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On the Estimation of the Distribution of Sample
Means Based on Non-Stationary Spatial Data
Magnus Ekstrom
Yuri Belyaev
Arbetsrapport 89 2001
Working Paper 89 2001
SW EDISH UNIVERSITY OF
ISSN 1401- 1204
AGRICULTURAL SCIENCES (SLU)
ISRN SLU- SRG- AR- - 89 -- SE
Department of Forest Resource
Management and Geomatics
S- 901 83
UMEA
Phone: 090- 786 58 25
Fax: 090- 14 19 15, 77 81 16
On the Estimation of the Distribution of Sample
Means Based on Non-Stationary Spatial Data
MAGNUS EKSTR O M and YURI BELYAEV
Swedish University of Agricultural Sciences
ABSTRACT. Two different methods of estimating the distribution of sample
means based on non-stationary spatially indexed data
a finite subset of the integer lattice
71},
are presented.
{Xi
:
i
E
I},
where
I
is
The information in the
different cells in the lattice are allowed to come from different distributions, but
with the same expected value or with expected values that can be decomposed
additively into directional components.
Furthermore, neighboring lattice cells
are assumed to be dependent, and the dependence structure is allowed to differ
over the lattice.
It will be shown under such rather general conditions that
the distribution of the sample mean can be estimated by resampling, as well
as by a normal approximation for which a non-parametric estimator of variance
is provided.
The developed methods can be applied in assessing accuracy of
statistical inference for spatial data.
Key words: resampling, m-dependent random variables, estimating distributions, spatial data
on integer lattices.
1.
Introduction
It is well known that Efron's (1979) bootstrap, based on independent and identically
distributed ( i.i.d. ) observations, provides good estimators in nonparametric statistical
analysis. However, Efron's bootstrap fails if the observations are not independent (cf.
Singh, 1981, Remark 2.1). For making the bootstrap suitable for dependent observa­
tions, various block resampling methods have been proposed. Instead of drawing one
sample observation at a time, as in Efron's bootstrap, a block of sample observations is
drawn where the size of the block should increase at some rate as the sample size gets
larger. In the present paper, some block resampling methods for spatially dependent
data are proposed. The specific use of blocks for such data was first studied by Hall
(1985). Other contributions have been made by Hall (1988), Garcia-Soidan & Hall
(1997), and Lahiri et al. (1999), among others. See also Chapter 5 in Politis et al.
(1999), and the references therein.
Block resampling methods, like those suggested in the articles mentioned above,
have been proposed to handle stationary spatial data. In this paper methods of es­
timating the distribution of sample means based on non-stationary spatial data are
proposed.
Assume that we have some spatially indexed data, i.e. {Xi : i E I}, where I is
a finite subset of the integer lattice 71}. Remote sensing data from satellites are, for
1
example, of this form. For simplicity, the case when I is rectangular will be consid­
ered, but extensions to non-rectangular subsets of 7!}, that possess some regularity, are
possible. The kind of data we consider is of the following type: The values registered
at different lattice cells are allowed to come from different distributions, and the de­
pendence structure is allowed to differ over the lattice. We assume that all observed
values are from distributions with the same expected value, or with expected values
that can be decomposed additively into directional components. Furthermore, it will
be assumed that observations separated by a certain distance are independent. The
distribution of the sample mean under such conditions does not necessarily converge
to a limit distribution. However, it will be shown that the sequence of distributions of
sample means weakly approaches a sequence of normal distributions, for which Ekstrom
(2001) provided consistent non-parametric variance estimators.
The concept of weakly approaching sequences was introduced by Belyaev (1995),
and further developed by Belyaev & Sjostedt-de Luna (2000). It is a natural general­
ization of weak convergence, without the need of a limit distribution.
A block resampling scheme is also suggested in the present paper, and it is shown
that the proposed sequence of resampling estimators weakly approaches the sequence
of distributions of sample means, in probability. The type of estimators considered
in this paper has been studied earlier by Belyaev (1996), who considered triangular
arrays of row-wise finitely dependent r.v.s, and proposed blockwise resampling schemes
for estimating distributions of sums of r.v.s. Lemma 1-2 and Theorem 1-3 in the next
section extend and modify results of Belyaev (1996) to spatial lattice data.
This paper is organized as follows: In the next section the estimators are introduced,
and the obtained theoretical properties are presented. In Section 3, a simulation study
is carried out in order to compare the different estimators. In Section 4 an approach,
developed by the second author, for estimation of accuracy of discretely colored maps
is considered. Proofs of lemmata are given in the Appendix.
We use the following notation: All sets are shown by calligraphic letters A, and
IAI is the number of elements in A; r.v.s are denoted by capital letters; point in index
position is used for sums, e.g. X .. = L:i L:j Xij; bold letters are used for vectors and
lists; £(Z ) is the distribution (probability measure) of a r.v. Z; Fz(-) and Fz(- I U)
are the distribution function of Z and the regular conditional distribution function
of Z given the r.v. U, respectively; N(fJ,, a2) is the normal distribution with mean /-l
and variance a2; a 1\ b = min{a,b}; a V b = max{a,b}; an;::::: bn is used for sequences
{an}n�l,{bn}n�l, for which there exist two constants 0 < c1 < c2 < oo such that
c1 :S an/bn :S c2, n = 1, 2, .... We use � for convergence in probability and � for
weak convergence. Two sequences of r.v.s {U�}n�l and {U�}n�l have weakly approach­
ing distribution laws { £(U�)}n�l, { £(U�)}n�l, if for any bounded continuous function
j(-), E[j(U�)]- E[f(U�)]---+ 0 as n---+ oo. Then we write £(U�) A £(U�), n---+ oo.
Let {U�, Zn}n�l be a sequence of pairs of r.v.s and {£(U� I Zn)}n�l be the sequence of
v;.a(P)
regular conditional distribution laws of u� given Zn. We write £(U� I Zn)
£(U�)
if E[f(U�) I Zn]-E[f(U�)] � 0, n---+ oo, for any bounded continuous function f(·).
2
2. Results
Assume we have spatially indexed data Xn = {Xi: i E In}, where In={i = {i1, i2}:
i1 =1, ..., n1, and i2 =1, ..., n2}, n = {n1, n2}. Observations separated by a certain
distance will be assumed to be independent, as formalized in the following definition.
Definition 1
The r.v.s Xi, i E In, are said to be spatially m-dependent if Xi' and Xi", i', i"
are independent whenever li�- i�l > m1 or li;- i�l > m2, m={ml , m2}.
E
In,
Remark. The results in this paper hold for more general data than indicated, i.e.,
the results are valid for arrays Xn= {Xi,n : i E In}, n1, n2=1, 2, ... , of collections of
r.v.s Xi,n, such that for each n, the r.v.s in Xn are m-dependent. To keep the notation
manageable, we will write Xi rather than Xi,n
·
Define m1,2=m1m2 and n1,2= n1n2. Also, let X .. = L:iEin Xi, X.. =X.. jn1,2, and
!n=n1,2Var[X..] . We are interested in the expected value p.. = E[X ..] . X .. is a point
estimator of p.. , and in order to do inference, e.g., confidence intervals, we need to
estimate the distribution law £(�(X .. -p..)). Below, we suggest different methods
for estimating this distribution. We introduce the following assumptions:
AM: For all
n,
E [Xi] = 0, i E In.
AD ( m): For all
AL ( 5): For all
n,
n,
the r.v.s Xi, i E In, are spatially m-dependent.
and some positive constants 5::;2 and T0, E[ ! Xi i2H]
, i E in
<T0
·
By AM and AD ( m), we have
(
L
m1i\(n1-il)
E [X i] + L 2E [XiXi1+h1h ]
l,2 iEin
h1=l
-i
-i
m1i\(n1 1) m2A(n2 2)
+
L
L 2E [XiXil+hl,i2+h2]
hl=l
h2=1
m1i\(n1-i1) m2A(i2-l)
+
L
L 2E [XiXil+hlh-h2 ] .
