S a is ics and P obabili y Le e s 83 (2013) 1580–1587
Con en s lis s a ailable a SciVe se ScienceDi ec
S a is ics and P obabili y Le e s
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On he op imal designs o he p edic ion o
O ns ein–Uhlenbeck shee s
Sándo Ba ana, Kinga Sikolyaa, Milan S ehlíkb,∗
aFacul y o In o ma ics, Uni e si y o Deb ecen, Kassai ú 26, H-4028 Deb ecen, Hunga y
bIns i u ü Angewand e S a is ik, Johannes Keple Uni e si y in Linz, Al enbe ge S aße 69, A-4040 Linz a. D., Aus ia
a icle in o
A icle his o y:
Recei ed 8 No embe 2012
Recei ed in e ised o m 5 Ma ch 2013
Accep ed 6 Ma ch 2013
A ailable online 15 Ma ch 2013
MSC:
p ima y 62K05
seconda y 62M30
Keywo ds:
O ns ein–Uhlenbeck shee
In eg a ed mean squa e p edic ion e o
En opy
Fishe in o ma ion
abs ac
Compu e simula ions a e o en used o eplace physical expe imen s o explo ing he
complex ela ionships be ween inpu and ou pu a iables. We s udy he op imal design
p oblem o he p edic ion o a s a iona y O ns ein–Uhlenbeck shee on a mono onic se
wi h espec o he in eg a ed mean squa e p edic ion e o c i e ion and he en opy
c i e ion. We show ha he e is a subs an ial di e ence be ween he shapes o op imal
designs o O ns ein–Uhlenbeck p ocesses and shee s. In pa icula , we show ha he
op imal p edic ion based on he in eg a ed mean squa e p edic ion e o does no
necessa ily lead o space- illing designs.
©2013 Else ie B.V. All igh s ese ed.
1. In oduc ion 1
A common p oblem in spa ial s a is ics is deciding how o choose a se o sample loca ions in o de o p edic a 2
andom p ocess in an op imal way. In he p esen pape we s udy he p oblem o op imal design o he p edic ion o an 3
O ns ein–Uhlenbeck (OU) shee on a mono onic se wi h espec o he in eg a ed mean squa e p edic ion e o (IMSPE) 4
c i e ion and he en opy c i e ion. 5
We conside he s a iona y p ocess 6
Y(s, )=θ+ε(s, )(1.1) 7
wi h he design poin s aken om a compac design space X= [a1,b1]×[a2,b2], whe e a1<b1and a2<b2and ε(s, ), 8
s, ∈R, is a s a iona y OU shee , ha is a ze o-mean Gaussian p ocess wi h co a iance s uc u e 9
Eε(s1, 1)ε(s2, 2)=σ2
4αβ exp (−α|s1−s2|−β| 1− 2|),(1.2) 10
whe e α > 0, β > 0,σ > 0. We ema k ha ε(s, )can also be ep esen ed as 11
ε(s, )=σ
2√αβ e−αs−β We2αs,e2β ,12
whe e W(s, ), s, ∈R, is a s anda d B ownian shee (Ba an e al.,2003;Ba an and Sikolya, 2012). 13
∗Co esponding au ho . Tel.: +43 732 2468 6806; ax: +43 732 2468 9846.
E-mail add ess: [email p o ec ed] (M. S ehlík).
0167-7152/$ – see on ma e ©2013 Else ie B.V. All igh s ese ed.
h p://dx.doi.o g/10.1016/j.spl.2013.03.003
S. Ba an e al. / S a is ics and P obabili y Le e s 83 (2013) 1580–1587 1581
In o de o apply he usual no a ion o spa ial modelling (Kiseľák and S ehlík, 2008) we in oduce σ:= σ/(2√αβ) and1
ins ead o (1.2) we in es iga e2
Eε(s1, 1)ε(s2, 2)=σ2exp (−α|s1−s2|−β| 1− 2|),(1.3)3
whe e σis conside ed as a nuisance pa ame e .4
Fu he , we equi e ha he shi ed OU shee (1.1) is measu ed a he poin s cons i u ing a mono onic se . A mono onic5
se can be de ined in an a bi a y Hilbe space H, wi h eal o complex scala s. Fo x,y∈H, we deno e by ⟨x,y⟩ he eal pa 6
o he inne p oduc . A se E⊂H×His called mono onic (see Min y (1962,1963)) p o ided ha o all (x1,y1), (x2,y2)∈E7
we ha e ⟨x1−x2,y1−y2⟩ ≥ 0. A p ac ical example o such a se is p o ided by he measu emen s on iso he ms o a8
s a iona y empe a u e ield, wi h se e al applica ions in he mal slab modelling (see e.g. Babiak e al. (2005)o Koizumi9
and Jin (2012)). In his pape we conside he ollowing e sion o a mono onic se :10
