ORIGINAL PAPER

Analysing space–time tree interdependencies based on individualtree growth functions

C. Comas • L. Mehtatalo • J. Miina

Published online: 8 April 2013

� Springer-Verlag Berlin Heidelberg 2013

Abstract We analysed the space–time structure of two

spatially explicit forest data sets considering the associated

growth function for each tree obtained from the annual

radial growth measured from increment cores bored at

breast height. We used a new second order formulation

based on the mark correlation function, the functional mark

correlation function, to analyse spatial pattern involving

functions to each spatial location. A decomposition of

individual growth function into spatial and non-spatial

components was considered and only the spatial compo-

nents were analysed. Our results confirm the usefulness of

these new approach compared with other well-established

spatial statistical tools such as the mark correlation func-

tion. In particular, the functional mark correlation function

of the spatial and temporal components of tree growth

determines the space–time structure of tree development

regardless of the non-spatial components contained in this

function. Moreover, this explicit temporal analysis detects

space–time interaction effects that are not evident when

analysing the spatial distribution of cumulative growth

measures such as the tree basal area.

Keywords Functional mark correlation function �Norway spruce � Point processes � Scots pine � Space–time

dynamics � Tree growth dynamics

1 Introduction

Forest science has applied numerous statistical methods

belonging to point processes (Diggle 2003; Illian et al.

2008) to tackle several important forest-ecological ques-

tions (Comas and Mateu 2007). This includes the study of

the spatial structure of pure and mixed forest stands

(Moeur 1993; Pelissier 1998; Park et al. 2005), the space–

time modelling of forest dynamics (Renshaw et al. 2009;

Comas 2009) and the estimation of stand conditions such

as timber volume and number of stems per hectare

(Comas et al. 2011a). Traditionally, point process meth-

odologies have been restricted to analyse forest patterns

with qualitative or quantitative characteristics (marks)

associated to each tree. For instance, Comas et al. (2009)

analysed the spatial relationships between Scots pine

(Pinus sylvestris L.), Back pine (P. nigra Arn.) Arnold.

and Aleppo pine (P. halepensis Mill) in a case study in the

central Catalonia (Spain), whilst Penttinen et al. (1992)

studied the spatial structure of tree diameters of a Norway

spruce stand in Saxonia (Germany). However, little

attention has been paid to the spatial analysis of tree

characteristics involving complete functions (curves)

instead of single marks such as the analysis of tree growth

functions. A probable reason for that it is that this type of

data is quite uncommon.

In fact, despite of the relatively long history of point

process theory (see, amongst others, Diggle 2003; Stoyan

et al. 1995; Daley and Vere-Jones 2004) few approaches

have been performed to analyse spatial point patterns

C. Comas (&)

Department of Mathematics, Agrotecnio Center, University of

Lleida, Campus de l’ETSEA, Av. Alcalde Rovira Roure, 191,

25198 Lleida, Spain

e-mail: carles.comas@matematica.udl.cat

L. Mehtatalo

School of Computing, University of Eastern Finland, P.O. Box

111, 80101 Joensuu, Finland

J. Miina

Finnish Forest Research Institute, Joensuu Unit, P.O. Box 68,

80101 Joensuu, Finland

123

Stoch Environ Res Risk Assess (2013) 27:1673–1681

DOI 10.1007/s00477-013-0704-3

where the features of interest are functions (i.e. curves)

instead of qualitative or quantitative variables. Comas et al.

(2011b) have developed a functional counterpart version of

the mark correlation function (see, Stoyan and Stoyan

1994) to analyse function-marked point patterns. These

authors prove the usefulness of their new approach com-

pared with other well-established spatial statistical tools

such as the mark correlation function, for a case study

involving two spatially explicit demographic functions,

namely the town population pyramids and the demographic

evolution under 121 distinct town positions.

This new statistical tool analyses spatial dependencies of

complete curves, for instance, growth and interaction

functions of individual trees, instead of single marks such

tree diameter at breast height and tree basal area at a given

year. This spatial analysis could provide useful insights

into the space–time interdependencies of tree growth

interaction dynamics. A tentative way to incorporate the

time domain in analysing the spatial structure of forest

patterns is therefore via the individual tree growth function,

usually given by the annual radial growth measured from

increment cores bored at breast height. Here, individual

trees are characterised by a function showing the growth

and interaction dynamics during the last years. The use of

growth functions to analyse space–time forest dynamics

can be an alternative to study the space–time structure of

forest stands in which complete time series of forest field

observations are not available. Obviously, the analysis of

growth functions of alive trees at a given sampling year

involves several important limitations. For instance, only

alive trees can explain growth and interaction dynamics of

neighbouring trees although the effects of past interaction

of already dead trees can also be apparent in the growth

functions of the alive ones.

