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OCCASIONAL PAPERS

of the

MUSEUM OF NATURAL HISTORY The University of Kansas Lawrence, Kansas

NUMBER 116, PAGES 1-18 24 MAY 1985

ROBUST STATISTICS FOR SPATIAL ANALYSIS: THE BIVARIATE NORMAL HOME RAI^GE MODEL APPLIED TO SYNTOPIC POPULATIONS OF TWO SPECIES OF GROUND SQUIRR^LS^^f^

By \/0/^ '^Vv.^^4

J. W. KoEPPL and R. S. Hoffman ^ ^^Ster-

INTRODUCTION

The bivariate normal home range models of Jennrich and Turner (1969), Mazurkiewicz(1969. 1971) and Koeppl et al. (1975) differ in only minor respects, and are useful in analyzing spatial data of diverse kinds of organisms (Adams, 1976, Hawes, 1977; Inglis et al., 1979; Johns and Armitage, 1979; O'Farrell, 1978, 1980; Wasserman. 1980; Zach and Falls, 1978). The models employ the center of activity (Hayne, 1949) which IS not robust because it is highly sensitive to outliers, bad data, or non-normal distributions (Koeppl et al. 1985). Yet failure to minimize the effect of outliers leads to erroneous conclusions about the location, shape, orientation and size of home ranges. Moreover, these models use the activity center in the computation of variances and covariances which further exacerbates these problems. The latter two statistics are used in the Cauchy-Schwarz inequality relationships to determine eigen-values and -vectors for calculating lengths and orientations of major and minor axes of the bivariate data scatter, which ultimately lead to determination of the useful home range parameters of size, shape and orientation in space. All the above statistics can also be used in the multivariate statistical technique of principal components analysis. Hawkins (1974) used the sensitivity of PC analysis to outliers to detect errors in multivariate data bases, thus supportmg the notion of non-robustness of the bivariate normal home range model.

' Museum of Natural History and Department of Systematics and Ecology, The University of Kansas, Lawrence, Kansas 66045

2 OCCASIONAL PAPERS MUSEUM OF NATURAL HISTORY

Because of the evident usefulness but non-robust nature of existing bivariate normal home range models, we propose several methods for making the computation of bivariate normal home range models more robust. We use methods for identifying and dealing with bivariate outliers which lead to robust estimates of the home range parameters mentioned previously. Initially we illustrate the methods on simulated bivariate data to which outliers have been adjoined, and on locational data obtained from vernal observations of an adult Columbian ground squirrel (Spermophilus columbianus). Then, having determined the utility of the algorithms, we apply one of them to a spatial analysis of two syntopic ground squirrel species.

SIMULATION STUDY

Computational Methods

Data generation. — Ninety-five bivariate normal location coordinates were generated in which the standard deviation was 1.0 value on the abscissa, while those on the ordinate were 0.5; the mean in each case was near 0.0. To these 95 coordinates, five random uniform coordinates were adjoined, but in the interval - 10.0 to -f 10.0 for both axes. Hence, 95 percent of the simulated data are from a bivariate normal distribution with an eccentricity of 50 percent, centered near the origin, and 5 percent of the data are from a square random uniform distribution, 10 units on a side. Robust bivariate algorithms.— Four ditferent types of robust algorithms have been tested; all use the methods described in Koeppl et al. (1975). with modifications suggested in Koeppl et al. (1977) and Madden and Marcus (1978).

Initially we tried a one-step bivariate trim algorithm (BVTIS) consist- ing of three steps for which (1) the elliptically standardized distance

De = { [X^ - x//S,^l + [( Y^ - yT^/%J ^ ''^

is computed for each observation (Koeppl et al., 1975) where Xj and Y

are the transformed observation coordinates which display zero covaria- tion, while X and y , S and S are their respective means and

standard deviations; (2) percentage of the observations (a) with the largest Dg are trimmed from the data set; (3) home range parameters are then

computed on the remaining observations.

