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Zucchini, Schmidt and von Gadow A Model for the Diameter-Height Distribution in an Uneven-Aged Beech Forest and ...

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A Model for the Diameter-HeightDistribution in an Uneven-Aged BeechForest and a Method to Assessthe Fit of Such Models

Walter Zucchini, Matthias Schmidt and Klaus v. Gadow

Zucchini, W., Schmidt, M. & Gadow, K. von. 2001. A model for the diameter-height distribu-tion in an uneven-aged beech forest and a method to assess the fit of such models. SilvaFennica 35(2): 169–183.

This paper illustrates the application of a mixture model to describe the bivariatediameter-height distribution of trees growing in a pure, uneven-aged beech forest. Amixture of two bivariate normal distributions is considered but the methodology isapplicable to mixtures of other distributions. The model was fitted to diameter-heightobservations for 1242 beech trees in the protected forest Dreyberg (Solling, Germany).A considerable advantage of the model, apart from the fact that it happens to fit this largedata set unusually well, is that the individual parameters all have familiar interpretations.The bivariate Johnson SBB distribution was also fitted to the data for the purpose ofcomparing the fits.

A second issue discussed in this paper is concerned with the general question ofassessing the fit of models for bivariate data. We show how a device called “pseudo-residual” enables one to investigate the fit of a bivariate model in new ways and inconsiderable detail. Attractive features of pseudo-residuals include the fact that they arenot difficult to interpret; they can be computed using generally available statisticalsoftware and, most important of all, they enable one to examine the fit of a model bymeans of simple graphs.

Keywords diameter-height distribution, mixture models, bivariate normal distribution,SBB distribution, goodness-of-fit, pseudo-residuals, beech forest.Authors’ addresses Zucchini: Georg-August-Universität Göttingen, Institute for Statisticsand Econometrics; Schmidt: Forest Research Station of Lower Saxony; von Gadow:Georg-August-Universität Göttingen, Institute for Forest Management and Yield SciencesFax +49 551 397 279 E-mail [email protected] 10 November 1999 Accepted 22 January 2001

Silva Fennica 35(2) research articles


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1 Introduction

One of the most important elements of foreststructure is the relationship between tree diame-ters and heights. Information about size-classdistributions of the trees within a forest stand isimportant for estimating product yields. The size-class distribution influences the growth potentialand hence the current and future economic valueof a forest stand (Knoebel and Burkhart 1991).More recently, a rather different need for diame-ter-height relations arose in response to an in-creasing interest in the structure of natural for-ests. Unmanaged forests are used as a standardfor comparison of different types of managedstands, using indices such as the Shannon-Indexfor analysis of vertical structure (Shannon 1948,Weber 1998). Thus, detailed modelling of thevariation of heights within diameter classes isrequired.

The product yield estimates are usually basedon an assessment of the distribution of diametersand a common practice is to fit a model, such asthe Weibull function, to the empirical diameterclass frequencies (Clutter and Allison 1974,Gadow 1987, Saborowski 1994). For improvedproduct estimates, tree heights are routinely as-sessed and a variety of models are used for de-fining the relationship between diameters andheights (Schmidt 1967, Curtis 1967). The tradi-tional method does not quantify the distributionof heights for a given diameter and one approachfor modelling the conditional height distributionfor the different diameters is to use the heightresiduals (Gaffrey 1996, p. 258). In our experi-ence, it is very seldom that the height residualsare homoscedastic and normally distributed. Inmost forest stands the variance about the diame-ter-height regression is heterogeneous.

Hence there has been considerable interest inidentifying suitable bivariate distributions to de-scribe diameter-height frequency data. It has beenreported that the SB distribution (Johnson 1949a)fits the marginal frequencies of both diameterand height consistently better than the Weibull,beta, gamma, lognormal and normal distribu-tions (Johnson 1949b, Hafley and Schreuder1976) and that the bivariate extension of the SB

distribution, the SBB, is more realistic and pro-vides more useful information than the currently

accepted approach for describing forest diame-ter-height data (Schreuder and Hafley 1977).

