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An Adaptive Generalized Likelihood Ratio Control Chart for Detecting an Unknown Mean Pattern

GIOVANNA CAPIZZI and GUIDO MASAROTTO Department of Statistical Sciences University of Padua Italy 2ND INTERNATIONAL SYMPOSIUM ON STATISTICAL PROCESS CONTROL Rio de Janeiro, Brazil July 13-14, 2011

An adaptive GLR control chart. . . 1/32

Outline

1

Problem/Reference model

2

Literature review

Known fault signature (CUSCORE, GLR, Optimal Linear Filter) Unknown fault signature (reference free CUSCORE, Weighted CUSUM,. . . )

3

A novel control chart

4

Comparisons

CUSUM Weighted CUSUM

5

Non parametric version

6

Conclusions

An adaptive GLR control chart. . . 2/32

Most statistical process control methods. . .

. . . are focused on the detection of a constant and persistent shift.

MEAN PATTERN ASSUMING A CONSTANT SHIFT AT t = 51

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

An adaptive GLR control chart. . . 3/32

However, it is more realistic. . .

. . . to consider also the possibility of a time varying mean after a fault.

SOME STYLIZED MEAN PATTERNS

t

−40 −20 20 40 20 40 60 80 100

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

Gradual degradation Intermittent fault

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

Partial recover (feedback) Vibration in a mechanical system An adaptive GLR control chart. . . 4/32

Data preprocessing. . .

. . . can introduce (or modify) a dynamic pattern.

EXAMPLES

autocorrelated data computation

• f

residuals from a time series model self-starting control chart trasformation of the origi- nal observations to elimi- nate the unknown parame- ters

An adaptive GLR control chart. . . 5/32

Data preprocessing (example)

MEANS OF OBSERVATIONS AND RESIDUALS

yt = 1.13yt−1 −0.64yt−2 +ut +0.9ut−1

(Model 1, Apley and Shi, IIE Trans., 1999)

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

Original observations Residuals An adaptive GLR control chart. . . 6/32

Data preprocessing (example)

MEANS OF OBSERVATIONS AND RESIDUALS

yt = 1.13yt−1 −0.64yt−2 +ut +0.9ut−1

(Model 1, Apley and Shi, IIE Trans., 1999)

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

Original observations Residuals An adaptive GLR control chart. . . 6/32

Reference model

INDEPENDENT DATA y1,y2,... Original observations, residuals from a suitable time series model,. . . DISTRIBUTION

yt ∼

• N(µ,σ2)

if t < τ (in control) N(µ +σγt,σ2) if t ≥ τ (out of control)

PROBLEM AT TIME t

H0 = {the process is in control} ⇐ ⇒ {t < τ} H1 = {the process is out of control} ⇐ ⇒ {t ≥ τ}

An adaptive GLR control chart. . . 7/32

Literature review: summary

Methods differ for the assumed level of knowledge on the

• ut of control mean pattern γt (t = 1,2,...).

SCENARIO SCHEMES Completely known pattern CUSCORE Partially known pattern (only the shape) Generalized Likelihood Ra- tio (GLR) and Optimal Lin- ear Filter Unknown one-sided pattern (all γi either greater

• r

lesser than zero) Adaptive CUSCORE (dif- ferent versions) Unknown

• scillatory

pat- tern ???

An adaptive GLR control chart. . . 8/32

CUSCORE

Box and Ramirez (1992),QREI, . . .

Sequential likelihood test for H0 : γi = 0 against H1 : γi = gi (i = 1,...,t) for a known sequence gi (and, hence, for a known τ). The control statistic is Ct = max

• 0,Ct−1 +log f1(yt)

f0(yt)

• where f0(·) and f1(·) are the densities computed

under H0 and H1, respectively. A “restarting” procedure can be used to avoid the assumption of known τ.

An adaptive GLR control chart. . . 9/32

Generalized Likelihood Ratio

Apley and Shi (1999), IIE Trans., . . .

It is assumed that γt = δst−τ (t ≥ τ) where δ is unknown while the “fault signature” si is a known sequence. At time t, the control statistic is max

t−M+1<τ≤t sup δ

log f1(yt,··· ,yt−M+1;τ,δ) f0(yt,··· ,yt−M+1) . where M is a suitable integer.

