RESEARCH PAPER
h ps://doi.o g/10.1071/WF22030
Non-pa ame ic compa a i e analysis o he spa io empo al
pa e n o human-caused and na u al wild i es in Galicia
M. F. Ma ey-Pé ez
A,*
, Isabel Fuen es-San os
B
, Paula Saa e a-Nie es
C
and Wenceslao González-Man eiga
C
ABSTRACT
Backg ound. Wild i e is a majo en i onmen al h ea wo ldwide and clima e change is
expec ed o inc ease i s se e i y. Galicia has su e ed high wild i e incidence du ing he
las decades, mos wild i es being om a son, in con as wi h he low a e o na u al wild i es.
Aim. This wo k aims o cha ac e ise he spa io empo al dynamics o human-caused and na u al
i es in Galicia. Me hods. We apply i s - and second-o de non-pa ame ic in e ence o
spa io empo al wild i e poin pa e ns. Key esul s. The dis ibu ion o na u al wild i es
emained s able o e yea s, wi h high incidence in summe and in he eas e n a ea o Galicia.
A son wild i es had agg ega ed pa e ns, wi h s ong in e ac ion be ween ou b eaks and i es,
and hei dis ibu ion a ied bo h o e and wi hin yea s, wi h high incidence shi ing be ween he
sou he n and wes e n a eas, and high haza d in ea ly sp ing and la e summe . Negligence wild i e
pa e ns showed sho -dis ance agg ega ion, bu la ge-dis ance agg ega ion be ween ou b eaks
and i es; hei spa ial dis ibu ion also a ied be ween and wi hin yea s. Conclusions. Di e en
models and co a ia es a e equi ed o p edic he haza d om each wild i e ype. Na u al i es
a e linked o me eo ological and en i onmen al ac o s, whe eas socioeconomic co a ia es a e
c ucial in human-caused wild i es. Implica ions. These esul s a e he basis o he u u e
de elopmen o p edic i e spa io empo al poin p ocess models o human-caused wild i es.
Keywo ds: inhomogeneous poin p ocesses, in ensi y unc ion, non-pa ame ic in e ence,
smoo h boo s ap, Spain wild i e, spa io empo al in e ac ions, wild i e haza d.
In oduc ion
Wild i e is a majo en i onmen al h ea a ound he wo ld ha causes subs an ial na u al
damage and socioeconomic losses (Zhang e al. 2016; Cos a eda-Aumedes e al. 2017;
Bowd e al. 2019; Ma s e al. 2019; Pausas and Keeley 2021). In he sou he n Eu opean
Union (EU) coun ies (Po ugal, Spain, F ance, I aly and G eece), 48 600 o es i es bu n
on a e age 447 800 ha (1980–2015) e e y yea , mos o his a ea being bu ned by a small
numbe (<15%) o la ge i es (San-Miguel-Ayanz e al. 2013). This high wild i e inci-
dence in Medi e anean Eu ope can be a ibu ed o clima e, en i onmen al and an h o-
pogenic ac o s. On one hand, clima e change has inc eased empe a u e and has educed
ela i e humidi y. On he o he hand, socioeconomic ac o s such as he abandonmen o
a ms and u al a eas ha e esul ed in an unusual accumula ion o o es uels (Vila del
Hoyo e al. 2011; A agó e al. 2016).
Un a elling wild i e pa e ns is c ucial o unde s and i e beha iou and o de elop
p e en ion and mi iga ion s a egies, as well as landscape es o a ion plans. The inc eas-
ing a ailabili y o geo e e enced wild i e da a has enhanced he de elopmen o a p oli ic
line o esea ch ocused on analysing he spa ial dis ibu ion o wild i e and on iden i y-
ing bio ic and abio ic isk ac o s. Spa io empo al clus e analysis has been p oposed as a
use ul ool o add ess impo an scien i ic challenges such as cha ac e ising obse ed
sho - e m clus e ing o igni ion poin s, de ec ing changes in he spa ial dis ibu ion o
igni ion poin s and unde s anding he causes o hose changes, and measu ing empo al
a ia ions in wild i e clus e s (P es emon e al. 2013). Clus e ing analysis applied o
Fo ull lis o au ho a ilia ions and
decla a ions see end o pape
*Co espondence o:
M. F. Ma ey-Pé ez
PROePLA Resea ch G oup, Depa men o
C op P oduc ion and Enginee ing P ojec s,
Uni e sidade de San iago de Compos ela,
Spain
Email: [email p o ec ed]
Recei ed: 15 Ma ch 2022
Accep ed: 22 No embe 2022
Published: 23 Decembe 2022
Ci e his:
Ma ey-Pé ez MF e al. (2023)
In e na ional Jou nal o Wildland Fi e
32(2), 178–194. doi:10.1071/WF22030
© 2023 The Au ho (s) (o hei
employe (s)). Published by
CSIRO Publishing on behal o IAWF.
wild i e esea ch has been conduc ed h ough he combina-
ion o classical s a is ics wi h GIS (Geog aphic In o ma ion
Sys em) ools (Rod igues e al. 2019), and by spa io empo al
poin p ocess analysis and modelling (Se a e al. 2013;
Wing and Long 2015; A agó e al. 2016; Díaz-A alos e al.
2016; Pa en e e al. 2016; Woo e al. 2017). In addi ion o
he analysis o in e ac ions be ween i e e en s, poin p o-
cess modelling also allows es ima ion o he e ec o en i -
onmen al and socioeconomic co a ia es on wild i e
incidence, and he c ea ion o i e haza d maps.
