Criminal Geographical Profiling: Using FCA for Visualization and Analysis of Crime Data

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1 Criminal Georaphial rofilin: Usin FCA for Visualization and Analysis of Crime Data Quist-Aphetsi Kester, MIEEE Leturer, Faulty of Informatis Ghana Tehnoloy University Collee Ara, Ghana / Abstrat fihtin riminal ativities in our modern soieties required the enaement of intellient information systems that an analyze rime data eoraphially and enable new onepts to be dedued from it. These information systems should be able to reate visualization of suh data as well as have the apability of ivin new inite of information, if data is updated whilst maintainin the previously predited patterns. This paper proposed the use of Formal Conept Analysis, or Galois Latties, a data analysis tehnique rounded on Lattie Theory and ropositional Calulus, for the visualization and analysis of rime data. This method onsidered the set of ommon and distint attributes of rimes in suh a way that ateorization are done based on related rime types, eoraphial loations and the persons involved. Keywords: Criminal Georaphial rofilin, FCA, Visualization, Analysis, Galois Latties I. INTRODUCTION To be able to fiht rime effetively or understand riminal ativities in the future, our information systems need to have the apability of analyzin and extratin sinifiant knowlede from data based on predefined rules with supervised or unsupervised learnin tehniques. Hene there is a need for us to use effetive omputational and mathematial models in the domain of data minin and mahine learnin in buildin our artifiial intelliene systems. These information systems should have the apability of usin eonomi data, eoraphial data, demoraphi data, soial networkin data et in analyzin and preditin behavior of modern soiety espeially in the domain of peae and seurity whih is one of the ruial platforms for effetive development. Modern soiety has been hallened with rise in riminal ativities and these rime rates vary enormously from one ountry to another and from one reion to another [].With the doumentation of riminal ativities and the use of omputerized systems to trak rimes, omputer data analysts have started helpin the law enforement offiers et to understand rime patterns []. These systems should be apable of atherin and interpretin intelliene so as to help ontrol of riminal ativities as well as influene effetive deision makin as in fiure below []. Sine riminal ativities have beome very omplex, its monitory with intellient systems has beome neessary by usin with eoraphial omponents. Interpret Intelliene /Analysis Fiure : attern analysis theory Criminal environment Influene Impat Deision Maker The most effiient and effetive way of fihtin rime today annot be resoureful without eoraphial profilin. Criminal ativities have beome very omplex in suh a way that rapid monitory an only be ahieved by usin intellient systems with eoraphial omponents. Georaphi profilin is a mathematial tehnique to derive information about a serial rime spree iven the loations and times of revious rimes in a iven rime series []. Georaphi profilers have aess to a olletion of strateies for preditin various attributes of rime suh as a serial offender s home loation, possible roups, relationship between events and rimes et. These strateies rane in omplexity, some involve more alulations to implement than others and the assumption often made is that more omplex strateies will outperform simpler strateies [5]. Over the years there have been developments and approahes in the analysis of rime data suh as the introdution of a raph based dataset representation that allows one to mine a set of datasets for orrelation [6], data minin tehniques usin lusterin alorithm to help detet the rimes patterns and speed up the proess of solvin rime [7], proedure for detetin hanes over time in the spatial pattern of point

2 events, ombinin the nearest neihbor statisti and umulative sum methods [8] et. Crime ativities are eospatial phenomena and as suh are eospatially, thematially and temporally orrelated and. disoverin these orrelations allows a deeper insiht into the omplex nature of riminal behavior. This paper used Formal onept analysis, or Galois Latties, a data analysis tehnique rounded on Lattie theory and propositional Calulus to disover the patterns and onepts within riminal data. This method onsidered the set of ommon and distint attributes of rime data. The oranization of this paper is as follows, Setion II proposes how the FCA was used to lassify and analyze rime data. Setion III provided appliation and results: provides the use of FCA rime analysis. The last setion of this paper, setion IV, Conludes the paper. II. METHODOLOGY In this paper, Formal Conept Analysis, or Galois Latties, a data analysis tehnique rounded on Lattie Theory and ropositional Calulus, was used for the visualization and analysis of rime data. This method onsidered the set of ommon and distint attributes of rimes in suh a way that ateorization are done based on related rime types, eoraphial loations and the persons involved. Formal onept analysis (FCA) is a method of data analysis with rowin popularity aross various domains. FCA analyzes data and desribes relationship between a partiular set of objets and a partiular set of attributes. Suh data ommonly appear in many areas of human ativities. FCA produes two kinds of output from the input data. The first is a onept lattie. A onept lattie is a olletion of formal onepts in the data whih are hierarhially ordered by a suboneptsuper onept relation [9][0]. In FCA, a formal ontext onsists of a set of objets, G, a set of attributes, M, and a relation between G and M, I G M. A formal onept is a pair (A,B) where A G and B M. Every objet in A has every attribute in B. For every objet in G that is not in A, there is an attribute in B that that objet does not have. For every attribute in M that is not in B there is an objet in A that does not have that attribute. A is alled the extent of the onept and B is alled the intent of the onept. If A and m B then (,m) I,or Im. A formal ontext is a tripel (G,M,I), where G is a set of objets, M is a set of attributes and I is a relation between G and M. (,m) I is read as objet has attribute m. For A G, we define A := {m M A:(,m) I }. For B M, we define dually B := { G m B:(,m) I }. For A, A, A G holds: A A A` A` A A`` A`= A``` For B, B, B M holds: B B B B B B`` B`= B``` A formal onept is a pair (A, B) where A is a set of objets (the extent of the onept), B is a set of attributes (the intent of the onept), A`= B and B`= A. The onept lattie of a formal ontext (G, M, I) is the set of all formal onepts of (G, M, I), toether with the partial order (A, B) (A, B): A A ( B B). The onept lattie is denoted by (G,M,I). Theorem: The onept lattie is a lattie, i.e. for two onepts (A, B) and (A, B), there is always reatest ommon subonept: (A A, (B B) ) and a least ommon superonept: ((A A), B B) More eneral, it is even a omplete lattie, i.e. the reatest ommon subonept and the least ommon superonept exist for all (finite and infinite) sets of onepts. Corollary: The set of all onept intents of a formal ontext is a losure system. The orrespondin losure operator is h(x):= X``. An impliation X Y holds in a ontext, if every objet havin all attributes in X also has all attributes in Y. Def.: Let X M. The attributes in X are independent, if there are no trivial dependenies between them. III. THE RESULTS AND ANALYSIS Considerin table below, we have the rows to onsist of persons and the olumns whih is the attributes, entails the ae, sex, rime type ommitted by this persons, and the eoraphial loation of the rime. Table onsists of the eoraphial loations and the eonomi fators, whih inludes inome index, Index of eduation, and population index, existin at these loations. Fiure two below is the eoraphial maps of the loations. From Table, Let a =aes <8, b=aes <0 and aes>0 For the sex, let m= male and f=female Let =drus, =rape, =burlary, =robbery Let the eoraphial loations be denoted by,.5. From Table, Let the eoraphial loations be denoted by,.5. For the index of inome, let a be index =< 0.5, b be index = < 0.5, be index =< 7.5, and d be index =< For the index of eduation, let e be index =< 0., f be index = < 0., be index =< 0.6, h be index =< 0.8, and i be index =<. Let that of population be {j, k, l, m, n}={0.,0.,0.6,0.8,}.

