towards efficient prospective detection of multiple spatio -temporal clusters

Towards efficient prospective detection of multiple

spatio-temporal clusters

Bráulio Veloso, Andréa Iabrudi and Thais Correa.Universidade Federal de Ouro Preto – UFOPNovember, 2013, Campos do Jordão, SP – Brazil

XIV Brazilian Symposium on GeoInformatics

Content

• Introduction• Method– STCD– Problem– STCD-Sim

• Metrics• Simulated Datasets• Results• Final Considerations

Introduction

• Technique to efficiently detect multiple emergent clusters in a space-time point process

Introduction

• Technique to efficiently detect multiple emergent clusters in a space-time point process– Surveillance Systems;• On-line;• Prospective;

Introduction

• Technique to efficiently detect multiple emergent clusters in a space-time point process– Surveillance Systems;– Applications:

Introduction

• Technique to efficiently detect multiple emergent clusters in a space-time point process– Surveillance Systems;– Applications:• Epidemic surveillance;

Introduction

• Technique to efficiently detect multiple emergent clusters in a space-time point process– Surveillance Systems;– Applications:• Epidemic surveillance;• Criminology behavior;

Introduction

• Technique to efficiently detect multiple emergent clusters in a space-time point process– Surveillance Systems;– Applications:• Epidemic surveillance;• Criminology behavior;• Traffic control;

Introduction

• Technique to efficiently detect multiple emergent clusters in a space-time point process– Surveillance Systems;– Applications:• Epidemic surveillance;• Criminology behavior;• Traffic control;• Social networks behavior;

Introduction

• Technique to efficiently detect multiple emergent clusters in a space-time point process– Spatio-temporal data are more available;

Introduction

• Technique to efficiently detect multiple emergent clusters in a space-time point process– Spatio-temporal data are more available;– Process with more then one cluster;

Introduction

• Technique to efficiently detect multiple emergent clusters in a space-time point process– Spatio-temporal data are more available;– Process with more then one cluster;– Need of computationally efficient approaches.

Introduction

• STCD– Point Process;

Introduction

• STCD– Point Process;– Earlier identification;

Introduction

• STCD– Point Process;– Earlier identification;– Fast Execution;

Introduction

• STCD– Point Process;– Earlier identification;– Fast Execution;– Efficient detection;

Introduction

• STCD– Point Process;– Earlier identification;– Fast Execution;– Efficient detection;

– But identifies only one cluster.

The Space-Time Cluster Detection

STCD – Space Time Cluster Detection

• Renato Assunção and Thais Correa. Surveillance to detect emerging space-time clusters. Computational Statistics and Data Analysis, 53(8):2817-2830, 2009.

• Surveillance Systems

• Surveillance Systems– Process: Under Control vs. Out of Control;– System: try to detected earlier a change in the

process

• Surveillance Systems;• Spatio-Temporal Events – Tuple: (id, x, y, t);– Order by time.

• Surveillance Systems;• Spatio-Temporal Events;• Alarm– Evidence that the process changed from in control

to out of control.

• Surveillance Systems;• Spatio-Temporal Events;• Alarm;• Space-Time Cluster– Cylindrical shape• Circular base in space• Temporal Height

• Surveillance Systems;• Spatio-Temporal Events;• Alarm;• Space-Time Cluster ;• Prospective Detection– Live Cluster

STCD – Space Time Cluster Detection– Ck,n : candidate cylinder to be a cluster, beginning

(centered) in event k and ending in the last event

STCD – Space Time Cluster Detection– Ck,n : candidate cylinder to be a cluster, beginning

(centered) in event k and ending in the last event;

– Lk : likelihood of the space-time Poisson process when there is a cluster Ck,n;

– L ∞ : likelihood of the space-time Poisson process when there is no cluster.

STCD – Space Time Cluster Detection• Cumulative Sum Statistic

)(exp1 ,)(

nk Cεε+ nk

• Each parcel k is related to a candidate cluster.

)(exp1 ,1

)( ,nk

CNn Cεε+=R nk

– ε: increase in the intensity inside the cluster Ck,n;

)(exp1 ,1

)( ,nk

CNn Cεε+=R nk

– N(Ck,n): number of events inside Ck,n;

)(exp1 ,1

)( ,nk

CNn Cεε+=R nk

– μ(Ck,n): expected number of events inside Ck,n.• non parametric estimate for μ(Ck,n).

