Basic Multivariate Process
Monitoring
(Section 4.3 of Vardeman and Jobe)
1
Realm of Application
• Where relationships between variables
matter
• Where there will be “automatic” calculation
2
“Obvious” Points About Such
Situations
• Separate monitoring of variables (e.g. using
multiple x charts) can never hope to detect
“unusual relationships” … a single
summary of all variables is needed
• Practical use of an “out of control” signal
will require digging into the data to track
down what has changed
• One must have (and use) some measure of
“relationship” between variables
3
Correlation (A Measure of Linear
Association Between x and y)
• Sample version is
rxy =
∑ ( x − x )( y − y )

2 
2
 ∑(x − x)  ∑( y − y) 



−1 ≤ rxy ≤ 1
and
• We’ll call the “population”/theoretical
version ρ xy and − 1 ≤ ρ xy ≤ 1
4
r ≈ −.1
r ≈ .8
r ≈ −.8
5
Use of Correlation (“Standard
Relationship”) Information for p
Variables??
• Collect correlations and variances in a
matrix
 σ 12

ρ 21σ 1σ 2

Vp× p =  M

 ρ p1σ 1σ p
ρ12σ 1σ 2
σ 22
M
ρ p 2σ 2σ p
L ρ1 pσ 1σ p 

L ρ 2 pσ 2σ p 

O
M

L
σ p2 
• Plot the summary statistic (4.24)
X
2
'



−1 
= n x − µ  V  x − µ 
 p×1 p×1  p× p  p×1 p×1 
6
More Use of Correlation
Information
• Use an upper control limit (only)
CLX 2 = p
UCLX 2 = p + 3 2 p
• Large values of the plotted statistic signal
– a change in one of the p means or standard
deviations, OR
– a change in the relationship between at least 2
of the p variables
7
??? Interpretation ???
• p=2 version of this can be written without
matrix notation
2
2

 x1 − µ1 
 x1 − µ1   x2 − µ2   x2 − µ2  
1
2

X =
 − 2 ρ12 

+
 
2
(1 − ρ12 )   σ 1 / n 
σ1 / n   σ 2 / n   σ 2 / n  



• in-control values of this statistic come from
vectors of sample means inside appropriate
p-dimensional “footballs” centered at the
vector of standard means with size and
8
orientation controlled by V
Example 4.5
0
n = 1 p = 2 µ =   σ 1 = 15 σ 2 = 18
2×1
0
2
2
x
x
x
x
X2 = 1 − 1 2 + 2
92.16 144 207.36
ρ12 = .6
“in-control” diameters
9
Retrospective Analyses
• Requires the same kind of pooling together
of information from r samples as for
retrospective x charts
• For a single sample (l)
 s12

¶ =  r21s1s2
Vp× p  M

 rp1s1s p
and
r12 s1s2
s22
M
rp 2 s2 s p
 x1 
x 
 2
=
x M
p×1
 
 x p 
L r1 p s1s p 

L r2 p s2 s p 
O
M 

2
s p 
L
10
Retrospective Analyses (cont’d)
• Sensible pooled estimators of µ and V are
p× p
p ×1
then
r
r




¶
µ =  ∑ nl x l  /  ∑ nl 
p×1  i =1

p ×1   i =1
and
 r
  r

¶
µ
V
=  ∑ (nl − 1)V
 /  ∑ (nl − 1) 
l
p× p  i =1

p
×
p

  i =1
(this is pooling of sample-by-sample
information … not pooling of r samples)
11
Workshop Exercise
• Consider two (A diameter, B diameter)
pairs for Example 4.5 (see slide 9):
(25,30) and (25,-30)
– plot them on slide 9
– compute X 2 for both of them
– determine whether either unit would cause an
out-of-control signal for either diameter alone
on an individuals chart (an n=1 x chart)
12