,

Sensor Drift Calibration via Spatial Correlation Model in Smart Building

Tinghuan Chen1, Bingqing Lin2, Hao Geng1, Bei Yu1

1 The Chinese University of Hong Kong2 Shenzhen University

,

Introduction

Sensor

Data Center

Part Number Temp. Range Accuracy Price

SMT172 −45 ∼ 130 °C ±0.25 °C $ 35.13AD590JH −50 ∼ 150 °C ±0.5 °C $ 17.91TMP100 −55 ∼ 125 °C ±2.0 °C $ 1.79MCP9509 −40 ∼ 125 °C ±4.5 °C $ 0.88LM335A −40 ∼ 100 °C ±5.0 °C $ 0.75

I Measurement from sensor without calibration is not accurate enough

I Looking for ways to do calibration

Spatial Correlation Model and Formulation

temperature

Iout

ideal transfer function

sensor 1 transfer function

sensor 2 transfer function

drift 1

drift 2

Sensor Drift CalibrationGiven the measurement values sensed by all sensors during severaltime-instants, drifts will be accurately estimated and calibrated.

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

Input:

I x(k)i : the measurement value sensed by ith sensor at kth time-instant.

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

Output:

I εi : a time-invariant drift calibration.

Spatial Correlation Model:

I drift-free model:x

(k)i ≈∑n

j=1,j 6=i ai ,jx(k)j + ai ,0, k = 1, 2, · · · ,m0.

I drift-with model:x

(k)i + εi ≈

∑nj=1,j 6=i ai ,j(x

(k)j + εj) + ai ,0, k = 1, 2, · · · ,m.

Further Assumption:

I Likelihood:P(x|a, ε) ∝exp(−δ0

2

∑ni=1∑m

k=1[x(k)i + εi −

∑nj=1,j 6=i ai ,j(x

(k)j + εj)− ai ,0]2).

I Prior distribution of model coefficients (Bayesian Model Fusion):

P(a) ∝ exp

(−∑n

i=1∑n

j=0,j 6=iλ

2a2i ,j

(ai ,j − ai ,j)2).

I Prior distribution of calibration:P(ε) ∝ exp(−δε2

∑ni=1 ε

2i ).

Maximum-a-posteriori:

mina,ε

δ0

n∑

i=1

m∑

k=1

[x(k)i + εi −

n∑

j=1,j 6=i

ai ,j(x(k)j + εj)− ai ,0]2

+ λn∑

i=1

n∑

j=0,j 6=i

1

a2i ,j

(ai ,j − ai ,j)2 + δε

n∑

i=1

ε2i .

Challenges:

I How to handle this Formulation

I How to determine hyper-parameters

Model Optimization

I With fixed one variable, Formulation is a convex unconstrained QPproblem w.r.t. another variable;

I With the first-order optimality condition, sub-Formulation can betransferred as linear equations.

Alternating-based Optimization

Require: Sensor measurements x, prior a and hyper-parameters λ, δ0, δε.1: Initialize a← a and ε← 0;2: repeat3: for i ← 1 to n do4: Fix ε, solve linear equations (1) using Gaussian elimination to

update ai ;5: end for6: Fix a, solve linear equations (2) using Gaussian elimination to up-

date ε;7: until Convergence8: return a and ε.

δ0

m∑

k=1

(x(k)t + εt)

n∑

j=1

ai ,j(x(k)j + εj) + ai ,0

+ λ

(ai ,t − ai ,t)

a2i ,t

= 0, (1)

δ0

n∑

i=1

m∑

k=1

ai ,t(

n∑

j=1

ai ,j(x(k)j + εj) + ai ,0)

+ δεεt = 0, (2)

Estimation of Hyper-parameters

Comparison of Estimation for Hyper-parameters

I Unsupervised Cross-validation:simple, accurate but time-consuming.

I Monte Carlo Expectation Maximization:fast, flexible but no-accurate.

Unsupervised Cross-validation

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Error Eatimation

Coefficient Eatimation

Run 1Run 1 Run 4Run 3Run 2

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Run n

n G

rou

ps

… …

… …

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Error Eatimation

Coefficient Eatimation

Run 1Run 1 Run 4Run 3Run 2

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Run n

n Groups

model training

error estimation

mina,

0

nX

i=1

mX

k=1

[x(k)i +i

nX

j=1,j 6=i

ai,j(x(k)j +j)ai,0]

2+

nX

i=1

nX

j=0,j 6=i

1

a2i,j

(ai,jai,j)2+

nX

i=1

2i .

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mina,

0

nX

i=1

mX

k=1

[x(k)i +i

nX

j=1,j 6=i

ai,j(x(k)j +j)ai,0]

2+

nX

i=1

nX

j=0,j 6=i

1

a2i,j

(ai,jai,j)2+

nX

i=1

2i .

