Design of Experiments with Spatially-Averaged Observations

NOT FOE QLJOTATIOK WITHOUT THE PERMISSION OF THE AUTHOR

DESIGN OF EXPERIkIENTS

WITH

SPATIALLY-AVERAGED OBSERVATIONS

VaLeri Fedorov

May 1986 W - 8 6 - 2 4

Working P a p e r s are interim r e p o r t s o n work of t h e I n t e r n a t i o n a l I n s t i t u t e f o r Applied Systems Analysis a n d h a v e r e c e i v e d only limited review. Views o r opinions e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e s e n t t h o s e of t h e Institute o r of i t s National Member Organizations.

iNTEENATIONAi INSTITUTE FOR APFLIED SYSTEMS ANALYSIS 2361 Laxenburg. Austria

I a m v e r y g r a i e f u l t o P r o f . A.B. Kurzhanski f o r encouraging discussions a n d Lo A . T e d a r d s f o r h e r p a t i e n c e in editing a n d proofing t h e p a p e r .

PREFACE

This paper deals with experiments when only some average values (over time or space intervals) can be measured. This kind o f experiment can be encountered in t h e areas o f remote sounding o f atmosphere, spectrometry, sample surveys, ra- dioactivity analysis, e t c . In these cases, the optimal experimental design means that the choice o f intervals o f observation (sometimes r e f e r r e d t o as "windows") and corresponding shares o f totally available time ( o r expenses) will maximize the final experimental information.

Formalization o f t h e problem leads t o a special class o f optimization problems closely related t o the classical "Markov's moments" problem. This paper contains new analytical and numerical results, together with a short but illuminative survey o f previous researches.

P r o f . M.A. Antonovsky Environmental Program

CONTENTS

1. Introduction

2. Experimental Probicrns 3. Optimization Problems in

Experimental Design Problem 4. Continuous Optimal Designs 5. Numerical Methods

6. S t r u c t u r e of S l i t Function 7. R e f e r e n c e s

DESIGN OF EXPE7UkiIENTS WITH SPATIALLY-AVERAGED OBSERVATIONS

VaLeri Fedorov

1. INTRODUCTION

A numner 01 puDllcaLlons c o n c e r n i n g oprimal aesign or experimenLs wnen con- t r o l s belong t o some functional s p a c e were published in t h e l a t e 1970's. Now i t i s evident t h a t t h e b a s i c i d e a s behind t h e s e t h e o r e t i c a l a p p r o a c h e s a r e t h e same a s in t r a d i t i o n a l e x p e r i m e n t a l design t h e o r y (e.g., Fedorov, Uspensky 1977; Mehra 1974; Kozlov 1981; Pazman 1986). The d i f f e r e n c e s become tangible in t h e applica- tion of g e n e r a l t h e o r e t i c a l r e s u l t s t o s p e c i f i c e x p e r i m e n t a l problems.

In t h i s p a p e r t h e s e d i f f e r e n c e s will b e t r a c e d f o r e x p e r i m e n t s with spatially- a v e r a g e d o b s e r v a t i o n s .

The simplest s t a t i s t i c a l model d e s c r i b i n g a t l e a s t a p a r t of t h e a b o v e men- tioned e x p e r i m e n t s i s t h e foliowing one:

yf

=

firzf + r i , i

=

1 . N

-

(1)

where +€Rm i s a v e c t o r of unknown p a r a m e t e r s ; ^ti a r e independent random values with z e r o means and f i n i t e v a r i a n c e s uf ( a more d e t a i l e d assumption wiil b e formu- lated l a t e r ) ; r s t a n d s f o r transposing. Variabies zi a r e defined by t h e i n t e g r a l

f ( v ) i s a v e c t o r of given b a s i c functions; h ( v ) a r e some functions which c a n b e chosen (controlled) by a n e x p e r i m e n t e r , O s h s l . In some c a s e s i n t e g r a l (2) must b e a Lebesque one. Function h ( v ) d e s c r i b e s t h e physical n a t u r e of a n e x p e r i m e n t a n d most typical examples will b e given below in S e c t i o n 2. If t h e l e a s t s q u a r e s es- timators

