Optimal designs for parameters of shifted Ornstein-Uhlenbeck sheets measured on monotonic sets

Op imal designs o pa ame e s o shi ed O ns ein-Uhlenbeck shee s measu ed on mono onic se s Ba an, S.a, S ehl´ık, M.b,c,∗ aFacul y o In o ma ics, Uni e si y o Deb ecen, Hunga y bDepa men o Applied S a is ics, Johannes Keple Uni e si y in Linz cDepa amen o de Ma em´a ica, Uni e sidad T´ecnica Fede ico San a Ma ´ıa, Valpa a´ıso, Chile Abs ac Measu emen on se s wi h a speci ic geome ic shape can be o in e es o many impo an applica ions (e.g., measu emen along he iso he ms in s uc u al enginee ing). The p ope ies o op imal designs o es ima ing he pa ame e s o shi ed O ns ein-Uhlenbeck shee s a e in es iga ed when he p ocesses a e obse ed on mono onic se s. Fo O ns ein-Uhlenbeck shee s mono onic se s ela e well o he no ion o non- e e sibili y. Subs an ial di e ences a e demons a ed be ween he cases when one is in e es ed only in end pa ame e s and when he whole pa ame e se is o in e es . The heo e ical esul s a e illus a ed by simula ed examples om he ield o s uc u e enginee ing. F om he design poin o iew he mos in e es ing inding o he pape is he possible loss o e iciency o he egula g id design compa ed o he op imal mono onic design. Keywo ds: D-op imali y, equidis an design, mono onic se s, op imal design, O ns ein-Uhlenbeck shee . 1. In oduc ion Measu emen on se s wi h a speci ic geome ic shape is o in e es o many impo an applica ions, e.g., measu emen along he iso he ms. S a ing wi h he undamen al wo k o Hoel (1958), he cen al impo ance o equidis an designs o he es ima ion o pa ame e s o co ela ed p ocesses has been ealized. Hoel (1958) compa ed he e iciencies o equally spaced designs o one-dimensional polynomial models o se e al design egions and co ela ion s uc u es. In his con ex by a design we mean a se ξ= {x1, x2, . . . , xn}o loca ions whe e he in es iga ed p ocess is obse ed. A compa ison in a mul i-dimensional se up including co ela ions can be ound, e.g., in He zbe g and Huda (1981). La e Kiseˇ l´ak and S ehl´ık (2008) p o ed ha equidis an design is op imal o es ima ing he unknown mean pa ame e o an O ns ein- Uhlenbeck (OU) p ocess, whe eas Zago aiou and Baldi An ognini (2009) also s udied shi ed s a iona y OU ∗Co esponding au ho . Depa amen o de Ma em´a ica, Uni e sidad T´ecnica Fede ico San a Ma ´ıa, Casilla 110-V, Val- pa a´ıso, Chile. Email: [email p o ec ed], Tel.:+4373224686808; Fax: +4373224686800 Email add esses: [email p o ec ed] (Ba an, S.), [email p o ec ed] (S ehl´ık, M.) P ep in submi ed o S a is ics and P obabili y Le e s Decembe 30, 2014 p ocesses. Howe e , in all abo e men ioned pape s on op imal design o OU p ocesses he design egions we e in e als o he eal line, bu a one-dimensional in e al is na u ally a di ec ed se induced by he o al o de ing o he eal numbe s. Ob iously, he e is a big di e ence in geome y be ween a plane and a line and hus OU shee s sampled on wo-dimensional in e als p o ide much mo e delica e design s a egies. In he p esen wo k we de i e op imal exac designs o pa ame e s o a shi ed OU shee measu ed in he poin s cons i u ing a mono onic se . A mono onic se can be de ined in a bi a y Hilbe space H, wi h eal o complex scala s. Fo x, y ∈H, we deno e by hx, yi he eal pa o he inne p oduc . A se E⊂H×His called mono onic (see Min y (1963) and e e ences he ein) i o all (x1, y1),(x2, y2)∈E we ha e hx1−x2, y1−y2i ≥ 0. A p ac ical example o such a se a e measu emen s on iso he ms o a 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)). Ano he impo an example in which mono onic measu emen s appea is mo i a ed by measu ing o me hane adso p ion (Lee and Webe , 1969) whe e keeping all measu emen s a iso he m dec eases he p oblems connec ed o s abili y. He e we conside he ollowing e sion o a mono onic se : Condi ion D The po en ial design poin s (s1, 1),(s2, 2),...,(sn, n)⊂ X, n ∈N, whe e Xdeno es a compac design space, sa is y 0< s1< s2< . . . < snand 0< 1< 2< . . . < n. We ema k ha he same obse a ion scheme is used in Ba an e al. (2013) whe e he au ho s deal wi h p edic ion o OU shee s and de i e op imal designs wi h espec o in eg a ed mean squa e p edic ion e o and en opy c i e ia. Condi ion D ela es he geome y o he unde lying se o poin s o he Ma ko ian p ope ies o OU shee s and co esponding Fishe in o ma ion ma ices. This geome y has di ec connec ion wi h he in e p e a ion o OU shee di usion. S anda d di usion is non- e e sible, and he hea pa ial di e en ial equa ion is no ime- e e sible. Thus, in some ealis ic physical si ua ions we canno s ep back in ime. In he modynamics, a e e sible p ocess is a p ocess ha can be “ e e sed” by means o in ini esimal changes in some p ope y o he sys em wi hou en opy p oduc ion (i.e., dissipa ion o ene gy, see, e.g., Sea s and Salinge (1986)). The e exis s a “ e e sible di usion”, which is a speci ic example o a e e sible s ochas ic p ocess, ha ing an elegan cha ac e iza ion due o Kolmogo o (1937). Thus, s a is ician shall decide, whe he he p ocess o be modelled is e e sible. I no , o es ima ing pa ame e s o an OU shee i is be e o conside a design sa is ying Condi ion D. We unde s and ha his does no necessa ily co e all applica ions, bu i is in e es ing o some o hem. We do no claim ha mono onic se designs should be used ou inely in enginee ing p ac ice. The aim o ou pape is me ely o show ha o an OU shee , in some scena ios, mono onic cu e could p o ide be e e iciency han simple g id designs. The e o e, he expe imen e is ad ised o in eg a e ca e ully he mono onic se design in o his/he po olio o candida e designs – especially in cases when he e is a s ong in ui ion/jus i ica ion o he Ma ko ian na u e o he p ocess. Being mo e pa icula , i is o en o e seen in p ac ice, ha in o ma ion inc eases wi h he numbe o poin s only in he case o independence (o 2 speci ic o m o dependence). Thus, gene al illing designs, gene a ed wi hou u he cau ion, may inc ease he a iance ins ead o in o ma ion. Fo a classical example see, e.g., Smi (1961). Ano he discussion o designing o co ela ed p ocesses in he con ex o space illing and i s limi a ions can be ound in M¨ulle and S ehl´ık (2009) and P onza o and M¨ulle (2012). The pape is o ganized as ollows. In his sec ion we in oduce he model o be s udied and ou no a ions. We also deal wi h an example which mo i a es he p esen s udy, namely, a design expe imen o measu ing on iso he ms o a s a iona y he mal ield. Sec ions 2, 3, and 4 deal wi h he op imal designs o he es ima ion o pa ame e s o ou model. We demons a e he subs an ial di e ences be ween he cases when one is in e es ed only in he end pa ame e and when he whole pa ame e se is o in e es . Sec ion 5 con ains an applica ion, whe eas o main ain he con inui y o he explana ion, he p oo s a e gi en in he Appendix. 