Applications of AEM modelling are presented in Blanchet et al. (2008a, b, 2011)
as well as in Gray and Arnott (2011), Sharma et al. (2011), and other papers.
7.5 Another Way to Look at Spatial Structures: Multiscale
Ordination (MSO)
7.5.1 Principle
Wagner (2003, 2004) took an entirely different path towards integration of spatial
information and MEM eigenfunctions into canonical ordination. Under the wellknown fact that autocorrelated residuals can alter the results of statistical tests, she
used geostatistical methods to devise diagnostic tools allowing (1) the partitioning of
ordination results into distance classes, (2) the distinction between induced spatial
dependence and spatial autocorrelation, and (3) the use of variograms to check
important assumptions such as independence of residuals and stationarity. The
principle of MSO is the following
6
:
• Analyse the species by RDA. The explanatory variables can be of any kind
(environmental, spatial, and so on). This provides the matrix of fitted values
and its eigenvectors, as well as the matrix of residuals and its eigenvectors.
• By way of a variogram matrix computed for the fitted values, obtain the spatial
variance profiles of the canonical ordination axes (see below).
• By way of a variogram matrix computed for the residuals, obtain the spatial
variance profiles of the residual ordination axes.
• Plot the variograms of the explained and residual variances. Permutation tests
may be used to identify significant spatial correlation in the distance classes.
A variogram matrix is a three-dimensional array containing a multivariate
variance-covariance matrix for each distance class (Wagner 2003 Fig. 2.2; Legendre
and Legendre 2012 Fig. 13.11). The diagonal of each matrix quantifies the contribution of the corresponding distance class to the variance of the data. MSO computes
a variogram matrix on the fitted values of a constrained ordination, thereby allowing
its spatial decomposition. Multiplying this variogram matrix with the matrix of
constrained eigenvectors provides the spatial decomposition of each eigenvalue
(variance profiles). The same holds for the residuals.
6 Wagner (2004) describes the method for CCA, but the principle is the same for RDA.
7.5 Another Way to Look at Spatial Structures: Multiscale Ordination (MSO)
355
as well as in Gray and Arnott (2011), Sharma et al. (2011), and other papers.
7.5 Another Way to Look at Spatial Structures: Multiscale
Ordination (MSO)
7.5.1 Principle
Wagner (2003, 2004) took an entirely different path towards integration of spatial
information and MEM eigenfunctions into canonical ordination. Under the wellknown fact that autocorrelated residuals can alter the results of statistical tests, she
used geostatistical methods to devise diagnostic tools allowing (1) the partitioning of
ordination results into distance classes, (2) the distinction between induced spatial
dependence and spatial autocorrelation, and (3) the use of variograms to check
important assumptions such as independence of residuals and stationarity. The
principle of MSO is the following
6
:
• Analyse the species by RDA. The explanatory variables can be of any kind
(environmental, spatial, and so on). This provides the matrix of fitted values
and its eigenvectors, as well as the matrix of residuals and its eigenvectors.
• By way of a variogram matrix computed for the fitted values, obtain the spatial
variance profiles of the canonical ordination axes (see below).
• By way of a variogram matrix computed for the residuals, obtain the spatial
variance profiles of the residual ordination axes.
• Plot the variograms of the explained and residual variances. Permutation tests
may be used to identify significant spatial correlation in the distance classes.
A variogram matrix is a three-dimensional array containing a multivariate
variance-covariance matrix for each distance class (Wagner 2003 Fig. 2.2; Legendre
and Legendre 2012 Fig. 13.11). The diagonal of each matrix quantifies the contribution of the corresponding distance class to the variance of the data. MSO computes
a variogram matrix on the fitted values of a constrained ordination, thereby allowing
its spatial decomposition. Multiplying this variogram matrix with the matrix of
constrained eigenvectors provides the spatial decomposition of each eigenvalue
(variance profiles). The same holds for the residuals.
6 Wagner (2004) describes the method for CCA, but the principle is the same for RDA.
7.5 Another Way to Look at Spatial Structures: Multiscale Ordination (MSO)
355