Beech forests often develop a single layer struc-ture when management stops, as a result of heavyshading and drought eliminating the smaller sup-pressed trees of this species. A variety of verticalstructures may develop in unmanaged beech for-ests, depending on the successional stage. Thereare numerous examples showing that virgin beechforests exhibit structures which include more thanone layer of tree heights (Korpel 1992, Kosir1966). In a managed beech forest, the verticalstructure depends on the type of thinning that isapplied. In a high thinning, which is generallypractised in Germany, only bigger trees are re-moved while the smaller ones may survive for avery long time, resulting in a typical pattern withtwo subpopulations. Usually, the gradient of thediameter-height regression is much steeper inthe smaller subpopulation. A possible explana-tion for this phenomenon may be the greatercompetition for light in the lower layer. Thus,trees in the lower layer have to support heightgrowth more than diameter growth. On the otherhand, because of static reasons, trees that havereached the upper layer have to invest more indiameter growth. The result of these differentstrategies are different diameter-height relationsand, for most managed beech forests, it would bebiologically plausible to find that the populationof trees is composed of a mixture of two subpop-ulations having different diameter-height distri-butions. The particular model that is discussedhere is specifically designed to describe suchpopulations.

The purpose of this paper is to illustrate howone can go about fitting a mixture of bivariatenormal distributions to diameter-height observa-tions. The bivariate Johnson SBB was also fittedto the data for the purpose of comparison. Sec-ondly we show how “pseudo-residuals” can beused to assess the fit of this, or any other, modelthat one has fitted to univariate or bivariate ob-servations.


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2 Material and Methods

2.1 Study Material

The models investigated in this paper were fittedto diameter-height measurements on 1242 beechtrees in the protected forest Dreyberg (Solling,Lower Saxony, Germany). The data were col-lected in 1995 by the Forest Research Station ofLower Saxony from 50 permanent 1000 m2 cir-cular plots on a 100 m × 100 m regular grid. Theage of the stand, between 113 and 120 years, canbe determined only approximately because thestand originated from natural regenerations. Somestand characteristics are given in Table 1.

The dominant species is beech and therefore thestand is close to the potential association of a lu-

zulo fagetum which is typical for extensive areasin the Southern Lower Saxony hill country.

A simple height-diameter curve as shown inFig. 1, describes how the mean height varieswith diameter (at breast height) but it does notquantify the complete distribution of heights foreach diameter.

Fig. 2 shows a nonparametric (kernel) esti-mate of the bivariate diameter-height distribu-tion (see for example Silverman 1986). This fig-ure indicates that there may be two subpopula-tions of trees and that the height-diameter rela-tionship differs in these subpopulations. The larg-er subpopulation (approximately 80% of the en-tire population) comprises larger trees in whichthe slope of the height-diameter regression isless steep than that for the smaller subpopulation(approximately 20% of the population).

One way of modelling such a distribution is touse a mixture of two bivariate normal distribu-tions described in Section 2.2). The resultingregression curve (the conditional expectation ofheight given diameter) for such a model is asmooth function of the two straight lines: steeperfor the smaller trees subpopulation and less steepfor the dominant trees subpopulation. The gener-al shape of this curve is similar to that shown inFig. 1.

Table 1. Mean stand characteristics with standard error(s.e.) for the whole stand and for beech trees of theDreyberg data.

Whole stand s.e. Beech

Trees/ha 273.4 10.1 269.6Stand basal area (m2 / ha) 28.4 0.5 28.0Standing volume (m3 / ha) 383.3 9.2 377.9

Fig. 1. Breast height diameter–height data for 1242 beech trees in the protected forestDreyberg (Solling). The height-curve has been drawn using h = eα+β/d with α = 3.6638and β = –12.601.


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Section 2.3 gives an outline of the Johnson SBB

model which we also fitted to the data for thepurpose of comparison.

