| madifa {adehabitatHS} | R Documentation |
The MADIFA: a Factorial Decomposition of the Mahalanobis Distances
Description
The MADIFA allows a factorial decomposition of the Mahalanobis distances. This method is presented here in the framework of niche-environment studies.
predict.madifa allows the computation of the Mahalanobis
Distances based on a restricted number of factorial axes.
All other functions allow various graphical displays of the results of
the MADIFA.
Usage
madifa(dudi, pr, scannf = TRUE, nf = 2)
## S3 method for class 'madifa'
print(x, ...)
## S3 method for class 'madifa'
scatter(x, xax = 1, yax = 2, pts = FALSE, percent = 95,
clabel = 1, side = c("top", "bottom", "none"),
Adensity, Udensity, Aangle, Uangle, Aborder,
Uborder, Acol, Ucol, Alty,
Ulty, Abg, Ubg, Ainch, Uinch, ...)
## S3 method for class 'madifa'
hist(x, scores = TRUE, type = c("h", "l"), adjust = 1, Acol,
Ucol, Aborder, Uborder, Alwd = 1, Ulwd = 1, ...)
## S3 method for class 'madifa'
predict(object, map, nf, ...)
s.madifa(x, xax = 1, yax = 2, cgrid = 1, clab = 1, ...)
## S3 method for class 'madifa'
plot(x, map, xax = 1, yax = 2, cont = FALSE, ...)
Arguments
dudi |
a duality diagram, an object of class |
pr |
a vector giving the utilization weights associated to each unit |
scannf |
logical. Whether the eigenvalues barplot should be displayed |
nf |
an integer indicating the number of kept factorial axes |
x, object |
an object of class |
xax |
the column number for the x-axis |
yax |
the column number for the y-axis |
pts |
logical. Whether the points should be drawn. If
|
percent |
100 minus the proportion of outliers to be excluded from the computation of the minimum convex polygons |
clabel |
a character size for the columns |
side |
if |
Adensity |
the density of shading lines, in lines per inch, for
the available pixels polygon. See |
Udensity |
the density of shading lines, in lines per inch, for
the used pixels polygon. See |
Aangle |
the slope of shading lines, given as an angle in degrees (counter-clockwise), for the available pixels polygon |
Uangle |
the slope of shading lines, given as an angle in degrees (counter-clockwise), for the used pixels polygon |
Aborder |
the color for drawing the border of the available pixels
polygon (or of the bars of the histogram) . See
|
Uborder |
the color for drawing the border of the used pixels
polygon (or of the bars of the histogram). See
|
Acol |
the color for filling the available pixels polygon.
if |
Ucol |
the color for filling the used pixels polygon.
if |
Alty |
the line type for the available pixels polygon, as in
|
Ulty |
the line type for the used pixels polygon, as in
|
Abg |
if |
Ubg |
if |
Ainch |
if |
Uinch |
if |
scores |
logical. If |
type |
what type of plot should be drawn. Possible types are: |
adjust |
if |
Alwd |
if |
Ulwd |
if |
cgrid |
a character size, parameter used with par("cex")*
|
clab |
if not NULL, a character size for the labels, used with
|
map |
an object of class |
cont |
logical. Whether contour lines should be added to the maps |
... |
additional arguments to be passed to the functions
|
Details
The Mahalanobis distances are often used in the context of
niche-environment studies (Clark et al. 1993, see the function
mahasuhab). Each environmental variable defines a dimension in
a multidimensionnal space, namely the ecological space. The
Mahalanobis distance between any resource unit in this space (e.g. the
point defined by the values of environmental variables in a pixel of a
raster map) and the centroid of the niche (the distribution of used
resource units) can be used to give a value of eccentricity to this
point.
For a given distribution of available resource units, for which a
measure of Mahalanobis distances is desired, the MADIFA (MAhalanobis
DIstances Factor Analysis) partitions the ecological space into a set
of axes, so that the first axes maximises the average proportion of
their squared Mahalanobis distances. Note that the sum of the squared
scores of any resource unit on all the axes of the analysis is equal
to the squared Mahalanobis distances for this resource unit. Thus,
the MADIFA partitions the Mahalanobis distances into several axes of
biological meaning (see examples). predict.madifa allows to
compute approximate Mahalanobis distances from the axes of the
MADIFA.
