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

L . T O R E L L I ( * ) 

R e c e i v e d o n A u g u s t 2 0 t h , 1 9 7 5 

SUMMARY. — T h i s a r t i c l e c o n t a i n s a s c h e m a t i c e x p o s i t i o n of t h e t h e o r e -
t i c a l f r a m e w o r k o n w h i c h r e c e n t t e c h n i q u e s of t r e n d a n a l y s i s a n d i n t e r -
p o l a t i o n r e s t . 

I t is s h o w n t h a t s u c h t e c h n i q u e s c o n s i s t i n t h e j o i n t a p p l i c a t i o n of 
Analysis of the Variance a n d of Multivariate Distribution Analysis. T h e 
t h e o r y of Universal Kriging b y G . M a t h e r o n is a l s o d i s c u s s e d a n d r e d u c e d 
t o t h e a b o v e t h e o r i e s . 

RIASSUNTO. — I l p r e s e n t e a r t i c o l o e s p o n e i n f o r m a s c h e m a t i c a i r i -
s u l t a t i t e o r i c i s u c u i si b a s a n o l e t e c n i c h e p i ù r e c e n t i d i i n t e r p o l a z i o n e e 
a n a l i s i d i t e n d e n z a . 

V i e n e d i m o s t r a t o c h e t a l i t e c n i c h e r i s u l t a n o d a l l ' a p p l i c a z i o n e c o n g i u n t a 
d i Analisi della Varianza e d i Analisi Multivariata. Si d i s c u t e i n o l t r e l a t e o r i a 
d e l " K r i g e a g e Universel" d i G . M a t h e r o n , r i d u c e n d o l a a l l e t e o r i e g i à m e n -
z i o n a t e . 

1 . - I N T R O D U C T I O N 

One deals with a function y(x) defined in a region of a real space. 
T h e value of t h e function is known at a few points, xt, of the region. 
One writes yi — y(xt). Interpolation means prediction of t h e value 
of t h e function a t points of t h e region where t h e value of t h e f u n c t i o n 
is unknown. Trend is a simplified f u n c t i o n capable to describe t h e 
"general behavior" of t h e original function. These concepts are b e t t e r 
defined when t h e theoretical approach to the problem is described. 

(*) I D R O T E C N E C O , R o m e . 



2 7 2 L. TOEELLI 

One considers t h e f u n c t i o n in question as a realization of a sto-
chastic f u n c t i o n Y(x) -with t h e following s t r u c t u r e : 

*(«) = /(*) + «(«); Ee(x)±0 [1] 

where j(x) is a deterministic f u n c t i o n describing t h e e x p e c t a t i o n 
EY(x) of Y(x) and e(x) is a stochastic residual with e x p e c t a t i o n zero. 

f(x) is w h a t is usually called t r e n d of y(x) and t h e problem of 
t r e n d analysis is to find an e s t i m a t e as good as possible of f(x). I t 
will be shown t h a t it is possible to find an unbiased e s t i m a t o r which 
has m i n i m u m v a r i a n c e in t h e class of t h e linear estimators of f(x). 

As to interpolation, one w a n t s to find an o p t i m a l predictor of 
Y{x), i.e. a predictor Y(x) such t h a t : 

E( Y(x) — Y(x)) = 0 

E()'(x)— Y(x))2 is m i n i m u m . 

Section 2 gives a schematic account of t h e t h e o r y which can be 
used to a t t a i n t h e above objectives. I t is a r t i c u l a t e d in t h r e e p a r t s , 
one a b o u t t r e n d analysis, one a b o u t interpolation when f(x) = 0 
and t h e last a b o u t interpolation in general. T h e t h e o r y used is t h e 
Analysis of the Variance ( S c h e i e , 1959) (8) and Multivariate Distribution 
Analysis (Anderson, 1958) (2). 

Section 3 discusses a t h e o r y elaborated in t h e l a t e sixties b y 
M a t h e r o n and co-workers called Universal Kriging (Matheron, 1969 (5), 
Matheron, 1971 (6), H u i j b r e g t s and Matheron, 1970 (4) and reduces it t o 
t h e theory presented in section 2. Conclusions are d r a w n in section 4. 

