I d
ð Þ ¼
1
W
P n
h¼1
P n
i¼1
w hi
À
y h À
y
ÁÀ
y i À
y
Á
1
n
P n
i¼1
À
y i À
y
Á 2
ð7:3Þ
The expected value of Moran’s I for no spatial correlation is
E I
ð Þ ¼
À1
n À 1
ð7:4Þ
Values below E(I) indicate negative spatial correlation, and values above E(I)
indicate positive correlation. E(I) is negative but close to 0 when n (the total number
of observations) is large.
Geary’s c is more akin to a distance measure:
c d
ð Þ ¼
1
2W
P n
h¼1
P n
i¼1
w hi y h À y i
ð
Þ
2
1
nÀ1
P n
i¼1
À
y i À
y
Á 2
ð7:5Þ
The expected value of Geary’s c for no spatial correlation is E(c) ¼ 1. Values
below 1 indicate positive spatial correlation, and values above 1 indicate negative
correlation.
y h and y i are the values of variable y at pairs of sites h and i. To compute spatial
correlation coefficients, one first constructs a matrix of geographical distances
among sites. These distances are then converted to classes d. Both formulas show
the computation of the index value for a class of inter-site distance d. The weights w hi
have value w hi ¼ 1 for pairs of sites belonging to distance class d, and w hi ¼ 0
otherwise. W is the number of pairs of points used to compute the coefficient for the
distance class considered, i.e., the sum of the w hi weights for that class.
A correlogram is a plot of the spatial correlation values against the distance
classes. Combined with statistical tests, a correlogram allows a quick assessment of
the type and range of the spatial correlation structure of a variable. A typical case is
spatial correlation that is positive at short distances, decreases to negative values, and
levels out to a point where it becomes nonsignificant. The corresponding distance
class sets the distance beyond which a pair of values can be considered as spatially
independent. It is important to note that spatial correlograms will display any kind of
spatial correlation, i.e. induced spatial dependence (Eq. 7.1) or spatial autocorrelation (Eq. 7.2); so the name “spatial autocorrelogram” which is often given to these
plots is too restrictive and therefore somewhat misleading.
Univariate spatial correlograms can be computed using the function
sp.correlogram()of package spdep. We can apply this function to the
variable “Substrate density” of the oribatid mite data set. We will first
define neighbourhoods of size 0.7 m around the points using the function
dnearneigh(). These links can be visualized using our function plot.links().
304
7 Spatial Analysis of Ecological Data
ð Þ ¼
1
W
P n
h¼1
P n
i¼1
w hi
À
y h À
y
ÁÀ
y i À
y
Á
1
n
P n
i¼1
À
y i À
y
Á 2
ð7:3Þ
The expected value of Moran’s I for no spatial correlation is
E I
ð Þ ¼
À1
n À 1
ð7:4Þ
Values below E(I) indicate negative spatial correlation, and values above E(I)
indicate positive correlation. E(I) is negative but close to 0 when n (the total number
of observations) is large.
Geary’s c is more akin to a distance measure:
c d
ð Þ ¼
1
2W
P n
h¼1
P n
i¼1
w hi y h À y i
ð
Þ
2
1
nÀ1
P n
i¼1
À
y i À
y
Á 2
ð7:5Þ
The expected value of Geary’s c for no spatial correlation is E(c) ¼ 1. Values
below 1 indicate positive spatial correlation, and values above 1 indicate negative
correlation.
y h and y i are the values of variable y at pairs of sites h and i. To compute spatial
correlation coefficients, one first constructs a matrix of geographical distances
among sites. These distances are then converted to classes d. Both formulas show
the computation of the index value for a class of inter-site distance d. The weights w hi
have value w hi ¼ 1 for pairs of sites belonging to distance class d, and w hi ¼ 0
otherwise. W is the number of pairs of points used to compute the coefficient for the
distance class considered, i.e., the sum of the w hi weights for that class.
A correlogram is a plot of the spatial correlation values against the distance
classes. Combined with statistical tests, a correlogram allows a quick assessment of
the type and range of the spatial correlation structure of a variable. A typical case is
spatial correlation that is positive at short distances, decreases to negative values, and
levels out to a point where it becomes nonsignificant. The corresponding distance
class sets the distance beyond which a pair of values can be considered as spatially
independent. It is important to note that spatial correlograms will display any kind of
spatial correlation, i.e. induced spatial dependence (Eq. 7.1) or spatial autocorrelation (Eq. 7.2); so the name “spatial autocorrelogram” which is often given to these
plots is too restrictive and therefore somewhat misleading.
Univariate spatial correlograms can be computed using the function
sp.correlogram()of package spdep. We can apply this function to the
variable “Substrate density” of the oribatid mite data set. We will first
define neighbourhoods of size 0.7 m around the points using the function
dnearneigh(). These links can be visualized using our function plot.links().
304
7 Spatial Analysis of Ecological Data