=
hl l
h2=1
1
In = --;;;:--
)
3
+
m2A(n2-i2)
L
h2=l
2E [XiXi1h+h2 ]
(1)
Theorem 1
If AM, AD ( m) , and AL ( 6 ) are valid, then
£(yfn1,2X .. ) A N(O,In), as n1,n2-+ oo,
i.e., the sequences of distrib ution laws {£(yfn1,2X..)}n1,n22:1 and {N(O, ln)}n1,n22:1 weakly
approach each other as n1,n2 -+ oo.
Remark. If in Theorem 1, In-+ 1 > 0, as n1,n2 -+ oo, then yfn1,2X .. converges
weakly to N(O, 1). This weak convergence also follows from more general results, e.g.,
from Theorem 6.1.2 in Lin & Lu (1996) on a-mixing random fields.
Let
52j-l,i = {(j -1)ki + (j - 1)mi + 1,...,(j ki + (j -1)mi) 1\ ni} ' i = 1, 2,
52j,i = {j ki + (j - 1)mi + 1,...,(j ki + j mi) 1\ ni} ' i = 1,2,
and define the following rectangular blocks of indices,
Tj(1) = {i: i1E 82)!-1,1 and i2E 82j2-1,2} , j E J (1) = {(1,1),...,( sl, s2)},
Tj(2) = {i: i1E 82j1-1,1 and i2E 8212,2} , j E J (2) = {(1,1),...,( s1,t2)},
Tj(3) = {i: i1E 52)1,1 and i2E 8212-1,2} , j E J(3) = {(1,1),...,(t1, s2)},
Tj(4) = {i: i1E 82j1,1 and i2E 8212,2} , j E J(4) = {(1,1),...,(t1,t2)},
where kh > mh, and sh, th, h = 1,2, are the largest integers such that,
kh + mh + nh-1 and th mh + nh-1 Sh-th _ 1.
sh<
<
<
- ---- k h + mh '
kh + mh ---l
. X-.. = L4 Xn
We can wnte
l=l ( ), where
Lemma 1
Under the assumptions of Theorem
1,
and if kh = o(nh) as
(i)
and
4
(ii)
kh,nh-+ oo, h = 1, 2,
,.,;r(n1)- ""rn-+ 0 '
(2)
Lemma 2 (Ekstrom, 2001)
Assume, for all n and i, that ai = ai( n) has an a b solute value less than or equal to 1.
Then, under the assumptions of Theorem 1, for some constant 77 2::1 and b h-ah 2:: mh,
h =1, 2,
[ t+l a,x, ]
[ E1 , �+1 a,X, ]
2+J
(i)
(ii)
E
E
,,
:S
T;'q(8rn,.(b"-"'")) I+J/2,
'+'
,
,
:S
h
= 1, 2,
Tory(6 4m1,2(b1-a1)(b2-a2))1+ 0 /2,
Proof of Theorem 1. The desired result is proved by Belyaev & Sjostedt (1996) in
the case when the r.v.s Xi, i E In, are independent. That is, if AM, AD(O) , and
AL(o) are valid, then
(3)
Without loss of generality we assume that m1,m2 2:: 1. By (1), the Cauchy-Schwarz,
and the Lyapunov inequalities, respectively, one can verify that
(4)
and therefore the collection of r.v.s {J7i1,2X ..}n1,n2;:::1 is tight. If the difference of ele­
ments of two random collections tends to zero in probability, and if one of the collec­
tions is tight, then the other collection is also tight and the collections weakly approach
each other in distribution (Lemma 7, Belyaev & Sjostedt-de Luna, 2000) . Thus, by
1
Lemma 1(i), it is enough to show that .C(J7i1,2X � )) A N(O, In), as kh/nh---+ 0 and
kh,nh---+ oo, h = 1, 2.
Consider the r.v.s,
By their construction, they are independent and have expectation zero. Further, by
Lemma 2(ii) and the upper bound (2) for sh, for any nh > kh V mh, h = 1, 2,
Hence, the r.v.s xy), j E .:J(1), satisfy assumptions AM, AD(O) , and AL(o) (with T5
replaced by T5TJ(576m1,2)1+512). Thus, by (3), with xjl) instead of xi, sh instead of nh,
5
.
-1
�
1
-1
.
and by notmg that vn02Xn( ) = L...J jE .J(lJ Xi( )/ -JSi82, 1t follows that £( vn02Xn( )) +---t
N( 0, In(1)), as s1, s2-+ oo.
Observe that inequality ( 4) implies that the collection of normal distributions
{N(O,ln)}n is tight as n1,n2 -+ oo. Two random collections, of which one is tight,
weakly approach each other in distribution if and only if the difference of their charac­
teristic functions tend to zero (Theorem 1, Belyaev & Sjostedt-de Luna, 2000) . From
Lemma 1(ii), ��) - In -+ 0, and so N(O, ��)) A N(O, In), as kh/nh -+ 0 and
kh,nh-+ oo, h = 1, 2. Thus, we have
£(Jn1,2X..) A £(Jn1,2X�1l) A N(O, ��1l) A N(O, In),
as kh/nh-+ 0 and kh,nh -+ oo, h = 1, 2, and this completes the proof.
wa
D
For making use of Theorem 1, we need to know or estimate the variance In· Define
rectangular blocks of indices
B;
=
{
(5)
and let
(6 )
Xj = 2)Xi- X ..) Ii,
i EBj
where Ii is equal to 1 if i E In, and zero otherwise . In Ekstrom (2001) the following
estimator of In is proposed:
(7)
where k1,2 = k1k2 = !BjI is the block size, and Jn = {j = {j1,j 2} :
2- k2, ...,n2}.
We make the following assumption on k1 and k2:
J1 = 2-k1, ...,n1, and J2 =
= 2, then kh = o(nh) as kh,nh -+ oo, h = 1, 2. If 0 < 5 < 2, then
(kl/k2)((k1k2/(n1n2)) log k2)15, (k2/k1)((k1k2/(n1n2)) log k1)15, and (kh/nh) log kh, h =
1, 2, all tend to zero as kh,nh-+ oo, h = 1, 2.
Remark. If n1 ::=:::: n2, k1 ::=:::: k2, and k1n11logk1 -+ 0, as min{n1,n2,k1,k2} -+ oo,
AK(5) : If 5
then AK(5) holds for any 0
<
5 S 2.
Theorem 2 (Ekstrom, 2001)
If AM, AD(m) , AL(5) , and AK(5) are valid, then
6
Remark. Alternatively, the estimators of variance suggested by Politis & Romano
(1993) and Sherman (1996), which are consistent under the assumptions of Theorem
2, can be used. Further, if 6 =2, then for some c 1 > 0, E[(i'n - In?] ::::; c1(k 1, 2nl,� +
k1 2 + k2 2). Thus, ifn1 ::=:: n2 and kh ::=:: y'nh, h = 1,2, then yn1,2E[(i'n-ln?J = 0(1).
See Ekstrom (2001).
