International Association of Mathematical Geology

2008 Distinguished Lecturer

MULTIVARIATE SPATIAL STATISTICS

Donald E. MyersUniversity of Arizona

http://www.u.arizona.edu/~donaldm

IAMG• www.iamg.org• Formed in Prague in 1968• Earth science in broad sense• Celebrated 25th anniversary in Prague• Publishes three journals• Annual conferences• Five previous Distinguished Lecturers

WHY MULTIVARIATE?• Physical reasons

– Air temperature, relative humidity, barometric pressure, wind velocity, wind direction, etc

– Hydraulic conductivity, head, porosity– Elevation vs precipitation

• Economic reasons– Ore usually contains multiple metals, value is

sum of values• Multiple hazards

– Multiple pollutants in soil, air, water

• Measurements at different scales– Ground level vs satellite– Ground level vs radar

• Proxy variables– Seismic readings vs porosity– Electric measurements vs hydrological

parameters• All variables of interest, correlated but no

known functional relationship• Response variable depends on multiple

variables

SOME QUESTIONS TO ASK

• Do any of the variables exhibit a trend?• Are any of the variables spatially

correlated?• Is there data for all variables at all

locations?• Is there a primary variable?• Is there less data for the primary variable?

QUESTIONS CONTINUED

• If there is a primary variable– Are the other variables to be used to

• (1) improve the estimation prediction of the primary?

• (2) to construct a functional dependence on some or all of the other variables

• (3) perhaps a combination of the two

VARIOGRAMS AND COVARIANCE FUNCTIONS

• Z(s) = m(s) + Y(s)– E[Z(s)] = m(s) deterministic part– Y(s) random part

• Covariance function• Cov{Z(s), Z(s+h)}

– 2nd order stat. m(x) = constant, Covariance depends only on h

• Variogram– γ(h) = 0.5Var[Z(s)-Z(s+h)

VARIOGRAMS vs COVARIANCE Functions

• Crossvariogram and cross-covariance function– C12(h) = Cov{Z1(s+h), Z2(s)}– γ12(h)= 0.5Cov {Cov{Z1(s+h)-Z1(s), Cov{Z2(s+h)- Z2(s)}

– γ12(h) = C12(0) – 0.5[C12(h) + C21(h)]– γ12(h) = γ12(h) , C12(0) = C21(0) but in general C12(h) ≠ C21(h)

Pseudo-cross-variogramγM

12(h) = 0.5Var[Z1(s+h)-Z2(s)]γM

12(h) ≠ γM12(-h)

Marcotte (Computers and Geosciences 22 (1996) 1175-1186) has given a MATLAB program to compute all three forms

REFERENCES• Clark, I., Basinger, K., and Harper, W.,

1989, MUCK--A Novel Approach to Co-Kriging.

• in B.E. Buxton (Ed.), Proceedings of the Conference on Geostatistical, Sensitivity, and Uncertainty:Methods for Ground-Water Flow and Radionuclide Transport Modeling: Batelle Press,Columbus, p. 473-494.

• Myers, Donald E.,1991, Pseudo-cross variograms, Positive Definiteness and cokriging.

• Mathematical Geology 23, 805-816

PSEUDO-CROSS VARIOGRAM CONTINED

• γij(h) = 0.5γijM(h) +0.5γji

M(h) - γijM(0)

• Several authors have used the latter as a way to estimate γij(h) . See use of LCM

• N. Cressie and C. Wilke (1998) The variance based cross variogram: You can add apples and oranges. Mathematical Geology 30, 789-799– Standardize each data set (subtract arithmetic

mean, divide by std deviation)

COKRIGING• A multivariate generalization of kriging

– Essentially everything in the kriging estimator and kriging equations is replaced by a vector or matrix

– Z*(s0) = ΣλiZ(si) vs – Zk*(s0) = Σλ1k

i Z1(si) + …+ Σλmki Zm(si)

• Model based – Some (untestable) statistical assumptions are

implicit– Model-Estimator-Equations.pdf

SOME PRACTICAL ASPECTS

• It is necessary to estimate and model – A variogram/covariance function for each

variable• There is a large class of known valid models

– A cross variogram/cross covariance function for each pair of variables

• There is no class of known valid models

• The problem is complicated by differences in the number of data locations for different variables

