FIRE OCCURRENCE PATTERNS AT LANDSCAPE LEVEL: BEYOND POSITIONAL ACCURACY OF IGNITION POINTS WITH...

NATURAL RESOURCE MODELINGVolume 17, Number 4, Winter 2004

FIRE OCCURRENCE PATTERNS ATLANDSCAPE LEVEL: BEYOND POSITIONALACCURACY OF IGNITION POINTS WITH

KERNEL DENSITY ESTIMATION METHODS

NIKOS KOUTSIASGeographic Information Systems Division (GIS)

Department of GeographyUniversity of Zurich

CH-8057 Zurich, SwitzerlandE-mail : [email protected]

KOSTAS D. KALABOKIDISDepartment of GeographyUniversity of the Aegean

GR-81100 Mytilene, GreeceE-mail : [email protected]

BRITTA ALLGOWERGeographic Information Systems Division (GIS)

Department of GeographyUniversity of Zurich

CH-8057 Zurich, SwitzerlandE-mail : [email protected]

ABSTRACT. Based on kernel density estimation methods,this paper introduces an alternative approach of fire occur-rence modeling that addresses the inherent positional inaccu-racies of recorded wildland fire ignition points. These obser-vations, recorded in longitude and latitude using only degreesand first minutes, contain positional inaccuracies of about±700 to ±925 meters in x and y axes. Kernel density estima-tion was applied to these historical fire observations recordedbetween 1985 and 1995 in Halkidiki peninsula, Greece, as wellas, to simulated inaccurate points into which positional in-accuracies of the same magnitude were randomly introduced.Substantial differences were observed when a regular grid ofquadrants was superimposed over the two point distributions.Although, at higher grid resolution these mismatches wereminimized, the problem of generalization appeared. Contrar-ily, the concept of “moving window” assisted to retain highgrid resolution and minimize the effect of inaccurate point ob-servations. In addition, the kernel approach, which considersalso the relative position of points within the “moving win-dow,” produced more realistic estimates.

KEY WORDS: Kernel density estimation, wildland fire oc-currence, wildland fire ignition points, positional inaccuracies.

Copyright c©2004 Rocky Mountain Mathematics Consortium

359

360 KOUTSIAS, KALABOKIDIS AND ALLGOWER

1. Introduction. Fire is one of the most important natural distur-bance factors in the Mediterranean Basin landscapes and constitutes acritical component of forest ecosystems by influencing their structure,function and dynamics. Maps of fire occurrence at multiple spatialand temporal scales based on historical fire records are important tofire managers and scientists. They provide information that is usefulfor studying the phenomenon itself and the mechanisms responsible forgenerating it. These fire occurrence maps can be used for fire planning,natural resource management, risk assessment and management, as-sessment of short- and long-term post fire effects across space and time,and for gaining knowledge on factors influencing fire regimes (Morganet al. [2001]).

Historical wildland fire occurrence records usually provide x and ycoordinates of fire ignition points, fire extent and perimeters, topo-graphic characteristics, environmental and climactic conditions of thefire location, occurrence time, extinction time, personnel used, as wellas damage assessment. These primary fire occurrence data, recorded foreach wildland fire event, provide the most detailed level of information.Using aggregation methods and spatial analysis tools, secondary data,e.g., fire statistics, are generated from these fire records to representwildland fire occurrence pattern at multiple hierarchical levels, e.g.,community to country, at multiple spatial scales, e.g., local to global,and for various thematic contents, e.g., total number of fires, density,total and mean area burned. Furthermore, descriptive statistics aregenerated to define spatio-temporal patterns, and multivariate statis-tical methods are also used for explaining underlying causal factors offire occurrence at multiple scales.

