* E-mail: mkhan@ulb.ac.be; mnikhan@yahoo.com Affiliations Laboratory of Systems Ecology and Resource Management, Département de Biologie des Organismes, Faculté des Sciences, Université Libre de Bruxelles–ULB, Bruxelles, Belgium, Institute of Forest Growth and Forest Computer Sciences, TU Dresden, Tharandt, Germany, Forestry and Wood Technology Discipline, Khulna University, Khulna, Bangladesh ⨯
Affiliations Biodiversity and Ecology Research Unit, Faculty of Sciences and Bio-engineering Sciences, Vrije Universiteit Brussel–VUB, Brussels, Belgium, Plant Production Systems, Wageningen University and Research Centre, Wageningen, Netherlands ⨯
Affiliation Institute of Forest Growth and Forest Computer Sciences, TU Dresden, Tharandt, Germany ⨯Affiliation Biodiversity and Ecology Research Unit, Faculty of Sciences and Bio-engineering Sciences, Vrije Universiteit Brussel–VUB, Brussels, Belgium ⨯
Affiliation Institute of Forest Growth and Forest Computer Sciences, TU Dresden, Tharandt, Germany ⨯ Affiliation Forestry and Wood Technology Discipline, Khulna University, Khulna, Bangladesh ⨯ Affiliation Forestry and Wood Technology Discipline, Khulna University, Khulna, Bangladesh ⨯Affiliations Laboratory of Systems Ecology and Resource Management, Département de Biologie des Organismes, Faculté des Sciences, Université Libre de Bruxelles–ULB, Bruxelles, Belgium, Biodiversity and Ecology Research Unit, Faculty of Sciences and Bio-engineering Sciences, Vrije Universiteit Brussel–VUB, Brussels, Belgium ⨯
In the Point-Centred Quarter Method (PCQM), the mean distance of the first nearest plants in each quadrant of a number of random sample points is converted to plant density. It is a quick method for plant density estimation. In recent publications the estimator equations of simple PCQM (PCQM1) and higher order ones (PCQM2 and PCQM3, which uses the distance of the second and third nearest plants, respectively) show discrepancy. This study attempts to review PCQM estimators in order to find the most accurate equation form. We tested the accuracy of different PCQM equations using Monte Carlo Simulations in simulated (having ‘random’, ‘aggregated’ and ‘regular’ spatial patterns) plant populations and empirical ones.
PCQM requires at least 50 sample points to ensure a desired level of accuracy. PCQM with a corrected estimator is more accurate than with a previously published estimator. The published PCQM versions (PCQM1, PCQM2 and PCQM3) show significant differences in accuracy of density estimation, i.e. the higher order PCQM provides higher accuracy. However, the corrected PCQM versions show no significant differences among them as tested in various spatial patterns except in plant assemblages with a strong repulsion (plant competition). If N is number of sample points and R is distance, the corrected estimator of PCQM1 is 4(4N − 1)/(π ∑ R 2 ) but not 12N/(π ∑ R 2 ), of PCQM2 is 4(8N − 1)/(π ∑ R 2 ) but not 28N/(π ∑ R 2 ) and of PCQM3 is 4(12N − 1)/(π ∑ R 2 ) but not 44N/(π ∑ R 2 ) as published.
If the spatial pattern of a plant association is random, PCQM1 with a corrected equation estimator and over 50 sample points would be sufficient to provide accurate density estimation. PCQM using just the nearest tree in each quadrant is therefore sufficient, which facilitates sampling of trees, particularly in areas with just a few hundred trees per hectare. PCQM3 provides the best density estimations for all types of plant assemblages including the repulsion process. Since in practice, the spatial pattern of a plant association remains unknown before starting a vegetation survey, for field applications the use of PCQM3 along with the corrected estimator is recommended. However, for sparse plant populations, where the use of PCQM3 may pose practical limitations, the PCQM2 or PCQM1 would be applied. During application of PCQM in the field, care should be taken to summarize the distance data based on ‘the inverse summation of squared distances’ but not ‘the summation of inverse squared distances’ as erroneously published.
