Sensor Drift Calibration via Spatial Correlation Model in Smart - - PowerPoint PPT Presentation

SLIDE 1

Sensor Drift Calibration via Spatial Correlation Model in Smart Building

Tinghuan Chen1 Bingqing Lin2 Hao Geng1 Bei Yu1

1Chinese University of Hong Kong, Hong Kong 2Shenzhen University, China

1 / 19

SLIDE 2

Smart Building and Cyber-Physical System

Sensor Data Center

2 / 19

SLIDE 3

Temperature Sensor

◮ Errors exist in senor output; ◮ Manufacturing defect, noise, aging... ◮ Cost varies significantly. Part Number

• Temp. Range

Accuracy Price SMT172

−45 ∼ 130 °C ±0.25 °C

$ 35.13 AD590JH

−50 ∼ 150 °C ±0.5 °C

$ 17.91 TMP100

−55 ∼ 125 °C ±2.0 °C

$ 1.79 MCP9509

−40 ∼ 125 °C ±4.5 °C

$ 0.88 LM335A

−40 ∼ 100 °C ±5.0 °C

$ 0.75

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SLIDE 4

Problem Formulation of Sensor Drift Calibration

◮ Several low-cost sensors are deployed to sense in-building temperatures; ◮ The sensor output deviates by a time-invariant drift.

temperature

Iout

ideal transfer function sensor 1 transfer function sensor 2 transfer function drift 1 drift 2

Sensor Drift Calibration

Given the measurement values sensed by all sensors during several time-instants, drifts will be accurately estimated and calibrated.

4 / 19

SLIDE 5

Basic Model

Spatial Correlation Model: ◮ Defines a linear correlation among different temperature values; ◮ drift-free model: x(k)

i

≈ n

j=1,j=i ai,jx(k) j

+ ai,0, k = 1, 2, · · · , m0. ◮ drift-with model: ˆ x(k)

i

+ ǫi ≈ n

j=1,j=i ˆ

ai,j(ˆ x(k)

j

+ ǫj) + ˆ ai,0, k = 1, 2, · · · , m.

Input:

◮ ˆ x(k)

i

: the measurement value sensed by ith sensor at kth time-instant.

◮ ai,j: the drift-free model coefficient.

Output:

◮ ǫi: a time-invariant drift calibration.

5 / 19

SLIDE 6

Further Assumption

Likelihood: P(ˆ x|ˆ a, ǫ) ∝ exp(−δ0 2

n

• i=1

m

• k=1

[ˆ x(k)

i

+ ǫi −

n

• j=1,j=i

ˆ ai,j(ˆ x(k)

j

+ ǫj) − ˆ ai,0]2). Prior for all model coefficients (Bayesian Model Fusion [Wang+,TCAD15]): P(ˆ a) ∝ exp  −

n

• i=1

n

• j=0,j=i

λ 2a2

i,j

(ˆ ai,j − ai,j)2   .

6 / 19

SLIDE 7

Mathematic Formulation based on MAP

Maximum-a-posteriori:

min

ˆ a,ǫ

δ0

n

• i=1

m

• k=1

[ˆ x(k)

i

+ ǫi −

n

• j=1,j=i

ˆ ai,j(ˆ x(k)

j

+ ǫj) − ˆ ai,0]2 + λ

n

• i=1

n

• j=0,j=i

1 a2

i,j

(ˆ ai,j − ai,j)2 + δǫ

n

• i=1

ǫ2

i .

Challenges:

◮ How to handle this Formulation ◮ How to determine hyper-parameters

7 / 19

SLIDE 8

Overall Flow

Drift-free Measurements Model Coefficients Measurements with Drift

Hyper-parameters Induction Option 1: Cross-validation Hyper-parameters Induction Option 2: EM with Gibbs Sampling Model Optimization: Alternating-based Optimization

Drift Calibration

8 / 19

SLIDE 9

Alternating-based Optimization

Require: Sensor measurements ˆ

x, prior a and hyper-parameters λ, δ0, δǫ.

1: Initialize ˆ

a ← a and ǫ ← 0;

2: repeat 3:

for i ← 1 to n do

4:

Fix ǫ, solve linear equations (1) using Gaussian elimination to update ˆ

ai;

5:

end for

6:

Fix ˆ

a, solve linear equations (2) using Gaussian elimination to update ǫ;

7: until Convergence 8: return ˆ

a and ǫ.

