Summary characteristics for multivariate function‐valued spatial point process attributes

Ecka d , Ma hias; Comas, Ca les; Ma eu, Jo ge
A icle — Published Ve sion
Summa y cha ac e is ics o mul i a ia e unc ion‐ alued
spa ial poin p ocess a ibu es
In e na ional S a is ical Re iew
P o ided in Coope a ion wi h:
John Wiley & Sons
Sugges ed Ci a ion: Ecka d , Ma hias; Comas, Ca les; Ma eu, Jo ge (2024) : Summa y cha ac e is ics
o mul i a ia e unc ion‐ alued spa ial poin p ocess a ibu es, In e na ional S a is ical Re iew,
ISSN 1751-5823, Wiley, Hoboken, NJ, Vol. 93, Iss. 1, pp. 150-178,
h ps://doi.o g/10.1111/ins .12582
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/319336
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Summa y cha ac e is ics o mul i a ia e
unc ion- alued spa ial poin p ocess
a ibu es
Ma hias Ecka d
1
, Ca les Comas
2
and Jo ge Ma eu
3
1
Depa men o S a is ics, Humbold -Uni e si ä zu Be lin, Spandaue S asse 1, Be lin, Ge many
2
Depa men o Ma hema ics, Uni e si a de Lleida, A . Alcalde Ro i a Rou e 191, Lleida, Spain
3
Depa men o Ma hema ics, Uni e si a Jaume I, E-12071 Cas ellón, Spain
Co espondence Ma hias Ecka d , Depa men o S a is ics, Humbold -Uni e si ä zu Be lin,
Spandaue S asse 1, Be lin, Ge many. Email: m.ecka [email p o ec ed]
Summa y
P omp ed by mode n echnologies in da a acquisi ion, he s a is ical analysis o spa ially dis ib-
u ed unc ion- alued quan i ies has a ac ed a lo o a en ion in ecen yea s. In pa icula , com-
bina ions o unc ional a iables and spa ial poin p ocesses yield a highly challenging ins ance o
such mode n spa ial da a applica ions. Indeed, he analysis o spa ial andom poin configu a ions,
whe e he poin a ibu es hemsel es a e unc ions a he han scala - alued quan i ies, is jus in
i s in ancy, and ex ensions o unc ion- alued quan i ies s ill emain limi ed. In his iew, we ex end
cu en exis ing fi s - and second-o de summa y cha ac e is ics o eal- alued poin a ibu es o
he case whe e, in addi ion o e e y spa ial poin loca ion, a se o dis inc unc ion- alued quan i ies
a e a ailable. P o iding a flexible ea men o mo e complex poin p ocess scena ios, we build a
amewo k o conside poin s wi h mul i a ia e unc ion- alued ma ks, and de elop se s o di e -
en c oss- unc ion (c oss- ype and also mul i- unc ion c oss- ype) e sions o summa y cha ac e is-
ics ha allow o he analysis o highly demanding mode n spa ial poin p ocess scena ios. We con-
side es ima o s o he heo e ical ools and analyse hei beha iou h ough a simula ion s udy and
wo eal da a applica ions.
Key wo ds: c oss- unc ion ma k co ela ion; o es moni o ing da a; unc ional-ma ked poin p ocesses;
ma k a iog am; ma k weigh ed second o de summa y cha ac e is ics; nea es neighbou ma k indices;
u ban economics.
1 In oduc ion
In oducing gene al ideas om unc ional da a analysis (Fe a y & Vieu, 2006; Ho á h &
Kokoszka, 2012; Hsing & Eubank, 2015; Ramsay & Sil e man, 1997) in o he field o spa ial
s a is ics, he s a is ical analysis o unc ional spa ial da a has a ac ed a lo o a en ion in ecen
yea s (see Delicado e al., 2010; Ma eu & Romano, 2017, Ma ínez-He nández & Gen on, 2020,
o a gene al e iew). Po en ial applica ions om he li e a u e include he analysis o egional
pene a ion esis ance p ofiles (Gi aldo e al., 2011), ai pollu ion moni o ing da a (Boho quez
e al., 2017) and egional g oss domes ic p oduc dynamics (Pineda-Ríos e al., 2019). Di e en
om mo e classical spa ial s a is ics (see, e.g. C essie, 1993), he objec s o in e es in any such
In e na ional S a is ical Re iew (2025),93, 1, 150–178 doi: 10.1111/ins .12582
© 2024 The Au ho (s). In e na ional S a is ical Re iew published by John Wiley & Sons L d on behal o In e na ional S a is ical Ins i u e.
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any me-
dium, p o ided he o iginal wo k is p ope ly ci ed.
