of chaotic
binary points (Boots & Getis, 1988). Statistical tests are only inferential, and
it may be necessary to corroborate results over time or examine higher-order
or 

k

-nearest (e.g., second-nearest) neighbour distances.

r

n A

e

=

( )

1 2

SE r

n A

e

=

0 26136

2

.