3 TABLE : ERSONS X CRIME DATA WITH GEO. LOCATION Ae Sex Crime Type Georaphial Loation a b m f 5 x x x x x x x x x x x x x x x x x x x x 5 x x x x x 6 x x x x x 7 x x x x 8 x x x x 9 x x x x x TABLE : GEOGRAHICAL LOCATIONS X ECONOMIC FACTORS Inome Index Index of Eduation opulation Index a b d e f h i j k l m n x x x x x x x x x x x x 5 x x x 5 Fiure : eoraphial map indiatin the loations Fiure : Galois latties of intents and extents from table

4 Fiure 5: A raph of rime ount aainst eoraphial loations. Fiure : Galois latties of intents and extents from table TABLE : GEOGRAHICAL LOCATIONS X CRIME TYES Total Total 6 6 The onept lattie of a formal ontext (G, M, I) as shown in fiure, is the Galois latties of intents and extents from table. This is the set of all formal onepts of (G, M, I), toether with the partial order (A, B) (A, B): A A ( B B). From the onepts, it an be observed that onept 57, 6, 69, and 7 have most of the attribute and most of the attributes of the rime types. This indiated the hih ourrene of rime types in. The onept lattie of a formal ontext (G, M, I) as shown in fiure,, is the Galois latties of intents and extents from table.. This is the set of all formal onepts of (G, M, I), toether with the partial order (A, B) (A, B): A A ( B B). From the onepts, it an be observed that onept, and 5 have most of the attribute with the least index. This indiated the hih ourrene of rime types in. Table onsists of the matrix of eoraphial loations ross rime types and fiure 5 is the raph of rime ount versus eoraphial loations. IV. CONCLUSION A formal onept analysis was used to analyze and visualize rime data. And relationships between the various rime types and different eoraphial areas were visualized. This method onsidered the set of ommon and distint attributes of rimes data in suh a way that ateorization was done based on related rime types and their eo-loations. This will help in buildin a more defined and oneptual systems for analysis of eoraphial rime data that an easily be visualized and intelliently analyzed by omputer systems. REFERENCES [] Kester, Quist-Aphetsi. "Computer Aided Investiation: Visualization and Analysis of data from Mobile ommuniation devies usin Formal Conept Analysis." arxiv preprint arxiv: (0). [] hillips,.; Lee, I.;, "Minin top-k and bottom-k or-relative rime patterns throuh raph representa-tions," tions," Intelliene and Seurity Informatis, 009. ISI '09. IEEE International Conferene on, vol., no., pp.5-0, 8- June 009.

5 [] Ratliffe, Jerry H. Intelliene-led poliin. Routlede, 0. [] Alex Leone, Chris Raastad, and Ior Tolkov. Yet Another Approah to Georaphi rofilin. Mathematial Contest in Modelin. Retrieved from: [5] Snook, B., Zito, M., Bennell, C., & Taylor,. J. (005). On the omplexity and auray of eoraphi profilin strateies. Journal of Quantitative Criminoloy, (), -6. [6] hillips,.; Lee, I.;, "Minin top-k and bottom-k orrelative rime patterns throuh raph representa-tions," Intelliene and Seurity Informatis, 009. ISI '09. IEEE International Conferene on, vol., no., pp.5-0, 8- June 009 [7] Shyam Varan Nath;, "Crime attern Detetion Usin Data Minin," Web Intelliene and Intellient Aent Tehnoloy Workshops, 006. WI-IAT 006 Work-shops. 006 IEEE/WIC/ACM International Confe-rene on, vol., no., pp.-, De. 006 [8]. Roerson, Y. Sun Spatial monitorin of eoraphi patterns: an appliation to rime analysis Computers, Environment and Urban Systems, Volume 5, Issue 6, November 00, aes [9] Radimbˇ Elohl Avek introdution to formal onept analysis Olomou 008. [0] Kester, Quist-Aphetsi. "Visualization and Analysis of Georaphial Crime atterns Usin Formal Conept Analysis." arxiv preprint arxiv: