Estimating a distance dependent contagion function using point sample data

Environ Ecol Stat (2014) 21:61–82DOI 10.1007/s10651-013-0244-5

Estimating a distance dependent contagion functionusing point sample data

Habib Ramezani · Sören Holm

Received: 29 November 2011 / Revised: 13 February 2013 / Published online: 27 March 2013© Springer Science+Business Media New York 2013

Abstract Natural events and human activities cause changes in landscape structure.Landscape metrics are used as a useful tool to study landscape trends and ecologicalprocesses related to the landscape structure. These metrics are commonly calculatedon wall-to-wall raster data from remote sensing. A recent trend is to use sample datato estimate landscape metrics. In this study, point sampling was used to estimate avector-based and distance dependent contagion metric. The metric is an extensionof the established contagion. The statistical properties, for both unconditional andconditional contagions, were assessed by a point (point pairs) sampling experiment inmaps from the National Inventory of landscapes in Sweden. Random and systematicsampling designs were tested for nine point distances and five sample sizes and for twoclassification systems. The systematic design showed slightly smaller root mean squareerror (RMSE) and bias than the random design. Both true and estimated values werecalculated using computer programs in FORTRAN, which was specifically writtenfor the purpose of the study. For a given sample size, RMSE and bias increased withincreasing point distance. The estimator of unconditional contagion had acceptableRMSE and bias for moderate sample sizes, but in the conditional case the bias (andthus the RMSE) was unacceptably large. The main reason for this is that small classes(by area) affect both the true value of the contagion and are often missing in thesample. The method proposed can be adopted in gradient-based model of landscapestructure where no distinct border is assumed between polygons. The method can alsobe applied in field-based inventories.

Handling Editor: Ashis SenGupta.

H. Ramezani (B) · S. HolmDepartment of Forest Resource Management, Swedish Universityof Agriculture Science, SLU, Umeå 901 83, Swedene-mail: [email protected]; [email protected]

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62 Environ Ecol Stat (2014) 21:61–82

Keywords Bias · Landscape metrics · Monte-Carlo simulation · Point pairs ·Point sampling · Root mean square error

1 Introduction

Natural events like fire and human activities, forest management and farming, causesignificant changes in landscape structure (composition and configuration of differentland cover types within landscape). It is recognized that some ecological phenomenacan be affected by landscape changes (Turner 2005). Further, since landscape changesmay lead to biodiversity loss (Fischer and Lindenmayer 2007; St-Laurent et al. 2009)landscape pattern analysis has received much attention in both ecological research andmanagement society over recent decades.

In order to understand and interpret ecological processes in relation to landscapestructure, as a first step landscape structure must be quantified (Li and Wu 2004).For this purpose a set of landscape metrics have been developed (e.g., McGarigal andMarks 1995). The metrics are based on measurable attributes of landscape units suchas number, size, and edge length of patches. Landscape metrics are increasingly usedas indicators in monitoring programs (Hunsaker et al. 1994; Dramstad et al. 2002) asthey are useful tools for comparing different landscapes and for following changes ina given landscape over time. Also, the metrics can provide quantitative informationabout biodiversity (Benitez-Malvido and Martinez-Ramos 2003; Bebber et al. 2005).

Traditionally, quantifying landscape pattern through landscape metrics is conductedusing complete land cover/use maps (wall-to-wall map) from remotely sensed data;computer software such as FRAGSTATS (McGarigal and Marks 1995) is frequentlyused for this purpose. In such procedures all polygons are delineated on remote sensingimagery, either manually or automatically. Manual approaches are time-consuming(e.g., Corona et al. 2004) and may be associated with polygon delineation errors. Withautomatic approaches, dissimilar adjacent land cover types may be merged into thesame land cover type; in addition, overall cost (time for delineation and correction)may be high due to the time required for correction (Wulder et al. 2008). As shown byFang et al. (2006) land cover/use maps based on medium-resolution satellite images(e.g., Landsat TM) may have low overall accuracy while maps based on images withhigh spatial resolution may imply high storage costs due to the large amounts of data(Lu and Weng 2007; Gardner et al. 2008).

In some recent studies, sampling data has been proposed as a promising alternativeto wall-to-wall data to derive landscape metrics (Corona et al. 2004; Ramezani et al.2010, 2011; Ramezani and Holm 2011b). It is shown that in some cases, sample-based assessment not only is a cost-efficient alternative to traditional wall-to-wallmapping but in addition it can provide reliable information. Further, time series ofmetrics can be derived from existing data sources, such as field-based national forestinventories (Kleinn 2000). When probability sampling methods (i.e., sampling basedon probability theory) are applied, the precision of the estimates can also be assessed.It is known that the precision of estimates generally is high in well-designed samplesurveys (Freese 1962; Raj 1968; Gregoire and Valentine 2008).

Point sampling is recognized as an efficient method to estimate areas of two-dimensional objects like polygons on categorical maps or in the field survey. To derive

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Environ Ecol Stat (2014) 21:61–82 63

basic landscape metrics, land cover/use categories at the sample points are assessed.In addition, variables such as distance to the closest polygon boundary make it pos-sible to derive metrics involving edge length. The efficiency of the method has beendemonstrated by Ramezani et al. (2010) in estimating Shannon diversity and edgedensity based on remotely sensed data.