2.2 A Mixture of Two Bivariate NormalDistributions

Let f(d, h) denote the bivariate probability densi-ty function of diameter and height. The proposedmodel is

f d h n d h n d h( , ) ( , ) ( ) ( , )= + −α α1 21 (1)

where α, a parameter in the interval (0, 1), deter-mines the proportion of trees belonging to eachof the two component bivariate normal distribu-tions n1(d, h) and n2(d, h). The parameters ofnj(d, h) are the expectations µdj, µhj; the varianc-es σdj

2 and σhj2, and the correlation coefficient, ρj,

j = 1, 2. Thus the full probability density func-tion of diameter and height is given by (1) with

n d h

J z z z z

j

j

dj j dj hj hj

j

( , )

exp( )

=

−−

− +−

2 1

2

2 12

2 2

2π ρ

ρρ

(2)

zdj = (d – µdj) / σdj, zhj = (h – µhj) / σhj andJ = (σdjσhj)–1, j = 1,2

The term J, the Jacobian of the transformationfrom the standard to the general bivariate nor-mal, is identified explicitly here in order to con-trast this distribution with the Johnson SBB inSection 2.3.

The marginal distributions for the diametersand heights are also mixtures of the correspond-ing component marginal distributions. They areof the form

f x n x n x( ) ( ) ( ) ( )= + −α α1 21 (3)

The components for the diameters and heightsare given, respectively, by

n dz

n hz

j

jdj

dj

jhj

hj

( )exp( / )

,

( )exp( / )

, ,

=−

=−

=

2

2

2

2

2

21 2

πσ

π σ

(4)

The conditional distributions of height given di-ameter, and of diameter given height are notmixtures of normal distributions but they can becomputed using expressions (1) to (4) from theirdefinitions:

Fig. 2. Perspective plot of a nonparametric (kernel) estimate of the bivariate densityfunction for the diameters and heights.


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f h d f d h f d

f d h f d h f h

( ) ( , ) / ( ),

( ) ( , ) / ( )

=

=(5)

The maximum likelihood estimators of the pa-rameters of this model are not available in ex-plicit form but the estimates are not difficult tocompute by directly maximizing the likelihoodfunction or (more conveniently) by minimizingthe negative log-likelihood function:

l( ;( , ), , ,..., )

log( ( , ))

Parameters d h i n

f d h

i i

ii

n

i

= =

−=∑

1 2

1

(6)

where (di, hi), i = 1, 2,..., n, are the observed val-ues of diameter and height, respectively. As someof the parameters are constrained (0 < α < 1;σdj, σhj > 0; –1 ≤ ρj ≤ 1) it is advisable to use aconstrained minimization procedure. If the soft-ware for this is not available then unconstrainedminimization will also work if one uses startingvalues that are sufficiently close to the final pa-rameter estimates. With a little practice it is notdifficult to find suitable starting values by inspect-ing a scatterplot of the diameter-height observa-tions.

2.3 Johnson’s SBB Distribution

The Johnson SBB distribution (Johnson 1949b,Elderton and Johnson 1969) is a bivariate ver-sion of the SB distribution (Johnson 1949a, seealso Hafley and Schreuder 1976, 1977, Schreu-der and Hafley 1977). It is based on a transfor-mation of a single bivariate normal distributionand has density function given by

f d hJ z z z zd d h h( , ) exp ,=−

− +−( )

2 1

2

2 12

2 2

2π ρ

ρρ

zd

d

zh

h

d d dd

d d

h h hh

h h

= + −+ −

= + −+ −

γ δ ξλ ξ

γ δ ξλ ξ

log ,

log ,

(7)

Jd d h h

d d g h

d d d h h h

=− + − − + −

δ λ δ λξ λ ξ ξ λ ξ( )( )( )( )

where ξd < d < ξd + λd, ξh < h < ξh + λh,λd, δd, λh, δd > 0 and –1 ≤ ρ ≤ 1.

The marginal distributions are Johnson SB:

f dz

d d

f hd z

h h

d d d

d d d

h h h

h h h

( )exp /

( )exp /

=−( )

−( ) + −( )

=−( )

−( ) + −( )

δ λ

π ξ λ ξ

δ

π ξ λ ξ

2

2

2

2

2

2

(8)

We note all the parameters in (8) are also presentin (7). Thus once a bivariate SBB distribution hasbeen fitted the two SB marginal distributions areimmediately available. The conditional distribu-tions f(h|d) and f(d|h) are not SB but they can becomputed using (7) and (8) in (5).

