The statistics of fields - pantherFILE

Geographic Information Science

Geography 625

Intermediate


Week 13: The Statistics of Fields

Instructor : Changshan Wu

Department of Geography

The University of Wisconsin-Milwaukee

Fall 2006

University of Wisconsin-Milwaukee


Outline

1. Introduction

2. Review of Regression

3. Trend Surface Analysis: Regression on Spatial

Coordinates

4. Statistical Interpolation: Kriging



1. Introduction

Previous methods for interpolation use specific mathematical functions (deterministic interpolation)

Problems

1) No environmental measurements can be made without error. It is illadvised 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.



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 nonindependent random variables (second order effect)



1. Introduction

Surface Trend Analysis (ArcGIS)



1. Introduction

Kriging (ArcGIS)



1. Introduction

Kriging (ArcGIS)



2. Review of Regression

Simple linear regression

Dependent variable: y

Independent variable: x y i

 i

 b

0

 y i



 b

1 x i

ˆ i



 y i

 i

 b

0

 b

1 x i

To obtain parameters b

0 and b the best-fit equation is the one

1

, that minimizes the total square error Σε i

2 x i and y i

.

for observed values of y

0

University of Wisconsin-Milwaukee x


2. Review of Regression

Solve the optimization problem

Minimize:



 i



 i

 y i

2

 i

( y i



2 b

0



 b

0 y i



 b

1 x i

2 b

1

)

2

 x i y i

 nb

0

2 

2 b

0 b

1

 x i

 b

1

2

 x i

2

  i



 b

0

 i



 b

1

 i i

2

2



0



0

Lagrangian algorithm

Any statistical software can calculate these parameters (e.g.

SPSS, S-Plus, R, SAS)



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. z i

 f ( s i

)

 f ( x i

, y i

)







The coefficient of multiple correlation: R 2

R

2 

1



 n i



1

 n i



1

( z i



 i

2 z )

2

Sum of squared errors

Sum of squared differences from mean

Different function forms: higher order polynomial




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.



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.



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 Krige



Theory of regionalized variables (Georges Matheron, 1960)



Statistic for Spatial Data (Noel A. C. Cressie 1993)

Three steps

1) Produce a description of the spatial variation in the sample control point data

2) Summarizing the spatial variation by a regular mathematical function

3) Using this model to determine the interpolation weights



4. Kriging

- Describing the spatial variation: the semi-variogram

Variogram cloud : a plot of a measure of height differences against the distance d ij points.

between the control points for all possible pairs of

P ij

(d) = ( z i

z j

) 2

P

4

8

2

10

0 2 d



4. Kriging


Example of variogram cloud

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.



4. Kriging


Spatial dependence can be described more concisely by the experimental semivariogram function as follows

2

 ˆ

( d )



1 n ( d ) d ij



 d

( z i

 z j

)

2 n(d) is the number of pair of points at separation d

 ˆ is the estimated semi-variogram



4. Kriging


2

 ˆ

( d )



1 n ( d ) d ij



 d

( z i

 z j

)

2

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 .



4. Kriging


In reality, variogram is estimated for different bands (or lags) rather than continuously at all distances.

2

 ˆ

( d )



1 n ( d ) d ij d



 d







/



2

(

/ z i

2

 z j

)

2

Δ is the lag width n ( d ) is the number of point pairs within (d- Δ/2, d+ Δ/2)



4. Kriging

- Describing the spatial variation: the semi-variogram b (12) a (10) c (8)

Distance matrix a b c d e















 2

1

1

.

a b c d e

0 1 1 3 2 .

5



3

5

0

2

2

3

2

0

3

1

2

3

0

3

3

1

3

0













 d (10) e (6)

Δ = 0.5

d = 0.5

What is the value of γ(0.5)?

What is the value of γ(1.5)?



4. Kriging




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.



4. Kriging


Nugget (c

0

): variance at zero distance

Range (a): the distance at which the semivariogram levels off and beyond which the semivariance is constant

Sill (c

0

+c

1

): the constant semivariance value beyond the range



4. Kriging


Mathematical Functions



Nugget model



Linear model



Spherical model



Exponential model

 Power model



Gaussian model



Others



4. Kriging


Nugget model : A constant variance model

γ

Nugget

( c

0

)

γ = c

0 d



4. Kriging


Linear model : Variances change linearly with the change of distance

γ a

γ = d When d < a d



4. Kriging


Spherical model starts from a nonzero variance ( c

0

) and rise as an elliptical arc to a maximum value ( c

0

+c

1

) at distance a.

If d <= a then



( d )

 c

0

 c

1







3 d

2 a



0 .

5

 d a



If d > a then

( d )

 c

0

 c

1

3









4. Kriging


Variogram model fitting methods

1) Interactive model fitting

2) Weighted least squares (R and Gstat)

3) Modified weighted least squares (ArcMap Geostatistics)

4) Others



4. Kriging


Typical spatial profiles and their associated semivariograms



4. Kriging


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



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



4. Kriging


Kriging z

ˆ s

 w

1 z

1

 w

2 z

2



...

 w n z n

Minimize E {[ z

ˆ s

 z s

]

2

}



2 i n 



1 w i



( d is

)

 i n n 



1 j



1 w i w j



( d ij

)

Subject to: w

1

 w

2



...

 w n



1



4. Kriging


Kriging

Solve the above equation w

1



( d

11

)

 w

2



( d

12

)



...

 w n



( d

1 n

)

   

( d

1 s

)

......

w

1

 w

1



( d n 1

) w

2

 w

2



....





( d n 2

) w n



1



...

 w n



( d nn

)

   

( d ns

) n+1 variables, n+1 linear equations



4. Kriging


Kriging s a (10)

3

1

2

2

4 c (8) b (8)



( d )



1 w a

 w a

 w a

 w a

( d aa

)



( d ba

)





( d ca

) w b



 w c w b

 w b

 w b



( d ab

)

( d bb

)

( d cb

)





 w c

 w c

 w c



( d ac

)

( d bc

)

( d cc

)















1





 





( d as

)

( d bs

)

( d cs

)

What is the value of w a

, w b

, w c

, and

λ?

What is the value of s?



4. Kriging


Kriging



( d )



( d )

 d



2 if d <=2 if d > 2 s a (10)

3

1

2

2

4 c (8) b (8)

What is the value of w a

, w b

, w c

, and

λ?

What is the value of s?

What is the value of s with IDW method?



4. Kriging


Kriging

Software

ArcMap Geostatistics

R Package

IDRISI (G-Stat)

GSLIB



4. Kriging


Kriging

Trend analysis

Semivariogram



4. Kriging


Kriging

Kriging



4. Kriging


Kriging

Conclusion

1) Kriging is computationally intensive

2) All the results depend on the model we fit to the estimated semi-variogram from the sample data

3) If the corrected model is used, the methods used in kriging have an advantage over other interpolation procedures



4. Kriging

Variations

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)


The statistics of fields - pantherFILE

Intermediate

Geographic Information Science

Variations

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The statistics of fields - pantherFILE

Intermediate

Geographic Information Science

Variations

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