The contagion metric has been used in many studies (e.g., O’Neill et al. 1988;Hunsaker et al. 1994; Ricotta et al. 2003) for quantifying landscape configuration.The metric measures the degree of clumping of patches within a landscape. This met-ric can indirectly provide information of landscape fragmentation. The contagion ishighly correlated with metrics of diversity, dominance, and patch richness (Riitterset al. 1995; Cain et al. 1997; Frohn 1998). Typically, contagion is defined and cal-culated based on wall-to-wall raster data of a landscape. A vector-based version ofthe contagion metric was proposed by Wickham et al. (1996). This definition is basedon the proportions of edge lengths between different land cover types for adjacentpolygons. In a related study by Ramezani and Holm (2011a), a distance dependentcontagion function is proposed. With this contagion definition, based on random pairsof points, the contagion metric can be estimated not only using raster maps, but alsowith vector-based maps and point sampling data. In that study it is shown that thecontagion function provides information about both landscape composition and con-figuration, and that a simple negative exponential function can be used as a usefulproxy function for the (unconditional) contagion function.

In this study we evaluate the usefulness of point sampling to estimate the distance-dependent contagion metric. The aim was to investigate the statistical properties ofthe contagion estimator for different point sampling designs (random and systematic),number of point pairs and distances between points. The study was conducted on realdata from landscapes in Sweden.

2 Material and methods

2.1 Materials

The study was conducted as a sampling experiment, with sample points selected fromalready photo-interpreted landscapes in vector format. Interpreted landscapes wereobtained from the National Inventory of Landscapes in Sweden (NILS) (Ståhl et al.2011), which is a major environmental monitoring program run by the Swedish Envi-ronmental Protection Agency. A 25 km2quadrate is used in order to capture the broadlandscape context. Within a 1 km2 centrally located quadrate, a detailed delineation ofpolygons is made every fifth year. To obtain a genuine sample of landscapes for ourstudy, we used data from 50 randomly selected NILS quadrates.

The aerial photographs, in which interpretations were made, were colour infraredand had a ground resolution of 0.4 m. Polygon delineation was made using the inter-pretation program Summit Evolution from DAT/EM and ArcGIS from ESRI. For thepurpose of the present study, the NILS variables were used together with two differentclassification systems (7 and 20 classes) in order to produce land cover maps. Theclassification systems are described in Table 1.

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Table 1 Classes according tothe two different classificationsystems (with 7 and 20 classes)

Level 1 (seven classes) Level 2 (twenty classes)

1. Forest 1-1. Coniferous-dense

1-2. Coniferous-sparse

1-3. Deciduous-dense

1-4. Deciduous-sparse

1-5. Mixed-forest-dense

1-6. Mixed-forest-sparse

2. Urban 2-1. Housing-areas

2-2. Urban-green-areas

2-3. Urban-forest

3. Cultivated fields 3-1. Crop fields

3-2. Grassland

4. Wetlands 4-1. Bog

4-2. Fen

4-3. Mixed-wetland

5. Water 5-1. Open-water

5-2. Water-vegetation

6. Pasture 6-1. Open-pasture

6-2. Pasture-sparse-trees

6-3. Wooded-pasture

7. Other land 7-1. Other land

2.2 Contagion metric

The contagion metric was first proposed by O’Neill et al. (1988) and was then improvedby Li and Reynolds (1993). The first definition was based on raster-based data wherethe data are assumed to be arrays of pixels with equal side lengths. There are sev-eral versions of the raster-based contagion, e.g., unconditional and conditional defin-itions (Riitters et al. 1996). Typically, calculation of the metric utilizes an adjacencymatrix which provides the proportion of all adjacency types on the raster-based map(Haralick et al. 1973). Contagion values range between 0 and 1. A low value indicates afragmented landscape with many small patches; a high value indicates that the classespresent are clumped into a few large patches.

The contagion metric for raster-based data was extended to vector-based data byRamezani and Holm (2011a). For a given distance d the unconditional contagion isdefined as

Cu(d) = 1 +∑s

i=1∑s

j=1 pi j (d) · ln(pi j (d))

2 ln(s)(1)

where pi j (d) is the probability that two randomly chosen points at distance d belongto the classes iand j and s is the number of classes in system (here 7 and 20 classes).For pi j (d) = 0, pi j (d) · ln(pi j (d))is set to zero.

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The conditional contagion function is (Ramezani and Holm 2011a)

Cc(d) = 1 +∑s

i=1∑s

j=1 p j/ i (d) · ln(p j/ i (d))

s ln(s)(2)

where conditional probability is p j/ i (d) = pi j (d)/pi (d), with pi (d) = ∑sj=1 pi j (d).

The conditional probability is the probability that the second point in a pair of arandomly chosen pair of points at distance d belongs to class j , given that the first pointbelongs to class i . Both (1) and (2) can be applied either with s equal to the numberof classes of the classification system (here s is 7 or 20), denoted “all classes” in whatfollows or with s equal to the number of classes actually present in the landscape.