In he analysis o wild i es, we should dis inguish
be ween na u al and human-caused i es, which, in addi ion
o di e en o igins, ha e di e en dynamics (González-
Olaba ia e al. 2015). Na u al i es, mainly caused by ligh -
ning, end o be concen a ed in summe mon hs, os e ed
by he high empe a u es, low ai humidi y and, hence, low
uel mois u e con en (Pineda e al. 2014; Nampak e al.
2021; He e al. 2022). On he o he hand, mo e han 90% o
i es wo ldwide a e di ec ly o indi ec ly linked o in en-
ional and unin en ional human ac i i ies (Cos a eda-
Aumedes e al. 2017). The spa io empo al dynamics o
human-caused i es ha e been s udied o a long ime
(C osby 1954; B uce 1963; Donoghue and Main 1985).
Recen s udies ound ha a son i es in he Ibe ian
Peninsula ha e agg ega ed pa e ns (Cos a eda-Aumedes
e al. 2016) and analysed he biophysical and socioeconomic
ac o s ha a ou he occu ence o a son and negligence
i es (Rod igues e al. 2016; Pa en e e al. 2018; Sil a e al.
2019; Dupuy e al. 2020), eaching complemen a y conclu-
sions. In pa icula , Rod igues e al. (2016) iden i ied
changes in ag icul u al ac i i y as he main d i e o a son
wild i es in Spain in he las decade o he 20 h and he i s
decade o he 21s cen u ies; Dupuy e al. (2020) and
Rod igues e al. (2018) linked human-caused wild i es
wi h clima e change, while Sil a e al. (2019) sugges ed
ha an h opogenic ac o s a e he main cause ha explains
a son i es in he Ibe ian Peninsula.
Galicia (no h-wes Spain, Fig. 1 (le )) has su e ed high
wild i e incidence o e se e al decades, al hough clima e-
ela ed i e dange in his egion is lowe han in he
Medi e anean and sou he n egions o Spain (Fig. 1,
igh ). This is due o he ac ha a la ge ac ion, abo e
75%, o wild i es in Galicia a e a son, 25% highe han he
na ional a e (MAGRAMA 2016). The pa icula cha ac e -
is ics o wild i es in Galicia, and hei se e e en i onmen al
and socioeconomic consequences, ha e mo i a ed o a la ge
body o esea ch wi h he aim o unde s anding i e dynam-
ics, iden i ying i e isk d i e s and p edic ing he occu -
ence o u u e e en s. Any p og ess in hese esea ch opics
will con ibu e o imp o ing he e ec i eness o cu en i e
p e en ion and supp ession plans. Di e en s udies ha e
p oposed he applica ion o a wide a ie y o s a is ical
analysis and modelling echniques. Boube a e al. (2015)
p oposed a semipa ame ic ime se ies model o p edic
he bu ned a ea in Galicia 1 week in ad ance. Boube a
e al. (2016, 2019) used a ea-le el mixed e ec s models
o p edic he numbe o i es; he o me showed good
pe o mance o Poisson mixed eg ession models and boo -
s ap echniques, and he la e inco po a ed ime e ec s
in o hese models. Wild i e occu ence and bu ned a ea
ha e also been es ima ed h ough s uc u ed addi i e eg es-
sion (STAR) ha inco po a es space- and ime- a ying
co a ia es, as well as spa ial and empo al co ela ions
(Ríos-Pena e al. 2017, 2018). Non-pa ame ic i s - and
Fi es/Yea 10 sqkm o wildland
0.00
0.01
0.02
0.03–0.05
0.06–0.08
0.09–0.12
0.13–0.18
0.19–0.26
0.27–0.60
0.61–24.98
No i e da a
Loca ion o Galicia
A e age me eo ological o es i e dange , 1981–2010
Seasonal se e i y a ing
<0.4
0.4–1
1–2
2–3
3–4
4–5
5–6
6–7
7–10
10–15
>15
Ou side
co e age
Fig. 1. Le : loca ion o Galicia and annual mean in ensi y o wild i e in Eu ope. Sou ce: Bi o e al. (2009). Righ : a e age
clima e- ela ed o es i e dange in Eu ope (Eu opean En i onmen al Agency).
www.publish.csi o.au/w In e na ional Jou nal o Wildland Fi e
179
second-o de analysis o spa ial poin p ocesses was used o
s udy he spa ial dis ibu ion o wild i e ac i i y in a o es
dis ic in Galicia by Fuen es-San os e al. (2013). In pa al-
lel, some s udies ha e de eloped new non-pa ame ic in e -
ence echniques o spa ial and spa io empo al poin
p ocesses and ha e used wild i e pa e ns in Galicia as
case s udies o illus a e hei applicabili y. Fuen es-San os
e al. (2016) p oposed a plug-in bandwid h selec o o he
ke nel in ensi y es ima o , which con ibu es o ob aining
mo e accu a e es ima o s o he spa ial dis ibu ion o wild-
i es. The non-pa ame ic i s -o de in ensi ies compa ison
by Fuen es-San os e al. (2017) can be used o es i i es
wi h di e en causes ha e he same spa ial dis ibu ion,
whe eas he log- a io based sepa abili y es p oposed by
Fuen es-San os e al. (2018) allows us o check i he spa ial
dis ibu ion o wild i es a ies o e ime.
