SOME COMMENTS ON DESIGN-BASED LINE-INTERSECT SAMPLING WITH - - PowerPoint PPT Presentation

SOME COMMENTS ON DESIGN-BASED LINE-INTERSECT SAMPLING WITH SEGMENTED TRANSECTS

LUCIO BARABESI UNIVERSITÀ DI SIENA

1 LINE-INTERSECT SAMPLING

• is a method for sampling units scattered over a

Line-intersect sampling planar region.

• In its

, a unit is if a given line-segment (the basic version sampled “ ”) intersects the unit. transect

• In forestry, line-intersect sampling has found widespread application for

the purpose of estimating and . plant abundance vegetative coverage

• Much recent attention has focused on line-intersect sampling for the

assessment of for monitoring biodiversity of coarse woody debris ecosystems.

2 DESIGN-BASED LINE-INTERSECT SAMPLING

• Let

be a

• f

(given by T ß á ß T R

" R

fixed population units connected compact sets) scattered over the study region . V

• If

represents a

• f

, in the C T

4 4

fixed measurable attribute design-based approach population total the target parameter is usually given by the 7 œ C

• 4œ"

R 4

• The design-based approach is convenient for line-intersect applications

in order to avoid

• n unit shape and placement

unrealistic assumptions under the . model-based approach

• Moreover, field researchers

really wish to do not identify the model parameters make predictions random total and

• f a

, but they seek to estimate a . fixed population total

3 UNIT SELECTION

• The intersection of a unit is

when the intersects the complete transect unit boundary as the containing the transect. In as many times line contrary, an intersection is . partial

• A unit is always

if the intersection is complete. selected

• Partial intersections are usually handled by assuming that a transect

endpoint is . In this case, units are by partial intersections signed sampled

• f the

. signed endpoint

• In some designs units may be sampled by partial intersections of both

transect endpoints.

4 UNIT SELECTION Complete intersections Partial intersections Partial intersections by non-signed endpoint by signed endpoint

5 DESIGN IMPLEMENTATION

• A transect of

is identified by a

• n

and a fixed length position P ? V direction on . ) 1 Ò!ß # Ò

• The

is practically implemented by selecting and . design ? )

• For each , an

for each , the unit is ) inclusion region is defined T4 i.e. sampled if is in the inclusion region ? .

6 INCLUSION REGION

• The following figure shows the
• f a unit by assuming

inclusion region that is the . In the ? transect midpoint yellow-colored complete sets intersections

• range-colored

partial intersections

• ccur, while in the

sets

• ccur. The unit in the inclusion region.

is

7 INCLUSION REGION

• The following figure shows the
• f a unit by assuming

inclusion region that is the . Partial intersections are ? non-signed endpoint with the dot selected by the . The unit in signed endpoint with the arrow is not inclusion region.

8 CONDITIONAL AND UNCONDITIONAL APPROACHES

• The “

” is achieved by fixing and obtaining as conditional approach ) ? the realization of a suitable random variable . Y

• The “

” is considered if and are realizations unconditional approach ? )

• f two suitable random variables

and . Y K

• Let us assume that D Ð?ß Ñ

4

) is a function

• utside the inclusion

vanishing region and exclusively on the depending transect position.

• The function D Ð?ß Ñ

T

4 4

) represents a

• n the unit

measurable quantity according to its with the intersection transect (for example the

• f

length the intersection).

9 KAISER'S ESTIMATORS

• The

for is given by the Kaiser's (1983) conditional estimator linear 7 homogeneous estimator 7 ) ) ) ) s ÐY ± Ñ œ C D ÐYß Ñ ÒD ÐYß Ñ ± Ó

Ð-Ñ 4œ" R 4 4 4

E

• The

for is given by the Kaiser's (1983) unconditional estimator linear 7 homogeneous estimator 7 K K K s ÐYß Ñ œ C D ÐYß Ñ ÒD ÐYß ÑÓ

Ð?Ñ 4œ" R 4 4 4

E

• The two estimators are obviously

. unbiased

10 AN EXAMPLE

• If

represents the

• f

, the target parameter is the . C T

4 4

area coverage

• According to the protocol suggested by Barabesi and Marcheselli

(2008), is the

• f

with the transect. D Ð?ß Ñ T

4 4

) length of the intersection

• The following figures represent the

and the inclusion region function D Ð?ß Ñ

4

) for a if is the . circular unit transect midpoint ?

