192
Elements of Spatial Data Analysis in Ecological Assessments
scale(s) of observation (Haining, 1990; Legendre
and Legendre, 1998). As discussed in Section
13.2.1, the observed spatial structure in a data set
may be due to different sources. Therefore, a useful description of approaches for spatial analysis
must begin with clear definitions of basic terms,
because inconsistent or imprecise use of terminology may lead to the appearance of conflict among
the available methods.
Spatial autocorrelation refers to the lack of independence among values of a variable of interest
due to error introduced by the effects of a process
operating within a zone of spatial influence (Legendre and Fortin, 1989; Legendre, 1993; Legendre
and Legendre, 1998). In contrast to spatial autocorrelation, spatial dependency results from the influence on a variable of interest of one or more explanatory variables that exhibit spatial structuring
(Legendre and Legendre, 1998). In practice, it is
difficult to determine the origin of spatial structure
(see Section 13.3.2). In some complex statistical
models, spatial structure is expressed as a combination of spatial autocorrelation in a response variable, the effects of explanatory variable(s), plus explanatory variable autocorrelation structure (Cliff
and Ord, 1981; Griffith, 1988; Legendre and Legendre, 1998).
Implicit in statistical analyses is the assumption
that the statistical properties of spatially correlated
variation are the same over the entire study area.
This assumption, known as the hypothesis of stationarity, is necessary for applying statistical methods to spatial analysis (Burrough, 1995). However,
in spatial analyses, the stationarity assumption is
usually replaced by second-order or weak stationarity, in which the mean is constant, covariance depends only on sampling interval, and the variance
is finite and constant (Legendre and Legendre,
1998). The spatial and non spatial definitions of stationarity differ in the recognition, for the spatial
case, that the variance includes a covariance term
that depends on sample spacing (Burrough, 1995).
For spatial data, a third, relaxed form of stationarity, the intrinsic assumption (Legendre and Legendre, 1998), may be used. This assumption specifies that the differences between successive points
must meet the stationarity hypothesis, rather than
the values of the points themselves; that is, it considers only the increments of the values of the regionalized variables. This condition must be met in
the use of surface pattern methods, such as the correlogram and semivariance (see Section 13.3.2)
(Burrough, 1995; Legendre and Legendre, 1998).
There are four aspects of nonstationarity to consider: nonnormality in the distribution of a variable,
non stationarity of the mean, non stationarity of the
variance, and anisotropy (variation in the value of
the variable as a function of direction). Nonnormal
distributions are common in EA data and require
transformation to approximate normality if possible. Nonstationarity of the mean is corrected for by
filtering out long-range trends. Nonstationarity of
the variance results in a meaningless expression for
autocorrelation, because a constant variance is required (Burrough, 1995). In this case, using semivariance rather than covariance is suggested, because semi variance does not require stationarity of
the variance (see discussion of the intrinsic assumption). Anisotropic variation occurs when there
is strong directional control on a variable of interest. Detection is necessary because anisotropy affects the interpretation and applicability of the
analyses. All techniques discussed in this section
are sensitive to one or more of these aspects of nonstationarity and to extreme values that can bias the
quantification of spatial patterns in the data.
The quantification and characterization of spatial structure can be accomplished using two distinct categories of methods, spatial statistics and
landscape metrics. Spatial statistics quantify spatial
structure based on sampled data, whereas landscape
metrics characterize the geometric and spatial properties of mapped data. Therefore, spatial statistics
estimate the spatial values of sampled variables,
and landscape metrics characterize the properties
of spatially homogeneous units (patches) or mosaics of patches (Fortin, 1999b). The two categories
of techniques are discussed in Sections 13.3.2 and
13.3.3.
13.3.2 Spatial Statistics
Three families of spatial statistics, point pattern
methods, surface pattern methods, and line pattern
analysis, are used in spatial analysis, based on the
type of data and objective of the analysis. Point pattern analysis is concerned with the spatial distribution of all individual objects (points) in a study
area. Its main objectives are to quantify the geographic distribution of the objects, test for randomness, and describe the type of pattern and scale.
The second family of methods, surface pattern
analysis, applies to the study of spatially continuous variables. It is based on the concept of regionalized variables (Matheron, 1965), which are assumed to be distributed as a continuous surface
(Oden et aI., 1993; Legendre and McArdle, 1997).