plot.madifa returns a graphical summary of the analysis: it
returns graphs of (i) the eigenvalues of the analysis (each eigenvalue
measures the average Mahalanobis distance explained by each factorial
axis); (ii) the scores of the habitat variables (i.e. the coefficients
associated to each environmental variable in the linear combination
defining the axes) - note that as the ecological space is distorted to
"sphericize" the niche, the factorial axes are no longer orthogonals,
and the scores of the variables are distributed within an ellipsoid
instead of an hypersphere of radius equal to one in classical PCA. The
limits of this ellipsoid is displayed on this graph, to see the amount
of distortion done by the analysis (further research needs yet to be
done on this graph); (iii) The projection of the available and used
points on the factorial plane of the MADIFA; (iv) The map of the
Mahalanobis distances computed from the original environmental
variables; (v) the map of the approximated Mahalanobis distances
computed from the two axes displayed in this plot; the correlations
between the original environmental variables and the factorial axes;
(v) the map of the first factorial axis and (vi) the map of the second
factorial axis.
hist.madifa returns a graph of the niche and the available
resource units on the factorial axes of the analysis.
Value
madifa returns a list of class madifa containing the
following components:
call |
original call. |
tab |
a data frame with n rows and p columns. |
pr |
a vector of length n containing the number of points in each pixel of the map. |
nf |
the number of kept factorial axes. |
eig |
a vector with all the eigenvalues of the analysis. |
lw |
row weights, a vector with n components. |
li |
row coordinates, data frame with n rows and nf columns. |
l1 |
row normed coordinates, data frame with n rows and nf columns. |
cw |
column weights, a vector with p components. |
co |
column coordinates, data frame with p rows and nf columns. |
mahasu |
a vector of length n containing the squared Mahalanobis distances for the n units. |
cor |
the correlation between the MADIFA axes and the original variable |
predict.madifa returns a matrix of class SpatialPixelsDataFrame.
Author(s)
Clement Calenge clement.calenge@ofb.gouv.fr
References
Clark, J.D., Dunn, J.E. and Smith, K.G. (1993) A multivariate model of female black bear habitat use for a geographic information system. Journal of Wildlife Management, 57, 519–526.
Calenge, C., Darmon, G., Basille, M., Loison, A. and Jullien J.M. (2008) The factorial decomposition of the Mahalanobis distances in habitat selection studies. Ecology, 89, 555–566.
See Also
mahasuhab for a detailed description of the
Mahalanobis distances, enfa and gnesfa for
closely related methods.
Examples
## Not run:
data(bauges)
map <- bauges$map
locs <- bauges$loc
## We prepare the data for the MADIFA
tab <- slot(map, "data")
pr <- slot(count.points(locs, map), "data")[,1]
## We then perform the PCA before the MADIFA
pc <- dudi.pca(tab, scannf = FALSE)
(mad <- madifa(pc, pr, nf=7,
scannf = FALSE))
#########################################
## ##
## Graphical exploration of the MADIFA ##
## ##
#########################################
hist(mad)
plot(mad, map)
## this plot represents:
## - the eigenvalues diagram
## - the scores of the columns on the axes
## - a graph of the niche in the available space
## - a map of the Mahalanobis distances computed
## using all environmental variables
## - a map of the Mahalanobis distances computed
## using the two factorial axes used in the
## previous graphs
## - the correlation between habitat variables
## and factorial axes
## - the geographical maps of the two
## factorial axes
## predict with just the first axis
pred <- predict(mad, map, nf=1)
image(pred)
#########################################
## ##
## Mathematical properties of MADIFA ##
## ##
#########################################
## mad$li is equal to mad$l1, up to a constant (mad$l1 is normed)
plot(mad$li[,1],mad$l1[,1])
## This constant is the square root of the corresponding eigenvalue:
## the variance of mad$l1 is equal to the eigenvalue
apply(mad$l1,2,function(x) sum(x^2))/nrow(mad$li)
## the variance of mad$l1 weighted by pr is equal to 1
apply(mad$l1,2,function(x) sum(mad$pr*x^2)/sum(mad$pr))
## Therefore, the eigenvalues are equal to the average of Mahalanobis
## distance for the available resource units on each axis
mean(mahalanobis(matrix(mad$l1[,1], ncol=1), 0, 1))
mad$eig[1]
## Computation of the Mahalanobis distances
ma1 <- mahasuhab(map, locs)
## The sum of squared scores for a given Resource unit is equal to the
## Mahalanobis distances
ma2 <- apply(mad$l1,1, function(x) sum(x^2))
plot(ma2, slot(ma1, "data")[,1])
## End(Not run)