Can Fuzzy Decision Support Link Serial Serious Crime?
Don Casey and Phillip Burrell
London South Bank University, Borough Rd, London, SE1 0AA, U.K.
Keywords: Crime Linkage, Decision Support, Fuzzy Clustering.
Abstract: The problem addressed is one of great practical significance in the investigation of stranger rape. The
linkage of these crimes at an early stage is of the greatest importance in a successful prosecution and also in
the prevention of further crimes that may be even more serious. One of the most important considerations
when investigating a serious sexual offence is to find if it can be linked to other offences; if this can be done
then there is a considerable dividend in terms additional evidence and new lines of enquiry. In spite of a
great deal of research into this area and the expenditure of considerable resources by law enforcement
agencies across the world there is no computer-based decision support system that assist crime analysts in
this important task. A number of different crime typologies have been presented but their utility in decision
support is unproven. It is the authors’ contention that difficulties arise from the inadequacy of the adoption
of the classical or ‘crisp set’ paradigm. Complex events like crimes cannot be described satisfactorily in this
way and it proposed that fuzzy set theory offers a powerful framework within which crime can be portrayed
in a sensitive manner and that this can integrate psychological knowledge in order to enhance crime linkage.
1 INTRODUCTION
The need for computerised systems to support the
work of crime analysts and investigators has been
recognised for some time. The authors of an
influential study into linking serious sexual assaults
remarked that:
The ultimate goal is to create a computer-based
screening system that will allow routine and
systematic comparison of serious offences on a
national basis , selecting cases on the basis of
their behavioural similarity that are appropriate
for more detailed attention by detectives or crime
analysts (Grubin et al., 2000)
And this view has been reinforced by an eminent
criminal psychologist:
The development and test of theories and
implementation of findings into computer-based,
decision-support systems … has to be the proper
basis for any professional derivation of
inferences about offenders. (Canter, 2000)
The problem at the heart of crime linkage resides in
the need for an adequate typology of offences but
the search for an effective system has proved
elusive.
The most influential crime classification system
has been that proposed in the Crime Classification
Manual (Douglas et al., 1992) which is the work of
senior Federal Bureau of Investigation agents. It
advances the notion of an organised-disorganised
dichotomy and was developed from interviews with
offenders (Ressler and Douglas, 1985). The basis of
this approach is that crimes can be differentiated by
the level of planning and organization associated
with them and the authors extend this to assert that
the dichotomy can be applied to the offender so that
organised and disorganised crimes are committed by
individuals who can be differentiated in discrete
groups with distinct characteristics. Very serious
objections have been made to the methodology
employed by the FBI. Only 36 offenders were
interviewed, no attempt was made to ensure this
group was representative and the interviews
conducted were not structured or consistent. An
evaluation of this typology (Canter et al., 2004)
applied to 100 serial murderers provided no support
for it.
In the most comprehensive research programme
into the linkage of serious sexual offences (Grubin et
al., 2000) the authors propose
Our starting premise is that rape attacks can be
organized into distinct types
383
Casey D. and Burrell P..
Can Fuzzy Decision Support Link Serial Serious Crime?.
DOI: 10.5220/0004332403830388
In Proceedings of the 5th International Conference on Agents and Artificial Intelligence (ICAART-2013), pages 383-388
ISBN: 978-989-8565-39-6
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
It is certainly the aim of investigation of any
field to initially classify the objects contained within
it but it is the hypothesis of this paper that although
rapes can be organized into types that these will be
far from ‘distinct’. And that the attempt to
discriminate between crimes in this way is likely not
only to be barren but actively misleading in that they
will be forced into mutually exclusive types that will
misrepresent their complexity; a perspective arrived
at after many years of research by one of the area’s
foremost investigators
… assigning criminals or crimes to one of a
limited number of ‘types’ will always be a gross
oversimplification. (Canter, 2000)
Canter and his associates who are identified with
Investigative Psychology have published numerous
studies (Hakkanen et al., 2004); (Santilla et al., 2003)
on sexual assault, homicide and other serious crimes
but have been unable in any of them to construct a
satisfactory typology with even the most relaxed
rules of assignment (Salfati and Canter, 1999).
