a machine learning approach for constrained …a machine learning approach for constrained sensor...

Problem formulationMotivation

ContributionFurther improvements

A machine learning approach for constrainedsensor placement

Kévin KASPER, Lionel MATHELIN, Hisham ABOU-KANDIL

ENS Cachan - CNRS

July 3, 2015

Kévin KASPER A machine learning approach for constrained sensor placement 1 / 27

GoalFind sensor locations that enable efficient field estimation

through the use of a linear estimator

Available DataSnapshot sequence Y(Machine Learning)Number of sensors : nSPossible sensor locations(constraint)

The sensor locations must :

Minimise εr = ||Y − Y ||F/||Y ||F“Work” with few sensorsNot overlap (constraint)

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Contents

1 Problem formulation

2 Motivation

3 Contribution

4 Further improvements

Learning SequenceSensorsConstraints

Contents

1 Problem formulationLearning SequenceSensorsConstraints

2 Motivation

3 Contribution

A snapshoty i ∈ Rnx

(nx = 3015)

A collection of nsnap ∈ N∗ pressurefield snapshots is formed(nsnap = 1875)

Each snapshot is spatiallydiscretized on the same grid

Learning Sequence

Y =[y iy2 . . . ynsnap

]∈ Rnx×nsnap

contains spatially and temporallydiscretized data

A snapshoty i ∈ Rnx

(nx = 3015)

A collection of nsnap ∈ N∗ pressurefield snapshots is formed(nsnap = 1875)

Each snapshot is spatiallydiscretized on the same grid

Learning Sequence

Y =[y iy2 . . . ynsnap

]∈ Rnx×nsnap

contains spatially and temporallydiscretized data

Choose wiselyThe learning sequence is critical tobe able to guarantee goodestimation performances→Requires expertise

−1 0 1

0 50 100 150 200 250 300 350 400 450 5001.2

Sensor specificitiesThe nS ∈ N∗ sensors :

are linked to specific grid points.give s i , the exact value of y i onthose grid points

−1 0 1

To manipulate the sensors, we use :the set containing the grid indexes of the sensors : posS orthe matrix C ∈ RnS×nx with s i = Cy i

C has a special structure → it contains only one 1 per line(indexed by posS)→ CY contains the posS row restriction of Y

Sensor specificitiesThe nS ∈ N∗ sensors :

are linked to specific grid points.give s i , the exact value of y i onthose grid points

−1 0 1

To manipulate the sensors, we use :the set containing the grid indexes of the sensors : posS orthe matrix C ∈ RnS×nx with s i = Cy i

C has a special structure → it contains only one 1 per line(indexed by posS)→ CY contains the posS row restriction of Y

ConstraintsSensor locations have to follow certain rules

Environment constraintsSensors cannot be setanywhere on the grid→ the search domain isrestricted.

Sensor geometry constraintsSensors cannot overlap→ a new scheme is introduced

ConstraintsSensor locations have to follow certain rules

Environment constraintsSensors cannot be setanywhere on the grid→ the search domain isrestricted.

Sensor geometry constraintsSensors cannot overlap→ a new scheme is introduced

Contents

2 Motivation

3 Contribution

In order to guarantee good recovery performances with few sensorsand in constrained cases, we cannot use methods such as :Effective Independence (EI) orFrameSense (FS)

Why? Because they rely on a reduced order basis such as UnD

UnD is the POD “basis” of order nD ∈ N∗Y → UΣV T (SVD) → UnD ∈ Rnx×nD (restriction of U)

What will be the price to pay for not using a reduced orderbasis ?

Optimization ProblemReducing computational costGreedy minimisation schemeResults

Contents

2 Motivation

3 ContributionOptimization ProblemReducing computational costGreedy minimisation schemeResults

The goal is to minimize ||Y − Y ||F where Y = RCY :R ∈ Rnx×nS is a linear estimatorCY is the measurement sequence

For a given C , ||Y − RCY ||F exhibits its lowest value for

R = Y (CY )+

(with the use of the Moore-Penrose Pseudo-Inverse)

Criterion

Find C ∈ argminC∈MC

||Y − Y(CY

)+︸︷︷︸

linear estimator

measurements︷︸︸︷CY ||F = εr

whereMC indicates the special structure of C (one 1 per line)

R = Y (CY )+

Criterion

||Y − Y(CY

)+︸︷︷︸

linear estimator

R = Y (CY )+

Criterion

||Y − Y(CY

)+︸︷︷︸

linear estimator

DrawbackThe criterion is computationally costly to evaluate

To overcome this, the invariance of the Frobenius norm underunitary transformation is used.A SVD decomposition of Y of order no ∈ N∗ is computed withno � nS → Y ≈ Uno ΣnoV T

εr (C ) ≈ ||UnoΣnoVTno −UnoΣnoV

Tno (CY )+ CUno ΣnoV

Tno ||F

= ||Σno − ΣnoVTno (CY )+ CUno Σno ||F

→ the error matrix Y − Y of size nx × nsnap is approximated by ano square matrix → considerable speed-up

DrawbackThe criterion is computationally costly to evaluate

To overcome this, the invariance of the Frobenius norm underunitary transformation is used.A SVD decomposition of Y of order no ∈ N∗ is computed withno � nS → Y ≈ Uno ΣnoV T

εr (C ) ≈ ||UnoΣnoVTno −UnoΣnoV

Tno (CY )+ CUno ΣnoV

Tno ||F

= ||Σno − ΣnoVTno (CY )+ CUno Σno ||F

→ the error matrix Y − Y of size nx × nsnap is approximated by ano square matrix → considerable speed-up