In he p esen wo k, we analyse a la ge da ase comp is-
ing he igni ion poin s and ime o occu ence o he wild-
i es eco ded in Galicia du ing 16 yea s (1999–2014). We
ocused on compa ing he pa e ns o na u al, a son and
negligence wild i es. Gi en ha o e 70% o wild i es in
Galicia bu n less han 1 ha, we analyse wild i es smalle
(ou b eaks) and la ge ( i es) han 1 ha bu ned sepa a ely.
The no el non-pa ame ic i s -o de in e ence echniques
o spa ial and spa io empo al poin p ocesses in oduced
abo e we e applied o desc ibe he dis ibu ion o a son,
negligence and na u al wild i e pa e ns, and es o di e -
ences be ween yea s in o de o iden i y a po en ial change
poin in wild i e pa e ns and be ween he spa ial dis ibu-
ions o wild i e causes. We also applied Mon e Ca lo es s
based on he inhomogeneous L- unc ion (Baddeley e al.
2000), he inhomogeneous L-c oss (Ripley 1981) and he
spa io empo al K- unc ion (Fuen es-San os e al. 2018) o
es o in e ac ions be ween i es.
S udy a ea and da ase
S udy a ea
Galicia, loca ed in he no h-wes o he Ibe ian Peninsula
(Fig. 1), has a su ace a ea o 29 574 km
2
, which ep esen s
5.78% o he su ace o Spain, and has a popula ion o
2.7 million (5.84% o he Spanish popula ion), which
implies a mean popula ion densi y o 94.1 inhabi an s/
km
2
, close o he mean popula ion densi y o Spain
(Spanish S a is ical O ice (INE) 2019, h ps://www.ine.es/).
Fo es s occupy 69% o he su ace in Galicia, a la ge p o-
po ion in compa ison wi h he Spanish a e age (55%,
MAGRAMA 2019). Fo es ed land in Galicia is mainly p i-
a e (97%) and dis ibu ed among a la ge numbe o small
landowne s a an a e age a e o 2 ha/owne (Ma ey-Pé ez
e al. 2012). The o es y sec o in Galicia gene a es 20 320
di ec and 50 000 indi ec jobs, and ep esen s 3.5% o he
egional g oss domes ic p oduc (GDP) (XERA 2018). Indeed,
Galicia, wi h an a e age income o EUR2652 million/yea , is
he majo imbe p oduce in Spain and he nin h in Eu ope
(Balsa-Ba ei o and He mosilla 2013).
As we can see in Fig. 1 (le ), Galicia is one o he egions
wi h highe wild i e incidence in Eu ope (Comas e al. 2014;
Ál a ez-Díaz e al. 2015; Da Pon e e al. 2019) despi e i s
ela i ely un a ou able i e wea he condi ions. In ac ,
Galicia su e ed mo e han 255 000 o es i es be ween
1961 and 2014, wi h an es ima ed a ec ed a ea o 1.83 mil-
lion ha (Ríos-Pena e al. 2017).
Wild i e da ase
This wo k analyses he 104 000 wild i es eco ded in
Galicia om 1999 o 2014. Da a we e ob ained om he
Wild i e Gene al S a is ics a ailable a he Spanish
Go e nmen Da a Po al (h ps://da os.gob.es/es/
ca alogo/e05068001-es adis ica-gene al-de-incendios- o e-
s ales). The da ase comp ises he spa ial loca ion and ime o
de ec ion o each igni ion poin oge he wi h addi ional
in o ma ion abou he sou ce o igni ion, supp ession and
damage caused by he wild i e. This wo k ocuses on he
bu ned a ea and he cause o i e. Al hough he Wild i e
Gene al S a is ics p o ide wild i e eco ds o m 1983, he
pe iod 1983–1998 was disca ded in his s udy a e conduc -
ing a da a quali y con ol o he geo e e enced in o ma ion
p o ided. Da a quali y con ol was conduc ed es ing
whe he he spa ial coo dina es o each igni ion poin
we e wi hin he epo ed municipali y and ype o land
assigned o he e en . Disc epancies ound in he pe iod
1999–2014 we e co ec ed, assigning new coo dina es o
he e en h ough he p ocedu e de ailed in Fuen es-
San os e al. (2013), and less han 10% o igni ion poin s
needed o be co ec ed.
Fo es i es in Spain ha e been classi ied wi h i e main
causes: na u al, a son, acciden al o negligence ( e e ed o
as negligence he ea e ), ep oduc ions ( eigni ing i es) and
unknown cause (MAGRAMA 2016). This wo k ocuses on
a son, negligence and na u al wild i es. Howe e , as in p e-
ious wo k (Fuen es-San os e al. 2013), o es i es we e
also di ided in o hese g oups acco ding o bu ned a ea (S),
ou b eaks (S < 1 ha), egula i es (1 ≤ S< 25 ha) and la ge
i es (S ≥ 25 ha).
Non-pa ame ic poin p ocesses analysis
Poin p ocesses a e ma hema ical models ha go e n he
occu ence o a andom numbe o e en s in a bounded
domain,
Wd
(wi h
d
being Euclidian space). I each
e en has any measu e o ma k associa ed wi h i , we ha e
a ma ked poin p ocess. A mul i ype poin p ocess is a
ma ked poin p ocess wi h ca ego ical ma ks ha de ine
di e en poin p ocesses acco ding wi h he ype o e en .