Grubin is obliged to propose a 256 element
taxonomy in which many of the elements are
redundant, a classification system in which many if
not most of the elements will never occur cannot be
satisfactory. The assumption of the crisp set
paradigm in this research appears to be the cause of
the problems relating to these difficulties. This can
be illustrated by a simple description of a crime such
as ‘a very violent assault on a middle-aged woman
by a young man’ which cannot be properly expressed
in terms of crisp sets. It can lead to either the
misallocation of fundamentally different offences to
the same place or to crimes that bear strong
resemblances to each other being regarded as entirely
unconnected, a phenomenon referred to as linkage
blindness (Egger, 1990) of which researchers are
fully aware but have been unable to address.
In the analysis of serious crime, particularly
sexual offences, there are two computer databases
that dominate: the Violent Criminal Apprehension
Program (ViCAP) introduced by the F.B.I and the
Violent Crime Linkage System (ViCLAS) first
developed by the Royal Canadian Mounted Police
(RCMP). ViCAP is used predominantly in the USA
while ViCLAS is employed throughout most of
Europe .Both systems are essentially repositories of
criminal records that analysts search using their
training and expertise in order to link offences. This
is achieved employing straightforward Boolean
searches on groups of variables deemed to be
significant. There has been no decision support
system devised to assist in this task and no attempt
has been made to incorporate the results from
psychological research into crime linkage.
2 DATA
We have been fortunate in being successful in
obtaining data on 574 serious sexual offences from
the Serious Crimes Analysis Section of the U.K
National Policing Improvement Agency. We have
excluded those offences that do not relate to serial
stranger rapes, by which we mean a set of rapes
committed by a single individual unknown to the
victim. This results in a much narrower dataset (n =
112), development set n = 83, test = 29). The
development set consisted of 28 series, mean length
2.96, while the test set comprised 11 series with a
mean length of 2.64.
The dataset made available contained 22 single
or ‘one off’ stranger rapes which allowed variants on
the set to be constructed that could be regarded as
more realistic in that they contained a mixture of
both serial and individual offences. In the first
instance these were added to the 29 offences in Test
Set 1 to produce Test Set 2 (n = 51). By using this
set of crimes the effect of a substantial group (>
40%) of unlinked offences in tests could be
observed. Both Test Set 1 and 2 used the value of
variables derived from Development Set 1. In order
to replicate the development of a crime database
over time the entire set of 112 linked crimes was
used as Development Set 2. As with the other
development set this pool of offences generated its
own value for variables; they were found to be
somewhat different but in line with the first set.
Finally as before and for the same reasons this set
was combined with the 22 single offences to
produce a composite set of 134 linked and unlinked
offences.
Table 1:Development and test datasets.
n=
Development 1 83 83 linked crimes
Test 1 29 29 linked crimes
Test 2 51 29 linked, 22 unlinked
Development 2 112 112 linked crimes
Test 3 134 112 linked crimes, 22 unlinked
ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence
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3 CRIME AND FUZZY SETS
Fuzzy set theory (Zadeh, 1965) allows us to
represent crimes and criminals as highly descriptive
objects in the concept space and to undertake
experimental procedures to discover what are the
most significant differentiating features using
mathematically and logically sound methods
The Grubin study was taken as a starting point
for modelling stranger rape linkage because it was
focussed on this crime, had used the most similar
data and was the most lucidly expounded. The basis
of this approach was to identify four ‘dimensions’ of
criminal behaviour: Control, Sex, Escape and Style.
The last of these was a late addition by the authors
and was found to be of very limited use. It was
therefore discarded and the first three employed. A
total 44 variables were found to directly or indirectly
correspond to those employed in the earlier research.
On inspection the Control and Sex dimensions both
appeared to have a natural division in their variables
in that Control consisted of overtly violent actions
such as assault and use of weapons and other more
enabling actions such as engaging the victim in
conversation. The Sex dimension similarly divided
into those actions that constituted rape and others
such as kissing. Consequently, in testing, offences
were characterised by the original dimensions (3D);
Control, Sex1, Sex2, Escape (4D2S); Control1,
Control2, Sex, Escape (4D2C); and all five
dimensions (5D). By extending the number of
dimensions it was hoped that an optimum
configuration would emerge
The number of variables in each dimension was
distributed as: Control (19), Sex (14), Escape (11),
Control1 (12), Control2 (7), Sex1 (6) and Sex2 (8).
