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Approaches for Testing Uniformity Hypothesis in Angular Data of Mega-Herbivores

Circular statistics is an area not used very much by ecologists to describe animal movement patterns. Nevertheless, the connection between the evaluation of temporal recurring events and the analysis of directional data have converged in several papers, and show circular statistics to be an outstanding tool for understanding animal movement better. The aim of this chapter is to evaluate the applications of circular statistical tests to check uniformity hypothesis in animal movement and its potential interpretation within the general framework of movement ecology. Four methods of circular statistics: Rayleigh’s, Watson’s, Rao’s spacing and Kuiper’s test based on the mean resultant length are applied to examine the uniformity hypothesis of GPS derived telemetry data of elephant movement collected from Kruger National Park(KNP) South Africa. Overall, circular statistical uniformity tests methods represent a useful tool for evaluation of directionality elephant movement with applications including (i) assessment of animal foraging strategies; (ii) determination of orientation in response to landscape features and (iii) determination of the relative strengths of landscape features present bin a complex environment.


Introduction
The elaboration of appropriate conservation management and protection of endangered species of vegetation cover should be based on accurate interpretation of data and knowledge of animal movement on its habitat. One metric in movement ecology critical to this understanding is animal turn angles. It is urged in the literature that animals turn more during the wet season and less during the dry season. This variation is attributed to heterogeneous food resources and landscape features (Duffy et al., 2011). The movement of animals in protected areas (PA) and distribution of artificial water points has great impact on the vegetation and ecological dynamics of the ecosystem in general.
Circular statistical test of uniformity provides an opportunity to ecologists for understanding the turning patterns of the animals. In most studies, unimodal orientation may be the expected outcome. In several situations where for instant species or population preferences have been studied or when compass cues were set in experimental conflict, bimodal or multi-modal orientation may be expected (Fisher and Lee, 1992). Simulation studies based on random walk and Lévy flight theory of animal movement assume that turn angles are uniformly distributed (Viswanathan et al., 1996). As in the case linear statistics, the main objective here is to draw objective, reliable and biologically meaningful statistical inference about the population parameters on the basis of samples. Observations are either geometric or temporal in nature, where time related distributions can be fitted into a circular or spherical pattern (Batschelet, 1981). Parametric and non-parametric statistical methods can be used to test hypotheses concerning angular data (Batschelet, 1981).
These methods are based on the assumption that observed angles are independent, a condition that may not be satisfied when multiple angles are recorded from a single individual. Among the distributions used to describe data on a circle are the circular uniform and Von Mises distributions. Goodnessof-fit tests exist for these distributions. Four widely known methods for carrying out such a task are the subject of the current paper. Biological applications related to the technique described in section2 and 3 illustrate the earlier predominant role of one particular life science in predicting the potential utility of circular statistics in other domains of scientific and technical endeavour.
In recent years new data sources and GIS tools have been increasingly used in ecological studies. A peculiar characteristic of these data sources is that, often, only information about the locations of the animal trajectory/path have been recorded over some sampling resolution. Methodologies targeted especially for these data need to be developed. Motivation of this work arises from the a study about the movement patterns of Elephant in Kruger National park, South Africa and the apparent lack of tangible awareness of circular statistics in the movement ecology literature. This study was performed in order to support decisions for the management, in particular the conservation strategies for Elephants.
A limitation of the most common approaches is that they treat turn angles qualitatively, that is, converting angular data as either north, south, east or west which may lead to loss of valuable information (Dai et al., 2007). The specific aim here is to demonstrate the potential usefulness of circular statistics to animal movement analysis, in deciding whether GPS derived telemetry data justify the inference of uniformity in animal movement. Following a short presentation of

Theory of Hypothesis Testing and Notation
Circular hypothesis testing for uniformity is a valuable tool in movement ecology (Fisher and Lee, 1992). Statistical test should depend on previous assumptions supported by the descriptive or exploratory analysis (Fisher, Lewis and Embleton, 1993). In applied research, critical decisions based on data depends on objective and reliable assumptions. Due to technological advances and huge data collected on animal movement, testing such assumptions on animal movement data requires the knowledge of hypothesis testing.
A common question in circular statistics is whether a sample of data is uniformly distributed around the circle or has a common mean direction (Berens, 2009 In all the testing methods, the null hypothesis H0 states that the population samples are uniformly distributed around the circle and the alternative hypothesis HA the population samples do not show a uniform (or random) circular distribution. Each method rejects H0 if its test statistic exceeds a critical value depending usually on sample size n, and level of significance α (Jammalamadaka and SenGupta, 2001). In the theory of statistics, α=0.05 regarded as significant and α=0.01 is highly significant, meaning that a Type 1 error, is made by rejecting H0 is 5% or 1%, respectively. In the theory of statistics, α=0.05 is regarded as significant and α=0.01 is regarded as highly significant, meaning that a Type 1 error, is made by rejecting H0 is 5% level of significance, respectively. The current approach is more flexible by allowing the test statistic to determine the rejection of H0 on the basis of the p-value. According to Hogg and Craig (1978), the p-value is the magnitude of the error committed in rejecting H0 in face of the computed test statistic.
Rejecting the null hypothesis thus implies that deviations from uniformity are too large to assign them to chance factor; hence they are of deterministic origin Pewsey and Aurthur (2013). Since rejection of a null hypothesis is statistically stronger than it's opposite, Rao's method is more inviting than any other test when at least medium-size deviations from non-uniformity are expected from prior inspection descriptive statistics or exploratory analysis (Mardia and Jupp, 2009). Pewsey and Aurthur (2013) emphasis the importance of testing uniformity hypothesis in circular. We note that if the data fits neither a von Mises distribution nor a uniform distribution and contains a single mode, then this data is said to follow a unimodal distribution (Fisher, Lewis and Embleton, 1993). In this case, although it is not possible to identify the actual distribution, the presence of a single mode not only indicates a preferred orientation in the sample, but also enables the use of non-parametric methods to estimate a mean direction with a confidence interval (Fisher, Lewis and Embleton, 1993). We then tested the uniformity hypothesis following the procedure outlined in figure 1.