δ0

m

• k=1

(ˆ x(k)

t

+ ǫt)  

n

• j=1

ˆ ai,j(ˆ x(k)

j

+ ǫj) + ˆ ai,0   + λ(ˆ ai,t − ai,t) a2

i,t

= 0,

(1)

δ0

n

• i=1

m

• k=1

 ˆ ai,t(

n

• j=1

ˆ ai,j(ˆ x(k)

j

+ ǫj) + ˆ ai,0)   + δǫǫt = 0,

(2)

9 / 19

SLIDE 10

Estimation of Hyper-parameters

Comparison of Estimation for Hyper-parameters ◮ Unsupervised Cross-validation:

simple, accurate but time-consuming.

◮ Monte Carlo Expectation Maximization:

fast, flexible but no-accurate.

10 / 19

SLIDE 11

Unsupervised Cross-validation

… … … … … … … … … … Run 1 Run 1 Run 4 Run 3 Run 2 … … … … … … … … … … … … Run n n Groups

model training error estimation

min

ˆ a,

δ0

n

X

i=1 m

X

k=1

[ˆ x(k)

i

+i−

n

X

j=1,j6=i

ˆ ai,j(ˆ x(k)

j +j)−ˆ

ai,0]2+λ

n

X

i=1 n

X

j=0,j6=i

1 a2

i,j

(ˆ ai,j−ai,j)2+δ

n

X

i=1

2

i .

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min

ˆ a,

δ0

n

X

i=1 m

X

k=1

[ˆ x(k)

i

+i−

n

X

j=1,j6=i

ˆ ai,j(ˆ x(k)

j +j)−ˆ

ai,0]2+λ

n

X

i=1 n

X

j=0,j6=i

1 a2

i,j

(ˆ ai,j−ai,j)2+δ

n

X

i=1

2

i .

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Training Set: Validation Set:

n

X

i=1 m

X

k=1

[ˆ x(k)

i

+ i −

n

X

j=1,j6=i

ˆ ai,j(ˆ x(k)

j

+ j) − ˆ ai,0]2

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n

X

i=1 m

X

k=1

[ˆ x(k)

i

+ i −

n

X

j=1,j6=i

ˆ ai,j(ˆ x(k)

j

+ j) − ˆ ai,0]2

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ˆ ai,j

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ai,j

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and i

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SLIDE 12

Monte Carlo Expectation Maximization

Maximum Likelihood Estimation:

max

δǫ,δ0,λ

P(ˆ x; δ0, λ, δǫ).

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SLIDE 13

Expectation Maximization

E-Step Q(Ω|Ωold) = P(ˆ a, ǫ|ˆ x; Ωold) ln P(ˆ x, ˆ a, ǫ; Ω)dˆ adǫ ≈ 1 L

L

• l=1

ln P(ˆ x, ˆ a(l), ǫ(l); Ω) M-Step

max

Ω

1 L

L

• l=1

ln P(ˆ x, ˆ a(l), ǫ(l); Ω).

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SLIDE 14

Benchmark

(a) (b)

Benchmark: (a) Hall; (b) Secondary School. choose building model and weather, set sensor parameters random drift

• btain golden

temperature drift calibration comparison MAPE noise The generated simulation data.

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SLIDE 15

Accuracy

5 10 15 20 25 30 0.5 1

# sensor MAPE

(a)

5 10 15 20 25 30 0.5 1

# sensor CV EM TSBL

(b)

Drift variance is set to (a) 2.25; (b) 2.78; Benchmark: (a,b) Hall;.

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SLIDE 16

Accuracy

5 10152025303540 0.5 1 1.5

# sensor MAPE

(a)

5 10152025303540 0.5 1 1.5

# sensor CV EM TSBL

(b)

Drift varians set to (a) 2.25; (b) 2.78; Benchmark: (a,b) Secondary School.

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SLIDE 17

Runtime

10 20 30 10 20 30 40 50 60 70

# sensor Runtime (s)

(a)

10 20 30 40 50 100 150 200

# sensor CV225 EM225 TSBL225 CV278 EM278 TSBL278

(b)

Runtime vs. # sensor (a) Hall; (b) Secondary School.

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SLIDE 18

Conclusion

◮ A sensor spatial correlation model has been proposed to perform drift calibration ◮ MAP estimation is then formulated as a non-convex problem with three

hyper-parameters, which is handled by the proposed alternating-based method.

◮ Cross-validation and EM with Gibbs sampling are used to determine

hyper-parameters, respectively.

◮ Experimental results show that on benchmarks simulated from EnergyPlus, the

proposed framework with EM can achieve a robust drift calibration and better trade-off between accuracy and runtime.

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SLIDE 19

Thank You

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