Sampling Theory for Forest Inventory: A Teach-Yourself Course

407 Pages·1986·8.431 MB·English

by  Pieter G. de Vries (auth.)

Pieter G. de Vries
Sampling Theory for
Forest Inventory
A Teach-Yourself Course
With·20 Figures
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo·
PIETER G. DE VRIES
Dept. of Forest Management
Wageningen Agricultural University
"Hinkeloord"
Gen. Foulkesweg 64
P.O.B.432
NL-6700 AH Wageningen
ISBN-13: 978-3-540-17066-2 e-ISBN-13: 978-3-642-71581-5
DOl: 10.1007/978-3-642-71581-5
This work IS subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of
the German Copyright Law where copies are made for other than private use a fee is payable to
"Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1986
The use of registered names, trademarks, etc. in this publication does not imply, even in the
absence of a specific statement, that such names are exempt from the relevant protective laws and
regulations and therefore free for general use.
Product Liability: The publisher can give no guarantee for information about drug-dosage and
application thereof contained in this book. In every individual case the respective user must check
its accuracy by consulting other pharmaceutical literature.
213113130-543210
PREFACE
Forest inventory may be defined as the technique of collecting, evaluating
and presenting specified information on forest areas. Because of the generally
la~ge extent of forest areas, data are usually collected by sampling, i.e. by
making observations on only part of the area of interest. As there are many
different sampling methods (e.g. Appendix 1), a choice must first be made as
to which method suits the given field and financial circumstances best. On
completion of the sampling procedure, the numerous data collected have next to
be condensed to manageable representative quantities. Finally, from these
quantities, inferences about the situation in the entire forest area are made,
preferably accompanied by an indication of their reliability.
This book is intended for students who want to know the whepefope of the
sampling techniques used in forest inventory. The danger of lack of knowledge
is a blind following of instructions and copying statistical formulae, or,
even worse, feeding data into a computer loaded with a program that is said to
print out the required information. In serious persons, such approaches may
leave a feeling of dissatisfaction or even of professional incompetence, be
cause of inability to direct or evaluate the procedure critically.
If a student tries to improve his or her situation, he/she will find that the
few existing forest inventory textbooks, though some with merit, either use
confusing statistical symbols or do not adequately cover theoretical principles.
As a result, complex formulae may drop out of a blue sky, and the student is
discouraged from trying to work out their principle and origin. On the other
hand, there are a number of excellent general textbooks on sampling theory,
though the lucidity of their symbolism may differ. The statistical sophistica
tion required of the reader, however, is often too high for even a graduate
forestry student, who experiences the gaps in his knowledge as unexplained
"jumps" in the statistical text.
The present book is an effort to evade the above drawbacks, by going step
by step, giving ample proofs in the text, and using a symbolism as clear as
possible. Moreover, the appendixes review many elementary statistical concepts
concisely and include some indispensable general statistical proofs. Numerical
examples are worked out with restricted simple data for each sampling method,
in order to illustrate the type of calculations involved.
All this, however, should not lead to the false impression that the student
VI
could manage without any previous mathematical and statistical knowledge. Apart
from being conversant with ordinary algebra, he should know the principles of
calculus (differentiation and integration), and he should have taken a course
in elementary statistics. Those acquainted with the principles of vectors and
vector spaces (linear algebra) will appreciate the theory of stochastic vectors
(Appendix 4), by which otherwise tedious proofs can be expressed elegantly.
Moreover, stochastic vector theory gives a clear insight into concepts such as
"degrees of freedom" and "analysis of variance".
Though this text can be used in regular courses, its primary purpose is for
self-teaching. The "normally intelligent" student should then realize that
relevant statistical knowledge cannot be assimilated in a hurry. He should take
his time (a year, say), during which the new matter can sink in. Further, he
should know that no part of the text, even the appendixes, can be skipped, and
that every phrase has its meaning and purpose. Just to complete or brush up his
statistical knowledge, the budding student should first read Appendixes 2 and 3
before starting with Chapter I. Then study Appendix 5, and (if you know a bit
about vectors), Appendix 4.
If the reader tackles the subject matter with this advice in mind, I guar
antee that he will experience the satisfaction of mastering some sampling
methQds widely used in forest inventory, that he will be able to read critically
more professional literature than before, and that he will possess a sound basis
on which to extend" his knowledge of sampling.
