Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

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Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design

Transcript of Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Page 1: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Dr. Miguel Bagajewicz

Sanjay Kumar

DuyQuang Nguyen

Novel methods for Sensor Network Design

Page 2: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Minimize cost of instrumentation while satisfying the constraints on attributes like

• Accuracy• Precision• Reliability• Residual Accuracy

etc…

The Sensor Network Design Problem

Page 3: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Minimize Cost of instrumentation

such that accuracy of

S3= 7%S7= 8%

Similarly we can have constraints on residual accuracy, reliability,

precision etc..

The Sensor Network Design Problem

Page 4: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Tree Enumeration Procedure

• At each node calculate accuracy (and other attributes mandated by the constraints) compare with thresholds.

• If node is feasible, stop; explore sister nodes.

• If infeasible, go down.

How to find optimal solution?

Page 5: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

• The Tree enumeration procedure can be made computationally effective by using cutsets instead of streams (Bagajewicz and Gala, 2006(a)).

• The efficiency is further more increased by decomposing the graph into subgraphs, (Bagajewicz and Gala, 2006(b))

• Gala M and M. Bagajewicz. (2006b). “Rigorous Methodology for the Design and Upgrade of Sensor Networks using Cutsets. Industrial and Engineering Chemistry Research”. Vol 45, No 21, pp. 6679-6686.

• Gala M and M. Bagajewicz. (2006b) “Efficient Procedure for the Design and Upgrade of Sensor Networks using Cutsets and Rigorous Decomposition”. Industrial and Engineering Chemistry Research, Vol 45, No 21, pp. 6687-6697.

Modified Tree Enumeration Procedure

Page 6: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

•Accuracy has been conventionally defined as the sum of absolute value of the systematic error and the standard deviation of the meter (Miller, 1996).

•Since the above definition is of very less practical value, accuracy of a stream can defined as the sum of the precision and the maximum induced bias in the respective stream, Bagajewicz (2005).

Software Accuracy

Page 7: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

-Software Accuracy -Precision -Maximum induced bias

• The maximum induced bias in a stream ‘i’ due a gross error in ‘s’ is given by, (using maximum power measurement test)

Where,

‘A’ is the incidence matrix and ‘S’ is the variance covariance matrix of measurements

Software Accuracyiiia ˆˆˆ

ia

i

i

ss

is

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Page 8: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

• In the presence of nT gross errors in positions given by a set T, the corresponding induced bias in variable ‘i’ is

• We have to explore all the possible combinations of locations of gross errors. Thus the problem can be stated using a binary vector as

Software accuracy in the presence of ‘nt’ gross errors

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)()( )(]][[ˆ pscritis

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Page 9: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

•When there is more than one gross error, two gross errors may be equal in magnitude but opposite in sign which tend to cancel each other.

Gross Error Equivalency

S1

S2

S3

Page 10: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

• Residual Accuracy of order ‘k’ is the software accuracy when ‘k’ gross errors have been found out and the measurements have been eliminated.

Residual Accuracy

Page 11: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Probability with which a variable ‘i’ can be estimated using its own measurement or through material balance equations in the time interval [0, t].

Estimation Reliability

Page 12: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Cutset is the set of edges (streams) when eliminated, separates the graph into two disjoint subgraphs. Deletion of a subset of the edges in cutset does not separate the graph into two subgraphs.

Streams 8, 6, 2 is a cutset. Streams 2, 3 is another cutset. There are several others.

Cutset

Page 13: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

xm = [1, 2, 3]; xm is also a cutsetP{S1}= P{S2}= P{S3}= 0.9

• Probability of estimating S1= Probability of S1 working or Probability of S2, S3 working simultaneously.

