PRINCIPAL COMPONENT ANALYSIS OF CRIME DATA IN GWAGWALADA AREA COMMAND, ABUJA FROM

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1 Vol.5, No.3, pp.30-40, June 07 PRINCIPAL COMPONENT ANALYSIS OF CRIME DATA IN GWAGWALADA AREA COMMAND, ABUJA FROM Nsru M.O, S.O. Adms nd A.I. Alumbugu Deprtment of Sttstcs, Unversty of Abu, Abu Deprtment of Mthemtcs nd Sttstcs, Federl Polytechnc Nsrw ABSTRACT: Ths pper nlyses Abu crme dt whch conssts of the verges of twenty mor crmes reported to the polce for the perod nlyss nd prncpl component nlyss (PCA) were employed to expln the correlton between the crmes nd to determne the dstrbuton of the crmes over the three Are Councls under the Gwgwld Are Commnd. The result hs shown sgnfcnt correlton between robbery nd rpe, grevous hurt nd wound (GHW), theft, ssult, murder nd unlwful escpe. Gwgwld Are Councl hs the overll crme rte n the Are Commnd. Rpe s more prevlence n Kwl Are Councl whle unlwful possesson nd escpe s more prevlence n Kue Are Councl Are. The PCA hs suggested retnng four components tht expln bout 94.3 percent of the totl vrblty of the dt set. KEYWORDS: Multvrte Anlyss, Fctor Anlyss; PCA, GHW, DPHs INTRODUCTION In recent tme there hs been n ncrese n the reported cses of crme cross the country consequently n the FCT Abu, thus rsng the ssue of wht ctegores of crme s commtted n the federl cptl terrtory. The scope of crme preventon hs grown consderbly n the lst few yers. Wht ws prevously the sole concern of the polce nd the prvte securty ndustry hs spred nto res from rel estte developer, cr mnufcturers, resdents' groups, buldng publc fcltes lke socety offces nd shoppng centers, ll these clls for contnuously usng mproved new wys to prevent crme. Crmnl vctmzton hs serous consequences for the ctzens nd socety becuse hgh stndrd of lvng s undermned by hgh level of crmnl vctmzton (Alemk nd Chukwum, 005). The ssue of crme n the cty of Abu hs ncrese snce the relocton of the federl cptl to Abu, the sttstcl nlyss of the crme rte n Abu wll be expln nd nlyzed s Crme s one of the contnuous problems tht bedevl the exstence of mnknd. Prncpl component nlyss (PCA) s very useful n crme nlyss becuse of ts robustness n dt reducton nd n determnng the overll crmnlty n gven geogrphcl re. PCA s dt nlyss tool tht s usully used to reduce the dmensonlty (number of vrble) of lrge number of nterrelted vrbles whle retnng s much of the nformton (vrton) s possble. Lterture hs shown tht prevous reserch on crme dt were nlyzed usng Prncpl Component Anlyss (PCA). (Kendll Wllms et l, 004), clssfed cty s sfe or unsfe n the US Ctes by usng multvrte methods of prncpl components, fctor nlyss, nd dscrmnnt nlyss to reduce the 4 dstnct vrbles tht cn ffect the crme rte of cty to 6 nd 7 mportnt vrbles tht show hgh correlton wth ll the vrbles. Prncpl component nlyss dd decrese the number of vrbles to 6 nd ccounted for 86% of the totl vrnce, whle fctor nlyss decresed the number to 7 nd ccounted for 79.7% of the totl vrnce. Prnt ISSN: ISSN (Prnt), Onlne ISSN: ISSN (Onlne) 30