The maximum likelihood estimators of the pa-rameters are not available in closed form butthey can be computed by numerical minimiza-tion of (6) using the density (7). Numerical meth-ods are needed even in the univariate case, the SB

(see Mønnes 1982, Siipilehto 1999). There arenumerous constraints on the parameters and, un-less software for constrained minimization isavailable, some trial and error experimentationwith the starting values, step size and scale fac-tors may be needed to achieve convergence. Analternative approach, outlined in Appendix 2, isto reparameterize the distribution in terms ofunconstrained parameters.

3 Results and Evaluation ofthe Models

3.1 Results

The mixture of bivariate normals model and theJohnson SBB model were each fitted to the obser-vations described in 2.1 using the method ofmaximum likelihood. The S-PLUS functions“nlmin” and “nlminb” (Mathsoft 1995) wereused to minimize expression (6) to compute theparameter estimates given in Tables 2 and 3.


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Maximization of the likelihood for the SBB dis-tribution led to some numerical difficulties be-cause, for these data, the maximum likelihoodestimate of ξh is effectively “minus infinity”which leads to a limiting form of (SB) marginaldistribution for the heights. To achieve conver-gence it was found convenient to reparameterizethe distribution as described in Appendix 2. Wealso fitted the SBB using the additional parameterconstraints ξd, ξh ≥ 0 but this led to substantialdeterioration in the fit – the negative log-likeli-hood increases from 7798.5 to 7848.8. The cor-responding value for the mixture of bivariatenormals model was 7678.5 which indicates abetter fit than the SBB model. However, this doesnot establish that the mixture model provides agood description of the data.

The contour plot of the fitted density and theobserved values given in Fig. 3 shows that the

model seems to capture the main features of thedata but clearly it is preferable to have a some-what more objective method to assess details ofthe fit.

The chi-squared and the Kolmogorov-Smir-nov goodness-of-fit-tests, which are routinelyapplied in the context of univariate models, canbe used to check the fit of the marginal distribu-

Table 3. Maximum likelihood estimates for the SBB distribution

Parameter ξd λd δd γd ξh λh δh γh ρ

Estimate 3.767 86.213 –1.408 0.901 –974.514 1027.804 5.723 –20.580 0.843

Table 2. Maximum likelihood estimates for the mix-ture of two bivariate normal distributions

Parameter µdi µhi σdi2 σhi

2 ρi α

Estimate(i = 1) 18.00 17.91 5.16 4.85 0.88 0.19Estimate(i = 2) 39.06 27.27 9.69 2.90 0.63

Fig. 3. Contour plot of the fitted density function for the diameters and heights. The observedvalues are shown as circles.


175

tions (see for example Reynolds et al. 1988).The chi-squared test is also applicable to bivari-ate distributions. One way to carry out the test isto partition the diameter-height plane by meansof a rectangular grid, to count the number ofobservations in each rectangle (thus obtainingthe “observed frequencies”), to compute the “ex-pected frequencies” under the model, and then tocarry out the well-known chi-squared test. Thegrid needs to be constructed in such a way thatnone of the expected frequencies are too small(five being generally regarded as large enough);otherwise the distribution of the test statistic ispoorly approximated by the chi-squared and thetest can be inaccurate. The main objection to thisprocedure, especially when it is applied to bivar-iate data, is that the value of the test-statisticdepends strongly on how one partitions the plane,that is how one constructs the grid. Indeed it isnot unusual to find that whereas one choice ofgrid leads to the rejection of the model an alter-native grid does not.

An alternative, and somewhat more versatiletool for assessing the fit, is to use so-called pseu-do-residuals (Zucchini and MacDonald 1999).These will now be defined for univariate contin-uous distributions and we will then illustratehow they can be used to assess the fit of bivariatedistributions, such as the above diameter-heightmodel.

