UNIV1"'RSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
CUMULATIVE SUM CONTROL CHARTS FOR THE FOLDED
NORMAL DISTRIBUTION
by
N. L. Jolmson
December 1962
Grant No. AFOSR -62..148
Methods of construction of cumulative sum control
charts for folded normal variates are described.
These charts are likely to be useful \-Then the
sign of an approximately normally distributed
quantity is lost in measurement. Some assessment
is given of the information lost by omission of
the sign.
This research was supported by the Air Force Office of Scientific
Research.
Institute of Statistics
Mimeo Series No. 346
CUMUIATIVE SUM CONTROL CHARTS FOR TEE FOlDED NORMAL DISTRIBUTION
by
N. L. Johnson
I
University of North Carolina
1.
Introduction.
The distribution of the modulus of a normal variable is known as the
'folded' normal distribution.
It is reasonable to apply this distribution when
tbe sign of a variable, which can be justifiably represented by a normally distributed random variable, is irretrievably lost in measurement.
given by Leone et al ~7_7.
normal distribution.
Some examples are
This paper also describes properties of the folded
It also includes a discussion of methods of estimating tbe
parameters, which are further discussed by Elandt
,['1_7
and Johnson ~3_7.
Many standard control chart procedures are based on the assumption that tbe
observations used (either individually or as sample arithmetic means) in plotting
the chart can be represented by normally distributed random variables.
It is
evident that, on occasion, it may be desirable to construct control chart procedures when the appropriate distribution is that of a folded normal variable.
this paper the construction of cumulative sum control charts for such variables
will be described.
2.
The methods used will be based on the ideas described by
Statement of Problem.
If the original variable (x', say) is normally distributed with expected
value £ and standard deviation
folded normal variable x
=
g,
then the probability density function of the
lx' lis
1. This research was supported by the Mathematics Division of the Air
Force Office of Scientific Research.
In
2
- '21[j2
G1 + (xl (J) 2J
e
where
9
=
F;,
I
cosh [o~1 C[
]-,
(J •
We will consider two situations (a) where we are trying to keep
F;,
=
0, so
that the original variable has zero mean (b) where we are trying to control
specified value
(J
o
at a
•
In case (a) we suppose
3.
(J
(J
is known;
in case (b) we suppose
F;,
is known.
Control of Mean.
If we have a sequence of m independent random variables xl' x ' ••• , x ,
m
2
each having a folded normal distribution, then the likelihood ratio of the hypo-
I = ~l)
thesis H : (IF;,
l
against the hypothesis H : (r;,
o
= 0)
is
Hence the sequential probability ratio test discriminating between these two
hypotheses has its continuation region defined by the inequalities
Ctl
II ( 1 _ a
"n
where a.1 = Pr
)
o
12m
II
+ '2 mQ 1 < . L"... "n
cos h [Qlx~/""
... v
raccept Hl
-
~=l
.
-1.
IH.1 ]
0
+ -21
-1"'1
!lJ.'el
2
1
~.
(i = 0, 1).
Now vie apply the method described in
(cscd
l-a
J < "\.h
dn(~)
["2_7,
regarding the cumulative sum
as a reversed sequential test, with a very small value for a ,
l
and using only the right-hand inequality of (3) which now becomes, effectively
control chart
( 4)
3
To apply this method we must (i) transform the observed values xi to
'scores' y.
1
= in
cosh rolx./u] and (ii) plot the cumulative sum ~.y .•
-
1
]. 1
Then
change in the value of Is I from zero to 0lu is indicated if any plotted points
1\
fall below the line PQ in
~ Q~ and OQ
=
Figu~e
1.
=
Here 0 is the last plotted point, tan OPQ
2
-in ao ' so that OP = (-2 Kn a o )Oi •
-----"'It------:/'p
Fig. 1
With this system of plotting Yi is never negative, so the value of
non-decreasing.
