=e indicators mining ex- ? total rich- xxlziJ0ns. Mo- the analytic creasing the t they waild

? assessment Lysis taxied cal gLlidance great value

1ut progrr?ss logic judge-

'alse preci- %ions, pro- - resolution discwered

I sxnarios, =-4'- ef iciencies usefulness

Some Aspec ts o f M u l t i v a r i a t e A n a l y s i s

Dona ld E. Myers Department o f Mathematics U n i v e r s i t y o f Ar izond Tucson, A 2 85721 U.S.A.

ABSTRACT. C l a s s i c a l m u l t i v a r i a t e methods f o r a n a l y z i n g da ta t ab leaux

do n o t e x p l i c i t l y i n c o r p o r a t e s p a t i a l c o r r e l a t i o n . I n c o n t r a s t geo-

s t a t i s t i c a l t echn iques such as k r i g i n g , dua l k r i g i n g and c o k r i g i n g a r e

p r i m a r i l y e s t i m a t i o n methods. The m u l t i v a r i a t e f o r m a t i o n s o f dua l

k r i g i n g and d i s j u n c t i v e k r i g i n g , dua l c o k r i g i n g and c o - d i s j u n c t i v e

k r i g i n g , p r o v i d e methods which b r i d g e t h e gap between t h e two

approaches. B a s i c p r o p e r t i e s o f b o t h a r e p resented .

INTRODUCTION

The impor tance o f m u l t i v a r i a t e techn iques has been recogn i zed f o r a

number o f yea rs . I n p a r t i c u l a r , C l u s t e r A n a l y s i s . P r i n c i p a l

Components and D i s c r i m i n a n t Ana l ys i s have had widespread use.

However, t hese t o o l s wh i ch had t h e i r o r i g i n i n o t h e r d i s c i p l i n e s

o the r t h a n t h e geosciences do n o t f u l l y r e s p e c t some c h a r a c t e r i s t i c s

o f m u l t i v a r i a t e d a t a s e t s a r i s i n g o u t o f problems i n t h e geosciences,

m in ing , env i r onmen ta l m o n i t o r i n g , hyd ro logy and remote sensing. I n

P a r t i c u l a r t h e models u n d e r l y i n g t hese do n o t s p e c i f i c a l l y

I n c o r p o r a t e s p a t i a l c o r r e l a t i o n . The d icho tomy between t hese

techn iques and e s t i m a t i o n techn iques such as k r i g i n g can be seen i n

the c o n t e x t o f two p e r s p e c t i v e s o f a da ta t a b l e a u . Cons ider

( ' (.hun,q PI uI. I ~ d y . ) , Qu~lntlta(llive An(11ys~ of Mtnrrul rind Eticrpy Rt..wrrrrcs. 669.487 ' ' hh? 1). Rt,dcl Puhlrrhrng Company.

I n gene ra l m u l t i v a r i a t e techn iques i n t e r p r e t t h e t a b l e a u as a

( random?) sample o f s i z e n from a v e c t o r va l ued random v a r i a b l e

whereas g e o s t a t i s t i c s a l a Matheron wou ld model t h e t a b l e a u as a

non-random sample f r om one r e a l i z a t i o n o f a v e c t o r - v a l u e d random I

f u n c t i o n where t h e s p a t i a l c o r r e l a t i o n i s o f p r i n c i p a l i n t e r e s t . The I

above m u l t i v a r i a t e t echn iques do n o t r e q u i r e p o s i t i o n c o o r d i n a t e s and i

do n o t r e f l e c t t h e suppo r t o f t h e sample. I n gene ra l t h e y a r e n o t

e s t i m a t i o n , t echn iques b u t r a t h e r seek t o i d e n t i f y u n d e r l y i n g

phenomena o r d i s c e r n p a t t e r n s . The v a r i o u s forms o f k r i g i n g have

been deve loped and used f o r e s t i m a t i o n a lmos t e x c l u s i v e l y . There

have been a few i n s t a n c e s where a m u l t i v a r i a t e t e c h n i q u e such as

p r i n c i p a l components has been a p p l i e d i n c o n j u n c t i o n w i t h k r i g i n g ,

