.
~.
This research was partially supported by National Institutes of Ilcalth.
Division of General Medical Sciences, Grant No. S RoOl CM 12868-05, and
by Army Research Office. Durham, Grant No. DA-ARO-D-3l-124-G910 •
"
OPTIMAL RESPONSE SURFACE TECHNIQUES USING
FOURIER SERIES AND SPHERICAL HARMONICS
by
LAWRENCE L. KUPPER
Department of Statistics
University of Nortll Carolina at Chapel Hill'
Institute of Statistics Mimeo Series No. 678
APRIL 1970
......
~
...
- .....,
_:.:---.
T gratefully ackJ.lo"'lcdge the gu1dtUH.:e and
enr~ouragement ~iven
to me by my adviser. ProfesRcT 1. M. ChAkravarti, dut'inr. tho p"oRrel!lS
of this
rt~caT.ch.
I also "-ish to thank the other 1Rembers of my doctoral cOJIIIIittee,
Professors ll. C. 'Bose, N. L. Johnson. H. A. David, J. E. Crizzl.!, and
Harry Smith, for their interest in my
..
e~pel'j.E:ncp.
WO'E"k..
It has been a stitlJu18t1ng
being associated with tho membet"8 of the faculty of
DepartmentR of
Sta~i8tics
and Biostatistics
~t
t:h~
the University of
North Carul1na.
For finandal ald. 1 acknov1.edge a National Science Foundation
Traineeship and support os a research assistant in the
D~partmp.nc
of
il!ostat:tstics.
1 want to express my appreciation to my wife, Margie, for her
patience
Ilnd
understanding tht"oughout the
year~
of my
~raduate
study,
and I also want to thank Mrs. Gay GOS9 for her efficient typing of
this manuscript.
Finally. I
~e
a very special debt of gratItude to Prof.ssor
William Mendenhall. Chairman of the Department of Statistics at the
University of Florida, who stimulated my initial interest in Statistics
and was
re~pot1sil,le
d~Jpartmcnt.
for my decision to pursue graduat.e studies in his
I shall alw.ays
his persowd :l.nterest in ·Me.
r~me'l1lbcr
boi.a cl'!thuni8t1lh 8'ld dedication and
y . ., -.
•
TABLF. Of CONTr.U1S
ACKNOWI.!Dm!F.m'S • • • • •
INTRODUCTION
A~~ 5t~niARY.
..
0
•
•
•
. .. . . . .
.... .
0
•
0
·.·..
·. .
•
~
1J
tv
Chapter
Io
f[ll.TRIER
l'Hf~IR
..
1
..
1
·
The
2.
Intel:ral Properties of. the F.S ••lnd S.H. Models.
6
OJ
Pol)111cml.al Rt:t>r'!sental"ions alld Add} tion Th(!ott~nlr:
f Qr the F.~:. .lnd S .H. lfodels. • • • • • • • • •
10
l~. S.
aI1d S. H. Mod,';ls. •
CmITRIBV'l'10NS "0 TI!F: TtiKORY OF rUE: OP'lTt'f..AL DESICN
O}!' tVERIMF.NTS • • • • • • • • • • • • • • • • •
4.
Exact ·md Approximate Des.i~n8 •
S.
Optimal ~"'t~ponse ~urf ,:.lee Designs. •
6.
A
~eneral
7.
'IllA
OPTIMAL
8.
·..
, •
0
AdmisslbiU ty of a
DESIG~!iFOR
RI'!':ipOllRe
••
22
Surface Design..
26
F.S. tUm S.H. RECRESfiION.
i:
rs
The Cbfiracteri:uttion of 0'>'3 and IS
Opt:l1l41 Properties of M(8
0
•
•
·....
."
)80d ~(6SH) •
sn
0
••
•
as Rotatable
RESPONSE Sm:FACE TECHNIQUES FOR F.S. AND S.H.
'tEGRES~:IllN • • • • • • • • • • • • • • • • •
lOt
LIST OF
·. ...
31
31
51
66
Anulysis of Fitted Second Order F.S. find S.H.
Madels. . . . • . " •
11.
17
Result on lhe ConstructiOTl of Weighted
ArrangelUent.r,. • • • • • • • • • • • • • • • • •
IV.
14
14
• • • • •
Bias Designs. . • • • . • • • • • • •
III.
·..
1.
.:J.
u.
S1'1RIl~S AUD SPHERICAL UA.T{.~~mnCS MODELS AND
PROPERr ll~:, • • • • • • • • • • • • • • • •
It
•
•
•
•
"
It.
•
•
..
•
••
Cunfidp.nce Reg:J.ons for. StationAry Points of Sp.coud
Order .F.S. and S ,R. l{1)dels. • • • • • , • • . ,
REFF.Rf~CES
••• •
•
• • • • •
........
...
66
74
78
84
INTRODUCTION AND SUMtIARY
..
'n\e general problem of rtlsronse surface cxplO%Qt:f.on may be
clcsc'tibed
88
follows: g.'ven that the form
r'Jlatioaship Ey(!) • 1'1 (~;~)
b(~tween
the
at
tho true functional
exp~eted
value of a raudUli
varLable y(!) and a k-ver.toT of non-ytocllastic factors ~' •
(xl' x 2 '···, ~) is unknown, the object 13 to approximate 8D
ac,:urately
89
k~dimeDsional
8raduatin~
pCtss1ble, within a given re~ion of interest. *.in the
factur space, the form of the surface
~(!j@)
by some
;..
function
,(!;~).
tn a given experimental situation, one
might be interested in using ;(~;E) to estimate the point !o in ~
at which 1'}(!;D) achieves a m:;ucilllum (or 1II1niJllum).
Sueh
I:l
situation
often occurs, for exal-pple. in the ellemleal industry where researchers
,
are interested in determining
.. those settings of the factorD (e.g.,
-
.
pressure, temperature, concentration) which gtve maximuM y1.eld or
higbnst pud ty of a chf.-'lldcal or wbic:h minimize the cost of production.
Invariably, 1 t bas been 8ssumad that the Ullkn'ND response fune1'}(!;~)
tion
could be
~ep~esented
at
ev~ry
point in a Riven region with-
in the factor space by a polynomial of degree d. say, in xl' x2' ••• '~
of the form
(1)
80
wh~re
n
ia a
2
2
+ B1xl +••• + f3kX k + PJlxl +••• + fl kk;.;, .. 131 2":1'.2 +•••
+ fik-l,k'1c:-l~ + eli.lx~ +••• ,
t
•
(a O' Bl'···'-~' all'···· ~, ~12'···' 6k- 1 ,k' 13111 , ••• )
1 (k:.s) row v~ctor of vn1tnmm p8T". .tttr
:l{
R •
The pr.blllry mc>tiva-
v
tion for considedng
to tbe parUal
BUIll
t~t:io
type of
cxpr~Hs1Cll'
of a TayJ.or'f.; series
is that i t corresponds
eXf).'1l1tdon
of
T\(~;§)
about the
origin of the factor levels, the derivativQB at the origIn being
,;imple IlUlt:l:ples uf the
~. s
•
responses are obseloved at a set of !
It
f
8
suitably
numerous
., .
OD
a circular or spherIcal conductor or in predicting the rate of
steady-state heat flow or the elevation at an arbitrary point on the
•
'.
surface of the earth.
However, it bas been
sh~
(see (5), )).217)
that it is not possible, if d >1, to obtain separate least squar.s
estimates of all the
.
parameee~8
-
in (1) if each of the x's must be
chosen to satisfy the restriction x'x
.... •
.~
(/0.
To consider yet Rnother situation, let us suppose that
that n(!;@) is periodie in 8t
leo~t ~nft
cleat' that an adequate approximation to
of the Xi'
T'l(!;~) ud,ng
;)
W~
know
In this case, it ie
a polynOlDial
\'1
would t'equire a very lar.r,e
nu~be,~
of tams.
The::;1J would oscillate in
Yign and the cancellation of lar~e posi tive and negnt.fvo values wO\1ld
thus g:l:ve poor precision in tho ustimation of
would certainl)"
periodic.
p~eferto
11(~;~).
HeTe:, one
use an appr('\x:hnatJng run~t1nn 'Wll'lch is itself
As a ,:;imple example, if
n(x;~)
..
cos 5x, then the partial
r:'
sum of a Fouriet' series, say aO +
l (am COR mx + bm s:10 m.x). would
be a gradtJating functiun of few tE:rms with smaller coefficients thaD
those of a polynOJaial.
Many natural phenomena are subjec.t to periodic
behavior and, in such casos, pulynomf.al-typp.
~laduatlng fun~ons
would certainly be unsAtisfactory.
.
Motivat.ed by the abc.wEl rema:rks. the author attempts in tbis
d:tssertation to broaden the area of respunse "uriaee methodology to
include the use of partial sums of lourier series and spherical
harmonics as response surfaee graduating functions.
Chapter I seyves as
~odels
that vill be under
ap
introduction to the types of regression
conBid~ratlon.
Several fmportmlt properties
of these functions which are needed for later work are diocusBed.
In the framelt.'Ork of the "approximate theory" of the optimal
design of experiments, a general approach to the problem of the construction of designs for response surface exploration in described in
Chapter II.
Various cTiteria are considered and
certain of these criteria are given.
.
s~ne
results relating
A general theorem on the mini-
mization. of average squared blas is presented and, based on this
re~
sult and some decision-theoretic notions, the new concept of tbe
"admissibility of a response sut ftt::e dfJsign" is introduced.
nle results contained in the first two chapters aTe used in
Chapter III t.o show that the desigt\8 proposed for the least squares
vii
fitting of the Fourier Hcri.ee anti spherical hC?rmonics regression
models ar':! optimal in n,;:tny impurtant rEspects.
ship which is found to exist
..
be~."een
these
The valuable relation-
desit~ns
and "rotatable
arrangements" of points (Box and Hunter, [.5]) is used to construct
"exact" designs for Fourier series regression of all orders and for
spherical harmonics rer-ressl"n of: orders one and two.
In Qlapter IV,techniqueR are
d~velQped
for locating the
stationary points of fitted second-order Fourier series and spherical
hamonic8 functions and arc used to con..qtruct confidence regions for
the corresponding points of
B~rfaces
under atudy.
Numerical
examples are included.
1' • • • •
CHAPTER 1
..
FOURIER SERIES AND SPHERICAL HARMONICS MODELS
AND nrnR 'P'ROPRRTlES
1.
:Fha .,.
S. and S• .!h1!oelels
In all that follows, we ghall boconcernecl "-ith two regres.ion
models.
The first is the Fourier sedea (F. S.) model
..,
where
(1.2)
un(;;Ain
: Fs» •
an cos n~ + en sin n" n • O,l, ••• ,el.
It will be both convenient and useful to write Un (;;~:»
where
D •
..
:rhus, we may write (1.1) compactly as
(1.5)-
1,2, ••• ,el •
as
•
2
where
and
Qt
(l.7)
~FS
• (Q(D)t
!:FS
QCI)'
t
~FS
•
... ,
Note that
(l.8)
and hence that
.
(l.9)
The second model that we will study is the spherical harmonics
(S.H.) model
(1.10)
where
unce ,ep ";a.. s(nl1 » •
(l.ll)
where r
IUD
rm-n
O(tt
nrn cos mP +
eIUD
sin arlJ) p
mn (c.),
(cosO) is the "associated Legendre function" defined as
(1.12)
Pn(cos9) being the Legendre polynomial of degree n wIth argument cose.
Aa for the F. S. model, it will be \lorthwhile to write (1.11) as
where
~
or
..
3
sin~P
. n1(CQs6) ••••• cos n~ Pnn (cose), sin n~ pnn (cose)]
for n • 1.2 ••.• ,d.
•
and
(1.1S)
.~
Q.
(0)
S1t
(n)
nPS ' • ( a o.a 1'~
• UOO ' ...
n
n C! n1,···,ann ,pa nn ) f or n· 1 • 2 , .•.• d •
H
Thus, we Ny write (1.10) in the form
(1.16)
..
where
(1.17)
and
~SH
(1.18)
•
(0)'
(l)'
.(d)'
(~SH ' gSlI . ' ••• , eSa ) •
When written as in (l.5) and (1.16), the F. S. and S. H. models
will be said to be of "order d". the vectors
!(~)
(2d + 1) and (d + 1)2 elements, respectively.
obvious. and the value (d +
(2n + 1) elements in
Remark 1.1.
L?
l)~
!(n)(e,~),
and
~(e,~)
having
The value (2d + 1) 18
follows from the fact that there are
n • O,l, ••• ,d.
The domain of definition of (1.1) can be taken to be
either the closed interval
in the latter case,
~
lO,2~]
or the circumference of a circle;
is the angle associated with
representation of a point on the circle.
~le
polar coordinate
4
Remark 1.2.
The domain of defInition of (1.10) can be taken to be
either the rectangle {(O,~):
..
a sphere; in the latter
cas~,
0 ~ 0 ~ w, 0 ~ ~ ~ 2w} or the surface of
D and
~
are the
~ngles
associated with
the spheri.cal coordinate representation of a point on the sphere.
..
The expression (1.1) is recognized to be the partial sum of a
Fourier series and is of period 211' in I).
, • 2w(x-a)/(b-a) indicates that
~'e
The linear transformation
F. S. regresBion model, in addition
to providing a natural way of representing a function whose domain of
definition i8 the circumference of a circle, can also be used either to
graduate an unknown response function
n(x;~)
on the finite interval
latb] for values of x on that interval or to graduate an unknown
periodic function
n(x;~)
of
k;L~
period (b-a) for all x e
(~,~).
Clearly, the 1. S. model cannot be used to represent a function on the
entire real line if this function is not periodic.
The expression (1.10) can be considered to be a generalization to
two
dimensions of (1.1); tbe quantity
face spherical harmonic of degree n".
tn(e,~;~~:» is known as a "surUsing (1.12) and the trigono-
metric functions for sums and differences of angles, one can easily
• (n)
• (n) )
verify that Un ( e"'~SH ) • Un(2~-e'~'~SH
for
0
~ e ~ wand
o ~ Q ~ ~, and that Un(9,~;~~~» • Un(2~-et$-~;~~:» for 0 ~ 9 ~ wand
w ~ $ ~ 2~.
Using those relations and the fact that nd(e'~;~SH) is of
period 27l' in both
nd(e'~;~SH)
e and +,
it tben follow8 that the behavior of
on the entire (a,.> - plane will be known once its behavior
on the rectangle {(e,~):
0~
e~
7l't 0 ~ • ~ 27l'} hag been determined.
In this light"the pair of linear transformations
e·
w(x1-a1)/(b1-a1 )
and • • 2'Tr(x -a )/(b -a ) indicates that the S. H. regression model,
2 2
2 2
....
5
TE~presenting .'1
while not onl1 pr()viding a natul-al bat;i", for
def lnod on the sur face Clf
Go
spiwre. could
ate an unknown re9pOl\lie function
a 1 ~ xl ~ ~l'
{(Yo ,x ):
l
2
3
2 :;.
4:
a1.,~o
n('lCl,Jt2;~)
'In
function
be used ei ther tl') grsduth~
fillite rect'11lg1e
~. b } for villues ot (x ,x ) on that.
Z
l 2
rectangle or to gt'adu:lte over tbc whole (x1Jx ) .- plane an unknown funcZ
tion Tl(X ,x ;~) which is of known period 2(bC31) in xl and of known
1 2
period (b -a ) in x
2
2
2
cnd which also s,ttisfiel3 ths quite restrictive
pair of relations
(1.19)
tn fact, the relations (1.19) are
60
limiting that one can feel justi-
fied in using an S. H. regression model to grauuate a function
n{xl'x ;f) on the (xl'x ) - plane poly when the region of :Interest has
2
2
finite area.
For graduating over the whole (x 1 ,x 2)
n(xl,x2;~)
.
s.
plane a rusponse function
which is periodic in xl and x 2 ' a possible regression model
is n(tbl,$2;~FS) • n
F.
~
dl
(41l;~FS)
regression functions.
as the sum of
Tl
d2
«j)2;~FS)'
the direct product of two
Clearly, rl(¢1.ep2;~FS)' which can be written
(2dl~1)(2d2+l) di~tinct terros~
possesses the desired
periodi\': properties without any Q! the limitati.ons suffered by the S. H.;
model.
A little more vill be said al'out these notions at the end of
the third chapter, but let us note now th~t thip "direct product" idea
6
clearly
2.
~eneralizes
to any number of
dim~nsions.
]..!!.~e..&!al 12~et;!.:~!!.~...E!.~hc:...¥_._ ::J. ;},!l.U~~~ ..1"!~.Eel.!
Let the " weighti':1g funr..«:1.011" w bQ an clernertt of the set
Where)y.*-
{w(~): w(~)
1.s a
rr~bE4bili.ty
example, the "uniform" wc1.ghting funct:ton
'\I;)~,
melJtJure on a space,*l.
118!tl~S
For
equal "7eigllt
-1
(*
I d!) to every point ~ f..;-t.
M~lt1un._~!J..
onal on a
f
~
A sequence of functionu {f (~), r2(~)' ... } is orthog-
spac~ ~ with rer;p6'(~t
fk(x) !n(X)
7.,....
dw(x) a
...,
iJIW
by dividl.ng f k by
1
to a weightIng function w
0 whenever k
IIfkll,
~~.
where f If k /1
n,c sequence is
2
til
YJ~ if
~ormaliaed
&
J~ f~(~) dw(~).
Now, consider the "Pourir,ir sequence"
(2.1)
{I,
C08$, 9in~,
••• , cos u~. sln n"
1
The relations cos m~ cos n¢ • Z[cos(m+n)$
1
2[8in(m+n)~
+
sin(m-n)~].
and sin
m~
cin
+
n~
••• }.
cos(m-n)~],
sin m~ cos n~ •
1
• I[cos(m-n)$ -
cos(m+n)~l
. imply that the sequence (2.1) Js orthogonal with respect to the uniform ,)
weightinr function
w(~)
.. ,/2r. on each I)f the two regions discussed in
Remark 1.1, and they also "how that
2w
2n
J cos2n~ d~ J sin2n~ d~
(2.2)
•
o
• W, n • 1,2, ••••
0
Next, consid&T' the 8equen~e of functions {1, x, x2 , ••• , xn ,
They are linearly independe~~
line.
on
... }.
every finite interval of the real
Let
be the sequence of functions, orthogonRl un the interval
[~.1]
with
7
respect to the unU'orm vcighting function ,.,,(x) • ~(1+Jl:), 'IIhich "is
..
} by the Cram-Schmidt ortbogonaliobtained from' {I. x, x 2 , •••• xn
, ••.
•
zatlon procedure.
Sq, by
P (x). I njOe jXj ,c
constTucti~n,
n
•
n
D11
> 0,
n • 0,1,2, ••• , and
1
plll(X)
f
(2.4)
r 11 (x)
dx • 0 If rn • n.