Condi ion D. The design poin s {(s1, 1), (s2, 2), . . . , (sn, n)}⊂X,n∈N,a e no o e lapping (obse a ions wi hou 11
epe i ions); mo eo e , 0<s1<s2<··· <snand 0< 1< 2<··· < nhold.12
Fo he shi ed OU shee (1.1) obse ed a poin s sa is ying Condi ion D,Ba an and S ehlík (submi ed o publica ion)13
de i ed he exac o m o he Fishe in o ma ion (ma ices) co esponding o he end pa ame e θand o co a iance14
pa ame e s =(α, β), and he au ho s in es iga ed he p oblem o D-op imal designs o he es ima ion o hese quan i ies.15
We de i e he exac o m o he IMSPE ecen ly used in se e al pape s (see e.g. C a y (2002) o Sacks e al. (1989)), and16
show ha in con as o he case o he OU p ocess on a compac in e al in es iga ed by Baldi An ognini and Zago aiou17
(2010), he equidis an design is usually no op imal. This is an impo an obse a ion, and also consis en wi h he esul s18
o P onza o and Mülle (2012), since s a is ical olklo e is ha space- illing designs ha e supe io p ope ies when i comes19
o he p edic ion o emula o unc ions. We also in es iga e he p ope ies o he op imal design wi h espec o he en opy20
c i e ion (Baldi An ognini and Zago aiou, 2010;Shew y and Wynn, 1987).21
Obse e ha by condi ion Condi ion D, he esul s p esen ed he e a e no applicable o space- illing designs. In he22
gene alcase he mainp oblemis hedi icul yo inding hein e se o heco a iancema ixo heobse a ions.Theau ho s23
ha e some p elimina y esul s o egula g id designs, whe e he co a iance ma ix is he K onecke p oduc o wo simple24
co a iance ma ices co esponding o he wo spa ial dimensions, and his special o m allows inding a easonable o m25
o he in e se and calcula ing he objec i e unc ion o he IMSPE c i e ion. Howe e , he p esen a ion o hese esul s is26
beyond he space limi a ions o his wo k.27
2. Op imal design wi h espec o he IMSPE c i e ion28
Suppose we obse e ou p ocess Y(s, )a he design poin s {(s1, 1), (s2, 2), . . . , (sn, n)}sa is ying Condi ion D. The29
main aim o he k iging echnique consis s in p edic ing he ou pu o he simula o on he expe imen al egion, and o 30
any un ied loca ion x=(x1,x2)∈X he es ima ion p ocedu e is ocused on he bes linea unbiased es ima o o Y(x)31
gi en by
Y(x)=
θ+Q⊤(x)C−1(n, )(Y−1n
θ), whe e Y=(Y(s1, 1), . . . , Y(sn, n))⊤is he ec o o obse a ions, 1nis he32
column ec o wi h en ies 1 o leng h n,C(n, )is he co a iance ma ix o Y,Q(x)is he ec o o co ela ions be ween33
Y(x)and Yde ined by Q(x)=(ϱ(x,s1, 1), . . . , ϱ(x,sn, n))⊤wi h ϱ(x,si, i):= exp (−α|x1−si|−β|x2− i|)and 
θis34
he gene alized leas squa es es ima o o θ, ha is
θ=1⊤
nC−1(n, )1n−11⊤
nC−1(n, )Y. Usually, co ela ion pa ame e s35
α, β a e unknown and will be es ima ed by he maximum likelihood me hod. Thus, he k iging p edic o is ob ained by36
subs i u ing he maximum likelihood es ima o s (MLE) (α,
β) o (α, β) and in such a case
Y(x)is called he MLE–empi ical37
bes linea unbiased p edic o (San ne e al., 2003).38
Thus na u al c i e ia will minimize sui able unc ionals o he mean squa ed p edic ion e o (MSPE) gi en by39
MSPE
Y(x):= σ21−1,Q⊤(x)01⊤
n
1nC(n, )−11,Q⊤(x)⊤.(2.1)40
Since he p edic ion accu acy is o en ela ed o he en i e p edic ion egion X, he design c i e ion IMSPE is gi en by41
IMSPE 
Y:= σ−2X
MSPE 
Y(x)dx.42
Wi hou loss o gene ali y, we can assume ha he design space X= [0,1]2. The e o e, as ex apola i e p edic ion is no 43
ad isable in k iging, we can se 1=s1=0 and n=sn=1.44
Theo em 2.1. In ou se -up,45
MSPE 
Y(x)=σ21−ϱ(x,sn, n)2−
n−1