6 =

Arg min

z

N ^{u;2)Yf +Tzi 1 2}

*

^{i = 1}

a r e used t o analyze e x p e r i m e n t s d e s c r i b e d by ( I ) , then t h e quality of t h e s e estima- t o r s is_ defined by t h e i r dispersion (variance-covariance) m a t r i c e s D

=

E)(fi-.IPt) ( t h i t a - T Y ~ ) ~ ] , where t h e s u b s c r i p t r s t a n d s f o r t r u e values. I t i s well known ( s e e Fedorov 1972) t h a t in r e g u l a r cases

where ?c couid depend upon h ( t ) also. Matrix M i s usuaily called "information ma- t r i x . "

The o b j e c t i v e of optimal experimental design i s t h e s e a r c h f o r c o n t r o l s h i ( v ) providing b e t t e r d i s p e r s i o n m a t r i c e s or (more a c c u r a t e l y ) some functions of tinem ( s e e S e c t i o n s 3, 4).

2. EXPERIMENTAL PROBLEMS

Example 1. S p e c t r o m e t r i c E x p e r i m e n t s

In t h e s e e x p e r i m e n t s t h e measurement tools (e.g., a s p e c t r o m e t e r , a radiome- ter, e t c . ) h a v e a f i n i t e s p e c t r a l resolution and can m e a s u r e only a s p e c t r u m (of ab- s o r b t i o n o r r a d i a t i o n ) a v e r a g e d o v e r some f r e q u e n c y i n t e r v a l which i s defined by t h e so-called "slit" function. Formally t h e model of a measurement p r o c e s s c a n b e d e s c r i b e d in t h e simplest c a s e (when B e e r ' s law i s still working a n d t h e s p e c t r u m i s linearly dependent upon c o n c e n t r a t i o n s of components) by t h e following model:

-

The t o t a l s p e c t r u m intensity l i n e a r l y depends upon t h e s p e c t r u m intensities of any component

w h e r e v i s f r e q u e n c y ; Br

=

(B1, .

.

.

.

^r9,) i s t h e v e c t o r of c o n c e n t r a t i o n s .

-

The o b s e r v e d sicnai yi i s a l i n e a r functional of t h e s p e c t r u m u n d e r analysis [ c o m p a r e with (2)J

w h e r e hi ( v ) i s a s l i t function ( o r resolution function) of t h e i - t h o b s e r v a t i o n ; V i s t h e f r e q u e n c y i n t e r v a l avaiiable f o r o b s e r v a t i o n s ; E, i s t h e e r r o r of ob- s e r v a t i o n .

iisually a s l i t function c a n b e s a t i s f a c t o r i l y approximated by

-

E r r o r s E , a r e assumed t o b e random, independent f o r d i f f e r e n t i and t h e i r v a r i a n c e a: c a n d e p e n d upon hi ( v ) a n d upon t h e time ti s p e n t on t h e i - t h ob- s e r v a t i o n . In p r a c t i c e o n e c a n f a c e s e v e r a l possibilities. F o r i n s t a n c e ,

(a) a:

= c?

( b ) a:

= c?

[ j h , ( v ) d v ]

v

( C ) a:

= S t ,

^{( d ) a:}

=

a2ti [ j h i (v ) d v

1.

₍₆₎

v

The l a t t e r two c a s e s c o r r e s p o n d usually t o t h e situation when a n e x p e r i m e n t e r can choose t h e d u r a t i o n of a n o b s e r v a t i o n . If instead of yf ( t h e s e values c a n b e called

"total radiation") o n e c o n s i d e r s a v e r a g e values ( " a v e r a g e radiation"):

=

yi l j h f (v ) d v

I-' v

t h e n t h e i r v a r i a n c e s will b e correspondingly equal:

( a ) of

= 02

( b ) a:

=

g ! J h f ( v ) d v 1-I

v

o r

( c ) a:

=

^{( d ) a: =} d?t,-'!Jhi(v)dv

v

and v a r i a b i e s zi wiii b e defined by t h e following formuiae:

In what follows below, t h e notation y i will b e used e v e r y w h e r e and t h e s t r u c t u r e of v a r i a n c e s of and vaiues zi will define t h e c a s e .