A simple modification of BVTIS produced an iterative bivariate trim (BVTI): Steps 1 and 2 are repeated, with the observation having the largest Dg deleted until a percent observations have been trimmed. Finally, step 3

is implemented.

The third robust algorithm (IWILKS) is also iterative and stems from a suggestion of Wilks (1963), who proposed using a ratio (Rj) of determi- nants for detecting single outliers where R- = A_-/A, for A, the sum of

HOME RANGES OF TWO GROUND SQUIRRELS 3

products matrix, and A_;. the same matrix but with the i-th observation

deleted. In this algorithm we use the equation above but A represents the area of the home range for n observations, while A _ : represents the area

of home range for n- 1 observations. For each iteration, the observation having the smallest R; is deleted until o; percent observations have been

trimmed from the sample. Finally, home range statistics are computed for the remaining observations.

The fourth procedure (KKSA) entails computing a weighting factor

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coordinate of the ordinate. The outlier with the smallest weight is deleted from the data set, and home range parameters are then computed for each iteration.

Five different variations of the fourth procedure were then used, each employing robust estimates of location adapted from Andrews et al. (1972). These variations are: "10%", the simple symmetrical trim; the restricted adaptive trimmed mean, "JBT"; the M-Estimates, "AMT" which employed sine function weighting, and "17A'" which used inde- pendent piecewise weighting; and "STl", the multiply-skipped mean, with (5K, 2 deleted).

We chose the algorithms described by Koeppl, et al. (1985) because they offered a variety of approaches to the problem of contaminated spatial data, but selection of the particular algorithms is partly intuitive.

Results and Simulations

Values for the home range parameters computed on a theoretically true bivariate normal distribution having a major axis (x) of 1.0 and a minor axis (y) of 0.5, should approximate (0.000, 0.000) for the activity center coordinates; and show an eccentricity (minor axis/major axis) of 50%, slope of major axis of 0.000, and area of the 95 percent confidence ellipse of 9.4248 (Table 1, bottom). Using traditional methods on the bivariate

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random normal distribution with a 5 percent contamination results in values of 0.2813, 0.2045, 49.00%, 0.4255, and 30.273, respectively (Table 1, top). In this case, only the eccentricity of the home range approximates the parametric value; the other sample values are consider- ably inflated. Results of the application of the robust bivariate algorithms revealed that they are less sensitive to the presence of the contaminated observations, even when the severity of trimming is twice the percentage of nominal contaminated values (i.e., at a 10 percent trim when 5 percent of the non-outliers in the data are trimmed) (Table 1). Robust analysis of home range data of an adult S. columbianus.— The effectiveness of the robust algorithms on home range data of a S. columbianus (Fig. 1) was determined by comparing them with home range parameters computed on the full data set by traditional computations designated as BVN (Table 2). Activity centers for BVN compared to the robust estimates generally differ more than the robust estimates differ among themselves; the same is true for eccentricities. Slopes of the major axes computed using BVN differ appreciably from the same statistics of the robust estimates, and of these, BVTIS differs considerably from the remaining robust estimates. For the home range areas, differences

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Fig. 1. Vernal spatial data of an adult Columbian ground squirrel (Spermophilus columbianus) used to test robust bivariate normal algorithms. The ellipse having the solid lines represents the 95% confidence ellipse determined for the full data set using the traditional bivariate normal model (BVN). The ellipse having dashed lines represents the 95% confidence ellipse determined for the same individual, but employing a 5% iterative bivariate normal trim (BVTI).

OCCASIONAL PAPERS MUSEUM OF NATURAL HISTORY

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HOME RANGES OF TWO GROUND SQUIRRELS 7

between 5 and 10 percent robust trims are less than the same percent of trim between the full data set and the 5 percent trimmed robust estimates. All of the above indicate the superiority of the robust methods over the traditional methods.