2 . - S C H E M E O F T H E T H E O R Y 

2.1. - Trend Analysis 

Consider equation [1]. If f(x) is f o r m u l a t e d as a linear f u n c t i o n , 
with known coefficients f u n c t i o n of x, of a n u m b e r of p a r a m e t e r s to 
be estimated, one can m a k e use of t h e Analysis of t h e Variance to 
solve t h e problem of e s t i m a t i n g f(x). 

I n f a c t s one constructs a linear model, which in m a t r i x n o t a t i o n 
is expressed as follows: 

y = X'ß + s, Ee = 0, £ = £ e = B [2] 



M O D E R N T E C H N I Q U E S OF T R E N D A N A L Y S I S A N D I N T E R P O L A T I O N 2 7 3 

•where y is t h e vector of t h e observations, ft t h e vector of t h e para-
m e t e r s t o b e e s t i m a t e d , e t h e vector of t h e stochastic residuals a n d X' 
a full r a n k m a t r i x whose ieth row is composed b y t h e coefficients of 
t h e linear expression of f(xt) in ft. ' m e a n s transposed and 
are t h e covariance matrices of e a n d y respectively and are assumed 
k n o w n . 

Given model 2, t h e following results follow (Scheffe, 1959) (8). 

1) ft = (XBxX')^XB Uj is an unbiased estimator of ft which 
h a s m i n i m u m v a r i a n c e in tlie class of t h e linear estimators of ft. I t 
should be noted t h a t ft is different f r o m t h e usual least s q u a r e esti-
m a t o r of ft, 

ft = (XX')-1 Xy 

ft is unbiased b u t has not m i n i m u m v a r i a n c e in t h e class of t h e linear 
estimators of ft. 

2) (Gauss Markov Theorem) Given a vector a if t h e r e exists a 
vector c such t h a t a = cX',aft is t h e u n i q u e unbiased estimator of 
aft which has m i n i m u m variance in t h e class of t h e linear estimators 
of aft. 

Since /(.») is linear in ft, we get 

f(x) = a(x) ft 

where a(x) satisfies t h e condition of t h e Gauss Markov Theorem be-
cause X' is full r a n k . Consequently, t h r o u g h t h e Analysis of t h e 
Variance, one obtains an e s t i m a t o r f(x) = a(x)ft of f(x) which is op-
t i m a l in t h e class of t h e linear estimators of f(x). Moreover, tlie t h e o r y 
gives t h e m a t r i x of t h e covariance of ft 

Xp = (XB-^X')-1 

so t h a t t h e variance of t h e estimator a(x) ft of f(x) is a{x) (XB-lX')~la(x)'. 

2.2. - Interpolation ivlien f(x) = 0 

To solve this problem one uses Multivariate Distribution Analysis. 
Let us consider t h e r a n d o m normal vector Y* = ( Y , Y i , Y 2 , . . . , Y „ ) ' 

where Y = Y(x), x being t h e point of interpolation and Y( = Y(xt), 
such t h a t BY* = 0. 2 „ * = -4 is assumed to b e known. 



2 7 4 L. TORELLI 

.1 m a y be partioned as follows 

A = 
c D' 
DB 

where B = 2 of t h e preceding section, c — v a r Y, D = (cov( Y, Yi), 
c o v ( r , r 2 ) , . . . , c o v ( Y , r „ ) ) ' . 
One lias t h e following result (Anderson, 1958) (2). 
Given Yt = yt, i — 1 , 2 , . . .,n, Y is normally d i s t r i b u t e d with m e a n 
E(YlYt = yr, i = 1 , 2 , . . . , » ) =-- D ' S ^ y 
and variance 

var( YI Yt = i = 1,2,. . .,») = c — D'B-W 

YM(X) — E(YjYi = ?/i, i = 1 , 2 . . .,»)) is obviously an optimal predictor 
in t h e sense of section 1. Subscript M refers to Multivariate. I n this 
method t h e assumption of t h e n o r m a l i t y of Y* is m a d e . Such as-
sumption can be considered safe and it is based on Central Limit 
Theorem considerations. 

