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

Approaches for Testing Uniformity Hypothesis in Angular Data of Mega-Herbivores International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 Approaches for Testing Uniformity Hypothesis in Angular Data of Mega-Herbivores 1, 3 1 2 Dr. Robert Mathenge Mutwiri , Prof. Henry Mwambi , Rob Slotow School of Mathematics, Statistics and Computer Sciences, University of KwaZulu Natal, South Africa School of Pure and Applied Sciences, Kiriinyaga University College, Kenya Professor, School of Mathematics, Statistics and Computer Sciences, University of KwaZulu Natal, South Africa Professor, School of Life Sciences, University of KwaZulu Natal, South Africa Abstract: 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. Keywords: Circular statistics, animal movement, turn angles, uniformity hypothesis These methods are based on the assumption that observed 1. Introduction angles are independent, a condition that may not be satisfied when multiple angles are recorded from a single individual. The elaboration of appropriate conservation management and Among the distributions used to describe data on a circle are protection of endangered species of vegetation cover should the circular uniform and Von Mises distributions. Goodness- be based on accurate interpretation of data and knowledge of of-fit tests exist for these distributions. Four widely known animal movement on its habitat. One metric in movement methods for carrying out such a task are the subject of the ecology critical to this understanding is animal turn angles. It current paper. Biological applications related to the is urged in the literature that animals turn more during the technique described in section2 and 3 illustrate the earlier wet season and less during the dry season. This variation is predominant role of one particular life science in predicting attributed to heterogeneous food resources and landscape the potential utility of circular statistics in other domains of features (Duffy et al., 2011). The movement of animals in scientific and technical endeavour. protected areas (PA) and distribution of artificial water points has great impact on the vegetation and ecological dynamics In recent years new data sources and GIS tools have been of the ecosystem in general. increasingly used in ecological studies. A peculiar characteristic of these data sources is that, often, only Circular statistical test of uniformity provides an opportunity information about the locations of the animal trajectory/path to ecologists for understanding the turning patterns of the have been recorded over some sampling resolution. animals. In most studies, unimodal orientation may be the Methodologies targeted especially for these data need to be expected outcome. In several situations where for instant developed. Motivation of this work arises from the a study species or population preferences have been studied or when about the movement patterns of Elephant in Kruger National compass cues were set in experimental conflict, bimodal or park, South Africa and the apparent lack of tangible multi-modal orientation may be expected (Fisher and Lee, awareness of circular statistics in the movement ecology 1992). Simulation studies based on random walk and Lévy literature. This study was performed in order to support flight theory of animal movement assume that turn angles are decisions for the management, in particular the conservation uniformly distributed (Viswanathan et al., 1996). As in the strategies for Elephants. case linear statistics, the main objective here is to draw objective, reliable and biologically meaningful statistical A limitation of the most common approaches is that they treat inference about the population parameters on the basis of turn angles qualitatively, that is, converting angular data as samples. Observations are either geometric or temporal in either north, south, east or west which may lead to loss of nature, where time related distributions can be fitted into a valuable information (Dai et al., 2007). The specific aim here circular or spherical pattern (Batschelet, 1981). Parametric is to demonstrate the potential usefulness of circular statistics and non-parametric statistical methods can be used to test to animal movement analysis, in deciding whether GPS hypotheses concerning angular data (Batschelet, 1981). derived telemetry data justify the inference of uniformity in animal movement. Following a short presentation of Volume 5 Issue 3, March 2016 www.ijsr.net Paper ID: NOV162109 1202 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 theoretical aspects, numerical examples illustrate calculations non-uniformity are expected from prior inspection required for drawing proper conclusions about the animal descriptive statistics or exploratory analysis (Mardia and movement patterns. 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 2. Theory of Hypothesis Testing and Notation uniform distribution and contains a single mode, then this data is said to follow a unimodal distribution (Fisher, Lewis Circular hypothesis testing for uniformity is a valuable tool in and Embleton, 1993). In this case, although it is not possible movement ecology (Fisher and Lee, 1992). Statistical test to identify the actual distribution, the presence of a single should depend on previous assumptions supported by the mode not only indicates a preferred orientation in the sample, descriptive or exploratory analysis (Fisher, Lewis and but also enables the use of non-parametric methods to Embleton, 1993). In applied research, critical decisions based estimate a mean direction with a confidence interval (Fisher, on data depends on objective and reliable assumptions. Due Lewis and Embleton, 1993). We then tested the uniformity to technological advances and huge data collected on animal hypothesis following the procedure outlined in figure 1. 