2.3 Contagion metric estimator

If no delineation of the landscape exists it is natural to estimate the probabilitiespi j (d)andp j/ i (d) from a sample of point pairs at distance d in the landscape, wherethe pi j (d)is estimated by the relative frequency of points in classes i and j (care takento boundary effects) and p j/ i (d) is estimated according to its definition above. Theestimators pi j (d)and p j/ i (d) are then inserted into the defining expressions (1) and(2) to obtain estimators of Cu(d) andCc(d) of the contagion functions.

2.4 Monte-Carlo sampling simulation

Monte-Carlo sampling simulation is recognized as a useful approach to study thestatistical performance of estimators in sampling surveys (e.g., Hazard and Pickford1986; Ståhl 1998; Ramezani et al. 2010). In this study, samples of pairs were takenwithin NILS landscape quadrates to estimate bias and root mean square error (RMSE)of the two contagion estimators under different point sampling designs. For each pointthe already delineated map indicated the class it belongs to.

Sampling simulations were conducted for each combination of simple randomsampling and systematic sampling, five sample sizes (25, 49, 100, 225 and 400 pointpairs), and nine different distances between point pairs (2, 5, 10, 20, 30, 60, 100, 150and 250 m). In all cases, estimates were derived for both conditional and unconditionalcontagion using both classification systems (7 and 20 classes), applying both thedefinition with all classes and the number of classes actually observed in sample. Atleast 550 samples were simulated for different sampling designs and combinations. InFig. 1 an example of a random sample of pairs is shown.

For the systematic design the first points were laid out systematically in the principaldirections with random start. The second point was taken at a random direction at

distance d from the first point, with modification if the second point fell outside thequadrate.

In principle the relative frequency of classes i and jamong the sampled pairsestimatespi j . However, care must be taken to avoid boundary effects. In this study, ifthe second point in the pair fell outside the quadrate then a new second point was taken

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Fig. 1 Illustration of random point sampling pairs on a 1 km2 NILS map

until it was inside. The estimators were modified accordingly by weighting each pairwith the probability that the second point should fall inside the quadrate (also when itactually fell inside at the first attempt). The resulting estimator pi j has a small bias sinceit is a ratio estimator with random denominator. Further the relation pi j = p ji wasused for the estimation. The conditional probability was estimated in accordance withits definition (following the Eq. 2).

Both true and estimated values were calculated using computer programs in FOR-TRAN (Ekman and Eriksson 1981), which was specifically written for the purpose ofthe study. Calculations of true values were based on the method described by Ramezaniand Holm (2011a). In this method a number of first points are sampled in each poly-gon. With the first point as the centre of a circle with radius dthe relative length of thecircle circumference falling in polygons of class j is determined for all j . The meanof the lengths is used as an estimator of the local pi j ’s for the given polygon. The finalglobal pi j ’s are determined by polygon area weighting over all polygons of class i .The true values were estimated with at least two decimals accuracy (Ramezani andHolm 2011a).

2.5 Efficiency evaluation

Properties of the contagion estimators were derived through a large number of sim-ulated samples, taken independently for all combinations of designs, distances andclassification systems. For an estimator, θ say, the expected value was estimated bythe mean over the simulations

E(θ) = 1

m

m∑

i=1

θi (3)

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where θi is the estimated value of the i th simulation and mis the number of simu-lations. The estimated bias isE(θ) − θ , where θ is the true value. The variance wasestimated analogously by the sampling variance. The root mean square error, RMSE,was estimated by

RM SE =√√√√

m∑

i=1

(θ−i θ)2/m (4)

For s equals to the number of classes observed in the sample the bias was estimatedwith respect to the true value with sequals to the number of classes present in thelandscape.

3 Results

Bias and RMSE of the two contagion estimators were estimated for all 45 combinationsof distance and sample size, two classification systems and for all classes and presentclasses. Some major results are presented here. More details about true value, bias,and RMSE of both 7 and 20 classification systems are provided in Appendices.

The RMSE and bias, averaged over the 50 landscapes, of the contagion estima-tors were compared for systematic and simple random sampling designs. The formersampling design showed a slightly smaller RMSE and bias than the latter one and,as expected, both RMSE and bias decreased with increasing sample size. This wastrue for all combinations considered. In Fig. 2 bias and RMSE of the unconditionalcontagion estimators are presented for the distance 20 m and with all classes in the 7classification system.

Both contagion estimators showed bias. Bias and RMSE of the conditional estimatorwere much larger than for the unconditional estimator. For all classes the bias ofthe estimator was positive whereas for the case of observed classes bias was mostlynegative. In Fig. 3 is shown the relationship between bias (left) and RMSE (right) andsample size for the unconditional and conditional estimators, for point distance 20 m,and for all classes.

Fig. 2 Bias (left) and RMSE (right) of the unconditional contagion estimator using a 20 m point distance,averaged over the 50 landscapes using the 7 classes system and for “all classes” (s = 7).

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Fig. 3 Relationship between the bias (left) and RMSE (right) with sample size of the contagion estimators,for point distance 20 m, the 7 classes system, and all classes (s = 7).