-1
The sequence (2.3) is called the "Legendre sequence" Clnd
, Pn (x) is the
"Legolldre polynomial of degree n in x".
P (x'
n '
A standard expro8sion for
is
p
(2.5)
n
(x) • (2n-1) (20-3) • •• 1. [xn _ n (!l-l:.L xn- 2
n'
2(2n-1)
I
xn- • - ••• ]
..
t
where the last term is a multiple of x if n is odd and is a multiple
of xO if n is even.
Now, (2.S) can be written as
and thus, from (2.6) we have
(2.7)
which is known as Rodl'iguGS I formula.
The relation
1
f
(2.8)
P;(x) dx • 2/(20+1)
-1
.
foll~s
directly by using (2.7) and performing an o-fold integration
by parts.
Nov, i.t will be shown that the "sssociaUd Legendre sequence"
8
{p."tm(:t), P.,..(x), ... , P , f.<), ... J.
.l
':'lil
nf 1
(2.9)
Wll€l'e P
(x) is given by (1.12) for x
J.t~gendre
functions with different val ues t,f mare, :f 11 general, not
nm
orthogonal.)
'£0
&:
do this. \Ie sh:tll make
case, is au urthog..,nal lJequence
URe
of the following recun1ve
relation for Legendre rolynomia]s:
(2.1.0)
+
m-l
(n+n)(n-~l)(d
/dx
m-l
2 11\-1
The multiplication of (2.10) by (I-x)
)Pn(x). O.
gives
(2.ll)
Now, from (1.12), we have
1
1
p (x) p , (x)
om
nm
J
. -1
dx •
f
(1_x 2)m (dm/dxm)p (x) (dM/dxm)P ,(x) dx
n
n
-1
1
· - J (dm-l/dXm-l)Pn,(x)
-1
or, using (2.11),
1
f
P
nm
(x) p , (x) dx
nrn
-1
I
1
• (n+m)(n-m+1)
(1_x 2)m-l
-1
1
• (n+m)('l-m+1)
1(x)
f Pn,m- lex) r.
n ,m-1
c'x.
9
this reduction fCI'Nmla m times and /-tupesl to (2.4) and
It we apply
(2.8), we obtain the lmportant reflult
I
1
(2.12)
~/ 2 (n-nn)! I (2n+1)Cn-m)!
.
F
if n • n'.
, (x) dx •
om (x) Pnm
-1
0 if
Using (1.12) and (2.S), we
n
~el
~
ute
a useful expansion for P tlm (x)
ne.eded in the next section:
(2.13)
where the expresr-tion in brackets has its
l-!l~t
term as a multiple of
x if (n-m) is odd and as a multiple of xo :If (n-m) is even.
From the integral properties we have Riven for the sequences
(2.1) and (2.9), i t follows that the "spherical harmonic sequence"
8in2¢ P22(x)~ ••• , coe¢ Fn1(X), sin¢ Pnl(x),
sin
m~
is an orthogonal
w(x.~)
•
P (x), ""
nm
s~quence
(1+x)¢/4~
cos
n~
P (x), sin
nn·
n~
•• • t
COB ~
F
nm
(x).
P (x) •..• }
.nn
with resrect to the uniform weighting function
on the rectangle {(x,¢): -1 ~ x ~ 1; 0 ~ ¢ ~ 2n}.
Tne transformation x • cos9 can now be employed to show that the
validity of the expressions (2.4), (2.8) and (2.12) nnd the orthogonality of the sequence (2.14) are stUl mai.ntafnE'd when the rE-gions
of
inter~st
are those r,iven in
Rem~rk
1.2.
It ie important to note,
however, that the weighting function refoJu1tinn from this transfonnation,
10
namely w(A .$) • (l-cosO)tt 14';', is dearly !!2! the uniform weighting function associated with the rectangle of
Rema~k
fom on the sutface of a sphere, since dxdlft
III
1.2, but is, in fact, uni-
de
sine
d4J is the element
of sutface area o{ a unit sphere.
Before proceeding, let us emphnsize that, with no loss in
generality, we may restrict our attention to the unit circle and unit
sphere.
Any adjustment to be made for a radius different
is simply a matter of multiplying by the
3. Polyocmial
Represent~~
and
a~rropriate
fro~
unity
scale factor.
for the ¥. S. and
Add}~!on T~)eorems
1.:..H. Models
The following well-known expansions will be needed:
(3.1)
cos
n~
• cos
+
!in-I)
n
ell -
21
cos
n-2
~
sin
2
~
n(n-l) (n-2) {n-3)
n-4~ i 4~ _
41
cos
~ s n ~
••• ,
and
sin ncll • n cosn- 1• sin' - n(n-1)(n-2)
(.3.2)
31
coso- 3• sin3•
(n-3).!n-4)
n-5... i 5...
+ n(n-l) (n-2)
51
cos
• s n • - •• ~,
where, in each expansion, only those terms with non-negative exponents
appear.
We now give two lemmas which will be quite useful in later wot"k.
~a 3.1. The expression Un(';~;;» can be written as a homogeneous
r3tional integral algebraic expression of degree n in the elementa of
~s' 'Where
(3.3)
,
.,
..
11
Thr. result follows direct ly
Pr(.t()f.
cos n.b 1s ()f the form C (n,k) cos
n~2k
4> ~dn
1
where 2k < (n+l) and C1(n,k) is
8
si,,~e,
[rom 0.1),
2k
.
~ach
n-2k
41" C (n,k) xl
1
l"onstant depending only
k. and, from (3.2), each tertii of sIn
n~
tet1ll of
Ott
x
2k
2
nand
is of the fonn
n f ' .. rv2 ( n,~II) Xln-2t-l x 222"1 t 211~ < n.
Analogously t we have
( II)
C2n,~
COR
n-U-l",~
S
i 2£"'1",
The surface spheri.;al harmonic U (0 ,4>;eS~}j~»
n
-.
can be wr1 tten
as a homogeneous rational integTctl algebr.1ic expression of degree n in
~SJt'
the elements of
(3.4) ~SH
where
•
(cose t cos¢ sinl), sin, sinO) • (x 3,x 1 ,x 2), ~S~'SH • 1.
Fraof. Using (2.5) and (3.4), we see that each term of P (cose) is
n
n-2j
2
2 2 j n-2j
of the form C(n,j) cos
6 • C(n,j)(x1 + x 2 + :Ie) x
' 2j <: (n+1).
3
From (2.13), (3.1), and (3.4), each term of (cos ~) Pnm(cos6> is of
m-2k
the form C (n,m,k,l,) cos
¢ sin 2k~ sinme cos n-21,-1Re
1
2
2
2 t m-2k 2k n-2t-m
• C1(n,m,k,t) (xl + x 2 + x3) Xl
x2 x 3
' 2k < (M+l) and
2t < (n-m+1).
term of (sin
And, similarly, from (2.13), (3.2), and (3.4), each
~)
P (cosO)
11m
c~n
be put in the form
2
2
2 t m-2k-l 2k+l n-2t-m
Cz(n,m,k,C) (xl + x 2 + x ) Xl
x2
x3
' 2k < m and 21 < (0-.+1).
3
This completes the proof.
Note:
The author makes no claim to originality tor the results of
Section 2 or
f~r
the contents of Lemmas 3.1 and 3.2.
These propp.rtles
of Fourier series and spherical harmonics are well known to those con-
versant in this subject area.
Remark 3..:1.
There are
above Iellllllas which vilt
two
very
tmporta~t
:implfc;ations contained in the
beq.~ite ll~ef,ul in ~~ter:.,.ork;:j)1i"1'l.t?ifirstone
; .. ~.
tf
12
:Is that
l")d(¢;~FS)
is a function of ¢ £..nJ): through thE'
e and
~le~lt!nt.s
of
~FS
.2!!..lJ.'. throllgh the elements
and that
nd(O~(p;~5tI) is
of ~SH.
Secondly, the ~xpressions PU(~;~FS) and nd(Ot~;~SH) can be
a luncHon of
¢
written as polynomials of degree d in the eicmeotg of
s and
~ ..
~SH'
respectively,
although such representa!.ions are not unique due to the
,.
restrictJon ~rs ~FS • ~SU ~SH • 1.
For exnmpltl, it is e8!l1y t.o show
that
where
and that
where
(3.8)
Expressions (3.S) and (3.7) w.Ul be used in Chapter IV.
Finally, there is an important "addition theorem" relating the
clements of f(
}(e,.> which will be needed for future work.
... 11
The general
result, the proof of which can be found in any standard reference on
spherical harmonira (e.a., see Hobson (17), PI"
141-143), is as followsr
13
p
n
(cosO cose' + dnO sfn'1'cos(¢>-¢')}
Setting 0 • t3' and. • " , and notIng that. 'P (1) • 1, we obtain a special
n
form, namely
(3.9)
p2(cos6) + 2tn
u
l.a.-I
(n-m)f p2 (cosO) • 1, 1 _<
(n+m) I
M
0
_< d.
This expression can be regarded 1n the same light as the relation
(1.8)
fo~ th~
F. S. model.
,.,
i.
.7'
'4
CHAPTER tl
CONTRIBUTIONS TC THE TUFQlll OF 'rJ-!E
I~PTIMAl.
.
DESIGN OF EXPERIMENTS
The tht"!ory (\f the opti.mal deflign of elCpedments
principally develoved
~y
Elfving, Kiefer and Wolfowitz
iniU 8tl!d and
~.•
a.,
see [121.
Let !'(~) .. (fl(:)tf2(~).".'
[20J, (221) fiu the fnllcwing fr.a1l1eworlt.
be a vector of m real·valued functions definrd on a given ,pace
f (x»
m ..
.¥-:.
aR
In most applications.
f(x) taken
......
*
is taken to be comract and t1'>e element!5 of
to be continuous (so that $\Il'rema of c.ertain functions are
attained, insuring that
(so that trivial
optl~um de$ign~
redund~ncie8
exist) and l1nearly independent
are avoided).
We assume that for each x...
or "c:ombinat.ton of factor level.s· t 1n ~ an nxperiment can t:e performed
wboBe ClUtcome 1s a random variable y(~), tlud that Var y(~) • (12 fot" every
~ e ~.
It i8 further aAswned that y(~)
holS
8n expected value of the
explicit fol'lQ
(4.1)
and that the functions fl(x). f (x), ••• , f mex), called the "regress Lon
2
~
functioftR".
~re
paranteter vector
~
known to the
~'
•
<e l ,
experim~nter
~
while the elements of the
82 , ••• , Bm> Rre unknowns to be estimated on
the basis of u finite number N of uncorrelated observations {Y(~1)1~.1.
1')
•.• , wp
rwi ntFi ~1'"
".... ......
,"
.\0
x , t',:<;tpncl
"1"
~vr.:
,p
are 'lntegera, and whe,.e Lt-l wt •
ly, wht're w
N
i
at the dlstjnct
"" n.J. , :l
.. 1, 2,
... , p,
C)r.41rly. tltl'! 'Ilt1al!lurf' (dc.'81sn)
)..
~
Rpecificr. the PQint':4 at l:hich experhlents take place, llalllelv the
{I.t }~.P and t~le
t'xperitllt!l'ts
nUl1lbEH' of
~i· The set {~l' ~2' "', x~." } where wi
°i at
r ~ldoT c.om.M n.lUon. nnmely
."It ell/·tt
ft
o(x ) > 0 for every i,
... t
1, 2, ••• , p, and wh.?r(l Ii.l wi • 1 i~ called the spect!.~ or t.hu
:L •
,lcsilJn 0, written 8(-5).
The th.:!ory ilwl.,lved in tl.c cnn.:;trur.:tion f)f
experim~ntal desi8r.~ ~i}1
exact
be
refer.r~d
to
a~
tha
!Xnc~ the~.
TI,e (mxm) matrix
(4.2)
wh~re X'
In
(! (~1)' ! (~2)'
••• ,
!l!l!!,;"di of the exact desigft 6.
! (~N»'
is known c.s the J.n~2.!!llilloE
If tbe para1\\etf'r ,'ector ...R is estimated
by fittlns (4.1) using the methorl
or
least
sq~ureR,
thus obtaining a
best linear unbiased estimate b , then the diRperflion matrix of h is
E(b-e)(b-G)']·. (X'X)-l
t"4
....
0
2•
_.."
More generally, if r:,. .. {~:6 :h an arbilt'ary probabillty measure
on the Borel sets
for each
~
£
~,
(4.3)
and then de fine
(4.4)
j30f :i,
let
u~
writp.
wh~rc 13includcs all one-point sets}, then,
16
The following letrdlla.
tak~n
th~
important prQperties of
..
L'~fI!ll•.£ Lid.'
fr.('Jm Karlin cmd
btl.1dJ~tl
f.J.9). l'f!cord9 several
matrix M(6).
Let HOi) be ddined
.:lA
in (4.3) 'tncJ (4.4).
Then •
(1) for each 5 t 4, H(6) 15 positive seroi-drfinttr;
(ii)
jH(O)
I•
0 wheneVt.~ ft(l;) c.ontrlins 1el'J$ than m pC'.lnts;
(iii) the filt1l11.y of matdces M(O). 15
(1v) for flach 0
£;
At the
The most import.lnt
(tv) wi1:f.ch permits us to
re~trict
compact set;
Cl"lnVQX,
can be uritten as
matT1.X M(t)
ctmsequ~n(;e
h. is a
F;
of thb 1eil1Jlla is cont'.\ine·,! in part
attenti':lT. to
tit
aSilres cont.:p.ntrfJted on
nltly a fJ.nitl1 set of pd.nts ~hen working W:J.I:~l thl~ CllG8 ~'f information
matricl!a lJ (~). 6
£
A.
In particular, if
n ~ t:.
(.orrespc1nds to au e:<act
dp.tdgn, th~n l'.'!of(l) • X' X.
Sin~'~ the ,t"malln('.ss" of U-1<O ')r the "largeness" of M(&) intuitivp..l:r Buggests thou b is "dose" t\l St i t is uraderstandtlble that ml.'.n)"
~
N
optimal design cr1 f'L~l'ia are based on lninJ.nb:Jn!J hy choit.e of 6 £: A
some 11)ean11'1gfn1 teal func:tional of N(6),
S8Y
1
IM- (0) I or tr
~C.l(\5).
Now, suppose that one fi.ndH a 6* E!l which millimizes some such function
flf K(6).
Clearly, it can happen that (~
multiples of lIN and thus
d~es
~
tAkes
£'11
values other than
not correspond to an exact design.
Neverthp.1eRs, it is worthwhile to con"'lder thh general npproach to
the construction of optimal design9, calling it the
and calling any mCdsure 6
£
A an
appr2~~. ~es~&~.
~roxi~~
theorY
The justification
and convenience in ·a.llowing this ~re.a.ter r.cuerllHty in the. choice of
mea~ures
(designs) iB that" in soae cl1ses. it permits one to give a
17
conlplet:e r.haToc.-.tedu.tt!otl t't certatn
opt'(tuiil desi~:ns
i~
Tt;'levant
d~tlign
for enoch
which
for aU N.
..
An ol\timlDn appr.oxiir.Atc
N
d~sirn
...· Ul givE' an c,&uct.
whi~h In optimunl to within or~1~r ~rl.
In some situat.lonst the optimum
approx{mate design will turn out to be an exact de&igll for many and
\ole will Reo truportl1nt examples l,f such phenomena
posslbly for all N.
in Ch.1pter III..
5. 9ptimClL~..!.'!l!S'nse S\1~~~.!1an..!
Let us recall frolic the lntroduct:f"n that the objective of an"
response tJurface investigation ca.n be briefly sununAriz(!d
itS
one vf
apJ'r.oximating over 8\)me rcgion:f; an unknown sur.face n (x;B) by sotae
. . . .y
graduating
fun~t1on
y(x;b).
w
If we agree to take y(x;b) to be the usual
~
.~
~
least-squ.:1.res t.stimator which possesses ccrtai.n well-known uropet" ties t
the.n the design problem can be re,'lsoncbly forlftulated as on•.! of ('hooRing
the tY.lE'ASUre 0
some
<.!•.!.,
meanln~ful
n(x;B)
... ...
88
the
~'s
and thl·ir currespl)ndirtg weights) to make
function of the absolute deviation of y(x;b)
....... from
small as possiblE!!.
One
e&pc~ially
relevant choice for such •
function is
~
E(y(x;h) - n(x;8»
(5.l)
f'\lt
....,
IIW
....J
2
•
,.
which is the ,xpectec! J!1!!.!n-sguare s.I.£!tt of y(~;~) at ~ £~
It is
easy to show that (5.1) can be written as the sum of two distinct error
terms, namely
"
2
Ey(x;b» •
(5.2)
...
'"
the yariance of y(x;b)
at x £~. and
'"
.
..
."
...
18
(5.J)
,.
the ~g"l.1_~:S! bi~p_ or ,(x;b)
at x E.'~ •
...
~
Now,
t!lt~
theor~
appT.'mti",ate
discusslHt in the pre'liN19 flection
th~ 0l-\t:fm.'lJ dr~~ign
of
~S9Um~$
of mcpcl"lments
that thp. t't'.f,ression function
(4 .1) t~xactl.y describes thl' tr~je rfJRpOnSe at every noint l}
given
('.lJos1dl':'ration h
t~l
E:
~ <!nd no
the plohli'm t)f design when thu l\('SA1bl1ity
(udsts that the incorrect modfll has been used.
HM<Yf.wer., 1.n reality •. it
ia quite possihle that thert! may be n"t en Iy one, but two,
~cur.::es
of
error, the first due to varIance (l:l,'lmvlinr) ert'o( and t.he second due to
the t"adequ.., cy
...
n(x;e).
~
('f.
the fitted model in
T~preser4tltlg
In a response surface setting,
measure of which 1s g-tven hy (5.3),
nltlV
thf~
the trUI3 response
1.lltler snurce nf error, a
far outweigh the fc·nner in
importance {'!!~'.8.'. see [2]. [3]. (8j, [91. [lOl. (11]).
Motivated by these remarks, this author llttempts to ('xtend the
concepts involved in the approximate theory of optimal design conatruction to permit the c.onsideration of bias error:.- as well as
errors.
ltalB
'rbe remaining
WQ't'k
in
t~lis
'lTari~mce
chapter is involved with this prob-
and import.ant appHcationR of the results developed herein are given
in Chapter III.
Now. let
e(~to)
represent one of the three functions (5.1)-(5.3)
sui tably nOl"ll1alized with respect to N.
In t.h:l.s rr9mework, we give the
following two definitions.
pefJltitton 5.1.
(5.4)
A design
.."
~
£ h is said to he a ~l~i~ax desisn Jf
19
f;l.juivalently, tS
'*
'is tt1inimax if
A dcaign 6
r~spect
with
to some
.. e 6
is said to be a :weighted..~t.:£!: des.!.8l!.
..
w~ight1ng
~
func-tion w
¥y* if
' \ ..'
0I\: minimi?t's the
.
quantity
(5.5)
A dcs1nn <5
.