The main purpose of this paper was to analyse the

space–time structure of two spatially explicit forest data

sets considering the associated growth function for each

tree obtained from the annual radial growth measured from

increment cores bored at breast height. To do so, we pre-

sented a new second order formulation based on the mark

correlation function formulated by Comas et al. (2011b),

and discussed the practical applicability of such formula-

tion in forestry applications. A decomposition of individual

growth function into spatial and non-spatial components

was considered.

2 Methodology

2.1 The functional mark correlation function

Comas et al. (2011b) develop a functional counterpart

version of the mark correlation function to analyse spatially

explicit functional point patterns, i.e. point patterns in

which associated to each spatial position there is a function

(curve) instead of a single mark. Although for certain

function-marked point patterns a partial analysis of such

configurations can be performed by, for instance, the point-

wise study of such functions, the use of such partial

information can be extremely inefficient for a space–time

process, say, involving large periods of time, since we have

to estimate second order characteristics for each time slice.

Moreover, this point-wise study may hide the time inter-

dependencies at different lags, where only a global study of

the time intervals can reveal the underlying space–time

structure.

In analysing marked point patterns, a well known and

extensively used statistical tool is the mark correlation

function (see Stoyan and Stoyan 1994; Illian et al. 2008).

This correlation function is defined as

jhmðx1; x2Þ ¼

E½hmðm1;m2Þ�E½hmðm�1;m�2Þ�

ð2:1Þ

where these expectations depend on spatial positions x1

and x2 with associated marks m1 * m1* and m2 * m2

*, and

being m1* and m2

* independent realisations of the marginal

distribution of marks, in which * stands for ‘‘is distributed

as’’, and hmð�; �Þ is a test function. Mark correlation

functions for quantitative marks were first defined and

discussed in Stoyan (1984) and Isham (1987). Further

development and forestry applications can be found in

Penttinen and Stoyan (1989), Cressie (1993), Gavrikov and

Stoyan (1995), Stoyan and Stoyan (1994) and Stoyan et al.

(1995). A functional counterpart version of this mark

correlation function can be defined as

ghðrÞ ¼E½hðf1; f2Þ�E½hðf �1 ; f �2 Þ�

ð2:2Þ

in which both expectations are conditioned to r ¼ kx1 �x2k; f 1 and f2 are the associated functions, f �1 � f1; f

�2 � f2;

with f �1 and f �2 independent, and hð�; �Þ is a test function.

Comas et al. (2011b) derive and define a functional

version of (2.1) based on (2.2) for homogenous function-

marked point processes via

ghðrÞ ¼chðrÞchð1Þ

; r� 0; ð2:3Þ

where

chðrÞ ¼kð2Þh ðrÞkð2ÞðrÞ

; kð2ÞðrÞ 6¼ 0:

Here k(2)(r) is the second order product density (Stoyan

and Stoyan 1994), which loosely speaking it is a density

function providing information about the spatial structures

at point distances r; kh(2)(r) is the functional counterpart

1674 Stoch Environ Res Risk Assess (2013) 27:1673–1681

123

version of this second order product density, and chð1Þ is

normalising constant assuming some mixing properties of

the functional mark point process. For further details for

this mathematical derivation see Comas et al. (2011b).

2.2 The test function

Given the potentially infinite number of combinations

involving two functions, the test function hðfx; fyÞ has to be

defined suitably to the problem and the type of functions

under analysis. Comas et al. (2011b) analysed the effect of

three different test functions on (2.5). These authors con-

sidered test functions reflecting distances between observed

functions: hðfx; fyÞ ¼ dðfx; fyÞ: In this case, ghðrÞ in (2.5) is

always positive and its interpretation is similar to that of

the mark correlation function. They assumed the distances

L1 and L2, and the symmetrised Kullback–Leibler diver-

gence function when density functions are observed.

Because these authors did not find significant differences

between these test functions, we shall consider the L2

distance between function as a tentative test function

hL2ðfx; fyÞ ¼

Zb

a

ðfxðtÞ � fyðtÞÞ2dt

0@

1A

1=2

; ð2:4Þ

for the simplicity of computation of this function. Note that

similar results are also obtained under the L1 distance as a

test function (not included).

