9.3 and 9.4
The Spatial Model
And
Spatial Prediction and the Kriging
Paradigm
9.3
The Spatial Model
The Spatial Model….
What is it?
• Statistical model decomposing the
variability in a random function into a
deterministic mean structure and one or
more spatial random processes
For Geostatistical data:
•
•
•
•
Z(s) = μ(s) + W(s) + η(s) + e(s)
Where
Large scale variation of Z(s) is expressed
through the mean μ(s)
W(s) is the smooth small-scale variation
η(s) is a spatial process with variogram γη
e(s) represents measurement error
Two Basic Model Types
Signal model
S(s)=μ(s) + W(s) + η(s)
Mean model
δ(s)= W(s) + η(s) + e(s)
Signal Model
S(s)=μ(s) + W(s) + η(s)
• Denotes the signal of the process
• Then:
– Z(s) = S(s) + e(s)
Mean Model
δ(s)= W(s) + η(s) + e(s)
• Denotes the error of process
• Then:
– Z(s) = μ(s) + δ(s)
• Model is the entry point for spatial regression and
analysis of variance
• Focus is on modeling the mean function μ(s) as a
function of covariates and point location
– δ(s) is assumed to have spatial autocorrelation structure
• Mean and signal models have different focal
points
• Signal model: we are primarily interested in
stochastic behavior of the random field and, if it is
spatially structured, predict Z(s) or S(s) at
observed and unobserved locations
– μ(s) is somewhat ancillary
• Mean model: Interest lies primarily in modeling
the large scale trend of the process
– Stochastic structure that arises from δ(s) is somewhat
ancillary
• For mean models, the analyst must decide
which effects are part of the large-scale
structure μ(s) and which are components of
the error structure δ(s)
• No solid answer
One modeler’s fixed effect is someone else’s
random effect
Cliff and Ord’s
Reaction and Interaction Models
• Reaction model: sites react to outside
influences
– Ex: Plants react to the amount of available
nutrients in the root zone
• Interaction model: sites don’t react to
outside influences, but rather with each
other
– Ex: Neighboring plants compete with each
other for resources
In general:
• When dominant spatial effects are caused
by sites reacting with external forces,
include the effects as part of the mean
function
• Interactive effects call for modeling spatial
variability through the spatial
autocorrelation structure of the error process
• Spatial autocorrelation in a model does not
always imply an interactive model over a
reactive one
– Can be spurious if caused by large-scale trends
or real if caused by cumulative small-scale,
spatially varying components
– Thus, error structure often thought of as the
local structure and the mean structure as the
glocal structure
9.4
Spatial Prediction and the
Kriging Paradigm
Prediction vs. Estimation
• Prediction is the determination of the value of a
random variable
• Estimation is the determination of the value of an
unknown constant
• If interest lies in the value of a random field at
location (s) then we should predict Z(s) or the
signal S(s). If the average value at location (s)
across all realizations of the random experiment is
of interest, we should estimate E[Z(s)].
The goal of geostatistical
analysis:
• Common goal is to map the random
function Z(s) in some region of interest
• Sampling produces observations
Z(s1),…Z(sn) but Z(s) varies continuously
throughout the domain D
• Producing a map requires prediction of Z()
at unobserved locations (s0)
The Geostatistical Method:
1. Using exploratory techniques, prior
knowledge, and/or anything else, posit a
model of possibly nonstationary mean plus
second-order or intrinsically stationary error
for the Z(s) process that generated the data
2. Estimate the mean function by ordinary least
squares, smoothing, or median polishing to
detrend the data. If the mean is stationary this
step is not necessary. The methods for
detrending employed at this step ususually do
not take autocorrelation into account
3. Using the residuals obtained in step 2 (or the
original data if the mean is stationary), fit a
semivariogram model γ(h;θ) by one of the
methods in 9.2.4
4. Statistical estimates of the spatial dependence in
hand (from step 3) return to step 2 to re-estimate
the parameters of the mean function, now taking
into account the spatial autocorrelation
5. Obtain new residuals from step 4 and iterate
steps 2-4, if necessary
6. Predict the attribute Z() at unobserved locations
and calculate the corresponding mean square
prediction errors
• In classical linear model, the predictor (Y)
and the estimator (X) are the same
• Spatial data exhibit spatial autocorrelations
which are a function of the proximity of
observations
– We must determine which function of the data
best predicts Z(s0) and how to measure the
mean square prediction error
Kriging
• These methods are solutions to the
prediction problem where a predictor is best
if it
– Minimizes the mean square prediction error
– Is linear in the observed values Z(s1),…,Z(sn)
– Is unbiased in the sense that the mean of the
predicted value at s0 equals the mean of Z(s0)
Optimal Prediction
• To find the optimal predictor p(Z;s0) for Z(s0)
requires a measure for the loss incurred by using
p(Z;s0) for prediction at s0.