An adaptive GLR control chart. . . 10/32

Optimal Linear Filter

Chin and Apley (2006), Tech.; Apley and Chin (2007), JQT

It based on the GLR assumptions (known fault signature, unknown direction and size). The control statistic is

t−M

∑

i=0

wiyt−i where the weights wi are optimal for detecting the specified fault signature.

An adaptive GLR control chart. . . 11/32

Adaptive CUSCOREs

SCHEME MEAN PATTERN ESTIMATE reference free CUSCORE (Han and Tsung, 2006, JASA) ˆ γt =

• yt − µ

σ

• , ˆ

γ−

t = −ˆ

γ+

t

Weighted CUSUM (Shu et al., 2008, JQT) ˆ γ+

t =

• EWMAt − µ

σ

• , ˆ

γ−

t = −ˆ

γ+

t

Adaptive CUSUM (Jiang et al., 2008, IIE Trans.) ˆ γ+

t = max

• d, AEWMAt − µ

σ

• ˆ

γ−

t = min

• −d, AEWMAt − µ

σ

• An adaptive GLR control chart. . .

12/32

Questions

1

How can an adaptive GLR control chart be defined?

2

How does it compare with the CUSUM (EWMA,. . . ) for detecting persistent mean shifts?

3

How does it compare with the adaptive CUSCORE control charts for detecting arbitrary one-sided patterns?

4

Can it be designed also for detecting arbitrary

• scillatory out-of-control mean patterns?

5

Can it be modified to cope with non Gaussian data?

An adaptive GLR control chart. . . 13/32

An adaptive GLR control chart

Control statistic aGLRt = max{GLRt(ˆ s1),GLRt(ˆ s2)} One-sided fault signature ˆ s1,t = max

• kq
• λ

2−λ ,

• EWMAt − µ

σ

• Generic

fault signature (yt,...,yt−M+1)

T

− → (ˆ s2,t,...,ˆ s2,t−M+1) T = (DWT)+(THRESHOLDING)+(IDWT)

An adaptive GLR control chart. . . 14/32

Comparisons: generalities

SCHEMES

1

adaptive GLR

2

standard CUSUM

3

weighted CUSUM

PERFORMANCE

IN CONTROL: ARL = 500; OUT OF CONTROL: E(RL−200|RL ≥ τ = 201) (It can be viewed as an approximation

• f the steady state out of control

ARL).

An adaptive GLR control chart. . . 15/32

Comparisons: CUSUM

aGLR(M = 128,λ = 0.1,kq = 1.5,kw = 3.5) vs. CUSUM(k = 0.5)

CONSTANT SHIFT

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

δ

1 2 5 10 20 50 100 200 400 1 2 3 4 5 6

Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 16/32

Comparisons: CUSUM

aGLR(M = 128,λ = 0.1,kq = 3.5,kw = 3.5) vs. CUSUM(k = 1)

CONSTANT SHIFT

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

δ

1 2 5 10 20 50 100 200 400 1 2 3 4 5 6

Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 17/32

Comparisons: CUSUM

aGLR(M = 128,λ = 0.1,kq = 3.5,kw = 3.5) vs. CUSUM(k = 1)

CONSTANT SHIFT

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

δ aGLR CUSUM 0.25 110.01 246.92 0.50 33.63 80.93 0.75 17.18 30.74 1.00 10.58 14.48

Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 17/32

Comparisons: CUSUM

aGLR(M = 128,λ = 0.1,kq = 1.5,kw = 3.5) vs. CUSUM(k = 0.5)

A ONE-SIDED SCENARIO

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

δ

1 2 5 10 20 50 100 200 400 1 2 3 4 5 6

Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 18/32

Comparisons: CUSUM

aGLR(M = 128,λ = 0.1,kq = 1.5,kw = 3.5) vs. CUSUM(k = 0.5)

A ONE-SIDED SCENARIO

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

δ aGLR CUSUM 0.50 160.6 218.5 0.75 56.8 94.1 1.00 15.2 25.0

Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 18/32

Comparisons: CUSUM

aGLR(M = 128,λ = 0.1,kq = 1.5,kw = 3.5) vs. CUSUM(k = 0.5)