.. c h satisfyinr;
(5.4) ....111 b'J C3Ut~tl
3
minimax vuri.-
ance (M\f), a minimax bias (MB), ur a mtn1.JJw'lC mean··square (HHS) design
depending on \\'hether e(~.o) has th13
t'el!lpectiv~l.~'.
A design.s
• r.
a weighted variance (W\'), a
(WMS) design if
~(~,~)
L~
(5.2). (5.3), or (5.l),
fOr'll
ndnimizing
w~iSthted
(5.~,)
has the form (5.2), (5.3),
Den~tion 5.3.
wei~hted
desi~n
error
A we1.ghcing function v,*
favorabl(! if MJ.N"""A e(w* ,0)
-_.-
O.. L\
ref~rred
to as
bias (lol8), or a weighted mean-square·
The following decision-the(.retic notions
a. situation in which a
wHl be
~r
(5.1),
~llHr.!e Un
r~spectively.
to characterize
is a1:3o a minimax design.
~'W* is
said to be least
.. MAX A.,M1N.r... C{W.O).
WE; "".
'1~.1.\
A least fl1vorable we!ghti11g fUHctJem is ch"l':Jcterized by the
fol1owiIlll lemmA.
Len~~5.1.
Suppose there exists a
* e:
Then, 0..
a wei3ht.ed erT'C'r design 0
> MAX
-
~
xE';;iIt:O
...
e(x,o *).
...
f:.
weightin~
Wit~l
function 107*
r.espect to
107
..
e~~ and
* *
such that e (w ,0 )
is n minimax design, and the inequality
above is actually an equality.
Pro£-!..
The conclusions of the lennna
follow df rect 1y sinr.e t for every
* *
• ?
~ e b, MAX:!e* e(~,c) ~ e("" * ,0) ~ ~(w JO ) ~ j>fAX::F.:.* e(J$.15 )
e(w
*to.. ).
20
i.;l?l~1
l1w conditions of Lemillil 5.1
~ieighting
fav·)r3blt.'
<:
-
is a l ..~ast
". E:'\V*,' HIN
*
*.
6c
l.\ e(w,6)
~ <H.A.X ..
* <. e(w ,:5)
k
c(w,6)
:'MIN~ . ! ..- (w ,6), so that
v e(x,'5)
~~~
0El
*
~
MAX eJ\.\ ~1JN,J.'." e(\\',ft) ... ~~lN\
Ill·
r,,:*
vt~.(.,
w~
now prove
5.2.
Lt'lIll1Vl
SUItPOSt! <5
* e; '\.,
yv.*,md
~,
e\1f~ry ~
for
dw ,0).
A
lJ E...,
some w
* c t.
is a \-leight'!d m·ror
)I(
SUPPI;tS~ e(~,o) me
.'Y~
f,::c.
'Ihen. IS
*
j
<
+
'»,
So, e(w*,~ '* ) •
J:v ~(x,6 *)
~
~
C~
* (x)
-
d~~si~n
'tiith respp.l".t to
C a const:.mt indepenc1ent of
*'*
s a minim,:lx tlesi~n and c lw ,c5 ) •
Since e(:,~'* ) ... C for every ~ r.;~. we
Proof.
...
f'~)r t~very
f.unrtion. since.
*
that w
~ C ~ MAX
-
MAX~e~I!(~.O*) •
III
v- e(x,o ).
..XF:'~
follows by appealing to Lemma 5.1.
h·1VP.
c.
c.
The result then
In Chapter III, an important situation will be presented where
Lemma 5.2 can be put to use.
Before proceeding wit.h the next ec:ction, let
Ut3
briefly discuss
some of the more important results that have been ohtalncd in the field
of optimal design of regression experiments using the approximate
and exact theot'1es.
As mentioned earlier, almost all the work done in
th1s area has involved the use of a polynomial-type m\."dnl where, in the
notation of Sect1.o~\ (4.1), thE' {fj(~)};=l are ~J1. the functions of the
k
rj
form ITj=l x j
the relation
Sptlce.
llot~
for which the {rj } are nonnegative integers satisfying
I~.l r,1 ~
d end where,* is a subset of ~, Euclidean k -
that. in thh case,
fJ ..
~:d).
Some elegant results blive been obtained in the framework of the
approximate theory when the regressi')l1 roode ls are polynomials (see [19]
for
11
area).
general treatise on the concributloM of wrious authors in thb
For example, it has
~'~e'l
shewn for p(tlync·mlal reRresdon of
. I.,; ...
1
order d that, when k .. It and:t.S "" [-1.1J. MA.'{ "f 1 1] 1:'(x)M- (o)f(x)
X~
j
-
s m.tnimJ zed Ot', b.Y' the Eq ui valence The,,! em c f 'Kiefer nOll Wo!f(jwitz
[~)J,
..
-,.
IM(6)
I
is max:1:u1zed by that 6* E: to, concentrating equal weights
?
l/(d+l) at th~ zeros of the polynomial 0-,,"') i'd(~)'
(D'1signs sotis-
fying the f.irst eri tart ,:.-n al)ovc nre said to be C;-optimnl and those
which satisfy
th~ se~ond
Kit~fer.
are D-optimal).
gent~rO\l result which shows when ~.. {~:~' ~ :: ]}
itt [21], proves a
(the ulli t
k-sllh~re) that
!IS rotatabl~ .~esi..sn.~,
some but not a.ll of the class of designs known
which were originiilly propofuJd by Box .:Iud Hunter [5 J and then cons Idl."red
by others
(~.&..
[l], [14]), arc D- and G-optimal.
In the framework of
the approximate theory, thls work by J:iefer t'EpresentB the only attempt
tbat has been made in the liternture to justify theintdtive appeAl of
these designs.
This author r,ives a Rflttle.what related result in Section
9 which characterizes
!2!~able ~r~~~~~.~ of
points as optimal
designs for F. S. and S. H. regression.
Using the exact theory, several aut.hors
~ave con~idered
the
problem of the construction of optimal regression designs when the
degree d]
degree el
2
of the least squares polynomial is exactly oue less than the
of the po1ynom.f.al repruseilting the true respl'nse.
[7J
both David and Arens
For eXRmple t
and B,x and Dr,1ft'r [2J consider the problem
of con:itructing WMS deRignR with respect to the uniform weighting fUl1ction on [-1,1)
~'hen
d
1
'" 1, d
Z
.. 2.
construct MMS designs in the above
The fvnner pair of authors also
s~tting
and are the only ones to do
so, mainly because, for general d and k, thp. conr.truction of MB and HMS
designs for pol)Ouomial regression
appN!Tf?
The latter pair investigate the l'ht':f ce cof
alou
011
t.
the un! t lc.-sphen., whn\ d
t
to l'te 'luite a formidable task.
rl,,~'d~nlJ
fC'T
polynomial regres-
... 1. d.2 ,.. 2 1\1 (2 J and ""hen d1 .. 2.
22
02
II
D ~;ellt:n11
3 in (3], by first finding
sl'!lectlng from
theSt~
a subc.lass C,r which the noo-c('nt'l;'<lLity term in
the expectation of the residual
.
~l~
of
squar~~
LawT,'ence, in [8], ['I] and [10], find \m
and d
l
das.3 \>.1. \~,.rs d~s:i.~ns .md then
"'" 2. di .. 3 whert>
:6 i~
d'~Rig[lS
is large.
when d
1
Draper Rnd
.... J, d
Z
• 2,
thn~e­
thp. uni t k-cl.lbe ion [e}, the
d:f.mensional simplex (equibternl triangle) in 19}, and the fourdimensional
8
implmc (tetrahedron) 'in [10], and,
tlf.!
clo the authorg in
[2] and [31, they show that the designs approi,riate \olhen hoth variance
l.md bias oeem: are
~H.'t V~l'Y
di ffer~llt frolr t!lesc "nll-bias tt c!es1 gns.
In (l1J, "3equential second-ordt'r \iB designs ara C".{'Mtructed
}loth t.he first-order portion and
l~le ~~cond-ord~t'
tion againot hiases of one order 'higher.
w~ighting
function, the
distributio'1
:1
wei~ht
~uthors u~e
Hen~.
{Jr.-rUon
~o
~i.ve
that
pTotec-
rathe:' than a uni.form
a family of syMmctric multivariate
funr:ttons depending on a single parameu'r to find
series of optimal Ilesicns for polynomial regression.
;.
Lt:>t Y(~;~l) ,.. !(l)(~)~1 and Ey(~) ... H(~:~1!~2) • !(l){~)~1
+
!(2)(~)~2 denote the forms of th~ ~stimateu and tru~ responses at ~ c~,
respectively; t.he vectors !<'t)(~)
f '2)(x) • (f
- t. ' "
M
..
(fl(~)' f 2 (:), .... , fm/~» and
+l(x),
f
,(x), .•• , f (x»
...
m~ -to", ~
m"l ...
....
....
are ass\~ed to be known
1
the
?J
of fl'U t'h.nk.
Tht'n, if tJ.l
l1~re.c
t·,) t':lk.::.! ~'l fifo:
,
-1 I
c "1'-:'~0r"if the form OC 1X ) Xl~' "lhe're
1
t'
In
L.lw standard leait squares
;Y(~l)' Y(~2), ••• , Y(~N»
is the vector of ollf:erv.tt:i.llus, 'It follows that f:V(~;~.l) .. nl)(?5)~l
+
!h) (~M~2'
the mntrtx A :: (:{i;'\ 1) -1 XiX:>. helng cOl'Il1l1only referred t<)
ae the "alias matrix",
Xi>~2 •
Ii'''l ni~ (1) (~i)
£1
0
£(2) (~i)'
at + !i?2'
A. then Eb i •
E21 •
when~ XiXl ..
ri"'l ni!(l)(:-::i)
(Note that if
i • 1, 2 •••.• m ;
1
11\
!(l)(~.t) and
!! iA
1>aJ:f:i~ulnr, if A • 0, then
)'
So, {rom these results, it is
t1squarcd b:faF."
n:;(~;~l)
-
~te~rlhdt
we
write thp.
~an
Tj(:;~1'~2»2 a~' [(fCt) (~)
A -
which is a fUIlC'tion of the choice of thp exact desigLl
the
l1UH rix
Now.
it is
the l-lh row of
!(2) (~»~2]2,
.l) .Q'!.).Y..
through
A.
s1.ncl~
me;m1n~ful
A can be written in the R>.Jggest:1.ve fonn M(XiXl)-l
tC' work in the more general setting of the approximate
theory and to conaide-r minblidng the integi"<1tcd -.qIJared bias
J Bo(~;~2) dW(~)
by choice of 6 Co /)., where w € ""'3Eand
*
2
Bo(~;~2) • ((!(l)(~) A(6) - !(2)(~»~21 ,
A(o) • U~~(tS) M12 (o),
(6.3)
and
(6.4)
~ XiX2'
....
24
the ge:n.~ral express"tllo for m (6) ht'lng Fiven hy (4.1).
ij
.lrt!
Notfl that we
taking HU(O) to bp. non-:i~;'i:~\llar.
So, pro.::p,eding tn this \"ay I
Wf:
have
n6(W;~2) ~ f~ ~Z[A'(6) !(l)(~) - !(2)(~)lf!il)(~) A(~)
• ~i r
with All·
and /\22·
~2' 1¥'heru
fil . !(l) (::)
r •.~t(li)lil1A{O) - r.it\(O) -
!ill (~)
dw(~).
-
!(2)(~)1~2 dW(~)
A'(6)"12 + /\22'
A,..\.4... '"' J""f(J')('()
clw(x),
...""f('O'\)(~)
t.
..,
..,
r
:£
J !(2){~) !(2j(~) ~w(~).
~
singular, we can write
Thus, we
~ave
the follow:i.ng t.eneral
regtllt~
no matter what the
value of ~2' B~{w;~2) is mtnimi?ec hy choice of 0 c ~ whon A(f,) •
-1
AllA ,
12
and,
ill
parti.cular. Wh~l1 M11 (·:n
...
All
aUf}
~112(O) -.. j'I2"
In fact, from tho structures of the pairs of mntrtccs M11 (o), All and
1'1
12
(0). 1\12' it follows dirp.I:t:1y th.'\t the latt'et' two matl'i,x: equalities
15
ahove can be satisfied by ch(,11ce of 0
~clghting
:;in{~(;'.
funct.ion w itself.
(C'It., it is chosen to be uniforil1
m~~cessary tt)
.
in ~ (see
f: f,
if.
t,
11$ taken to bt." th(·
in gt'neral, w· is flat disc-n·te'
o,,:f::.. tn many
instanccR). it is
find n .mensure concentrated onl>· on a f:lnite set of poinLs
part (tv) of Lemma .'&.1) which is equivalent to w in the
scnf;.,"! that the pair of l'J'Ii1trice$ AU and "12 lire tdentical fo·r bot.:h 'v
and
j
ts discrete (;ounterpart.
Thh result follows
8S H
direct generali~atl(m'of work by n"x
and Draper In (2], \-,ho restrict their attenCicfJ to the conatructJon
of exact wB desir,ns where tht" fittp.d and true models an, polynomials
and the
assod.~ted
weighting funcUon is always uniform on the
region of interust ~.
~iVCll
The new t reatntent given here allows one to
consider more general types of regression functtons for both the fitted
and true linear models and enables one to d18T.3cterizt;! the WB design
with respect to any given w
~'0~ aa
an optiw81 fiJ
pro"Xima~
design.
That such a glmeralization 1.s meullingfl',l and necessary will berome
apparent bo:h in the next section and in the next chapter.
•
Motivated by the Eox-Draper and Draper-Lawrence
result~
die-
cussed in Section 5 vhieh emphasize the dominating influonce of bias
cfmdderfltions in the construction (If WMS designs for pol)nominl
regression, Hader, Karson
~nd ~anso~
in [15j consider using a
~ethod
nf estimation of the coefficients of the fitted polynomial which is
aimed direct.:l~' at mini.mi :.:ing lntegrllted stJuD.red bias with respect to
a unifoIl'Il weighting function, rather th.1n aSRumin.g, as we h3.vedl'ne,
tbat the coefficients must be estimated hy the met.hod of least Rquares
anti that the lntegrated squared
design 6.
~ince
a WR deaign is
bh~
is then minimized by choic'! of
Cl~'ta1nE'd
by choice "f estimator using
to' ..
•·
26
this now approach, the authors call and do use the
;\ddit'l~m;Jl
flexib:Ulty
that remains in the choice of theloeatirm t'f the desill:u points in
UB~,ng
to construct WV designs.
this tcr.hnique. they are able to find
designs witb SllIaHer integrated mean-square error than
tl}~S
'*'
corres~·onding
den1.cns that assume least squares estimation of the coefficients
in the fitted polynomial.
In Cha.pter III, it \fill be shown that the class uf optimal
df;signlJ proposed for th(! least squat'es fitting of the F. S. and S. H.
regress ton mode.ls minimiae integrated
variaD~e
simult 3!"leoulfly ODd lhus
nO
squat~d
bias and
intr.g~ated
"better" claf.l:l of deugns could be
found by using the H:.tder.-lVtrson-Ml.1nson "ninimum bias estinaation tt
technique.
7. 'The Admlssibility of a•. ~e.s£'0.t]s.e Surface Design
The approximate theory approach to thp- optimal design of cxperi. ment8 88
described in Section 4 characterizes an optimal design 0
1\
c A
as one which mln:l.mizes some appropriate real function of the :Jnform.-
tion qatrix
M(~)
defined by the relations (4.3) and (4.4).
light, the standard concepts that have
80
In this
far been developed concerning
the admissibility of an experimental design, (see [19], pp. 808··812)
have been based on tho following defInition.
~~J.OD
~
7.1.
A design ·S c A is l1aid to be ~~!.sible if there does
exist a design 0'
E
t such that M(6')
signifies that the matrix
M(~')
~
M{c), where this inequality
- M(o) is positive
8emi-d~f.inite and
K(6') j. K(5).
It is Important to realizE'! that this notion 9f
~1ssibility
is
based solely on vadanee considerations and :Implicit in Definitlon 7.1.
21
is the aSHumptioll t.ha t the
1CSpOnAf'! a r: evt>ry point
:~
C
for the posatble ocC'urrf:!nc.e
inCt1rrt!::t modpl.
*;
l:lode 1 (If.
rei' l:p.~(~nts the. true
,in other 'Words. no allCt'1'inCe is lIIade
nr
bias ern,rR
re~ul tit1g
However.. Buell as assU'llption is
in a response surface setting where a
polynomi."l will alwnys fail, at
l:eApons~
~':i'.'.:l.~
1)
surface.
!.lQt
f-rCl'll1 the use of an
pndrely realistic
grHc'JuC)t.I ng function SUdl as ~
l~ast tf'J
Home E')j;t.ent. to d,!scr1be a
'(he neec to consider on alUrnative to the approxJ-
tIUlte theory concept of admissibil:itj :,a
{~Xl)r.es$~d
in DeUnition 7.1 is
even further m!lphus1.zed by re,qults mt:tntJr.med e.,rlier. ([2], [3], (8}, [91.
r10],
[11]) which illustrate in typical s1t'tatiOlt.9 the overr.tdlng
importance of bias considerations when snIp.cUng a response surface
design Co minimize
lntag~3tp.d
mean-square error.
Motivated by these considerations, we
no~
proceed to define the
new concepts of V-, B-. and MS- admissihi1ity, which have particular
appeal in a response surface framework.
let
e(~.6)
Tn the notation of Section 6,
denote any of the folluwing three functions:
(7.1)
(7.2)
and..
(7.3)
where
n&(~;@2)'
II
-1'0 en , we
~inition·1.2.
A(6), M11 (O) and M12 (oi are
t1v~n
by (6.1)-(6.4).
have
A
d~sJ8n
0 c 6 is said
t~
be variance (V-), bias
(1->.. cor mean-square (MS-) adll:i.ss1t-le, depending on vhether
e(~t6)
is
equal to Vo(~). B~(~;@2)' ot' N~o(~;~2)' reape1;tively, if thel'~ doe& .!El
..
23
mdst a design t;t (;
e(~O,6')
<
e(~O'o)
l!,
ftucl! th"'l.t
for SOmp.
e(~,c.I) :~ e(~,0)
for
eVHV
C:t and
~O c~.
It 1s e3sy to see- that the prnrwrties of V-. B'-,
b:Uity are certainly dll'siral.le ones fer <1
possess.
x
r.~::lponse
~nd
NS- aclmissi-
surf'ice design to
In fact, one shnuld fee! sc·melo'!lat disaproint<:,d in learning
that a response surface design which minimized some meaningful function
like
max~mum
or integt'ated mean-square error. say, was not MS-a.tm:f.sslble.
Unfortunately, such a situation is not out of the realm of possibility;
for example,
sari1y B-
...
