Diapositive 1

1
Introduction to kriging The Best Linear
Unbiased Estimator (BLUE) for space/time mapping
2
Definition of Space Time Random Fields

• Spatiotemporal Continuum
• p(s,t) denotes a location in the space/time
  domain ESxT
• Spatiotemporal Field
• A field is the distribution c across space/time
  of some parameter X
• Space/Time Random Field (S/TRF)
• A S/TRF is a collection of possible realizations
  c of the field, X(p)p, c
• The collection of realizations represents the
  randomness (uncertainty and variability) in X(p)

3
Multivariate PDF for the mapping points

• Defining a S/TRF at a set of mapping points
• We restrict Space/Time to a set of n mapping
  points, pmap(p1,, pn)
• Each field realization reduces to a set of n
  values, cmap(c1,, cn)
• The S/TRF reduces to set of n random variables,
  xmap (x1,, xn)
• The multivariate PDF
• The multivariate PDF fX characterizes the joint
  event xmap cmap as
• Prob.cmaplt xmaplt cmap dcmap fX(cmap)
  dcmap
• hence the multivariate PDF provides a complete
  stochastic description of trends and dependencies
  of the S/TRF X(p) at its mapping points
• Marginal PDFs
• The marginal PDF for a subset xa of xmap (xa,
  xb) is
• fX(ca) ? dcb fX(ca , cb)
• hence we can define any marginal PDF from
  fX(cmap)

4
Statistical moments

• Stochastic Expectation
• The stochastic expectation of some function
  g(X(p), X(p), ) of the S/TRF is
• E g(X(p), X(p), ) ? dc1 dc2 ... g(c1, c2
  , ...) fX(c1, c2 , ... p p , ...)
• Mean trend and covariance
• The mean trend
• mX(p) E X(p)
• and covariance
• cX(p, p) E (X(p)-m(p)) (X(p)-m(p))
• are statistical moments of order 1 and 2,
  respectively, that characterizes the consistent
  tendencies and dependencies, respectively, of X(p)

5
Homogeneous/Stationary S/TRF

• A homogeneous/stationary S/TRF is defined by
• A mean trend that is constant over space
  (homogeneity) and time (stationarity)
• mX(p) mX
• A covariance between point p (s,t) and p
  (s,t) that is only a function of spatial lag
  rs-s and the temporal lag t t-t
• cX(p, p) cX ( (s,t), (s,t) ) cX(
  rs-s , tt-t )
• A homogeneous/stationary S/TRFs has the following
  properties
• Its variance is constant, i.e. sX2(p) sX2
• Proof sX2(p) E(X(p)- mX(p))2 cX(p, p)
  cX( r0, t0 ) is not a function of p
• Its covariance can be written as
• cX(r , t) EX(s,t)X(s,t) s-s r,
  t-t t- mX2 ,
• ? This is a useful equation to estimate the
  covariance

6
Experimental estimation of covariance

• When having site-specific data, and assuming that
  the S/TRF is homogeneous/stationary, then we
  obtain experimental values for its covariance
  using the following estimator
• where N(r,t) is the number of pairs of points
  with values (Xhead, Xtail) separated by a
  distance of r and a time of t.
• In practice we use a tolerance dr and dt, i.e.
  such that
• r-dr shead-stail rdr
• and t-dt thead-ttail tdt

7
Spatial covariance models

• Gaussian model cX(r) co exp-(3r2/ar2)
• co sill variance
• ar spatial range
• Very smooth processes
• Exponential model cX(r) co exp-(3r/ar)
• more variability
• Nugget effect model cX(r) co d(r)
• purely random
• Nested models cX(r) c1(r) c2(r)
• where c1(r), c2(r), etc. are permissible
  covariance models
• Example Arsenic cX(r) 0.7sX2 exp-(3r/7Km)
  0.3sX2 exp-(3r/40Km)
• where the first structure represents variability
  over short distances (7Km), e.g. geology, the
  second structure represents variability over
  longer distances (40Km) e.g. aquifers.

8
Space/time covariance models

• cX(r,t) is a 2D function with spatial component
  cX(r,t0) and temporal component cX(r0,t)
• Space/time separable covariance model
• cX(r,t) cXr(r) cXt(t) , where cXr(r) and
  cXt(t) are permissible models
• Nested space/time separable models
• cX(r,t) cr1(r) ct1(t) cr2(r)ct2 (t)
• Example Yearly Particulate Matter concentration
  (ppm) across the US
• cX(r,t) c1 exp(-3r/ar1-3t/at1) c2
  exp(-3r/ar2-3t/at2)
• 1st structure c10.0141(log mg/m3)2, ar1448 Km,
  at11years is weather driven
• 2nd structure c10.0141(log mg/m3)2, ar117 Km,
  at145years due to human activities

9
The simple kriging (SK) estimator

• Gather the data chardc1, c2, c3 , T and
  obtain the experimental covariance
• Fit a covariance model cX(r) to the experimental
  covariance
• Simple kriging (SK) is a linear estimator
• xk(SK) l0 l T xhard
• SK is unbiased
• Exk(SK) Exk -? xk(SK) mk l T (xhard
  - mhard)
• SK minimizes the estimation variance sSK2 E(xk
  - xk(SK) )2
• ?sSK2 / ?lT 0 -? lT Ck,hard Chard,hard-1
• Hence the SK estimator is given by
• xk(SK) mk Ck,hard Chard,hard-1 (xhard. -
  mhard) T
• And its variance is
• sSK2 sk2 - Ck,hard Chard,hard-1 Chard,k

10
Example of kriging maps
Run Kriging Example introToKrigingExample.m
11
Example of kriging maps

• Observations
• Only hard data are considered
• Exactitude property at the data points
• Kriging estimates tend to the (prior) expected
  value away from the data points
• Hence, kriging maps are characterized by
  islands around data points
• Kriging variance is only a function to the
  distance from the data points
• Limitations of kriging
• Kriging does not provide a rigorous framework to
  integrate hard and soft data
• Kriging is a linear combination of data (i.e. it
  is the best only among linear estimators, but
  it might be a poor estimator compared to
  non-linear estimators)
• The estimation variance does not account for the
  uncertainty in the data itself
• Kriging assumes that the data is Gaussian,
  whereas in reality uncertainty may be
  non-Gaussian
• Traditionally kriging has been implemented for
  spatial estimation, and space/time is merely
  viewed as adding another spatial dimension (this
  is wrong because it is lacking any explicit
  space/time metric)