Zbornik gozdarstva in lesarstva, 38, 1991, s. 257 - 273 
GDK 561.015.5 
Math. Subj. Class. (1985) 62102, 90C30 
EKSTREMNE KORELACIJE 
Anton CEDILNIK* 
Izvleček 
V članku je definirana količina, ki meri kvaliteto aproksimacije tabelarično podane funkcije s 
funkcijo iz neke družine D nekonstantnih funkcij. V ta namen je treba poiskati infimum in su-
premum množice korelacijskih koeficientov med dano funkcijo in funkcijami iz D. Tu je to 
storjeno za družino vseh nekonstantnih funkcij in za družino rastnih funkcij. 
Ključne besede: aproksimacija, ekstremna korelacija, nelinearni program 
EXTREME CORRELATIONS 
Anton CEDILNIK* 
In the article we define a quantity which measures the quality of the approximation of a 
function, given by a table, with a function from a certain family D of nonconstant functions. 
For this purpose one has to find the infimum and the supremum of the set of correlation coef-
ficients between the given function and functions from D. We do this for the family of ali 
nonconstant f unctions and for the famiJy of growth functions. 
Key words: approximation, extremal correlation, nonlinear programming 
Dr., docent, Biotehniška fakulteta, Oddelek za gozdarstvo, Univerza v Ljubljani, 
61000 Ljubljana, Večna pot 83, Slovenija 
258 
Zbornik gozdarstva in lesarstva, 38 
CONTENTS 
I. lntroduction .............................. 
II F . . . ,, - approx1mat1on ....................... . 
III. R~ - approximation (n > 2) ................ . 
Appendix: a nonlinear program ................. . 
I. INTRODUCTION 
In table 1 and table 2 we have measurings of function values of a function y (x), 
which we approximate with the function w1 = 0.5 x + 3.0. lts correlation coeffi-
cient with data from table 1 is O. 7048 and with data from table 2 is 0.9986. In the 
first case the correlation is weak and in the second one very strong. Nevertheless, it 
can be shown that, contrary to the expectations, in the first case this approximation 
is the best (in the sense that the correlation cannot be increased), but in the second 
case we easily find a better approximation, for instance w2 = O.OS x2 + 0.35 x + 
+ 3.10, for which the correlation coefficient is 1. Therefore, the correlation coeffi-
cient is not necessarily a good measure of the quality of approximation. 
X y X y 
l 3.3 3.4 3.5 3.5 3.8 1 3.5 
2 3.0 4.0 4.1 4.9 2 4.0 
3 4.1 4.4 4.7 4.8 3 4.6 
---
Table 1 Table 2 
The purpose of this article is to introduce a better measure of the quality of approxi-
mation (definition 1) and t.o calculate it in two special cases. 
But, firstly, some conventions and definitions! 
We deal with a functiony(x) for which the measurings of function values are in 
able 3. The table is ordered: X1 < X2 < ... < Xn. 
X y 
XJ Y11 Y12 Yts, 
X2 Y21 Y22 Y2s2 
Table 3 
x„ Ynt Yn2 ... Ynsn 
259 
CedilnikA.: Ekstremne korelacije. 
n 5k 
sk 2. l (k = l , ... ,n). E ,_ E ' E ,.._, r (k and / Will be the summation indexes 
k k= 1 I I= 1 
only in these two sums). 
. (1) 
(2) 
s:= Esk ';?:_n 
k 
- 1 '{' Yk:= - ~Ykl sk , (k= l, ... ,n) 
The values Yk form a new function y. 
(3) - 1 '{' -y := s fSkYk 
(4) -: : = [ ~ ; (vk/ -y)it 2 = [ ~ ;Yi1 - syi•l/2 
(S) w : =,: [ 1tsAJf - syi]-112 
Presumption: 
(P) There is at least one k with Yk:l=y. 
Obviously, n>2, -r >o, w ~ l. w = 1 if and only if s1 = ... = sn = 1, hence if s = n. 