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). A multiple of test statistics have been designed for testing this hypothesis. These methods includes: (i) Rao spacing test; (ii) Kuiper’s test, (iii) Rayleigh test and (iv) Watsons test. These four tests can be used to assess the evidence for a uniform, unimodality and the goodness-of-fit for the von Mises distributions respectively (Mardia and Jupp, 2009). Previous investigations shows that the Rao’s spacing method is more susceptible to rejecting H0 than the kuiper’s test and the Rayleigh test in the face of a small data sample, unless data distribution is appreciably uniform at least in some of its sub- domains (Russell and Levitin, 1995). This implies that the Rao’s spacing test carries a similar Type 1 error compared to Figure 1: Flowchart representing the sequence of hypothesis other test; however, we point out that from the nature of the tests based on circular distributions data, rejection of H0 cannot be absolutely certain (Mardia and Jupp, 2009). 2.1 Rayleigh’s Test In all the testing methods, the null hypothesis H0 states that The Rayleigh’s test is based on the intuitive idea of rejecting the population samples are uniformly distributed around the uniformity when the vector sample mean , is far from 0, circle and the alternative hypothesis HA the population when R is large (Fisher, Lewis and Embleton, 1993). The samples do not show a uniform (or random) circular Rayleigh’s test is the score test of uniformity within the von distribution. Each method rejects H0 if its test statistic Mises model (Mardia and Lee, 2009). Put w= (cos exceeds a critical value depending usually on sample size n, (),cos())', the log likelihood Von Mises based on and level of significance α (Jammalamadaka and SenGupta, circular observations  ,, is 2001). In the theory of statistics, α=0.05 regarded as 1 n significant and α=0.01 is highly significant, meaning that a l( : , , ) n nlogI (k), 10 n Type 1 error, is made by rejecting H0 is 5% or 1%, respectively. In the theory of statistics, α=0.05 is regarded as where I () is the modified Bessel function and significant and α=0.01 is regarded as highly significant, meaning that a Type 1 error, is made by rejecting H0 is 5% = (cos ,sin ) is the sample mean vector. The score n i i level of significance, respectively. The current approach is i=1 more flexible by allowing the test statistic to determine the is: rejection of H0 on the basis of the p-value. According to l Hogg and Craig (1978), the p-value is the magnitude of the U n - nA( )(cos , sin ) (1) error committed in rejecting H0 in face of the computed test statistic. From the moments properties ofR, Mardia2009 it is possible to note that the score statistic is Rejecting the null hypothesis thus implies that deviations 1 2 U'var(U) U=2nR . (2) from uniformity are too large to assign them to chance factor; From the general theory of score test, Mardia2009 the large hence they are of deterministic origin Pewsey and Aurthur (2013). Since rejection of a null hypothesis is statistically sample asymptotic distribution of 2nR under uniformity is a stronger than it’s opposite, Rao’s method is more inviting 2  with two degrees of freedom: than any other test when at least medium-size deviations from Volume 5 Issue 3, March 2016 www.ijsr.net Paper ID: NOV162109 1203 Licensed Under Creative Commons Attribution CC BY   International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 2 2 It follows from this definition that the Watson’s statistic is 2nR  (3) invariant under rotation and reflections. As for the Kuiper’s where n is the sample size. It has been demonstrated, also test, it is useful to consider the following modified statistic: that the Rayleigh’s test coincides with the likelihood test of 2 2 0.1 0.1 0.8 uniformity within the von Mises family. U =(U  + )(1+ ). (8) n n n 2 n 2.2 Kuiper’s Test Statistic Stephens (1970) provides the in a tabular form the quantiles of the Watson 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 2.4 Rao’s Spacing Test Kolmogorov-Smirnov (KS) test, as both compare cumulative distributions (Berens, 2009). For the one-sample test, the Rao’s Spacing test is a useful and powerful statistic for empirical cumulative distribution is compared to a theoretical testing uniformity of circular data. As with other circular cumulative distribution. As for circular case, this test statistics, Rao’s Spacing test is applicable for analysis of measures the deviation between empirical distribution, S (x) angular data, in studies of movement and spatial trends in geographical research (Pewsey, Neuh¨auser and Ruxton, and the Uniform cumulative distribution functions (cdf), 2013). In many cases, particularly with an underlying F(x)= . In the case of circular data, the definition of distribution that is multimodal, it is more powerful than the 2 popular Kuiper’s Test and Rayleigh Test. cumulative distribution is not obvious and is quite different from the in line cdf (Pewsey, Neuh¨auser and Ruxton, 2013). Rao’s Spacing Test is based on the idea that if the underlying In the circular data case, in fact, we first have to choose the distribution is uniform, successive observations should be circle zero point and orientation, then we need to augment approximately evenly spaced, about 360/N apart. Large the ordered observations,  ,, of x =0 and  =2. 