# Factorial Analysis of the Specialization in Habitat Selection Studies. Unpublished Work of James Dunn (University of Arkansas) `dunnfa` performs a factorial decomposition of the Mahalanobis distances in habitat selection studies (see details). ```r dunnfa(dudi, pr, scannf = TRUE, nf = 2) ## S3 method for class 'dunnfa' print(x, ...) ``` ## Arguments - `dudi`: an object of class `pca` - `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 axes - `x`: an object of class `dunnfa` - ``...``: additional arguments to be passed to the function `print` ## Details This analysis is in essence very similar to the MADIFA (see `?madifa`). The Mahalanobis distances are often used in the context of niche-environment studies (Clark et al. 1993, see the function `mahasuhab`). Each resource unit takes a value on a set of environmental variables. Each environmental variable defines a dimension in a multidimensionnal space, namely the ecological space. A set of points (resource units) describes what is available to the species. For each point, a "utilization weight" measures the intensity of use of the point by the species. The set of points for which the utilization weight is greater than zero defines the "niche". 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. James Dunn (formerly University of Arkansas) proposed the analysis programmed in the function `dunnfa`, as an alternative to the MADIFA (unpublished results). This analysis is closely related to both the ENFA (Ecological niche factor analysis, Hirzel et al. 2002) and the MADIFA. The analysis proposed by James Dunn searches, in the multidimensional space defined by environmental variables, synthesis variables which maximise the ratio (variance of the scores of available resource units) / (variance of the scores of used resource units). This ratio is sometimes called "specialization" in the ecological literature (Hirzel et al. 2002). It is therefore very similar to the ENFA (which also maximises the specialization), except that the factorial axes returned by this analysis are not required to be *orthogonal to the marginality axis*. James Dunn demonstrated that this analysis also partitions the Mahalanobis distances into uncorrelated axes, which makes it similar to the MADIFA (the difference is that the MADIFA maximises the mean squared Mahalanobis distances on the first axes, whereas the DUNNFA maximises the specialization on the first axes). Therefore, as for the MADIFA, the DUNNFA can be used to build reduced rank habitat suitability map. Note that although this analysis could theoretically be used with all kinds of variables, it it currently implemented only for numeric variables. ## Returns `dunnfa` returns a list of class `dunnfa` containing the following components: - **call**: original call. - **tab**: a data frame with n rows and p columns (original data frame centered by column for the uniform weighting). - **pr**: a vector of length n containing the number of points in each pixel of the map. - **nf**: the number of kept axes. - **eig**: a vector with all the eigenvalues of the analysis. - **liA**: row coordinates (centering on the centroid of the cloud of available points), data frame with n rows and nf columns. - **liU**: row coordinates (centering on the centroid of the cloud of available points), data frame with p rows and nf columns. - **mahasu**: a vector of length n containing the reduced-rank squared Mahalanobis distances for the n units. - **co**: column (environmental variables) coordinates, a data frame with p rows and nf columns - **cor**: the correlation between the DUNNFA axes and the original variable ## 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. Hirzel, A.H., Hausser, J., Chessel, D. & Perrin, N. (2002) Ecological-niche factor analysis: How to compute habitat-suitability maps without absence data? **Ecology**, 83 , 2027--2036. 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. ## Author(s) Clement Calenge clement.calenge@ofb.gouv.fr ## Note This analysis was developed by James Dunn during an e-mail discussion on the MADIFA, and is still unpublished work. Implemented in adehabitatHS with his autorization. ## See Also `madifa`, `enfa` and `gnesfa` for related methods. `mahasuhab` for details about the Mahalanobis distances. ## Examples ```r ## Not run: data(bauges) map <- bauges$map locs <- bauges$loc ## We prepare the data for the analysis tab <- slot(map, "data") pr <- slot(count.points(locs, map), "data")[,1] ## We then perform the PCA before the analysis pc <- dudi.pca(tab, scannf = FALSE) (dun <- dunnfa(pc, pr, nf=2, scannf = FALSE)) ## We should keep one axis: barplot(dun$eig) ## The correlation of the variables with the first two axes: s.arrow(dun$cor) ## A factorial map of the niche (centering on the available points) scatterniche(dun$liA, dun$pr, pts=TRUE) ## a map of the reduced rank Mahalanobis distances ## (here, with one axis) dun2 <- dunnfa(pc, pr, nf=1, scannf = FALSE) df <- data.frame(MD=dun2$mahasu) coordinates(df) <- coordinates(map) gridded(df) <- TRUE image(df) ## Compute the specialization on the row scores of ## the analysis: apply(dun$liA, 2, function(x) { varav <- sum((x - mean(x))^2) / length(x) meanus <- sum(dun$pr*x)/sum(dun$pr) varus <- sum(dun$pr * (x - meanus )^2)/sum(dun$pr) return(varav/varus) }) ## The eigenvalues: dun$eig ## End(Not run) ```

dunnfa function