Correlation coefficient r (v, u) between the function y f rom table 3 and some other 
function u with values uk (k = 1, ... ,n) is the correlation coefficient for table 4: 
y Y11, .. ,, Y1s1, Y21, .. ,, Y2s2, .. ·,YnJ, .. ,, Ynsn 
u UJ,, ... , UJ, U2, .. ,., u2, .. ,., Un, .... , Un 
Table 4 
(6) 
L L (vk1-.Y)(uk-ii) 
1„ k / r V', u): = ~-------------
[ }: }: ()'k1-y)2 · L L (u - ii)2) J/2 
k I k / k 
where 
(7) 
260 
Zbornik gozdarstva in lesarstva, 38 
Let us approximate the f unction y from table 3 with a f unction w from some family 
D of nonconstant functions, defined on the set (x., ... , xnl. lr(y, u)I ueDJ is a subset 
of the interval [-1, 1) with 
(8) A(v) := inf(r(y, u)lueDI , 
(9) B(y) : = sup lr(y, u) 1 u e DJ , 
(10) A(v) ::S, r(y, w) ,:5 B(v). 
Dejinition J. Quality oj D-approximation oj f unction y with f unction w e D is 
(11) K, .. weD)·- 2r(v,w)-B(y)-A(v) v-, · - B(v) - A(v) 
The definition is based on fig. 1. 
-1 
o 
/ I \ 
/ I \ 
/,, I \ 
/ / \ ,, . , I :,,.., 
-1 K 
Figure l 
1 
' 
1 
The quality of D-approximation is again a number between -1 and 1, and 
(12) K(v, w e D) = K(ay + /3, yw + t5 ED) 
for ali such a, /3, y, t5 that ay > O and still is yw + t5 e D. 
Obviously, the family D must be at least so extensive that f or each y from table 3 
and with the presumption (P), the denominator B(v)-A(y) in (11) is > O. We shall 
name such family sufficient. 
Examples. 
1. The family Fn (n ~ 2) of all nonconstant functions on the set lxt,··•,xnl is suffi-
cient. For proving this, calculate r(y, u), where uk = O (k ::/- i), u; = + 1, and de-
mand that the.result is always the same for al\ i and both signs. 
261 
Cedilnik A.: Ekstremne korelacije. 
2. A function u on lx1, ... ,xnJ is growthfunction if O< u1 ~ ... ~ Un and is non-
constant if u1 t: Un, The family R~ (n ~ 3) of such functions is sufficient. To see 
this, calculate r(v, u), where Ut = ... = u;-1 = O, u; = ...l., u;+ 1 = ... = Un = 1 
(2 Si< n-1), and demand that for each i and any le(0,1): -;,r/tJA = O. The family 
R~ is not sufficient. 
3. A function u on (xJ,,,,,xnl is nonconstant linear if uk = axk + /3 (k = l, ... ,n) 
and a f:; O. The family Ln (n ~ 3) of these functions is not sufficient, since for the 
function y from table 5 we have: A (v) = B(v) = O. 
1 X y 
Xn-X 
O (2 ~ k :5, n-1) 
x-xi 
Table 5 
We already said that the family L2 = F2 is sufficient. 
(n the next two sections we shall find A (v) and B(y) and so also K(y, w r:D) for the 
families D = Fn (n ~ 2) and R~ (n >- 3). 
I. Fn - APPROXIMATION 
A(v) = min (r(y, u)I u eFnJ = -w-1 
B(y) = max (r(v, u)I u eFnl = w -t = r (v, y). 
or any w e Fn there is 
K(y, we Fn) = w .r (v, w). 
roof. lt's easy to verify (using (l)-(6)) that 
6) 
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Zbornik gozdarstva in lesarstva, 38 
The numerator in (16) can be ragarded asa scalar product, therefore, by Cauchy-
-Schwartz inequality, r(y, u) is the biggest for uk - i1 = A.OiA, - y) 
(A>O, k= l , ... ,n), and for such u we have: r(y, u) = w -J. 