i=1
(ϱ(x,si, i)−ϱ(x,si+1, i+1)qi)2
1−q2
i
46
+1+
n−1

i=1
1−qi
1+qi−11−ϱ(x,sn, n)−
n−1

i=1
ϱ(x,si, i)−ϱ(x,si+1, i+1)qi
1+qi2
,47
1582 S. Ba an e al. / S a is ics and P obabili y Le e s 83 (2013) 1580–1587
Fig. 1. IMSPE co esponding o he op imal design o he p edic ion and Fishe in o ma ion on θas unc ions o co ela ion pa ame e s (α, β) o n=3.
whe e qi:= exp(−αdi−βδi)wi h di:= si+1−siand δi:= i+1− i,i=1,2,...,n−1. Fu he , 1
IMSPE 
Y=1−An+1+
n−1

i=1
1−qi
1+qi−1
Bn,(2.2) 2
whe e 3
An=R1,1+
n−1

i=1
Ri,iq2
i−2Ri+1,iqi+Ri+1,i+1
1−q2
i
,(2.3) 4
Bn=1−2ωn+Rn,n+
n−1

i=1Ri,i−2Ri+1,iqi+Ri+1,i+1q2
i
(1+qi)2+2(Rn,i−ωi)−(Rn,i+1−ωi+1)qi
1+qi
+2
n−1

i=2
i−1

j=1
Ri,j−Ri+1,jqi−Ri,j+1qj+Ri+1,j+1qiqj
(1+qi)(1+qj),(2.4) 5
wi h 6
ωi=1
αβ 2−e−α(d1+···+di−1)−e−α(di+···+dn−1)2−e−β(δ1+···+δi−1)−e−β(δi+···+δn−1),(2.5) 7
Ri,j=1
2α2e−α(dj+···+di−1)−e−α(2d1+···+2dj−1+dj+···+di−1)−e−α(dj+···+di−1+2di+···+2dn−1)
+(dj+···+di−1)e−α(dj+···+di−1)
×1
2β2e−β(δj+···+δi−1)−e−β(2δ1+···+2δj−1+δj+···+δi−1)−e−β(δj+···+δi−1+2δi+···+2δn−1)
+(δj+···+δi−1)e−β(δj+···+δi−1),i,j∈N,1≤j≤i≤n.(2.6) 8
Example 2.2. Conside a h ee-poin design, ha is n=3, s1= 1=0,s2:= d, 2:= δ, s3= 3=1. Fo his pa icula 9
design we ob iously ha e d1=d,d2=1−d, δ1=δ, δ2=1−δ. By analysing he pa ial de i a i es o IMSPE 
Yone 10
can easily show ha he minimum o IMSPE is eached a d=1/2, δ =1/2, so o n=3 he equidis an design is op imal. 11
In Fig. 1, IMSPE 
Yco esponding o he op imal design and he Fishe in o ma ion on he shi pa ame e θa e plo ed 12
as unc ions o co ela ion pa ame e s (α, β). Ha ing a look a he igu es o IMSPE and Mθone can obse e ha he e is
S. Ba an e al. / S a is ics and P obabili y Le e s 83 (2013) 1580–1587 1583
Fig. 2. The alues o d∗
1,d∗
2and δ∗
1, δ∗
2gi ing he op imum o IMSPE as unc ions o co ela ion pa ame e s (α, β) o n=5.
