HOME RANGE: AUTOCORRELATION, MODELLMG. AND LIMITATIONS OF

THE CONCEPT

A thesis

Presented to

The Faculty of Graduate Studies

of

The University of Guelph

by

SHANE R. DE SOLLA

In partial fulfillment of requirernents

for the degree of

Master of Science

December. 1997

O Shane R. de Solla. 1997 National Library Bibliothèque nationale du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON K1A ON4 WwaON K1A ON4 Canada Canada

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HOME RANGE: AUTOCORRELATION. MODELLING. AND LIMITATIONS OF THE CONCEPT

Shane R. de Solla Advisor: University of Guelph. 1997 Professor R. J. Brooks

Home range estimators measure different things. and so an al1 encompassing definition of home range is ambiguous. Statistical home range estimators are ofien considered to be violated by autoconelation arnong the locational observations. Using both field data and a Monte Carlo simulation, I demonstrated that the accuracy and precision of home range estimates improved with larger sample sizes and thus with stronger autocorrelation. therefore destructive subsampling or restrictive sarnpling should not be used before estimating home range. Incorporating information on how animals detect resources into home range analyses may improve the consistency of the estimators. This procedure may add biological relevance. but home ranges estimated in this manner will be dependent on the scale of investigation, and thus produce multiple values. Differences among studies in home range estimators and sarnpling regimes may prevent testing for ueatment effects due to methodological artifacts. Acknowledgements

1 would like to thank my advisory cornmittee for their helpfbl comments and recommendations on my ciraft. 1 would like to thank Ron Brooks. my supervisor. for offenng a scholarship-less student a place in his lab. and for giving me license to pursue my weird ideas, which hopefùlly won't destroy his career when (if?) 1 publish. Tom

Nudds 1 thank for his enthusiasm and statistical advice. as well as philosophical debates on the precise meaning of accuracy. Glen Van Der Kraak 1 thank for giving a non- physiological ecologist access to his laboratory.

Considering Ron's propensity to hire entire brigades of graduate and undergraduate students, this next section is rather long. The following, AI. Barb M. Ben.

Cam. Dan. Dave. Elaine, Heather. Holly. Jackie, Janet. Jon. Kevin. Kym. Le~y.Lisa B..

Lisa E., Meg, Mel, Nicki, Rob V., Rob W.. Russell. Sema. Steve H.. and Steve W.. al1 were valuable for thoughtfùi discussions. Company. and times at the Albion. 1 would particularly like to thank Ben Porchuk for his enthusiasm and original impetus into investigating autocorrelation ("That can't be right!"). and David Cu~ington.Nicki

Koper, Albrecht Schulte-Hostedde. and Erran Seaman for their statistical advice and helpful comments. 1 appreciate my discussions with Carolyn Callahan. whose many statistical questions she asked me undoubtedly helped me much more than her.

Although 1 was not blessed by a pnstine study site like my more fortunate colleagues, rny stay in the city of Hamilton was nevertheless refieshing due to the presence of my two field assistants: Melinda Portelli and Holly Spiro. Oh yeah. thanks for helping me catch turtles. Ditto for Leslie Frye. Christine Bishop and Karen Pettit were invaluable as fiiends, logistic support. and for showing me what a snapping turtle is.

Russell Bonduriansky loaned me some of his data, allowing me to catch a glimpse of real science.

1 thank my parents and brother for their support. and watching me playing in the mud and catching gross things as a kid. Apparently some things are hard to grow out of.

1 was partially funded through The Great Lakes Action Plan. Environment

Canada. Tri-council Eco-Research Program, and NSERC grant A5990 to RJB. The

Ministry of Natural Resources. Royal Botanical Gardens. and the Central Lake Ontario

Conservation Authority gave me permission to kidnap turtles.

1 would also like to thank snapping turtles evevhere, for (involuntarily) allowing me to poke and prod them. and for bringing a bit of the Cretaceous to the present-

Finally. the misceIlaneous thanks: Barb T.. Nicole. and Tana.

Special no thariks to:

i) Rampant pesticide use. chemical dumping. and waste production which has tumed these snapping turtles into walking/swimrning toxic reservoirs. and tumed Cootes

Paradise into a cesspool.

ii) Those who advocate unlirnited development (and population growth) which will hasten the demise of snapping turtles (and other ugly critters). and those like myself who study them.

iii) Microsoft, just because 1 don't like them. Table of Contents

Acknowledgements ...... i

ListofTables ...... vi

List of Figures ...... vii

Chapter 1 The home range concept ...... -7

Abstract ...... 3

introduction ...... 3

Thehomerangeconcept ...... 4

Definition of home range ...... -5

Nonstatistical measurement of home range: 1) discontinuous home ranges .....6

Nonstatistical measurement of home range: II) macroscale habitat selection ....7

Nonstatistical measurement of home range: III) track pattern ...... 10

Statistical measurement of home range ...... 12

Kemel density estimators ...... 14

Estimating the underlying distribution of home range ...... 18

Autocorrelation ...... -21

A small note on bias ...... 23

References ...... 27

Chapter 2 Sacrificing statistical rigidity for biological relevance: spatial autocorrelation

and home range estimates ...... JJ9

Abstract ...... 34

Introduction ...... 34 Methods ...... 37

Monte Cario simulation ...... -37

Field studies ...... -38

Density estimation ...... 39

Other home range estimators ...... 40

Measure of autocorrelation ...... -41

Results ...... 44

Monte Car10 simulation ...... -44

Other home range estirnators ...... -45

Field studies ...... 53

Discussion ...... 68

Monte Car10 Simulation ...... 68

Other home range estimators ...... -70

Field studies ...... 71

General Conclusions ...... -75

References ...... 81

Chapter 3 Using search radii to choose the bandwidth of kernel estimators for home

rangeanalysis ...... 87

Abstract ...... 88

Introduction ...... 89

Kemel estimators ...... 90

Defining the population of observations ...... 91

List of Tables

C hapter 2

2.1 Regression summary of number of observations within each grid ce11 with the respective probability density. for each time interval for the male antler fly.

2.2 Regression sumrnary of number of observations within each grid ce11 with the respective probability density. for each tirne interval for the male snapping turtle.

2.3 Regression of home range size and time interval of four antler flies.

2.4 Regression of home range size and e/&,,, of four antler flies.

Chapter 3

3.1 Relationship between time interval and il? with home range size for LSCV and invariant kernel home range estimates.

3.2 Relationship between time interval and ?/?,,, with home range size for LSCV and invariant kemel home range estirnates.

Chapter 4

4.1 Summary of home range estimators found in articles published in the Joumal of Mammalogy from 1992 to 1996.

4.2 Summary of home range estimators found in articles published in the Joumal of Wildlife Management from 1992 to 1996.

4.3 Summary of home range estimators found in articles published in the Journal of Animal Ecology from 1992 to 1996.

4.4 Summary of home range estimators found in articles published in the Journal of Zoology, London from 1992 to 1996. List of Figures

C hapter 1

1.1 Representation of a discontinuous estimate of home range, surrounded by "dead space".

1.2 Representation of an estimate of home range as macroscaie habitat selection.

1.3 Representation of an estimate of home range using a track pattern.

1.4 Kemel estimate of 5 observations. showing individual kemels.

1.5 The effect of changing h on the shape of a normal kemel.

1.6 Discrepancy between the path of the animal and the estimated home range.

Chapter 2

Mean and standard deviation of r'li of simulated home ranges at different time intervals. The expected value of independent observations is 2.

Mean and standard deviation of home range estimates (kernel density) from the Monte Car10 simulation at different time intervals. The unbiased home range size is 10000 units. and there are 100 samples per time interval.

Mean and standard deviation of home range estimates (harmonie means) from the Monte Car10 simulation at different time intervals. The unbiased home range size is 10000 units, and there are 20 sarnples per tirne interval.

Mean and standard deviation of home range estimates (MCP) fiom the Monte Car10 simulation at different time intervals. The unbiased home range size is 10000 units. and there are 20 samples per time interval.

Mean and standard deviation of home range estimates (95% probability ellipse) from the Monte Car10 simulation at different time intervals. The unbiased home range size is 10000 units. and there are 20 sarnples per time interval.

Mean and standard deviation of home range estimates (75% probability ellipse) from the Monte Car10 simulation at different time intervals. The unbiased home range size is 10000 units. and there are 20 samples per time interval.

Estimated total distance travelled as a function of increasing time intervals

vii between observations. The true total distance travelled is estimated as the y- intercept. Polynomial regression of total straight line distance and time interval of four antler flies. 0 antler fly 1: ? = 0.9755. F = 16-85, df = 3. 1 1. P < 0.000 1; y - int = 92.67 cm antlerfly2: ?= 1.0; y-int = 11.57cm + antler fly 3: 9 = 0.9769. F = 4.27, df = 3. 9. P = 0.0389: y - int = 66.89 cm antler fly 4: i = 0.9745, F = 1 14.54. df = 3.9, P < 0.000 1; y - int = MO. 19 cm

2.8 Estimated total distance travelled as a Function of increasing time intervals between observations. The true total distance travelled is estimated as the y- intercept. Polynornial regression of total straight line distance and time interval of five snapping turtles. snapping turtle 1: ? = 0.9789. F = 154.64. df = 3. 10. P 1 0.0001 ; y - int = 7.48 km + snapping turtle 2: i = 0.9026. F = 2 1-61. df = 3.7. P = 0.0006: y - int = 3.44 km snapping turtle 3: i = 0.6105. F = 4.70. df = 3.9. P = 0.0307; y - int = 2.38 km A snapping turtle 4: i = 0.965 1. F = 64.58, df = 3. 7. P c 0.0001; y - int = 3.83 km 0 snapping turtle 5: r' = 0.9634. F = 82.78. df = 3.9. P < 0.000 1 : y - int = 2.93 km

2.9 Relationship between z?? with time interval between observations. Polynomial regression of I'/$ and time interval of four antler flies. 0 antler fly 1: y = 0.077 + 0.035~-0.0004 1x-' + 0.00000 15x3 antler fly 2: y = 0.14 + 0.0 10s + 0.00032~'- 0.0000 1 0x3 + antler fly 3: y = 0.082 + 0.044~- 0.00040~'+ 0.00000 1 1 x' antler fly 4: y = 0.23 + 0.076~- 0.00 12x' + 0.000006~' The expected value of independent observations is 2.

2.10 Relationship between ?/i,,,with time interval between observations. Polynomial regression of r'l?,, and time interval of four antler flies. 0 antler fly 1: y = 0.17 + 0.020~- 0.00020x2 + 0.00000069x3 antler fly 2: y = 0.03 1 + 0.028~- 0.00080~'+ 0.000010x3 + antler fly 3: y = O. 12 + 0.030~- 0.00024~'+ 0.0000064x3 antler fly 4: y = 0.20 + 0.05 1x - 0.00082~'+ 0.000004 1x' The expected value of indeoendent observations is 2.

2.1 1 Relationship between $/? with time interval between observations. Polynornial regression of ?/>' and time interval of five snapping turtles. a snapping turtle 1 : y = 0.77 + 0.23~- 0.024~' + 0.00086x3 + snapping turtle 2: y = 1-57 - 0.057~+ 0.024~'- 0.0020~' snapping turtle 3: y = 2.06 - 0.048~+ 0.0084~'- 0.00035x3 A snapping turtle 4: y = 0.75 + 0.30~- 0.063~'+ 0.003 1x3 0 snapping turtle 5: y = 0.032 + 0.32~- 0.060~" 0.0034xx' The expected value of independent observations is 2.

2.1 2 Relationship between r'lr',,, with time interval between observations. Polynornial regression of r'/?,,, and time interval of five snapping turtles. 0 snapping turtle 1 : y = 0.47 + 0.17~- 0.000 17x' + 0.00000066x3 + snapping tutle 2: y = 2.06 - 0.043~+ 0.00082~'- 0.0000046x3 snapping turtle 3: y = 1.33 - 0.026~+ 0.00040~'+ 0.00000 16x3 A snapping turtle 4: y = - 0.80 + 0.086~- 0.0014x2 + 0.0000063~' osnapping tutle 5: y = - 0.79 + 0.061~- 0.00092~' + 0.0000044x3 The expected value of independent observations is 2.

2.13 Cornparison of MCP home range of a male antler fly and snapping turtle for al1 observations. and first statistically independent ($/?) subsarnple.

2.14a.b Adaptive kemel density estimates. (a) Antier fly home range with 10 sec. intervals between observations (h = 2.6, n = 143) (b) Antler fly home range with 70 sec. intervals between observations (h= 2.6, n = SI ). 2.15a.b Adaptive kernel density estimates. (a) Snapping turtle home range with 2 day intervals between observations (h = 1.2. n = 43). (b) Snapping nirtle home range with 6 day intervals between observations (h = 1.2. n = 17).

Chapter 3

3.1 The relationship between scan time and distance to the target on the probability of detection.

3.2 Relationship between sample size and autocorrelation with time interval of simulated home ranges. The dotted line (?l? = 2) is the expected value of if the observations are independent.

3.3 Mean size of invariant and LSCV home range estirnates at different tirne intervals.

3.4 Variance in size of invariant and LSCV home range estimates at different time intervals.

Chapter 4 4.1 Mean and standard deviation of home range size estimates of the first 20 samples fkom the Monte Carlo simulation presented in Chapter 2. using the time interval of 1. K = Fixed kernel (LSCV). H = Harmonic means. E95 = 95% probability ellipse, E75 = 75% probability ellipse, M = MCP. Every cornparison is significantly different from each other (P < 0.000 1 ).

4.2 Mean and standard deviation of home range size estimates of the first 20 samples from the Monte Car10 simulation presented in Chaprer 2. using the time interval of 15. K = Fixed kemel (LSCV), H = Harmonic means. E95 = 95% probability ellipse, E75 = 75% probability ellipse. M = MCP. Every comparison is significantly different from each other (P < 0.02). " Hypotheses non fingo." Issac Newton Chapter 1

The home range concept Abstract

Home range is a poorly defined concept. and this lack of clarity is reflected in the determination and interpretation of the home range estimates. Typically home range is viewed in tems of macroscale habitat selection, and is estimated either by a boundary. which does not give any information on the interna1 composition of the home range. or a utilization distribution. which does not give any tme boundaries. Using the "classic" definition of home range results in inconsistent home range estimates, as any nurnber of home range estimators will satisfi the definition, yet they produce substantially different estimates. Furthemore. if the prevailing wisdom on autocomelation is correct. there will be an upper limit to the accuracy with which home range can be estimated due to the trade off between sample size and autocorrelation.

Introduction

The home range concept is ofien used to describe or infer animal behaviour and resource use from the rnanner in which animais occupy their living space. The study of home range is a growing field. and is no longer limited to those researchers who are interested in predominately descriptive or survey type studies. For example, home range analyses are used to infer reproductive strategies (M1C1oskeyet al. 1990. Wolff and

Cicirello 1990. Brooks 1993), community structure (Basset 1995), habitat fragmentation

( Wauters el al. 1994) and resource dispersion (Macdonald 1 983. Geffen et ai. l992), and rate of resource acquisition (Swihart et a!. 1988), among other relationships. There has been a similar growth in the technological methodology of coilecting data. from trapping and visual observations to radiotelemetry and satellite tracking. There has also been a large expansion of analytical techniques used to estimate home range size and shape - harmonic means (Dixon and Chapman l98O), R method (Schoener 198 1 ), Fourier transformations (Anderson 1982), multinuclear mode1 (Don and Rennolls 1983), probability polygons and cluster analysis (Kenward 1987). kemel densities (Worton 1987.

l989), fractds (Loehle 1990). and Dirichlet tessellations (Byers 1992. Wray et ~1.1992).

These analyses are in addition to the multitude of home range estimators that have appeared in the literature from 1940 to 1980. However. our ability to take advantage of the additional information avaiiable fiom these new technologies and analytical techniques has not improved (Kenward 1992).

The home range concept

It is convenient (and usually necessary) to define home range in a way to allow it to be mapped or othewise illustrated. and this has lead to some difficulties in marking boundaries. Despite how it has been estimated. researc hers typically assume that the home range concept is suficiently well understood within the scientific community to an extent that most recent home range papers no longer bother to define it. Review articles ofien use the sarne archaic references to define home range, such as Burt (1943). and

Mohr ( 1947), and some dont even define home range at al1 (e.g. Rose 1982). There has been a shift from these descriptive type of definitions to those of a statistical nature

(Jennrich and Turner 1969. Van Winkle 1975. Worton 1989, Andreassen et al. 1993). but discrepancies between what we perceive as the population of locational observations and the sample of observations makes the use of these methods problematic. Such ambiguity in home range definition is likely to be reflected in the measurement and interpretation of home range estimates, and especially in cornparisons arnong studies.

Although at the end of this chapter and throughout Chapter 2 and 3.1 use a statistical estimator (kemel density), 1 introduce the two main types of home range estimators. After outlining the different ways of defining home range, I attempt to demonstrate the discrepancy between the definition of home range and what the home range estimators actually measure. Then I tackle the problern of autocorrelation in

Chapter 2; if autocorrelation violates assumptions of home range estimators. then there will be an upper Limit on the ability to accurately estimate home range. In Chapter 3.1 concentrate on kemel estimators, and develop an alternative to the normal method of determining the bandwidth (h).which deterrnines the degrer of smoothing for the estimate. By incorporating perceptive ability of animals into the form of the kemel function. kernel density estimaton become potentially consistent estimators which are more comparable to the actual definitions of home range. Finally. in Chapter 4. 1 critique the use of home range in the literature. and suggest that home range estimates. as currently rneasured. are not comparable arnong studies.

Definition of home range

The most commonly cited definition of home range is. "that area traversrd by the individual in its normal activities of food gathering. mating, and caring for young," and

"the area, usually around a home site. over which the animal norrnally travels in search of food." (Burt 1943). The author asserted that animals could occasionally venture from this area, but that exploratory sallies were not part of the home range. Home range is often defined in ternis of resource use. or the area used for feeding. reproduction. and other purposes (Tinkle 1969). These definitions have been repeated by others without much refinement: home range is the area "over which an animal normally travels in pursuit of its routine activities" (Jewell 1966). or the area in which an animal nomally moves. exclusive of migration. emigration, or unusual erratic wanderings (Brown and Orians

1970). The size and shape of the home range is defined by a boundary in which the area inside is used by the animal, and the area outside is either unused. or rarely visited.

Apart fiom the dificulty in defining "normal" movement. these definitions are not conducive to precise measurement. There are numerous ways that "home range" can be measured, each of which is compatible with the basic definition of home range.

Nonstatistical measurement of home range: 1) discontinuous home ranges

Home ranges are not necessarily continuous. and this has long been recognized

(e-g.Burt 1943, Jewell 1966). Migration routes are not usually considered part of any home range. and home range can change over an animal's lifetime. However. it is among more recent work that discontinuous home ranges have been recognized more. due to newer analyses that readily produce discontinuous home ranges (ie harmonic means. kemel density, cluster analysis).

Nevertheless, resources are not evenly distributed and are ofien found in patches. and the distribution of these patches may determine home range size and shape. Home range size increases with decreasing resource drnsity (Mares and Lacher 1987. Galeotti

1994, Yosef and Grubb 1994. Storch 1995, Tufto et al. 1996). Since animals reduce their home range when resources become more abundant (e-g.Yosef and Gmbb 1994). it seems likely that at least sorne of the home range consists of "dead space." These areas contribute little or no resources, yet they are either near areas that are used, or they are in the space between resources. and so are sometimes travelled through (Geffen et al. 1992).

The home range, then. consists of patches of resources, used for foraging, lekking. basking, nesting, etc., with areas of dead space dispersed among or surrounding the patches (Fig. 1.1). If home ranges are subdivided into regions of resource use - foraging regions, for exarnple - the resulting home range is likely to be discontinuous.

Discontinuous measurements of home range are consistent with the definition of home range.

