Line transect sampling is a common method used in ecology to sampling the sample required. It is an important procedure for estimating the population density of objects in interested study area. There are mainly two ways to estimate the population density which a parametric and nonparametric estimation methods. In this paper, we present kernel estimation method to estimate new estimator of the propose population density. Kernel estimation method is used due to avoid the assumption about the shape of the unknown detectable functions. We investigate the performance of the new estimator using simulation study and compared with the existing estimators. Based on the simulation study, the results show that the proposed estimator preforms better than other well-known estimator

Noryanti, Muhammad

Saeed, Gamil Abdulraqeb Abdullah

Wan Nur Syahidah, Wan Yusoff

UMP Institutional Repository

*Corresponding author’s e-mail: noryanti@ump.edu.my ASM Sc. J., 12,  Special Issue 5, 2019 for ICoAIMS2019, 137-142  Kernel Estimation in Line Transect Sampling for Parametric Model  Gamil A.A. Saeed1,  Noryanti Muhammad1* and  Wan Nur Syahidah Wan Yusoff1  1Centre for Mathematical Sciences, College of Computing and Applied Sciences,  Universiti Malaysia Pahang, 26300, Gambang, Pahang, Malaysia   Line transect sampling is a common method used in ecology for sampling the sample required. It is an important procedure for estimating the population density of objects in interested study area. There are  two main ways to estimate the population density which are parametric and nonparametric estimation methods. In this paper, we present kernel  method to propose new estimator of the propose population density. Kernel estimation method is used due to avoid the assumption about the shape of the unknown detectable functions. We investigate the performance of the new estimator using simulation study and compared with the existing estimators. Based on the simulation study, the results show that the proposed estimator preforms better than other well -known estimators. Keywords:  line transect sampling; kernel method; nonparametric estimation method; parametric model   I. INTRODUCTION  There are many ways to estimate the population abundance, one of the methods is line transect sampling. In line transect sampling technique, the density (𝐷) of the objects on interested area depends on the measured distances between the detected objects and the line of transect, 𝐿 . Two key assumptions of line transect distances sampling (DS) are all objects on the line are certainty detected and the objects must be detected at their original location. If these two assumptions are valid, the DS gives unbiased estimators of the population   Density (Buckland et al., 2001). These assumptions lead us to define the important concept in the DS, also called a detection function. Detection function plays the central role to the line transect sampling technique which can be defined as  g(y) = P(observing object / its perpendicular distances y from center line).   (1)  It is reasonable to assume that the detectability function g(y) is nonincreasing function on [0; ∞), that means the detection function have a monotone decreasing curve, the probability of detection should close to one as distance from the line increases from zero, that means the detection function satisfied shoulder property which is (𝑔(0) =  1). Let 𝑌1,…,𝑌𝑛 donated to the nonpooled sighting perpendicular distances which selected randomly and independently from the transect strips, with common density function 𝑓(𝑦) , defined on [0; 𝑤], 𝑓(𝑦) was considered by Burnham and Anderson (1976) as     𝑓(𝑦) =𝑔(𝑦)∫ 𝑔(𝑡)𝑑𝑡𝑤0 ; 0 ≤ 𝑦 ≤ 𝑤.   (2)  By assuming that all objects on the line and have perfect probability, Burnham and Anderson (1976) showed that the density 𝐷 of the objects in surveyed area related to the probability density function (pdf) 𝑓(𝑦) which evaluated at 𝑦 =  0 as 𝐷 =𝐸(𝑛)𝑓(0)2𝐿   (3)  where 𝐸(𝑛) is the expected value of sighted objects. Since 𝐷  depends to 𝑓(0) , the density 𝐷  estimated by  𝑓 (0) . Based on Burnham and Anderson (1976) and Buckland et al., (1993; 2015), the density 𝐷 can be estimated by  ?̂? =𝑛2𝐿𝑓(0)  (4) ASM Science Journal, Volume 12,  Special Issue 5, 2019 for ICoAIMS2019   138 Let  𝑌1,…,𝑌𝑛    be a set of perpendicular distances which are usually assumed to be a random sample, having a density function 𝑓(𝑦;  𝜃)  depends on unknown parameter 𝜃, where 𝜃  may one parameter or vector of parameters. Since the 𝑓(0)  is function of the parameter  𝜃 therefore, the estimate of 𝜃  lead us to estimate  𝑓(0) =   𝑓(0; 𝜃)  . The exponential detection function is presented by Gates et al. (1968) given as  𝑔(𝑥; 𝜃) =  𝑒−𝑦 𝜃⁄ ;  𝑦 ≥ 0 , 𝜃 > 0  (5)  with corresponding pdf, 𝑓(𝑥; 𝜃) =1𝜃 𝑒−𝑦 𝜃⁄ ;  𝑦 ≥ 0 , 𝜃 > 0  (6) The maximum likelihood estimator (MLE) of 𝑓(0) is, 𝑓𝑀𝐿𝐸(0) =1?̅? ,  (7)  where ?̅? is the sample mean. It is important to refer that the negative exponential (NE) model does not satisfy ?́?(0) = 0  while half normal (HN) model satisfies ?́?(0) =  0. This means half normal detection deals with the property of the shoulder. In contrast, the NE detection 𝑔(𝑦)  does not achieve the shoulder condition. Hemingway (1971) suggested the half normal model with pdf  𝑓(𝑦; 𝜎2) = √2𝜋𝜎2 𝑒−𝑦2 2𝜎2⁄ ; 𝑦 ≥ 0 , 𝜎2 > 0 ,  (8) and the half normal detection function is, 𝑔(𝑦; 𝜎2) =  𝑒−𝑦2 2𝜎2⁄ ; 𝑦 ≥ 0 , 𝜎2 > 0 .  (9) The main estimator to estimate  𝜎2  for density in equation (8) is 1𝑛∑ 𝑦𝑖2𝑛𝑖=1 , given by using MLE to estimate 𝑓(0). Since  𝑓(0) = √2𝜋𝜎2 , the MLE of 𝑓(0) is given by 𝑓𝑀𝐿𝐸(0) = √2𝜋 (1𝑛∑ 𝑦𝑖2𝑛𝑖=1 )−1 2⁄  (10)  II. MATERIALS AND METHOD  Let 𝑌1,…,𝑌𝑛  random variables of size n representing the perpendicular distances have probability density function (pdf) 𝑓(𝑦), and independent and identically distributed (iid) have a detection function 𝑔(𝑦 ). Saeed et al., (2017) introduced 𝑔(𝑦; 𝜎2)  which is given as  𝑔(𝑦; 𝜎2) = (2 − 𝑒−𝑦2 2𝜎2⁄ )𝑒−𝑦2 2𝜎2⁄ ; 𝑦 ≥ 0 , 𝜎2 > 0 ,  (11) and the first derivative of 𝑔(𝑦; 𝜎2)  is given as 𝑔′(𝑦; 𝜎2) =2𝜎2 𝑦𝑒−𝑦2 2𝜎2⁄ (𝑒−𝑦2 2𝜎2⁄ − 1). (12)  For all 𝑦 ≥ 0; 𝜎2  >  0 , we can easily observe that 0 ≤𝑒−𝑦2 2𝜎2⁄ ≤ 1 , then equation (12) can be shown that 𝑔′(𝑦; 𝜎2) < 0, therefore, the detection function 𝑔(𝑦; 𝜎2)   is monotone decreasing function on  [0; ∞ ). In addition, 𝑔(𝑦; 𝜎2)     continuous function satisfies the condition 𝑔′(0; 𝜎2) = 1, which means that the probability of sighted object on the line equals to one. Figure 1 shows the shapes of the detection function for certain values of 𝜎2. Moreover, 𝑓(𝑦; 𝜎2)   is continuous function and proportional to the 𝑔(𝑦; 𝜎2),  𝑓(𝑦; 𝜎2) is decreasing function and related with 𝑔(𝑦; 𝜎2)   as  𝑔(𝑦; 𝜎2) = 𝜇𝑓(𝑦; 𝜎2),  (13)  where 𝜇 = ∫ 𝑔(𝑦; 𝜎2)𝑑𝑦∞0.  (14)  Then, the corresponding pdf of 𝑔(𝑦; 𝜎2), given by Saeed et al., (2017) as  𝑔(𝑦; 𝜎2) = 𝑓(𝑦; 𝜎2)(2 − 𝑒−𝑦2 2𝜎2⁄ )𝑒−𝑦2 2𝜎2⁄ ; 𝑦 ≥ 0 , 𝜎2 > 0 (15)  By solving the integration in equation (14), the 𝑓(0; 𝜎2)  is given as (Saeed et al., 2017)  𝑓(0; 𝜎2) =  2(2√2−1)√𝜋𝜎2  (16)  Equation (16) shows that the 𝑓(0; 𝜎2)  is a function with the parameter 𝜎2 . Therefore, it is enough to estimate 𝜎2  for estimating  𝑓(0; 𝜎2) . The considered model in equation (15) was studied by Noryanti et al. (2018), the maximum likelihood (MLE) method is used to estimate the proposed estimator, the performance of the proposed estimator is evaluated by simulation study. In next section, for this purpose, kernel method with the proposed model are used to compute the smoothing parameter and studied the proposed estimator performance. ASM Science Journal, Volume 12,  Special Issue 5, 2019 for ICoAIMS2019   139 Figure  1. The detection function 𝑔(𝑦)  graph of the proposed model for different parameter 𝜎2   III. KERNAL ESTIMATORR  Consider 𝑌1,…,𝑌𝑛  be random sample of size 𝑛, distributed from continuous probability density function (pdf) 𝑓(𝑦). According to in Silverman (1986) and (2018), the kernel estimate  𝑓(𝑦) of 𝑓(𝑦) supported on [0; ∞), is given by  𝑓(𝑥) =1𝑛ℎ∑ (𝐾 (𝑦−𝑌𝑖ℎ) + 𝐾 ((𝑦+𝑌𝑖ℎ)))𝑛𝑖=1 ;  𝑦 ≥ 0  ,  (17)  where ℎ is the bandwidth, which controls the smoothness of the fitted function shape and 𝐾 is  a kernel function, assumes to be symmetric function and satisfies the following  ∫ 𝐾(𝑡)𝑑𝑡 = 1∞−∞, ∫  𝑡𝐾(𝑡)𝑑𝑡 = 0∞−∞, ∫ 𝑡2𝐾(𝑡)𝑑𝑡 = 𝑐∞−∞≠ 0  . (18)  the standard kernel estimate 𝑓(0)  of 𝑓(0)  is given by Chen (1996) as, 𝑓(0) =2𝑛ℎ∑ 𝐾 (𝑌𝑖ℎ)𝑛𝑖=1 .  (19)  By assuming 𝑛ℎ →  ∞ as 𝑛 → ∞, the bias and variance of  𝑓𝑘(0) can be represented as  𝐵𝑖𝑎𝑠 (𝑓𝑘(0)) = 2ℎ𝑓′(0) ∫ 𝑢𝑘(𝑢)𝑑𝑢 ∞0+ℎ2𝑓′′(0) ∫ 𝑢2𝑘(𝑢)𝑑𝑢 + 𝑜(ℎ2)∞0  (20) and 𝑉𝑎𝑟 (𝑓𝑘(0)) =  4𝑓(0)𝑛ℎ∫ 𝐾2(𝑢)𝑑𝑢∞0+  𝑜(𝑛−1 ℎ−1 )   (21) Depending on the assumption that the shoulder condition is true, and the function 𝑓(0) has continuous derivative. Thus, lead us to introduce the following estimator.  𝑓(0) = 𝑓𝑘(0) − ℎ2 𝑓′′(0) ∫ 𝑢2𝐾(𝑢)𝑑𝑢 ∞0 . (22)  The 𝑓′′(0) is the second derivative of 𝑓(𝑦) at 𝑦 =  0.  Based on equation (22) and for underlying 𝑓(𝑦) which satisfies the shoulder condition, the bias of  𝑓(0) is given as   𝐵𝑖𝑎𝑠 (𝑓(0)) =2ℎ3𝑓(3)(0)6∫ 𝑢3𝐾(𝑢)𝑑𝑢∞0+2ℎ4𝑓(4)(0)24∫ 𝑢4𝐾(𝑢)𝑑𝑢 ∞0+ 𝑂(ℎ5) .  (23)  and the variance of  𝑓(0) is  𝑉𝑎𝑟 (𝑓(0)) =  4𝑛ℎ𝑓(0) ∫ (𝐾(𝑢))2𝑑𝑢∞0+  𝑜(𝑛−1 )   (24)  From equations (23), (24) and the assumption that  𝑓′(0) = 𝑓′′′(0) = 0 , the asymptotic mean square error (AMSE) can be written as 𝐴𝑀𝑆𝐸 ( 𝑓(0)) =4𝑛ℎ𝑓(0) ∫ (𝐾(𝑢))2𝑑𝑢∞0+2ℎ8576(𝑓(4)(0) ∫ 𝑢4𝐾(𝑢)𝑑𝑢 ∞0)2  (25)  The key aspect in nonparametric kernel method the value ℎ which plays the major milestone in the performance of  𝑓𝑘(0). A large smoothing parameter ℎ leads to an estimate with small variance and large bias while a small h produces a large variance and small bias. Then, the minimisation of the AMSE in equation (25) leads to compute the optimal value of ℎ, which is given by  ℎ = (72𝑓(0) ∫ (𝐾(𝑢))𝟐𝑑𝑢 ∞0(𝑓(4)(0) ∫ 𝑢4𝐾(𝑢)𝑑𝑢 ∞𝟎)𝟐)1 9⁄𝑛−1 9⁄   . (26)  The smoothing parameter h in equation (26) depends on 𝑓(0) , 𝑓(4)(0) and the kernel function 𝐾(𝑢). The negative exponential, half normal models and the proposed model in equation (15) are used as references to compute the smoothing parameter ℎ. In this paper for all models, we use standard normal 𝑁(0;  1)  as a kernel function. The optimal values ℎ  for the different estimators have been computed and give ℎ1  =  1.18229 ?