2 Vol.5, No.3, pp.30-40, June 07 (Olufolbo et. l, 05), nlyzed Oyo Stte crme dt whch conssts of 8 mor crme reported to the polce between the perod of They employed correlton nlyss nd prncpl component nlyss to expln the correlton between the crmes n the stte. Ther results showed sgnfcnt correlton, the prncpl component nlyss hs suggested retnng 6 components tht explned bout 83.79% of the totl vrblty of the dt set. (Yusuf Bello et l, 04), n ther reserch employed the method of fce-to-fce personl ntervew usng strtfed multstge rndom selecton procedure. They ppled PCA to nlyze the sptl pttern of crmnl vctmzton n the Locl Government Ares n Kstn Sentorl Zone. They found out tht Btsr hs the overll hghest verge vctmzton whle Rm hs the lowest verge vctmzton n the zone. (Shehu et l, 0), used ktsn stte dt whch consst of the verge of eght mor crmes reported to the polce for the perod nlyss nd prncpl component nlyss (PCA) were employed to expln the correlton between the crmes nd to determne the dstrbuton of the crmes over the locl government of the stte. The result hs shown sgnfcnt correlton between robbery, theft nd vehcle theft. The PCA hs suggested retnng four components tht expln bout percent of the totl vrblty of the dt set. Ths pper explores the use of correlton nlyss nd PCA for effectve crme control nd preventon. PCA offers tool for reducng the dmensonlty of very lrge dt set nd n determnng the res wth overll crme rte. These f properly mplemented, wll successvely solve mny of the complex crmnl problems tht hve bedevled the country n generl nd Abu n prtculr. The obectves of ths reserch nclude: () () () () to determne the bvrte ssocton exstng between prs of crme types. to dentfy the crme tht s domnnt n Gwgwld Are Commnd, FCT Abu to reduce the dmenson of the dt usng Prncpl Component Anlyss (PCA). to clssfy the crme type Dt Descrpton The totl number of crme commtted yerly wthn the perod for the 3 Dvsonl Polce Hedqurters (DPHs) or the 3 selected Are Councls n Abu ws collected offclly from the FCT Polce Hedqurters, Abu. For esy sttstcl nlyss nd nterpretton, the 3 Are Councls (AC) were ctegorzed ccordng. The three (3) re councls under the Gwgwld Are Commnd re Gwgwld, Kue nd Kwl. The dt conssts of thrteen mor crmes reported to the polce wthn the perod The crme nclude; rpe, robbery, Grevous Hurt nd Wound (GHW), theft, Vehcle theft, ssult, murder, cr stelng, unlwful possesson, brech of pece, broken store nd unlwful escpe. Frequences of crmes for ech ctegory were verged over the ten yers n the study perod to control for nomlous yers when there my hve been n unexplned spke or fll n crme levels pror to the sttstcl nlyss. The vlue for ech crme ws converted to crme rte per 00,000 popultons of the Are Councl (AC) whch ws clculted s (Kpedekpo nd Ary, 98). Prnt ISSN: ISSN (Prnt), Onlne ISSN: ISSN (Onlne) 3

3 Vol.5, No.3, pp.30-40, June 07 Number of Crmecommtted Crme Rte 00,000 Popultonof the ACs METHODOLOGY Prncpl Component Anlyss Prncpl component nlyss (PCA) s one of the most frequently used multvrte dt nlyss nd t cn be consder s proecton method whch proect observtons from p- dmensonl spce wth p-vrble to k-dmensonl spce (where K<P) so s to conserve the mxmum mount of nformton (nformton s mesured here through the totl vrnce of the sctter plots) from the ntl dmensons. Ths method trnsforms set of hghly correlted vrble nto the new sets whch re uncorrelted but contn lmost ll nformton n the orgnl dt. The new set of vrbles re often fewer n number thn orgnl vrbles. The fetures of PCA re; t ncorporte no error terms n ts structure nd hence t s mthemtcl procedure, It trnsforms set of vrble nto new set of uncorrelted vrbles, It s pproprte when vrbles re on equl footng. Dervton of PCA Gven P dmensonl rndom vector,,..., p., wth men vector,,.. p nd dsperson mtrx, PCA seeks new set vrble Z, Z,..., Z p (Pˈ often fewer thn P) so tht: Z... p p (3.) Where;,,..., p nd,,...,, combnton wth coeffcent,.., nd p p re coeffcents;thus; Z s re lner Z,...,, p.. p (3.) And Z, Z,..., Z p p (3.3) The procedures try to obtn so tht Z, the th PC of wll hve the followng propertes ()The Z s re orthogonl ()Ech Z cpture the mxmum vrble remnng n, hence we mxmze the vrton n Z subect to the constrnt Prnt ISSN: ISSN (Prnt), Onlne ISSN: ISSN (Onlne) 3

