A Model of Offender Profiling Don CaseyPhillip Burrell Knowledge-based Systems Centre Knowledge-based Systems Centre London South Bank University London South Bank University Metropolitan Police Service A Model of Offender Profiling

“ What sorts of people carry out what sorts of actions ? The question at the heart of ‘profiling’” - Canter 2000 What sets of people carry out what sets of actions ? Use of the language of sets throughout the literature Always figuratively – never literally in terms of set theory

A Model of Offender Profiling OFFENDERSVICTIMS RAPE The set of offenders S O = { O1, O2, O3, O4 } The set of victims S V = ( V1, V2, V3, V4,V5 }

A Model of Offender Profiling The Cartesian product S O x S V of the offender and victim sets is the set of all possible combinations of these sets and the set of rapes R can be defined as a subset of this product : S R  S O x S V And any individual rape is defined the same way : R i  S O x S V R i = { (O1,V4) } O1O2O3O4 V1 V2 V3 V4 V OFFENDERS VICTIMS S R = { (O1,V4), ( O2,V2), (O2,V5), ( O3,V1 ), ( O4, V3) }

A Model of Offender Profiling Having defined this relation others can be defined such as TiesUp = { ( 01, V4), (O2, V2)…. } UsesKnife = { (03, V1), (02, V5) …… } These sets can then be combined to produce new relations. TiedUpAndKnife = TiesUp  UsesKnife or TiedUpOrKnife = TiesUp  UsesKnife TiedUpNoKnife = TiesUp - UsesKnife Complex relations can easily be expressed

A Model of Offender Profiling Canter ( 1995 ) A → C A – salient actions at crime scene C – distinguishing characteristics of offender function f(a) to map an element or elements of S A to S C

A Model of Offender Profiling AC f ( a )

A Model of Offender Profiling ‘All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life but only to an imagined celestial existence’ -Bertrand Russell ‘Precision is not truth’ -Matisse cited in Fuzzy Logic ( Ross 2004 ) It doesn’t describe the real world

A Model of Offender Profiling No single point at which a day becomes cloudy or sunny - at which it can be assigned to the set of sunny or cloudy days Same problem exists with crime : ‘ assigning criminals or crimes to one of a limited number of ‘types’ will always be a gross oversimplification’ ( Canter 2000 ) Data is partial, contradictory, uncertain. Dependencies and relationships are not clear

A Model of Offender Profiling Area of Artificial Intelligence – Reasoning under uncertainty Fuzzy mathematics … a methodology for dealing with phenomena that are vague, imprecise, or too complex or too ill defined to be susceptible of analysis by conventional mathematical means Kandel (1986) Reference the proposition of fuzzy set theory ( Zadeh 1965 ) In this the rigid prescription of set membership in terms of in or out, 0 or 1 is changed so that elements of a set have a degree of membership of that set. They can be partial members of sets they would otherwise be excluded from

A Model of Offender Profiling The mechanism used to assign degrees of membership is a membership function. The function which is used depends largely on the area of application and may be result of statistical analysis, established knowledge or expert opinion Unlike classical set theory an element has a degree of membership in the interval 0 to 1 and not 0 or 1 When looking at an individual’s criminal histories it can be difficult to decide what type of criminal they are. Offending histories are usually very varied A description like ‘He’s a burglar who steals cars to order and deals drugs on the side’ cannot be represented in classical set theory but can in fuzzy sets

A Model of Offender Profiling Below is a membership function called an ‘S’ curve often used in fuzzy logic. This relates to the offenders membership of the set ‘vehicle criminal’ and puts his membership at 0.4 If we use similar functions for burglary and drug crime are used then we could define his offending history as a set ‘O’ where S O = {0.4/V, 1/B, 0.3/D } Convictions for vehicle crime

A Model of Offender Profiling Fuzzy logic/set theory is a rigorous mathematical discipline that deals with imprecise, ‘fuzzy’ concepts It is not itself an any way imprecise or fuzzy A method of approximate reasoning about imprecise propositions Applications throughout industry/research : Electrical / household goods, engineering control systems, medical diagnostics, economics/ decision support systems Possibility for a mutually enriching relationship with psychology Membership functions often depend on judgement/perception Appears suitable to a system of offender profiling

A Model of Offender Profiling 4 models Crime linkage - Grubin ( 2000 ) – linking serious sexual assaults using a 4 x 4 taxonomy Criminal history - Davies ( 1998 ) - whether offender’s previous criminal history could be inferred from his behaviour during a stranger rape Geo – profiling Investigative Psychology

A Model of Offender Profiling F( Db ) Psychological theory/hypothesis Statistical Method A X Db X Generic model

A Model of Offender Profiling F( Db ) Cognitive/ mental maps Distance decay function Db X Geo-profiling A X

A Model of Offender Profiling X dB2 dB1 dB3 dB4 Grubin A1 = f1 ( dB1 ) Investigative  A2 = f2 (dB2 ) Geo-profiling A3 = f 3( dB3 ) Davies A4 = f 4( dB4 ) Profiling Techniques are a series of functions on B The universe of discourse B = U Db 1- n X  ∩ Db 1 - n Y = ∩ A 1 - n Y A1 A2 A3 A4 X  Y

A Model of Offender Profiling ‘ We must exploit our tolerance for imprecision’ Lofti Zadeh founder of fuzzy set theory