̅?𝑛−1 9⁄   for  𝑓1;𝑘(0), ℎ2 =ASM Science Journal, Volume 12,  Special Issue 5, 2019 for ICoAIMS2019   140  0.949713𝑇𝑛−1 9⁄  for 𝑓2;𝑘(0)  and ℎ3  =  0: 904949?̅?𝑛−1 9⁄  for the proposed estimator  𝑓3;𝑘(0) , respectively, where  ?̅?  is sample mean and 𝑇 = √1𝑛∑ 𝑦𝑖2𝑛𝑖=1 .  IV. SIMULATION RESULT AND DISCUSSION  A simulation study is performed in order to investigate the performances of the considered estimators in Section 3. The data is simulated using one of commonly models used in line transect which is Hazard-Rate (HR) model (Hemingway (1971) given as 𝑓(𝑥) =1𝛤(1−1 𝛽)⁄(1 − 𝑒−𝑥−𝛽) ,    𝑥 ≥ 0 , 𝛽 ≥ 1.        (27)    For this purpose, we used HR model to generate  400 samples of sizes 𝑛 =  50, 100  and 𝑛 =  200  of perpendicular distances data set. Four HR models is selected with parameter values 𝛽  and the corresponding truncated value 𝑤  are given as (𝛽; 𝑤) ={(1.5, 20), (2, 12), (2.5, 8); (3, 6)}. The Relative Mean Error (RME) and Relative Bias (RB) be estimated to evaluate the performance of the estimators 𝑓1;𝑘(0), 𝑓2;𝑘(0),   and  𝑓1;𝑘(0). The RME and RB is given by  𝑅𝑀𝐸 =√𝑀𝑆𝐸(?̂?(0))𝑓(0) ,  (28) and 𝑅𝐵 =𝐸(?̂?(0))−𝑓(0)𝑓(0) .  (29)  The results of RME and RB are summarised in Table 1. Table  1. Simulated values of RB and RME for the different estimators.                    Table 1 shows that the results of RB and RME in of the three considered estimators which demonstrated in Section 3. The estimator 𝑓1;𝑘(0)  is performed better than  𝑓2;𝑘(0)  for all considered cases regardless of sample size n. For each sample                      size and for all considered cases, the estimator 𝑓1;𝑘(0) provides the smallest value of RME and RB comparing to the estimators 𝑓1;𝑘(0)and  𝑓1;𝑘(0). Indeed, the significant result is that the performance of proposed estimator 𝑓3;𝑘(0)is outperformed well-known considered estimators.  𝒏  𝒘  𝜷   ?̂?𝟏;𝒌(𝟎)  ?̂?𝟐;𝒌(𝟎)  ?̂?𝟑;𝒌(𝟎) 50   RB -0.40688 -0.50084 -0.30589    RME 0.42316 0.50911 0.33455 100 1.5 20 RB -0.38122 -0.38122 -0.27824    RME 0.39244 0.48645 0.29807 200   RB -0.35529 -0.45676 -0.25086    RME 0.36101 0.45947 0.26129 50   RB -0.26845 -0.35539 -0.17278    RME 0.29379 0.37279 0.21703 100 2 12 RB -0.24750 -0.33672 -0.15292    RME 0.26519 0.34771 0.18513 200   RB -0.22132 -0.31187 -0.12957    RME 0.23039 0.31683 0.14770 50   RB -0.14835 -0.20491 -0.06831    RME 0.18622 0.23302 0.14057 100 2.5 8 RB -0.13294 -0.18801 -0.05846    RME 0.15791 0.20408 0.11091 200   RB -0.10605 -0.15896 -0.03845    RME 0.12484 0.17095 0.08168 50   RB -0.09536 -0.12424 -0.03392    RME 0.14688 0.16463 0.13001 100 3 6 RB -0.07614 -0.10291 -0.02344    RME 0.10982 0.12760 0.09334 200   RB -0.05828 -0.08309 -0.01275    RME 0.08253 0.09942 0.06988 ASM Science Journal, Volume 12,  Special Issue 5, 2019 for ICoAIMS2019   141 Another noticeable results, the values of RME decrease as the sample size increases for all estimators which indicates that the consistency of the proposed estimators for 𝑓(0) . This result can be shown in Figures 2 and 3.   