1. Department of Computer Science Old Dominion University
Strategies for Sensor Data Aggregation in Support of Emergency Response
X. Wang A. Walden
M. Weigle
S. Olariu
Department of Computer Science Old Dominion University
October 7, 2014

2. Table of Contents 2 Information Discounting 4 Type 1 Operators
Outline
2 Information Discounting
Aggregating Time-discounted Information
4 Type 1 Operators
Aggregation Strategies for Type 1 Operators 3 5
A Fixed Aggregation Strategy
An Adaptive Aggregation Strategy
6 Simulation 7
Conclusions and Future Work

3. Table of Contents 2 Information Discounting 4 Type 1 Operators
Outline
2 Information Discounting
Aggregating Time-discounted Information
4 Type 1 Operators
Aggregation Strategies for Type 1 Operators 3 5
A Fixed Aggregation Strategy
An Adaptive Aggregation Strategy
6 Simulation 7
Conclusions and Future Work

4. Key Problem in Emergency Response
In emergency situations such as fire,
how to aggregate the information collected by sets of sensors in a timely and efficient manner?

5. Challenges in Emergency Response
The perceived value of the data collected by the sensors decays often quite rapidly;
Aggregation takes time and the longer they wait, the lower the value of the aggregated information;
A determination needs to be made in a timely manner;
A false alarm is prohibitively expensive and involves huge overheads.

6. Our Solution Aggregation usually increases the value of information;
We provide a formal look at various novel aggregation strategies;
Our model suggests natural thresholding strategies for aggregating the information collected by sets of sensors;
Extensive simulations have confirmed the theoretical predictions of our model.

7. Table of Contents 2 Information Discounting 4 Type 1 Operators
Outline
2 Information Discounting
Aggregating Time-discounted Information
4 Type 1 Operators
Aggregation Strategies for Type 1 Operators 3 5
A Fixed Aggregation Strategy
An Adaptive Aggregation Strategy
6 Simulation 7
Conclusions and Future Work

8. Information Value Discounting
Information is a good that has value;
The value of information is hard to assess;
Measuring the value of the aggregated information remains challenging;
The value of information is subject to rapid deterioration over time.

9. General Time-discounted Functions
Mathematical Description:
Assumptions:
The phenomena we discuss occur in continuous time;
The value of information is taken to be a real in [0, ∞);
The value of information decreases with time; 
X (t) ≥ 0
t ≥ 0
0 ≤ r ≤ t
X (t) = X (r )g (t, r )
X (t) ≤ X (r ) where
g : R+ ∪ {0} × R+ ∪ {0} → [0, 1] is
referred to as a discount function.

10. A Special Class of Time-discounted Functions
This discount function depends on the difference t − r only;
The value of information vanishes after a very long time;
value
Mathematical Description:
(X (t) = X (r )δ(t − r ), 0 ≤ r ≤ t
δ : R+ ∪ {0} −→ [0, 1]
X(r)
X(t) r t
time

11. Exponential Time-discounted Funtions
No discount at the beginning: X (r ) = X (r )δ(0) implies δ(0) = 1
Functional equation:
δ(t − r ) = δ(t − s )δ(s − r ), ∀ 0 ≤ r ≤ s ≤ t
Exponential discount function:
δ(t − r ) = e−µ(t−r ) ∀ 0 ≤ r ≤ t
where
µ = − ln δ(1) > 0
Exponentially discounted value of information:
X (t) = X (r )e−µ(t−r ),
∀ 0 ≤ r ≤ t

12. Table of Contents 2 Information Discounting 4 Type 1 Operators
Outline
2 Information Discounting
Aggregating Time-discounted Information
4 Type 1 Operators
Aggregation Strategies for Type 1 Operators 3 5
A Fixed Aggregation Strategy
An Adaptive Aggregation Strategy
6 Simulation 7
Conclusions and Future Work

13. Expected Effect of Aggregation
value 
X1 (t)♦X2 (t)
X2 (t)
X1 (t)
X1 (t1 )
X2 (t2 )
time t1
t2 t

14. Algebra of Aggregation
Aggregation operator ♦ integrates values of sensor data such as X and Y
to get an aggregated value X♦Y .
♦ is an application-dependent operator;
♦ can be extended to an arbitrary number of operands:
♦i =1Xi ≡ X1♦X2♦ · · · ♦Xn;
♦ is assumed to have the following fundamental properties:
Commutativity: X♦Y = Y ♦X, ∀X, Y ; Associativity: [X♦Y ] ♦Z = X♦ [Y ♦Z ] , ∀X, Y , Z ; Idempotency: If Y = 0 then X♦Y = X . n