By Theorem 1 and 2, N(O,i'n) can be used as an estimator of the distribution
£(yn1,2(X .. - [j..)). Alternatively, a different, possibly non-normal (except in the
asymptotics), approximation can be used, as will be seen in the next theorem. Let �k
denote the collection of blocks { Bj : j E Jn}, and let b1, 2 = ( n1 + k1- 1)( n2 + k2- 1)
denote the total number of blocks in �k· Draw randomly d1, 2 blocks with replacement
from �k and let Mj equal the number of times the block Bj appears in the set �Z of re­
sampled blocks. Hence, {Mj, j E Jn} follow a multinomial distribution and E*[Mj] =
d1, 2bl,�, E*[(Mj)2] = d1, 2bl,�(1- bl,� + d1, 2bl,�), and E*[Mj,Mj"] = d1, 2bl,�(-1 + d1 , 2),
j ' # j", where E* denotes the expectation over the resampling distribution.
Let
1/ 2
1_
b1, 2
_
'"" M':X?.
X*n =
�
3
3
n1, 2 k1, 2d1, 2
j E.:Jn
(
)
The distribution of X� depends both on the original randomness of Xn and on ran­
domness generated by resampligg from the collection of blocks �k· We will consider
the conditional distribution of X� given the data Xn.
From ( 6) we have that
(8)
j E.:Jn
and thus E*[X�IXn]
= 0.
Moreover, from (8) we see that
j E.:Jn
and so
b1, 2
n1, 2k1, 2d12
,
( d1,2 (1b1 , 2
)
)
_1_ + d1, 2 - d1, 2 (-1 + d 1 2) '"" (X?)2
J
'
�
b12, 2
b 1 , 2 b1 , 2
jE.:Jn
Thus, the conditional variance of
A .
= In
yn1,2X� given Xn approaches In as n1, n2 -7 oo.
7
Theorem 3
Under the assumptions of Theorem
2,
and if d1,2----+ oo as
a(P)-'--+ £(y'n1,2X ..),
£(y'n1,2X- n1* Xn) wf---
as
n1,n2 ----+ oo,
n1,n2----+ oo,
i.e., the sequence of conditional distribution laws {£( yn1,2X �
proaches {£(yn1,2X ..) }n1,n22:1, in probability, as n1,n2 ----+ oo.
IX,. )}n1,n22:1 weakly ap­
Remark. From Theorem 3 above, together with Lemma 9 in Belyaev & Sjostedt-de
Luna (2000) , it follows that supx IF
(xi:X,.)- Fyfn1,2X ..(x)l � 0, as n1,n2 ----+ oo.
Let
W
yfn1,2X�
(z) = ez - 1- z-z2 /2.
Lemma 3
Under the assumptions of Theorem
d
R n = � "" \lf
b1,2 j�
E.:ln
(
(
'tX'?
'L
J
3,
)
1/2
b1 '2
n1,2k1,2d1,2
)
Proof of Theorem 3. Without loss of generality, we assume that m1 /\ m2 :2: 1. Note
that n1 2(k1 2d1 2/b1 2)112X� is a sum of d1 2 i.i.d. r.v.s Xf*, ...,X�* , where Xf* takes
the values Xj, j E Jn, with probability b;;}n2 on each value. By this fact and (8), the
_
conditional characteristic function tp�J
of X� given the data
can be written
as
'
'
'
'
'
nl nz
IXn)
tp�(ti:X,.) = E* [eity!n1,2X;*.IXnJ =
X,.
( L f--1,2 (
exp itXj
jE.:Jn
By Theorem 2 and Lemma 3, for any given t
)
1/2
1 '2
n1,2 1,2d1,2
( �
) ) d1,2
E JR,
( 9)
where exp(-t2 1n/ 2) is the characteristic function of the normal distribution N(O, In).
Inequality (4) implies that {N(O,In)}n is tight. Hence, by Theorem 2 in Belyaev &
IXn ��
Sjostedt-de Luna (2000) , £( yn1,2X�
) (a( N(O, In), as n1,n2 ----+ oo.
Denote the characteristic function of yn1,2X.. by tpn( ) By Theorem 1 in this paper
and by Theorem 1 in Belyaev & Sjostedt-de Luna (2000) , tpn(t) - exp(-t2 1n/2) ----+ 0,
·
8
.
for any given t E JR. From this result, together with ( 9), we get that
rp�(tJXn,)- rpn(t) � 0, for any given t, as n1, n2-+ oo. Hence, Theorem 2 in Belyaev
D
& Sjostedt-de Luna (2000) implies the desired result.
as n1, n2 -+
oo,
Similar results can be obtained under the following more general assumption on the
first moment:
AM': For all
n,
E[Xi]=f.L, iEin, where
11
may depend on
n.
Corollary 1
If AM', AD(m) , and AL(o) are valid, then
(i)
£(yn1,2(X.. - M)) A N(O, In), as n1, n2-+
oo.
If, in addition, AK ( 6) is valid, then
""" )
( """
�
wa(P
*I""
n1, 2x n
n1, 2 (x .. - f1 )), as n1, n2-+
'-' ( y;;:n-:-::
.mn ) f---)----'--t '-' ( y;;:n-:-::
{'
{'
oo.
Proof. Note that the r.v .s Xf =Xi- f.L, i E In, satisfy the assumptions of Theorem
1, and that X?.=n l,� L.:iEin Xf =X ..- f.L· This implies that £(yff1'0(X ..- M)) A
N(O,In), as n1, n2-+ oo, and thus (i) is proved. Since
we see that neither :Yn nor X� depend on M· Hence, Theorem 2 and 3 implies (ii) and
D
(iii), respectively.
Below we consider a case when observed r.v.s do not have a constant mean.
AM": For all n, we have Yi = f.Li +Xi, where Xi, i E In, satisfy assumption AM.
f.Li decomposes additively into directional components, i.e., f.Li=f1 +ri2 +Ci1, where f1
is the overall mean, ri2, i2 =1, ... , n2, are the row effects, and ci1, i1 =l, ... , n1, are the
column effects. All effects, f.L, ri2, i2 =1, ... , n2, and ci1, i1 =1, ... , n1, may depend on n.
It should be noted that the row and column effects are defined as deviations from the
2
1
overall mean so that L'.:� = 1 ri2 =0 and L'.:� =1 Ci1 =0.
9
Theorem 4
If AM", AD(m) , and AL ( 5) are valid, then
£(yfn1,2(Y.. - M)) A N(O, In), as n1, n2 -+
oo.
The result is an immediate consequence of Theorem 1 and the fact that
- Proof.
D
X . . =Y.. -tJ.
The ordinary-least-squares estimators of the effects are
Thus, we estimate Mi with fli = fl +fi2 +cii, and we can define residuals, ei = Yi - Jli,
i E In.
Next we want to estimate the variance ln=n1,2 Var[Y.. ]=n1,2 Var[X ..]. It should
be noticed that the r.v.s Xi, i E In, are not observable, and that we therefore cannot
use the "old" estimator in of the variance In· Further, we cannot replace the Xis with
the }is in the formula for in, since the varying mean values of the }is will then ruin
the estimate of the variance In· We can, however, replace the Xis with the residuals,
and so we obtain the following estimator of variance:
Theorem 5 (Ekstrom, 2001)
If AM", AD(m) , AL(o) , and AK(o) are valid, then
Remark. If 5=2, then for some c2 > 0, E(i�-In?:::; c2(k1,2n l,� +k12 +k22 +kin12 +
k�n22). Thus, if n1 :o::: n2 and kh :o::: vn:h, h=1, 2, then yfn1,2E(i�- In?= 0(1). See
Ekstrom (2001).
10
By Theorems 4 and 5,
N(O, 'Y�) can be used as an estimator of the distribution of
-Jnl,2(Y.. - J.L). Alternatively, the following technique can be used. Let Yn = {Yi : i E
In}, and define
Yn* =
(
)
12
b1,2 /
_ 1_
"" M�Y?
