Page 370 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

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Detecting Outliers and Level Shifts 355
data; and Zhang, 25 Noorossana and Vaghefi 30 and Kalgonda and Kulkarni 31 who
looked into control charts for multivariate autoregressive processes.
Although much work has been done on statistical control of autocorrelated pro-
cesses, there has hitherto been no single method that is widely accepted or proven to
be effective in most situations. It is our opinion that the most promising approach is the
use of a newly conceived statistic. In the following, we present the scheme proposed
15
by Atienza et al. for monitoring autocorrelated processes. We focus on studying the
behavior of the statistic λ LS,t used for detecting level shifts in an AR(1) process. The
performance of the scheme is evaluated and a comparison with Alwan and Roberts’s
SCC is made.

23.2 DETECTING OUTLIERS AND LEVEL SHIFTS

Consider the ARMA model
φ(B)Z t = φ 0 + θ(B)ε t , (23.1)

where Z t is a stationary time series representing the process measurements,
2
φ(B) = 1 − φ 1 B − φ 2 B − ··· − φ p B p
is an autoregressive polynomial of order p,
2
θ(B) = 1 − θ 1 B − θ 2 B − ··· − θ q B q
is a moving average polynomial of order q, B is the backshift operator, and {ε t } is
a sequence of normally and independently distributed random errors with mean
2
zero and constant variance σ . Without loss of generality, we assume that the level
ˆ
φ 0 of the time series {Z t } is zero. If we let Z t represent the predicted value ob-
tained from an appropriately identified and fitted ARMA model, then the residuals
ˆ
ˆ
ˆ

e 1 = Z 1 − Z 1 , e 2 = Z 2 − Z 2 ,..., e t = Z t − Z t ,... will behave like independent and
identically distributed (i.i.d.) random variables. 32
Let
θ(B)
Y t = f (t) + ε t , (23.2)
φ(B)
where Y t and f (t) represent the ‘contaminated’ series and the anomalous exogenous
disturbances such as outliers and level shifts, respectively. The function f (t) may be
deterministic or stochastic, depending on the type of disturbance. For a deterministic
model, f (t) is of the form
ω(B) (d)
f (t) = ω 0 ξ t , (23.3)
δ(B)
where

(d) 1, if t = d,
ξ t =
0, if t = d,
is an indicator variable for the occurrence of a disturbance at time d, ω(B) and δ(B)
are backshift polynomials describing the dynamic effect of the disturbance on Y t , and