Discussion: Robust Bivariate Normal Models

Robust parameters for bivariate normal home range data initially focused on detecting and dealing with outliers, just as in many robust univariate estimates of location (Andrews et al., 1972). It is a logical extension of univariate analysis to compute robust independent estimates of location for each coordinate array and combine them to provide bivariate location estimates (Gnandesikan. 1977). This procedure pro- duces a bias in statistics computed for outliers lying along the reference axes of the grid, as has been shown for the sensitivity surfaces of the activity centers, but not for the arithmetic mean (Koeppl et al. 1985). Hence, although the traditionally computed activity center lacks robust- ness, it does exhibit the desirable property of invariance to rotation (Koeppl et al. 1985) which we wish to preserve. Moreover, because other statistics of the bivariate home range model are based on the activity center, they lack robustness and are invariant to rotation of the data when based on the traditional activity center, but are robust and vary with rotation of the data when the activity center is robust. Variation with rotation is due to independent consideration of the x and y coordinates when coordinates are not equally weighted, so that outliers are recognized by extreme X or Y coordinates, but not by intermediate values of either (Koeppl et al. 1985). To minimize the influence of outliers in a truly bivariate sense, allowing for the possibility of non-zero covariation, robust statistics must be computed on the data bivariately. To accomplish this, it is necessary to use the traditional non-robust mean. This dilemma has been solved by approaching the computation of the home range parameters iteratively, eliminating an outlier at a time. In the first few iterations, when the covariation is not so important, the more blatant outliers are trimmed. Later iterations approach the "core'' of the distribution where covariation from one iteration to the next is more stable (if significant covariation exists) and where observations not representative of the principal di.stribu- tion are more apparent. Final computation of the home range parameters is thus less aff'ected, even when lesser outliers remain.

Devlin et al. (1975:537) suggested three different types of metrics upon which to trim observations: (1) hyperbolic, because it is the shape suggested by the influence function (= sensitivity surface); (2) elliptic, which Devlin (1975:537) states can be the squared distance of the i-th observation [(Y: - m ) V~ (Y:-m )], which is related to the formula of Koeppl et al. (1975:87 and this paper) or the squared distance [(Xi-x_j)^A_j(Xi-Xj)] discussed by Rohlf (1975:98); (3) rectangular metric, which is a compromise between the hyperbolic and elliptical metrics. In the present paper we use the elliptical distance of Koeppl et al.

8 OCCASIONAL PAPERS MUSEUM OF NATURAL HISTORY

(1975). The distances so computed allow consideration of bivariate as well as multivariately derived distances in a univariate array. We take advan- tage of this to compute the algorithms of KKSA, which begin with robust estimates of location cxT, JBT, AMT, 17A, 5T1 (see Computational Methods, and Koeppl et al. 1985). These distances, when computed for each observation, allow consideration of the data on a univariate scale, corrected, or ostensibly influenced by the outlier weights. Outliers accumulate in the tails of the array and many were trimmed as discussed previously.

Metzgar (1972) and Koeppl et al., (1975) have shown empirically that a large size (n>20) is preferred for the bivariate normal model. Use of sample sizes of less than 20 after trimming make it impossible to determine the covariation in the sample with reasonable confidence.

The question arises as to which of the algorithms tested is the "best" one to use. This question has no simple answer, because depending on the data (how they are collected, the species concerned, and the question(s) asked) any method of home range analysis can be adequate. Some methods yield satisfactory results to a number of problems, yet none is the "best" is all situations. When selecting home range methods, one must try to choose the one adequate for the task at hand. In this light, the home range algorithms presented here are appropriate when the data at the core of the distribution are approximately bivariate normally distributed, when the sample size is large (n = 20), and when one would like to minimize the effect of outliers in the data. Shifts of home range (Cooper, 1978) and occasional forays (Burt, 1943) fall in the latter category. All the robust estimates seem to yield more meaningful results than the traditional computation for the bivariate normal model under these circumstances. BVTIS estimated home range location, eccentricity and size moderately well, but is less good at providing consistent estimates of the slopes of the ellipse axes. The KKSA methods consistently provide good estimates of axis slope across all home range parameters. Although the KKSA methods begin with a robust estimate of location, they provide no appreciable advantage over the estimates BVTI and IWILKS, which do not. The latter two estimates are, moreover, simpler and more economical to compute. We used the BVTI method, instead of the others discussed here, when we extended our analysis to actual ground squirrel data (see below).