Utmost care has been taken to avoid typographic errors and errors of calcu
lation. But to be human is to be imperfect, so errors will remain. I will be
grateful for any suggestion for improvement.
ACKNOWLEDGEMENT
I am greatly indebted to Dr A.C. van Eijnsbergen, Department of Mathematics,
Agricultural University, Wageningen, for expert advice. Without his construc
tive criticism and encouragement, this book would never have been completed.
For the opinions expressed, I assume full responsibility.
Wageningen, The Netherlands Pieter G. de Vries
February 1986
VII
CONTENTS
CHAPTER PAGE
SIMPLE RANDOM SAMPLING WITHOUT REPLACEMENT
1.1 Introduction ••••••••• 1
1.2 Expected Value. Estimators for Population Mean and Total 4
1.3 Population and Sample Variance •••••••• 7
1.4 Variances of Estimated Population Mean and Total 10
1.5 Confidence Interval and Confidence Statement 13
1.6 Estimation of Proportions ••• 18
1.7 Required Sample Size ••••• 20
1.8 Some General Remarks on Sample Plots 25
1.9 Numerical Examples. 29
2 STRATIFIED RANDOM SAMPLING 31
2.1 Introduction ••••• 31
2.2 Unbiased Estimators for Population Mean and Total. Variances 33
2.3 Some Special Cases ••••••••• 37
2.4 Optimization of the Sampling Scheme •••••••• 39
2.5 Confidence Intervals. Behrens-Fisher Problem ••• 41
2.6 Gain in Precision Relative to Simple Random Sampling 44
2.7 Numerical Examples ••••••••• 46
3 RATIO ESTIMATORS IN SIMPLE RANDOM SAMPLING 56
3.1 Introduction. Population Ratio. Ratio Estimators for Total and Mean 56
3.2 Variances. • • • • • • • • • • • • • • • • 59
3.3 Confidence Interval. Precision versus SRS. Required Sample Size. 62
3.4 Bias of the Ratio Estimator. • • • • • • • 63
3.5 Ratio Estimator per Species Group in Mixed Forest. • 67
3.6 Numerical Example. • • • • • • • • • • • 69
3.7 Combining Results of Different Samples to Obtain New Information 72
4 RATIO ESTIMATORS IN STRATIFIED RANDOM SAMPLING 75
4.1 Introduction •••••••• 75
4.2 The Separate Ratio Estimator 75
4.3 The Combined Ratio Estimator 78
4.4 Illustrations ••• 81
4.5 Numerical Example. 85
5 REGRESSION ESTIMATOR • 88
5.1 Introduction • • • • 88
5.2 Unbiased Estimator of Population Regression Line from Sample Data. 90
5.3 Linear Regression Estimator and its Variance • • • 94
5.4 Regression Estimator in Stratified Random Sampling 101
5.5 Numerical Example ••••••••••••••••• 101
VIII
6 TWO-PHASE SAMPLING or DOUBLE SAMPLING • • • 104
6.1 Introduction. • • • . • . . • • . • • • 104
6.2 The Ratio Estimator in Double Sampling. 105
6.2.1 Ratio Estimator in Double Sampling - Dependent Phases 105
6.2.2 Ratio Estimator in Double Sampling - Independent Phases 108
6.3 The Regression Estimator in Double Sampling. ....••. 109
6.3.1 Regression Estimator in Double Sampling - Independent Phases. 109
6.3.2 Regression Estimator in Double Sampling - Dependent Phases .• 114
6.3.3 Numerical Example - Dependent Phases.. • •.••..•.• 117
6.4 Optimization in Double Sampling with Ratio and Regression Estimators 121
6.5 Double Sampling for Stratification. . : .•.••• ' ••• 122
6.5.1 Introduction ••••••.•••.••••••••• 122
6.5.2 Unbiased Estimator for Population Mean. Variance Expression 123
6.5.3 Variance Estimator .••••.••• 127
6.5.4 Optimization of the Sampling Scheme •..•.••••••• 129
6.5.5 Numerical Example • • • • • • • . • • • • . • • • • • • • • 131
6.6 Correction for Misinterpretation in Estimating Stratum Proportions
from Aerial Photographs • • • 132
6.6.1 Derivation of Formulas ••. 132
6.6.2 Numerical Example ••••• 135
6.7 Volume Estimation with Correction for Misinterpretation 137
6.7.1 Derivation of Formulas. 1.37
6.7.2 Numerical Example ••••. 139
7 CONTINUOUS FOREST INVENTORY WITH PARTIAL REPLACEMENT OF SAMPLE PLOTS. 141
7.1 Introduction. • • • • • • • • • • • • . • • . • . • • 141
7.2 Definition of Symbols. • • • • • • • • • • • • . • • • • • • 142
