土木学会論文集 D3( 土木計画学 ), Vol. 71, No. 2, 31-43,

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1 murakami.daisuke@nies.go.jp tsutsumi@sk.tsukuba.ac.jp Key Words: sampling design, geostatistics, officially assessed land price, prefectural land price pp / sampling design Wang et al i design based approach ii model based approach i simple random sampling systematic sampling stratified random sampling two step sampling Gruijter Ripley design based approach 31

2 ii ii p p median p center space filling design Nychka and Saltzman ii 2 e. g., Silvey e. g., Cressie e. g., Zimmerman Brus and Heuvelink Zidek and Zimmerman Wang et al ii ii / ii / 4 1 ii

3 3 1 D R s n D n 1, N z s n 1 z=xβ+ε ε~n 0,C 1 z N 1 X N K β K 1 ε N 1 0 N 1 C N N d n, n ' 1 Xβ ε 2 + c d = τ σ if d =0 τ + σ f d if 0<d <r 2 0 otherwise f d = 3d 2r d 2r τ σ r nugget partial sillrange s 0 D z s z s 3 z s = s β+ε s 3 ' s s 0 K 1 ε s s 0 z s Best Linear Unbiased Predictor BLUP 4 z s = s β +c C z Xβ 4 β = X C X X C z c s 0 N 1 z s 5 Ez s z s =σ +τ c C c+ c C X X C X c C X Cressie Schabenberger and Gotway a Zhu and Stein min C S =min w c S 6 i i min C S =min max w c S 7 i i S S i i 1, 2,... I S S i S {S 1, S 2,...S I } m {m 1, m 2,...m M } D R M 1 w m m c S S i Ez s z s w c S S i m 6 7 w c S C S = w c S C S =max w c S 1 Brus and Heuvelink

4 S i S 10 C Si* M 30 w m c m S i 67 M w m c m S i 4 S i * 8 S =arg min i g {1, 2} b C S 8 S i * S NP NP hard S * i Simulated Annealing SA : Kirkpatrich et al SA i S 0 T T 0 iiii a ii b k ii a S i_a 1 S i_b ii b S i_a S i_b S i_a 1 9 S i_a S i_b exp CS _ CS _ T 9 iii T pt ii p 0 p 1 T p 1 SA Kirkpatrich et al p 1 T 0 10 Brus and Heuvelink T CS C S = 10 M log 0.8 S ii b SA 10 T 0 80% C S S i c m S i Ez s z s 7 D n c n c S 0 S 0 S i_a 1 n c 1 ii D n c n c n c S 0 n c ii 1 SA p 1 Kirkpatrich et al SA 1 1 SA n c D 34

5 D D D' D D' D D' Cressie A A 4 1 D D Brus and Heuvelink T 0 10 k 100 p c S τ σ r τ σ r M 1 w m 2 a 1 A B C 2 D C S C S A B B C 2 C 1 C 2 C 3 Zidek and Zimmerman C 3 2 C C 2 C 2 C 2 35

6 C 1 D A C 3 C 2 4 B C 1 D D C 1 2 a w m D 1 D 2 D 3 D 1 D 3 D 2 w m w m D 1 D 2D 3 StatTopPortal.do D 1 w m , 557 3, Clayton and Kaldor m t m 11 t ~Poisson θ t 11 t m m t m m 12 θ ~Gamma a,b 12 a b m 13 θ = t +a t +b 13 θ m θ t b t m a θ t Clayton and Kaldor b 36

7 1 max w c S b A C 3 b 6 7 D D 1 D 2 D 1 D 2 D 1 D 2 D 1 D D 3 c 1 w m c S w c S m w c S 6 4 1w m C 1D C 2 B C z z z z z z % 79 30% % c S w m 3, 943 c S 1/m 2 km km 1km 2 m WLS&EGLS Schabenberger and Gotway VIF Variance Inflation Factor 37

3 2 4 3 2 /m 2 VIF 10 1997 19 1% 10% 1% 1. 15 10 8 1.