Nonstatistical measurement of home range: II) macroscale habitat selection

Home range can also be viewed as the general area or habitat in which contains the entire path of the animal. Thus. the home range does not just include the animal's irnrnediate surroundings as it meanden across the landscape. but the habitat that contains the path the animal follows (Fig. I 2).Johnson (1980) outlined different orders of habitat selection; first-order habitat selection is the geographic range of the population or species. second-order selection is the home range. and third-order selection is the use of habitat components within the home range. Animais sarnple from the available habitat. which cmbe considered a resource pool. as the animals trace a path as they forage for food or other resources. The home range can be show by a boundary surrounding the resource pool area. The most common type of boundary estimators are the polygon methods, such as the minimum convex polygon (MCP. Mohr 1 947). although there have been numerous modifications of this method (Harvey and Barbour 1965. Kenward 1987). A polygon is drawn around the outermost observations, sometimes excluding the fanhest 5% from Figure 1.1. Representation of a discontinuous estimate of home range, surrounded by "dead space". Figure 1.2. Representation of an estirnate of home range as macroscale habitat selection.

9 the center of the home range (Kenward 1987), and the area within the polygon is the home range. This method of measuring home range is also consistent with the basic de finition.

Nonstatistical measurement of home range: III) track pattern

Animal movements are constrained to a path from which the animal traveis as it conducts its normal activities (Andreassen eï al. 1993). but the manner in which we choose to measure the area from which the animals gain resources is not. The most obvious pattern of resource use is what the animal can physically reach from its path. If we assume that an animal can only use resources that are within its reach or its ability to manipulate from a distance. then the home range area is the area within its reach dong the path. The area in which an animal can gather resources can be estimated by drawing a circle around the centre of the animal. with a radius equal to the reach of the animal. This area is then superimposed upon the path of the animal. and the total area is the home range (Fig. 1.3). This method of measunng home range is essentially nonexistent. but it is consistent with the basic definition of home range.

The track pattern view of home range is intuitively the easiest to measure. but requires a very large sample size, and would be very inaccurate at time intervals which are typically used in home range studies (e.g. Reynolds and Laundre 1990). The macroscale habitat view is the most common in the literature but since it includes areas which the animal never visits. and thus there are no observations. it is diffrcult to objectively determine the boundary of the home range. The discontinuous view of home range is the hardest to estimate, due to the difficulty of estimating resource use. Without Figure 1.3. Representation of an estimate of home range using a track pattern.

11 estimating resource use, how can one Say if an animal is using a particular area, even if it has been observed in that area? This may be a usehl technique to divide the home range into regions by activity type - foraging region, basking region. and so on. For examples of these different views, see Benhamou ( l990), Johnson et al.( 1992). Kenward ( 1987) and

Storch (1 995). Boundary methods have one large disadvantage: they do not produce any information on the intemal configuration of the home range.

These three views are not necessarily mutually exclusive - they may al1 represent different scaies of habitat selection and rnovement patterns. but they cannot be rneasured sirnultaneously. Either different techniques have to be used, or the same mode1 cm be used. but with different parameters. To add more confusion to this issue. for each different view of home range a number of home range estimators can be used. In light of the difficulties in measuring home range. the basic definition of home range is insuficient for home range studies to be comparable with each other. More specific. and objective. definitions of home range are needed.

Statistical measurement of home range

Despite the differences arnong views of boundary type home ranges. they al1 have sornething in common - a definite boundary dividing home range area from non home range area. There is another definition of home range which doesn't produce true boundaries, but is much more specific than the basic definition of home range (e-g.Burt

1943). Home ranges cm be estimated as utilization distributions. ofien using probability distributions. Utilization distributions are bivariate frequency distributions which estirnate how much time an animal spends at any point in space (Van Winkle 1975). While polygon methods have definite edges. "statistical" or utilization-based estirnates have no red boundary, but instead have declining slopes of fiequency along the periphery of the home range. Most utilization distributions that are currently used have a non zero value for al1 (x, y) locations; thus, they cover an infinitely large area (ie. kemel density, Fourier analysis, probability ellipses. harmonic rnean). Nevertheless. they have usefui properties for estimating home range use (Although the harmonic means analysis does not produce a true probability distribution. it produces a similar fimction. and can be considered a form of a kernel estimator [Worton 19871. Therefore 1 will consider it along with estimators that generate a true probability function.)

The utilization distribution may be estimated by a probability distribution. and home range can be described by

where P(S) is the probability of locating an animal within the area S. and f(x. y) is the probability distribution function (Andreassen 1993). Home range is often defined as the area that contains either 95% of the observations (Dixon and Chapman 1980. Spencer and

Barrett 1984) or 95% of the probability distribution (Ackerman 199 1, Searnan and Powell

1996), which is what 1 shall use. The home range size then is

where Sa is the area contained within a (ie. 95%). but under the condition so that s is the minimum intensity of use (Andreassen et ~1.1993).Excluding the lower

5% of the probability distribution results both in removing outlying observations as occasional sallies (Bondmp-Nielsen 1985) or non-normal movements (Burt 1943). and in restticting home range to a finite area. By choosing a particular percentage of the probability distribution a home range boundary can be made. This boundary is more a convention than a reality in that there is no a priori reason that the 95% isopleth has any more justification than any other percentage of the distribution.

The analyses that produce this function are varied. and have much different assumptions and produce different results. but they al1 are used to estimate the same thing. In essence. they al1 estimate the probability of locating an animal at a particular location at a random point in time. Except for probability ellipses, which assume a bivariately normal probability distribution (Jennrich and Turner 1969). most of these analyses are nonpararnetric. and they are considered to make no assumptions about the shape of the home range (Worton 1 987).

Kernel density estirnators

From this point. 1 shall be using primarily the kemel density estimator for estimating home range (Wonon 198% 1989). Density estimation allows us to constnict an estimate of the distribution of the population by using the sarnpled observations. A kemel estimator takes the generai form where K is a unirnodal symmetncd bivariate probability density (Silveman 1986), and the total volume of the kernel is 1 (100% of the probability density). A more specific definition is

where h is the window width (also know as the smoothing bandwidth of parameter). X, is a random sample of n independent points from the unknown density function (Silverman

1986). and K is a usually a syrnrnetrical probability density function such as the normal distribution for a given gnd point x (Worton 1989). A univariate example of estimating a population density by swingindividual kemels is displayed in Fig. 1.4. and the process is identical for bivariate data. The window width h changes the variance of the kernel. such that low values of h produce kernels with smaller variances. and high values of h produce kemels with larger variances (Fig. 1.5). aithough the volume of the kemel will remain unchanged. The value of h wiil change the total area contained within a utilization distribution estimated by a kernel estimator. and thus change home range size. The smallerr the value of h, the smaller the home range estimate.

While the actual form of the kemel does not greatly affect the eficiency of the kernel, die choice of h will have a large effect upon the estimated density (Silveman Figure 1.4. Kemel estimate of 5 observations, showing individual kernels.

16

1986). Although subjective choosing of h has its merits. automatic choosing is required for a standardized (and thus repeatable) procedure for density estimation. Generally, most automatic cnteria for estimating h atternpt to minimize the integrated squared error. which is

where 7 is estimated density. and f is the true density. Since the last term, f ', does not depend upon the estimated density. then the ideal h can be estimated by minimizing

(Silverman 1986). Since f is unknown, least-squares cross validation (LSCV) uses the observations to constmct an estimate off(Si1verman 1986). For examples of automatic methods for choosing h. see Silverman ( 1986). Wonon (1 989). and Searnan and Powell

( 1996). Typically. as the sarnple size approaches m. h approaches zero. While the selection of h may prove to be troubiesome. it also gives a larger amount of flexibility that is missing with any boundary method and most statistical methods.

Estimating the underlying distribution of home range

While many have recognized that locational observations are sarnpled fiom a statistical population of fixes when estimating home ranges (e.g. Harris el al. 1990). it is important to understand the nature of the population. Most validation or randomization simulations involving statistical home range analyses assume that the animal has a non- zero probability of being located at every point within the home range (Swihart and Slade 1 98Sb. Boulanger and White 1990. Cresswell and Smith 1992. Worton 1995. Seaman and

Powell 1996). In other words, they assume that the animal actually crosses every point within the home range.

The actual popuiation from which locational observations are sampled fiom is the path of the animal; outside this path, there is no possibility of locating the animal. Unless the animal's path crosses every single point within the area specified in these validation simulations, the simulated home range estimator will overestimate the probability of locating the animal off the path. and thus underestimate the probability of locating the animal at any point on the path (Fig. 1 A). Since these simulations typically use Monte

Carlo rnethods, they validate the home range estimator with the expected mean distribution of observations derived fiom a prespecified population distribution. usually a normal or uniform distnbution. Since the locational observations are sampled from a distribution of locations from the path, the real population cannot be a normal or uniform distribution that covers an unbroken area.

This may not seem like a problem. but some of the difficulties associated with estimating home range that have &sen may have been due to attempting to measure a phenornenon that is at a different scale than what the analyses are capable of estimating.

Kemel densities have the assumption that the observations taken from the population are random and independent (Silverman 1986). However. anirnals move in a path. from which it should be obvious that the observations cannot be truiy random. Locational observations are constrained in that they must appear between the previous and subsequent observations - there is an order to the locational observations. If the sarnpling Theoretical elliptical home range

Figure 1-6. Discrepency between the path of the animal and the estimated home range.

20 interval between observations is sufficiently short, then the value of one observation is dependent upon and is positively correlated with the previous observation.

Autocorrelation

Autocorrelation. or the degree of (dis)similarity among pairs of observations taken at a certain distance apart, has been a problem in describing animal movement patterns

(e.g. Swihart and Slade 1985a). When using automatic procedures to estirnate h, such as

LSCV. as the degree of positive autocorrelation increases. h decreases (Altman. 1990).

Although not stated explicitly, it is likely that as autocorrelation tends towards a. h tends towards zero. As a result. home range size tends to decrease with increasing autocorrelation. although this may be an artifact of the method used to mode1 autocorrelation (Swihart and Sade 1985a Cresswell and Smith 1992) or the method used to estimate home range (LSCV. Seaman and Powell 1996).

Increasing sample size is known to increase both the precision and the accuracy of the estimate at least within certain limits. Many criticize the rneager sample sizes in many published home range studies (Rose 1982. Harris el ul. 1990). Using a greater sample size reduces the likelihood that certain movements will be undetected. and allows a more accurate estimate of an animal's path (Reynolds and Laundre IWO). Sampling error is more relevant at smaller sample sizes, where rare observations would likely be excluded fiom the set of observations. Unlike sampling error when calculating means. this type of sampling error is not "random" for home range estimates. Rare observations tend to be at the edges of the distribution, so these errors are "averaged out" when calculating indices of central tendency, such as means. Home ranges are calculated as the area within the distribution, and so are underestimated when these rare observations are missing.

However, by increasing the number of observations, the location of each observation will becorne more autocorrelated, or more similar, to subsequent observations. It is a widely held view that autocorrelation among sequential observations biases estimates of home range use (Van Winkle 1975. Dunn and Gipson 1977. Schoener

1981. Anderson 1982, Don and Rennolls 1983. Slade and Swihart 1983. Laundre and

Keller 1984, Swihart and Slade 1985a. b. 1986,1987, Samuel and Garton 1987, Worton

198% Ackerman et al. 1990, Harris et ul. 1990. Spencer et al. 1990. White and Garrott

1990, Cresswell and Smith 1992. Kenward 1992, Swihart 1992. Andreassen et al. 1993.

North and Reynolds 1996.). If autocorrelation causes bias in home range estimates. the accuracy of the home range estimate cm be maximized by increasing the time interval between observations until autocorrelation is no longer present.

It is apparent that. assurning both sample size and autocorrelation affect accuracy. an unbiased home range estimate is impossible. Reducing bias by eliminating autocorrelation is only possible by reducing sample size. but reducing sample size increases the sampling error. and thus reduces accuracy of the estimate. Since bias due to autocorrelation and sarnple size has to be balanced. there would be an upper limit on the obtainable accuracy of any home range estimate, regardless of the accuracy of the sampling regime. Thus. unless there is a home range estimator that is unbiased by either autocorrelation or sample size. home range estimates will always be indeterminant. Any estimator that is independent of the observations used is not likely to be useful (or even possible), so home range estimators have to be unbiased with respect to autocorrelation. while the bias due to sarnple size has to be minimized.

The next chapter examines the effect of subsarnpling locational observations to

eliminate autocorrelation. Ultimately, 1 hope to show if autocorrelation does violate home

range estimators by increasing the bias or imprecision of home range estimates.

A small note on bias

There must be some caution when attempting to minimize bias. Obviousiy bias is

an undesirable element. but attempts to eliminate it. whether by changing estimators or by

manipulation of the observations used in the analyses. may prove to be more damaging

than the onginal bias. Biased estimates, for example, have been shown to be more usehl

in multiple regression than unbiased estimators in some circumstances. such as the use of

ridge regressions when there are correlated regressors (Vinod and Ullah 198 1 ). Also. if

the main concem is to get an estimate as close as possible to the true value. then a precise

but biased estirnated may be preferable than an imprecise but unbiased estimate. because

the mean (absolute) error may be less (Garthwaite et al. 1995).

Generally. researchers are interested in getting home range estimates that are as

close as possible to the true value. The mean integrated squared error (MISE). which is

the quantity used to estimate the optimal value of h. is the sum of the integrated square bias and integrated variance (Silverman 1986). The mean squared error (MSE). from which MISE is derived. is the sum of the rnean squared bias and the variance of the estimate. Minimizing the MSE requires a compromise between the bias and variance. The trade-off between bias and variance means that bias can only be reduced by increasing the variance of the estimate through increasing h (Silverman 1986). Thus, by minimizing the bias of the kemel density estimate, the imprecision of the home range would increase as would mean squared error of the estimate. There cm be too much emphasis on bias at the expense of other factors. References

Ackermax B. B., F. A. Leban. M. D. Samuel, and E. O. Garton. 1990. User's manual for

program home range. Second edition. Technical Report 15. Forestry. Wildlife and

Range Experimental Station. University of Idaho, Moscow, Idaho, USA.

Altmann. N. S. 1990. Kernel smoothing of data with correlated errors. J. Amer. Stat.

Assoc. 85: 749-759.

Andreassen, H. P. . Ims, R. A.. Stenseth. N. C.. and Yoccoz. N. G. 1993. Investigating

space use by means of radiotelemetry and other methods: a methodological guide.

In: The biology of lemmings. London. Limean Society of London.

Andersen. D. E. a~dRongstad. O. J. 1989. Home range estirnates of red-tailed hawks

based on random and systematic relocations. J. Wildl. Manage. 53: 802-807.

Anderson. D. J. 1982. The home range: a new nonpararnetric estimation technique.

Ecology. 63: 103- 1 12.

Basset. A. 1995. Body size-related coexistence: an approach through allometric

constraints on home-range use. Ecology. 76: 1027- 1035.

Benhamou, S. 1990. An analysis of movernents of the wood rnousr Apodemus syhuticzrs

in its home range. Behav. Processes. 22: 235-250.

Bondrup-Nielsrn. S. 1985. An evaluation of the effects of space use and habitat patterns

on dispersa1 in small rnammals. Ann. Zool. Fernici. 22: 373-383.

Boulanger, J. G. and White. G. C. 1990. A cornparison of home-range estimators using

Monte Carlo simulation. J. Wildl. Manage. 54: 3 10-3 15.

Brooks, R. J. 1993. Dynarnics of home range in collared lemmings. In: The biology of lemmings. London. Limean Society of London.

Brown, C. H. 1989. The active space of blue monkey and grey-cheeked mangabey

vocalizations. Anim. Behav. 37: 1023- 1034.

Brown, J. L. and Orians, G. H. 1970. Spacing patterns in mobile animals. Ann. Rev. Ecol.

Syst. 1: 239-262.

Burt. W. H. 1943. Territoriality and home range concepts as applied to mammals. J.

Mamm. 24: 346-352.

Byers. J. A. 1992. Dirichlet tessellations of bark beetle spatial attack points. J. Anim.

Ecol. 6 1 : 759-768

Cressweil. W. J. and Smith. G. C. 1992. The effects of temporally autocorrelated data on

methods of home range analysis. In: Wildlife telemetry Remote monitoring and

tracking of animals. Priede. I. G. and Swift. S. M. (Eds.) Ellis Howood Ltd..

England. pp 372-284.

Dixon. K. R. and Chapman. J. A. 1980. Harmonic mean measure of animal activity areas.

Ecology. 6 1: 1040- 1044.

Don. B. A. C. and Rennolls. K. 1983. A home range mode1 incorporating biological

attraction points. J. Anim. Ecol. 52: 69-8 1.

Doncaster, C. P. and Macdonald. D. W. 199 1. Drifting temtoriality in the red fox. Vulpes

vulpes. J. Anim. Ecol. 60: 423-439.

Dunn, J. E. and Gipson. P. 1977. Analysis of radio telemetry data in studies of home

range. Biometrics. 33: 85- 101.

Galeotti, P. 1994. Patterns of temtory size and defence Ievei in rural and urban tawny owls (StrLr aluco) populations. J. 2001. Lond. 234: 64 1 -658.

Garthwaite, P. H.. Jolliffe. 1. T., and Jones. B. 1995. Statistical inference. Prentice Hall.

UK. 290 pp.

Geffen, E.. Hefner. R.. Macdonald. D. W.. and Ucko, M. 1992. Habitat selection and

home range in the Blanford's fox. Vzdpes cana: compatility with the resource

dispersion hypothesis. Oecologia. 9 1 : 75-8 1.

Harris. S.. Creswell. W. J.. Forde, P. G.. Trewhella W. J.. Woollard, T., and Wray, S.

1990. Home-range anaiysis using radio-tracking data - a review of problems and

techniques particularly as applied to the study of mammals. Mamrnal Rev. 20: 97-

123.

Harvey. M. J. and Barbour. R. W. 1965. Home range of Microtus ochrogrlsier as

determined by a modified minimum area method. J. Mamm. 49: 398402.

Jennrich. . R. 1. and Turner. F. B. 1 969. Measurernent of non-circular home range. J.

Theor. Biol. 22: 227-237.

Jeweil. P. A. 1966. The concept of home range in mammals. In: Play. exploration. and

temtoriality in mammals. Symp. Zool. Soc. Lond. No. 18. pp 85- 109. Jewell. P.

A. and Loizos. C. (Eds.) Academic Press. London. England.

Johnson, A. R., Minle. B. T.. and Wiens. J. A. 1992. Diffusion in fractal landscapes:

simulations and experimental studies of tenebrionid beetle movements. Ecology:

73: 1968-1983.

Johnson, D. H. 1980. The cornparison of usage and availability measurernents for

evaluating resource preference. Ecology. 6 1 : 65-7 1. Kenward. R. E. 1987. Wildlife radio tagging. Equipment. field techniques and data

analysis. Biological techniques series. Academic Press. London. England. pp 222.

Kenward. R. E. 1992. Quantity versus quality: prograrnmed collection and analysis of

radio-tracking data. In: Wildli fe telemetry Remote monitoring and tracking of

animals. Priede. 1. G. and Swift. S. M. (Eds.) Ellis Horwood Ltd.. England. pp

332-246.

Laundre, J. W. and Keller. B. L. 1984. Home-range size of coyotes: a critical review. J.

Wildl. Manage. 48: 12% 139.

Loehle. C. 1990. Home range: a fractal approach. Landscape Ecology. 5: 39-52.

Macdonald. D. W. 1983. The ecology of carnivore social behaviour. Nature. 30 1 : 379-

3 84.

Mares. M. A. and Lacher. T. E. Jr. 1987. Social spacing in small marnmals: patterns of

individual variation. Amer. Zool. 27: 293-306.

MICloskey, R. T. Deslippe. R. J.. Szpak. C. P.. and Baia. K. A. 1990. Ecological

correlates of the variable mating system of an iquanid lizard. Oikos 59: 63-69.

Mohr. C. 0. 1947. Table of equivalent populations of North American small marnmals.

Am. MidI. Nat. 37: 223-249.

North, M. P. and Reynolds, J. H. 1996. Microhabitat analysis using radiotelemetry

locations and polytomous logistic regression. S. Wildl. Manage. 60: 639-653.

Reynolds. T. D. and Laundre. J. W. 1990. Time intervals for estimating pronghorn and

coyote home ranges and daily movements. J. Wildl. Manage. 54: 3 16-322.

