Functional Kriging Uncertainty Assessment

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ISBN 978-88-6493-035-0

9 788864 930350 ?iiT,ffrrrXbBKBX2m

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ζn×M = (ζ(s1), . . . , ζ(sn)) = ˆLn−1×n(Yn×M − ˆμn×M) .

jX :2M2`i2 B #QQibi`T bKTH2b rBi? bBx2 n+ 1- ζn∗₊₁ = (ζ∗(s1), . . . , ζ∗(sn), ζ∗(sn+1))

7`QK ζ(s1), . . . , ζ(sn)X

9X *`2i2 i?2 m;K2Mi2/ +Qp`BM+2 Ki`Bt ˆΛ = ˆΣ ˆcTn

ˆcn ˆσ2



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-ˆ

C Bb i?2 2biBKi2/ +Qp`BM+2 7mM+iBQM M/ ˆσ2 = ˆC(0) Bb i?2 2biBKi2/ bBHHX lb2

*?QH2bFv /2+QKTQbBiBQM bQ i?i ˆΛ = ˆR ˆRT _{M/ i`Mb7Q`K i?2 #QQibi`T bKTH2b}

ζ_n∗₊₁ b

(e∗_{(s1), . . . , e}∗_(s

n), e∗(s0)) = ˆR(n+1)×(n+1)ζ_(n+1)×M∗ .

8X h?2 }MH #QQibi`T bKTH2 Bb /2i2`KBM2/ b Ys∗_i(t) = ˆμs_i(t) + e∗s_i(t)- i = 1, . . . , n

M/ Ys∗₀(t) = ˆμs₀(t) + e∗s₀(t)X

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BMi2`pH 7Q` Ys0(t) +M #2 r`Bii2M b ( ˆYs0(t)−q∗<sub>1−α/2</sub>, ˆYs0(t)−qα∗<sub>/2</sub>)- rBi? q∗α i?2 α@T2`+2MiBH2

Q7 ˆF_n∗- i?i +M #2 Q#iBM2/ Q`/2`BM; i?2 +m`p2bX h?2`2 Bb MQ ;QH/ biM/`/ 7Q` Q`/2`BM; 7mM+iBQMH /iX >2`2 r2 +QMbB/2` irQ Q`/2`BM; i2+?MB[m2b #b2/ QM #M/ /2Ti? M/ L2 /BbiM+2X "M/ /2Ti? (N) +M #2 /2}M2/ 7Q` Mv b2i Q7 k +m`p2b U?2`2 k = 2VX h?2 bKTH2 #M/ /2Ti? U".V Q7 y(t) +M #2 +H+mHi2/ b i?2 T`QTQ`iBQM Q7 #M/b /2HBKBi2/ #v irQ

