Evaluation of Sampling with Partial Replacement and Double Sampling in a Managed Forest in the Brazilian Amazon

Reategui-Betancourt, Jorge L.; Mazzei, Lucas; Rezende, Alba Valéria; Briceño, Guido; Oliveira, Carlos Magno Moreira de; Gaui, Tatiana Dias; Khan, Salman; Figueiredo, Axa Emanuelle S.

doi:10.1590/2179-8087-FLORAM-2023-0011

person Jorge L. Reategui-Betancourt

investigation (equal) · resources (equal) · writing - original draft (equal) · writing - review & editing (equal)

schoolUniversidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.

0000-0002-1999-3001

person Lucas Mazzei

writing - original draft (support) · writing - review & editing (support) · formal analysis (support)

schoolUniversidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.schoolEmpresa Brasileira de Pesquisa Agropecuária (EMBRAPA) Amazônia Oriental, Belém, PA, Brasil.

0000-0002-1757-9472

person Alba Valéria Rezende

writing - original draft (support) · writing - review & editing (support) · formal analysis (support)

schoolUniversidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.

0000-0002-1167-9798

person Guido Briceño

conceptualization (equal) · methodology (equal) · writing - review & editing (equal) · supervision (equal)

schoolCentre International de Recherche Agronomique pour le Développement (CIRAD), The French National Centre for Scientific Research (CNRS), Montpellier, France.

0000-0003-2194-4716

person Carlos Magno Moreira de Oliveira

conceptualization (equal) · methodology (equal) · writing - review & editing (equal) · formal analysis (equal) · supervision (equal)

schoolUniversidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.

0000-0002-2694-7388

person Tatiana Dias Gaui

investigation (support)

schoolUniversidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.

0000-0002-1793-3275

person Salman Khan

investigation (support)

schoolCentre International de Recherche Agronomique pour le Développement (CIRAD), The French National Centre for Scientific Research (CNRS), Montpellier, France.

0000-0002-2460-8390

person Axa Emanuelle S. Figueiredo

investigation (support)

schoolUniversidade de Brasília (UnB), Departamento de Engenharia Florestal, Brasília, DF, Brasil.

0000-0002-7276-905X

CORRESPONDENCE TO Jorge L. Reategui-Betancourt Campus Universitário Darcy Ribeiro, Asa Norte, 70910-900, Brasília, DF, Brasil. e-mail:
jorgereategui91@gmail.com

Associate editor: Rafaella de Angeli Curto http://orcid.org/0000-0001-5509-4655

CFI		DS		SPR
First occasion	Second occasion	First occasion	Second occasion	First occasion	Second occasion
P1 (m)	P1 (m)	P1 (u)		P1 (u)
P2(m)	P2(m)	P2(u)		P2(u)
P3(m)	P3(m)	P3(u)		P3(u)
P4(m)	P4(m)	P4(u)		P4(u)
P5(m)	P5(m)	P5(u)		P5(u)
P6(m)	P6(m)	P6(u)		P6(u)
P7(m)	P7(m)	P7(u)		P7(u)
P8(m)	P8(m)	P8(u)		P8(u)
P9(m)	P9(m)	P9(u)		P9(m)	P9(m)
P10(m)	P10(m)	P10(u)		P10(m)	P10(m)
P11(m)	P11(m)	P11(m)	P11(m)	P11(m)	P11(m)
P12(m)	P12(m)	P12(m)	P12(m)	P12(m)	P12(m)
P13(m)	P13(m)	P13(m)	P13(m)	P13(m)	P13(m)
P14(m)	P14(m)	P14(m)	P14(m)		P14(n)
P15(m)	P15(m)	P15(m)	P15(m)		P15(n)
P16(m)	P16(m)	P16(m)	P16(m)		P16(n)
P17(m)	P17(m)	P17(m)	P17(m)		P17(n)
P18(m)	P18(m)	P18(m)	P18(m)		P18(n)

