sensor drift calibration via spatial correlation model in smart...
TRANSCRIPT
,
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>
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>
and i<latexit sha1_base64="Gr1j39PpeXpCWjIQ4sd311y4/mg=">AAAB9XicbVDLTgJBEOxFRcQX6tHLRjDxRHa56JHEi0dI5JEAktmhFybMzmxmZjVkw3948aAxXv0Fv8Gb8WccHgcFK+mkUtWd7q4g5kwbz/tyMhubW9nt3E5+d2//4LBwdNzUMlEUG1RyqdoB0ciZwIZhhmM7VkiigGMrGF/P/NY9Ks2kuDWTGHsRGQoWMkqMle5KXYw141L0UzYt9QtFr+zN4a4Tf0mK1Wz9+wMAav3CZ3cgaRKhMJQTrTu+F5teSpRhlOM03000xoSOyRA7lgoSoe6l86un7rlVBm4olS1h3Ln6eyIlkdaTKLCdETEjverNxP+8TmLCq17KRJwYFHSxKEy4a6Q7i8AdMIXU8IklhCpmb3XpiChCjQ0qb0PwV19eJ81K2ffKft0vViuwQA5O4QwuwIdLqMIN1KABFBQ8wjO8OA/Ok/PqvC1aM85y5gT+wHn/AUXhlIM=</latexit><latexit sha1_base64="X88qtlMNfsbe7q8R7NFs+863whI=">AAAB9XicbVA9TwJBEJ3DL8Qv1NLmIphYkTsaLUlsLDGRjwROsrfMwYa93cvunoZc+B82Fhpj63+x89+4wBUKvmSSl/dmMjMvTDjTxvO+ncLG5tb2TnG3tLd/cHhUPj5pa5kqii0quVTdkGjkTGDLMMOxmygkccixE05u5n7nEZVmUtybaYJBTEaCRYwSY6WHah8TzbgUg4zNqoNyxat5C7jrxM9JBXI0B+Wv/lDSNEZhKCda93wvMUFGlGGU46zUTzUmhE7ICHuWChKjDrLF1TP3wipDN5LKljDuQv09kZFY62kc2s6YmLFe9ebif14vNdF1kDGRpAYFXS6KUu4a6c4jcIdMITV8agmhitlbXTomilBjgyrZEPzVl9dJu17zvZp/51ca9TyOIpzBOVyCD1fQgFtoQgsoKHiGV3hznpwX5935WLYWnHzmFP7A+fwBT/WSUQ==</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: [email protected] http://www.cse.cuhk.edu.hk