Fig. 4 Bias (left) and RMSE (right) of conditional estimator with all (s = 20) and observed number ofclasses for sample size 25, for the 20 classes classification system

A comparison was also made for the contagion estimators, in terms of RMSE andbias, for all and observed classes. The contagion estimators with all classes showedsmaller RMSE and bias than with observed classes, see Fig. 4.

4 Discussion

In this study, bias and RMSE of the contagion estimators have been assessed for pointsampling data (point pairs) on vector format maps. Statistical properties of this metricwere previously discussed by Hunsaker et al. (1994), but for raster format maps and byusing plot sampling (hexagon shape) and a direct comparison of the results is difficultsince a different (conditional) contagion formula than here were used.

In contrast to the vector version of Wickham et al. (1996), with the vector versionused here there is no necessity to delineate borders between polygons. Furthermore,the new vector version is an extension of standard raster-based formula which hasbeen developed by Li and Reynolds (1993).

The comparison of the contagion estimators in terms of RMSE and bias indicatesthat a systematic sampling design is slightly more accurate than a random one. Forunconditional contagion the RMSE and bias have acceptable levels for sample sizes 49and larger. However, for the estimator of the conditional contagion the RMSE is verylarge for the same sample size as unconditional case and to achieve in acceptable levelsof accuracy (RMSE) a very large sample size is needed. This holds especially for the

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Fig. 5 An example of landscapewith small class

Table 2 True values, absolute bias (×100) and absolute RMSE (×100) of conditional and unconditionalcontagion estimators of a landscape with small class for system with 7 classes and point distance 10 m andsample size 25

Estimator True value Bias RMSE

Unconditional 0.975 0.4 5.4Conditional 0.997 35.6 35.7

Table 3 High, moderate, and low RMSE of contagion estimator for the 7 classes system, for point distance20 m and for all classes. The RMSE values are estimated for the landscapes in figure 5. Sample size 25and true contagion values in parenthesis.

Landscape fragmentation Unconditional Conditional

High 0.0985 (0.4145) 0.3644 (0.4405)

Moderate 0.0623 (0.7379) 0.1358 (0.8176)

Low 0.0242 (0.9782) 0.0209 (0.9476)

Sample size was 25 and true values of contagion are given within parenthesis

case of observed classes (i.e., the number of classes actually observed in sample). Thefigures indicate that the bias is main reason of the large RMSE; hence bias discussedin some details below.

Landscape metrics often are defined based on more than one component of land-scape units (patch) such as size, edge length, and the number of patches. In general,landscape metrics can at best be estimated without bias when their components areunbiasedly estimated. The results obtained show that both contagion estimators arebiased. There are several causes for bias: (a) the estimator pi j (d) is a ratio estimatorand as such has a certain bias. The bias is likely to be increasing for increasing distance

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Unconditional Conditional

High

Moderate

Low

Fig. 6 High, moderate, and low fragmented landscapes for unconditional and conditional definition,7 classes system, and point distance 20 m

since the denominator in pi j (d) (the sum of the weights) will have increasing variance(more second points can fall outside the quadrate); (b) the estimator p j/ i (d) is alsoa ratio estimator, with the biased estimator pi (d) as denominator; (c) the contagiondefining function, of the formt ln(t), is non-linear in its argument. Even with unbi-ased estimators of its argument some bias is produced; (d) the number rof classes

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observed in the sample can never exceed the number r of classes actually presentin the landscape. Especially small classes will be missing in the sample. As an esti-mator, the observed r underestimates the true r . The contagion functions will all beaffected by (a), (c) and (d) and the conditional also by (b). The effect of (a), (b),and (c) are likely to be small or moderate for the reasonable large sample sizes. Theeffect of (d) is more dramatic and explains the difference in bias between uncondi-tional and conditional contagion. The reason is that for conditional contagion the sums∑s

j=1 p j/ i (d) · ln(p j/ i (d))for i = 1, 2, . . . are equally important when summing overthe classes i , but for the unconditional contagion the same sums are weighted bypi (d)

(Ramezani and Holm 2011a). Hence, missing a small class, with smallpi (d), affectsthe conditional contagion much more than the unconditional. In Fig. 5 and Table 2 isprovided an example of landscape with small class and true values of conditional andunconditional and their corresponding absolute bias.

To understand the difference of the bias between all classes and classes observedwe have the relation

C (r)u (d) − C (r)

u (d) = ln(s)

ln(r)

(Cu(d) − Cu(d)

)+

(1 − C (r)

u (d)) (

1 − ln(r)

ln(r)

)

obtained by simple algebra. Here C (r)u (d) is written for unconditional contagion with

classes present and C (r)u (d)is its estimator. In this expression ln(s)/ ln(r) is a factor

that in general is larger than 1 by which the (positive) bias of Cu(d)is multiplied. Inthe second term the first factor is positive and the second is negative why the secondterm in general is negative. When the bias of Cu(d)is small the bias of C (r)

u (d) tendsto be negative (especially ifC (r)

u (d)is small). (Since r and Cu(d)are correlated wecannot give an exact and useful expression of the biases by taking the expectation whythe reasoning is somewhat intuitive). For the conditional contagion we have the samerelation except that ln(s), ln(r)and ln(r)are replaced bys ln(s), r ln(r)andr ln(r). Forconditional contagion the first (positive) term dominates.