da a a e he ealised ajec o ies, ha is, cu es, o some unde lying con inuous mechanism
which a e collec ed o e some spa ial domain X⊂ℝd, usually d¼2. As such, he unc ional
obse a ions hemsel es a e assumed o be spa ially dependen ela i e o he dis ance be ween
he spa ial en i ies, which needs o be accoun ed o in any s a is ical analysis. The obse ed
unc ions a nea by s a ions migh be mo e simila o dissimila in shape depending on whe e
he measu emen s a e eco ded. Clea ly, in an u ban con ex he amoun o gases and pa icu-
la es in he ai depends on he loca ion o he moni o ing s a ions, and nea by s a ions a e likely
o ha e simila ai pollu ion p ofiles. This spa ial s uc u e among he unc ions is, howe e , no
accoun ed o in any non-spa ial me hods yielding po en ially biased o misleading esul s. To
his end, a ious app oaches om classical spa ial s a is ics we e ex ended o unc ional ou -
comes yielding an e e -inc easing me hodological oolbox o di e en unc ional spa ial da a
analysis echniques. P ede e mined by he exac na u e o X, hese echniques help o in es iga e
he spa ial in e ela ions be ween he indi idual unc ional objec s in geos a is ical, a eal o
poin p ocess da a con ex s. While a ela i ely la ge body o con ibu ions exis s on
geos a is ical unc ional da a and unc ional a eal da a, he analysis o spa ial andom poin con-
figu a ions is jus in i s in ancy. In pa icula , di e en om geos a is ical unc ional
da a/ unc ional k iging app oaches (see Boho quez e al., 2022; F anco-Villo ia &
Ignaccolo, 2022; Ne ini e al., 2022, o gene al e iews) o unc ional a eal da a eg ession
models (Aw & Cab al, 2020; Pineda-Ríos e al., 2019; Zhang e al., 2016), he poin loca ions
a e ea ed as andom and he a ibu es hemsel es a e unc ions a he han scala - alued
quan i ies.
Despi e he no able p og ess in spa ial poin p ocess me hodology wi h ex ensions o mo e
challenging non-Euclidean domains o he poin s including he sphe e, linea ne wo ks and
g aphs wi h Euclidean edges, ex ensions o mo e complica ed non-scala ma ks ha e no been
co e ed much so a . Fo he scala - alued ma ks se ing, he e al eady exis a ious ma k sum-
ma y cha ac e is ics and nea es -neighbou e sions (S oyan & S oyan, 1994). He e, p ominen
ools o he cha ac e isa ion o eal- alued ma ks include he ma k co a iance (S oyan, 1984),
ma k co ela ion (Isham, 1985; S oyan & S oyan, 1994), ma k weigh ed K(Pen inen
e al., 1992), ma k a iog am (C essie, 1993; S oyan & Wälde , 2000; Wälde & S oyan, 1996),
and ma k di e en ia ion (Hui & Pomme ening, 2014; Pomme ening e al., 2011) unc ions.
Li e a u e cu en ly o e s me hodology o (ma ked) spa io- empo al poin p ocesses
(González e al., 2016; Ra hbun, 1993; Ve e-Jones, 2009) whe e di e en clus e ed poin p o-
cess (González e al., 2016), Gibbsian p ocesses (Redenbach & Sä kkä, 2013; Renshaw
e al., 2009; Renshaw & Sä kkä, 2001; Sä kkä & Renshaw, 2006), log-Gaussian (Se a
e al., 2014; Siino e al., 2018) and sho -noise (B ix & Chadœu , 2002; Mølle & Díaz-
A alos, 2010) Cox model specifica ions a e de eloped. We can also find co esponding
second-o de summa y cha ac e is ics (I imi e al., 2019; S oyan e al., 2017) used o cha ac e -
ise he empo al e olu ion o a se o (ma ked) poin loca ions. Howe e , con ibu ions o
unc ion- alued ma ks emain elusi e. In pa icula , ad ances o se s o dis inc
unc ion- alued a ibu es, ha is, mul i a ia e cu es, do no exis .
The ma k co ela ion unc ion o unc ion- alued poin a ibu es o igina es om he pio-
nee ing pape o Comas e al.(2008) and subsequen wo ks by Comas e al.(2011); Comas
e al.(2013). Gho bani e al.(2021) we e he fi s o p o ide a ma hema ically igo ous ea -
men on he subjec . Ins ead o he se ximxi
ðÞðÞ
g
n
i¼1wi h poin s xi∈Xand scala - alued
ma ks mxi
ðÞ on some sui able ma k space M, hese au ho s conside ed he se
xi; xi
ðÞ
ðÞ
;lxi
ðÞðÞðÞ
g
n
i¼1whe e each poin xiis augmen ed by a unc ion- alued quan i y
xi
ðÞ ðÞ∈Fand, po en ially, an addi ional Euclidean auxilia y ma k lxi
ðÞli ing on a sui able
la en ma k space L. As such, apa om he i ial case when no auxilia y ma k is a ailable,
151Summa y cha ac e is ics o mul i a ia e unc ion- alued spa ial poin p ocess a ibu es
In e na ional S a is ical Re iew (2025),93, 1, 150–178
© 2024 The Au ho (s). In e na ional S a is ical Re iew published by John Wiley & Sons L d on behal o In e na ional S a is ical Ins i u e.
his o mula ion allows o (i) unc ion- alued ma ked mul i a ia e poin pa e ns, whe e di e -
en ypes o poin s wi h one unc ion- alued poin a ibu e a e obse ed, and (ii)
unc ion- alued ma ked poin pa e ns whe e a each poin addi ional eal- alued in o ma ion
is a ailable. While p o iding a flexible ea men o mo e complex poin p ocess scena ios
which include unma ked ( esp. scala - alued ma ked) poin p ocesses as special case when
he xi
ðÞ ðÞ;lxi
ðÞðÞ( esp. xi
ðÞ ðÞ) a gumen is igno ed, ex ensions o poin s wi h mul iple dis-
inc unc ion- alued ma ks ha e no been co e ed o he e y bes o ou knowledge. Consid-
e ing a leas wo dis inc unc ion- alued poin a ibu es o each poin loca ion, his pape
aims o fill his gap. In pa icula , se s o di e en c oss- unc ion summa y cha ac e is ics o
poin s wi h wo dis inc unc ion- alued ma ks a e in oduced and ex ended o c oss- ype and
also mul i- unc ion c oss- ype e sions. O e all, he p oposed ools will allow o he analysis
o highly demanding mode n spa ial poin p ocess scena ios. All da a and R code o ep oduce
he p oposed au o- and c oss- unc ion ma k cha ac e is ics a e made publicly a ailable in a
gi hub eposi o y h ps://gi hub.com/ca lescomas/SppFDA.