Example 2. Remote S o u n d i n g o f A t m o s p h e r e

The s a t e l l i t e measurements of t h e outgoing r a d i a n c e in t h e i n f r a r e d s p e c t r u m band become r o u t i n e f o r d i s t a n t r e t r i e v a l of d i f f e r e n t physical p a r a m e t e r s of t h e a t m o s p h e r e ( f o r i n s t a n c e , t h e v e r t i c a l p r o f i l e of t e m p e r a t u r e , humidity, ozone c o n c e n t r a t i o n , e t c . ; see Condratjev and Timofeev 1970, S t r a n d a n d Westwater 1968, Twomey 1966). In a simplified form, t h e measured r a d i a t i o n u ( v ) d e p e n d s upon t h e v e r t i c a l t e m p e r a t u r e p r o f i l e T ( z ) , w h e r e z i s a n a l t i t u d e , and t h e t r a n s m i t t a n c e function of t h e a t m o s p h e r e p ( v , z ) :

w h e r e Z c o r r e s p o n d s t o t h e a l t i t u d e of a measurement tool; B ( v , T ) i s t h e P l a n c k ' s function. Both p ( v , z ) and B ( v , T ) are assumed t o b e known.

The most c r u c i a l assumption i s t h e possibility Lo a p p r o x i m a t e t h e function T ( z ) by some p a r a m e t r i c function T(z.19) with subsequent linearization of t h e in- Legral equation in t h e vicinity of a n initial estimate of t h e t e m p e r a t u r e p r o f i i e T ( z ) s o t h a t

a n d i n t e g r a t i o n witin a weight function h ( v ) o v e r f e a s i b l e f r e q u e n c y band V l e a d s t o t h e model d e s c r i b e d by ( I ) , (2), (6) o r (7) and (8).

3. OPTIMIZATION PROBLEMS IN EXPERIMENTAL DESIGN PROBIJW

Following S e c t i o n 1, l e t u s t r y t o formalize a design problem f o r e x p e r i m e n t s from S e c t i o n 2 . A s with t r a d i t i o n a l design t h e o r y ( s e e a b o v e c i t e d publications), t h e set of values

where t h e weights pi are t h e s h a r e s n i

/ N

of t o t a l number of measurements ( o r t h e s h a r e s ti / T of t o t a l Lime available), which h a v e t o b e done u n d e r t h e condi- tions zi ( o r a t t h e s u p p o r t i n g points z i ) , will b e called a design (of a n experiment).

In t h e t r a d i t i o n a l c a s e , t h e set XERm of f e a s i b l e c o n t r o l s ( o p e r a b i l i t y r e g i o n ) i s explicilly given. In t h e c o n s i d e r e d c a s e , X i s t h e mapping of a f e a s i b l e set

H

(in some functional s p a c e ) of c o n t r o l s h i (v ), and usually t h e c o n s t r u c t i o n of X (say, i t s boundaries) is a problem of g r e a t difficulty. T h e r e f o r e , i t could b e u s e f u l t o c o n s i d e r designs in t h e o r i g i n a l s p a c e aiso:

From (3) i t i s evident t h a t f o r model ( I ) , (2) t h e information matrix (and subse- quently d i s p e r s i o n m a t r i x ) d e p e n d s upon t h e location of t h e s u p p o r t i n g points z i , a n d if a t some points s e v e r a l measurements ni are d o n e , t h e n on functions hi a l s o , but d o e s not depend upon t h e r e s u l t s of measurements. In o t h e r words, t h e infor- mation matrix M d e p e n d s upon a design (, ( o r (,,).

Due t o t h i s f a c t , t h e design problem c a n b e formulated as t h e following minimi- zation problem

('

=

A r g min

@@(()I

t

where can n a v e b o t h possible s u b s c r i p t s . The function @ (optimality c r i t e r i a ) d e s c r i b e s t h e o b j e c t i v e s of a n e x p e r i m e n t e r .

If t h e whole set of p a r a m e t e r s p r e s e n t some i n t e r e s t , t h e n i t i s r e a s o n a b l e t o minimize t h e volume of t h e ellipsoid of c o n c e n t r a t i o n which i s p r o p o r t i o n a l t o iM((); -'I2 and one c a n u s e @[MI

=

M ( ( )

1

-'/'(everywhere / A I means t h e d e t e r - minant of m a t r i x A).

