the changepoint approach to spc douglas m. hawkins, peihua qiu university of minnesota chang-wook...
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The Changepoint Approach to SPC
Douglas M. Hawkins, Peihua QiuUniversity of Minnesota
Chang-Wook KangHanyang University
Changepoint approach to SPC 2
Background to SPC
• Have stream of process readings X1, X2,…Xn,….
• Need to decide whether all follow common statistical model, versus
• Isolated (transient) special causes (affect individual readings) or
• Persistent special causes that remain until detected and fixed.
Changepoint approach to SPC 3
The simplest statistical model
• In control the Xn are iid N(2)
• Isolated special causes change mean and/or variance then revert.
• Persistent special cause shifts the mean and/or variance.
• For example, step change in mean to
Changepoint approach to SPC 4
Standard SPC methods
• Shewhart Xbar and R/S chart used for isolated special causes.
• Persistent causes need memory – cumulative sum (cusum) or exponentially weighted moving average (EWMA) chart.
• For now we concentrate on latter.
Changepoint approach to SPC 5
Designing a chart
• An upward cusum is defined by
where K is ‘reference value’ or ‘allowance.
The chart signals a change if
where H is the ‘decision interval’.
0;0
max(0, )1
S
S S X Kn nn
nS H
Changepoint approach to SPC 6
The things you need to know
• Cusum is the optimal way to detect step shift if K is halfway between in-control and out-of-control means.
• So you must know and • You decide H by setting acceptable in-
control average run length (ARL).
• To do this, you also need to know
Changepoint approach to SPC 7
Who told you the Greek stuff?
• Very rarely, you do actually know it.
• More commonly, – do a Phase I study to estimate and – carefully check data for control (can use fixed-
sample-size methods for this)– pick a big enough to matter, small enough
not to be easy to see.
Changepoint approach to SPC 8
An estimate is not a parameter
• But sample estimates are not population parameters.
• So you have a target ARL, but your actual ARL will be a random variable.
• For sensitive methods like cusum with small K, EWMA with small , resulting uncertainty in your ARL can be large.
Changepoint approach to SPC 9
What cusum optimality?
• On top of this, cusum is optimal only for shift it is tuned for. Get a much different shift, you lose performance.
• Similarly for EWMA.
Changepoint approach to SPC 10
The changepoint-in-mean model
• For this model– Xi ~ N(2) for i <=
~ N(for i >
• None of the Greeks is known a priori.
• Suppose we are at observation number n.
Changepoint approach to SPC 11
Likelihood approach
• Write
• If we knew changepoint was (say) k then MLE’s for would be
2 MLE would be Sk,n = (V0,k + Vk,n)/(n-2)
(after the usual bias adjustment
, ,
2,1 1
/( ), ( )k j k j
j j
i k j ii k kX X j k V X X
0, ,k k nX X
Changepoint approach to SPC 12
…. continued
• Two-sample t for H0: =0 (no change) is
• Finally, estimate as k maximizing |Tk,n|
• And diagnose step change if Tmax,n > hn
, 0 ,,,
( )( )k n kk n
k n
k n kT X X
nS
Changepoint approach to SPC 13
Phase II use
• Changepoint formulation for fixed-sample (Phase I setting) is classical.
• For Phase II SPC use n is not constant. Modify the procedure to:– If Tmax,n, < hn, diagnose in control, continue
– If Tmax,n, > hn, conclude out of control. Use the MLE’s to diagnose time of change and pre- and post-change means.
Changepoint approach to SPC 14
Getting the control limits
• We need sequence of control limits hn.
• Fixed-sample theory not much help.
• A conceptual objective: Pick the hn so that
Pr[Tmax,n > hn | no signal before time n] = .
• With such a sequence, in-control RL would be geometric (like Shewhart), and with – In-control ARL = 1/
Changepoint approach to SPC 15
How to get the hn
• Big simulation: 16 million data sets.
• Estimated hn for several values.
• All on web site www.stat.umn.edu/hawkins
Changepoint approach to SPC 16
So why have a Phase I?
• Don’t need in-control parameter estimates, and so don’t need Phase I data gathering,
• Can get up and running in Phase II.
• As time goes by in control, ever-growing data base gives ever-better estimates (unlike conventional Phase I/II dichotomy)
Changepoint approach to SPC 17
….continued
• But most folk would ‘dry run’ at least some readings before turning on testing.