The emainde o he pape is s uc u ed as ollows. A e a gene al in oduc ion o spa ial
poin p ocesses wi h mul i a ia e unc ion- alued poin a ibu es, Sec ion 2es ablishes di e -
en c oss- unc ion ma k cha ac e is ics and po en ial ex ension o mul i ype poin p ocesses.
In pa icula , ex ensions o classical es unc ions o he unc ion- alued ma k se ing a e
discussed in Sec ion 2.2. Es ima o s o he p oposed ma k cha ac e is ics a e p esen ed in
Sec ion 3. The p oposed cha ac e is ics a e e alua ed h ough a simula ion s udy in Sec ion 4.
Sec ion 5p esen s an applica ion o he p oposed ools o wo di e en da a sou ces o igina ing
om o es y and u ban economic con ex s. The pape concludes wi h a discussion in Sec ion 6.
2 Spa ial Poin P ocesses Wi h Mul i a ia e Func ion-Valued Ma ks
To ex end he heo y and me hodology o unc ion- alued ma ked spa ial poin p ocesses o
mul i a ia e unc ion- alued poin a ibu es, le Xdeno e a subse o ℝ2equipped wi h Bo el
se s BXðÞ, and dðÞan Euclidean me ic on X.OnX,define ΨG¼xi
g
n
i¼1as g ound, ha
is, unma ked, spa ial poin p ocess wi h in ensi y measu e ΛG. As such, ΨGis well embedded
in o he heo y o spa ial poin p ocesses and a ich body o di e en ools can di ec ly be ap-
plied o in es iga e he s uc u al p ope ies o he poin s (see Mølle & Waagepe e sen, 2004;
Illian e al., 2008; Chiu e al., 2013, o gene al e e ences o spa ial poin p ocesses). Associ-
a ed wi h ΨG, deno e by Ψ¼xi x
i
ðÞ ðÞ
g
n
i¼1a ma ked spa ial poin p ocess on XFpwi h lo-
ca ions xi∈Xand p- a ia e associa ed unc ion- alued ma ks x
i
ðÞ ðÞ¼
1xi
ðÞ ðÞ…; pxi
ðÞ ðÞ

on Fpwhe e each hxi
ðÞ ðÞ:T⊆ℝ↦ℝ;h¼1;…;pwi h T¼
a;bðÞ;∞≤a≤b≤∞. In gene al, Fpis assumed o be a Polish, ha is, comple e sepa able
me ic space equipped wi h σ-algeb a Fp¼⊗p
h¼1Fh(Daley & Ve e-Jones, 2008). Fo Ψ,
he expec ed numbe o poin s NhðÞin B∈BXðÞwi h unc ion- alued a ibu e in
Fh∈Fhco esponds o he in ensi y measu e ΛhBFh
ðÞwhich simplifies o
ENhBFh
ðÞ½¼ΛhBFh
ðÞ¼∫BFhλGxðÞdxPdF
h
ðÞ
o fixed Fhin Fhwi h λGbeing he in ensi y unc ion o ΨGand PdF
h
ðÞa e e ence measu e on
FpFp
ðÞ. Fo s a iona y Ψ, ha is, i Ψ¼Ψxwi h Ψx¼xiþx; x
i
ðÞ ðÞðÞ
g
n
i¼1 o any ansla ion
xand fixed h ðÞ, he in ensi y measu e ΛhBFh
ðÞequals λhνBðÞ, wi h λhdeno ing he in ensi y
o Ψwi h espec o Fhand νðÞ he Lebesgue measu e, ha is, he olume, o i s a gumen .
Simila ly, Ψis called iso opic i Ψ¼ Ψwi h Ψ¼ xi; x
i
ðÞ ðÞðÞ
g
n
i¼1 o any o a ion .
152 ECKARDT ET AL.
In e na ional S a is ical Re iew (2025),93, 1, 150–178
© 2024 The Au ho (s). In e na ional S a is ical Re iew published by John Wiley & Sons L d on behal o In e na ional S a is ical Ins i u e.
To accoun o addi ional in ege - alued ma ks, ha is, when di e en ypes o poin s a e
a ailable, Ψcan be gene alised o a m- a ia e (i.e. mul i ype) spa ial poin p ocess Ψwi h n¼
n1þ…þnmpoin s and mul i a ia e unc ion- alued poin a ibu es on XmFpwi h co e-
sponding componen p ocesses Ψd;d¼1;…;mand associa ed g ound p ocess ΨG. We no e
ha he abo e poin p ocesses could also be ex ended by addi ional eal- alued ma k in o ma-
ion, o example, h ough addi ional auxilia y ma k e ms lxi
ðÞ∈ℝ, which allows o he o -
mula ion o doubly-ma ked (mul i ype) poin p ocesses whe e each poin is augmen ed by mul-
i a ia e unc ion- alued ma ks and one ( esp. wdis inc ) eal- alued ma k li ing on
XmFpℝ( esp. XmFpℝw).
