# Estimation of Kernel Home-Range The function `kernelUD` estimates the UD of one or several animals. `plotLSCV` allows to explore the results of the least-square cross-validation algorithm used to find the best smoothing value. `image` allows a graphical display of the estimates. `getvolumeUD` and `kernel.area` provide utilities for home range and home-range size estimation. `getverticeshr` stores the home range contour as an object of class `SpatialPolygonsDataFrame` (package sp), with one row per animal. `estUDm2spixdf` can be used to convert the result into an object of class `SpatialPixelsDataFrame` `as.data.frame.estUD` can be used to convert an object of class `estUD` as a data frame. ```r kernelUD(xy, h = "href", grid = 60, same4all = FALSE, hlim = c(0.1, 1.5), kern = c("bivnorm", "epa"), extent = 1, boundary = NULL) ## S3 method for class 'estUDm' print(x, ...) ## S3 method for class 'estUD' image(x, ...) ## S3 method for class 'estUDm' image(x, ...) ## S3 method for class 'estUD' as.data.frame(x, row.names, optional, ...) plotLSCV(x) getvolumeUD(x, standardize = FALSE) kernel.area(x, percent = seq(20, 95, by = 5), unin = c("m", "km"), unout = c("ha", "km2", "m2"), standardize = FALSE) estUDm2spixdf(x) ``` ## Arguments - `xy`: An object inheriting the class `SpatialPoints` containing the x and y relocations of the animal. If `xy` inherits the class `SpatialPointsDataFrame`, it should contain only one column (factor) corresponding to the identity of the animals for each relocation. - `h`: a character string or a number. If `h` is set to `"href"`, the ad hoc method is used for the smoothing parameter (see details). If `h` is set to `"LSCV"`, the least-square cross validation method is used. Note that `"LSCV"` is not available if `kern = "epa"`. Alternatively, `h` may be set to any given numeric value - `grid`: a number giving the size of the grid on which the UD should be estimated. Alternatively, this parameter may be an object inheriting the class `SpatialPixels`, that will be used for all animals. For the function `kernelUD`, it may in addition be a list of objects of class `SpatialPixels`, with named elements corresponding to each level of the factor id. - `hlim`: a numeric vector of length two. If `h = "LSCV"`, the function minimizes the cross-validation criterion for values of h ranging from `hlim[1]*href` to `hlim[2]*href`, where `href` is the smoothing parameter computed with the ad hoc method (see below) - `kern`: a character string. If `"bivnorm"`, a bivariate normal kernel is used. If `"epa"`, an Epanechnikov kernel is used. - `extent`: a value controlling the extent of the grid used for the estimation (the extent of the grid on the abscissa is equal to `(min(abscissa.relocations) + extent *diff(range(abscissa.relocations)))`, and similarly for the ordinate). - `same4all`: logical. If `TRUE`, the same grid is used for all animals. If `FALSE`, one grid per animal is used. Note that when `same4all = TRUE`, the grid used for the estimation is calculated by the function (so that the parameter `grid` cannot be a `SpatialPixels` object). - `boundary`: If, not `NULL`, an object inheriting the class `SpatialLines` defining a barrier that cannot be crossed by the animals. There are constraints on the shape of the barrier that depend on the smoothing parameter `h` (***see details***) - `x`: an object of class `estUD` (UD for one animal) or `estUDm` (UD for several animals). For the function `estUDm2spixdf`, an object of class `estUDm` only. For the function `as.data.frame.estUD`, an object of class `estUD` only. - `percent`: for `kernel.area`, a vector of percentage levels for home-range size estimation. For `getverticeshr`, a single value giving the percentage level for home-range estimation. - `standardize`: a logical value indicating whether the UD should be standardized over the area of interest, so that the volume under the UD and *over the area* is equal to 1. - `unin`: the units of the relocations coordinates. Either `"m"` for meters (default) or `"km"` for kilometers - `unout`: the units of the output areas. Either `"m2"` for square meters, `"km2"` for square kilometers or `"ha"` for hectares (default) - `row.names`: unused argument here - `optional`: unused argument here - ``...