Strategies for Sensor Data Aggregation in Support of Emergency Response
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This document discusses strategies for aggregating sensor data to enhance emergency response efficiency, particularly in scenarios like fires. It addresses the challenge of time-discounted information, proposing both fixed and adaptive aggregation strategies, validated through simulations. The findings emphasize the importance of timely aggregation and suggest potential future research directions in sensor data cooperation and varied discounting methods.
![General Time-discounted Functions
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;
Mathematical Description:
X(t) ≥ 0 t ≥ 0
X(t) = X(r)g(t, r) 0 ≤ r ≤ t
X(t) ≤ X(r) 0 ≤ r ≤ t
where
g : R+ ∪ {0} × R+ ∪ {0} → [0, 1] is
referred to as a discount function.](https://image.slidesharecdn.com/presentationmilcom2014-150624131647-lva1-app6892/85/Strategies-for-Sensor-Data-Aggregation-in-Support-of-Emergency-Response-9-320.jpg)
![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;
Mathematical Description:
X(t) = X(r)δ(t − r), 0 ≤ r ≤ t
δ : R+ ∪ {0} −→ [0, 1]
value
time
X(t)
X(r)
r t](https://image.slidesharecdn.com/presentationmilcom2014-150624131647-lva1-app6892/85/Strategies-for-Sensor-Data-Aggregation-in-Support-of-Emergency-Response-10-320.jpg)
![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:
♦n
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.](https://image.slidesharecdn.com/presentationmilcom2014-150624131647-lva1-app6892/85/Strategies-for-Sensor-Data-Aggregation-in-Support-of-Emergency-Response-14-320.jpg)
![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)](https://image.slidesharecdn.com/presentationmilcom2014-150624131647-lva1-app6892/85/Strategies-for-Sensor-Data-Aggregation-in-Support-of-Emergency-Response-15-320.jpg)
![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)] .](https://image.slidesharecdn.com/presentationmilcom2014-150624131647-lva1-app6892/85/Strategies-for-Sensor-Data-Aggregation-in-Support-of-Emergency-Response-16-320.jpg)
![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
[♦n
i=1Xi (t)] δ(τ − t) < ♦n
i=1Xi (τ).
Theorem
Assuming that the Type 1 aggregation operator ♦ is associative and
commutative, the discounted value of the aggregated information at time
t is upper-bounded by ♦n
i=1Xi (t), regardless of the order in which the
values were aggregated.](https://image.slidesharecdn.com/presentationmilcom2014-150624131647-lva1-app6892/85/Strategies-for-Sensor-Data-Aggregation-in-Support-of-Emergency-Response-18-320.jpg)
![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 ♦n
i=1Xi has the close
algebraic form:
♦n
i=1Xi = 1 − Πn
i=1(1 − Xi )](https://image.slidesharecdn.com/presentationmilcom2014-150624131647-lva1-app6892/85/Strategies-for-Sensor-Data-Aggregation-in-Support-of-Emergency-Response-19-320.jpg)
![Fixed Thresholding
Fixed Thresholding Criterion:
[♦n
i=1Xi (t)] δ(τ − t) > ∆
Latest Aggregation Time:
τ < t +
1
µ
ln
♦n
i=1Xi (t)
∆
Time Window for Aggregation:
t, t +
1
µ
ln
♦n
i=1Xi (t)
∆](https://image.slidesharecdn.com/presentationmilcom2014-150624131647-lva1-app6892/85/Strategies-for-Sensor-Data-Aggregation-in-Support-of-Emergency-Response-22-320.jpg)
![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
If Xi1 , Xi2 , · · · , Xim , m > 1, satisfy Xij
> 1 − m
√
1 − ∆, j = 1, 2, · · · , m,
then ♦m
j=1Xij
> ∆.](https://image.slidesharecdn.com/presentationmilcom2014-150624131647-lva1-app6892/85/Strategies-for-Sensor-Data-Aggregation-in-Support-of-Emergency-Response-23-320.jpg)
![Application-dependent Aggregation Operator
Value of Information:
Xi = Pr[Ti ∈ K|F] =
0.9 Ti ∈ K
0 Ti /∈ K
K = [100℃, 1000℃] is the critical temperature range.
