Spatial Statistics in Ecology: Area Data

Lecture Four

Recall and Introduction to Area Data

• Recall that we have visualized, explored and modeled point patterns and continuous processes

• Many of the same concepts that you have previously learned apply to area data

• Think of area data like a continuous process that is separated in zones

Can you think of some types of area data that ecologists might use?

Types of Area Data

• Recall that area data is either grid – raster format or irregular – vector format

• Area data must be continuous like states in a country – this is called “contiguity”

• Areas must be contiguous to create the spatial weight matrix that is used to study first order effects

• Areas that are not contiguous can still be studied for second order effects

Patterns in zonation• Each area is assumed to

have a fixed value or mean value

• We are not interested with areal data in predating the value of an attribute in areas which it has not been sampled

• Instead we are interested in how the zones vary and the type of pattern that arises across zones

Proportionalsymbols map

Visualizing area dataArea data is mapped using the choropleth map or using proportional symbols

Grid or lattice data Irregular zone data

Exploring Area data: Proximity

• The first issue to deal with when using area data is PROXIMITY

• Proximity refers to how to measure how close a unit is to another unit when they are irregularly shaped and do not have obvious grid like centroids

Proximity

Some examples are:1. Centroids2. Common boundaries3. Length of boundaries4. Hybrid measures such as shared

boundary + distance between centroids

centroids

Commonboundary

Length of boundary

Spatial Weight Matrices

• Proximity is used to create a spatial weight matrix

• (n x n) proximity matrix W, each of whose elements wij, represents a measure of the spatial proximity of areas Ai and Aj

Kernel Estimation

• For area data Kernel estimation requires a fixed centroid as it cannot be used to look at irregular areas.

• Thus, kernel estimation is more appropriate for grids and lattice data or data with obvious centroids

• In this case Kernel estimation is almost the same as it was for spatially continuous data

Second-order: AUTOCORRELATION

• Remember the variogram? The correlogram in typically used to explore spatial dependence of area data

• When applied to area data it is formally called spatial autocorrelation

• Autocorrelation involves correlation between values of the SAME variable at different spatial locations

AutocorrelationConceptual Geary’s c Moran’s I

Similar, regionalized, smooth, clustered

0 < C < 1 I > 0*

Independent, uncorrelated, random

C = O I < 0*

Dissimilar, contrasting, checkerboard-like

C > 1 I < 0*

Note 0* = 1/n(n-1) where n is the number of objects

Moran’s I represents the overall agglomerative patterns of areas (are areas of like value clumped together or dispersedGeary’s c explains similarity or

dissimilarity (is it possible to predictfrom one area what the value will be at a neighboring area).

The correlogramBoth Moran’s I and Geary’s C an be used to create correlograms showing the lag to lag (distance) spatial dependence between areas

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

lag distance

-0.4

0.2

rho

(a) Desiccation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

lag distance

0.0

0.5

rho

(b) pH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

lag distance

0.0

0.5

1.0

rho

(c) Salinity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

lag distance

-0.1

0.4

rho

(d) Elevation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

lag distance

-0.2

0.6

rho

(e) Depth

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

lag distance

-0.4

0.2

rho

(f) Temperature

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

lag distance

0.0

0.5

rho

(g) Oxygen

Figure 4.0(a-g) Correlograms for environmental variables indicating lag to lag distance

Example correlograms

What can you tellabout the spatial dependence of these 3 variables?

Modelling area data

Non-spatial regression models

• Suitable if there is no spatial dependence

• First produce a correlogram

• If there is no dependence proceed with regression

• Spatial regression models

• If there is spatial dependence than use a spatial regression model

• SAR or CAR models

Example: Ireland Blood Type A

• How is blood type A distributed in Ireland? Samples were taken from 55000 people in 26 counties

• How does the proportion of people with blood type A vary across counties?

Step One: Produce Map

Step Two: run SAR model

Step Three: map residuals and produce correlogram

Summary: Lecture Four

• Area data differs from continuous processes in that we look at variation in the mean value of an area to discover patterns between zones

• First-order effects are studied by kernel estimation or spatial autoregressive models

• Second-order effects are considered using Moran's I and Geary’s and by producing correlograms