SOME COMMENTS ON DESIGN-BASED LINE-INTERSECT SAMPLING WITH SEGMENTED TRANSECTS LUCIO BARABESI UNIVERSITÀ DI SIENA
LINE-INTERSECT SAMPLING • Line-intersect sampling is a method for sampling units scattered over a planar region. • In its basic version , a unit is sampled if a given line-segment (the “ transect ”) intersects the unit. • In forestry, line-intersect sampling has found widespread application for the purpose of estimating plant abundance and vegetative coverage . • Much recent attention has focused on line-intersect sampling for the assessment of coarse woody debris for monitoring biodiversity of ecosystems. 1
DESIGN-BASED LINE-INTERSECT SAMPLING T ß á ß T R • Let be a fixed population of units (given by connected " R V compact sets) scattered over the study region . C T • If represents a fixed measurable attribute of , in the design-based 4 4 approach the target parameter is usually given by the population total � R 7 œ C 4 4œ" • The design-based approach is convenient for line-intersect applications in order to avoid unrealistic assumptions on unit shape and placement under the model-based approach . • Moreover, field researchers do not really wish to identify the model parameters and make predictions of a random total , but they seek to estimate a fixed population total . 2
UNIT SELECTION • The intersection of a unit is complete when the transect intersects the unit boundary as many times as the line containing the transect. In contrary, an intersection is partial . • A unit is always selected if the intersection is complete. • Partial intersections are usually handled by assuming that a transect endpoint is signed . In this case, units are sampled by partial intersections of the signed endpoint . • In some designs units may be sampled by partial intersections of both transect endpoints. 3
UNIT SELECTION Complete intersections Partial intersections Partial intersections by non-signed endpoint by signed endpoint 4
DESIGN IMPLEMENTATION P ? V • A transect of fixed length is identified by a position on and a ) Ò!ß # Ò 1 direction on . ? ) • The design is practically implemented by selecting and . ) T 4 i.e. • For each , an inclusion region is defined for each , the unit is ? sampled if is in the inclusion region . 5
INCLUSION REGION • The following figure shows the inclusion region of a unit by assuming ? that is the transect midpoint . In the yellow-colored sets complete intersections occur, while in the orange-colored sets partial intersections occur. The unit in the inclusion region. is 6
INCLUSION REGION • The following figure shows the inclusion region of a unit by assuming ? that is the non-signed endpoint with the dot . Partial intersections are selected by the signed endpoint with the arrow . The unit is not in inclusion region. 7
CONDITIONAL AND UNCONDITIONAL APPROACHES ) ? • The “ conditional approach ” is achieved by fixing and obtaining as Y the realization of a suitable random variable . ? ) • The “ unconditional approach ” is considered if and are realizations Y K of two suitable random variables and . • Let us assume that D Ð?ß Ñ ) is a function vanishing outside the inclusion 4 region and depending exclusively on the transect position. • The function D Ð?ß Ñ ) represents a T measurable quantity on the unit 4 4 according to its intersection with the transect (for example the length of the intersection). 8
KAISER'S ESTIMATORS • The 7 Kaiser's (1983) conditional estimator for is given by the linear homogeneous estimator � E R D ÐYß Ñ ) 4 7 ÐY ± Ñ œ ) C s Ð-Ñ 4 ÒD ÐYß Ñ ± Ó ) ) 4 4œ" • The 7 Kaiser's (1983) unconditional estimator for is given by the linear homogeneous estimator � E R D ÐYß K Ñ 4 7 ÐYß K Ñ œ C s Ð?Ñ 4 ÒD ÐYß K ÑÓ 4 4œ" • The two estimators are obviously unbiased . 9
AN EXAMPLE C T • If represents the area of , the target parameter is the coverage . 4 4 • According to the protocol suggested by Barabesi and Marcheselli D Ð?ß Ñ ) T (2008), is the length of the intersection of with the transect. 4 4 • The following figures represent the inclusion region and the function D Ð?ß Ñ ) for a ? circular unit if is the transect midpoint . 4 10