Exponential Discount Rate: Value discount constant
µ = 1.25 × 10−3
s−1
Aggregation Operator:
Xi ♦Xj = Pr[{Ti ∈ K} ∪ {Tj ∈ K}|F]
= Xi + Xj − Xi Xj](https://image.slidesharecdn.com/presentationmilcom2014-150624131647-lva1-app6892/85/Strategies-for-Sensor-Data-Aggregation-in-Support-of-Emergency-Response-29-320.jpg)
Strategies for Sensor Data Aggregation in Support of Emergency Response
- 1. 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 1 Outline 2 Information Discounting 3 Aggregating Time-discounted Information 4 Type 1 Operators 5 Aggregation Strategies for Type 1 Operators A Fixed Aggregation Strategy An Adaptive Aggregation Strategy 6 Simulation 7 Conclusions and Future Work
- 3. Table of Contents 1 Outline 2 Information Discounting 3 Aggregating Time-discounted Information 4 Type 1 Operators 5 Aggregation Strategies for Type 1 Operators 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 1 Outline 2 Information Discounting 3 Aggregating Time-discounted Information 4 Type 1 Operators 5 Aggregation Strategies for Type 1 Operators 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 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; Mathematical Description: X(t) ≥ 0 t ≥ 0 X(t) = X(r)g(t, r) 0 ≤ r ≤ t X(t) ≤ X(r) 0 ≤ r ≤ t 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; Mathematical Description: X(t) = X(r)δ(t − r), 0 ≤ r ≤ t δ : R+ ∪ {0} −→ [0, 1] value time X(t) X(r) r t
- 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 1 Outline 2 Information Discounting 3 Aggregating Time-discounted Information 4 Type 1 Operators 5 Aggregation Strategies for Type 1 Operators A Fixed Aggregation Strategy An Adaptive Aggregation Strategy 6 Simulation 7 Conclusions and Future Work
- 13. Expected Effect of Aggregation time value ∆ t2t1 X1(t1) X2(t2) t X1(t)♦X2(t) X2(t) X1(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: ♦n 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.
- 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 1 Outline 2 Information Discounting 3 Aggregating Time-discounted Information 4 Type 1 Operators 5 Aggregation Strategies for Type 1 Operators 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 [♦n i=1Xi (t)] δ(τ − t) < ♦n i=1Xi (τ). Theorem Assuming that the Type 1 aggregation operator ♦ is associative and commutative, the discounted value of the aggregated information at time t is upper-bounded by ♦n i=1Xi (t), regardless of the order in which the 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 ♦n i=1Xi has the close algebraic form: ♦n i=1Xi = 1 − Πn i=1(1 − Xi )
- 20. Table of Contents 1 Outline 2 Information Discounting 3 Aggregating Time-discounted Information 4 Type 1 Operators 5 Aggregation Strategies for Type 1 Operators 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. Fixed Thresholding Fixed Thresholding Criterion: [♦n i=1Xi (t)] δ(τ − t) > ∆ Latest Aggregation Time: τ < t + 1 µ ln ♦n i=1Xi (t) ∆ Time Window for Aggregation: t, t + 1 µ ln ♦n i=1Xi (t) ∆
- 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 If Xi1 , Xi2 , · · · , Xim , m > 1, satisfy Xij > 1 − m √ 1 − ∆, j = 1, 2, · · · , m, then ♦m j=1Xij > ∆.
- 24. Illustration of Our Adaptive Aggregation Strategy A B C D G F E t1 t2 t3 t4 t5 0 t1 t2 t4 t5t3 X1(t1) X2(t2) X3(t3) X4(t4) X5(t5) 1 − √ 1 − ∆ 1 − 3√ 1 − ∆ 1 − 4√ 1 − ∆
- 25. Table of Contents 1 Outline 2 Information Discounting 3 Aggregating Time-discounted Information 4 Type 1 Operators 5 Aggregation Strategies for Type 1 Operators 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: 0.99.
- 29. Application-dependent Aggregation Operator Value of Information: Xi = Pr[Ti ∈ K|F] = 0.9 Ti ∈ K 0 Ti /∈ K K = [100℃, 1000℃] is the critical temperature range. Exponential Discount Rate: Value discount constant µ = 1.25 × 10−3 s−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 0 2 4 6 8 10 12 14 16 0 200 400 600 800 1000 Temperature/°C Decay and Renewal of Value of Temperature 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 Value Time/s Temperature C Temperature B Temperature F Temperature E Temperature A Temperature D Value C Value B Value F Value E Value A Value D
- 31. Renewal and Decay of Value of Temperature — 1 Sensor
- 32. Result for Fixed Thresholding 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 Value Fixed Aggregation of 3 Sensors’ Values 0 2 4 6 8 10 12 14 16 0 0.5 0.99 Threshold Time/s Value C Value B Value F Value E Value A Value D Aggregation
- 33. Result for Adaptive Thresholding 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 Value Addaptive Aggregation of Sensors’ Values 0 2 4 6 8 10 12 14 16 0 0.5 0.99 Threshold Time/s Value C Value B Value F Value E Value A Value D Aggregation
- 34. Comparison of Two Strategies
- 35. Table of Contents 1 Outline 2 Information Discounting 3 Aggregating Time-discounted Information 4 Type 1 Operators 5 Aggregation Strategies for Type 1 Operators 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-1116238
- 39. Questions?