4 Vol.5, No.3, pp.30-40, June 07 For nstnce f Z, s the frst PC we seek for, such tht Vr Z VrZ Vr, s mxmum If Z s the second PC fter determnng Z, nd t s uncorrelted wth Z, we seek for such tht; Vr( Z ) Vr( ) s mxmum 0, tht s, Z nd Z re orthogonl The procedure contnue n ths wy to select the Vr Z ) Vr ( s mxmum, m m 0.e., Z s re orthogonl The procedure of fndng the frst pc To fnd the frst PC, we seek for, such tht th PC such tht; Z... p p (3.4) s the PC of subected to the constnt. Vr( Z ) Vr( ), s mx mum To mxmze vr (Z) subect to, we defne the lngrngn functon. L( ) ( ) (3.5) λ s the lngrngn multpler. To mxmze L() we dfferentte L() prtlly wth respect to nd equte the result to zero Thus, L( ) And I 0 0 (3.6) (3.7) Prnt ISSN: ISSN (Prnt), Onlne ISSN: ISSN (Onlne) 33

5 Vol.5, No.3, pp.30-40, June 07 Where: λ = Egen vector of s the correspondng Egen vector of λ The soluton = 0 s trvl soluton nd snce cnnot be zero (.e. 0) to hve nontrvl soluton then I 0 (3.8) nd mples tht I (3.9) If (3.7) s to hve non-trvl soluton, then becuse of (3.8), λ must be the chrcterstcs root of. Hence we wll hve P chrcterstc root nd P s whch re vector snce s P x P dmensonl mtrx lets λ, λ..., λp be the chrcterstc roots of, then vr(z) = λ = λ hencemx vr (Z), s equvlent to mx (λ). Tht s, f we hve Pλ s we choose the mxmum nd Vr (Z) = λ. The Procedure for Fndng the Second PC Let Z be the second PC of, then We seek such tht; ( ) Vr( Z ) s mx mum ( ) ( ) 0 The lngrngn functon s Z. ) ( ) (3.0) L( Thus; L( ) 0 I 0 (3.) Multply by to get; 0 0 ˊˊ λ- Ө=0 0 Prnt ISSN: ISSN (Prnt), Onlne ISSN: ISSN (Onlne) 34

6 Vol.5, No.3, pp.30-40, June 07 ( I) 0 ( I) 0 (3.) Interpretton of the prncpl components The lodng or the egenvector α =α, α, αp, s the mesure of the mportnce of mesured vrble for gven PC. When ll elements re postve, the frst component s weghted verge of the vrbles nd s sometmes referred to s mesure of overll crme rte. Lkewse, the postve nd negtve coeffcents n subsequent components my be regrded s type of crme components (Rencher, 00 nd Prntcom, 003). The plot of the frst two or three lodngs gnst ech other enhnces vsul nterpretton (Soren, 006). The score s mesure of the mportnce of PC for n observton. The new PC observtons Y re obtned smply by substtutng the orgnl vrbles nto the set of the frst PCs. Ths gves Y=α + α +.+ α pp (3.3) =,,., n, =,,,p The proporton of Vrnce The proporton of vrnce tells us the PC tht best explned the orgnl vrbles. A mesure of how well the frst q PCs of Z expln the vrton s gven by: A cumultve proporton of explned vrnce s useful crteron for determnng the number of components to be retned n the nlyss. A Scree plot provdes good grphclrepresentton of the blty of the PCs to expln the vrton n the dt (Cttell, 966). ANALYSIS AND RESULTS In ths secton, we present the results of the nlyss of the dt on the totl number of crme commtted yerly from 995 through 04 n Gwgwld Are Commnd, Abu, Nger. The dt ws collected from the polce hedqurters, Abu. Prnt ISSN: ISSN (Prnt), Onlne ISSN: ISSN (Onlne) 35

