binary mosaics and the spatial pattern of heather

Binary Mosaics and the Spatial Pattern of HeatherAuthor(s): Peter J. DiggleSource: Biometrics, Vol. 37, No. 3 (Sep., 1981), pp. 531-539Published by: International Biometric SocietyStable URL: http://www.jstor.org/stable/2530566 .

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BIOMETRICS 37, 531-539 September 1981

Binary Mosaics and the Spatial Pattern of Heather

Peter 1. Diggle Department of Statistics, University of Newcastle upon Tyne,

Newcastle upon Tyne NE1 7RU, England

S UMMARY

An idealized model for the spatial pattern of heather in a planar region is proposed; in this model the subregiorl occupied by heather is represented by the union of randomly located discs of randomly varying radii. Some previous applications of the model are noted. Techniques for parameter estimation and for testing goodness of fit are suggested; these techniques are applied to data showing the incidence of heather in a lOmX20m rectangle at Jadraas, Sweden, thereby demonstrating the extension to binary mosaics of methodology previously illustrated with spatial point processes.

1. Introduction

A binary mosaic is a partition of a planar region into two qualitatively different subregions. In ecological work this partition might, for example, represent the incidence of a particular species of vegetation over a study region. Figure 1 shows such a representation of the incidence of heather, Calluna vulgaris, over a 10 m x 20 m rectangle at Jadraas, Sweden. These data were collected by G. Agren, T. Fagerstrom and the author as part of a general investigation into the natural development of a young coniferous stand in which heather and pine, Pinus sylvestris, were the two predominant species. A description of the spatial pattern of the heather was required as a preliminary to the assessment of small-scale spatial association between the two species. The map was prepared in the field at a scale of 1:100 and subsequently coded as a lOOx200 binary matrix in which each element represents an area 10 cm square. This was thought to be a realistic assessment of the accuracy achieved in the field. The study area, which formed part of the experimental site at Jadraas operated by the Swedish Coniferous Forest Project, contained no obvious inhomogeneities and had been allowed to regenerate naturally following clear-cutting some 25 years previously.

Pielou's (1964) approach to the analysis of binary mosaics in ecology was to treat successive observations at discrete intervals along a line transect as a realization of a two-state Markov chain. Kershaw (1957) used a modification of Greig-Smith's (1952) technique for contiguous quadrat counts. Neither author referred to possible underlying models, although Switzer (1965) showed that a Markov chain analysis is appropriate if the partition is effected by random lines in the plane, as defined by Miles (1964). In the present paper it is shown how binary mosaic data like Fig. 1 can be analysed in terms of an explicit stochastic model, using the methodology described by Diggle (1979) for the analysis of spatial point patterns.

2. A Model for the Spatial Pattern of Heather

The growth process of heather was described in detail by Gimingham (1972). Individual heather plants grow from seedlings into hemispherical bushes, reaching a maximum radius

Key words: Binary mosaic; Ecology; Random set; Spatial pattern. 531



532 Biometrics? September 1981

-Pt-b* +% g f -{ O m m-eS

Figure 1. Incidence of Calluna vulgaris over a 10 m x 20 m rectangular area at Jadraas, Sweden.

of about 50cm at age 20-25years. As the individual bushes expand, their branches intermingle and they occupy overlapping areas of ground. Very old bushes enter a degenerate phase during which they die away from their centres to leave rings of live plant material. This degenerate phase was not evident in the study region from which Fig. 1 was derived.

The above description suggests the following as a provisionalv and obviously idealized, model. Individual bushes occupy circular areas whose radii are determined as independent realizations of a non-negative random variable. The centres of these circular bushes form a realization of a homogeneous, planar Poisson process, and their union represents the region of the plane occupied by heather.

This model has appeared previously in a variety of contexts. Armitage (1949) used it to describe the random clumping of dust particles; Roach (1968) gave a review of this type of application. The case of constant radii was referred to by Matern (1960, Ch. 3) as a 'bombing process', and in photographic science it appears as the 'random-dot model' (Marchant and Dillon, 1961). Finally, the model appears in the theory of random sets as a special case of Matheron's 'Boolean schemes' (Matheron, 1975).

3. Statistical Methods

The basis of our statistical analysis will be the comparison of functional summary descriptions of the model with the corresponding empirical descriptions of the data. This follows recent developments in the analysis of spatial point patterns; we refer the reader to Ripley (1977) or Diggle (1979) for methodological details.

Two potentially useful functional summary descriptions can be deduced from the general theory of random sets developed by Matheron (1975), or by direct probability arguments of the type used by Bartlett (1975, Ch. 1) in his discussion of spatial point processes. In the model, let R denote the radius of a bush, with probability density function J(r), and let A denote the mean number of bushes per unit area. Define G(u) to be the distribution function of the distance from an arbitrary point in the plane to the nearest point occupied by heather. Then

G(u)= 1-exp{- 7rA 0(u+r)2t(r) dr}, u¢0. (1)

Note that G(0)-1-exp{-rA E(R2)} is the proportion of the plane occupied by heather,



Binary Mosaics and Spatial Pattem 533

and that G(u) depends on the distribution of R only through its first two moments. Now define ty(u) to be the covariance between values from points a distance u apart, coding 1 for a point occupied by heather, otherwise 0. Then

y(u)=exp{-27rAE(R2)}[exp{A | V"(r)f(r) dr}-1] u¢0> (2)

where Vu (r) = 2r2[cos-1{u/(2r)} - {u/(2r)}{1 - u2/(4r2)}2]

is the area of intersection of two discs with common radius r and centres a distance u apart.

Empirical versions of (1) and (2) are calculated by sampling the data over a fine grid of points. The parameters of the model are then estimated by a 'least squares' method to mlnlmlze, ror examp e,

ro: {G(u)- G(u)}2 du (3)

o

(a) Plot 1 O. 1 § - f , . i _

0.0 V\ S -0.1 +0.1 (b) Plot 2

-0.1 +0.1 Figure 2. Two-dimensional correlation functions for the heather data, contoured at intervals of 0.1

(origin bottom centre).



534 Biownetrics, September 1981

which can be considered as a function of the model parameters (cf. Bartlett and

Macdonald, 1968). Finally, simulations of the fitted model are used to assess goodness of

fit, in the manner of Barnard's (1963) Monte Carlo testing procedure.

4. Analysis of the Data

Figure 1 is split into two square plots, which we take to be of unit side, in order to provide

the opportunity for some cross-validation of results. The two-dimensional empirical correlation functions for the two plots are given in Fig. 2. The general appearance of the

two correlation functions, and in particular their close similarity, support the assumptions of stationarity and isotropy which are implicit in the model. We therefore fit the model

to the data, using (3). For Plot 1, the estimated parameter values give the mean number of

bushes per unit area as A = 397, and the mean and the standard deviation of bush-radius, R, as 0.0120 and 0.0203, respectively. To simulate the fitted model we need to adopt a

specific distribution for R. Accordingly a Weibull distribution

t(r)= kprk-1 exp(-prk) r ¢ 0,

in which the parameters k and p have been chosen to give the appropriate mean and

standard deviation, is employed in Fig. 3. A glance at Figs 1 and 3 shows how G(u) fails

to reflect the detailed geometry of small isolated patches in a mosaic. The covariance function y(u) provides an objective measure of this discrepancy between model and data

in that, as indicated in Fig. 4, the fitted covariance functions decline too slowly as u

increases. We therefore repeat the fitting procedure, using y(u) instead of G(u) in (3)

and integrating from 0 to 0.1. The upper limit of 0.1 is somewhat arbitrary; it is imposed

to make the estimation procedure insensitive to the precise values of the empirical covariances for large u, but for these data, its exact choice turns out not to be critical. We

also add a third parameter to the distribution of R, so that

f(r) = kp(r-a)k-l expt-p(r _ a)k}, r ¢ b. (4)

v^|j *:>

Figure 3. Realization of the model with Weibull distribution for R and with parameters (A, k, p)= (397, 0.618, 19.37).



-

l l | - - iK q

Binary Mosaics and Spatial Pattern 535

This gives a flexible class of distributions in which 6 represents a minimum value of R, whilst k and p allow a range of reverse J-shaped or unimodal distriblltions for R-b. Estimates of (A, 6, k, p) are (221, 0.0281, 0.8471, 144.7) for Plot 1, and (211, 0.0226, 1.011, 128.4) for Plot 2.

(a) Plot 1

z (u)

\ - \ \

\ \ \\

\ \ \ \

\ \

C ' -

\ -

- >

- t

l l l - l

0.254

0.20 -

0.15 -

0.10 -

- -

0.05 -

u

0.02 0.04 0.06 Q.08 0.10 (b) Plot 2

z (u)

0.1

0.1(

u

0.08 0.10 0.02 0.04 0.06 Figure 4. Observed and fitted covariance functions for the heather data. Key: @ observed;

fitted by G(@): --- fitted by yt ).



536 Biometrics, September 1981

Figure 4 shows that these parameter values now give very close agreement between empirical and fitted covariance functions. In addition, Fig. 5 shows that the fit remains good when assessed by the distribution function G(u). If we use (3) as a goodness-of-fit statistic for the model, we can rank the data amongst 99 simulated realizations of the model, with high ranks indicative of lack of fit. In the event, the ranks obtained for the two plots are 39 and 919 which seem quite satisfactory. Note that the data from Plot 1 are compared with simulations using parameter values estimated from the Plot 2 datas and

(a) Plot 1 data, parameters estimated from Plot 2

G (u)

1"

u

(b) Plot 2 data, parameters estimated from Plot 1

G (u}

oFe

u

0.02 0.04 0.06 0.08

Figure 5. Goodness-of-fit tests based on G(@). Key: O data; extremes from 99 simulations of fitted model.



