I
I
I "I
I
I
I
I
I
I
"I
I
I
I
I
I
I
,.
I
SOME RESULTS IN ROTATABLE DESIGNS OF ORDER TWO
by
Philip V. Piserchia
University of North Carolina
Institute of Statistics Mimeo Series No. 501
January 1967
This investigation was supported
by the Air Force Office of Scientific Research Contract Number
AF-AFOSR-760-65.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
I
I
.'I
I
I
I
I
I
.,
I
I
I
I
I
I
I
,.
I
ACKNOWLEDGEMENT
I would like to thank Dr. I. M. Chakravarti for his suggestions,
encouragement and guidance during the preparation of this thesis.
I
I.
I
I
I
I
I
I
I
.-
I
I
I
I
t
I
t
,.
I
-.'
~
iii
This thesis is an attempt both to explore some general properties of
rotatable des:igns of order two and to construct some new designs by a generalization of a method given by M. N. Das and V. L. Narasimham in [9].
Chapter II is concerned with observing properties that transformed
rotatable designs of order tvl0 possess. By a transformed rotatable design
we simply mean that every point in the design is obtained by a transformation
of a point in a rotatable design of order two.
MOre specifically, section 2.2 gives a method of obtaining constancy
for the variance of
y(!), where
y(~) is the estimated response of a second
degree polynomial in k factors at a point x, on k-dimensional ellipses •
Section 2.3 deals with the use of rotatable designs when there are linear
restrictions among the factors and section 2.1 gives a theorem showing that
every rotatable design of order two in k factors has, in a sense, embedded
within it ( k ) rotatable designs of order two in k-r factors.
r
In Chapter III
we consider the rows of association matrices of a
partially balanced association scheme as points in a v-dimensional factor
space and, by appropriate algebraic manipulation, are able to construct
rotatable designs of order two based on these matrices.
I
I
I'
t
I
I
iv
Table of Contents
Chapter
ACKNOWLEDGEMENTS
SUMMARY • • •
I
1.2.1
ii
iii
1
1
....
Definition of Schlaflian Vectors and Matrices •• . . . .
. . . . . ..
2
:;
1.2.2 Some Theorems in Schlaflian Vectors and Matrices •
4
1.3
The MOdel Expressed in Terms of Schlaflian Vectors •
4
1.4 Necessary and Sufficient Conditions for Second Order
Rotatability • • • • • • • . • • • • • •
6
1.5 Some Well-Known Results in Rotatability• • • • • • • •
7
II SOME: THEOREM3 CONCERNING THE STRUCTURE OF ROTATABLE DESIGNS
9
..
2.1 An Embedding Theorem • • • • • • • • • • • • • •
2.2
• • •
Constancy of var ~~) on k-dimensional Ellipses
2.3.1 The
~del
. ..
and the Problem.
....
CONSTRUCTION OF ROTATABLE ARRANGEMENTS FROM THE ASSOCIATION
MATRICES OF PARTIALLY BALANCED ASSOCIATION SCHEMES. • • • • . .
2.3.2 Solution of the Problem. • • • •
III
3.1 Partially Balanced Association Schemes and Association
Matrices . . • • . • . •
. • .
3.2 Notation and Definitions • • •
3.3
The Union Method of Construction •
9
11
2.3 Rotatable Arrangements and Quadratic Response Surfaces
with Linear Restrictions among the Factors •
t
I
. ..... ... . . . . .... .
1.2 Schlaflian Vectors and Matrices. •
I
,
,
t
,.
....
,
I
I
I
I
mTRODUCTION•••
.. .. .. . . . . . .. .. .. .. .. . . . . .. . . . .
1.1 Definition of Rotatable Designs of Order Two
,
"
Page
. . • . . .
...
...
13
13
14
18
18
19
20
3.4 The Union Method and Some Two-Class Association Schemes. • 22
3.4.1 Two-Class Group
Divisibl~
Association Schemes • • • • • •
23
t
I
,I'
t
I
,
t
,I
t
I"
I
I
,
t
I
,.
I
v
The Triangular Association Scheme •
..
Pseudo-Cyclic Association Schemes • . . .
· . . . . . . · 24
The L Association Scheme • • • • • •
• • 25
r
· .. ·
27
3.5 The Union Method and Some Three-Class Association Schemes •• 29
3.5.1 Three-Class Group Divisible Association Schemes. • • •
3.5.2 The Rectangular Association Scheme.
3.5.3 The Cubic Association Scheme.
3.5.4 The Tetrahedral Scheme.
.
•
..•
3.6 Construction by Linear Combinations
.
.
29
· • · . · · • · · · 30
· • · • · · 32
·•
···
·33
•
3.7 The Method of Linear Combinations and Some Two-Class
Association Schemes • • • • • • • • • • • • • •
Two-Class Group Divisible Association Schemes •
The Triangular Association Scheme • • • • •
· · · ·34
·.• .
·.·.
• 37
37
· . . · . . . • 38
3.7.3 Pseudo-Cyclic Two-Class Association Scheme ••
· · . - . 39
3.8 Linear Combinations and the Three-Class Group Divisible
Association Scheme. • • • • • • • • • • • • • • • • • •
39
3.9 Concluding Remarks Concerning the Two Methods of Construction 40
BIBLIOGRAPHY • • • • • • • • • • • • • • . • • • • • • • • • . • 42, 43
,
t
I'
Chapter I
t
•I
"
t
Introduction
1.1 Definition of Rotatable Designs of Order Two
Suppose we are interested in fitting a quadratic response surface in k
factors to N experimental points.
th
Let the w experimental point be denoted by ~l (w) = (x lw, ~, ••• , xkw )
and define the matrix of experimental points D.k by:
x 21 •••
.
t
I!..' (N)
IN
Also let Yw be the observed response corresponding to the point
~l
(w)'
Hence, we take as our model:
k
e(y(x»
J;
I
=
x
I
,
,
I
,.
.
Dk =
I"
t
~' (1)
~l
-
k
= ~ + Z ~. x.
0
i=l J. J.
+
~ii
Z
i=l
x2.i
+
Z
Z
~.
~J.'J'
xiwjw
x + ew ,
1 Si<j Sk J.j
x.x ,
J. j
with observations
(1.1.2)
Yw = ~o
k
Z
+ . 1 ~.J.
J.=
where it is assumed that
e
x..J.W
k
+ Z
. 1 ~J.'
J.=
(e ) = 0, var
w
2
Z
J.·W + Z
1 S i <j;S, k
(tw) =
aF
for
w = 1,2, ••• ,N and
t
t
2
that the
I'
I
I
"
I
t
t
t"
t
I
I
,
I
t
,.
I
ew
are independently distributed.
We are interested in estimating the response surface 1.1.1 and the
variance of this estimate at all points
in our k dimensional
factor space.
A matrix of experimental points D which has the property that the vark
iance of the estimated response at any point, ~ in our factor space is a
function only of the distance,
a rotatable design.
~I~
of that point from the origin is called
MOre specifically, if our assumed response surface is
quadratic then D is called a rotatable design of order two. That is, if
k
e(y(2S» is a quadratic response surface a.nd if 3.1 and ~ are both points in
our factor space (not necessarily experimental points) and if
implies that var
y (~l)
response at the point
~
= var
P (~),
3.1 2£1 = ~ ~
where Y(3) represents the estimated
then Dk is said to be a rotatable design of order
two.
In what follows we shall refer to rotatable designs of order two and
rotatable arrangements of order two (defined in section 1.4) as rotatable
designs and rotatable arrangements respectively.
It is the intention of this thesis to present new methods of using
rotatable designs and to construct some new designs by a generalization of
a method due to M. N. Das and V. L. Narasimham in [9J.
The remaining portion of this chapter is purely expository and is given
in order to facilitate the reading of the following chapters.
1.2 Schlaflian Vectors and Matrices [5,8]
Although the notion of Schlaflian vectors and matrices are not original
in references 5 and 8, excellent
di~cussions
given therein and are hence reccommended.
and fUrther references are
t
!.