3
�
J '
n1,2 k 1,2d 1,2
jE.:Tn
Theorem 6
Under the assumptions of Theorem
5,
and if d 1,2----+oo as n1, n2----+oo,
Lemma 4
Under the assumptions of Theorem 6,
d
Rln = _!.2_ L .T,
b1,2 jE.:Jn
'±'
(
(
,;ty?
"
3
b1,2
n1,2k 1,2d 1,2
) 1/2)
p
--+
0 , as n1, n2 ----+ oo.
Proof of Theorem 6. The proof is almost identical to the proof of Theorem 3. Only
some changes of notation are needed, and the references to Theorem 1 and 2, and
Lemma 3, should be changed to Theorem 4 and 5, and Lemma 4, respectively.
D
Remark. It is easily seen that the results in this paper hold also for rectangular index
sets I expanding in all directions, i.e. for I = { i : i1 =-n1, ... , n1 and i2 =-n2, ... , n2}.
Moreover, the assumption on the index set to be of rectangular shape can be relaxed
by using "subshapes" , as described in, for example, Sherman (1996).
3. A simulation study
In this Monte Carlo study, non-stationary spatially m-dependent data Xi, i E In, are
generated, where each Xi is a weighted average of independent and skewly distributed
r.v.s such that Xi has a small variance if both i1 and i2 are small, whereas the variance
of Xi is large when i1 and i2 are large. To be more specific, let mh = 2lh for some
integer lh 2: 0, h = 1, 2, and define weights w (i) = vi/ L_��=-h L.��=-b Vj , where vi =
((1 + li1l)(1 + li2l))-1, i1 = -h, ..., h, i2 =-l2, ... , Z2. Define
iz+b
L
w ( { li1 - J1l, li2 - J2l}) (Zj- E[Zj]) ,
11
where the r.v.s Zi are independent and log-normal with parameters e=E(log Zi)=0,
and
2
.
1 2:: Jh + lh
O'j= Var(logZi)=, for allJ.
2 h= nh + 2 lh + 1
V
l
For judging the performance of an estimate Fn, of the distribution Fn of y'n1,2(X .. JJ), we calculate the maximum absolute difference supx IFn(x)- Fn(x)l, where Fn will
be represented by the empirical distribution of 1 million replicates of y'n1,2(X .. - JJ).
Three different estimators of Fn will be considered:
NA: Normal approximation with the variance estimated by i'n;
R1: Resampling estimator y'nl,2X� with d1,2 = b1,2 (this choice of d1,2 corresponds
to the number of resampled blocks used by Belyaev (1996) & Sjostedt (2000) in
their resampling schemes for sequences of m-dependent r.v.s); and
R2: Resampling estimator y'nl,2X� with d1,2 = In1,2/k1,2l + 1, where Ix l denotes the
smallest integer greater than or equal to x.
In our study, 250 samples of Xn, are generated for each choice of rectangular shapes of
the index and blocks sets, N=n1 x n2, K=k1 x k2, and m = { m1, m2 } . The maximum
absolute differences for each of the three estimators of Fn are calculated. In Table
1-7, the mean (Mean) and the standard deviation (StDev) of the maximum absolute
differences are presented for each estimator.
Block size
(K)
5X5
10 X 10
15 X 15
Table 1. N =25
Block size
(K)
5X5
10 X 10
15 X 15
Table 2. N =25
Block size
(K)
5X5
10 X 10
15 X 15
Table 3. N =50
NA
Mean
StDev
Mean
R1
StDev
Mean
0.029
0.037
0.046
0.079
0.089
0.108
StDev
Mean
0.039
0.044
0.052
0.104
0.106
0.113
0.059 0.031 0.078
0.065 0.040 0.085
0.079 0.048 0.097
x 25 and m = {1, 1}.
NA
Mean
StDev
Mean
R1
0.085 0.041 0.102
0.080 0.047 0.098
0.084 0.055 0.102
x 25 and m = {2, 2}.
NA
Mean
StDev
Mean
R1
0.045 0.022 0.067
0.037 0.025 0.061
0.046 0.032 0.070
x 50 and m = {1, 1}.
12
StDev
Mean
0.021
0.024
0.029
0.069
0.063
0.071
R2
StDev
0.030
0.038
0.043
R2
StDev
0.037
0.043
0.050
R2
StDev
0.021
0.023
0.031
Jl
Block size
5X5
(K)
NA
Mean
StDev
Mean
R1
StDev
Mean
0.025
0.027
0.031
0.097
0.074
0.079
0.078 0.025 0.098
0.049 0.029 0.070
15 X 15
0.053 0.034 0.075
Table 4. N =50 x50 and m = {2, 2}.
lOxlO
Block size
NA
(K) Mean StDev Mean
15 X 15
0.017 0.009 0.045
20 X 20
0.016 0.010 0.045
25 X 25
0.018 0.011 0.045
30 X 30
0.017 0.012 0.045
Table 5. N =250 x250 and m = {1, 1}.
Block size
NA
(K) Mean StDev Mean
15 X 15
0.025 0.012 0.050
20 X 20
0.021 0.012 0.046
25 X 25
0.020 0.013 0.047
30 X 30
0.020 0.013 0.048
Table 6. N =250 x250 and m = {2, 2}.
Block size
NA
(K) Mean StDev Mean
15 X 15
0.033 0.013 0.057
20 X 20
0.026 0.013 0.051
25 X 25
0.023 0.014 0.049
30 X 30
0.024 0.015 0.051
Table 7. N =250 x250 and m = {3, 3}.
R1
StDev
Mean
0.014
0.012
0.013
0.014
0.045
0.045
0.046
0.047
R1
StDev
Mean
0.015
0.013
0.015
0.016
0.052
0.048
0.047
0.049
R1
StDev
Mean
0.016
0.015
0.015
0.017
0.057
0.052
0.051
0.051
R2
StDev
0.025
0.027
0.032
R2
StDev
0.014
0.014
0.015
0.014
R2
StDev
0.014
0.016
0.014
0.016
R2
StDev
0.014
0.015
0.016
0.016
In Table 1-7 we see that the normal approximation performs �e best, even for rather
small values of N. Further, for the resampling estimator �X� there is no real gain
in choosing the number d1,2 of resampled blocks as large as b1,2; d1,2 = In1,2/k1,2l + 1
seems quite enough. To state guidelines on how to choose an optimal block size in a
given situation is a difficult task under the assumptions given in this paper, and will
not be discussed here. It is clear, however, that the block size should increase with n1
and n2. Also, the optimal block size depends on the strength of dependence, as seen
in the tables above.
The estimators N(O, ry�), £(�Y�) with d1,2 = b1,2, and £(�Y�) with d1,2 =
In1,2/k1,2l + 1, gave less satisfactory results in our simulations, and we do not recom­
mend these estimators unless n1 and n2 are larger than in the simulations in this paper.
An alternative to these estimators will be presented in a forthcoming paper.
13
4.