Spa ial poin p ocesses gene a e a andom numbe o e en s
M. F. Ma ey-Pé ez e al. In e na ional Jou nal o Wildland Fi e
180
X x x= { , …, }
1 N
in a plana egion
W2
wi h a ea
|W| > 0. Spa io empo al poin p ocesses comp ise he
loca ion and ime o occu ence o a andom numbe o
e en s,
, ,Sx x = {( ), …, ( )}
1 1 N N
, i egula ly placed in
W T× ×
2 +
. Th oughou his pape , poin p ocesses
and pa e ns a e deno ed in bold capi als and e en s a e
deno ed in bold.
The wild i es da ase in oduced in he p e ious sec ion,
comp ising he spa ial loca ion o he igni ion poin s
and da e and ime o de ec ion o wild i es in Galicia, can
be seen as a ealisa ion o a spa io empo al poin p ocess.
Each e en is ma ked by he cause o i e and i s
bu ned a ea.
Non-pa ame ic in e ence echniques ecen ly de eloped
o i s - and second-o de analysis o spa ial and spa io-
empo al poin p ocesses allow us o answe impo an
ques ions ega ding he dis ibu ion and beha iou o di e -
en ypes o o es i es in Galicia. Ke nel es ima o s o he
i s -o de in ensi y unc ions (Diggle 1985; Fuen es-San os
e al. 2016; an Lieshou 2020) cha ac e ise he spa ial
dis ibu ion o a son and na u al i es, and he non-
pa ame ic compa ison o i s -o de in ensi ies es s
(Fuen es-San os e al. 2017), o ins ance, whe he a son
and na u al i es had he same spa ial dis ibu ion o
whe he wild i e dis ibu ion a ied o e yea s. A i s -
o de spa io empo al sepa abili y es (Fuen es-San os
e al. 2018) checks whe he he spa ial dis ibu ion o wild-
i e pa e ns a ies o e ime. Second-o de cha ac e is ics,
which analyse pai wise in e ac ions be ween e en s, allow
us o es i wild i e pa e ns a e clus e ed, egula o an-
dom, i.e. whe he a wild i e a a gi en loca ion enhances
(clus e ed) o inhibi s ( egula ) wild i e occu ence in i s
neighbou hood. Second-o de cha ac e is ics o mul i ype
poin p ocesses allow us o analyse he ela ionship be ween
di e en ypes o e en s. We can es , o ins ance, i a son
ou b eaks and i es appea close o u he apa han
would be expec ed i hey we e independen .
Fi s -o de analysis
One o he main issues in he analysis o any obse ed
pa e n is es ima ing i s i s -o de in ensi y unc ion, λ(·),
which de e mines he expec ed numbe o e en s pe uni
domain (Diggle 2013). In his wo k, we es ima e he i s -
o de in ensi y o he a son and na u al i e pa e ns unde
s udy h ough ke nel smoo hing wi h ull bandwid h ma ix
(Fuen es-San os e al. 2016).
x p x H k H x x
ˆ( ) = ( ) | | ( ( ))
HHi
N
i
1 1/2
=1
1/2
(1)
whe e k(·) is a bi a ia e Gaussian ke nel, H is a posi i ely
de ined bandwid h ma ix and |H| deno es he de e minan
o H·
p x H k H x y y( ) = ( ( ))d
HW
1/2 1/2
is he edge-
co ec ion e m ha educes he bias nea he bounda y o
he obse a ion domain W, and gua an ees he consis ency
o he ke nel es ima o o he densi y o e en loca ions,
x x m( ) = ( )/
0
whe e
m x x= ( )d
W
is he expec ed
numbe o e en s in W. The bandwid h pa ame e , which
de e mines he deg ee o smoo hing o ke nel es ima o s, is
c ucial o he pe o mance o he ke nel in ensi y es ima-
ion. Di e en bandwid h selec o s, such as he mean nea -
es dis ance used in he semina pape o Kou sias e al.
(2004), ha e been used in he analysis o wild i e pa e ns.
In he p esen wo k, bandwid h selec ion was conduc ed
h ough he wo-s age plug-in algo i hm in oduced by
Fuen es-San os e al. (2016), which minimises he boo s ap
es ima e o he AMISE (asymp o ic mean in eg a ed squa e
e o ) o he ke nel densi y o e en loca ions. As can be seen
in Fuen es-San os e al. (2016), la ge bandwid hs a e
ob ained o small poin pa e ns o na u al wild i es, and
small ones o la ge poin pa e ns o a son wild i es.
Taking ad an age o he consis ency o he ke nel densi y
o e en loca ions, Fuen es-San os e al. (2017) p oposed a
o mal es o compa e he spa ial dis ibu ion o wo
obse ed pa e ns. This es elies on he ac ha wo spa ial
poin p ocesses, X
1
and X
2
, wi h he same spa ial dis ibu-
ion ha e he same densi y o e en loca ions. Thus, we can
use an L
2
disc epancy measu e o es he null hypo hesis,
x x x: ( ) = ( ) = ( )
0 01 02 0
, and de ine he es s a is ic
T x x x
ˆ= ( ˆ( ) ˆ( )) d = ˆ+ˆ
(ˆ+ˆ)
W01 02
2
1 2
12 21
(2)
whe e
ˆ
ij
and
ˆ
i
a e es ima o s o
x x x= ( ) ( )d
ij Wi j0 0
o i, j = 1, 2 and
x x= ( ) d
iWi02
, ob ained by ke nel
smoo hing wi h plug-in bandwid h (Chacón and Duong
2010). The es was calib a ed h ough smoo h boo s ap
wi h B = 200 ealisa ions o he null hypo hesis. We use his
p ocedu e o compa e he in ensi y o a son and na u al
wild i es and o es i he spa ial dis ibu ion o each ype
o i e a ied ac oss yea s. This p ocedu e is e e ed as he
T- es he ea e .