a subs an ial di e ence be ween he a ia ions o he es ima ion and he p edic ion c i e ion wi h espec o co ela ion1
pa ame e s (α, β). The equila e al lines o Mθdi e conside ably om equila e al cu es o IMSPE, which shows he2
di e en le el o sensi i i y o hese c i e ia o he misspeci ica ion o co ela ion pa ame e s. Fo mo e in o ma ion on3
he sensi i i y o op imal design c i e ia wi h espec o pa ame e misspeci ica ion, see e.g. S ehlík (2004).4
Theo em 2.3. IMSPE gi en by (2.2) is symme ic in he sense ha he in e change o pai s o dis ances (di, δi)and (dn−i, δn−i),5
i=1,2,...,n−1, does no change he alue o IMSPE 
Y.6
Theo em 2.3 gi es us some idea o he beha iou o he op imal design wi h espec o he IMSPE c i e ion.7
Co olla y 2.4. The op imal design wi h espec o he IMSPE c i e ion is symme ic, ha is i he op imum o IMSPE 
Yis eached8
a (d∗
1,d∗
2,...,d∗
n−1, δ∗
1, δ∗
2, . . . , δ∗
n−1) hen d∗
i=d∗
n−iand δ∗
i=δ∗
n−i,i=1,2,...,n.9
Example 2.5. Conside a i e-poin design, ha is n=5 and s1= 1=0,s5= 5=1. By Co olla y 2.4, o he op imal10
design we ha e d∗
1=d∗
4,d∗
2=d∗
3and δ∗
1=δ∗
4, δ∗
2=δ∗
3. In Fig. 2 he alues o d∗
1,d∗
2and δ∗
1, δ∗
2a e plo ed as unc ions o 11
co ela ion pa ame e s αand βbo h a ying be ween 0 and 3. We ema k ha by symme y, d∗
1+d∗
2=δ∗
1+δ∗
2=1/2.12
3. The op imal in o ma ion gain o a shi ed OU shee 13
Ano he app oach o op imal design is o ind loca ions which maximize he amoun o in o ma ion ob ained. Following14
he ideas o Shew y and Wynn (1987), one has o maximize he en opy En (Y)o he obse a ions co esponding o he15
chosen design, which in he Gaussian case o m an n-dimensional no mal ec o wi h co a iance ma ix σ2C(n, ), ha is16
En (Y)=n
21+ln(2πσ2)+1
2ln de C(n, ).17
Theo em 3.1. Unde he condi ions o Theo em 2.1, en opy En (Y)has he o m18
En (Y)=n
21+ln(2πσ2)+1
2
n−1