3.1 Membership Functions
We can define the universe of crimes as a data set
(X) of n elements
,
,
,….
(1)
Where each crime (
) is defined by j features or
variables
,
,
,….
(2)
The variables constitute behavioural dimensions as
above. A problem arises with these dimensional
concepts because they cannot be incrementally or
hierarchically scaled. This makes the use of a
conventional membership function difficult. In order
to overcome this we have proposed that the amount
of these activities can be measured, i.e. the number
of separate sexual, controlling or escape-centred
actions within each crime,
.These variables have
dichotomous values that do not reflect their value in
contributing to the distinctiveness of the crime in
which they occur. However it can be posited that
each variable,
be associated with a value
that
represents a weighting based upon its prevalence in
the dataset of n crimes. In order to assign a value to
each variable that reflects this frequency the
reciprocal of the sum of its occurrences is taken. As
a result if the variable were to occur in every crime it
would have a value of 1/n while if it were to occur
only once its value would be equal to 1 with all
intermediate frequencies being assigned
corresponding values. This simple calculation
assigns an appropriate value to each variable; thus
the variable
is assigned a value or weight
by
w
1
∑
x
(3)
Once each variable has a value it is a simple matter
to sum the values of all the variables found in each
crime to give it a score,S
, n that dimension
.
(4)
The degree of membership can then be calculated by
normalising this score by dividing it by the highest
score encountered in the dimension
max
:1,….
(5)
The result is intuitively satisfying in that the score
attained is related directly to the most controlling or
sexually demeaning, etc., crime encountered up to
that point. It also means that a rudimentary form of
learning can take place in that as more crimes are
added the scores across dimensions for each crime
will be liable to change and its position in the sample
space will move. This could be taken to replicate the
manner in which experience affects the performance
of skilled users.
The membership function also derives closely
from the techniques used in Investigative Psychology
(Canter et al., 2003) which emphasizes the frequency
of variables and their co-occurrence within crimes.
An example of degrees of membership for two series
in three dimensions is given at table 2.
3.2 Fuzzy C-means Clustering
Having established the degrees of membership from
the development dataset for each crime in all four of
CanFuzzyDecisionSupportLinkSerialSeriousCrime?
385
the dimensional structures it was possible to
investigate the relationship between them and how
they were distributed in the concept space through
clustering.
Table 2: Three Dimensional (3D) memberships.
Crime series Control Sex Escape
1 1 0,38 0,17 0,15
2 1 0,38 0,22 0,08
3 1 0,32 0,24 0,07
4 1 0,34 0,34 0,37
5 1 0,34 0,10 0,14
6 2 0
,
32 0
,
06 0
,
00
7 2 0,27 0,14 0,16
8 2 0,38 0,03 0,09
Fuzzy c-means clustering (Bezdek, 1981) is the
most widely used fuzzy clustering strategy and
effectively addresses the difficulties raised by Canter
of exclusive types of crime. It does this by defining a
set of fuzzy sets on the universe X so that the sum of
degrees of membership of all the classes of any
datapoint is unity, there will be no empty classes and
no class that contains all the datapoints. This is an
iterative optimisation technique of the objective
function below where a degree of fuzziness 1 ≤ m <
∞ is specified and elements are assigned degrees of
membership of the clusters until some termination
criterion has been reached.
(6)
is the degree of membership of
in cluster j and
is the cluster centre
Cluster centres are initially distributed randomly
and are guaranteed to converge however there is no
technique to determine the optimum number for any
application. Therefore the number of clusters (j) was
specified from 2 to 5, 6 clusters produced
inconsistent results. It was also possible to vary the
degree of fuzziness (m) from 1.25 to 3 in increments
of 0.25. A value of m = 1 equates to a crisp partition
of the data which becomes correspondingly fuzzier
as it increases. There is no agreed best value for m
although around 2 is often cited (Ross, 2004).
3.3 Fuzzy Similarity
Clustering returned the membership of j fuzzy
clusters for each offence. Two similarity methods
were used to evaluate the strength of the
relationship,
, between the objects
and
.
Cosine amplitude reflects the size of the angle
between them; where they are colinear the value is
unity and when they are most dissimilar, i.e., at right
angles the relationship has a value of 0.