2.3 An edge corrected estimator

Comas et al. (2011b) provide an edge corrected estimator

based on the ratio of estimators in (2.3) in an observation

window W in R2 with area |W| as

ghðrÞ ¼1

2prk2pjW j

X6¼

ðx;fxÞ;ðy;fyÞ2wf

hðfx; fyÞjðkx� yk � rÞE½hðfx; fyÞ�eðx; kx� ykÞgðrÞ

ð2:5Þ

where wf is the observed functional mark point pattern,

gðrÞ is an estimator of the pair correlation function, kp is an

estimator of the point intensity,P

= stands for the

summation over all pairs such that x 6¼ y; jð�Þ is a kernel

function which is non-negative and symmetric with respect

to the origin, eð�Þ is a factor to correct edge-effects and E½��is the expectation of hð�; �Þ; which can be obtained as

E½hðfx; fyÞ� ¼ 1=ðn2 � nÞX6¼

ðx;fxÞ;ðy;fyÞ2wf

hðfx; fyÞ:

in which n is the number of points of the functional mark

point pattern. Here we shall consider the Ripley’s factor

(Ripley 1976) to correct for edge effect and the well-known

Epanechnikov kernel function, already used to analyse

forest patterns in Penttinen et al. (1992) and Baddeley

et al. (2000) amongst others. Broadly speaking, if functions

are independently located over spatial positions gh(r) = 1,

whilst under gh(r) [ 1 pairs of locations result in hðfx; fyÞwith larger values than the average value of E½hðf �x ; f �y Þ�;suggesting negative correlation (i.e. dissimilarities)

between functions, and for gh(r) \ 1 we obtain the oppo-

site case.

2.4 A Monte Carlo approach to test for functional

spatial dependencies

As in (2.5) there can be inter-dependencies between marks

(functions) and point locations, and that this function can

not completely disentangle functions from the function-

point interdependent structure, we consider a simple Monte

Carlo approach proposed by Comas et al. (2011b) to

overcome this problem. These authors proposed a method

similar to that considered for the mark correlation function

where the fifth largest and smallest envelop values for the

functional mark correlation function are obtained based on

199 random permutations of the functions over (fixed)

point positions. Then if the estimated functional mark

correlation function lies outside the fifth largest and/or

smallest envelopes we reject the null hypothesis of inde-

pendency at the exact significant level 2 9 5/

(199 ? 1) = 0.05. This approach permits to (at some

extent) obtain the function-marked structure independently

of the point domain. This Monte Carlo approach is based

on simulation from the null hypothesis. This principle was

developed independently by Barnard (1963) and Dwass

(1957) and has been applied extensively in spatial statis-

tical analysis (e.g. Besag and Diggle 1977; Ripley 1977;

Diggle 2003) where other statistical tests cannot be used.

Grabarnik et al. (2011) discussed about the number of

simulations from the null hypothesis related to the type I

error committed for marked point patterns. These authors

proposed the use of the envelope test considered here and a

deviation test to testing spatial independency.

3 Modelling data

Space–time interdependencies based on spatially explicit

tree growth functions are studied in natural Scots pine (P.

sylvestris) (165 trees) and Norway spruce [Picea abies (L.)

Karst.] (195 trees) stands located in North Karelia (63�6N,

30�40E) in Finland. Tree ages vary a lot within plots, the

range being 26–144 for the Scots pine plot and 57–290 for

the Norway spruce plot, both having an uneven-aged forest

structure. Here, the Scots pine stand is growing on a poor

site (Calluna type) and the Norway spruce stand on a

Stoch Environ Res Risk Assess (2013) 27:1673–1681 1675

123

medium site (Myrtillus type) in Finnish classification

(Cajander 1926). The field measurements were recorded in

1990 on two rectangular plots of sizes 50 9 50 and

50 9 40 m2 for the Scots pine and the Norway spruce

stands, respectively. The first year where we have therefore

a complete record of forest measurements is the year 1989,

and then having records until the year 1715. Data on tree

locations, dbh, annual radial growth, tree age and tree

species for each plot were recorded only for trees with

diameters [3 cm. In particular, the annual radial growth

was measured from increment cores bored at breast height,

and the age of each trees was determined from the cores.