3.2 Pseudo-Residuals

Suppose that the observations x1, x2,..., xn areassumed to be independently distributed accord-ing to the probability density function f(x) anddenote the corresponding probability distribu-tion function by F(x). The issue here is to assessthe fit of the model f(x). The pseudo-residual, ri,corresponding to the observation xi is defined as

ri = Φ–1(F(xi)), i = 1,2, ..., n

where Φ denotes the distribution function of thestandard normal distribution:

Φ( ) /z e dxxz

= −

−∞∫

1

2

2 2

π

Its inverse Φ–1(p), where 0 < p < 1, is the solu-tion to the equation (in z): p = Φ(z). Most stand-ard statistical software packages have a routineto compute these values – for example the S-PLUS function “pnorm” computes Φ and“qnorm” computes Φ–1. Alternatively tables ofthe standard normal distribution can be used.

It is shown in the Appendix that, under thehypothesis that f(x) is indeed the correct modelfor the observations, the pseudo-residuals ri, i =1,2 ..., n, follow the standard normal distribu-tion. Thus, no matter how complicated f(x) mightbe, we can assess it’s fit by computing the pseu-do-residuals and checking how well they arefitted by a familiar N(0,1) distribution. One ofthe most sensitive methods for doing this is toexamine a “normal probability plot of the pseudo-residuals”. This is constructed as follows:

The pseudo-residuals are ordered from thesmallest to largest. Let r(i) be the i-th largestvalue of r1, r2, ..., rn. The r(i) are plotted againsttheir so-called “plotting positions”. The figuresbelow are based on the plotting positions de-fined by

qi

ni ni = +

=−Φ 1

11 2, , ,..., .

For alternative types of plots see, e.g., Chamberset al. (1983) or Hoaglin et al. (1983). The normalprobability plot of the pseudo-residuals is a plotof the qi (on the horizontal axis) against the r(i)

(on the vertical axis).If the model fits the data then the points on this

plot should lie close to the straight line of unitslope through the origin. Deviations from thisline indicate lack of fit and also show whichaspect of the model fails to fit the data.

For example the bivariate mixture distributionfitted to the diameter-height data implicitly mod-els the marginal distribution of the diameter as amixture of the two univariate normal distribu-tions given in (3) and (4) with the parameterestimates given in Table 2. This density functiontogether with a histogram of the diameters isshown in Fig. 4. Also shown is the normal prob-ability plot of the corresponding pseudo-residu-als. It can be seen that the fit is generally goodbut that about 14 trees (approximately 1% of thepopulation) are somewhat larger than expectedunder the fitted model. These correspond to the


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two slightly raised boxes in the histogram overthe intervals 65–70 and 70–75 [cm]. We notethat in general the points on the extreme left andextreme right of a plot tend to be more variablethan those in the middle even when the model iscorrect – they have a larger standard error.

Fig. 5 shows that the fitted marginal distribu-tion of the heights marginal distribution fits unu-sually well. The 5 trees that are identified in thenormal probability plot as being slightly higherthan expected correspond to the small box in thehistogram over the interval 36–38 [m].

We emphasise that the marginal distributionsdisplayed in Fig. 4 and 5 were not fitted sepa-rately to the diameters and to the heights – theywere derived from the fitted bivariate distribu-tion thus making the accuracy of the fits to thetwo marginal distributions remarkable. By wayof comparison Fig. 6 shows the normal probabil-ity plots of the pseudo-residuals for the marginaldistributions of the diameters and heights associ-ated with the SBB distribution (7) with the param-eter estimates given in Table 3. The lack of fit isevident.

Fig. 4. Histogram of diameters and the fitted marginal density (left); normal plot of the pseudo-residuals (right).

Fig. 5. Histogram of heights and the fitted marginal density (left); normal plot of the pseudo-residuals (right).


177

That the model fits the marginal distributionswell is a necessary but not a sufficient conditionthat it describes the data as a whole well.