2.1 Func ional Summa y Cha ac e is ics o Unma ked Poin P ocesses
Be o e discussing a ious ma k cha ac e is ics, a succinc o e iew o he s anda d unc ional
summa y cha ac e is ics commonly associa ed wi h unma ked poin pa e ns is p esen ed. In he
absence o any ma k in o ma ion, he spa ial a angemen o poin s is ou inely examined ia
he assump ion o comple e spa ial andomness, whe ein he absence o disce nible s uc u e
is assessed, agains clus e ing o egula i y o he poin s. Using emp y-space,
nea es -neighbou o pai wise dis ances, commonly applied cha ac e is ics include he emp y
space unc ion F ðÞ, he nea es neighbou dis ance dis ibu ion unc ion G ðÞ, he pai co ela-
ion unc ion g ðÞand Ripley’sK ðÞ unc ion (Ripley, 1977). The emp y space unc ion F ðÞ
conside s he dis ance om a ypical poin o a poin in he pa e n, whe eas he nea es neigh-
bou dis ance dis ibu ion unc ion G ðÞis cons uc ed om he dis ance o any neighbou ing
poin s. Unde comple e spa ial andomness, co esponding o a homogeneous Poisson p o-
cesses, bo h he F ðÞand G ðÞ unc ions a e equal o 1 eλπ 2whe e λis he in ensi y o
he poin s. The pai co ela ion unc ion is cons uc ed om he pai wise dis ance be ween
he loca ions and becomes equal o 1 unde he comple e spa ial andomness assump ion. The
eade is e e ed o he comple e essays o Illian e al.(2008) and Chiu e al.(2013).
2.2 C oss-Func ion Second-O de Ma k Summa y Cha ac e is ics and Nea es Neighbou
Indices
Apa om he fi s -o de p ope ies, a a ie y o second-o de ma k summa y cha ac e is ics
and hei ela ed nea es -neighbou e sions ha e become use ul me hodological ools o he
analysis o classical ( eal- alued) ma ked spa ial poin p ocess scena ios. They help o in es i-
ga e he he e ogenei y and in e ela ion be ween he obse ed poin a ibu es, and decide on he
independen ma k hypo hesis as a unc ion o he dis ance be ween pai s o wo poin s. To ex-
end he me hodological oolbox o he unc ion- alued ma ks se ing and define sui able
c oss- unc ion cha ac e is ics, le hxðÞ ðÞ and lx’
ðÞ ðÞ deno e wo dis inc unc ion- alued
ma ks o a pai o dis inc poin loca ions in Ψwi h in e poin dis ance dx;x’
ðÞ
¼ ; >0.
Adop ing he co e p inciples om classical ma k cha ac e is ics and applying a poin wise e al-
ua ion fi s , di e en c oss- unc ion ma k cha ac e is ics can be defined by in oducing a es
unc ion (Pen inen & S oyan, 1989), ha is, a map :FF→ℝþ, which i sel akes he
ma ks h ðÞand l ðÞ o a pai o dis inc poin s in Ψas a gumen s. In wha ollows, we assume
Ψ o be second-o de s a iona y and iso opic such ha he cha ac e is ics solely depend on
he dis ance ; >0 and i su fices o conside he ma ks a he o igin ∘and he dis ance
whe e d∘ ðÞ¼ . Depending on he p ecise specifica ion o , di e en ma k cha ac e is ics
can be cons uc ed by aking he expec a ion E∘; o unde he condi ion ha Ψhas indeed
poin s a loca ions ∘and . W i ing h∘ðÞ ðÞ and l ðÞ ðÞ o deno e he h- h and l- h
unc ion- alued ma ks a he o igin ∘and a a dis ance ∥ ∥¼ apa , di e en summa y
153Summa y cha ac e is ics o mul i a ia e unc ion- alued spa ial poin p ocess a ibu es
In e na ional S a is ical Re iew (2025),93, 1, 150–178
© 2024 The Au ho (s). In e na ional S a is ical Re iew published by John Wiley & Sons L d on behal o In e na ional S a is ical Ins i u e.