Figure  2. RME values for the different estimators  V. CONCLUSION  In this paper, parametric model is used to construct the nonparametric kernel estimator 𝑓(0).  Moreover, the smoothing parameter of the kernel estimator is computed for considered estimators, which plays a major milestone in the performance of the kernel estimator. Simulation study is constructed to compare the performance of the proposed estimator with other existing estimators. The simulated results of RB and RME indicate that the proposed estimator is performed better than other estimators considered. In general, the proposed model be recommended to estimate 𝑓(0) and then to estimate the population density 𝐷.  VI. ACKNOWLEDGEMENT  We gratefully thanks to Universiti Malaysia Pahang for financial support under Doctoral Research Scheme (DRS) and RDU170359. We also thanks Ministry of Yemen Higher Education for sponsor this research study.  Figure  3. RB values for the different estimators                               ASM Science Journal, Volume 12,  Special Issue 5, 2019 for ICoAIMS2019   142 VII. REFEREENCES  Buckland, S. T., Anderson, D. R., Burnham, K. P. and Laake, J. L. (1993). Distance sampling. Chapman and Hall, London.  Buckland, S. T., Anderson, D. R.,Burnham, K. P., Laake, J. L. Borchers, D.L., and Thomas, L. (2001). Introduction to distance sampling. Oxford university Press,Oxford. Buckland, S. T., Rexstad, E. A., Marques, T. A. and Oedekoven, C. S. (2015), Distance sampling: methods and applications, Springer.  Burnham, K. P., Anderson, D. R. (1976).Mathematical models for nonparametric influences from line transect data.Biometrics, 32, 325-336. Burnham, K. P., Anderson, D. R. and Laake, J. L. (1980). Estimation of density from line transect sampling of biological populations. Wildlife Monograph,72,1-202.  Chen, S. X. (1996). A kernel estimate for the density of a biological population by using line transect sampling. Applied Statistics, pages 135150. Gates, C, E., Mashall, W. H. and Olson, D. P. (1968). Line transect method of estimating grouse population densities. Biometrics, 24, 135-145. Hayes, R. J. and Buckland, S. T. (1983). Radial distance models of line transect method. Biometrics, 39,29-42. Hemingway, P. (1971). Field trials of the line transect method of sampling large populations of herbivores. pp 405-411. The scientific management of animal and plant communities for conservation. Blackwell Sci. Publ. Oxford. Noryanti Muhammad, Gamil.A.A.Saeed and Wan Nur Syahidah binti Wan Yusoff, Performance of Parametric Model for Line Transect Data, ASM Sci. J.Special Issue 2018(1). Saeed, G. A. A., Muhammad, N., Liang, C. Z., Yusoff, W. N. S. W. and Salleh, M. Z. (2017), Model for estimating of population abundance using line transect sampling, in ‘Journal of Physics: Conference Series’, Vol. 890, IOP Publishing, p. 012153. 85. Silverman, B. W. (1986). Density estimation for statistics and data analysis.Chapman and Hall, London. Silverman, B. W. (2018). Density estimation for statistics and data analysis. Routledge.    

Kernel Estimation in Line Transect Sampling for Parametric Model

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Kernel Estimation in Line Transect Sampling for Parametric Model

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