Continuous Forest Inventory (CFI)
Estimators	First occasion	Second occasion	Growth
Sampling error	Absolute: $E_{a} = t \cdot S_{\bar{x}}$	Absolute: $E_{a} = t \cdot S_{\bar{y}}$	Absolute: $E_{a} = t \cdot S_{{\bar{d}}_{m}}$
Sampling error	Relative: $E_{r} = t \cdot \frac{S_{\bar{x}}}{\bar{x}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{\bar{y}}}{\bar{y}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{{\bar{d}}_{m}}}{{\bar{d}}_{m}} \cdot 100$
Standard error of the mean	$S_{\bar{x}} = \sqrt{{S^{2}}_{\bar{x}}}$	$S_{\bar{y}} = \sqrt{{S^{2}}_{\bar{y}}}$	$S_{{\bar{d}}_{m}} = \sqrt{{S^{2}}_{{\bar{d}}_{m}}}$
Variance of the Mean	${S^{2}}_{\bar{X}} = \frac{{S^{2}}_{X}}{n_{1}} \cdot f c;$	${S^{2}}_{\bar{y}} = \frac{{s^{2}}_{y}}{n_{1}} \cdot f c;$	${S^{2}}_{{\bar{d}}_{m}} = {S^{2}}_{\bar{x}} + {S^{2}}_{\bar{y}} - \frac{2 \cdot S_{x y}}{m}$
Variance of the Mean	Where: $f c = (1 - \frac{n_{1}}{N_{1}})$	Where: $f c = (1 - \frac{n_{1}}{N_{1}})$	Where: S_xy = Covariance between the two occasions
Variance	${S^{2}}_{x} = \frac{\sum_{i = 1} n_{1} {(X i - \bar{X})}^{2}}{n_{1} - 1}$	${S^{2}}_{y} = \frac{\sum_{i = 1} n_{2} {(Y i - \bar{Y})}^{2}}{n_{2} - 1}$
Mean	${\bar{X}}_{m} = \frac{\sum_{i = 1} n_{1} X_{i}}{n_{1}}$	${\bar{Y}}_{m} = \frac{\sum_{i = 1} n_{2} Y_{i}}{n_{1}}$	${\bar{d}}_{m} = {\bar{Y}}_{m} - {\bar{X}}_{m}$
Double Sampling (DS)
Estimators	First occasion	Second occasion	Growth
Sampling error	Absolute: $E_{a} = t \cdot S_{{\bar{x}}_{1}}$	Absolute: $E_{a} = t \cdot S_{\bar{y}}$	Absolute: $E_{a} = t \cdot S_{{\bar{d}}_{r}}$
Sampling error	Relative: $E_{r} = t \cdot \frac{S_{{\bar{x}}_{1}}}{{\bar{x}}_{1}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{\bar{y}}}{\bar{v}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{{\bar{d}}_{r}}}{{\bar{d}}_{r}} \cdot 100$
Standard error of the mean	${S_{\bar{x}}}_{1} = \sqrt{{S^{2}}_{{\bar{x}}_{1}}}$	$S_{\bar{y}} = \sqrt{{S^{2}}_{\bar{y}}}$	$S_{{\bar{d}}_{r}} = \sqrt{{S^{2}}_{{\bar{d}}_{r}}}$
Variance of the Mean	${S^{2}}_{{\bar{x}}_{1}} = \frac{{S^{2}}_{x}}{n_{1}} \cdot f c;$	${S^{2}}_{\bar{y}} = \frac{{S^{2}}_{y} \cdot (1 - ρ^{2})}{n_{2}} + \frac{ρ^{2} \cdot {S^{2}}_{y}}{n_{1}}$	${S^{2}}_{{\bar{d}}_{r}} = {S^{2}}_{y} \cdot (\frac{{(1 - ρ)}^{2}}{n_{1}} + \frac{1 - ρ^{2}}{n_{2}})$
Variance of the Mean	Where: $f c = (1 - \frac{n_{1}}{N_{1}})$	Where: $ρ = \frac{C o ν (x y)}{S_{x} S_{y}}$
Variance	${S^{2}}_{x_{1}} = \frac{\sum_{i = 1} n_{1} {(X_{i} - \bar{X})}^{2}}{n_{1} - 1}$	${S^{2}}_{y} = \frac{\sum_{i = 1} n_{2} {(Y_{i} - \bar{Y})}^{2}}{n_{2} - 1}$
Mean	${\bar{x}}_{1} = P u {\bar{x}}_{u} + P u {\bar{x}}_{m}$	$y_{r} = y_{m} + b \cdot (x_{1} - x_{m})$	${\bar{d}}_{r} = {\bar{y}}_{r} - {\bar{x}}_{1}$
		Where: b =
		b = angular coefficient of the regression
Sampling with Partial Replacement (SPR)
Estimators	First occasion	Second occasion	Growth
