Abstract
The line transect survey method is commonly used to estimate the population size. However, recent developments in this field have tended to prefer practical mathematical models for this purpose. In this study, a new model called the weighted Weibull detection model was introduced to specifically address certain criteria related to line transect data. This study thoroughly explored the characteristics of this new model, including its shapes, moments, and probability density functions. The maximum likelihood estimation and Bayesian estimation methods were employed to ensure accurate parameter estimation. To gauge the performance of this model, population size estimates were generated and compared with various existing parametric estimation methods that are commonly used in the field. Through simulations, the resulting estimates were assessed and compared to widely adopted approaches for estimating population size. Additionally, this mathematical model was applied to real-world data involving perpendicular distances, allowing for a direct comparison of its performance against both traditional and contemporary methods using various measures of goodness of fit. Moreover, the study calculates statistical metrics, such as the variance–covariance matrix, parameter confidence intervals, and estimated population size, were obtained using the proposed detection model. These metrics provide valuable insights into the precision and uncertainty associated with estimated parameters and population size estimates. Confidence intervals offer a range of plausible parameter values, whereas the variance–covariance matrix quantifies the relationships and uncertainties between the estimated parameters.
Reviewing Editor:
1. Introduction
In place of plot or strip sample techniques, the line transect approach has been employed over the past 50 years to determine population density or abundance. It can take a lot of time to establish a plot and then count every object of interest in it. It can also be difficult to define plots in some habitats, such as those with fast-moving species or at sea. In ecology and wildlife biology, the line transect method is widely used for determining the density or abundance of a specific species in a given area. Using this method, a straight line or transect is drawn through the research area, and the number of individuals or signs of the target species that are seen along that line are counted and recorded in a methodical manner. For instance, see Burnham et al. (Citation1980). A retrospective study of this topic is proposed below, with the main references provided.
The critical component for estimating population abundance is the detection function, which can be evaluated using nonparametric, semiparametric, or parametric methods.
Gates et al. (Citation1968) introduced a function characterized by an exponential distribution and a single scaling parameter, while Hemingway (Citation1971) developed a function based on the half-normal distribution with a shoulder function as its foundation.
Parameter representing the probability density function (PDF) at distance 0, it also plays a central role in these functions. Consequently, under the shoulder condition, the population abundance, often denoted as D in the literature, was calculated using the various approaches described in the following sections.
We will cover the parametric estimation methods which are proposed by Burnham and Anderson (Citation1976), Pollock (Citation1978), Ramsey (Citation1979), Karunamuni and Quinn (Citation1995), Buckland (Citation1985), Eberhardt (Citation1978), Eidous (Citation2004), Ababneh and Eidous (Citation2012), Quinn and Gallucci (Citation1980), Eidous and Al-Eibood (Citation2018), Ameeq et al. (Citation2023), Muneeb Hassan et al. (Citation2023), Naz et al. (Citation2023), and Hassan et al. (Citation2023). Al-Hussaini (Citation1991), Fewster et al. (Citation2008), Laake et al. (Citation2008), Porteus et al. (Citation2011), Rodriguez (Citation1977), Schmidt et al. (Citation2012), and Naz et al. (Citation2023).
Buckland and Turnock (Citation1992) employed primary and secondary observation platforms in a whale survey to challenge the assumption that every whale can be perfectly detected. Mark–recapture distance sampling (MRDS) was developed by Amstrup et al. (Citation2001) represent to estimate the population of polar bears (Ursus maritimus) in northern Alaska. Their methodology also considers population estimation variables such as group size.
Quang and Becker (Citation1997) introduced an MRDS model using a stratified Lincoln–Petersen estimate and data obtained from a line transect. Becker and Quang (Citation2009) opted for an iteratively reweighted least-squares approach instead of maximum likelihood when fitting the model, and they incorporated a logistic detection technique that considered contour transect factors. They also utilized the Horvitz–Thompson estimate (Horvitz & Thompson, Citation1952) to determine the size of the brown bear population, sometimes referred to as grizzly bears.
To estimate the population of harbor porpoises, Borchers et al. (Citation1998) developed an MRDS model with various variables and relied on the Horvitz–Thompson estimate. In their data collection efforts related to harbor porpoises, Amstrup et al. (Citation2001), Becker and Quang (Citation2009), and Borchers et al. (Citation1998) utilized survey designs that involved two observers on a single aerial survey platform.