)(exp1 ,1

)( ,nk

CNn Cεε+=R nk

– μ(Ck,n): expected number of events inside Ck,n.• non parametric estimate for μ(Ck,n).

)(exp1 ,1

)( ,nk

CNn Cεε+=R nk

ttANttkBNC nkn

],(.],(),()(ˆ 0

• Alarm or not?

– A and‘

knkn =R

Timetactual

tactual

Timetactual

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ)• Used in the definition of spatial neighborhood for each

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ)• ρ

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ)• ρ ↑

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ)• ρ ↑↑

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ)– Increase in the intensity inside the cluster (ε);

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ);– Increase in the intensity inside the cluster (ε)• ε

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ);– Increase in the intensity inside the cluster (ε)• ε ↑

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ);– Increase in the intensity inside the cluster (ε)• ε ↑↑

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ);– Increase in the intensity inside the cluster (ε);– Threshold (A)

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ);– Increase in the intensity inside the cluster (ε);– Threshold (A)• A↓• Faster Detection

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ);– Increase in the intensity inside the cluster (ε);– Threshold (A)• A↓↓• Increase the number of false alarms

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ);– Increase in the intensity inside the cluster (ε);– Threshold (A)• A ↑• Slower Detection

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ);– Increase in the intensity inside the cluster (ε);– Threshold (A)• A ↑↑• No Detection

STCD – Space Time Cluster Detection• Parameters: – Spatial Radius (ρ);– Increase in the intensity inside the cluster (ε);– Threshold (A)• How much events the user wants to wait before a false alarm.

STCD – Space Time Cluster Detection• Operations at time i:

N(C1,12)

μ(C1,12)

N(C1,12)

μ(C2,12)

N(C2,12)

• Alarm!

• Identifying the cluster– a

– Cylinder:

nknknk 1,max ,*,

nkC *,

Problem• Is there more than one cluster?

Problem• How to identify these two clusters

simultaneously?

STCD-Sim

Our extension

STCD-Sim• Our extension:– Simultaneous Space-Time Clusters Detection

(STCD-Sim);

(STCD-Sim);– Same parameters and data type;

(STCD-Sim);– Same parameters and data type;– Automatically identifies as many clusters as there

are in the database.

STCD-Sim• a

– N(Ck*,n): number of events inside the detected cluster;

– μ(Ck*,n): number of events expected in detected cluster.

)(exp1 *,)(

nk Cεε+ nk

STCD-Sim• a

– N(Ck*,n): number of events inside the detected cluster;

– μ(Ck*,n): number of events expected in detected cluster.

• Excess of events

)(exp1 *,)(

nk Cεε+ nk

)()()( *,*,*, nknknk CCNC

STCD-Sim• Excess of events– A ;)()()( *,*,*, nknknk CCNC

STCD-Sim• Excess of events– A ;

• Delete excess of events inside Ck*,n – Random way;

• Adjusted threshold– A ;

)(' *,nkCAA

• Adjusted threshold– A ;

• Re-run the method with the reduced database and adjusted threshold A’ (same ρ and ε).

)(' *,nkCAA

Evaluation Metrics

• For an unique cluster case– No Alarm– Incorrect Alarm– Correct Alarm

Evaluation Metrics

• For an unique cluster case– No Alarm• a

– Incorrect Alarm– Correct Alarm

niARi ,..,1

Evaluation Metrics

• For an unique cluster case– No Alarm– Incorrect Alarm• A and

– Correct Alarm niARi ,..,1 TRUEik CC *,

Evaluation Metrics

• For an unique cluster case– No Alarm– Incorrect Alarm– Correct Alarm• A and niARi ,..,1 TRUEik CC *,

Evaluation Metrics

• For an unique cluster case

Evaluation Metrics

• For an unique cluster case

Evaluation Metrics

• For two simultaneous clusters case– No Alarm– Single Alarm– Double Alarm

Evaluation Metrics

• For two simultaneous clusters case– No Alarm• a

– Single Alarm– Double Alarm

Evaluation Metrics

• For two simultaneous clusters case– No Alarm– Single Alarm• A and

– Double AlarmARi '' AR i

Evaluation Metrics

• For two simultaneous clusters case– No Alarm– Single Alarm– Double Alarm• A andARi '' AR i

