A **home range** is the area in which an animal lives and moves on a periodic basis. It is related to the concept of an animal's territory which is the area that is actively defended. The concept of a home range was introduced by W. H. Burt in 1943. He drew maps showing where the animal had been observed at different times. An associated concept is the utilization distribution which examines where the animal is likely to be at any given time. Data for mapping a home range used to be gathered by careful observation, but nowadays, the animal is fitted with a transmission collar or similar GPS device.

The simplest way of measuring the home range is to construct the smallest possible convex polygon around the data but this tends to overestimate the range. The best known methods for constructing utilization distributions are the so-called bivariate Gaussian or normal distribution kernel density methods. More recently, nonparametric methods such as the Burgman and Fox's alpha-hull and Getz and Wilmers local convex hull have been used. Software is available for using both parametric and nonparametric kernel methods.

The concept of the home range can be traced back to a publication in 1943 by W. H. Burt, who constructed maps delineating the spatial extent or outside boundary of an animal's movement during the course of its everyday activities.^{ [1] } Associated with the concept of a home range is the concept of a utilization distribution, which takes the form of a two dimensional probability density function that represents the probability of finding an animal in a defined area within its home range.^{ [2] }^{ [3] } The home range of an individual animal is typically constructed from a set of location points that have been collected over a period of time identifying the position in space of an individual at many points in time. Such data are now collected automatically using collars placed on individuals that transmit through satellites or using mobile cellphone technology and global positioning systems (GPS) technology, at regular intervals.

The simplest way to draw the boundaries of a home range from a set of location data is to construct the smallest possible convex polygon around the data. This approach is referred to as the minimum convex polygon (MCP) method which is still widely employed,^{ [4] }^{ [5] }^{ [6] }^{ [7] } but has many drawbacks including often overestimating the size of home ranges.^{ [8] }

The best known methods for constructing utilization distributions are the so-called bivariate Gaussian or normal distribution kernel density methods.^{ [9] }^{ [10] }^{ [11] } This group of methods is part of a more general group of parametric kernel methods that employ distributions other than the normal distribution as the kernel elements associated with each point in the set of location data.

Recently, the kernel approach to constructing utilization distributions was extended to include a number of nonparametric methods such as the Burgman and Fox's alpha-hull method ^{ [12] } and Getz and Wilmers local convex hull (LoCoH) method.^{ [13] } This latter method has now been extended from a purely fixed-point LoCoH method to fixed radius and adaptive point/radius LoCoH methods.^{ [14] }

Although, currently, more software is available to implement parametric than nonparametric methods (because the latter approach is newer), the cited papers by Getz et al. demonstrate that LoCoH methods generally provide more accurate estimates of home range sizes and have better convergence properties as sample size increases than parametric kernel methods.

Home range estimation methods that have been developed since 2005 include:

- LoCoH
^{ [15] } - Brownian Bridge
^{ [16] } - Line-based Kernel
^{ [17] } - GeoEllipse
^{ [18] }^{ [19] } - Line-Buffer
^{ [20] }

Computer packages for using parametric and nonparametric kernel methods are available online.^{ [21] }^{ [22] }^{ [23] }^{ [24] } In the appendix of a 2017 JMIR article, the home ranges for over 150 different bird species in Manitoba are reported.^{ [25] }

A **histogram** is an approximate representation of the distribution of numerical data. It was first introduced by Karl Pearson. To construct a histogram, the first step is to "bin" the range of values—that is, divide the entire range of values into a series of intervals—and then count how many values fall into each interval. The bins are usually specified as consecutive, non-overlapping intervals of a variable. The bins (intervals) must be adjacent, and are often of equal size.

**Statistics** is a field of inquiry that studies the collection, analysis, interpretation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities; it is also used and misused for making informed decisions in all areas of business and government.

In geometry, the **convex hull** or **convex envelope** or **convex closure** of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.

**Nonparametric statistics** is the branch of statistics that is not based solely on parametrized families of probability distributions. Nonparametric statistics is based on either being distribution-free or having a specified distribution but with the distribution's parameters unspecified. Nonparametric statistics includes both descriptive statistics and statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are violated.

In probability and statistics, **density estimation** is the construction of an estimate, based on observed data, of an unobservable underlying probability density function. The unobservable density function is thought of as the density according to which a large population is distributed; the data are usually thought of as a random sample from that population.

**Mathematical statistics** is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.

In statistics, **kernel density estimation** (**KDE**) is a non-parametric way to estimate the probability density function of a random variable. Kernel density estimation is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the **Parzen–Rosenblatt window** method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating it in its current form. One of the famous applications of kernel density estimation is in estimating the class-conditional marginal densities of data when using a naive Bayes classifier, which can improve its prediction accuracy dramatically.

In statistics, a **parametric model** or **parametric family** or **finite-dimensional model** is a particular class of statistical models. Specifically, a parametric model is a family of probability distributions that has a finite number of parameters.