f o r example Borgman and Frahme (1976) , t lyers and Ca r r ( 1984 ) , Flyers

(1984) . Dav is and Greenes (1983) . I n each o f t hese t h e o b j e c t i v e was

t o s i m p l i f y o r t o a v o i d c o k r i g i n g . A d d i t i o n a l i n t e r e s t i n g

a p p l i c a t i o n s a r e g i v e n i n Bopp and B iggs (1981) o r Hopke e t a1 (1976)

n e i t h e r o f whom used g e o s t a t i s t i c s b u t s i m p l y con tou red t h e d e r i v e d

f a c t o r s t o i d e n t i f y p a t t e r n s . A l though p a t t e r n r e c o g n i t i o n i s a t o o l

much d e s i r e d i t i s as y e t n o t f u l l y developed.

The r e s u l t s p resen ted he re i n c o r p o r a t e some aspec t s o f c l a s s i c a l

mu1 t i v a r i a t e methods as w e l l as i n c o r p o r a t i n g s p a t i a l c o r r e l a t i o n and

a r e i n t e n d e d t o s t i m u l a t e f u r t h e r thought .

I. POSSIBLE GENERALIZATION

A . Cons ider a d a t a t a b l e a u w i t h an i n f i n i t e number o f rows,

i .e . one f o r each p o i n t i n a domain o f i n t e r e s t , i.e., each column i s

a f u n c t i o n , n o t j u s t a f i n i t e number o f p o i n t s and w h i c h i s w r i t t e n

s i m p l y as

For a f i n i t e number o f p o i n t s , i .e. a f i n i t e number o f l i n e s , a

u n i f o r m d i s t r i b u t i o n i s assumed, t h a t i s each l i n e i s e q u a l l y

weighted. For t h e i n f i n i t e case a p r o b a b i l i t y d i s t r i b u t i o n must be

assumed f o r each column. The ana logy t o choos ing a v e c t o r wh i ch

c a r r i e s t h e l a r g e s t amount o f v a r i a n c e i s t o choose a random v a r i a b l e

such t h a t t h e sum o f t h e squares o f t h e p r o j e c t i o n i s maximized. In

i a

! r i a b l e

u a s a

a ndom

re not

have

There

h a s

. ig ing,

, tlyers

c t i v e was

l ass ica l

t ion and

ows , )lumn Is - i t t e n

t h e f i n i t e version t h i s i s given as the inner product, f o r random

var iab les i t i s the conditionpl expectation. Let t h e conditional

expectat ion E ( Z i / U ) be wr i t t en in operator form DU Z i which

e a s i l y extends t o the vector Z = [Z1 , ... , Zm]. Writing the inner

product of two random var iables V , V as < V , W > = E ( V W ) , the

problem i s t o maximize

which i s a p o s i t i v e quadratic form. However, unl ike t h e f i n i t e

dimensional case where the b ivar ia te d e n s i t i e s a r e implied by the

d a t a , in t h i s case the operator D U i s character ized by b ivar ia te

d e n s i t i e s t h a t a r e unknown. One possible reso lu t ion of t h i s problem

i s t o assume t h a t each pa i r Zi, U i s b i v a r i a t e Gaussian and then i t

i s s u f f i c i e n t t o know t h e c o r r e l a t i o n f o r each pair . This i s

e s s e n t i a l l y t h e d i s junc t ive kr iging approach but r e c a s t in

mu1 t i v a r i a t e form, and will be discussed i n Sect ion IV.

6. Because the above formulation does not lead t o a numerical

solut ion without the use of s t rong d i s t r i b u t i o n a l assumptions i t i s

useful to ' r e -cons ider the problem from another perspect ive.

Principal components produces a representat ion of t h e data in t h e

following form

where FTF i s diagonal and the columns of F a r e ordered by percent

o f variance explained. One of the ob jec t ives of t h i s transformation

is t o remove i n t e r - v a r i a b l e c o r r e l a t i o n however t h i s method only

incorporates the i n t e r - v a r i a b l e c o r r e l a t i o n a t t h e same locat ion. I t

would be more appropria te t o remove s p a t i a l i n t e r - v a r i a b l e

correla t ion.