AN EXAMPLE (continues) • By assuming that Y and K are independent uniform random variables V Ò!ß # Ò 1 , the on and conditional and the unconditional estimators require the same field measurements since ÑÓ œ PC 4 ÒD ÐYß Ñ ± Ó œ ) ) ÒD ÐYß K E E 4 4 E • Hence � R E 7 ÐY ± Ñ œ ) D ÐYß Ñ ) s 4 Ð-Ñ P 4œ" while � R E 7 ÐYß K Ñ œ D ÐYß K Ñ s 4 Ð?Ñ P 4œ" 11
DESIGN-BASED LINE-INTERSECT SAMPLING WITH SEGMENTED TRANSECTS O • A segmented transect is a fixed set of oriented line segments of total P length . A segmented transect may not be connected . • Segmented transects include radial transects (such as L-shaped or Y- shaped transects) and polygonal transects (such as triangular or squared transects), which are adopted in many national forest inventories. • The Forest Inventory and Analysis (FIA) program of the U.S.D.A. Forest Service assumes a non-connected segmented transect consisting of a symmetric arrangement of four Y-shaped transects . • Field scientists adopt this sampling protocol on the basis of the false belief that segmented transect may capture anisotropy in the population. 12
UNIT SELECTION AND DESIGN IMPLEMENTATION • A population unit is sampled if it is selected by at least a line segment of the transect. ? ) • A segmented transect is identified by a position and a direction . ? • As an example, radial transects may be characterized by the position ) of their common vertex and by the direction of the leading line segment. selecting ? ) • The design is implemented by and . • The inclusion region turns out to be the union of the inclusion regions corresponding to each line segment of the transect (which may overlap ). 13
INCLUSION REGION USING L-SHAPED TRANSECTS • The following figure shows the inclusion region of a unit when a L- ? shaped transect is adopted by assuming the common vertex as . Partial intersections are selected by the transect endpoints . The unit is not in the inclusion region. 14
INCLUSION REGION USING Y-SHAPED TRANSECTS • The following figure shows the inclusion region of a unit when a Y- ? shaped transect is adopted by assuming the common vertex as . Partial intersections are selected by the transect endpoints . 15
INCLUSION REGION USING TRIANGULAR TRANSECTS • The following figure shows the inclusion region of a unit when a ? triangular transect is adopted by assuming the bottom left vertex as . Partial intersection are selected by the transect endpoints with the arrows . The unit is in the inclusion region. 16
17 FIA-PROGRAM TRANSECTS
EXTENSIONS OF KAISER'S CONDITIONAL ESTIMATOR • Let us assume that the function D Ð?ß Ñ ) represents a measurable 45 T 4 quantity on the unit according to its intersection with the k -th line segment. • Two unbiased extensions of Kaiser's conditional estimator are � � O R � D ÐYß Ñ ) 45 5œ" 7 ÐY ± Ñ œ ) C s 4 Ð-Ñ" O ÒD ÐYß Ñ ± Ó ) ) E 45 4œ" 5œ" � � E O R D ÐYß Ñ ) " 45 7 ÐY ± Ñ œ ) C s 4 Ð-Ñ# O ÒD ÐYß Ñ ± Ó ) ) 45 4œ" 5œ" 18
EXTENSIONS OF KAISER'S UNCONDITIONAL ESTIMATOR • Two unbiased extensions of the Kaiser's unconditional estimator are � � O R � D ÐYß K Ñ 45 5œ" 7 ÐYß K Ñ œ C s 4 Ð?Ñ" O ÒD ÐYß K ÑÓ E 45 4œ" 5œ" � � E O R D ÐYß K Ñ " 45 7 ÐYß K Ñ œ C s 4 Ð?Ñ# O ÒD ÐYß K ÑÓ 45 4œ" 5œ" 19
EXTENSIONS OF KAISER'S ESTIMATORS first-type extensions i.e. 7 7 s s • The ( and ) include the estimators Ð-Ñ" Ð?Ñ" proposed by Affleck (2005). et al. • The Forest Inventory and Analysis of the U.S.D.A. Forest Service utilizes estimators which are special cases of the second-type extensions i.e. 7 7 ( s and s ). Ð-Ñ# Ð?Ñ# E ÒD ÐYß Ñ ± Ó ) ) • The two conditional estimators coincide if does not 45 5 4 depend on for each , as well as the two unconditional estimators E ÒD ÐYß K ÑÓ 5 4 coincide when does not depend on for each . This feature 45 is common to most of the estimators usually adopted in the practice of segmented-transect sampling. E ÒD ÐYß Ñ ± Ó œ ) ) • For some important protocols it may even occur that 45 E ÒD ÐYß K and this quantity does not depend on for each and hence ÑÓ 5 4 45 the four estimators are equivalent from the sampling effort perspective. 20
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