i=1
ln 1−q2
i.(3.1)19
Fo any sample size, he equidis an design when alues αdi+βδi,i=1,2,...,n−1(and in his way qi), a e cons an is op imal20
wi h espec o he en opy c i e ion.21
Rema k 3.2. Obse e ha he equispaced mono onic design d1=d2= ···dn, δ1=δ2= ··· = δnclea ly op imizes he22
en opy c i e ion.23
Rema k 3.3. I he design space Xis he uni squa e hen he op imal alue o he en opy equals24
n
21+ln(2πσ2)+n−1
2ln (1−exp (−(α +β)/(n−1))) → −∞ as n→ ∞.25
1584 S. Ba an e al. / S a is ics and P obabili y Le e s 83 (2013) 1580–1587
Table 1
IMSPE alues co esponding o he op imal and o he equispaced mono onic design and ela i e e iciency o he equispaced mono onic design.
nα=0.5, β =0.8α=1, β =1α=1, β =10 α=2.5, β =1.5α=3, β =3
4
Op imal 0.2602 0.4008 0.9266 0.6549 0.8487
Equispaced 0.2693 0.4010 0.9326 0.6598 0.8493
Rel. e . (%) 96.62 99.95 99.36 99.26 99.93
5
Op imal 0.2309 0.3699 0.8290 0.5981 0.7866
Equispaced 0.2473 0.3700 0.8409 0.6065 0.7873
Rel. e . (%) 93.37 99.97 98.58 98.62 99.91
6
Op imal 0.2130 0.3529 0.7593 0.5640 0.7502
Equispaced 0.2351 0.3530 0.7766 0.5763 0.7509
Rel. e . (%) 90.60 99.97 97.77 97.87 99.91
7
Op imal 0.2007 0.3423 0.7066 0.5241 0.7269
Equispaced 0.2274 0.3424 0.7288 0.5571 0.7275
Rel. e . (%) 88.26 99.97 96.95 94.08 99.92
8
Op imal 0.1692 0.3351 0.6655 0.5007 0.7111
Equispaced 0.2222 0.3352 0.6918 0.5441 0.7115
Rel. e . (%) 76.15 99.97 96.20 92.02 99.94
9
Op imal 0.1620 0.3300 0.6325 0.4858 0.6997
Equispaced 0.2184 0.3301 0.6626 0.5348 0.7001
Rel. e . (%) 74.18 99.97 95.46 90.84 99.94
10
Op imal 0.1570 0.3262 0.6057 0.4756 0.6912
Equispaced 0.2155 0.3262 0.6390 0.5278 0.6915
Rel. e . (%) 72.87 99.98 94.79 90.11 99.95
Table 2
Op imal ou -poin designs wi h espec o heIMSPEc i e ion.
α=β=0.5α=β=1α=β=1.5
d1=δ10.3372 0.3401 0.3421
d2=δ20.3256 0.3199 0.3158
d3=δ30.3372 0.3401 0.3421
4. Nume ical expe imen s 1
4.1. Compa ison o he IMSPE c i e ion and he en opy c i e ion 2
In o de o compa e he pe o mances o he wo c i e ia we compa e he op imal alues o IMSPE(
Y)calcula ed using 3
he mincon unc ion o Ma lab o i s alues co esponding o he equispaced mono onic design which is op imal o he 4
en opy c i e ion. In Table 1 he alues o IMSPE a e gi en o bo h designs oge he wi h he ela i e e iciency o he 5
equispaced mono onic design wi h espec o he op imal one o a ious sample sizes and combina ions o pa ame e s 6
(α, β). Un o una ely, la ge sample sizes cause echnical p oblems in op imiza ion, since o ndesign poin s one has o ind 7
nume ically he cons ained minimum o unc ions wi h 2n−2 pa ame e s. 8
Obse e ha o symme ic models when α=β, he e iciency o he equispaced mono onic design is nea ly 100%, 9
while o di e en co a iance pa ame e s i educes signi ican ly. Howe e , e en in his special case he op imal designs 10
wi h espec o he IMSPE and en opy c i e ia do no coincide. As an example, conside Table 2 whe e dis ances diand δi11
co esponding o he op imal ou -poin design a e gi en o h ee di e en alues o α=β. Ob iously, in all h ee cases we 12
ha e d1+δ1= d2+δ2, so acco ding o Theo em 3.1 hese designs canno be op imal wi h espec o he en opy c i e ion. 13
4.2. IMSPE on a mono onic se and on a egula g id 14
In Theo em 2.1, he exac o m o IMSPE is de i ed only o designs sa is ying Condi ion D. Howe e , one migh ask wha 15
is he ela i e e iciency o he op imal alue o IMSPE on mono onic se s con aining n=m2design poin s compa ed o he 16
IMSPE o a egula g id wi h he same numbe o poin s. Table 3 gi es he op imal alues o IMSPE on mono onic se s, IMSPE 17
alues o egula designs and he ela i e e iciencies o he op imal IMSPE alues on mono onic se s o di e en sample 18
sizes and combina ions o pa ame e s (α, β). Obse e, ha o ou poin s he op imal mono onic design gi es much be e 19
IMSPE alues han he egula g id and o α=β=3 he ela i e e iciency is sligh ly abo e 100% e en in he case n=9. 20
5. Conclusion 21
We de i e he exac o m o he IMSPE o an OU shee on a mono onic se . We show ha he op imal design o he 22
p edic ion based on IMSPE may di e subs an ially om he equidis an one. This is in con as bo h o he op imal design 23