2. Materials and methods.

2.1 Study area and wildland fire ignition records. The fire databaseused in this study consists of 218 ignition points that occurred between1985 and 1995 in the Halkidiki peninsula located in northern Greece,Figure 1. Their x and y coordinates were recorded in longitude andlatitude using only degrees and first minutes resulting in positionalinaccuracies of about ±700 to ±925 meters in x and y axes. Forstatistical analysis, another set of simulated inaccurate points wasestablished based on those fire ignition points, into which positional

FIRE OCCURRENCE PATTERNS 361

8888 8 8

888 8 8 88 8

8 8 8 888 8 8 8

88 8 88 8 88 8 8 888 8

8 8888 888 888888 888 8 8 8 88 8

8 8 8 88 8 88 8 8 8 88 888 8 8 8 88 88 8 8

8 8 8 888888 888

8888 88

8 8888 8 8 888 888 8

8 88 8888888

88 88888 88888

8888 88 888888 888

88 8

8 888 88888888888 888888888 8888 88888 88 8 8

8888 8888888 888 8 888

888 8 8

8

N

20Km

8 Fire ignition pointsMin : 0 m Max : 2033 m

Elevation

NNI : 0.7576clumped

n = 218 events

8

8 88 8

8

#

# #

##

#

X

Y Grid size : 250 by 250 m

Y

X

Y

8 Ignition points # Simulated inaccurate

Number of points in each grid cell0 1 2

< <

FIGURE 1. Wildland fire ignition points superimposed on a digital elevationmap in Halkidiki peninsula, Greece. A regular grid of 250 by 250 meters su-perimposed over the wildland fire ignition points and the simulated inaccuratepoints revealed significant differences in the fire occurrence pattern.

inaccuracies of the same magnitude were randomly introduced. Thelatter were further used to compare the methods for defining fireoccurrence pattern and check how this pattern is influenced by theinaccurate fire ignition point observations.

2.2 Problem statement. Due to fire characteristics and logistic prob-lems, the precise detection of the actual ignition points is difficult andthe recorded fire ignition locations contain positional inaccuracies. Po-sitional as well as attribute uncertainties may result from factors suchas small-scale or inaccurate, non-updated maps used to read the x andy coordinates or large interval resolutions, e.g., coordinates given onlyin degrees and minutes. In this case the assumption of exact locations,which point-based statistics require, is violated (Jacquez and Waller[2000]). If the aim is to explain the spatial pattern of landscape fireregimes and /or the underlying causal factors these inaccurate pointrecords may, introduce substantial errors. Especially, if explanatoryvariables are extracted from other geo-referenced data layers using spa-tial overlay techniques, these records may lead to serious inaccuracies.

In pattern analysis, very frequently a regular grid of quadrants issuperimposed over the event distribution to allow the study of point


Y Y

X

X

Y

X

Grid size : 2000 by 2000 meters Grid size : 4000 by 4000 meters

Grid size : 6000 by 6000 meters Grid size : 8000 by 8000 meters

R = 0.767

R = 0.962

R = 0.932

R = 0.972

a. b.

c. d.

XY

FIGURE 2. Grids of different resolution superimposed over the point dis-tributions. Each circle represents the relative number of ignition points andsimulated inaccurate points for each grid cell. As the grid size increases thedifferences between the patterns tend to be minimized followed, however, byloss of spatial variability.

pattern (Gatrell et al. [1996]). A regular grid of quadrants of 250 by250 meters superimposed over the wildland fire ignition point as wellas over the simulated inaccurate points revealed significant differences,Figure 1. These two grids of quadrants that were completely uncorre-lated, R = −0.006, indicate the different spatial pattern that resultswhen using the two point datasets. This is reasonable because the gridsize was much less than the magnitude of the positional inaccuracies.To overcome this undesirable situation one solution would be to increasethe size of the superimposed grid, however, this creates generalized andover-smoothed results (associated to grid size).

In Figure 2, which shows the different patterns resulting when usingvarious quadrant sizes, the problem seems to become less as the gridsize increases. The correlation coefficient between the grids of ignitionpoints and the simulated inaccurate points for the whole study areaincreases to 0.767 for the 2000 meters, 0.932 for the 4000 meters,0.962 for the 6000 meters and 0.972 for the 8000 meters. However, as


previously mentioned, the increment of the grid size induces a certaindegree of generalization that is followed by loss of spatial variabilityand hence does not allow the detailed description of the fire occurrencepattern.

To overcome the deficiencies resulting from superimposing a regulargrid of quadrants over a point distribution, a “moving window” of fixeddimensions can be used as an alternative for estimating the intensityat each grid cell. In this case the intensity at each grid cell is estimatedusing the number of points falling within the size of the “movingwindow” centered on each grid cell (Bailey and Gatrell [1995]). Kernelestimation is an extension of the “moving window” concept where thefixed-size window is replaced by a three-dimensional function (Gatrellet al. [1996]), Figure 3.