An application to assessing characteristics of accuracy of discretely col­
ored digital maps
The methods considered in this paper can be applied to assessing characteristics of
accuracy of discretely colored maps created by using remote sensing data ( Belyaev
2000a, b ) . For example, different types of landscapes can be shown in maps by using
different colors. Let JC = {1, 2, ... , q} be the number of colors used. A digital discretely
colored map is a collection of small colored squares called pixels. We consider the lattice
Z2 = { i = { il, i2}: il, i2 = 0, 1, 2, ... }, and the i-pixel is the square with the vertices
{ i1, i2}, { i1 + 1, i2}, { i1 + 1, i2 + 1}, { i1, i2 + 1}, whose color is coded by a digit Ci E JC. We
denote pixels as { i, ci}, i E Z2. The same area can be shown by maps having different
number of pixels. We consider a rectangular digital map 9Jtn = { { i, ci} : i E In}
where as above In = { i = { i1, i2} : i1 = 1, ... , n1, i2 = 1, ... , n2}. Suppose that all
pixels are correctly colored, i.e. 9Jtn is the true map with n1,2 = n1n2 pixels. Usually
it is impossible or too expensive to create the true maps. Instead remotely sensed
data §, e.g. data registered by satellite sensors, are used to obtain approximations
to the true maps. Suppose that some classification based on § is used in selecting
colors of pixels. Denote by ci the ( number of the ) color obtained after classifying a
part of the data si E § related to the i-pixel. Assume that the cross-classification
probabilities Pkz(s) = P(ci = l I Ci = k, si = s) are known. Here, Pkz(s) is the
conditional probability to select color ci = l given the true color ci = k and the related
remotely sensed data s = si. For sake of simplicity we assume that l E JC and square
matrices IP'( s) = (Pkt(s)), s E §, have full rank. Then there exist inverse matrices
IP'-1 (s) = (pkt(s)), s E §. The numbers Nk,z of pixels which have the true color k and
have been classified as having the color l , are important characteristics of accuracy of
the created map 9J1� = { { i, ci} : i E In}· By using the indicator functions we can
write Nk,1 as follows:
N;z = L I(ci = k)I(ci = l ).
(10 )
iE'In
Its mathematical expectation is
E[N;z] =
L
I(ci = k)Pkt(si)·
(11)
iE'In
Both values Nk,z and E[Nk,z] are not feasible. However, there exist the unbiased estima­
tors E[Nk,z]
(12)
fl;t = L f(ci = k)Pkz(si),
iE'In
where
k
I(ci = k) = "'""'
� p l (s)I(ci = l).
A
•
lEJC
14
(13)
We will consider digital maps with very large numbers of pixels n. For simplicity, we
will consider the asymptotic behavior as n1,n2 -+ oo. We can consistently estimate
variances and distributions of normed deviations (N;1 - E[N;1 ])/ ytnlj, k,l E JC. Let
g1,2 be the number of all boundary pixels, i.e., the pixels whose colors differ from the
colors of some of their neighboring pixels. We need the following assumption:
In words this assumption means that the number of pixels in 9J1n, constituting the
boundaries dividing subsets of pixels having the same true colors, is comparable with
the minimal diameter of In. Heuristically, this means that the boundaries dividing
subsets of pixels having the same colors in the true map 9J1n can be well approximated
by smooth curves.
The r.v.s Xi considered in Section 2 will be defined here by the following relation:
(14)
We will use the same collection IJ3k of blocks Bi defined by (5) and suppose that the
results of classification {ci : i E In } are m-dependent, m = {m1,m2} . Then AD(m)
holds for Xn. We also introduce the following assumptions:
From AC it follows that all i Xi l are uniformly bounded and AL(o) holds for any 5 > 0.
If AK' holds then AK(2) also holds. Hence, all results stated in Section 2 are valid
for r.v.s Xi, i E In defined by (14). But we can not directly use them in assessing
distribution of deviations
(15)
because the values of indicators I(ci = k) are not observable. Henceforth, in order to
be brief, we suppose that m1,m2,k1,k2 are even.
We can solve the problem of assessing variance and the distribution function of
deviations (15) by using instead of X'j the following r.v.s:
Xj =
Let
2::(-1)I(ii>ii+kl/2)+I(iz>h+kz/2)j(ci = k)pkz(s).
iEBj
x; = 2::(-1)I(il>h+h/2)+I(i2>Jz+k2/2l(xi _ x ..)Ii,
iEBj
15
and
�o
fn
1 � �2
(X )
k1,2n1,2 j�
E:Tn J
=
·
Theorem 7
Suppose that the random pixels' colors C� { ci, i E In} in the created map 9J1� are
m-dependent and assumptions AB, AC and AK' hold. Then
=
P
�o
fn- In --t 0 ,
(16)
The proof of (16) is rather long. Here we give a short sketch of it. In the proof we use the
equality XJ Xj if Bj is inside of In and it does not contain boundary pixels in 9J1n.
Assumption AB implies that the share of such blocks tends to one as n1, n2 -t oo. Let
A(1l(j, k, m) {i: j :::; i:::; j + (k- m)/2} and A(2l(j, k, m) {i: j + (k + m)/2 <
i :S j + k} and let BJs,t) { i: i1 E A(s)(j1, k1, m1), i2 E A(t)(j2, k2, m2)}, s, t 1, 2. We
define the following r.v.s:
=
=
=
=
xjs,t)
=
(-1)s+t
L (Xi - X ..)h
iEBJ(s,t)
Further we prove that
�a
fn
=
1
k1,2n1,2
�
�
�
�
(X\s,t))2 + Ro
'E v n s,t-1
- ,2 J
3
n'
-r
-t oo. We then use Theorem 2 in Ekstrom (2001) for each sum
,
1
t
)
(k�,2n1,2)- L,jE:Tn (X? )2 with k�,2 (kl/2)(k2/2), and we obtain relation (16). The
random sampling of d1,2 blocks Bj E s:Bk, j E :ln, can also be applied here. Let
1/2
b
1
2
,
1
_
_
� M': X�.
j(a*
�
n1,2 n1,2k1,2d1,2
3
3
j E.Jn
where R� � 0 as n1, n2
=
=
n
)
(
Theorem 8
Under the assumptions of Theorem
7,
(17)
The proof of (17) can be obtained by using the same method as in Section 2. The
detailed proof of Theorem 7 and 8 will be given in a separate paper with several numer­
ical examples. Presently we do not know how large n1, n2 should be so that it would
be possible to apply the suggested asymptotic approach.
16
5. Appendix
Two useful inequalities:
Inequality A: For any positive numbers z1,... , Zr and A 2:
inequality,
1 we have,
from the Jensen
( Z1 + ... + Zr )A :::; rA-1(z1A + ... + zrA).
Inequality B: For any independent random variables Z1,... ,Zr and A2: 1, with E[Zh] =0
and E[IZhiAJ < oo, h =1, . . . , r, it holds that
E[IZ1 + ... + Zr lA] :::; W(A/2-l)vo(E[IZliAJ + ... + E[IZr lA]),
<
where 1 :::; 77 = TJ ( A)
oo
is a constant, and 77 :::; 2 if A:::; 2 ( Petrov
1995, p.
82-83).
Proof of Lemma l(i) . Note that Xi' and Xi" are independent when i' and i" belong
to two different blocks TP)
and T�?).
Thus,
J
J
Define
s2j-l,i,h = {(j-1)ki + (j-1)mi + (h-1)mi + 1,. . ,
((j-1)ki + (j-1)mi + h mi) 1\ (j ki + (j-1)mi) 1\ ni} .
.
Further, define
,(2h)- { ..�1E
z . s211-1,1,hand�2. E s212,2} ,J E J (2) and h-- 1,..., l ,
'i,
where l:::; (k1 + m1- 1)/m1, and
•
.
{
uj(2,g)- h .. 0(,2h) E
.
.