2.3. — Interpolation in general 

I t is common sense to use t h e following p r o c e d u r e : 
one gets an o p t i m a l e s t i m a t e / (.r) of f(x) using t h e t h e o r y of 
section 2.1. Then one applies t h e m e t h o d described in section 2.2 to 
t h e r a n d o m f u n c t i o n Z(x) = Y(x) — f(x). T h e final i n t e r p o l a t o r is 
t h e n f (x) + ZM{x). 

3 . - U N I V E R S A L K R I G I N G 

This t h e o r y of interpolation and t r e n d analysis has been developed 
b y Matheron and co-workers in t h e l a t e sixties. A great mass of lit-
e r a t u r e on t h e t h e o r y has been produced b y this school. Only t h e 
essential references are given here. I n t h e p r e s e n t discussion of Uni-
versal Kriging, c o n s t a n t reference is m a d e to t h e article b y H u i j b r e g t s 
and Matheron, in which this t h e o r y is concisely and clearly exposed. 



M O D E R N T E C H N I Q U E S OF T R E N D A N A L Y S I S A N D I N T E R P O L A T I O N 2 7 5 

The assumption of H u i j b r e g t s and Matheron is t h a t one deals 
with a realization of a stochastic f u n c t i o n Z(x) for which t h e m o m e n t s 
of 1st and 2nd order are defined: 

EZ(x) = m(x) 

EZ(x)Z(y) = C(x,y) + m(x)m(y) 

where G(x,y) — cov(Z(x),Z(y)). 

Moreover 

m(x) = a, /' (x) (x e 7 ) 

(I.H-M) 

(1.H-M) 

where V is some neighborhood of t h e points of observation. 
I n other words, m(x) is a linear f u n c t i o n with known coefficents 

f u n c t i o n of x, viz f'(x), of a n u m b e r of p a r a m e t e r s <u to be e s t i m a t e d . 
A p p a r e n t l y t h e s t r u c t u r e described b y (I,H-M) and (1,II-M) is 

coincident with t h e linear model of equation [2]. 
As f a r as trend analysis is concerned, t h e proclaimed goal of 

H u i j b r e g t s and Matheron (4) is to find a linear e s t i m a t o r of m(x) un-
biased and with m i n i m u m variance. Now, given t h e f a c t t h a t t h e 
p r o b a b i l i t y s t r u c t u r e considered b y H u i j b r e g t s and Matheron (4) is a 
linear model and given t h e unicity p r o p e r t y of t h e e s t i m a t o r of t h e 
Gauss Markov theorem, it follows t h a t t h e H u i j b r e g t s and Matheron (4) 
solution is coincident with t h e s t a n d a r d solution of Analysis of the 
Variance given in section 2.1. 

As t o t h e problem of interpolation in absence of t r e n d , t h e pre-
dictor proposed b y H u i j b r e g t s a n d Matheron is 

Z k = X a f Z a , « = 1 , 2 , . . . , « 

a 
with coefficients ?.k verifying t h e following s y s t e m 

2 / j i °afl = C (y.a, xo) (S.K., H-M) 

where Za = Z(Xa) are t h e observations, h refers to t h e p o i n t of in-
terpolation, i.e. x0, oap = C (xa, xp) and a runs, like /?, on t h e n u m b e r 
of observations n. 

If we set Ik = (At1, A-2, . . . , h " ) , a n d write, according to t h e 
n o t a t i o n of section 2, y = (Zi, Z°, . . ., Z„) and I) = (G(xi, xa), 



6 L. TORELLI 

C(x2, x0), • . •, C{x„, x0)) and we let B be t h e covariance m a t r i x of y, we 
can write H u i j b r e g t s and M a t h e r o n ' s expressions (J) in m a t r i x n o t a t i o n : 

Z, = h y Xk B = D' 

from which 

Zk = D'B-iy 

coincident with t h e solution of section 2.2. 
The H u i j b r e g t s a n d M a t h e r o n ' s solution to t h e problem of in-

terpolation in general is given b y : 

Z* = M* (Xo) + 2 a (Za — M* (xa)) (5,II-M) 

where M*(x0) is t h e optimal t r e n d e s t i m a t o r a n d 2 « — (xa)) 

is the solution discussed above for t h e stochastic f u n c t i o n w i t h no 
t r e n d , Z(x) — M*(x). This is again coincident with t h e solution of 
section 2.3. 

