The Dynamic Spatial Disaggregation Approach: A Spatio-Temporal Modelling of Crime

Transcription

1 The Dynamic Spaial Disaggregaion Approach: A Spaio-Temporal Modelling of Crime Chrisian Ivaha 1, Hasan Al-Madfai 2, Gary Higgs 3 and Andrew Ware 4 Absrac The spaio-emporal modelling and forecasing of incidences of crimes have now become a rouine par of crime prevenion operaions. However, obaining reliable forecass for cases of changing spaial boundaries using models which ake he influences of exogenous facors on he spaial and emporal dynamics of crime ino accoun remains a challenge. The proposed Dynamic Spaial Disribuion Approach (DSDA) is a modelling approach ha provides spaio-emporal forecass ha incorporae he influences of salien weaher condiions ha have a bearing on crime dynamics a boh he emporal and spaial levels. The DSDA sars wih modelling and forecasing he daase of he enire area of ineres as a whole. Given reliable forecass, he DSDA spaially disaggregaes hem by exrapolaing he spaial paerns idenified from he hisoric daa. This is carried ou by firsly idenifying he spaial paern and hen defining a se of weighs, one for each of he clusers making up his paern. A number of mehods o deermining hese weighs, including a benchmark, univariae and causal approaches are proposed. The DSDA was developed wihin a GIS and classical saisics environmen. Applied o he daily number of criminal damage incidences in he Ciy of Cardiff, UK, he DSDA yielded a ciy-wide forecas error of 3.51 crimes per day. The spaial disaggregaion using weighs idenified using Leas Squares esimaes of expeced crime percenages in each cluser yielded an average of crimes per day per cluser. The model used incorporaed he influences of weekdays and ambien condiions on he spaial disribuion of crime Index Terms Geographical Informaion Sysem (GIS), Spaial disaggregaion, Spaio-Temporal forecasing model. I. INTRODUCTION The main purpose of he implemenaion of he Dynamic Spaial Disaggregaion Approach (DSDA) is o provide he Police Crime Prevenion Uni of Souh Wales Police wih an elaborae spaio-emporal crime forecasing model o aid in opimising he use of heir resources. In comparison wih esablished approaches in spaio-emporal forecasing (see, for example, [1-3]), DSDA incorporaes exogenous informaion boh a he emporal and spaial level of is implemenaion. In addiion, wih DSDA, he emporal forecas of crime was 1 Faculy of Advanced Technology, Universiy of Glamorgan, UK, CF37 1DL. civaha@glam.ac.uk 2 Senior Lecurer in he Faculy of Advanced Technology, Universiy of Glamorgan, UK, CF37 1DL. hmadfai@glam.ac.uk 3 Professor in he Faculy of Advanced Technology, Universiy of Glamorgan, UK, CF37 1DL. ghiggs@glam.ac.uk 4 Professor in he Faculy of Advanced Technology, Universiy of Glamorgan, UK, CF37 1DL. jaware@glam.ac.uk produced using he enire ciy as opposed o modelling each se of idenified cluser independenly, as demonsraed in [4] for example. DSDA herefore offers major improvemens on exising approaches since modelling he clusers independenly ignores he evoluion in he spaial disribuion of he daa over ime and i is liable o produce daases in which he scales and variabiliy are oo limied for reliable forecass o be produced. The forecasing mehod used in his projec was he Hierarchical Profiling Approach (HPA) which was demonsraed o provide superior forecasing resuls in ime series, and in crime specifically, compared o radiional approaches as discussed in [5]. The crime daa uilised in his paper enclose he criminal damage incidences in he Ciy of Cardiff, wih spaial (x, y) coordinaes and ime of occurrence of he assauls. DSDA is a generalisaion of he Simple Spaial Disaggregaion Approach (SSDA) which was developed in [6]. SSDA is consruced wihin a GIS environmen, and uses STAC as he cluser deecion rouine echnique. STAC, using an ellipsoid-shaped cluser algorihm, offers he main advanage of calculaing clusers wihou being resriced o any se boundaries. This way, whils SSDA is only concerned wih spaial paerns purely based upon calendar evens, DSDA also includes weaher parameers in is spaio-emporal implemenaion. In his paper, SSDA will be briefly inroduced in Secion III o provide he grounds for undersanding he basis of his approach. Then DSDA is deailed in Secion IV, followed by Conclusion and Fuure Work in Secion V. The nex secion gives an overview of he HPA and is applicaion in modelling he Criminal Damage ime series. A. An Overview of HPA II. CRIME MODELLING OF CARDIFF Profiling, which idenifies and models he change of behaviour of a imes series during disinc periodic and aperiodic evens, maches he observed level shifs in he daa o evens salien o he ime series. HPA decomposes variabiliy in ime series daa ino hree componens: deerminisic, sochasic and noise. Based on his, he general form of he HPA model is y = f ( ) + Z, (1) where f ( ) is he deerminisic profile funcion and Z is he

