# Diapositive 1

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```					      Introduction to kriging:
The Best Linear Unbiased Estimator
(BLUE)
for space/time mapping
Definition of Space Time Random Fields
• Spatiotemporal Continuum
   p=(s,t) denotes a location in the space/time domain E=SxT
• 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}
X(p)
Realization c(1)
Time t
Space s

X(p)                     Realization c(2)
Time t
Space s
   The collection of realizations represents the randomness (uncertainty and
variability) in X(p)
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.[cmap< xmap< 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)
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)
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 r=||s-s’|| and the temporal lag t = |t-t’|
cX(p, p’) = cX ( (s,t), (s’,t’) ) = cX( r=||s-s’|| , t=|t-t’| )

• A homogeneous/stationary S/TRFs has the following properties
   It’s variance is constant, i.e. sX2(p)= sX2
Proof: sX2(p)= E[(X(p)- mX(p))2] = cX(p, p) = cX( r=0, t=0 ) is not a function of p
   It’s covariance can be written as
cX(r , t)= E[X(s,t)X(s’,t’)] ||s-s’|| =r, |t-t’| =t- mX2 ,
 This is a useful equation to estimate the covariance
Experimental estimation of covariance

• When having site-specific data, and assuming that the S/TRF is
homogeneous/stationary, then we obtain experimental values for it’s
covariance using the following estimator

1
i 1 X head ,i X tail ,i  mX
N ( r ,t )
c X ( r ,t ) 
ˆ                                                          2

N ( r ,t )

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
and           t-dt ≤ ||thead-ttail|| ≤ t+dt
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.
Space/time covariance models

cX(r,t) is a 2D function with spatial component cX(r,t0) and temporal component cX(r=0,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 c1=0.0141(log mg/m3)2, ar1=448 Km, at1=1years is weather driven
   2nd structure c1=0.0141(log mg/m3)2, ar1=17 Km, at1=45years due to human activities
The simple kriging (SK) estimator
• Gather the data chard=[c1, 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
   E[xk(SK) ] = E[xk]        ═►          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
Example of kriging maps

Run Kriging Example

introToKrigingExample.m
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)

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