Sampling error	Absolute: $E_{a} = t \cdot S_{\bar{x}}$	Absolute: $E_{a} = t \cdot S_{\bar{y}}$	Absolute: $E_{a} = t \cdot S_{{\bar{d}}_{p}}$
Sampling error	Relative: $E_{r} = t \cdot \frac{S_{\bar{x}}}{\bar{X}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{\bar{y}}}{\bar{Y}} \cdot 100$	Relative: $E_{r} = t \cdot \frac{S_{{\bar{d}}_{p}}}{{\bar{d}}_{p}} \cdot 100$
Standard error of the mean	$S_{\bar{x}} = \sqrt{{S^{2}}_{\bar{x}}}$	$S_{\bar{y}} = \sqrt{{S^{2}}_{\bar{y}}}$	$S_{{\bar{d}}_{p}} = \sqrt{{S^{2}}_{{\bar{d}}_{p}}}$
Variance of the Mean	${S^{2}}_{\bar{x}} = \frac{{s^{2}}_{x}}{n_{1}} \cdot f c;$	${S^{2}}_{\bar{x}} = a^{2} \cdot {S^{2}}_{x} (\frac{1}{u} + \frac{1}{m}) + c^{2} \frac{{S^{2}}_{y}}{m} + {(1 + c)}^{2} \cdot \frac{{S^{2}}_{y}}{n} - - 2 \cdot a \cdot c \cdot r \cdot \frac{S_{x} S_{y}}{m}$	${S^{2}}_{\bar{x}} = a^{2} \cdot {S^{2}}_{x} (\frac{1}{u} + \frac{1}{m}) + c^{2} \frac{{S^{2}}_{y}}{m} + {(1 + c)}^{2} \cdot \frac{{S^{2}}_{y}}{n} - - 2 \cdot a \cdot c \cdot r \cdot \frac{S_{x} S_{y}}{m}$
Variance of the Mean	Where: $f c = (1 - \frac{n_{1}}{N_{1}})$	Where: $r = \frac{C o v (x y)}{S_{x} S_{y}}$	Where: $A = \frac{m}{n_{2} - P_{u} \cdot n \cdot r^{2}} + \frac{n \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{x}}{S_{y}} B = \frac{- m \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{y}}{S_{x}} - \frac{n_{2} \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}}$
Variance	${S^{2}}_{x} = \frac{\sum_{i = 1} n 1 {(X_{i} - \bar{X})}^{2}}{n_{1} - 1}$	${S^{2}}_{y} = \frac{\sum_{i = 1} n 1 {(Y_{i} - \bar{Y})}^{2}}{n_{2} - 1}$
Mean	$\bar{X} = (1 - b) \cdot {\bar{x}}_{u} + b \cdot {\bar{x}}_{m} + c \cdot {\bar{y}}_{m} - c \cdot {\bar{y}}_{n};$	$\bar{Y} = a \cdot {\bar{x}}_{u} - a \cdot {\bar{x}}_{m} + c \cdot {\bar{y}}_{m} + (1 - c) \cdot {\bar{y}}_{n};$	${\bar{d}}_{p} = A \cdot {\bar{y}}_{m} + B \cdot {\bar{x}}_{m} + (1 - A) \cdot {\bar{y}}_{n} - (1 + B);$
Mean	Where: $b = \frac{n_{2} \cdot P_{m}}{n_{2}} - P_{u} \cdot n \cdot r^{2} c = \frac{- n \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{y}}{S_{x}}$	Where: $c = \frac{m}{n_{2}} - P_{u} \cdot n \cdot r^{2} a = \frac{m \cdot P_{u}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{y}}{S_{x}}$	Where: $A = \frac{m}{n_{2} - P_{u} \cdot n \cdot r^{2}} + \frac{n \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{x}}{S_{y}} B = \frac{- m \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}} \cdot r \cdot \frac{S_{y}}{S_{x}} - \frac{n_{2} \cdot P_{m}}{n_{2} - P_{u} \cdot n \cdot r^{2}}$

{S^{2}}_{\bar{x}} = a^{2} \cdot {S^{2}}_{x} (\frac{1}{u} + \frac{1}{m}) + c^{2} \frac{{S^{2}}_{y}}{m} + {(1 + c)}^{2} \cdot \frac{{S^{2}}_{y}}{n} - - 2 \cdot a \cdot c \cdot r \cdot \frac{S_{x} S_{y}}{m}

{S^{2}}_{\bar{x}} = a^{2} \cdot {S^{2}}_{x} (\frac{1}{u} + \frac{1}{m}) + c^{2} \frac{{S^{2}}_{y}}{m} + {(1 + c)}^{2} \cdot \frac{{S^{2}}_{y}}{n} - - 2 \cdot a \cdot c \cdot r \cdot \frac{S_{x} S_{y}}{m}