Rose, B. 1982. Lizard home ranges: methodology and functions. J. Herpetol. 16: 253-269. Samuel, M. D. and Garton. E. 0. 1987. Incorporating activity time in harmonic home

range analysis. J. Wildl. Manage. 5 1: 254-257.

Schoener, T. W. 198 1. An empirically based estimate of home range. Theor. Pop. Biol.

20: 28 1-325.

Seaman. D. E. and Powell, R. A. 1996. An evaiuation of the accuracy of kemel density

estimators for home range analysis. Ecology. 77: 2075-2085.

Silverman, B. W. 1986. Density estimation for statistics and data analysis. Chapman and

Hall. London. 175pp.

Slade. N. A. and Swihart. R. K. 1983. Home range indices for the hispid cotton rat

(Sigrnodon hispidus) in northeastem Kansas. J. Mamm. 64: 580-590.

Spencer. S. R., Carneron. G.N.. and Swihart, R. K. 1990. Operationally defining home

range: temporal dependence exhibited by hispid cotton rats. Ecology. 7 1 : 1 8 17-

1822.

Swihart. R. K. 1992. Home range attributes and spatial structure of woodchuck

populations. J. Marnm. 73 : 604-6 1 8.

Swihart, R. K. and Slade. N. A. 1985a. Influence of sampling intemal on estimates of

home range size. J. Wildl. Manage. 49: 10 19- 1025.

Swihart. R. K. and Slade. N. A. 1985b. Testing for independence of observations in

animal movements. Ecology. 66: 1 1 76- 1 184.

Swihart, R. K. and Slade, N. A. 1986. The importance of statisticai power when testing

for independence in animal rnovements. Ecology. 67: 255-25 8.

Swihart. R. K. and Slade, N. A. 1987. A test for independence of movements as shown by live trapping. Am. Midl. Nat. 1 17: 204-207.

Swihart. R. K., Slade, N. A. and Bergstrom. B. J. 1988. Relating body size to the rate of

home range use in marnmals. Ecology. 69: 393-399.

Tinkle, D. W. 1969. The life and demognphy of the side-blotched lizard. Uirr

siunsburiana. Misc. Publ. Mus. 2001.. Univ. Mich. 132: 1-182.

Tufto, J., Andersen, R.. and Linnell, J. 1996. Habitat use and ecological correlates of

home range size in a small cervid: the roe deer. J. Anim. Ecol. 65: 71 5-724.

Van Winkle, W. 1975. Cornparison of several probabilistic home-range models. J. Wildl.

Manage. 39: 118-123.

Vinod. H. D.. and Ullah. A. 198 1. Recent advances in regression methods. Statistics:

textbooks and monographs. vol. 41. Marcel Dekker. Inc. USA. 36 lpp.

Wauters. L.. Casale. P.. and Dhondt. A. A. 1994. Space use and dispersal of red squirrels

in fragmented habitats. Oikos. 69: 140- 146.

White. G. C. and Garrott. R. A. 1990. Estimating home range (Chapter 7). In: Analysis of

wildlife radio-tracking data. White. G. C. and Garrott. R. A. Academic Press. Inc.

London. 383 pp.

Wolff. J. 0. and Cicirello. D. M. 1990. Mobility versus temtoriality: alternative

reproductive strategies in white-footed mice. Anim. Behav. 39: 1222-1224.

Worton. B. J. 1987. A review of models of home range for animal rnovement. Ecol.

Modelling. 38: 277-298.

Wonon, B. J. 1989. Kemel methods for estimating the utilization distribution in home-

range studies. Ecology. 70: 164- 168. Worton, B. J. 1995. Using Monte Car10 simulation to evaluate kemel-based homc range

estimates. J. Wildl. Manage. 59: 794-800.

Wray, S., Cresswell. W. J., White, P. C. L.. and Harris, S. 1992. What. if anythmg, is a

core area? An analysis of the problems of descnbing intemal range configurations.

in: Wildlife telemetry Remote monitoring and tracking of animals. Priede. 1. G.

and Swift, S. M. (Eds.) Ellis Horwood Ltd.. England. pp 256-271.

Yosef, R. and Gmbb Jr.. T. C. 1994. Resource dependence and territory size in

loggerhead shrikes (Lanius ludoviciunus). Auk. 1 1 1 : 465-469. "Statisticsare no substitute for judgement." Henry Clay Chapter 2

Sacrificing statistical rigidity for biological relevance: spatial autocorrelation and home range estimates Abstract

Destructive subsampling and restrictive sampling are ofien standard procedures used to obtain independence arnong spatial observations in home range analyses. 1 examined whether kemel density and other home range analyses require sed independence of observations using a Monte Car10 simulation and antler flies and snapping nirtles as models. Home range size. time partitioning, and total straight line distances travelled were tested to determine if subsampling improved kernel performance and estimation of home range parameters. The accuracy and precision of home range estimates from the simulated data set improved at shorter time intervals with autocorrelated observations. Subsampling did not reduce autocorrelation arnongst locational observations of snapping turtles or antler flies, and home range size. time partitioning and total distance travelled were better represented by autocorrelated observations. Autocorrelation , especially at longer time intervals. is primarïly a property of the animal's behaviour. ratlier than a fünction of the sarnpling regime. I found that home range estimators do not require senal independence of observations, and 1 recornrnend that researchers maximize the number of observations using constant time intervals to increase the accuracy and precision of estirnates.

Introduction

It is cornmonly asserted that most analyses of home range require independence of observations for an unbiased estimate (Swihart and Slade 1985a, 1985b, Worton 1937,

Harris et al. 1990, White and Garrott 1990. Cresswell and Smith 1 992, Kenward 1992).

Ecological relationships ofien depend, either directly or indirectly, on their underlying spatial or temporal structure. and thus autocorrelation can be a tool to understand underlying causes of such relationships. Unfortunately, autocorrelation can also be a barrier in ecological studies, as it interferes with standard statisticd hypothesis testing.

Lack of independence arnong observations increases the probability of a type 1 error. by inflating the degrees of fieedom (Legendre 1993). Animals typicdly move in a non- random fashion, and thus ecologists are fiequently faced with strongly autocorrelated data sets, particularly when frequent observations are collected using radiotelemetry. Although fiequent monitoring of individuals should not be discouraged (Swihart and Slade 1985b). a common procedure is to eliminate autocorrelation before estimating home range size, either by subsampling (Worton 1987. Ackerman et al. 1990. Kenward 1992) or by restricting the sampling regime afier a pilot study (Swihart and Slade 198%). It is from these truncated data sets that hypotheses are tested and conclusions are drawn.

Eliminating autocorrelated fixes from the data set not only reduces the sample size but may also limit the biological significance of the analysis. Some statistical methods of home range analyses produce home range sizes that are inversely proportional to the degree of temporal dependence between observations (Swihart and Slade 198%.

1985b). For example, restricting sampling effort to statistically independent time intervals underestimated the home range size of pronghoms (Antilocapra amerkana) and coyotes

(Canis latrans) (Reynolds and Laundre 1990). It is questionable whether a statistically independent data set can sufficiently descnbe what is essentially a non-independent phenornenon, as autocorrelated observations may reveal better behavioural information than would independent observations (Lair 1 98 7). There have been many studies comparing home range estimates of independent and autocorrelated data sets (Swihart and Slade 1985a, 1985b. White and Garrott 1990, Cresswell and Smith 1992). but few that have examined which method is more appropriate.

The objective of this study was to compare how well statistically independent and autocorrelated observations represented home range use. First I used a Monte Car10 simulation to generate locational observations within a bounded area and changed the degree of autocorrelation among the observations to determine the effect of autocorrelation on home range size. Second. I cornpared space use within the home range.

1 assumed that the probability of detecting an animal within a given area of a home range. as estimated from an utilization distribution. represents the tirne spent in that area

(Samuel and Garton 1987, Seaman and Powell 1996). 1 compared the probability of locating an animal within an area to the actual time spent in that area. both before and afier reducing autocorrelation to determine how eliminating autocorrelation affects how well the estimate of home range correlates with the tnie home range. Third, 1 examined the total distance travelled within the home range. Increasing the time interval between observations has been shown to underestimate the distance travelled by an animal

(Reynolds and Laundre 1990). 1 repeated their test to determine the extent to which reducing autocorrelation affected the total distance travelled between consecutive observations. Finally, 1 tested if reducing autocorrelation affected the accuracy of the home range size estimate. 1 compared the total area traversed by the animal with the home range estimate after increasing the time interval between observations. 1 used data from a computer simulation, and two species as models: antler Ries (Profopiophilalifigata) and snapping turtles (Chelydra serpentifla).

Male antler flies, which mate on abandoned cervid antlers. are aggressive insects

that defend well-defined lek temtories (Bonduriansky 1995). Males defend their

temtories with frequent agonistic contests. and have a high site fidelity (Bonduriansky

1996). Consequently, their small territories are well defined. and their time budgets easily

measured. Snapping turtles are omnivores t hat inhabit shallow wetlands and marshes. and

have overlapping home ranges. although male snapping turtles rnay temporally. if not

spatially, avoid each other (Ga1braith er al. 1986). Since they rarely bask. and are highly

cryptic. their home ranges are diffcult to define and their activity patterns are hard to

establish. Antier flies represent ideal study organisms for home range analyses. whereas

snapping turtles represent a more typical study animal because of their enigmatic

rnovement patterns.

Methods

Monte Car10 simulation

To examine the effect of a correlation on home range size. paths were

constructed using highly autocorrelated observations. and then autocorrelation (estirnated

using c'lr'. see next section) was reduced by subsampling. I generated paths by randomly

selecting a location using a unifonn random number generator. Each subsequent location

was estimated using X, = 4-,+ E, and Y,= Y,-,+ E!, where X, and Y, are Cartesian

coordinates at time i, and E, and E, are nomally distributed randorn error terms (Swihart and Slade 1985a). The error terms have a mean of zero, with a constant variance. Al1 the observations were limited to a 100 x 100 unit square home range. 1 generated 100 paths of 500 observations each by using a smail variance terrn relative to the size of the home

range. where E, = E, = 12. I chose this value so that the paths were highly autocorrelated at

shorter time intervals, yet more independent at longer time intervals. These paths were

subsampied using time intervals of 2 through 15 units, producing 1500 paths in total.

Estimates of home range sizes were calculated for al1 paths for each tirne interval using a

kemel density anaiysis. I used least squares cross validation (LSCV) to estimate the

optimal value of h (Searnan and Powell 1996). 1 used a multiple regression to determine

the effect of time interval and autocorrelation on home range size estimates.

Field studies

In 1995. antler nies were observed at the Wildlife Research Station. Algonquin

Park. Ontario (45O35'N. 780401W).A 2x2 cm grid was drawn on the upper surface of a discarded moose antler that was occupied by flies. Male flies were caught. placed in a restraining device. and an individual code was painted on the notum of each fly

(Bonduriansky and Brooks 1996). The locations of several temtorial males were recorded at 10 sec intervals for the duration of their temtory defense. which may last for over 30 minutes (Bonduriansky 1996). The movement of four antler flies were included in the analyses.

Snapping turtles were trapped at Cootes Paradise. Hamilton. Ontario (43' 17'N.

79O53'W). Cootes Paradise is a highly eutrophic 90-ha marsh, which is surrounded by emergent vegetation but little submergent vegetation. Snapping turtles were caught using unbaited hoop traps in the summers of 1994 and 1995. Radiotransmitters (Holohil

Systems Inc., 1 12 John Cavanagh Rd.. Carp, Ont.) were attached to adult male snapping turtles by drilling two small holes in the postenor marginal scutes. and tied on using

0.0 12 gauge steel trolling wire. Turtles were tracked from a canoe using a hand held receiver (Wildlife Materials Inc, Carbondale. Il.) and yagi antema. They were usually tracked up to 5 times a week. either on consecutive or altemate days, fiorn May to late

August 1995. The radiolocation were recorded on a 10x 1 O meter gnd. The rnovernents of five snapping turtles were included in the analyses. The total distance travelled by both antler flies and snapping turtles were also estimated at time intervals of varying length.

The true total distance covered would be measured by length of a Iine joining continuous observations separated by O time. 1 estimated this distance as the y-intercept of a polynomial regression of the total straight line distance between consecutive observations. and the tirne intervat between the observations.

Density estimation

1 used kemel estimators to measure home range because they are among the more reliable home range estimators (Worton 198% 1995. Searnan and Powell 1996). A fixed kemel density estimate is calculated by

where K is a unimodal syrnmetncal bivariate probability density for a given grid point x, h is the smoothing parameter. and X, is a random sample of n independent points from the unknown utilization distribution (Worton 1989). A utilization distribution is generated by making a surface plot of the kernel densities for al1 of the grid points. Seaman and Powell (1 996) found that home range size estimates were more accurate when they used fixed kemels. The fixed kemel estimates may fom spurious noise at the edges of long-tailed distributions (Silverman 1986). and so may be biased as there are ofien areas within a home range that receive little use by the animal. Adaptive kernels Vary the smoothing parameter with the estimated density. such that noise at long- tail distributions is smoothed without "over smoothing" areas of high density (Silverman

1986). The equation to measure the adaptive kemel is identical to Eq. 2.1. except h is replaced by h,, where h, varies with the density estimated by a "pilot" estimate. such as the fixed kemel estimate (Worton 1989). 1 used fixed kemels to estimate home range size. which does not involve the three dimensional shape of the distribution. but merely produces and outline of the home range. and I used adaptive kemels to estimate the shape of the probability distribution.

Other home range estirnators

Although most of the focus in this chapter is upon kemel density estimates. 1 also examined the accuracy and precision of other home range estimators. I estimated the home range size of the first 20 replicates from the Monte Carlo simulation using MCP. hmonic means analysis. and the Jennrich-Turner probability ellipse (Jennrich and

Turner 1969).

The home range size of a MCP is calculated by drawing a polygon using the

outmost locations without using any interval angles that are greater than 1 80 O. The area within the polygon is given by: where (x,,t,,) are the coordinates of the polygon ordered in a ciockwise sequence. and n is the number of coordinates (White and Gmoa 1990). The home range size of a probability ellipse cm be calculated by:

where s,' and s: are the variances of the x and y coordinates. and s, is the covariance

(White and Grnon 1990). A 95% confidence ellipse is obtained if a= 0.05. and f(I+Y21 =

5.99. The home range size estimated by harmonic means is calculated by estimating the utilization distribution, rnuch in the same manner as kernel estimators. However. instead of a kemel. the inverse first areal moment is calculated at each grid intersection that is superimposed over the observations. The harmonic mean d,, is calculated by

where (x,,yx2) is a grid point. and (x,.y,) are the coordinates (Dixon and Chapman 1980).

Measure of autocorrelation

I used Schoener's ratio (el$) to estimate temporal autocorrelation (Schoener

198 1), where r' is the mean squared distance between successive observations. and is defined by where rn is the number of successive observations. The mean squared distance between each observation and the center of activity is defined as:

where n is the number of observations, and (X. Y) is the arithmetic mean of the observations (Schoener 198 1). The "time to independence" (TTI) was estimated by subsampling sets of observations. and thus increasing the time intervals between observations, and calculating Tl$. 1 then compared the observed r'l? values with critical values calculated for bivariate uni form distributions (~~0.25)(Swihart and Slade 1%Sb). and rejected the nul1 hypothesis of independence if the observed t'l? lay outside the cntical values around the expected value of 2. Since the arithrnetic mean of X and Y. which represents the center of activity. would change with subsampling, autocorrelation may be underestimated if the animal exhibits cyclical movement. If. through subsampling, a disproportionate number of observations fa11 within a small portion of the total home range, the mean squared distance between each observation and the true center of activity would be underestimated. and thus inflate r'/>".

To cornpensate for this. 1 chose a static center of activity to estimate p. thus reducing sarnpling error fiom subsampling. A minimum convex polygon (Mohr 1947) was constructed of al1 the observations. and the arithmetic mean of the corners defining the polygon was calculated. I used the corners of a minimum convex polygon to derive a static center of activity. because this type of home range analysis is not directly affected by autocorrelation (Swihart and Slade I985a. Harris et al. IWO). The mean value of the corners was used for al1 subsequent estimations of i (henceforth called ?,,,). Both r'l? and r'l?,, were calculated for each time interval. until there were fewer than ten observations. I calculated Niand r'lr?,,, for different time intervals for both snapping turtles and antler flies, and a nonlinear regression of best fit was calculated to determine the relationship between autocorrelation and time interval. Independence between observations was assurned when there were three consecutive nonsignificant el? values.

The first of the three nonsignificant values was taken to represent the TT1 (Swihart and

Slade 1985b).

For each time interval. a utilization distribution was estimated using an adaptive kernel and a grid was supenmposed over the observations. The relative time spent throughout the home range was estimated by summing the nurnber of observations within each grid sector, where each observation represented the time span between successive observations at the shortest tirne interval. Thus. each antler fly observation represented 10 seconds, while each snapping turtle observation represented 2 days. The kemel densi- estimate was also surnmed within each grid sector. and a regression of density vs. tirne spent was calculated for each time interval.

Initially, 1 used a LSCV approach to determine the optimal value of h for the kernel density estimate (Seaman and Powell 1996), but 1 foound that this approach underestimated the home range size. There were ofien multiple. identical observations. which caused LSCV to produce an overly small value of h. and thus underestimate the home range (Searnan and Powell 199 1 ). 1 chose to estimate h by comparing the estimate uith a minimum home range size calculated by grid cells. 1 used the complete data set. and for each animal 1 counted the number of grid cells that contained observations. This defined the minimum home range size. Still using the complete data set. 1 calculated the value of h that produced a home range estimate equal to the minimum home range size. 1 used the same value of h for ail subsequent time intervals to keep al1 home range estimates comparable. Home ranges were not calculated when there were fewer than 10 observations. 1 used regression to examine the effect that time interval and autocorrelation have upon home range size. First 1 compared the mean home range size with time interval, as then with the mean il? for each time interval. In case averaging obscured individual trends. 1 perf'orrned the same regressions with each individual.

Statistical analyses were calculated using the program Statistica v 5.0 (StatSofi

1995). except for the kemel probability densities. which were generated by Kemel Home

Range Program v. 4.2 1 (Searnan and Powell 1991). and the MCP. hmonic means. and probability ellipses, which were generated by CALHOME (Kie ri uZ. 1996). The autocorrelation statistics were calculated and the kemel density estimates were plotted using Mathcad Plus v.6.O 1 (Math Sofi 1995).

Results

Monte Carlo simulation

There was a significant effect of both autocorrelation (b = 0.1 12 1. P = 0.005) and tirne interval (b = 0.4585, P < 0.000 1 ) on home range size (adj. ? = 0.3097. F = 335.8 1.

df = 2. 1497. P < 0.000 1). Home range size shrank with both increasing autocorrelation

and decreasing time interval. Autocorrelation among locational observations declined as

the time interval between observations increased. until independence was nearly reached

at the longest time interval (Fig. 2.1). The estimates of autocorrelation were less variable

at shorter time intervals (Fig. 2.1 ). where sample sizes were larger and thus perhaps less

sampling error occurred. Home range estimates were least biased and most precise at the

shortest time intervals. and thus at the highest levels of autocorrelation (Fig. 2.1. 2.2). The

magnitude of the absolute errors (between each replicate and the true value) of the home

range estimates increased with the time interval (adj. i = 0.3056. F = 660.79. df = 1.

1498, P < 0.000 1). Unfortunately the residuals were strongly heteroscedastic. and the

relationship was not quite linear. I was unable to use a transformation that simultaneously

reduced heteroscedasticity and linearized the relationship, so the regression was

unreliable. Instead, I regressed the mean error (bias) of the home range estimates with the time interval. Both the variance of the home ranges size estimates (adj. 9 = 0.8928. F =

1 17.55, df = 1. 13. P < 0.0001). and the squared bias of the home ranges estimates (adj. i

= 0.9290. F = 184.32. df = 1.13. P < 0.000 1 ) were positively related to the time interval.

Sample size was not included in the regression because it was redundant with the time

interval. 1 kept the total sampling period constant. so sarnple size is a (nonlinear) function of the time interval.

Other home range estimators

Both the harmonic mean and MCP estimates improved with a decreasing time 12 3 4 5 6 7 8 9101112131415 Time interval

Figure 2.1. Mean and standard deviation of r'lT of simulated home ranges at different time intervals. The expected value of independent observations is 2.