APPLICATION OF ROBUST STATISTICS

Spatial Analysis of Syntopic Populations of Ground Squirrels

During the last half-century, Spermophilus columhianus has extended its range southward and eastward in certain areas of the northwestern United States. East of Glacier National Park, S. hchardsonii has retreated before this advance and the ranges of the two species have been shown to exhibit both parapatry and sympatry (Hoff"mann et al., 1969; Howell, 1938; Michener, 1977). S. columhianus also displaces 5. elegans, the sibling species and southern ecological equivalent of S. hchardsonii , in

HOME RANGES OF TWO GROUND SQUIRRELS 9

southwestern Montana (Koeppl et al.. 1978; Koeppl and Hoffmann, 1981; Nadler et al., 1971 ; Robinson and Hoffmann, 1975). We conducted spatial analysis of syntopic populations (populations inhabiting the same locality) of S. columhianus and S. elegans near Harrison, Montana. Our purpose was to learn how the available space was used within and between these species.

Materials and Methods

Description of the study area.— The study was conducted in a pasture at the Bohrman Ranch, 2 mi. N., 2'/: mi. E. of Harrison, Madison County, Montana, at an elevation of 1463-1478 m. Duration of the study was 157 days, from 20 May to 18 August 1971, and 5 April to 19 June 1972. Data reported here only cover the latter vernal period, about two weeks subsequent to emergence of the first squirrels from hibernation, and just prior to emergence of the young of the year.

Prominent topographic features include three slopes, facing northwest, northeast and southwest respectively, separated by a shallow ravine. All three slopes and the ravine drain into a marshy floodplain of South Willow Creek (Figs. 2, 3A-D).

Three types of substrate are found in the study area. Those on the northwest- and northeast-facing slopes are similar and consist of approxi- mately 25% sand, 50% organic matter and 25% clay. The substrate of the southwest-facing slope, in contrast, is composed largely of sand, gravel.

Fig. 2. View of study area taken in late July; camera facing south. Tree lines in foreground border tributaries of Willow Creek. Dark areas traversing study area indicate drainage sites for irrigation water from hay fields above. The blind from which observations were made is marked by an arrow.

10

OCCASIONAL PAPERS MUSEUM OF NATURAL HISTORY

Fig. 3. View from observation blind taken in late May. A, South view: B. Southwest view; C, West view; D, Northwest view. Also shown are grid stakes.

and many large stones, with lesser amounts of clay and organic matter. The floodplain substrate is largely silt and organic matter.

The northwest- and northeast-facing slopes are more mesic than the southwest slope, and this difference accentuates the differences among the three substrate types. The sandier soil has more effective drainage, as well as being exposed to greater insolation. The marshy floodplain is the most mesic, being saturated with water throughout the period of the study, principally because of spring flooding, but also due to irrigation water runoff from fields above the study area. Photographs of the study area taken in late May (Fig. 3A-D), compared with the photograph taken of the same area in August (Fig. 2), show the relative moisture differences in the study area during the period of above-ground activity of the squirrels. Trapping, handling, and preliminary data collection.— ^quxxroX^ were trapped in National live-traps using peanut butter or rolled oats as bait. Nembutal (sodium pentobarbital in concentrations of 50 mg/ml) injected into the muscles of the thigh in dosages of O.lcc/100.0 g body weight proved satisfactory for anesthetizing squirrels. Species, sex and age (adult or yearling) were recorded. Females were examined for parity (promi-

HOME RANGES OF TWO GROUND SQUIRRELS 1 1

nence of mammae) and lactation by gently squeezing the mammae to express milk.