~Yi
is
To reduce the amount of paper needed for the chart it may some-
times be preferred to use, instead of y., the modified score
1
Yi'
Then ~
= y.].
2
- 1:
2 01
y! is plotted and we simply use a horizontal line, (- in a ) below the
1
0
last plotted point, as boundary.
The need to transform observed vlaues xi to scores, y.]. or y!,
makes the
1
construction of the CSCC a little troublesome.
However a
t~le
of values of
in cosh X as a function of X makes the task qUite easy, while if a considerable
amount of work is to be done for well-established values of 0l( = sl/U) and u, a
special table of y
= in
COSh(Olx/u) can be constructed once for all.
noted that when x is large, y - 0lx/u - fn2.)
(It may be
4
4.
Control of Mean-Average Sample Number.
We will restrict ourselves to consideration of the situation when the true
mean has in fact shifted to
= 0la.
~l
The expected number of observations before
this will be detected is approximately (-in a )E- l where
o
1
=--0
2
2 1
-1
+a
!(2!n)
• dx
r
11"\2
= -
1 2
- 0 +
2 1
e
!(2!n)
CD
'2.. . 1
v
o
The integral can be evaluated as an infinite series using the expansion
= I;) 1 Y
Hence
and
-
"n 2
fI
+
co
E
. 1
J=
(_l)j+l J.-l e-
2jO Y
-1,
5
( 6)
;:; ~l
{ 2F( Q1) - l} ] - in 2 + I ( ~ 1)
[J{2/rr.)
-1
(Xl
+ I: (_l)j+l {j(j+l)}
j=l
I«2j+l)01)
where
_ 1:02
I(rO )
l
=J(2/rr.)
e
2 1.
~
(])
J
o
Finally we obtain
- 1:,l
( 7)
E ;:; 01 [J{2/rr.} e
2 1 + 01 {2F(Ol) - ~
f-
] -
in 2 + 1 - F(~l)
An approximate formula for E can be obtained by using the approximation to
Mills' ratio
The resulting formula
( $)
- 1:rl
E ;" <_1_ e
.
J21c
21)
gives useful results for &1 larger than 3/4.
6
Using further terms in the asymptotic expression
(a
o
for i ~ 1)
= 1;
for MillIs ratio we get the formal expression
where E. is the j-th Euler number.
J
Evaluating the coefficients in the asymptotic expansion we obtain
... ]
(8)'
Some values of E- l as a function of 01 are shown in Table 1.
Values given
by the approximate formulas (8) and (8)' (excluding the term 0.006030i5) are
also shown.
TABLE 1.
Values of E- l
= (-in
-1
°1
E
a )-1 x (Expected number of observations)
o
Values from
(8)
(8)'
-1
Value from
°1
E
1.25
2.95
2.93
(8) or (8)'
0.25
(980)
0.50
73·5
(26.4 )
(1.12)
1.50
1.67
1.67
0.75
16.6
14.6
i'7'~
1. 75
1.06
1.06
61'7
2.00
0.73
0.73
1.00
6.13
6.01
{~
J~
I
7
When
91 is large, E... '21 B2l -
d
~n 2 (for ~l = 2, E =
1.36735 and
( ~ 9~ - in 2 = 1.30685). The average number of observations is then approximately
(-2 in a ) [~~ - in 4 J-l, as compared with (-2 in ao)~i2 (= OP) for the CSCC for
o
means of normal variables (see ["2_7).
The average sample number for the folded normal distribution is thus
greater by a factor of
approximately.
This may be regarded as reflecting the loss in information about
the signs of the variables in the folded normal.
As would be expected, the ratio
tends to 1 as 01 increases.
5. Control of Standard Deviation.
We now suppose that
at
is known and that we wish to control the value of rr
~
(for del
finiteness we will suppose rr > rr ) then the appropriate likelihood ratio is
l
o
(Jo'
If we desire to detect a-'change in the value of rr from rr
o
to rr
m
-2 -2)
2l( ms 2+ . l.:1 x.2)(
Cl -CTl
CT m
k
0
= (~)
Cl
e
k=
m cosh(x.s/rri) ]
IT,
i=l
l
We will infer that there has been a change in
Cl
1
2
cosh(X.S/CT
-
1
0
)
if the inequality
]
( 10)
< - in exo
is not satisfied (the
tion).
