Lecture 14 cusum and ewma

Lecture 14: CUSUM and EWMA

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CUSUM, MA and EWMA Control Charts

Increasing the sensitivity and getting ready for automated control:

The Cumulative Sum chart, the Moving Average and the Exponentially Weighted Moving Average Charts.


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Shewhart Charts cannot detect small shifts

Fig 6-13 pp 195 Montgomery.

The charts discussed so far are variations of the Shewhartchart: each new point depends only on one subgroup.Shewhart charts are sensitive to large process shifts.The probability of detecting small shifts fast is rather small:


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Cumulative-Sum ChartIf each point on the chart is the cumulative history (integral) of the process, systematic shifts are easily detected. Large, abrupt shifts are not detected as fast as in a Shewhart chart.

CUSUM charts are built on the principle of Maximum Likelihood Estimation (MLE).


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The "correct" choice of probability density function (pdf) moments maximizes the collective likelihood of the observations.

If x is distributed with a pdf(x,θ) with unknown θ, then θ can be estimated by solving the problem:

This concept is good for estimation as well as for comparison.

Maximum Likelihood Estimation

max θ

m

Πi=1

pdf( xi,θ)


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Maximum Likelihood Estimation ExampleTo estimate the mean value of a normal distribution, collect the observations x1,x2, ... ,xm and solve the non-linear programming problem:

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛Σ

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

Π⎟⎠⎞

⎜⎝⎛ −

−

=

⎟⎠⎞

⎜⎝⎛ −

−

=

22 ˆ21

1ˆ

ˆ21

1ˆ 21logmin

21max σ

μ

μσ

μ

μ πσπσ

ii xm

i

xm

ieore


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MLE Control Schemes

If a process can have a "good" or a "bad" state (with the control variable distributed with a pdf fG or fB respectively).This statistic will be small when the process is "good" and large when "bad":

m

Σi=1

logfB(xi)fG(xi)

∑=

==Π

m

iii

m

ipp

11)log()log(


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MLE Control Schemes (cont.)

Sm =m

Σi=1

logfB(xi)fG(xi)

- mink < m

k

Σi=1

logfB(xi)fG(xi)

> L

or

Sm = max (Sm-1+logfB(xm)fG(xm), 0) > L

Note that this counts from the beginning of the process. We choose the best k points as "calibration" and we get:

This way, the statistic Sm keeps a cumulative score of all the "bad" points. Notice that we need to know what the "bad" process is!


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The Cumulative Sum chart

Sm =m

Σi=1

(xi - µ0)

AdvantagesThe CUSUM chart is very effective for small shifts and when the subgroup size n=1. DisadvantagesThe CUSUM is relatively slow to respond to large shifts. Also, special patterns are hard to see and analyze.

If θ is a mean value of a normal distribution, is simplified to:

where μ0 is the target mean of the process. This can be monitored with V-shaped or tabular “limits”.


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Example

-15-10-505

1015

20 40 60 80 100 120 140 160

µ0=-0.1

LCL=-13.6

UCL=13.4

-60-40-20

020406080

100120

0 20 40 60 80 100 120 140 160

0

Shew

hart

smal

l shi

ftC

USU

M s

mal

l shi

ft


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The V-Mask CUSUM design for standardized observations yi=(xi-μo)/σ

Figure 7-3 Montgomery pp 227

Need to set L(0) (i.e. the run length when the process is in control), and L(δ) (i.e. the run-length for a specific deviation).


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The V-Mask CUSUM design for standardized observations yi=(xi-μo)/σ

d = 2δ 2

ln1- β

α

θ = tan -1 δ2A

δ = Δσ x

dθ

δ is the amount of shift (normalized to σ) that we wish to detect with type I error α and type II error β.Α is a scaling factor: it is the horizontal distance between successive points in terms of unit distance on the vertical axis.


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ARL vs. Deviation for V-Mask CUSUM


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CUSUM chart of furnace Temperature difference

I

IIIII

Detect 2Co, σ =1.5Co, (i.e. δ=1.33), α=.0027 β=0.05, A=1

=> θ = 18.43o, d = 6.6

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0

10

20

30

40

50

100806040200-3

-2

-1

0

1

2

3


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Tabular CUSUMA tabular form is easier to implement in a CAM system

Ci+ = max [ 0, xi - ( μo + k ) + Ci-1

+ ]

Ci- = max [ 0, ( μo - k ) - xi + Ci-1

- ]

C0+ = C0

- = 0

k = (δ/2)/σ

h = dσxtan(θ)

dθ

h


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Tabular CUSUM Example


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Various Tabular CUSUM Representations


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CUSUM Enhancements

Other solutions include the application of Fast Initial Response (FIR) CUSUM, or the use of combined CUSUM-Shewhart charts.

To speed up CUSUM response one can use "modified" V masks:

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0

5

10

15


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General MLE Control Schemes

Since the MLE principle is so general, control schemes can be built to detect:• single or multivariate deviation in means• deviation in variances• deviation in covariancesAn important point to remember is that MLE schemes need, implicitly or explicitly, a definition of the "bad" process.The calculation of the ARL is complex but possible.


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Control Charts Based on Weighted Averages

The 3-sigma control limits for Mt are:

Mt = xt + xt-1 + ... +xt-w+1w

V(Mt) = σ2n w

UCL = x + 3 σn w

LCL = x - 3 σn w

Small shifts can be detected more easily when multiple samples are combined.Consider the average over a "moving window" that contains w subgroups of size n:

Limits are wider during start-up and stabilize after the first w groups have been collected.


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Example - Moving average chart

-10

-5

0

5

10

15

0 20 40 60 80 100 120 140 160

sample

w = 10


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The Exponentially Weighted Moving Average

If the CUSUM chart is the sum of the entire process history, maybe a weighed sum of the recent history would be more meaningful:

zt = λxt + (1 - λ)zt -1 0 < λ < 1 z0 = x

It can be shown that the weights decrease geometrically and that they sum up to unity.

zt = λ ( 1 - λ )j xt - j + ( 1 - λ )t z0Σj = 0

t - 1

UCL = x + 3 σ λ( 2 - λ ) n

LCL = x - 3 σ λ( 2 - λ ) n


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Two example Weighting Envelopes

0.00.10.20.30.40.50.60.70.80.91.0

0 10 20 30 40 50

EWMA 0.6EWMA 0.1 -> age of sample

relativeimportance


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EWMA Comparisons

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-5

0

5

10

15

0 20 40 60 80 100 120 140 160sample

-10

-5

0

5

10

15

0 20 40 60 80 100 120 140 160

λ=0.6

λ=0.1


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Another View of the EWMA• The EWMA value zt is a forecast of the sample at the t+1

period. • Because of this, EWMA belongs to a general category of

filters that are known as “time series” filters.

• The proper formulation of these filters can be used for forecasting and feedback / feed-forward control!

• Also, for quality control purposes, these filters can be used to translate a non-IIND signal to an IIND residual...

xt = f ( xt - 1, xt - 2, xt - 3 ,... )xt- xt = atUsually:

xt = φixt - iΣi = 1

p+ θjat - jΣ

j = 1

q


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Summary so far…While simple control charts are great tools for visualizing the process,it is possible to look at them from another perspective:

Control charts are useful “summaries” of the process statistics.

Charts can be designed to increase sensitivity without sacrificing type I error.

It is this type of advanced charts that can form the foundation of the automation control of the (near) future.

Next stop before we get there: multivariate and model-based SPC!

Lecture 14 cusum and ewma

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