1 n 0 n+1 deviations from this distribution, resulting from unusually Then S is then defined by large or unusually short spaces between observations, are evidence for directionality. It is related to the general class of linear statistical tests that are based on successive order S ()   (ii )  (  1) n statistics and spacing. The spacing tests sample arc lengths if i=0,1,…, n (4) T ,,T defined as: 1 n Just as in Kolmogorov-Smirnov’s test for in line distribution Ti  ( ) (i 1), Tn  2 ( ( )  (1)). i=1…n-1, i n Durbin1973, the following quantities are defined: (9) D sup {S () F()}, D sup {F() S () }. nn  n n 2 Under uniformity E[T ]= . Hence, it is reasonable to reject +  i n To overcome the dependence of D and D on the choice of n n uniformity for large values of the initial direction, (Kuiper,1960) defined 1 2 L= |T  | (10) (5) 2 i n i=1 The statistic (5) has been demonstrated (Mardia and Jupp, Large values of L indicate clustering of observations (Russell 2009) to be invariant under the change of initial direction. and Levitin, 1995). An extensive table of quantiles of L is The null hypothesis of uniformity is rejected for large values given (Russell and Levitin, 1995), while (Sherman, 1950) of V . Moreover, the Kuiper’s test is consistent against all shows that a suitable transformation of L is asymptotically standard normally distributed. alternative to uniformity (Pewsey, Neuh¨auser and Ruxton, 2013). For practical purposes, the following modification of V is used: 3. Application to Elephant Movement Data 0.24  0.155 1/2 V =n V (1+ + ). (6) 3.1 Ethics Statement n n Elephant capture and handling was conducted in strict 2.3 Watson’s Test accordance with ethical standards. Specific approval for this particular research project was obtained through the Another common test of uniformity in circular statistics is the University of KwaZulu-Natal Animal Ethics sub-committee (Ref. 009/10/Animal). This research also forms part of a Watson U statistic (Watson, 1961) which is a modification registered and approved SANParks project, in association of the Cramér-von Mises test (Durbin, 1973). This test is with Kruger National Park and Scientific Services (Ref: used as a goodness-of-fit statistics for the von Mises BIRPJ743) (Birkett et al., 2012). distribution Mardia1972. As a test of goodness of fit for circular data, it is invariant to the choice of the origin. The 3.2 Study area and GPS Data on a elephants watson test statistic is defined as 2 2   The methods described were applied to GPS location data for   U = (S ()F()) (S ()F())dF dF (7)     n n n   three female elephant herds in Kruger national park of South 0  0  Africa. Elephant movement have been previously studied by Volume 5 Issue 3, March 2016 www.ijsr.net Paper ID: NOV162109 1204 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 Vanak2010. The elephant population in KNP was estimated The GPS locations provided data every 30 minutes during an to be 14,000 individuals during 2010 (SANParks, entire day with an accuracy of the locations within 50 meters. unpublished data). From 2006 to 2010, we collected This information was sent via cellular phone (GSM) network geographical location data, downloaded from GPS/GSM to a website from where the information was downloaded. Collars (Africa Wildlife Tracking cc., South Africa), fitted to three elephant cows from different herds. To ensure the 4. Results independence of sampling, a single female in each herd was selected and collared. Because GPS coordinates were Angular data of three elephant herds was used to construct measured continuously at frequent time intervals, the circular histograms that depict the mean and frequency of trajectory we obtained was almost smoothly connected movement orientations of animals Figure (2) and (3). The (Vanak et al., 2010). The GPS data are freely available upon rose diagram indicates that elephants move in a non-random request. manner during the dry and the wet seasons. This variation is attributed to uneven distribution of resources during dry The turning patterns of elephants were monitored by season than in the wet season. Similarly, the linear computing the angle θ between two consecutive relocations histograms in Figure (3) shows that elephants turn angles are over 30 Minutes interval of time between May, 2006 and concentrated around zero degrees which indicate that the June 2009. The data set consists of 36395, 29221 and 29908 movement is target oriented. The tendency is known as the observations for herd AM108, AM307 and AM308 unimodal movement pattern better described by unimodal respectively recorded at an interval of 30 minutes. The distributions. In all the four herds, the assumption of turning angle (θ) was computed for the three herds as the Unimodality is fulfilled, and is statistically significant results change in the direction of movement made by an each are obtained based on the conventional circular statistical individual elephant tagged from one location to the next. The methods. turning angle is a right-hand turn that ranged from -π to π. Table 1: Descriptive analysis of the turn angle data of three elephant herds collected from Kruger National Park, South Africa Statistic AM108 AM307 AM308 Seasons Seasons Seasons Variable Angle Wet Dry angle Wet Dry angle Wet Dry Number of Observations 36395 17736 18659 29221 12652 16569 29908 13550 16358 Mean Vector (µ) o o o o o o o o o 359.425 359.434 359.418 0.706 0.84 0.609 359.957 359.7 0.16 Length of Mean Vector (r) 0.481 0.464 0.498 0.434 0.418 0.446 0.482 0.471 0.492 Concentration 1.096 1.046 1.144 0.962 0.92 0.995 1.099 1.065 1.127 Circular Variance 0.519 0.536 0.502 0.566 0.582 0.554 0.518 0.529 0.508 Circular Standard Deviation o o o o o o o o o 69.292 71 67.686 74.044 75.65 72.833 69.19 70.336 68.248 Standard Error of Mean o o o o o o o o o 0.414 0.618 0.556 0.519 0.821 0.668 0.455 0.695 0.602 Table 2: Tests for uniformity for three elephant herds turn angles derived from GPS tracking data. Kuiper’s Rayleigh Rao spacing Watson’s statistic p-value statistic p-value statistic p-value statistic p-value AM108 61.6574 < 0.01 0.4813 0.000 166.2602 < 0.001 452.2261 < 0.01 AM307 50.7271 < 0.01 0.4339 0.000 162.4811 < 0.001 301.1729 < 0.01 AM308 56.3221 < 0.01 0.4823 0.000 166.2796 < 0.001 376.4143 < 0.01 The distribution of the turning angles between successive uniformity by Rao’s test. The results of the Rayleigh tests in positions