1.1. S a is ical Model Conside he s a iona y p ocess Y(s, ) = θ+ε(s, ) (1.1) wi h design poin s aken om a compac design space X= [a1, b1]×[a2, b2], whe e b1> a1and b2> a2 and ε(s, ), s, ∈R, is a s a iona y O ns ein-Uhlenbeck shee , ha is a ze o mean Gaussian p ocess wi h co a iance s uc u e Eε(s1, 1)ε(s2, 2) = eσ2 4αβ exp −α| 1− 2|−β|s1−s2|,(1.2) whe e α > 0, β > 0,eσ > 0. We ema k ha ε(s, ) can also be ep esen ed as ε(s, ) = eσ 2√αβ e−α −βsWe2α ,e2βs, whe e W(s, ), s, ∈R, is a s anda d B ownian shee (Ba an e al., 2003; Ba an and Sikolya, 2012), i.e., a cen e ed Gaussian andom ield wi h co a iances EW(s1, 1)W(s2, 2) = min(s1, s2)·min( 1, 2). Co a iance s uc u e (1.2) implies ha o d= (d, δ), d ≥0, δ ≥0, he a iog am 2γ(d) := Va ε(s1, 1)−ε(s2, 2)= eσ2 2αβ 1−e−αd−βδ, whe e now |s1−s2|=d, | 1− 2|=δ, and he co ela ion be ween wo measu emen s depends on he dis ance h ough he semi a iog am γ(d). In o de o apply he usual app oach o design in spa ial modeling (Kiseˇ l´ak and S ehl´ık, 2008) we in oduce σ:= eσ/(2√αβ) and ins ead o (1.2) we in es iga e Eε(s1, 1)ε(s2, 2) = σ2exp −α| 1− 2|−β|s1−s2|,(1.3) whe e σis conside ed as a nuisance pa ame e . In an unco ela ed model he pa ame e σin luences nei he he es ima ion o he mean alue pa ame e s, no he op imal design. In he p esen pape we assume σ o be known bu a aluable di ec ion o he u u e esea ch will be he in es iga ion o models wi h unknown 3 nuisance pa ame e σ. Mo eo e , he assump ion o known σis easonable when αand βa e known as well. Fo mos o he ealis ic si ua ions, whe e he pa ame e s o he co ela ion s uc u e a e no known, he e is no op imal design, as we show, e.g., in Sec ions 3 and 4. Howe e , we hink ha all ecen de elopmen s on op imal design s a egies o es ima ion o pa ame e s should mos ly be conside ed as benchma ks in mo e ealis ic se ups o op imal design (e.g., like geome ic p og ession ones, as discussed in Sec ion 4, o in Zago aiou and Baldi An ognini (2009) o a one-dimensional design space). These benchma ks should always be con on ed di ec ly wi h a subjec science, e.g., wi h me hane modelling wi h he help o he modi ied A henius model in Rod ´ıguez-D´ıaz e al. (2012). Ne e heless, o m (1.3) o he co a iance s uc u e is mo e sui able o s a is ical applica ions, while (1.2) i s be e o p obabilis ic modelling. Fu he , we equi e Condi ion D o be hold on he design poin s because unde his condi ion we may use he cons uc ion o Kiseˇ l´ak and S ehl´ık (2008) o ob ain he in e se o he co a iance ma ix o obse a ions which is idiagonal. Mo eo e , in case o an equidis an design he co a iance ma ix is Toepli z. He e we conside D-op imali y, which co esponds o he maximiza ion o objec i e unc ion Φ(M) := de (M), he de e minan o he s anda d Fishe in o ma ion ma ix. This c i e ion, “plugged” om he widely de eloped unco ela ed se up, o e s conside able po en ial o au oma ic implemen a ion, al hough u he de elopmen is needed be o e i can be applied ou inely in p ac ice. Theo e ical jus i ica ions o using Fishe in o ma ion o D-op imal designing unde co ela ion can be ound in Ab and Welch (1998) and P´azman (2007). The concep o uni o m designs has now gained popula i y and p o ed o be e y success ul in indus ial applica ions and in compu e expe imen s (M¨ulle and S ehl´ık, 2009; San ne e al., 2003). I has become s anda d p ac ice o selec he design poin s such as o co e he a ailable space as uni o mly as possible, e.g., o apply he so called space- illing designs. In highe dimensions he e a e se e al ways o p oduce such designs. In his pape we illus a e ha o he OU shee he design sa is ying mono onici y Condi ion D could be possibly supe io o he space illing g id designs. The idea o choosing a mono onic se is mainly mo i a ed by Ma ko ian p ope ies o he OU shee . 