I
I
3
1.2.1 Definition of Schlaflian Vectors and Matrices
t
I
lilt
t
il
I
I,
I
"I
,.
I
Then the SchJaflian vector
x(2] of x is defined such that:
(1.2.1.1)
I
I
I
Xk).
Let ~'= (xO,xl ' ••• ,
~[2]' ~(2]
=
k
(~'
3,)2
= !: ~~
i=o
+ 2!:
a :s i<
~
For example, if k=l then ~[2]' ~(2] = (~,~)2 =
=
X~
+
i;. + 2 {xi
and we take
!:
j
.:s k
<{
{~.
+ xi)2
.!:C2 ] , = (x;, ~,J2X·o ~).
i
In general, if X' = (x, , xl' ... , x ) then x[2] is the
-
'k-
0
2
2
dimensional vector consisting of all powers xo' xl ••• ,
2
~~
(k+l)(k+2)
and all
cross-products of the form~ x. ~., i < j = 0,1, ' •• , k, and is
~
J
written as:
(1.2.l.2)
4[2]' =
(~, ~,
... , {,.j2"x x , ... ,.J2"x X'k,[2x x , ••• ,.{2X _ X ).
o l
l 2
O
Suppose H is a matrix defining the transformation
~ =~.
Then the
Schlaflian matrix H[2] of H is defined such that
(1.2.1.3)
H[2] ~(2] =1,[2].
For example, if k = 1 and H=[:OO
10
then H[2] is given by:
.
2
H[2]
=
:o~]
11
h
00
2
hlO
~hoo h lO
2
hal
.{2 hOO hal
2
J2 1'\0 hu
h1l
:12 hOlhll
•
(hal h lo + hOO h ll )
k
l k
I
t
.1
I
I
4
1.2.2 Some Theorems in Schlaf1ian Vectors and Matrices
~
The following theorems will be convenient for our purposes and are
stated without proof.
Theorem 1.2.2.1
If H and K are matrices then
{HK)[2] = H[2] K[2].
Theorem 1.2.2.2
If H is a matrix then
I
I
I
I
"
I
it
I
I
I
Theorem 1.2.2.3 If H is a matrix then
{H-1 )[2] = (H[2])-1.
Theorem 1.2.2.4
,.
I
=I
then H[2] H[2]'
= I[2]
where I[2] is the identity matrix of proper order.
1.3
The MJdel Expressed in Terms of Schlaflian Vectors
In 1.1.1 we saw our model expressed as:
k
k
2
e{Y{x) ) = ~O + E ~. x. + E ~ .. X. + E
E
~'j x. x j •
i=l 1 1
i=l 11 1 1 ~ i< j ~ k 1
1
Rewriting 1.1.1 we have:
e
k
(Y{2f) )
2
= i~ ~ii Xi
+ E
E (~ij/12
o ~ i< j ~ k
Where x 0 == 1 and ~oi =
=
~0
if i =
0
f
0
~. if i
1
) .f2 Xi
Xj
Hence we may write our model as:
I
I
If H is a matrix and HH'
where
R'
t::.
=
~
(~00, "., ~kk,f301/ J2 , ... , ~Ok/.J2" J2/ ,/2,
and x (2] is the Schlaflian vector of
(~~.)
= (••::.).
x
x
~
'.', k-1-,k/l2)
I
5
I.
I
Similarly we write our observations as:
[2] ,
,
I
I
I
I
I
.,
I
I
I
I
I
I
I
,.
I
~ + e1
Yl = A.(l)
..
where
[2]
~(i)
1
... --, [2]
= ( -~(i)
or equivalently
(1.3.2)
where y..'
e elJ = x'
= (Yl'
~
,
••• , .Yn) and X'
[2] ,
?S.(l)
=
.[2]. ,
!.(n) .
Now, assuming the matrix XX' to be non-singular the least squares
estimates of the
~'s
are given by:
~ = (xx') -1 Xz
(1.3.3)
,
and the estimated response y(~) at any point,
Xl
= (xl' ••• '.~)
in our factor
space is given by:
where
x r2 ] =
The variance of ~(~) is seen to be:
Hence it is seen that the problem of constructing rotatable designs
reduces to the problem of finding points in our k dimensional factor space
such that XX' is non-singular and that the quadratic form x[2]' (XX,)-l ~(2]
is constant on hyperspheres defined by
~'~ =
const.
I
I
.1
I
I
I
6
1.4 Necessary and Sufficient Conditions for Second Order !otatability.
Let us define the "moments" of our experimental points as:
N
"I
I
I
,.
I
a
JSrnk
w=l
with the convention that if a.
al
a2
, 2
'\
, ... , k
~
=0
we shall drop the symbol
.
(for example ~f a
)
l
= 4 and a
2
= a
3
a.
i ~
from
= ••• =
C1t
=0
then
With the "moments" of our experimental points so defined, necessary
conditions for second order rotatability are given in [8] as:
(i)
~
M(l
a
, 2 2, ••• , k
Ok
)
=0
if a
1
+ ••• +
ak -< 4
and at least one of the a. is odd.
~
(ii)
I
I
I
I
I
•••
M(l
I
I
I
a
X 1
lw
2
M(i , j2) =A4 > 0 (say) for
i
f
j
and i,j = 1,2, ••• , k
4
M(i ) = 3A4 for i = 1,2, ••• , k
2
M(i ) =
1.
2 (say) for
i = 1,2, ••• , k.
If in addition to (1.4.2) being satisfied we have:
then the conditions (1.4.2) and (1.4.3) are both necessary and sufficient [8]
for a set of experimental points to be
A
a rotatable design.
set of points which satisfies (1.4.2) will be called a rotatable
arrangement of order two.
I
I
7
1. 5
"
I
Some Well-Known Results in Rotatability
The notion of rotatable arrangements is introduced in section 1.4
because of the following simple but important theorem.
Theorem 1.5.1
I
I
I
I
I
made into a rotatable "design by the addition of n center points.
o
In the above, a center point is nerely the point
!' = (0,0, .•• , 0) = Q'
and n is completely arbitrary.
o
Another theorem which we will find useful is:
Theorem 1 • 5 • 2
,.
I
" a rot a t able arrangement th en 7= 7\J.J..
If Dk ~s
1/\ 2
2
> k/k+2
7\4 and 7\ 2 are as defined in (1.4.2).
For completeness the following two results are inculded [5]:
"
I
I
I
I
I
I
I
Given a rotatable arrangement with N points, it can be
1
= if
-1:
°
°
° i 2" ~~i
°
-------~--------------.----------lI
U,
-------~--------------~----------I
1
I
°
l
0:
I
I
1
.
2' \ Ik(k-l)
2
where
-1
U =
2(k+2)y2
a
-2Y/A2
···
-2Y/A2
-2Y/A2
{(k+i)Y-(k-l)}/A~
..
(l-Y)/A~
I
I
I
I
I
1
I
-2Y/A2
(l-Y)/A~
I
I
I;
I
I
I
1
1
I
1
I
1
(l-Y)/A~
I
I
I
·...
·...
I
I
I
I
I
1
-2Y/A2
(l-Y)/A~
··•
·
·... {(k:l)Y-(k-l)J/A~
I
I
I
I
I
I
I
.'
I
I
I
I
I,
I
•I
I
I
I
I
I
I
,.
I
8
and
4
"2
a == - - - - - - - - - -
The matrices I
k
and I
are the identity matrices of order k and
k(k-l)/2
k(k-l)/ 2 respectively.
(1.5.4)
N/ >2
cr
var
~
=
<.~)
2(k+2) ./ + 2(ki-2) '1 ('1-l){p2;A2) + [(k+l)'1 _(k_l)](p4jA2 )
- __2~_
2'1[(k+ 2) '1-k]
'Where
p
2
,
2
=~2S.=~
+~
2
+ ...
+~
2
•
I,
I
9
.1
I
I
I
I
I
I
'.I
I
I
I
I
I
I
,.