If i t i s n e c e s s a r y t o know a b e h a v i o u r of some l i n e a r function of T9: +?(z)T9, then i t i s r e a s o n a b l e t o minimize t h e a v e r a g e v a r i a n c e of t h i s function o v e r some region of i n t e r e s t . In t h i s c a s e

R a t h e r d e t a i l e d l i s t s of t h e most p o p u l a r optimality c r i t e r i a c a n b e found in Fedorov 1972; Silvey 1980.

In t h e f u t u r e i t i s r e a s o n a b l e t o distinguish between two t y p e s of designs:

continuous a n d d i s c r e t e o n e s .

In t h e f i r s t c a s e , weights pi c a n v a r y continuously between 0 and 1. This t a k e s p l a c e when t h e weight i s p r o p o r t i o n a l to t h e time of measurement. We c a n g o f u r t h e r a n d assume t h a t any p r o b a b i l i s t i c m e a s u r e (,

=

( ( d z ) or

tH =

( ( d h ) d e s c r i b e s some design. In e x p e r i m e n t a l p r a c t i c e , i t could b e impossible t o r e a l i z e , f o r i n s t a n c e , continuous m e a s u r e s . But f o r t u n a t e l y f o r any design with continuous m e a s u r e , i t i s possible to find t h e design with t h e same information m a t r i x , but with m e a s u r e c o n c e n t r a t e d in t h e f i n a l number of s u p p o r t i n g points. I t i s e a s y t o see t h a t in t h e c o n s i d e r e d s i t u a t i o n

In what follows below, t h e s u b s c r i p t z o r h will b e omitted without a n y comments if i t will not l e a d t o confusion.

Assuming t h a t @(NM)

=

a(N)*(M) (and i t i s t r u e f o r t h e majority of optimality c r i t e r i a used in p r a c t i c e ) minimization problem (11) can b e r e p l a c e d by

#' = A r g min *[M (()

t ⁽¹²⁾

w h e r e n o values depend upon t h e toLa1 time o r t h e t o t a l number of available meas- urements. This means t h a t a continuous optimal design d o e s n o t depend upon them also.

This useful p r o p e r t y i s n o t valid in t h e d i s c r e t e c a s e ,

4 . CONTINUOUS OPTIMAL DESIGNS

F o r t h e s a k e of simplicity in t h i s section and a l l subsequent s e c t i o n s only t h e c a s e when

* [ M i

=

M - ' 1 and (' = A r g m a x M I t

will b e c o n s i d e r e d . O t h e r c r i t e r i a c a n b e handled in a similar way ( s e e Fedorov 1980; Silvey 1980). I t i s most convenient t o start with continuous v e r s i o n of designs (, defined by (9). Then t h e c e i e b r a t e d Kiefer-Woifovitz equivaience theorem ( s e e Fedorov 1972) can b e used and only o n e assumption i s n e c e s s a r y f o r i t s fulfillment:

(a) Operability r e g i o n X i s compact

Theorem 1

(1) T h e r e e x i s t s a n optimal design ^(; containing no more t h a n m ( m + 1 ) / 2 s u p p o r t i n g points.

(2) The following problems are equivalent:

-

maximization of

/

M ((, )

I .

-

minimization of max X ( z ) d ( z ,(,), zEX

-

m a x X ( z ) d ( z . ( , ) = m , z4Y

w h e r e X(z,)

=

m i 2 a n d d ( z , ( , ) = Z ~ M - ~ ( ( , ) ~ .

(3) A t t h e s u p p o r t i n g points of a n optimal design ^(; t h e function d ( z ,(,) ap- p r o a c h e s i t s maximum.

(4) The set of optimai designs i s convex.

In a number of comparatively simple situations Theorem 1 g i v e s a c h a n c e t o c o n s t r u c t optimal design analytically. F o r more compiicated models i t h e l p s t o develop numerical p r o c e d u r e s a n d t o u n d e r s t a n d some g e n e r a l f e a t u r e s of optimal designs.

F o r i n s t a n c e , if one manages t o p r o v e t h a t o p e r a b i l i t y r e g i o n X i s compact.

t h e n h e c a n b e s u r e t h a t ( s e e point (3) of t h e t h e o r e m ) a l l s u p p o r t i n g points of a n optimal design are some boundary points of X.