differed significantly from a uniform distribution Table 2 indicates that elephant movement turn angles are not for the three herds (Kuiper’s test: p<0.001). Turning angles evenly distributed in all directions and concentrated around were backward oriented in the three herds at 359.43, 0.706 o 0 (θ>0). The uniformity and von Mises provide more and 359.96 (mean direction of resultant vector, Rayleigh test: information as the Rayleigh and Rao’s test confirm that p<0.001) indicating a tendency of elephants to make a U- azimuths are not uniformly distributed (see, Table 2). As turns while foraging (Dai et al., 2007). There distribution was azimuths do not follow a random distribution, we can look not different between the wet and dry season 1. Kuiper’s and for a preferred direction; therefore, two tests for von Mises Rayleigh’s test exhibit small p-values although the former is distribution were performed. less realistic with respect to the powerful rejection of Volume 5 Issue 3, March 2016 www.ijsr.net Paper ID: NOV162109 1205 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 Figure 2: Circular histograms of elephant movement turn angles θ. Circular bars indicate the number of observations within each class range and have been centred on 0 . Figure 3: Histogram of herd AM108, AM307 and AM308 GPS-telemetry derived turn angles data collected from Kruger National Park South Africa (May 2006 -April 2009) suggest that elephant turn angle data is not uniformly 5. Discussion distributed and showed no seasonal variation. The four statistical tests reject the uniformity hypothesis and conclude This study focuses on the test of uniformity hypothesis for that animal movement turn angles data is not uniform. These the circular animal movement data. The primary objective applications, in summary entail calculation of animal was to determine whether elephant orientation patterns were orientation variables mean turn angle, concentration uniform. We also investigated the hypothesis that elephant parameter, distribution, and determination of modality and orientation patterns would vary between seasons. Testing the testing hypothesis about uniformity of animal turn angles. It null hypothesis can be accomplished by comparing any test is important to note that the circular statistics applies to any statistic for uniformity to a reference distribution obtained by level and scale of analysis, from individuals to several simulation. This is appealing in that precise distributions species of animals, and that the descriptive measures consistent with turn angle data need not to be assumed for the procedure to have the proper type I error rate. Our results Volume 5 Issue 3, March 2016 www.ijsr.net Paper ID: NOV162109 1206 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 calculated are easily compared statistically by means of a a highly mobile mega-herbivore? Biological variety of two or more multi-sample tests. Conservation, pages 2631–2637. [20] Viswanathan, G. M., Afanasyev, V., Buldyrev, S. V., Murphy, E. J., Prince, P. A., and Stanley, H. E. (1996). References L’evy flight search patterns of wandering albatrosses. Nature, 381(6581):413–415. [1] Batschelet, E. (1981). Circular Statistics in Biology. [21] Watson, G. S. (1961). Goodness-of-fit tests on a circle. Mathematics in Biology. Academic Press. Biometrika, pages 109–114. [2] Berens, P. (2009). Circstat: A matlab toolbox for circular statistics. Journal of Statistical Software, 31(10):1–21. [3] Birkett, P. J., Vanak, A. T., Muggeo, V. M. R., Ferreira, S. M., and Slotow, R. (2012). Animal perception of seasonal thresholds: Changes in elephant movement in relation to rainfall patterns. PLoS ONE, 7(6):e38363. [4] Dai, X., Shannon, G., Slotow, R., Page, B., and Duffy, K. J. (2007). Short-duration daytime movements of a cow herd of African elephants. Journal of Mammalogy, 88(1):151–157. [5] Duffy, K. J., Dai, X., Shannon, G., Slotow, R., and Page, B. (2011). Movement patterns of African elephants (Loxodonta africana) in different habitat types. South African Journal of Wildlife Research, 41(1):21–28. [6] Durbin, J. (1973). Distribution theory for tests based on sample distribution function, volume 9. Society for Industrial Mathematics. [7] Fisher, N. and Lee, A. (1992). Regression models for an angular response. Biometrics, pages 665–677. [8] Fisher, N., Lewis, T., and Embleton, B. (1993). Statistical analysis of spherical data. Cambridge University Press. [9] Hogg, R. V. and Craig, A. T. (1978). Introduction to mathematical statistics. New York: Macmillan. [10] Jammalamadaka, S. R. and SenGupta, A. (2001). Topics in Circular Statistics,. World Scientific, New York. [11] Kuiper, N. H. (1962). Tests concerning random points on a circle. Nederl. Akad. Wetensch. Proc. Ser. A, 63:38–47. [12] Mardia, K. (1972). Statistics for directional data. Academic press. [13] Mardia, K. and Jupp, P. (2009). Directional statistics, volume 494. Wiley. [14] Pewsey, A., Neuh¨auser, M., and Ruxton, G. D. (2013). Circular statistics in R. Oxford University Press. [15] Russell, G. S. and Levitin, D. J. (1995). An expanded table of probability values for rao’s spacing test. Communications in Statistics-Simulation and Computation, 24(4):879–888. [16] Sherman, B. (1950). A random variable related to the spacing of sample values. The Annals of Mathematical Statistics, 21(3):339–361. [17] Stephens, M. A. (1970). Use of the kolmogorov- smirnov, cram´er-von mises and related statistics without extensive tables. Journal of the Royal Statistical Society. Series B (Methodological), pages 115–122. [18] Tracey, J. A., Zhu, J., and Crooks, K. (2005). A set of nonlinear regression models for animal movement in response to a single landscape feature. Journal of Agricultural, Biological, and Environmental Statistics, 10(1):1–18. [19]Vanak, A., Thaker, M., and Slotow, R. (2010). 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Approaches for Testing Uniformity Hypothesis in Angular Data of Mega-Herbivores