I
Chapter II
Some Theorems Concerning the Structure of Rotatable Designs
2.1 An Embedding Theorem
The following theorem gives some insight into the structure of
rotatable arrangements by showing that every rotatable arrangement in
k factors has, in a sense, embedded within it rotatable arrangements in
k-r factors.
Theorem 2.1
If Dk
=
~l
x ••• x
21
kl
~2
~2··
.
x
2n
x ,~
•
...
\2
b
is a rotatable arrangement in k factors then the set of points
x
D
k-r
=
11
~21·· • xk-r,l
x
'12
x22 •• • xk-r,2
x·
x
'In
"2n
...
x
'k-r,n
is a rotatable arrangement (in fact a rotatable design) in k-r factors.
Proof:
2
) ' \ - r ) 1S
. such that
Suppose M( 1~ ,2Q , ••• , (k-r
l + Q2 + ••• + '\-r ~ 4 and at least one Q is odd. Then define
Q
Ql
Q2
'\-r
Q
k
l
~-r+l = •.• = '\ = 0 and we have M(l , 2 , ••• , (k-r)
) = M(l , ••• , k )
Q
= 0
since Dk is a rotatable arrangement and
on
Q.
1
Q
l
+
Q
2
+ ••• +
~
:: 4 with at least
odd.
Now, i f D is a rotatable arrangement then by 1. 4.2 we have:
k
I
I
10
.1
I
I
I
I,
I
I
I"
I
I
I
I
I
I
,.
I
But if this is true then certainly
2
M(i ) = A. > 0 for i = 1, 2,
k-r
2
2
M(i , j2) = A.
for i .;. j and i, j = 1, 2,
4
... ,
4
M(i )
=3
A.
4
for i
= 1,
2,
... , k-r
... , k-r
Hence, 1.4.2 is satisfied and Dk
is a rotatable arrangement.
-r
The fact that D
is a rotatable design is shown by observing that
k -r
>
k-r/
(k-r +2)
That is, by using theorem 1.5.2 condition 1.4.3 is also satisfied.
It will be noticed that D
was defined by excluding the last r
k -r
columns of the matrix D • However, this was done merely for notational
k
convenience and, in fact, any r columns of D may be omitted and the
k
theorem will still stand.
Hence, we conclude that any rotatable arrange-
ment in k factors has embedded within it (k) rotatable arrangements in
r
k-r factors.
A natural question, it seems, now to ask is:
Is every rotatable arrange-
ment D consisting of N points embedded in some arrangement D " k' > k,
k
k
also consisting of N points?
The answer to this question is, "not in general."
For consider the
following.
We lmow by theorem l.5.1 that if D is any rotatable arrangek
ment then by adding a center point the arrangement is a rotatable design;
that is, the matrix XX' is non-singular.
But, if D with a center point
k
is a rotatable design then N + 1 ~ 1/2 (k+2)(k+l) or N ~ ~ (k+3)
I
I.
I
I
I
I
I
I
I
•I
I
I
I
I
I
I
fI
I
11
for any rotatable arrangement in k factors.
for k' > 2 'We must have N 2: 9.
No'W, for k=2 'We have N 2: 5 and
Ho'Wever, there does exist a rotatable
arrangement in two factors 'With exactly five points (the vertices of a regular
pentagon centered at the origin).
Hence, this arrangement is not embedded
in any arrangement D ,., k' > 2, and the question is ans'Wered.
k
2.2
~(x) on k-dimensional Ellipses
Constancy of var
Part of the appeal of rotatable designs stems from the fact that quite
often the statistician does not kno'W the orientation of the assumed response
surface.
Because of this lack of kno'Wledge having the var
function of the distance of
~
1\
y(~)
be only a
to the origin has a great deal of heuristic appeal.
If, ho'Wever, the statistician kno'Ws that the assumed response surface is
constant on contours other than hyperspheres then rotatability no longer has
quite the same appeal •
The follo'Wing theorem gives a method of utilizing rotatable design when,
in fact, the statistician kno'Ws the response surface is constant on a
particular family of concentric and coaxial k dimensional ellipses.
,
is, the statistician knows that if
!i A ~l = ~
That
A ~ then e(Y~l) ) =
e(y (~2) ) where A is a kno'Wn positive definite matrix and x~ A x = const. is
the particular family of ellipses.
Theorem 2.2
family of ellipses
Suppose A is a kno'Wn positive definite matrix defining the
~'
Ax = const.
Further suppose that D =
k
z'
~(l)
~N)
is any rotatable design in k - factors.
that the set of experimental points Ek =
Then there exists a matrix B such
';(1)
'Where x(.)
-
.!~N.)
J.
= B~
(i)
I
12
II
I
I
I
I
I
I
has the property that if
Proof:
::~
A ~1
= ~~
-1
B x = z.
Ax =
-
x' A x
~(~1) = var ~(~2).
If A is a positive definite matrix then by [lJ there exists a
non-singular matrix B such that BlAB
Xl
A ~2 then var
= I.
Now consider the transformation
We llllIllediately see that if x satisfies
- = -- = c;
BI ABz
-
Zl
ZIZ
Xl
A x = c then
that is, all points on the ellipse
= c are mapped into the sphere z' z = c.
Now suppose Dk
=[i~l) 1 is any r~t:table design in
~(n)1
k
:f'actors.
Then define:
(i)
\=
[!:(l)l
where ~(i)
= B~(i)
-(n)
(ii) x [2 J
-
Ie
= ~ J._ ) [2 J ,
x
(iii)
I
I
I
I
I
I
I
.I
x'
(iv)
=
[2J'
-~1)
···
[2J'
x[2J
-=(i)
= (
[2J
g(i)
=(
...1
~(i)
~(n)
~I
=
[2J'
~(l) ;
.•[2 J'
)[2J
...1z
)[2J
~(n)
,
Now assuming ~1 and ~2 both satisfy ~ A?: = c we write ~l
~
= B-1 ~"2
where
~l
and
~2
both lie on the sphere
,
~ ~
= c•
= B-1
:'1 and
I
I.
I
I
I
I
I
I
I
13
Hence we write:
var
I
I
I
I
I
,.
I
(X )
-1
= xf2J' (X
-l
= !~2J'
= -2
z'
~[2J
1
rr2
R[2J.' [R[2J zz , R[2J' JR[2J.~2Jrr2
= -2
z[2J' (z
z' -1
z
-1
X' )-1
~[iJ' ( z Z,)-l ~~2J rr2
=
_z2
Z,)-l ~2J rr2 since
~
and Dk is a rotatable design
= ~2J'
(R[2J f l' (Z
= x[2J' (R[2J Z
-2
z'
z' f l (R[2J r l .:~2Jrr2
R[2J' )-1 [2J rr2
!2
= ~2J' (X X,)-l ~2J' rr2
Ie
I
I
~
= var
2.3
1\
y (~) and the theorem is proved.
Rotatable Arrangements and Quadratic Response Surfaces with Linear
Restrictions among the Factors
2.3.1 The Model and the Problem
In this section we deal with the model:
with the r linear restrictions among the factors A ~ =
o.
Here it is assumed that A is an r x k matrix (r < k) of full rank
with orthogonal rows and ].2J'
~
is defined as in 1.3.L
I
I.
I
I
I
I
I
I
I
14
It is our intended purpose to find designs
x1l
D
k
=
I
xln
~n·· •
t
~(l)
~
=
.
A ~(i)
=0
x'
·-(n)
~
which satisfy the following two criteria:
(2.3.1.2)
If lE is any point in our factor space then the response surface
That is, if A ~ = .2 then D should have
k
1\
[2JI
[2J'
e(y(~) ) = ~
~. Again ~
~ is as rlefined in
must be estimable at the
the property that
point~.
1.3.1 and ~(!) represents the estimated response surface at the point ~.
(2.3.1.3)
If!l and !2 are both in our factor space and if ~l ~l
1\
then the design Dk should be such that var Y(~l)
= var
= ~2 ~2
1\
Y(~2).
