Geographic Information Science Geography 625 Intermediate Geographic Information Science Week 13: The Statistics of Fields Instructor: Changshan Wu Department of Geography The University of Wisconsin-Milwaukee Fall 2006

Outline Introduction Review of Regression Trend Surface Analysis: Regression on Spatial Coordinates Statistical Interpolation: Kriging

1. Introduction Previous methods for interpolation use specific mathematical functions (deterministic interpolation) Problems 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 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 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) Kriging: attempts to make optimum use of the underlying phenomenon as a spatially continuous field of non-independent 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 y Dependent variable: y Independent variable: x y To obtain parameters b0 and b1, the best-fit equation is the one that minimizes the total square error Σεi2 for observed values of xi and yi. x

2. Review of Regression Solve the optimization problem Minimize: 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.

3. Trend Surface Analysis

3. Trend Surface Analysis The coefficient of multiple correlation: R2 Sum of squared errors Sum of squared differences from mean Different function forms: higher order polynomial

3. Trend Surface Analysis Problems It is not reasonable to assume that the phenomenon of interest varies with the spatial coordinates in such a simple way The fitted surfaces do not pass exactly through all the control points 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 Produce a description of the spatial variation in the sample control point data Summarizing the spatial variation by a regular mathematical function 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 dij between the control points for all possible pairs of points. Pij(d) = (zi-zj)2 P 4 8 2 10 2 d

4. Kriging - Describing the spatial variation: the semi-variogram 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 - Describing the spatial variation: the semi-variogram Spatial dependence can be described more concisely by the experimental semivariogram function as follows n(d) is the number of pair of points at separation d is the estimated semi-variogram

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.

4. Kriging - Describing the spatial variation: the semi-variogram In reality, variogram is estimated for different bands (or lags) rather than continuously at all distances. 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 a b c d e a b c d e a (10) Distance matrix b (12) c (8) e (6) d (10) Δ = 0.5 d = 0.5 What is the value of γ(0.5)? What is the value of γ(1.5)?

4. Kriging - Describing the spatial variation: the semi-variogram

- Summarize the spatial variation by a regular mathematical function 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.

- Summarize the spatial variation by a regular mathematical function 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

- Summarize the spatial variation by a regular mathematical function 4. Kriging - Summarize the spatial variation by a regular mathematical function Mathematical Functions Nugget model Linear model Spherical model Exponential model Power model Gaussian model Others

- Summarize the spatial variation by a regular mathematical function 4. Kriging - Summarize the spatial variation by a regular mathematical function Nugget model: A constant variance model γ γ = c0 Nugget (c0) d

- Summarize the spatial variation by a regular mathematical function 4. Kriging - Summarize the spatial variation by a regular mathematical function Linear model: Variances change linearly with the change of distance γ γ = d When d < a a d

- Summarize the spatial variation by a regular mathematical function 4. Kriging - Summarize the spatial variation by a regular mathematical function Spherical model starts from a nonzero variance (c0) and rise as an elliptical arc to a maximum value (c0+c1) at distance a. If d <= a then If d > a then

- Summarize the spatial variation by a regular mathematical function 4. Kriging - Summarize the spatial variation by a regular mathematical function Variogram model fitting methods Interactive model fitting Weighted least squares (R and Gstat) Modified weighted least squares (ArcMap Geostatistics) Others

- Summarize the spatial variation by a regular mathematical function 4. Kriging - Summarize the spatial variation by a regular mathematical function Typical spatial profiles and their associated semivariograms

- Summarize the spatial variation by a regular mathematical function 4. Kriging - Summarize the spatial variation by a regular mathematical function Problems with variogram estimation The reliability of the calculated semivariance varies with the number of point pairs used in their estimation Spatial variation may be anisotropic (varies with directions), favoring change in a particular direction It assumes there is no systematic spatial change in the mean surface height (first order effect) The experimental semivariogram can fluctuate greatly from point to point Many functions are non-linear

- Use the model to determine interpolation weights by Kriging Assumptions The surface has a constant mean, with no underlying trend The surface is isotropic, having the same variation in each direction The semivariogram is a simple mathematical model with some clearly defined properties The same variogram applied over the entire area

- Use the model to determine interpolation weights by Kriging Minimize Subject to:

- Use the model to determine interpolation weights by Kriging Solve the above equation n+1 variables, n+1 linear equations

- Use the model to determine interpolation weights by Kriging 3 b (8) 1 s 2 2 4 c (8) What is the value of wa, wb, wc, and λ? What is the value of s?

- Use the model to determine interpolation weights by Kriging if d <=2 if d > 2 a (10) 3 b (8) 1 s 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? 2 2 4 c (8)

- Use the model to determine interpolation weights by Kriging Software ArcMap Geostatistics R Package IDRISI (G-Stat) GSLIB

- Use the model to determine interpolation weights by Kriging Trend analysis Semivariogram

- Use the model to determine interpolation weights by Kriging

- Use the model to determine interpolation weights by Kriging Conclusion Kriging is computationally intensive All the results depend on the model we fit to the estimated semi-variogram from the sample data If the corrected model is used, the methods used in kriging have an advantage over other interpolation procedures

4. Kriging Variations Simple kriging (the summation of the weights does not equal to one) Ordinary kriging (taught in this class) Universal kriging (combine trend analysis with ordinary kriging) Co-kriging (more than one variable)