JBMBbvKTQbBmK 9d 0.0 0.2 0.4 0.6 0.8 1.0 0246 Station 1 t ●● ●● ●●●●●●● ● ● ● ● ● ● ●● ● ●●●●●●● ● ● ●●●●● ● ●● ● ●●●●●● ● ● ●●●●●●●●●●● ●●● ● ● ● ●●● ● ●●●● ●●●●● ●● ●●●● ●●●●●●●● ● ● ● ● ● ● ●●●●● 0.0 0.2 0.4 0.6 0.8 1.0 0246 Station 2 t ●● ● ● ●●●●●●●●● ● ●● ●●● ●● ● ● ● ● ●●●●● ● ● ● ● ●●●●●●●●●●● ●●●●●●●● ● ●● ● ● ● ● ●●●● ●● ● ● ● ●●●●● ● ● ● ● ● ●●●●●● ● ●● ● ●● ●● ● ●●●●● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0246 Station 3 t ●●●●●● ●●●●●●● ● ●●● ● ● ●● ●●●●●● ● ●●● ●● ● ● ●● ●●●●●● ● ●●●●● ● ● ● ● ●●● ● ●●● ● ● ●●●● ●●●● ● ●●● ●●● ●●●● ● ●● ●● ●●● ●● ●●●● ● ●● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0246 Station 4 t ● ●●● ● ● ●●●●●●●● ● ● ● ● ●● ●●●● ● ●● ● ● ● ● ● ● ●● ● ● ●● ●●●●●●●●●●● ● ●● ● ● ●● ● ● ● ●●●●●●●●●●● ● ●● ●●●●●● ● ●● ● ● ●●●● ●●●●●●● ● ●● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0246 Station 5 t ●●●●●●● ●●●● ● ● ●●●●●●● ● ● ●● ●● ● ●● ●● ●● ●● ● ● ●● ●● ● ●●●●●● ● ●● ● ● ● ●●●●●●●●●●●● ● ● ● ● ● ●●● ● ●● ●●● ● ● ● ● ●● ● ● ●● ●● ●● ● ● ●●●● ● 0.0 0.2 0.4 0.6 0.8 1.0 0246 Station 6 t ● ● ● ● ● ●●●● ● ● ● ●●● ● ● ●●●●●●●● ● ● ● ● ●●●● ● ● ● ● ●● ● ●●●●● ● ● ●● ● ●● ● ●● ● ●●●●● ● ● ●● ● ● ●●●●●● ● ● ●●●●●●●● ●● ● ●●●●●●●● ● ● ● ●● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0246 Station 7 t ●● ●● ● ●●● ●●●●●●●● ● ● ● ● ●● ● ● ● ●● ● ●● ● ●● ● ● ● ●●● ● ●● ● ● ●●●● ●●● ●●●●●●●● ●● ●●● ●●●● ● ● ● ●●●● ● ● ● ● ● ● ●●● ● ● ● ●●●●● ● ● ●●●● ● ●● 0.0 0.2 0.4 0.6 0.8 1.0 0246 Station 8 t ● ● ●● ● ● ● ● ●●●● ● ●●●●●●● ●● ● ● ● ●●●●●● ●●●●● ● ●●●●● ● ● ●●● ● ● ●● ● ● ●● ● ●●● ● ●●●●●●●●●●●● ●● ● ● ●● ●●● ●●● ● ● ● ● ●●●●● ●●● ● ●●● ● 0.0 0.2 0.4 0.6 0.8 1.0 0246 Station 9 t ●● ● ●● ● ●●●●●●● ● ● ●●● ● ● ● ● ●● ●●●●●●● ● ● ● ●●●● ● ●● ●●● ● ●●●● ● ●●●● ● ● ● ● ●●●● ●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●●● ●● ● ● ● ●● ●●●●●● 0.0 0.2 0.4 0.6 0.8 1.0 0246 Station 10 t ● ●●●●●● ● ● ●● ● ●● ●●●●●●● ● ● ● ● ● ● ● ● ●●●●●●● ● ●● ● ● ●● ● ● ● ● ● ●● ●●●●● ● ●●●● ●●●● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ●● ● ●● ● ●●● ● ● ●● ● ● ● ●● ● ● ●

6B;m`2 R, P`B;BMH /i U#H+F /QibV- 6E1. T`2/B+i2/ +m`p2 UbQHB/ `2/ HBM2V- N8W T`2/B+iBQM #M/ UTBMFV #b2/ QM L2 /BbiM+2 UTBMFV M/ QM J". U#Hm2V 7Q` n = 50- σ2 =

0.25-φ= 1.5X φ=0.5 width 246 σ2=0.25 φ=1 φ=1.5 width 246 σ2=0.5 t width 246 0 0.2 0.40.6 0.8 1 σ2=0.75 t 0 0.2 0.4 0.6 0.8 1 t 0 0.2 0.40.6 0.8 1 φ=0.5 width 246 σ2=0.25 φ=1 φ=1.5 width 246 σ2=0.5 t width 246 0 0.20.4 0.6 0.8 1 σ2=0.75 t 0 0.2 0.4 0.6 0.8 1 t 0 0.2 0.4 0.6 0.8 1 φ=0.5 width 246 σ2=0.25 φ=1 φ=1.5 width 246 σ2=0.5 t width 246 0 0.2 0.4 0.6 0.8 1 σ2=0.75 t 0 0.2 0.4 0.6 0.8 1 t 0 0.2 0.4 0.6 0.8 1

6B;m`2 k, "M/ rB/i? U/2Ti?V 7Q` n= 25 UH27iV- n = 50 UKB//H2V M/ n = 90 U`B;?iVX

+m`p2b +QMiBMBM; i?2 r?QH2 +m`p2 y(t) (N)c ?2`2 r2 mb2 i?2 KQ/B}2/ #M/ /2Ti? UJ".V-i?i iF2b BMiQ ++QmMi r?2i?2`  TQ`iBQM Q7 i?2 +m`p2 Bb BM i?2 #M/ U7Q` /2iBHb b22 (N)VX h?2 HQr2`fmTT2` HBKBib Q7  U/2Ti? #b2/V N8W T`2/B+iBQM #M/ `2 Q#iBM2/ #v iFBM; i?2 TQBMirBb2 UrX`XiX tV KBMBKmKfKtBKmK Q7 i?2 N8W /22T2bi +m`p2b UBM i?2 +b2 Q7 #M/ /2Ti?V Q` Q7 i?2 N8W +m`p2b +HQb2bi iQ i?2 x2`Q +m`p2 UBM i?2 +b2 Q7 mbBM; L2 /BbiM+2V

j aBKmHiBQM bim/v

q2 BK iQ MHvb2 i?2 BKT+i i?i i`2M/ +QKTH2tBiv- bTiBH bi`m+im`2 UpB i?2 +Qp`BM+2 7mM+iBQM T`K2i2`b Q7 i?2 7mM+iBQMH `2bB/mH `M/QK }2H/V M/ Q`/2`BM; i2+?MB[m2 ?p2 QM i?2 T2`7Q`KM+2 Q7 i?2 #QQibi`TTBM; K2i?Q/ r?2M BM+`2bBM; i?2 MmK#2` Q7 bBi2bX .i r2`2 bBKmHi2/ mbBM; +m#B+ "@bTHBM2b QM  bTiBH B``2;mH` ;`B/ Un HQ+iBQMbV QM