S. Ba an e al. / S a is ics and P obabili y Le e s 83 (2013) 1580–1587 1585
Table 3
IMSPE alues co esponding o he op imal mono onic and o he egula g id design and ela i e e iciency o he op imal mono onic design.
nα=0.5, β =0.8α=1, β =1α=1, β =10 α=2.5, β =1.5α=3, β =3
4
Mono onic 0.2602 0.4008 0.9266 0.6549 0.8487
Regula 0.3580 0.5389 1.1449 0.8370 1.0094
Rel. e . (%) 137.59 134.46 123.56 127.81 118.93
9
Mono onic 0.1620 0.3300 0.6325 0.4858 0.6997
Regula 0.1527 0.3018 0.5762 0.4850 0.7011
Rel. e . (%) 94.26 91.45 91.10 99.84 100.20
o es ima ion on a mono onic se (Ba an and S ehlík, submi ed o publica ion) and o he op imal design o he p edic ion1
o he OU p ocess on a compac in e al in es iga ed by Baldi An ognini and Zago aiou (2010). We also in es iga e he2
p ope ies o he op imal design wi h espec o he en opy c i e ion, whe e cons aining sample poin s om a ec angle3
o a mono onic se , as expec ed, dec eases he en opy o he Gaussian ield. Simula ions illus a e selec ed cases o op imal4
designs o small numbe o sampling loca ions. Since he abo e discussed designs depend on he alues o he co ela ion5
pa ame e s, he op imal designs ob ained a e only locally op imal. We b ie ly s udy he dependence o he designs ob ained6
on hese pa ame e s, oo. Such knowledge may be c ucial o an expe imen e o inc easing he e iciency o a design in a7
p ac ical se -up.8
Acknowledgemen s9
The au ho s a e g a e ul o Lenka Filo á o help ul commen s du ing he p epa a ion o he manusc ip . This esea ch10
was suppo ed by he Hunga ian Scien i ic Resea ch Fund unde G an s Nos OTKA T079128/2009 and OTKA NK101680/201211
and by he Hunga ian–Aus ian in e go e nmen al S&T coope a ion p og am TÉT_10-1-2011-0712, and pa ially suppo ed12
by he TÁMOP-4.2.2.C-11/1/KONV-2012-0001 p ojec . The p ojec was suppo ed by he Eu opean Union, wi h co- inancing13
om he Eu opean Social Fund. The hi d au ho acknowledges he suppo o he p ojec DESIRE. The au ho s a e indeb ed14
o he unknown e e ees o hei use ul ema ks ha helped a lo wi h imp o ing he manusc ip .15
Appendix16
A.1. P oo o Theo em 2.117
To sho en ou o mulae, in wha ollows ins ead o ϱ(x,si, i)we a e using simply ϱi,i=1,2,...,n. Conside i s 18
MSPE 
Y(x)gi en by (2.1). Sho ma ix algeb aic calcula ions show ha 19
01⊤
n
1nC(n, )−1
=00⊤
n
0nC−1(n, )−1⊤
nC−1(n, )1n−1
1−C−1(n, )1n⊤
−C−1(n, )1nC−1(n, )1nC−1(n, )1n⊤
,20
and acco ding o he esul s o Ba an and S ehlík (submi ed o publica ion) we ha e 1⊤
nC−1(n, )1n=1+n−1
i=11−qi
1+qi. We21
ema k ha 1⊤
nC−1(n, )1nis he Fishe in o ma ion on θbased on Y. In his way, we ob ain22
MSPE 
Y(x)=σ21−Q⊤(x)C−1(n, )Q(x)+1⊤
nC−1(n, )1n−11−Q⊤(x)C−1(n, )1n2
23
=σ21+ϱ2
1−2ϱ1ϱ2q1
q2
1−1+ϱ2
n
q2
n−1−1−
n−1