Dead trees were not measured, since it was impossible to

determine the year of death [for more details see Miina

(1993, 2000)].

3.1 Removing age trend effect

We considered the basal area increment of individual trees

as a convenient growth function because it is highly related

to volume growth. This basal area increment, BAi(t), con-

sists of four components

BAiðtÞ ¼ TAðtÞ þ aðtÞ þ bi þ eiðtÞ ð3:6Þ

where TA(t) is the deterministic age trend effect at the

cambial age t, a(t) is the random effect related to calendar

year at the cambial age t, bi is the tree i random effects, and

ei(t) is the unexplained residual variation. Here calendar

year is the year when the observed basal area growth

occurred and the cambial age is the number of years

elapsed at the year of growth since the tree reached the

height of measurement. From (3.6), only bi and ei(t) are

spatial dependent, and TA(t) and a(t) are non-spatial

components depending only on the cambial age t. Whilst bi

summaries individual tree random variation and it incor-

porates, for instance, genetic tree variation and/or local site

inhomogeneity, ei(t) incorporates tree and cambial age

unexplained variations. As a consequence, only the random

function ei(t) contains interaction and random effects that

are expected to evolve through time and between trees. We

therefore focused on this last function to analyse the spatial

dependence of tree growth dynamics. Notice that a direct

comparison of basal area increments may result in dis-

similarities between functions only due to the different

growth development (i.e. age) of each tree.

The age trend was modelled using a restricted cubic

spline regression (Harrell 2001; Gort et al. 2011). We

considered a spline regression since this is a flexible and

smooth curve that does not make any strict mathematical

assumption on the shape of the trend while removing this

age trend. The restricted spline function with pj knots

(j ¼ 1; . . .; k), placed within the range of cambial age, is

given by

TAðtÞ ¼ b0 þ b1t1 þ b2t2 þ . . .þ bk�1tk�1 ð3:7Þ

in which t1 = t, i.e. the cambial age, and

tjþ1 ¼ ðt � pjÞ3þ þ ðt � pk�1Þ3þðpk � pjÞ=ðpk � pk�1Þþðt � pkÞ3þðpk�1 � pjÞ=ðpk � pk�1Þ:

ð3:8Þ

We used 6 knots at the 5th, 23th, 41th, 59th, 77th and,

99th percentiles of the cambial age to model BA(t) (Harrell

2001, p. 23). Then the resulting mixed model (3.6)

assuming the splines regression for the age trend was

fitted to isolate the spatial dependent components of

BAi(t), i.e.

eiðtÞ ¼ BAiðtÞ � TAðtÞ � aðtÞ � bi: ð3:9Þ

Writing (3.7) in (3.6) yields a linear mixed-effect model,

with crossed calendar year [a(t)] and tree effects (bi) and a

random residual [ei(t)]. The model was fitted using

restricted maximum likelihood, assuming that these three

effects have normal distribution with mean zero (Pinheiro

and Bates 2000). After model fitting, the random effects for

each calendar year and tree were predicted using the best

linear unbiased predictor (BLUP).

3.2 Adapting the test function to discrete time

increments

Because the growth and interaction tree dynamics are

based on records of m annual radial growth increments (or

basal area increments) c1; . . .; cm; we need to adapt the L2

distance function for these discrete increments. Under the

Norway spruce stand we have at most m = 275 radial

growth increments (either cambial or calendar years), and

for the Scots pine stand we have at most m = 139. A

function f is therefore represented as the collection

f ðc1Þ; . . .; f ðcmÞ: Another problem is that we do not have

the same number of radial basal area increments for all the

trees. Here we compare trees that where born at different

years but that they were all alive at the year 1990. To

overcome this problem we compare pairwise individual

tree functions only for the calendar years in which the two

trees have records. This therefore ensures that only time

overlapping functions are compared. As a consequence, for

discrete growth functions based on calendar years, the L2

distance function (2.4) is estimated as

hL2ðfx; fyÞ ¼

Xminðmx;myÞ

j¼1

ðfxðcjÞ � fyðcjÞÞ2 !1=2

; ð3:10Þ

where mi is the number of annual radial basal area

increments for the tree x and y: In particular, if we consider

the random function ei(t) (3.9), this test function can be

written as

1676 Stoch Environ Res Risk Assess (2013) 27:1673–1681

123

hL2ðfx; fyÞ ¼

Xminðmx;myÞ

j¼1

ðexðcjÞ � eyðcjÞÞ2 !1=2

; ð3:11Þ

where c1; . . .; cm are calendar years, starting at the calendar

year c1 = 1989. Notice that effectively in (3.11) we only

consider the absolute difference between functions when

ei(t) functions coincide at the same time interval. We

therefore compare tree growth functions sharing the same

calendar years. This ensures that these trees were alive and

could interact to each other at the same time period (cal-

endar year). Otherwise, if we compare functions sharing

the same cambial year interval, this does not ensure

interaction between trees because trees having the same

cambial age not necessarily have to be alive at the same

time period (i.e. calendar year).