2.3 Kernel density estimation. The implementation of kernel densityestimation is based on the estimation of the density at each intersectionof a grid of quadrants superimposed on the data, after placing aprobability density (kernel) over each event (fire ignition points). Theestimated density results from the sum of the densities of all the kernelsoverlapping at each grid cell (Levine [2000], Seaman and Powell [1996]),Figure 4. In the present study, kernel density interpolation has beenimplemented using CrimeStat c©, Ver. 1.1, which is a spatial statisticsprogram for the analysis of crime incident location (Levine [2000]).

Kernel density estimation, a nonparametric statistical method for es-timating probability densities, has been widely used for home range es-timation in wildlife ecology (Worton [1989], Seaman and Powell [1996],Tufto et al. [1996]). It has the advantage of directly producing densityestimates that are not influenced by grid size and localization effects.Furthermore, it produces densities of any shape and analyzes any datadistributed multi-modally or non-normally (Seaman and Powell [1996]).

The bivariate kernel density estimator is mathematically defined as(Silverman [1986], Worton [1989], Seaman and Powell [1996]):

f(x) =1

nh2

n∑i=1

K

{(x − Xi)

h

}

where n is the number of points, h is the smoothing parameter or thebandwidth, K is a kernel density function, x is a vector of coordinates


#

# ## #

#

X

Y

G(a) rid size : 1000 by 1000 m

#

# ## #

#

X

Y

G(b) rid size : 250 by 250 m

Moving windows1750 m1250 m750 m

#

#

#

(C)

−4

−2

0

2

4

−4

−2

0

2

40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

FIGURE 3. Different approaches to study the pattern of wildland fire occur-rence: (a) by a regular grid superimposed over the point observations; (b) bythe “moving window” concept of fixed dimensions; and (c) by the kernel estima-tion where the fixed-size window is replaced by a three-dimensional function.

that define the location where the function is estimated, and Xi arevectors of coordinates that define each observation i.

According to Levine [2002], besides the normal distribution that is themost commonly used kernel function (Kelsall and Diggle [1995]), othersinclude the triangular function (Burt and Barber [1996]) or the quarticfunction (Bailey and Gatrell [1995]). The normal distribution functionthat has been used in the present study is given by the followingfunctional form (Levine [2002]):

g(xj) =∑ {

[WiIi]1

h22πe[d2

ij/2h2]

}

where dij is the distance between a point observation (i) and anylocation in the region where the function is estimated (j), h is the


(a)

(b)

FIGURE 4. The density at each intersection of a grid superimposed on thedata, after placing a probability density (kernel) over each event (a), resultsfrom the sum of densities of all the kernels that overlap that point (b).

bandwidth which represents the standard deviation of the normaldistribution, Wi and Ii represents weight and intensity factors at thepoint location, respectively.

Two types of kernel methods, depending on whether constant ormultiple adaptive values are used for the smoothing parameter overthe entire region, constitute the two alternative methods that can beapplied in kernel density estimation; the fixed and adaptive method,respectively. In the fixed kernel mode, the smoothing parameter, whichis defined in distance units, is constant over the entire area of interest.In the adaptive mode the smoothing parameter, which is defined byusing a minimum number of point observations found under the kernel,varies depending on the concentration of point observations. Thismeans that in areas of low concentration the smoothing parameter takeshigher values than in areas of high concentration (Worton [1989]).


An important issue, however, and rather difficult to define when im-plementing kernel density interpolation, is the choice of the smoothingparameter of the kernel, both in the fixed and in the adaptive mode.Narrow bandwidths allow nearby observations to dominate the den-sity estimate, while wide bandwidths favor distant locations (Worton[1989], Seaman and Powell [1996]). According to Silverman [1986], thechoice of the bandwidth depends mostly on the purpose for which thedensity estimate is used. If the aim is to explore the data and suggestmodels and hypotheses about them, it would be sufficient to choose thesmoothing parameter subjectively by visual inspection.