Binary Mosaics and Spatial Pattern 537

vice versa. Finally, Fig. 6 shows a realization of the fitted model for each of the two plots and this may again be compared with the data in Fig. 1.

5. Discussion

Whilst it is encouraging that such a simple model appears to fit the data well, the possible limitations of the available statistics must not be forgotten. For example, a comparison between Figs 1 and 6 suggests that the data contain distinctly fewer separate patches than

(a) Plot 1, 01 = (221, 0.0281, 0.8471, 144.7)

(b) Plot 2, 09 = (21 1 ? 0.0226, 1.01 1, 128.4)

Figure 6. Realizations of fitted models.



538 Bioncetrics, September 1981

are generated by the fitted models. The significance of this effect is confirmed by further simulationX but the obvious conclusion must be qualified by the limited resolution in Fig. 1, which involves some implicit smoothing of the underlying patterla. Certainly, the strict circularity of the areas occupied by individual bushes and the precise form (4) for f(r) are no more than convenient mathematical fictions. Nevertheless, the fitted models do permit a reasonable biological interpretation. The estimated parameter values translate to an intensity of about two individual heather bushes per square metre, with a mean and a standard deviation for bush radius of about 30 and 6 cm, respectively. Figure 6 shows how the model produces patches of overlapping bushes which are both larger and very irregular in shape.

Two reasonable modifications to the model would be firstly to allow some other point process to define the bush centres, and secondly to incorporate some form of dependency between the radii of adjacent bushes. These modifications appear to be intractable, but could be used in simulation studies when the basic model proves inadequate.

ACKNOWLEDGEMENTS

The author began this work during a visit to the Swedish Coniferous Forest Project, Uppsala. I thank colleagues there, especially Goran Agren and Tobbe Fagerstrom, for helpful discussions. I also thank Professor C. H. Gimingham for information on the growth process of heather.

RESUME

On propose un modele idealise de la repartition spatiale de la bruyere dans une region plane; dans ce modele, la sous-region occupee par la bruyere est representee par la reunion de disques de centre et de rayon aleatoires. Quelques applications anterieures du modele sont soulignees. On suggere des techniques d'estimation de parametres et de test de la qualite d'adjustement; ces techniques sont appliquees a des donnees relatives a la repartition de la bruyere dans un rectangle de 10 m sur 20 m a Jadraas, Suede, ce qui constitue une extension aux mosaiques binaries d'une methodologie illustree auparavant avec des processus ponctuels spatiaux.

R EFER EN C ES

Armitage, P. (1949). An overlap problem arising in particle counting. Biometrika 36, 257-266. Barnard, G. A. (1963). Discussion of Professor Bartlett's paper. Joumal of the Royal Statistical

Society, Series B 25, 294. Bartlett, M. S. (1975). The Statistical Analysis of Spatial Pattem. London: Chapman & Hall. Bartlett, M. S. and Macdonald, P. D. M. (1968). 'Least squares' estimation of distribution mixtures.

Nature 247, 195-196. Diggle, P. J. (1979). On parameter estimation and goodness-of-fit testing for spatial point patterns.

Biometrics 35, 87-101. Gimingham, C. H. (1972). Ecology of Heathlands. London: Chapman & Hall. Greig-Smith, P. (1952). The use of random and contiguous quadrats in the study of the structure of

plant communities. Annals of Botany 16, 293-316. Kershaw, K. A. (1957). The use of cover and frequency in the detection of pattern in plant

communities. Ecology 38, 291-299. Marchant, J. C. and Dillon, P. L. P. (1961). Correlation between random-dot samples and the

photographic emulsion. Joumal of the Optical Society of America 51, 641-644. Matern, B. (1960). Spatial Variation, Bd. 49, No. 5. Stockholm: Meddelanden fran statens

skogsforskningsinstitut. Matheron, G. (1975). Random Sets and Integral Geometry. New York: Wiley.



Banary Mosaics and Spatial Pattem 539

Miles, R. E. (1964). Random polygons determined by random lines in a plane. Proceedings of the National Academy of Sciences 52, 901-907.

Pielou, E. C. (1964). The spatial pattern of two-phase patchworks of vegetation. Biometrics 20, 156-167.

Ripley, B. D. (1977). Modelling spatial patterns (with discussion). Journal of the Royal Statistical Society, Series B 39, 172-212.

Roach, S. A. (1968). The Theory of Random Clumping. London: Methuen. Switzer, P. (1965). A random set process in the plane with a Markovian property. Annals of

Mathematical Statistics 36, 1859-1863.

Received January 1980; revised August 1980



binary mosaics and the spatial pattern of heather

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