Example 1 Let

f T ( v )

=

( 1 . ~ ) . v < I , *(M)

= IMI-'",

The set X can b e easily c o n s t r u c t e d b e c a u s e of t h e simplicity of i n t e g r a l (2) a n d i t s boundary i s d e s c r i b e d b y t h e following c u r v e s :

From point 2 of Theorem 1 i t follows t h a t t h e s u p p o r t i n g points of a n optimal design must coincide with t h e points where t h e ellipse zrM-l((;)z i s t a n g e n t to X.

This ellipse must h a v e t h e l e a s t a r e a ( o r IM I ) between a l l ellipses containing X.

The simple calculations show t h a t t h e points with c o o r d i n a t e s (2;0), ( 6 - 1 . 2 6 - 4 ) . ( 6 - 1 , 4 - 2 6 ) could b e supporting ones.

From symmetry of X and point 4 of Theorem 1 i t follows t h a t t h e weights of t h e two l a s t points must b e equal. S t r a i g h t f o r w a r d maximization of t h e d e t e r m i n a n t

i M

i

gives

Finally, from t h e simple i n t e g r a l equations defined at t h e beginning of t h e example i t i s e a s y t o find t h a t

with t h e information m a t r i x

F o r comparison of t h e t r a d i t i o n a l design with two A-windows at t h e points v =*I, t h e same matrix e q u a l s

This means t h e r a t i o o f s t a n d a r d e r r o r s will b e 0.5A ( f o r GI) and 5 . 5 ~ ~ ( f o r Z,). S o t h e optimal design i s essentially more e f f e c t i v e t h a n t h e t r a d i t i o n a l a p p r o a c h , especially f o r small A.

Example 2.

The c h a r a c t e r i s t i c s of optimal designs ( f o r instance, t h e location of s u p p o r t - ing points) essentially depend upon t h e c h o s e n b a s i c function. To illuminate t h i s , l e t us consider t h e r e g r e s s i o n problem formulated in t h e p r e v i o u s example with a new b a s i c function f l ( v )

=

( s i n n v , c o s n v ) . I t c a n b e p r o v e d t h a t f o r any s l i t func- tion h ( v ) t h e v e c t o r (Krein, Nudelman 1973. VII:3)

must belong t o t h e c i r c l e

{x:z:

+ zZ2 ~ 2 1

The optimal designs f o r t h i s o p e r a b i l i t y region a n d r e s p o n s e function v r z

=

1 9 ~ 2 ~

+

v 2 z 2 c a n b e easily c o n s t r u c t e d . F o r i n s t a n c e , optimal design c a n consist of t h e s u p p o r t i n g points coinciding with a l l v e r t e x e s of any r e g u l a r polygon refined to t h e c i r c l e X and t h e i r weights must b e equal. One of t h e simplest optimal designs i s

To find o u t t h e c o r r e s p o n d i n g optimal design in s l i t function s p a c e t h e i n t e g r a l equations

1 1

zii ⁼

^Jhi ( v ) s i n r v d v a n d

zit ⁼

^Jht ( v ) c o s r v d v

-1 -1

must b e solved. One of t h e solutions i s

and

I t i s worthwhile t o n o t e t h a t t h e widths of s l i t function "windows" have t h e same o r d e r as i n t e r v a l s of typical v a r i a t i o n s of b a s i c functions.

5. NUMERICAL METHODS

If assumption (a) holds a n d t h e r e i s a way t o find X t h e n t h e following numeri- c a l c a n b e used f o r optimal design c o n s t r u c t i o n (Fedorov 1972):

Cs =

(1-a, )C, + a, t ( z , ) (13)

where t ( z s ) i s t h e design c o n c e n t r a t e d at t h e single point zs

=

A r g maxX(z)d ( z .(,)

S E X (14)

where

d ( 2 . t s )

=

z r M ) z

The s e q u e n c e la, j c a n b e , f o r instance:

s =' X(z,)d ( z , , t s ) *

( a )

z a , = -

^{, lim a, =O ( b ) as}

⁼

s =o ^s -.- ~ ~ < ~ , ) d ( ~ , , t ~ ) - l ~ m

Both of them g u a r a n t e e t h a t

lim M(C)l = m i n M(C)l

2 -.' _t

More s o p h i s t i c a t e d v e r s i o n s of t h i s method are discussed in Ermakov 1983;

F e d o r o v a n d Uspensky1975.