7.3 Most Precise Unbiased Linear Estimator for Population Mean on the
Second Occasion • • • • • • • • • • • • • • • 143
7.4 Optimization of Sampling for Current Estimate 150
7.5 Estimation of Change (Growth or Drain). 153
7.t A Compromise Sampling Scheme. 156
7.7 Numerical Example • • • • • . • 158
8 SINGLE- AND MORE-STAGE CLUSTER SAMPLING 161
8.1 Introduction. • • . • • • • • • • 161
8.2 Estimators in Two-Stage Sampling. • 168
8.2.1 Definition of Symbols. • • • 168
8.2.2 Unbiased Estimators for Population Total and Mean per SUo 169
8.2.3 Unbiased Estimators in Special Cases. 171
8.2.3.1 Single-Stage Cluster Sampling. 171
8.2.3.2 Primary Units of Equal Size. • 172
8.2.3.3 Equal Within-Cluster Variances. 172
8.2.3.4 Relation to Stratified Random Sampling. 173
8.2.4 Ratio Estimator for Population Total and Mean per SUo 173
8.3 Optimization of the Two-Stage Sampling Scheme 175
8.4 Three- and More-Stage Sampling. • • • • 178
8.5 Numerical Example of Two-Stage Sampling • • • 182
IX
9 SINGLE-STAGE CLUSTER SAMPLING AS A RESEARCH TOOL 183
9.1. Introduction ••••••••••••• 183
9.2. Intracluster Correlation Coefficient. 183
9.3. Variance and Intracluster Correlation 185
9.4. Measures of Heterogeneity ••••• 190
9.4.1. The Intracluster Correlation Coefficient 190
9.4.2. The C-Index ••••••• 192
9.4.3. The Index of Dispersion •••••••• 192
9.4.4. Numerical Example ••••••••••• 194
9.5. Intracluster Correlation Coefficient in Terms of Anova Quantities 197
9.6. About the Optimum Sample Plot Size •• 198
10 AREA ESTIMATION WITH SYSTEMATIC DOT GRIDS. 204
10.1. Random Sampling with n Points ••• 204
10.2. Systematic Sampling with n Points. 206
10.3. Numerical Example ••• 210
II SAMPLING WITH CIRCULAR PLOTS 212
11.1. Sampling from a Fixed Grid of Squares. 212
11.2. Sampling from a Population of Fixed Circles. 213
11.3. Sampling with Floating Circular Plots. 215
11.4. Comparison of Variances. 218
12 POINT SAMPLING • 223
12.1. General Estimator •• 223
12.2. Specific Estimators. 227
12.3. Variances •••••• 228
12.4. Sampling Near the Stand Margin 231
12.5. Required Sample Size. Choice of K. Questionable Trees. 234
12.6. Numerical Example ••••••••• 237
12.7. A More General View at PPS-Sampling, wtr 238
13 LINE INTERSECT SAMPLING ••• 242
13.1. Introduction. • • • • 242
13.2. BUFFON's Needle Problem and Related Cases 244
13.3. Total-Estimator Based on One-line Data. 249
13.4. Variance in Case of One-Line Data • • • 251
13.5. Sampling with More Than One Line •••• 253
13.6. Required Number and Length of Transects 256
13.7. Estimating Properties of Residual Logs in Exploited Areas 258
13.8. Estimators Based on Circular Elements • • • • • • • • • 262
13.8.1. Generalization of STRAND's Estimator •••••• 262
13.8.2. Density Estimation of Mobile Animal Populations 263
13.8.3. Biomass Estimation in Arid Regions. 264
13.9. Bias in Oriented Needle Populations • • • • 266
13.10. Generalization of LIS Theory •••••••• 269
13.10.1. KENDALL Projection and Expected Number of Intersections269
13.10.2. General LIS Estimator and its Variance 273
13.10.3. Applications ••• 275
13.11. Line Intersect Subsampling. • • • • • • • • • • 276
x
14 LIST SAMPLING ••••••••••••••••• 280
14. I. Introduction . • • • • • • • . • . • • . 280
14.2. Estimation of Population Total. Variance 282
14.3. Optimum Measure of Size. Comparison with Simple Random Sampling. 284
14.4. Numerical Example. • . • 287
14.5. Two-Stage List Sampling. 292
15 3-P SAMPLING •••• 296
15.1. Introduction. 296
15.2. The Principle of 3-P Sampling. 297
15.3. Variance and Expected Value of Sample Size and its Inverse 301
15.4. Considerations about the Sample Size 302
15.5. GROSENBAUGH's 3-P Estimators 307
15.6. Summary and Conclusions •• 311
15.7. Numerical Example. 313
15.8. List of Equivalent Symbols 317
APPENDIX