Rather than choosing an arbitrary value for the smoothing parameter,either fixed or adaptive, it is better to adopt a more rigorous way. Inthe literature, many different methods have been proposed to definesmoothing parameters (for further details see Silverman [1986], Worton[1989], Jones et al. [1996], Seaman and Powell [1996], Grund andPolzehl [1998], Katkovnik and Shmulevich [2002]). In the present study,kernel density estimation of fire ignition points has been applied usingthe fixed kernel method, while the choice of the bandwidth of the kernelwas based on the value of the mean nearest distance (Clark and Evans[1954]) when an equal number of randomly distributed points as thoseof fire ignition points are considered over the area of interest.

3. Comparative evaluation of the three methods. To com-pare the three methods mentioned for studying wildland fire occur-rence pattern (a. regular grid of quadrants superimposed over the fireignition points, b. “moving window” of fixed dimensions and c. kernelapproach), these methods were applied to the original fire ignition loca-tion records as well as to the simulated inaccurate ignition points. Thefirst method, employed by superimposing a regular grid of quadrantsover the point distribution, Figure 5a, revealed substantial differenceswhen the grid resolution was relatively small compared to the mag-nitude of the positional errors. Although these differences tended tobe minimized at higher grid resolutions, another problem, that of gen-eralization and loss of spatial variability, appeared. The dilemma ofkeeping these mismatches by retaining the high grid resolution or min-imizing the effect of inaccurate fire ignition points by losing the spatialvariability seems to be overcome by the other two techniques. Theconcept of “moving window” using either a circle of fixed dimensions


X

Y

X

Y

X

Y

X

Y

X

Y

X

Y

Fire

Igni

tion

Poi

nts

Sim

ulat

ed In

accu

rate

250 by 250 meters 500 by 500 meters 1000 by 1000 metersG r i d s i z e

) R

(a)

= - 0.005 R = - 0.021 R = 0.541))

X

Y

#

# ## #

#

X

Y

X

Y

X

Y

#

# ## #

#

X

Y

X

Y

Fire

Igni

tion

Poi

nts

Sim

ulat

ed In

accu

rate

250 meters 500 meters 1000 metersR a d i u s d i s t a n c e

) R

(b)

= 0.0275 R = 0.1463 R = 0.4212))

Grid size : 250 by 250 m

X

Y

X

Y

X

Y

X

Y

X

Y

X

Y

Fire

Igni

tion

Poi

nts

Sim

ulat

ed In

accu

rate

250 meters 500 meters 1000 metersK e r n e l b a n d w i d t h

) R

(c)

= 0.058 R = 0.3629 R = 0.7068))

Grid size : 250 by 250 m

FIGURE 5. Patterns of wildland fire occurrence when applying the three approaches tothe original ignition points as well as to the simulated inaccurate points: (a) regular gridof quadrants superimposed over the point observations; (b) the “moving window” conceptof fixed dimensions; and (c) the kernel approach where the fixed-size window is replacedby a three-dimensional function.


k(a)ernel density estimation

(bandwidth size : 2000 meters)moving window density estimation

(r

(b)

adius size : 6000 meters)

N10 0 10 20 30 Km

FIGURE 6. In addition to fixed window size (a), the kernel approach (thatweighs the point observations according to their distance from the grid cellat which the intensity is estimated) produces more smoothed and realisticestimates (b).

(Figure 5b) or a three-dimensional function (Figure 5c) assists so thatto retain the high grid resolution and minimize the effects of inaccuratepoint observations. The correlation coefficient (R) between the gridsresulted from the original ignition points and the simulated inaccuratepoints increased considerably at higher circles or kernel sizes, while thespatial resolution of the grids kept very high. One problem that stillexists in the “moving window” concept of fixed dimensions is that therelative location of point observations within the window is not takeninto consideration (Bailey and Gatrell [1995]).

The results of the application of these two techniques to wildlandfire ignition points over the entire study area are depicted in Figure 6.The effect of fixed window size, which does not consider the relativeposition of point observations within the window, is obvious on theoutput image. The resulting spatial pattern that follows the shape ofthe circles used to estimate the density values, presents an unrealisticoutput surface. In addition, the kernel approach, that weighs the pointobservations according to their distance from the grid cell at which theintensity is estimated (Gatrell et al. [1996]), produces more smoothedand realistic estimates.