Obviously, r(v, u) = -r(y, -u), from where we get (13). Then, (15) follows from 
the definition (11). QED. 
-Notes. 
l. For s1 = ... = Sn = 1 we have: K(y, w e Fn) = r(y, w). 
2. The f unction y has not only the highest correlation with y but also the lowest 
sum of squares of declinations from y (even if the presumption (P) is not valid). The 
proof is simple. 
3. For n > 2 we can continuously deform y to -y not going through the constant. 
Therefore: 
(17) lr(v, u)I u EFnl = [-w-1, w - 1]. 
III. R~ -· APPROXIMATION (n > 2) 
This tirne we'll approximate the function y with a function u e R~: 
O < u1 S ... S Un -1 u1• Let us change y with a new function p: 
. (18) (k = l , ... ,n) . 
For this function it holds 
(19) LPk = o, 
k 
(20) [p7/sk = W -l ~ l. 
k 
Let us also change u with q: 
(21) Qk : = (uk - u,)!t5(u), 
where 
(22) c5(u): = ~ ~ sk (Uk- Ut)> o. 
For q it holds: 
(23) 
(24) 
O = Q1 < Q2 < •·· < Qn, - - -
263 
CedilnikA.: Ekstremne korelacije. 
Because of q e R~ and Qk = (qk - q 1)/o(q) for each k, we conclude that while u 
having traversed ali the set R~, q traverses ali that subset defined with (23) and (24). 
Easy computation shows: 
(25) r(y, u) = L PkQk · [r SkQI - s]· - 112• 
k k 
The maximum of this function is f ound in Appendix,. If we consider also that 
minr = -max(-r), the equations (26) and (27) of the next theorem f ollow. 
Theorem 3. Leta : = lP2,•••,Pn1 and b : = [s2, ... ,snJ- Then: 
(26) A(y) =-Hn-t (-a, b, s, -s), 
(27) B(y) = Hn-1 (a, b, s, -s), 
(28) lr(y, u) 1 u E R~J = [A (J'), B(y)]. 
Proof. We have to prove (28) else. The family R~ is convex. If min r(y, u) = 
= r(y, u') and max r(y, u) = r(y, u "), then A ~ r(y, Ju" + (1-,l)u') is a continu-
ous function, which maps the interval [O, 1) onto the interval [A(y), B(y)J. QED. 
For the illustration we'll examine a special case: R~ - approximation of a noncon-
stant increasing onevalued function. Instead of table 3 we have much more special 
table 6, for which we demand: 
Y Y1, Y2, .•• , y„ 
Table 6 
l. n > 2, 
2. XJ < X2 < ... < X 11 , 
3. Yt ~ Y2 :::; ••• :5 Y11 f= Yt · 
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Zbornik gozdarstva in lesarstva, 38 
The function u: = y -Y1 (uk:= Yk -Y1 (k = 1, ... ,n)) is from R~. 
Because of r(v, u) = l we already know: 
(29) B(v) = l. 
According to (26) we also have: 
(30) A (v) = -Hn-1 (-a, b, n, -n) 
where 
(31) 
(32) 
(33) 
(34) 
a; = (v;+ 1 -Y)/ -r (i = l, ... ,n-1), 
b; = 1 (i = 1, ... ,n-1), 
- l r Y = - 1.JYk, n k 
With the extension of (31) to index i = O we have: 
(35) ao S 01 ~ ••• S On-J, 
n-1 
(36) E a; = o. 
i=O 
In the notation from Appendix there is 
(37) 
j-1 
E; = ~ E a; U = 1, ... ,n-1), 
. J i=O 
(38) 
Therefore, P = n - 1 and 
a-1 
(39) Hn-l (-a, b, n, -n) = max a(n~a·) E a;. 
o<a<n i=O 
From (30) we endly get 
(40) A(y) =....!-min-./ n (uy- Ey;). 