This suggests the following: i f

I S a vector of second order s t a t i o n a r y func t ions , f i n d

672

- Y ( x ) = [ Y 1 ( x ) , ..., Y ( x ) ] and C s u c h t h a t

P

- a n d t h e componen ts o f Y ( x ) a r e u n c o r r e l a t e d . W r i t t e n i n t e r m s o f

t h e v a r i o g r a m s a n d c r o s s - v a r i o g r a m s t h i s becomes

and 7 ( h ) i s t o be a d i a g o n a l m a t r i x . H e r e r Z ( h ) , Y y ( h ) d e n o t e Y

a n d

v Z ( h ) c o r r e s p o n d s t o a c o r r e l a t i o n mat r ix . , The r e p r e s e n t a t i o n

i s a n a l o g o u s t o t h e d i a g o n a l i z a t i o n o f t h e c o r r e l a t i o n m a t r i x as i n

P r i n c i p a l Components A n a l y s i s .

I f a l l o f t h e v a r i o g r a m s o f t h e componen ts o f Z ( x ) h a v e s i l l s t h e n

t h e t o t a l i n e r t i a o f i ( x ) i s t h e sum o f t h e s i l l s , i . e .

2 T r C Z ( - ) = I ( I C j k ) ~ y ( - ) k (9)

The c o l u m n s o f C c a n be n o r m a l i z e d b y r e q u i r i n g

t h e n t h e componen t o f P ( x ) w i t h t h e l a r g e s t p e r c e n t o f v a r i a n c e i s

t h e componen t o f Y o ( ) whose v a r i o g r a m h a s t h e l a r g e s t s i l l .

When t h e o b j e c t i v e i s p a t t e r n r e c o g n i t i o n a s o p p o s e d t o c o -

k r i g i n g , t h e n t h e s a m p l e v a r i o g r a m m a t r i x may s u b s t i t u t e f o r t h e t r u e

v a r i o g r a m m a t r i x . A f t e r t h e samp le v a r i o g r a m m a t r i x i s c o m p u t e d f o r

~ a r i c

each

s i n g ~

can t

and h

d e t e r - y ( x

d e c o ~

the c

Wac ke

Wacke

o f t h .

compo - Y ( x )

t h e v

r ange ' - r z ( h ) .

c o n s t !

i s mot

v a r i o l

11. I

t

t he fc

!jot or

i n t e r C

the s,

F

t e rms o f

(5

d e n o t e

( 7 )

i o n

(8)

I( as i n

1s t h e n

0-

he true ed f o r

\ a r i o u s l a g s a n d d i a g o n a l i z e d t o f i n d t h e e i g e n v a l u e s , t h e s i l l s f o r

each v a r i o g r a m a r e e s t i m a t e d by t h e e i g e n v a l u e s . When C i s non-

s i n g u l a r t h e e q u a t i o n

can be s o l v e d as

and hence a t l e a s t a t d a t a l o c a t i o n s t h e v a l u e s o f Y ( x ) a r e

d e t e r m i n e d . T h i s w o u l d p e r m i t s e p a r a t e k r i g i n g o f t h e components c f - - Y(x ) w i t h s u b s e q u e n t r e c o n s t r u c t i o n o f k r i g e d v a l u e s o f Z ( x ) . T h i s

d e c o m p o s i t i o n w o u l d a l s o a l l o w s i m u l a t i o n o f Z ( x ) by s i m u l a t i o n o f - t h e components o f Y ( x ) .

The model f o r m u l a t i o n g i v e n by ( 1 1 ) i s s i m i l a r t o t h a t used by

Wackernagel (1985) b u t t h e method o f a c h i e v i n g i t i s d i f f e r e n t . I n