The
Ii
design which mini.mizes maximum !llquared
bia~
is not neces-
8clMissibl~.
f~.,llC1d.ng
theon·ro. a tal{e-off on a rf!.slIlt in (13], p. 62,
attempts to shed some llght en this rroblem hy
C'harl!ctet'izill~
nn 1mpor-
tant situation 1.n which a response surface design can he said to
pnsse.ss one of the types of adl'1J.e·"'ihili.ty it.trod.uced in Defjnit1(ln 7.2.
First, a preliminary definition •
.Q!~i.nitj.on..L:l.
Let
is said to be in the
*
be n nletric space wt th h1Ct.riC d.
~ort
of a
wci~ht1ng
functi.on w e::
every £ > O. the neighborhood Nc(~O) ~ {x t~:
r
"positive weight" tn the sellS"'~ that)
dW(~)
A point ~O e::
'h>x if,
*
for
d(~,~O) < e::) has
> O.
NF;(:O)
Then, we have
~HEO~EM
7.1.
Let
~
be a metric space with metric d dnd assume that
...
the functions (7.1), (7.2), and (7.)) are each cont-Lnuous in x... for all
Then, a design 6* e: 6. is V-. I1-, or MS- admhsihle depending
on whether e(ji,6) h8S the form (1.1), (7.2), or (7.3), n:·.epectively, if
c5
" minimizes
*
e(w ,0) . '
1
*
e{~,a) (h, (~) with respect to Borne
~
* £ '\..yy~
,
w
29
and if the su-pport of w* i6 ~
~!',9..~f·
.
Assume th::tt (,
'Inaa. t1u_re
~ F;;t and
whi~h n
for
e(~.tS)
ueeause
t
) -
e(~o'~")
>
~
''W
-
e(~O'o ')
-
.....
> 0 for som'!
~O
£;
*.
- e(~O'o') I
It
lnus, for ~ c N&(:O) , e(~.o ) - e(~,~') ~
-
•
J(e{x,o
' " ) - ..
~(x.cSt»
~....
e(~,ot» dW*(~) ~ ¥
N&(~O)
!o
7.2..
n/4) ... (e(!O,d') + n/ 4 ) • e(~o,oit} - -a(~OfcS') - '1/2 • n/2 > 0..
(e(~,o*)
f
Deftnit~oa
for \.i\i.ch e(x.6') < e(x,o ) for all
Rence, we have e(w""
,~ ) - u(w*
,6, )
.
of
III
'\' I '
o(~O,o)
~ ~/4 au~ le{~,o')
Ie(~,t).
n/4 if ~ & N£(~O).
(e(?50,l5
•
I: ~
U~J1se
IB t:Ol1tiuuous in .!. for each tS c t:, there exists an
£ > 0 such that
~
" is not admissible in thu
exist a 6'
1tl\llft
1tSf~lf.
J
dW·(~).
4wit (x)
.
~
But. since
N&(~O)
is in the support of w*, the last
greater than
ZCI'O.
e~p~cssion
above is strictly
This cCl'ntr.:\dicts the assumption that 0
" Ulin:lmbes
"
c(.., ,0) and completel» tbe proof.
It is clear from tbe
structt.1r~
of. the fuuctions (7.1)-(/.3) that
the continuity 8asumptions in Theorem 7.1 are not the least bit
restrictive, and, in fact,
obvious~y
hold for the standard regression
models such as polynomials and various tr:J.gonOllletric functions like
(1.1) and (1.10).
Theorem 3.1 pro'ddes grE-at incentive for constructing and using
responae surface designs which lllin1mize a given Dleasure of error bo;ween
the fitted and true l1IOdels tllat is ave:t'aged rather than, say. IIlUblized
over ~; in particular, it l~nds support to the work of Box and Draper
and Draper and Lawrence and shows that their WV, W, and \-,1118 designs
are V-,
]1-,
propose for
aad MS- admissible,
F~
r~Rpectively.
8. ancl S. H. regl'essiun will be
and HS- adaissible).
(The designs that we will
!imultane~}1s1.I V-,
B-,
. i.
.30
Finally, it is i1"lportant to tL'.lntion tho face thnt aUl1llssibility
in the ~:ense of Def.iuinon 7 • .L Is 'l$tr.on1~er" t.han V- admissibility
.
since a design 0 £ A ".thich is acJmi.ef::lble i.n the former
v- admissible, while the cCJn-verse
..
:£8
SentHl
not necessarily true •
is clearly
..
.
.
OPTIMAL
DEStGl~S
FOR F. S. AND S. 11. REGRESSION
Working in the framework of the approximate theory and 'Using
the results in Chapters I and II. we will, in
..
t~l,1e
section, ohare.c-
• and 6* for F. S. and S. H.
terize tIle optimal des1gns CiS
sa
re~ressiont
respectively. of order d in terms of their corresponding infClnnat..LOQ
..
..
matrices M(Oyf,) and M(SSH)'
The mavy optimal propertieQ possessed
..*
J:*
by the approxiroate designs 0pS and usn will be ewphasized.
important question of
wl~ther
The
"
It
0FS
and 0SH
correspond to exact
in certain cases will be answered in the
n~t
dQsig~s
section.
Now, for the S. U. medel, we define frOll (1.14) tile quantit1es
f
2
~1/2
\[t<n+i»
I
cos~ Pnl(cos6),
2 \ 1/2
..• , \(2u)l~
n • 1, 2, ""
I..
2
~ 1/2
,i"(n+lj)
sin41 Pn1(COS&), ""
I U
cos n~ Pnn(COsO) , {~2n)~
d.
1/ 2
J
sin D~ Pnn(cos9)
t
32
Correspondingly, ",e let
{nJ
~SU
be thl.l vector of elQl!lent:s of
witb suitAhle 'multiplierti Attached so that
..
i~ parti~u1ar,
( 8.2)
lin (e,~;es(:)
• ft[n)(9,;)p~nHJ;
... ,,'"....,
...
~
it follOWN from (1.15) that
B[O]. D(O) •
...Sll
eSH
a . a[n)' -[a
N
,
/Un)j\
1/2
;. -) B
(
\~1Dg
"'nl' \: 2
1..1f!::!:!!'lD1/2
n~t \2(n-m)ij Bam'
···'~(n·m)1
]
nn
J
In~n+l~ 1/2a
ll}t1l.+.ll) 1/2
nO t~: 'J
00' ...58
L(n-hn) !')Ina.
So,
~(n)
£~B
'"}
!(2n)
···t \
2
'nl'
i\ 1/2
I unn '
n • 1, 2, ... , d.
(8.1) and (8.2), we are able to
w~it.
It follows from the addition theoram (3.9)
(1.10) a8
t~at
, (8.6)
and bence that
(8.7)
The
~ations
(8.6) and (8.7) are an31ogous to (1.8) and (1.9) fnr the
F. S. regression model.
First, coneider the probln of ftndtns WV designs for F. S. aud
s. H. regrea.ioa of order d.
If the model to be fitted by least squares
33
.La of the geneTal forra (4 .. 1),
th~n,
in thf! frnmewnrk of the approximate
.*
tb1iory, 11.: iii desiTed to Und a meallure 0
2 Vo(w) •
f f'
I~.lWi~(~1>!.'(!i)
and '"
'luantity
L
(1
.
'"
:t
y~ V6 (w)
ven.1ent to write
(':
.
E:
A ~'itlch m1.n1r.dzea the
I
. N(~)
(~)M-1 (.s)~(~) d,,'(~~' ~'l\ere
&t\y~.
tn what !olJ.ovs, we shall find it
as - N2 V6 (w) • tr[M- 1 (o)
t1
Nw, for S. 11. :(".&1.'.881011 of
Ordt~T
J!IE!(!)
!t(~)
1;011-
clw(!)].
d J ve wi&b to find a cteo:l.att
where
and where the inteK!"aticn call be considered to be perfomed either with
respect to the
m~1form welghti~
with respect to the
wei~ht'ing
rectRgle {(~.,): O:i 6 ~
.!!2!!.:
iii
~'fact
*
""
the 6urface of a sphere or
w~at(O,¢I)
• (1-C090),/411 on the
U ~ ,t ~ 2'11}.
'It
f
(Of "1) wi.ll not affttct the charactertzat:1on of the
More will be !:aid ,lbuut this later in the aection.
'rom (8.4). it is easy
..
fum'Uon
OD
that we are loIork1:ng uith .M (0811) instead of M(iS SH)
I~.l"i £(6 1 '.1)
WV design .sSU.
'R,
function
til
see that ...f lit (e,f) ...f '* '(9,,) 1. a
(d+l)2 )( (d+1)2 block matrix with He typical clement being the
(20+1)
D' •
x
(2n'+1) matrix !(n) <8,,)
0, 1, ••• , d.
!(n,]<9'9>'
n • 0, 1, ••• , d.
Using (2.2), (2.8), (2.12) and the o~thogona11ty
34
propertieR of the "spherical IHlrtnonic sequence" di.scussed in Sectl.on 2,
. we can write
0,
1l
ntn'
sine dad,) . . .
(S.9)
,
(2;'1) 12n+l·..... •
where In denotes the identity matTix of order n.
So, from (8.9), it
follws that
(S.lO)
where
(8 ..11)
'It
[11111111
DSH • diag 1; 3'3'"3; 5'5'5'"5'5•
Thus,
N
'2
V
cS
o
.•. ;
]
1/(23+1), ••• , 1/(2d+l) •
'It
sa
(w ) is a function only of the diaBollal elements of
SH
Similarly, for F. S. regression of order d, we are searching for
where
(8.12)
and where integration can be looked·upon
to the uniform weighting
iun~tion
•
w (4!')
rs
a8
III
being performed with respect
4>/2Tr either on the circum-
feTence of a eircle or on the closed 111terval [O,2Tr]. By appealing to
(2.2) anel to the ortbogonality prOl)ertieli of the "Fourier sequence"
and by noting, from (1.6), that
!(~)
!'(¢,) is a (20+1) x (2d+l) block
35
matrix witht v pical clenamt L
"
n' .. 0, 1, ... ,
-
u,
,(}) f~ , ... ~tt·)J
~.n)
-\.11} .
{'If
J
10,n 1 o·
!(u) (¢)
f(n t) (In
o
..
an.d
benCE!
0,1, .•. , d,
r
it follows that
2n
(8.13)
t).",
d.~ •• ) l~ n ... n r
....
0
)
'l
,,1 n-n'>O
l ,tI
. l" ..
thot;
(8.14)
where
.
(a.1S)
Note that (8.14) involves only the diagonal elements of M-1 (t5
)'
FS
The implications of the {\')llow:!ng two remarks wi 11 be quite
useful.
!emark 8.1.
The set of (2d+l)
i.nt~gel."s {O, 1, .... 2d} can
be par-
titioned into (d+1) disjoint sets AO' AI' ••• , Ad' where AO • {a}
~nd
An • {2u-l. 2n} for n • 1, 2•••• , d.
~~ark S.2. The set of (d+1)2 integers fO, 1, •.• , d(d+2)} c~n be
partitioned into (d+l) disjoint sets BO' BI , ... , Ed' where BO .' (0)
and Bn • . {2
n ,n~ +·2m - 1 for m· 1, 2, ••• ,0, and n 2 + 2m for m. 1,
2, ••• , oJ, n • 1, 2, ••• , d.
Note that B has (2n+I) elements,
n
n • 0, 1, ••• , d.
given by (8.S), then the form of (S.4) and the. partition dbcu8sec:l
in Remark 8.2 allow us to make the following identifications:
(S.16)
36
(8.1i)
.... C
'''0')
\. 1""
,,-, P • (""~""')
[...
".v.I>';.
:to n
.l
Il1'll 1....
C(,,~
.d
. i ,"
v_ ",
....
1
Pt'_'lt'(~o
....._t!.:l,).
"0,,,..
\1
"
(S.lS)
I i) si n .......
'. C1lIll"'J.e::
)~ 1."4.1. 'Po' ("'06
("o... t:;,)'
....
.
.,''t'i f om....,c
i t
J'
(S.l!1)
• Cnm C,
I
,(cos9 t ),
Li • IV,!. cos m¢i Pnm (cos0 1) cos mt~i Pnm
um ,rP
(8.20)
."
(8.21)
~ 2
?
n +2m.u'''+2m'
In a
simila~
(OSH)
manneT, a labeling of the elements of M(OFS)
• «mi.j (~FS»)~~j.O can be made by using the. partitioning of Remark 8.1
and by notin¥ the torm of (l.6).
following relations bold:
(8.22)
(8.23)
In fact, it is easy to see that the
31
..
(8.24)
lUO,2n(6F~)
(8.25)
m2n-l,2n'-lCors)
1';1
I?1.-1
,. I P:t"'l
.
~
\\1.
J.
fan H'l'i'
to:
eo!>
i
n$1 cos
n'9 i
,
m2n- 1 ,2n' (oFS) • ri.lwi . cos n¢.l. sin n'~i'
(8.2.6)
m2'11, 211' (oF~)
(8.27)
:For notational
• 1-1-1
yP wt .sin 1\9.1.
sin n''it
r
'*
\\Ie shall 30metimcs write mJ.,j for
~onv,-,niencc,
'*
mi,j(oSU) and mi,j for mi,j(oFS)'
Using (8.16), (8.l9), (8.21) and (3.9). we can show after a
little work that the following rclattons hold among the
..
..-
of U (6
dia~onal
elements
sn ):
* • 1; m*2 2 + £m_l(m
t n ·2
(8.28) MO,O
2
+ m*2
2 ) - 1,
n ,n
n +2m-l,n +2m-l
n +2m,n +2m
n .. 1, 2, •••• d.
And, from (8.22), (8.25) and (8.27), it is obvious that
(8.29)
mO,O· 1; m2n- l ,2D-l ~ m2n ,2n • 1, n • 1, 2, ••• , d.
Now, the form of (8.11) permits
UR
to express (8.10) as
(8.30)
-1
Sind.laTly, if M
(~FS)·
i,j
«m
(~FS
» )i,jaO'
2d
then. from (8.15),
we can put (8.14) in the form
(8.31)
!- V
2
G
~
( . ) _ 0,0 +
WF~··
\).~
FS
(2n-l,20-1 + 2n,2n)
2"n-1.
m.
~~d
1
38
From the structurus of the ':!)o;;prt:ssions (8.30) find (8.31), it
follows that we can minimize
N
-'2
o
"0
(w
Sk
*
SH
N V (w * ,) by prupcrly
1r
0- oFS
~
and~"";j
)
*-1
specifyIng the values of the diagonal clem;mtl:1 of M
ltOlvever, the optimal choices for these elements cannot
r·.Hlpect·lve1y.
be made
due to the relations (8.28) and (8.29).
fr~ely,
restrictior~into Account, YO UR6 L~grange'a
minimize by
(osn) and M-1 (OFS),
prope~
To t&ke these
technique and attempt to
clloices of M* (oSH) anJ M(OFS) the quantities
2 2
*0,0 + ~d
1
*n ,n
(8.32) f(oSH) • m.
LO"'l (:~n+l) [ m
+
I:'1 (m*n2+2m-l,02+2m-l +
i1'!
*n 2+2m. t1 2.J.-.'2111)] + ~Oif (D'lo,o-l)
*
*[It
ttd
\*n
*
.
+ Ln=lA
n mn 2 ,n 2 + ~.l(mn2+2m-1,n 2+2m-l
..
+m2
]
2) - 1 ,
n +2m,n +2m
and
+ L~du-! ~ n (m2n-1.2n-l + m2n,2n
- 1)
t
"
respectively.
The A 's and A'9 appearing in (8.32) and (8.33) are
Lagrange multipliers.
Before we proceed with the
the
foll~'ing
~t"k 8.3.
result wUl be
J.et A •
m1ni~tzation of
ne.~ded.
«aij»~,j"l'
taken to be symmetric and
fC6 sn) and {Cops)'
A··
l
non-sln~llar.
...
«a:tj»~,j.l' where
A is
-1
h"omA A-I ,wehave
D
39
Thus. it follows
t~at
r-l\E('X(l'
'.
0.
a.\I3a~ti "" -A('JA- l n;}J,f,) A • ~
;;D
13
lI-'A(E~n"+E,,pG)A,
where 'E
aB
is t.he
n)~n
Using
(~.32)
a
Hore specificsl1:.-. if ~~ • (Gal ,sea' .. · ,a
represEnts t.hi! atl' row (·f A, then 'aA/'da
-Scd 8
~
matrix with a one in the (OI.tB)th pos1t"lon and
zeros everywhere else.
48ijtaaatJ. •
a
I.:tC
aj , i • 1. 2. _... n t
<: -
j
0:
un )
~~ ~~, or
1, 2, ... , n.
and appealing to RemaTk 8.3, we have:
.
... r::.l
(m";
,2+ , 1 ... m;r~
2
>],
n.n 2m n ,n' +2m'
n •
.
(8.35)
'/(
O. 1, ••• , d;
*2
-).Om 2
n +2m-l t O
~d
* [ *2
~n'
*2
). t m
2 + 1..' (m.
n -1 n n 2+2m-,n
l'
m -1 n 2+2m-,
1 n ,2+ 2!l\ , - 1
- L'
+
(8.36)
m*;n +2m-l,n '''''+2m'
~ )]~ m· 1,
2•••• , a,
.
n• 1. 2•••••
dj
40
Equating the (d+l)
2
partial derivatives (8.34) - (8.36) to zero and
Rolvtng the resulting
of
~~t
Jlon-linea~
equations in conjunction with
the (d+l) relatiuns giv3n by (8.28), we obtain the following solution
* in
wbich characterizes the WV design 0SH
tor~s
*
of M*(oSH):
* • 1; m*2 2· m*2
(8.37) .0,0
2
• m*2
2
• 1/(2n+1),
n ,n
n +2m-l,n +2l:n··l
n ·+2m,n +2111
... ,
1, 2,
m •
if
An •
and,
Similarly, it
n • 1, 2,
11,
it
• • • J
d; IDt,j • 0 if 1 ; j;
(2~1),
n • 0, 1) ••• , d.
f~llows
from (8.33) and Remark 8.3 that
..
(8.39)
3£(0
(8 •40)
v uYS
FS
~f(~
)/~2n-l,2n-l. 1
2
)/~ 2n,2n
om
•
21 -
_~
2
fd
A (2
0 -2n-1,0 - [,n'-1 n' JD2n- l ,2n'-1
•
2
td
'( 2
+ 2
)
AO m2n ,O - [,n'-1 An' m2n ,2n'-1 m2n ,2n""
n .. 1, 2, ••• , d.
As before, we set the (2d+l) partial derivatives (8.38) - (8.40)
equal to
~ero
and solve along with (8.29) to obtain the specification
of 6ifFS in terms of
*
M(~FS);
(8.41) mO,O • 1; m.2n-l,2n-1
namely,
= m2n, 2
·
14
if i , j; and, hO • 1,
~n •
1/2, n • 1, 2, ••• , dj m~~
• 0
....,
2) n • 1. 2, ••• , d.