Up o now, we ha e ocused on he spa ial dis ibu ion o
wild i e, o e looking he empo al dimension. To ill his
gap, we es ima e he spa io empo al in ensi y o he wild i e
pa e ns unde s udy by ke nel smoo hing, and apply he
log- a io based sepa abili y es p oposed by Fuen es-San os
e al. (2018) o check whe he he spa ial dis ibu ion o
wild i es a ies o e ime. The ke nel in ensi y es ima o is
gi en by
x p x H h
ˆ( , ) = ( , ) | |
H h H h
,,1s1/2 1
s s
(3)
k H x k h x ( ( )) ( ( ))
i
N
i i
=1
ss
1/2
1
whe e k
s
(·) and k
(·) a e bi a ia e and uni a ia e Gaussian
ke nels, H
s
is he wo-dimensional bandwid h ma ix o he
www.publish.csi o.au/w In e na ional Jou nal o Wildland Fi e
181
spa ial componen , and h
is he bandwid h o he empo al
componen .
p x k x u k u ( , ) = ( ) ( )d d
H h T W H h
,s, ,
s s
is he spa io empo al edge-co ec ion ac o .
A spa io empo al poin p ocess is i s -o de sepa able i
i s in ensi y can be exp essed as he p oduc o i s spa ial
and empo al ma ginals:
x x ( , ) = ( ) ( )
1 2
and, conse-
quen ly, he log- a io be ween he spa io empo al and spa-
ial in ensi ies,
x x x( , ) = log( ( , )/ ( ))
1
, does no
depend on he spa ial loca ions x o any
T
.
Conside ing his p ope y, Fuen es-San os e al. (2018)
p oposed using a no-e ec es o check he sepa abili y
assump ion. To implemen he es , we i s es ima e
ρ(x, ) as he log- a io o he ke nel spa io empo al (Eqn 3)
and spa ial (Eqn 1) in ensi y unc ions wi h diagonal band-
wid h ma ices selec ed by leas -squa es c oss- alida ion.
Once
x ˆ ( , )
is ob ained, we conside a eg ession p oblem
wi h a esponse a iable,
,Y y i nx = { = ˆ ( ), =1, … , }
ii i
,
ha may depend on he spa ial co a ia e
X x x x i n= { = ( , ), =1, … , }
i i i1 2
comp ising he e en loca-
ions, and we es o he e ec o X on Y. Thus, ollowing
Bowman and Azzalini (1997), we check whe he
µE Y X[ | ] =
o
E Y X m X[ | ] = ( )
h ough he es s a is ic
F=(RSS RSS )/(d d )
RSS /d
0 1 1 0
1 1
(4)
whe e RSS
0
and RSS
1
a e he esidual sum o squa es o
non-pa ame ic es ima o s o he null and al e na i e mod-
els, and d
0
, d
1
deno e hei espec i e deg ees o eedom.
The es is calib a ed h ough a pe mu a ion es , which
elies on he ac ha unde
0
, he pai ing o any pa icula
X and Y is comple ely andom. This p ocedu e is e e ed o
as he F- es he ea e .
Second-o de analysis
Once he spa ial and spa io empo al dis ibu ion o wild i e
pa e ns a e cha ac e ised, we apply Mon e Ca lo es s
based on he spa ial and spa io empo al inhomogeneous
K- unc ions o check o in e ac ion be ween igni ion
poin s. The K- unc ion o a homogeneous spa ial poin
p ocess is de ined as
K N ( ) = [ ( )]
10
whe e N
0
( ) is
he numbe o u he e en s wi hin dis ance o an a bi-
a y e en (Ripley 1977). Gi en ha he homogenei y
assump ion can be e y es ic i e and un ealis ic when
dealing wi h eal da a, Baddeley e al. (2000) ex ended
his unc ion o he case o inhomogeneous poin p ocesses,
K
inhom
( ), and p oposed an empi ical es ima o
¸ ¸
,
K W
I
x x
x x
ˆ( ) = 1
| |
( )
ˆ( ) ˆ( ) ; 0
W
i j
i j
x x X
i j
i j
inhom ,1
max
i j
(5)
whe e I(·) is he indica o unc ion,
max
is he maximum
dis ance a which he unc ion is e alua ed, w
ij
is Ripley's
edge-co ec o , and
x
ˆ( )
is he ke nel in ensi y es ima o
de ailed in Eqn 1. The L- unc ion (Besag 1977), de ined
as
L K ( ) = ( )/
imhom inhom
, is a ans o ma ion o he
K- unc ion widely used in p ac ice. The inhomogeneous
L- unc ion o a spa ial Poisson poin p ocess is
L ( ) =
inhom
, whe e alues below o abo e his h eshold
indica e, espec i ely, inhibi ion o clus e ing be ween
e en s a dis ance . This p ope y sugges s es ing o
in e ac ion be ween e en s h ough a Mon e Ca lo es ,
which compa es he L- unc ion o he obse ed pa e ns,
X, wi h he uppe and lowe en elopes o B − 1 ealisa ions
o he inhomogeneous spa ial Poisson poin p ocess wi h
he same i s -o de in ensi y as X.