}
u
0(,2g)+2(f-l) .
f:g+2(f-l)�l, f?l
By Inequality A with A= 2 and r = 2, Inequality B with A= 2 and
Inequality A with A=2 and r :::; M, respectively,
17
r
= IU}�j I, and
<
� ;!;,, t hE/ [ C�, X, ) ']
n ,,
J ,h
J ,g
<
2
4:1 2
� L L L L E[Xf]
' j E.:J(2) g=1 h UC2l i T(2)
E
E
J ,g
J ,h
4m1,2(k1 + m1 + n1)(m2 + n2)k1m2T02/(2+J)
(18)
'
n1,2(k1 + m1)(k2 + m2)
where the last inequality follows from (2) and the inequality l7j��� :s; k1m2 . Likewise,
2/(2+/i)
< 4m1,2(m1 + n1)(k2 + m2 + n2)k2m1T0
3
)
(19)
,
?J
n1 '2E[(X� n1,2(k1 + mi)(k2 + m2)
2/(2+J)
2
< 4m1,2(m1 + n1)(m2 + n2) 'a
2
4
(
)
(20)
n1 '2E[(Xn ) ] n1,2(k1 + m1)(k2 + m2)
By the assumptions on kh and nh, h 1, 2, we see that n1,2E[(X(1))2] -----t 0, l 2, 3, 4,
D
and so �X�) � 0, l 2, 3, 4, as kh/nh -----t 0 and kh,nh -----t oo, h 1, 2.
<
=
=
=
=
Proof of Lemma l(ii). By the Cauchy-Schwarz inequality, and inequalities (4) and
(18)-(20),
l'l'n- 'l'};lI n,,, E
=
[ (t )']
X�l
- E[(X).'l)']
4
4 4
2
n1,2 L E[(X�)) ] - 2 L L E[X�)X�)]
l=2
l=1 j=l+1
4
4 4
2 1/2
2
)
:s; n1,2 L E[(X� ) ] + 2n1,2 L L (E[(X�)?JE[(X�)) ] )
l=2
l=2 j=l+1
'
)
E x- x� E[X!Pi'
+2n,,
=
t, ( [ t, J
f'
4
4 4
12
2
)
�
2n
]
n
E[X
+ 1,2 L L (E[X�)j2E[X�)J 2 ) 1
:s; 1,2 L
l=2
l=2 j=l+1
18
D
Proof of Lemma
Jez -1J::; JzJ,
3.
By the inequalities
Jw(z)l
<
lzJ2+P, if lz l < 1, 0 < p < 1, and
�)
(
2
z
l
2
+
z
P
I{lzl2:1}::; 4JzJ2+P.
Jw(z)l::; lzJ J{Izl<1} + le -11 + lz l +
(21)
Thus, by (21) and Inequality A,
2+p
2b1,2 l+p/2
2k1,2b1,2 x?. l+p/2
d
6t
6t
1
1
""
"""'
"'
.
.
< �
+ d1,2
� �XJ
- b1,2 n1,2k1,2d1,2
n1,2d1,2
jE.:Tn
= R� ) + R� l, say.
If we choose p such that 0 < p < 6 1\ 1, then by the Lyapunov inequality, E[JXiJ2+P]::;
(E[JXiJ2H])(2+P)/(2H)::; TJ2+p)/(2H). This implies that a modified version of Lemma
2(ii) can be used, i.e., we can use Lemma 2(ii) with 6 and To replaced by p and
TJ2+p)/(2H), respectively. Hence, if kh 2:: mh, h = 1,2,
- 2 -+
l2 2
2 o2 2
< J16tJ +pT ( +p)/( H)'Ilm1,+p/ (b1,2/n1,2)l+P/2d1,p/ - 0 '
(22)
E[R(1)] 2
(
)
�
�
(
)
'I
n
2 o2 2
< J16tJ +pT ( +P)/( H)'Ilm +P/ (k1,2b1,2 jn2 )l+P /2d p 1 --+ 0 '
E[R(2)] 1,l 2 2
1,2
1,-2 2
n
'I
(23 )
D
Proof of Lemma 4. Define P, = fl- J1 = X .., Ti2 = i\2- ri2 = X.i2/n1 - P,, i2 = 1,...,n2,
and ci1 = ci1-Ci1 = Xi1.jn2-P,, i1 =1 , ...,n1. By inequality (21 ) and Inequality A,
19
2+p
From (22) and (23) we see that Ri!:) � 0, h
and Lemma 1 we have, for i1 1, ...,n1,
=
=
1,2, as n1,n2 ----+ oo.
By Inequality A
(24)
and thus, by the Lyapunov inequality, Eli\21 2+P :::; (Ts1J)(2+p)/(2+5)(256m1,2/n1)l+P/2 .
By Inequality A and Lemma 2(i), with p instead of 6, fi2 instead of Xi, and with
(Ts7J)(2+p)/(2Hl(256m1,2/n1)l+P/2 instead of 18, we get (assuming k2 � m2),
)
(
kl-1 J2+'"'
k2-1 -. . 2+P
l+p 16t2b1,2 l+p/2 '"' )l+'"'
d
k
1,
2
1
<
E[R(3l] � �
� r�2 J
b1,2 n1,2k1,2d1,2
�l-]l �2-]2
J"Err
1
2
2
'11)(2+p)/(2+5)(b1,2k1/(n1,2n1))l+P/2d1,-p2 12 ----+ 0 '
< (2 5t m2m1,2)l+P/ '11( 7:J.,
2
n
·
Jn
_
·
·
-
·
'/
which implies that Rf:l � 0 as n1,n2
result follows.
----+
oo.
Likewise,
R�) � 0, and so the desired
0
Acknowledgements
This work of both authors is a part of the Swedish research programme Remote Sens­
ing for the Environment, RESE ( home page http:/ / rese.satellus.se ) , supported by the
Swedish Foundation for Strategic Environmental Research, MISTRA. For the second
author this work is also supported by The Bank of Sweden Tercentenary Foundation.
We thank Heather Reese for improving the English language.
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21
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Forfattarna svarar sjalva for rapporternas vetenskapliga innehall.
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37
Odell, G. & Stahl, G. VegetationsfOrandringar i svensk skogsmark mellan 1980- och
90- talet. - En studie grundad pa Standortskarteringen. ISRN SLU- SRG- AR-- 37-- SE.
38
Lind, T. Quantifying the area of edge zones in Swedish forest to assess the impact of
nature conservation on timber yields. ISRN SLU- SRG- AR-- 38-- SE.
1999
50
Stahl, G., Walheim, M. & Lofgren, P. Fjallinventering. - En utredning av innehall och
design. ISRN SLU- SRG-- AR-- 50-- SE.
52
Riksskogstaxeringen infor 2000- talet. - Utredningar avseende innehall och omfattning i
en framtida Riksskogstaxering. RedaktOrer: Jonas Fridman & Goran Stahl.
ISRN SLU- SRG- AR-- 52-- SE.
54
Fridman, J. m.fl. Sveriges skogsmarksarealer enligt internationella agoslags­
definitioner. ISRN SLU- SRG- AR-- 54-- SE.
56
Nilsson, P. & Gustafsson, K. Skogsskotseln vid 90- talets mitt - lage och trender.
ISRN SLU- SRG- AR-- 56-- SE.
57
Nilsson, P. & Soderberg, U. Trender i svensk skogsskotsel - en intervjuundersokning.
ISRN SLU- SRG- AR-- 57-- SE.
1999
61
Broman, N & Christoffersson, J. Matfel i provtradsvariabler och dess inverkan pa
precision och noggrannhet i volymskattningar. ISRN SLU- SRG- AR- - 61-- SE.
65
Hallsby, G m. fl. Metodik for skattning av lokala skogsbransleresurser.
ISRN SLU- SRG- AR-- 65-- SE.
75
von Segebaden, G. Komplement till "RIKSTAXEN 75 AR".
ISRN SLU- SRG- AR-- 75-- SE.
2001
86
Lind, T. Kolinnehall i skog och mark i Sverige - Baserat pa Riksskogstaxeringens data.
ISRN SLU- SRG- AR-- 86-- SE
Planering och inventering:
1995
3
(Forest inventory and planning)
Holmgren, P. & Thuresson, T. Skoglig planering pa amerikanska vastkusten - intryck
fran en studieresa till Oregon, Washington och British Columbia 1- 14 augusti 1995.