"t o<a<n v -a(n-a) i= 1 
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CedilnikA.: Ekstremne korelaciie. 
It is not difficult to _see that A (y) = r(y, u), _where u is a func~ion such as: u1 = ... = 
= u x = O, u x+ 1 = ... = Un = 1 (for certam x ef 1, .... ,n-l}J. Hence: A (y) > A (u ). 
From (40) we find out: 
(41) A(u) = min { V- ,. (~I) • }(_x)(n-1) } · 
Minimum is got for x = 1 andx = n-1 and its value is n.:_l · So we have found 
found the infimum of the numbers A(y): 
(42) 1 A(y) > n-1 . 
Note. The results of this section are because of (12) valid also for the Rn-ap-
proximations, where Rn is the family 0f nonconstant increasing functions. 
APPENDIX: A NONLINEAR PROGRAM 
We shall solve the next nonlinear program. 
There are given a = [ak1mxJ, b = [bk1mxJ and numbers c and d. 
These parameters answer the following four conditions: 
me(l,2,3, ... 1, 
bk > (k = 1, ... , m). 
C> 0, 
( 1) € : = c2 + d L b k > O, 
k 
III 
where, E ,_ E and - avoiding the triviality - at least one ak is -1= O. 
k k=I 
266 
Zbornik gozdarstva in lesarstva, 38 
Maximize the objective function 
(2) z = [Zk1m-!1 - h,n (z; a, b, C d) : = t akZk . [ ~ bkz'i + d ]- 112• 
subject to the constraints 
(3) 0 ~ Z t < . . . ~ Z1n, 
(4) L bkZk = C. 
k 
The function hm is well defined: 
The constraints (3) and (4) determine a nonvoid convex compact set S, so the func-
tion hm, which is contionuous, for certain has the maximum 
(6) H,,, (a, b, c, d) : = max lh111 (z: a, b, c, d) 1 z e SJ. 
Form = 1 there is only one point z: z1 = clb1 in Sand so: 
(7) 
ca1 
H 1 (a, b, c, d) = ~ · 
veu1 
The case m = 2 demands much more work - though quite elementary - than the 
previous one. lf the condition 
(8) 
is fulfilled, then 
forJ: 
{\O) (k = 1,2). 
lf the condition (8) is not true, then 
267 
CedilnikA.: Ekstremne korelacije. 
(11) 
for z: 
(12) Zt = O, Z2 = clb2 
(if the first term in. (11) is greater) or 
(if the second term in (11) is greater). 
From now on we shall suppose that m > 2. The Lagrangian function: 
m-1 
(14) L = hm + E lk (Zk+ 1 - Zk), 
k=I 
where J-s are the Lagrange multipliers. The corresponding Kahn-Tucker system 
U = l, ... ,m-1): 
(15) Zj ~ 0, 
(16) t1 LI a Zj < O, 
(17) Zj • ( ali tJZj) = 0, 
(18) A.j > o, 
(19) aLI a)i.j = Zj+I -Zj ~ o, 
(20) Aj · la LI i1 A-j} = ).J (Zj + 1 - ZJ) = O, 
considering (4) and 
(21) a Zml a Zj = -bjlbm, 
Let us sum up the equations (17), using (4) and (20): 
EakZk 
E * 1 ] . (E akzi + d)-112. 
bkz + d * 
(22) Am-1 = -[am -bm (Z,n + ~) 
k 
We shall change A.-s: 
(23) (j = 1, .... ,m-1). 
268 
Zbornik gozdarstva in lesarstva, 38 
Putting (22) and (23) in ( 16) and ( 17) we get: 
(24) 
(25) Zj- [left side of (24)] = O. 
-µ, u = 1) 
µj (1 <.j<m) $ O, 
U = m) 
We now have to solve the system (3), (4), (24), (25), (26) and (27): 
(26) µj > o, 
(27) µj (Zj + 1 - Zj) = 0 
(for j= l, .... ,m-1). 