Here there are n objects (crimes) represented in
m-dimensional space
∑
∑
∑
(7)
where i, j = 1, 2 … n
The max-min method is simpler than cosine
amplitude and uses max, min operations on pairs of
datapoints to establish similarity in a straightforward
manner.
∑
,
∑
,
(8)
where i, j = 1, 2 … n
As a result an n x n similarity matrix was
generated for all values of fuzziness (m) and number
of clusters (j) for both methods. The values in each
row (n) were then rank-ordered to show the relative
closeness of all the other crimes to the
offence.
For the development set of 83 crimes these values
range from 1 (closest) or most similar to 82 for the
most dissimilar.
4 RESULTS
The rank-ordered similarity distance for each dataset
was computed to produce an n x n matrix and the
mean and median for the total of those comparisons
between serial offences recorded. It should be noted
that these distances were not symmetrical: if
and
have a similarity of 0.8 it does not follow that
they are equally distant from each other.
These results should show distances of ≈ n/2
between linked crimes if they are randomly
distributed. However it was found that for virtually
all combinations of dimensions, clusters and
fuzziness values that distances were consistently
below this level and often considerably so. Table 3
shows the best results for each set; medians are
shown as they would best represent the search
strategy of analysts in searching for matches. In
addition in nearly all results and particularly the
more successful ones there was an evident positive
skew indicating that successful matching was
concentrated towards the low distance.
The exceptional performance obtained with Test
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Set 2 suggests that the environment of this dataset in
which serial offences are a bare majority may
substantially enhance the outcome. This is an
interesting result in that although there are no
figures to indicate the ‘mix’ between serial and
individual offences because of partial detection
rates; it is likely to replicate ‘real life’ in which a set
of stranger rapes is composed of both serial and
single offences.
Results show that although there is a marked
advantage in using the methodology and techniques
outlined here there is no convincing combination of
number of clusters or levels of fuzziness that
consistently returns optimum distances.
Table 3: Median distances between linked crimes.
n=
lowest
median %
n/2
dimensions clusters m =
Dev Set 1 83 < 50% 3D 2
low to
medium
Test Set 1 29 <35% 3D 3 all vals
4D2C 4,5
low to
medium
4D2S 4 all vals
Test Set 2 51 <25% 3D 4,5
low to
medium
4D2S 4,5 all vals
Dev Set 2 112 <55 % 3D 2 all vals
5
med to
high
Test Set 3 134 ≈ 50% 4D2C 4
low to
medium
5 low
4D2S 5 medium
5 CONCLUSIONS
A recent report into rape investigation in the U.K.
(HMIC/HMCPSI, 2012) detailed the problems of
productivity in terms of analysis in that only around
25% of suitable crimes submitted were analysed and
that a backlog was building up that might never be
cleared. These are some of the most serious crimes
that occur in society and the need for more advanced
techniques to assist investigators is clear.
This research has demonstrated that fuzzy
methodology can be used successfully in
representing serious crimes in a sensitive manner
that reflects their complexity and builds on insights
from criminal psychology.
Although attempts have been made to associate
stranger rapes in order to enable linkage there has
been no input from Artificial Intelligence or
Decision Support and they have been purely
psychology-based. This is surprising in view of the
clearly stated views of leading researchers. The
research presented here has for the first time
endeavoured to find the strength of similarity
between offences in the way that crime analysis
requires.
The problem of rigid typology that has hampered
this area of research is precisely the one that fuzzy
sets avoid. Because of the nature of the area under
investigation any crisp classification method is
bound to fail. Either a large number of crimes elude
classification as in Investigative Psychology or a
highly redundant typology of stranger rape, which is
itself a small subset of rape has to be proposed.
Reference has been made to the possibility of
increasing the productivity of analysts but it is also
possible and perhaps probable that given assistive
technology that the quality of analysis would
improve. Given new, and better, ways of achieving
their goals expert users are highly likely to adapt and
improve their expertise.
The results obtained here strongly suggest that
the methods used can be of considerable value in
crime linkage and that further research may well
refine dimensionality and clustering to produce even
more useful inferences.
If this can be done then the consequences may
feed back into criminal psychology in that cluster
centres can be regarded as prototypes of criminal
action and be interpreted as a usable typology. And a
circle of effective advance and cross-fertilisation
result.
It may be this approach can yield the
psychologically valid and meaningful set of numbers
(Canter, 1985)
called for in the earliest days of
research into this area .
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