ISRN SLU- SRG- AR-- 3-- SE.
4
Stahl, G. The Transect Relascope - An Instrument for the Quantification of Coarse
Woody Debris. ISRN SLU- SRG- AR-- 4-- SE
1996
15
van Kerkvoorde, M. A sequential approach in mathematical programming to include
spatial aspects of biodiversity in long range forest management planning.
ISRN SLU- SRG- AR-- 15-- SE.
1997
18
Christoffersson, P. & Jonsson, P. Avdelningsfri inventering - tillvagagangssatt och
tidsatgang. ISRN SLU- SRG- AR-- 18-- SE.
19
Stahl, G. , Ringvall, A. & Lamas, T. Guided transect sampling - An outline of the
principle. ISRN SLU- SRGL- AR-- 19-- SE.
25
Lamas, T. & Stahl, G. Skattning av tillstand och forandringar genom inventerings­
simulering - En handledning till programpaketet "NV SIM".
ISRN SLU- SRG- AR-- 25-- SE.
26
Lamas, T. & Stahl, G. Om dektering av forandringar av populationer i begransade
omraden. ISRN SLU- SRG- AR-- 26-- SE.
1999
59
Petersson, H. Biomassafunktioner fOr tradfraktioner av tall, gran och bjork i Sverige.
ISRN SLU- SRG- AR-- 59-- SE.
63
Fridman, J. , Lofstrand, R & Roos, S. Stickprovsvis landskapsovervakning - En
forstudie. ISRN SLU- SRG- AR-- 63-- SE.
2000
68
Nystrom, K. Funktioner fOr att skatta hojdtillvaxten i ungskog.
ISRN SLU- SRG- AR-- 68-- SE.
70
Walheim, M. & Lofgren, P. Metodutveckling fOr vegetationsovervakning i fjilllen.
73
ISRN SLU- SRG- AR-- 70-- SE.
Holm, S. & Lundstrom, A. Atgardsprioriteter. ISRN SLU- SRG- AR-- 73-- SE.
76
Fridman, J. & Stahl, G. Funktioner for naturlig avgang i svensk skog.
ISRN SLU- SRG- AR-- 76-- SE.
2001
82
Holmstrom, H. Averaging Absolute GP S Positionings Made Underneath Different
Forest Canopies - A Splendid Example of Bad Timing in Research.
ISRN SLU- SRG- AR-- 79-- SE.
Biometri:
(Biometrics)
1997
22
Ali, Abdul Aziz. Describing Tree Size Diversity. ISRN SLU- SEG- AR-- 22-- SE.
1999
64
Berhe, L. Spatial continuity in tree diameter distribution.
ISRN SLU- SRG- AR-- 64-- SE
2001
88
Ekstrom, M. Nonparametric Estimation of the Variance of Sample Means Based on
Nonstationary Spatial Data. ISRN SLU- SRG- AR-- 88-- SE.
89
Ekstrom, M. & Belyaev, Y On the Estimation of the Distribution of Sample Means Based on
Non- Stationary Spatial Data. ISRN SLU- SRG- AR-- 89-- SE.
Fjarranalys:
1997
28
29
(Remote Sensing)
Hagner, 0. Satellitfjarranalys for skogsforetag. ISRN SLU- SRG- AR-- 28-- SE.
Hagner, 0. Textur till flygbilder for skattning av bestandsegenskaper.
ISRN SLU- SRG- AR-- 29-- SE.
1998
32
Dahlberg, U., Bergstedt, J. & Pettersson, A. Faltinstruktion fOr och erfarenheter fran
vegetationsinventering i Abisko, sommaren 1997. ISRN SLU- SRG- AR- - 32-- SE.
1999
43
Wallerman, J. Brattakerinventeringen. ISRN SLU- SRG- AR-- 28-- SE.
51
Holmgren, J., Wallerman, J. & Olsson, H. Plot - Level Stem Volume Estimation and
Tree Species Discrimination with Casi Remote Sensing.
ISRN SLU- SRG- AR-- 51-- SE.
53
Reese, H. & Nilsson, M. Using Landsat TM and NFI data to estimate wood volume,
tree biomass and stand age in Dalarna. ISRN SLU- SRG- AR- - 5 3-- SE.
2000
66
LOfstrand, R., Reese, H . & Olsson, H. Remote Sensing aided Monitoring of Non­
Timber Forest Resources - A literature survey. ISRN SLU- SRG- AR-- 66-- SE.
69
TingelOf, U & Nilsson, M.Kartering av hyggeskanter i pankromaotiska SPOT -bilder.
ISRN SLU-SRG-AR--69 --SE.
79
Reese, H & Nilsson, M. Wood volume estimations for Alvsbyn Kommun using SPOT
satellite data and NFI plots. ISRN SLU-SRG-AR--79--SE.
Kompendier och undervisningsmaterial:
1996
14
(Compendia and educational papers)
Holm, S. & Thuresson, T. samt jagm.studenter kurs 92/96. En analys av skogstill­
standet samt nagra alternativa avverkningsberakningar for en del av Ostads sateri.
ISRN SLU-SRG-AR--14--SE.
21
Holm, S. & Thuresson, T. samt jagm.studenter kurs 93/97. En analys av skogsstill­
standet samt nagra alternativa avverkningsberakningar fOr en stor del av Ostads
sateri. ISRN SLU-SRG-AR--21--SE.
1998
42
Holm, S. & Lamas, T. samt jagm.studenter kurs 93/97. An analysis of the state of the
forest and of some management alternatives for the Ostad estate.
ISRN SLU-SRG-AR--42--SE
1999
58
Holm, S. samt studenter vid Sveriges lantbruksuniversitet i samband med kurs i strate­
gisk och taktisk skoglig planering ar 1998. En analys av skogsstillstandet samt nagra
alternativa avverknings berakningar fOr Ostads sateri. ISRN SLU -SRG-AR--58--SE.
2001
87
Eriksson, 0. (Ed.) Strategier fOr Ostads sateri: Redovisning av planer framtagna under
kursen Skoglig planering ur ett foretagsperspektiv HT2000, SLU Umea.
ISRN SLU -SRG-AR--87--SE.
Examensarbeten:
1995
5
(Theses by Swedishforestry students)
Tornquist, K. Ekologisk landskapsplanering i svenskt skogsbruk - hur borjade det?.
Examensarbete i amnet skogsuppskattning och skogsindelning.
ISRN SLU-SRG-AR--5--SE.
1996
6
Persson, S. & Segner, U. Aspekter kring datakvalitens betydelse for den kortsiktiga
planeringen. Examensarbete i amnet skogsuppskattning och skogsindelning.
ISRN SLU -SRG-AR--6 --SE.
7
Henriksson, L. The thinning quotient - a relevant description of a thinning?
Gallringskvot - en tillforlitlig beskrivning av en gallring? Examensarbete i amnet
skogsuppskattning och skogsindelning. ISRN SLU-SRG-AR--7--SE.
8
Ranvald, C. Sortimentsinriktad avverkning. Examensarbete i amnet skogsuppskattning
och skogsindelning. ISRN SLU-SRG-AR--8--SE.
9
Olofsson, C. Mangbruk i ett landskapsperspektiv - En fallstudie pa MoDo Skog AB,
Ornskoldsviks forvaltning. Examensarbete i amnet skogsuppskattning och skogs­
indelning. ISRN SLU-SRG-AR--9 --SE.
10
Andersson, H. Taper curve functions and quality estimation for Common Oak (Quercus
Robur L. ) in Sweden. Examensarbete i amnet skogsuppskattning och skogsindelning.
ISRN SLU-SRG-AR--10--SE.