To explain the above perhaps a geometric argument will suffice.
Ie
I
I
I
I
I
I
I
I-
~l·"
Our factor space can be looked at as a k-r dimensional plane passing
through the origin.
Since this is so, we are restricted in our choice of
experimental points; that is, each experimental point must lie on the plane
A x = O.
Now, condition 2.3.1.2 simply says that the response must be
estimable for every point in the plane while condition 2.3.1.3 requires
1\
that the var y(~) be constant on the family of k-r dimensional spheres
defined by A 3£ =
.2,
3£1 3£ = const.
Having explained the problem we now give the solution.
2.3.2
Solution of the Problem
Theorem 2.3.2
such that AB 1
Let B be the orthogonal completion of A; that is let B be
=0
and BB 1
= I.
Also let Sk-r
=
zll
z21
z
~12
z22
z
k-r,2
z2n
z
k-r,n
.zln
k-r,l
=
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
15
=
be any rotatable design in k-r factors.
Then the matrix of experimental
points
=
x'
-(n)
defined by ~(i)
Proof:
= B'
~(i) satisfies conditions 2.3.1.2 and 2.3.1.3.
First we must show that every experimental point
~(i)
does)
indeed) lie in our factor space.
However) this is obvious since B is the orthogonal completion of A and we have
AX{.)
-\1.
= AB' -z(.) = oz(.)
= -O.
~
~
Now) before proceeding with the proof we make the following definitions:
x[2J'
(i)
Xt
=
-(1)
··• [2 J'
=(.L
!.ti)
x(n)
Z'
=
[2J'
~(l)
··• [2 J'
~(n)
(ii)
-t -:_2'_]
C' - 0
-
I
B'
)[2J
;
I
16
II
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
II
(iii)
Continuing with the proof we know from [3J that condition 2.3.1.2 is
satisfied if and only if the set of equations X ~
= ~[2:A ~ = O,has
a solution.
However;
X
= (~~~~ , •.• ;
= (e
~~~?
)
[2J' [2J
[2J'
~(l) , ••• ; C
= C[2J'Z.
Also, for any
~
such that A x
-Z ' -- (z l'
J
Zk-r ).
• ••
=
° we. can wri te x = B'
~
for some
Therefore, 2.3.1.2 is satisfied if and only if the set of equation
e[2J' Z c = C[2J' z[2J
ee'
=I
has a solution.
But, BB' = I implies
and by theorem 1.2.2.4 we have e[2J c[2J'
= 1[2J.
Hence,
condition 2.3.1.2 is satisfied if and only if the set of equations
Z c
= ~[2Jhas
However, Sk-r is a rotatable design.
[2J
Therefore, Z is of full rank which implies that Z ~ = z
does have
a solution.
a solution.
Hence, condition 2.3.1.2 is satisfied.
Now let us consider condition 2.3.1.3.
First let us notice that if ~1
~l
= B~l
and. !2
= B~2
= ~2 = Q and
have the property that ~i ~l
This is easily shown by the following:
= -Xl 1-1
x
since ('A~') is an orthogonal matrix.
~i ~l
= ~2
= ~2
~2'
~2
then
I
II
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
II
17
Similarly, z2 z2
= ~2
~2 and we have ~i ~l
= z2
~2·
let us consider the variance of ~(~l)
2
= X[2J' (XX I )* x[2] rr where (XX I )* is the conditional inverse
-1
-1
.
I
[3J of
XX~
Hence, va,r
~(~l) = ~~2 J (XX *
= ~~2J
= ~~2J'
t )
C[2J
!l[2]
(C[2J' ZZ'C[2J)
*
(l
C[2J' ~~2J rr2
C[2J C[2J' (ZZ,)-l C[2J C[2J' ~i2J rr 2
= ~l[2J (ZZI ) -1 ~l[2J.rr2 from theorem 1.2.2.4
and Sk_r is a rotatable design.
Th.erefore,
var
~(~l) = Z~2J'
= ~;2J'
(ZZ,)-l z;2J
rr 2
C[2J' (ZZ,)-l C[2J' ~~2J rr2
= ~~2J' (XX I )* ~~2J rr2
= var
1\
Y(~2)
and the theorem is proved.
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
,I
I
I
I
I·
I
18
Chapter III
Construction of Rotatable Arrangements from the Association Matrices of
Partially Balanced Association Schemes
3.1 Partially Balanced Association Schemes and Association Matrices [4J
Given v objects, 1,2, ••• , v, a relation satisfying the following
conditions is said to be an m.-class partially balanced association scheme:
(1)
Any two objects are either 1st, 2nd, ••• , or m-th associates, the
relation of association being symmetric; that is, if the object a is the
i-th associate of the
object~,
then
~
is the i-th associate of the
object a.
(11)
Each object has n
i
i-th associates, the number n
i
being independent
of a.
(iii)
If any two objects are i-th associates then the number of objects
which are j-th associates of a and k-th associates of
independent of the pair of i-th associates a and
~
i
is Pjk and is
~.
The parameters of the association scheme are given by:
(3.1.1)
where
(v, n , Pi) ; i
i
1 =(
= 1,2, ••• ,
m
(P~k) ) ; j, k = 1,2, ••• , m.
The following indentity is easily established
i
Pjk
if
"l,
= Pkj
or any
J, k •
We now define the association matrices of a partially balanced
association scheme as:
~ A
and ""',f-'
~ ... ,
= 1 ,c,
otherwise.
i
v where b~
=1
if
a,~
i
are i-th associates and bap
=0
I
I
19
.e
The following two results are easily verified:
If B.
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
1
bI
•
= (b- l 1.,
b(3' = P.~
- en -
1
••• , b .) is an association matrix then
-V1
It is assumed 0: ~ (3.
if 0: and (3 are l-th associates.
11
+ ••• + c
wbere the
are association matrices then
B~
1
m
b' b
~-=-(3
=
B
m m
L:
j-l
:2 l
c. Pjj + :2
J
L:
L:
1:: k< j <m
l
c k c j Pkj
if 0:~(3 are l-th associates.
In this chapter we consider the rows of tbe matrices
v dimensional factor space; that is, as v-tuplets.
B~
1
as points in a
By generalizing a
method given in [9J (where the authors consider a more restrictive kind
of matrix) we use these v-tuplets to construct rotatable arrangements in v
factors.
Notation and Definitions
is a v-tuplet consisting of k:: v non
a:2'
-w
w ••• a vw )
l w,
k
k
zero co-ordinates then the notation a' x:2 refers to the:2 distinct
I f a'-: (a
-w
points of the form ( + a 1 w' -+ a:2w' ••• , -+ a vw ).
If
]-
!~
A:
is a n x v matrix
-n
each of whose rows contain exactly k non-zero elements, then A x :2 k is the
set of N
a'
- w
k
=n
x:2, w
x :2
= 1'
k
:2 ,
k
k
points {!:i x :2 , !:2 x :2 , ••• ,
..., n,
!:~ x :2k J where
is defined above •
... , xvw ) is a typical member of this
set tben
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
20
al
N
1
~
X
N w=llW
av
a2
X-
X
vw
-"""2 w
is the (a , a , ••• , a ) -th "moment" of that
l
2
v
set of points where "moment" is defined in 1.4.1 and the summation extends
over all members of the set.
3.3 The Union Method of Construction
The following theorem gives a method of constructing rotatable
arrangements which we call the union method.
Theorem 3.3
Suppose we have an m-class association scheme with parameters
= 1,2, ••• ,
(v, n , Pi)'
i
i
m, and association matrices B ,
l
Then if there
4 k
a l Pll +
the following
exist real number a l , a , ••• , am' not all zero, such that
2
4 k
4 k
a 2 P22 + ••. + am Prom = P (say) for k = 1,2, ••• , m then
sets of points form rotatable arrangements:
... ,
m
2
-n
/4
m
B JX2
m
nm
}
... ,
(if)
• •• + a
4 n
m
(iii)
[a
B •
m
([a
l
2-
nl
/4 B1Jx 2
nl
m
, ••• , [am 2- nm/4 BmJX 2
nm
,
V
(b, b, ... , b) X 2 }
.