1.2. Mo i a ing example: measu emen o a s a iona y he mal ield Tempe a u e dis ibu ion calcula ions du ing he p ocess o designing a building is a necessa y pa o es ing he c i ical places a he building en elope. The aim is o inc ease he minimal su ace empe a u e, and o p edic he possible he mal b idges which a e possible loca ions o mould g ow h in he building. Figu e 1a displays he composi ion o ma e ials o he 2D sec ion o a he mal b idge wi hin he building cons uc ion. Da a a e aken om Min´a o ´a (2005), whe e a ini e elemen me hod o compu a ion o he empe a u e ield is applied using so wa e package ANSYS. Figu e 1b illus a es he iso he ms o he he mal ield which i well o measu emen s o ming a mono onic se sa is ying Condi ion D. Da a poin s in which we measu e he empe a u e a e plo ed on Figu e 2. We assume ha he co a iance pa ame e s αand βa e gi en and we a e in e es ed in he es ima ion o he end pa ame e θo he model 4 (a) (b) Figu e 1: 2D sec ion o a agmen o he building en elope nea he he mal b idge (Min´a o ´a, 2005). (a) Composi ion o a ma e ial; (b) Iso e ms o he he mal ield. (1.1). Table 1 lis s he ela i e e iciency, he in o ma ion Mθgained in he da a poin s and he op imal in o ma ion gain (max Mθ) o he da a om Figu e 2 o h ee choices o known co ela ion pa ame e s α, β. He e Mθis e alua ed on he gi en obse a ions and max Mθis he heo e ical maximal alue eachable a he gi en numbe o poin s, ajec o y leng h and gi en alues o pa ame e s. Ob iously, he ela i e e iciency o he gi en da a poin s a ies wi h hese pa ame e s. 2. Es ima ion o end pa ame e only Assume i s ha pa ame e s α, β and σo he co a iance s uc u e (1.3) o he OU shee εa e gi en and we a e in e es ed in he es ima ion o he end pa ame e θ. In his case he Fishe in o ma ion on θbased on obse a ions Y(si, i), i = 1,2, . . . , nequals Mθ(n) = 1> nC−1(n, )1n, whe e 1nis he column ec o o ones o leng h n, = (α, β)>, and C(n, ) is he co a iance ma ix o he obse a ions (P´azman, 2007; Xia e al., 2006). Fu he , le di:= si+1 −siand δi:= i+1 − i, i = 1,2, . . . , n −1, be he dis ances be ween wo adjacen design poin s. Wi h he help o his ep esen a ion one can p o e he ollowing heo em. Co ela ion Pa ame e s Mθmax MθE iciency (Mθ/max Mθ) α= 1, β = 1 1.4816 1.4817 0.990 α= 1, β = 10 4.9726 5.0813 0.978 α= 10, β = 1 2.2124 2.2129 0.999 Table 1: E iciency depending on co ela ion pa ame e s. 5 Figu e 2: Obse a ion poin s on Iso he ms. Theo em 1. Conside he model (1.1) wi h co a iance s uc u e (1.3) obse ed in poin s (si, i), i = 1,2, . . . , nsa is ying Condi ion D and assume ha he only pa ame e o in e es is he end pa ame e θ. In his case, he equidis an design sa is ying αd1+βδ1=αd2+βδ2=. . . =αdn−1+βδn−1is op imal o es ima ion o θ. Acco ding o Theo em 1 he op imali y holds o αdi+βδi=λ n−1, whe e λis he “skewed size” o he design egion, i.e., λ:= αPn−1 i=1 di+βPn−1 i=1 δiand Pn−1 i=1 di< b1−a1,Pn−1 i=1 δi< b2−a2. Se e al si ua ions may appea in p ac ice. As now we conside he co a iance pa ame e s α, β o be ixed and make in e ence only on he unknown end pa ame e θ, om he p oo o Theo em 1 we ob ain Mθ(n) = 1 + n−1 X i=1 1−qi 1 + qi ,(2.1) whe e qi:= exp(−αdi−βδi). Thus, o an op imal design we ha e Mθ(n) = Mθ(n;λ) = 1 + (n−1)1−exp(−λ/(n−1)) 1 + exp(−λ/(n−1)), which is an inc easing unc ion o bo h he numbe o design poin s nand he “skewed size” λ. Fu he , Mθ(n;λ)→λ/2 + 1 as n→ ∞ and Mθ(n;λ)→nas λ→ ∞, which alues a e bounds o in o ma ion inc ease in expe imen s. To illus a e he la e ac le us conside he design egion X= [0,1]2and a ou -poin design, and assume ha he co ela ion pa ame e s a e α=β= 1. As a compa ison we conside a egula g id design which pu s he ou poin s in o he e ices o he ec angle X( his design does no sa is y Condi ion D). The in o ma ion co esponding o his la e design is Mθ= 2.13. Ha ing he same design egion we canno each such an e iciency, because λ= 2 and Mθ(n;λ)< λ/2 + 1. Indeed, he maximal in o ma ion gain can 6 be Mθ(4; 2) = 1.965 which gi es us an e iciency o 0.919. I we