University of Wisconsin-Milwaukee
Geographic Information Science
Geography 625
Intermediate Geographic Information Science
Instructor: Changshan WuDepartment of GeographyThe University of Wisconsin-MilwaukeeFall 2006
Week 13: The Statistics of Fields
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Outline
1. Introduction
2. Review of Regression
3. Trend Surface Analysis: Regression on Spatial Coordinates
4. Statistical Interpolation: Kriging
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1. Introduction
Previous methods for interpolation use specific mathematical functions (deterministic interpolation)
Problems1) No environmental measurements can be made without error. It is ill-
advised to try to honor all the observed data without recognizing the inherent variability
2) Deterministic methods assume that we know nothing about how the variable being interpolated behaves spatially. However, the observed control point data may provide useful information.
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1. Introduction
1) Trend surface analysis: specified functions are fitted to the locational coordinates (x,y) of the control point data in an attempt to approximate trends in field height (first order effect)
2) Kriging: attempts to make optimum use of the underlying phenomenon as a spatially continuous field of non-independent random variables (second order effect)
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1. Introduction
Surface Trend Analysis (ArcGIS)
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1. Introduction
Kriging (ArcGIS)
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1. Introduction
Kriging (ArcGIS)
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2. Review of Regression
x
ySimple linear regression
Dependent variable: yIndependent variable: x
iiiii
iii
xbbyyy
xbby
10
10
ˆ
To obtain parameters b0 and b1, the best-fit equation is the one that minimizes the total square error Σεi
2 for observed values of xi and yi.
0
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2. Review of Regression
Solve the optimization problem
Minimize:
22110
2010
2
210
222
)(
iiiiii
iii
ii
xbxbbnbyxbyby
xbby
0
0
1
2
0
2
b
b
ii
ii
Lagrangian algorithm
Any statistical software can calculate these parameters (e.g. SPSS, S-Plus, R, SAS)
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3. Trend Surface Analysis
The trend of a surface is any large-scale systematic change that extends smoothly and predictably across the region of interest.
It is an exploratory method to give a rough idea of the spatial pattern in a set of observations.
),()( iiii yxfsfz
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3. Trend Surface Analysis
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3. Trend Surface Analysis
The coefficient of multiple correlation: R2
n
i i
n
i i
zzR
1
2
1
22
)(1
Sum of squared errors
Sum of squared differences from mean
Different function forms: higher order polynomial
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3. Trend Surface Analysis
Problems
1. It is not reasonable to assume that the phenomenon of interest varies with the spatial coordinates in such a simple way
2. The fitted surfaces do not pass exactly through all the control points
3. Other than simple visualization of the pattern they appear to display, the data are not used to help select this model.
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4. Kriging
Mathematical methods of interpolation (e.g. local spatial average, IDW) determine the distance weighting function and neighborhood definition based on expert knowledge, not from the data
Trend surface analysis uses the sampling data, but it only consider the first-order effect
Kriging estimates the choice of function, weights, and neighborhood from the sampling data, and interpolate the data with these choices.
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4. Kriging
Kriging is a statistical interpolation method that is optimal in the sense that it makes best use of what can be inferred about the spatial structure in the surface to be interpolated from an analysis of the control point data
Methods used in the South African mining industry by David KrigeTheory of regionalized variables (Georges Matheron, 1960)Statistic for Spatial Data (Noel A. C. Cressie 1993)
Three steps1) Produce a description of the spatial variation in the sample control
point data2) Summarizing the spatial variation by a regular mathematical function3) Using this model to determine the interpolation weights
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4. Kriging- Describing the spatial variation: the semi-variogram
Variogram cloud: a plot of a measure of height differences against the distance dij between the control points for all possible pairs of points.
2
10
8
Pij(d) = (zi-zj)2
P
d20
4
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4. Kriging
Example of variogram cloud
- Describing the spatial variation: the semi-variogram
There is a trend such that height differences increase as the separation distance increases
Indicating the farther apart two control points are, the greater is the likely difference in their value.
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4. Kriging- Describing the spatial variation: the semi-variogram
Spatial dependence can be described more concisely by the experimental semivariogram function as follows
dd
ji
ij
zzdn
d 2)()(
1)(ˆ2
n(d) is the number of pair of points at separation d is the estimated semi-variogram ̂
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4. Kriging- Describing the spatial variation: the semi-variogram
This is the theoretical equation for variogram estimation and it is not straightforward in applications
E.g. for a given distance d, it is more likely that there will be no pair of observations at precisely that separation.
dd
ji
ij
zzdn
d 2)()(
1)(ˆ2
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4. Kriging- Describing the spatial variation: the semi-variogram
2/
2/
2)()(
1)(ˆ2
d
ddji
ij
zzdn
d
In reality, variogram is estimated for different bands (or lags) rather than continuously at all distances.