Rayleigh's Test
The Rayleigh's test is based on the intuitive idea of rejecting uniformity when the vector sample mean , is far from 0, when R  is large (Fisher, Lewis and Embleton, 1993 T is the sample mean vector. The score is: From the moments properties ofR , Mardia2009 it is possible to note that the score statistic is

Kuiper's Test Statistic
Kuiper's test is used to determine if a given set of data can be a sample from a specific distribution. It is similar to the Kolmogorov-Smirnov (KS) test, as both compare cumulative distributions (Berens, 2009). For the one-sample test, the empirical cumulative distribution is compared to a theoretical cumulative distribution. As for circular case, this test measures the deviation between empirical distribution, S n (x) and the Uniform cumulative distribution functions (cdf), . In the case of circular data, the definition of cumulative distribution is not obvious and is quite different from the in line cdf (Pewsey, Neuh¨auser and Ruxton, 2013).
In the circular data case, in fact, we first have to choose the circle zero point and orientation, then we need to augment the ordered observations,  1 ,, n of x 0 =0 and  n+1 =2.
Then S n is then defined by ( ) (4) Just as in Kolmogorov-Smirnov's test for in line distribution Durbin1973, the following quantities are defined: . For practical purposes, the following modification of V n is used:

Watson's Test
Another common test of uniformity in circular statistics is the Watson U 2 statistic (Watson, 1961) which is a modification of the Cramér-von Mises test (Durbin, 1973). This test is used as a goodness-of-fit statistics for the von Mises distribution Mardia1972. As a test of goodness of fit for circular data, it is invariant to the choice of the origin. The watson test statistic is defined as It follows from this definition that the Watson's statistic is invariant under rotation and reflections. As for the Kuiper's test, it is useful to consider the following modified statistic: Stephens (1970) provides the in a tabular form the quantiles of the Watson test statistic.

Rao's Spacing Test
Rao's Spacing test is a useful and powerful statistic for testing uniformity of circular data. As with other circular statistics, Rao's Spacing test is applicable for analysis of angular data, in studies of movement and spatial trends in geographical research (Pewsey, Neuh¨auser and Ruxton, 2013). In many cases, particularly with an underlying distribution that is multimodal, it is more powerful than the popular Kuiper's Test and Rayleigh Test.

Rao's
Under uniformity E[T i ]= 2 n . Hence, it is reasonable to reject uniformity for large values of Large values of L indicate clustering of observations (Russell and Levitin, 1995). An extensive table of quantiles of L is given (Russell and Levitin, 1995), while (Sherman, 1950) shows that a suitable transformation of L is asymptotically standard normally distributed.

Ethics Statement
Elephant capture and handling was conducted in strict accordance with ethical standards. Specific approval for this particular research project was obtained through the University of KwaZulu-Natal Animal Ethics sub-committee

Study area and GPS Data on a elephants
The methods described were applied to GPS location data for three female elephant herds in Kruger national park of South Africa. Elephant movement have been previously studied by  ., 2010). The GPS data are freely available upon request.
The turning patterns of elephants were monitored by computing the angle θ between two consecutive relocations over 30 Minutes interval of time between May, 2006 and June 2009. The data set consists of 36395, 29221 and 29908 observations for herd AM108, AM307 and AM308 respectively recorded at an interval of 30 minutes. The turning angle (θ) was computed for the three herds as the change in the direction of movement made by an each individual elephant tagged from one location to the next. The turning angle is a right-hand turn that ranged from -π to π.
The GPS locations provided data every 30 minutes during an entire day with an accuracy of the locations within 50 meters. This information was sent via cellular phone (GSM) network to a website from where the information was downloaded.

Results
Angular data of three elephant herds was used to construct circular histograms that depict the mean and frequency of movement orientations of animals Figure (2) and (3). The rose diagram indicates that elephants move in a non-random manner during the dry and the wet seasons. This variation is attributed to uneven distribution of resources during dry season than in the wet season. Similarly, the linear histograms in Figure (3) shows that elephants turn angles are concentrated around zero degrees which indicate that the movement is target oriented. The tendency is known as the unimodal movement pattern better described by unimodal distributions. In all the four herds, the assumption of Unimodality is fulfilled, and is statistically significant results are obtained based on the conventional circular statistical methods.

Discussion
This study focuses on the test of uniformity hypothesis for the circular animal movement data. The primary objective was to determine whether elephant orientation patterns were uniform. We also investigated the hypothesis that elephant orientation patterns would vary between seasons. Testing the null hypothesis can be accomplished by comparing any test statistic for uniformity to a reference distribution obtained by simulation. This is appealing in that precise distributions consistent with turn angle data need not to be assumed for the procedure to have the proper type I error rate. Our results suggest that elephant turn angle data is not uniformly distributed and showed no seasonal variation. The four statistical tests reject the uniformity hypothesis and conclude that animal movement turn angles data is not uniform. These applications, in summary entail calculation of animal orientation variables mean turn angle, concentration parameter, distribution, and determination of modality and testing hypothesis about uniformity of animal turn angles. It is important to note that the circular statistics applies to any level and scale of analysis, from individuals to several species of animals, and that the descriptive measures