Wackernagel t h e p r o c e d u r e i s t o model t h e v a r i o g r a m ( o r c o v a r i a n c e s )

o f t h e componen ts o f ? (x ) by n e s t e d models . It i s assumed t h a t t h e

components o f t h e s e n e s t e d mode ls r e p r e s e n t t h e components o f - Y ( x ) t h e n ' t h e c r o s s - v a r i o g r a m s a r e mode led as 1 i n e a r c o m b i n a t i o n s o f

t he v a r i o g r a m m o d e l s . I n p a r t i c u l a r i t i s o f t e n assumed t h a t t h e

p r i n c i p a l d i f f e r e n c e i n t h e v a r i o g r a m mode ls i s r e f l e c t e d i n t h e

ranges. The a p p r o a c h p r o p o s e d above a l l o w s a m o r e g e n e r a l model f o r - . iZ (h ) . N o t a l l s u c h m o d e l s c a n be d i a g o n a l i z e d whereas t h e

c o n s t r u c t i o n i n Wackernage l e n s u r e s a d i a g o n a l i z a b l e model and hence

i s more r e s t r i c t i v e i n p a r t i c u l a r i t assumes a l l v a r i o g r a m s and c r o s s

v a r i o g r a m s a r e p r o p o r t i o n a l t o t h e same model .

11. DUALITY - AN ALTERNATIVE

B e f o r e i n t r o d u c i n g a mu1 t i v a r i a t e v e r s i o n i t i s u s e f u l t o r e c a l l

t he f o r m u l a t i o n o f t h e u n i v a r i a t e k r i g i n g e s t i m a t i o n i n d u a l form.

' lot o n l y i s t h i s f o r m a n a l o g o u s t o t h e u s e o f s p l i n e s f o r

' " t e r p o l a t i o n b u t i t a l s o p r o v i d e s a me thod f o r r e m o v i n g o r r e d u c i n g

'he s p a t i a l c o r r e l a t i o n .

R e c a l l t h e e q u a t i o n s d e t e r m i n i n g t h e o r d i n a r y K r i g i n g

e s t i m a t o r . L e t xl, ..., xn be d a t a l o c a t i o n s w i t h d a t a Z ( x l ) ,

.. . . Z ( x n ) . Then t h e k r i g i n g e q u a t i o n s a r e w r i t t e n as

where

I n m a t r i x fo rm we have

wh ich we can w r i t e a s

where

f o r

a n c

i n t

not

t h e n

L e t

If u n i q u e neighborhood k r i g i n g i s used t h e n [ i b l T need o n l y be

computed once s i n c e a l l t h e i n f o r m a t i o n about t h e p o i n t t o be T

e s t i m a t e d i s c o n t a i n e d i n [ K O 11. Y i s a d a t a v e c t o r " d u a l " t o

Z. F i r s t i n t r o d u c e d by Matheron, Dubru le (1983) showed t h a t t h i s

f o r m u l a t i o n a l l owed f o r t h e use o f a un ique ne ighborhood i n c ross -

va l i d a t i o n and more r e c e n t l y Dubru le and Kostov (1986) have shown how

i t f a c i l i t a t e s t h e i n c o r p o r a t i o n o f i n e q u a l i t y c o n s t r a i n t s . Royer

and V i e i r a (1984). Ga l l i, M u r i l l o and Thomann (1984) have suggested

i n t e r p r e t a t i o n s o f Y.

Y , b have some p r o p e r t i e s wh ich have perhaps n o t been

noted. For example

A

YF = 0 and E(Y) = [O, 0,--0, 01

So t h a t Y i s cen te red i n two d i f f e r e n t senses.

I f E ( Z ( x ) ) = m t hen ~ ( i ) = m ~ ~ and E (b ) = m hence b i s

an unb iased e s t i m a t o r o f m. By w r i t i n g

then we o b t a i n

In a c r u d e sense Z i s r ep laced by new v e c t o r i n wh i ch each e n t r y i s - a l i n e a r c o m b i n a t i o n b u t t h e components o f Y a r e l e s s c o r r e l a t e d .

I f we compute

676

t h e n

w h i c h i s ana logous t o a c o r r e l a t i o n m a t r i x .

111. CO-DUALITY

A l t h o u g h d u a l k r i g i n g i s n o t q u i t e a m u l t i v a r i a t e t e c h n i q u e i t

sugges ts an e x t e n s i o n o f C o k r i g i n g t h a t can p r o p e r l y be c a l l e d m u l t i -

v a r i a t e .