allow he g ow h o he design egion, e.g. X= [0, x]2, o a ou -poin design, unde he abo e condi ions we ob ain Mθ=4 1+exp(−2x)+exp(−x)→4 o x→ ∞ a a egula g id design wi h e ices. 3. Es ima ion o co a iance pa ame e s only Assume now ha we a e in e es ed only in he es ima ion o he pa ame e s αand βo he OU shee . Acco ding o he esul s o P´azman (2007) and Xia e al. (2006) he Fishe in o ma ion ma ix on = (α, β)> has he o m M (n) =  Mα(n)Mα,β(n) Mα,β(n)Mβ(n) ,(3.1) whe e Mα(n) := 1 2 C−1(n, )∂C(n, ) ∂α C−1(n, )∂C(n, ) ∂α , Mβ(n) := 1 2 C−1(n, )∂C(n, ) ∂β C−1(n, )∂C(n, ) ∂β , Mα,β(n) := 1 2 C−1(n, )∂C(n, ) ∂α C−1(n, )∂C(n, ) ∂β , and C(n, ) deno es he co a iance ma ix o he obse a ions Y(si, i), i = 1,2, . . . , n. No e, ha he e Mα(n) and Mβ(n) a e Fishe in o ma ion on pa ame e s αand β, espec i ely, aking he o he pa ame e as a nuisance. The ollowing heo em gi es he exac o m o M (n) o he model (1.1). Theo em 2. Conside he model (1.1) wi h co a iance s uc u e (1.3) obse ed in poin s (si, i), i = 1,2, . . . , nsa is ying Condi ion D. Then Mα(n) = n−1 X i=1 d2 iq2 i(1 + q2 i) (1 −q2 i)2, Mβ(n) = n−1 X i=1 δ2 iq2 i(1 + q2 i) (1 −q2 i)2, Mα,β(n) = n−1 X i=1 diδiq2 i(1 + q2 i) (1 −q2 i)2,(3.2) whe e di, δiand qideno e he same quan i ies as in he p e ious sec ion, i.e. di:= si+1 −si, δi:= i+1 − i and qi:= exp(−αdi−βδi), i = 1,2, . . . , n −1. Using Theo em 2 one can o mula e he ollowing s a emen on he op imal design o he pa ame e s o he co a iance s uc u e o he OU shee . Theo em 3. The design which is op imal o es ima ion o he co a iance pa ame e s α, β does no exis wi hin he class o admissible designs. 7 4. Es ima ion o all pa ame e s Conside now he mos gene al case, when bo h α, β and θa e unknown and he Fishe in o ma ion ma- ix on hese pa ame e s equals M(n) =  Mθ(n) 0 0M (n) ,whe e Mθ(n) and M (n) a e Fishe in o ma ion ma ices on θand = (α, β)>, espec i ely, see (2.1) and (3.1). Theo em 4. The design which is op imal o es ima ion o he co a iance pa ame e s α,βand o he end pa ame e θdoes no exis wi hin he class o admissible designs. Loosely speaking, he op imal designs o he end ha e he endency o mo e he design poin s as a as possible, while he op imal designs o he co a iance s uc u e ha e he endency o sh ink he se o design poin s. Howe e , we can choose a comp omise be ween es ima ing he end and co ela ion pa ame e s. The e o e, simila ly o Zago aiou and Baldi An ognini (2009), we may conside he so-called geome ic p og ession design, which is gene a ed by he ec o s o dis ances dn, 1:= (k, k 1, k 2 1, . . . , k n−2 1),δn, 2:= (`, ` 2, ` 2 2, . . . , ` n−2 2), whe e 0 < 1, 2≤1. Assume Pn−1 i=1 di= 1 and Pn−1 i=1 δi= 1, o 1= 1, 2= 1 bo h cons an s kand `a e equal o (n−1)−1, while o 1<1 and 2<1 we ge k=1− 1 1− n−1 1 and `=1− 2 1− n−1 2 , espec i ely. The uning pa ame e s 1, 2 can be a ied acco ding o he desi ed e iciency o he es ima ion o he end o co ela ion pa ame e s. No e, ha case 1= 1, 2= 1 co esponds o he equidis an design, which we ha e p o ed o be op imal o es ima ion o he end pa ame e , whe eas o 1→0, 2→0, ec o s dn, 1and δn, 2 end o he bes design o he es ima ion o αand β. The ollowing heo em desc ibes he beha io s o Mθ(n) and de M (n)as unc ions o he uning pa ame e s 1and 2. Theo em 5. Fo any ixed n > 2,α > 0,β > 0, he in o ma ion Mθ(n)o he end is inc easing wi h espec o 1, 2, while he de e minan o he Fishe in o ma ion M (n)o co a iance pa ame e s has a global minimum a 1= 2. We ema k ha he i s s a emen o Theo em 5 is a s aigh o wa d ex ension o he co esponding pa o Theo em 5.1 o Zago aiou and Baldi An ognini (2009). Fu he , obse e ha Theo em 5 ob iously implies ha he o al in o ma ion de M(n)has he same beha io as de M (n), ha is i has a global minimum a 1= 2. 