Calculation of Estimation Reliability- Example

RS1= P{S1} υ [P{S2}∩P{S3}]

RS1= P{S1} υ [P{S2}×P{S3}]

RS1= P{S1}+ [P{S2}∩P{S3}]- [P{S1}×P{S2}×P{S3}]

RS1= 0.9+0.81-0.9×0.81

When xm = [2, 3]; S1 becomes non redundant and so it can be estimated

only by its material balance relations. Thus, RS1= P{S1} .P{S2}= 0.81

S2 S5

1

2

3S1

S3S4

4

S6

Page 14: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

• If the variable is measured, then its estimation is directly the service reliability of the sensor measuring it.

• If the variable is not measured,

Estimation Reliability for Non Redundant Variable

smssv RRRR ........... 21

Page 15: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

• Generate all the cutsets that has the variable of interest ‘i’.

• Removing the variable ‘i’ from those yields the reduced cutsets.

Estimation Reliability for Redundant Variable

S2

S5

1

2

3S1

S3

S44

S6

xm = [1, 2, 3]; Since the variable of interest is S1, the reduced cutset would be [2,3]. Let this be denoted by Zj(i), where ‘i’ is the variable of interest- here it is S1.

Page 16: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

xm = [1, 2, 3, 4, 5]; [1, 2, 3], [1, 4, 5] are two cutsets.

[2, 3] and [4,5] are reduced cutsets.

P{S1}= P{S2}= P{S3}= P{S4}= P{S5}= 0.9

• Probability of estimating S1= Probability of S1 working or Probability of S2, S3 working simultaneously or P { S4 and S5} working simultaneously

Calculation of Estimation Reliability- Example

RS1= P{S1} υ [P{S2}∩P{S3}] υ [P{S4}∩P{S5}]

RS1= P{S1} υ [P{S2}×P{S3}] υ [P{S4}×P{S5}]

RS1= [P{S1}+ [P{S2}∩P{S3}]- [P{S1}×P{S2}×P{S3}] ] υ [P{S4}×P{S5}]

RS1= 0.981+0.81- 0.981×0.81

When xm = [1, 3]; S1 becomes non redundant and so it can be estimated only

by its direct measurement. Thus, RS1= P{S1} = 0.9

Z1(1)- reduced cutset

Z2(1)

S2 S5

1

2

3S1

S3S4

4

S6

ENV

Page 17: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

• For a measured variable,

•For a unmeasured variable,

Estimation Reliability for Redundant Variable

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Page 18: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Estimation Reliability for Redundant Variable

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Page 19: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Computation of estimation reliability of unmeasured variable- Sum of disjoint products

)}().........()({)( 21 iZiZiZPtR nkui

It can be proved that,

}1}....{1}.{1{)}({

)}({)}.({)}...({)}.({

)()(.....)()()(

,

)}({)}....({)}({)}().........()({

21

121

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sjm

sj

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iZPiZPiZPiZP

iZiZiZiZiZ

where

iZPiZPiZPiZiZiZP

Page 20: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Input Data:1. Binary vector of measured streams at each node.2. Service reliability of sensors.3. Variables of interest.

Steps to be performed:4. Generate all the cutsets that has the variable of interest.

5. Choose only those reduced cutsets that have measured streams for reliability calculation. Other cutsets are useless as they do not make the variable of interest observable.

6. If no such cutset for unmeasured variable exist, then node is infeasible.

Implementation in the Program

Page 21: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Check if the variables of interest are non redundant. If so we got three cases.

• Case 1:The variable is measured, then estimation reliability is the sensor service reliability itself.

•Case 2:The variable is not measured, the estimation reliability is product of service reliabilities of sensors in the reduced cutset.

•Case 3:The variable of interest is not observable, then node is infeasible, go down the tree.

Implementation in the Program Non redundant variable

smssv RRRR ........... 21

Page 22: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Case 1: variable is measured too.

Case 2: unmeasured variable.

We have already discussed the computational method for above equations.

Implementation in the ProgramRedundant variable

)}().........()({)( 21 iZiZiZSPtR nkivi

)}().........()({)( 21 iZiZiZPtR nkui

Page 23: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Compare the obtained reliability with the specifications/ requirements/ thresholds,

• If node is feasible, transfer control to appropriate statement, which explores sister nodes

• If infeasible, go down the tree.