In order to estimate the unidentified population abundance, represented by D, an observer walks a distance L that is divided into K randomly spaced transects that cover the target region of interest. During the survey, the observer records the number of objects encountered and measures the perpendicular distance x from the centerline to each object’s location. Let’s assume there are n identified objects, and their distances from each other are denoted as These distances are used to estimate the population density. The detection function quantifies the conditional probability of observing an object at a specific perpendicular distance x from the transect line. It provides information about the likelihood of detecting an object at different distances. The detection function represents the probability of detecting an object given its perpendicular distance x from the line transect. It can be mathematically defined as:
In order to assess the population abundance D of unidentified items in a research area, an observer walks a distance L divided into K randomly distributed transects using the line transect sampling method. The observer records the perpendicular distance x from the centerline to each object found, and the detection function represents the probability of detecting an object at distance x from the line transect. If an object is found at a perpendicular distance x, there exists a PDF that has the same shape as but scaled. For this PDF can be written as where is the scaling constant. Since not all objects in a given region are detected using the line transect technique, detection probabilities can be estimated. The likelihood of identifying an object increases with proximity to the center of the transect line. Mathematically, if and are observed perpendicular distances such that then Considering that the probability of spotting an object at zero distance is one (), the population density can be estimated using where n is the number of identified objects and is an approximate sample estimate of The equation can be used to determine the projected population size N in the target area of area A. Indicating that detection is still certain or virtually definite at a close proximity to the line transect’s center, the detection function is anticipated to come from a shoulder ().
Various references provide in-depth discussions on line transect sampling, including Marques et al. (Citation2001), Barabesi (Citation2000), Eidous (Citation2015), Eidous and Al-Salman (Citation2016), Jang and Loh (Citation2010), Seber (Citation1982), Pollock (Citation1978), Strindberg and Buckland (Citation2004), Eberhardt (Citation1978), Quang and Becker (Citation1997), Drummer and McDonald (Citation1987), Burnham and Anderson (Citation1984), Routledge and Fyfe (Citation1992), Southwell et al. (Citation2008), Brockelman and Srikosamatara (Citation1993), Chen (Citation1996), Melville and Welsh (Citation2001), Anderson et al. (Citation2001), and Naz, Al-Essa, et al. (Citation2023). Line transect sampling can underestimate the true PDF for known-sized populations, as observed in kangaroo populations by Southwell (Citation1994). This underestimation may occur when animals on the centerline become disturbed and move before being observed, violating a fundamental assumption. Mark–recapture experiments and line transect estimation can be combined to directly test the detectability.
The article is organized as follows. The proposed detection function is discussed in Section 2, which also shows that it complies with all requirements for a shoulder function. This section also includes plots of various detection functions for different parametric values. The characteristics of the density function related to the suggested detection function are presented in Section 3, and the rth moments, which can be used to determine the mean and variance, are specifically explored. In Section 4 of the referenced article, the population abundance and PDF at a distance of 0 are calculated using the maximum likelihood method. The simulation is used in Section 5 to evaluate the performance of the calculated parameters. The simulation was performed using Mathematica 10, and plots and graphs were used to provide visual insight. Section 6 describes the application of the proposed function to practical perpendicular distance datasets. The authors demonstrated how the model performs by computing the related measurements for these datasets. Finally, Section 7 concludes the article. This study combines mathematical analysis, simulation, and practical data application to present and evaluate a new detection function based on the Weibull distribution.
2. The suggested detection method
The one-parameter model exhibits a lack of flexibility, which may result in a rigid shape that does not adequately capture the detection curve. Consequently, the model estimator lacked robustness. However, we propose a weighted Weibull model (WWM) with two parameters and a user-supplied detection function to solve this problem. As a result, the adaptability of WWM was higher than that of the one-parameter model. Therefore, the robustness of the estimator and the ability of the model to conform to the shape of the detection curve can be improved by adding more parameters. Additionally, customization is made possible by the user-supplied detection function, which enables the model to consider particular aspects of the detection process. By combining the Weibull distribution and the detection function, WWM provides a more accurate representation of the detection curve. The Weibull distribution is a commonly used model for survival data, and we modified it by multiplying it by a detection function. This integration ensures that the detection curve aligns more closely with the observed data, resulting in a more reliable estimation.
By utilizing the line transect method, estimate the population density D. Any reliable detection function should satisfy three main assumptions (Burnham et al. Citation1980). Those assumptions are justified for the proposed as follows:
that is the objects with perpendicular distance 0 will never be missed.
where is the first derivative of at i.e. the shoulder function of the detection function ought to be close to the origin (Burnham et al., Citation1980). Specifically, this means that within a relatively small distance of the line transect, the detection probability should continue to be nearly certain.
for all this shows that while the detection function monotonically decreases in x, the likelihood of detecting an object decreases as its perpendicular distance x increases.