R e c a l l f r o m Myers (1982, 1983, 1984) t h e g e n e r a l f o r m o f

c o k r i g i n g . As above xl, .. ., x n deno te sample l o c a t i o n s , - - Z(x l ) , ... , Z ( x n ) a r e t h e d a t a v e c t o r s where - Z ( x ) = [ Z 1 ( x ) , .. . , Z,(x)]. Then t h e c o k r i g i n g e s t i m a t o r i s g i v e n by

where

and

t h e n

- * U n l i k e t h e k r i g i n g e s t i m a t o r z t ( x 0 ) , Z ( x 0 ) i s n o t a s c a l a r and

hence i s n o t equa l t o i t ' s t r a n s p o s e b u t s i n c e

T T -*. L[~*(x,)I I = z ( x 0 ) (28)

Let

w i t h

we s h a l l work w i t h [ ? * ( x 0 ) l T u s i n g n o t a t i o n ana logous t o t h a t above

f o r d u a l k r i g i n g

Then

t h e n t h e c o k r i g i n g s y s t e m becomes

We h a v e t h e n

where

L e t

w i t h

A

Then 7 i s t h e " d u a l " t o 7 i n t h a t i t i s a v e c t o r o b t a i n e d f rom

b y i n c o r p o r a t i n g t h e s p a t i a l a n d i n t e r v a r i a c e c o r r e l a t i o n

q u a n t i f i e d by t h e v a r i o g r a m m a t r i x . We s e e i m m e d i a t e l y t h a t

and

o r a l t e r n a t i v e l y

The o r i g i n a l d a t a t a b l e a u i s t h e n g i v e n as a l i n e a r c o m b i n a t i o n o f

t h e e n t r i e s i n t h e new d a t a t a b l e a u . M o r e o v e r , B i s an e s t i m a t o r

o f E [ ~ ( X ) ] ~ when i(l;) i s second i y d e r s t a t i o n a r y and

T r a c e E[B - l ( i T ) ] [ B - E(? ' ) ] = T r [C(O) - K - ' F ( F ~ K - ~ F ) - ' ] ( 38 ) wh

6 . i s t h e sum o f t h e v a r i a n c e s o f t h e e s t i m a t i o n e r r o r s .

I V BIVARIATE GAUSSIAN MODEL o r . The f i r s t model p roposed t o g e n e r a l i z e p r i n c i p a l components

w o u l d r e q u i r e m i n i m i z i n g t h e q u a d r a t i c f o r m g i v e n b y e q u a t i o n ( 1 )

w h i c h i s d e t e r m i n e d by t h e c o n d i t i o n a l e x p e c t a t i o n o p e r a t o r . I n t u r n

t h i s w o u l d r e q u i r e knowledge o f ( a t l e a s t ) b i v a r i a t e d e n s i t i e s .

M a t h e r o n ( 1 9 7 6 ) e x t e n d e d t h e k r i g i n g e s t i m a t o r f r o m a l i n e a r t o a

n o n - l i n e a r f o r m by u t i l i z i n g a b i v a r i a t e Gauss ian d e n s i t y f u n c t i o n .

S p e c i f i c a l l y t h e d i s j u n c t i v e k r i g i n g e s t i m a t o r i s w r i t t e n i n t h e form

The f j l s a r e o b t a i n e d b y H e r m i t e p o l y n o m i a l e x p a n s i o n s w h i c h i n

t u r n a r e t h e e i g e n f u n c t i o n s o f t h e c o n d i t i o n a l e x p e c t a t i o n

O p e r a t o r . By u t i l i z i n g t h e g e n e r a l c o k r i g i n g f o r m u l a t i o n g i v e n i n

Eq. ( 2 6 ) , t h e DK e s t i m a t o r c a n b e f u l l y g e n e r a l i z e d .