In s p i t e of t h e f o r m a l simplicity of i t e r a t i v e p r o c e d u r e (13), (14), i t s p r a c t i c a l usefullness i s r a t h e r r e s t r i c t e d : one must find o u t t h e way t o c o n s t r u c t X b e f o r e using t h i s p r o c e d u r e .

P r o c e d u r e (13), (14) c a n b e r e p l a c e d by equivalent p r o c e d u r e s in t h e s p a c e of functions h ( v ) . F o r t h a t o n e h a s t o r e p i a c e t h e v e c t o r X1"(z)z with t h e vec- t o r [ h ( v ) f ( v ) d v ( f o r cases ( a ) , ( c ) from (6) and (a)), a n d with t h e v e c t o r

v

Jh (v)f (V /

4 7

( f o r cases (b).(d) from (6) a n d (8))

v

F o r t h e s a k e of simplicity, t h e i t e r a t i v e p r o c e d u r e will b e c o n s i d e r e d f o r t h e f i r s t cases when i t c a n b e p r e s e n t e d in t h e following form:

t,.l =

( 1 - a , ) ( , + a , t ( h , ) . h,

=

Arg max d ( h , ( , ) ,

h € H where

d ( h , [ , )

=

j j h ( ~ ) h ( v ~ ) f ~ ( v ) M - ~ ( t , ) f ( v ' ) d v d v ' ,

v v

Unfortunately, maximization problem ( 1 7 ) is more complicated than (14). One o f t h e simplest could be t h e following one:

-

discretisize t h e set V (and t h e r e f o r e VxV' also), say with interval A;

-

collect all points v j on t h e corresponding grid which positively contribute t o t h e sum

d ( ~ , )

=

z f ( ~ ~ ) ~ - ~ ( [ , ) f ( v , , ) ( 1 8 ) jd'

-

put h , ( v j )

=

1 i f v j was chosen on t h e previous stage; otherwise h , ( v j )

=

0

-

t h e fulfillment o f t h e inequality (which is corollary t o Theorem 1 ) d (t,h,)zzm tests that h , can be used in ( 1 6 ) .

This procedure is admissible f o r applications, f o r instance, when V is one- dimensional and that is t h e case f o r a number o f applied problems ( s e e Section 2 ) .

N o t e 1.

Iterative procedure ( 1 6 ) will converge in sense ( 1 5 ) ( f o r both original and discrete versions) i f instead o f ( 1 6 ) only t h e inequality

d ( h , , t , ) > m will take place on e v e r y step.

N o t e 2.

The sequences i j

!

and f j ' j must be identical. Therefore i f j = j o is included in sum

( l a ) ,

then j ' = j o must be included also.

N o t e 3.

When t h e slit function h ( v ) can change its value at t h e modes o f A-grid, then one can tell about A-optimal designs

(ti)

which can be considered as some approxi- mation o f optimal designs defined by (12). The iterative procedure ( 1 6 ) , (17) and ( 1 8 ) guarantees that

The idea o f A-optimal designs can be advanced f u r t h e r . As it was observed in Example 2 , t h e widths o f slit function windows were related t o t h e intervals o f vari- ation o f basic functions. T h e r e f o r e , it is reasonable t o decompose t h e set V into a comparatively moderate number o f subsets Aj ( j = G ) , f o r instance, coinciding with t h e most typical fluctuations o f basic functions. Assume that integrals

can b e c a l c u l a t e d (numerically or analytically). Then t h e o p e r a b i l i t y region X c a n b e approximated by XA with elements

where F = ( F 1 ,..., F, ), u

=

( u l ,

. . . .

^uk ) c U A a n d u j

=

1, if h ( v ) = l , v €Aj, and u . _J =0; o t h e r w i s e , j

=l.k.

Observing t h a t drz =grPu. =o,u a n d t h e information ma- t r i x equals

w h e r e ( ( d u ) d e s c r i b e s a design f u with supporting points in UA , o n e c a n conclude t h a t r a n k F must b e equal t o m (number of estimated p a r a m e t e r s ) . T h e r e f o r e , t h e decomposition of V should contain at l e a s t m s u b s e t s A j ( k a m ) .