4. Implementing the kernel approach to fire occurrencemodeling. To implement kernel density estimation in wildland fire


occurrence, the fixed mode approach has been adopted in order tokeep the smoothing parameter of the kernel constant over the entirestudy area. The incentive behind this decision was to avoid differenttreatments of the point observations over the areas with differentdegrees of concentration. In addition to the choice of the kerneltype, which might not be so important, the choice of the smoothingparameter is very crucial since it controls the amount of variation ofthe estimates (Worton [1989]). To define the size of the bandwidthboth the positional inaccuracies of wildland fire ignition points, andmostly the mean random distance of an equal number of randomlydistributed points as those of fire ignition points, were considered. Thepositional inaccuracies of fire ignition points indicate that their actualpositions lie inside an area of about 1400 and 1800 meters in the x andy dimensions, respectively.

The mean random distance depends on the total number of pointobservations and the area of the region. Keeping the area constant,the increase of the number of point observations will cause a decreaseof the mean random distance. The association of the size of thesmoothing parameter of the kernel to the mean random distance isreasonable, since the amount of point observations that is going to beinterpolated is related to the information content. If the sample sizeis large, which corresponds to more informative datasets, then a finerinterval resolution would be more suitable to avoid over-smoothing andloss of the variability in the estimates. For small size samples, whichcorrespond to less informative datasets, a large bandwidth would bemore appropriate, since a fine interval would lead to density estimationsthat may be perceived as nothing more than a random variation. Thesize of the smoothing parameter also defines the level of scale of theestimates; large intervals diminish local variability and their estimatesare associated with global patterns, while narrow intervals preservedata variability and are associated to local patterns.

Given that the area of the digital coverage of the study area is3,257,633,184 m2, the mean random distance of the 218 randomlydistributed points is 1933 meters. Kernel density interpolation ofrandomly distributed points using a bandwidth of 2000 meters, whichcorresponds to the mean random distance, produced density estimateswhose values tend to follow, as indicated from their histogram dataplot, a normal distribution, Figure 7. A Kolmogorov-Smirnov test was


Bandwidth : 2000 m

Kernel density estimation based on randomly distributed points4500

4000

3500

3000

2500

2000

1500

1000

500

0

Histogram

High density areasLow density areas

N

20 Km

FIGURE 7. Kernel density estimation based on 218 randomly distributedpoints (simulated) using a bandwidth of 2000 meters (that corresponds to themean random distance of these 218 points) produces density estimates whosevalues follow a normal distribution.

implemented in the analysis to test whether the estimates follow anormal distribution. Since the value of the p-level is less than 0.05,the estimates can be considered, at least marginally, following an evendistribution, Table 1. The normally distributed estimates suggestthat the bandwidth of 2000 meters is appropriate to be used for thenumber of points that will be interpolated in the given area. First,we avoid over-smoothing that results from relatively large bandwidth,and second, we do not produce estimates that would be considered asnothing else but random variation.

Although that, based on both the positional inaccuracies of wildlandfire ignition points and the mean random distance, the most appro-priate size of the bandwidth seems to be that of 2000 meters, kerneldensity estimation was applied using four alternative bandwidths; thoseof 1000, 2000, 4000 and 6000 meters, Figure 8.


TABLE 1. Kolmogorov-Smirnov goodness of fit test: the zero hypothesis

states that the kernel density estimates follow a normal distribution.

Kernel density estimates maxD p

Random points, 2000 m bandwidth 0.074501 < 0.05Fire ignition points, 1000 m bandwidth 0.263590 < 0.01Fire ignition points, 2000 m bandwidth 0.174723 < 0.01Fire ignition points, 4000 m bandwidth 0.117032 < 0.01Fire ignition points, 6000 m bandwidth 0.097352 < 0.01

Histogram data plots (Figure 8) of density estimates of each kernelsize suggest the existence of favorable areas that present a high con-centration of wildland fires, as well as the existence of large areas withvery low concentration. In fact, this clustered spatial arrangement ofwildland fires is responsible for the non-normal distributed output ofthe kernel density estimates of fire ignition points. The majority of thedensity values, as shown in the histogram data plots (Figure 8), areaccumulated on the left side of the histogram. Kolmogorov-Smirnovgoodness of fit test at p < 0.01 revealed also significant differencesbetween the distribution of the kernel estimates and the normal distri-bution, Table 1.