46 Time interval

Figure 2.2. Mean and standard deviation of home range estimates (kemel density) fiorn the Monte Carlo simulation at different time intervals. The unbiased home range size is 10000 units, and there are 100 samples per time interval. interval, despite the increased autocorrelation (Fig. 2.3,2.4). The magnitude of the errors

(between each replicate and the true value) of the home range estimates increased linearly with the time interval (harmonie means. adj. 6 = 0.8494. F = 1687.9, df = 1.298, P <

0.0001 ; MCP adj. ? = 0.6985. F = 693.77. df = 1.298, P < 0.0001). However, the rnean home range size estimates from the 95% probability ellipse did not change with the time interval (Fig. 2.5). The magnitude of the errors did not change with the time interval for the 95% probability ellipse (adj. ? = 0.0072. F = 3.16 19, df = 1. 298. P = 0.0764).

Although the bias remained unchanged, the home range estimates were biased for al1 time intervals. 1 repeated the analysis using a 75% probability ellipse, and found that mean home range size did not change with the time interval (Fig. 2.6), and the magnitude of the errors did not change with the tirne interval (adj. 6 = 0.0065, F = 2.985. df = 1,298, P =

0.085 1).

There was significant heterscedasicity of the residuals for both the 95% and 75% probability ellipse home range estimates. but 1 was unable to use data transformations to equalize the variance. Regression coefficients are unbiased and consistent if unequal variance exists arnong the residuals, but the resulting F and I tests are affected (Marshall et ai. 1995). The estimated variances of the regression coefficients are underestimated if there is a positive correlation between the variance of the residuals and the independent variable (Krnenta 1986). As a result. the calculated P-values would underestimate the true

P-values. The F-test and t-tests associated with the regression coefficients of the 95% and

75% probability ellipse home range estimates resulted in P-values that were too small, but nevertheless they were non-signifiant. Thus, the bias of the home range estimates Time interval

Figure 2.3. Mean and standard deviation of home range estimates (harmonie means) from the Monte Car10 simulation at different time intervals. The unbiased home range size is 10000 units. and there are 10 sarnples per time interval.

49 Figure 2.4. Mean and standard deviation of home range estimates (MCP) from the Monte Carlo simulation at different tirne intervals. The unbiased home range size is 10000 units. and there are 20 samples per time interval.

50 ------12 3 4 5 6 7 8 9101112131415 Time Interval

Figure 2.5. Mean and standard deviation of home range estimates (95% probability ellipse) from the Monte Carlo simulation at different tirne intervals. The unbiased home range size is 10000 units. and there are 20 samples per time interval. Figure 2.6. Mean and standard deviation of home range estimates (75% probability ellipse) fiom the Monte Car10 simulation at different time intervals. The unbiased home range size is 10000 units. and there are 20 sarnples per tirne interval. was independent of the interval using the 95% and 75% probability ellipse estimates.

The standard deviation of the home range estimates using harmonic means increased with the time interval (adj i = 0.5749, F = 19.935, df = 1.1 3, P = 0.0006).

Similarly, the square root of the standard deviation of the 95% and 75% probability ellipse and MCP home range estimates increased with the time interval (95% ellipse, adj. i = 0.4487, F = 12.396, df = 1.13, P = 0.0038; 75% ellipse, adj. ? = 0.4665. F = 13.240. df= 1.13, P =0.003; MCP. adj. 3 =0.5055. F = 15.311, df= 1.13. P = 0.0018). A square root transformation was used with the 95% and 75% probability ellipses and the MCP home range estimates to reduce heterscedasicity of the residuals.

Field studies

Consecutive observations of antler fly movement were recorded every 10 sec for a maximum of 25.5 min. yielding a total of 423 observations for four antler flies. Agonistic contests with rival male flies bccurred for 18 of the observations for one of the flies. when the fly would attack and bnefly leave its defended territory to chase the intruder.

Because these agonistic contests occurred over a much larger area than the area normally patrolled, observations during agonistic behaviour were excluded from the home range analyses. There were 134,42, 1 16. and 1 13 observations for the four antler flies. The largest gap between observations after excluding agonistic contests was 40 sec, although most of the gaps were 10 to 20 sec. The mean time interval between observations was

10.43 sec (S.D. = 3.65, n = 41 8). As the time interval between observations increased, the estimated total distance travelled declined (Fig. 2.7). The tme straight line distances travelled were estimated by extrapolation to be the y-intercepts, although these were O 20 40 60 80 1O0 120 140 Time interval (sec)

Figure 2.7. Estimated total distance travelled as a function of increasing time intervals between observations. The true total distance travelled is estimated as the y-intercept. Polynomial regression of total straight Iine distance and time interval of four antler flies. 0 antler fiy 1 : ? = 0.9755, F = 145.85, df = 3. 1 1. P < 0.000 1; y - int = 92.67 cm antler fly 2: i = 1.0; y - int = 11.57 cm + antler fly 3: 1 = 0.9769, F = 4.27. df = 3, 9, P = 0.0389; y - int = 66.89 cm antler fly 4: 1 = 0.9745, F = 114.54, df = 3, 9, P < 0.0001; y - int = 140.19 cm underestirnates as animais tended not to travel in straight lines.

The time intervals between observations of snapping turtle were varied, and so observations were excluded to give a minimum of 1 observation every 2 or 3 days. The sarnpie size of the 5 sets of observations of snapping mies was 35,43,38,4 1, and 28. which were recorded over 4 months. The mean time between observations was 3.12 days

(S.D.= 1.87. n = 178). excepting the first observations. which were collected a month earlier than the rest when the turtles were hibemating. Deleted or missing observations increase the mean time interval between observations. and so autocorrelation would be slightly underestimated. As the time interval between observations increased, the estimated total distance travelled deciined (Fig. 2.8). Again, the tme straight line distances travelled were estimated by extrapolation to the y-intercepts.

I averaged the r'/>' and r'l?,,, values of the antler flies for each time interval.

There was a non significant positive trend between the average 8li values of antler nies and the time interval (adj. i = 0.2 149. F = 4.559. df = 1. 12, P = 0.0540). but there was a positive relationship between the average r'l?,,, and the time interval (adj. 2 = 0.2623. F

= 5.623. df = 1. 12. P = 0.0353). Generally. independence was not reached despite increasing the time interval of individual antler flies (Fig. 2.9.2.10). Only 14.29% of the

$/>2 values and none of the t'l~,,, values reached independence (a = 0.25). Similady. 1 averaged the pl>' and r'/i,,, values of the snapping turtles for each time interval. There was a significant relationship between the averaged r'/+ and the time interval (adj. ? =

0.2464, F = 5.250, df = 1. 12. P = 0.0408), and between the averaged I'/?,, and the time interval (adj. i = 0.4427, F = 1 1.326, df = 1, 12. P = 0.0056). Although the values of r'l? Time interval (days)

Figure 2.8. Estimated total distance travelled as a fùnction of increasing time intervals between observations. The true total distance travelled is estimated as the y-intercept. Polynomial regression of total straight Iine distance and time interval of five snapping turtles. CI snapping turtle 1 : ? = 0.9789, F = 154.64, df = 3, 16, P < 0.0001 ; y - int = 7.48 km + snapping turtle 2: 9= 0.9026. F = 2 1.6 1. df = 3. 7. P = 0.0006: y - int = 3-44 km snapping turtle 3: ? = 0.6105. F = 4.70. df = 3,9. P = 0.0307; y - int = 2.38 km A snapping turtle 4: ? = 0.965 1, F = 64.58, df = 3, 7, P < 0.0001; y - int = 3.83 km 0 snapping turtle 5: 9 = 0.9654, F = 82.78, df = 3.9. P < 0.000 1; y - int = 2.93 km O 20 40 60 80 1O0 120 140 Time interval (sec)

Figure 2.9. Relationship between r'lT with tirne interval between observations. Polynomial regression of il>"and time interval of four antler flies. 0 antler fly 1 : y = 0.077 + 0.035~-0.00041 x' + 0.00000 1 5x3 antler fly 2: y = 0.14 + 0.0 10x + 0.00032~'- 0.0000 10x3 + antler fly 3: y = 0.082 + 0.044~- 0.00040~'+ 0.00000 1 lx3 antler fly 4: y = 0.23 + 0.076~- 0.0012~'+ 0.000006x3 The expected value of independent observations is 2. Tirne Interval (sec)

Figure 2.10. Relationship between r'l?,, with time interval between observations. Polynomial regession of r'/$,c, and time interval of four antler flies. 0 antler fly 1 : y = 0.1 7 + 0.020~- 0.00020x2 + 0.00000069x3 antler fly 2: y = 0.03 1 + 0.028~- 0.00080~'+ 0.000010~~ + antler fly 3: y = 0.12 + 0.030~- 0.00024~' + 0.0000064x3 antler fly 4: y = 0.20 + 0.05 1x - 0.00082~'+ 0.0000041 x' The expected value of independent observations is 2. and r'/$,,, of individual snapping ~iesinitially increased, they then appeared to reach an asymptote and for two turtles the values declined as the time interval between observations increased (Fig. 2.1 1.2.12). Only 26.23% of the tl? values (most from one turtle) and 0.03% of r'lr',, reached independence (a = 0.25).

Subsampiing affected the distribution of observations within the home range.

While there were sometimes drastic reductions in antler tly home range size with subsarnpling as rneasured by MCP (e.g. Fig. 2-13), the snapping turtle home range size did not change much with subsarnpling (Fig. 2.1 3). In both cases, the subsampled home ranges were constmcted From statistically independent observations.

Single examples of an antler fly. at 1O-sec and 70-sec intervals. and a snapping turtle at 2-day and 6-day intervals, illustrate the effects of subsampling on the probability distribution estimated by kemels. Although the locations of the peak densities remained the same. the relative heights were different. and the distribution at low densities (ie.. around the perimeter) were different (Figs 2.14% b, 2.15a, b). Every regression was significant, but as the time interval between observations increased. the relationship between the number of observations and probability decreased (Tables 2.1.7.2).

The mean home range sizes of al1 antler flies showed no significant relationships with time interval (adj. ? = -0.0490. F = 0.3932. df = 1. 12. P = 0.5424), but individual home ranges showed two significant negative relationships, and two non significant trends (Table 2.3). The mean home range size for al1 antler nies showed no significant relationship with the mean r'l?,,, (adj. i = 0.041 8, F = 1.567. df = 1. 12. P = 0.2345). nor were there any significant relationships between home range size and &r',,, of individual O 2 4 6 8 10 12 14 16 Time interval (days)

Figure 2.1 1. Relationship between r'l? with time interval between observations. Polynomial regression of r'li and time interval of five snapping turtles. snapping turtle 1 : y = 0.77 + 0.23~- 0.024~' + 0.00086x3 + snapping turtle 2: y = 1.57 - 0.057~+ 0.024x2 - 0.0020x3 snapping turtle 3: y = 2.06 - 0.048~+ 0.0084~'- 0.00035x3 A snapping turtle 4: y = 0.75 + 0.30~- 0.063~' + 0.003 1x3 0 snapping turtle 5: y = 0.032 + 0.32~- 0.060~' + 0.0034x3 The expected value of independent observations is 2. Time Interval (days)

Figure 2.12. Relationship between ?/r',,, with time interval between observations. Polynornial regression of r'li,,, and time interval of five snapping turtles. snapping turtle 1: y = 0.47 + 0.1 7x - 0.000 17x2 + 0.00000066x3 + snapping nirtle 2: y = 2.06 - 0.043~+ 0.00082~'- 0.0000046x3 snapping turtle 3: y = 1.33 - 0.026~+ 0.00040x2+ 0.00000 16x' A snapping turtle 4: y = - 0.80 + 0.086~- 0.0014x2 + 0.0000063x3 osnapping turtle 5: y = - 0.79 + 0.06 1x - 0.00092~'+ 0.0000044x3 The expected value of independent observations is 2. al t observations n = 138

---A* independent

Antler fly

al 1 observations n=43

---a-.-** independen t n= IO

Snapping turtle

Figure 2.13. Cornparison of MCP home range of a male antler fly and snapping turtle for ail observations. and first statistically independent (W)subsample.

62 Figure 2.14.Adaptive kemel density estimstes. (a) Antier fly home range with 10 sec. intervals between observations (h = 2.6, n = 143) (b) Antier fly home range with 70 sec. intervals between observations (h = 2.6, n = 21) Figure 2.1S. Adaptive kemel density estimates. (a) Snapping turtle home range with 2 day intervals between observations (h = 1.2, n = 43). @) Snapping turtle home range with 6 &y intervals between observations (h = 1.2, n = 17). Table 2.1. Regression summary of number of observations within each grid ce11 with the respective probability density. for each time interval for the male antler fly.

Time interval / # of observations adj. i F P-value Sample size 10 0.5743 134 0,9793 4642.4 1

-- - - Time intervai WmCp# of observations adj. ? F P-value Sample size -7 1 .O97 43 0.9308 1 170.92

adj. # Slope F Sample size P-value fl~1 O. 1786 -0.1523 3.826 14 0.0742 fly 2 0.8433 -0.1620 17.143 4 0.053 7 fl~3 0.450 1 -0.1335 10.002 12 0.0 1O 1 fly 4 0.7785 -0.3935 39.662 12 0.000 1 antler flies (Table 2.4). 1 chose not to estimate snapping turtle home range sizes, because counting the number of grid cells underestimated the minimum home range size. Many of the grid cells were isolated, thus excluding areas which the turtles must have used, and so

1 could not accurately describe the reiationship between home range size and the time interval or autocorrelation.

Discussion

Monte Car10 Simulation

Decreasing the time interval between successive observations improved the estimates of the simulated home ranges. Although autocorrelation had a small (but significant) effect on home range size. even after the effect of the iime interval was removed, this effect nevertheless reduced the bias of the home range estimate. Similarly, after the effect of autocorrelation was removed, shorter time intervals reduced the bias of the estimates. The precision of the estimates also improved at shoner time intervals. although 1 do not know if this is due mainly to reduced sarnpling error associated with larger samples sizes or due to an irnproved performance of the kemel estimator.

Similady. the mean squared error (MSE) also irnproved, although 1 do not show the results here. Since the (MSE) is equivalent to the squared bias plus the variance, and both variance and bias decreased with shorter time intervals, then the MSE also decreased with shorter time intervals. Minimizing the MSE will minimize the expected error of each estimate of home range.

At al1 time intervals. simulated home ranges were overestimated. Worton (1 995) suggested multiplying the h estimated by LSCV by 0.6 to 0.8, which would reduce the Table 2.4. Regression of home range size and &?,, of four antler flies.

adj. ? F Sample size P-value fly 1 -0.0818 0.0 168 14 0.8989 home range size and thus reduce the bias. It appears that this correction factor should be inversely proportional to the time interval. where a correction factor should be considerably smailer than O at larger time intervals, and approach 1 at smail time intervals.

Other home range estimators

At ail time intervais. harmonic means and MCP home range estimators underestirnated the home range size. Home range estimates were less biased and more precise as the time interval decreased, despite increased autocorrelation among the observations. Surprisingly. the harmonic means and kemel density estimators were biased in different directions, considenng that the harmonic means analysis cm be considered to be a form of a kemel density estimator (Worton 1989) and both were estimated using grids supenmposed over the observations. However, these differences may be due to discrepancies in the computational methods of calculating the home ranges. rather than intrinsic differences in the estimators. For example. kemel density estimates that are based upon contours that bound 95% of the observations rather than 95% of the probability distribution (e.g.. Kie et al. 1996 and Seaman and Powell 199 1 ) underestimate instead of overestimate the home range size (unpublished data).

The probability ellipses improved with decreasing time intervals. Although the bias did not change. the precision of the estimates improved at shorter time intervals. The

95% probability ellipse overestirnated the home range size at al1 tirne intervals. while the

75% probability ellipse underestimated the home range size. The change in bias is not surprising, since decreasing a results in a smaller area for al1 probabilistic or pseudo- probabilistic home range estimators. However. while the harmonic means and kemel density estimators improve with changes in time interval, the only way to change bias or the probabilistic estimaton in my simulation is to change a. (See Chapter 4 for discussion on the unsuitability of probability ellipses).

It has been suggested that equivalent levels of autocorrelation (Swihart and Slade

1985a) or sample size (Harris et al 1990) are required before some home range parameters can be compared among different animals. It is also likeiy that the time interval has to be the same among animals to conectly compare home range estimates. meaning that sample sizes have to be the sarne as well if the total sampling period is to rernain the same.

Field studies

Increasing the time intervals between observations failed to generate consistently independent data sets for either antler nies or snapping turtles. The degree of autocorrelation cannot be reduced by increasing sample size because sample size could only be increased by reducing the time interval between samples. Antler flies defend temtones for only a short time before they are overcome by compeiitors or leave to copulate. Because their temtories are so transient, the only way to significantly increase sample size is by reducing the time interval between observations. The sampling regime used to estimate snapping turtles home ranges encornpassed alrnost the entire annual active season, so again sample size could only be increased by reducing the time interval between observations.

As an index of autocorrelation, ?/&, appears to be better than r'l?, because as the time interval increases, it is not as susceptible to sampling errors, especially at srnalier sample sizes. Due to sampling error, observations fiom small sections of the home range were over-represented. Estimating $/,' using a static center of activity would detect autocorrelation present when al1 or most of the observations were centered around a subsection of the home range. Clwnping of observations at certain intervals suggests repeated activity, such as those associated with foraging, mating, resting, or similar activities (e.g. Swihart and Slade 1985b). Using a variable center of activity to estimate autocorrelation might not detect a lack of independence arnongst observations, and so would reduce the ability to detect cyclical behaviour. 1 used the corners of a minimum convex polygon to denve a static center of activity; however, other methods of estimating the center of activity may also be appropriate. Lair (1 987), for exarnple, found the harmonic mean center (Dixon and Chapman 1980) to have advantages over the arithrnetic mean center (Hayne 1947) and bivariate median center (Neft 1966).

Increasing the time interval between observations of antler nies or snapping turtles did not reliably reduce autocorrelation. If the main cause of autocorrelation is due to short term movements. 1 would expect that, as the time interval between observations increased. i'l~,,, would also increase. The initial increase in independence arising fiom increasing the time interval was likely due to the elimination of short-term dependence between observations. but independence was not reached even after long time intervals. bkrvations may remaid~utocorrelatedif tk tlme interval mroaches an integer multiple of the period length for animals with cyclical behaviour, the animal shifts its home range, or moves along a path in a temporally predictable manner (Swihart and Slade 1985b). Although 1did not find evidence of periodicity in the relationship between

$1,' or r'/?,, and time interval. an asymptote was reached, where fiirther changes in the time interval did not strongly affect autocorrelation. Much of the autocorrelation present. particularly at the longer tirne intervals, is likely an invinsic property of the home range behaviour.

Antler Bies defend a stable temtory, but use short range search patterns to detect fivals or potential mates. Patrolling for mates or intruders should entail short but fiequent visits by the resident animal (Shewin and Nicol 1996). Similady. areas in which the turtles spent prolonged penods of time were also sites to which they frequently retumed. suggesting that these areas were refùges from which the turtle would occasionally venture. Any autocorrelation present at longer tirne intervals, while statistically dependent, were likely biologically independent (Lair 1987). The assumptions of homogeneous spatial and temporal sampling of the r'li statistic is violated when animals move systematically in a temporally predictable manner, habitat use is constrained by spatial heterogeneity, or there are shifts in the animals activity pattern (Minta 1992). The relevance of a repeated behaviour would be underestimated by eliminating autocorrelation, not overestimated by incorporating it in the analyses.

As expected. as the tirne interval between antler fly observations increased. the relationship between the nurnber of observations and the corresponding probability density weakened. Although the relationship was still significant. at 70-sec intervals only

71% of the variation in the probability density was explained by the number of observations, as opposed to 98% at IO-sec intervals. Yet even at 70-sec intervals, independence was not achieved, and was reached only sporadically at a 160-sec interval

for ?/?. and not at al1 for ?/$,,,. With such a long time interval. there were only 2 1 observations, or 15.7% of the total observations taken. Similady. the relationship between the number of observations and the corresponding probability density weakened as the time interval between snapping turtie observations increased. At 8-day intervais. only

56% of the variation in the probability density was explained by the number of observations. Independence was only erratically achieved at 15-day intervals. at which point there were only 17 observations. or 30.4 % of the initia! number of ~hservations.