A difficulty that arises when assessing the fitof the conditional density, say f(h|d) is that itsparameters are a function of d. This is also thecase in the familiar simple regression model basedon a single bivariate normal distribution as op-posed to the mixture of two such distributions asused here. However, in the simple regressioncase only the conditional expectation is a func-tion of d; the conditional variance does not de-pend on d. (The term homoscedasticity is used torefer to this property.) Consequently the ordi-nary residuals – obtained by subtracting the ob-served heights from the regression line – areidentically distributed. They can therefore becompared for the purposes of detecting outliersand for assessing the fit of the model.

This property does not hold for more complexbivariate distributions such as the Johnson SBB orfor the mixture model considered here. For suchdistributions the conditional mean, the condi-tional variance, and even the shape of the densi-ty function depends on d. In other words they areheteroscedastic. Consequently the ordinary re-siduals all have different distributions; they arenot comparable and thus cannot be used to as-sess the fit. For example, it would not makesense to construct a histogram of these residualsbecause there is no single density with which it

can be matched. Pseudo-residuals were devel-oped to overcome this problem, they are compa-rable because they all have the same distribu-tion, namely the standard normal.

Fig. 7 shows a histogram of the pseudo-residu-als for the conditional distribution of height giv-en diameter derived from the fitted bivariate mod-el. Under the model these should have a standardnormal distribution (whose density function isalso shown in the figure). This, together with thenormal probability plot also given in Fig. 7, con-firms that this conditional distribution is fittedvery well.

Pseudo-residuals also enable one to assess thefit of a bivariate model in additional detail. Forexample we can examine the fit of its projectionsonto lines other than the horizontal axis (margin-al distribution of the diameter) or the verticalaxis (marginal distribution of height). This ena-bles us to examine the fit from additional pointsof view, thereby make it possible to detect out-liers or lack of fit that might not be less evident ifone only considers the usual marginal distribu-tions. Such projections are illustrated in Figs 8and 9.

Fig. 8 shows a plot of the standardized diame-ter and height data, that is after subtracting theirrespective means and then dividing by their re-spective standard deviations. It also shows anumber of straight lines at various angles throughthe origin. By projecting the points onto one of

Fig. 6. Normal probability plot of the pseudo-residuals for the marginal SB distribution ofdiameters (left) and of heights (right) obtained by fitting a bivariate SBB model.


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these lines (for example that at 45%) we obtainthe observed values of the marginal distributionof a linear function of the original observations:li = α0 + α1di + α2hi, i = 1,2, ...,n, where α0, α1

and α2 are determined by the particular projec-tion that we wish to examine, determined by theangles shown in Fig. 9. We can now compare thedistribution of the li with that implied by themodel fitted to the original data.

The normal plots of pseudo-residuals associat-ed with the li enable us to assess the fit of themodel from a variety of different angles. Suchplots are shown in Fig. 9. (The plots for 0° andfor 90° have already been given in Fig. 4 and 5,respectively.)

The plots in Fig. 9 show that, on the whole, themodel fits these data very well. (Compare theseplots with those in Fig. 6.) The projection at the

Fig. 7. Histogram of the pseudo-residuals and the standard normal density (left); normalprobability plot of the pseudo-residuals (right) for the conditional distribution of heightsgiven diameters.

Fig. 8. Contour plot of the fitted bivariate density function for the standardized diameters andheights and selected lines through the origin.


179

angle –30° shows a slight lack of fit in the lowertail of the distribution. The plots reveal the 14exceptionally large trees that were evident in theordinary marginal distribution of diameter in Fig.4. These show up clearly on the upper right-handcorners of the plots for the projections corre-sponding to the angles between 60° and –30°.Regarding the 14 exceptionally large trees thedata were derived from a systematic sample withplots spread over a large area, covering severalhectares, which includes sections exhibiting dif-ferent tree dimensions. Two plots are situated insmall section where the trees are much largerthan the rest. Although these exceptions show upprominently on the plots, it needs to be kept inmind that they represent about 1% of the pointson each plot, the remaining 99% (1228 points)fall either on, or very close to the main diagonal.