5. Applica ion o De e io a ion o highways Typically, enginee s a e using egula g ids o es ima ion o he pa ame e s o a andom ield. E.g., in Mohapl (1997) he de e io a ion o a highway in New Yo k s a e is in es iga ed whe e da a we e collec ed 8 in ou successi e yea s a dis ances o 0.2 miles om each o he o ming a 4 ×16 able, Based on hese da a he au ho es ima ed he pa ame e s o he unde lying s ochas ic p ocess. Wha is he e iciency o such a design? The design egion has he na u al o m [0,4] ×[0,3.2] and he numbe o obse ed poin s is 64. In he case α=β= 1 design sa is ying Condi ion D and ha ing 64 poin s in such a egion has Mθ(64,7.2) = 4.596. Now, le us ha e 16 ime coo dina es uni o mly gene a ed om ime egion [0,4] and 16 loca ion coo - dina es gene a ed om space egion [0,3.2]. Then o ime poin s 1.35,3.66,1.86,0.996,0.89,1.56,3.37,2.189,0.5157,2.58, 0.058,0.32,0.58,1.4,0.36,1.82 and leng hs 0.64,0.37,1.2,0.91,1.34,2.82, 2.56,2.44,0.257,2.568,2.223,0.66,2.298,2.814,2.75,1.61 we ob ain Mθ= 5.2 in he case bo h pa ame e s αand βa e equal o 1. Acco ding o Sec ion 2 he maximal in o ma ion gain wi h Condi ion D o n= 64, λ = 5.12 equals 3.56, hus he ela i e e iciency is 0.68. Howe e , he e is an open ques ion, how o es ima e pa ame e s in case o his pa icula se ing o poin s, which is a no i ial. Since he obse a ions o m a Gaussian andom ec o , one can de i e he likelihood unc ion and ind he ML es ima es a leas nume ically. Fo a egula g id design Ying (1993) p o ed consis ency and asymp o ic no mali y o he ML es ima o s, bu acco ding o he au ho s bes knowledge his is he only esul in his di ec ion. The p oblem is ha in he gene al case he dependence o he likelihood unc ion on he pa ame e s and design poin s is oo complica ed o ind i s asymp o ic p ope ies. When one uses egula g ids o Mohapl (1997), hen he ollowing si ua ion occu s: ime is measu ed in 16 equispaced momen s s a ing om 0, un il 3.75 by 0.25, while he de e io a ion o he highway is measu ed in 16 poin s (by 0.2 miles). Then Mθ= 4.32 (in he case α=β= 1) wi h ela i e e iciency o 0.827. Table 2 is e ealing an in e es ing ac , ha egula g id design (wi h 256 = 162poin s) has a los o e iciency wi h espec o he op imal design sa is ying Condi ion D wi h he same numbe o poin s in he same design egion. This loss can be subs an ial depending on he alues he co ela ion pa ame e s. A simula ion compa ison be ween mono onic and La in hype cube designs (LHS) has been made by S ehl´ık e al. (2014). Fo a speci ic se up, e.g., α= 1, β = 10, σ = 1 and a small numbe o design poin s, i.e., n < 15, D-op imal designs ha e be e e iciency han bo h implemen a ions o LHS designs (i.e., S-op imal and Euclidean dis ance) and ac o ial design. Howe e , o n > 15 he LHS and ac o ial designs a e mo e e icien . Acknowledgmen s Au ho s a e g a e ul o Lenka Filo ´a o he help ul commen s du ing he p epa- a ion o he manusc ip . We acknowledge M´a ia Min´a o ´a o p o iding us simula ed da a o he mal ields. This esea ch was suppo ed by he Hunga ian Scien i ic Resea ch Fund unde G an No. OTKA NK101680/2012 and by he Hunga ian –Aus ian in e go e nmen al S&T coope a ion p og am T´ ET 10-1- 9 Zago aiou M, Baldi An ognini A (2009) Op imal designs o pa ame e es ima ion o he O ns ein-Uhlenbeck p ocess. Appl S och Models Bus Ind 25:583–600 Xia G, Mi anda ML, Gel and AE (2006) App oxima ely op imal spa ial design app oaches o en i onmen al heal h da a. En i onme ics 17:363–385 Ying Z (1993) Maximum likelihood es ima ion o pa ame e s unde a spa ial sampling scheme. Ann S a is 21:1567–1590 16