cha ac e is ics can be ob ained by gene alising hei classic o m o he unc ion- alued ma ks
se ing. Focusing on he mos p ominen es unc ions om he li e a u e, we yield he ollow-
ing fi e specifica ions, no ing ha each has as inpu a gumen s h∘ðÞ ðÞ; l ðÞ ðÞðÞ: 1¼
1=2 h∘ðÞ ðÞ l ðÞ ðÞðÞ
2and 2¼min h∘ðÞ ðÞ; l ðÞ ðÞðÞðÞ=max h∘ðÞ ðÞ; l ðÞ ðÞðÞðÞ, which
a e based on he di e ence o he a io be ween he pai o dis inc ma ks, and 3¼
h∘
ðÞ
ðÞ l
ðÞ
ðÞ
; 4¼ h∘
ðÞ
ðÞand 5¼ l
ðÞ
ðÞ
, which a e based on he p oduc o he a gu-
men s (see Schla he , 2001; Illian e al., 2008, o de ailed discussion). All o he abo e es
unc ions add ess only ce ain aspec s o he ma k dis ibu ion. Taking he condi ional expec a-
ion o es unc ion 1yields he ma k a iog am which depic s he a ia ion o he ma ks as a
unc ion o he dis ance . The ma k a iog am could be used o decide on he a e age (dis)sim-
ila i y o pai s o ma ks a dis inc poin s. I he ma ks a e on a e age simila , hei a ia ion is
on a e age also small leading small alues o he ma k a iog am. A simila cha ac e is ic, he
ma k di e en ia ion unc ion, is ob ained by aking he condi ional expec a ion o 1  2.I
he ma ks a e simila in alues, he a io in 2will become close o one such ha 1  2is
also close o ze o. Di e en om hese wo es unc ions, 3and 4o 5can be used o
compu e S oyan’s ma k co ela ion and he -ma k unc ions. Al hough di e en om Pea son’s
co ela ion, S oyan’s ma k co ela ion (i.e. he condi ional expec a ion o 3) eflec s he a e age
pai wise associa ion o he ma ks a any wo dis inc poin s. Unde independence o he
ma ks, his cha ac e is ic coincides wi h he squa ed mean ma k such ha no malising S oyan’s
ma k co ela ion by he squa ed mean ma k yields a alue o one. I he ma ks a e on a e age
la ge ( esp. smalle ) han he squa ed mean ma k, he es ima ed p oduc s will also be clea ly
di e en om he squa ed mean ma k. Finally, he condi ional expec a ion o ei he 4o 5
ela es o he condi ional mean o he fi s o he second ma k o any pai o poin s gi en
ha he e a e poin s a bo h loca ions, espec i ely. The condi ional mean could be used o de-
ec dependence be ween he ma ks and he poin s. We no e ha only unde independen ma ks,
he condi ional expec a ion o 4o 5coincides wi h he ma k mean. Ob iously, he abo e
o mula ions include au o-ma k cha ac e is ics as special cases o h¼l. We no e ha apa
om a concu en se ing, al e na i e poin wise es unc ions may be defined o he ma ks
h ðÞand lsðÞwi h s< .
2.2.1 C oss- unc ion a ia ion and di e en ia ion cha ac e is ics
As a fi s c oss- unc ion ma k cha ac e is ic, he poin wise c oss- unc ion ma k a iog am
γhl ; ðÞ, which helps o in es iga e he s eng h and ange o he a ia ion in he ma k di e -
ences wi h espec o he dis ance , can be de i ed by aking he condi ional expec a ion E∘;
o 1. This poin wise cha ac e is ic can hen be u ned in o a global c oss- unc ion ma k
a iog am γhl ðÞ, which co esponds o he L2me ic, by in eg a ing γhl ; ðÞo e T,
γhl ðÞ¼∫TE∘; 1 h∘ðÞ ðÞ; l ðÞ ðÞðÞ½d ≕∫Tγhl ; ðÞd :
The limi o his cha ac e is ic equals he non-spa ial a iance, and he no malised e sion
yields a s aigh line ha is cons an ly one unde he independen ma k assump ion. In con as ,
la ge alues o his c oss- unc ion cha ac e is ic will indica e a s ong he e ogenei y be ween he
unc ion- alued a ibu es a a dis ance .
Di e en om he c oss- unc ion ma k a iog am, aking 3as a gumen o E∘; yields a
poin wise c oss- unc ion e sion o S oyan’s ma k co a iance unc ion (S oyan, 1984),
o S o
hl ; ðÞ¼E∘; 3
½μh ðÞμl ðÞ
whe e μh
ðÞ¼E h
ðÞ½
and μl
ðÞ¼E l
ðÞ½
a e he non-spa ial means o h
ðÞ and l
ðÞ
,
154 ECKARDT ET AL.
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espec i ely. Again, a co esponding global cha ac e is ic is achie ed as o S o
hl ðÞ¼
∫o S o
hl ; ðÞd . Al e na i ely, a c oss- unc ion ma k co a iance can also be ob ained by ew i -
ing C essie’s (C essie, 1993) co a iance unc ion o C e
hl as o C e
hl ; ðÞ¼E∘; 3
½
E∘; 4
½E∘; 5
½, wi h o C e
hl ðÞ¼∫o C e
hl ; ðÞd .
Inse ing he a io o 2ins ead o he di e ence be ween he pai ed ma ks in o he condi ional
expec a ion E∘; ½yields a poin wise c oss- unc ion e sion o he ma k di e en ia ion unc-
ion (Hui & Pomme ening, 2014; Pomme ening e al., 2011)τhl ;
ðÞ
o unc ion- alued ma ks,
defined by τhl ; ðÞ¼1E∘; 2
½wi h global cha ac e is ic τhl ðÞ¼∫τhl ; ðÞd . Ob iously,
alues o τhl ; ðÞequal o close o ze o imply ha he unc ion- alued poin a ibu es a
dis ance a e equal o almos iden ical while inc easing non-ze o alues indica e an inc ease
in he e ogenei y o he ma ks.