Δ is the lag widthn(d) is the number of point pairs within (d- Δ/2, d+ Δ/2)
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4. Kriging- Describing the spatial variation: the semi-variogram
a (10)
b (12) c (8)
d (10) e (6)
03135.2
30323
13021
32201
5.23110
Distancematrix
Δ = 0.5d = 0.5
What is the value of γ(0.5)?What is the value of γ(1.5)?
a
b
c
d
e
a b c d e
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4. Kriging- Describing the spatial variation: the semi-variogram
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4. Kriging- Summarize the spatial variation by a regular
mathematical function
Having approximated the semivariograms by mean values at a series of lags, the next step is to summarize the experimental variogram using a mathematical function.
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4. Kriging- Summarize the spatial variation by a regular
mathematical function
Nugget (c0): variance at zero distance
Range (a): the distance at which the semivariogram levels off and beyond which the semivariance is constant
Sill (c0+c1): the constant semivariance value beyond the range
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4. Kriging- Summarize the spatial variation by a regular
mathematical function
Mathematical Functions
Nugget modelLinear modelSpherical modelExponential modelPower model Gaussian modelOthers
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4. Kriging- Summarize the spatial variation by a regular
mathematical function
Nugget model: A constant variance model
d
γ
Nugget(c0)
γ = c0
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4. Kriging- Summarize the spatial variation by a regular
mathematical function
Linear model: Variances change linearly with the change of distance
d
γ
a
γ = d When d < a
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4. Kriging- Summarize the spatial variation by a regular
mathematical function
10
3
10
)(
5.02
3)(
ccd
a
d
a
dccd
If d <= a then
If d > a then
Spherical model starts from a nonzero variance (c0) and rise as an elliptical arc to a maximum value (c0+c1) at distance a.
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4. Kriging- Summarize the spatial variation by a regular
mathematical function
Variogram model fitting methods
1) Interactive model fitting 2) Weighted least squares (R and Gstat)3) Modified weighted least squares (ArcMap Geostatistics)4) Others
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4. Kriging- Summarize the spatial variation by a regular
mathematical function
Typical spatial profiles and their associated semivariograms
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4. Kriging- Summarize the spatial variation by a regular
mathematical function
Problems with variogram estimation
1. The reliability of the calculated semivariance varies with the number of point pairs used in their estimation
2. Spatial variation may be anisotropic (varies with directions), favoring change in a particular direction
3. It assumes there is no systematic spatial change in the mean surface height (first order effect)
4. The experimental semivariogram can fluctuate greatly from point to point
5. Many functions are non-linear
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4. Kriging- Use the model to determine interpolation weights by
Kriging
Assumptions
1) The surface has a constant mean, with no underlying trend
2) The surface is isotropic, having the same variation in each direction
3) The semivariogram is a simple mathematical model with some clearly defined properties
4) The same variogram applied over the entire area
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4. Kriging- Use the model to determine interpolation weights by
Kriging
nns zwzwzwz ...ˆ 2211
n
i
n
i
n
jijjiisi
ss
dwwdw
zzE
1 1 1
2
)()(2
}]ˆ{[
Minimize
Subject to: 1...21 nwww
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4. Kriging- Use the model to determine interpolation weights by
Kriging
Solve the above equation
1....
)()(...)()(
......
)()(...)()(
21
2211
11122111
n
nsnnnnn
snn
www
ddwdwdw
ddwdwdw
n+1 variables, n+1 linear equations
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4. Kriging- Use the model to determine interpolation weights by
Kriging
1
22
s
a (10)b (8)
c (8)
3
4
1)( d
1
)()()()(
)()()()(
)()()()(
cba
csccccbbcaa
bsbccbbbbaa
asaccabbaaa
www
ddwdwdw
ddwdwdw
ddwdwdw
What is the value of wa, wb, wc, and λ?What is the value of s?
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4. Kriging- Use the model to determine interpolation weights by
Kriging
1
22
s
a (10)b (8)
c (8)
2)(
)(
d
dd
if d <=2
if d > 2
3
4
What is the value of wa, wb, wc, and λ?What is the value of s?What is the value of s with IDW method?
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4. Kriging- Use the model to determine interpolation weights by
Kriging
Software
ArcMap GeostatisticsR PackageIDRISI (G-Stat)GSLIB
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4. Kriging- Use the model to determine interpolation weights by
Kriging
Trend analysisSemivariogram
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4. Kriging- Use the model to determine interpolation weights by
Kriging
Kriging
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4. Kriging- Use the model to determine interpolation weights by
Kriging
Conclusion1) Kriging is computationally intensive2) All the results depend on the model we fit to the
estimated semi-variogram from the sample data3) If the corrected model is used, the methods used in
kriging have an advantage over other interpolation procedures
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4. Kriging
1) Simple kriging (the summation of the weights does not equal to one)
2) Ordinary kriging (taught in this class)3) Universal kriging (combine trend analysis with ordinary
kriging)4) Co-kriging (more than one variable)
Variations