In statistics, **resampling** is any of a variety of methods for doing one of the following:

- Estimating the precision of sample statistics by using subsets of available data (
**jackknifing**) or drawing randomly with replacement from a set of data points (**bootstrapping**) - Exchanging labels on data points when performing significance tests
- Validating models by using random subsets

**Nonparametric regression** is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. Nonparametric regression requires larger sample sizes than regression based on parametric models because the data must supply the model structure as well as the model estimates.

In statistics, **semiparametric regression** includes regression models that combine parametric and nonparametric models. They are often used in situations where the fully nonparametric model may not perform well or when the researcher wants to use a parametric model but the functional form with respect to a subset of the regressors or the density of the errors is not known. Semiparametric regression models are a particular type of semiparametric modelling and, since semiparametric models contain a parametric component, they rely on parametric assumptions and may be misspecified and inconsistent, just like a fully parametric model.

**Truncated regression models** are a class of models in which the sample has been truncated for certain ranges of the dependent variable. That means observations with values in the dependent variable below or above certain thresholds are systematically excluded from the sample. Therefore, whole observations are missing, so that neither the dependent nor the independent variable is known. This is in contrast to censored regression models where only the value of the dependent variable is clustered at a lower threshold, an upper threshold, or both, while the value for independent variables is available.

**Local convex hull (LoCoH)** is a method for estimating size of the home range of an animal or a group of animals, and for constructing a utilization distribution. The latter is a probability distribution that represents the probabilities of finding an animal within a given area of its home range at any point in time; or, more generally, at points in time for which the utilization distribution has been constructed. In particular, different utilization distributions can be constructed from data pertaining to particular periods of a diurnal or seasonal cycle.

In probability theory, **heavy-tailed distributions** are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distribution that is of interest, but a distribution may have a heavy left tail, or both tails may be heavy.

The term **kernel** is used in statistical analysis to refer to a window function. The term "kernel" has several distinct meanings in different branches of statistics.

In statistics, **Kernel regression** is a non-parametric technique in statistics to estimate the conditional expectation of a random variable. The objective is to find a non-linear relation between a pair of random variables * X* and

A **utilization distribution** is a probability distribution giving the probability density that an animal is found at a given point in space. It is estimated from data sampling the location of an individual or individuals in space over a period of time using, for example, telemetry or GPS based methods.

**Polynomial chaos (PC)**, also called **Wiener chaos expansion**, is a non-sampling-based method to determine the evolution of uncertainty in a dynamical system when there is probabilistic uncertainty in the system parameters. PC was first introduced by Norbert Wiener using Hermite polynomials to model stochastic processes with Gaussian random variables. It can be thought of as an extension of Volterra's theory of nonlinear functionals for stochastic systems. According to Cameron and Martin such an expansion converges in the sense for any arbitrary stochastic process with finite second moment. This applies to most physical systems.

In statistical signal processing, the goal of **spectral density estimation** (**SDE**) is to estimate the spectral density of a random signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

Kernel density estimation is a nonparametric technique for density estimation i.e., estimation of probability density functions, which is one of the fundamental questions in statistics. It can be viewed as a generalisation of histogram density estimation with improved statistical properties. Apart from histograms, other types of density estimators include parametric, spline, wavelet and Fourier series. Kernel density estimators were first introduced in the scientific literature for univariate data in the 1950s and 1960s and subsequently have been widely adopted. It was soon recognised that analogous estimators for multivariate data would be an important addition to multivariate statistics. Based on research carried out in the 1990s and 2000s, **multivariate kernel density estimation** has reached a level of maturity comparable to its univariate counterparts.

