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 nicheenvironment 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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17  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 xaxis 
yax 
the column number for the yaxis 
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 (counterclockwise), for the available pixels polygon 
Uangle 
the slope of shading lines, given as an angle in degrees (counterclockwise), 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
nicheenvironment 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 asc
.
Author(s)
Clement Calenge clement.calenge@oncfs.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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78  ## 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)

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