• The system of equations is larger, greater chance of numerical instability

• Ordinary cokriging assumes that the mean of each random function is constant (with respect to location), estimating the trend is not the same as estimating the mean– The sample variogram only estimates the

variogram is the mean is a constant– In practice one often uses the residuals from

a trend surface or a regression to estimate the variograms. This is sub-optimal

LCM• Linear Coregionalization Model

– Each Zk(s) = ΣblkYl(s); Yl(s)’s uncorrelated, l=1,…,p ; γl(h) variogram of Yl(s)

– G(h) = ΣBlγl(h), Bl positive definite matrices

• Fit variogram of each Zk(s) as positive linear combination of variograms

SOFTWARE• gstat (R package)

• gstat tutorial.pdf• gstat.pdf

– cran.r-project.org/web/packages/gstat/vignettes/gstat.pdf

– cran.r-project.org/web/packages/gstat/gstat.pdf• GSLIB

– FORTRAN codes– pangea.stanford.edu/ERE/research/scrf/software/gslib/help/progr

am/cokb3d.html– www.uofaweb.ualberta.ca/ccg/pdfs/Vol1-IntroCCGSC.pdf

• FAO-Rome– www.enge.ucl.ac.be/recherche/projets/agromet/man0.htm

• SGeMs – http://sgems.sourceforge.net/

• Cokriging with Matlab– Computers & Geosciences 17 (1991) 1265-

1280

MORE ON SOFTWARE

Overview gstat and geoR.pdf

Simplied form of cokriging demo in gstat.pdf

• 1997 A.E. Long and D.E. Myers, A new form of the Cokriging equations. Math. Geology 29, 685-703

SOME EXAMPLES• Precipitation estimation in mountainous terrain

using multivariate geostatistics– J. Applied Meteorology 31 (1992) 661-676 & 677-678

• Cokriging estimation of daily suspended sediment loads – J. Hydrology 327 (2006) 389-398

• A Statistical-Topographical model for mapping climatological precipitation in mountainous terrain – J. of Applied Meteorology 33 (1994) 140-158

MORE EXAMPLES• Flood estimation using radar and rain

gage data. – J. Hydrology 239 (2000) 4-18

• Spatial interpolation of climatic Normals: test of new method in the Canadian boreal forest. – Agric. and Forest Met. 92(1998) 211-225

• Estimating soil water content using cokriging. – Soil Science Soc. Amer. J. 51(1987) 21-30

MORE EXAMPLESRemote sensing based-geostatistical based modeling of forest canopy structure

www.kars.ku.edu/forest/asprs2000_cb.pdf Design of a Low-Boom Supersonic Business Jet Using Cokriging Approximation Models9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization September 4–6, 2002/Atlanta, GeorgiaAerodynamic Optimization of Rocket Control Surfaces Using Cartesian Methods and CAD Geometry23rd AIAA Applied Aerodynamics Conference, Jun. 6–9, 2005, Toronto, Ontario

Nonlocal problems involving spatial structure for coupledreaction-diffusion systemsApplied Mathematics and Computation 185 (2007) 449-463

MORE EXAMPLES• Application of Cokriging in iron ore

evaluation: Iron ore quadrangle-Brazilin Geostatistics Wollongong ‘96 (eds) E. Baafi

and N. Schofield, Kluwer Academic publishers 1997

• A comparison between cokriging and ordinary kriging: Case study with a polymetallic deposit. Mathematical Geology 25 (2004) 377-398

• 3-D seismic porosity modeling using a new form of cokriging. World Oil May 1999

MORE EXAMPLES

• Climate spatial variability and data resolution in a semi-arid watershed, south-eastern Arizona

Susan M. Skirvin et al (2003) Journal of Arid Environments 54, 667–686

• Cokriging Optimization Of Monitoring Network Configuration Based On Fuzzy And Non-fuzzy Variogram Evaluation – Environmental Monitoring and Assessment 82

(2003) 1-21

MORE EXAMPLES• Detecting and modeling spatial and

temporal dependence in conservation biology – Conservation Biology 14(2000) 1893-1897

• A cokriging method for estimating population density in urban areas.– Computers, Environment and Urban Systems

29 (2005)• Analysis of microarray gene expression

data – Current Bioinformatics 1 (2006) 37-53

SPATIAL-TEMPORAL DATA AND COKRIGING II

• Practical problems with spatial temporal data– More time points than spatial– Not the same spatial locations for different

times– Complicates modeling a spatial temporal

variogram• Treat values at different times as different

variables

SPATIAL-TEMPORAL DATA AND COKRIGING II

• Use cokriging– Use pseudo-cross variogram – Doesn’t allow prediction at intermediate times