Kernel density estimation was applied also to the simulated inaccu-rate points of the whole study area using the same four alternativebandwidths (Figure 9). Simple descriptive statistics were derived, andthe correlation coefficient between the kernel densities of the two pointdatasets derived under the same bandwidth was estimated in order toevaluate the differences that arise because of the inaccurate location ofthe point observations (Table 2). Graphically, the output surfaces donot present a distinct differentiation, especially at higher bandwidthsizes (Figure 9). The correlation coefficients range above 0.95 (Table2), indicating an evident spatial agreement of those surfaces. Specifi-cally, the correlation coefficient even increased considerably from 0.95to 0.99 when increasing the bandwidth size from 1000 to 2000 me-ters. The bandwidth size of 2000 meters covers the uncertainty levelof the original wildland fire ignition point observations (±750 m and±925 m in x and y axes, respectively). In addition, it corresponds to


Bandwidth : 1000 m Bandwidth : 2000 m


Kernel density estimation based on fire ignition points30000

25000

20000

15000

10000

5000

0

Histogram12000

10000

8000

6000

4000

2000

0

Histogram

4000

3500

3000

2500

2000

1500

1000

500

0

Histogram3500

3000

2500

2000

1500

1000

500

0

Histogram


N

20Km

N

20Km

N

20Km

N

20Km

FIGURE 8. Kernel density estimation based on fire ignition points using fourdifferent bandwidths.

the mean random distance that results when a corresponding numberof randomly distributed (simulated) points is considered over the studyarea.

TABLE 2. Simple descriptive statistics and correlation coefficients between

the kernel estimates of the two point datasets using the same bandwidth size.

Kernel Standardsize Kernel estimates Mean Deviation Correlation

(meters)

1000 Ignition points 0.00383676 0.0065293 0.952448

Simulated inaccurate points 0.00375742 0.0061031

2000 Ignition points 0.00347149 0.0039997 0.991617


4000 Ignition points 0.00293487 0.0023342 0.998679


6000 Ignition points 0.00254760 0.0015971

Simulated inaccurate points 0.00253913 0.0015730 0.99964




Kernel density estimation based on simulated inaccurate points30000

25000

20000

15000

10000

5000

0

Histogram12000

10000

8000

6000

4000

2000

0

Histogram

4000

3500

3000

2500

2000

1500

1000

500

0

Histogram3500

3000

2500

2000

1500

1000

500

0

Histogram


N

20Km

N

20Km

N

20Km

N

20Km

FIGURE 9. Kernel density estimation based on the simulated inaccuratepoints using the same four alternative bandwidths.

Considering that we should avoid over-smoothing and not produceestimates generated by local variability and random processes, thekernel size of 2000 meters seems to be an appropriate bandwidthsize. However, additional research, under a wider range of conditionscovering any particularities found in various study cases, is needed toadditionally evaluate and support the proposed methodology.

5. Concluding remarks. In this study we introduced an alter-native approach for fire occurrence analysis based on kernel densityestimation of wildland fire ignition points. The inherent positional in-accuracies of ignition points prohibit the use of certain techniques fordefining and explaining the spatial pattern of fire occurrence. Espe-cially, if explanatory variables are extracted from other geo-referenceddata layers using spatial overlay techniques, these inaccurate locationsof fire ignition points may lead to serious inaccuracies. By smooth-


ing the influence of the individual fire ignition point, density surfacescan be created and used to delineate critical zones of fire occurrenceat landscape level. Moreover, the output of kernel density estimationis a continuous surface instead of a binary one that would restrict theanalysis to specific statistical methods.

The continuous output surface allows the identification of “hot spot”areas by setting up certain threshold values corresponding to differentlevels of densities. In addition, contour lines at various probabilitylevels can be extracted to delineate critical zones of fire occurrence.The resulting probabilistic pattern of wildland fire occurrence can beutilized from risk management, and hazard mitigation, to landscapeecology studies.

Environmental heterogeneity (that implies the existence of favorableand non-favorable areas for having a fire) is the fundamental character-istic of the landscape, based on which we can further test hypothesesfor explaining the processes responsible for generating these patterns(Boots and Getis [1988]). The study of potential interactions betweenenvironmental factors, either natural or social, and wildland fire oc-currence is very critical so as to better understand and conceptualizethe driven processes. The detection of spatial dependencies and thestudy of scale effects that govern such processes are other critical is-sues that need to be examined. To move from purely point patternanalysis techniques to surface pattern analysis techniques, which implythe use of semi-variograms and/or correlograms, the point data shouldbe converted to continuous surfaces (Davis et al. [2000]). Using kerneldensity estimation, point observations (given in x and y coordinates)are efficiently converted to continuous density surfaces by addressingsimultaneously some of their inherent positional inaccuracies.