The likelihood of detecting an object given its perpendicular distance on the line transect is one, hence this detection function states that the probability of detection on the line transect center is assured. The detection function’s first derivative with respect to x is, which gives results for and the is uniformly monotonically declining Various shapes confirm this shoulder condition once more (see ). The PDF related to is possible by averaging the detecting function as where Then the PDF is (2.1) (2.1)
In , plots of the PDF are given for various parametric values.
The shoulder condition assumptions are satisfied since shows similar behavior to but scaled to be a PDF, and it is monotonically decreasing in x. Also, has the same shape as with different scales.
Since for the parameter is (2.2) (2.2)
Then, from EquationEquation (2.2)(2.2) (2.2) , replacing this value with in EquationEquation (2.1)(2.1) (2.1) , Then, the PDF of the WWM can be stated as such that (2.3) (2.3) (2.4) (2.4)
This is determined by the two parameters and The estimation of the parameter’s maximum likelihood will be based on this formula.
With regard to x, the first derivative of the PDF (2.1) is
It is clear from statement that is immediately connected to As a result, and share some characteristics, including and the monotone decreasing shape property. and exhibit these features. The proposed model would then provide a reliable estimate for known as the Shape Criterion (see Burnham et al., Citation1980). The detection function, which adjusts and has parameters for scale and for shape, can have a wide range of shapes, as can be seen when carefully examining these pictures. A desirable property for the detection model under consideration is that all plots steadily fade to 0 as An integrated estimator of corresponds to and if they are the estimators of the parameters and respectively. (2.5) (2.5) where the estimate for population density (abundance) is
To calculate the values of the parameters and we shall employ the maximum likelihood method. Thus, we calculate and D represents the population abundance. Section 4 will go into further detail.
3. A few statistical characteristics
In this section, we seek immediate outcomes that will aid in the calculation of parameter and, ultimately, population abundance. These instant results offer insightful statistical data regarding the distribution of random variables. These moments can be used to estimate the population abundance parameter, By calculating and analyzing various moments, such as the mean, variance, and higher-order moments, we gain insight into the central tendency, dispersion, and shape of the distribution. These moment results aid in understanding the underlying characteristics of the population and enable estimation and inference about the parameter of interest. The specific calculations of moments for the PDF given in EquationEquation (2.1)(2.1) (2.1) would require the functional form of Using this information, the integrals can be solved to obtain the moment results for a given random variable.: where E denotes the expectation operator. After some algebra, we have (3.1) (3.1)
By using and to determine the mean and variance, it can be calculated that (3.2) (3.2) (3.3) (3.3) and
Based on EquationEquation (3.1)(3.1) (3.1) , the skewness and kurtosis of X are calculated using the following formulas.
Ghitany et al. (Citation2013) and Wackerly et al. (Citation2014) conducted studies investigating the relationship between the model parameters and and their impact on the statistical properties of the random variable X. In , various parameter options are presented alongside the corresponding mean, variance, skewness, and kurtosis values for X. It is observed that as the parameter values increase, these statistical measures decrease. The skewness and kurtosis values, however, remain constant for fixed values. The 3D plots shown in and illustrate the relationships between the mean, variance, skewness, kurtosis, and the values of and within the model. These plots visually depict how changes in and significantly affect the skewness and kurtosis characteristics of the parameter X. The results indicate that adjusting the values of and allows for control over and a better understanding of the skewness and kurtosis behaviors of the parameter X, in addition to its mean and variance.
4. Inference
In this section, we focus on making statistical inferences about the parameter when the values of and are unknown. To achieve this, we employed maximum likelihood and Bayesian estimation (BE) methods, which involve determining the values of and that maximize the likelihood function. Consider the PDF connected to the suggested detection function. The observed values of a random sample are given by equation
4.1. Maximum likelihood estimator of
To simulate an n-dimensional random sample of perpendicular distances, denoted as from a random variable defined by EquationEquation (2.5)(2.5) (2.5) , we can generate random numbers using the specified PDF with parameter values and The maximum likelihood estimators (MLEs) and for the parameters and respectively, may be estimated once we have the observed sample. The approach outlined in Burnham et al. (Citation1980) can then be used to compute the MLE of (4.1) (4.1) (4.2) (4.2)
Differentiating with respect to and respectively, we get and where and
The MLEs and of and respectively, are the nonlinear equation’s solutions and They can be calculated mathematically with the use of any statistical program, such as Mathematica, Python, R, etc. The MLE of is then obtained by substituting the values of and in EquationEquation (2.5)(2.5) (2.5) , which is where we get the population abundance estimator D. We consider the observed Fisher information matrix for calculating confidence intervals for the model parameters about given by where and So, V, which is the inverse of I, is the variance–covariance matrix connected to On the other hand, it is simple to derive the estimate of from EquationEquation (2.5)(2.5) (2.5) . Furthermore, based on given by EquationEquation (2.1)(2.1) (2.1) , we may calculate the MLEs numerically and then the confidence intervals for and
Based on the above discussion, It is simple to calculate the approximate large sample confidence interval for and as where and are the MLEs of the variances of the estimators behind and respectively.