11 AN EXAMPLE (continues)

• By assuming that Y and

are K independent uniform random variables

• n

and V Ò!ß # Ò 1 , the and the require conditional unconditional estimators the since same field measurements E E ÒD ÐYß Ñ ± Ó œ ÒD ÐYß ÑÓ œ PC E

4 4 4

) ) K

• Hence

7 ) ) s ÐY ± Ñ œ D ÐYß Ñ E P

Ð-Ñ 4œ" R 4

• while

7 K K s ÐYß Ñ œ D ÐYß Ñ E P

Ð?Ñ 4œ" R 4

12 DESIGN-BASED LINE-INTERSECT SAMPLING WITH SEGMENTED TRANSECTS

• A

is a fixed set of

• f total

segmented transect

• riented line segments

O length . A segmented transect . P may not be connected

• Segmented transects include

(such as L-shaped or Y- radial transects shaped transects) and (such as triangular or squared polygonal transects 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 consisting of non-connected segmented transect a symmetric arrangement of . four Y-shaped transects

• Field scientists adopt this sampling protocol on the basis of the false

belief anisotropy that segmented transect may capture in the population.

13 UNIT SELECTION AND DESIGN IMPLEMENTATION

• A population unit is

if it is by a line segment of sampled selected at least the transect.

• A

transect is by a and a . segmented identified position direction ? )

• As an example,

may be characterized by the position radial transects ?

• f their

and by the direction

• f the

common vertex leading line ) segment.

• The design is implemented by

and . selecting ? )

• The

turns out to be the

• f the inclusion regions

inclusion region union corresponding to each line segment of the transect (which may ).

• verlap

14 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 . The unit in the transect endpoints is not inclusion region.

15 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

16 INCLUSION REGION USING TRIANGULAR TRANSECTS

• The following figure shows the inclusion region of a unit when a

triangular transect bottom left is adopted by assuming the vertex as . ? Partial intersection are selected by the . transect endpoints with the arrows The unit is in the inclusion region.

17 FIA-PROGRAM TRANSECTS

18 EXTENSIONS OF KAISER'S CONDITIONAL ESTIMATOR

• Let us assume that the function D Ð?ß Ñ

45

) represents a measurable quantity intersection

• n the unit

according to its with the T4 k-th line segment.

• f Kaiser's

are Two unbiased extensions conditional estimator 7 ) ) ) ) s ÐY ± Ñ œ C D ÐYß Ñ ÒD ÐYß Ñ ± Ó

Ð-Ñ" 4œ" R 5œ" O 45 5œ" O 45 4

• E

7 ) ) ) ) s ÐY ± Ñ œ C " O ÒD ÐYß Ñ ± Ó D ÐYß Ñ

Ð-Ñ# 5œ" O R 4œ" 45 45 4

E

19 EXTENSIONS OF KAISER'S UNCONDITIONAL ESTIMATOR

• Two unbiased extensions

unconditional estimator

• f the Kaiser's

are 7 K K K s ÐYß Ñ œ C D ÐYß Ñ ÒD ÐYß ÑÓ

Ð?Ñ" 4œ" R 5œ" O 45 5œ" O 45 4

• E

7 K K K s ÐYß Ñ œ C " O ÒD ÐYß ÑÓ D ÐYß Ñ

Ð?Ñ# 5œ" O R 4œ" 45 45 4

E

20 EXTENSIONS OF KAISER'S ESTIMATORS

• The

( and ) include the estimators first-type extensions i.e. 7 7 s s

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 ( and ). i.e. 7 7 s s

Ð-Ñ# Ð?Ñ#

• The two conditional estimators

if does not coincide EÒD ÐYß Ñ ± Ó

45

) ) depend on for each , as well as the two unconditional estimators 5 4 coincide when does not depend on for each . This feature EÒD ÐYß ÑÓ 5 4

45

K is to most of the estimators usually adopted in the practice of common segmented-transect sampling.

• For

it may even occur that some important protocols EÒD ÐYß Ñ ± Ó œ

45

) ) EÒD ÐYß ÑÓ 5 4

45

K and this quantity does not depend on for each and hence the four estimators are from the sampling effort perspective. equivalent

21 THREE IMPORTANT ISSUES

• While boundary effects

straight- may be easily managed in the case of line transects serious more complex transect shapes , the problem is with .