XIS
being numbered backwards, starting from the last observa-
Examination of (10) is more troublesome than applying the condition (4)
for the control of mean value.
However, (10) can be written in the form
8
(11)
So if tables of 'scores'
(j
==
0, 1)
are available we can plot E(y(l) - y(o»against number of observations to form a
CSCC.
To check whether there is evidence of a change in the value of cr we see
whether any points fall below the line PQ (see Figure 1) where, as before, 0 is
1\
the last plotted point
and OQ
== (-
x{/ n ( (Jo / crl ) - "21 ~ 2( cr-o2 - cr-l 2) •
It should be noted that tan
in a).
o
O~
In the present case tan OPQ
can be negative.
If s2 >
==
@Kn(crO/crl )]
( cr-2 - cr-2)-1 then the point P will be to the left of 0, as in Figure 2, which
o
l
exhibits a situation where a change in the value of cr would be indicated at T
"""=-------jl{O
Q
o b se ,-v o..ti..oY\.
yu)..
W\.6 e -{
Figure 2.
In the special case where we can assume that the mean (s) is controlled at
s ==
0, the problem reduces to that of constructing a CSCC for the variance of a
normal population with known mean (in this case zero).
This case can be covered
by the methods similar to those described by Johnson and Leone ~5_7.
It should be noted, however, that the methods described in~5_7 which use
sample range cannot be employed when the sign of each observation is lost.
There
9
is a tecbnically correct way of introducing range, even in tbis case, by assigning
positive and negative signs at random (witb probability 1/2 eacb) to tbe observations.
However, apart from tbe labor involved, tbe element of arbitrariness
militates against acceptance of tbis metbod as a practical procedure.
6. Average Sample Number under More General Conditions.
Tbe average number of observations needed for tbe control of mean procedure
(described in section 3) indicates a cbange in ~ wben tbe true value of ~ is g
I
(a general value) is (- in a )E , - l wbere
o
E' ::::
_12
ci1
_1012Joo
+ ~(2rIr.) e
2
1 2
- -y
2
e
o
cosb ~ly in cosb ~lY dy
"I : : 1/
and
~ (J
Tbe integral in tbis expression can be evaluated in a similar way to tbat in
(5),
tbougb tbe result is a little more complicated.
Numerical evaluation of tbis quantity, and of similar quantities needed to
assess tbe performance of tbe procedure described in section
elsewbere.
5, will be discussed
Tbe present paper is intended to sbow bow cumulative sum control cbarts
can be constructed for certain types of control based on signless observations.
10
REFERENCES
Elandt, Regina C.
The folded normal distribution:
~lO
methods of
estimating parameters from moments", Technometrics, Vol. 3 (1961),
pp. 551- 562 •
Johnson, N. L.
"A simple theoretical approach to cumulative sum control
charts", J. Amer. Statist. Ass., Vol. 56 (1961), pp. 835-840.
Johnson, N. L.
"The folded normal distribution:
Accuracy of estimation
by maximum likelihood", Technometrics, Vol. 4 (1962), pp. 249-256.
Johnson, N. L. and Leone, F. C.
"Cumulative
SlJlU
control charts:
Mathematical principles applied to their construction and use. I",
Indust. Qual. Control, Vol. l8(12} (1962}~ pp. l5-2l.
"
do
,
II lI ,
"
do
,
III", dO , Vol. 19(2) (1962), 'pp. 22-28.
Leone, F. C., Nelson
do , Vol. 19(1) (1962) .. pp. 29-36.
and Nottingham
( 1961) "The folded normal
distribution", Technometrics, Vol. 3 (1961), pp. 543~550.