One more definiton (k= l , .... ,m): 
(28) 
Let P be the index of the greatest Ek: 
(k > P ~ Ek < Ep) /\ (k s; P ~ Ek < Ep). 
Suppose that a is the adjacent index, for which Za> O, Za-1 = O (or a = 1 if z1 > O). 
a could be any number in (1, ... ,mL Because of (27) it is for a:f= 1: . 
µa-1 = O. 
The inequalities (24) are because of (25) the equations, as well, for a <i < m. Sum-
ming up these equations we get: - -
Ea = L OkZk I (L bkZk + d), 
k k 
and (24) can be written in this way: 
-µ1 (i= l) < O (O<.;<a) 
(32) 
Uj-bj(Zj + ~Ea + µj-1 -µj (1 <.i<m) 
µm-1 U=m) = O (a9"5:_m) 
269 
Cedilnik A.: Ekstremne korelacije. 
Suppose that a> 1 and i<a. Summing up the inequalities (31) for i~ <a we find 
out 
(33) 
d m 
(Ea - E;) · (c + - I: b·)-c j=i J 
and therefore 
(34) E; S Ea (i<a), 
hence 
(35) a $ p, 
For a>l and 
(36) 
{ 
O (i = 1) } 
µJ-1 (i> l) 
~ o, 
OSi<a> 
the inequalities (31) are fulfilled, so we can forget them. But from the equations (32) 
we can calculate ali other µ-s: from the first equation we get µa, from the second one 
µ«+ 1, and so on. Summing up ali this equations, we get the identity, which means 
that these equations are dependent and the system is surely solvable. We find out: 
(a~i<m). 
ow we shall prove that the system (27) is already fulfilled too. This system is 
quivalent with 
111-l 
8) I: µ; (Z;+ 1 - Z;) = 0, 
i=a 
f, of course, a<n; for a=n there is no equations (27) at ali. Firstly we shall write 
37) in another way: 
39) 
270 
Zbornik gozdarstva in lesarstva, 38 
An auxiliary· formula: 
111-1 ;. 
E (Z;+ 1-Z;) · E <Tj = (Za+ 1-Za)<Ta + (Za+2 - Za+ 1) (<Ta + <Ta+ 1) + •·· + 
i=a j=a 
+ (Zm - Zm-1) (<Ta + •·· + <Tm-1) = 
= <Ta (Z,n - Za) + <Ta+ 1 (Zm - Za+ 1) + •·· + <Tm-1 (Zm - Zm-1) = 
111-I m-1 
= Z,n E <T; - E <T;Z;. 
i=a i=a 
111-I i m m 
(40) E (Z;+ J - Z;) E <Tj = Zm E <T; - E <T;Z; 
j=a j=a i=a i=a 
Putting (39) in (38) with the help of (4) and (40) we rediscover (30). 
The previous Kuhn-Tucker system is now reduced to the following one: 
m 
(41) 
E U;Z· fi·----. a '} m 
Hm (a,.b, c, d) = ✓m ,. = Ea ~ b;Zfk + d, 
. ~ b r,T + d . 1 = a 
1=a 
(42) 0 < Za $ . . . ~ Z,n , 
"' 
(43) E b;Z; = C' 
i=a 
(44) 
d III ; 
(Ea-E;+1> (c + c ~ .~i) > Ea ~ bp.j (a$j<m). 
J=I+ J ]=O 
Now we shall consider the sign of Ep. 
Case 1. Ep = O. 
111 
For the function z': Z; = O (i<P), z; = c!E bj (i2P), 
i=P 
we easily find: h,n(z'; a, b, c, d) = O. From (41) it follows then: Ea-:ZO. 
But since Ep ~Ea, there must be Ea= O and 
(45) Hm(a, b, c, d) = hm (z'; a, b, c, d) = O. 
271 
CedilnikA.: Ekstremne korelacije. 
Case 2. Ep < O. 
