A Bayesian mixture model for detecting unusual time trends

Modelling burglary counts in Cambridge

Guangquan (Philip) Li4th ESRC Research Methods Festival

July 5-8, 2010

Joint work with Nicky Best, Sylvia Richardson and Robert Haining

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Outline

1. Motivations

2. A Bayesian mixture model for detecting unusual time trends

3. Preliminary results from analysis of burglary data in Cambridge

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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

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• 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 (1993-1994), 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 CambridgePolice research Series paper 108

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Trend comparisons

Cambridge city as a whole

Two targeted wards

Need modelling

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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.

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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

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Specific model components (1)

• A conditional autoregressive (CAR) model is used to impose the spatial correlation.

Spatial smoothing

Specific model components (2)

• A random walk of order 1 is used to define the temporal structure.

Temporal smoothing

tt-1 t+1

Time

Non-informative priors are assigned to other parameters in the model

1. S varies2. S fixed (>1)

Classification• For each area, the posterior mean of zi (denoted by

pi) presents evidence for area i to follow the common trend pattern

a small value of pi 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.,

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The idea of classification

Unusual Usual

pi

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.

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A simulation study

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Simulation results

Scenario 1

Scenario 2

Scenario 3Small 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

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Simulation results

s-vary

S-vary

S=5

SaTScan

Reference: S = 2

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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.

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Burglary data in Cambridge

• Geo-referenced offence records in Cambridgeshire (2001-2008) 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).

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Overall spatial/temporal pattern

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Detected LSOA (FDR=0.01)

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High risks and unusual LSOA

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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.

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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.

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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.

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ReferencesSaTScan

• 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, 2:216-224, 2005.

False discovery rate (FDR)

• Storey J. A direct approach to false discovery rates. JRSS(B), 64: 479-498, 2002.

• Newton M, Noueiry A, Sarkar D, Ahlquist P. Detecting differential gene expression with a semiparametric hierarchical mixture method. Biostatistics, 5:155-176, 2004.

Crime

• Bennett T. and Durie L. Preventing residential burglary in Cambridge: From crime audits to targeted strategies. Police research series Paper 108, 1998.

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False discovery rate• The FDR measures the percentage of areas that

are identified as unusual but being truly usual.

• The FDR is a trade-off between the sensitivity and the specificity.

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A simulation approach to control for the FDR

• There are methods to approximate the FDR and hence the cutoffs based on posterior probabilities (e.g., Storey 2002 and Newton et al. 2004).

• However, these methods are not applicable in the current situation as here each area has its own alternative hypothesis.

• We have proposed a simulation based approach to estimate area-specific cutoffs that achieve the required level of confidence in the detection.

• If the distribution of the probability pi under the null hypothesis is known, then we can work out a cutoff value that corresponds to a pre-defined level of FDR, 5%, say.

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• H0 : the area follows the common trend. That is,

• We then fit the above model to real data to get the estimates for the intercept, α*, the space, ηi* , and the time, νt* ,components

Approximating the distribution of pi under the null hypothesis

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Approximating the distribution of pi under the null hypothesis

• Step 1: generate data, ysimi,t , from

0 1 pi

• Step 3: repeat Steps 1 and 2 many times (≈200)

The null: No areas are unusual!

• Step 2: fit the full detection model to the simulated data to get

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Classification with control for FP

We are with 95% confidence in our classification if the cutoff is obtained by controlling the false positive at the 5% level.

pcut=0.2

Unusualpi=0.1

Usualpi=0.35

Obtain the test statistic, pi by fitting the full detection model to real data

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