i=22ϱiϱi+1qi
q2
i−1+ϱ2
i(1−q2
iq2
i−1)
(q2
i−1)(q2
i−1−1)
24
−1+
n−1

i=1
1−qi
1+qi−1
25
×1+ϱ1−(ϱ1+ϱ2)q1
q2
1−1+ϱn
q2
n−1−1−
n−1

i=2(ϱi+ϱi+1)qi
q2
i−1+ϱi(1−q2
iq2
i−1)
(q2
i−1)(q2
i−1−1)2

26
=σ2
1−ϱ2
n−
n−1

i=1
(ϱi−ϱi+1qi)2
1−q2
i+1+
n−1

i=1
1−qi
1+qi−11−ϱn−
n−1

i=1
ϱi−ϱi+1qi
1+qi2
.27
Fu he , acco ding o he de ini ion o he IMSPE c i e ion, we can w i e28
IMSPE 
Y=1−An+1+
n−1

i=1
1−qi
1+qi−1
Bn,29
1586 S. Ba an e al. / S a is ics and P obabili y Le e s 83 (2013) 1580–1587
whe e 1
An:= X
Q⊤(x)C−1(n, )Q(x)dx= C−1(n, )R,2
Bn:= 1−21⊤
nC−1(n, )W+1⊤
nC−1(n, )RC−1(n, )1n,3
wi h 4
W=(ω1, . . . , ωn)⊤:= X
Q(x)dxand R=Ri,j:= X
Q(x)Q⊤(x)dx.5
Ob iously, 6
ωi=1
αβ 2−e−αsi−e−α(1−si)2−e−β i−e−β(1− i)7
=1
αβ 2−e−α(d1+···+di−1)−e−α(di+···+dn−1)2−e−β(δ1+···+δi−1)−e−β(δi+···+δn−1),8
Ri,j=1
2α2e−α|si−sj|−e−α(si+sj)−e−α(2−si−sj)+|si−sj|e−α|si−sj|9
×1
2β2e−β| i− j|−e−β( i+ j)−e−β(2− i− j)+| i− j|e−β| i− j|10
=1
2α2e−α(di∧j+···+di∨j−1)−e−α(2d1+···+2di∧j−1+di∧j+···+di∨j−1)−e−α(di∧j+···+di∨j−1+2di∨j+···+2dn−1)11
+(di∧j+···+di∨j−1)e−α(di∧j+···+di∨j−1)12
×1
2β2e−β(δi∧j+···+δi∨j−1)−e−β(2δ1+···+2δi∧j−1+δi∧j+···+δi∨j−1)−e−β(δi∧j+···+δi∨j−1+2δi∨j+···+2δn−1)13
+(δi∧j+···+δi∨j−1)e−β(δi∧j+···+δi∨j−1),14
whe e i∧j:= min{i,j},i∨j:= max{i,j},i,j∈N, and he emp y sum is de ined o be ze o. In his way, we ob ain 15
An=Rn,n+
n−1

i=1
Ri,i−2Ri+1,iqi+Ri+1,i+1q2
i
1−q2
i=R1,1+
n−1

i=1
Ri,iq2
i−2Ri+1,iqi+Ri+1,i+1
1−q2
i
,16
Bn=1−2ωn−2
n−1

i=1
ωi−ωi+1qi
1+qi+Rn,n
+2
n−1

i=1
Rn,i−Rn,i+1qi
1+qi+
n−1

i=1
n−1

j=1
Ri,j−Ri+1,jqi−Ri,j+1qj+Ri+1,j+1qiqj
(1+qi)(1+qj)
=1−2ωn+Rn,n+
n−1

i=1Ri,i−2Ri+1,iqi+Ri+1,i+1q2
i
(1+qi)2+2(Rn,i−ωi)−(Rn,i+1−ωi+1)qi
1+qi
+2
n−1

i=2
i−1

j=1
Ri,j−Ri+1,jqi−Ri,j+1qj+Ri+1,j+1qiqj
(1+qi)(1+qj).17
A.2. P oo o Theo em 2.3 18
Le 
di:= dn−i,
δi:= δn−i,
qi:= exp −α
di−β
δi,i=1,2,...,n−1, and deno e by ωkand
Rk,ℓ he alues calcula ed 19
om (2.5) and (2.6), espec i ely, using dis ances (
di,
δi), i=1,2,...,n−1. Fu he , le qi, ωkand Rk,ℓ deno e he 20
co esponding quan i ies calcula ed om he o iginal dis ances (di, δi), i=1,2,...,n−1. I is easy o see ha 21