Spa ial in e ac ions be ween wo ypes o i es, e.g. a son
and na u al i es in a gi en yea , we e es ed h ough Mon e
Ca lo es s based on he inhomogeneous L-c oss unc ion,
which is he na u al ex ension o he p e ious p ocedu e o
he mul i ype amewo k (Ripley 1981; Baddeley
e al. 2015).
The Mon e Ca lo p ocedu es in oduced abo e es o
spa ial in e ac ion in wild i e pa e ns o e looking he em-
po al lag be ween e en s. Howe e , es ing o spa io-
empo al in e ac ion, i.e. whe he he occu ence o a
wild i e inc eases o educes he p obabili y o new e en s
in i s neighbou hood du ing he nex hou s and days, is
c ucial o a p ope analysis o i e beha iou . To add ess
his issue, we applied a Mon e Ca lo es based on he
spa io empo al inhomogeneous K- unc ion (Gab iel and
Diggle 2009), as o a Poisson poin p ocess
K ( , ) =
ST,inhom 2
, whe e la ge and smalle alues indi-
ca e clus e ing and inhibi ion, espec i ely. To implemen
he spa io empo al K- es , we i s ob ained he empi ical
es ima o o
K ( , )
ST,inhom
as ollows
¸ ¸
, ,
, ,
K W T
I x x
x x
ˆ( , ) = 1
| × |
( < , | | < )
ˆ( ) ˆ( )
W T
i j
i j
x x
i j i j
i i j j
ST,inhom
,1
i j i j
(6)
0 0
max max
whe e I(·) is he indica o unc ion, ω
ij
is Ripley's spa io-
empo al edge-co ec o ,
max
,
max
a e he maximum spa ial
and empo al dis ances a which he unc ion is e alua ed,
and
x
ˆ( , )
is an es ima o o he spa io empo al i s -o de
in ensi y unc ion. Gab iel and Diggle (2009) assume i s -
o de sepa abili y, i.e. hey use a sepa able es ima o o he
spa io empo al in ensi y,
x x
ˆ( , ) = ˆ( ) ˆ( )
1 2
, and a i-
bu e any non-sepa able e ec o he second-o de s uc u e.
We used sepa able o non-sepa able ke nel in ensi y es i-
ma ed in ag eemen wi h he esul s o he i s -o de sepa-
abili y es in oduced in he p e ious sec ion. As well as o
he spa ial amewo k, he Mon e Ca lo es compa es he
spa io empo al K- unc ion o he obse ed pa e n,
K x
ˆ( , )
ST,inhom
, wi h uppe and lowe en elopes de e mined
M. F. Ma ey-Pé ez e al. In e na ional Jou nal o Wildland Fi e
182
by Mon e Ca lo ealisa ions o a spa io empo al Poisson
poin p ocess wi h he same in ensi y as he obse ed pa -
e n. The maximum alues and o which
K x
ˆ( , )
ST,inhom
alls abo e o below he en elopes de e mine he spa ial and
empo al in e ac ion adius, espec i ely.
So wa e
The s a is ical analysis o he spa ial and spa io empo al
wild i e pa e ns was conduc ed wi h he R s a is ical so -
wa e (R Co e Team 2018). The Spa s a package (Baddeley
e al. 2015) was used o ke nel in ensi y es ima ion and
second-o de analysis o spa ial poin pa e ns, whe eas
plug-in bandwid hs we e ob ained wi h he help o he ks
package (Duong 2018). The kde. es unc ion in he ks pack-
age was adap ed by Fuen es-San os e al. (2017) o imple-
men he L
2
- es o compa ison o i s -o de in ensi ies.
Ke nel log- ela i e isk unc ions and hei ole ance con-
ou s we e es ima ed using he spa package (Da ies e al.
2018). Fuen es-San os e al. (2018) ex ended some unc ions
in he spa package (Da ies e al. 2018) o es ima e he
spa io empo al in ensi y and log- a io unc ions o imple-
men he sepa abili y es wi h he help o he sm package
(Bowman and Azzalini 2014).
Resul s
Explo a o y analysis
Fig. 2 shows he numbe o wild i es and a ea bu ned by
cause o i e and yea (see Table A1 in Supplemen a y
Appendix A o u he de ails). The majo i y o wild i es
eco ded we e a son (>80%), while he incidence o hose
caused by negligence (~3%)) and na u al causes (~1%) was
e y small. The yea 2006 egis e ed a dec ease in he
numbe o wild i es in compa ison wi h p e ious yea s bu
had he la ges a ec ed a ea, and we obse ed a signi ican
educ ion in he numbe o wild i es and a ec ed a ea om
2007 onwa d. The co espondence analysis conduc ed as a
p elimina y analysis o wild i e beha iou iden i ied 2006
as a change poin , as we ound associa ion be ween a son
i es and he yea s 1999–2006, and be ween negligence i es
and 2007–2014 (see de ails in Supplemen a y Appendix A).