To conclude, 'ns we have dene, that
~he
*
I':
designs 0SH
and O,S'
*
1\
characterized in terms of their information motTices Mit (~SH) and M(OFS)
41
by the soluti4,lns (8.37) and (8.41). r03pectivcly, are :f.ndeed {IV
designs, it :i.s
neCF,:i~sa:ry
are minimized,
t'ubj~ct
to shew th·'1t
th~..?
expression1] (8.30) and (8.31)
th~ 't".~strict'.Qt'l.S
to
*
(8 .. 2Jl) i.lnd (8.29). at
aSH
if
and '\'S'
Let the (d+l) 2 x (d+l)4"
To do this. we proceed as follows.
..
..
d(d+2)
~ave. as its
matrix FS11 .. «fij)}i,jaO
.*
L
ij
In
.2 .
(d 1( 15
. *i 1
sa )/3m
tive of f(oSH)
.
~
(lm
j
li': j
the solution
*.
'Ie
(!.~.,
) (10\.37)
evaluat~d at
typical eloment tll'l quantity
the sec.oud
partial deriva-
(8.31», and let the (d+l)2
d(d+2) d
x (d+l) matrix GSH • «gin»i-O,n.O be defined such that
..
2
*11 *
Sin • (0 f(oSU)Jam
a"n)(8.37)·
Then, a necessary and sufficient
N V (W* ) to be minimized at 0SH
* subject to the
condition for -;
6SH Sn
(1'"
conditions (3.28) is that the roots of the d(d+l)th degree polynomial
in ).. obt:ainod by a.,.··q>sudini
• 0 are all positive.
(See Hancock (l6), Cha.pter VI).
Similarly,
N
'2'
Vf,
J
ita minimum at
*
~FS
(ow
FS
#I
FS ) achieves
subject to the conditiQns (8.29) if the roots of the
.. 0
dth degree polynotUia1 in ). obtained by expanding
G~s
ure
.ill positive,
where the (2d+J.) x (2d+l) matrix FFS
bas as its typical element f ij •
the (2d+l) )( (d+l) matr:ix G~S •
I;
2
element gin • (3 £(~Fs)/am
ii
0
2d
\11
«fij»i.j_O
<a 2£(OFS)/ami i ~jj)(8.41) and
«r...
.:1-
whore
»:l2dO
,d 0 has as its typi.cul
.~n'"
3).n)(S.41)·
4.2
'It
Now, lhe dt!te.l'udnation ofthu elements of t'le lr.atrlces lSHt 'pst
*
GSij and r·,CS is tedious but.
..,
st'tf1i~h-tfoTV.:lrd.
2
a
and ~}Wark 8.3, it follows directly that
For exalt.ple, from (8.34)
2 2 * ,2 ,2
f(~ )/&m n ,n am n tn
SH
*
~ * *
•
m*,2 + 2 iLn "-1
.A
O mn 2 ,0 mn ,22
,n n ,0
d
. .- ..
n""
."."
m 22 m 2
2
+ .) .". .I'mn2
2
,n" '+2nl' '_I n'.n n' ,n" +2m' '-1
+ m*2
2
n ,n" ..2m"
m~f2
l\
2 m* 2
,0
0'
1.
,ll"
'+2m"
>], and
80,
using (8.37), ¥e
C if n r1 n l
'It
have f 2
11
'"'
2·
,n'
~
for 0
n, n'
~
d.
It is just as
2/(Zn+J)2 if nnn'
* * 2, f 'it 2 2·
straightforward to show that, in aentira1, fO,O
11 ,n
."
I 2
"'l
.
~
• f 2
2
• 2/(ln+l)"', m • 1, 2, ... ,
+2m-l,n "'2m-1
D, +2.,n +2m
2
II
n • 1, 2, ••• , d, and f jj
•
a if
where D;JI is given by (8.11).
0t
'It
i ~ j.
*2
In ~'thcr words, 'SU • lDSli '
Also, from (8.34) - (8.36), i.t is
...
* • -1 and g. 0 • 0 for 1 • 1, 2, ••• , d(d+2)~
readily verlfled that '0.0
*'
land that. ror each f 1xed n, n • 1, 2, ••• , d, g 2 • g 2
n ,n
n +2~1,n
It
• S*2
• -1/(20+1)2 for m -.1,2, ••• , n and gin
* • 0 fur all
n +2JD,n
Thus. it follows that 2/(2n+l)2 is a r?ot of multiplicity 2n of
i.
.....
~ther
*
F -).1
SB
.t
*
Gsa
• 0 for n • 1. 2, •••• d; since these d(d+l) roots
G
SII
0
II
are all p08:t.tive. we e.an conclude that 0 SlJ Iii a WV design for S. 11.
regression of order d.
that
Similarly, using (8.38) - (8.40), we can show
2
'rs • 2Dys
• where DpS is
given
b~
(8.1S); a180, it is clear that
"
43
'0,0 • -1 and ni,O • 0 for 1 • 1, 2, ••• , 2d, and that, for each fixed
n. n • 1, 2, ••• , d, g2n"'1~n • "20,n • -1/4 and gin· 0 for all
other 1.
Sine. i t follow. f'roan the•• r••ults that 1/2 1s a root of
"
multiplicity d of
• O. 'oS,S is a 'WV dellign for F. S.
FS
C
0
rei;l:ti8sion of order d.
Using (8.37), we can then say that a WV design 6
regression of order d can be charact.erized as
~ue
"SH for S.
H.
whose information
.
* "'ltere M" (6 SR ) 1s given by (8.8) and
lDatrix has the fom 14""
(tS ) • USB'
sn
"
...
DSH by ·(8.11); also, from (8.41), it follows that a Wi design ~FS for
F. S. regression of oTder d has the information 'J:l4trix M(6 pS ) • D ,
FS
"
where M(6FS ) is given by. (8.12) and DpS by (8.lS).
We can now appeal directly to Lemma 5.2 to show that
"
~SR
are also MY designs, since, from (8.6),
fd
2
• Ln_OC2n+l) • (d+1) ,
and, from (1. 8).
Also, we must have
(8.44)
N
6la(w"sn ) •
~ V
(d+l)
2
N
"
:and eS pS
*
and tJ2 V6:S (v,s) • (2d+l);
(8.44) is obvious frca the form of expressions (8.10) and (8.14).
44
Thus,
have been able to characteri.e. in tenDS of their cor-
WEl
•
resl)ondtnginfom&tiou matrices, th•. optimal approx1lllate del1gns 6
SH
*
and. 61'S for S. H. and F. S. nar.8.ion of order d which 8J.mult.eoualy
minimize both a waighted aver4ge variance and maximum variance over
certain specified regiona.
The geaeTal di8cussion given in Section 6 will now he used to
•
Itt
8how that aSH and 6ps are also
~'B
In particular. 8uppose that
de81gns.
the S. B. model to be fittea by lea8t squares i8 of order d. while the
true response is l'ld' (9,4H@SU)' where d' > d
\Ie
~
1.
Then, in this setti...
know frOll the results of Section 6 that a WB design can be charac-
..
terizeel as one which makes the (d+l)
•
22·
x (d+l)
matrix
!1.1(osa)
(or,
*
equivalently, M (oSB» equal to A11 (again, no pt'oblem will result frca
•
working V1th M (oSH) instead of M(I$SU»
(d'-d)(4 '+d+2) IIBtr:1x
*
M~2(OSH)
•
and which make8 the (d+l)
I~.lwi ~*( ei'~i)
!:'
2
x
(6 i '+:1) teks the
•
form AJ:2; where M (oSH)' is g1vu by (8.8),
!;'(9,.) .• (!ld+ll(O,4!>' ...•
A~l
A~2
•
•
ir
lr
211'
J
o
n
J
!*(8,.)
!id'l(9,'~, .
~*t(9,.) sine
dad',
0
21r ""
f
o
f !*(e,~) £;'(9,,)
sine d9d',
0
and where the integTala are to be liven the 8ame interpretations here
as they were for the WV desian problem discussed earlier in this seed.on.
• .. DSH
* and A12
* • 0, wbere DSH
*
Now, it:1s easy to see from (8.9) that All
18 &ivan by (8.11).' Hence, the WB design for S. H. regression, which
is 0811 approprla"tewheB tbe biu ..nor 18 due solely to the fitting
Oty leutsquan.) of an S. R. lIOdel of too 8mall a degree, is
8 imp ly
L ..
4S
* , characterized earlier in this section, which
the WV and MV design 68B
. ..
* * ) • D* of
8a
sa
...
has its associated infoTmationmatrix M (6
"lerge enough
size" to insure that the orthogonality conditions 1.mplied by the matrix
..
• *
equation M12 (6 SH ) • 0 are. satisfied.
This last notion will be made more
precise in the next section.
~
'In a completely analogous JaaDnel', it follows that a
design
•
6FS tor P. S. regression haa an information matrix of the geueral
fo~ Dps ' since we are to take M
ll (6 FS )
21f
i; J
o
equal
!(.) !'C.) d~ •
=K(6 rs ) equal
D15 and M12 (6 FS ) •
!r.1W1
2'If
~o ~
Io
!<;> £.<.)
· •. , !(d') ('~.
Again,
Drs
d+ • O. where
aust be chosen
!B<.) •
80
!(+i)
(r
requirements given 'by K: (6 FS ) • 0 are fulfilled.
12
to mention that the WB designs
SH
and
6*
FS
!~('i)
(dU)
<;).
that the orthogonality
Ie
6•
to
It 18 important
also pORsess another
desirable property which was discussed in Section 6; namely. the
." *
*
* ) M (6 * ) and A(6Ie )
alias matrices A (oSH) • M*-1 (6 S8
rs
12 SH
-1
• M
*
."
(61'S) ~2C6FS) are null matrices.
Since cS
•
*
su anc1 6FS are also WKS designs in the
ab~e
framework,
it follows directly from theorem 1.1 that they are each si1lult8D80usly
V-, )-, and HS- admissible.
And, by appealing to theorem 7.1. p. 808,
of [19] and to the relations (8.42) and (8.43), we can show faiTly
*
directly that 6SH and
*
~FS
are also ac!missible in the sense of Defini-
tion 7.1.
* and 61'S
* that we have discussed
Now, the optimal pl'OPerties of 6SB
so far are relevant mainly in a reeponse surface setting and
th~t
is
why they are concerned with eBttmatlon of the entire regl'88s1cm function
. '"
46
~ather
than of specific parameters.
In what now follows, we will show
information m4trlces D*SlJ and D,S also possess several important properties which aro :f.ntimately related to the optimal estimation
that
~he
of individual regrossion coefficients in the S. H. and F. S. models and
of meaningful functions of the coefficients.
First of all. since
~
~SH
..
and.c5 rs are MV designs (or IIG-optimal"
designs in approximate theory terminology), the Equivalence Theorem
(see (23]) ~ntioned in Section 5 allows us to conclude that IM*(~SH)I
and IM(~FS)I are maximized at 6;11 mld at o~, respeetively; in other
words,
6;s
6;H are both
and
"D-optimal".
Before considering some other design criteria, let us digress for
just a moment to make the following remarks.
In all the work that we
have done in this section up to now, the optimal approximate destgn
c5 ~SH has buen characterized using M* (6
•
I~.lw.i !(8 i
"i)
f
(6 i "1)' where
*
(8.45)
SH) Jnstead of M(6 sB)
1/2
1/2
M (oSH) • GSH . M(6 SlI) GSll '
GSB being the (d+l)2 x (d+1)2 diagonal matrix
(8.46)
where
(8. 47)
a
~O
•
l'
t:! t
'2n
•
[
2.. _2
1. n(o+l)' n(n+l)' ••••
(~)I' (2~)TJ. n· 1,
?: (n-m) I l{n-m) !
(n+m)I'
(n+m)I' ••• ,
2, •••• 4.
.
.,.
The expression (8.45) giving the relationship between M (&su) and
M(6
) follows directly by noting {rom (8.1) ·that
SH
47
(8.48)
Similarly, from (8.2), we can write
-1/2
II
(S.49)
~SH • Gsn
NOW', one can easily
vf~rify
~SH'
daat the des:f.CD criteria for S. II.
regression so far considered are invariant under the pair of transformations (8.48) and (8.49) in the senae that, for any of these cr1-
.
* .
*
teria~ the optimal design ~Sll characterit:ed by .H (6~) is exactly the
same as the optimal design ~~H characterized by M(O~H).
The re~on
we bring this point up at this pa'rticular time is that the design criteria now about to be dlscus!:"ed do not possess this desirable invariance
property and hence, for thesu criteria. the designs
in general, be different.
1\
0sa
0
and 0SHwill,
In what follows, we shall continue as before
to wurk with M*(oSH).
Techniques
complet~ly
analogous to thoRe used earlier in finding
WV designs for S. H. and F. S. Tsgression can be applied to sbow that
".*
.
15SB and 0PS are both "A-optimal' designs; io other words, tr M*-1 (6 sa)
and tr M-1 (O'S) are lldn:1mized at
i and
tat"
diS'
O~H
respectively.
In
* follows
brief, the verification of the A- optimality of 8IeSH ana 6FS
directly by using the differentiation formulae of Remark 8.3 to miui2 2
--1
~O
a
yd
mize the quantities tr M (tS sn )· 111 ' + l.n-l m t
[*n n
2
2
2 2..] and tr M-l(c5F~) • mO,O
+ I:.l (m*0 +2m-l,n +2m-l + ra*n +2m,n +"JD)
+
I:.
l
(m2o- l ,2n-l + m2u ,2n) subject to the restrictions (8.28) and
(8.29) t respectively, whl.:h are introcluced via lHagrangian multipliers
as in (8.32) and (8.33).
48
Now, let M(4) be tbe information
correapondingto some
A, where K(~) bas the typical elem8Dt "1j(6) :; 1I ; a1Ailarl" let
1j
.1j (6) e . i j be the U ,j )thentl'Y 111 )(-1(6) • Inthil franlwork, we
15
..
matri~
£
give the follawins definition (see llfving, {12).
Definitigp 8.1.
"
A design 4
&
A is said to be m.s.p. (1d.n1aax with
respect to sinale paranteters) when it minimizes MAX m11 on A; it ia
i
said to be a.st.f. (minimax viUl respect to standard parametric fnrms)
when it adnimt&es
MAX
c I M" l (l5) c
c,c ' c-1 ~
~
.....
....
A.
Oft
-
When all the par_tera tn the fitted model are to be lncluclecJ
in the 1IIinimax approach defined nbove, we sball refer to this ••
"total minimaxityll; when on1y a subset of these J'ar_etere is of
intereat, we will be concerned with "part1al lllinbaax1ty".
In the
• MIN11l:u.'
~bove
aetting, let h be a subscript for which 'bb
hh'
~ ("hb)
Then, since 111
-1
, we heva
(8.50)
where H(4) is taken to be (kXlt).
equality ia tltat H(o) • A~ for
Clearl"
'01118
a sufficieut condition for
A > O.
If A(K) represents a cbaracteristic root of M(4), tben it follows
that
(8.51)
e' M-1 (S)
c,c'c-l
MAX
.........
~
• Amax(H-l) -
1/~in(H) ~
k/cr
M(~);
again, a suffici-ent condition for equality i.s that M(4) be of the forti
Alk for some
coD8tan~
A > O.
111 tbose situations when
tt" M(~)
1s either C01l8taDt
OIl
IJ. ox has
an obvious upper bowel, it ia clear tbe.t (8.50) and (8.51) will then
1; ...
49
i1
provide lower bounds for MA.'<tUl
.
-1
HAX c' M (6) c, respectively.
c.ct~.l ~
... ...
tJ
tr M (oSH) • tr M(&FS) • (d+1). it followR that the quantities
~i..~~e
11
MAXiand
*
.
MAX
-1
c' M (&) c have
c,c'e-l ...
"I
and
-
lower bound
A
-
ot
(d+1) for
............
d
M (osa) and a lower bound of [1 + (d+l») for M(~FS)' Except in the
".
uninteresting case d .. 0, these lower boun«UJ are not attained by 0SH
"
•
*i1 *
and ~FS.' since for d ~ 1, MAXim
(&SH)·
~~
c,ctc-l
..........
restrictiQns (8.28) and (8.29) prohibit us from
H(~FS)
aDd
to be multiples of
an
*
id~ntity
matrices.
£,
*-1
H
Cho~9ing
*
(8 SB ) ...
c •
M*(~SU) and
*
Thus, the designs aSH
arll:l neither m.s.p. nor m.st.f. in the context of "total
* and ~FS'
lIlinimaxity" with respect to ~SH
In the framework of "partial udnimaxlty", one can develop inequalities completely analngous to (8.50) and (8.51) by introducing the
notion of a "partial information matrix".
By appealing directly to
Theorem 3.4 of Elfviug [12], we may conclude that, for each fixed n,
.-
0SH
is both m.s.p. "and m.st.f. with respect to the (2n+l) elementB of
[n] given by (8.2) and that 6* is both m.s.p. and m.st.f. with
!SH
FS
respect to the elements of ~~) given in (1.4), n • 1, 2, ' •• f d.
The partial information matrices corresponding to p.ln]
...sa and BCD)
... FS are
1
~(~2It+~1":"")
1211+1 aDd
'21
12 t respectively.
We may summarize the more important results of this section in
tbe followic, theorem.
THE01m'M8.1.
*
The appro:dTaate designa 6
SR
and
a•rs
characterized by
50
.
•
A
•
"
the information matTiceG M (oSH) • DSH and M(OFS) • Drs' respectively,
possess the following optimal properties:
*
(i) they are orthogonal dcsi~s (i.~., D
alid
SH
Drs
are diagonal);
(11) they areW'll, WB, and WMS designs with respect to the least
favorable veighting functions
o~
•
~ ~ 2~,
and wrs (9) •
W;a(9,t) •
~/2w,
0
~
•
~
(1-cose),/4w, 0 ~
a ~ w,
2. (see Remark 5.1);
(:l.ii) they are V-, B-. and HS- admissible. and they are also admissible
in the sense of Definition 1.1:
(iv) they are HV and D- optimal designs;
..
(v) they are A- optunal designs; and,
(vi) they are m.B.p. and m.st.f. with reupect to the elements of
[n]
~SH and
(n)
@Ps '
n • 1, 2, "', d.
'l'hia author feels that
J:lIt
u
.. 'it:
and 0FS are also MB and MMS
SH
designs but has not been able to justify this belief.
.Qonjp.cture:
*
* characterized by
Let us emphasize again that the design 6SH
#I
;.
M (~SU) • DSlt can be claimed to be identi cally the same as the design
*
*
.
0
-1/2
characterized
by ll(OSH) • DSH • GSH
DSH G-1/2
SH ' where DSH is
given by (8.11) and where G is given by the relations (8.46) and
SH
(8.47), only when re£er~ing to parts (i) - (iv) (and not to parts (v)
o
~Sll
aud (vi»
of Theorem 8.1.
One final word is in order concerning the minimax1ty criteria
of Definition 8.1.
Suppose that the model under study is of the form
(4.1) and that there
e~ists
a design 0~ & l for whiCh
diagonal ftatrix D, where M(o) •
Ii.l Wi
that 0
~
'" is both a.8.p.