A. P r e l i m i n a r i e s

2 L e t L . . d e n o t e t h e space o f f u n c t i o n s f , s u c h t h a t 1 J

f o r

t h e

f u n

i n

C.

r a n

d i s

i s

tha

for

f ( z i ( x j ) ) i s s q u a r e I n t e g r a b l e , i.e. E [ ~ ( z ~ ( x ~ ) ) ] ~ < -. Then f o r m arc

f o r

2 2 t h e sum L = L,,, wh ich i s n o t a d i r e c t sun s i n c e t h e

0 f

t o r

t u r n

n.

form

' i n

2 c o n s t a n t f u n c t i o n s a r e i n each Lij. I f I I ( I i i s t h e norm i n

2 m n 112-

Li t h e n [.I I I 1 i s a norm on L ~ . The non-1 i n e a r 1 = 1 * ~ = l

e s t i m a t o r f o r Z . ( x o ) i s t h e n t h e p r o j e c t i o n o f Z j (xo) o n t o t h e J

l i n e a r subspace o f L' genera ted by t h e e lements o f t h e L ? t h a t l k '

i s .

k 2 where f i j E L . l k ' 0 . The P r o j e c t i o n s

S i n c e Z * ( X ) i s a p r o j e c t i o n , t h e e r r o r s Z . ( x ) - Z * ( X ) a r e J 0 k J O j 0

or thogona l t o a l l t h e fi(zi ( x k ) ) and hence we have

* f o r a l l is J, k. S i n c e Z . ( x ) i s a 1 i n e a r combina t ion , each o f

J 0 these e x p e c t a t i o n s o n l y r e q u i r e s a b i v a r i a t e d e n s i t y . However t h e

k f unc t i ons fi a r e i n gene ra l unknown and t h e problem i s i n s o l u a b l e

i n t h i s g e n e r a l i t y .

C. T rans fo rma t i ons

As w i t h s i n g l e v a r i a b l e d i s j u n c t i v e k r i g i n g we assume t h a t each

random f u n c t i o n may be t r a n s f o r m e d t o one w i t h a Gaussian

d i s t r i b u t i o n . More s p e c i f i c a l l y we assume t h a t f o r each Zj t h e r e

i s a random f u n c t i o n Y j wh ich i s Gaussian and a f u n c t i o n 4 such j

t h a t Z . ( x ) = Q . ( Y . ( x ) ) f o r each j, x. Moreover , we assume t h a t J J J

f o r each i, j and each p a i r x, y

a r e b i v a r i a t e Gaussian. If i = j t h e n t h i s i s t h e usual assumpt ion

d i s j u n c t i v e k r i g i n g .

0. Hermi t i a n Expansions

I f Y(x) i s a Gaussian random f u n c t i o n and f i s a f u n c t i o n

such t h a t E [ f ( y ( x ) ) l Z < - then f may be rep resen ted by an k expans ion i n te rms o f H e r m i t e p o l y n o m i a l s . I n p a r t i c u l a r i f fij i s

a f u n c t i o n i n Li 2 t hen

k m

f . . [ Y ( x ) ) = 1 fk? H ( Y ( x ) ) 1 J p= 0 1J P

where f . . kP = E [ ~ ~ . ( Y ( ~ ) ] H ~ ( Y ( X ) ) ] 'J 1 J (431

T h i s p r o v i d e s a method f o r r e p r e s e n t i n g t h e unknown f u n c t i o n s k f . . Moreover t h e f u n c t i o n s . can be rep resen ted as 1 ~ ' J

E. B i v a r i a t e D e n s i t i e s

Under t h e assumpt ions t h a t each p a i r Yi(x) , Y j (y ) i s b i -

v a r i a t e Gaussian, t h e j o i n t d e n s i t i e s can be w r i t t e n i n t h e f o rm

where g ( x ) , y ( y ) a r e s tandard Gaussian d e n s i t i e s , pi: i s t h e

c o r r e l a t i o n c o e f f i c i e n t o f Yi(x), Y . (y ) . As w i t h u n i v a r i a t e J

d i s j u n c t i v e k r i g i n g we c o u l d a l s o cons ide r more gene ra l b i v a r i a t e

d e n s i t i e s such as t h e Hermitean models.