Citation: Khan MNI, Hijbeek R, Berger U, Koedam N, Grueters U, Islam SMZ, et al. (2016) An Evaluation of the Plant Density Estimator the Point-Centred Quarter Method (PCQM) Using Monte Carlo Simulation. PLoS ONE 11(6): e0157985. https://doi.org/10.1371/journal.pone.0157985
Editor: Kurt O. Reinhart, USDA-ARS, UNITED STATES
Received: December 1, 2015; Accepted: June 8, 2016; Published: June 23, 2016
Copyright: © 2016 Khan et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting Information files.
Funding: The study was undertaken as part of the project ‘Ecological functionality and stability of mangrove ecosystems: a modelling approach’ under the grant type ‘Mandat d’ Impulsion Scientifique’ (MIS ID 1765914) of the National Science Foundation (FNRS), Belgium. This study was also funded by Khulna University, Bangladesh. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Density estimators are used in a wide variety of fields ranging from plant ecology, forestry and demography studies to medical sciences and astronomy. Density of plant populations is generally defined as the number of plants per unit area, which can be estimated by counting plants in plots with a known area. Instead of using plots, density of plant populations can however also be estimated using plotless methods, e.g. Point-Centred Quarter Method (PCQM) [1, 2] among other plotless methods [3–5]. In PCQM, the distance of plants to random sample points is converted to plant density. To address a number of practical problems that arise in some fields, such as mangroves (multiple-stemmed trees, quadrants where no trees are immediately present) the PCQM+ protocol was proposed [6]. The PCQM serves as a suitable method in vegetation study [7] especially when there is an accessibility issue as commonly observed in mangroves [8–11].
Plotless methods are preferred when plot-based (quadrat) sampling would be difficult or too costly [12, 13]. Plotless methods are faster, less laborious and require less equipment. Comparisons of various plotless methods [3, 4, 14] reveal that they have statistical uncertainty and there is no uniformly best plotless method for all types of spatial patterns in vegetation. Although a new composite k-tree estimator has been reported to mitigate the statistical bias [15], this still suffers from implementation issues concerning the spatial pattern of plants. As previous studies exist which compare various plotless methods, this review rather focuses on reviewing the different equations used for PCQM.
In vegetation study, there are many approaches, each having its strengths and weaknesses, making them more or less suitable for achieving a given objective. When difficult field conditions exist which make it challenging to access sites and trees (for example mangroves), using PCQM methods is an excellent option providing speedy sampling while requiring few logistics. PCQM allows estimation of plant densities based on scattered points over a larger geographic area than is possible for quadrat sampling. Its main limitation, however, is its bias or statistical uncertainty like any other plotless methods, which is partly related to the number of sample points. In this study, we thus focus on the optimization of PCQM methods varying in the order of considered trees per quadrant and the estimator equation used. Their performance is related to costs and effort, to data quality, and to statistical accuracy and precision.
The pioneer work on PCQM by Cottam and Curtis [2] was further modified by Pollard [16], which improved the statistical bias with PCQM estimator and later on Beasom and Haucke [17] found this method as the best plotless density estimator. In PCQM, the mean distance of the first nearest plant in each of four quadrants of a random sample point is converted to density. The accuracy of PCQM has also been explored through the second order distance (PCQM2 –distance of second nearest plant in each quadrant is measured) as well as 3rd order distance (PCQM3– distance of third nearest plant in each quadrant is measured). It has been argued that higher order PCQM offers better accuracy of density estimation [3, 4]. Based on the first order PCQM estimator [16] and the concept of the k- th nearest plant in a circular distance from sample point described by Pollard [16], higher order PCQM density estimators has also been derived, as reported in Engeman et al. [3] and White et al. [4] where the performance of various plotless density estimators have been compared. However, the estimators for simple PCQM (PCQM1) and the higher order ones (PCQM2 and PCQM3) need to be clarified further because of ambiguity in the equations used for PCQM in recent publications [3, 4, 16, 18, 19].