2.2.2 C oss- unc ion co ela ion cha ac e is ics
Di e en om he di e ence and a io based cha ac e is ics, an al e na i e se o
c oss- unc ion ma k cha ac e is ics can be defined h ough he p oduc o he unc ion- alued
ma ks. Taking 3as a gumen o E∘; yields a poin wise c oss- unc ion e sion o he condi ional
mean p oduc o ma ks chl ; ðÞwi hin a dis ance a ∈T. We no e ha chl ; ðÞ ansla es
again in o a global c oss- unc ion cha ac e is ic chl ðÞby in eg a ion o he poin wise one o e
T,
chl ðÞ¼∫TE∘; h∘ðÞ ðÞ l ðÞ ðÞ½d ≕∫Tchl ; ðÞd :(1)
Fu he , no malisingchl ; ðÞby μh ðÞμl ðÞ, ha is, he p oduc o non-spa ial means, yields a
c oss- unc ion poin wise e sion o S oyan’s ma k c oss-co ela ion unc ion κhl ; ðÞ
(S oyan, 1987) om which he global cha ac e is ic κhl ðÞ ollows analogous o (1) by in eg a-
ion o κhl ; ðÞo e T. We no e ha S oyan’s ma k co a iance unc ion is indeed a linea ans-
o ma ion o he ma k co ela ion unc ion such ha o S o
hl ðÞand κhl ðÞa e essen ially he
same (Schla he , 2001).
Apa om S oyan’s ma k co ela ion unc ion, Isham (1985) and Beisba and
Ke sche (2000) in oduced wo al e na i e ma k co ela ion unc ions ha could also be ex-
ended o he unc ion- alued ma k se ing. Beisba and Ke sche (2000) p oposed a simple
e sion o he abo e o mula ion o he ma k co ela ion unc ion in which he p oduc o he
ma k alues is eplaced by he no malised sum o ma ks. Using his o mula ion allows o a
s aigh o wa d ex ension o a poin wise e sion o unc ion- alued ma ks h∘ðÞ ðÞ and
l ðÞ ðÞdefined by
κBei
hl ; ðÞ¼
h∘ðÞ ðÞþ l ðÞ ðÞ
μh ðÞþμl ðÞ
wi h κBei
hl ðÞ¼∫κBei
hl ; ðÞd . As he nomina o app oaches he p oduc ( esp. he sum) o means
μh ðÞand μl ðÞas limi s, bo h κkl and κBei
kl a e cons an ly equal o one o all ∈Tin case o
independen ma ks. In cons as , posi i e o nega i e ma k co ela ions could easily be iden i-
fied by posi i e o nega i e de ia ions om one, espec i ely. Opposi e o he abo e o mula-
ions, Isham (1985) in oduced a di e en ype o ma k co ela ion unc ion which is closely
ela ed o Pea son’s co ela ion. Using he c oss- unc ion e sion o C essie’s ma k co a iance
unc ion, a poin wise c oss- unc ion analogue o Isham’s ma k co ela ion unc ion can be de-
fined as
155Summa y cha ac e is ics o mul i a ia e unc ion- alued spa ial poin p ocess a ibu es
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o Ish
hl ; ðÞ¼ o C e
hl ; ðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Va hh ; ðÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Va ll ; ðÞ
p
whe e Va hh ; ðÞ¼E∘; h∘ðÞ ðÞ h ðÞ ðÞðE∘; h∘ðÞ ðÞ½E∘; h ðÞ ðÞ½
and Va ll ; ðÞis
defined analogous, and o Ish
hl ðÞ¼∫o Ish
hl ; ðÞd .
Apa om he ex ended c oss- unc ion ma k co ela ion cha ac e is ics ou lined abo e, ak-
ing 4o 5as a gumen s o E∘; leads o poin wise -ma k unc ions ch• ; ðÞand c•l ; ðÞ, e-
spec i ely, whe e ch• ; ðÞ¼c•l ; ðÞ. As be o e, bo h poin wise -ma k unc ions ansla e in o
global cha ac e is ics ch• ðÞand c•l ðÞby in eg a ion o ch• ; ðÞand c•l ; ðÞo e T, espec i ely.
Fu he , no malisa ion o ch• ; ðÞand c•l ; ðÞby μh ðÞand μl ðÞyields he poin wise -ma k
co ela ion unc ionsκh• ; ðÞand κ•l ; ðÞ, espec i ely, whe e κh• ðÞ¼∫κh• ; ðÞd and κ•l ðÞ¼
∫κ•l ; ðÞd .
We no e ha he c oss- unc ion ma k co ela ion and -co ela ion unc ions can also be used
o define a coun e pa e sion o he U ðÞ unc ion o unc ion- alued ma ks, his U ðÞbeing
he mean p oduc o ma ks si ed a dis ance apa ,
U ðÞ¼∫λ2g ðÞκhl ðÞdada0;(2)
whe e λ≡λGis he in ensi y o he poin s, g
ðÞ
he pai co ela ion unc ion, and aand a0a e wo
infini esimal small a eas con aining poin s xand x’which a e sepa a ed by a dis ance
(Capobianco & Renshaw, 1998; Renshaw, 2002). Including second-o de summa y cha ac e is-
ics o bo h he poin s and he unc ion- alued ma ks, hese cha ac e is ics accoun join ly o
spa ial a ia ion o he poin loca ions and he ma ks. Unde he independen ma ks assump ion,
κhl ðÞ¼1 whe eas g ðÞ¼1 unde he comple e spa ial andomness hypo hesis, ha is, he ho-
mogeneous Poisson poin p ocess case. Al e na i e o mula ion o U ðÞcan be achie ed by
subs i u ing κhl ðÞby he -co ela ion unc ions κh• ðÞand κ•l ðÞ, he ma k a iog am γhl ðÞ
and he ma k di e en ia ion unc ion τhl ðÞ, o al e na i ely by ew i ing U ðÞin o pola coo -
dina es allows o handling aniso opic beha iou .
