ANALYZING SPACE-USE OAT A WITH A HARMONIC MEAN ESTIMATOR IN SAS R

Transcription

1 ANALYZING SPACE-USE OAT A WITH A HARMONIC MEAN ESTIMATOR IN SAS R larry J. Layne and Paul F. Steblein Syracuse University ABSTRACT Examinat~on of the utilization of geographic space is of importance to such disciplines as ecology. social science. and policy and planning. Characterizing the intensity of space use fram point data is central to this process. A harmonic mean estimator is one technique Ior estimating space-use, and is less biased than other estimators which use arithmetic means. A series of SAS programs have been developed which read geographic locations as Cartesian coordinates, identifies the coordinates for the harmonic-mean, and projects a three-dimensional representation of the intensity of use (from the first areal moments). Application of the programs are demonstrated using locational data from research projects on bighorn sheep and broad-winged hawks. Utility programs were also developed for managing data prior to the analysis. and exporting harmonic mean results to other systems for further spatial and statistical analyses, INTRODUCTION In ecology. the activity centers and boundaries estimated from a set of coordinate loci are useful for est1mating animal activity areas such as home range. territory. foraging. mating. and corridor areas. SUch information is useful for comparing intensity of land use for groups, such as between sexes, comparing among discrete groups across a specified geographic area, or even among species. Evaluation of intensity of use may lead to further insights in how an antmal utilizes the area within which it lives. both among different habitats and through time. All of this information may be derived merely from a set of locational observations of an animal and a reasonable estimate of the features found within the area frequented by.the animal. For the harmonic mean measure,to yield valid estimates of animal activity areas, several assumptions should be met by the data. The first of these assumptions is conformation of the data to a bivariate normal distribution~ which is not robust for the model. Another necessary assumption that con orms to parametric statistical estimators is the independence of observations. When this assumption is not met. variances may be underestimated and result in biased confidence intervals. Lastly, loci coordinates should be measured without error, although the amount of error encountered is dependent on the resolution of the grid system and data collection techniques used. Despite these limitations of the model. the harmonic mean measure of animal activity provides a better esttmate than other models developed for analysis of animal activity (Dixon and Chapman 1980). The objective of this paper is to present a series of programs for analyzing spatial use data using the harmonic mean estimator. These programs estimate the harmonic mean of spatial activity and provide estimates of area included within the boundaries d rived from the estimate of dispersion of the loci. Graphic visualizations are provided to depict the distribution of the data loci. the overlay grid used to calculate the inverse first areal moments. isopleths which comprise the boundaries of the loci under consideration. and 3-dimensional representation of the overlay grid and inverse first areal moments. In addition. several utility programs were developed for managing spatially referenced point data and exporting the results of the harmonic mean analysis in.a format appropriate for further analysis with other software. DESCRIPTION OF PROGRAMS DATA ENTRY AND MANIPULATION The data can be introduced to the harmonic program from a variety of sources. The harmonic program requires the X and Y coordinates (Cartesian coordinates) in a permanent SAS data set <HARMONIC.LOCI}. Full-screen data entry templates were created with FSEDIT for directly entering X and Y coordinates. and 1349

2 information associated with the location. Text files with the same type of data can also be read into a permanent SAS dataset using a standard input statement. The data set created from these operations can be used directly in the harmonic program or subset for a group of observations. Location coordinates (or activity loci) often need to be estimated from remote positions. Locations of free-ranging animals are often triangulated from a pair of known positions with the aid of a compass and radio telemetry equipment. Data entry screens were prepared for coordinates of the reference positions. and for the directional information on the animal; Because of varying env-ironmental conditions, equipment error, and individual bias. there is a calculable error associated with triangulation estimates. A program was written that calculates the coordinates for the animal location and an error polygon. The permanent SAS data set that contains the loci for the harmonic program (HARMONIC.LOCI) should only contain the group of observations that are to be analyzed. All observations will be included in some cases, whereas other situations require subsets of observations for intergroup comparisons. The infonmation that is recorded with the activity loci can be utilized for subsetting the observations. DATA ANALYSIS Estimation of the intensity of space use is accomplished with several steps. all of which originate from a matrix of the location data. The first step is to calculate an overlay grid that includes all of the location data points. This is performed by taking the difference of the maximum and minimum abscissa values and the difference of the maximum and minimum ordinate values. The shorter of these two differences is then divided by 10 to form the grid interval. The corners of the grid are then calculated using combinations of the maxima and minima of tlte ordinat~ and.. abscj.ss.a values. The grid is then laid over the location loci. Inverse first areal moments are calculated for each grid intersection- by estimating the harmonic mean of the distances from the intersection to all location loci. The harmonic mean center is the location of the maximum value of the inverse first areal moment. Other peaks may exist on the resulting surface of the grid intersections and inverse first areal moments. The location of these peaks are also saved. Isolines. the lines which comprise boundaries of the location loci by including a specified percentage of the loci points with the harmonic mean center as the mean, are estimated as the confidence ellipses of the principle and minor axes (Sokal and Rohlf 1981). Isoplet hs, lines formed as the average distance from the isoline to the location loci. are estimated to determine the areal patterns of activity. The area within this isopleth boundary is then estimated using Simpson's rule. Further analyses can be conducted to elucidate how other factors may relate to the distribution of activity in geographic space. Other types of data are cofijlionly collected with the loci or are available for the entire geographic area. The association of these factors with distribution of activity can be analyzed by statistical analysis or spatial coincidence. The simplest approach would be to statistically characterize the original loci by data collected at the same locations. If geographically continuous data is available for the study area. the data can be extr'8cted for the coordinates of the overlay grid and utilized in a statistical model with the first areal moments. Finally. the overlay grid coordinates and first areal moments can be ported to a geographic information system for further spatial analyses. DATA VISUALIZATION Several graphical outputs 1'rere developed to aid in the interpretation of th~ activity data and harmonic analysis. A scattergram of the activity loci is produced. and geographically referenced to the intersections of the overlay grid. The surface of the first areal moments and inverse first areal moments can be viewed with 3-dimensional plots (G3D) and contour plots (GCONTOUR). Both types of plots aid in identifying and interpreting activity centers. Isopleth plots provide a standardized approach for identifying an area of activity. Criteria commonly employed include 95% and 75% isopleths. which respectively include 95% and 75% of the observations. The area enclosed by the isopleth can be totaled for intergroup comparisons or analyzed tor geographic makeup. EXAMPLE APPLICATIONS The programs that were developed in this manuscript were tested with two widely disparate sets of animal location data. Free-ranging bighorn sheep were monitored with radio-telemetry (Layne 1987), and geographic location determined from visual observations, Universal Transmercator (UTM) coordinates were recorded from USGS 'ropographic maps. Figure 1 shows the visual results of the analysis conducted on locational information for a herd of ewes. Note the two centers of activity (or peaks) on the 3-dimensional plot of the inverse first areal moments. The spatial activity of broad-winged hawks was also monitored with radiotelemetry equipment (Steblein 1989). However, the geographic location of the 1350

