p
Yin
X in
p
2
2
10 15
RASTER
-Data Postings => symbol maps -Contour Maps •Moving Windows => “heteroscedasticity” •Spatial Continuity h-scatterplots
3
12
Definitions
Variograms: What are they?
Covariance C (h) cov( Z (s), Z (s h)) Autocorrelation (h) C (h) / C (0) Variogram 2 (h) var( Z (s) Z (s h))
•Correlogram = p(h) = the relationship of the correlation coefficient of an h-scatterplot and h (the spatial lag) •Covariance = C(h) = the relationship of the coefficient of variation of an h-scatterplot and h •Semivariogram = variogram = (h) = moment of inertia moment of inertia =
When first proposed (O’Neill 1988) proved incorrect, Li & Reynolds (1993) alternative Based upon the product of two (2) probabilities
1 n 2n i 1
x y
i i
2
OR: half the average sum difference between the x and y pair of the h-scatterplot
OR: for a h(0,0) all points fall on a line x=y
Spatial Description
Spatial lag = h = (0,1) = same x, y+1
h=(0,0)
h=(0,3)
h=(0,5) * (0,0)
tj
hij=tj-ti * Xi,Yi
ti
correlation coefficient
(i.e the correlogram, relationship of p with h
Introduction to Geostatistics
D
Z(s)
• D is the spatial domain or area of
interest
• s contains the spatial coordinates • Z is a value located at the spatial
coordinates
{Z(s): s D} Geostatistics: Z random; D fixed, infinite, continuous Lattice Models: Z random; D fixed, finite, (ir)regular grid Point Patterns: Z 1; D random, finite
5
1
11
4 Spatial Lag = h = distance
Lag bins
1 2 3 4
Values at locations that are near to each other are more similar than values at locations that are farther apart. = Autocorrelation
•Spatial Description
Univariate
•One Variable •Frequency (table) •Histogram (graph) •Do the same thing (i.e count of observations in intervals or classes •Cumulative Frequency (total “below” cutoffs)
GeoStatistics
-A way of describing the spatial continuity as an essential feature of natural phenomena. - The science of uncertainty which attempts to model order in disorder. - Recognized to have emerged in the early 1980’s as a hybrid of mathematics, statistics, and mining engineering. - Now extended to spatial pattern description •Univariate •Bivariate
IQR
Q Q
3
1
CS
3 1 n xi n i 1

2
CV
Bivariate
Scatterplots
p
Yin
Correlation
1 n
p
p
y x
n i 1 i x i y
X in
Linear Regression
Area Metrics Patch Density, Size and Variability Edge Metrics Shape Metrics Core Area Metrics Nearest-Neighbor Metrics Diversity Metrics Contagion and Interspersion Metrics
Configuration = The physical distribution in space and
spatial character of elements.
Isolation, placement, adjacency ** some metrics do both **
Types of Metics
- Data Postings = symbol maps (if only 2 classes = indicator map - Contour Maps - Moving Windows => “heteroscedasticity” (values in some region are more variable than in others) - Spatial Continuity (h-scatterplots * Xj,Yj



Double log fractal dimension (DLFD) Mean patch fractal (MPFD) Area-weighted mean patch fractal dimension (AWMPFD)
Contagion, Interspersion and Juxtaposition
Applied Geostatistics
Miles Logsdon mlog@
Mimi D’Iorio mimid@
•"An Introduction to Applied Geostatistics" by Edward H. Isaaks and R. Mohan Srivastava, Oxford University Press, 1989.
•"Spatial Data Analysis: Theroy and Practice" by Robert Haining, Cambridge University Press, 1993. •"Statistics for Spatial data" by Noel a. c. Cressie, Wiley & Sons, Inc. 1991.
Physiognomy / Pattern / structure
Composition = The presence and amount of each
element type without spatially explicit measures.
Proportion, richness, evenness, diversity
Fractal Dimension (D), or (FRACT)
log P = 1/2D*log A; P = perimeter, A = area P = sq.rt. A raised to D, and D = 1 (a line) as polygons move to complexity P = A, and D -> 2 A few fractal metrics
variance standard deviation interquartile range
1 / n xi
2
n
2
st . d . Measurements of shape (symmetry
& length
coefficient of skewness coefficient of variation
Shape Metrics
perimeter-area relationships
Shape Index (SHAPE) -- complexity of patch compared to standard shape