I. A Family of Sampling Schemes • • • • • 319
2. Permutations, Variations, Combinations 320
3. Stochastic Variables • • • • • • • • • 322
3.1. Stochastic Variables in General. Normal and Standard Norm. Variable 322
3.2. The Chi-Suare Distribution. 325
3.3. STUDENT's t-Distribution. 326
3.4. FISHER's F-Distribution • 327
4. Stochastic Vectors and Some of their Applications 328
Introduction 2 328
Appl.l. Distribution of the Sample Variance s in Simple Random Sampling 331
Appl.2. Distribution of the Pooled Variance in Stratified R.S. 333
Appl.3. Analysis of Variance in Stratified Random Sampling • . 335
Appl.4. Analysis of Variance in 2-Stage Sampling • • • • • • • 336
Appl.5. Proof of STEIN's Method for Estimating Required Sample size. 339
5. Covariance, Correlation, Regression. • • • • • • • • • • • 341
6. The LAGRANGE MUltiplier Method of Optimization • • . • . . 348
7. Expected Value and Variance in Multivariate Distributions. 351
8. Hypergeometric, Multinomial and Binomial Distributions • • 357
9. The Most Precise Unbiased Linear Estimator of a Parameter X, based on
a Number of Independent Unbiased Estimates of Different Precision. 362
10. Variance Formulas for Sums, Differences, Products and Ratios 367
II. The Random Forest (POISSON FOREST) • • • • • • • 371
12. Derivation of the Identity used in List Sampling 372
13. Expanding a Function in a TAYLOR Series. 373
14'. About Double Sums. 376
15. Exercises. 378
REFERENCES 393
INDEX. • • 395
CHAPTER 1
SIMPLE RANDOM SAMPLING WITHOUT REPLACEMENT
1.1. Introduction
We will consider a population consisting of N elements (named population
elements or sampling elements), numbered i I, ••• ,N, from which a sample of
size n is drawn. The sample is a subset of size n from N, i.e. the order in
which the elements occur in the sample is irrelevant, and "doubles" are not
allowed to occur.
Examples: a sample of n trees from a stand containing N trees; a sample of n
squares from a stand area divided in N equally-large squares; a sample of n
subareas from a stand area divided in N subareas of different sizes; a sample
of n stands from a forest area consisting of N stands; a sample of n days from
a period of N days.
In practice the elements for the sample generally are selected suaaessively
from the population. This implies that an ordered sample is obtained; subse
quently however, this order is ignored.
Once drawn, an element is not returned to the population, i.e. the drawing
is without replaaement (wtr), in order to avoid "doubles".
The elements for the sample (sample elements) seldom are drawn directly from
the population, as in doing so the selection may be influenced by their location
in the population, the sampler's personal preference, and so on. Generally, in
drawing a sample either use is made of a table of random numbers, or the N sam
pling elements are substituted e.g. by N numbered marbles or tickets.
=
To execute the sampling procedure, we will use N marbles, numbered i I, ••
• ,N , which are thoroughly mixed in an urn. From the urn, n marbles are selected
in succession and without replacement. Then the sample consists of the n popula
tion elements that bear the same number as the marbles selected.
The selection is random. That is: each of the N marbles has equal probability
(viz. lIN) of being the first one to be selected. Once selected, the first sam
ple element is not returned to the urn. So each of the remaining N-I marbles
has equal probability (viz. I/(N-I» to be selected as the second sample element,
and so on. Finally, when we come to selecting the n-th sample element, each of
the remaining N-(n-I) marbles in the urn has equal probability of being selec
ted, viz. I/(N-n+l) • So the probability of a sample that consists of a specific
set of numbers (population elements) arranged in the order in which they were
selected, is: