2018
Relationship between plot size and the variance
of the density estimator in West African
savannas
N. Picard, Y. Nouvellet, and M.L. Sylla
Abstract: The relationship between the variance of the density estimator and the plot size in forest inventories based
on fixed area plots was characterized in West African savannas. Nine sites ranging from dry to moist savanna were
surveyed, and the variance of the density estimator was assessed for varying sample plot sizes. An approximate theoretical
expression of the variance was derived, taking into account the uncertainty on plot limits. In six sites, a Matérn process
was fitted to the spatial pattern of trees. The point process was used to generate spatial patterns, yielding simulated
values of the variance. A power function was also fitted to observed and simulated data. The theoretical expression and
simulations showed that the contribution of uncertain plot limits to the variability of the density estimator was negligible
with respect to the contribution of the spatial pattern of trees. The theoretical expression matched the data for small areas,
but was inaccurate for large areas. The power function provided better fits. Nevertheless, the theoretical expression did not
require any statistical fit to data and established a clear link between the spatial pattern of trees and the variance of the
density estimator, with interpretable parameters.
Résumé : La relation entre la variance de l’estimateur de la densité et la taille des placettes dans les inventaires forestiers
fondés sur des placettes de taille fixe est caractérisée dans des savanes d’Afrique de l’Ouest. Neuf sites, allant de la savane
sèche à humide, ont été inventoriés et la variance de l’estimateur a été estimée pour des tailles de placette variables.
Une expression théorique approchée de la variance a été établie, en prenant en compte les erreurs dans la délimitation
des placettes. Dans six des sites, un processus de Matérn a été ajusté à la répartition spatiale des arbres. Ce processus
ponctuel est utilisé pour générer des répartitions spatiales, ce qui donne des valeurs simulées de la variance. Une fonction
puissance a aussi été ajustées aux données observées et simulées. L’expression théorique et les simulations montrent que
la contribution des limites variables des placettes à la variabilité de l’estimateur de la densité est négligeable par rapport
à celle de la répartition spatiale des arbres. L’expression théorique s’accorde aux données pour des surfaces faibles, mais
est inexacte pour des surfaces élevées. La fonction puissance donne de meilleurs ajustements. Cependant l’expression
théorique ne requiert aucun ajustement statistique à des données, et elle établit une relation claire entre la répartition
spatiale des arbres et la variance de l’estimateur de la densité, avec des paramètres interprétables.
Introduction
Estimating tree density with fixed area sample plots is one of
the most common methods in forestry. Given n sample plots of
area s and the counts N
i
of trees in each plot (i = 1,...,n), an
unbiased estimator of the density λ is simply
ˆ
λ =
n
i =1
N
i
/ns
If the spatial pattern of trees is homogeneous, the shape of the
plot has no impact on the properties of
ˆ
λ (Schreuder et al. 1987).
However, the distribution of
ˆ
λ depends on the count N of trees
in an arbitrary plot, that in turn depends on the kind of spatial
Received 24 November 2003. Accepted 20 April 2004. Published
on the NRC Research Press Web site at http://cjfr.nrc.ca/ on 12 Oc-
tober 2004.
N. Picard
1
and Y. Nouvellet. Cirad, BP 1813, Bamako, Mali.
M.L. Sylla. Institut Polytechnique Rural de Formation et de
Recherche Appliquée (IPR-IFRA), BP 6, Koulikoro, Mali.
1
Corresponding author (e-mail: nicolas.picard@cirad.fr).
pattern of the trees. In particular, the variance of
ˆ
λ is
[1] Var(
ˆ
λ) =
1
ns
2
Var(N)
The question then is to work out an optimal size s of the sample
plots (Savage 1956). This question has triggered many studies
(Johnson and Hixon 1952; Bormann 1953; Hebert et al. 1988;
Grenier et al. 1991; Gray 2003). It may be expressed in sev-
eral ways: Bormann (1953) and Schreuder et al. (1987) seek
to minimize the variance of the estimator at a fixed sampling
rate; Zeide (1980), Wiant and Yandle (1980), and Gambill et
al. (1985) seek to minimize the total time (or cost) of inventory
at a given precision on
ˆ
λ; Johnson and Hixon (1952) seek to
maximize the precision of
ˆ
λ for a total time of inventory (it ac-
tually yields the same optimum as before); Hebert et al. (1988)
seek to minimize the variance of the estimator for a total time
of inventory; etc.
In all these optimization issues, a relationship between the
variance of N and the size s of the plots is required. Let A be the
area of the forest and τ = ns/A be the sampling rate. Equation
1 may be rewritten as
[2] Var(
ˆ
λ) =
1
τA
Var(N)
s
Can. J. For. Res. 34: 2018–2026 (2004) doi: 10.1139/X04-079 © 2004 NRC Canada