4
4
4
if 3p < a n + a n + ••• + a n where b in (ii) and (iii) is
l l
m m
2 2
a real number determined by the moment conditions for rotatable arrangements.
Proof:
al
In all three cases
if the v-tuplet (~,
~,
~
2
av
XlW X2w
X
vw
a
... , - x.,
J.
=0
if any a. is odd since
J.
••• , x v ) appears in the set then so does
the v-tuplet (xl' x2 ' ••• ,
x.,
....,.xV ).
J.
Case (i)
4
+ ••• + a m nm
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
We have:
I:
w
2
X" = a
1W
2 2n'l/2 n
l
2
+ ••• + a 2ll.m/2 n
m
l
m
4
+ a m nm for all i
+ a 4 pk 1"f i , j are k - th associat es
m mm
from 3.1.4.
Hence, by assumption we conclude that I: xiw ~w
4
4
w
However, if 3p = a n + ••• + am nm then
l l
x2 jW
3 I: i!-iW
w
=
1::
W
X\cw
for all
i, j, k, i
f
=p
for all i, j, i ~ j.
j.
Hence, 1.4.2 is satisfied and the set of points is a rotatable
arrangement.
Case (ii)
+ n
m
+
a~
nm + 2 b
4
for all i and any b.
_2
4 k
4 k + ••• + am4 Pmm
k 1"f"1, J" are k-th associates
Xjw
= a l Pll + a 2 P22
But, by assumption we conclude I: ~w ~w = p for all i, j.
4
4
w
Now, . if 3p > a n + •••
+ am nm then there exists a real number b
.
l l
by 3.1.4.
such that
3p = a
3 I:
'w
Jt Y?:
1W
JW
= I:w ~
for all i, j, k, i
~
j.
That is,
444
m + ••• + am n + 2 b has a real solution in b and we conclude
m
l l
that the set of points in case (ii) with b so determined form a rotatable
arrangement.
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
22
We have:
,,_2
LJ
w
X1
_
W -
2,..,n l / 2
al
c:
nl
+
...
+ n
a
4
m m
4v
+ b 2 for all i and any b.
4 k
4 k
Pll + a 2 P- 2 + •••
associates by 3.1.4
4 v
Hence, 1: ~ ~j = P + b 2 by assumptio~.
lW
w
w
444
However, if 3p < n a + ~ a + ••• + n am then there exists a
2
m
l l
_22
1: Xfw Xjw
w
=al
real number b such that 3 1:
w
v 4
3(p + 2 b )
in b.
= n1
a
X~w X~w = 1: ~ for all i,j,k,i 1= j. That is,
w
4
4
4
v 4
+ n a + ••• + n am + 2 b has a real solution
2 2
l
m
Hence, we conclude that the set of points in case (iii) with b so
determined form a rotatable arrangement.
It is to be noticed that if a. = 0 then the set of points
1
[a
i
2
-no /4
1
B
i
Jx
2
ni
need not be included for the entire set to be a
rotatable arrangement.
With the theorem now proved, it will be our concern in sections 3.4
and 3.5 to find real numbers a , a , ••• , am such that:
l
2
4 k
4 k
4 k
a l Pll + a 2 P22 + ••• + am Prom
=P
(say) for all k, for particular
association schemes.
3.4
The Union Method and Some Two-Class Association Schemes.
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
23
3.4.1 Two-Class Group Divisible Association Schemes [6J
In this scheme v
= mn,
and the objects are divided into m groups of
n each such that any two objects of the same group are first associates
while any two objects from different groups are second associates.
Here we have an assoQ1ation scheme with parameters given by:
(3.4.1.1) n l = n-l, n2 = n(m-l) , v = mn,
n - 2
0
0
n(m-l)
o
n-l
,
Pl =
n(m-2)
n-l
Hence, we wish to find real numbers a , a such that:
l
2
a
4 (n-2) + a 4 [n(m-l) J = a 4 (0) + a 4 [n (m-2) J.
2
l
2
l
Equivalently, we wish to find real numbers a , a such that:
2
l
a
For n
4 (n-2) + a 4 (n)
2
l
= 2,
2
=0
and a
l
arbitr~y
= 1, n2 = 2(m-l).
l
Immediately we see that p = 0 and we determine the number b as in
Then, v
= 2m
choose a
=0
(m is ofcourse arbitrary) and n
case (iii) of the theorem.
Also, no other rotatable arrangement is possible by the union method
from this scheme.
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
3.4.2
The Triangular Association Scheme [6J
An association scheme is said to be triangular if it is an array of n
rows and n columns with the following properties:
(i)
The positions in the principal diagonal (rurming from upper left hand
to lower rigbt hand corner) are left blank.
(ii)
The n(n-l)/2 positions above the principal diagonal are filled by
the numbers 1, 2, ••• , n(n-l)/2
(iii)
which are the objects.
The n(n-l)/2 positions below the principal diagonal are filled so
that the array is symmetric about the principal diagonal.
(iv)
For any object i the first associates are exactly those objects
which lie in the same row (or in the same column) as i.
The second
associates of i are merely those objects which are not its first associates.
We then obtain an association scheme with parameters:
l
n1
= 2n
-4, n2
:~3~(n_4)!J '
n - 2
n -
3
4
p
2
= (n-2)(n-3)/2,
2n-8
=
2n-8
(n-4)(n-5)/2
=4
For example, with n
*
1
2
1
*
4
2
4
*
3
5
6
3
5
6
*
!
we have
v
= n(n-l)/2
and,
I
I
•I
I
I
I
I
I
Ie
I
I
I·
I
I
I
I.
,I
25
and the first associates of 2, for instance, are 1,3,4,6 •
Now we find real numbers a , a such that:
1
2
(3.1~.2.2)
a~
ai (n-2) + 1/2
(n-3)(n-4) =
4a~
+ 1/2
a~
(n-4)(n-5),
or equivalently:
(3.4.3.3)
ai (n-6) +
a~
= o.
(n-4)
Consequently we see for n > 7 or n
= 4,
However, for n
~
3 no rotatable arrangements are possible.
5, 6 we do get the rotatable arrangements given
in the following:
n
2
a
4
1
0
arbitrary
5
6
3
arbitrary
a
6
8
6
arbitrary
0
v
n
n
6
4
10
15
3.1~.3
1
a
1
2
1
p
b
0
case (iii)
4
4a
1
)+a 4
1
case (ii)
case (ii)
The L Association Scheme [6J
r
Consider v
= k2
objects set forth in a k.xk square.
Now take r-2
mutually orthogonal squares (if such'a set eXists) and define the L
r
association scheme by the following:
Two objects are first associates if they o'ccur in the same row or
column of the square scheme or correspond to the same symbol of one of the
latin squares.
Otherwise two objects are defined to be second associates.
The parameters of the scheme are:
v
= k2 ,
n1
= r(k-1),
n2
= (k-1)(k-r+1),
(k-2)+(r-1)(r-2)
(r-1) (k-r+1)
(r-1) (k-r+1)
(k-r) (k-r+1)
I
26
I.
I
I
I
I
I
I
I
at
I
I
I
,.
I
I
(k_r)2 + (k-2)
r(k-r)
= 4,
For example, with k
°4
8
12
1
2
3
5
6
7
9
10
11
13
14
15
=4
r
we have:
with the two mutually orthogonal
1
2
3
2
°
3
3
2
°1
latin squares
3
2
°2
and
1
°1
3
1
°
Here, for instance, the first associates of 6
1
2
3
2
°1
°
3
3
1
°2
are,
4,5,7,2,10,14,3,9,12,0,11,13.