,n 
For the function z" : z; = O (i<a), z; = c/ E bj (i>a), 
j=a 
we get 
,n 
(46) Hm (a, b, c, d) > hm (z"; a, b, c, d) = Ea [d + c2/ E bJ 112 
i=a 
On the other side, by (41): 
m m 
(47) Hm (a, b, c, d) < max Ea ( E b;z; + d) 112 = Ea · (min E b;ZT + d) 112, 
i=a i=a 
where we determine the extreme for ali z -s satisfying (43) only. It is easy to see that 
the extreme is gotten for z= z" and so: 
(48) Hm (a, b, c, d) = hm (z"; a, b, c, d). 
The question remains, how to find a. With some effort we transform (44) into 
(49) 
"' m m . 
~ a/ E b· > E a·; i: b· 
;=a j=a J j=i J j=i J 
(a<i<m) 
But this condition is complicated and perhaps it would be simplier to say: 
(50) H,n(a, b, c, d) = max Ea t + c2; f b; 
15a5p ;=a 
Case 3. Ep> O. 
For the function z' from case 1 there is hm (z'; a, b, c, d) > D. Hence 
Hm (a, b, c, d) > O and Ea > O because of (41). From (44) it follows then that 
(51) a = p, 
If P = m, we have 
(52) H / b CO,n 
m \a, ' c, d) = y b,n(c2 + dbm) · 
From now on let P < m. It is easy to verify that 
272 
Zbornik gozdarstva in lesarstva, 38 
for all z which correspond the conditions (42) and (43). The right side of this equa-
tion is minimal exactly for that z for which the function hm is maximal. Hence, if: 
(54) 
then (53) is equivalent to 
(55) M = ~ [-H;,,J - 1 
or, explicitely, 
(56) Hm (a, b, c, d) = Ep y-d - 2M. 
The determination of M is a problem of quadratic programming and there is no 
need to discuss it here. 
POVZETEK 
če je tabelarično podana neka funkcija in jo aproksimiramo s funkcijo iz neke iz-
brane družine nekonstantnih funkcij, ustrezni korelacijski koeficient v splošnem ne 
more biti katerokoli število med -1 in 1. V tem sestavku zato s pomočjo ekstremnih 
korelacijskih koeficientov definiramo mero za kvaliteto aproksimacije (definicija 
1). Ekstremne korelacije izračunamo najprej za družino vseh nekonstantnih funk-
cij, nato pa še za družino rastnih funkcij. Medtem ko je prva naloga preprosta, pa 
pri rastnih funkcijah zahteva po monotonosti povzroči, da je iskanje ekstremnih ko-
relacij reševanje nelinearnega programa. Tega eksaktno rešimo v Dodatku. 
SUMMARY 
If a function given by a table is approximated by a function from a certain family of 
non-constant functions, then, in general, the correlation coefficient cannot be any 
number between -1 and 1. The aim of the study was to define a quantity that mea-
sures the quality of approximation of a function by calculating extreme correlation 
coefficients (Definition 1 ). Extreme correlations were calculated first f or the family 
of ali non-constant function~ and \\\en fot tne fa.m\\"y oY growtn Yunctions. The for-
mer is an easy task whi\e the \atter, due to the demand f or monoto~ici~y, ~ecoT?est~: 
f act a so\ving of a non-\inear program, the exact so\ution of whtch 18 gwen m 
Appendix. 
273 
CedilnikA.: Ekstremne korelacije. 
REFERENCES 
CEDILNIK A., 1986. Optimalna aproksimacija rastnih funkcij. Zb. gozd in les. 
28, 5-16. 
CHIANG, A. C., 1974. Fundamental Methods of Mathematical Economics. 
McGraw-Hill, Inc., Tokyo. · 
JETER, M. w·., 1986. Mathematical Programming. Marcel Dekker, Inc., New York. 
LUENBERGER, D. G., 1969. Optimization by Vector Space Methods. John Wiley 
& Sons. Inc., New York . 
. PETRIC, J., ZLOBEC, S., 1983. Nelinearno programiranje. Naučna knjiga, 
Beograd.