• Rouhani, S. and D.E. Myers, D.E. (1990) Problems in Space-Time Kriging of Hydrogeological data. Mathematical Geology, 22, 611-623

Pseudo-cross variogram examples I

Estimation of soil nitrate distributions using cokriging with pseudo-cross variograms

J. Environmental Quality 28 (1999) 424-428Robust estimation of the pseudo-cross variogram for cokriging soil properties

European J. of Soil Science 53 (2002) 253-270

Pseudo-cross variogram examples II

• Ortiz and Emery (J. South African Institute of Mining and Metallurgy 106 (2006) 577-584) use the pseudo-cross variogram to fit an LCM for drill hole and blast hole data for a Porphry copper deposit

• Vanderlinden et al (J. Environmental Quality 35 (2006) 21-36) used the pseudo cross variogram in mapping non-collocated spatial temporal mine spill data

Pseudo-cross variogram examples III

• Geostatistical analysis of stereoscopic pairs of satellite images– In P. Monestiez et al (eds) geoENV III

Environmental Applications of geostatistics (2001)

• Registering two different images of the same terrain

• Lark et al (Geoderma 133 (2006) 363-379) used pseudo cross variograms to map non –collocated spatial temporal soil data

VARIATIONS ON COKRIGING• UNDERSAMPLED CASE

– Most common application– Can be accommodated in the software

• COLLOCATED COKRIGING– Data on secondary variables only at

estimation points• FACTORIAL KRIGING

– Estimate data on Yl(s)’s in LCM• UNIVERSAL COKRIGING or

DETRENDED COKRIGING

ALTERNATIVES CONTINUED

• Which Models for Collocated Cokriging? Jacques Rivoirard (2001) Mathematical Geology 33, 117-131

• Estimating Monthly Streamflow Values by Cokriging . Andrew R. Solow and Steven M. Gorelick (1986) Mathematical Geology 18 , 785-809

FACTORIAL KRIGING EXAMPLES

• Scale Matching with Factorial Kriging for Improved Porosity Estimation from Seismic Data– Math. Geology 31 (1999) 23-46

• FACTOR2D: a computer program for factorial cokriging– Computers & Geosciences 28 (2202) 857-875

COLLOCATED EXAMPLES• Mapping Soil Carbon Using Collocated

Cokriging with Wetness Index – 12th Conference of Int. Association for Mathematical

Geology, Beijing, China, August 26-31, 2007, 391-395

• 3-D Seismic porosity modeling using a new form of cokriging (collocated cokriging)– World Oil 220 (1999) 77-84

ALTERNATIVES TO COKRIGING• KRIGING WITH EXTERNAL DRIFT

– E[Z1(s)] = f(Z2(s)), usually a polynomial• Assumes data for Z2(s) available at estimation points

– Geoadditive Models• J. of the Royal Statistical Society: Series C (Applied

Statistics) 52 (2003)1-18

• KRIGING LINEAR COMBINATIONS– Math.Geology 15 (1983) 633-637

• USE OF PCA TO SEPARATE VARIABLES– Math. Geology 15 (1983) 287-300

EXTERNAL DRIFT EXAMPLES I• Ahmed, S. and de Marsily, G. (1987).

Comparison of geostatistical methods for estimating transmissivity using data on transmissivity and specific capacity. Water Resources Research 23, 1717-1737

• Yates, S. and Warrick, A. W. (1987). `Estimating soil water content using cokriging'. Soil Science Society of

• America Journal 51, 23-30

EXTERNAL DRIFT EXAMPLES II

• Kriging with an external drift versus collocated cokriging for water table mapping – Applied Earth Science (Trans. Inst. Min. Metall. B)

115 (2006) 103-111

• Application of kriging with external drift to estimate hydraulic conductivity from electrical-resistivity data in unconsolidated deposits near Montalto Uffugo, Italy – Hydrogeology J. 8 (2000) 356-367

Generalized Linear Models• Geoadditive Models (Geographic variability of

reproductive health outcomes, e.g. birth weight). J. Royal Stat. Soc. Series C 52(1) 1-18

• Yi = β0 + f(si) + g(ti) + β1Txi + S(xi) + εi

– f, g unknown smooth functions fit by penalized splines

• This can be re-written in the form– Y = Xβ + Zb + ε– See lme4 package in R

http://www.ic.arizona.edu/ic/geog657/

IAMG

http://www.iamg.org