``: additionnal parameters to be passed to the generic functions `print` and `image` ## Details The Utilization Distribution (UD) is the bivariate function giving the probability density that an animal is found at a point according to its geographical coordinates. Using this model, one can define the home range as the minimum area in which an animal has some specified probability of being located. The functions used here correspond to the approach described in Worton (1995). The kernel method has been recommended by many authors for the estimation of the utilization distribution (e.g. Worton, 1989, 1995). The default method for the estimation of the smoothing parameter is the **ad hoc** method, i.e. for a bivariate normal kernel $$ h = \sigma n^{- \frac{1}{6}}h = Sigma*n^(-1/6) $$ where $$ \sigma^2 = 0.5 (var(x)+var(y))Sigma^2 =0.5*(var(x)+var(y)) $$ which supposes that the UD is bivariate normal. If an Epanechnikov kernel is used, this value is multiplied by 1.77 (Silverman, 1986, p. 86). Alternatively, the smoothing parameter h may be computed by Least Square Cross Validation (LSCV). The estimated value then minimizes the Mean Integrated Square Error (MISE), i.e. the difference in volume between the true UD and the estimated UD. Note that the cross-validation criterion cannot be minimized in some cases. According to Seaman and Powell (1998) **"This is a difficult problem that has not been worked out by statistical theoreticians, so no definitive response is available at this time"** (see Seaman and Powell, 1998 for further details and tricky solutions). `plotLSCV` allows to have a diagnostic of the success of minimization of the cross validation criterion (i.e. to know whether the minimum of the CV criterion occurs within the scanned range). Finally, the UD is then estimated over a grid. The default kernel is the bivariate normal kernel, but the Epanechnikov kernel, which requires less computer time is also available for the estimation of the UD. The function `getvolumeUD` modifies the UD component of the object passed as argument: that the pixel values of the resulting object are equal to the percentage of the smallest home range containing this pixel. This function is used in the function `kernel.area`, to compute the home-range size. Note, that the function `plot.hrsize` (see the help page of this function) can be used to display the home-range size estimated at various levels. The parameter `boundary` allows to define a barrier that cannot be crossed by the animals. When this parameter is set, the method described by Benhamou and Cornelis (2010) for correcting boundary biases is used. The boundary can possibly be defined by several nonconnected lines, each one being built by several connected segments. Note that there are constraints on these segments (not all kinds of boundary can be defined): (i) each segment length should at least be equal to `3*h` (the size of "internal lane" according to the terminology of Benhamou and Cornelis), (ii) the angle between two line segments should be greater that `pi/2` or lower that `-pi/2`. The UD of all the pixels located within a band defined by the boundary and with a width equal to `6*h` ("external lane") is set to zero. ## Returns The function `kernelUD` returns either: (i) an object belonging to the S4 class `estUD` (see `?estUD-class`) when the object `xy` passed as argument contains the relocations of only one animal (i.e., belong to the class `SpatialPoints`), or (ii) a list of elements of class `estUD` when the object `xy` passed as argument contains the relocations of several animals (i.e., belong to the class `SpatialPointsDataFrame`). The function `getvolumeUD` returns an object of the same class as the object passed as argument (`estUD` or `estUDm`). `kernel.area` returns a data frame of subclass `hrsize`, with one column per animal and one row per level of estimation of the home