F. The O r t h o g o n a l i t y C o n d i t i o n s

Combining Eq. ( 4 1 ) , (42) we w i l l have

then \

where

f o r a l l j, E, r

U s i n g t h e t r a n s f o r m a t i o n s Z X ) = 4 . ( y . ) ) we o b t a i n J J

k S i n c e b o t h t h e $ .'I a n d t h e f i j l s a,re unknown we r e p l a c e k J k k f i j j j k b y s i m p l y f ( x k ) w h e r e o f c o u r s e t h e f . . I s

1 J J a r e n o t t h e o r i g i n a l ones a n d t h e n E q u a t i o n ( 4 0 ) becomes

1 J

If we write i t ( x ) = [ z * ( x ) , . .. , z:(x0)] and 0 1 0

t h e n we h a v e

where

Now i f we l e t F/ be w r i t t e n as

t h e n

and t h e e s t i m a t i o n v a r i a n c e i n ~2 i s g i v e n by

I = I Trace (tp) w C P

p= 0 P

- H

F s o l

Since t h e c P ' s a r e de te rm ined o n l y by t h e t r a n s f o r m a t i o n s t o

m i n i m i z e t h e e s t i m a t i o n v a r i a n c e i t i s s u f f i c i e n t t o m in im i ze , f o r

each p s e p a r a t e l y ,

f.loreover, f o r each p, i t i s s u f f i c i e n t t o c o n s i d e r

But m i n i m i z i n g

i s e x a c t l y t h e p rob lem o f c o k r i g i n g R (x ) u s i n g t h e d a t a - - P 0 H ( x ) . ... . H ( x ). Hence t h e m a t r i c e s BY. ... , B: a r e t h e

P 1 P n s o l u t i o n s o f t h e (simp1 e) c o k r i g i n g systems

a s f o rmu la ted i n Myers (1982)

For t h e b i v a r i a t e Gaussian case t h e e n t r i e s i n G~ a r e j k

i J. where p jk = c o r r e l a t i o n o f Y i ( x j ) YI(xk)

6 . The Undersampled' Case

~f f i s under-sampled, i .e . if f o r i and some k. Zi(xk) i s .k m i s s i n g t h e n i n e q u a t i o n (10) f o r a l l j, f . . E 0 and hence f o r a l l 1 J

k p p, f . . = 0. I n t h e c o n t e x t o f e q u a t i o n ( 2 7 ) t h i s i s e x a c t l y t h e 1 J

u n d e r - s a m p l e d f o r m u l a t i o n o f t h e c o k r i g i n g p r o b l e m a s g i v e n i n Myers

( 1 9 8 4 ) and i n C a r r . Myers and G lass (1985) hence t h e a l g o r i t h m

g i v e n t h e r e s t i l l s u f f i c e s .

V. DATA o r MODEL DRIVEN

P r i n c i p a l Components i s e s s e n t i a l l y a d a t a d r i v e n t e c h n i q u e ,

t h a t i s , n o p a r a m e t e r s need be s e p a r a t e l y f i t t e d n o r a r e t h e r e

a s s u m p t i o n s s u c h a s n o r m a l i t y o r s t a t i o n a r i t y . Some o f t h e

i n t e r p r e t a t i o n s o f t h e s e components and t h e i r e i g e n v a l u e s f o l l o w o n l y

frorn t h e g e o m e t r i c f o r m u l a t i o n w i t h o u t t h e u s e o f e x t e r n a l models.

C o k r i g i n g , d u a l k r i g i n g and c o k r i g i n g i n c o n t r a s t r e q u i r e f i t t i n g

v a r i o g r a r n s and c r o s s - v a r i o g r a r n s , t h e s e f i t t i n g t e c h n i q u e s a r e n o t on

c o m p l e t e l y s o l i d g r o u n d s t a t i s t i c a l l y and a l s o r e q u i r e i m p l i c i t

model 1 i ng o f t h e phenomena b y random f u n c t i o n s a n d t h e varograrns must

s a t i s f y c e r t a i n c o n d i t i o n s . The e x t e n s i o n f r o m l i n e a r t o n o n - l i n e a r

t e c h n i q u e s r e q u i r e v e r y s p e c i f i c s t a t i s t i c a l m o d e l s and assumpt ions

t h e s e a d d i t i o n a l c o n d i t i o n s a r e o f t e n u n t e s t a b l e . I t w o u l d appear , . t h a t m u l t i - v a r i a t e t e c h n i q u e s t h a t t r u l y i n c o r p o r a t e s p a t i a l

c o r r e l a t i o n a r e e i t h e r n o t p o s s i b l e o r a t l e a s t n o t a s y e t deve loped.

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