After some simplification, the above scheme gives a rotatable
arrangement if the equation
I
t
r(k-r)
r(r-l)
4
4
a 1 (k-2r) + a 2 ( 2-r) =
°
has a real solution in aI' a 2 •
Some values of aI' a
along with the associated value of p and the
2
method of determination of b are given below for particular values of
k and r.
4
1
4
2
v
k
r
a
9
3
2
0
arbitrary
4
4
16
h
2
arbitrary
arbitrary
6
9
25
5
2
arbitrary
8
16
4-9
7
2
°
°
arbitrary
12
36
a
n
1
n
2
p
4
2
4
4
2a + 6a
1
2
4
12a
2
4
3 0a2
2a
b
case (ii)
case (ii )
case(ii)
case (ii)
I
I
,,I
I
I
I
I
t
Ie
I
I
I
I
I
I
I
.-
I
27
~-
4
2
4
a
l
b
n
2
18
30
14
49
21
42
0
28
35
0
arbitrary
16
64
3
arbitrary
4
3a l
21
59
135a l
case (ii )
l~.
2a
arbitrary
28
52
4
5 6a2
case (ii )
n
k
r
a
49
7
3
arbitrary
64
8
2
0
arbitrary
64
8
3
arbitrary
2a
64
8
4
arbitrary
81
))
2
81
a-'
81
9
3.4.L~
p
l
a
v
l
L~
2
4
l
4
l
4
42a
2
4
68a
l
4
12a
l
4
56a2
26a
L~
case (ii )
case (ii )
case (ii )
case (ii )
case (ii)
Pseudo-Cyclic Association Schemes (6J
Let the objects be denoted by the v integers, 1, 2,
we can find a set of n
l
••• J
v.
Suppose
integers d , d , ••• , dn satisfying the following
l
l
2
conditions:
(i)
(ii)
The d t s are all different,
o<
d. < v.
J
Among the nl(nl-l) differences d j - d j, (j, j I
reduced mod v, each of the nLwbers d , d , ••• ,
l
2
d~
= 1,
2, ••• , n
1
and j ~ j I
occurs g timesj
whereas, each of the numbers e , e , ••• e
occurs h times, where
l
2
n2
d , d , ••• , d n ' e , e , ••• , e
are all the different integers
l
2
2
l
1
n2
1, 2, •.. , v-l.
Then, the first associates of the object i are defined to be i + d ,
l
i + d , •.• , i + d • The remaining objects are said to be second
2
n1
associates.
~le
(3.4.4.1)
parameters of such a scheme are:
v, n , n ( all defined above)
l
2
)
I
I
28
I'I
n
g
p
1
n -g-l
1
nl-b
Ie
I
I
I
Now, all tbe known cyclic association scbemes are sucb that
v
= 4w+l,
n
l
= n2 = 2w
and g
v
p
p
1
2
=
= 4w+l,
l:-1
n
l
= n2
Hence,
=
2w,
:]
=
I
or eQuivalently:
I
for some positive integer w.
association scbeme:
numbers a , a sucb that:
l
2
.-
= w-l
we call any association scbeme witb the following parameters a pseudo-cyclic
I
I
I
]
n2 -n l+b-l
'
I
I
-g-l
=
I
I
I
1
Hence, for any pseudo-cyclic association scheme we wish to find real
I
I
,I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
II
29
That is, we must merely choose a
4 = a 4 and a pseudo-cyclic association
l
2
IB ]
scheme gives rise to a rotatable arrangement.
However, in [2] it is shown that the matrix N =
matrix of a balanced incomplete block design.
n;- Is the incidence
Now, the case of obtaining
rotatable arrangements from the incidence matrices of balanced incomplete
block designs is covered in [9J; hence, this scheme is no longer of any interest
to us.
3.5
The Union Method and Some Three-Class Association Schemes
3.501 Three-Class Group Divisible Association Schemes [lOJ
Consider ordered 3-tuplets (Xl' X2 , X ), Xi
3
and define a relation of association as follows:
= 1,
2, "0' Ni ,
(Xl' X2 , X ) and (Y , Y , Y ) are i-th associates if they have the first
2
l
3
3
3-i co-ordinates alike.
We then obtain a three-class association scheme with the following
parameters:
o
o
p
1
=
o
o
0
P2 =
0
N -1
3
N -1
3
N (N2-2)
3
0
0
0
N N2 (N -l)
l
3
I
I
.e
30
=
P3
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
,e
I
0
0
0
0
N -1
3
N (N -1)
3 2
N -1
3
N (N2 -1)
3
N N (N -2)
3 2 l
After some algebraic manipulation it is seen that 3.5.1.1 gives a
rotatable arrangement if the set of equations
4
a (N -2) + a
l
4
3
4
2 (N3 )
=0
4
a [N (N -2) J+ a (N N ) = 0
2 3
2 3 2
3
has a real solution in a , a •
2
l
Therefore, we see that if N > 2 no arrangements are possible while
3
if N = 2 we may take a arbitrary and a = a = O.
2
l
3
3
Notice that if N = 2 then p = 0 and b is determined as in case (iii)
3
of theorem 3.3.
3.5.2
The Rectangular Association Scheme [12J
Consider a rectangle with n columns andl rows.
Let an object
correspond to one of the .t n cells.
Now define:
(i)
(ii)
(iii)
Two objects are first associates if they are in the same row.
Two objects are second associates if they are in the same column.
Two objects are third associates if they are not first or second associates.
We then obtain a three-class association scheme with the following
parameters:
I
I
.e
31
p
1
=
I
I
I
I
I
p 2 --
p
3
I
Ie
I
I
I
I
I
I
I
•e
I
=
n-2
0
0
o
o
l-l
o
l -1
o
o
n-l
o
l -2
o
n-l
o
(e -2) (n-l)
o
1
n-2
1
o
.t -2
n-2
l-2
(l-l) (n-2)
Ct-2) (n-2)
It is seen that the rectangular association scheme gives rise to a
rotatable arrangement by the union method if the equations:
a
4
4
4
(n-2)-a2 (l-2) + a (n-l)
l
3
4
=0
4
a (l-2) + a (l-2) = 0
2
3
have a real solution in a , a , a •
l
2 3
We now notice that if.t > 2 iole must take a 2 = a = o. Hence, we get
3
a non-trivial solution for n = 2,J > 2 where a = a = 0 and a is chosen
2
l
3
in an arbitrary fashion. Also, for n = 2,J > 2 we see that p = 0 and b
must be determined as in case (iii) of theorem 3.3.
Now, if J
=2
we take a l
= a3 = 0
and a 2 arbitrary.
Again we see p = 0
and b is determined as in case (iii).
Concluding we state that the rectangular association scheme gives
rotatable arrangements for any v dimensional factor space where v j.s even,
v
= 2l (say) •
I
I
.e
I
I
I
I
I
I
32
3.5.3
The Cubic Association Scheme [llJ
Let each object correspond to the ordered triplet (Xl'
Xi
= 0,
.e
I
X ),
3
1, ... , 8_1.
Define (Xl' X , X ) and (Yl , Y2 , Y ) to be i-th associates if they
2 3
3
have i co-ordinates different.
He then get an association scheme with parameters:
(3 .5 •3 • 1)
v = S3 , nl
Ie
I
I
I
I
I
I
I
~,
= 3 ( S-l),
n2
) 2 ,n
= 3 (S-l
3
= (S-l) 3 ,
S-2
2(S-1)
o
2(S-1)
2(S-1)(S-2)
(S_1)2
o
(S_1)2
(S_1)2(S_2)
2
2(S-2)
(S-l)
2(S-2)
2(S_1)+(S_2)2
2(S-1) (S-2)
(S-l)
2(S-1)
(S_1)(S_2)2
0
3
3(S-2 )
3
6(S-2 )
3(S_2)2
3(S-2)
3(S_2)2
(S_2)3
,
After some calculation we see the above scheme gives us a rotatable
arrangement if the equations
(3.5.3.2)
4
4
2
4
a (S-4) + a 2 (S -4s + 2) + a (S-1)(S-2)
l
3
=0
4
4
4
a (S-2) + 2a2 (8-2)(8-4) + a (8-2)(28- 3 ) = 0
l
3
have a real solution in a , a , a •
l
2
3
Now, for 8 > 4 it is easy to verify that no real non-trivial solutions
are possible.