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Training Set:

Validation Set:nX

i=1

mX

k=1

[x(k)i + i

nX

j=1,j 6=i

ai,j(x(k)j + j) ai,0]

2

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nX

i=1

mX

k=1

[x(k)i + i

nX

j=1,j 6=i

ai,j(x(k)j + j) ai,0]

2

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ai,j<latexit sha1_base64="32AeZ0xVi6nxY1o/Hvg40wcHKDI=">AAAB+HicbVDLSsNAFL2pr1ofjY+dm8FWcCEl6UaXBTcuK9gHtCVMppN27GQSZiZCDQH/w40LRdwK/og7P8S908dCWw9cOJxzL/fe48ecKe04X1ZuZXVtfSO/Wdja3tkt2nv7TRUlktAGiXgk2z5WlDNBG5ppTtuxpDj0OW35o8uJ37qjUrFI3OhxTHshHggWMIK1kTy7WO4OsU5x5qXs7DYre3bJqThToGXizkmpdvj98QAAdc/+7PYjkoRUaMKxUh3XiXUvxVIzwmlW6CaKxpiM8IB2DBU4pKqXTg/P0IlR+iiIpCmh0VT9PZHiUKlx6JvOEOuhWvQm4n9eJ9HBRS9lIk40FWS2KEg40hGapID6TFKi+dgQTCQztyIyxBITbbIqmBDcxZeXSbNacZ2Ke+2WalWYIQ9HcAyn4MI51OAK6tAAAgk8wjO8WPfWk/Vqvc1ac9Z85gD+wHr/AZrQlTw=</latexit><latexit sha1_base64="cu3CzfMOFQ9SN0Eg+/YW9bW+5R8=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoZbEVPEhJetFjwYvHCvYD2hA22027drMJuxuhhvwSLx4U8epP8ea/cdvmoK0PBh7vzTAzL0g4U9pxvq3SxubW9k55t7K3f3BYtY+OuypOJaEdEvNY9gOsKGeCdjTTnPYTSXEUcNoLpjdzv/dIpWKxuNezhHoRHgsWMoK1kXy7Wh9OsM5w7mfs8iGv+3bNaTgLoHXiFqQGBdq+/TUcxSSNqNCEY6UGrpNoL8NSM8JpXhmmiiaYTPGYDgwVOKLKyxaH5+jcKCMUxtKU0Gih/p7IcKTULApMZ4T1RK16c/E/b5Dq8NrLmEhSTQVZLgpTjnSM5imgEZOUaD4zBBPJzK2ITLDERJusKiYEd/XlddJtNlyn4d65tVaziKMMp3AGF+DCFbTgFtrQAQIpPMMrvFlP1ov1bn0sW0tWMXMCf2B9/gA9eJK+</latexit>

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Maximum Likelihood Estimation:

maxδε,δ0,λ

P(x; δ0, λ, δε).

E-Step

Q(Ω|Ωold) =

∫ ∫P(a, ε|x; Ωold) lnP(x, a, ε; Ω)d adε

≈ 1

L

L∑

l=1

lnP(x, a(l), ε(l); Ω)

M-Step

maxΩ

1

L

L∑

l=1

lnP(x, a(l), ε(l); Ω).

Conditional distribution of each variable

ap,q ∼ P(ap,q|ε, a/ap,q, x; δε, δ, λ) = N (µap,q, σ−1ap,q

),

εt ∼ P(εt|ε/εt , a, x; δε, δ, λ) = N (µεt , σ−1εt ),

Monte Carlo Expectation Maximization

Require: Sensor measurements x, prior a;1: Initialize hyper-parameters Ω;2: repeat3: Initialize samples Ψ(0) by Alternating-based Optimization;4: for l ← 1 to L do5: for i ← 1 to n2 + n do6: Sample ψ

(l)i from the desired conditional distribution

N (µψi , σψi) with ψ(l)1 , · · · , ψ(l)

i−1, ψ(l−1)i+1 , · · · , ψ(l−1)

n2+n;

7: end for8: end for9: Update hyper-parameters Ω;

10: until Convergence11: return hyper-parameters Ω.

Experimental Results

BenchmarkI Hall

I Secondary School

The generated simulation data

choose building model and weather, set sensor

parameters

random drift

obtain golden temperature

drift calibrationcomparisonMAPE

noise

Accuracy

I Drift variance is set to 2.25; Benchmark: Hall.

Accuracy

5 10 15 20 25 300

0.5

1

# sensor

MAP

E EMEM

TSBL

22 / 56

I Drift variance is set to 2.78; Benchmark: Hall.

Accuracy

5 10 15 20 25 300

0.5

1

# sensor

MAP

E CVEM

TSBL

23 / 56

I Drift variance is set to 2.25; Benchmark: Secondary School.

Accuracy

5 10 15 20 25 30 35 400

0.5

1

1.5

# sensor

MAP

E CVEM

TSBL

24 / 56

I Drift variance is set to 2.78; Benchmark: Secondary School.

Accuracy

5 10 15 20 25 30 35 400

0.5

1

1.5

# sensor

MAP

E CVEM

TSBL

25 / 56

Runtime

I Runtime vs. # sensor; Benchmark: Hall.

Runtime

10 20 30010203040506070

# sensor

Runt

ime

(s)

CV225EM225

TSBL225CV278EM278

TSBL278

26 / 56

I Runtime vs. # sensor; Benchmark: Secondary School.

Runtime

10 20 30 400

50

100

150

200

# sensor

Runt

ime

(s)

CV225EM225

TSBL225CV278EM278

TSBL278

27 / 56

Tinghuan Chen – CSE Department – The Chinese University of Hong Kong E-Mail: thchen@cse.cuhk.edu.hk http://www.cse.cuhk.edu.hk