11
Djurberg, H. Den skogliga informationens roll i ett kundanpassat virkesflode. - En
bakgrundsstudie samt simulering av inventeringsmetoders inverkan pa noggrannhet i
leveransprognoser till sagverk. Examensarbete i amnet skogsuppskattning och
skogsindelning. ISRN SLU-SRG-AR--11--SE.
12
Bredberg, J. Skattning av alder och andra bestandsvariabler - en fallstudie baserad pa
MoDo: s indelningsrutiner. Examensarbete i funnet skogsuppskattning och
skogsindelning. ISRN SLU-SRG-AR--14--SE.
13
Gunnarsson, F. On the potential of Kriging for forest management planning.
Examensarbete i funnet skogsuppskattning och skogsindelning.
ISRN SLU-SRG-AR--13--SE.
16
Tormalm, K. Implementering av FSC-certififering av mindre enskilda markagares
skogsbruk. Examensarbete i amnet skogsuppskattning och skogsindelning.
ISRN SLU-SRG-AR--16--SE.
1997
17
Engberg, M. Naturvarden i skog lamnad vid slutavverkning. - En inventering av upp
till 35 ar gamla fOryngringsytor pa Sundsvalls arbetsomsade, SCA. Examensarbete i
amnet skogsuppskattning och skogsindelning. ISRN-SLU-SRG-AR--17 --SE.
20
Cedervind, J. GP S under krontak i skog. Examensarbete i amnet skogsuppskattning
och skogsindelning. ISRN SLU-SRG-AR--20--SE.
27
Karlsson, A. En studie av tre inventeringsmetoder i slutavverkningsbestand.
Examensarbete. ISRN SLU-SRG-AR--27--SE.
1998
31
Bendz, J. SODRAs grona skogsbruksplaner. En uppfOljning relaterad till SODRAs
miljoma.I, FSC's kriterier och svensk skogspolitik. Examensarbete.
ISRN SLU-SRG-AR--31--SE.
33
Jonsson,
b.
Tradskikt och standortsforhallanden i strandskog. - En studie av tre backar
i Vasterbotten. Examensarbete. ISRN SLU-SRG-AR--33--SE.
35
Claesson, S. Thinning response functions for single trees of Common oak (Quercus
Robur L. ) Examensarbete. ISRN SLU-SEG-AR--35--SE.
36
Lindskog, M. New legal minimum ages for final felling. Consequences and forest
owner attitudes in the county of Vasterbotten. Examensarbete.
ISRN SLU-SRG-AR--36--SE.
40
Persson, M. Skogsmarksindelningen i grona och bla kartan - en utvardering med
hjalp av riksskogstaxeringens provytor. Examensarbete. ISRN SLU-SRG-AR--40--SE.
41
Eriksson, F. Markbaserade sensorer for insamling av skogliga data - en forstudie.
Examensarbete. ISRN SLU-SRG-AR--41--SE.
45
Gessler, C. Impedimentens potientiella betydelse for biologisk mfmgfald. - En studie av
myr- och bergimpediment i ett skogslandskap i Vasterbotten. Examensarbete.
ISRN SLU- SRG- AR-- 45-- SE.
46
Gustafsson, K. Umgsiktsplanering med geografiska hansyn - en studie pa B racke
arbetsornrade, SCA Forest and Timber. Examensarbete. ISRN SLU- SRG- AR-- 46-- SE.
47
Holmgren, J. Estimating Wood Volume and B asal Area in Forest Compartments by
Combining Satellite Image Data with Field Data. Examensarbete i amnet Fjarr analys.
ISRN SLU- SRG- AR-- 47-- SE.
49
Hardelin, S. Framtida fO rekomst och rumslig fordelning av gamm al skog.
- En fallstudie pa ett landskap i B racke arbetsornrade. Examensarbete SCA.
ISRN SLU- SRG- AR-- 49-- SE.
1999
55
Imamovic, D. Simuleringsstudie av produktionskonsekvenser med olika miljomal.
Examensarbete for Skogsstyrelsen. ISRN SLU- SRG- AR-- 55-- SE
62
Fridh, L. Utbytesprognoser av rotstaende skog. Examensarbete i skoglig planering.
ISRN SLU- SRG- AR-- 62-- SE.
2000
67
Jonsson, T. Differentiell GPS-matning av punkter i skog. Point- accuracy for
differential GP S under a forest canaopy. ISRN SLU- SRG- AR-- 67-- SE.
71
Lundberg, N. Kalibrering av den multivariata variabeln tradslagsfOrdelning.
Examensarbete i biometri. ISRN SLU- SRG- AR-- 71-- SE.
72
Skoog, E. Leveransprecision och ledtid - tva nyckeltal fOr stym ing av virkesfl odet.
Examensarbete i skoglig planering. ISRN SLU- SRG- AR-- 72-- SE.
74
Johansson, L. Rotrota i Sverige enligt Riksskogstaxeringen. Examens arbete i amnet
skogsindelning och skogsuppskattning. ISRN SLU- SRG- AR-- 74-- SE.
77
Nordh, M. Modellstudie av potentialen for renbete anpassat till komrn ande slut­
avverkningar. Examensarbete pa jagmastarprogramrnet i amnet skoglig planering.
ISRN SLU- SRG- AR-- 77-- SE.
78
Eriksson, D. Spatial Modeling of Nature Conservation Variables useful in Forestry
Planning. Examensarbete. ISRN SLU- SRG- AR- - 74-- SE.
81
Fredberg, K. Landskapsanalys med GIS och ett skogligt planeringssystem.
Examensarbete pa skogsvetarprogrammet i amnet skogshushallning.
ISRN SLU- SRG- AR- - 81-- SE.
83
Lindroos, O. Underlag for skogligt lansprogram Gotland. Examensarbete i amnet skoglig
planering. ISRN SLU- SRG- AR--83- SE.
84
Dahl, M. Satellitbildsbaserade skattningar av skogsornraden med rojningsbehov.
Examensarbete pa akogsvetarprograrnm et i arnn et skoglig planering.
ISRN SLU- SRG- AR- - 84-- SE.
85
Staland, J. Stym ing av kundanpassade timmerfloden -Inverkan av traktbankens stlorlek och
utbytesprognosens ti llfOrlitlighet. Examensarbete i amnet skoglig planering.
ISRN SLU-SRG-AR--85--SE.
Internationellt:
1998
39
(International issues)
Sandewall, Ohlsson, B & Sandewall, R. K. People' s options on forest land use - a
research st udy of land use dynamics and socio- economic condit ions in a hist orical
perspective in t he Upper Nam Nan Water Cat chment Area, Lao PDR.
ISRN SLU-SRG-AR- -39 --SE.
44
Sandewall, M. , Ohlsson, B. , Sandewall, R. K. , Vo Chi Chung, Tran Thi Binh & Pham
Quoc Hung. People' s options on forest land use. Government plans and farmers
intentions - a strategic dilemma. ISRN SLU-SRG-AR--44--SE.
48
Sengthong, B. Estimat ing Growing Stock and Allowable Cut in Lao PD R using Data
from Land Use Maps and the National Forest Inventory (NFI) . Master thesis.
ISRN SLU-SRG-AR-- 48- -SE.
1999
60
Inter-active and dynam ic approaches on forest and land-use planning - proceedings
from a training workshop in Vi etnam and Lao PDR, Apri l 12-30, 1999.
Edited by Mats Sandewall ISRN SLU-SRG- AR--60--SE.
2000
80
Sawathvong. S. Forest Land Use Planning in Nam Pui National Biodiversit y
Conservation A rea, Lao P. D. R. ISRN SLU- SRG-AR- - 80-- SE.