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– 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

Space

Time



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

Space

Time


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.

• a


LLk

STCD – Space Time Cluster Detection• Cumulative Sum Statistic

n

knk

n

k

kn L

L=R1

,1


STCD – Space Time Cluster Detection• Cumulative Sum Statistic

n

knk

n

k

kn L

L=R1

,1

)(exp1 ,)(

,,

nkCN

nk Cεε+ nk



• Each parcel k is related to a candidate cluster.

)(exp1 ,1

)( ,nk

n

=k

CNn Cεε+=R nk




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

)(exp1 ,1

)( ,nk

n

=k

CNn Cεε+=R nk





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

)(exp1 ,1

)( ,nk

n

=k

CNn Cεε+=R nk






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

)(exp1 ,1

)( ,nk

n

=k

CNn Cεε+=R nk






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

)(exp1 ,1

)( ,nk

n

=k

CNn Cεε+=R nk

n

ttANttkBNC nkn

nk

],(.],(),()(ˆ 0

,



• Alarm or not?

– A and‘

ARn

n

knkn =R

1,



Space

Timetactual



tactual

Space

Time



Space

Timetactual


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

event


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:

i

Space

Time



N(C1,12)

i

Space

Time



μ(C1,12)

i

Space

Time

N(C1,12)



i

Space

Time

μ(C2,12)

N(C2,12)



i

Space

Time



• Alarm!

– A

ARn



• Alarm!

– A

• Identifying the cluster– a

– Cylinder:

ARn

nknknk 1,max ,*,

nkC *,



Space

Time


Problem• Is there more than one cluster?


Problem• Is there more than one cluster?

Space

Time


Space

Time

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 *,)(

*,*,

nkCN

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 *,)(

*,*,

nkCN

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


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


Evaluation Metrics

• For two simultaneous clusters case– No Alarm• a

– Single Alarm– Double Alarm

ARi


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

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

– Double Alarm• Correct• Incorrect• Half-Correct

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| Introduction | STCD | Problem | STCD-Sim | Metrics | Datasets | Results | Final Considerations |


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


Final Considerations

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

88%;




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!

[email protected]@gmail.com

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