Moreover, for the purpose of illustration, we computed

an estimator of the pair correlation function (Illian et al.

2008) for tree positions via

gðrÞ ¼ 1

2prk2jW j

X 6¼

ðx;yÞ2w

jðkx� yk � rÞeðx; kx� ykÞ ð3:12Þ

in which w is a realisation of a point process. Broadly

speaking, this function indicates point inhibition when

g(r) \ 1, g(r) = 1 denotes the Poisson case (i.e. a random

point process) with no interaction between points, and

g(r) [ 1 implies point clustering. This function has been

applied to a wide range of forest studies including the

spatial analysis of even-aged forests (Penttinen et al. 1992;

Gavrikov and Stoyan 1995), tropical forest (Pelissier

1998), tree interaction of unmanaged forests (Leemans

1991; Szwagrzyk and Czerwczak 1993; Moeur 1993) and

the development of a selfthinning approach in even-aged

tree populations (Gavrikov 1995). Also to analyse the

spatial dependence of tree basal area at the year 1989, we

used an estimator of the mark correlation function (Stoyan

and Stoyan 1994)

kmmðrÞ ¼1

2prk2jW j

�X6¼

ðx;mxÞ;ðy;myÞ2wm

hmðmx;myÞjðkx� yk � rÞE½hmðmx;myÞ�eðx; kx� ykÞgðrÞ

ð3:13Þ

where wm is a realisation of a marked point process and mx

is the basal area of a tree at the spatial location x in R2:

Forest examples of this function can be found in Penttinen

et al. (1992) and Gavrikov et al. (1993).

4 Results

Figure 1 shows the spatial position of pine and spruce

stands along with the annual basal area increments for each

tree. Visual inspection of these spatial patterns does not

provide much information about the spatial dependence of

these spatially explicit functional data sets. Before ana-

lysing the spatial dependence of the function-marked point

patterns, we studied the unmarked structure of the point

patterns since, as explained, there can be inter-dependence

between marks (functions) and point locations, that can

completely affect the empirical mark (functional) spatial

structure.

The spatial point patterns of trees for both species are

shown in Fig. 2, and it does not highlight any apparent

structure for the point domain. The resulting estimator of

the pair correlation functions (3.12) and their respective

fifth largest and smallest envelopes based on 199 complete

(Poisson) point randomisations are shown in Fig. 3. This

suggests that both point configurations are at random, i.e.

trees are independently located from each other. Then the

spatial structure of tree positions should not affect greatly

the empirical mark and functional spatial configurations.

Note that under both tree species the empirical pair cor-

relation lies outside their respective simulated envelopes in

some specific points. Probably these outside excursions can

be interpreted as statistical artefacts, rather than significant

effects.

Figure 4a shows the empirical functional mark correla-

tion function for the interaction and random effects func-

tion ei(t) (3.9) assuming calendar years increments for the

Scots pine together with the fifth largest and smallest

envelopes for the functional mark correlation function

based on 199 random permutations of the functions over

(fixed) point (i.e. tree) positions. It can be seen that trees

have grown dependently from each other as the this

empirical function lies outside these upper and lower

envelopes. It shows that for trees at distances of\2.5 m, we

have similarities between growth functions. This suggests

that neighbouring trees share similar growth patterns and

that when the distance between trees increases this positive

correlation structures becomes less consistent.

Moreover, similar results are also obtained from the

Norway spruce stand, where there are positive spatial

correlation effects between tree growth patterns at dis-

tances in the range of 2–7 m (Fig. 3b). This last result is not

easy to interpret since it suggests that tree growth patterns

for trees in this distance range are more similar than those

for trees at distance \2 m (for instance).