• for a

may be expressed since the No preference particular estimator variances unknown population frame

• f the estimators depend on the

. The conditional estimator ) s (as well as the unconditional estimators require the since they involve the same field same sample effort

• measurements. The

could be preferred to the conditional approach unconditional approach measurement (or ) on the basis of vice versa complexity.

• for a

may be asserted since the No superiority particular transect shape variances of estimators based on different transect types (of the same total length) depend on the . unknown population frame Straight-line transects should be since their field implementation is easier and edge preferred effects are easily handled.

22 AN EXAMPLE

• As an example for which

does not E E ÒD ÐYß Ñ ± Ó œ ÒD ÐYß ÑÓ

45 45

) ) K depend on for each , let us assume that represents the

• f

, in 5 4 C T

4 4

area such a way that the reduces to the . target parameter coverage

• Following the protocol suggested by Barabesi and Marcheselli (2008), if

D Ð?ß Ñ T

45 4

) is the

• f

with the -th line segment length of the intersection k

• f the transect, it turns out that

E E ÒD ÐYß Ñ ± Ó œ ÒD ÐYß ÑÓ œ PC OE

45 45 4

) ) K

23 AN EXAMPLE (continues)

• The conditional estimators coincide

7 ) 7 ) ) s s ÐY ± Ñ œ ÐY ± Ñ œ D ÐYß Ñ E P

Ð-Ñ" Ð-Ñ# 4œ" R O 5œ" 45

and they are require the same sampling effort of the unconditional estimators which in turn coincide 7 K 7 K K s s ÐYß Ñ œ ÐYß Ñ œ D ÐYß Ñ E P

Ð?Ñ" Ð?Ñ# 4œ" R O 5œ" 45

• It is apparent that the

with the sampled units is solely total intersection needed for computing the estimators.

24 REPLICATED TRANSECTS

• As is usual in environmental protocols, 8
• f the line-

replications intersect design are usually performed. The goal is the selection of an appropriate strategy replicates for the placement of the . 8

• By assuming the

, the target parameter may be conditional approach 7 expressed as an integral 7 œ " E (

V

7 ) s Ð? ± Ñ ?

Ð-Ñ6

d

• Accordingly, the estimation reduces to a Monte Carlo integration.

25 SIMPLE RANDOM SAMPLING OF REPLICATES

• The

(equivalent to the ) random sampling crude Monte Carlo integration is achieved by generating independent transect locations 8 Y ß á ß Y

" 8

uniformly distributed on . V

• Two

may be obtained

• verall estimators

7 7 ) ) – – ,

Ð-Ñ6 Ð-Ñ6 " 8 3 3œ" 8

œ ÐY ß á ß Y ± Ñ œ ± Ñ 6 œ "ß # " 8 7 s ÐY

Ð-Ñ6

• The estimators are

, while – unbiased VarÒ7Ð-Ñ6Ó œ SÐ8 Ñ

" .

• The

are variance estimators Var s Ò ÐY s 7 7 7 – – ,

Ð-Ñ6 Ð-Ñ6 3œ" 8 3 #

Ó œ Ð ± Ñ  Ñ 6 œ "ß # " 8Ð8  "Ñ

Ð-Ñ6

)

• It is easily proven that

as . – – Var s Ò7 7

Ð-Ñ6 Ð-Ñ6 "Î# .

Ó Ð  Ñ Ä Ð!ß "Ñ 8 Ä ∞ 7 a

26 NON-ALIGNED SYSTEMATIC SAMPLING OF REPLICATES

• An alternative suitable strategy is given by non-aligned systematic

sampling ( . equivalent to the ) modified Monte Carlo integration

• Non-aligned systematic sampling involves covering the study region by

means of a

• f equal rectangles and generating independent

partition 8 8 random transect locations in these rectangles. Z ß á ß Z

" 8

Random sampling Non-aligned systematic sampling

27 NON-ALIGNED SYSTEMATIC SAMPLING OF REPLICATES

• The strategy provides the two overall estimators

7 7 ) ) – – ,

Ð-Ñ6 Ð-Ñ6 ‡ ‡ " 8 3 3œ" 8

œ ÐZ ß á ß Z ± Ñ œ ± Ñ 6 œ "ß # " 8 7 s ÐZ

Ð-Ñ6

• The estimators are

. unbiased

• Moreover,

, – – Var Var Ò Ò 7 7

Ð-Ñ6 ‡ Ð-Ñ6

Ó Ÿ Ó i.e. non-aligned systematic sampling is always to preferable simple random sampling.