range. `getverticeshr` returns an object of class `SpatialPolygonsDataFrame`. `estUDm2spixdf` returns an object of class `SpatialPixelsDataFrame`. ## References Silverman, B.W. (1986) **Density estimation for statistics and data analysis**. London: Chapman and Hall. Worton, B.J. (1989) Kernel methods for estimating the utilization distribution in home-range studies. **Ecology**, 70 , 164--168. Worton, B.J. (1995) Using Monte Carlo simulation to evaluate kernel-based home range estimators. **Journal of Wildlife Management**, 59 ,794--800. Seaman, D.E. and Powell, R.A. (1998) **Kernel home range estimation program (kernelhr)**. Documentation of the program. Benhamou, S. and Cornelis, D. (2010) Incorporating Movement Behavior and Barriers to Improve Biological Relevance of Kernel Home Range Space Use Estimates. **Journal of Wildlife Management**, 74 , 1353--1360. ## Author(s) Clement Calenge clement.calenge@ofb.gouv.fr ## See Also `mcp` for help on the function `plot.hrsize`. ## Examples ```r ## Load the data data(puechabonsp) loc <- puechabonsp$relocs ## have a look at the data head(as.data.frame(loc)) ## the first column of this data frame is the ID ## Estimation of UD for the four animals (ud <- kernelUD(loc[,1])) ## The UD of the four animals image(ud) ## Calculation of the 95 percent home range ver <- getverticeshr(ud, 95) ## and display on an elevation map: elev <- puechabonsp$map image(elev, 1) plot(ver, add=TRUE, col=rainbow(4)) legend(699000, 3165000, legend = names(ud), fill = rainbow(4)) ## Example of estimation using LSCV udbis <- kernelUD(loc[,1], h = "LSCV") image(udbis) ## Compare the estimation with ad hoc and LSCV method ## for the smoothing parameter (cuicui1 <- kernel.area(ud)) ## ad hoc plot(cuicui1) (cuicui2 <- kernel.area(udbis)) ## LSCV plot(cuicui2) ## Diagnostic of the cross-validation plotLSCV(udbis) ## Use of the same4all argument: the same grid ## is used for all animals ## BTW, we indicate a grid with a fine resolution: udbis <- kernelUD(loc[,1], same4all = TRUE, grid = 100) image(udbis) ## Estimation of the UD on a map ## (e.g. for subsequent analyses on habitat selection) ## Measures the UD in each pixel of the map udbis <- kernelUD(loc[,1], grid = elev) image(udbis) ########################################## ## ## Estimating the UD with the presence of a barrier ## The boars are located on the plateau of Puechabon (near ## Montpellier, France), and their movements are limited by the ## Herault river. ## We first map the elevation: image(elev) ## Then, we used the function locator() to identify the limits of the ## segments of this barrier. BEWARE! The boundary should satisfy the two ## constraints: (i) segment length > 3*h, (ii) no angle lower than pi/2 ## between successive segments. We choose a smoothing parameter of 100 ## m, so that no segment length should be less than 300 m. ## Because the resolution of the map is 100 m, this means that no ## segment should cover less than 3 pixels. We have used the function ## locator() to digitize this barrier and then the function dput to ## have the following limits: bound <- structure(list(x = c(701751.385381925, 701019.24105475, 700739.303517889, 700071.760160759, 699522.651915378, 698887.40904327, 698510.570051342, 698262.932999504, 697843.026694212, 698058.363261028), y = c(3161824.03387414, 3161824.03387414, 3161446.96718494, 3161770.16720425, 3161479.28718687, 3161231.50050539, 3161037.5804938, 3160294.22044937, 3159389.26039528, 3157482.3802813)), .Names = c("x", "y")) lines(bound, lwd=3) ## We convert bound to SpatialLines: bound <- do.call("cbind",bound) Slo1 <- Line(bound) Sli1 <- Lines(list(Slo1), ID="frontier1") barrier <- SpatialLines(list(Sli1)) ## estimation of the UD kud <- kernelUD(loc[,1], h=100, grid=100, boundary=barrier) ## Result: image(kud) ## Have a closer look to Calou: kud2 <- kud[[2]] image(kud2, col=grey(seq(1,0,length=15))) title(main="Home range of Calou") points(loc[slot(loc,"data")[,1]=="Calou",], pch=3, col="blue") plot(getverticeshr(kud2, 95), add=TRUE, lwd=2) lines(barrier, col="red", lwd=3) ```

kernelUD function