Home range analyses cannot accurately estimate the home range size with so few observations (Harris et al. 1 990. White and Garrott IWO).

My results support Reynolds and Launcire's ( 1990) conclusion that increasing the

length of tirne intervals between observations underestirnates the tnie distance traveiIed.

The estimate of total distance travelled by antler flies decreased precipitolisly as sampling

rate was increased from IO to 50-sec intervals. then slowly reached an asymptotic

minimum. There was a similar decline in the estimate of total distance travelled by snapping turtles. from 2 to 8-day intervals. 1 extrapolated from a polynomial regression to represent the actual distance travelled. but this is likely an underestirnate since the regressions were based on straight line movement, rather than non linear movements typical of animals. Shorter time intervals than those used here would result in higher estimates. although at the expense of reducing independence between observations.

Nevertheless. this method may be an improvement on the method used by Reynolds and

Laundre ( 1990). who assumed that the shortest time intervals used represented the actual distance moved. Prediction loses accuracy with extrapolation unless the relationship does not change outside of the range in which the observations were made. However. a shorter time interval may have produced an estimate that better represents the tnie distance moved because the extrapolation is over a shorter distance.

General Conclusions

Our results contradict the conclusions of Swihart and Slade (1 985a) and Cresswell and Smith (1 992). First. 1 did not End any relationship between home range size and

I'/>Zm,, fiom my field data, and although 1 found a negative relationship between +/? and kernel density estimates of home range size from rny simulation, the effect reduced bias.

Also, other home range estimators such as harmonic means and MCP. and to a smaller degree probability ellipses. improved with shorter tirne intervals and higher autocorrelation. Instead. my attempt to reduce autocorrelation by increasing the time

interval between observations resulted in srnaller home range sizes. which is the opposite of previous predictions. Secondly, Swihart and Slade (1 985a) used a Monte Car10 simulation to mode1 the effect of autocorrelation on home range size. in which they kept the sarnple size constant. but the total sampling the penod was proportional to the time interval between observations. They found that using n nurnber of autocorrelated observations within a short period of tirne resulted in a smaller home range size than using n nurnber of independent observations over a much longer penod of time. I do not dispute their result that sampling within a shorter tirne frarne would result in a smaller home range estimate. but I disagree with their conclusion that autocorrelation is the cause.

If I had used a shorter total sampling period, my home range estimates would also have been smaller. Instead, 1 interpreted their results as suggesting that sarnpling penods should be as long as possible. so as to minimize the risk of any information loss, as well as to incorporate their entire home range behaviour.

TT1 is supposed to estimate the time required by the animal to traverse its home range, where its current position at time r is a function of home range use, rather than a function of its position at t- 1 (Swihart et al. 1988). However. this index of statistical independence is not the same as biological independence, because the animal may choose to move in a non-independent fashion (Lair 1987). Antler flies could traverse the length of their home ranges within seconds, and often did. Similarly. snapping turtles at Cootes

Paradise could traverse the pond which contains their entire home range within minutes.

That they usually did not is probably a better reflection of their pattern of home range use than is TTI.

It has generally been assumed that autocorrelation is caused by short time intervals (Swihart and Slade l985b. Worton 1987. Harris et al. 1990. White and Garrott

1990). rather than an inherent pattern of home range use. At shorter time intervals than

TTI, observations may not be statistically independent, but an animal's position at t may not be a function of its position at t- 1. Instead. the positions at t and f- 1 may both be a hction of a third factor. TT1 is generally considerably longer than the length of time required for the animal to travel between any two points within the home range (this study, Lair 1987). It should not be surprising that areas with autocorrelated observations are ofien associated with important resources (e.g. Swihart and Slade l985b. Lair 1 987.

Weatherhead and Robertson 1990). If an animal's resource use changed such that the animal spent more time at a resource. this would be interpreted as autocorrelation. The time spent at a resource was a better indication of resource use than the frequency of visitations (Shenvin and Nicol

1996). Schoener's ratio can be used to demonstrate how resource use is partitioned by rate of visitations and total length of time spent at the resource. If autocorrelation is caused primarily by time partitioning. rather than overly short time intervals. then the lack of independence among consecutive observations would be more apparent than real.

Eliminating the autoco~elationamongst the observations would alter the underlying spatial structure which is partially responsible for the values of any measurements taken

(Griffith 1992). The prime concem of rnost studies using probabilistic or pseudo- probabilistic home range analyses. such as kernel estimators, harmonic mean, Fourier. and probability ellipses, is to estimate the likelihood of finding the animal ai any point within the home range. or. altematively, time partitioning. which is essentially the same thing. One of the arguments for reducing autocorrelation arnong observations is that dependent data are redundant and yield less information than independent observations

(Swihart and Slade 1 985a. White and Gmott 1990). However. each observation. independent or not, holds the same information when used to determine time partitioning.

Given this objective. the researcher should be less concerned with the autocorrelation among consecutive observations, but instead should be mostly concerned with the autocorrelation between the time spent at adjacent areas within the home range.

Autocorrelation arnong consecutive observations is analogous to the problem of pseudoreplication, which is often defined as the use of inferential statistics to test for treatment effects using replicates that are not statistically independent (Hurlbert 1984).

The lack of independence among observations generally inflates the degrees of freedom for rnost statistical tests (Legendre 1993, Lombardi and Hurlbert 1996), and prohibits knowledge of the actual a value (Hurlbert 1984). However. using non-independent replicates or observations is not invalid. as long as the replicates are pooled to estimate a mean value for an experimental unit. and the correct degrees of freedom describing the number of experimental units, are used (Hurlbert 1984). Replication of samples within treatments increases precision by reducing "noise" or random error (Hurlbert 1984).

Individuai observations in home range analyses are not treated as independent replicates to compare treatment effects. but rather a single value, home range size, is estimated by using the observations as replicate samples. Each home range can be then treated as a single experimental unit if the treatment is specific to each animal. or if the treatrnent is specific to certain sites. the home ranges of each animal are also pooled to compare mean home range sizes among sites. In either case. the number of observations used to estimate each home range are not used to represent the degrees of freedorn for inferential statistics to test for treatment effects. Increasing the number of observations used to calculate home range size or shape increases the accuracy and precision of the home range estimate, but does not inflate the degrees of freedom used for inferential statistics. Consequently. the assumption of independence among sequential locational observations for home range analyses is not relevant.

There is an important exception to my conclusion about autocorrelation. As long as the time interval between successive observations remain relatively constant, autocorrelation should not be a problem. However, uneven sampling does bias home

range estimates. "Bursts"of sarnpling, where clusters of observations are closely spaced

in time but are separated from other clusters by a long time interval, will overestimate the

probability distribution at any area associated with the bursts. Sarnple size should not

necessarily be maximized at the cost of grossly unequal sampling intervais.

The nurnber of observations should be maximized for home range or time budget

analysis using constant time intervals, even at the expense of increasing autocorrelation

between observations. My findings demonstrate that shorter time intervals better estimate

parameters such as the accuracy and precision of home range size estimates. time partitioning, and distance moved. Furthermore, including autocorrelation in valid

statistical models is a preferable way to deal with autocorrelation rather than to attempt to eliminate it by restricting data pnor to analysis (Griffith 1992. Legendre 1993). Even if it was desirable to remove autocorrelation. this study shows that it may not always be possible to do so. This stresses the importance of combining models and empirical data to examine home range charactenstics. Field studies may reveal patterns not discernable

from simulations. while simulations allow proper replication and experimental control. I am not arguing that tests for independence are unimportant, because they can be used to infer movement patterns; however, eliminating autocorrelation reduces statistical power. reduces the accuracy of home range analyses, and destroys biologically relevant information. Few animals move in a random or ternporally independent fashion (but see

Loreau and Nolf 1993), so autocorrelated data are required to sufficiently mode1 animal movement and space use. Finally? if the prime concem of the researcher is to estimate time partitioning within the home range. then the Iack of spatial independence arnong observations does not violate assumptions of home range analyses. References

Ackerrnan, B. B., F. A. Leban, M. D. Samuel, and E, O. Garton. 1990. User's manual for

program home range. Second edition. Technical Report 15. Forestry. Wildli fe and

Range Experimental Station. University of Idaho. Moscow. Idaho. USA.

Bonduriansky, R. 1995. A new Nearctic species of Prolopiophila Duda (Diptera:

Piophilidae), with notes on its behaviour and cornparison with P. Ialipes

(Meigen). Canadian Entomologist 1 27:859-863.

Bonduriansky, R. 1996. Effects of body size on mate choice and fecundity in the antler

fly, Protopiophila litigata (Diptera: Piophilidae). M. Sc. Thesis. University of

Guelph. Guelph. Canada.

Bondunansky. R. and R. J. Brooks. 1997. A technique for measuring and marking live

flies. Canadian Entomologist.

Cresswell, W. J. and G. C. Smith. 1992. The effects of temporally autocorrelated data on

methods of home range analysis. In: Wildlife telemetry Remote monitoring and

tracking of animais. Priede, 1. G. and Swift, S. M. (Eds.) Ellis Horwood Ltd., UK.

Dixon. K. R. and J. A. Chapman. 1980. Harmonic mean measure of animal activity areas.

Ec010gy 6 1 :1040- 1044.

Galbraith, D. A., M. W. Chandler, and R. J. Brooks. 1986. The fine structure of home

ranges of male Chelydra serpentinu: are snapping turtles territorial? Canadian

Journal of Zoology 65:2623-2629.

Griffith, D. A. 1992. What is spatial autocorrelation? Reflections on the past 25 years of

spatial statistics. L'Espace geographique 3:265-280. Harris. S.. W. J. Cresswell. P. G. Forde, W. J. Trewhella. T. Woollard. and S. Wray.

1990. Home-range analysis using radio-tracking data - a review of problems and

techniques particularly as appiied to the study of marnrnals. Mammal Review

2O:W- 123.

Hayne, D. W. 1949. Calculation of size of home range. Journal of Mammology 30: 1-18.

Hurlbert. S. H. 1984. Pseudoreplication and the design of ecological field experiments.

Ecological Monographs 54: 187-2 1 1.

Jennrich, . R. 1. and Turner. F. B. 1 969. Measurernent of non-circular home range. J.

Theor. Biol. 22: 227-237.

Kie, J. G., Baldwin. J. A.. and Evans. C. J. 1996. Calhome: a program for estimating

animal home ranges. Wild. Soc. Bull. 24: 342-344.

Kenward. R. E. 1992. Quantity versus quality: programmed collection and analysis of

radio-tracking data. In: Wildlife teiemetry Remote monitoring and tracking of

animals. Priede. 1. G. and Swift. S. M. (Eds.) Ellis Honvood Ltd.. UK.

Krnenta. J. 1986. Elements of econometrics. Second Edition. Macmillan Publishing

Company. New York.

Lair, H. 1987. Estimating the location of the focal center in red squirrel home ranges.

Ecology. 68: 1092- 1 10 1.

Legendre. P. 1993. Spatial autocorrelation: trouble or new paradigm? Ecology. 74: 1659-

1673.

Lombardi, C. M. and S. H. Hurlbert. 1996. Sunfish cognition and pseudoreplication.

Animal Behaviour 27:4 19-422. Loreau, M. and C.-L. Nolf. 1993. Occupation of space by the carabid beetle Abmr afer.

Acta Oecologica I4:M7-258.

Marshall, P., Szikszai, T., LeMay, V.. and Kozak, A. 1995. Testing the distributional

assumptions of least squares linear regression. For. Chron. 7 1 : 2 13-2 18.

Math Soft, Inc. 1995. Mathcad. User's guide. Mathsoft inc.. Massachusetts, USA.

Minta S. C. 1992. Tests of spatial and temporal interaction among animals. Ecological

Applications 2: 1 78- 1 88.

Mohr, C. 0. 1947. Table of equivalent populations of North Arnerican mamrnais.

Arnerican Midland Naturalist 3 7:223-249.

Nef?. D. S. 1966. Statistical analysis for areal distributions. Monograph Series Nurnber

Two. Regional Science Research Institute. Philadelphia, Pennsylvania USA.

Reynolds, T. D. and J. W. Laundre. 1990. Time intervals for estimating pronghom and

coyote home ranges and daily rnovements. Journal of Wildlife Management

54:3 16-322.

Samuel. M. D. and E. O. Garton. 1987. Incorporating activity time in harmonic home

range analysis. Journal of Wildlife Management 5 1 34-257.

Schoener. T. W. 198 1. An empirically based estimate of home range. Theoretical

Population Biology 20:28 1-325.

Seaman, D. E. and R. A. Powell. 1991. Kemel home range estimation program. North

Carolina State University. North Carolina USA.

Seaman, D. E. and R. A. Powell. 1996. An evaluation of the accuracy of kernel density

estimators for home range analysis. Ecology 77:2075-2085. Sherwin. C. M. and C. J. Nicol. 1996. Reorganization of behaviour in laboratory mice.

Mus musculus. with varying cost of access to resources. Animal Behaviour

5 1: 1087-1093.

Silverman, B. W. 1986. Density estimation for statistics and data analysis. Chapman and

Hall, London. UK-

StatSoft, Inc. 1995. Statistica for windows. General conventions and Statistics 1. StatSoft.

Inc., Tulsa USA.

Swihart, R. K. and N. A. Slade. 1985a. Influence of sarnpling interval on estimates of

home range size. Journal of Wildlife Management 49: 10 19- 1025.

Swihart, R. K. and N. A. Slade. 1985b. Testing for independence of observations in

animal movements. Ecology 66: 1 1 76- 1 184.

Swihart, R. K.. N. A. Slade. and B. J. Bergstrom. 1988. Relating body size to the rate of

home range use in marnmals. Ecology 69:393-399.

Weatherhead. P. J. and 1. C. Robertson. 1990. Homing to food by biack rat snakes

(Elaphe obsolera). Copeia 1990: 1 164- 1 165.

White. G. C. and R. A. Gmott. 1990. Analysis of wildlife radio-tracking data. Academic

Press. San Diego. California. USA.

Worton, B. 1. 1987. A review of models of home range for animal movement. Ecological

Modelling 3 8 :277-298.

Worton. B. J. 1989. Kemel methods for estimating the utilization distribution in home-

range studies. Ecology 70: 164- 168.

Worton, B. J. 1995. Using Monte Car10 simulation to evaluate kemel-based home range estimates. Journal of Wildlife Management 59:794-800. "What we observe is not nature itself. but nature exposed to our method of questioning." Werner Heisenberg

"In al1 scientific fields, theory is frequently more important than experimental data. Scientists are generally reluctant to accept the existence of a phenornenon when they do not know how to explain it. On the other hand. they will often accept a theory that is especially plausible before there exists any data to support it." Richard Moms Chapter 3

Using search radii to choose the bandwidth of kernel estimators for home range analysis Abstract

The usual biological definition of home range is the area in which an animai

travels as it goes about its normal activities. This definition does not coincide with the

definition of a utilization distribution. which is oflen used as an estimate of home range.

Attempts to estimate the area of a utilization distribution from locational observations

taken from the animal's path converge to a line of zero area as the sarnple size is

increased. I suggest 2 methods to estimate home range which use the animal's search area

to determine the boundary of the home range. instead of the outer limits of the

observations. A naive estimator cmbe used in conjunction with the maximum detection

distance to estimate the area that the animal cm detect targets. The cumulative detection

function cmbe used as the kernel to estimate the probability of detection for any given

area. I simulated 100 animal paths within a bounded region and then subsampled the

paths at difierent tirne intervals. Home ranges were then estimated for each interval using

kemel density analysis. The bandwidth of the kemels were selected using least-squares cross-validation (LSCV) to estimate the utilization distribution. and also using various

invariant kemels to estimate the total search area. The use of LSCV led to inconsistent and highly variable estimates, while the invariant kemels tended to reach asymptotes and

had much lower variances. The limit of LSCV kernels (and thus home range size) tended to zero as the sample size tended to infinity. while the limit of invariant kemels was an

asymptote which depended upon the search radius. The use of search radii calls into question the concept of a single boundary home range: instead home ranges rnay be multi-dimensional areas that depend upon interactions between search patterns and resource types, and varies with spatial scale.

Introduction

Some recent difficulties in delineating home range have arisen due to the dichotomy that exists between the biological and statistical definitions of home range.

Home range is ofien defined as the area in which an animal travels as it goes about its normal activities, and is usually considered in terms of foraging and habitat selection (see

Burt 1943, JewelI 1966, Brown and Orians 1970). Home range is often estimated using a utilization distribution, which descnbes the relative time an animal spends in any particular area (Van Winkle 1975), and the home range is defined as the area of the utilization distribution that contains 95% of the observations.

Several different methods have been used to estimate the utifization distribution

(Calhoun and Casby 1958, Jennrich and Turner 1969. Dixon and Chapman 1980.

Anderson 1982. Don and Remolls 1983). Kemel estimators have attracted a lot of attention for home range estimation (Worton 1987. 1989, 1995, Naef-Daenzer 1993.

Seaman and Powell 1996. Tufio et al. 1996). Kemel estimation has some advantages over other methods because they produce true probability distributions. they have few assumptions about the underlying distribution or shape of the home range, and they are quite flexible. However. the main disadvantage of the kemel estimators is their extreme sensitivity to the use of sub-optimal bandwidths, or h (Silverman 1986, Marron 1988,

Hall et al. 199 1 ). Misinterpreting the nature of the population fiom which the observations are sampled may contribute to the selection of an inappropriate bandwidth. 1 shall demonstrate that this statistical definition does not coincide with the standard biological definition. The purpose of this chapter is to conectly identiQ the nature of the

population of values, and describe new approaches to choose the bandwidth.

Kernel estimators

The bivariate kemel takes the form

where K is a unimodal symmetrical bivariate probability density for a given grid point x, h is the

bandwidth, and X, is a random sarnple of n independent points fiom the unknown utilization

distribution (Worton 1989). The actual form of the kemel K does not greatly affect the efficiency

of the kernel (Silverman 1986), so the choice of the kernel is not as important as the choice of the

bandwidth (Epanechnikov 1969). The bandwidth. also known as the window width or smoothing

factor, changes the smoothness, bis. and variance of the estimate by changing the width of the kemels. One of the most cornrnon methods for determining the optimal baridwidth is to minimize the mean integrated square error (MISE). which is:

wherefis the estimated density, and f is the true density. Since the last term, f ',does not depend upon the estimated density, then the ideal h can be estimated

.ci,=j-f2 - 2s' (Silverman 1986). Automatic procedures such as least-squares cross-validation (LSCV,

Bowman 1984) optimize the selection of h to detemine the best estimate of the underlying distribution. Since f is unknown, LSCV uses the observations to constnict an estimate off (Silverman 1986). The optimal h is the one that gives a minimum score

M(h) for the estimated error:

where K* = Kc'- 2K. and K") is the bivariate normal density with a 2 x 2 variance- covariance matrix, whose left to right diagonal elements are 2. and the right to left diagonal elements are 0. (Worton 1989). Minimizing M(h) approximates rninimizing

MISE (Worton 1989).

Defining the population of observations

Kemel density estimators attempt to estimate the underlying distribution from which a set of observations are sampled. A kemel is a probability distribution which is placed over each observation. and the underlying distribution is estimated by summing al1 the kernels. The width of the kemel is determined by the bandwidth. h. In tems of home range. kemels are used to estimate the probability that the animal will be located at any point in space within any time period. and it is from this that the home range size is estimated. However. the actual population fiom which locational observations are sarnpled is the path of an animal; outside this path. there is no possibility of locating the animal. Thus, the underlying distribution of locational observations is the time the animal spent at points throughout the path. However, estimating the path is not the same as estimating home range. The most

cornmon biological definition of home range is, "that area traversed by the individuai in

its normal activities of food gathering, mating, and caring for young," (Burt 1943), which

implies a larger area than the animal's path. Most validation or randomization simulations

involving statistical home range analyses treat the population as if it were bounded within

a continuous area that contains the path (Swihart and Slade 1985a Boulanger and White

1990, Cresswell and Smith 1992. Worton 1995, Seaman and Powell 1996). In other

words, the home range forms a continuous distribution. of which there is a non-zero

probability of locating the animal at every point within the home range. Attempting to estirnate the underlying distribution of locationai observations is insufficient for estimating home range, since the home range includes areas which the path may never cross.

