The function `gnesfa`

allows to perform a general
niche-environment system factor analysis.

1 2 3 4 5 |

`dudi` |
an object of class |

`Focus` |
a vector containing the focus weights |

`Reference` |
a vector containing the reference weights |

`centering` |
a character string indicating the type of centering (see details) |

`scannf` |
a logical value indicating whether the eigenvalues bar plot should be displayed |

`nfFirst` |
the number of first axes to be kept |

`nfLast` |
the number of last axes to be kept |

`x` |
an object of class GNESFA |

`...` |
further arguments to be passed to other functions |

The GNESFA is an algorithm which generalises several factor analyses of the ecological niche. A table X gives the values of P environmental variables in N resource units (e.g. the pixels of a raster map). A distribution of weights D describes the availability of the resource units to the species (if not specified, these weights are considered to be uniform). Another distribution of weights Dp describes the use of the resource units by the species (for example the proportion of relocations in each pixel of a raster map).

Each environmental variable defines a dimension in a multidimensional space, the ecological space. The N resource units define a cloud of points in this space. Each point is associated to two weights. The GNESFA finds, in the ecological space, the directions on which these two distributions of weights are the most different.

The GNESFA relies on a choice of the analyst, followed by three steps. Before all, the analyst has to choose one distribution of weights as the Reference distribution, and the other one as the Focus distribution; (i) The first table X is centred on the centroid of the Reference distribution; (ii) a principal component analysis of this Reference distribution is performed; (iii) the cloud of points is distorted, so that the Reference distribution takes a standard spherical shape; (iv) a non centred principal component analysis of the Focus distribution allows to identify the directions of the ecological space where the two distributions are the most different.

Depending on the distribution chosen as Reference, this algorithm returns results with different meanings (see examples). This algorithm is closely related to several common analyses of habitat selection/niche (ENFA, MADIFA, Mahalanobis distances, selection ratios, etc.). The examples below give some examples of the mathematical properties of this algorithm.

Note that the function takes a parameter named `centering`

.
Indeed, two types of centering can be performed prior to the GNESFA.
The choice `"single"`

consists in the centering of the cloud of
point in the ecological space on the centroid of the Reference
distribution. The choice `"twice"`

consist to center the cloud
of points on both the centroid of the Reference distribution and the
centroid of the Focus distribution. This is done by projecting the
cloud of points on the hyperplane orthogonal to the marginality vector
(the vector connecting the two centroids. If this choice is done, the
GNESFA is identical to the commonly used Ecological Niche Factor
Analysis (see examples).

`gnesfa`

returns a list of class `gnesfa`

containing the
following components:

`call` |
original call. |

`centering` |
The type of centering required. |

`tab` |
a data frame with n rows and p columns. |

`Reference` |
a vector of length n containing the Reference weights. |

`Focus` |
a vector of length n containing the Focus weights. |

`nfFirst` |
the number of kept first axes. |

`nfLast` |
the number of kept last axes. |

`eig` |
a vector with all the eigenvalues of the analysis. |

`li` |
row coordinates, data frame with n rows and nf columns. |

`l1` |
row normed coordinates, data frame with n rows and nf columns. |

`co` |
column scores, data frame with p rows and nf columns. |

`cor` |
the correlation between the GNESFA axes and the original variables |

Clement Calenge clement.calenge@oncfs.gouv.fr

Calenge, C. and Basille, M. (2008) A General Framework for the
Statistical Exploration of the Ecological Niche. *Journal of
Theoretical Biology*, **252**: 674-685.

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.