15. The Interaction between Aggregation and Discounting
The aggregated value of X (r ) and Y (s ) at time τ , with 0 ≤ r ≤ t ≤ τ
and 0 ≤ s ≤ t ≤ τ , is X (τ )♦Y (τ ):
X (τ )♦Y (τ ) = [X (r )δ(τ − r )] ♦ [Y (s )δ(τ − s )]
= [X (t)δ(τ − t)] ♦ [Y (t)δ(τ − t)]
The discounted value of the aggregated value X (t)♦Y (t) at time τ is:
(X (t)♦Y (t)) δ(τ − t)

16. A Taxonomy of Aggregation Operators
Three distinct types of the aggregation operator ♦ (0 ≤ t ≤ τ ):
Type 1: if
[X (t)♦Y (t)] δ(τ − t)< [X (t)δ(τ − t)] ♦ [Y (t)δ(τ − t)] ;
Type 2: if
[X (t)♦Y (t)] δ(τ − t)= [X (t)δ(τ − t)] ♦ [Y (t)δ(τ − t)] ;
Type 3: if
[X (t)♦Y (t)] δ(τ − t)> [X (t)δ(τ − t)] ♦ [Y (t)δ(τ − t)] .

17. Table of Contents 2 Information Discounting 4 Type 1 Operators
Outline
2 Information Discounting
Aggregating Time-discounted Information
4 Type 1 Operators
Aggregation Strategies for Type 1 Operators 3 5
A Fixed Aggregation Strategy
An Adaptive Aggregation Strategy
6 Simulation 7
Conclusions and Future Work

18. General Properties of Type 1 Operators
Lemma
Assume an associative Type 1 operator ♦. For all t, τ with
max1≤i≤n{ti } ≤ t ≤ τ we have
[♦ X (t)] δ(τ − t) < ♦ Xi (τ ).
n n
i =1 i
i =1
Theorem
Assuming that the Type 1 aggregation operator ♦ is associative and commutative, the discounted value of the aggregated information at time n
t is upper-bounded by ♦ Xi (t), regardless of the order in which the
i =1
values were aggregated.

19. A Special Type 1 Operator
Definition
X♦Y = X + Y − XY ; X, Y ∈ [0, 1]
This operator satisfies the associativity, commutativity and idempotency properties and is a Type 1 operator.
Lemma
Consider values X1, X2, · · · , Xn in the range [0, 1] acted upon by the
operator defined above. Then the aggregated value ♦ Xi has the close n
i =1
algebraic form:
n n
♦i =1Xi = 1 − Πi =1(1 − Xi )

20. Table of Contents 2 Information Discounting 4 Type 1 Operators
Outline
2 Information Discounting
Aggregating Time-discounted Information
4 Type 1 Operators
Aggregation Strategies for Type 1 Operators 3 5
A Fixed Aggregation Strategy
An Adaptive Aggregation Strategy
6 Simulation 7
Conclusions and Future Work

21. Aggregation Strategies for Type 1 Operators
Scenario: In an emergency n (n ≥ 2) sensors have collected data about an event at times t1, t2, · · · , tn and let t = max{t1, t2, · · · , tn}.
Further, let X1(t1), X2(t2), · · · , Xn(tn) be the values of the data collected by the sensors.
Problem: How to aggregate these values at current time τ (τ ≥ t) and trigger an alarm?
Solution: THRESHOLDING
Two Classes of Aggregation Strategies: Aggregation strategy with fixed threshold; Aggregation strategy with adaptive threshold;

22. ln i =1 ln i =1 Fixed Thresholding Fixed Thresholding Criterion:
[♦ Xi (t)] δ(τ − t) > ∆
i =1
Latest Aggregation Time:
1 ♦n Xi (t)
τ < t +
ln i =1 ∆
Time Window for Aggregation:
1 ♦n Xi (t) l
t, t +
ln i =1 ∆