Figure 5 shows the empirical mark correlation function

(3.13) for the BLUPS’s of the individual random effect bi

[see expression (3.6)] for the Scots pine and the Norway

spruce. Here, we analysed individual random effect

bi, summarising, for instance, genetic tree variation and/or

local site inhomogeneity, as being a mark associated to

each tree with test function hmðmx;myÞ ¼ 0:5ðmx � myÞ2

Stoch Environ Res Risk Assess (2013) 27:1673–1681 1677

123

(see Stoyan and Stoyan 1994) to avoid negative mark

effects. The resulting mark correlation kt(r) is usually

called mark variogram and typically is not normalized by

its expectation (Illian et al. 2008). It can be seen that these

random effects are spatially uncorrelated under the Scots

pine and present negative correlation effects for the

(a)

0 10 20 30 40 500

10

20

30

40

50

(b)

0 10 20 30 40 500

10

20

30

40

Fig. 1 Function-marked point

patterns involving the individual

basal area increments and tree

positions (dots) for two natural

forest stands; for each function

the x-coordinate (in meters)

starts at the calendar year 1989

and finish at the year 1715. Here

time runs backwards for x-axes

of individual tree growth

functions starting at the year

1989

(a)

0 10 20 30 40 500

10

20

30

40

50

(b)

0 10 20 30 40 500

10

20

30

40

Fig. 2 Locations of trees for athe Scots pine and b the Norway

spruce stands located in North

Karelia in Finland

(a)

r

g(r)

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

2.5

(b)

r

g(r)

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

2.5

Fig. 3 Empirical estimator of

the pair correlation function

(3.12) (given in meters) together

with the fifth largest and

smallest envelopes based on 199

complete (Poisson) point

randomisations

1678 Stoch Environ Res Risk Assess (2013) 27:1673–1681

123

Norway spruce at distance in the range of 2–7, virtually the

same range as the spatial dependencies of the ei(t) function.

Notice the random effect bi does not affect ei(t), since bi is

eliminated from this growth function [see expression (3.9)].

Finally, an additional analysis was also performed

considering the basal area spatial structure at the year 1989

for each tree species; this resulted in a standard marked

point pattern considering the product of marks as test

function (see Stoyan and Stoyan 1994). Figure 6 shows the

basal area point pattern and the mark correlation functions

(3.13) of these two marked point patterns along with the

largest and smallest envelopes based on 199 random

permutations of tree basal area over the fixed tree positions.

Now tree basal area is a quantitative variable summarising

growth dynamics of each tree in a single value. For the

Scots pine stand, this function suggests that tree basal areas

at distances of \2 m are negatively correlated (Fig. 6b),

thereby suggesting that trees located at short distances have

smaller tree sizes than they should have under the

hypothesis of random marking. Notice that this result is

virtually the same as that found under the functional mark

correlation. Under the Norway spruce the empirical mark

correlation lies slightly outside the simulated envelopes

around 2 m, thereby suggesting a weak negative correlation

(a)

r

g f(r)

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0(b)

r

g f(r)

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

Fig. 4 Empirical estimator of

the functional mark correlation

(2.5) and test function (3.11)

(black line) (given in meters)

together with the fifth largest

and smallest envelopes based on

199 random permutations of the

ei(t) function (3.9) over fixed

tree positions for a Scots pine,

and b Norway spruce

(a)

r

k t(r)

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

(b)

r

k t(r)

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Fig. 5 Empirical estimator of

the mark variogram (black line)

(given in meters) for the

BLUP’s of the individual

random effect bi (3.6) together

with the fifth largest and

smallest envelopes based on 199

random permutations of the

resulting marks over fixed tree

positions for a Scots pine, and bNorway spruce

Stoch Environ Res Risk Assess (2013) 27:1673–1681 1679

123

effects for this specific inter-tree distance. Thus, it is not

clear if this spatial dependence is either a significant effect

or a statistical artefact.

5 Discussion and conclusions

We have analysed the space–time structure of two spatially

explicit forest data sets by using the associated growth

function for each tree obtained from the annual radial

growth measured from increment cores bored at breast

height. To do so, we considered a new second order for-

mulation defined by Comas et al. (2011b) based on the

mark correlation function. In particular, we analysed the

space–time dependent component of BAi(t), i.e. ei(t).