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I-
33
However, for S = 3 the equations becotne:
4
1.j.
4
(3.5.3.3) -a - a + 2a = 0
123
444
a l - 2a2 + 3a3 = O.
By substitution we see that a real solution exists if we choose a
3
4
4
4
1/
4
4 1 /. 4 .
arbitrary and a = 3 a wlth a = 2a -a = 3 a •
2
2
1
3
3
3
Here, we get a rotatable arrangement with v = 27, n
n
3
= 8.
Also, p = 11
a~
while n1ai + n2
a~
+ n
3
a~
= 30
l
= 6, n
a~.
2
= 12, and
Hence,
b is determined as in case (ii).
For S = 2 the equations reduce to:
(3.5.3.4)
4
4
-2al -2a2 = 0
Hence, for S = 2 we choose a
l
= a
2
= 0 and a
3
arbitrary and we obtain
a rotatable arrangement with v = 8, n = 1, p = 0 and b is determined
3
as in case (iii).
3.5.4
The
Tetrahedral Scheme [7J
Consider the integers 1, 2, •.• , m and the v = (;) sets of size 3
that can be formed from them.
Let an object correspond to each one of these sets and define
[Xl' X2 , ~} and [Y , Y2 , Y } as i-th associates if their intersection
l
3
has (3-i) integers in conunon.
We then get an association scheme with parameters.
(3.5.4.1)
v = (~), n l = 3(m-3),
n2 =3/2 (m-3)(m-4), n =1/6 (m-3)(m-4)(m-5),
3
m-2
2(m-4)
2(m-4)
(m-4)
o
1/2 (m-4) (m-5)
2
o
1/2 (m-4) (m-5)
1/6 (m-4) (m-5)(m-6)
I
I
34
.e
p
2
=
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
4
2(m-4)
(m-5)
2(m-4)
1/2 (m-5) (m+2)
(m-5) (m-G)
(m-5)
(m-5) (m-6)
1/6(m-5)(m-6)(m-7)
o
9
3(m-6)
9
9(m-6)
3/2 (m-6)(m-7)
3(m-6)
3/2 (m-6) (m-7)
1/6 (m-6) (m-7)(m-8)
The above association scheme gives us a rotatable arrangement if the
equations:
L~
4
4
a (m-6) + 1/2 a 2 (m-6)(m-7) + 1/2 a (m-5)(m-6) = 0
l
3
4
4
4
4 a + 1/2 a (m-7)(m-14) + 1/2 a (m-6)(m-7) = 0
2
l
3
have a real non-trivial solution in a , a , a •
l
2 3
It is obvious that for m ~ 7 or m ~ 5 that no real solutions are
possible.
However, for m
Hence, for m
=6
=6
the equations reduce to:
if we choose a
rotatable arrangement in v
m
=6
3.6
implies p
=0
=
(~)
l
= a2 = 0
= 20 factors.
and a
arbitrary we get a
3
Again we notice that
and b is determined as in case (iii)
Construction by Linear Combinations
In the theorem which follows, we derive a method of constructing
rJtatable arrangement from association matrices which we call the method
of linear combinations.
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
35
Theorem 3.6
Suppose we have an m-class association scheme with parameters
(v, n , Pi)' i = 1,2, ••. , m, and association matrices B , B ,
l
2
i
i f there exist real numbers
... , a.It
non-zero (a. , a. ,
II
l2
+ 2
L:
1
8.,;
<
<' i
L:
j
2
... , am'
m
Then,
t of which are
(say) ) , such that:
a
<m
a ,
B •
2 2 k
a Pij
i j
=P
(say) for k
= 1,
2, .•. , m,
then the following sets of points form rotatable arrangements:
(i)
t2- k / 4 (alB
if 3p
..
( ll,~
r
l2
= a 4l
-k/4 (
alB
+ a 2B +
2
l
n
+ a
l
k
+ a B ) X2 }
mm
4
4
n + ••• + a n
2 2
m m
+ a 2B2 + ... + a B ) X 2
mm
l
i f 3p > ai n
l
+ a
k
,
b~ X2}
4
4
n + • .• + a n
2 2
m m
k
(b, b, ... , b) X2 V}
+ a B ) X2
mm
'
(iii)
•.• + a
where k
= n.
II
+ n.
l2
4
n
m m
+ •.. + n. and b in cases (ii) and (iii) is determined
It
by the moment conditions for rotatable arrangements.
a
Proof:
As is explained in theorem 3.3,
L:
w
a
Xl~ X2~
if any a. is odd.
l
Now, we show that the other necessary moment conditions are satisfied.
Case
(l') 3P
4
= aln
l
4 n + ••. + am4 n
+ a2
2
m
Here:
4
+ ••• + a n ) for all i
m m
= 1,
2, ••. , v.
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
e
I
I
36
2
_2
~ ~
~
w lW JW
4
= m~
k
aj P . . + 2
JJ
j=l
= p
1
~
2 2 k
~
< £ <
aUajPdj if 1, j are K-th associates by 3.1.5.
m}..
}..
by assumption.
4
But, if 3p = a n + • •• + a 4 n
then
11m m
3
~
=~
i! JtJW
Dl
W
W
4
f
for all i, j, k, i
'X.-lew
j and we conclude that the set of
points form a rotatable arrangement.
Case (ii)
4
> a l nl
3p
+ a
4
2
n
4
2
+ ••• + am n
m
We have:
2
~ X.
lW
W
= 2k/2
2
(a n +
1
1
... + a2mnm) + 2b2 for all
+ a
2_2
~ ~
W
X.
lW
JW
4k
m
=~
a j P'j + 2 ~
j=l
=P
4
n + 2b
m m
4
S.e <
n + 2h
m m
~
j.:sm
and any
4 for all i
22k
at a j Ptj
if
b•
and any
i, j are
b.
k-~
associates by 3.1.5
by assumption
But, if 3p > ai n
+ a
1
J.
4
i
l
+ ••• + a: n then 3p
m
has a real solution in
b
+ a
= ai
n
l
+
a~
n2 + ...
and the arrangement is rotatable.
4
n
m m
Here:
~
_2
X1"T -_
2
k/2
W
~
W
x.4lill = a4l
n
l
2
2
2)
v 4
(al n l + a 2 n2 +••• + am nm + 2 b for all
+ a
4
4
v 4
n +••• + am nm + 2 b for all
2 2
i
i
and any b.
and any b.
then
a real solution
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
•e
I
37
in b and we conclude that the set of points in (iii) with b so determined
form a rotatable arrangement.
With the theorem now proved) we will try to obtain in sections 3.7
and 3.8 real numbers such that:
m
l:
for all k
a
4 k
i=l
i
= 1)
2)
PH + 2 . l:
l:
1 < i .< j < m
... , m)
a.2 a.2 p.k .
1
J
=p
lJ
( say)
for particular association schemes.
However) the equations for determining al' a ) ••• ) am are qUite
2
often intractable and) as a result) not many examples are given.
Also) it is worth noting that if we choose a
l
= a 2 = ••. = am = a
(say)
then we always get a rotatable arrangement from the linear combination
method.
This is apparent since the matrix N
= Jv
= Bl
+ B + ••• + B
2
m
=
- I v ) where J v is the v x v matrix
whose elements are all 1 and Iv is the v x v identity matrix) is the
incidence matrix of a balanced incomplete block design [2J.
Since this
case is covered in [9J) such solutions will not be of interest to us.
3.7 The Method of Linear Combinations and Some Two-Class Association Schemes
3.7.1 Two-Class Group Divisible Association Schemes
If 'ole consider the association scheme explained in section 3.4.1 of this
thesis) we see a rotatable arrangement can be obtained from it by the method
of linear combinations if the equation:
4
22
4
(n-2) -2a a 2 (n-l) + a (n) = 0
l
2
l
has areal solution in al' a 2 ) a .
