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Spatial Statistics
Jonathan Bossenbroek, PhD
Dept of Env. Sciences
Lake Erie Center
University of Toledo
What is Spatial Statistics?
The quantitative study of phenomena located in space.
Spatial patterns
Autocorrelation Semivariance
Example – Moose on Isle Royale
Where are people in Bowman-Oddy?
Are they randomly distributed?
Point-to-Point Nearest-Neighbor Analysis Uses distances between points as its basis. The distance observed between each point
and its nearest neighbor is compared with the expected mean distance that would occur if the distribution were random.
G Statistic
nyG ydi
1
)(ˆ
•di is the distance of point i to its nearest neighbor
•y is distance
•n is the number of points
0.00 0.02 0.04 0.06 0.08
0.0
0.2
0.4
0.6
0.8
r
G(r
)Distance
Examples: paint splatters, dandelions in a field, …
0.00 0.05 0.10 0.15
0.0
0.2
0.4
0.6
0.8
r
G(r
)
Distance
Examples: breeding birds, beach blankets,
0.00 0.01 0.02 0.03 0.04 0.050
.00
.20
.40
.60
.8
r
G(r
)
Distance
Examples: Buffalo at a watering hole, fast food restaurants, …
How old are those people in Bowman-Oddy?
Are they randomly distributed?
Geostatistical Tools For Modeling And Interpreting Ecological Spatial Dependence Ecological Monographs 62(2). 1992. pp. 277-3146 1992 by the
Ecological Society of America Richard E. Rossi et al.
“…geostatistics is never a replacement for sound ecological reasoning”
Geostistical Tools Spatial and temporal dependence are the
norm in natural systems: Different plant species are often different on north
and south facing slopes. Grasshoppers are more dense during hot dry
periods. Spatial dependence is particularly important
in analysis of spatially varying organisms and environmental variables.
Spatial statistics can test for independence!
Always know your data!
Rossi et al. suggest always beginning with exploratory data analysis. Histograms, regressions, scatter plots etc.
From Rossi et al 1992
From Rossi et al 1992
From Rossi et al 1992
Basic statistics do not tell the story Two statistical tools:
h-scatterplots Variography
h-scatterplots displays the degree of spatial continuity or correlation at
some lag distance h Variograms
Variograms model the average degree of similarity between the values of a variable as a function of distance.
Scatter plots
Typical scatter plot compares measurement of two parameters at the same location or of the same object.
h-scatter plots compares measurement of the same parameter at a certain distance apart.
h-scatterplot: if distance (h) = 1
h-scatterplot
h-scatterplot: if distance (h) = 2
How do you measure variance?
semivariance
A variogram summarizes all h-scattergrams for all possible pairings of the data or rather distributes variance across space.
•y(h) is the estimated semivariance for lag h
•N(h) is number of pairs of points separated by lag h
•Z(xi) is the value of variable Z at location xi
•Z(xi + h) is the value of variable Z at location xi + h
Looking back at h-scatter plots… What is the variance
at h = 1?
Is the variance at h = 2 > or < h = 1?
Sem
ivar
ian
ce
Sill
Nugget
Range
semivariogram
Distance (h)1 2
Sem
ivar
ian
ce
Sill
Nugget
Range
semivariogram
Distance
Semivariogram
Sill Variance level equivalent to the global variance of
the area Range
Distance at which data are no longer spatially autocorrelated.
Patch size? Nugget
Represents micro-scale variation or measurement error.
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Temp.dbf# 690 - 740# 740 - 793# 793 - 841# 841 - 890# 890 - 960
0 1 2 3 4 5 6 7
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Distance
Other topics in spatial statistics. Kriging: an interpolation method for obtaining stastically
unbiased estimates for field attributes (yield, nutrients, elevation) from a set of neighboring points.
Other topics in spatial statistics. Correlogram: a
measure of spatial dependence (correlation) of a regionalized variable over some distance
Other topics in spatial statistics. Metapopulation Models: A set of partially isolated populations
belonging to the same species. The populations are able to exchange individuals and recolonize sites in which the species has recently become extinct.
Spatial patterns in the moose-forest-soil ecosystem on Isle Royale, Michigan USA – J. Pastor et al.
Observations:
Hypotheses:
Results:
Spatial patterns in the moose-forest-soil ecosystem on Isle Royale, Michigan USA – J. Pastor et al.
Observations: Moose preferentially forage on aspen and avoid
conifers. Hypotheses:
If moose browsing causes a shift in dominance from hardwoods to conifers across adjacent areas, we should expect corresponding changes in soil nutrient availability over the landscape.
Results:
Spatial patterns in the moose-forest-soil ecosystem on Isle Royale, Michigan USA – J. Pastor et al.
What was the study about?
Examine the largescale landscape distribution of moose browsing intensity in relation to plant community composition and size structure, as well as soil nitrogen availability. Do moose control plant community composition
and soil nitrogen at large scales?
What did they measure?
Available browse. Annual consumption by moose. Soil nitrogen availability.
What did Pastor conclude?
No differences in nitrogen availability or consumption due to slope or aspect. Spatial patterns not caused by topographic relief.
Patterns are a result of dynamic interactions between moose foraging and plant communities.
Uncommonly strong impact for a large mammal.
This patterns has occurred in less than 50 generations.
Why are things spatially autocorrelated? Environment
Examples: soil, climate, moisture, ... Interactions
Examples: competition, herbivory, mutualism