8 /m 2 VIF % 10% 1% / % 6. 48km

9 4 Mean Percentage Error MPE 14 MPE= 100 N z s z s z s 14 z s n z s MPE % 3 2 a b a b 13 a / b fold cross validation 4 3 w c S c S w m c S w m c S w m w m c S w m c S c S w m I IIIII w c S 39

8 5 7 50% w m c m S i w c S w m c S / 50% 5 I II III 1

10 % w m c m S i w c S w m c S / 50% 5 I II III 1 III 50% III I II I II III 12 10% 30% 50%

12 12 Fuentes et al trans gaussian kriging e. g., Cressie geoadditive model Kammann and Wand z z z z z Japan Geoscicence Union Meeting ) ) Wang, J. F., Stein, A., Gao, B. B. and Ge, Y. : A Review of spatial sampling, Spatial Statistics, Vol.2,pp.1 14, ) Gruijter, J. : Spatial sampling schemes for remote sensing, In Stein, A., Meer, F. and Gorte, B. eds., Spatial Statistics for Remote Sensing, pp , ) Ripley, B. D. : Spatial Statistics, JohnWiley& Sons, ) Vol. 47 pp ) Nychka, D. and Saltzman, N. : Design of air quality 42

13 monitoring designs, In Nychka, D., Piegorsch, W. W. and Cox, L. H. eds., Case studies in Environmental Statistics, New York: Springer, pp , ) Silbey, S. D. : Optimal Design, London: Chapman& Hall, ) Cressie, N. : Statistics for Spatial Data. RevisedEdition, John Wiley & Sons, ) Zimmerman, D. L. : Optimal network design for spatial prediction, covariance parameter estimation, and empirical prediction, Environmentics, Vol.17,pp , ) Brus, D. J. and Heuvelink, G. B. M. : Optimization of sample patterns for universal kriging of environmental variables, Geoderma, Vol.138,No.1 2, pp , ) Zidek, J. V. and Zimmerman, D. L. : Monitoring network design, In Gelfand, A. E., Diggle, P. J., Fuentes, M. and Guttorp, P. eds., Handbook of Spatial Statistics,CRCPress, pp , ) p/chika/kouji/2012/01.html 2014/ 05/ ) 23 GIS Vol. 17 No. 1 pp ) GIS Vol. 22 No. 2 pp ) Schabenberger, O. and Gotway, C. A. : Statistical Methods for Spatial Data Analysis, ChapmanandHall/CRC, ) Zhu, Z. and Stein, M. L. : Spatial sampling design for parameter estimation of the covariance function, Journal of Statistical Planning and Inference, Vol.134,No.2,pp , ) Kirkpatrich, S., Gelatt, C. D. and Vecchi, M. P. : Optimization by simulated annealing, American Association for the Advancement of Science, Vol.220,No.4598,pp , ) Clayton, D. and Kaldor, J. : Empirical Bayes estimator of age standardized relative risks for use in disease mapping, Biometrics, Vol.43,No.3,pp , ) ) Fuentes, M., Chaudhuri, A. and Holland, D. M. : Bayesian entropy for spatial sampling design of environmental data, Environmental and Ecological Statistics,Vol.14,No.3,pp , ) Kammann, E. E. and Wand, M. P. : Geoadditive models, Journal of the Royal Statistical Society: Series C (Applied Statistics), Vol.51,No.1,pp.1 18, GEOSTATISTICS FOR THE ASSESSED SITE ALLOCATION PROBLEM IN OFFICIALLY ASSESSED LAND PRICE AND PREFECTURAL LAND PRICE Daisuke MURAKAMI and Morito TSUTSUMI Designing sampling strategy is a major concern in statistics. The same also holds for study fields that discuss spatial data modeling, including geostatistics; and, recently, sample site selection design is intensively discussed in these fields. This study applies geostatistics to the assessed site allocation problem in the officially assessed land price data and prefectural land price data in Japan. Firstly, we explain these land price data, while mainly focusing on basic rules of their assessed site allocation. Then, studies discussing sampling design problems are briefly summarized. Subsequently, points, which we must consider, are clarified, and geostatistical approaches for the land price assessed site allocation problem are developed based on the points. Finally, the developed approaches are used to the assessed site reduction problem in Ibaraki prefecture, and their effectiveness is examined. 43

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