Evaluation Metrics

• For two simultaneous clusters case– No Alarm– Single Alarm• Correct• Incorrect

– Double Alarm• Correct• Incorrect• Half-Correct

Evaluation Metrics

No Alarm

Incorrect Alarm

IncompleteAlarm

Complete Alarm

Evaluation Metrics

No Alarm

Incorrect Alarm

IncompleteAlarm

Complete Alarm

Evaluation Metrics

No Alarm

Incorrect Alarm

IncompleteAlarm

Complete Alarm

Evaluation Metrics

No Alarm

Incorrect Alarm

IncompleteAlarm

Complete Alarm

Evaluation Metrics

No Alarm

Incorrect Alarm

IncompleteAlarm

Complete Alarm

Evaluation Metrics

No Alarm

Incorrect Alarm

IncompleteAlarm

Complete Alarm

Evaluation Metrics

• Delay Time

Evaluation Metrics

• Delay Time– elapsed time between the cluster start and its

actual identification;

Evaluation Metrics

• Delay Time– elapsed time between the cluster start and its

actual identification;

Delay Delay

Datasets

Simulated Databases

Simulated databases

• Homogeneous Poisson Point Process– Space:• X: [0, 10];• Y: [0, 10];

– Time:• T: [0, 10];

– Databases with one and two clusters.

• Clusters:– ε = 1.0, 3.0 e 10.0;– ρ = 0.5, 1.0, 1.5 e 2.0;– Δt = [5, 10], [7, 10] e [8, 10];• The process begin under control in time 0 and one or

two clusters start at time 5, for example.

• Running the STCD:– Input parameters (equal to the true values);– A = n (total number of events).

Simulated databases

Results

Percentage of AlarmsDelays

Results – Alarms unique cluster

• No Alarm

• Incorrect Alarm

• Correct Alarm

Results – Alarms unique cluster

• No Alarm

• Incorrect Alarm

• Correct Alarm

General Mean:1.37%; 3.39%; 95.24%

Results – Alarms two clusters

• No Alarm

• Incorrect Alarm

• Incomplete Alarm

• Complete Alarm

• No Alarm

• Incorrect Alarm

• Complete Alarm

General Mean:• 1.00%• 1.69%• 63.68% • 33.63%

• No Alarm

• Incorrect Alarm

• Complete Alarm

General Mean:• 1.00%• 1.69%• 63.68% • 33.63%

Our extension reached a complete Alarm in 88.2% of cases in database

Results – Delay

• Delay 1C.

• 2C. Delay 1st

• 2C. Delay C1

• 2C. Delay C2

• 2C. Delay Duplo

Results – Delay

• Delay 1C.

• 2C. Delay 1st

• 2C. Delay C1

• 2C. Delay C2

• 2C. Delay Duplo

Final Considerations

• Our extension for multiple cluster– Percentage of detection for both clusters around

88%;– Delay for two clusters slightly larger than delay for

one cluster;

88%;– Delay for two clusters slightly larger than delay for

one cluster;– Promising method.

• Future works– Evaluate the impact of changing ρ and ε;

• Future works– Evaluate the impact of changing ρ and ε;– Apply to a real database and benchmark;

• Future works– Evaluate the impact of changing ρ and ε;– Apply to a real database and benchmark;– Compare with others approaches;

• Future works– Evaluate the impact of changing ρ and ε;– Apply to a real database and benchmark;– Compare with others approaches;– Remove the restriction of the cylindrical shape for

the cluster.

References[1] Renato Assunção and Thais Correa. Surveillance to detect emerging space-time clusters. Computational Statistics and Data Analysis, 53(8):2817-2830, June 2009.

[2] B. Veloso, A. Iabrudi and T. Correa. Localização em tempo real de acontecimentos através de vigilância espaço-temporal de microblogs. In IX Encontro Nacional de Inteligência Artificial, 12 pages, Curitiba - PR, Brazil, October 2012.

[3] C. Sonesson and D. Bock. A review and discussion of prospective statistical surveillance in public health. Journal of the Royal Statistical Society: Series A (Statistics in Society), 166(1):5–21 , 2003.

[4] M. Höhle. surveillance: An R package for the monitoring of infectious diseases. Computational Statistics, 22:571–582, 2007.

Thank you!

brauliocic091@gmail.comandrea.iabrudi@gmail.com

thaiscorrea@iceb.ufop.br

towards efficient prospective detection of multiple spatio -temporal clusters

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