Under the Scots pine the empirical functional mark

correlation function suggests similar growth dynamics for

trees at distance of\2.5 m. This is virtually the same range

of correlation as that found for tree basal area. This sug-

gests that, for this species, spatial growth dynamics are

well explained by tree basal area. In the relatively even-

aged pine stand, the results indicate that neighbouring trees

having the same age have also similar growth dynamics,

although they are affected by strong competition effects

(negative basal area dependences). This may be due to low

genetic variation in growth properties of trees and low

variation in soil and site conditions. Thus, no one of the

neighbouring trees is winning and all of them are suffering

from competition. This last point of low genetic variation

and low variation in soil and site conditions are also

observed in the spatial distribution of tree random effects

bi. The empirical mark correlation function of these tree

effects suggested that either these variations are at random

or else they are too low to have spatial dependencies.

Furthermore, under the Norway spruce, the functional

correlation function shows similar growth dynamics for

trees at distances in the range of 2–7 m, whilst the mark

correlation function for tree basal area only detects such

spatial configurations around a distance of 2 m between

trees. This suggests that although the final tree size (basal

area) shows basically spatial independence, the growth

dynamics to obtain such final size are spatial dependent. So

tree-to-tree interaction effects are basically reflected in tree

dynamics but not in the final tree size. Because tree basal

area is a cumulative measure of tree growth it may fail

detecting interaction dynamics that happens over time.

This is so since the basal area measure may not only

account for tree-to-tree interaction effects but also for

deterministic effects such as the total age trend and the

total calendar year effects. These cumulative effects in the

basal area may hide the space–time interaction dynamics

that rule such systems. In fact it is possible having two

trees with very different basal areas, but sharing similar

growth dynamics. Moreover, the spatial distribution of tree

random effects bi suggested that under the spruce stand

random tree effects, such as genetic and site variation, are

spatially correlated stretching the idea that neighbouring

trees share similar growth patterns.

Miina (1993) studied the time series of residuals of

diameter growth models having fixed tree and stand char-

acteristics as predictors for the same two data sets. He

showed that the annual residuals were autocorrelated

(autoregressive of first order) and that in general time

patterns and trends were more variable for the spruce than

(a)

0

10

20

30

40

50

(b)

0

10

20

30

40

(c)

r

k mm(r)

0.0

0.5

1.0

1.5(d)

r

k mm(r)

0 10 20 30 40 50 0 10 20 30 40 50

0 2 4 6 8 10 0 2 4 6 8 100.0

0.5

1.0

1.5

Fig. 6 Mark point patterns of

tree basal area for a Scots pine

and b Norway spruce, and the

resulting estimator of mark

correlation function (3.13) and

test function hmðmx;myÞ ¼mxmy (black line) (given in

meters) for the tree basal

together the fifth largest and

smallest envelopes based on 199

random permutations of the

resulting marks over fixed tree

positions for c Scots pine and

d Norway spruce; bubble plot

radius is proportional to tree

basal area

1680 Stoch Environ Res Risk Assess (2013) 27:1673–1681

123

they were for the pine stand. This autocorrelation structures

are therefore spatial dependent and that for trees located at

short distances these patterns are positively correlated.

Therefore, the combined space–time analysis of the ran-

dom residual function ei(t) can be crucial to understand

fully complex forest space–time dynamics.

Our results confirm the usefulness of the functional

mark correlation function compared with other well-

established spatial statistical tools such as the mark corre-

lation function. In particular, the functional mark correla-

tion function of the spatial and temporal component of the

tree growth function permits to analyse the space–time

structure of tree development regardless of the non-spatial

components contained in this function. Moreover, this

explicit temporal analysis detects space–time interaction

effects that are not evident when analysing the spatial

distribution of cumulative growth measures such as the tree

basal area.

Acknowledgements We are grateful to the Editor, AE and two

anonymous referees whose comments and suggestions have clearly

improved an earlier version of the manuscript.

References

Baddeley A, Møller J, Waagepetersen R (2000) Non- and semi-

parametric estimation of interaction in inhomogeneous point

patterns. Stat Neerl 54:329–350

Barnard G (1963) Contribution to discussion of ‘‘The spectral analysis

of point processes‘‘ by M.S. Bartlett. J R Stat Soc B 25:294

Besag J, Diggle PJ (1977) Simple Monte Carlo tests for spatial

pattern. Appl Stat 26:327–333

Cajander AK (1926) The theory of forest types. Acta For Fenn 29:l–

108

Comas C (2009) Modelling forest regeneration strategies through the

development of a spatio-temporal growth interaction model.