4 Discussion

The type of model discussed in this paper isappropriate in situations where there is reason tosuppose that the population under considerationcomprises two subpopulations. This was the casefor the example considered here and it was foundthat a mixture of two bivariate normal distribu-

tions fits the data unusually well. The fitted modelis easily interpretable and reflects something realabout the population, namely that it comprisestwo distinct subpopulations whose diameter-height distributions differ. The larger one (ap-proximately 80% of the trees) has a differentdiameter-height relationship to the smaller (ap-proximately 20% of the trees). In the smallersubpopulation the diameter-height relationshipis steeper. This finding would lead one to expectthat the distribution of other tree attributes ofthis forest (for example maximum crown width,height to crown base or dead branch height)could also be appropriately modelled by a mix-ture of distributions.

Of course it is also possible to improve the fitof such models by using a mixture of three ormore distributions instead of just two. However,unless there are persuasive biological groundsfor believing that the population comprises threeor more distinct subpopulations the parametersof the fitted model lose their simple interpreta-bility. Furthermore the resulting increase in thenumber of parameters erodes the precision withwhich it is possible to estimate them.

For special investigations like structural anal-ysis in natural and virgin forests it is generallyno problem to obtain funds for measuring a suf-

Fig. 9. Normal probability plots of pseudo-residuals for the marginal distributions of selectedlinear combinations of the diameters and heights (cf. Fig. 8).


180

ficient number of heights to fit a bivariate diam-eter-height distribution. However, in regular man-agement inventories, foresters cannot afford tomeasure more than say 20–30 heights per beechstand. Thus, if the application of bivariate tech-niques is to become common practice in forestinventories, it will be necessary to develop meth-ods for estimating the parameters of the distribu-tion from a relatively small number of measuredtrees (Schmidt and Gadow 1999). Such samplesizes are too small to fit the mixture of twobivariate normal distributions outlined here –one needs about 50 observations to obtain usea-ble estimates.

One approach might be to investigate whethersome of the parameters of the model might beestimated by relating them to available data, suchas dominant height or basal area. A second ap-proach, currently under investigation, is to de-velop a method of estimation from larger sam-ples of approximate height measurements thatwould be less expensive to make than precisemeasurements.

The second issue discussed in this paper wasthe use of pseudo-residuals for assessing the fitof a model.

A convenient feature of pseudo-residuals isthat one only needs to think of them as observa-tions from a standard normal distribution – if themodel fits. This applies irrespective of whetherone is examining the fit of a marginal distribu-tion, a conditional distribution, the distributionof a linear combination, or some more complexfunction, of the variables. Thus, for example,histograms of pseudo-residuals are always com-pared with the familiar bell-shaped N(0,1) densi-ty function; normal probability plots of the pseu-do-residuals are always plotted on the same stand-ard scale.

Finally, the fit of the model can be assessed bysimply examining plots. Of course such plots donot replace formal goodness-of-fit tests; theyshould be regarded as additional tools that areeasy to apply and that can reveal not only wheth-er or not a model fits, but also details regardingthe lack-of-fit.

The above examples illustrate just some of theways in which pseudo-residuals can be used toassess the fit of a particular bivariate model. Ofcourse they can be applied to any of the other

bivariate models that are of interest in a forestrycontext. Bivariate distributions have been usedfor some time to describe the relationship be-tween diameters and heights (Hafley andSchreuder 1976, Warren et al. 1979, Tewari andGadow 1999) and more recently also to includeother tree attributes, such as height to maximumcrown width or height to crown base (Uusitaloand Kivinen 1998). More exact estimation ofheight of maximum crown width, crown baseand dead branch height can improve the estima-tion of branchiness and wood quality (Collin andHoullier 1992, Maguire et al. 1994, Seifert 1999).Another application of a bivariate distribution isto model height and density of natural regenera-tion in stands (Schweiger et al. 1997). Growthmodels are an important tool for long- and medi-um-term simulation studies and Knoebel andBurkhart (1991), for example, presented an ap-proach for modelling the bivariate distributionof diameter at two points in time. A related ap-plication involves the simultaneously estimationof mechanical wood properties (Pearson 1980,Warren et al. 1979) and the estimation of con-comitant strength wood properties related toknown nondestructive wood properties (Pelli-cane 1993).