Acco ding o he o icial epo s, a la ge p opo ion o he
wild i es, 72.5%, bu ned less han 1 ha, and jus 2% a ec ed
mo e han 25 ha. Al hough we do no see la ge di e ences
be ween causes, a son wild i es ha e he lowes a e o
e en s wi h less han 1 ha bu ned (see Supplemen a y
Table A1). Taking in o accoun hese pa icula ea u es o
wild i e in Galicia, his wo k conside s all e en s, no only
12 000
8000
No. i es
6000
4000
2000
0
80 000
Bu ned a ea (ha)
60 000
40 000
20 000
0
1999 2000 2001 2002 2003
Yea
Yea
2004 2005 2006 2007 2008
1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
A son
Na u al
Negligence
Rep oduc ion
Unknown
Fig. 2. Numbe o igni ion poin s egis e ed in Galicia om 1999 o 2014 ( op); and bu ned a ea (bo om) by
cause o i e.
www.publish.csi o.au/w In e na ional Jou nal o Wildland Fi e
183
hose wi h mo e han 1 ha bu ned as is usual in wild i e
esea ch. He ea e , 'wild i es' e e s o all e en s, 'ou b eaks'
o wild i es wi h S < 1 ha, and ' i es' o hose wi h S ≥ 1 ha.
Spa ial and spa io empo al dis ibu ion o
wild i e
We analysed he spa ial and spa io empo al dis ibu ion o
a son, negligence and na u al wild i es, dis inguishing
be ween ou b eaks (S < 1 ha) and i es (S ≥ 1 ha) when
he size o he obse ed pa e n was la ge enough. Gi en
he la ge numbe o analyses conduc ed, de ailed esul s o
he spa ial and spa io empo al analyses a e p o ided as
Supplemen a y Appendices B and C, espec i ely.
The ke nel in ensi y es ima o s o a son, negligence and
na u al wild i es shown in Figs 3–5 show clea di e ences in
he spa ial pa e ns o wild i e be ween he h ee causes.
These di e ences we e con i med by he non-pa ame ic
compa ison be ween in ensi y unc ions (T- es , P alue
< 0.005 o all yea s wi h mo e han 20 na u al i es).
Now, we ocus on es ing o in e annual a iabili y in
he spa ial pa e n o each cause o wild i e, and on es ing
o di e ences be ween ou b eaks and i es o he wo ypes
o human-caused wild i es. The non-pa ame ic es ound
ha he spa ial dis ibu ion o a son wild i es, ou b eaks and
i es a ied ac oss yea s (T- es , P alue < 0.005). In pa ic-
ula , Fig. 3 sugges s a end wi h wild i e ho spo s sp eading
om he sou h o Galicia, on he bounda y wi h Po ugal, o
he wes coas e e y 4 yea s. We also ound di e ences
(T- es , P alue < 0.005) be ween he spa ial pa e ns o
ou b eaks (Supplemen a y Fig. B1 in Supplemen a y
Appendix B) and i es (Supplemen a y Fig. B2); o ins ance,
in 2006, we obse ed a low in ensi y o ou b eaks, bu a
high incidence o i es in he SE o Galicia.
The spa ial dis ibu ion o wild i es caused by negligence,
ou b eaks and i es a ied o e yea s, al hough pai wise
1999 2000 2001 2002
0.2
0.5
1
1.5
2
0.5
0.2
0.6
1
1.4
1
1.5
2
0.6
1
2003 2004 2005 2006
2007 2008 2009 2010
2011 2012 2013 2014
0.2
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.1
0.3
0.5
0.1
0.3
0.5
0.1
0.1
0.2
0.3
0.4
0.1
0.2
0.02
0.06
0.3
0.4
0.3
0.5
0.5
1
1.5
2
0.5 0.2
0.4
0.6
0.8
1
1
1.5
2.5
2
0.6
1
Fig. 3. Fi s -o de in ensi y es ima o o he spa ial pa e ns o a son wild i es by yea (di e en scales).
M. F. Ma ey-Pé ez e al. In e na ional Jou nal o Wildland Fi e
184
compa ison did no always epo signi ican di e ences
(see Table S3). Fig. 4 shows negligence i es occu ed mainly
in he A lan ic coas al a ea o Galicia, wi h high-incidence
a eas mo ing be ween he no he n and sou he n coas al
a eas, and some ho spo s in he inne egion o Galicia.
Compa ison be ween ou b eaks and i es ound signi ican
di e ences be ween hem in 1999, 2000, 2002, 2003, 2007
and 2008. As obse ed o a son i es, he inne and sou h-
e n a eas end o ha e la ge i e han ou b eak (de ails a e
gi en in Supplemen a y Table B2, Supplemen a y Figs
B3, B4).
The spa ial dis ibu ion o na u al wild i es was mo e
s able o e yea s, wi h highe incidence in he eas o
Galicia (Fig. 5), al hough he non-pa ame ic compa ison
de ec ed di e ences be ween some pai s o yea s
(Supplemen a y Table B3). Ou b eaks and i es we e no
analysed sepa a ely owing o he small numbe o e en s
obse ed.
Supplemen a y Appendix C shows he ke nel es ima o o
he spa ial and spa io empo al in ensi ies, and he empo al
densi y o igni ion poin s o a son, negligence and na u al
i es by yea . As o he spa ial dis ibu ion, we obse ed
di e ences be ween he empo al pa e ns o human-caused
and na u al wild i e pa e ns. The empo al pa e n o a son
wild i es a ied o e yea s, ea ly sp ing and la e
summe –au umn being he seasons wi h he highes i e
incidence. Fi es caused by negligence occu ed mainly in
sp ing and summe , bu he highes -incidence pe iod also
a ied o e yea s. Finally, na u al i es occu ed mainly in
summe . The spa io empo al sepa abili y es ejec s he null
hypo hesis o a son and negligence wild i es (F- es , P alue
<0.005), i.e. he spa ial dis ibu ion o wild i e a ied o e
ime, as can be seen in he spa io empo al in ensi y es ima-
o s (see anima ions in Supplemen a y Appendix C).