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International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 Approaches for Testing Uniformity Hypothesis in Angular Data of Mega-Herbivores 1, 3 1 2 Dr. Robert Mathenge Mutwiri , Prof. Henry Mwambi , Rob Slotow School of Mathematics, Statistics and Computer Sciences, University of KwaZulu Natal, South Africa School of Pure and Applied Sciences, Kiriinyaga University College, Kenya Professor, School of Mathematics, Statistics and Computer Sciences, University of KwaZulu Natal, South Africa Professor, School of Life Sciences, University of KwaZulu Natal, South Africa Abstract: 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. Keywords: Circular statistics, animal movement, turn angles, uniformity hypothesis These methods are based on the assumption that observed 1. Introduction angles are independent, a condition that may not be satisfied when multiple angles are recorded from a single individual. The elaboration of appropriate conservation management and Among the distributions used to describe data on a circle are protection of endangered species of vegetation cover should the circular uniform and Von Mises distributions. Goodness- be based on accurate interpretation of data and knowledge of of-fit tests exist for these distributions. Four widely known animal movement on its habitat. One metric in movement methods for carrying out such a task are the subject of the ecology critical to this understanding is animal turn angles. It current paper. Biological applications related to the is urged in the literature that animals turn more during the technique described in section2 and 3 illustrate the earlier wet season and less during the dry season. This variation is predominant role of one particular life science in predicting attributed to heterogeneous food resources and landscape the potential utility of circular statistics in other domains of features (Duffy et al., 2011). The movement of animals in scientific and technical endeavour. protected areas (PA) and distribution of artificial water points has great impact on the vegetation and ecological dynamics In recent years new data sources and GIS tools have been of the ecosystem in general. increasingly used in ecological studies. A peculiar characteristic of these data sources is that, often, only Circular statistical test of uniformity provides an opportunity information about the locations of the animal trajectory/path to ecologists for understanding the turning patterns of the have been recorded over some sampling resolution. animals. In most studies, unimodal orientation may be the Methodologies targeted especially for these data need to be expected outcome. In several situations where for instant developed. Motivation of this work arises from the a study species or population preferences have been studied or when about the movement patterns of Elephant in Kruger National compass cues were set in experimental conflict, bimodal or park, South Africa and the apparent lack of tangible multi-modal orientation may be expected (Fisher and Lee, awareness of circular statistics in the movement ecology 1992). Simulation studies based on random walk and Lévy literature. This study was performed in order to support flight theory of animal movement assume that turn angles are decisions for the management, in particular the conservation uniformly distributed (Viswanathan et al., 1996). As in the strategies for Elephants. case linear statistics, the main objective here is to draw objective, reliable and biologically meaningful statistical A limitation of the most common approaches is that they treat inference about the population parameters on the basis of turn angles qualitatively, that is, converting angular data as samples. Observations are either geometric or temporal in either north, south, east or west which may lead to loss of nature, where time related distributions can be fitted into a valuable information (Dai et al., 2007). The specific aim here circular or spherical pattern (Batschelet, 1981). Parametric is to demonstrate the potential usefulness of circular statistics and non-parametric statistical methods can be used to test to animal movement analysis, in deciding whether GPS hypotheses concerning angular data (Batschelet, 1981). derived telemetry data justify the inference of uniformity in animal movement. Following a short presentation of Volume 5 Issue 3, March 2016 www.ijsr.net Paper ID: NOV162109 1202 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 theoretical aspects, numerical examples illustrate calculations non-uniformity are expected from prior inspection required for drawing proper conclusions about the animal descriptive statistics or exploratory analysis (Mardia and movement patterns. 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 2. Theory of Hypothesis Testing and Notation uniform distribution and contains a single mode, then this data is said to follow a unimodal distribution (Fisher, Lewis Circular hypothesis testing for uniformity is a valuable tool in and Embleton, 1993). In this case, although it is not possible movement ecology (Fisher and Lee, 1992). Statistical test to identify the actual distribution, the presence of a single should depend on previous assumptions supported by the mode not only indicates a preferred orientation in the sample, descriptive or exploratory analysis (Fisher, Lewis and but also enables the use of non-parametric methods to Embleton, 1993). In applied research, critical decisions based estimate a mean direction with a confidence interval (Fisher, on data depends on objective and reliable assumptions. Due Lewis and Embleton, 1993). We then tested the uniformity to technological advances and huge data collected on animal hypothesis following the procedure outlined in figure 1. 