Acknowledgments. This work is funded by the EC project ‘For-est Fire Spread and Mitigation (SPREAD),’ EC-Contract Nr. EVG1-CT-2001-00027, and the Federal Office for Education and Science ofSwitzerland (BBW), BBW-Contract Nr. 01.0138.

REFERENCES

T.C. Bailey and A.C. Gatrell [1995], Interactive Spatial Data Analysis, LongmanScientific & Technical, Burnt Mill, Essex, England.


B.N. Boots and A. Getis [1988], Point Pattern Analysis, Sage Publications, Inc.,California.

J.E. Burt and G.M. Barber [1996], Elementary Statistics for Geographers, TheGuilford Press, New York.

P.J. Clark and F.C. Evans [1954], Distance to Nearest Neighbor as a Measure ofSpatial Relationships in Populations, Ecology 35, 445 453.

J.H. Davis, R.W. Howe and G.J. Davis [2000], A Multi-Scale Spatial AnalysisMethod for Point Data, Landscape Ecol. 15, 99 114.

A.C. Gatrell, T.C. Bailey, P.J. Diggle and B.S. Rowlingsont [1996], Spatial PointPattern Analysis and Its Application in Geographical Epidemiology, T.I. Brit.Geogr. 21, 256 274.

B. Grund and J. Polzehl [1998], Bias-Corrected Bootstrap Bandwidth Selection,J. Nonparametr. Stat. 8, 97 126.

G.M. Jacquez and L.A. Waller [2000], The Effect of Uncertain Locations onDisease Cluster Statistics, in Quantifying Spatial Uncertainty in Natural Resources:Theory and Applications for GIS and Remote Sensing (H. Todd Mowrer and RussellG. Congalton, ed.), Ann Arbor Press, Chelsea, Michigan, pp. 53-64.

M.C Jones, J.S. Marron and S.J. Sheater [1996], Progress in Data-Based Band-width Selection for Kernel Density Estimation, Computation. Stat. 11, 337 381.

V. Katkovnik and I. Shmulevich [2002], Kernel Density Estimation with AdaptiveVarying Window Size, Pattern Recogn. Lett. 23, 1641 1648.

J.E. Kelsall and P.J. Diggle [1995], Kernel Estimation of Relative Risk, Bernoulli1, 3 16.

N. Levine [2000], CrimeStat: A Spatial Statistics Program for the Analysis ofCrime Incident Locations (Version 1.1), Ned Levine and Associates, Annandale,VA and The National Institute of Justice, Washington, DC.

N. Levine [2002], CrimeStat II: A Spatial Statistics Program for the Analysis ofCrime Incident Locations (Version 2.0), Ned Levine and Associates, Annandale,VA and The National Institute of Justice, Washington, DC.

P. Morgan, C.C. Hardy, T.W. Swetnam, M.G. Rollins and D.G. Long [2001],Mapping Fire Regimes across Time and Scale: Understanding Coarse and Fine-Scale Fire Patterns, Int. J. Wildland Fire 10, 329 342.

D. Erran Seaman and R.A. Powell [1996], An Evaluation of the Accuracy ofKernel Density Estimators for Home Range Analysis, Ecology 77, 2075 2085.

B.W Silverman [1986], Density Estimation for Statistics and Data Analysis,Chapman and Hall, London, England.

J. Tufto, R. Andersen and J. Linnell [1996], Habitat Use and Ecological Correlatesof Home Range Size in a Small Cervid: The Roe Deer, J. Anim. Ecol. 65, 715 724.

B.J. Worton [1989], Kernel Methods for Estimating the Utilization Distributionin Home-Range Studies, Ecology 70, 164 168.

FIRE OCCURRENCE PATTERNS AT LANDSCAPE LEVEL: BEYOND POSITIONAL ACCURACY OF IGNITION POINTS WITH...

Documents

Transcript of FIRE OCCURRENCE PATTERNS AT LANDSCAPE LEVEL: BEYOND POSITIONAL ACCURACY OF IGNITION POINTS WITH...