Exponentially Weighted Moving Average Chart Given a sequence of - - PowerPoint PPT Presentation

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Exponentially Weighted Moving Average Chart

Given a sequence of observations x1, x2, . . . , xn, the exponentially weighted moving average (EWMA) is defined recursively by zi = λxi + (1 − λ)zi−1, i = 1, 2, . . . , n where 0 < λ ≤ 1 is a constant, and the starting value is the process target: z0 = µ0. Successive substitution shows that zi =

i−1

• j=0

λ(1 − λ)jxi−j + (1 − λ)iz0, a weighted average of xi, xi−1, . . . , x1, z0.

1 / 8 Cusum and Moving Average Control Charts EWMA Control Chart

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

So if E(Xi) = µ0, then E(Zi) = µ0, and if the observations Xi are uncorrelated random variables with variance σ2, then Var(Zi) = σ2

• λ

2 − λ 1 − (1 − λ)2i . We can set up an L-sigma control chart for zi: UCL = µ0 + Lσ

• λ

2 − λ

• [1 − (1 − λ)2i]

Center line = µ0 LCL = µ0 − Lσ

• λ

2 − λ

• [1 − (1 − λ)2i]

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

In R: Use the simulated data used as an example of the cusum chart; recall that µ0 = 10 and σ = 1; use λ = 0.1 and L = 2.7:

library(qcc) summary(ewma(x, center = 10, std.dev = 1, lambda = 0.1, nsigmas = 2.7))

Notes The control limits change with i, but approach asymptotes. The same two observations fall outside the control limits as in the Cusum control chart.

3 / 8 Cusum and Moving Average Control Charts EWMA Control Chart

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Design Like the cusum chart, the EWMA chart’s properties are difficult to calculate, so the focus is on the ARL. The performance of the chart is determined by L and λ. Suppose we want an EWMA chart with a specified ARL0, say 370, and a low ARL1 for a one-sigma shift. For any choice of λ, the function xewma.crit(l, L0) in the spc package computes the L that gives the desired ARL0, L0, for a given λ, l.

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Optimal design

• <- optimize(function(lambda)

xewma.arl(l = lambda, c = xewma.crit(lambda, 370, sided = "two"), mu = 1, sided = "two"), interval = c(0, 1)) print(o) xewma.crit(o$minimum, 370, sided = "two")

The optimal λ = 0.14 and control limits L = 2.79, with ARL1 = 9.58. The performance is very similar to, but slightly better than, the

• ptimal cusum chart (with no head start), for which the ARL1

was 9.92.

5 / 8 Cusum and Moving Average Control Charts EWMA Control Chart

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Moving Average Control Chart

The EWMA chart is easy to set up recursively, since the current value

• f z depends on only the current x and the previous z.

However, because it is a (exponentially) weighted average of all past

• bservations, it is impacted, at least a little, by the entire past.

An alternative, with finite memory, is the moving average (MA) chart with span w, based on the unweighted average mi = xi + xi−1 + · · · + xi−w+1 w i = w, w + 1, . . . .

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

For i < w, we define mi to be the average of x1, x2, . . . , xi. If E(Xi) = µ0 and the Xs are uncorrelated with variance σ2, then E(Mi) = µ0, and Var(Mi) = σ2 min(w, i)

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

So we can construct a control chart for mi with UCL = µ0 + Lσ

• 1

min(w, i) Center line = µ0 LCL = µ0 − Lσ

• 1

min(w, i) Like the EWMA chart, the limits change with i, but are constant after i ≥ w, instead of asymptoting to a constant. See Figure 9.8; the qcc package does not provide MA control charts.

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