4.2. Bayesian estimator of
This section deals with the BE of the unknown parameters and respectively. For the Bayesian parameter estimation, loss functions, including squared error loss (SEL) functions, can be taken into consideration by Tolba et al. (Citation2023), Alsadat et al. (Citation2023), Bhat et al. (Citation2023), and Chinedu et al. (Citation2023). We can consider applying independent gamma priors for the variables and with PDF in the parameter prior detection function. (4.3) (4.3) where the hyper-parameters are selected to reflect the prior knowledge about the unknown parameters. The joint prior for is given by (4.4) (4.4)
The corresponding posterior density, given the observed data is given by (4.5) (4.5)
Consequently, the posterior density function is denoted by (4.6) (4.6)
Given any function, such as under the SEL function, the Bayes estimator is given by (4.7) (4.7)
They can be calculated mathematically with the use of any statistical program, such as Mathematica, Python, R, etc. The Bayesian estimator of is then obtained by substituting the values of and in EquationEquation (2.5)(2.5) (2.5) , which is where we get the population abundance estimator D.
5. Simulation research and findings
To assess how well the WWM estimate, performs in comparison with other available estimates for the function we perform a simulation exercise. We consider the estimates from the negative exponential model (NEM), half-normal model (HNM), and weighted exponential model (WEM) (see Saeed, Citation2013). We evaluated two different target detection functions, coupled with observations of with the sample sizes of and to duplicate the perpendicular distances. These identification strategies were selected to highlight the range of shapes that might be present in a specific field (see Eidous, Citation2015). The preferred models for replicating the perpendicular distances are as follows:
Detection of exponential power (EP) (see Pollock, Citation1978): (5.1) (5.1) where is the standard gamma function defined as
The beta exponential (BE) detection function is described by Eberhardt (Citation1978): (5.2) (5.2)
Each model is shortened at specific distances w in order to simulate the data. In contrast to the BE detection function, which is truncated at and the EP detection function is truncated at and 2. A number of arbitrary values are selected for both models. We consider sample sizes of and generate 1500 samples of perpendicular distances for each model randomly. and present the relative bias (RB) and the relative mean square error (RME) for the different estimators. Based on these results, we draw the following conclusions:
As the sample size n increases, the RB and RME of all estimators decrease. This suggests that the estimators are reliable as the sample size grows.
The suggested estimate exhibits lesser RB and RME compared to the NEM, HNM, and WEM estimators for all studied target detection functions, which shows that the proposed estimator outperforms the competing models, as shown in and .
depict the RME for and respectively.
6. Model compatibility and real-world data application
The primary subjects of discussion in this section are model selection and validation. However, choosing these models is the most difficult step in creating a good model. It was chosen after carefully weighing all the available information. The chosen model must be flexible enough to consider the trade-off between model complexity and ease of assessment while accurately reproducing the provided data. Additionally, it takes a lot of effort to simulate behaviors for extreme values of the relevant variable. In this situation, a variety of visualizations and goodness-of-fit checks are required as part of the statistical process step for validating the model.
6.1. Alternative models
Using some goodness-of-fit analyses, the performance of the proposed detection model is compared to a variety of other current detection models. For comparison, the following detection models are utilized:
‘New two-parameter detection model (NDM)’ (Bakouch et al., Citation2022):
‘Model (2013)’ (Saeed, Citation2013):
‘Generalized exponential model (GEM)’ (Quinn & Gallucci, Citation1980):
‘Exponential power series model (EPSM)’ (Pollock, Citation1978):
‘Reverse logistic model (RLM)’ (Eberhardt, Citation1978):
‘Exponential quadratic model (EQM)’ (Burnham et al., Citation1980):
‘Weighted half-normal model (WHNM)’ (Eidous & Al-Salman, Citation2016):
‘Model (2015)’ (Eidous, Citation2015):
‘Weighted exponential model (WEM)’ (Ababneh & Eidous, Citation2012):
‘Negative exponential model (NEM)’ (Gates et al., Citation1968):
To compare the MLEs with the data being discussed, we consider the pertinent PDF of each detection model. The effectiveness of the detection model is strongly correlated with its PDF, as shown previously.