`madifa`

, `mahasuhab`

,
`enfa`

, `wi`

for closely related methods (see
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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 | ```
## Not run:
################################################################
##
## Study of the habitat selection by the chamois in the French
## mountains of Les Bauges
## Loads the data
data(bauges)
names(bauges)
map <- bauges$map
locs <- bauges$locs
## displays the data
mimage(map)
image(map,1)
points(locs, pch = 3)
## Prepares the data for the GNESFA:
tab <- slot(map, "data")
Dp <- slot(count.points(locs,map), "data")[,1]
pc <- dudi.pca(tab, scannf = FALSE)
## Example of use with Dp = Reference
gn <- gnesfa(pc, Reference = Dp, scannf=FALSE)
## One main axis:
barplot(gn$eig)
## The correlation with variables indicate that
## the elevation, the proximity to grass and to
## deciduous forests:
s.arrow(gn$cor)
## The factorial map of the niche...
scatterniche(gn$li, Dp, pts = TRUE)
## The chamois is rather located at high elevation,
## in the grass, far from deciduous forests
##########################################################
##########################################################
##
##
## Some interesting properties of the GNESFA
##
##
##########################################################
##########################################################
################################
################################
##
## Interesting properties of the
## choice: Dp as Reference
## identical to the MADIFA
## (Calenge et al. 2008),
## See the help page of the function madifa
## for other properties)
gn <- gnesfa(pc, Reference = Dp, scannf=FALSE,
nfFirst = 7)
gn
## This is the same as the MADIFA:
mad <- madifa(pc, Dp, scannf=FALSE)
## Indeed:
plot(gn$li[,1], mad$li[,1])
cor(gn$li[,1], mad$li[,1])
## And consequently the sum of the squared scores,
## On the axes of the GNESFA...
su <- apply(gn$l1,1,function(x) sum(x^2))
## ... is equal to the Mahalanobis distances between
## the points and the centroid of the niche
## (Clark et al. 1993, see the help page of mahasuhab)
su2 <- slot(mahasuhab(map, locs), "data")[,1]
## Indeed:
all(su - su2 < 1e-7)
plot(su, su2)
################################
################################
##
## Centering twice is identical to
## the ENFA (Hirzel et al. 2002, see the help
## page of the function enfa)...
#######
##
## ... If Dp is the Reference:
gn <- gnesfa(pc, Reference = Dp, center = "twice", scannf = FALSE)
gn
enf <- enfa(pc, Dp, scannf = FALSE)
plot(enf$li[,2], gn$li[,1])
cor(enf$li[,2], gn$li[,1])
## The first specialization axis of the ENFA
## is the first axis of the GNESFA!
#######
##
## ... If Dp is the Focus:
gn <- gnesfa(pc, Focus = Dp, center = "twice",
scannf = FALSE, nfFirst = 6)
plot(enf$li[,2], gn$li[,6])
cor(enf$li[,2], gn$li[,6])
## The first specialization axis of the ENFA
## is the last axis of the GNESFA!
#######
##
## Whatever the distribution chosen as Reference,
## projecting the cloud of points on the hyperplane
## orthogonal to the marginality axis, and performing
## a GNESFA in this subspace is identical to an ENFA!
## The marginality axis of the ENFA is identical
## to the component "projmar" of the GNESFA
plot(enf$li[,1],gn$projmar)
cor(enf$li[,1],gn$projmar)
################################
################################
##
## Interesting properties of the
## case: Dp as Focus, one categorical
## variable. Relationships with the selection
## ratios of Manly et al. (1972, see the
## help page of wi)
## For example, take the Elevation, and
## define a factor with 4 levels
elev <- data.frame(el = cut(slot(map, "data")$Elevation, 4))
## Now, compute the complete disjonctive table
dis <- acm.disjonctif(elev)
head(dis)
## Now perform the GNESFA with Dp as Focus:
pc <- dudi.pca(dis, scannf = FALSE)
gn <- gnesfa(pc, Dp, scannf = FALSE, nfFirst = 3)
#######
##
## This analysis is closely related to the concept of
## selection ratios
## Compute the percentage of use of each level:
us <- apply(dis, 2, function(x) sum(x*Dp)/sum(Dp))
av <- apply(dis, 2, function(x) sum(x)/length(x))
## The selection ratios
wi <- widesI(us, av)$wi
## Compute the sum of the eigenvalue
sum(gn$eig)
## Compute the sum of the selection ratios - 1
sum(wi) - 1
## In other words, when the GNESFA (Dp as Focus) is
## applied on only one categorical variable, this
## analysis finds a set of axes which partition the
## sum of the selection ratios so that it is maximum
## on the first axes!!
## End(Not run)
``` |

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