When k =m a n d of c o u r s e IF

I

+O, t h e n

1

M

i

=IF

1 1 M, 1

and t h e design problem i s r e d u c e d to t h e maximization of

1 ^M, 1 .

The l a t t e r problem coincides with t h e rou- tine problem of "optimal weighting," (See Ermakov 1983)

If k >m t h e n i t e r a t i v e p r o c e d u r e (13),(14) can b e used with t h e r e p i a c e m e n t

k

of t h e v e c t o r A ~ ' ~ ( Z ) Z with t h e v e c t o r

Pu.

[ o r

Pu.

/

/

u , ; c o m p a r e with comments

j =1

t o (16), (17)]:

u,

=

A r g max u F T M -'((,)Fu.

U (20)

The maximization probiem (20) i s a d i s c r e t e one a n d at e v e r y s - t h s t a g e i t demands n o more t h a n 2' calculations of uFrM-'((,)Fu

.

6. STRUCTURE OF SLIT FUNCTlON

In t h e p r e v i o u s s e c t i o n s , i t w a s assumed t h a t t h e s l i t function c a n equal 1 or 0 . Some "physical" a r g u m e n t s were behind t h i s assumption. The compactness of o p e r - ability region X ( s e e S e c t i o n 4) w a s a l s o a n e s s e n t i a l assumption which was done t o slimplify a l l final r e s u l t s . If o n e r e f u s e s t h i s assumption, t h e n instead of optimal designs, so-called optimal s e q u e n c e s ( s e e , f o r example. Ermakov 1983) must b e c o n s i d e r e d and t h a t l e a d s t o some t e c h n i c a l difficulties. The following r e s u l t s (which are s t r a i ~ h t f o r w a r d c o r o l l a r i e s of well-known r e s u l t s from c l a s s i c a l a p - proximation t h e o r y ; s e e , f o r example, Karlin and Studden 1966, C h a p t e r VIII) il- luminate t h a t both a b o v e mentioned assumptions a r e n o t v e r y r e s t r i c t i v e . F o r t h e s a k e of simplicity, w e c o n s i d e r a one-dimensional c a s e (V€R1):

Assume now t h a t :

( a ) OSh ( v ) S l , f o r a n y v € ( a , b )

(b) Functions f ( v ) c o n s t i t u t e a Tchebysheff system on t h e open i n t e r v a l ( a , b ) . w h e r e a a n d b are possibly infinite. This assumption r e q u i r e s t h a t t h e func- tions J' ( v ) b e continuous on ( a , b ) a n d t h e determinants

J ' ~ ( t r n )

I

. . .

J ' ~ ( t r n )

I

. . .

^'

^I

I ^'

. . . ' I

' I

i.rm(t,) f m ( t z )

. . .

J'm (tm

1

I t , € ( a . b ) , i = l , m

-

a r e positive.

T h e o r e m 2.

The o p e r a b i l i t y region

b

x =

lz

= f

f ( v ) h ( v ) d v : O s h ( v ) s l j

0

i s a c o m ~ a c t convex set in R m

.

From Theorem 1 it i s c l e a r t h a t a l l supporting points of any optimal design must b e boundary points of X. T h e r e f o r e , only t h e s e points had t o b e c o n s i d e r e d in t h e p r e v i o u s s e c t i o n s , a n d f o r them t h e following r e s u l t t a k e s place:

T h e o r e m 3.

The n e c e s s a r y and sufficient condition f o r z t o b e a boundary point of X i s t h e fulfillment of t h e condition

h ( u ) [ l - h ( v ) j

=

⁰ (21)

almost e v e r y w h e r e in ( a , b ) .

Let h ( v ) b e a function satisfying (21) and l e t I ( z ) b e t h e number of s e p a r a t e n o n d e g e n e r a t e i n t e r v a l s (windows of a s l i t function) where h ( v ) = l with t h e s p e c i a l convention t h a t a n i n t e r v a l whose c l o s u r e contains point a o r b , i s counted as 1 / 2 . F o r a n y point z CY. I' ( z ) s t a n d s f o r t h e l e a s t possible I ( z ) ( i t could b e s e v e r a l dif- f e r e n t functions h ( v ) g i v i n ~ t h e same z ) .