However, t h e article of H u i j b r e g t s a n d M a t h e r o n (4) offers a few 
interesting marginal results, since it shows t h a t Zi.- is an o p t i m a l linear 
predictor i n d e p e n d e n t l y of the n o r m a l i t y of Z(x) and t h a t t h e general 
interpolator (5,H-M) which is t h e sum of t h e o p t i m a l linear e s t i m a t o r 
of t h e trend and of t h e optimal linear p r e d i c t o r of t h e residuals is 
itself an o p t i m a l linear p r e d i c t o r . Moreover, it provides t h e overall 
v a r i a n c e of predictor (5,II-M). 

4 . - C O N C L U S I O N 

This article is m e a n t to be a schematic exposition of t h e theoretical 
f r a m e w o r k on which recent techniques of trend analysis and inter-
polation rest. I t is shown t h a t such techniques consists in t h e joint 
application of t h e Analysis of the Variance and of Multivariate Dis-
tribution Analysis. I t is also shown t h a t t h e b u l k of Universal Kriging 
is reducible to these long established and well k n o w n theories. I n 
this we t h i n k to h a v e given a c o n t r i b u t i o n to t h e unification a n d 
clarification of t h e developments in t h e field. 

I n t h e p a p e r , t h e basic p r o b a b i l i t y s t r u c t u r e , i.e. t h e linear model, 
is assumed as given. The construction of a suitable model, which 



M O D E R N T E C H N I Q U E S OF T R E N D A N A L Y S I S A N D I N T E R P O L A T I O N 2 7 7 

implies t h e determination of t h e f o r m of t h e linear expansion of f(x) 
and of t h e covariance s t r u c t u r e of t h e residuals is n o t discussed, being 
b e y o n d t h e scope of t h e present article. 

I n a n y case, this is a r a t h e r i n t r i c a t e m a t t e r , in which t h e Analysis 
of the Variance is again of great help. F o r an example of i n q u i r y on 
t h e covariance s t r u c t u r e of a stochastic f u n c t i o n defined in a unidi-
mensional discrete space, t h e reader is referred to Torelli (10) and 
Torelli and Chow (»). 

R E F E R E N C E S 

( ! ) A G T E R B E R G F . P . , 1 9 7 4 . - " G e o m a t h e m a t i c s " , E l s e v i e r , A m s t e r d a m . 
( 2 ) A N D E R S O N T . W . , 1 9 5 8 . - An introduction to Multivariate Statistical 

Analysis. W i l e y , N e w Y o r k . 
( 3 ) D E L F I N E R P . a n d J . P . D E L H O M M E , 1 9 7 3 . - Optimum interpolation by 

Kriging, i n Display and Analysis of Spatial Data. W i l e y , N e w Y o r k . 
( 4 ) H U I J B R E G T S C . a n d G. M A T H E R O N , 1 9 7 0 . - Universal Kriging: an 

Optimal Method for Estimating and Contouring in Trend Surface 
Analysis. C I M M , I X I n t e r n . S y m p . , M o n t r e a l . 

( 5 ) M A T H E R O N G., 1969. - Le Krigeage Universel. " C a h i e r s d u C e n t r e d e 
M o r p h o l o g i e M a t h é m a t i q u e s " , 1, F o n t a i n e b l e a u . 

(®) M A T H E R O N G . , 1971. - The Theory of Regionalized Variables and its 
Applications. " C a h i e r s d u C e n t r e d e M o r p h o l o g i e M a t h é m a t i q u e " , 5, 
F o n t a i n e b l e a u . 

(7) OLEA R . A . , 1974. - Optimal Contour Mapping Using Universal Kriging. 
" J . J . R . " . 

( 8 ) S C I I E F F É H . , 1959. - The Analysis of the Variance. W i l e y , N e w Y o r k . 
( " ) T O R E L L I L . a n d V. T . C H O W , 1972. - Tests of Stationarity of Hydrologie 

Time Series. « I n t . S y m p . on U n c e r t a i n t i e s i n H y d r o l o g i e a n d W a t e r 
R e s o u r c e S y s t e m s », T u c s o n , D e c e m b e r . 

( 1 0 ) T O R E L L I I;., 1973. - The Analysis of Monthly Hydrologie Time Series. 
I ' l l . D . T h e s i s , U n i v e r s i t y of Illinois.