3 hawks was largely determined by triangulation. Harmonic mean analysis was conducted on location data from a male hawk during the breeding season~ The contour and 3-dimensional plots of'the inverse areal moments (Figure 2) indicate a dominant peak and very minor sub-peak. DISCUSSION The emphasis in this manuscript has been with application of the harmonic mean estimator to the spatial distribution of vertebrate activity. The original description of the technique was by Nett (1966) and seems appropriate for characterizing areal distributions in many other disciplines. LITERATURE CITED Dixon. K. R. and J. A. Chapman Harmonic mean measure of animal activity areas. Ecology 61: Layne. L. J Habitat selection and sexual segregation of in Rocky Mountain bighorn sheep {Ovis canadensis) in Custer State Pa~South Dakota. M.S. Thesis. South Dakota,State Univ. Brookings. 120 pp. Neft. D. S Statistical analysis for areal distributions. Monograph Series No.2. Regional Science Research Institute. Philadelphia. Sokal. R. R.. and F. J. Rohlf Biometry: the principles and practice of statistics in biological research. W.H. Freeman and Co.. San Francisco. 859 pp. Steblein. P. F Foraging ecology of broad-winged hawks (Buteo plattpterus): test of a patch select~an mode w1th central place constraints. Ph.D. Dissertation. College of Environmental Science and Forestry, State Univ. of New York. Syracuse. APPENDIX The formulae for calculating the harmonic mean estimator (Sakal and Rohlf 1981) and inverse first areal moment (Dixon and Chapman 1980) are listed below: Harmonic Mean Estimator H n Inverse First Areal Moment -1/r:f -.,j" -1 p " p x=l r Jx where H is the harmonic mean estimator. n is the number of observations, and X is the variate; and P is the number of loci. r is the distance between the grid intersections and loci. j is the grid coordinate. and x is the loci number. SAS is a registered trademark of SAS Institute Inc. Cary. NC. 1351

4 FI<P 6<335.2 Ftgure L Grid int.j"".eet1cn~ OtlQttl!ld till X-V and fir-at li('l!isl Moment Z GX1S for bl9hom sheep (oms ca.fl4<tensis). plotted as y < Figure 2a. 4fr694;} "1 COil tour' plot o.f inverse first lireal moment fer Broad-winged HaWk x (Buteopl4typteru.s). 1352

5 Figut'/:'l 21'1. Gr"i<l illter,ectlqna plotted &6 X-Val'll;! fif"st areal I\'j(Jcnent plotte<l 86 Z axis for Broad-winged Hawk (Buteo pl<ttypterus) "" t III <II Jl... "" ~ 562:) 'I iii'\ c ~ "lii" ' WI + O' + +" li:+ it ~ 562MO e ' +..l " SHIOO ' +' +++ t ;H ,"~~rn,,~rn~~rn~~~T"~~~'T~~~~C'~~~T"~~~rr~~~", l!OO Ae70000 L0fl91tud1nal \lalv/:,:$ Figure 2c. Grid overloy, location loci, an.;! h1jrmonic mean center plotted oos Universal Transmercator coordinates for Brood-winged Hawk (Buko pletypterus). 1353