3
a
Now) if we choose a
are possible •
l
=0
then we immediately conclude no arrangements
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
38
If 'I-Ie choose a
2
= 0 with a
arbitrary then a solution exists if n = 2.
l
However) the arrangement obtained is identical with that given in section
3.4.1 and is of no interest here.
Now) assuming al' a 2
f
0 let y
Then the equations 3.1.1 become:
y2(n_2) -2y(n-l) +"n
=0
Solving 3.7.2 for y we get:
(3.7.1.3)
y
= 1)
y
The solution y
= n/(n_2).
=1
corresponds to a
is not of much interest here.
However) if we choose y
2
l
~/
= a 22
and as was explained before
"
= ajra~ = ;!n-2
with n > 2 then we obtain
rotatable arrangements in any number of factors.
k
= nl
3.7.2
+ n2
Here it is noticed that
= nrn-l.
The Triangular Association Scheme
The relation of association and the parameters" for this scheme are
given in section 3.4.2.
The problem of finding numbers al' a
2
so that 3.6.1 will hold reduces
to solving the equation:
(3.7.2.1)
4
4
2 2
a (n-6) + a (n-4) -2a a (n-5)
1
2
1 2
Notice that for n
3.6.1 and for n
=5
= 6}
4 we obtain the same arrangements as in section
we must have a
Now, assuming a , a 2
1
=0
F0
2
= a~
which is of no interst.
and :etting y =
a~~
the equation 3.7.2.1
becomes:
2
y (n-6) - 2y(n-5) + (n-4)
Upon solving for y} we get y
= 1}
=0
as expected} and y
__ n-4/ _ •
In 6
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
39
= n(n-l)/2
Renee, for n > 6 and v
we can always obtain a rotatable
arrangement by the method of linear combinations from this scheme.
3.7.3
Pseudo-Cyclic Two-Class Association Scheme
The relation of association and the parameters of this scheme are
given in section
3.4.4~
Now, we wish to find a , a 2 such that
l
+ 2a
2
a
l
+ 2 a
2
l
2
2
a
(w)
2
2
(w).
This is equivalent to:
4
4
-a + a
l
2
(3.7.3.2)
=0
and is of no interest to us as was explained
previously.
3.8 Linear Combinations and the Three-Class Group Divisible Association
Scheme
In section 3.5.1 the relation of association and the parameters for this
scheme are given.
After some algebraic manipulation it is seen that 3.6.1 is satisfied
if we can find a real solution to the equations:
(3.8.1)
a
4
4
2 2
(N -2) + a 2 (N ) - 2a a (N -l)
l
l 2
3
3
3
=0
a 4[ N (N -2) ] + a 4 (N2N ) + 2 a 2 a 2 (N -1)
2 3 2
l 2 3
3
3
- 2 a 2a2 ( N -l) -2 a 2a2 [N( N -l) ] = 0
2 3
l 3
3
3 2
=0
We see that if a
(3.8.2)
3
1 + 2 a~ (N -1)
3
Renee, assuming a
the equations become:
3
then the second equation reduces to:
=0
which has no real solutions.
~ 0 and letting
2
2
ai/2
X = /a
3
any Y
a
2!2
= /
a3
I
I
40
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
a_
I'
I(N3-2) + i?(N3 ) - 2XY(N3 -1) = 0
v2CN3(N2-2)] + 2 XY(N -1) -2X(N -1)
3
3
-2Y[N (N -1)]+ N2 N = 0
3 2
3
Solving the
firs~
equation of 3.8.3 for X in terms of Y we get as
N
solutions X = Y,
Y
3
X = N -2 ' N3 > 2.
3
Substituting X = Y in the second equation of (3.8.3) we obtain the two
solutions X = Y = 1 and X = Y =
N N
2
j
(N N -2)'
2 3
The solution X = Y = 1 implies ai = a~ = a~
and as explained
previously, is not of much interest.
N2 N /
3
I (N2 N3 -2),
N > 2, then we
3
obtain a rotatable arrangement in any number of v = N N N factors.
l 2 3
N Y
I f we substitute X = N3 _2' N > 2, in the second equation of
3
3
3.8.3, the expression becomes unmanageable and is omitted.
However, if we choose X = Y =
3.9
Concluding Remarks Concerning the Two Methods of Construction.
The methods of construction given in theorems 3.3 and 3.6 although
interesting mathematically do not always lead to arrangements with a small
number of experimental points.
If we call N the number of points obtained by using the methods of
theorems 3.3 and 3.6 and if a. , a. , ••• ,a. are the non-zero numbers of
1
1
1
2
t
1
'.', am} where a , a , ••• , am satisfy 3.3.1 or 3.6.1
l
2
then we see that by the union method we bave:
.
n.
n.
1
1
Case (i)
N = v x 2 1 + v x 2 2 + ••• + v x
n.1
Case (ii)
N=vx2
n.
1
1
+vx2
2
n.1
+ ... +vx2
t
+ 2v
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
-I
.e
I
41
n.
N
Case (iii)
1
n
=vx~
n.
1
n
1n
n
+vx~
~
+ ••.
n
it
+vx~
+
n
~
V
and by the method of linear combinations we have:
Case (i)
N = v x 2
Case (ii)
N = v x 2
Case (iii)
where k = n.
1
k
+ 2v
k
v
N = v x 2 + 2
+ n.
1
k
1
2
+ ••• + n. •
1
t
Hence, we see that if t is large and if n. , j = 1, ••• , t, are large
1
j
then N is extremely large and not of much practical value.
Therefore, it seems that any future work employing these two methods
should, of course, deal with finding more arrangements from association
schemes other than those already discussed and, perhaps more importantly,
finding methods of reducing the number of points in the arrangements already
constructed.
Some of the notions used in fractional replication seem to be
of some use in doing this; however, we will not discuss these methods in
this thesis.
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I_
I
BIBLIOGRAPHY
[lJ
Anderson, T. W. (1958), An Introduction to MUltivariate Statistical
Analysis.
[2J
Wiley, New York.
Blackwelder, W. C. (1966), "Construction of balanced incomplete block
designs from association matrices."
Institute of Statistics,
Mimeo Series No. 481.
[3J
Bose, R. C., Unpublished notes on the analysis of variance.
[4]
Bose, R. C. and Nair, K. R. (1939), "Partially balanced incomplete
block des5.gns." SankhYa, 4, 337-372.
[5J
Bose, R. C. (1958), "Rotatable designs" Notes taken by the statistics
staff of the Bettis Plant, Atomic Power Division, Westinghouse
Electric Corporation.
[6]
Bose, R. C.;Clatworthy, W. H. and Shirkande, S. S. (1954), "Tables
of partially balanced designs with two associate classes."
North Carolina Agricultural Experiment Station Technical
Bulletin No. 107.
[7]
Bose, R. C. and Laskar, Renu (1966), I~ characterization of tetrahedral
graphs." Unpublished paper.
[8J
Box, G. E. P. and Hunter, J. S. (1957), "MUlti-factor experimental
designs for exploring response surfaces." Annals of Mathematical Statistics, 28, 195-241.
[9]
Das, M. N. and Narasimham, V. L. (1962), "Construction of rotatable
designs through balanced incomplete block designs." Annals
of Mathematical Statistics, 33, 1421-11~39.
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I_
I
[10] Raghavarao, D. (1960), "A generalization of group divisible designs."
Annals of Mathematical Statistics, 31, 756-771.
[11] Raghavarao, D. and Chandrasekhararao, K. (1964), "Cubic designs."
Annals of Mathematical Statistics, 35, 389-397.
[J2]
Vartak, M. N. (1959), "The non-existence of certain partially
balanced incomplete block designs."
matical Statistics, 30, 1051-1062.
Annals of Mathe-
© Copyright 2024 Paperzz