Both RMSE and bias of the contagion estimator presented are averaged over the 50landscapes. For single landscape, for both unconditional and conditional definitions,the lowest accuracy was obtained for highly fragmented landscape with small andscattered patches. For landscapes with a few and large patches the accuracy was higher.In Fig. 6 and Table 3 are provided examples of high, moderate, and low fragmentedlandscapes and their RMSE’s of contagion estimator.

5 Conclusion

This study shows the potential of sampling data (point pairs) for estimating contagionmetric. With such sampling method there is no need for delineated map; as a con-sequence the error of polygon delineation can be eliminated. The proposed methodcan, at least for unconditional contagion, be implemented both on raw (non-delineated)remotely sensed data and in field-based inventories in operation such as national forestinventory (NFI). Furthermore, the method applied in this study appears to be adapted

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to a gradient-based model of landscape structure (McGarigal and Cushman 2005)where no distinct border is assumed between landscape patches.

Acknowledgments We are grateful to Prof. Göran Ståhl for constructive comments and Mats Högströmfor help with GIS at SLU in Umeå, Sweden.

6 Appendix 1

See Table 4.

Table 4 True values, averaged over the 50 landscapes, of contagion for nine distances for the 7 classificationsystem

Point distance (m)2 5 10 20 30 60 100 150 250

Unconditional

All classes 0.777 0.759 0.734 0.701 0.680 0.646 0.625 0.613 0.602

Observed classes 0.695 0.669 0.636 0.590 0.560 0.514 0.486 0.466 0.452

Conditional

All classes 0.951 0.902 0.849 0.795 0.773 0.753 0.746 0.741 0.742


7 Appendix 2

See Table 5.

Table 5 True values, averaged over the 50 landscapes, of contagion for nine distances for the 20 classifi-cation system

Point distance (m)2 5 10 20 30 60 100 150 250

Unconditional

All classes 0.732 0.710 0.680 0.638 0.611 0.569 0.546 0.532 0.520


Conditional

All classes 0.968 0.934 0.893 0.843 0.817 0.785 0.772 0.764 0.760


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8 Appendix 3

See Table 6.

Table 6 Absolute estimated RMSE (×100), averaged over the 50 landscape, of the contagion estimatorsin systematic design and system with 7 classes

Point distance (m) Unconditional Conditional25 49 100 225 400 25 49 100 225 400

All classes

2 4.2 2.9 2.0 1.3 0.9 4.4 4.1 3.6 2.9 2.4

5 5.0 3.5 2.3 1.4 1.0 8.0 7.0 5.8 4.2 3.3

10 5.6 3.9 2.5 1.5 1.1 11.4 9.6 7.5 5.1 3.7

20 6.2 4.1 2.6 1.6 1.1 14.0 11.4 8.5 5.7 3.9

30 6.3 4.2 2.6 1.5 1.1 14.5 11.5 8.5 5.6 3.9

60 6.1 4.1 2.6 1.7 1.3 13.8 10.7 7.7 5.2 3.8

100 6.1 4.1 2.9 1.8 1.4 13.1 10.1 7.6 5.1 3.9

150 6.4 4.6 3.0 2.0 1.6 13.0 10.3 7.6 5.3 4.0

250 6.8 4.5 3.1 2.2 1.8 12.9 10.0 7.5 5.2 4.0

Observed classes

2 8.1 6.0 4.2 2.7 1.9 10.8 9.5 8.0 6.3 5.3

5 8.8 6.5 4.5 2.9 2.0 17.1 14.4 11.6 8.5 6.6

10 9.4 7.0 4.7 3.0 2.1 22.4 18.2 14.1 10.0 7.4

20 10.1 7.2 4.8 3.0 2.1 25.2 20.1 15.5 11.1 8.0

30 10.3 7.2 4.8 3.0 2.2 25.2 19.8 15.2 10.9 8.2

60 10.4 7.2 4.9 3.3 2.5 22.9 17.9 13.5 9.9 7.8

100 10.3 7.4 5.4 3.4 2.6 21.8 17.1 13.1 9.5 7.5

150 10.7 8.1 5.4 3.7 2.8 21.6 17.5 13.2 9.7 7.6

250 11.3 8.1 5.9 3.9 3.1 21.7 17.5 13.5 9.9 7.8

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9 Appendix 4

See Table 7.

