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 (hrr06@mails.tsinghua.edu.cn, hgwang@tsinghua.edu.cn, xiaoyc10@mails.thu.edu.cn, xuwl@tsinghua.edu.cn)

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

[1] S. J. Qin, "Statistical process monitoring: basics and beyond,"

JOURNAL OF CHEMOMETRICS, vol. 17, pp. 480-502, 2003. [2] , . . :

2000.

[3] G. Spitzlsperger, C. Schmidt, G. Ernst, H. Strasser, and M. Speil,

"Fault detection for a via etch process using adaptive

multivariate methods," Semiconductor Manufacturing, IEEE

Transactions on, vol. 18, pp. 528- 533, 2005.

[4] S. Wold, "Exponentially weighted moving principal

components analysis and projections to latent structures,"

Chemometrics and Intelligent Laboratory Systems, vol. 23, pp.

149-161, 1994.

[5] V.B. Gallagher, R.M. Wise, S.W. Butler, D.D. White, G.G.

Barna, "Development and benchmarking of multivariate

statistical process control tools for a semiconductor etch

process; improving robustness through model updating, " in:

Proc. of ADCHEM 97, Ban, Canada, pp. 78-83,1997.

[6] W. Li, H. H. Yue, S. Valle-Cervantes, and S. J. Qin, "Recursive

PCA for adaptive process monitoring," Journal of Process

Control, vol. 10, pp. 471-486, 2000.

[7] X. Wang, U. Kruger and G. W. Irwin, "Process Monitoring

Approach Using Fast Moving Window PCA," Industrial &

Engineering Chemistry Research, vol. 44, pp. 5691-5702, 2005.

[8] S.Wold, "Cross validatory estimation of the number of

components in factor and principal component analysis,"

Technometrics , vol. 10, 397-406,1978.

[9] P. Hall and R. Martin, “Incremental eigenanalysis for

classification,” in Proc. British Mach. Vis. Conf., 1998, vol. 1,

pp. 286–295.

[10] B. M. Wise, N. B. Gallagher, S. W. Butler, D. D. White, and G.

G. Barna, "A comparison of principal component analysis,

multiway principal component analysis, trilinear decomposition

and parallel factor analysis for fault detection in a

semiconductor etch process," JOURNAL OF CHEMO-

METRICS, vol. 13, pp. 379-396, 1999.

3427