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.

##

## #

##

#

##

#

#

##

#

# # ##

##

#

# #

#

# #

#

#

#

##

## #

# #

##

# #

##

#

#

#

#

#

#

#

#

#

Temp.dbf# 690 - 740# 740 - 793# 793 - 841# 841 - 890# 890 - 960

0 1 2 3 4 5 6 7

02

00

40

06

00

80

01

00

0

xp

yp

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