Acknowledgements

The data were provided by the Forest ResearchStation of Lower Saxony. The constructive sug-gestions of two anonymous referees are grateful-ly acknowledged.

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Appendix 1. Proofs of Claims RegardingPseudo-Residuals.

We first show that if a continuous random varia-ble X has distribution function F(x) then the cor-responding pseudo-residual r = Φ–1(F(X)) is dis-tributed as N(0,1).

We note that since X is a continuous randomvariable its distribution function F(x) is mono-tone strictly increasing and therefore has an in-verse F–1(x) which is also monotone strictly in-creasing. The distribution function Φ of a stand-ard normal random variable and its inverse Φ–1

also have these properties.The random variable U = F(X) is uniformly

distributed on the interval (0,1). This followsfrom

P{U ≤ u} = P{F(X) ≤ u} = P{X ≤ F–1(u)}= F(F–1(u)) = u, for 0 < u < 1,

which is the distribution function of a uniformlydistributed random variable.

The distribution function of the pseudo-resid-ual, r, is given by

P{r ≤ z} = P{Φ–1(F(X)) ≤ z}= P{F(X) ≤ Φ(z)}= P{U ≤ Φ(z)} = Φ(z),

which is the distribution function of a N(0,1)random variable.

Suppose now that X1,X2, ..., Xn are independ-ently distributed and that distribution function ofXi is Fi(x), i = 1,2, ..., n. (Note that the Xi are notassumed to be identically distributed.) Then, bythe result above, each pseudo-residualri = Φ–1(Fi(Xi)), i = 1,2, ..., n, is distributed asN(0,1). Finally, as the Xi are independently dis-tributed, the ri, are also independently distribut-ed.

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Total of 36 references


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Appendix 2. A Reparameterization of theJohnson SBB Distribution for UnconstrainedMaximization of the Likelihood Function.

The nine parameters of the SBB are subject toseveral constraints (see Section 2.3) which needto be respected in the numerical minimization ofthe negative log-likelihood function. If softwarefor constrained numerical minimization is notavailable it is convenient to (temporarily) ex-press the likelihood function in terms of differ-ent parameters that are unconstrained. Such areparameterization and its inverse is given be-low.

New parameters Original parameters

θ1 = log(dmin – ξd) ξd = dmin – exp(θ1)θ2 = log(ξd + λd – dmax) λd = dmax – ξd + exp(θ2)θ3 = log(δd) δd = exp(θ3)θ4 = γd γd = θ4θ5 = log(hmin – ξh) ξh = hmin – exp(θ5)θ6 = log(ξh + λh – hmax) λh = hmax – ξh + exp(θ6)θ7 = log(δh) δh = exp(θ7)θ8 = γh γh = θ8θ9 = tanh–1 (ρ) ρ = tanh(θ9)

Here dmin, dmax, hmin, hmax denote the smallest andlargest of the observed diameters and heights,respectively.

If one wishes to impose the additional con-straints ξd > 0 and ξh > 0 this can be achieved byreplacing θ1 and θ5 in the above table with thefollowing:

New parameters Original parameters

θ1 = log(ξd / (dmin – ξd)) ξd = dmin ⋅exp(θ1) / (1 + exp(θ1))

θ5 = log(ξh / (hmin – ξh)) ξh = hmin ⋅exp(θ5) / (1 + exp(θ5))

Of course these transformations can also be ap-plied for estimating the parameters in the univar-iate case, that is to fit a SB distribution. For thedata considered in this paper, and for the particu-lar software that we used, namely the S-PLUSfunctions “nlmin” and “nlminb” (Mathsoft1995), unconstrained maximization of the repa-rameterized likelihood required fewer iterationsthan did constrained maximization with respectto the original parameters.

a model for the diameter-height distribution in an … schmidt and von gadow a model for the...

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