Using search areas as kernels

Home range cm be viewed as the general area or habitat which encloses the entire path of the animal. Thus. the home range is not just an animal's imrnediate surroundings as it rneanders across the landscape, but also the surrounding habitat that contains the path the animal follows. Habitat selection is a scale-dependent phenomenon.

Macrohabitat selection, at the level of the home range, is determined by the ability to disperse and relocate the home range (Morris 1992), and is likely dependent upon overall vegetative structure of the habitat (Laundre and Keller 1984. Bersier and Meyer 1995) and interactions with other animals (Galbraith et al. 1986, Homer and Powell 1990,

Galeotti 1994). Microhabitat selection, at the level of the path of the animal, is determined by differential use of foraging locations within a home range (Morris 1992).

The proportion of habitat used at smaller scales, at the level of the path, is likely deterrnined by taxonomie composition (Bersier and Meyer 1995). or similar immediate environmental cues. Ultimately. the population of values is the intensity of habitat selection. which is at the level of the perception. or search area of the animal (e.g. Loehle

1990. 1994). This is outside of our ability to estimate using only locational observations.

Loehle (1990. 1994) constmcted fracta1 home range estimates using the size of the search pattern of a hawk. He stated that home ranges consisted of search areas in which the animal perceived habitat, temtondity, and food resources, and that the search area can be estimated through experirnentation or observation. For exarnple, audible signals have been hypothesized to affect home range behaviour and temtoriaiity (Waser 1975).

Brenowitz (1 982) hypothesized that the active space (Le. effective range) of temtorid vocalizations should be on average twice the diameter of home ranges. An animal's perceptive range of acoustic cues may be important in determining home range or temtory (Waser 1975. Robertson 1984. Brown 1989). Similarly, chernical and visual cues are important for designating home range (Graves 1994, Lima and Zoliner 1996). Once the search range or perceptive abilities of animais are estimated, kernel density estimators cm easily be rnodified to incorporate such information.

The search area can be modelled either as a radius (see naive estimator below), or as a detection fûnction (see kemel estimator below). In either case. the search area is the area in which the animal is capable of detecting a target. The target, depending upon the question in which the investigator is interested, can be a prey item (e.g. insects, Tye 1989; seeds, Getty and PuIliam 199 l), small scale landscape structure (host trees and fit clusters, Roitberg 1985), or larger-scale habitat structure (forested habitat, Zollner and

Lima 1997). For exarnple. the search for host trees in the parasitic tephritid 8y

(Rhagoletis pomonella) is dependent on visual and chernical cues, hitclusters are identified primaily visually. but ideal oviposition sites are detennined using chemosensory information (Roitberg 1985). The selection of the target depends upon the scale of habitat selection in which the researcher is interested.

Using naive estimators to measure search areas

Perceptive abilities have been used to mode1 foraging or search dynarnics

(Hofhan 1983. Roitberg 1985, Nottingham 1988. Tye 1989). A cornmon method of incorporating perceptive abilities is to use a search radius. in which the animal is 100% successful in detecting a target within a radius r, (Stone 1975, Dusenberry 1989. Tye

1989). and 0% successful in detecting a target outside of this radius. The sweep width is thus W = 2 rd,and if the animal travels a straight line distance L. the total area searched is

WL. Although this method can be modified for nonlinear travel. it may be easier to use an estimator that is independent of any linear assumptions. If a cylinder of radius r is placed over each observation. then the density can be calculated by

which is a modification of a naive estimator. where w is the weight function (see naive estimator. Silverman 1986, pg 12). The value rd would be the maximum detection distance of the animal, and the area searched would be the area containing

100% of the density estimate. The value r, in Eq. 3.5 is analogous to the bandwidth h used in kemel estimates (Eq. 3.1). Unlike most probability densities which contain infinite area, the density estimate of the naive estimator is finite, so the area incorporating

100% of the estimated density will also be tinite. The advantages of this method is its simplicity, and ease of computation. and because it is similar in form to kemel estimators

(Silverman 1986), naive estimators have similar properties as kernels.

Using kernel estimators to measure searcb areas

A disadvantage of the naive estimator is that it assumes that the animal is 100% successful in detecting the target within r,,. However, animals do not detect everything within their search area. but instead will miss targets. The likelihood of an animal detecting a target declines with distance (generally considered a negative exponentiai function. Stone 1975. Getty and Pulliam 199 1 ), and changes with the inherent detectability of the target. Large. conspicuous objects would be easier to detect at al1 distances than smaller. cryptic objects. Also. the longer an animal scans an area. the more likely it will detect the target. The detection function. which is the cumulative probability that the animal detects the target i within time t. takes the form where r is the time spent searching. k is the detectability of the target. r, is the distance

between the animai and the target, and 6 is the distance decay parameter (Getty and

Pulliarn 199 1 ). nie distance decay parameter determines the rate that the probabiliv of

detection declines with distance - the larger the value of 6. the faster the detectability

declines with distance (Getty and Pulliam 1993). Obstructions within the line of sight. for

example, increase the rate at which detectability declines. Fig. 3.1 shows the relationship

between scan time and distance on the probability of detection of a target. assuming that k

= 0.0025, and 6 = 2 (arbitrarily chosen for an exarnple. and adapted from Getty and

Pulliarn 199 1 ).

It should be apparent, though, that as the scan tirne increases, the probability of detection will increase at al1 distances. Although this is fine for modelling short time scans, as the time of scanning increases towards infinity, because the exponentiai distribution is continuous and infinite. the probability of detecting a target will. unrealistically. tend towards 100% for al1 distances from the animal. One solution would be to tmcate Eq. 3.7 at the maximum detection distance (rd).which would restrict the probability of detection to O at distances longer than r,. Furthemore, the probability of detection can be restricted by substituting an asymptotic fùnction that places an upper limit on the effect that time has on the probability of detection into Eq. 3.7 such as a negative exponential fùnction of time, in place of time (t)(unpublished data). This would Probability of detection

Target distance Scan time

Figure 3.1. The relationship between scan time and distance to the target on the probability of detection. increase the probability of detecting a target at any given point in space as the scan time increases, but decrease the rate that time affects the probability at longer scan times.

However, an easier, if somewhat inaccurate, method of modelling the detection function is simply to use a normal distribution. As can be seen from Fig. 3.1, for any given time penod the relationship between the probability of detection and iarget distance roughly resernbles a normal distribution, and the standard deviation increases with increasing scan time. Rather than denving a complicated (and computationally expensive cornputer algorithm), 1 used a normal distribution as the kemel to sirnulate the perceptive abilities for the rest of this chapter. Fortunately, the shape or form of the kemel does not greatly affect the efficiency estimate (Silverman 1986), assuming that the kernel is bivariately symmetrical.

In order to incorporate perceptive range of an animal into the kemel method. the basic format of the kemel estimator can remain unchanged; only h need be altered.

Automatic methods for finding the optimal values of h. although highly usefùl. are not the only acceptable rnethod of determining h. Subjective choice is still suggested (Silverman

1986. Altman 1990), depending on the purpose of the estimate. if the detection function of an animal can be estimated. the shape of the kemel can be determined by the decay in perception (ie. by tmncating Eq. 3.7) or by using a normal distribution as the kemel. The bandwidth of the kemel. h. is determined by the standard deviation of the probability of detecting a target. Although estimating the perceptive abilities of animals is difficult. there are exarnples in the literature where the probability of detecting a target has been estimated over different distances (Roitberg 1985. Zollner and Lima 1997). Once we know the detection fûnction of the animal, the same h is used regardless of sample size, thus forming an invariant kernel that does not change with sample size.

Autocorrelation

Locational observations fiom an animal's path are constrained in that they must appear between the previous and subsequent observations - there is an order to the locational observations. If the sampling interval between observations is sufficiently low, then the value of one observation is dependent upon, and is more similar to. the previous observation. Autocorrelation. or the degree of (dis)sirnilarity among pairs of observations taken at certain distances apart. has been a recurring problem in descrïbing animal rnovement patterns, and is widely believed to violate assurnptions of home range models

(Van Winkle 1975. Schoener 198 1, Swihart and Slade 198%~b, Samuel and Garton 1987

Cresswell and Smith 1992). When using automatic procedures such as LSCV to estimate h. the degree of positive autocorrelation and h will be inversely related (Altman IWO). As autocorrelation tends towards m. h tends towards zero. As a result, home range size tends to decrease with increasing autocorrelation. 1 examined the effect of autocorrelation upon both invariant and LSCV kernels.

Consistency of home range estimators

Consistency is one of the most important properties of an estimator. For an estirnator to be at least weakly consistent, it must converge on the true value as the sample size approaches infinity. Thus. a necessary condition for a consistent estimator is that it mut converge upon an asymptote. Home range estimators are often unstable and may not reach a true asymptote within reasonable sample sizes. Some boundary methods. such as MCP, sometimes do not reach asymptotes even after 1000 observations have been collected (Doncaster and Macdonald 199 1, Gautestad and Mysterud 1993). Similady, anempts to estimate the underlying distribution of the locational observations are estimates of the path of the animal, and thus will converge towards a zero area. Since these analyses are inconsistent. we cannot guarantee a good estimate even if the sample included the entire population of values. Using invariant kernels may resuit in estimates that do converge upon a finite positive number, which satisfies at least one necessary condition for consistency.

1 compareci the performance of invariant kernels with kemels estimated by LSCV using a simulated data set, by comparing the precision and asyrnptotic behaviour of the estimators. I used Monte Car10 simulations in which locational observations were sarnpled at different time intervals from randomly generated paths, producing data sets with different sample sizes. Asymptotic limits of the kemel estimates for both invariant and LSCV kernels were calculated to determine whether the minimum requirements for consistency were met. It is not important for Our purposes to determine the actual limit to home range size. but merely to deterrnine if there is an finite asymptote. Since h does not

Vary with invariant kemels, the limit of the home range size cm be detemined by forcing the sample size to infinity. The limits of LSCV kemels. since h varies. can be detemined by forcing h to zero and infinity.

Methods

To examine the effect of autocorrelation and time interval on home range size. paths were constmcted using highly autocorrelated observations, and then autocorrelation was reduced by subsampling. Ninty-nine paths were generated by randomly selecting a location using a uniform random number generator within a 100 x 100 unit square home range. Each subsequent location was estimated using X, = 4-,+ E, and Y, = Y,-, + G, where

X, and Y, are Cartesian coordinates at time i, and E, and E, are normally distributed random error terms. 1 generated 99 paths of 500 observations each by using a small variance term relative to the size of the home range, where E, = E, = 12. These paths were subsampled. using time intervals of 1 through 15 units. Estimates of home range sizes were calculated for al1 paths for each time interval using LSCV, and with invariant h values of 1 5 to 45. in steps of 5.

1 used Schoener's ratio (z??) as well as my modification (I'/?,,,) to estimate temporal autocorrelation (Schoener 198 1, Swihart and Siade 1985b. Chapter 2). For each path, r'/? and t??,,, were calculated for every time intewal. A backward stepwise multiple regression was used to detemine the relationship between the sample size. time interval. and autocorrelation on the home range size for each invariant h. and for LSCV.

1 used a linear regression to determine if the variance arnong the estimates of the home range sizes declined with increasing sarnple size for each invariant kemel and

LSCV. I determined if the estimates converged on an asymptote within the range of sarnple sizes used in the simulation.

To further examine the asymptotic properties of invariant kemels, 1 determined the limits of the kemel functions. Ideally, 1 would calculate the limit of the home range size as the sample size increased towards infinity, for both invariant and LSCV kemels.

However, since the home range size estimate wouid depend partially on the location of the obsentations, die limit on the home range size would Vary and thus be in part a function of the spacing of observations. Since 1 was interested in the intrinsic properties of the kemel estimators to determine if they have a finite asymptote, 1 attempted to remove the effects of the variability of the location of the observations on the limits.

Instead of determining if the home range estirnates converge upon a finite value. 1 decided to focus on the kernet itself. Although 1 was interested in the area contained within the kemel. a computationally less intensive method is to estimate the limit of the height of the kemel as the sample size of h tends towards infinity. Assurning that al1 of the locational observations are confined within a finite space, the home range estimate can only be infinitely large if each kemel is as well. Sirnilarly, the home range estimate can only have zero area if the kemel also has zero area. Since each kemel always has a volume of one. the height of the kernel will approach infinity as its area tends towards zero, and vice versa. As long as the limit of the height of the kemel is finite, the limit of the home range area also has to be finite.

The maximum height of the kernel (using a bivariate normal distribution) would be at the centre. thus the point of evaluation would also be at the centre of the kemel. By substituting a bivariately normal distribution for the kemel K, Eq. 3.1 expands to

(x -,Y,)'(x -X,) > 1 2h'

(Worton 1989). If the location of the observations (4)is the sarne as the point of evaluation (x) such that 1 x-X, 1 = O, then Eq. 3.7 simplifies to The limit of the height of the kernels cm then be calculated fiom Eq. 3.9.1 used the

Kemel Home Range Program (v. 4.2 1 ) to estimate home range sizes (Seaman and Powell

1991).

Results

1 forced the total sampling penod to remain constant as would be expected in most practical situations, so as the time interval between successive observations increased. the number of observations in each path decreased (Fig. 3 -2). Similady. as the time interval between observations decreased, the degree of positive autocorrelation (&?) increased

(Fig. 3.2). Therefore. there is a positive relationship between sample size and the degree of autocorrelation. Independence of observations was largely reached at longer time intervals. although there does not seem to be strong autocorrelation until time intervals of less than 15 units (sample size = 67).

There was a significant effect of $12and t'l?,,, on the LSCV estimates of home range size (Tables 3.1, 3.2). As the degree of autocorrelation increased. the mean home range size decreased afier the effect of time interval was removed; f2l? and ~lr',,,were inversely related to the home range size of the invariant kemel estimates when the time interval was included in the multiple regression (Tables 3.1. 3.2). There was a significant effect of time interval on the home range size for both the invariant and LSCV kernels

(Tables 3.1.3.2) after removing the effect of autocorrelation. The home range size had a positive relationship with time interval for the LSCV estimates. but had negative Figure 3.2 (Monte Carlo) Table 3.1. Relationship between time interval and r'l? with home range size for LSCV and invariant kemel home range estimates.

Partial R'

tl? Time Interval h R' s n 15 20 25 30 35 40

45 LSCV Note: * significant with P < 0.001

* * significant with P < 0.0 1 Table 3.2. Relationship between time intervai and I'/i,,, with home range size for LSCV and invariant kemel home range estimates.

Partial R? Time Interval h p * n 15 20 25 30 35 40 45 LSCV Note: * significant with P < 0.001

** significant with P < 0.01

*** significant with P < 0.05 relationships with the time interval for al1 the invariant kernels (Fig. 3.3). However, when

I regressed the natural log of the i with the natural log of time interval using each invariant kernel, there was a strong relationship between the strength of the relationship between time intervai and home range size (?) and h (9= 0.9948. F = 95 1.50. d.f. = 1. 5).

In other words, time interval has a greater effect on home range size when smaller ban~widthsare used for invariant kemels. Sarnple size was never significant and so was removed as an independent variable for each multiple regression.

Invariant kemels were more consistent estimators than LSCV kemels in my simulation. For every tirne interval. the variance of the home range estimates produced by

LSCV was higher than that of the estimates produced by invariant kemels. and the discrepancy increased at longer time intervals (Fig. 3.4). Similady. the mean home range size estimated by LSCV did not converge at any time interval. and the rate of change increased at shorter time intervals (Fig. 3.3). The time interval had a smaller effect on mean home range size when using invariant kemels (Fig. 3.3). At larger values of h (15). the mean home range size approached an asymptote by the tenth time interval. while at smaller values of h (1 5). it is not clear that there was any convergence towards an asymptote (Fig. 3.3). Nevertheless. invariant kemels performed better than LSCV kernels.

From Eq. 3.8 the limit of the height of the kemel as sample size tends to infinity is

such that the limit of the kemel depends upon the value of h. The limit of invariant kemels therefore will be a finite nurnber because the search radius of an animal must also O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Time interval

Figure 3.3. Mean size of invariant and LSCV home range estimates at different time intervals. Time interval

Figure 3.4. Standard deviation of invariant and LSCV home range estimates at different time intervals. be finite. Since the height of invariant kemels is finite, the area of the kemels mut be as

well. The limit of the height of the LSCV kemels as h + O was m. and as h -+ was 0.

Since h calculated by LSCV can vary 60m infïnity to O, then the kernel height can range fiom O to -, and thus the home range area can also range from O to a.

Discussion

The performance of invariant kemels improved with decreasing time intervals, despite increasing autocorrelation. After controlling for the effect of the time interval upon the home range estimates, home range size increased with increasing autocorrelation, regardless of the value of h. Depending upon whether the value of h was overestimated or underestimated (and thus overestimate or underestimate the home range size), increased autocorrelation would increase or decrease the bias respectively. The effect of autocorrelation is dependent upon the appropriate selection of h, so autocorrelation by itself cannot be said to bias the home range estimate. The mean home range size increased with the sarnple size, until it approached an asymptote. Similarly, as the mean home range size estimates approached the asymptote. the variance among the home range estimates declined. These results partly contradict the conclusions reached by

Swihart and Slade (1985a), who stated that autocorrelation caused underestirnation of home range size. This discrepancy cmbe attributed to their model. which forced the sample size to remain constant even as they simulated changing the time interval between observations. Sample size can only be constant if the total sampling period varied with the tirne interval. In other words. the shorter the time interval between observations. the shorter the total sampling period. It should not be surprising that home ranges are estimated to be smaller when they are sarnpled for a shorter penod of time (which was associated with high autocorrelation in the mode1 by Swihart and Slade 1985a). It is more realistic to have a constant total sampling period, with varying time intervals and thus varying sampie sizes.

I found that the opposite was meof LSCV kemel estimates of home range. As the time interval becarne shorter, and thus the degree of autocorrelation increased, the home range size decreased. This effect upon home range size was evident, even afler the effect of time interval was taken into account. Nevertheless. the performance of LSCV kernels was ambiguous with decreasing time intervals between observations. The performance improved with decreasing time interval because the variance of the estimates declined drarnatically. The high variance associated with LSCV density estimates is one of the strongest drawbacks to the technique (Marron 1988, Hall et al. 199 1). The rate of convergence of the LSCV to the h associated with MISE is "excruciatingly slow" (Hall and Marron 1987). and it is not unknown for LSCV to produce different estimates from the same underlying distribution (Marron 1988). The dramatic reduction in variance arnong the LSCV home range estimates that I found with decreasing time intervals is therefore a marked improvement.

However, since the optimal h will tend to O as n tends to =, and the underlying distribution of locational observations is from a one dimensional line, the home range size estimates will also tend to O with increasing sample size. As a result, LSCV cannot converge to the tme home range size for any range of n. and so cannot be an even weakly consistent home range estimator. Invariant kernels, as long as a reasonable h is used to approximate the animal's search radius, is weakly consistent. At n = -, where every possible observations is used, the variance will tend to O (as will LSCV), but the home range estimate does converge to a finite value. Since the asymptote of the home range depends on the value of h. there is always some value of h for which the home range estimate will converge upon the tme home range value.

LSCV also performs badly when multiple observations take the sarne value. which often happens when the animal has not moved dunng 2 or more sampling periods.

This is analogous to discretization errors. where the values of observations are "rounded" or are collected only within certain grid cells. If the degree of discretization is above a certain threshold, then the LSCV score M(h) tends to - - as h tends to 0. and so LSCV will choose h O as the optimal bandwidth (Silverman 1986). Discretization thus causes undersrnoothing, and may reduce home range size. Reducing the time interval would increase the proportion of observations with identical values. and thus cause a reduction in h. Invariant kemels are not adversely affected by discretization or multiple identical values; the probability density is strictly proportional to the number of identical observations at that location. As a result. the estimation of the probability distribution elsewhere is unaffected. except by rescaling to keep the volume of the probability distribution to 1.