6.2. Perpendicular distance datasets
Dataset 1: Information about wooden stakes parallel distances in meters
The dataset comprises perpendicular distances between a sagebrush meadow and wooden stakes. These distances were obtained by walking a 1000 m path and selecting a sample of 67 stakes from a total population of 150 stakes. The true density of the stakes is defined as stakes/meter. The recorded distances, denoted as represent the perpendicular distances associated with the selected stakes along the single 1000 m path. The estimated value of given the sample size n and stake density D, is determined as 0.11029 (refer to Karunamuni & Quinn, Citation1995; Zhang, Citation2001 for more details). In addition, the zone of interest’s dimensions, notably the sagebrush meadow east, are calculated as using the equation where (see Karunamuni & Quinn, Citation1995; Zhang, Citation2001). Burnham et al. (Citation1980) and Barabesi (Citation2000) have both reviewed and reported on the data in great detail. The perpendicular distance value of 31.31 in the accompanying table significantly deviates from the transect path in contrast to other values, as is evident when looking at the wooden stake dataset. The accuracy of the estimation will therefore be increased by removing this outlier (as suggested by Zhang, Citation2001). As a result, the truncation point is removed.
presents the important statistics of the wooden stakes dataset, including the relevant theoretical metrics of the WWM (Wooden Stakes Analysis). Additionally, the accompanying box plot, shown in , provides further visual assistance. It is worth noting that the box plots display the minimum score of a dataset, first quartile (lower quartile), median, third quartile (upper quartile), and maximum score as part of the five-number summary.
Dataset 2: The Hemingway’s perpendicular distance data
The second dataset in comes from Hemmingway’s research on several ungulates in Africa (see Burnham et al., Citation1980). Seventy-three animals were found along a single-line transect that traveled for 60,000 m. Using the sighting method, the perpendicular distances are calculated. Sighted angles and distances. Hemmingway’s dataset does not include the true density D or the area A of the discovered items (not recorded during the survey). However, the TRANSECT software, which adapts the estimate of and D, which are, respectively, 0.0065 and 0.0396 animal per hectare (or 3.096106 animal per meter), creates the estimation technique. presents the important statistics of Hemmingway’s dataset, including the relevant theoretical metrics of the WWM. Additionally, the accompanying box plot is shown in .
The following common criteria are applied in order to compare the fit performance of the models that were taken into consideration for the study of Datasets 1 and 2: the Kolmogorov–Smirnov test p value, the Bayesian information criterion (BIC), the Hannan–Quinn information criterion (HQIC), the Akaike information criterion (AIC), and the corrected AIC (CAIC). Their values are shown in . It is evident that our model has the lowest information criterion values and the greatest p value for the Kolmogorov–Smirnov test. We can therefore conclude that the suggested model works well in this situation.
The CIs of the PDF at 0 based on the MLEs are obtained in and (with appropriate truncation for the bottom bound of the intervals) based on Datasets 1 and 2 and the proposed model.
The MLEs of all the models were used to produce the estimated population abundance, denoted as and the maximum likelihood estimate of the PDF at 0, for the analyzed data. The theoretical standard deviation of the model is represented by SD, but the standard deviation of the sample is represented by and display the findings for and According to the tables, the WWM exhibits the lowest absolute difference, indicating a closer match between the estimated and theoretical standard deviations. Furthermore, among the examined detection models, the WWM yields and values that closely resemble the genuine values of and D for the considered data.
7. Concluding remarks
This study introduced and examined a novel two-parameter detection model. The proposed model is beneficial for line transect data. This model ensures that it meets the shoulder requirement and that its perpendicular distance monotonically decreases. The proposed WWM is adaptable to suit line transect data because it depends only on two parameters. However, statistical techniques to estimate these parameters, such as MLEs, do not exist in closed form. Consequently, in this context, a numerical method, such as the Newton–Raphson method, is required. According to the simulation results from this study, the estimators obtained under the suggested model are promising for estimating population abundance using line transect approaches. Favorable properties of the proposed estimator, which were obtained using the suggested model, are shown in the simulation results. The performance of our model in comparison with other models shows how the fit can be improved using a valuable collection, such as a table of perpendicular distances and some information on goodness of fit. In addition, investigating the inclusion of a scale parameter in the detection function is a potential extension; however, more work needs to be done on the definition of the normalization constant and the suggested approach needs to be modified.
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