A Bayesian mixture model for detecting unusual time trends Modelling burglary counts in Cambridge Guangquan (Philip) Li 4 th ESRC Research Methods Festival July 5-8, 2010 Joint work with Nicky Best, Sylvia Richardson and Robert Haining

Outline 1.Motivations 2.A Bayesian mixture model for detecting unusual time trends 3.Preliminary results from analysis of burglary data in Cambridge 2

Reasons for detecting unusual time trends Emergence of local risk factors? Change of population composition? Impact of a new policing scheme? Modelling Highlighting areas deserving of further scrutiny Identifying possible risk factors Informing policy making Assessing effect of policy 3

The report describes the work of the Domestic Burglary Task Force (DBTF) in Cambridge, which was established to examine the nature of residential burglary in Cambridge and to design and implement initiatives to prevent it. Analysis of the burglary counts ( ), the DBFT identified the largest ‘hot spot’ in the north of the City, and the two wards which contained the ‘hot spot’, as the targeted area. After a series of seminars, a number of burglary prevention strategies were identified and implemented. Question: whether the strategies helped to reduce residential burglary rates? Preventing residential burglary in Cambridge Police research Series paper 108 4

Trend comparisons Cambridge city as a whole Two targeted wards 5 Need modelling

A Bayesian detection model We have proposed a detection method that, for each area, provides estimates independently from the common trend component and the area-specific trend component and selects estimates between the two to describe the observed data. For each area, the posterior probability of selecting the common trend component is used to classify the area/trend as “unusual” or not. 6

A schematic diagram of the detection model Space-Time variations Common time trend Area-specific time trends Common spatial pattern Area-specific time trends Common spatial pattern Common time trend Space-time separable Space-time inseparable 7

Specific model components (1) A conditional autoregressive (CAR) model is used to impose the spatial correlation. Spatial smoothing 8

Specific model components (2) A random walk of order 1 is used to define the temporal structure. Temporal smoothing tt-1t+1 Time Non-informative priors are assigned to other parameters in the model 1.S varies 2.S fixed (>1)

Classification For each area, the posterior mean of z i (denoted by p i ) presents evidence for area i to follow the common trend pattern  a small value of p i suggests that the area is unlikely to follow the common trend The area is unusual if the above probability is less than some threshold, i.e.,

The idea of classification Unusual Usual pipi Prob (An area follows the common trend pattern) Choose cutoff to achieve pre-specified false detection rate (FDR) Cutoff values cannot be obtained using conventional approaches such as Storey 2002 since null hypothesis is specific to each areas. We have proposed a novel simulation approach to obtain area-specific cutoffs so that we can maximize the sensitivity while controlling for FDR. Choose cutoff to achieve pre-specified false detection rate (FDR) Cutoff values cannot be obtained using conventional approaches such as Storey 2002 since null hypothesis is specific to each areas. We have proposed a novel simulation approach to obtain area-specific cutoffs so that we can maximize the sensitivity while controlling for FDR. ⌘ 11

A simulation study 12

Simulation results Scenario 1 Scenario 2 Scenario 3 Small departures Large departures 15 (out of 354) areas were selected according to the population sizes and spatial risks and assigned the unusual trend. Comparing the gain/loss of sensitivity amongst the following 4 models 1. S-vary 2. S=2 (the optimal setting, the reference) 3. S=5 4. SaTScan (space-time permutation test) S-vary S=5 SaTScan Scenario 1 13

Simulation results s-vary S-vary S=5 SaTScan Reference: S = 2 14

Summary: Key features of the model The comprehensive simulation study has shown some key features of our model: 1.Our model can detect various realistic departure patterns; 2.The performance is robust over different model settings; 3.Our model outperforms the popular SaTScan; 4.Our detection model works relatively well on sparse data. 15

Burglary data in Cambridge Geo-referenced offence records in Cambridgeshire ( ) are made available by the Cambridgeshire Constabulary; In this analysis, we focus on the burglary counts in Cambridge at the Lower Super Output Area (LSOA) level for each quarter from 2001 to 2002 (2584 reported burglary cases). Numbers of houses were taken from the 2001 Census then aggregated to LSOA level (≈600 houses). 16

Overall spatial/temporal pattern 17

Detected LSOA (FDR=0.01) 18

High risks and unusual LSOA 19

Future work We are currently working closely with the Cambridgeshire Police to assess effectiveness of possible policing schemes; The framework can be extended to a prospective surveillance system by applying the detection model sequentially to observed data; Incorporation of time-varying covariates (e.g., unemployment from surveys) can enrich the detection analysis. 20

Summary We have proposed a Bayesian mixture model for detecting unusual time trends; The extensive simulation study has shown the superior performance of the model in detecting various “real” departures; Applying the model to the offence data can assist/inform policy making (by identifying abrupt changes) and help to assess policy. 21

Acknowledgement Funded by ESRC The BIAS project (PI Nicky Best), based at Imperial College London, is a node of the Economic and Social Research Council’s National Centre for Research Methods (NCRM) The offences data are kindly provided by the Cambridgeshire Constabulary. 22

References SaTScan Kulldorff M, Heffernan R, Hartman J, Assunção RM, Mostashari F. A space-time permutation scan statistic for the early detection of disease outbreaks. PLoS Medicine, 2: , False discovery rate (FDR) Storey J. A direct approach to false discovery rates. JRSS(B), 64: , Newton M, Noueiry A, Sarkar D, Ahlquist P. Detecting differential gene expression with a semiparametric hierarchical mixture method. Biostatistics, 5: , Crime Bennett T. and Durie L. Preventing residential burglary in Cambridge: From crime audits to targeted strategies. Police research series Paper 108,