There is a difficulty that aises from the inherent differences between the underlying population that LSCV and invariant kernels attempt to estimate. LSCV is used to determine the underlying distribution from which the observations are sampled.

Invariant kemels. in the marner that 1 am suggesting that they be used, attempt to estimate the intensity of habitat selection through the overlap of the animal's search radius as it moves through the landscape (see Loehle 1990, 1994). This means that the probability distribution would extend to include areas in which the animal had never travelled, as long as that area is within the perceptive field. This, of course, makes it dificult to simultaneously mode1 LSCV and invariant kemels using the same simulation.

1 forced the observations to be contained within a 10000 unit area, thus this is the "true" home range which LSCV should correctly estimate. If the home range was modelled using the animal's search radius. then the home range would be larger than the 10000 unit are%and the size would depend upon the search radius. Due to these differences. 1 was not directly able to compare which method more closely estimated the true home area, since the true area differs between methods.

Another problem with using invariant kemels based upon the perceptive abilities of the animal is that the estimate of MISE (using Eq. 3.4) is no longer appropriate. This equation uses locational observations to estimate the true distribution. The estimate of

MISE will be very high, and will increase as the sarnple size increases, because the estimated probability distributions of invariant kemels includes areas in which there are no observations. Direct observations of the perceptive ability of the animal are diffïcult to obtain, and there are no data driven methods to minimize the MISE or MSE for invariant kemels. 1 am unaware how Eq. 3.4 could be modified to include the perceptive abilities of the animal.

A consequence of using the perceptive radius to form the kemels is that home ranges would be multifaceted. The perceptive field, or search radius, depends upon which senses are considered, and what the target is. For example, a nochunal animal rnay have a short visual perceptive field but a large audible field, so its ability to detect resources wouid vary with each sense. Home range size rnay not take on a single value, but depend instead on the sensory capabilities in question and the corresponding resource being searched for. An animal, for example. rnay have a small home range in terrns of foraging for food, which may be limited to olfactory ability, but rnay have a much larger one in terms of territoriality, which may depend upon visual or audible senses.

Varying the scale of investigation rnay allow researchers to determine the hierarchical nahue of habitat selection (Johnson 1980). The response of animals rnay be dependent upon the scale of perception of landscape structure (Zollner and Lima 1997).

Estimating the area in which the animal cmperceive large habitat patches can be used to measure second order habitat selection (Johnson 1980). whiie estimating the area in which the animal can detect prey items through foraging pertains to third or fourth order habitat selection (Johnson 1980). Since animals' perceptions vary with Landscape scale. it rnay not be sensible to envision home range as a single, scale-independent concept.

Multifaceted home range estimates rnay capture features of habitat selection and resource use that rnay be hidden by using estimators that produce a single value. Furthemore. they allow the researcher to focus on specific behaviours or resources instead of having to use a generic home range estimate for all analyses.

1 have not analyzed the characteristics of the naive estimator in detail, although it is very similar to kemel estimators (compare Eq. 3.1 with Eq. 3.5). Because the naive estimator uses cylinders with radius r, it is also at least potentially consistent. As long as the locational observations are confined to a limited area, the home range estimate must converge upon a finite area (unlike LSCV kemels, this chapter. and harmonic means, unpublished data). They have an advantage over invariant kemels in that they require less

information. Maximum detection distance is much more cornmon in the literature than knowledge of the detection Cunction (Brenowitz 1982, Nottingham. 1988, Hoffinan 1990,

Browman and 0' Brien 199 1), and is much easier to estimate. Consequently. naive estimators rnay be more generdly applicable than invariant kemels.

Although invariant kernels may be dificult to apply because of the problems in determining the perceptive abilities of the animal, they also have some desirable advantages. First, the kemel gains some biological significance. rather than being an abstract tool with no direct comection with the behaviour of the animal. This method allows researchen to incorporate biologically relevant factors. such as the search patterns of the animal, into estimating the movement panerns of animals, instead of using obscure mathematical formulae for which the researcher has little feel. Secondly. since the bias of a kemel density estimate is connected with the value of h, and h is determined independently of sample size or distribution of observations. it becomes more dificult to argue that autocorrelation will bias the density estimate. Thirdly, using the perceptive field to estimate h wiII establish a Iink between the locational observations and macrohabitat selection. Home range estimators that are based solely upon locational observations attempt to estimate a quantity, macrohabitat selection, that is not directly observable from the observations. Lastly, this method is more consistent and less variable than using LSCV, and likely similar automatic techniques, to optimize h. References

Altman, N. S. 1990. Kemel smoothing of data with correlated errors. J. Amer. Stat.

ASSOC.85: 749-759.

Anderson, D. J. 1982. The home range: a new nonparametric estimation technique.

Ecology. 63: 103- 1 12.

Bersier, LF., and Meyer. D. R. 1995. Bird assemblages in mosaic forests: the relative

importance of vegetation structure and floristic composition along the

successional gradient. Acta Oecologica. 15: 56 1-576.

Boulanger. J. G. and White. G. C. 1990. A cornparison of home-range estimators using

Monte Carlo simulation. J. Wildl. Manage. 54: 3 10-3 1 5.

Bowrnan. A. 1984. An alternative method of cross-validation for the smoothing of density

estimates. Biometrika 65: 52 1-528.

Brenowitz E. A. 1982. The active space of red-winged blackbird song. J. Comp. Physiol.

147: 5 1 1-522.

Browman. H. 1. and 0' Brien. W. J. 1 99 1. Foraging and prey search behaviour of golden

shiner (Noremigonus crysoleucas) larvae. Can. J. Fish. Aquat. Sci. 49: 8 13-819.

Brown, C. H. 1989. The active space of blue monkey and grey-cheeked mangabey

vocalizations. Anim. Behav. 37: 1 023- 1034.

Brown, J. L. and Orians. G. H. 1970. Spacing patterns in mobile animals. AM. Rev. Ecol.

Syst. 1 : 239-262.

Burt, W. H. 1943. Temtoriality and home range concepts as applied to marnmals. J.

Mamm. 24: 346-352. Calhoun, J. B. and Casby. J. U. 1958. Calculation of home range and density of small

mamals. Public Health Monographs No. 45.

Cresswell, W. J. and Smith. G. C. 1992. The effects of temporally autocorrelated data on

methods of home range analysis. In: Wildlife telemetry Remote monitoring and

tracking of animals. Priede, 1. G. and Swift, S. M. (Eds.) Ellis Honvood Ltd.,

England. pp 272-284.

Dixon, K. R. and Chapman. J. A. 1980. Harmonic mean measure of animal activity areas.

EcoIo~~.6 1 : 1040- 1044.

Don, B. A. C. and Remolls. K. 1983. A home range mode1 incorporating biological

attraction points. J. Anirn. Ecol. 52: 69-8 1.

Doncaster. C. P. and Macdonald. D. W. 1 99 1. Drifting temtoriality in the red fox. Vulpes

vufpes.J. Anim. Ecol. 60: 423339.

Dusenbeny. D. B. 1989. Ranging strategies. J. Theor. Biol. 136: 309-3 16.

Epanechnikov. V. A. 1969. Non-parametric estimation of a multivariate probability

density. Theory of probability and its applications. 14: 153-1 58.

Galbraith, D. A.. Chandler. M. W.. and Brooks, R. J. 1986. The fine structure of home

ranges of male Chelydru serpentinu: are snapping turtles temtorial? Can. J. Zool.

65: 2623-2629.

Galeotti. P. 1994. Patterns of territory size and defence level in rural and urban tawny

owls (Strix aluco) populations. J. Zool. Lond. 234: 64 1-658.

Gautestad, A. 0. and Mysterud, 1. 1993. Physical and biological mechanisms in animal

movement processes. J. Appl. Ecol. 30: 523-535. Getty. T. and Pulliam, H. R. 199 1. Random prey detection with pause-travel search. Am.

Nat. 138: 1459-1477.

Getty, T. and Pulliam, H. R. 1993. Search and prey detection by foraging sparrows.

Ecology. 74: 734-742.

Graves, B. M. 1994. The role of nasolabial grooves and the vomeronasal system in

recognition of home area by red-back salarnanders. Anim. Behav. 47: 12 16- 12 19.

Hall. P. and Marron. I. S. 1987. Extent to which least-squares cross-validation minimizes

integrated square error in nonparametnc density estimation. Probability Theory

and Related Fields. 74: 567-581.

Hall. P.. Sheather. S. J.. Jones. M. C.. and Marron. J. S. 199 1. On optimal data-based

bandwidth selection in kemel density estimation. Biometrika. 78: 263-269.

Hoffman. G. 1983. The search behavior of the desert isopod Hemilepistur reaumziri as

compared with a systematic search. Behav. Ecol. Sociobiol. 1 3 : 93- 1 06.

Homer, M. A. and Powell. R. A. 1990. Intemal structure of home ranges of black bean

and analyses of home-range overlap. J. Marnm. 7 1 : 402-4 10.

Jemich, . R. 1. and Turner, F. B. 1969. Measurement of non-circula home range. J.

Theor. Biol. 22: 227-237.

Jewell, P. A. 1966. The concept of home range in mammals. In: Play. exploration. and

temtoriality in mammals. Syrnp. 2001. Soc. Lond. No. 18. pp 85-1 09. Jewell, P.

A. and Loizos. C. (Eds.) Academic Press. London. England.

Johnson. D. H. 1980. The cornparison of usage and availability measurements for

evaluating resource preference. Ecology. 6 1: 65-7 1. Laundre, J. W. and Keller. B. L. 1984. Home-range size of coyotes: a critical review. J.

Wildl. Manage. 48: 127-139.

Lima, S. L. and Zollner, P. A. 1996. Towards a behavioral ecology of ecological

landscapes. TUE. 11: 131-135.

Loehle. C. 1990. Home range: a fractal approach. Landscape Ecology. 5: 39-52.

Loehle, C. 1994. Home ranges reconsidered. A reply to 'Are home ranges fractais?'

Landscape Ecol. 9: 147- 149.

Marron. J. S. 1988. Automatic smoothing parameter seiection: a survey. Empir. Econ. 13:

187-208.

Moms, D. W. 1992. Scales and costs of habitat selection in heterogeneous landscapes.

Evol. Ecol. 6: 412-432.

Naef-Daenzer. B. 1993. A new transmitter for srna11 animals and enhanced methods of

home-range analysis. J. Wildl. Manage. 57: 680-689.

Nottingham, S. F. 1988. Host-plant finding for oviposition by adult cabbage mot fly.

Deiia radiam. J. Insect. Physiol. 34: 227-234.

Robertson. J. G. M. 1984. Acoustic spacing by breeding males of Ilperoieiu rugosu

(Anura: Leptodactylidae). Z. Tierpsychol. 64: 283-297.

Roitberg, B. D. 1985. Search dynarnics in fruit-parasitic insects. J. Insect. Physiol. 3 1:

865-872.

Samuel, M. D. and Garton. E. 0. 1987. lncorporating activity time in harmonic home

range analysis. J. Wildl. Manage. 5 1: 254-257.

Schoener, T. W. 198 1. An empirically based estimate of home range. Theor. Pop. Biol. 20: 281 -325.

Seaman, D. E. and Powell, R. A.. 199 1. Kemel home range estimation program. North

Carolina State University. North Carolina USA.

Seaman, D. E. and Powell. R. A. 1996. An evaluation of the accuracy of kemel density

estimators for home range analysis. Ecology. 77: 2075-2085.

Silverman, B. W. 1986. Density estimation for statistics and data analysis. C hapman and

Hall. London. 175pp.

Stone. L. D. 1 975. Theory of optimal search. Academic Press. New York.

Swihart. R. K. and Slade. N. A. 1985a. Influence of sarnpling interval on estimates of

home range size. J. Wiidl. Manage. 49: 10 19- 1025.

Swihart. R. K. and Slade. N. A. 1985b. Testing for independence of observations in

animal movements. Ecology. 66: 1 176-1 184.

Tufio. J.. Andersen. R.. and Linnell. J. 1996. Habitat use and ecological correlates of

home range size in a smali cervid: the roe deer. J. Anirn. Ecol. 62: 7 15-724.

Tye, A. 1989. A mode1 of search behaviour for the northern wheatear Oenanthe oenanthe

(Aves. Turdidae) and other pause-travel predators. Ethology. 83: 1-1 8.

Van Winkle, W. 1975. Cornparison of several probabilistic home-range models. J. Wildl.

Manage. 39: 1 18-1 23.

Waser. P. M. 1975. Experimental playbacks show mediation of intergroup avoidance in a

forest monkey. Nature. 255: 56-58.

Worton. B. J. 1987. A review of models of home range for animal movement. Ecol.

Modelling. 38: 277-298. Worton. B. 1. 1989. Kernel methods for estimating the utilization distribution in home-

range studies. Ecology. 70: 1 64- 168.

Worton, B. J. 1995. Using Monte Carlo simulation to evaluate kemel-based home range

estimates. I. Wildl. Manage. 59: 794-800.

Zollner, P. A. and Lima, S. L. 1997. Landscape-level perceptual abilities in white-footed

mice: perceptual range and the detection of forested habitat. Oikos. 80: 5 1-60. "There is something fascinating about science. One gets such wholesorne retums of conjecture out of such trifling investrnent of fact." Mark Twain Chapter 4

Sense and nonsense: a skeptic's view of science and home rafige Introduction

The currency of science is the peer-reviewed publication. Aside from the social aspects of publication. scientists publish their results so that othen can use this information. Scientists use the literature as a foundation to advance their own work, and as a source to compare with their own results or conclusions. In the eye of the realist. this process maintains (or at least maximizes) the objectivity of hypothesis and theory testing, and allows for advancement of science. The peer-review process exposes the manuscript to intense scrutiny and constructive criticism. The review of manuscripts is supposed to eliminate falsehoods. logical and computational errors. and unsupported claims. such that the resulting work becomes a valuable contribution to science even if it is uitirnately found to be wrong. At worst. the peer-reviewed manuscnpt should be more readable and arnbiguities of the analyses should be clarified. Unfominately. the peer-review process of researchon home range seems to lack a cntical edge. and, in rny opinion. has failed to eliminate errors in estimation and hypothesis testing. Logical errors sometimes persist without refutation for decades. partially because there is disproportionate effort devoted

10 probing for violations of assumption of models, and not enough effort to eliminating bad logic.

Errors in statistical tests

A type of error in decision making that has received much focus are errors involving estimation of type I or type 11 errors. Inappropriate use of statistics. or violation of assumptions will result in over- or underestirnates of type 1 and II errors. which cm negatively affect decision making by the author or reader. Since most statistical hypothesis testing by biologists revolves around the ubiquitous P - value of 0.05, and

occasionally power (1-B), errors in estimating the P - value or P may result in improper

acceptance or rejection of the nul1 hypothesis. Certainly there have been many

publications by biologists on inappropriate use of statistical tests in the Iiterature, ofien causing replies from the offending parties (e.g. Johnson 1995, Steward-Owens 1995,

Lombardi md Hurlbert 1996 and Wilson and Dugatkin 1996, Thomas and Juanes 1996

and Johnsson 1996). Although not quite as cornmon in the home range literature. there certainly are accounts where the vdidity of statistical tests conceming home range are questioned (Johnson 1980. Lindstedt and Calder 198 1, Samuel and Garton 1985.

Aebischer et 01. 1993).

However. errors in estimating P - values or p are not the most senous type of statistical error in the scientific literature. Al1 too ofien biologists, statisticians. and over- zealous graduate students pounce upon perceived or real biases in a statistical analysis.

Even if tests are biased or are othenvise suboptimal. usehl information can still be harvested from results through statistical "salvaging". In my second chapter. for example. the residuals of some of the regressions were severely heteroscedastic. Any tests involving the standard enors of the regression coefficients. including the F and t-tests. would be invalid because the P-values would be biased (Marshall et al 1995).

Nevertheless, the test was still usefùl because the estimates of the regression coefficients were unbiased despite heteroscedasticity (or autocorrelation) among the residuals.

Statistical "salvaging" of a less than ideal situation allowed me to arrive at a valid conclusion regardless of the statistical violation. An overemphasis on P-values rather than the size of effect or other critena (e.g. Steward-Owens 1995) has led to the loss of much usefùl information as researchers reject the faulty analysis. Even pseudoreplication. which results in inflated degrees of Freedom and so underestimates P -values, cm still be a source of valuable information (Hurlbert 1984).

Logical Erron

A more insidious form of miscornmunication are errors in logic, where the statistical tests may be valid, but the reader misunderstands or misuses the author's conclusion. This type of enor is very difficult to control, and has a larger potential impact on decision making by the reader than errors in statistical tests do. The simulation used by

Swihart and Slade ( 1985). for example. effectively models what happens to home range estimates if the time interval among locational observations and total sampling period are simultaneously shortened while maintaining a constant sample size. There are MO main effects: first, the degree of autocorrelation among observations increases: secondly. the home range estimates become smaller. Since this was a computer simulation and thus other confounding effects are not present. there was a causal link between the treatment and the two observed effects. Thus one valid conclusion is that reducing the time interval and total sampling penod increases the positive autocorrelation among observations, and a second valid conclusion is that reducing the time interval and total sampling penod reduces the size of home range estimates. However, the authors and numerous readers improperly interpreted the results of the simulation and conc!uded that autocorrelation caused the reduction in home range size (e.g. Worton 1987. Ackerman et al. 1990, Harris et al. 1990, see Chapter 1 for more references). There has been little evidence to support this claim, which was based upon an invalid conclusion that was made without establishing any logical or experimental causation.

The claim that autocorrelation causes home range estimators to underestimate home range size lias spread throughout the literature, fvst appearing in articles by authors who cited (incorrectly) Swihart and Slade (e.g. Worton 1987. Harris et al. 1990). Like

Dawkins' anaiogy of a meme, the rate of infection of this successful idea increased as authors started citing secondary sources (e.g. Bright and Moms 1993 citing Harris et al. 1990 ). and some even considered the idea as common knowledge and cited no one

(e-g. Breitenrnoser et al. 1993. Kauhala et al. 1993). Furthemore. there have been few challenges to this idea, except for those who argued for biological independence as opposed to statistical independence (Lair 1987, Minta 1992). Ironically, one reason for the lack of critical examination, of course. is that the actual analyses of Swihart and Slade

(1985) are statistically valid. and so their conclusions have not been seriously questioned.

If it appears that I am making Swihart and Slade into scapegoats, the impression is unintentional and unwarranted; they merely provide a convenient and relevant example. It is not that logical errors do not exist elsewhere in the scientific literature, but that the use of the tools of inquiry within the field of home range perpetuates these logical errors.

Scientific literature is the foundation on which further scientific work, whether contradictory or complementary, is built. For progress to occur, published results must be comparable, and aiso must be reproducible for tùture research. The home range literature is contains many nonsensical daims because home range analyses are not comparable among and sornetimes even within studies. Incompatibility among home range estimators

Different home range estimators give different values, which restricts valid comparisons (other than showing that there are differences in the home range estimators)

(Jemerich and Turner 1969, Boulanger and White 1990, Hamis et al. 1990, White and

Gmoa 1990). Many others have tested for differences among estimators (Van Winkle

1975, Swihart and Slade 1985, Jaremovic and Crofi 1 987, Boulanger and White 1990,

White and Garrott 1990). I will use the data fiom the Monte Carlo simulation presented in

Chapter 2 to determine if there are any differences among the home range esfimators I used: kemel density, harmonic means, 95% and 75% probability ellipse. and MCP. 1 used t-tests to compare the mean home range size among estimators. using both the shortest (1) and longest (15) time intervals. The true home range size was 10000 units. For each home range estimator. 1 used the first 20 samples from the Monte Carlo simulation. 1 tested to determine if there were unequal variances between each pair of home range estimators using Levene's test for homogeneity of variances.

There was unequal variance among 80% of the 10 comparisons using the shortest time interval. while 70% of the IO comparisons using the longest time interval had unequal variances. To account for unequal variance, I used a t-test for unequal variances

(Blalock 1972) for each comparison. The mean home range size was significantly different for al1 estimators for both the longest and shortest time intervals (Figs. 4.1,4.2).