D= [0, 2] × [0, 3] M/ +m`p2 /QKBM T = [0, 1]X h?2 `2bB/mH 7mM+iBQMH `M/QK }2H/ rb

#mBHi b es(t) =

₁₀

j=1ξj(s)Bj(t)- r?2`2 Bj(t) Bb i?2 j

th _{#bBb 7mM+iBQM 2pHmi2/ i t∈ T X}

h?2 bTiBHHv +Q``2Hi2/ bTHBM2 +Q2{+B2Mib {ξj(s), s ∈ D} r2`2 ;2M2`i2/ 7Q` 2+? j BM

1, . . . , 10 mbBM; i?2 bK2 2tTQM2MiBH +Qp`BM+2 7mM+iBQM rBi? `M;2 M/ b+H2 T`K2i2`b

φ∈ (0.5, 1, 1.5) M/ σ2 ∈ (0.25, 0.50, 0.75) `2bT2+iBp2Hv- `2bmHiBM; BM N /Bz2`2Mi b+2M`BQbX

h?2 /`B7i rb Q#iBM2/ b ms(t) = α(t) + β1(t)lon + β2(t)lat- r?2`2 lon M/ lat `2 i?2

bTiBH +QQ`/BMi2b- α(t) Bb  7mM+iBQMH BMi2`+2Ti M/ β1(t), β2(t) `2 7mM+iBQMH +Q2{+B2Mib i?i +M #2 2tT`2bb2/ BM i2`Kb Q7 "@bTHBM2 #bBb Ur?Qb2 +Q2{+B2Mib +M #2 +?Qb2M iQ /2i2`KBM2 i?2 +QKTH2tBiv Q7 i?2 /`B7iVX 6BMHHv- bBKmHi2/ Q#b2`piBQMb r2`2 #mBHi b

JBMBbvKTQbBmK 9d

r?2`2 ξ(t) = {ξs<sub>1</sub>(t), . . . , ξs<sub>n</sub>(t)} ∼ Nn(0, 0.09) Bb  p2+iQ` Q7 `M/QK 2``Q`b 7Q` 2+? }t2/

t ∈ [0, 1]X 6Q` 2+? bBKmHiBQM b+2M`BQ- r2 ;2M2`i2/ 7mM+iBQMH /i i n = 25, 50

M/ Ny M2bi2/ HQ+iBQMbX //BiBQMHHv /i r2`2 ;2M2`i2/ i Ry KQ`2 bBi2b mb2/ b pHB/iBQM biiBQMbX h?2 6E1. KQ/2H Ua2+iBQM RV rb TTHB2/ iQ 2+? bBKmHi2/ /i b2i iQ T`2/B+i +m`p2b i i?2 Ry pHB/iBQM bBi2bX 6Q` 2+? pHB/iBQM biiBQM B = 500 T`2/B+iBQMb r2`2 Q#iBM2/ 7QHHQrBM; a2+iBQM k M/ N8W T`2/B+iBQM #M/b r2`2 T`Q/m+2/ mbBM; #Qi? /BbiM+2 M/ J".X M 2tKTH2 +M #2 b22M BM 6B;m`2 RX hQ 2pHmi2 i?2 T2`7Q`KM+2 Q7 i?2 T`QTQbH- r2 +QMbB/2` irQ /Bz2`2Mi BM/B+iQ`b, i?2 rB/i? Q7 i?2 `2bmHiBM; N8W T`2/B+iBQM BMi2`pH M/ i?2 T`QTQ`iBQM Q7 i?2 bBKmHi2/ +m`p2 rBi?BM i?2 BMi2`pHX 6B;m`2 k bmKK`Bx2b U/2Ti?V #M/ rB/i? 7Q` HH bKTH2 bBx2b M/ bBKmHiBQM b+2M`BQbX b QM2 rQmH/ 2tT2+i- #M/ rB/i? /2+`2b2b rBi? BM+`2bBM; bKTH2 bBx2X JQ`2Qp2`- #M/ rB/i? BM+`2b2b rBi? BM+`2bBM; σ2 M/ /2+`2b2b bHB;?iHv rBi? BM+`2bBM; φ 7Q`  }t2/ pHm2 Q7