Implementation in the ProgramComparison with threshold

Page 24: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

• Let there be ‘n’ sensors when calculating reliability. Assume one of the sensors has malfuntioned and the measurement eliminated, the estimation reliability we now have is “Residual Reliability of order one”

• The sensor that has a gross error or the malfunctioned sensor can be identified. This helps to know which measurement is eliminated.

Residual Reliability

Page 25: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Input Data:1. Binary vector of measured streams at a node. Say

‘ns’ streams are measured.

2. Sensor Service Reliability

3. Reduced Cutset Information

Calculation of Residual Reliability

Page 26: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Steps Involved:1. Choose reduced cutsets from already available

information. Eliminate those who have streams with the malfunctioning sensor.

2. Calculate Reliability the same way.

Calculation of Residual Reliability- Order One.

Page 27: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Example

1 2 3 4 5

6 7 10 11 8 9

5

11

16

21

22 23 24 12 14

13

19 2017

15

6 7 9 4

18 103

1

2

8

Madron and Veverka (1992)

Page 28: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Instrumentation Details- Madron and Veverka (1992)

Stream Flow Sensor cost Sensor Precision

(%)

Stream Flow Sensor Cost Sensor

Precision

1 140 19 2.5 13 10 12 2.5

2 20 17 2.5 14 10 12 2.5

3 130 13 2.5 15 90 17 2.5

4 40 12 2.5 16 100 19 2.5

5 10 25 2.5 17 5 17 2.5

6 45 10 2.5 18 135 18 2.5

7 15 7 2.5 19 45 17 2.5

8 10 6 2.5 20 30 15 2.5

9 10 5 2.5 21 80 15 2.5

10 100 13 2.5 22 10 13 2.5

11 80 17 2.5 23 5 13 2.5

12 40 13 2.5 24 45 13 2.5

Page 29: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Software Accuracy when all streams are measured

Stream Software Accuracy

1 6.9201

2 27.8008

3 5.6573

4 15.1941

5 52.7085

6 15.1186

7 36.4044

8 52.6420

9 52.6686

10 8.1288

11 7.4660

12 14.2693

Page 30: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Software Accuracy when all streams are measured

13 51.7501

14 51.7501

15 6.6338

16 5.9791

17 101.7665

18 5.4671

19 12.7847

20 18.6534

21 7.4659

22 52.6091

23 101.7665

24 12.7847

Page 31: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Requested Software Accuracy

• The software accuracy requested were.

•Three gross errors were allowed and no feasible nodes were found.

• Computed Reliability values were 90% for all streams when two gross errors are allowed.

Stream Threshold Accuracy

10 10%

16 8%

18 10%

19 15%

24 19%

Page 32: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Solution of Madron and Veverka (1992)

Cost of the Node Streams MeasuredValues of Accuracy of requested streams in percentage

185 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 19, 20, 23, 24

S10= 9.00S16= 5.46S18=7.53S19=13.85S24=13.85

202 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 19, 20, 23, 24

S10= 9.00S16= 5.46S18=7.53S19=13.85S24=13.85

220 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 23, 24

S10= 7.63S16= 5.48S18=5.49S19=12.91S24=12.91

227 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 23, 24

(node- a)

S10= 9.00S16= 5.46S18=7.53S19=13.85S24=13.85

Page 33: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

227 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 19, 20, 23, 24

(node- b)

S10= 9.00S16= 5.46S18=7.53S19=13.85S24=13.85

239 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 23, 24

S10= 9.00S16= 5.46S18=7.53S19=13.85S24=13.85

257 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24

S10= 7.64S16= 5.48S18=5.49S19=12.91S24=12.91

275 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 23, 24

S10= 8.91S16= 5.48S18=7.00S19=13.20S24=13.20

293 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24

S10= 7.62S16= 5.48S18=5.49S19=12.20S24=12.20

Page 34: Dr. Miguel Bagajewicz Sanjay Kumar DuyQuang Nguyen Novel methods for Sensor Network Design.

Thank You