T h e o r e m 4.

A n e c e s s a r y a n d sufficient condition t h a t z belongs t o t h e boundary of X i s t h a t I ' ( z ) s ( m - 1 ) / 2 . Moreover, e v e r y boundary point c o r r e s p o n d s t o a unique h ( u ) with I ( z ) = I ' ( z ) .

Theorems 3 and 4 allow f o r t h e development of a comparativeiy simple algo- rithm of optimal design c o n s t r u c t i o n .

Let ;=(a . v l , . .

.

.urn b ) , where a i v l S

. . .

S u m - l s b . According t o Theorem 4, t h e r e e x i s t optimal designs with a l l s u p p o r t i n g points (in t h e o p e r a b i l i t y r e g i o n H ) which h a v e t h e following s t r u c t u r e s :

a n d h ( v ) = 1 - h ( v ) .

That aliows f o r t h e modification of t h e i t e r a t i v e p r o c e d u r e (16).(17) Lo t h e p r o c e d u r e with maximization in s p a c e which dimension i s l e s s o r e q u a l (m - l ) , w h e r e m i s a number o f b a s i c functions:

h,

=

Arg max d (z7,(,), 7 = 1 , 2 , (23)

7.1

and

P r o c e d u r e (22),(23) in a computational s e n s e coincides with i t e r a t i v e pro- c e d u r e s used f o r t r a d i t i o n a l design problems and c a n b e handled with software deveioped f o r t h e latter o n e ( s e e Fedorov. Uspensky 1975).

REFERENCES

Condratjev, K.J. and Timofeev (1970) The Thermal Remote Sounding of Atmosphere from S a t e l l i t e s . Leningrad, 410.

Ermakov, S.M., e d i t o r (1983) Mathematical Theory of Experimental Design. Mos- cow: Nauka, 3 9 1 (In Russian).

Fedorov. V.V. (1972) Optimal Design of Experiments. New York: Academic P r e s s , 292.

Fedorov. V . V . a n d A.B. Uspensky (1975) Numerical Aspects of t h e Least S q u a r e s Methods. Moscow: Moscow S t a t e University, 168.

Fedorov, V.V. a n d A.B. Uspensky (1977) On t h e Optimal Condition Choice of Spec- t r o s c o p i c Measurements. Moscow: Proc. of S t a t e Scient. a n d R e s e a r c h C e n t e r f o r t h e Study of E a r t h ' s Environment and Natural R e s o u r c e s . 4:42-53 (In Russian).

Feciorov, V.V. (1980) Convex Design Theory. Math. Oper. S t a t . . 11:403-413.

Karlin, S. and W.J. Studden (1966) Tchebycheff Systems: With Applications in Analysis a n a S t a t i s t i c s . New York: Wiley & Sons, 586.

Kozlov, V . P . (1981) Design of Regression Experiments in Functional S p a c e s , in

"Mathematical Methods in t h e Design of Experiments", ed. Penenko. V.V. Mos- cow: Nauka, Novosibirsk, 74-101 (In Russian).

Krein. M.G. a n d A.A. Nudelman (1973) Markov Moment Problem and E x t r e m a l P r o b - lems. Moscow: Nauka, 552.

Mehra, R.K. (1976) Optimal Input Signals f o r P a r a m e t e r Estimation in Dynamic Sys- tem. IEEE T r a n s . on Autom. Control AC-19:753-768.

Pazman, A. (1986) Foundations of Optimum Experimental Design. D o r d r e c h t : D.

Reidel Publishing Company, 228.

Silvey, S.D. (1980) Optimal Design. London: Chapman a n d Hall. 86.

S t r a n d , O.N. and E.R. Westwater (1968) Minimum RMS-Estimation of t h e Numerical Solution of a Fredholm l n t e g r a l Equation of t h e F i r s t Kind. SIAM J. Numer.

Anal. 5:287-295.

Twomey. S . (1966) I n d i r e c t Measurements of Atmospheric T e m p e r a t u r e P r o f i l e s f r o m Satellites: Mathematical Aspects of t h e Inversion Problem. Monthly Weather Review, 99:363-366.