Some New Spatial Statistical
Models for Stream Networks
By
Jay M. Ver Hoef
Alaska Dept. of Fish and Game
1300 College Road
Fairbanks, AK 99701
ffjmv@uaf.edu
Co-Authors
Erin Poston
Department of Geosciences, CSU
David M. Theobald
Natural Resources Ecology Lab, CSU
Stream Chemistry Example
Literature
• Curriero, F. C. (1996). The Use of Non-Euclidean Distances in
Geostatistics. Ph.D. Dissertation, Kansas State University.
• Little, L. S., Edwards, D., and Porter, D. E. (1997). Kriging in
•
•
•
estuaries: as the crow flies, or as the fish swims? Journal of
Experimental Marine Biology and Ecology 213:1-11.
Rathbun, S. L. (1998). Spatial Modelling in Irregularly Shaped
Regions: Kriging Estuaries. Environmetrics 9:109-129.
Gardner, B., Sullivan, P. J., and Lembo, A. J. ( 2003). Predicting
stream temperatures: geostatistical model comparision using
alternative distance metrics. Canadian Journal of Aquatic Science
60:344-351.
Yuan, L. L. (2004). Using Spatial Interpolation to Estimate
Stressor Levels in Unsampled Streams. Environmental Monitoring
and Assessment 94:23-38.
Spatial Linear Model
 z observed 

  Xβ  ε, var(ε)  Σ(θ)
 z unobserved 
Prediction
Estimation
Valid Models in 1-D
Valid Models for Linear
Networks
Valid Models for Stream
Networks
+ 1 layer
+ 2 layers
+ 3 layers
Moving Average Construction




Z (s)    g (x  s; θ)Y (x)  U (s)  
where
Y (s) and U (s) are white noise processes
with expection  0 and variance  1.
Moving Average Construction
2
0
-2
0
20
40
60
80
100
0
20
40
60
80
100
2
0
-2
Valid Covariance




C (h)    g (x; θ) g (x  h; θ)dx  I (h  0)
2




Z (s)    g (x  s; θ)Y (x)  U (s)  
where
Y (s) and U (s) are white noise processes
with expection  0 and variance  1.
Moving Averages in 1-D
Literature
•
•
•
•
Barry, R.D. and Ver Hoef, J.M. 1996. Blackbox kriging: spatial
prediction without specifying the variogram. Journal of
Agricultural, Biological, and Environmental Statistics 1: 297 –
322.
Ver Hoef, J.M. and R.D. Barry. 1998. Constructing and fitting
models for cokriging and multivariable spatial prediction.
Journal of Statistical Planning and Inference 69: 275 – 294.
Cressie, N. and Ver Hoef, J.M. 2001. Multivariate geostatistics
for precision agriculture (with discussion). Bulletin of the
International Statistical Institute, Invited Papers Volume 59,
Book 1, 407 – 410
Ver Hoef, J.M., N. Cressie, and R.P. Barry. 2003. Flexible
Spatial Models for kriging and cokriging using moving averages
and the Fast Fourier Transform (FFT). In Press in Journal of
Computational and Graphical Statistics.
Kriging on Streams
Flow
Constructing a Valid Covariance
Matrix
Σ  [ R(D /  )   I]  F
2
2
where Σ is covariance matrix
  partial sill,   range,   nugget
2
2
R is matrix spatial correlatio n function
e.g., circular, spherical, exponentia l, etc.
D is all pairwise stream distances

is Hadamard (element - wise) product
F is " Flow" matrix
Flow Matrix Example
F
1
2
Flow
3 4
5
6
Flow
Flow
Choosing a Covariance Model
Estimation
 z observed 

  Xβ  ε, var(ε)  Σ(θ)
 z unobserved 
Estimating Fixed Effects
Ikaluruk
River
Flow
Estimation
Flow
 z observed 

  Xβ  ε, var(ε)  Σ(θ)
 z unobserved 
Flow
Prediction
 z observed 

  Xβ  ε, var(ε)  Σ(θ)
 z unobserved 
Prediction
Discontinuity at Nodes
Flow
Flow
Flow
Prediction Variances
Fish Example
Pure Distance Models
Flow
Predictions