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). A multiple of test statistics have been designed for testing this hypothesis. These methods includes: (i) Rao spacing test; (ii) Kuiper’s test, (iii) Rayleigh test and (iv) Watsons test. These four tests can be used to assess the evidence for a uniform, unimodality and the goodness-of-fit for the von Mises distributions respectively (Mardia and Jupp, 2009). Previous investigations shows that the Rao’s spacing method is more susceptible to rejecting H0 than the kuiper’s test and the Rayleigh test in the face of a small data sample, unless data distribution is appreciably uniform at least in some of its sub- domains (Russell and Levitin, 1995). This implies that the Rao’s spacing test carries a similar Type 1 error compared to Figure 1: Flowchart representing the sequence of hypothesis other test; however, we point out that from the nature of the tests based on circular distributions data, rejection of H0 cannot be absolutely certain (Mardia and Jupp, 2009). 2.1 Rayleigh’s Test In all the testing methods, the null hypothesis H0 states that The Rayleigh’s test is based on the intuitive idea of rejecting the population samples are uniformly distributed around the uniformity when the vector sample mean , is far from 0, circle and the alternative hypothesis HA the population when R is large (Fisher, Lewis and Embleton, 1993). The samples do not show a uniform (or random) circular Rayleigh’s test is the score test of uniformity within the von distribution. Each method rejects H0 if its test statistic Mises model (Mardia and Lee, 2009). Put w= (cos exceeds a critical value depending usually on sample size n, (),cos())', the log likelihood Von Mises based on and level of significance α (Jammalamadaka and SenGupta, circular observations  ,, is 2001). In the theory of statistics, α=0.05 regarded as 1 n significant and α=0.01 is highly significant, meaning that a l( : , , ) n nlogI (k), 10 n Type 1 error, is made by rejecting H0 is 5% or 1%, respectively. In the theory of statistics, α=0.05 is regarded as where I () is the modified Bessel function and significant and α=0.01 is regarded as highly significant, meaning that a Type 1 error, is made by rejecting H0 is 5% = (cos ,sin ) is the sample mean vector. The score n i i level of significance, respectively. The current approach is i=1 more flexible by allowing the test statistic to determine the is: rejection of H0 on the basis of the p-value. According to l Hogg and Craig (1978), the p-value is the magnitude of the U n - nA( )(cos , sin ) (1) error committed in rejecting H0 in face of the computed test statistic. From the moments properties ofR, Mardia2009 it is possible to note that the score statistic is Rejecting the null hypothesis thus implies that deviations 1 2 U'var(U) U=2nR . (2) from uniformity are too large to assign them to chance factor; From the general theory of score test, Mardia2009 the large hence they are of deterministic origin Pewsey and Aurthur (2013). Since rejection of a null hypothesis is statistically sample asymptotic distribution of 2nR under uniformity is a stronger than it’s opposite, Rao’s method is more inviting 2  with two degrees of freedom: than any other test when at least medium-size deviations from Volume 5 Issue 3, March 2016 www.ijsr.net Paper ID: NOV162109 1203 Licensed Under Creative Commons Attribution CC BY   International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 2 2 It follows from this definition that the Watson’s statistic is 2nR  (3) invariant under rotation and reflections. As for the Kuiper’s where n is the sample size. It has been demonstrated, also test, it is useful to consider the following modified statistic: that the Rayleigh’s test coincides with the likelihood test of 2 2 0.1 0.1 0.8 uniformity within the von Mises family. U =(U  + )(1+ ). (8) n n n 2 n 2.2 Kuiper’s Test Statistic Stephens (1970) provides the in a tabular form the quantiles of the Watson 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 2.4 Rao’s Spacing Test Kolmogorov-Smirnov (KS) test, as both compare cumulative distributions (Berens, 2009). For the one-sample test, the Rao’s Spacing test is a useful and powerful statistic for empirical cumulative distribution is compared to a theoretical testing uniformity of circular data. As with other circular cumulative distribution. As for circular case, this test statistics, Rao’s Spacing test is applicable for analysis of measures the deviation between empirical distribution, S (x) angular data, in studies of movement and spatial trends in geographical research (Pewsey, Neuh¨auser and Ruxton, and the Uniform cumulative distribution functions (cdf), 2013). In many cases, particularly with an underlying F(x)= . In the case of circular data, the definition of distribution that is multimodal, it is more powerful than the 2 popular Kuiper’s Test and Rayleigh Test. cumulative distribution is not obvious and is quite different from the in line cdf (Pewsey, Neuh¨auser and Ruxton, 2013). Rao’s Spacing Test is based on the idea that if the underlying In the circular data case, in fact, we first have to choose the distribution is uniform, successive observations should be circle zero point and orientation, then we need to augment approximately evenly spaced, about 360/N apart. Large the ordered observations,  ,, of x =0 and  =2. 