2 sochasic componen, conaining he noise, o be modelled. The HPA combines differen profiling levels hierarchically in is applicaion. For he Criminal Damage daa, his was carried ou as follows: Level 1: This level modelled level shifs and salien profiled evens ha occur wihin he year, as displayed in Table 1, and Level 2 was used o model he general shape of he series all year round, including rend, seasonaliy and oher periodic disurbances. Table 1: Caalogued Evens used in he applicaion of HPA Evens Criminal Damage Profiles Halloween Day Six Naions Rugby Tournamen Main Cardiff Universiy Sudens Union Evens Concers a Cardiff Inernaional Arena Fooball maches a Millennium Sadium Good Friday SpeedWay a Millennium Sadium Oher Rugby maches a Millennium Sadium Fooball maches a Ninian Park Day Afer Halloween S Parick's Day 9.97 Cricke Mach ( ) 9.82 ½ Half Term / Bank Holidays 9.72 Bonfire Nigh 9.43 Easer Weekend 8.81 Rugby maches a Arms Park 7.99 Boxing Mach 7.85 B. Concluding Model The sochasic Z series was modelled using ARIMA (2,0,2)(2,0,0) as follows ( )( ) 2 ( B + B ) e B B B + B Z = , (2) Hence, he one sep-ahead forecass, y (1) were obained by aggregaing he profiles back ino he sochasic forecass as y (1) = f ( ) + Z (1) (3) 1 1 The RMSE obained from his model was 4.401, he MAPE was found o be 20.64% wih a saisical mode of 17 crimes for he enire daa series. Therefore, he average crime predicion error (i.e., MAPE muliplied by he saisical mode) was found o be ± 3.51 crimes per day for he ciy as a whole. Fig. 1 displays he observed values of he Criminal Damage daase alongside he one-sep ahead forecass obained from he HPA, presened in (3) above. III. THE SIMPLE SPATIAL DISAGGREGATION APPROACH (SSDA) In is simples form, he spaial disaggregaion of crime aims o divide he daily forecass obained wih HPA and allocaes hem o he clusers idenified wih STAC as displayed in Fig. 2. Upon furher invesigaion of he spaial paerns in he daa, spaial dependency beween weekdays and Cardiff s 29 Census Wards was idenified using a coningency able 2 ( χ = , d.f=168, p<0.05). Hence, he spaial disaggregaion of crime was revised accordingly, so ha each weekday presened is own spaial paern and hence is own se of clusers. Fig. 3 shows he weekday-relaed spaial disaggregaion diagrammaically. Figure 1: Daily Criminal Damage Daa wih One-sep ahead forecass

3 NFM is non rivial and provides a benchmark for he oher mehods inspeced in his work. The Ordinary Leas Square mehod on Number of Incidences (OLS-NI) Regression analysis was performed on he Number of Incidences per cluser per weekday so ha he number of incidences per cluser on a, say, Monday was reaed as a series wihin each cluser. Therefore, each cluser, per weekday, was allocaed wih a Leas Square Fi from which an esimae of expeced number of fuure crimes is obained. Wih he sum of he esimaes se o 100%, he SDF weigh for each cluser is se o be equal o he percenage of he cluser s leas squares esimae o he oal. Figure 2: Illusraion of Spaial Disaggregaion on Cardiff Ciy In order o incorporae he spaial dimension ino forecas evaluaions, he novel Spaio-Temporal Mean Roo Square Esimae (STMRSE) was developed o evaluae he spaio-emporal forecas errors. The STMRSE is wrien as follows The Arihmeic Mean (AM) In his mehod, he SDF weighs were se o be equal o he AM of he Percenages of he overall Incidences per cluser per weekday The Ordinary Leas Square mehod on Percenages of Crime (OLS-PC) OLS was used on he daily percenages of crime occurrences per cluser. Regression analysis allowed he implemenaion of Leas Square Fis for all clusers per weekday as previously inroduced in he OLS-NI measure wih he SDF weighs se o be equal o he esimaes of he percenages of crimes obained from he OLS fi of he regression line. STMRSE ( ε ) ( j ) m 1 j = n n O O m j 2, (4) Table 2 displays he STMRSE resuls calculaed using (4) for he four SDF weighs esimaion mehods used in disaggregaing he forecass wihin SSDA. Table 2: STMRSE for SSDA wih n being he oal number of days, O j being he crime forecas for cluser j a day and m he oal number of clusers a day. In order o calculae he proporions of he forecas o be allocaed o each cluser, a se of weighs was defined so ha he forecas for each cluser, O j, is obained from y (1) 1 O j = β, wih β 1 j = j m, (5) where β are he Spaial Disaggregaion Forecas (SDF) weighs. Four mehods o idenify suiable SDF weighs were j implemened o obain he crime forecass O j. These are: The Naïve Forecasing Mehod (NFM) NFM uses pas values o predic fuure weekday-relaed values, so ha he SDF weighs of, say, Monday of week (n+1) will be hose of Monday of he week before (i.e., week n). The NFM OLS-NI AM OLS-PC I can be seen from Table 2 ha OLS-NI was superior, wih STMRSE of crimes per weekday per cluser followed closely by OLS-PC and AM. IV. THE DYNAMIC SPATIAL DISAGGREGATION APPROACH (DSDA) DSDA generalises he idea of SSDA o he inclusion of exogenous variables in he disaggregaion of forecass so ha spaial paerns can be allowed o vary in response of exogenous salien variables. The exploraory daa analysis revealed ha Rainfall influences he spaial dynamics of crime in such a way ha shifs of clusers, emergence of new clusers and diminuion of criminal aciviies wihin idenified clusers would be observed. Comap, inroduced by [7], was uilized o idenify he exen of he influence of ambien condiions on Cardiff Ciy. Comap is based on condiional ploing (i.e., Coplo) and Kernel Densiy Esimaion procedures (i.e., Kernmap). Fig. 4 displays