2.2.3 C oss- unc ion nea es -neighbou indices and k-nea es neighbou cha ac e is ics
While second-o de c oss-cha ac e is ics p o ide unc ional summa y cha ac e is ics o he
pai wise in e ela ions be ween he unc ion- alued poin a ibu es agains he dis ance ,
nea es -neighbou indices a e essen ially nume ical ma k summa y cha ac e is ics which help
o quan i y he local a ia ion be ween he ma ks o a pai o nea es -neighbou ing poin s. Sim-
ila o he p e ious sec ions, di e en c oss- unc ion nea es -neighbou cha ac e is ics can be
cons uc ed by aking he condi ional expec a ion o pa icula es unc ions. In con as o
he abo e sec ions hese, howe e , only conside he unc ion- alued ma ks h∘ðÞ ðÞ and
lz∘ðÞðÞ ðÞa he o igin ∘and i s nea es neighbou ing poin z∘ðÞ(S oyan & S oyan, 1994).
Rew i ing he es unc ion 3in o a nea es -neighbou e sion nn
3¼ h∘
ðÞ
ðÞ lz∘
ðÞðÞ
ðÞand
aking he condi ional expec a ion E∘;z∘ðÞ nn
3
leads o a poin wise c oss- unc ion nea es -
neighbou ma k p oduc index cnn
hl ðÞ. The co esponding poin wise nea es -neighbou ma k
p oduc co ela ion index κnn
hl ðÞde i es di ec ly om cnn
hl ðÞby no malising cnn
hl ðÞby he p oduc
o means μh ðÞμl ðÞ. Likewise, aking he condi ional expec a ion o nn
4¼ lz∘ðÞðÞ ðÞyields a
poin wise nea es -neighbou ma k index cnn
•;l ðÞ which ans o ms in o he poin wise
nea es -neighbou ma k co ela ion index by no malising cnn
•;l ðÞ by μl. Simila ly, poin wise
c oss- unc ion nea es neighbou ma k a iog am and ma k di e en ia ion indices γnn
hl ðÞand
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τnn
hl ðÞcan be cons uc ed by aking he condi ional expec a ion E∘;z∘ðÞo he es unc ions nn
1¼
1=2 h∘ðÞ ðÞ lz∘ðÞðÞ ðÞðÞ
2and nn
2¼minnn=maxnn whe e minnn ¼min h∘ðÞ ðÞ; lz∘ðÞðÞ ðÞðÞ
and maxnn ¼max h∘ðÞ ðÞ; lz∘ðÞðÞ ðÞðÞ, espec i ely. We no e ha all poin wise indexes
ansla e in o global nume ical summa y cha ac e is ics by in eg a ion o he poin wise e sion
o e T.
Apa om conside ing only he unc ion- alued ma k o he nea es neighbou ing poin lo-
ca ionz∘
ðÞ
, he nea es neighbou indices can also be used o compu e cumula i e c oss- unc ion
k-nea es neighbou summa y cha ac e is ics om he ma ks hand la he o igin ∘and i s k- h
nea es neighbou ing poin z ∘ðÞwi h ¼1;…;k. Subs i u ing z∘ðÞby z ∘ðÞ, a co esponding
cumula i e ma k co ela ion index can be compu ed om he poin wise c oss- unc ion k- h
nea es neighbou ing index Kk ðÞ,
Kk ðÞ¼1
kE∘;z X
k
¼1
h∘ðÞ ðÞ lz ∘ðÞðÞ ðÞ
!
=μh ðÞμl ðÞ;
wi h Kk¼∫Kk ðÞd . Likewise, a cumula i e ma k a iog am index can be defined as Γk¼
Γk ðÞd wi h
Γk ðÞ¼1
kE∘;z X
k
¼1
1
2 h∘ðÞ ðÞ lz ∘ðÞðÞ ðÞðÞ
2
!
:
In addi ion, a poin wise coun e pa e sion o Hui’s ma k dominance index Dk(Hui
e al., 1998) o unc ion- alued ma ks can be defined as
Dk ðÞ¼1
kE∘;z X
k
¼1
1 h∘ðÞ ðÞ > lz ∘ðÞðÞ ðÞðÞ
!
which ansla es in o a global cha ac e is ics by compu ing Dh¼∫Dk ðÞd .
2.3 C oss-Func ion Ma k-Weigh ed Summa y Cha ac e is ics
A di e en use ul se o c oss- unc ion cumula i e summa y cha ac e is ics can be defined by
adjus ing classical unc ional poin p ocess summa y cha ac e is ics o he unc ion- alued
ma ks by in oducing a es unc ion as weigh in o he specific unc ional poin p ocess sum-
ma y exp ession. Al hough he p incipal idea also applies o he emp y space and nea es
neighbou con ac dis ibu ion unc ions and ela ed quan i ies, we explici ly only co e ex en-
sion o second-o de summa y cha ac e is ics o he unc ion- alued ma k scena io including
he ma k-weigh ed pai co ela ion, Kand L unc ions.