1 n 0 n+1 deviations from this distribution, resulting from unusually Then S is then defined by large or unusually short spaces between observations, are evidence for directionality. It is related to the general class of linear statistical tests that are based on successive order S ()   (ii )  (  1) n statistics and spacing. The spacing tests sample arc lengths if i=0,1,…, n (4) T ,,T defined as: 1 n Just as in Kolmogorov-Smirnov’s test for in line distribution Ti  ( ) (i 1), Tn  2 ( ( )  (1)). i=1…n-1, i n Durbin1973, the following quantities are defined: (9) D sup {S () F()}, D sup {F() S () }. nn  n n 2 Under uniformity E[T ]= . Hence, it is reasonable to reject +  i n To overcome the dependence of D and D on the choice of n n uniformity for large values of the initial direction, (Kuiper,1960) defined 1 2 L= |T  | (10) (5) 2 i n i=1 The statistic (5) has been demonstrated (Mardia and Jupp, Large values of L indicate clustering of observations (Russell 2009) to be invariant under the change of initial direction. and Levitin, 1995). An extensive table of quantiles of L is The null hypothesis of uniformity is rejected for large values given (Russell and Levitin, 1995), while (Sherman, 1950) of V . Moreover, the Kuiper’s test is consistent against all shows that a suitable transformation of L is asymptotically standard normally distributed. alternative to uniformity (Pewsey, Neuh¨auser and Ruxton, 2013). For practical purposes, the following modification of V is used: 3. Application to Elephant Movement Data 0.24  0.155 1/2 V =n V (1+ + ). (6) 3.1 Ethics Statement n n Elephant capture and handling was conducted in strict 2.3 Watson’s Test accordance with ethical standards. Specific approval for this particular research project was obtained through the Another common test of uniformity in circular statistics is the University of KwaZulu-Natal Animal Ethics sub-committee (Ref. 009/10/Animal). This research also forms part of a Watson U statistic (Watson, 1961) which is a modification registered and approved SANParks project, in association of the Cramér-von Mises test (Durbin, 1973). This test is with Kruger National Park and Scientific Services (Ref: used as a goodness-of-fit statistics for the von Mises BIRPJ743) (Birkett et al., 2012). distribution Mardia1972. As a test of goodness of fit for circular data, it is invariant to the choice of the origin. The 3.2 Study area and GPS Data on a elephants watson test statistic is defined as 2 2   The methods described were applied to GPS location data for   U = (S ()F()) (S ()F())dF dF (7)     n n n   three female elephant herds in Kruger national park of South 0  0  Africa. Elephant movement have been previously studied by Volume 5 Issue 3, March 2016 www.ijsr.net Paper ID: NOV162109 1204 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 Vanak2010. The elephant population in KNP was estimated The GPS locations provided data every 30 minutes during an to be 14,000 individuals during 2010 (SANParks, entire day with an accuracy of the locations within 50 meters. unpublished data). From 2006 to 2010, we collected This information was sent via cellular phone (GSM) network geographical location data, downloaded from GPS/GSM to a website from where the information was downloaded. Collars (Africa Wildlife Tracking cc., South Africa), fitted to three elephant cows from different herds. To ensure the 4. Results independence of sampling, a single female in each herd was selected and collared. Because GPS coordinates were Angular data of three elephant herds was used to construct measured continuously at frequent time intervals, the circular histograms that depict the mean and frequency of trajectory we obtained was almost smoothly connected movement orientations of animals Figure (2) and (3). The (Vanak et al., 2010). The GPS data are freely available upon rose diagram indicates that elephants move in a non-random request. manner during the dry and the wet seasons. This variation is attributed to uneven distribution of resources during dry The turning patterns of elephants were monitored by season than in the wet season. Similarly, the linear computing the angle θ between two consecutive relocations histograms in Figure (3) shows that elephants turn angles are over 30 Minutes interval of time between May, 2006 and concentrated around zero degrees which indicate that the June 2009. The data set consists of 36395, 29221 and 29908 movement is target oriented. The tendency is known as the observations for herd AM108, AM307 and AM308 unimodal movement pattern better described by unimodal respectively recorded at an interval of 30 minutes. The distributions. In all the four herds, the assumption of turning angle (θ) was computed for the three herds as the Unimodality is fulfilled, and is statistically significant results change in the direction of movement made by an each are obtained based on the conventional circular statistical individual elephant tagged from one location to the next. The methods. turning angle is a right-hand turn that ranged from -π to π. Table 1: Descriptive analysis of the turn angle data of three elephant herds collected from Kruger National Park, South Africa Statistic AM108 AM307 AM308 Seasons Seasons Seasons Variable Angle Wet Dry angle Wet Dry angle Wet Dry Number of Observations 36395 17736 18659 29221 12652 16569 29908 13550 16358 Mean Vector (µ) o o o o o o o o o 359.425 359.434 359.418 0.706 0.84 0.609 359.957 359.7 0.16 Length of Mean Vector (r) 0.481 0.464 0.498 0.434 0.418 0.446 0.482 0.471 0.492 Concentration 1.096 1.046 1.144 0.962 0.92 0.995 1.099 1.065 1.127 Circular Variance 0.519 0.536 0.502 0.566 0.582 0.554 0.518 0.529 0.508 Circular Standard Deviation o o o o o o o o o 69.292 71 67.686 74.044 75.65 72.833 69.19 70.336 68.248 Standard Error of Mean o o o o o o o o o 0.414 0.618 0.556 0.519 0.821 0.668 0.455 0.695 0.602 Table 2: Tests for uniformity for three elephant herds turn angles derived from GPS tracking data. Kuiper’s Rayleigh Rao spacing Watson’s statistic p-value statistic p-value statistic p-value statistic p-value AM108 61.6574 < 0.01 0.4813 0.000 166.2602 < 0.001 452.2261 < 0.01 AM307 50.7271 < 0.01 0.4339 0.000 162.4811 < 0.001 301.1729 < 0.01 AM308 56.3221 < 0.01 0.4823 0.000 166.2796 < 0.001 376.4143 < 0.01 The distribution of the turning angles between successive uniformity by Rao’s test. The results of the Rayleigh tests in positions