Table 7 Absolute estimated bias (×100), averaged over the 50 landscape, of the contagion estimators insystematic design and system with 7 classes


All classes

2 2.0 1.1 0.6 0.3 0.2 3.8 3.2 2.5 1.7 1.1

5 2.4 1.4 0.8 0.3 0.2 7.3 6.0 4.6 2.9 1.9

10 2.8 1.6 0.8 0.4 0.2 10.6 8.6 6.3 3.9 2.5

20 3.2 1.8 0.9 0.4 0.3 13.1 10.4 7.5 4.6 2.9

30 3.3 1.9 1.0 0.5 0.3 13.7 10.6 7.5 4.6 3.0

60 3.5 2.0 1.1 0.7 0.5 13.1 9.9 6.9 4.3 2.9

100 3.6 2.2 1.4 0.9 0.7 12.4 9.4 6.7 4.2 2.9

150 3.9 2.6 1.6 1.1 0.9 12.3 9.6 6.8 4.3 3.1

250 4.4 2.7 1.8 1.3 1.1 12.2 9.3 6.7 4.3 3.1

Observed classes

2 −4.2 −2.8 −1.6 −0.8 −0.5 6.5 5.4 4.2 2.8 1.9

5 −3.7 −2.5 −1.5 −0.7 −0.5 11.7 9.3 7.1 4.4 2.7

10 −3.3 −2.3 −1.3 −0.7 −0.5 16.2 12.8 9.5 5.9 3.5

20 −2.9 −2.0 −1.2 −0.5 −0.4 18.7 14.8 10.1 7.0 4.1

30 −2.8 −1.9 −1.0 −0.5 −0.3 18.4 14.6 10.9 6.9 4.3

60 −2.9 −1.7 −0.9 −0.4 −0.2 15.0 12.0 8.9 5.7 3.7

100 −2.9 −1.8 −0.9 −0.2 0.0 12.8 10.1 7.7 5.3 3.5

150 −2.9 −1.7 −0.6 0.0 0.3 11.5 9.6 7.7 5.3 3.9

250 −2.7 −1.7 −0.4 0.3 0.6 10.7 9.0 7.4 5.3 3.9

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10 Appendix 5

See Table 8

Table 8 Absolute estimated RMSE (×100), averaged over the 50 landscape, of the contagion estimatorsin random design and system with 7 classes


All classes

2 4.9 3.5 2.5 1.7 1.2 4.4 4.1 3.6 2.9 2.5

5 5.6 4.1 2.8 1.9 1.4 8.1 7.1 5.8 4.3 3.4

10 6.4 4.5 3.1 2.1 1.5 11.5 9.7 7.7 5.4 4.0

20 7.0 4.9 3.4 2.2 1.6 14.1 11.6 8.9 6.0 4.4

30 7.1 5.0 3.4 2.3 1.7 14.7 11.8 8.9 6.1 4.4

60 7.3 5.1 3.5 2.3 1.7 14.2 11.2 8.2 5.6 4.1

100 7.4 5.1 3.5 2.3 1.8 13.6 10.6 7.9 5.4 4.1

150 7.2 5.0 3.4 2.3 1.8 13.3 10.5 7.8 5.4 4.1

250 7.2 5.1 3.5 2.4 1.9 13.0 10.3 7.7 5.3 4.1

Observed classes

2 8.9 6.8 4.9 3.3 2.5 10.9 9.6 8.1 6.4 5.4

5 9.6 7.3 5.4 3.6 2.7 17.2 14.5 11.6 8.6 6.8

10 10.4 7.9 5.7 3.9 2.9 22.5 18.2 14.2 10.2 7.7

20 11.1 8.3 6.0 4.0 3.0 25.5 20.4 15.7 11.3 8.7

30 11.4 8.5 6.0 4.1 3.0 25.6 20.3 15.6 11.4 8.8

60 11.6 8.6 6.1 4.1 3.1 23.9 18.8 14.3 10.3 8.2

100 11.8 8.7 6.2 4.2 3.2 23.0 18.1 13.8 9.1 7.9

150 11.8 8.7 6.2 4.2 3.3 22.3 17.8 13.7 10.0 7.9

250 11.9 8.9 6.4 4.4 3.5 22.1 18.0 14.1 10.2 8.1

123


11 Appendix 6

See Table 9.

Table 9 Absolute estimated bias (×100), averaged over the 50 landscape, of the contagion estimators inrandom design and system with 7 classes