A/«//:///^?.— Squirrels were ear-tagged and toe-clipped for permanent iden- tification when initially captured. The most useful visual mark entailed sewing a small color-coded tag to the nape of the neck with fine nylon filament. Tags were manufactured by folding pieces of colored tape between strips of clear plastic tape. A thin monel wire at the point of the fold served to strengthen the points of attachment of the filament to the tag. Color stripes were arranged vertically, each color corresponding to an integer. Numbers with consecutive repeating digits (e.g., 33, 100. 166, etc.) were eliminated; hence different combinations often colors taken 1, 2. and 3 at a time supplied 900 unique numbers. Depending on the thickness of skin, tags survived for a few weeks to several months, and were replaced when necessary to routine trapping. Squirrels seemed to suffer no ill effects from the marking, nor did their behavior appear to have been altered.

Locational data collection.— A. grid was established in the study area, using numbered stakes at 10 m intervals. A tent was erected as a blind on the top of the northwest-facing slope where the entire grid could be seen (Figs. 2, 3A-D). Locational data of marked squirrels were obtained by periodically scanning the study area and noting the number and location of each squirrel; binoculars and a spotting scope aided observation. To this observational data we added the trapping locational data to form the basis of locational data for each squirrel used in the analysis. Computational methods. —To determine the effects of increasing degrees of data trimming using the robust bivariate normal home range model, "BVTF', we plotted mean values of estimates of location, shape, orientation in space and size of home range for the two species of ground squirrels. Trim values ranged from 0 to 50%, in 1 % increments. Because of variations in sample size, the values of home range parameters for a specified percent of trim have been interpolated from the two nearest values, and means have been computed from the interpolated values. Actual values plotted consist of: (1) shifts in statistical estimate of squirrel activity centers, determined as the mean distance between the squirrel's activity center computed from the full data set (0% trim), and that computed for each increasing increment of trim, squared and multiplied by 10; (2) variations in shape of the locational data scatter, computed as the mean of the ratio of the minor axis of the data ellipse divided by its major axis multiplied by 100; (3) shifts in orientation of the data ellipse, measured by mean angular degrees of difference between the major axis of the full data ellipse and that axis for each increasing increment of trim; (4) changes in home range size, measured by percent changes in home range size for the full data set compared with sizes computed for increasing increments of trim. Subsequent home range analyses employ a 10% trim unless otherwise noted. Estimates of activity centers computed from the bivariate home range model are robust, and are therefore considered good estimates of location. When sample size is not sufficient for this analysis

12

OCCASIONAL PAPERS MUSEUM OF NATURAL HISTORY

(n< 18). we employ STl (the multiply-skipped mean. Max SK. 2 deleted) to compute activity centers (Andrews et al., 1972; Koeppl et al. 1985).

Results: Further Analysis on Ground Squirrel Data

Effects of trimming on home range parameters. —Figures 4A-D revealed the effects of increasing percentages of bivariate trimming on the four home range parameters. In Figure 4A, the mean shift in location shown for S. columbiamis increases uniformly, while that of S. elegans shows an abrupt change in the 0 to 10% interval before stabilization. The mean shape of home ranges of the two species is markedly elliptic and becomes more so for trimming between 15% to 30%, but this effect diminishes with more severe trimming (Fig. 4B). Orientation of the major axis shows concomitant abrupt shifts with light trimming in S. columbianus before it stabilizes, but the curve for S. elegans stabilizes at the level of a 15% trim (Fig. 4C). For both species the relative decrease in home range size is greatest at about 10% trim (Fig. 4D). These results indicate that a light trim provides the most reasonable estimate of the home range of squirrels; we chose a 10% trim for subsequent analyses.

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HOME RANGES OF TWO GROUND SQUIRRELS

13

Home ranges.— V^hcn calculated by BVTI with 10% trim, the 95% confidence ellipses tor home ranges of three adult male S. columhianus were found to overlap somewhat, but each occupies a different area of the grid (Fig. 5A). Home ranges of adult females (Fig. 5A) also overlapped but were smaller in area; characteristically, several of their home ranges were encompassed by the home range of one or more adult males. Home ranges of yearling male S. columhianus are relatively large (Fig. 5B), but home ranges of yearling females plotted on the same graph (Fig. 5B) more nearly approximate those of aduh females. Home ranges of the three adult female 5. elegans are mostly spatially separate (Fig. 5C). No adult male S. elegans have been observed in the study area. However, two yearling males were observed, and exhibit home ranges of roughly the same size as those of the adult females (Fig. 5D). The home range of one yearling female is the smallest of any S. elegans.