7 Vol.5, No.3, pp.30-40, June 07 Tble : mtrx of Crme Types (per 0,000 Populton) n Gwgwld, Abu Rpe Person Rpe Robb ery GHW Thef t.883 ** ** ** V. Theft Ass ult Murd er Burgl ry C/ Unlw Stelng ful posses son Brech of Publc Pece Brok en Store Unlw ful Escpe **.857 ** ** Sg. (-tled) Robbery Person GHW Theft.883 ** ** ** **.767 ** ** Sg. (-tled) Person.930 **.965 **.980 ** **.8 ** ** Sg. (-tled) Person.909 **.95 **.980 ** **.803 ** ** Sg. (-tled) V. Theft Person Assult Murder ** Sg. (-tled) Person.9 **.97 ** ** ** ** ** Sg. (-tled) Person.857 **.767 ** ** ** ** ** Sg. (-tled) Burglry Person C/Steln g Unlwfu l possess on Sg. (-tled) Person ** Sg. (-tled) Person **.744 **.30 Sg. (-tled) Brech Person of Publc **.849 **.090 Pece Sg. (-tled) Broken Store Unlwfu l Escpe Person **.849 **.054 Sg. (-tled) Person.837 **.985 ** ** **.5.88 **.738 ** Sg. (-tled) **. s sgnfcnt t the 0.0 level (-tled). Prnt ISSN: ISSN (Prnt), Onlne ISSN: ISSN (Onlne) 36

8 Vol.5, No.3, pp.30-40, June 07 Source: Derved from Sttstcs Deprtment of the Nger Polce Force, Abu Tble : Egenvlues Fg. : Scree Plot Intl Egenvlues Component Egenvlues Proporton Cumultve Prnt ISSN: ISSN (Prnt), Onlne ISSN: ISSN (Onlne) 37

9 S e c o n d C o m p o n e n t S e c o n d C o m p o n e n t Europen Journl of Sttstcs nd Probblty Vol.5, No.3, pp.30-40, June 07 Tble 3: Egenvectors Vrble PC PC PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC0 PC PC PC3 Rpe Robbery GHW Theft V. theft Assult Murder Burglry Cr stelng Unlwful Possesson Brech of pece Broken store Unlwful escpe Kwl Kwl Kwl Kwl6 Kwl Kwl3 Kwl4 Gwgwld Gwgwld GHWrobbery Rpe Cr stelng Murder v. thefttheft Burglry Assult Kue Kue Kue -3-4 Brech of pece possesson unlwful Frst Component Frst Component 0 () (b) Fg. : Lodng nd score plot for the frst nd second Prncpl Component (PC) DISCUSSION OF RESULTS The correlton mtrx n Tble dsplyed dfferent levels of correlton between the crmes. There s strong postve reltonshp n between robbery, rpe, grevous hurt nd wound (GHW), theft, ssult, murder, nd unlwful escpe, unlwful possesson nd brech of publc pece, c/stelng nd v.theft. Ther reltonshp were lso sgnfcnt t 5% sgnfcnce level., whch mens tht ther vrbles cn be used to predct (expln) one nother. Smlrly, t ws observed tht the correltons between robbery nd burglry, brech of publc pece nd broken store, rpe nd burglry, GHW nd burglry, theft nd burglry, ssult nd burglry, murder nd broken store, c/stelng nd unlwful possesson were negtve nd nsgnfcnt. Prnt ISSN: ISSN (Prnt), Onlne ISSN: ISSN (Onlne) 38