Point distance(m) Unconditional Conditional25 49 100 225 400 25 49 100 225 400

All classes

2 2.3 1.5 0.8 0.4 0.3 3.8 3.2 2.6 1.7 1.2

5 2.8 1.7 1.0 0.5 0.3 7.3 6.1 4.6 3.0 2.0

10 3.3 1.9 1.1 0.5 0.3 10.7 8.7 6.5 4.1 2.7

20 3.7 2.2 1.2 0.6 0.4 13.3 10.6 7.8 4.9 3.3

30 3.8 2.3 1.3 0.7 0.5 13.9 10.9 7.9 5.0 3.3

60 4.1 2.5 1.4 0.8 0.6 13.5 10.4 7.4 4.6 3.1

100 4.2 2.6 1.6 1.0 0.8 12.9 9.8 7.0 4.4 3.1

150 4.2 2.7 1.7 1.1 0.9 12.6 9.7 6.9 4.5 3.2

250 4.4 2.9 1.9 1.3 1.1 12.4 9.5 6.8 4.4 3.2

Observed classes

2 −4.0 −2.8 −1.7 −0.9 −0.5 6.5 5.3 4.1 2.7 1.7

5 −3.4 −2.5 −1.6 −0.9 −0.6 11.5 9.1 6.7 4.2 2.6

10 −3.0 −2.3 −1.4 −0.9 −0.6 16.1 12.6 9.2 5.7 3.6

20 −2.7 −2.0 −1.2 −0.7 −0.5 18.6 14.6 10.8 6.9 4.4

30 −2.6 −1.9 −1.1 −0.6 −0.4 18.3 14.4 10.8 7.0 4.6

60 −2.4 −1.8 −0.9 −0.4 −0.2 15.6 12.3 9.1 5.9 3.9

100 −2.4 −1.7 −0.8 −0.2 0.0 13.2 10.3 8.0 5.3 3.8

150 −2.8 −1.8 −0.9 −0.2 0.0 11.8 9.5 7.4 5.1 3.7

250 −2.8 −1.7 −0.7 0.1 0.3 10.9 8.9 7.1 5.1 3.8

123


12 Appendix 7

See Table 10.

Table 10 Absolute estimated RMSE (×100), averaged over the 50 landscape, of the contagion estimatorsfor the systematic design, and 20 classes system


All classes

2 4.4 2.8 1.8 1.1 0.8 2.8 2.5 2.1 1.7 1.5

5 5.3 3.5 2.3 1.4 0.9 5.5 4.8 3.9 2.8 2.2

10 6.3 4.2 2.7 1.5 1.0 8.5 7.2 5.6 3.8 2.8

20 7.2 4.7 2.9 1.6 1.1 11.8 9.7 7.3 4.7 3.2

30 7.7 4.9 3.0 1.6 1.1 13.4 10.9 8.0 5.0 3.4

60 8.2 5.2 3.1 1.9 1.3 15.0 11.9 8.6 5.6 3.9

100 8.4 5.5 3.6 2.0 1.4 15.4 12.3 9.2 5.9 4.2

150 8.9 6.0 3.7 2.2 1.6 15.8 12.8 9.4 6.2 4.4

250 9.4 6.1 3.9 2.4 1.8 16.1 12.9 9.6 6.5 4.7

Observed classes

2 6.1 4.3 3.0 2.0 1.5 7.9 6.9 5.8 4.7 3.9

5 6.5 5.0 3.8 2.7 2.2 13.3 11.2 9.2 6.8 5.3

10 7.0 5.5 4.0 2.9 2.3 19.1 15.7 12.1 8.9 6.7

0 7.4 5.9 4.4 3.2 2.5 24.3 20.0 15.5 10.8 7.8

30 8.0 6.0 4.4 3.1 2.4 27.5 22.0 16.7 11.2 8.0

60 8.3 6.2 4.6 3.4 2.8 29.5 23.3 17.3 11.8 8.7

100 8.5 6.5 5.0 3.6 3.0 29.3 23.4 17.8 12.2 9.1

150 8.1 6.2 4.4 3.1 2.4 29.4 23.6 18.2 12.7 9.6

250 8.5 6.2 4.5 3.2 2.6 29.6 24.2 18.8 13.5 10.3

123


13 Appendix 8

See Table 11.

Table 11 Absolute estimated bias (×100), averaged over the 50 landscape, of the contagion estimators forthe systematic design, and 20 classes system


All classes

2 3.2 1.8 1.0 0.5 0.3 2.7 2.3 1.9 1.4 1.1

5 4.1 2.4 1.4 0.7 0.4 5.3 4.6 3.6 2.4 1.8

10 5.0 3.0 1.8 0.8 0.5 8.4 7.0 5.3 3.5 2.5

20 6.0 3.6 2.1 1.0 0.5 11.7 9.5 7.1 4.4 2.9

30 6.5 4.0 2.2 1.1 0.6 13.3 10.7 7.8 4.8 3.2

60 7.2 4.3 2.5 1.3 0.9 14.9 11.7 8.5 5.4 3.7

100 7.5 4.7 2.9 1.6 1.1 15.3 12.1 9.0 5.7 4.0

150 8.0 5.3 3.1 1.8 1.2 15.7 12.6 9.2 6.0 4.3

250 8.6 5.5 3.4 2.0 1.5 16.0 12.8 9.5 6.3 4.6

Observed classes

2 −3.5 −2.5 −1.4 −0.6 −0.4 5.9 5.1 4.2 3.2 2.5

5 −3.2 −2.2 −1.3 −0.8 −0.7 10.7 8.9 7.2 5.0 3.5

10 −2.0 −1.5 −0.9 −0.7 −0.7 16.2 13.1 9.9 6.7 4.6

20 −1.0 −1.0 −0.7 −0.8 −0.8 21.2 17.4 13.1 8.3 5.3

30 −0.1 −0.2 −0.3 −0.5 −0.5 24.4 19.5 14.4 9.0 5.9

60 0.5 0.1 −0.2 −0.4 −0.5 25.9 20.6 15.1 9.7 6.7

100 0.8 0.3 −0.1 −0.2 −0.3 25.9 20.5 15.2 10.1 7.1

150 1.8 1.1 0.9 0.7 0.6 25.9 20.5 15.7 10.4 7.6

250 2.2 1.6 1.3 1.1 0.9 26.5 21.7 16.6 11.6 8.5

123


14 Appendix 9

See Table 12.