Other researchers have atm come to the conclusion that different home range estimators give different results when using identical data sets (Jennrich and Turner 1969, Boulanger and White 1990, Harris et al. 1990). Statistical hypothesis testing ofien involves means. Means have the forninate property (among others) of being unbiased by sample size, which allows us to compare among studies despite differences in sarnple size. However. applying the same approach to home range estimators is nonsensical. The mean and variance of the estimated home range sizes are dependent on the time interval, and thus sample size (Chapter 2). If someone was interested in determining whether the home ranges of animals in two contrasting treatrnents were different. but had unequal sample sizes among the home range estimates, standard hypothesis testing becomes very difficult. Without going into more detailed analyses. it is impossible to determine if any observed effect (or apparent lack of effect) is due to the treatment or to differences in sample size. Not only are different home range estimators not comparable among each other. but the same home range estimator cannot be compared with itself if the tirne intervals among observations are different. Although Jennrich and Turner (1 969) attempted to devise a method to reduce the bias and thus "equalize" different home range estimators. there is no reason to believe that the relationships they found using bivariate normal models are applicable when home ranges aren't bivariately normal. which they almost never are.

Nevertheless. it is not uncommon for researchers to compare home range estimates among published papers. A few authors will make nonstatistical comparisons among studies without attempting to infer any treatment effect (Carey et a[. 1990, Cal1 el al. 1992). More comrnon are comparisons among studies that attempt to infer treatment effects (Reinert and Kodrich 1982. Laundre and Keller 1984, Mace et al. 1984. Shine and

Larnbeck 1985, Badyaev et al. 1996). There are few attempts to separate the treatment effects with the confounding effects of unequal sample sizes. unequal total sampling penods, or artifacts of the rnethod of home range analysis that was used for each study. 1 am not claiming that comparisons cannot be made. but that comparisons are only usefùl if the confounding factors are removed. or if the treatment effects are considerably larger than methodological artifacts. Exarnining changes in home range structure or use with changes among taxa can be very usefbl (Harestad and Bunnell 1979. Garland 1983.

Swihart et al. 1988) even in the presence of the rnethodological artifacts. Differences in home range size among taxa may change by 5 orders of magnitude (Swihart et al. 1988). and thus the treatment e ffect would ovenvhelm any rnethodological e ffec ts. However. home range estimates may change by at least a factor of 3 (Figure 4.1.4.2). Small or even moderate treatments effects cmot normally be tested for in the presence of substantial methodological artifacts.

A surplus of home range estimators

What confounds the incompatibility of home range estimators is the multitude of methods that are used. 1 did a small survey of four ecological journals which often publish home range studies. and grouped the estimators into categories (Table 4.1.4.2.4.3.4.4).

More than one home range estimator was used in some studies, so the total number of papers using home range is smaller than the total nurnber of home range estimators. The nurnber of different estimators are actualiy larger than the number of categories, because I grouped similar estimators into single categories. such as the various polygon techniques.

It should be clear the results from the home range analyses among most of these papers cannot be compared to each other. Home Range Estimator

Figure 4.1. Mean and standard deviation of home range size estimates of the first 20 sarnples from the Monte Car10 simulation presented in Chapter 2. using the time interval of 1. K = Fixed kemel (LSCV), H = Harmonic means, E95 = 95% probability ellipse, E75 = 75% probability ellipse. M = MCP. Every cornparison is significantly different fkom each other (P < 0.0001). 1 K H E95 E75 M Home Range Estimator

Figure 4.2. Mean and standard deviation of home range size estimates of the fint 20 sarnples from the Monte Carlo simulation presented in Chapter 2, using the time interval of 15. K = Fixed kernel (LSCV). H = Harmonic means. E95 = 95% probability ellipse, E75 = 75% probability ellipse. M = MCP. Every comparison is significantly different from each other (P < 0.02). Table 4.1. Summary of home range estimators found in articles published in the Journal

Year Home Range Estimator Kernei * Harmonic Polygon** Grid Ellipse 1996 1 2 3 O O

Home Range Estimator Length* * * Unknown Dirichlet or Total

other

* includes adaptive, fixed, and tucated kernels, and using 95% of utilization distribution

or the area containing 95% of the observations

** includes MCP, MMA, restricted polygons, probability polygons, cluster analyses. and

others

* ** includes radii. longest linear distance, and others Table 4.2. Summary of home range estimators found in articles published in the kumal

of Wildlife Management fiom 1992 to 1996.

Y ear Home Range Estimator Kemel* Harmonic Polygon** Grid Ellipse 1996 2 1 4 1 O

--- Total 4 16 29 1 O Home Range Estimator Length* * * Unknown Dirichlet or To ta1

other

* includes adaptive, fixed. and truncated kemels, and using 95% of utilization distribution

or the area containing 95% of the observations

** includes MCP, MMA, restricted polygons, probability polygons, cluster analyses. and

others

*** includes radii, longest linear distance. and others Table 4.3. Summary of home range estimators found in articles published in the Journal

of Animal Ecology fiom 1992 to 1996.

Y ear Home Range Estimator Kemel * Harrnonic Po lygon** Grid Ellipse

Home Range Estimator Length* * * Unknown Dirichlet or Total

other

* includes adaptive, fixed, and truncated kemels, and using 95% of utilization distribution

or the area containing 95% of the observations

* * includes MCP. MMA, restricted polygons. probability polygons, cluster analyses. and

others

* * * includes radii, longest linear distance, and others Table 4.4. Summary of home range estimaton Found in articles published in the Journal

of Zoology, London from 1992 to 1996.

Y ea. Home Range Estimator Kernel* Harmonic Po tygon* * Grid Ellipse

------Home Range Estimator Length* * * Unknown Dirichlet or Total

other

Total 8 5 O 65

* includes adaptive. fixed, and truncated kernels, and using 95% of utilization distribution

or the area containing 95% of the observations

** includes MCP. MMA, restncted polygons. probability polygons. cluster analyses. and

O thers

*** includes radii, longest linear distance, and others Ironically, despite the disparity in methodology among the articles, they generally use the

same definitions of home range (e.g. Burt 1943).

Since the fint estimators were published, new estimators have appeared at a

steady rate: MCP (Dalke and Sime 1938. Mohr 1947), bivariate normal (Caihoun and

Casby 19%). grid cells (Siniff and Tester l965), probability ellipse (Jennrich and Turner

1969). multivariate Ornstein-Uhlenbeck diffùsion process (Dmand Gibson 1977),

population utilization distribution (Ford and Knimme 1979). harmonic means (Dixon and

Chapman l98O), R method (Schoener 198 1 ). Fourier transformations (Anderson 1982).

multinuclear mode1 (Don and Rennolls 1983). probability polygons and cluster analysis

(Kenward 1987). kemel densities (Worton 1987. l989), fiactals (Loehle 1990). Dirichlet tessellations (Byen 1992, Wray el al. 1992). buffer zone (Boitani et al. 1994), and naive estimators (Chapter 3). This is not an exhaustive list. and it is biased towards the more recent home range estimators. Some of the earlier ones are no longer being used and thus are harder to find. and so there is likeiy a disproportionate number of recent home range estimators in the list. Also. this list does not include the various modifications of these estirnators. Although kemel estimators are relatively new, for example, many modifications have already appeared (fixed, Worton 1987: adaptive. Worton 1989. tnincated. Naef-Daenzer 1993, utilization based on 95% of observations, Kie et al. 1996: imariant kemels. Chapter 3). There are even more modifications to older estimaton, such as the probability ellipses and MCP.

The MCP is the best candidate as a standard home range estimator; not because it is the best estimator, but because of a historical accident. The MCP is the oldest home range estirnator (Dalke and Sime 1938, Mohr 1947), and so for years was the only one available, and it is still commonly used because previous studies have used it.

Researchers often state that they used a particular home range estimator. usually MCP. to compare with other studies (Carey et al. 1990, Harris et al. 1990, Geffen et al. 1992,

Andreassen et al. 1993. Crooks and Van Vuren 1995. Badyaev et al. 1996). The use of the most cornmon home range estimator gives a false sense of security. MCP is strongly affected by sample size (as are other estimators), and often incorporates unused areas within the estimate (Chapter 2. Jedchand Turner 1969. Harris et cil. 1990). Statistical tests of home range cornparisons among studies (and within studies) cannot distinguish between treatrnent effects and treatrnent arti facts.

To test hypotheses involving home ranges properly. the home range estimates fiom each treatment must have the same number of observations, collected with sirnilai- time intervals from an identical total sampling period. and using the sarne home range estimator. Since most home range studies involve wild animals in natural or pseudo- natural conditions where the investigator has little ability to control extraneous factors. the researcher has to worry about covariates that are not of direct interest yet affect the statistical test. Finally, researchers have to be etemally vigilant to ward off demonic intrusion that may contaminate the treatrnent effect (Hurlbert 1984). 1 am not optimistic that most of these demands can be met.

This does not mean. of course, that home range estimation should be avoided.

Home range analysis can be consistent within a study. as long as the researchers control the sampling regime so that the statistical testing of treatment effects are not confounded by methodological artifacts. Strict adherence to equal total sarnpling periods and time intervals among and within treatments would eliminate most problems. Cornparisons arnong studies. in which the researchers do not have control over the sarnpling regimes. cm be valid if it can be demonstrated that the treatment eEects are considerably larger than artifact effects. Nevertheless. failure to adhere to reasonably strict protocols wili prevent meaningful comparisons.

In rny first chapter 1 demonstrated that typical definitions of home range are not measurable, and that if autocorrelation does affect home range estimates, then we cannot accurately estimate home range. In the second chapter, I demonstrated that autocorrelation does not violate assumptions of home range estimators. yet they are strongly affected by changes in the time interval arnong observations. In the third chapter.

1 argued that more information other than locational observations may be necessary to meaningfully estimate home range. If Cartesian coordinates are not sufficient for accurately estirnating home range, then most home range studies lack meaning. As

Kenward (1992) alluded to, Our ability to meaningfully study home range has lagged behind our technoiogy to collect the observations. References

Ackerman, B. B.. F. A. Leban, M. D. Samuel, and E. O. Garton. 1990. User's manual for

program home range. Second edition. Technical Report 15. Forestry, Wildli fe and

Range Experimental Station. University of Idaho. Moscow, Idaho, USA.

Aebischer. N. J., Robertson. P. A.. and Kenward. R. E. 1993. Compositional analysis of

habitat use fiom animal radio-tracking data. Ecology. 74: 13 13- 1 325.

Anderson, D. J. 1982. The home range: a new nonparametric estimation technique.

Ecology. 63 : 103- 1 12.

Andreassen. H. P. . Ims. R. A.. Stenseth, N.C., and Yoccoz, N. G. 1993. Investigating

space use by means of radiotelernetry and other methods: a methodological guide.

In: The biology of lemmings. London. Limean Society of London.

Badyaev. A. V., Etges, W. J.. and Martin. T. E. 1996. Ecological and behavioral

correlates of variation in seasonal home ranges of wild turkeys. J. Wildl. Manage.

60: 154-1 64.

Btalock. H. M. 1972. Social statistics. Second edition. New York:McGraw-Hill.

Boitani. L.. Mattei. L.. Nonis. D.. and Corsi. F. 1994. Spatial and activity patterns of wild

boars in Tuscany. Italy. J. Mammal. 75: 600-612.

Boulanger, J. G. and White, G. C. 1990. A comparison of home-range estirnators using

Monte Car10 simulation. J. Wildl. Manage. 54: 3 10-3 15.

Breitenmoser, U., Kavezensky. P. Dotterer, M, Breitenmoser-Würsten. C., Cart, S.,

Berhart, F., and Liberek. M. 1993. Spatial organization and recruitment of lynx

(Lymlynx) in a re-introduced population in the Swiss Tura Mountains. J. Zool. Lond. 23 1 : 449-464.

Bright. P. W. and Morris, P. A. 1993. Foraging behaviour of dormice Muscardinus

mellanarius in two contrasting habitats. J. 2001. Lond. 230: 69-86.

Burt. W.H. 1943. Temtoriality and home range concepts as applied to marnmals. J.

Mamrn. 24: 346-352.

Byers. J. A. 1992. Dirichlet tessellations of bark beetle spatial attack points. J. Anim.

Ecol. 6 1 : 759-768.

Calhoun, J. B. and Casby, J. U. 1958. Calculation of home range and density of srna11

marnmals. Public Health Monographs No. 45.

Call, D. R., Gutierrez R. J.. and Vemer. J. 1992. Foraging habitat and home-range

charactenstics of Califomia Spotted Owls in the Sierra Nevada. Condor. 94: 880-

888.

Carey, A. B.. Reid, J. A.. and Horton. S. P. 1990. Spotted owl home range and habitat use

in southem Oregon Coast ranges. J. Wildl. Manage. 54: 1 1- 17.

Crooks. K. R. and Van Vuren. D. 1995. Resource utilization by two insular endemic

mammalian carnivores. the island fox and island spotted skunk. Oecologia. 104:

30 1-307.

Dalke. P. D. and Sime, P. R. 1938. Home and seasonal ranges of the eastem cottontail in

Connecticut. Transcripts of the North Amencan Wildlife Conference. 3: 659-669.

Dixon, K. R. and Chapman, J. A. 1980. Harmonic mean measure of animal activity areas.

Ec010gy. 6 1 : 1040-1044.

Don, B. A. C. and Rennolls, K. 1983. A home range mode1 incorporating biological attraction points. J. Anim. Ecol. 52: 69-8 1.

Dm,3. E. and Gipson. P. 1977. Analysis of radio telemetry data in studies of home

range. Biometrics. 33: 85- 101.

Ford, R. G. and Knimme, D. W. 1979. The analysis of space use patterns. J. Theor. Biol.

76: 125-1 55.

Garland Jr, T. 1983. Scaling the ecological cost of transport to body mas in terrestrial

mamrnals. Am. Nat. 12 1 : 57 1-587,

GeEen. E., Heher, R., Macdonald, D. W., and Ucko. M. 1992. Habitat selection and

home range in the Blanford's fox. VuZpes cana: cornpatibility with the resource

dispersion hypothesis. Oecologia. 9 1 : 75-8 1.

Harris, S.. Cresswell. W. I.. Forde. P. G.. Trewhella. W. J.. Woollard, T.. and Wray. S.

1990. Home-range analysis using radio-tracking data - a review of problems and

techniques particularly as applied to the study of mamrnals. Mammal Rev. 20: 97-

123.

Harestad. A. S. and Bunnell. F. L. 1979. Home range and body weight - a reevaluation.

Ecology. 60: 389-402.

Hurlbert, S. H. 1984. Pseudoreplication and the design of ecological field experiments.

Ecol. Monogr. 54: 187-2 1 1.

Jaremovic. R. V. and Croft, D. B. 1987. Cornparison of techniques to determine eastem

grey kangaroo home range. J. Wildl. Manage. 5 1 : 92 1-93 0.

Jermich, .R. 1. and Turner. F. B. 1969. Measurement of non-circular home range. J.

Theor. Biol. 22: 227-337. Johnson, D. H. 1980. The cornparison of usage and availability measurements for

evaluating resource preference. Ecology. 6 1 : 65-7 1.

Johnson, D. H. 1995. Statisticai sirens: the allure of nonparametrics. Ecology. 76: 1998-

2000.

Johnsson, J. 1. 1996. Statistics and biological sense: a reply to Thomas & Juanes. Anim.

Behav. 52: 860.

Kauhaia, K., Helle. E., and Taskinen, K. 1993. Home range of the raccoon dog

(Nyctereutesprocyonoides) in southem Finland. J. 2001. Lond. 23 1: 95- 106.

Kie, J. G., Baldwin. J. A.. and Evans. C. J. 1996. Calhome: a program for estimating

animal home ranges. Wild. Soc. Bull. 24: 342-344.

Kenward, R. E. 1987. Wildlife radio tagging. Equipment, field techniques and data

analysis. Biological techniques series. Academic Press. London. England.

Kenward. R. E. 1992. Quantity versus quality: programmed collection and analysis of

radio-tracking data. In: Wildli fe teiemetry Remote monitoring and tracking of

animals. Priede. I. G. and Swift. S. M. (Eds.) Ellis Howood Ltd.. England. pp

232-246.

Lair, H. 1987. Estimating the location of the focal center in red squirrel home ranges.

Ecology. 68 : 1092- 1 10 1.

Laundre, J. W. and Keller, B. L. 1984. Home-range size of coyotes: a critical review. 3.

Wildl. Manage. 48: 127-139.

Lindstedt, S. L. and Calder III. W. A. 198 1. Body size, physiological time, and longevity

of homeothermic animals. Quart. Rev. Biol. 56: 1 - 16. Loehle, C. 1990. Home range: a fiactai approach. Landscape Ecol. 5: 39-52.

Lombardi, C. M. and Hurlbert, S. H. 1996. Sunfish cognition and pseudoreplication.

Anim. Behav. 27: 4 19-422.

Mace. G. M.. Harvey. P. H.. and Clutton-Brock. T. H. 1984. Vertebrate home range-size

and energetic requirements. In Swingland, 1. R. and Greenwood, P. J. (eds.) The

ecology of animai movement, pp 32-53. Clarendon Press, Oxford.

Marshall, P.. Szikszai. T.. LeMay, V.. and Kozak. A. 1995. Testing the distributionai

assumptions of least squares linear regression. For. Chron. 7 1 : 2 13-21 8.

Minta, S. C. 1992. Tests of ~patialand temporal interaction among animals. Ecol. Appl.

2: 178-188.

Mohr. C. 0. 1947. Table of equivalent populations of North Arnencan smali mammals.

Am. Midl. Nat. 37: 223-349.

Naef-Daenzer, B. 1993. A new transmitter for srnall animals and enhanced methods of

home-range analysis. J. Wildl. Manage. 57: 680-689.

Reinert. H. R. and Kodrich, W. B. 1982. Movements and habitat utilization by the

massasauga Sisirurus ratenarus catenaius. J. Herpetol. 16: 162- 17 1.

Samuel. M. D.and Garton. E. 0. 1985. Home range: a weighted nomal estimate and

tests of underlying assumptions. J. Wildl. Manage. 49: 5 1 3-5 19.

Schoener. T. W. 198 1. An ernpirically based estimate of home range. Theor. Pop. Biol.

20: 28 1-325.

Shine. R. and Larnbeck, R. 1985. A radiotelemetric study of movernents,

themoregulation and habitat utilization of Arafura fiiesnakes (Serpentes: Acrochordidae). Herpetologia 4 1: 3 5 1-361.

Skiff, D. B. and Tester, J. R. 1965. Cornputer Analysis of animal movement data

obtained by telemetry. Bioscience. 15: 104- 108.

Steward-Owens, A. 1995. Rules and judgements in statistics: three exampies. Ecology.

76: ZOO 1-2009.

Swihart, R. K. and Slade, N. A. 1985. Influence of sampling interval on estimates of

home range size. J. Wildl. Manage. 49: 10 19- 1025.

Swihart. R. K., Slade, N. A. and Bergstrom. B. J. 1988. Relating body size to the rate of

home range use in mamals. Ecology. 69: 393-399.

Thomas. L. and Juanes. F. 1996. The importance of statistical power analysis: an exarnple

fiom Animai Behaviour. Anim. Behav. 52: 856-859.

Van Winkle, W. 1975. Cornparison of several probabilistic home-range models. J. Wildl.

Manage. 39: 118-123.

White, G. C. and Garros. R. A. 1990. Analysis of wildlife radio-tracking data. Academic

Press, Inc. London.

Wilson, D. S. and Dugatkin, L. A. 1996. A reply to Lombardi & Hurlbert. Anim. Behav.

52: 423-425.

Worton, B. J. 1987. A review of models of home range for animal movement. Ecol.

Modelling. 38: 277-298.

Worton. B. J. 1989. Kemel methods for estimating the utilization distribution in home-

range studies. Ecology. 70: 164- 168.

Wray, S., Cresswell, W. I.. and Rogers, D. 1992. Dirichlet tesselations: a new, non- parametric approach to home range analysis. In: Wildlife telernetry Remote monitoring and tracking of animais. Pnede. 1. G. and Swifi. S. M. (Eds.) Ellis

Horwood Ltd., England. pp 247-255. IMAGE WALUATION TEST TARGET (QA-3)

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