4 Figure 3: Illusraion of weekly-relaed Spaial Disaggregaion on Cardiff Ciy he condiional disribuion of crime a differen levels of Rainfall, wih he same sampling size in each drawn panel. In his simple case, no evidence of changes in he spaial dynamics was observed. However, furher deailed Figure 4: Comap Plo of All Rainfall Daa invesigaions revealed ha a relaionship beween Rainfall and he disribuion of crime does exis. These invesigaions led o he implemenaion of he Revelaory Index (RI) on Rainfall. A. Revelaory Index (RI) on Rainfall The Revelaory Index (RI) generally aims o reveal he varying spaial dynamics of crime in relaion o ambien condiions by eliminaing values and/or observaions which mask he relaionships beween he daases. This allows for rends and concealed crime dynamics, which would oherwise be hidden, o be sudied. In Cardiff Ciy s Rainfall daase, 493 observaions ou of 850 (i.e., around 58 %) had a value of 0.00 mm. This clearly biases he daase owards hese values and complicaes invesigaions ino he non-zero values. The implemenaion of RI on Rainfall was herefore o only consider he non-zero values in he Rainfall daa. The daa hence varies from mm o mm. Fig. 5 displays he RI on Rainfall Comap plo ono which changes in spaial crime dynamics in response o differen levels of rainfall can now be observed.

5 Rainfall condiions were considered in he implemenaion of DSDA. These were {Rainfall} = [0.00, 0.01[ and {Rainfall} = [0.01, 3.160]. In he execuion of DSDA, he four above-menioned cluser forecas approximaion mehods (i.e., NFM, OLS-NI, AM and OLS-PC) were individually calculaed for each Rainfall se and was hen combined wihin each mehod o mainain he inegriy of he crime daa. B. Resuls Fig. 6 illusraes he spaio-emporal implemenaion of DSDA wih any weaher parameers and weekly paerns inegraed in he algorihm. The resuls from applying he DSDA using he four differen ses of weighs discussed in Secion III are given in Table 3. Table 3: STMRSE for DSDA Figure 5: Revelaory Index on Rainfall Comap Plo I can be seen from Fig. 5 ha changes in he spaial dynamics of crime occurred when Rainfall 0.01 mm, hence 2 ses of NFM OLS-NI AM OLS-PC Figure 6: Illusraion of DSDA on Cardiff Ciy

6 I can be seen ha he forecass obained using OLS-PC were superior, wih STMRSE of crimes per weekday per cluser followed closely by he AM mehod. V. CONCLUSION AND FUTURE WORK The resuls obained using DSDA are superior o hose obained by SSDA, wih a STMRSE of crimes per weekday per cluser compared o a STMRSE of Boh hese spaio-emporal forecasing models are easy o implemen and can be exended o oher fields of sudy such as pahological and environmenal modelling and forecasing. Fuure work will be looking a oher significan weaher parameers o crime disribuion in order o improve he forecas accuracy of DSDA. Fuure work can also include he opimisaion of he SDF weigh β by uilizing opimisaion procedures such as he Kalman Filer. j REFERENCES [1] Ackerman, W.V. and A.T. Murray, Assessing spaial paerns of crime in Lima, Ohio. Ciies, (5): p [2] Brown, D.E. and H. Liu, Criminal inciden predicion using a poin-paern-based densiy model. Inernaional Journal of Forecasing, (4): p [3] Rossmo, D.K., Geographic profiling. 1999, Boca Raon, Fl: CRC Press. [4] Xue, Y. and D.E. Brown, Spaial analysis wih preference specificaion of laen decision makers for criminal even predicion. Decision Suppor Sysems, (3): p [5] Al-Madfai, H. Hierarchical Profiling of daily Crime Time Series Daa as a Precursor o Modelling. in ISF San Anonio, Texas. [6] Ivaha, C., e al., The Simple Spaial Disaggregaion Approach o Spaio-Temporal Crime Forecasing. Inernaional Journal of Innovaive Compuing, Informaion and Conrol (IJICIC), (3): p [7] Brunsdon, C., The comap : exploring spaial paern via condiional disribuions. Compuers, Environmen and Urban Sysems, : p