From Right to Left it Had No End: “Good”, “Poor”, “Dangerous”, “Tough”

 

Since the variables and factors that assist the police in establishing the modus operandi are imprecise in nature, it is difficult to develop any of the standard mathematical techniques to profile the offender. However, the new kind of mathematics based upon fuzzy logic appears to be useful in creating templates of offenders. We will develop a mathematical routine that models the modus operandi procedure followed by the police investigators. This fuzzy logic-based mathematical technique can assist in making sense of the evidence provided by the witnesses. As a practical example, we will use the case of motor vehicle theft in which multiple offenses by a single individual are more probable, but the technique could be applied to any type of a offense. We know, and the pattern theory asserts, that despite such individual differences there are set patterns that can be seen. Police detective work is indeed dependent on deciphering patterns determined by habitual actions of offenders. In the commission of almost any kind of crime, every offender adopts a fixed mode of behavior in terms of chosen time period, target preference, region of operation, and even the manner of committing the crime. In police terminology, this behavior is described as the “modus operandi” of the offender, and a good detective attempts to establish this by looking for recognizable patterns in the commission of a crime. Thus, in burglary cases, the pattern sought is the time, place, mode of entry into the premises, and items stolen or left behind. In serial killings, apart from the place, time and mode of killing, characteristics of the victim, nodes of the residence, workplace and acquaintances of the offender may form the set pattern, or modus operandi. 

 

Let Ω be the set of auto suspects. An auto thief (suspect) p = Ω, can be categorized by assigning to it the values of a finite set of fuzzy parameters relevant to him or her. Examples of such parameters may include places or times of operation, preferred vehicle type, busy or isolated road conditions of theft sites, value of the vehicle or the goods inside, mode of getting into the car, purpose of theft, and so on, where the highlighted parameters are fuzzy in concept. Each parameter is specific to some feature of the offender p in question. Thus, p can be associated with a mathematical object Fk = [m1(p), m2(p), m3(p), … mr(p)] where mi(p) is the measurement procedure of parameter i and mi (p) is that particular value. For example, we may have mj(p) = time period, i.e. day or night; or mk (p) = place, which refers to the boundary, limits of some particular neighborhood; ml (p) = value in terms of costly or low-priced car, and so on.

 

Here Fk will be called the pattern class, and many such pattern classes Fk ∈ I of mathematical objects could be associated with p. This will depend upon the various combinatorial values of mi(p) where I = 1 … r. The set F of all such mathematical objects will be called the pattern space. The objective is then to assign a given object to a class of objects similar to it, having the same structure. Such a class is often a fuzzy set F y. A recognition algorithm when applied to it yields the grade of membership MF(p) of p in the class F. In case the parameter is exactly known, such as the time of theft (someone may have noticed the car being driven away), then the grade of membership in time parameter will be 1, in accordance with the definition of fuzzy set. We will first define a fuzzy pattern class F based upon the parameters in question. The easiest way of doing this is to assign this class a “deformable” prototype constructed through the information available from convicted and old suspects. The assignment can be done by giving an interval of measures to each of the selected variables. Thus, young may mean 15 to 19 years of age, costly may imply a dollar value of around $5,000, etc. Other features such as ethnicity, casually dressed, tall, or local could also be added, based upon the information made available from victims’ statements or detectives’ knowledge about the active suspects. The measurement of these variables could be carried out through some form of smaller or larger scale developed for this purpose. A prototype may then be something like: {young, Asian, smart looking, Robson/Granville street areas, evening, (prefers) Japanese cars, medium valued, lighted locations, (uses) duplicate keys… and so on}. All these are fuzzy variables with a range of membership values. However, with larger data sets of suspects, and over the years, more and more information gets built into the system which would help in reducing some of the fuzzy measurements or in building more representative prototypes. Finally, a new auto theft offense will be analyzed about its attributes and for its membership values in each parameter. Some definite information will always be available, such as the make of the car and place of theft. Based upon these values and the information provided by the complainant or witness, the investigating officer can then assign the values of 1 or 0 or decide upon the grade of membership into other parameters. Mathematically, let Fk ∈ I be a fuzzy prototype pattern class defined by the fuzzy features f1 … fr , where fi is the fuzzy values of feature i. Symbolically, Fk = {f1 … fr} where f1 is (tall), f2 is (Chinese looking), f3 is (…around Robson street) f4 is (busy street…), f5 is (shabby clothes) and so on. Each fi will be having an interval of values. For example, busy may imply a situation when 15 to 25 cars pass a street crossing per minute, information about some suspect hanging around Robson street could mean the area is within four blocks on either side of Robson street, and so on. Fk will have a minimum value n obtained by aggregating all the minimum values of fi , and similarly a maximum value m. An object p, who is a suspect of this theft, will be characterized with respect to the class Fk by the r membership values– –fk mi(p), i = 1 … r. The value of p, denoted by MFk (p) will be constructed by aggregating the mi(p)s in some manner. This MFk (p) can then be compared to the maximum and minimum values of different prototype pattern classes Fk ∈ I which provides a numerical measure of the likelihood of a suspect belonging to a specific pattern class Fk (a group of suspects or a particular gang).