A statistical summary of home range sizes for each species by sex-age class (Table 3) reveals that mean home ranges of male yearling S.

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Fig. 5. Home ranges computed using the BVTI method with a 10% trim. A, adult S. columhianus males (solid) and females (dotted ellip.ses). B. yearling 5. columhianus males (solid) and females (dotted). C, adult S. elegans females (dotted). D, yearling males (solid) and female (dotted).

14 OCCASIONAL PAPERS MUSEUM OF NATURAL HISTORY

Table 3. Results of the Home range analysis for the iterative bivariate normal 10% trim (BVTI) versus the same algorithm with 0% trimming.

			BVTI (10% trim)	BVN (07c trim)
		n	Area (m^)±SE	Area (m2)±SE
S. coliimbianus	Adult males	3	3823 ±938.8	7134 ±1694.4
	Yr. males	3	13651 ±6220.9	18206 ±6737.6
	Adult females	10	1343±241.3	3301 ±920.0
	Yr. females	8	1291 ±252.5	2262 ±537.3
S. elegans*	Yr. males	2	3353 ±1827.0	7134 ±4379.3
	Adult females	3	2 179 ±698.5	6578 ±3797.9
	Yr. females	1	682	1744

* 1 adult female was an S. elegans X 5. richardsonii hybrid

columhiamis are about four times larger than those of adult males, while home ranges of adult and yearling females are approximately one-third the size of those of adult males. A similar, but less extreme, pattern was noted for S. elegans. Adult and yearling females exhibit smaller home ranges than yearling males. For comparison, the statistics for non-robust bivariate normal home range estimates (0% trim) are also shown in Table 3.

Activity centers computed for all individuals observed in the study (Fig. 6) reveal a marked tendency for interspecific segregation, with S. coliimbianus tending to occupy the mesic lower slopes, and S. elegans utilizing the more xeric upper slopes. The line drawn on Fig 6 was eye- fitted and denotes a hypothetical boundary between the two species.

Discussion: Ground Squirrel Spatial Analysis

In his monograph dealing with the daily and seasonal movement of vertebrates. Fitch (1958) noted that home ranges tend to be random centralized or random uniform. The two species of ground squirrels studied here clearly follow the former pattern; hence a bivariate normal home range model is a good choice for analysis of the spatial data.

We noted apparent outliers in the bivariate data. Moderate iterative, asymmetrical trimming with "BVTI" is efficacious because it tends to stabilize the mean estimates of home range location, shape, orientation and size. A 109^ trim was selected for later analyses.

Home ranges of male adults tend to be larger than for females. These results are reasonable because females need to be near the natal burrow when nursing and because male ground squirrels are polygamous and tend to include the ranges of several families within their own range. Yearlings tend to exhibit similar spatial patterns. However, yearling males have the largest home ranges of all and this may be due to frequent shifts of home range, probably because of their low status in the social hierarchy (Michener. 1972. 1973. 1977. 1979: Michener and Michener. 1977. 1979).

Home ranges of both species overlap, but centroids resulting from the robust algorithm clearly show that there is a tendency for the two species to segregate. S. coliimbianus occupies the more mesic slopes while S.

HOME RANGES OF TWO GROUND SQUIRRELS

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Fig. 6. Activity centers of individual squirrels for the time interval of the study, including squirrels whose home ranges were not plotted due to insufficient sample size. Broken line separating species was fitted by eye.

elegans occupies the xeric ones. This pattern is consistent with their macrogeographic distributions. S. columhianus generally occupies mesic montane regions while S. elegans tends to occur in more xeric intermon- tane valleys.

The methods of computing robust estimates of home range parameters presented here employ slight modifications in several

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