This family of methods includes trend surface
analysis, a special case of traditional multipleregression techniques (Turner et aI., 1991; Bur-
Elements of Spatial Data Analysis in Ecological Assessments
scale(s) of observation (Haining, 1990; Legendre
and Legendre, 1998). As discussed in Section
13.2.1, the observed spatial structure in a data set
may be due to different sources. Therefore, a useful description of approaches for spatial analysis
must begin with clear definitions of basic terms,
because inconsistent or imprecise use of terminology may lead to the appearance of conflict among
the available methods.
Spatial autocorrelation refers to the lack of independence among values of a variable of interest
due to error introduced by the effects of a process
operating within a zone of spatial influence (Legendre and Fortin, 1989; Legendre, 1993; Legendre
and Legendre, 1998). In contrast to spatial autocorrelation, spatial dependency results from the influence on a variable of interest of one or more explanatory variables that exhibit spatial structuring
(Legendre and Legendre, 1998). In practice, it is
difficult to determine the origin of spatial structure
(see Section 13.3.2). In some complex statistical
models, spatial structure is expressed as a combination of spatial autocorrelation in a response variable, the effects of explanatory variable(s), plus explanatory variable autocorrelation structure (Cliff
and Ord, 1981; Griffith, 1988; Legendre and Legendre, 1998).
Implicit in statistical analyses is the assumption
that the statistical properties of spatially correlated
variation are the same over the entire study area.
This assumption, known as the hypothesis of stationarity, is necessary for applying statistical methods to spatial analysis (Burrough, 1995). However,
in spatial analyses, the stationarity assumption is
usually replaced by second-order or weak stationarity, in which the mean is constant, covariance depends only on sampling interval, and the variance
is finite and constant (Legendre and Legendre,
1998). The spatial and non spatial definitions of stationarity differ in the recognition, for the spatial
case, that the variance includes a covariance term
that depends on sample spacing (Burrough, 1995).
For spatial data, a third, relaxed form of stationarity, the intrinsic assumption (Legendre and Legendre, 1998), may be used. This assumption specifies that the differences between successive points
must meet the stationarity hypothesis, rather than
the values of the points themselves; that is, it considers only the increments of the values of the regionalized variables. This condition must be met in
the use of surface pattern methods, such as the correlogram and semivariance (see Section 13.3.2)
(Burrough, 1995; Legendre and Legendre, 1998).
There are four aspects of nonstationarity to consider: nonnormality in the distribution of a variable,
non stationarity of the mean, non stationarity of the
variance, and anisotropy (variation in the value of
the variable as a function of direction). Nonnormal
distributions are common in EA data and require
transformation to approximate normality if possible. Nonstationarity of the mean is corrected for by
filtering out long-range trends. Nonstationarity of
the variance results in a meaningless expression for
autocorrelation, because a constant variance is required (Burrough, 1995). In this case, using semivariance rather than covariance is suggested, because semi variance does not require stationarity of
the variance (see discussion of the intrinsic assumption). Anisotropic variation occurs when there
is strong directional control on a variable of interest. Detection is necessary because anisotropy affects the interpretation and applicability of the
analyses. All techniques discussed in this section
are sensitive to one or more of these aspects of nonstationarity and to extreme values that can bias the
quantification of spatial patterns in the data.
The quantification and characterization of spatial structure can be accomplished using two distinct categories of methods, spatial statistics and
landscape metrics. Spatial statistics quantify spatial
structure based on sampled data, whereas landscape
metrics characterize the geometric and spatial properties of mapped data. Therefore, spatial statistics
estimate the spatial values of sampled variables,
and landscape metrics characterize the properties
of spatially homogeneous units (patches) or mosaics of patches (Fortin, 1999b). The two categories
of techniques are discussed in Sections 13.3.2 and
13.3.3.
13.3.2 Spatial Statistics
Three families of spatial statistics, point pattern
methods, surface pattern methods, and line pattern
analysis, are used in spatial analysis, based on the
type of data and objective of the analysis. Point pattern analysis is concerned with the spatial distribution of all individual objects (points) in a study
area. Its main objectives are to quantify the geographic distribution of the objects, test for randomness, and describe the type of pattern and scale.
The second family of methods, surface pattern
analysis, applies to the study of spatially continuous variables. It is based on the concept of regionalized variables (Matheron, 1965), which are assumed to be distributed as a continuous surface
(Oden et aI., 1993; Legendre and McArdle, 1997).
This family of methods includes trend surface
analysis, a special case of traditional multipleregression techniques (Turner et aI., 1991; Bur-