Table 12 Absolute estimated RMSE (×100), averaged over the 50 landscape, of the contagion estimatorsfor the random design, and 20 classes system


All classes

2 5.1 3.5 2.3 1.5 1.1 2.8 2.5 2.2 1.7 1.5

5 6.0 4.2 2.8 1.8 1.3 5.5 4.8 3.9 2.9 2.3

10 7.0 4.8 3.2 2.0 1.4 8.5 7.3 5.7 4.0 3.0

20 8.1 5.5 3.6 2.2 1.5 12.0 9.9 7.6 5.1 3.6

30 8.6 5.8 3.8 2.3 1.6 13.6 11.1 8.4 5.6 3.9

60 9.3 6.2 3.9 2.4 1.7 15.2 12.3 9.2 6.0 4.2

100 9.5 6.3 4.0 2.4 1.7 15.8 12.7 9.4 6.2 4.4

150 9.6 6.4 4.1 2.5 1.8 16.0 12.9 9.6 6.4 4.6

250 9.8 6.6 4.3 2.7 2.0 16.2 13.1 9.8 6.7 4.9

Observed classes

2 6.5 4.8 3.5 2.5 2.0 8.0 6.9 5.8 4.7 3.9

5 7.0 5.6 4.3 3.2 2.6 13.2 11.2 9.0 6.8 5.3

10 7.6 6.1 4.7 3.5 2.8 18.9 15.6 12.3 8.9 6.8

20 8.3 6.7 5.2 3.9 3.2 24.6 19.9 15.5 11.0 8.3

30 8.9 6.9 5.3 3.9 3.1 27.1 21.8 16.8 11.8 8.8

60 9.3 7.3 5.5 4.0 3.2 29.1 23.6 18.0 12.4 9.2

100 9.6 7.4 5.6 4.2 3.4 29.6 23.9 18.3 12.7 9.5

150 8.9 6.8 4.9 3.6 2.8 29.6 24.0 18.4 13.0 9.9

250 8.9 6.7 5.0 3.6 2.9 29.6 24.2 18.8 13.6 10.4

123


15 Appendix 10

See Table 13.

Table 13 Absolute estimated bias (×100), averaged over the 50 landscape, of the contagion estimators forthe random design, and 20 classes system


All classes

2 3.7 2.3 1.3 0.7 0.4 2.7 2.3 1.9 1.4 1.1

5 4.6 2.9 1.7 0.9 0.6 5.3 4.6 3.6 2.5 1.9

10 5.5 3.6 2.1 1.1 0.6 8.4 7.1 5.5 3.7 2.7

20 6.6 4.2 2.5 1.3 0.8 11.8 9.7 7.4 4.8 3.3

30 7.2 4.5 2.7 1.4 0.9 13.4 10.9 8.2 5.3 3.7

60 7.9 5.0 3.0 1.6 1.0 15.1 12.2 9.0 5.8 4.0

100 8.3 5.3 3.2 1.7 1.2 15.6 12.5 9.2 6.0 4.2

150 8.4 5.4 3.3 1.9 1.3 15.9 12.8 9.5 6.2 4.4

250 8.8 5.7 3.6 2.1 1.5 16.1 13.0 9.7 6.5 4.7

Observed classes

2 −3.4 −2.5 −1.5 −0.8 −0.5 5.7 4.9 4.0 3.1 2.3

5 −3.2 −2.2 −1.5 −1.0 −0.8 10.3 8.6 6.7 4.7 3.4

10 −2.0 −1.5 −1.1 −0.8 −0.7 15.6 12.7 9.8 6.6 4.5

20 −1.0 −1.0 −0.9 −0.8 −0.9 21.1 16.8 12.7 8.3 5.6

30 −0.2 −0.4 −0.4 −0.5 −0.5 23.5 18.8 14.2 9.3 6.4

60 0.6 0.2 −0.1 −0.3 −0.3 25.5 20.4 15.2 10.0 7.0

100 1.0 0.4 0.0 −0.2 −0.3 25.8 20.6 15.4 10.2 7.3

150 1.8 1.2 0.8 0.6 0.5 25.8 20.7 15.6 10.5 7.7

250 2.1 1.4 1.1 0.9 0.8 26.2 21.3 16.4 11.4 8.5

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Author Biographies

Habib Ramezani is a PhD in forest science (Forest Inventory). He received his PhD from the SwedishUniversity of Agricultural Sciences (SLU). He received MSc in Forestry from the Tarbiat ModaresUniversity of Tehran (Iran). His current research focuses on forest inventory methods and landscapepattern analysis through landscape metrics.

Sören Holm is senior lecturer in forest inventory at the Swedish University of Agricultural Sciences(SLU). He received MSc in Mathematics in 1966 and was employed at the department of Mathematicsin 1966, from 1971 at the section of Mathematical statistics as researcher. He attended SLU in 1976 asresearcher and consultant in statistical matters and holds his present position since 1989.

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Documents

Estimating a distance dependent contagion function using point sample data