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Incremental PCA based online model updating for multivariate process monitoring Ranran Hou, Huangang Wang, Yingchao Xiao and Wenli Xu
Department of Automation, Tsinghua University, Beijing, China ([email protected], [email protected], [email protected], [email protected])
Abstract—Principle Component Analysis (PCA) has been used widely for process monitoring in industry systems. But the data
drifting problem, which commonly exists in the actual process, disables the monitoring model, and subsequently makes the monitoring
system come out with plenty of false alarm. Therefore the efficiency of PCA based process monitoring is degraded in practical use. This
paper presents an incremental PCA based online model updating method for multivariate process monitoring. The proposed method is
based on the characteristic that industry processes preserve the correlation between variables under normal production conditions,
which enables the method update the direction of loading vectors as well as the mean value and the standard deviation of the model
automatically. Our method has low computational complexity, limited storage demand and robust to normal data drifting. Finally, the
performance of the proposed algorithm is compared with conventional PCA and EWMA-PCA methods on a benchmark dataset of
semiconductor etch process, through which our method is proved to be efficient.
Keywords—Multivariate process monitoring, Incremental PCA, Online model updating
PCA
Principle Component Analysis, PCA
PCA
PCA
Benchmark PCA PCA EWMA-PCA
PCA 1
PCA[1,2]
SPE T2
(No.2009ZX02001), 863(No.2011AA060203) 973
(No.2009CB320602)
[3]
PCA
PCA
Proceedings of the 10th World Congress on Intelligent Control and Automation July 6-8, 2012, Beijing, China
3422 978-1-4673-1398-8/12/$31.00 ©2012 IEEE
[5-7]
[4][5]EWMA PCA
Li[6] PCA Wang[7]PCA
PCA
benchmark
2 PCA
3 PCA
4 benchmarkPCA 5
2 PCA 2.1
PCA[2] 1 , 1...n
ix R i N×∈ =
n NPCA
N nR ×∈X (1) nnC
PCA
1 T T Tnn nk kk nkN
= = ≈C X X P P P P (1)
kk k
nkP k
[8] PCA ( )nk kkx , , , ,NσΩ = P
1N
iix x / N== 21 1N
ii ( x x ) / ( N )σ == − −
nkP kk N
SPE T2
SPE[2]
(2) T2
[2] (3) 2T T
nn nk nkSPE ( )x= −I P P (2) 2 1 T T
nk kk nkT x x−= P P (3)
(2)(3) x2SPE αδ≤ 2 2T Tα≤ 2
αδ2Tα α SPE T2
SPE T2
[2]
PCA
2.2
3423
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
x
y
• • • • • • •
• • • • • • •
-4 -2 0 2 4-4
-3
-2
-1
0
1
2
3
x
y
• • • • • • •
• • • • • • •
1(a) 1(b)
1(a)(b)
xy
1(a)
1(b)
PCA 3 PCA 3.1 PCA
PCA
P. Hall[9] 1998
SPE
2.2 :
PCA
PCA
3.2 PCA
N PCA( )nk kkx , , , ,NσΩ = P 1Nx + PCA
PCA( 1)nk kkx , , , ,Nσ′ ′ ′ ′ ′Ω = +P
[4] x′ σ ′
11 Nx x ( )xμ μ +′ = + − (4) 2 2
11 N( )( x x )σ μσ μ +′ = + − − (5)
μ
PCA PCA
PCA [9]
1Nx + g
1Nx + h
11 1
Tnp N Ng ( x x ) −
+ +′= −P (6) 1
1 1N N nph ( x x ) g−+ +′= − − P (7)
1N diag( )σ+ ′Σ = σ ′
h
h
h
3424
, if0 , otherwiseh / h h
hη>
= (8)
η
nk′P
nkP h
[9] R 0h ≠nkP h
nk nkˆ[ ,h ]′ =P P R (9)
0h =
nk nk′ =P P R (10)
R N+1
1 11 1
1 11 1 1 11
nn N N nn N NT
N N N N( ) ( x x )( x x )
μμ
− −+ +
− −+ + + +
′ =
+ − − −
C C (11)
N diag( )σΣ = σ
1 1
1 1 1 11 Tnn nn N N N N
Tnk kk nk
( ) ( x x )( x x )μ μ − −+ + + +′ = + − − −
′ ′ ′≈
C C
P P (12)
(9)(10) (12) 0h ≠
2
01
0 0
Tkk T
kkT T
qq q( )q
ρμ μρ ρ
′+ − ≈ R R (13)
0h =
( )1 T Tkk kk( )qqμ μ ′+ − ≈ R R (14)
11 1
TN N
ˆp h [ ( x x )]−+ += − 1
1 1T
nk N Nq [ ( x x )]−+ += −P
10 0 0 T k[ ,... ] R ×= ∈ (13)(14)
k
kk′ R h
R (9) (10)nk′P
SPE T2
SPE
SPE
SPEkk′
kkkCum( ) k
kkCum( )′ kk′
i ,i i ,i kk kkCum( ) / Cum( )′ ′ ′= × (15)
i ,i′ kk′ (i,i)
kSPE
PCA T2 F
2
2 1k ,N k ;
k( N )T FN ( N k )α α′−
′ −≡′ ′ −
(16)
k 1N N′ = +N ′
2aT
( 1)nk kkx , , , ,Nσ′ ′ ′ ′ ′Ω = +PSPE T2
3.3
PCA
1) N
( )nk kkx , , , ,NσΩ = P SPE T2
2αδ 2Tα
2) (2)(3)SPE T2
3425
3) SPE T2
PCA( 1)nk kkx , , , ,Nσ′ ′ ′ ′ ′Ω = +P
2) 4) SPE T2
PCA
( 1)nk kkx , , , ,Nσ′ ′ ′ ′ ′Ω = +P
2) 5)
PCA 4
Lam 9600 benchmark [10]PCA
PCA [1] EWMA-PCA [4]
34~37 6~9 10721 19
19
PCA 30
PCA EWMA-PCAPCA
PCA99% EWMA-PCA99% 0 95.μ = PCA
99%0 95.μ = 1η =
2 3 4SPE T2
2 PCA
0 20 40 60 80 100 1200
20
40
60
80
100
wafer
SP
E
0 20 40 60 80 100 1200
50
100
150
200
250
300
wafer
T2
3 EWMA-PCA
0 20 40 60 80 100 1200
20
40
60
80
100
wafer
SP
E
0 20 40 60 80 100 1200
50
100
150
200
250
300
wafer
T2
4 PCA
3426
1~128
SPE T2
1
2 PCA 21
97.4%PCA
3 EWMA-PCA20
76.2%
EWMA-PCAEWMA-PCA
4 PCA
21SPE T2
benchmarkPCA
1
PCA 97.4% 0%
EWMA-PCA 76.2% 4.76% PCA 23.4% 0%
5
PCA
benchmarkPCA EWMA-PCA
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3427