differed significantly from a uniform distribution Table 2 indicates that elephant movement turn angles are not for the three herds (Kuiper’s test: p<0.001). Turning angles evenly distributed in all directions and concentrated around were backward oriented in the three herds at 359.43, 0.706 o 0 (θ>0). The uniformity and von Mises provide more and 359.96 (mean direction of resultant vector, Rayleigh test: information as the Rayleigh and Rao’s test confirm that p<0.001) indicating a tendency of elephants to make a U- azimuths are not uniformly distributed (see, Table 2). As turns while foraging (Dai et al., 2007). There distribution was azimuths do not follow a random distribution, we can look not different between the wet and dry season 1. Kuiper’s and for a preferred direction; therefore, two tests for von Mises Rayleigh’s test exhibit small p-values although the former is distribution were performed. less realistic with respect to the powerful rejection of Volume 5 Issue 3, March 2016 www.ijsr.net Paper ID: NOV162109 1205 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 Figure 2: Circular histograms of elephant movement turn angles θ. Circular bars indicate the number of observations within each class range and have been centred on 0 . Figure 3: Histogram of herd AM108, AM307 and AM308 GPS-telemetry derived turn angles data collected from Kruger National Park South Africa (May 2006 -April 2009) suggest that elephant turn angle data is not uniformly 5. Discussion distributed and showed no seasonal variation. The four statistical tests reject the uniformity hypothesis and conclude This study focuses on the test of uniformity hypothesis for that animal movement turn angles data is not uniform. These the circular animal movement data. The primary objective applications, in summary entail calculation of animal was to determine whether elephant orientation patterns were orientation variables mean turn angle, concentration uniform. We also investigated the hypothesis that elephant parameter, distribution, and determination of modality and orientation patterns would vary between seasons. Testing the testing hypothesis about uniformity of animal turn angles. It null hypothesis can be accomplished by comparing any test is important to note that the circular statistics applies to any statistic for uniformity to a reference distribution obtained by level and scale of analysis, from individuals to several simulation. This is appealing in that precise distributions species of animals, and that the descriptive measures consistent with turn angle data need not to be assumed for the procedure to have the proper type I error rate. Our results Volume 5 Issue 3, March 2016 www.ijsr.net Paper ID: NOV162109 1206 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 calculated are easily compared statistically by means of a a highly mobile mega-herbivore? Biological variety of two or more multi-sample tests. Conservation, pages 2631–2637. [20] Viswanathan, G. M., Afanasyev, V., Buldyrev, S. V., Murphy, E. J., Prince, P. A., and Stanley, H. E. (1996). References L’evy flight search patterns of wandering albatrosses. Nature, 381(6581):413–415. [1] Batschelet, E. (1981). Circular Statistics in Biology. [21] Watson, G. S. (1961). Goodness-of-fit tests on a circle. Mathematics in Biology. Academic Press. Biometrika, pages 109–114. [2] Berens, P. (2009). Circstat: A matlab toolbox for circular statistics. Journal of Statistical Software, 31(10):1–21. [3] Birkett, P. J., Vanak, A. T., Muggeo, V. M. R., Ferreira, S. M., and Slotow, R. (2012). Animal perception of seasonal thresholds: Changes in elephant movement in relation to rainfall patterns. PLoS ONE, 7(6):e38363. [4] Dai, X., Shannon, G., Slotow, R., Page, B., and Duffy, K. J. (2007). Short-duration daytime movements of a cow herd of African elephants. Journal of Mammalogy, 88(1):151–157. [5] Duffy, K. J., Dai, X., Shannon, G., Slotow, R., and Page, B. (2011). Movement patterns of African elephants (Loxodonta africana) in different habitat types. South African Journal of Wildlife Research, 41(1):21–28. [6] Durbin, J. (1973). Distribution theory for tests based on sample distribution function, volume 9. Society for Industrial Mathematics. [7] Fisher, N. and Lee, A. (1992). Regression models for an angular response. Biometrics, pages 665–677. [8] Fisher, N., Lewis, T., and Embleton, B. (1993). Statistical analysis of spherical data. Cambridge University Press. [9] Hogg, R. V. and Craig, A. T. (1978). Introduction to mathematical statistics. New York: Macmillan. [10] Jammalamadaka, S. R. and SenGupta, A. (2001). Topics in Circular Statistics,. World Scientific, New York. [11] Kuiper, N. H. (1962). Tests concerning random points on a circle. Nederl. Akad. Wetensch. Proc. Ser. A, 63:38–47. [12] Mardia, K. (1972). Statistics for directional data. Academic press. [13] Mardia, K. and Jupp, P. (2009). Directional statistics, volume 494. Wiley. [14] Pewsey, A., Neuh¨auser, M., and Ruxton, G. D. (2013). Circular statistics in R. Oxford University Press. [15] Russell, G. S. and Levitin, D. J. (1995). An expanded table of probability values for rao’s spacing test. Communications in Statistics-Simulation and Computation, 24(4):879–888. [16] Sherman, B. (1950). A random variable related to the spacing of sample values. The Annals of Mathematical Statistics, 21(3):339–361. [17] Stephens, M. A. (1970). Use of the kolmogorov- smirnov, cram´er-von mises and related statistics without extensive tables. Journal of the Royal Statistical Society. Series B (Methodological), pages 115–122. [18] Tracey, J. A., Zhu, J., and Crooks, K. (2005). A set of nonlinear regression models for animal movement in response to a single landscape feature. Journal of Agricultural, Biological, and Environmental Statistics, 10(1):1–18. [19]Vanak, A., Thaker, M., and Slotow, R. (2010). 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