9.3 and 9.4

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9.3 and 9.4

• The Spatial Model
• And
• Spatial Prediction and the Kriging Paradigm

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9.3

• The Spatial Model

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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

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For Geostatistical data

• Z(s) µ(s) W(s) ?(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
• ?(s) is a spatial process with variogram ??
• e(s) represents measurement error

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Two Basic Model Types

• Signal model
• S(s)µ(s) W(s) ?(s)
• Mean model
• d(s) W(s) ?(s) e(s)

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Signal Model S(s)µ(s) W(s) ?(s)

• Denotes the signal of the process
• Then
• Z(s) S(s) e(s)

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Mean Model d(s) W(s) ?(s) e(s)

• Denotes the error of process
• Then
• Z(s) µ(s) d(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
• d(s) is assumed to have spatial autocorrelation
  structure

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• 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 d(s) is
  somewhat ancillary

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• 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 d(s)
• No solid answer
• One modelers fixed effect is someone elses
  random effect

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Cliff and Ords 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 dont react to outside
  influences, but rather with each other
• Ex Neighboring plants compete with each other
  for resources

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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

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• 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

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9.4

• Spatial Prediction and the Kriging Paradigm

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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 EZ(s).

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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)

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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

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• 3. Using the residuals obtained in step 2 (or the
  original data if the mean is stationary), fit a
  semivariogram model ?(h?) 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

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• 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

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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)

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Optimal Prediction

• To find the optimal predictor p(Zs0) for Z(s0)
  requires a measure for the loss incurred by using
  p(Zs0) for prediction at s0.
• Different loss functions result in different BEST
  predictors
• The loss function of greatest importance in
  statistics is squared-error lossZ(s0)-p(Zs0)2
• Mean Square prediction error is the expected
  value\
• Conditional mean minimizes the MSPE
• EZ(s0)Z(s)

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Basic Kriging MethodsOrdinary 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

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Classical Kriging techniques

• Methods for predicting Z(s0) based on combining
  assumptions about the spatial model with
  requirements about the predictor p(Zs0)
• p(Zs0) is a linear combination of the observed
  values Z(s1),,Z(sn)
• p(Zs0) is unbiased in the sense that
  Ep(Zs0)EZ(s0)
• p(Zs0) minimizes the mean square prediction error

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Method Assumption about u(s) Delta (s)
Simple Kriging U(s) is known Second order or intrinsically stationary
Ordinary Kriging U(s) u, u unknown Second order or intrinsically stationary
Universal Kriging U(s) x(s)B, B unknown Second order or intrinsically stationary
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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-cepsilon-1c
• Useful in that it determines the benchmark for
  other kriging methods

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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

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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 shouldnt be done on a process with
  linear drift and exponential semivariogram
• Use trend removal, use a method that does not
  require S

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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)

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• Advantages
• Computational efficiency
• Reasonable to assume that the mean is at least
  locally stationary

• Disadvantages
• User needs to decide on the size and shape of the
  neighborhood
• Local kriging predictors are no longer best

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Kriging Variance

• Variance-covariance matrix S is usually unknown
• An estimate Sˆ of S 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

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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

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• Cokriging is a multivariate spatial prediction
  method that relies on the spatial autocorrelation
  of the primary and secondary attributes as well
  as the cross-covariances 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

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Homework

• Read the applications section Spatial
  prediction-Kriging 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
• spechauer_at_luc.edu
• Thanks!