Seeing as nS � nsnap, CY is highly likely to be full row rank

(CY )+ = (CY )T(CY (CY )T

By using the previous reduced SVD

(CY )+ = Vno ΣTnoU

T(CYY TCT

The criterion becomes :

εr (C ) ≈ ||Σno − Σno ΣTnoU

T(CYY TCT

)−1CUno Σno ||F

→ Once and for all pre-computation of the blue terms→ considerable speed-up (C acts as a restriction operator)

Seeing as nS � nsnap, CY is highly likely to be full row rank

(CY )+ = (CY )T(CY (CY )T

By using the previous reduced SVD

(CY )+ = Vno ΣTnoU

T(CYY TCT

The criterion becomes :

εr (C ) ≈ ||Σno − Σno ΣTnoU

T(CYY TCT

)−1CUno Σno ||F

→ Once and for all pre-computation of the blue terms→ considerable speed-up (C acts as a restriction operator)

→ Greedy SchemeMade viable becausefew sensors are used

InitializationSelect nS admissiblegrid points at randomto initialize the sensorconfiguration

Main loop1→ select a randomsensor (red)

Main loop2→ remove theselected sensor anddefine the new possiblesearch domain (green)

Main loop3→ replace the sensorin the location thatminimizes the criterion

Main loop4→ select a sensor atrandom among theremaining ones andrepeat the above steps

Exit the loop whennS − 1 consecutiveupdates do not reducethe criterion

UnconstrainedCase

ConstrainedCase

Number of Sensors

RelativeErrorǫr(in%)

1 2 3 4 50

FSEISS

Number of Sensors

1 2 3 4 50

Goal-OrientedNoisy measurements

Contents

2 Motivation

3 Contribution

4 Further improvementsGoal-OrientedNoisy measurements

Goal-OrientedHow to place sensors that retrieve data from Y in order to recover

H ∈ Rnh×nsnap , nh ∈ N∗ ?CY → H

Example : Recover the velocity field from pressure measurementsonly

εr (C ) = ||H − H (CY )+ CY ||F

+ computational cost reduction through the reduced order SVD ofH

Goal-OrientedHow to place sensors that retrieve data from Y in order to recover

H ∈ Rnh×nsnap , nh ∈ N∗ ?CY → H

Example : Recover the velocity field from pressure measurementsonly

εr (C ) = ||H − H (CY )+ CY ||F

+ computational cost reduction through the reduced order SVD ofH

Noise RobustnessHow to place sensors that allow for noise robust estimation ?

The sensors deliver : s = Cy + ξ with ξ ∈ RnS

Each sensor is plagued with additive white noise assumedstationary, statistically independent and drawn from the sameprobability distribution.

εr (C ) = ||Y − Y (CYξ)+ CYξ||F

where Yξ = Y + Ξ and Ξ ∈ Rnx×nsnap contains elements drawnfrom above distribution (generating sensor noise).

Redundancy can be introduced to further increase robustness :s i + ξi ,1 → y i , s i + ξi ,2 → y i , s i + ξi ,3 → y iduplicate snapshots can be added in Y (Ξ is generated accordingly)

Noise RobustnessHow to place sensors that allow for noise robust estimation ?

The sensors deliver : s = Cy + ξ with ξ ∈ RnS

Each sensor is plagued with additive white noise assumedstationary, statistically independent and drawn from the sameprobability distribution.

εr (C ) = ||Y − Y (CYξ)+ CYξ||F

where Yξ = Y + Ξ and Ξ ∈ Rnx×nsnap contains elements drawnfrom above distribution (generating sensor noise).

Redundancy can be introduced to further increase robustness :s i + ξi ,1 → y i , s i + ξi ,2 → y i , s i + ξi ,3 → y iduplicate snapshots can be added in Y (Ξ is generated accordingly)

Two noise distributions are considered for the training and the testsequence →both zero-mean Gaussian but with different variance

Training NoiseUsed by thealgorithm→ Variance σ2

Testing NoiseUsed to testthe locations→ Variance σ2

Noise Variance σ2b

ErrorExpectationE[ǫ

0 0.02 0.05 0.1 0.220

EISS (σ2

SS (σ2n= 0.5)

SS (σ2n= 0.05)

SS (σ2n= 0.01)

Two noise distributions are considered for the training and the testsequence →both zero-mean Gaussian but with different variance

Training NoiseUsed by thealgorithm→ Variance σ2

Testing NoiseUsed to testthe locations→ Variance σ2

b Noise Variance σ2b

ErrorExpectationE[ǫ

0 0.02 0.05 0.1 0.220

EISS (σ2

SS (σ2n= 0.5)

SS (σ2n= 0.05)

SS (σ2n= 0.01)

Thank you for your [email protected]

Because most methods rely on UnD , the sensor placement andsubsequent estimation are limited by using nD = nS modes.

This originates from the neccessity of having a well-posed problem.Let ynD

be the nD POD approximation of y , a snapshot.With snD = CynD

, we have s = snD + δs.

Sensor placement methodsrely on measuring snD

(implicitly though theFisher Information Matrix).For “low” values of nD ,||δs||2 can be problematicto correctly estimate y .

1 2 3 4 50

Because most methods rely on UnD , the sensor placement andsubsequent estimation are limited when sensors are constrained.

By relying on UnD that is optimal in the `2 sense, such methodsimplicitly rely on being able to set the sensors freely to achieve thelowest estimation error.

To deal with prohibited sensorlocations, most methodsrestrict UnD

→ UnD is no longer optimal

To prevent overlapping sensors,most methods require are-work of their minimizationprocedure. Such overlapping iscaused by UnD

a machine learning approach for constrained …a machine learning approach for constrained sensor...

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