~
~
-
*)
is a
!(~i) £'(~i) for any
Then, if we write f'(x)6 • (D-1/2f(x»'(D1/2S) •
~
M(~
-
f*'Cx>a*,
-
~
&& h.
it follows
~
and m.st. f. with respect to the elements of tbe
-a
parameter vector a* • nl/2 since M*(o~) • D-l/~(o*)n-1/2 • I, where
...
51
M*(O) ... r~-l wi (Jr(~i)
usinC
pTeV~()US
f*' (~1)
notation, that
fOT
~SH i~
~1/2"
~1/2 -1/2
a
.
D
·..SH
sa G8M a
-SH
to Dsa
1/2
respect to D"S
~FS'
.
~iny 0
E
1':..
This result i.mpliea,
m.s.p. and m.st.f. with respect
and that
~
..
FS
is m.s.p. and m.st.f. with
FrOUl theRe CrJfia:tentSt it 1s clear that one need
only search for a diagonal lnfCtrmation matrix to concl'lde t118t the
aSflociated design is both m.8.p. and lIl.st.f. with re9pect to .!2!l.!. set
of parameters.
.. and o,.1S. as Rotatable Arrangements
--;;;-.'._-.-....;;-_.....-_------~SB
9. The Characterization of
Consider the estimated response Yd(~;~P) • !d(~)
2
2
..
2p •
b O + b1x1 + bi 2 + ... + bk1Ck + bJ1x l + ••• + bkk~ + b12x 1x 2 +
'
3
'" + bk-l,k~4l~ + bl11xl + .... ·which 1s a polynomial of degree d
in the k variablu xl' x 2 ' '"''
squares using a design
.Ii·
~?
~
obtained by the method of least
cOhslstlng of
th~
N points
(-x11 ' x 21 ' ••• , ~). i · 1,2, ••• , N.
1 l'
a l a2
If we write H(al , G2 , ••• , ~) •
~li
x21
i 1
NI
a.K
••• ~i' then
the set of design points {lSi):.l is snid to iorm. a "~otatable arrangeDent" (8ee (5» if it satisfie.s the moment conditiQD8
o if
one or JllI)t'e a
k
"here 0
~ ai ~ 2d for every
1
are odd
~\~ni.lai I
if all a i are even
a/2-k
2 IT i _J ("'i/2)I
\
i and where 0
,
! a • I~.lai ~ 2d, a being
the order of the moment; Aa is a positive coustant depending only on a.
These moment conditioDB are equivaleDt to the ccndition that
N
~
-1
~Var Yd(!;~P} • !d(~) H
c:J
(6 p> !d(~) be a function only of ~'~, where
52
M:(op)
II:
~ L~.l !cl(~i) !d(~1); however, they are
not, in goneral, suffi-
cient to make M(op) non-sluRulart and, in this CAse, all the
(k~)
polynOtllial coefficients will not bo individually est~able. In partic-
..
ular, when d > 1, this Ai~ular case will arise when all the design
points are equidistant from the origin of the factor levels.
Using the notions just developed and tile results of the previous
section, we prove the following theorem.
* eharacterized by the
TH!OR!M ~.l. The opti~l approximate design ~SB
2 *
(d+l) 2
)( (d+l) information
matrix D is exactly' that Bet of points
SH
•
.
{el'.i}~.l with p.qual weights Wi • liN for every
i which satisfies the
moment conditioos (9.1) or a rotatable arrangement of order cl when
transformed to the set of points
{~~~)}~.lt where !SH is given by (3.4);
•
similarly, the optimal appro1tmate design &pg characterized in terms of
the (24+1) )( (2d+l) information matrix Drs is exactly that set of points
(ti}~.l with equal weights Wi • lIN for every
i which
satisf1~8 the
moment conditions (9.1) of a rotatable arrangement of order d when. transformed to the Bet of polnts
{~~~)}~.l' ~ere !FS 1s given
by (3.3).
• and 6* which are optiIn particular, then" it follows that, designs 6SB
FS
mal for S. H. and 1. S. regression of order d, respectively, are also
optimal for all lower oTder regressions •
Proof.
Let
....
Yd(a";~SH)
denote the estimated response at the point
(e,,>
obtained by the least squares fitting of an S. H. model of order d;
'"
further, let Yd(~SH;~P) be a dth degree polynomial representation of the.
response y"d{e,ep;~SH) at the point !SB
corre~!ponding
to
(e,~),
a representation is described in Lemma 3.2 and bmark 3.1.
obvioUIJ that the vadance of
'"
Yd{a,.;~S.H)
at (a,4')
I1N8t
where such
It 1s
be exactly the
..
53
S. H. model, it follows from (8.42) that Var
Yd(~SH;~P)
same as the variance of
u~ed
to fit
• (d+l)
2 2
(1
~le
"
Yd(~SH;~P)
at
IN, which is independent of
~S1i'
~SH'
Assuming that 5SH has been
...
Since ~SU:SH
• 1,
the
design points {(a1 , tPi ) } N1-1 m.ust, by definition t fornl a rotatable
(1) N
arrangement of or-del' d when expressed as the set of points (~SH }i-l'
.
A completely analogous argument can be used to characterize &FS in a
similar manner.
Since a rot3table
arrangem~nt
of order d is also a
rotatable arrangement of any order d' < d, the last statement in the
theorem is true.
•
This completes the proof.
Now, there ure aome interesting
itnplications of the ebove theorem.
c~'ent&
to be made concerniug the
First: of all, the moments of a
rotatable design of oIcler d gi.ven by (9.1) are actually the moments
..
..2 - .
up to order 2d of a general spherical distribution p(x) • cf(x'x),
i.~., a distribution which 1s a function only of ~'~ - L~.l Xi and
lienee is constant (or "uniform") on hyperspbct'es cente~ed at the origin
of the xi' a.
In this ligllt, rotatable designs can be and are usually
formed by combining one or 1ftC,.re "rotatable arrangements" of different.
radii, a rotatable arrangement consisting of a set of points uniformly
spaced on the surface of a hypersphere.
An
analytical argument advo-
cating the use of an aestbetically appealing "rotatable arrangement"
of points as a design in ita own right haa not been forthcoming until
the present work.
Using the results concerning rotatable arrangements and designs
that are already available in the literature, we will be able to specify
optimal
exac~~esign8
construct optimal
..
(6 pg ) for F. S. regression of
~xact
any
order d and to
designs (6 *
5ft) for S. H. regression of orders one
54
and two.
E!Jsentially, the rO.Olson f;)r the limit<.;d range of d In the
latter. case above is due to the fact that. then are only ftvc regular
figures in three dimeusions, none pf "'hLeh has ver'!:tces
. moment conditions (9.1) for any d ?- 3.
r21), it is
of points
~
the
In fact, to paraphrase Kiefer
generally possible to conbtruct a
li.!..,
satisfyin~
rotatabl~
arrangement
a unift'nll discrete measuru on an appropriately choBen
set of "equally spaced" points on a k-din:eusional hyperliphere) when both
d > 2 and k ) 2.
So, we have
Th~re
does
.~
*
exist an S. H. optinwll ~~~" design 05B
for S. lIe regressJ.on of any order' d > 2.
As promised in the preYious lH1!ction, we will now pr~cisely sp~cify
the W. B.
desi~n8
for S. H. and F. S. regression using the cllaracteriza-
tion given in Ibeorem 9.1.
In particu13r, it is fairly easy to see, if (d+d') • 2d
a
(using
the notation of Section 8), that a WB design is a rotatable arrangement
of order do ; if (d+d') • 2d0 + 1, then a WB de8i~n can be either a
rotatable arrangement of order (d +1) or a rotatable arrangement of
o
order do with moments of order (2d0 +1) equal to zero.
P"rom the fact that we can fit an S. H. model. of order d • 2 and an
F. S. model of any order d
~
2. but cannot fit a polynomial of degree
d > 1, by using a rotatable arrangement of design points, we might say
..
that, heuristically, the "loss of information" suffered by not using
a rotatable design to fit the
pol~'nomial
is measured, in some sense, by
the "non-uniqueness" of the artificial polynomial response that can be
constructed from the fitted S. R. or F. S. model using the techniques
in Section 3.
One final comment is in oTder bere before proceeding with the
•
l'
•.
ss
construl~tion
of opt:11Ml .!!2im
Ul~!1trms
£,,1:' S. H. and F. S. regre1sion.
Intf:lret\tint1;r enough, Theorem 8.1 il1ustrt:ltes the only sltufJtion In
the HteratUl,-e ,,1I.'re a class of designs possesSiO!' so many optJ\l'lal
prorert5.c~
simultaneously.
Howev<"!. thu results are not raBUy a~
J~Sl1fyiD~
as they mlght seem at
fir~t
glance if one takes note of
nleoreJl1 9.1 in conjunc.tion with the geometrical interpretation of a
rotatable arrangement
~ld
theory given in Section 6
thpn remembers the implication of the general
(Wl1j~h ~ssent1ally
implies that a uniform
weighting function dictat.es a uniform. w:B design) as well as the fact
that designs constructed to mlnimize variance are, in general, "spread
out" as much as possible owr and extend to the limits of thQ region
of interest.
Hoel [18] is the sole autbor in tbe literature to st.ud.y F. S. and
S. H. regression.
He conoider.
~_n~
G-optimality and characterizes the
correspondlng MV designs in tenus of certa.in n!)rmalizing c:oefficients
* and M-1 (oYS)'
*
which are, in faet, the d.iagonal elements of M-1 (oSH)
,
*
However, he does not suspoct the valuable chaxacter1zation of 6SB and
0FS emphasized in Theorem 9.1 .Dor does hO mention the many optimal rro-
"
.
parties possessed by these designs.
developed for these
fUtlctloujl)
the response
8u~face
teclluiques
in the next chapter are also new and
qui.te useful.
If we WTite MpS(~1,a2) MSH(C&l' a 2 , Q3)
1
C
N
J. N
N
Ii - l
N I i - l (cos~1 sin8i )
(cos~i)
a1
a1
(sin~i)
"2
a,
and
(sin~i 8inSi ) • (cosS t )
a3
,
then, by evaluating the necessary surface inlegrals on the unit circle
and unit. sphere, respectively, we are able to show that the moment
requirements for
o:S in te1'll8 of tbe points (~~)}~.1 and for cS:H in
S6
terms of the points'
(9.2)
{!~~)}~.l are as follows:
M,s(Ql,02) •
r
re:~ r~:~ Iff ~ + Ql:9
if all the Qj are .even and is zero otherwise, where 0
j • 1, 2, and 0 .! OIl + 01 2 .! 2d;
~
OJ .!
2d,
~("l'''2'''~) • rt:~ rr.:~ rt:~ /2" r~ + li<ll:2i<1~
(9.3)
if all the Qj are even and is zero otherwise, where 0 .!
j ·,1, 2, 3, and 0 ~ OIl + 01 + a .! 2d.
2
3
OJ
~
2d,
Another way to verify the values of the non-zero moments above is to
expand ;
.
j •
r~.l (xii + xii + .•• + ~i)j
• 1 sequentially for
1, 2, ••• , d using either k • 2 or 3, and, at each stage, to
introduce the relationships between moments of the same order which
are given by (9.1).
Now, consider an arrangement of p points equally spaced on the
circumference of a circle.· It has been shown in [5] that p > 2d is
sufficient for this arrangement to be rotatable of order d; the
"necessity"
ar~nt
can be found in [14].
Thus, after appealing to
Theorem 9.1, we have the general result which now follows.
THEOREM 9.2.
•
~FS
For F. S. regression of order d, an optimal exact design
88signs equal weight IIp to each of the distinct points 4>i • 2wi/pt
i • 1, 2, ••• , p, where p > 2d.
Note that it was not necessary to appeal to (9.2) in order to
identify the optimal spacing given in the above theorem.
On the other
hand, we will need to use (9.3) in the construction of optimal exact
designs for S. B. regression of orders one and two.
57
ttow, adopting the notation of Bose and Draper in [1], we let the
set of p • 24 points G{a,b,c) be
(9.4)
define~
as
G(a,b,c) • {(±a,±b,±c),(±c,±a,±b),(±b,±c,±a)},
where the restriction a 2 + b 2 + c 2 • 1 is required since, by Theorem
9.1, each point of G{e,b,c) is to have the general representation !SH'
Without loss of generality, we can take a
e and
~
0, b
~
0, c
~
0.
If
, are the colati.tude and azimuth, respectively, of a point on the
1
surface of a unit sphere, then lG{l,O,O)
will represent the six vertices
of an octahedron, ic(l/J3,
11lS, 1113)
the eight vertices of a cube,
1
and 2G{a,b,O) the twelve vertices of an icosahedron.
Now, from the fom of (9.4), it is clear that the "odd moment"
requirements of (9.3) are satisfied by the set of points G(a,b,c),
where MSH(al,a2,a3) is an "odd moment" if at least one a i is odd.
Naturally, then, MSH «l1'(l2,(l3) is said to be an "even moment" if all
(Ii are even.
From (9.3), the even moment requirements for d· 1 are
MSH(O,O,O) • 1 (which i8 always satisfied) and MSH (2,0,O) • MSu(O,2,0)
• MSH (O,O,2) • 1/3.
1
So, it is easy to see that 4G(1,O,O) and
~(l//3, 1/13, 1/13) are S. H. optimal exact designs of order one. The
four vertices of a tetrahedron (which is the G-optimal design of order
one suggested by Hoel [18]) is yet another example of an S. H. optimal
exact design of order one, but it is useleas without augmentation except
for purposes of estimation since there will be no degrees of freedom for
error after fitting the first order S. H. model using this design.
lor S. H. regression of order d • 2, the corresponding optimal
exact designs must satisfy, in addition to the even moment requirements
.•.•.
58
for d • 1 given earlier, the following relations:
Msn (0,4,O) • MSH (0,0,4)
• 1/15.
..
= 1/5
MSu (4,O,O).
and l1SH (2,2,O) .. MSH (2,O,2) • MSH (O,2,2)
One can readily verify that the p • 20 vertices of·the dodecahedron {1c(a,b,O) + f;(l/I3, 1/13, l/13)}, where a 2 • ~(l + 1513) and
2
1
r.:'
b • ~l - vS13), provide an exact S. H. optimal design of ~rder two.
The values given for a 2 and b2 are the solutions of the two equations
2
2
a + b • land a 4 + b 4 • 7/9; it can be shown that these equations
determine the values of a
conditions are satisfied.
2
and b
2
BO
that all the necessary even moment
Similarly, it is easy to show that the 28
point design {ic<a,b,o) + jc<l/l3, 1/13, 1/13) +
.
jcU/I3, 1/13, l/fl)}
;(l
is an S. H. optimal exact design of order two if a 2 ...
+ ¥41f5) and
211
b • 2(1 - ~).
Thisde.sign is an example of a s.eguential
design in the sense that the two '3c(l/l3, 1/13, 1/13) designs could
initially be used together to fit a first-order S. H. model, and, if
subsequently, a test of lack of fit (based on 4 degrees of freedom) was
. found ·to be significant. the remaining 12 points of ~(a,b,O) (where
the appropriate values of a 2 andb 2 are as given) could be added to
provide the complete 28 point design of order two.
It is also possible to construct infinite classes of second
order exact S. H. optimal designs.
For example. consider the set of
p • 24 points {~(a1tbl'O) + ~(a2,b2'O)}.
Values of a l , bl , a 2 and b 2
that satisfy the required moment conditions can be found by solving the
..
2
2
2
2
4
set of simultaneous equations a l + hl • 1, a 2 + b2 • 1, and a l +
4
4
8
2 + b2 • 6/5. These equations generate a quadratic equation in
422
2
tbe form a 1 - a - a 2 (1-a2) + 21S • O. which bas the two roots
1
2
2
1/2
.
2
1/2 ± [a2 (1-82) - 3/20]
• Since 0 ! a 1 ~ 1, we require
4
bl +
2
a l of
S9
The upper bound is obviously satisfied since a 22 must lie between
· 4 2
zero and one, and so we need only have 8 2 - a + 3/20 ~ 0 which is
2
satisfied if (1/2 - 11M) ~ ai:s. (1/2 + l/ffO). Thus, we have
succeeded in constructing an infinite class of exact S. H. optimal
designs of order two, each design in the class consisting of the 24
. {1
1
points 2G(a1 ,b1 ,0} +jG(a 2 ,b 2 ,o)} and depending only on the single
2~
2~
2
2
parameter a 2 , wh~re (1/2 - 1/... 101~ a 2 ~ (1/2 + 1/r10J, b 2 • (1 - 8 2),
2 1
2
2
1/2 2
2
a 1 • "2 + [a 2 (1 - a 2) - 3/20]
, b i • (1 - a 1)·
As another example, consider the set of 24 points G(a,btc).
..
Values of at b and c that satisfy the appropriate moment requirements
are found by solving the pair of equations a 2 + b2 + c 2 • 1 and
a 4 + b4 + c 4 • 3/5, which together yield the quadratic in a 2 of the
42224
form a - (1 - c ) a - (c - c - 1/5) • 0, whose two roots expressed
2
1
2
2
4
. 1/2
as a function of care 2[(1-c ) ± (2c - 3c + 1/5)
].
Now, we
require firstly that (2c2 - 3c4 + 1/5) ~ 0 which implies that
1
_~
2
2
4
1/2
o ~ c 2 ~ 3(1
+ 2... 2,5), secondly that (1 - c ) - (2c - 3c + 1/5)
which is equivalent to either 0 ~ c 2 ~ ;<1 - 1/1S) or
~O
2
2
2
4
1, and finally that (1 - c ) + (2c - 3c + 1/st/ ~ 2
2
.
which is satisfied if c ~ O•. Combining these restrictions, we have
~(1 + 1/~ ~
c2
~
another infinite class of exact S. H. optimal designs of order two. each
design in this class consisting of the 24 points G(a,b,c) and depending
1
~
only on the one parameter c2 , where either 0 <c 2 :s. "2(1
- 1/l'5)
or
2
2
4
+ 1/.(J) ~ c2 ~ ;<1 + 21215>, a 2 •
(l - c ) + (2c - 3c + 1/5)1/2],
and b2 • (1 - a 2 - c 2). In particular, the special cases
t(l
2
i[
1
.
1
c • 0, '2(1 - 111$) and '2(1 + 1/-/5). give the twelve vertices of an
60
icosahedron.
Other second-order exact S. H. optimal designs with as
many points as desired can similarly be constructed.
An
example of an
S. H. optimal approximate design of order three is given in [18].
For any of the previously constructed exact S. H. optimal designs,
to convert a design point
use (3.8).
~~i) to a point of the form (ai"i)' one would
However, it is necessary to make this transformation to
locate in terms of a and , the points at which responses are to be
observed only when the region of interest is two-dimensional (see
Section One), for, in this case, the elements of
P~SH
are artificial
and do not represent the rectangular coordinates of a point on the
surface of a sphere of radius p.