10 Vol.5, No.3, pp.30-40, June 07 The egenvlues, proporton nd the cumultve proportons of the explned vrnce re dsplyed n Tble. Consderng the egenvlue-one crteron nd the Scree plot n fgure, t would be resonble to retn the frst four PCs. A commonly ccepted rule sys tht t suffces to keep only PCs wth egenvlues lrger thn. However, the fourth egenvlues nd re pproxmtely close to, so tht the frst 4 PCs cn be retn to expln up to percent of the totl vrblty. From Tble 3, the frst PC combnes the number of ll the crmes wth pproxmtely postve constnt ( ) weght, nd s nterpreted s the overll mesure of crme. From fgure, Gwgwld Are Councl hs the overll crme rte, whle Kwl hve the lowest crme rte. The second PC on fgure (b) hs clssfed the crmes nto groups: () the concentrted crmes consstng of rpe, GHW, cr stelng, robbery, murder, rpe nd vehcle theft () Assult, burglry, brech of pece nd unlwful possesson. From Fgure (b), rpe s n outler nd s locted t the upper sde, nd G.H.W. s locted t the rght prt. Therefore, from Fgure (), the Are Councl t the upper sde show tendency towrds rpe, therefore Kwl Are Councl hs the hghest prevlence for rpe. Gwgwld Are Councl shows tendency towrds G.H.W., robbery, Assult, murder, cr stelng, Vehcle theft nd theft whle brech of pece nd unlwful possesson re locted t the lower prt of fgure,.e. Kue Are Councl. CONCLUSION The followng re the conclusons deduced from the nlyss. There s strong postve reltonshp between robbery nd rpe, grevous hurt nd wound (GHW), theft, ssult, murder nd unlwful escpe, the reltonshp were lso sgnfcnt, whch mens tht ther vrbles cn be used to predct (expln) one nother. It ws lso observed tht the correltons n between robbery nd burglry, brech of publc pece nd broken store were negtve nd nsgnfcnt. The Are Councl wth the hghest crme rte s Gwgwld, Kue hs moderte crme rte, whle Kwl Are Councl hve the lowest crme rte n Abu. Four PCs tht explns bout per cent of the totl vrblty of the dt set re suggested to be retned. The second component hs clssfed the crmes nto two, nmely, () concentrted offences: Rpe, G.H.W., vehcle theft, theft, Cr stelng, robbery, Murder () less concentrted: murder nd robbery. Bse on ths, the component hs geogrphclly dvded Gwgwld Are Commnd between the north nd south n relton to the crme clssfctons. The southern prts of A Contn more ssult, brech of pece, theft, vehcle theft, whle the northern prt hs the prevlence of the concentrted crmes lke rpe, GHW, robbery, murder, cr stelng. Ths wll help ndentfyng the dstrbuton of crmes n Gwgwld, Abu, llowng the nvestors to mesure the level of rsk nd to pln how preventve mesures for sfegurdng ther nvestments. Prnt ISSN: ISSN (Prnt), Onlne ISSN: ISSN (Onlne) 39

11 Vol.5, No.3, pp.30-40, June 07 REFERENCES Alemk E.E.O nd I.C.Chukwum (005), Desgnng ndctors of sfety nd ustce: lesson from the clen foundton s Ntonl Crme Vctms Surveys n Nger. Cttell, R.B. (966), The Scree test for the number of fctors. Multvrte Behvorl Reserch,, Kendll Wllms et l (004), A Multvrte Sttstcl Anlyss of Crme Rte n US Ctes. Kpedekpo, G.M.C. nd Ary, P. L. (98),Socl nd Economc Sttstcs for Afrc. George Allen nd Unwn, London. Olufolbo O.O. et l (05), Anlyzng the Dstrbuton of Crmes n Oyo Stte Usng Prncpl Component Anlyss. IOSR Journl of Mthemtcs (IOSR-JM) Vol. Issue 3 Rencher, A.C. (00), Methods of Multvrte Anlyss.nd edn, John Wley & Son, New York. Shehu U. et l. (0), Anlyss of Crme dt usng Prncpl Component Anlyss. A cse study of Ktsn stte CBN Journl of Appled Sttstcs vol.3no:39 Soren, H. (006). Exmple of multvrte nlyss n R Prncpl Component Anlyss(PCA). Yusuf Bello et l (04), Prncpl Component Anlyss of Crme Vctmztons n Kstn Sentorl Zone Interntonl Journl of Scence nd Technology Vol.3 No.4 Prnt ISSN: ISSN (Prnt), Onlne ISSN: ISSN (Onlne) 40