After Cottam [1] and Cottam and Curtis [2] the density (ρ) estimator of PCQM [16] stands as (1)
Where Rij = the distance from the i th random point to the closest individual in the j th quadrant; N is the number of random points used; 4 is the number of equiangular sectors about the random sample point and 4N is the number of distances measured. After the work of Pollard [16], Engeman et al. [3] followed by White et al. [4] described the second and third order PCQM density (ρ) estimators using the following general formula: (2)
Where k the number of equiangular sectors (quadrants) about the random sample point (k is always 4 for PCQM); g the number of individuals located in each quadrant and other notations are same as Eq 1. Solving this general equation (Eq 2) for PCQM1, PCQM2 and PCQM3, Engeman et al. [3] and White et al. [4] came to the following equations: (3) (4) (5)
Since the publication from Engeman et al. (1994), these equations have been widely used [3, 4, 20]. In further sections, we will refer to these three equations as the published estimators. For PCQM1, it is obvious that the formula (Eq 3) deviates from the one proposed by Pollard [16], who did not propose any formula for PCQM2 and PCQM3. However, based on appropriate interpretation of PCQM1 in Pollard [16] the PCQM2 and PCQM3 can be expressed by the following general equation: (6)
Where Rg(ij) is the distance from the i th sample point to the g th individual in the j th quadrant and other notations are same as mentioned above. Solving this general equation for PCQM1, PCQM2 and PCQM3, we come to the following new equations: (7) (8) (9) where the notations are the same as mentioned above. For more clarity, the terms ‘4’, ‘8’ and ‘12’ in the Eqs 7, 8 and 9 represent the four, eight and 12 plants encountered with the PCQM1, PCQM2 and PCQM3, respectively (Fig 1). In further sections, we will refer to Eqs 7, 8 and 9 as the corrected estimators, which is based on appropriate interpretation of the equation for PCQM1 as given byPollard [16]. In our more recent work [21], we have used these equations without any detailed description on PCQM formulae. Comparing the effects of the different formulas on accuracy of PCQM is the focus of this study. As expressed in the Eq 7, PCQM1 stands the same as Pollard [16], which differs with published equation (Eq 3). However, the formulae for PCQM1, PCQM2 and PCQM3 (Eqs 7, 8 and 9) differs from published estimators (Eqs 3, 4 and 5) depending on the number of random sample points N and the multiplying constants used. For example, when N = 10, the numerator in the equations of PCQM1, PCQM2 and PCQM3 stands for 120, 280 and 440, respectively in published estimators, i.e. Eqs 3, 4 and 5 but for 156, 316 and 476, respectively in corrected estimators, i.e. Eqs 7, 8 and 9. The original concept of PCQM suggests that at least 30 random sample points are required to obtain acceptable results in density estimation through PCQM [2]. In recent publications PCQM1, PCQM2 and PCQM3 have been applied using some constants (12 for PCQM1, 28 for PCQM2 and 44 for PCQM3) in the equations (Eqs 3, 4 and 5). However, in our judgment, there must be 4 objects falling in the imaginary circle of PCQM1, 8 objects in PCQM2 and 12 objects in PCQM3. In the corrected versions of the PCQM equations, we kept these numbers 4, 8 and 12 in the equations for PCQM1, PCQM2 and PCQM3, respectively instead of using those constants (12, 28 and 44). Therefore, in this study, we explore the performance of the corrected and published estimators for PCQM1, PCQM2 and PCQM3 in plant density estimation. For this purpose, we use some simulated and empirical datasets of plant positions. We hypothesize that the corrected estimators are more robust than the published estimators and that the higher order PCQM (PCQM2 and PCQM3) shows higher accuracy in the density prediction over first order PCQM.