Stoch Env Res Risk Assess 23(8):1089–1102

Comas C, Palahı M, Pukkala T, Mateu J (2009) Characterising forest

spatial structure through inhomogeneous second order charac-

teristics. Stoch Env Res Risk Assess 23:387–397

Comas C, Mateu J (2007) Modelling forest dynamics: a perspective

from point process methods. Biom J 49(2):176–196

Comas C, Mateu J, Delicado P (2011a) On tree intensity estimation

for forest inventories: some statistical issues. Biom J

53(6):994–1010

Comas C, Delicado P, Mateu J (2011b) A second order approach to

analyse spatial point patterns with functional marks. TEST

20:503–523

Cressie N (1993) Statistics for spatial data. Wiley, New York

Daley DJ, Vere-Jones D (2004) An introduction to the theory of point

processes, 2nd ed. Springer, New York

Diggle PJ (2003) Statistical analysis of spatial point patterns. Hodder

Arnold. London.

Dwass M (1957) Modified randomization tests for nonparametric

hypotheses. Ann Math Stat Statistics 28:181–187

Gavrikov VL (1995) A model of collisions of growing individuals: a

further development. Ecol Model 79:59–66

Gavrikov VL, Stoyan D (1995) The use of marked point processes in

ecological and environmental forest studies. Environ Ecol Stat

2:331–344

Gavrikov VL, Grabarnik P, Stoyan D (1993) Trunk-top relations in a

Siberian pine forest. Biom J 35:487–498

Gort J, Mehtatalo L, Peltola H, Zubizarreta-Gerendiain A, Pulkkinen

P, Venalainen A (2011) Effects of spacing and genetic entry on

radial growth and ring density development in Scots pine (Pinus

sylvestris L.). Ann For Sci 68(7):1233–1243

Grabarnik P, Myllymaki M, Stoyan D (2011) Correct testing of mark

independence for marked point patterns. Ecol model

222:3888–3894

Harrell FE (2001) Regression modeling strategies. Springer, New

York.

Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis

and modelling of spatial point patterns. Wiley, London.

Isham V (1987) Marked point processes and their correlations. In:

Droesbeke F (ed) Spatial processes and spatial time series

analysis, proceedings of the 6th Franco-Belgian meeting of

statisticians, Rouen

Leemans R (1991) Canopy gaps and establishment patterns of spruce

(Picea abies (L.) Karst.) in two old-growth coniferous forest in

Central Sweden. Vegetation 93:157–165

Miina J. (1993) Residual variation in diameter growth in a stand of

Scots pine and Norway spruce. For Ecol Manag 58:111–128.

Miina J (2000) Dependence of tree-ring, earlywood and latewood

indices of Scots pine and Norway spruce on climatic factors in

eastern Finland. Ecol Model 132:259–273

Moeur M (1993) Characterizing spatial patterns of trees using stem-

mapped data. For Sci 39:756–775

Park A, Kneeshaw D, Bergeron Y, Leduc A (2005) Spatial

relationships and tree species associations across a 236-year

boreal mixedwood chronosequence. Can J For Res 35:750–761

Pelissier R (1998) Tree spatial patterns in three contrasting plots of a

southern Indian tropical moist evergreen forest. J Trop Ecol

14:1–16

Penttinen A, Stoyan D (1989) Statistical analysis for a class of line

segment processes. Scand J Stat 16:153–161

Penttinen A, Stoyan D, Henttonen HM (1992) Marked point processes

in forest statistics. For Sci 38:806–824

Pinheiro JC, Bates DM (2000) Mixed-effects models in S and

S-PLUS. Statistics and computing series. Springer, New York.

Renshaw E, Comas C, Mateu J (2009) Analysis of forest thinning

strategies through the development of space–time growth–

interaction simulation models. Stoch Env Res Risk Assess

23:275–288

Ripley BD (1976) The second-order analysis of stationary point

processes. J Appl Probab 13:255–266

Ripley BD (1977) Modelling spatial patterns (with discussion). J R

Stat Soc B 39:172–212

Stoyan D (1984) On correlations of marked point processes. Math

Nachr 116:197–207

Stoyan D, Stoyan H (1994) Fractals, random shapes and point fields:

methods of geometrical statistics. Wiley, Chichester.

Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its

applications. Wiley, New York

Szwagrzyk J, Czerwczak M (1993) Spatial patterns of trees in natural

forests of East-Central Europe. J Veg Sci 4:469–476

Stoch Environ Res Risk Assess (2013) 27:1673–1681 1681

123