• For lack of obvious best choice, suggest starting testing at n=10 (but Web tables give cutoffs for starts of n=3 through 21)
• For example, for =0.005:
Changepoint approach to SPC 18
1.0 1.3 1.6 1.9 2.2
3.2
4.0
4.8
5.6
Plot of h_n for alpha=0.005
h_n
LOG10N
Changepoint approach to SPC 19
The 0.005 cutoff
• The cutoffs seem to tend to around 3.2
• This corresponds roughly to the two-sided 0.001 point of a N(0,1)
• This ‘Bonferroni multiplier’ of 5 is what you pay for the multiple testing.
Changepoint approach to SPC 20
Do we need the Shewhart?
• Changepoint formulation with compares latest X with mean of all previous data; this includes Shewhart I chart as one of its tests. Asymptotic cutoff of 3.2 is close to European standard.
• and tests the newest mean against grand mean of all previous data; this includes Shewhart Xbar chart for rational groups of any and all sizes.
Changepoint approach to SPC 21
How does method perform?
• Compared to what? Methods that fix IC ARL with unknown parameters scarce.
• Self-starting cusum doesn’t need IC parameter values. Also seamless from Phase I to Phase II.
• Does however need size of shift for tuning purposes.
Changepoint approach to SPC 22
A method comparison
• Three cusums, k=0.25, 0.5, 1
• (tuned for shifts of 0.5, 1, 2 sd’s)
• Two in-control ARL’s – 100, 500
• Shift occurring early (observation 10) or later (observation 100)
• a: ARL 100, early; b: ARL 100, later
• c: ARL 500, early; d: ARL 500, later
Changepoint approach to SPC 23
Changepoint approach to SPC 24
Results
• Changepoint is sometimes best.
• Mostly is second best (no surprise, given cusum’s theoretical optimality).
• Where not best, it is a close second best and has by far most robustly good performance.
Changepoint approach to SPC 25
Example – triglyceride data
• Data set kindly supplied by Dr. Dan Schultz, Rogasin Institute, New York.
• Assay triglyceride standard every week. Use as a QC check on unknowns. Triglyceride reading should be constant (doesn’t much matter what its value is).
• Here’s one year of data (given as I chart):
Changepoint approach to SPC 26
Outlier? Upward shift at end?
0 9 18 27 36 45 54
109
114
119
124
129
I Chart for TRIG
Case Number
119.35
128.79
109.91
sigma 3.1463 E(MR bar) 3.5490 Exceptions: 34
Changepoint approach to SPC 27
First clear exceedance is at week 40
10 20 30 40 50
1
3
5
7
9
T_max,n and h_n
WEEK
T_MAX
H_N
Changepoint approach to SPC 28
What are estimates of the changepoint?
10 20 30 40 50
8
17
26
35
MLE of changepoint
CU
T
WEEK
Changepoint approach to SPC 29
and of the before- and after-change means
10 20 30 40 50
109
115
121
127
Estimates of pre- and post-change mean
WEEK
XB_0_K
XB_K_N
Changepoint approach to SPC 30
Focus
• Don’t interpret estimate of changepoint or of separate means in non-significant bit.
• First signal is 5 weeks after apparent shift.
• Pre-change mean estimate is 117 mg/dL
• Post-change mean estimate is124 mg/dL
• Right from first signal, all three estimates highly stable.
Changepoint approach to SPC 31
Conclusions
• Conventional Shewhart, cusum, EWMA calibrated assuming known parameters.
• Random errors of estimation in parameters become systematic distortions in run distribution of any particular chart
• making IC and OOC ARL’s random.
• Ugly tradeoff between Phase I sample size and control over IC RL distribution.
Changepoint approach to SPC 32
…
• The unknown-parameter changepoint formulation lets you fix in-control run length distribution exactly, with or without sizeable Phase I sample.
• Furthermore, interval alternative means performance competitive regardless of size of the shift.
Changepoint approach to SPC 33
References
Hawkins, D. M., Qiu, P., and Kang, C.-W. (2003) The Changepoint Model for Statistical Process Control to appear in Journal of Quality Technology.
Pollak, M. and Siegmund, D., (1991), 'Sequential Detection of a Change in a Normal Mean When the Initial Value Is Unknown', Annals of Statistics, 19, 394-416.
Siegmund, D, (1985), Sequential analysis : tests and confidence intervals, Springer-Verlag, New York.
Siegmund, D. and Venkatraman, E. S., (1995), 'Using the Generalized Likelihood Ratio Statistic for Sequential Detection of a Change-point', Annals of Statistics, 23, 255-271.
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