σ2X h?2 /2Ti?@#b2/ #M/ Bb T`+iB+HHv Hrvb rB/2` i?M i?2 /BbiM+2@#b2/ QM2X AM i2`Kb Q7 +Qp2`;2 U};m`2 MQi b?QrM ?2`2V- i?2 T2`7Q`KM+2 TT2`b ;QQ/ M/ BKT`Qp2b rBi? BM+`2bBM; bKTH2 bBx2X

9 .Bb+mbbBQM

q2 T`QTQb2  b2KB@T`K2i`B+ #QQibi`T TT`Q+? i?i HHQrb i?2 +QMbi`m+iBQM M/ 2pHmiBQM Q7 bBKmHiM2Qmb T`2/B+iBQM #M/b @ Qp2` T @ 7Q` i?2 7mM+iBQMH F`B;BM; T`2/B+iQ` rBi?  MQM@+QMbiMi rB/i?X h?2 bBKmHiBQM bim/v b?Qrb i?i i?2 T`QTQb2/ i2+?MB[m2 ?b  ;QQ/ T2`7Q`KM+2X q2 `2 +m``2MiHv BMp2biB;iBM; i?2 2z2+i Q7 KQ`2 +QKTH2t /`B7ib b r2HH b Hi2`MiBp2 rvb Q7 2pHmiBM; i?2 T2`7Q`KM+2 Q7 i?2 T`QTQbHX

_272`2M+2b

(R) *#HH2`Q- qX- :B`H/Q- _X M/ Ji2m- CX kyRj  mMBp2`bH F`B;BM; TT`Q+? 7Q`

bTiBH 7mM+iBQMH /iX aiQ+?X 1MpX _2bX _BbF bb2bbX kdUdV- R88jĜR8ejX

(k) .2HB+/Q- SX- :B`H/Q- _X- *QKb- *X M/ Ji2m- CX kyRy aiiBbiB+b 7Q` bTiBH

7mM+iBQMH /i, bQK2 `2+2Mi +QMi`B#miBQMbX 1MpB`QMK2Mi`B+b- kR- kk9ĜkjNX

(j) :B`H/Q- _X- .2HB+/Q- SX M/ Ji2m- CX kyRR P`/BM`v F`B;BM; 7Q` 7mM+iBQM@pHm2/

bTiBH /iX 1MpB`QMK2MiH M/ 1+QHQ;B+H aiiBbiB+b- R3UjV- 9RRĜ9keX

(9) :QmH`/- JX M/ oQHix- JX RNNj :2QbiiBbiB+H BMi2`TQHiBQM Q7 +m`p2b,  +b2 bim/v BM bQBH b+B2M+2X AM :2QbiiBbiB+b h`QB ǶNkU2/X X aQ`2bV- pQHX k- TTX 3y8Ĝ3Re- EHmr2` +/2KB+- .Q`/`2+?iX

(8) A;M++QHQ- _X- Ji2m- CX M/ :B`H/Q- _X kyR9 E`B;BM; rBi? 2ti2`MH /`B7i 7Q` 7mM+iBQMH

/i 7Q` B` [mHBiv KQMBiQ`BM;X aiQ+?X 1MpX _2bX _BbF bb2bbX k3U8V- RRdRĜRR3eX

(e) A`MTM?- LX- JQ?KK/x/2?- JX M/ hvHQ`- *X*X kyRR  +QKT`BbQM Q7 #HQ+F

M/ b2KB@T`K2i`B+ #QQibi`T K2i?Q/b 7Q` p`BM+2 2biBKiBQM BM bTiBH biiBbiB+bX

*QKTmiX aiiBbiX .i MHX 88- 8d3Ĝ83dX

(d) J2M7Q;HBQ- X- a2++?B- SX M/ .HH _Qb- JX kyRj  lMBp2`bH E`B;BM; T`2/B+iQ` 7Q`

bTiBHHv /2T2M/2Mi 7mM+iBQMH /i Q7  >BH#2`i aT+2X 1H2+i`QMX CX aiiX d- kkyNĜkk9yX

(3) L2`BMB- .X- JQM2biB2x- SX M/ JMiû *X kyRy *QF`B;BM; 7Q` bTiBH 7mM+iBQMH /iX CX JmHiBp`Bi2 MHX RyR- 9yNĜ9R3X

JBMBbvKTQbBmK 9d

(N) GQT2x@SBMi/Q- aX M/ _QKQ- CX kyyN PM i?2 +QM+2Ti Q7 /2Ti? 7Q` 6mM+iBQMH .iX CX K2`X aiiBbiX bbQ+X Ry9U93eV- dR3Ĝdj9X

(Ry) a+?2HBM- GX M/ aDƺbi2/i@/2 GmM- aX kyRy E`B;BM; T`2/B+iBQM BMi2`pHb #b2/ QM