 

Consider the situation in which an investigator obtains some fuzzy information about the suspect from the descriptions provided by few eyewitnesses. In such a hypothetical situation, the fuzzy terms could be analyzed following the technique as mentioned above. For instance, suppose the witnesses mentioned that the offender was tall, with brown color hair, wearing dirty clothes and was a young person. As indicated, these are fuzzy terms that mean different characteristics to different witnesses. To determine the overlapping range of these characteristics, the investigator could hold an in-depth examination of their perceptions to fix a range within which they could be describing these characteristics. Consider the fuzzy characteristic tallness. It is fuzzy because for one witness 166 cms. and above is the height that makes a person tall. For a second witness, only a height of 172 cms. and above is tall, while for the third witness, a person is tall if he/she is over 170 cms. How about 168 cms. or 166 cms.? Is this height “tall” for the first witness? A detailed examination of this witness’ perception may suggest that for him/her, any person of height 165 cms. Or below is definitely not tall (membership value is 0), while 166 to 167 is tall, perhaps with 0.15 membership value. It is possibly 0.7 for 169 to 170 and 1 for over 173 cms. Thus for each witness, there is a minimum range of height of membership value <0.1 and a maximum range of value >0.9 for describing the fuzzy characteristics of tallness. As suggested above, the police investigator could obtain this information and possible ranges with membership values by a detailed examination of perceptions of each of the witnesses. 

 

Fuzzy logic techniques can also be useful to police managers for analyzing other kinds of data that is non-dichotomous and fuzzy in nature. For instance, fuzzy logic can be a promising tool for improving decision making in the police department. Job evaluations that involve ratings such as “good”, “poor”, “average” are difficult to interpret since different supervisors have different perceptions of these ratings. However, each of these ratings could be assigned a range of values and then reconciled through fuzzy logic methods to judge what constitutes “good” for all the supervisors. Similarly, rather than classifying neighborhoods as “dangerous”, “tough”, or “troublesome” for extra deployment and special attention, the patrol officers may be trained to provide a range of gradation for these areas in order to make deployments more appropriate and cost effective. Thus, areas victimized by gang activity have boundaries that are fuzzy in nature. Instead of designating the whole region as the “turf” of some particular gang, the patrol officers could be trained to see that the region could be divided into a range of dominance. Using some measurement scale, portions of the region could be identified as being partially a turf (say, 0.2 inclusion value) or overwhelmingly (0.9 inclusion value) under the dominance of that particular gang. This may be useful in planning cost-effective resources for surveillance or patrolling purposes.

 

 

 

Above all, the understanding that certain variables are fuzzy in nature will enhance police capabilities and can also improve our understanding of police behavior, action, and organizational culture. The discretion used by officers in making arrests, in stopping and questioning people is commonly based upon fuzzy factors. Thus, variables like “race”, “age”, “socioeconomic status”, “appearance”, that influence discretion exercised by the police The application of fuzzy logic-based techniques could be useful in examining the range that begins to affect officer’s perceptions. The reasons police officers stopped and questioned a particular “dark”, “lower class” person may perhaps be explained by realizing that these are fuzzy parameters. The police officers may be using a graded scale to make their decisions in which not every dark, lower class person is a suspect but some particular ones are, for whom these parameters have a high inclusion value in the fuzzy perception of the officers. Fuzzy logic can also provide a powerful method for applications in other criminal justice fields. A large category of data such as citizen responses, attitudes, and opinions are generally fuzzy in nature where possibilities for the utilization of this form of mathematics are extensive. A possible application could be in the field of comparative studies in law. It is generally acknowledged that international comparative legal studies are difficult since the meaning of offenses differs considerably.

 

 

 

Acknowledgements:

www.aivd.nl    AIVD ©

Algemene Bestuursdienst – @Erik Akerboom ©

www.politie.nl  Politiekorpschef  @Janny Knol©

www.politie.nl WEB Politie - @Henk van Essen©

https://english.marechaussee.nl  Royal Netherlands Marechaussee    @Willem-Alexander King of the Netherlands©

PolitieAcademie - https://www.politieacademie.nl/en©

 Author: W.R. Łaskawiec ©

 

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