• Different loss functions result in different BEST
predictors
• The loss function of greatest importance in
statistics is squared-error loss{Z(s0)-p(Z;s0)}2
– Mean Square prediction error is the expected value\
• Conditional mean minimizes the MSPE
– E[Z(s0)|Z(s)]
Basic Kriging Methods
Ordinary and Universal Kriging
• Kriging predictors are the best linear
unbiased predictors under squared error loss
• Simple, ordinary, and universal kriging
differ in their assumption about the mean
structure u(s) of the spatial model
Classical Kriging techniques
• Methods for predicting Z(s0) based on
combining assumptions about the spatial
model with requirements about the predictor
p(Z;s0)
– p(Z;s0) is a linear combination of the observed
values Z(s1),…,Z(sn)
– p(Z;s0) is unbiased in the sense that
E[p(Z;s0)]=E[Z(s0)]
– p(Z;s0) minimizes the mean square prediction
error
Method
Assumption
about u(s)
Delta (s)
Simple Kriging
U(s) is known
Ordinary Kriging
U(s) = u, u
unknown
Universal
Kriging
U(s)= x’(s)B,
B unknown
Second order or
intrinsically
stationary
Second order or
intrinsically
stationary
Second order or
intrinsically
stationary
Simple Kriging
• Unbiased
• The optimal method of spatial prediction (under
squared error loss) in a Gaussian random field
• The minimized mean square prediction error of an
unbiased kriging predictor is often called the kriging
variance or the kriging error.
• For simple kriging, the kriging variance is:
sigma2SK(s0)= sigma2-c’epsilon-1c
• Useful in that it determines the benchmark for other
kriging methods
Universal and Ordinary Kriging
• Mean of the random field is not known and
can be expressed by a linear model
• Ordinary kriging predictor minimizes the
mean square prediction error subject to an
unbiasedness constraint
Notes on Kriging
• Kriging is not a perfect interpolator when
predicting the signal at observed locations
and the nugget effect contains a
measurement error component
• Kriging shouldn’t be done on a process with
linear drift and exponential semivariogram
– Use trend removal, use a method that does not
require Σ
Local Kriging and the Kriging
Neighborhood
• Kriging predictors must be calculated for
each location at which predictions are
desired.
– Matrix can become formidable
• Solution: Consider for prediction of Z(s0)
only observed data points within a
neighborhood of s0, called the Kriging
Neighborhood (aka local kriging)
Advantages
Disadvantages
• Computational
efficiency
• User needs to decide
on the size and shape
of the neighborhood
• Reasonable to assume
that the mean is at
least locally stationary
• Local kriging
predictors are no
longer best
Kriging Variance
• Variance-covariance matrix Σ is usually unknown
• An estimate Σˆ of Σ is substituted in expressions
• However, the uncertainty associated with the
estimation of the semivariances or covariances is
typically not accounted for in the determination of
the mean square prediction error
• Therefore, the kriging variance obtained is an
underestimate of the mean square prediction error
Cokriging and Spatial Regression
• If a spatial data set consists of more than
one attribute and stochastic relationships
exist among them, these relationships can
be exploited to improve predictive ability
• One attribute, Z1(s), is designated the
primary attribute and Z2(s),…,Zk(s) are the
secondary attributes
• Cokriging is a multivariate spatial prediction method
that relies on the spatial autocorrelation of the primary
and secondary attributes as well as the crosscovariances among the primary and the secondary
attributes
• Spatial regression is a multiple spatial prediction
method where the mean of the primary attribute is
modeled as a function of secondary attributes
• An advantage of spatial regression over cokriging is
that it only requires the spatial covariance function of
the primary attribute
• A disadvantage is that only colocated samples of Z1(s)
and all secondary attributes can be used in estimation
– If only one of the secondary attributes has not been observed
at a particular location the information collected on any
attribute at that location will be lost
Homework
• Read the applications section: Spatial predictionKriging of Lead Concentrates (9.8.3):
– Why are the estimates of total lead conservative?
– Why would an estimate of total lead based on the
sample average be positively biased?
– What does the top panel of Figure 9.41 tell us?
– Why do we perform universal kriging?
– What does figure 9.43 represent? Does it show the total
lead concentration?
– What is the estimate of the total amount of lead
obtained by universal block-kriging?
– Please email the answers to
• [email protected]
• Thanks!