.
Also, since
the elements of
nd{a,.;~SU)
~SH
is a function of a and , only through
(see Remark 3.1), we can write (1.11) for n • 1 and
2, respectively, as
and
So, computationally-wise, we are able to fit S. H. models of orders
one and two by least squares using the exact S. H. optimal designs constructed earlier in this section without having to convert the points
from the form !SU to the form (a,t) in order to be able to perform the
necessary calculations.
61
Since we will be concerned. in the next chapter with studying the
properties of S. H. and F. S. models which have been fitted by least.
squares hopefully employing the optimal exact designs presented earlier .
in this section, let
of these designs.
In particular, let
~FS
(9.7)
discuss the methodology associated with the use
US
(0)'
(1)'
• (2'8 ' ~FS
' ••••
and
(0)'
(9.8)
(~SH
2SB •
•
(1)'
' 2SH
(d)'
' .. .,
~SH
).
denote the vectors of least squares estimates of the elements of @FS
...
and
@SU'
respectively, where
and
If
We
use the optimal spacing advocated in Theorem 9.2 to fit an
F. S. aodel of order d, then it is easy to see that
(9.11)
'<l
ao
•
%r~.l
(9.12)
an -
(9.13)
bn • ;
r~.l
lIPi-I
N
I rj-l 1 j (2'll'i/P.) •-y,
cos(2ntti/p) y(2'11'i/p), n • 1, 2, •••• d,
s:Ln(2n1fi/p) y(2"'i/p), n - 1, 2,
• • • t
d,
where Y (21f1!p) is the observed response for the jtb replication of
j
.the ith point '1 • 2wi/p, j • I. 2, ••• , r, 1 - 1, 2, ••• , p,
62
...
1 rr
y(2wi/p) - r 4j-l Yj(2Wi/p), and N • pro Also, it follows from what
-1 2.
has been done that Var(eFS) • Dps a IN and that Var Yd (9;eFS) •
(2d+l) a2/N •
ft
•
Similarly, for S. H. regression of order two, we use the computational forms (9.5) and (9.6) to write
aOO •
1 rp
Iri
(i)N Li-l j-1Yj{~SH) m Y,
3 p
(1)
",ri
a 10 - N I i a1 x3i L.j-l Yj (3SSH ),
~
t
3 rP
rri
(i)
all • N Li-1 xli L.j a 1 Yj(~SH ).
_ 3 rp
..
rri
(i)
i L.i-1 x2i L.j-l Yj(~SH ),
b ll
a 20
IS rp
2 Iri
(i)
S
- 2N L.i_l x 3i j-l Yj (~SH ) - '2 aOO '
a 21
•
b 21
_~N IPi-I x
5
IP
rr
i
(1)
ii i-l x lix 31 j-l Yj (~SH ) ,
rr1
(1)
x
L.j-l
Yj(~SH
),
2i 31
5 \"p
2 Iri
(1)
1
&22· 2N L.i-l xli j-l Yj(~SH ) +6(a20 - 5aOO )'
b 22 -
where
5
I
rri
(i)
P
2N .i-l
x1ix2i L.jal Yj (~SH ),
Yj(!~i» is the observed response for the jth replication of the
(i)'
ith point ISH - (xli' xli' x2i ) - (cose i , cos~i sinei , sin~i sinei ),
.
rp
j - 1, 2, ••• , r i , i • 1, 2, ••• , p, and N • Li-l rio Also it follows
-1 2
1/2 *-1 1/2 a2
that Var(2SH) - DSH a IN • GSH DSH GSH
IN and that Var OftYd(9";2su)
• (d+l)2 a 2/N. In particular, for d - 2, we have
.
"
~
..
.
,
63
In the above framework, it is easy to form the analysis of
variance tables associated with these designs •
.
*
,!NOVA, TABLE FOR F. S. ~RESSION OF ORDER d USING 0FS
.
'Source
D.r.
·0
1
a
1
bI
1
8
1
1
2
b2
1
•
•
•
•
..
ad
I
b
1
d
Lack of fit
,Pure. error
Total
....
•
·
Sum of Squares
pr a 2
O
pr 8 2/2
pr b2 /2
1
1
pr 8 2/2
2
pr b~/2
·••
p-(2d+l)
pr a 2/2
d
pr b 2/2
d
By difference
per-I)
Ir.1 Ij.l (Yj (2Wi/P)-y(2Wi/p»2
pr
Ii.1 Ij.I Y~(2'1fi/P)
64
it
ANOVA TABLE FOR S .H. BEGRESSION OF .ORDER TWO USING. 6SH
·Souree
..
...
D.F.
2
a OO
1
NaoO
a lO
1
all
1
b
ll
1
2
Na lO
/3
2
Na 1l /3
2
Nb11/J
a 20
1
Na20 16
a 21
1
b
21
1
22
1
. l2Na~2/5
1
12Nb 222 /5
8
b
e
Sum of Squares
22
Lack of fit
(p-9)
Pure error
(N-p)
2
2
3Na 2l
/5
2
3Nb /5
2l
By difference
ri
I
(i)
- (i) 2
i-l 1-1 (Yj(!SH ) - Y(!SH »
IP
N·r~.lri
Total
1 2 (1)
IPi=l !rj-l
Yj(~SH )
Finally, as promised in Section 1, we will make some brief comments here concerning the use of direct products of F. S. regression
fuactions as models for response functions which are known to be
periodic in more than one variable.
function Tl(xr x2 ' ••• t
periodic in xl' x 2 t ""
~; ~)
~
In generalt to graduate a response
on the k-dimens1onal hyperplane which is
for k
~
2, we can take as a model the
(4)1; @PS) l'ld ($2; ~FS) '"
dl
2
n~ ($k; ~FS)' where 0 S..i ~ 2'JT for i - 1,2, ••• , k, which can be
expression TlC.1 , .2 t
••• ,
4>k; ~) • Tl
written as the sum of (2di+1)(2d2+1) ••• (2d +l) distinct terms. We
tt
ean appeal to Theorem 9.2 of this section and to Theorem 2, p. 1099,
65·
of [18J to conclude that an MV (or G-optimal) exact design for fitting
n($l' ....
~; ~)
-by least squares is one that assigns equal weight to
each of the Pl P2 ••• ~ points ($1' ~2' ••• , ~k)' where ~i is allowed
..
to run over the set of values 2~j/Pi' j • 1, 2, ••• , Pi' Pi > 2di for
every i, i-I, 2, ••• , k. nlese notions have not been pursued further •
.
"" -..
CHAPTER IV
RESPONSE SURFACE TECHNIQUES FOR F. S. AND S. H. REGRESSION
10.
Analysis of Fitted Second Order F. S. and S. H. Models
As
stated in the introduction, the objective of any response
4surface.study is to approximate as accurately as possible using some
graduating function the form of an unknown response surface on some
region of interest
...
ing
*.
'*
One is then interested in using this graduat-
fun~tion to estimate the locations in
at which the surface
has a maximum, minimum, or saddle point so that, for example, optimal
settings of the factor levels can be specified.
(Henceforth, such
locations will be referred to as "stationary points".)
Techniques
for studying polynomials of degree two in this light were developed
by Box and Wilson [6].
In this section, we present a general method for locating the
stationary points of fitted F. S. and S. H. models of order two.
Of
course, it is suggested that the least squares fitting of these models
be done using the designs presented in the previous section •
By noting that
•
elements of
!Fs
....
Yd(~;2FS).
....
is a functionaf
~
only through the
and that Yd(e'~;~SH) is a function of
e and
~
only
through the 'elements of !SH (again, see Remark 3.1), we can conclude
that the determination of the stationary points of fitted F. S. and
s.
H. models could be made by differentiating these functions with
67
respect to the elements of !FS and !SH' respectively, subject to
the restrictions !FS!FS • 1 and ~~~SH • 1, rather than by perform...
,iug the differentiations directly with respect to
F. S. model and with respect to a and
~
~
for the fitted
for the fitted S. H. model.
As a matter of fact, an attempt to determine the stationary points
"
"
by solving aYd(a'~;~SH)/ae
• 0 and dYd(e'~;2SH)/a~
• 0 in the case
"
of S. H. regression or by solving dYd(~;eFS)/d~
• 0 in the case of
F. S. regression will not meet with any noticeable success, except
possibly for the case d • 1.
On the other hand, the technique we
propose, which involves 'Working with fitted F. S. and S. B. models
of order two in polynomial form, 1s quite easy to use and yields
results which lend themselves to direct interpretation.
The rela-
tions (3.6) and (3.8) can then be used to convert stationary points
~~)
and
!~~)
to points of the form <Po and
(eo'~o)'
respectively.
Now, using (3.7), we can write a fitted S. H. model of order
two in the form
(10.1)
where !SB is given by
(10.2)
~3.4),
(1)
ESB by (9.10), and where
•
Similarly, we may write (3.5) as
(l0.3)
"(!Fs'...
·b)
' bel)
' B18 IpS'
'2
PS - ·0 +~s
... F5 + !FS
68
where
~FS is given by (3.3), 9~;) by (9.9), and where
(10.4)
•
It will be shown later (Lemma 10.1) that the validity of the
proposed techniques will not be affected by the non-uniqueness of the
polynomial representations (10.1) and (10.3).
.
Now, from the above, it follows that, in general, we are
..... .
interested in determining the stationary points of the function
(10.5)
subject to the restriotion !' ! • 1, where bO is a known real constant
and where and B are, respectively, a (k x 1) vector and a (k x k)
E
symmetric matrix of known real constants.
This general description
covers the situation of interest to us, namely the case when bot
:e
and B are stochastic and (10.5) is a specific realization of the
random variable f(x);
in particular, the work in Section 11 is based
..,
solely on this notion.
So, let Q(!) • bO + !'E + !'~ - U(!'! - 1), where U is a Lagrange multiplier. Then, it is a straightforward exercise to verify
that a vector
'0 which satisfies the equation dQ(!)/d!- 9 is a solu-
tion of the pair of equations
(10.6)
where
0.0.7)
c! -
-.~ ~
and !'!. 1,
f.
.
69
Clearly, a different solution'~o of (10.6) will be obtained
for each different value assigned to p.
for the mOment and
where
adj(B-p~)
(B-l.I~),
Keeping p arbitrary in (10.6)
~ememb~ring t~at. (B-PIk)-l • adj(B-PIk)/IB-~Ikl,
is the transpose of the matrix of cofactors of
we see that the equation determining the permissible set of
values of p is
(10.8)
A2 ,.·., ~ are the k characteristic roots of B. Note
that some of the ~'s may be zero since B has not been assumed to be
where
~l'
of full rank here.
....
The second equality in (10.8) follows directly by
pre- and post-multiplying IB-PIkI2 by Ip12, where P is a real orthog-
onal matrix such that PDP' • diag(A 1 ,
this particular matrix P again.)
~2' •••
'
~k).
(We will refer to
Now, (10.8) can equivalently be written as a polynomial of
degree 2k in lH moreover, this polynomial will have at least two real
*.'
roots since (lO.S) is a continuous real function on the compact set
{!: ~'! .l} and hence there must exist points in,* at which
(lO.S) achieves its maximum and its minimum.
If a real root, Po say.
of (10.8) is not a characteristic root of B as well, then the
sol~tion
!o of (10.6) corresponding to Po is simply J o • - ~(B-l\fk)-te. On
the other hand, if R'[adj(B-~jIk)]~ is zero for some ~j' j • l •••• ,k,
then ~j is a root of lIlultiplicity two of (l0.8), IB-Aj1kl- 0, and
the two equations (B-Ajlk)! • simultaneously.
Remark 10.1.
i hand !'! • 1 must be solved
It will be shawn later in this section that the solution
70
Ie
of (10.6) using the largest real root of (10.8) is that value of
~
which maximizes (lO.S) subject to the constraint !'! - 1, and the
solution of (10.6) using the smallest real root of (10.8) is that
It
value of ¥ which minimizes (10.5) subject to !'! • 1.
Now, if we set
h • e~;)
and B • BFS 1n (10.8), the resulting .
expression can be written in the form
(l0.9)
Similarly, if we set
e• h~)
considerable amount of algebra gives
and B • BSH in (l0.8) J then a
71
•
The determination of the values of the coefficients in (10.9)
and especially in (10.10) is best and most easily done by writing a
simple computer program to perform the necessary arithmetical manipulations.
This method of approach attempts to avoid possible round-off
errors which seriously affect the values of the roots of these polynomials (which are also easily obtained by standard computer techniques), thus altering the corresponding stationary points found by
solving the equations (10.6).
Since it is reasonable to assume that
the random matrices (BFS-P I 2) and (BSU-P I 3) will be non-singular
o
o
with probability one, a check on the accuracy of the calculation of
the coefficients and roots of (10.9) and (10.10) is to see how close
!~
1
-1-
!o is to one, where, :In our general notation, !o • - i(B-Polk ) 2(The case when IB-lJo1k I is close to zero will be discussed in the
next section.)
72
The following lemma removes the doubt raised by the nonuniqueness of the representations (10.1) and (10.3).
kmma 10.1.
•
.
The method proposed for determining the set of stationary
points of a fitted F. S. or S. H.. model of order two will generate the
correct set of points regardless of the choice of the second degree
polynomial representation of
Y2(~;~FS)
or
Y2(e'~;~SH)
used to employ
(10.6) and (10.8).
Let fl(~) • hI + ~'21 + ~'Bl~ and f2(~) • b2 + ~'22 + ~'B2~
denote two different second degree polynomi~l representations for
Proof.
either Y2(~;2FS) or'Y2(a,~;2sH)' where ~'!. 1.
It is clear from the
way in which these representations are formed that the relations
(10.11)
hold, where c 1s a non-zero real, constant.
So, using (10.11), we can
write the equations (10.6) and (l0.8) associated with the polynomial
representation f2(~).in the form [Bl-(v+c)Ik]~·- ~l' ~'~. 1,
~i{adj[Bl-(~+C)Ik]}~l• 4IBl-(~C)IkI2. and these are easily seen to
be equivalent to (lO.6) and (10.8) under the representation
wi th
(~) replaced by
lJ... , say.
fl(~)
Thus. the set of solutions t {x(I)}
say, obtained using the representation
-0
fl(~)
set of solutions {x(2)}
obtained using f 2 {x).
-0
...
proof.
is identical to the
This completes the
In particular, it follows from the above argument and from the
.
form of (10.6) that the same set of matrices· {cl, where C is given by
(lO.7), will be obtained regardless of the choice of the polynomial
representation.
Briefly, what is happening here is that a change
in the polynomial representation affects both the form of B
.73
and the set of roots of (10.8) in stich a way that the set of matrices
. {e}, the vector
..
..
2'
and hence tho set of stationary points: {~o} all
remain invariant •
Now, suppose
x'x·
1, where C0 •
... ...
is
x
"'0
a
(B-~ o-k
L)
solution of the equations e 0'"x
= --2~...
and
and where P0 is a real root of (10.8) •
Then, since x'x .. x'x ,. 1, we have
... -
"'0"'0
.
f(!) - f(!o) • (bO + !'2 + !'B!) - (b O + !~2 + !~B!o)
• (x-x )'b + x'c x - x'e x
-
-0
-
...
0'"
-0 0-0
• (x-x )'b + 2(x-x )'C x + (x-x )'C (x-x )
"''''0
-
... -0
0-0
-
-0
0 -
"'0
• (x-x
)'C0"''''0
(x-x ).
-"'0
If the orthogonal matrix P is chosen as before, then it is clear
that we can write
f(x)
- f(! 0 ) • (x-x
)'P'[diag(Al-p0 , A2-P0 , •.• , Ak-V 0 )]P(x-x
), or
...
... "'0
- "'0
(10.12)
2
f(x)
- f(x-0 ) • L~~
l(Ai-P0 )wi , (wI' w2 " " t wk ) • (x-x
... "'0)'P'.
1_
Geometrically, the expression (10.12) can be considered to be
the result of shifting the origin
~o
Qof
the
~'s
to the stationary point
and of rotating the coordinate axes in terms of xl' x 2 , ••• ,
that they become axes expressed in terms of wI' w2 ' ••• , wk'
easy to see that the quantities (AI-Po), (A2-Po)'.'"
~
so
It is
(Ak-Po ) are the
characteristic roots of . Ca • (B-p0 I k ) •
Note that the expression (10.12) illustrates the change in
response on moving away from the stationary point !o'
In particular,
if the characteristic roots of Co are all non-negative (non-positive),
then (lO.S) takes on its minimum (maximum) value at·x in the set
"'0
~ c" {~: !'! • d. If ·at least two of the characteristic roots of Co
are of different sign, tben!o is said to be a saddle point.
It is
. \';
'.'
74
important to realize that a stationary point x....0 (whether it be a
maximum, minimum, or saddle point) will be ynigue only when lcol =\: O.
Since the maximum (minimum) of (10.5) subject to
•
..
~'~
• I
must be found using this procedure, it follows that there will be a
~o
greater (smaller) than or equal to all the A's, and this
obviously be the largest (smallest) root of (10.8).
~o
will
In a given
situation, then, one would really need to solve (10.6) using only
the largest and smallest roots of (10.8) since the extremum points
(and not the saddle points) are of main interest.
One can now
appreciate the import of Remark 10.1.
These techniques can be used for all the regions on which we
have considered these F. S. and S. B. functions to be defined,
although the transformations, etc., actually have a geometrical
interpretation when the regions of interest are the circumference
of a eirc1e and the surface of a sphere.
. For completeness, let us characterize the extremum points
x
and X_of
-max
-uu.n for the following special cases. If b... Jlp 0 and B • 0 t
1/2
1/2
then !max • ~/(2'~)
and ~min· -~/(,2'~)
, where we agree to take
the positive square root of b'b.
!min are the
If b... • o and B "'r 0, then x....max and
charac~eristic vectors
of unit length associated with
the largest and smallest characteristic roots of B, respectively.
11.
Confidence B;egions for Stationary Points of Second Order F. S.
and S. H. Models
If x denotes a solution of the equations (10.6) corresponding
"'0
to a real root
~
o of .(10.8), then it follows, ignoring bias effects,
.-
that the true (but unknown) value, x-0 , of this stationary point must
. i ... ..
15
satisfy the matrix equation Hx .: ~
" '" • k
+ Ece0 )x*
• 0,
where
~
~o
~
2 (b)
/Jx" • 1. Clearly, Hx'" is in the form of a general linear hypo~o
~o~o
~o
1
~o
thesis ana any procedure for testing HX : ""X
0 • -2E(b)
+ B(Co~
)x • 0~ for
-
..
to construct a confidence region for x.
...
any given x..., (in our case, for any x...., for which x'x... • 1) can be used
~o
along this line, see [4] and [24].
~
(For general discussions
An application relatea to rotat-
able designs is given in [5].)
Now, let d
~1.
• -21b + C x be the estimate of IS.