28 NON-ALIGNED SYSTEMATIC SAMPLING OF REPLICATES

• The estimators

are . 7 –

Ð-Ñ6 ‡

very efficient

• Barabesi and Marcheselli (2003) show that

if VarÒ s 7 7 –

Ð-Ñ6 ‡ Ó œ SÐ8

Ñ

# Ð-Ñ6

is a Lipschitz function.

• Barabesi and Pisani (2004) show that

if VarÒ s 7 7 –

Ð-Ñ6 ‡ Ó œ SÐ8

Ñ

$Î# Ð-Ñ6 is an

elementary function.

• Barabesi and Marcheselli (2008) show that

with VarÒ7 –

Ð-Ñ6 ‡ Ó œ SÐ8

Ñ

α

" Ÿ Ÿ $Î# α if 7 sÐ-Ñ6 is a function and with piecewise Sobolev $Î# Ÿ Ÿ # α if 7 sÐ-Ñ6 is a function. Sobolev

29 NON-ALIGNED SYSTEMATIC SAMPLING OF REPLICATES

• I

is a Lipschitz function f , a for 7 7 s Ò

Ð-Ñ6

consistent estimator Var – is

Ð-Ñ6 ‡ Ó

Var s Ò ÐZ s 7 7 7 – – – ,

Ð-Ñ6 ‡ # # 3œ" 8 3 Ð-Ñ63 Ð-Ñ6

Ó œ Ð ± Ñ  Ñ 6 œ "ß # % &8

Ð-Ñ6

) 7 where is the average of the quantities – 7 )

Ð-Ñ63

7 s ÐZ ± Ñ 4

Ð-Ñ6 4

• ver the indexes

corresponding to the rectangles adjacent to the -th rectangle. i

• Moreover,

as – – Var s Ò s 7 7 7

Ð-Ñ6 Ð-Ñ6 ‡ "Î# ‡ .

Ó Ð  Ñ Ä Ð!ß "Ñ 8 Ä ∞ 7 a if

Ð-Ñ6 is a

Lipschitz function.

• The

turns out to be for less regular variance estimator conservative functions and the previous pivotal quantity produces large-sample conservative confidence intervals even if is an elementary function 7 sÐ-Ñ6

• r a Sobolev function.

30 FINAL REMARKS

• Barabesi and Marcheselli (2005, 2008) have proven that a further

strategy, i.e. locally antithetic non-aligned systematic sampling, may be more effective than non-aligned systematic sampling. However, locally antithetic non-aligned systematic sampling may be more involved in the field.

• The efficiency of

under non-aligned systematic sampling the unconditional approach worse conditional approach is than under the . This problem occurs since integrals of 3-dimensional functions must be evaluated in this case and the “ ” takes places with curse of dimensionality respect to the unconditional approach. This drawback may motivate the preference conditional approach

• f the
• ver the unconditional approach

when the sampling effort for collecting field measurements is the same.

31 REFERENCES

Affleck, D.L.R., Gregoire, T.G. and Valentine, H.T. (2005) Design unbiased estimation in line intersect sampling using segmented transects, Environmental and Ecological Statistics , 139-154. 12 Barabesi, L. (2007) Some comments on design-based line-intersect sampling by using segmented transects, Environmental and Ecological Statistics 14, 483– 494. Barabesi, L. and Marcheselli, M. (2003) A modified Monte Carlo integration, International Mathematical Journal , 555-565. 3 Barabesi, L. and Marcheselli, M. (2005) Some large-sample results on a modified Monte Carlo integration method, Journal of Statistical Planning and Inference , 420-432. 135 Barabesi, L. and Marcheselli, M. (2008) Improved strategies for coverage estimation by using replicated line-intercept sampling, Environmental and Ecological Statistics . 15 Barabesi, L. and Pisani, C. (2004) Steady-state ranked set sampling for replicated environmental sampling designs, . Environmetrics 15, 45-56 Kaiser, L. (1983) Unbiased estimation in line-intercept sampling, Biometrics 39, 965-976.