23. Motivation for Adaptive Thresholding
Assume sensor readings about an event were collected and the resulting values X1, X2, · · · are reals in [0, 1] and one of the network actors (e.g., a sensor) is in charge of the aggregation process and an aggregation operator ♦ is employed in conjunction with a threshold ∆ > 0.
Theorem √m
If Xi1 , Xi2 , · · · , Xim , m > 1, satisfy Xij > 1 − 1 − ∆,
j = 1, 2, · · · , m,
then ♦ Xi > ∆. m
j=1 j

24. Illustration of Our Adaptive Aggregation StrategyC D t1
X1 (t1 ) B
1 − √1 − ∆ t3 G
X2 (t2 )
X3 (t3 )
X4 (t4 )
X5 (t5 )
1 − √3 1 − ∆
1 − √4 1 − ∆ E t4 A t2 F
t1 t2
t3 t4 t5

25. Table of Contents 2 Information Discounting 4 Type 1 Operators
Outline
2 Information Discounting
Aggregating Time-discounted Information
4 Type 1 Operators
Aggregation Strategies for Type 1 Operators 3 5
A Fixed Aggregation Strategy
An Adaptive Aggregation Strategy
6 Simulation 7
Conclusions and Future Work

26. Emergency Scenario
A fire just broke out on a ship instrumented by a set of relevant sensors.

27. Emergency Model
Temperature distribution: linear model with a plateau temperature of 1000( and an ambient temperature of 20(;
Fire propagation: dot source model with an isotropic spreading speed of 1m/s ;
The fire source location is randomly generated.

28. Sensor Network Configuration
The temperature sensors are deployed in a rectangular lattice of size 3 × 2 in a plane with every side of 3m ;
The sensors are asynchronous; Sensor sampling period: 2s ; Threshold to trigger an alarm:

29. Application-dependent Aggregation Operator
Value of Information:
(0.9 Ti ∈ K
Xi = Pr[Ti ∈ K|F ] =
0 Ti ∈/ K
K = [100(, 1000(] is the critical temperature range.
Exponential Discount Rate: Value discount constant
µ = 1.25 × 10−3s−1
Aggregation Operator:
Xi ♦Xj = Pr[{Ti ∈ K} ∪ {Tj ∈ K}|F ]
= Xi + Xj − Xi Xj

30. Renewal and Decay of Value of Temperature — 6 Sensors
Decay and Renewal of Value of Temperature
1000
800
600
Temperature B
Temperature C 1
0.8
0.6
Temperature F
Temperature E
Temperature A
400
Value C
Temperature D
0.4
Value B
Value F
Value E
200
Value A
0.2
Value D
Temperature/°C
Value
0 2 4 6
Time/s 8 10 12 14 16

31. Renewal and Decay of Value of Temperature — 1 Sensor

32. Result for Fixed Thresholding
Fixed Aggregation of 3 Sensors’ Values 1
0.99
0.8
0.6
Value
Value C
Value B
Value F
Value E
Value A
Value D
Aggregation
Threshold
0.5
0.4
0.2 2 4 6 8
Time/s 10 12 14 16

33. Result for Adaptive Thresholding1
Addaptive Aggregation of Sensors’ Values
0.99
0.8
0.6
Value
Value C
Value B
Value F
Value E
Value A
Value D
Aggregation
Threshold
0.5
0.4
0.2 2 4 6 8
Time/s 10 12 14 16

34. Comparison of Two Strategies

35. Table of Contents 2 Information Discounting 4 Type 1 Operators
Outline
2 Information Discounting
Aggregating Time-discounted Information
4 Type 1 Operators
Aggregation Strategies for Type 1 Operators 3 5
A Fixed Aggregation Strategy
An Adaptive Aggregation Strategy
6 Simulation 7
Conclusions and Future Work

36. Conclusions
Offered a formal model for the valuation of time-discounted information;
Provided a formal way of looking at aggregation of information;
Found that the aggregated value does not depend on the order in which aggregation of individual values take place;
Suggested natural thresholding strategies for the aggregation of the information in support of emergency response.

37. Future Works
How to aggregate data across various types of sensors in a cooperative fashion?
How about discounting regimens other than exponential discounting?
Step function? Linear function? Polynomial function?
How to retask the sensors as the mission dynamics evolve?

38. THANK YOU
Work supported by NSF grant CNS

39. Questions?