Some remarks about Lab 1

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Some remarks about Lab 1
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• image(parana.krige,valsqrt(parana.krigekrige.var
  ))
• contour(parana.krige,locloci,addT)

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• likfit(parana,nugget470,cov.model"gaussian",ini.
  cov.parsc(5000,250))
• kappa not used for the gaussian correlation
  function
• --------------------------------------------------
  -------
• likfit likelihood maximisation using the
  function optim.
• likfit estimated model parameters
• beta tausq sigmasq phi
• " 260.6" " 521.1" "6868.9" " 336.2"
• Practical Range with cor0.05 for asymptotic
  range 16808.82
• likfit maximised log-likelihood -669.3

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Nonstationary variance

• Let ?(x) be a Gaussian process with constant mean
  ?, constant variance ??????, and correlation
  .
• f is the same deformation as for the covariance
  modelling.
• Define the variance process
• Its distribution at gauged sites is

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Moments of the variance process

• Mean
• Variance
• Covariance
• Correlation

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Priors

• N(?,?)
• The full conditional distributions are then of
  the same form (Gibbs sampler).
• To set the hyperparameters we use an empirical
  approach Let Sii be the sample variance at site
  i.

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Method of moments

• Setting the sample moments (over the sites) equal
  to the theoretical moments we get
• and let that be the prior mean. The prior
  variance is set appropriately diffuse.

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French precipitation

• Constant variance Nonconstant
• variance

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Prediction vs estimation

• Leave out 8 stations, use remaining 31 for
  estimation
• Compute predictive distribution for the 8
  stations
• Plot observed variances (incl. nugget) vs.
  estimated variances
• and against predictive distribution

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Estimated variance field
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Global processes

• Problems such as global warming require modeling
  of processes that take place on the globe (an
  oriented sphere).
• Optimal prediction of quantities such as global
  mean temperature need models for global
  covariances.
• Note spherical covariances can take values in
  -1,1not just imbedded in R3.
• Also, stationarity and isotropy are identical
  concepts on the sphere.

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Isotropic covariances on the sphere
Isotropic covariances on a sphere are of the
form where p and q are directions, ?pq the angle
between them, and Pi the Legendre polynomials
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Some examples

• Let ai?i, o?lt1. Then
• Let ai(2i1)ri. Then
• Given C(p,q)

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Global temperature

• Global Historical Climatology Network 7280
  stations with at least 10 years of data. Subset
  with 839 stations with data 1950-1991 selected.

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Isotropic correlations
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Spherical deformation

• Need isotropic covariance model on transformation
  of sphere/globe
• Covariance structure on convex manifolds
• Simple option deform globe into another globe
• Alternative MRF approach

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A class of global transformations

• Deformation of sphere g(g1,g2)
• latitude def
• longitude def
• Avoid crossing of latitudes or longitudes
• Poles are fixed points
• Equator can be fixed as well

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Simple latitude deformation
knot
Iterated simple deformations
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Two-dimensional deformation

• Let b and ? depend on longitude ?
• Alternating deform longitude and latitude.

location
scale
amplitude
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Three iterations
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Resulting isocovariance curves
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Comparison

• Isotropic Anisotropic

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Assessing uncertainty
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Another current climate problem

• General circulation models require accurate
  historical ocean surface temperature records
• Data from buoys, ships, satellites