qi=qn−i,i=1,2,...,n−1,and ωk=ωn−k+1,
Rk,ℓ =Rn−k+1,n−ℓ+1,k, ℓ =1,2,...,n,(A.1) 22
and in his way we ob ain 23
n−1

i=1
1−
qi
1+
qi=
n−1

i=1
1−qi
1+qi
.24
S. Ba an e al. / S a is ics and P obabili y Le e s 83 (2013) 1580–1587 1587
Now, deno e by
Anand
Bn he alues calcula ed om (2.3) and (2.4), espec i ely, using dis ances (
di,
δi), i=1,2,...,n−1.1
Then wi h he help o (A.1), a e s aigh o wa d bu edious calcula ions one can show ha 
An=Anand
Bn=Bn, which2
comple es he p oo . 3
A.3. P oo o Theo em 3.14
Acco ding o he esul s o Ba an and S ehlík (submi ed o publica ion),5
C(n, )=
























1q1q1q2q1q2q3··· ···
n−1

i=1
qi
q11q2q2q3··· ···
n−1

i=2
qi
q1q2q21q3··· ···
n−1

i=3
qi
q1q2q3q2q3q31··· ··· .
.
.
.
.
..
.
..
.
..
.
.....
.
.
.
.
..
.
..
.
..
.
....qn−1
n−1

i=1
qi
n−1

i=2
qi
n−1

i=3
qi··· ··· qn−11
























.6
Hence, one can use he same ype o decomposi ion as Baldi An ognini and Zago aiou (2010, Lemma 3.1), ha is7
C(n, )=LDL⊤,8
whe e Lis a lowe iangula ma ix wi h en ies 1 in i s main diagonal and Dis a diagonal ma ix whe e in he diagonal we9
ha e 1,1−q2
1,1−q2
2,...,1−q2
n−1. In his way10
de C(n, )=de D=
n−1

i=1
(1−q2
i),11
which p o es (3.1).12
In o de o ind he op imal design one has o ind he cons ained maximum o 13
F(q1,q2,...,qn−1):=
n−1

i=1
ln 1−q2
iunde condi ion
n−1

i=1
ln qi= −α−β.14
By analysing he i s pa ial de i a i es and he Hessian o he Lag ange unc ion15
Λ(q1,q2,...,qn−1;λ):=
n−1

i=1
ln 1−q2
i+λn−1

i=1
ln qi+α+β
16
one can easily see ha he maximum is eached when q1=q2= ··· = qn−1, which comple es he p oo . 17
Re e ences18
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Kiseľák, J., S ehlík, M., 2008. Equidis an D-op imal designs o pa ame e s o O ns ein–Uhlenbeck p ocess. S a is . P obab. Le . 78, 1388–1396.29 Koizumi, H., Jin, Y.H., 2012. Pe o mance enhancemen o a la en hea he mal ene gy s o age sys em using cu ed-slab con aine s. Appl. The m. Eng. 37,30 145–153.31 Min y, G.J., 1962. Mono one (non-linea ) ope a o s in Hilbe space. Duke Ma h. J. 29, 341–346.32 Min y, G.J., 1963. On a mono onici y me hod o he solu ion o nonlinea equa ions in Banach spaces. P oc. Na l. Acad. Sci. US 50, 1038–1041.33 P onza o, L., Mülle , W.G., 2012. Design o compu e expe imen s: space illing and beyond. S a . Compu . 22, 681–701.34 Sacks, J., Schille , S.B., Welch, W.J., 1989. Design o compu e expe imen s. Technome ics 31, 41–47.35 San ne , T.J., Williams, B.J., No z, W.I., 2003. The Design and Analysis o Compu e Expe imen s. Sp inge -Ve lag, New Yo k.36 Shew y, M.C., Wynn, H.P., 1987. Maximum en opy sampling. J. Appl. S a . 14, 165–170.37 S ehlík, M., 2004. Some p ope ies o D-op imal designs o andom ields wi h di e en a iog ams. Resea ch Repo Se ies Repo No. 4, Depa men o 38 S a is ics and Ma hema ics, Wi scha suni e si ä Wien, Aus ia.39