Al hough da a spa seness and he small numbe o na u al
wild i es eco ded each yea do no allow he applica ion o
1999 2000 2001 2002
0.002
5e–04
0.0015
0.005
0.015
0.005
0.015
0.01
0.006
0.01
0.01
0.001
0.005
0.01
0.015
0.005
0.01
0.015
0
0.001
0
0
0.004
0.001
0.003
0.008
0.012
4e–04
0.001
0.004
0.007
0.003
0
0.005
0.01
0.015
0
0.005
0.015
0.005
0.002
0.008
0.014
0.015
0.03
0.05
2003 2004 2005 2006
2007 2008 2009 2010
2011 2012 2013 2014
Fig. 5. Fi s -o de in ensi y es ima o o he spa ial pa e ns o na u al wild i es by yea (di e en scales).
www.publish.csi o.au/w In e na ional Jou nal o Wildland Fi e
185
he sepa abili y es o hese pa e ns, isual inspec ion o
he ke nel es ima o s o he spa io empo al in ensi y sug-
ges s ha hei dis ibu ion also a ied o e ime.
Spa ial and spa io empo al in e ac ion be ween
wild i es
Table 1 shows he maximum in e ac ion a es ound wi h
he inhomogeneous L- es s o a son, negligence and na u al
wild i e pa e ns, and o in e ac ions in he ou b eak
and i e pa e ns o he wo ypes o human-caused i es.
The inhomogeneous L- es de ec ed clus e ing in he a son
wild i e pa e ns o all yea s bu 2007 wi h an in e ac ion
adius be ween 1 km in 2004 and 6 km in 2014, and
inhibi ion a la ge dis ances (see Supplemen a y Fig. B4 in
Supplemen a y Appendix B). We also ound in e ac ions
be ween he ou b eaks (Supplemen a y Fig. B6) and i es
(Supplemen a y Fig. B7); as obse ed o he whole wild i e
pa e ns, he in e ac ion adius a ied o e yea s. The
inhomogeneous L-c oss es ound e idence o posi i e in e -
ac ion be ween a son ou b eaks and i es (Table 2,
Supplemen a y Fig. B8). A son wild i es also show agg ega-
ion be ween a son e en s o e yea s (Supplemen a y
Fig. B17), i.e. a highe p obabili y o a son i es in a eas
ha eco ded a son i es in p e ious yea s. In con as , we
did no ind e idence o in e ac ions be ween i es o e
sequen ial yea s (Supplemen a y Fig. B18).
The spa io empo al inhomogeneous K- es s (Fig. 6),
which es o in e ac ions be ween e en s up o 15 km
and 15 days a e he occu ence o a gi en e en , show
ha he p obabili y o new wild i es in he neighbou hood
o an obse ed e en dec eases o e ime. In his sense,
excep o 2009 and 2012, which had long- e m agg ega ion
up o 15 km, he in e ac ion adius dec eases wi h ime.
In pa icula , in 1999, 2005 and 2006, he spa io empo al
K- es de ec ed long- e m in e ac ion up o 6–10 km, an
1999 2000 2001 2002
0.005
0.01
0.02
0.03
0.01
0.01
0.03
0.01
0.01
0.02
0.005
0.015
0.03
0.05
0.07
0.02
0.03
0.04
0.01
0.01
0.006
0.002
0.014
0.008
0.002
0.02
0.03
0.04
0.015
0.025
0.005
0.015
0.005
0.015
0.025
0.005
0.015
0.005
0.015
0.005
0.015
0.002
0.006
0.025
0.005
0.015
0.025
2003 2004 2005 2006
2007 2008 2009 2010
2011 2012 2013 2014
Fig. 4. Fi s -o de in ensi y es ima o o he spa ial pa e ns o negligence wild i es by yea (di e en scales).
M. F. Ma ey-Pé ez e al. In e na ional Jou nal o Wildland Fi e
186
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1155501
Da a a ailabili y. Wild i e da a used in his we e ob ained o m he Wild i e S a is ics da ase o he Wild i e Gene al S a is ics a ailable a he Spanish
Go e nmen Da a Po al (h ps://da os.gob.es/es/ca alogo/e05068001-es adis ica-gene al-de-incendios- o es ales).
Con lic s o in e es . Thea au ho s decla e no con lic s o in e es s.
Decla a ion o unding. This wo k was suppo ed by P ojec s MTM2016-76969-P and PID2020-116587GB-I00 (AEI/FEDER, UE), and p ojec s IN607B:
Modalidad B – G upos con po encial c ecemen o, GPC GI-1716 PROXECTOS E PLANIFICACIÓN (PROEPLA), 2018-PG078 (Re . ED431B 2018/23) om
he Galician Go e nmen (Xun a he Galicia).
Acknowledgemen s. We would like o acknowledge Luis F anco o his echnical suppo in he access and p e-p ocessing o wild i e da a.
Au ho a ilia ions
A
PROePLA Resea ch G oup, Depa men o C op P oduc ion and Enginee ing P ojec s, Uni e sidade de San iago de Compos ela, Spain.
B
Ma ine Resea ch Ins i u e, Spanish Na ional Resea ch Council, Spain.
C
Depa men o S a is ics, Ma hema ical Analysis and Op imisa ion, Uni e si y o San iago de Compos ela, Spain.
M. F. Ma ey-Pé ez e al. In e na ional Jou nal o Wildland Fi e
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