~
Then, it is
~
0'"
easy to see that the elements of d~X are linear
combinations of the
'
elements of b ana B if one notes that Po •
~~'2
+
~'B!.
enables us to write dawn the dispersion matrices V(d
~x-
~ES
.
This fact
) ana V(d-x )
.... s~
of!x- and!x corresponding to the special cases!· ~FS' ~. 2~)'
...,"S
-SH(l)
B • Irs and! • !SH' ~ • 2SH ' B • BSH ' respectively. In particular,
-1 2
using (9.14) and the fact that Var(~FS) • DFSO IN and also employing
the relations !Fs!FS • 1 and !SH!SH • 1 at the proper times, we can
S
2
show after some labor that V(!x- ) • 2(1 2 - !FS!FS'o IN and V(!x )
~~S
.... SH
~(I3 - !S~SH)o2/N. It is important to realize that the matrices
•
(1 2 - !Fs!Fs' ana (1 3 - !SH!SH) are both idempotent and are of ranks
one and two, respectively. The reason that the dispersion matrices
V(d!Fg')
and V(d .) turn out to be singular is due to the fact that
...
...
"'SH
the general relation !'!x • !'
+ (B-Polk )!] • l!'2 + !'~ - lJo • 0
...
holds among the elements of d for any x for which XiX • 1.
~x
(h
...
~x
~
~
~
If we agree to make the standard assumptions in least squares
..
theory concerning the independence and normality of errors, then it
follows from what has been done· so· far that, under the hypotheses
H ,d
and d
will have singular multinormal clistd.x~ SH "'x"'x~ SH
~E~
but:1cns with 0 mean vectors and dispersion matrices V(d
) and
H
?"""1
aDd
~~~
~~s
...... ,.-
76
V(d
), respectively.
"'~SH
In particular, it can be shown fairly directly
that the quantities
2
(I27~FS!FS)2
and (2N/9a )d' (1 -x x' )d
~S
!:Fos
. . ~SH 3 ...SH...SH "'~SH
2
(2NISa ) 2,
are distributed as central X2 variates with
freedom, respectively.
~ne
and two degrees of
Of course. the idempotency of
(I2-~S~Fs)
and (I3-!SH~SH) is crucial to the argument.
Now, let s2 be an estimate of (12 based on V degrees of freedom
which has been obtained either from a residual sum of squares
(assuming an adequate model) or from a "pure error" sum of squares
obtained by replication; moreover, we take s 2 to be distributed as a
2 2
a X I'\J random variable independently of d
in the case of F. S.
.
.
"'~FS
regression or of d
in the case of S. H. regression. Then, it
"'~SH
follows that T
!Fs
•
(2N/Ss2)!~ (I2-!FS~FS)2x
2
• (N/9s )d'
...FS
is distributed as
...FS
is distributed as F(2,v),
F(l,'\J) and T
(13-xS.~S'H)d
!SH
"'~SH'" ti...
-~SH
where F(y,v) denotes the F distribution with y and V degrees of freedom.
If we now make use of the relations xF'Sd
- -!Fs
• 0 and x su
' d
..
-xS
• 0
to note that T
can be written as (2N/5s 2)d'''' d
and that; H
!rS
"'~FS"'~FS
!SH
can be written as (N/9s 2)d' d
, then it is reasonable to reject
-~SH"'!SH
.
the hypotheses H : 0
• 0 and H
: IS
• 0 at the a level of
!Fs '-!FS'"
!SH -!SH ...
significance if T
> F (l,v) and T
> F (2,v), respectively, where
::FS - a
!SH - a
F (y,v) is the upper lOOaX point of the F(y,v) distribution.
a
Finally, it follows from the structure of the above tests that
..
the associated lOO{l-a)% confidence regions for x"'0'* (corresponding
to
.
po) for F. S. and S. H. regression have the following forms:
(11.1)
R-_ .' {x- : d' d
-75
...¥ s "'!FS"'~S
< 2.Ss2p (l,v)!N},
a
77
RSH -{!SU: ~~ ~x
(11.2)
"'SH "'SH
where
~!SH • ~~~}
+
~ 9S2Fa(2,~)/N},
(BSH-PoI3)~SH·
In what follows, we will attempt to provide a geometrical
R,s
and RSH • Now, if ~o is a solution
of (10.6) corresponding to a real root po of (10.8), then 9x • Q by
.
"'0
definition. So, the same procedures which led to (10.12) can also be
interpretation for the regions
applied here to obtain the expression .
(11.3)
..
where the row vector 1!' • (wl ' w2 ' ••• , wk ) is defined in (10.12).
Hence, from thefom of (11.3), it follows that if Icol
k
• ni_l(Ai-Po) is non-zero, then g~g! - z represents the locus of an
ellipsoid in the space of xl' x2 ' ••• , lCk for any fixed value of z, the
center of this ellipsoid being at the stationary point !o and its
principal axes lying directly on the set of coordinate axes wI' w2 '
In general, then, 8 point !FS will be in
R.,s
if and only if
it liea not only on the unit circle but 81soon or within the ellipse
2 2
222
(AI-Po) wl + (A 2- PO) w2 - 2.Ss Fa(l,V)!N, where (wl ' w2)
- (!Ps-~~»tPFS' PFS being the orthogonal matrix satisfying P,SBpsP
rs
• diag(A l , A2)· Similarly, 8 point !SH will be in RsB if and only if
it lies not only' on the uuit sphere but also on or within the
2 2
2 2
222
ellipsoid (Al-po) wl + (A 2- Po) w2 + (A 3-p o) w3 • 98 Pa(2,~)/N,where
(Wl , w2 ' W3 ) •
satisfying
(!SH"-!~:»tPSH'
PSUS.SH • diag(>"l'
PSH being the orthogonal matrix
A2 , ).3)·
78
o), (~2-~0 ), ••• , (A r -~ 0 ),
r < k, are small compared to the remaining (k - r) characteristic
Now, in (11.3), suppose that
,
(ll-~
Then, the ellipsoid "'x"'x
d'd • z will be elongated in the
direction of the axes w1 ' w2 '···, wr • So, choices of x • x-va + p'w
roots of Co.
.....
#Itt
IV
such that !'! • 1 possibly differing.markedly from the true value
x* of the stationary point will give small values of d'd over a
......
"'0
"'X"'X
wide range of values of the coordinates wl ' w2 t " • • , wr ; such points
would then most likely be included (and undesirably so) in confidence regions with structures like (11.1) and (11.2).
limiting case when
(~l-~o)'
(A 2- Po)'···'
(~r-po)
In the
are exactly zero,
C will be of rank (k . - r), and any point x... • x...0 + P'w... which satisfies
·0
(10.6) will do so independently of the values assigned to the
coordi~
..
An example of the construction of confidence regions of the
type discussed in this section will be forthcoming.
12.
Numerical Examples
Here, vewill present two examples illustrating the use of the
designs and techniques developed in the three previous sections.
The
first example is completely artificial.
Example I.
Suppose that an S. H. model of order two has the form
2
cos. Pl1(cose) + 3COS2~ P22 (cosS) • cos. sine + cos2. 81n 9. To find
:::.::t::::-'::1:t:.O:s:~.(~::-:::a:':-B::r:t[gotifgr.m.(9.10) end
o
·w
0
-lJ
Using these results, it is easy to show that expression (10.10)
takes the form ~6 - !p4 - ~~3 + ~lJ2 • 0, which has double roots at 0
and -1 and single roots at 1/2 and 3/2.
Since 3/2 is the largest root
<
79
in this set and is obviously not a
root of B above,
SH
the theory previously developed tells us that the maximum value of
~haracteristic
(cos. sine + cos2~sin2a) over 0 ~ a ~ ~, 0 ~ 9 ~ 2~, expressed in
terms of the elements of
~
V
cose
cos, sina •
sin9 sine
or, in terms of
~SH,occurs
at the unique point
0
(0) (0)
0 ]-1
~ [3/2
0 1/2 0
1 • 1 ,
0
e anq 9,
at
0 5/2
e•
~/2
and
0
0
9 • O. Since the smallest
root in this set, namely -1, also turns out to be a characteristic
2
:~O:yB::l::::-:h:h:a::-::U:q::t::::' [~in~ +gfo:::-:n_e~'([)i~i.
!SH!SH • 1 simultaneously, occurs at the two distinct points
1<0,
'''1,
±1i5), or, in terms of a and 9J at e • Tr/2 and
tan-1(!1i5').
4> •
The minimum is attained at more than one point due
to the fact that ~
~ ~ ia aingular.
This example illustrates the situation when a "singularity"
arises (i.e., When a root of (10.8) is also a characteristic root
of B), and, except for the purpose of crystallizing the general
concepts presented in Section 10, is only of pedantic interest.
The
folloWing example will be somewhat more realistic.
Example II.
It was assumed that the true function representing a
response surface was of the form n(x1 ,x2 .8) = 2 +3x l - %2 + Sxi
+ 5X~ + 6x l x 2 and that the region of interest ~ was the circumference of a circle (scaled to unit radius).
It was desired to
graduate n(x ,x ;B) by fitting by least squares an F. S. model of the
1 2
appropriate degree. This fitting was to be accomplished by taking
.e
r • 2 observations at each of the p • 6 points' ~i • ~i/3, i • 1, 2,
•• 'J
6.
Note that this design is an F. S. optimal exact design of
• 4.• ,.
80
order two (see Theorem 9.2).
Random normal deviates with
n(xltX2;~)
were added to the values of
C1
=1
at the points of this design
to generate the following set of observations: 13.620 t 14.400 at
$ • 0; 10.552, 10.602 at
~
•
~/3;
~
7.250 at • • 'IF; 8.854 t 10.684 at
~
2.196 t 3.696 at
~
= 2n/3;
6.390 t
• 41£/3; and, 5.408 t 8.488 at
The formulas (9.11)-(9.13) were then applied to this data
• 5,,/3.
to yield the estimates a O • 8.512, a1 • 3.198, and b i • -.922.
,.
However, the sum of squares due to lack of fit of Y1 ($;12FS ) t wbich was
based on three degrees of freedom, was found to be significantly large
when
compared to the "pure error" sum of squares having six d. f.
So, second-order
te~~s
were added in an attempt to provide a better
graduatiDn of the true response function and the resulting F. S. model
..
.
of order
two
took the form
+ 1.903coa2. +
3.017sin2~.
,.,...
Y2(~;2FS)
• 8.512 +
3.198cos~
-
.922sin~
As can be seen from the ANOVA table below,
there is no evidence of a lack of fit (F(1,6) - 1.345) of Y2(~;2FS).
SOURCE
a
a
b
8
O
1
l
2
b2
Lack of Fit
Pure Error
Total
D.F.
MEAN SQUARE
1
1
869.450
61.363
5.100
21. 736
54.614
1
1.842
6
1.370
1
1
1
12
Confidence sets for those points on the circle at which
n(X1'X2;~)
achieves its maximum and minimum are found
8S
follows.
For the values of aI' bl , 8 2 , b 2 above, the expression (10.9) evaluates to lJ4 - 2S.21672ll2 - .02654ll + 126.65651 • 0, which has the four
81
real roots -4.75491, -2.36725, 2.36529, and 4.75647.
Since the two
1. 903 3.0171 . + 2
characteristic roots of BFS - [3.017 -1.90~ are _(a 2 +
• !3.S67, it follows that Y2(~;gFS) attains its uaximum
at the unique points
x
-min
!max
• _
-.6020\
( .7985)
4 75647 I ]-1 (3.198) •
2
-.922
.9351)
( .3544
\3.017 -1.903/
. _ ! [/1.903 3.01"'_
2
and minimum
)-1 (3.198)
•
-.922
! [/1.903 3.01'i\+ 4.75491
2
2)1/2
b2
I
2
t3.017 -1.903)'
and
.respectively; in terms of $, then, the estimated locations of the
points on the circle at which
minimum are at
c1J •
n(xl,x2;~)
attains its maximum and
.11531r and • 79441r, respectively.
Now, it can be seen from the form of (11.1) that the
confidence sets for the extremum points of
n(xl,x2;~)
100(1~)%
can be specified
exactly by determining the points of intersection of the ellipse
d' d
• 2.58 21' (l,v)/N and the unit circle. Tbe8e~oints of inter-
-!FS-!Fs
a
section are given by the real roots of the polynomial in Xl • cos,
of the general form
(12.1)
..
where k • 2.5S 2F (l,v),tN. This polynomial will have only two real roots
a
(except in the rare case when the ellipse is attenuated in such a way
that it intersects the circle in more than two points), each root
lying between -1 and +1.
82
e'
,
Now, the value of k for a 95% confidence region in this
example is k
= 2.5(1.370)(5.99)/12
-1.71.
For this value of k and
for the values of aI' b l , a 2 , b 2 given earlier, expression (12.1)
takes the form 4602.75S8x~ + 2145.4602xi - 1576.4737xi - 128.2284x1
+ 285.4421
= 0 with
roots-.469212, -.722356, aud .362721 ± .226723i
if ~o • -4.75491, and it takes the form 4605.7765xi - 950.32l3xi
2
- 685l.0484x l + 5l4.4296xl + 2722.0831 • 0 with roots .979843,
.867891, and -.820701 ± .146414i if ~ • 4.75647. Converting to
o
radian measure~ we find that the 95% confidence set for the maximum of
n(xl,x2;~) on the circle is {$: .064tt! $ !
it is" {$: •743w ~ ~ ~ .845v}.
true
lo~ations
.166~} and for the minimum
These sets do, in fact, include the
of the points on the circle at which
n(xl,x2;~)
achieves
its maximum and minimum, namely .l26w and •783'1l', respectively.
We can slightly modify the above example to demonstrate the
procedures that are to be used for the case of a response function
defined on the real line.
+
3cos~
-
sin~
In particular, suppose that
+ Bcos 2~ + 5sin2W+
6s1nwcos~,
n(x;~)
where w·
• 2
2tt(x-a)/(b-a),
represents a response function which is to be graduated over the
entire real line by an appropriately chosen F. S. model.
We can take
the region of interest to be the closed interval [a,b] since n(x;B>
....
is of (assumed to be known) period (b-a).
One can then show, if the
random normal deviates are assigned as in the first part of this
example, that exactly the same set of responses as before will be
obtained, but, in this case, they will be associated with the six
points Xi • a + (b-a);i/2w • a + (b-a)i/6, i • 1, 2, ••• , 6, in the
inter'lal [a,b].
The fitted F. S. model of order
two
will have
'"
exactly the same form as Y2($;eFS)
above with $ replaced by
~.
. ..
83
The 95: confidence intervals for the locations of the maximum and
minimum of
n(x;~)
on [a,b] will be
. {x: a + .032(b-a) ~ x ~ a
+ .083(b-a)}, and
. {x: a + .37lS(b-a) ~ x ~ a + .422S(b-a)}, respectively.
The procedures developecl for using S. H. models as graduating
functions can be similarly applied to study response functions defined
on the surface of a sphere or on the plane (see Section 1).
The
corresponding confidence regions for the stationary points of the
response functions under study Will. however, be somewhat more
difficult to appreciate •
.
. t •. '.
i
LIST OF REFERENCES
, 1. BOSE, R.C. and DRAPER, N.R. (1959) •. Second order rotatable
designs in three dimensions.
1097-1112.
Ann.~.
Statist.
~
2. BOX, G.E.P. and DRAPER, N.R. (1959). A basis for the selection
of a response surface design.
622-654.
3.
BOX, G.E.P. and DRAPER, N.R.
order rotatable design.
J.~.
(l9~3).
Biometrika
Statist. Assoc.
~
The choice of a second
50 335-352.
-.,
4. BOX, G.E.P. and HUNTER, J.S. (1954). A confidence region for
.
the solution of a set of simultaneous equations with an application to experimental design. 'Biometrika 41 190-199.
-
S.
BOX, G.E.P. and HtlNTER, J .S. (l957). Multifactor experimental
designs for exploring response surfaces. Ann.~. Statist.
,t§ 195-241.
6.
BOX, G.E.P. and WILSON, K.B. (1951). On the experimental
attainment of optimum conditions. J. ROI' ,Statist. Soc.
Sere B. 13 1-45.
7.
DAVID, H.A. and ARENS, B.E. (1959).
-
-
-
. regression analysis.
~.
Optimal spacing in
Math. Statist. ~ 1072-1081.
8.
DRAPER. N.R. and LAWRENCE, W.E, (1965). Designs which minimize
model inadequacies: cuboidal regions of interest. Biometrika
~ 111-118.
9.
DRAPER, N.R. and LAWRENCE, Tol.!. (1965). Mixture designs for
three factors. J. Roy. Statist. !2.£. Sere B.
450-465.
10.
DRAPER, N.R. and LAWRENCE, W.E. (1965). Mixture designs for
four factors. d. ROI· Statist. !Q.£. !!!:. A· ,l! 473-478.
11.
DRAPER, N.R. and LAWRENCE, W.E. (1967). Sequential designs for
spherical weight functions. Technometrics 9 517-529.
12.
ELP'VING, G. (1959). Design of linear experiments. Probability
and Statistics (Cr8lll'r Volume). Wiley, New York. 58-74.
£
-
85
13.
FERGUSON, THO}~S s. (1967). Mathematical Statistics: A Decision
Theoretic Approach. Academic Press, New York.
14.
GARDINER, D.A., GRANDAGE, A.H.E., and HADER, R.J. (1959). Third
order rotatable designs for exploring- response surfaces.
Ann.
Math.
30 1082-1096.
- Statist. ~
15.
HADER, R.i., KARSON, M.J., and MANSON, A.R. (1969). Minimum bias
. estimation and experimental design for response surfaces.
Technometrics
461-475.
11
16.
HANCOCK, HARRIS (1917).
and Company, Boston.
Theorz of lfaxima and Minima.
17.
HOBSON, B.W. (1931). !i.l£ Theory of Spherical and Ellipsoidal
Harmonics. Cambridge University Press, London.
18.
HOEL, P.G. (1965). Minimax designs in two-dimensional regression.
An~. ~. Statist. l§ 1097-1106.
19.
KAlU.IN, SAMUEL and Sl'UDDEN, WILLIAM, J. (1966). Optimal experimental designs. Ann. Math. Statist. - ill 783-815.
20.
KIEFER, J. (1959). Optimal experimental designs.
Statist. Soc. !!!..!. ~ 273-319.
21.
l<IUER, J. (1960). Optimal experimental designs y,with
applications to systematic and rotatable designs. ~.
Fourth Berkeley~•
381-405.
:!.
Ginn
Roy.
.!
22.
KIEFER, J. and WOLFOWITZ, J. (1959). Optimum designs in
regression problems. Anrt. Math. Statist. ~ 271-294.
23.
KIEFER, J. and WOLFOWITZ, J. (1960). The equivalence of two
extremum problems. Canad. d. Math. ,...,
12 363-366.
24.
WALLACE, .0. (1958). Intersection region confidence procedures
with an application to the locati"on of the maximum in quadratic regression. ~. ~., Statist. ~ 455-47S.