statmos journal club at ncsu an-ting jhuang 2016.9guinness/readinggroup/classicalgeo... ·...
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STATMOS Journal Club at NCSU An-Ting Jhuang
2016.9.8
Background
Geostaitistical model
Provisional estimation of the mean function
Estimation of the semivariogram
Modeling the semivariogram
Re-estimation of the mean function
Kriging
Objective: use the sampled realization to
make inferences about the process
Data type: point-referenced data
Spatial region: usually
Spatial process:
2RD
}:)({)( DYY ss
The model can be expressed as
Observation = Mean + Error
where Y(s) is the response process,
is the mean process, and
is the error process.
)()()( sss eY
)(s
)(se
Two types of stationarity: 1. Second-order stationary 2. Intrinsically stationary : the semivariogram 2nd-order stationary intrinsically stationary
)()](),([Cov tsts Cee D ts,
)()]()([Var2
1tsts ee D ts,
)( ts
In regression, we consider , then
.
Now, we take spatial variation into account. Let
then
eβXYT
YXX)(XβT1T ˆ
,)();( βsXβsT
).()()]()([)(ˆ sYsXsXsXsβT1T
Define the lag , u=1, …, k.
The empirical semivariogram is
: no. of times that lag occurs among the data locations
2)](ˆ)(ˆ[
)(2
1)(ˆ
uii
ji
u
u eeN
hss
ssh
h
Tuuu hh ),( 21h
)( uN h uh
When data locations are irregularly spaced, we can partition the lag space H into lag classes H1, …, Hk.
Common choices:
polar partition and rectangular partition
2)](ˆ)(ˆ[
)(2
1)(ˆ
uii H
ji
u
u eeHN
ss
ssh
Several issues:
1. How to choose number of bins
2. Sensitive to outliers
Cressie and Hawkins (1980) provided a
more robust estimator.
,30)( uHN .||||max5.0|||| uu
u hh
To smooth the empirical semivariorgram because
a. often bumpy
b. often fail to be conditionally nonpositive
definite
c. need estimates of the semivariogram at
lags not included in the lag bins
Three sufficient and necessary conditions for a semivariogram model:
1. Vanishing at 0.
2. Evenness
3. Conditional negative definiteness
Five commonly-used types:
Spherical, exponential, Gaussian, Matern, and
power semivariograms.
From http://www.intechopen.com/books/advances-in-agrophysical-research/model-averaging-for-semivariogram-model-parameters
Next, estimate the parameters in a chosen semivariogram.
Weighted least squares (WLS):
Likelihood-based method will be discussed next time.
2
2)];()(ˆ[
)];([
)(minargˆ θhh
θh
hθ uu
Uu u
uN
Use estimated generalized least squares (EGLS)
We have , then
))(ˆ(ˆjiC ssΣ
YTT 111EGLS
ˆ)ˆ(ˆ ΣXXΣXβ
To predict the value of at desired locations
First developed and applied by D. G. Krige
Two properties of predictor
1. Linearity
2. Unbiasedness
)(Y
)(ˆ0sY
Universal kriging predictor
In practice, two possible modifications:
1. reduce the computing load
2. use the empirical and
YΓXΓXXγs111
0 ])([)(ˆ TTY
γ Γ