AN OSCILATORY FAULT SIGNATURE

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

δ

1 2 5 10 20 50 100 200 400 1 2 3 4 5 6

Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 19/32

Comparisons: Weighted CUSUM

aGLR(M = 128,λ = 0.1,kq = 1.5,kw = 3.5) vs. four WCUSUMs

CONSTANT SHIFT

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

δ

1 2 5 10 20 50 100 200 400 1 2 3 4 5 6

Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 20/32

Comparisons: Weighted CUSUM

aGLR(M = 128,λ = 0.1,kq = 1.5,kw = 3.5) vs. four WCUSUMs

A ONE-SIDED SCENARIO

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

δ

1 2 5 10 20 50 100 200 400 1 2 3 4 5 6

Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 21/32

Comparisons: Weighted CUSUM

aGLR(M = 128,λ = 0.1,kq = 1.5,kw = 3.5) vs. four WCUSUMs

AN OSCILLATORY FAULT SIGNATURE

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

δ

1 2 5 10 20 50 100 200 400 1 2 3 4 5 6

Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 22/32

Comparisons: Weighted CUSUM

aGLR(M = 128,λ = 0.1,kq = 1.5,kw = 3.5) vs. four WCUSUMs

ANOTHER OSCILLATORY FAULT SIGNATURE

t

−1.0 −0.5 0.0 0.5 1.0 20 40 60 80 100

δ

1 2 5 10 20 50 100 200 400 1 2 3 4 5 6

Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 23/32

A non parametric version: framework

REFERENCE MODEL y1,y2,... are indipendent such that P(yt ≤ a) =

• F(a)

if t < τ (in control) F(a−γt) if t ≥ τ (out of control) τ and γt (t ≥ τ) are unknown. INFORMATION ON F(·) F(·) is unknown but a reference sample of n preliminary in control observations, say y(IC)

1

,...,y(IC)

n

, is available.

An adaptive GLR control chart. . . 24/32

A non parametric version: key idea

CONTROL CHART Detect the change point applying an adaptive GLR to y⋆

t = φ−1

• nF (IC)

n

(yt)+1 n +2

• where F (IC)

n

(·) is the empirical c.d.f. of the reference sample and φ(·) is the c.d.f. of a standard normal r.v. REMARK nF (IC)

n

(yt)+1 is the rank of the current observation yt in the “combined sample” (yt,y(IC)

1

,...,y(IC)

n

).

An adaptive GLR control chart. . . 25/32

Four distributions

0.0 0.1 0.2 0.3 0.4 −4 −2 2 4

In each case, the mean and the variance are equal to 0 and 1, respectively. An adaptive GLR control chart. . . 26/32

Effect of the reference sample size

In control run length distributions

a b c d a b c d a b c d a b c d all 200 400 600 800 1000 1200

n = 100 n = 250 n = 500 n = 1000 n = ∞

In every case, the in control ARL is equal to 500 An adaptive GLR control chart. . . 27/32

Effect of the reference sample size

Out of control performance 1

For some mean pattern, the reference sample size only slightly affects the performance. E(RL−200|RL ≥ τ = 201) - γt = 1 for t ≥ 201 Reference Sample Size Distribution 100 250 500 1000 Normal 11.52 10.98 10.88 10.71 Skew Normal (λ = −5) 7.76 6.92 6.57 6.26 Skew Normal (λ = 20) 13.01 12.50 12.50 12.40 t of Student (df = 3) 8.79 8.54 8.55 8.49

An adaptive GLR control chart. . . 28/32

Effect of the reference sample size

Out of control performance 2

But in other cases, performance increases as n increases. E(RL−200|RL ≥ τ = 201) - γt = 2cos(2πt/8) for t ≥ 201 Reference Sample Size Distribution 100 250 500 1000 Normal 22.32 16.12 14.46 13.72 Skew Normal (λ = −5) 16.72 11.18 9.44 8.40 Skew Normal (λ = 20) 14.75 10.11 8.63 7.87 t of Student (df = 3) 16.75 13.13 12.40 12.38

An adaptive GLR control chart. . . 29/32

A non parametric version: recommendations

1

Monitoring can be started with as few as 50/100 in control observations.

2

The reference set should be updated as more observations are gathered.

An adaptive GLR control chart. . . 30/32

Conclusions

1

Adaptive GLR control charts perform well in a variety

• f out of control scenarios
• ne-sided patterns both constant and time-varying;
• scillatory patterns;

different shift sizes.

2

A distribution free version is available.

3

It is relatively simple to design since a single scheme provides a good performance in a variety of out of control scenarios.

An adaptive GLR control chart. . . 31/32

THANKS...

SINCE 1222

“UNIVERSA UNIVERSIS PATAVINA LIBERTAS”

(Paduan Freedom is Complete and for Everyone) Observatory (1374) Palazzo Bo (1405)

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