32 SPAZI DI SOBOLEV

• Lo spazio di Sobolev

, dove , contiene funzioni [ Ð Ñ : − Ò"ß ∞Ñ

"ß: ‘

0 − P Ð Ñ 1 − P Ð Ñ

: :

‘ ‘ per cui esiste una funzione (la cosiddetta derivata debole di ordine ) per la quale : ( (

‘ ‘

0ÐBÑ .B œ  .B : :

wÐBÑ

1ÐBÑ ÐBÑ per ogni funzione test . : ‘ − G Ð Ñ

"

• Equivalentemente,

se e solo se è 0 − [ Ð Ñ

"ß: ‘

quasi ovunque differenziabile in modo che per ogni si ha 0 − P Ð Ñ Bß C −

w : ‘

‘ 0ÐBÑ  œ .> 0ÐCÑ 0 Ð>Ñ (

C B w

33 SPAZI DI SOBOLEV

• Lo spazio di Sobolev [

Ð Ñ

"ß" ‘ coincide con lo spazio delle funzioni

assolutamente continue e quindi può contenere funzioni molto irregolari. Una funzione di Lipschitz è equivalente ad una funzione in [ Ð Ñ

"ß∞ ‘ .

• Una estensione di [

Ð Ñ [ Ð Ñ

"ß: =ß:

‘ ‘ è data dallo spazio di Sobolev dove è un intero. = #

• Lo spazio

è definito in modo ricorsivo, ovvero i membri di [ Ð Ñ

=ß: ‘

[ Ð Ñ 0ß 0 − [ Ð Ñ [ Ð Ñ

#ß: w "ß: $ß:

‘ ‘ ‘ sono tali che , i membri di sono tali che , e così via. 0ß 0 ß 0 − [ Ð Ñ

w ww #ß: ‘

34 SPAZI DI SOBOLEV

• Supponiamo un insieme centrato in

la cui frontiera soddisfa Ð ß Ñ

• "

#

l'equazione ± ?  ±  ± ?  ± œ

" " # # ; ; ;

• 3

con e . ; − Ò"ß ∞Ñ  ! 3 La lunghezza dell'intersezione fra un transetto in ? e l'insieme è data da

• Ð?Ñ œ #Ð

 ± ?  ± Ñ Ð?Ñ 3

• ;

; "Î; " Ò  ß  Ó

I -

3 - 3

" "

• Evidentemente

e

• −

 [ Ð Ñ ? œ

"ß: ‘ , dal momento che nei punti

• 3

"

? œ 

• 3

"

la funzione non è differenziabile, ma ammette solamente una derivata debole di opportuno ordine . Questa semplice funzione soddisfa : la condizione di solo per . Lipschitz ; œ "

35 SPAZI DI SOBOLEV

• Nella seguente figura sono riportati i grafici dell'insieme e della relativa

funzione per , e . Si può verificare

• ; œ "ß #ß $

œ œ "Î# œ "Î$

• 3

" #

che Lip se , se e

• − [

œ ; œ "

• − [

; œ #

"ß∞ "ß#

Ð Ñ Ð Ñ Ð Ñ ‘ ‘ ‘

%

• − [

; œ $  !

"ß$Î#%Ð Ñ

‘ se (per ogni ). %

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8

q=1 cHuL

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8

q=2 cHuL

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8

q=3 cHuL

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

36 SPAZI DI SOBOLEV

• La famiglia [

Ð

"ß: ‘#Ñ può essere definita in modo analogo. Una

generalizzazione della funzione del precedente esempio è data da

• Ð? ß ? Ñ œ #Ö

 ÒÐ?  Ñ  Ð?  Ñ Ó × Ð? ß ? Ñ

" # " " # # " # ; # # ;Î# "Î;

3

• IW

dove W œ ÖÐ Ñ À ? ß ? Ð?  Ñ  Ð?  Ñ Ÿ ×

" # " " # # # # #

• 3

.

• Questa funzione è continua, ma non è differenziabile sulla frontiera di
• W. Si può provare che -

"Î; soddisfa la condizione di Hölder di ordine ,

• vvero

.

• − G

Ð Ñ

!ß"Î; ‘#

37 SPAZI DI SOBOLEV

• Nella seguente figura sono riportati i grafici di per

,

• ; œ "ß #ß $
• 3

" # "ß∞

œ œ "Î# œ "Î$

• − [

œ ; œ " e . Risulta Lip se , Ð Ñ Ð Ñ ‘ ‘

# #

• − [

; œ #

• − [

; œ $  !

"ß# "ß$Î# % %

Ð Ñ Ð Ñ ‘ ‘

# #

se e se (per ogni ). %

cHu1, u2L q=1 cHu1, u2L q=2 cHu1, u2L q=3