- ↑ Burt, W. H. (1943). "Territoriality and home range concepts as applied to mammals".
*Journal of Mammalogy*.**24**(3): 346–352. doi:10.2307/1374834. JSTOR 1374834. - ↑ Jennrich, R. I.; Turner, F. B. (1969). "Measurement of non-circular home range".
*Journal of Theoretical Biology*.**22**(2): 227–237. doi:10.1016/0022-5193(69)90002-2. PMID 5783911. - ↑ Ford, R. G.; Krumme, D. W. (1979). "The analysis of space use patterns".
*Journal of Theoretical Biology*.**76**(2): 125–157. doi:10.1016/0022-5193(79)90366-7. PMID 431092. - ↑ Baker, J. (2001). "Population density and home range estimates for the Eastern Bristlebird at Jervis Bay, south-eastern Australia".
*Corella*.**25**: 62–67. - ↑ Creel, S.; Creel, N. M. (2002).
*The African Wild Dog: Behavior, Ecology, and Conservation*. Princeton, New Jersey: Princeton University Press. ISBN 978-0691016559. - ↑ Meulman, E. P.; Klomp, N. I. (1999). "Is the home range of the heath mouse
*Pseudomys shortridgei*an anomaly in the*Pseudomys*genus?".*Victorian Naturalist*.**116**: 196–201. - ↑ Rurik, L.; Macdonald, D. W. (2003). "Home range and habitat use of the kit fox (
*Vulpes macrotis*) in a prairie dog (*Cynomys ludovicianus*) complex".*Journal of Zoology*.**259**(1): 1–5. doi:10.1017/S0952836902002959. - ↑ Burgman, M. A.; Fox, J. C. (2003). "Bias in species range estimates from minimum convex polygons: implications for conservation and options for improved planning" (PDF).
*Animal Conservation*.**6**(1): 19–28. doi:10.1017/S1367943003003044. - ↑ Silverman, B. W. (1986).
*Density estimation for statistics and data analysis*. London: Chapman and Hall. ISBN 978-0412246203. - ↑ Worton, B. J. (1989). "Kernel methods for estimating the utilization distribution in home-range studies".
*Ecology*.**70**(1): 164–168. doi:10.2307/1938423. JSTOR 1938423. - ↑ Seaman, D. E.; Powell, R. A. (1996). "An evaluation of the accuracy of kernel density estimators for home range analysis".
*Ecology*.**77**(7): 2075–2085. doi:10.2307/2265701. JSTOR 2265701. - ↑ Burgman, M. A.; Fox, J. C. (2003). "Bias in species range estimates from minimum convex polygons: implications for conservation and options for improved planning" (PDF).
*Animal Conservation*.**6**(1): 19–28. doi:10.1017/S1367943003003044. - ↑ Getz, W. M.; Wilmers, C. C. (2004). "A local nearest-neighbor convex-hull construction of home ranges and utilization distributions" (PDF).
*Ecography*.**27**(4): 489–505. doi:10.1111/j.0906-7590.2004.03835.x. - ↑ Getz, W. M; Fortmann-Roe, S.; Cross, P. C.; Lyonsa, A. J.; Ryan, S. J.; Wilmers, C. C. (2007). "LoCoH: nonparametric kernel methods for constructing home ranges and utilization distributions" (PDF).
*PLoS ONE*.**2**(2): e207. Bibcode:2007PLoSO...2..207G. doi:10.1371/journal.pone.0000207. PMC 1797616 . PMID 17299587. - ↑ Getz, W. M.; Wilmers, C. C. (2004). "A local nearest-neighbor convex-hull construction of home ranges and utilization distributions" (PDF).
*Ecography*.**27**(4): 489–505. doi:10.1111/j.0906-7590.2004.03835.x. - ↑ Horne, J. S.; Garton, E. O.; Krone, S. M.; Lewis, J. S. (2007). "Analyzing animal movements using Brownian Bridges".
*Ecology*.**88**(9): 2354–2363. doi:10.1890/06-0957.1. PMID 17918412. S2CID 15044567. - ↑ Steiniger, S.; Hunter, A. J. S. (2012). "A scaled line-based kernel density estimator for the retrieval of utilization distributions and home ranges from GPS movement tracks".
*Ecological Informatics*.**13**: 1–8. doi:10.1016/j.ecoinf.2012.10.002. - ↑ Downs, J. A.; Horner, M. W.; Tucker, A. D. (2011). "Time-geographic density estimation for home range analysis".
*Annals of GIS*.**17**(3): 163–171. doi:10.1080/19475683.2011.602023. S2CID 7891668. - ↑ Long, J. A.; Nelson, T. A. (2012). "Time geography and wildlife home range delineation".
*Journal of Wildlife Management*.**76**(2): 407–413. doi:10.1002/jwmg.259. hdl: 10023/5424 . - ↑ Steiniger, S.; Hunter, A. J. S. (2012). "OpenJUMP HoRAE – A free GIS and Toolbox for Home-Range Analysis".
*Wildlife Society Bulletin*.**36**(3): 600–608. doi:10.1002/wsb.168. (See also: OpenJUMP HoRAE - Home Range Analysis and Estimation Toolbox) - ↑ LoCoH: Powerful algorithms for finding home ranges Archived 2006-09-12 at the Wayback Machine
- ↑ AniMove - Animal movement methods
- ↑ OpenJUMP HoRAE - Home Range Analysis and Estimation Toolbox (open source; methods: Point-Kernel, Line-Kernel, Brownian-Bridge, LoCoH, MCP, Line-Buffer)
- ↑ adehabitat for R (open source; methods: Point-Kernel, Line-Kernel, Brownian-Bridge, LoCoH, MCP, GeoEllipse)
- ↑ Nasrinpour, Hamid Reza; Reimer, Alex A.; Friesen, Marcia R.; McLeod, Robert D. (July 2017). "Data preparation for West Nile Virus agent-based modelling: protocol for processing bird population estimates and incorporating ArcMap in AnyLogic".
*JMIR Research Protocols*.**6**(7): e138. doi:10.2196/resprot.6213. PMC 5537560 . PMID 28716770.

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