2.3.1 Ma k-weigh ed cha ac e is ics o uni ype poin p ocesses wi h mul i a ia e
unc ion- alued ma ks
To define a sui able pai co ela ion unc ion o unc ion- alued ma ks hx
ðÞ
ðÞand lx’
ðÞ
ðÞ
inΨ, le α2ðÞ
ðÞdeno e he poin wise c oss- unc ion second-o de ac o ial momen measu e wi h
densi y ϱ2ðÞ
ðÞ, ha is, he poin wise c oss- unc ion second-o de p oduc densi y unc ions. Fo
¼ 3,α2ðÞ
hl becomes
157Summa y cha ac e is ics o mul i a ia e unc ion- alued spa ial poin p ocess a ibu es
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o nega i e in e ac ion be ween cu es (cen al and igh panels, espec i ely), hese es ima o s
lie ou side his g ey shading a ea, confi ming he p esence o in e - unc ion dependencies. In
pa icula , unde posi i e co ela ion o he unc ion- alued ma ks, he empi ical
c oss- unc ion ma k a iog am lies ou side his g ey shading a ea wi h alues smalle han
he smalles en elope alues, o small alues. This sugges s ha he posi i e in e ac ing
unc ion- alued ma ks ha e less a iabili y han unde he independen ma k se ing. Simila ly,
unde nega i e co ela ion be ween unc ions, es ima o s o bo h he c oss- unc ion ma k
a iog am and he ma k co ela ion lie ou side he g ey shading a ea wi h alues la ge han
Figu e 1. C oss- unc ion ma k summa y cha ac e is ics o a simula ed homogeneous Poisson p ocess on he uni o us wi h
poin in ensi y λ¼200 . C oss- unc ion ma k a iog am ( op) and c oss- unc ion ma k co ela ion (bo om) wi h
no-in e ac ion e ec s ( JhðÞ¼JlðÞ¼JðÞ, wi h c = 0) (le ), posi i e in e - unc ion in e ac ion ( JhðÞ¼JlðÞ¼
JðÞ, wi h c ¼0:5) (cen al), and nega i e in e - unc ion co ela ion JhðÞ¼JðÞðand JlðÞ¼0wi h c ¼0:5) ( igh ).
Empi ical e sions o bo h cha ac e is ics a e highligh ed in ed, heo e ical alues in black. G ey shading shows he
fi h-la ges and smalles en elope alues based on 199 andom simula ions acco ding o he null hypo hesis o andom la-
beling o unc ions o e fixed poin loca ions.
164 ECKARDT ET AL.
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he la ges en elope alues, o small alues, sugges ing nega i e in e ac ions be ween
unc ions.
Simila esul s can be ound o he Thomas (Figu e 2) and he S auss p ocess scena ios
(Figu e 3). In absence o in e - unc ion dependencies bo h es ima o s (c oss- unc ion ma k
a iog am and ma k co ela ion) lie wi hin he g ey shading a ea, whils o posi i e o nega i e
co ela ion e ec s be ween unc ions hese unc ions lie ou side hese en elopes. This confi ms
ha he new c oss- unc ion ma k summa y cha ac e is ics can de ec spa ial dependencies be-
ween unc ions o dis inc ype independen ly o he spa ial s uc u e o he unde lying poin
pa e n.
Figu e 2. C oss- unc ion ma k summa y cha ac e is ics o a simula ed Thomas p ocess wi h o sp ing dispe sion pa ame e
σ¼0:04, pa en in ensi y λp¼40, and μ¼4expec ed o sp ings pe pa en . C oss- unc ion ma k a iog am ( op) and
c oss- unc ion ma k co ela ion (bo om) wi h no-in e ac ion e ec s (JhðÞ¼JlðÞ¼JðÞ, wi h c = 0) (le ), posi i e
in e - unc ion in e ac ion ( JhðÞ¼JlðÞ¼JðÞ, wi h c ¼0:5) (cen al), and nega i e in e - unc ion co ela ion
JhðÞ¼JðÞðand JlðÞ¼0wi h c ¼0:5) ( igh ). Empi ical e sions o bo h cha ac e is ics a e highligh ed in ed, heo-
e ical alues in black. G ey shading shows he fi h-la ges and smalles en elope alues based on 199 andom simula ions
acco ding o he null hypo hesis o andom labeling o unc ions o e fixed poin loca ions.
165Summa y cha ac e is ics o mul i a ia e unc ion- alued spa ial poin p ocess a ibu es
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Fo comple eness, we conside a simula ion s udy based on he simula ion o 100 da ase s o
he abo e scena ios o illus a e he pe o mance o ou second-o de cha ac e is ics. In pa ic-
ula , we simula ed 100 ealisa ions o he scena io defined abo e o show he pe cen age o
imes he esul ing empi ical unc ion lies ou side he simula ed en elopes based on 199 andom
ealisa ions acco ding o he null hypo hesis o andom labeling o unc ions o e fixed poin
loca ions. Table 1shows he pe cen age o imes he simula ed empi ical c oss- unc ion ma k
a iog am and co ela ion lie ou side he en elopes o he scena ios defined abo e. This high-
ligh s ha unde he scena ios wi h absence o in e ac ion be ween unc ions, he pe cen age o
imes bo h empi ical unc ion lie ou side he simula ed en elopes is a ound 25% o he
Figu e 3. C oss- unc ion ma k summa y cha ac e is ics o a simula ed S auss p ocess wi h in e ac ion dis ance Rin ¼0:05
and in e ac ion pa ame e q ¼0:05. C oss- unc ion ma k a iog am ( op) and c oss- unc ion ma k co ela ion (bo om) wi h
no-in e ac ion e ec s (JhðÞ¼JlðÞ¼JðÞ, wi h c = 0) (le ), posi i e in e - unc ion in e ac ion (JhðÞ¼JlðÞ¼JðÞ,
wi h c ¼0:5) (cen al), and nega i e in e - unc ion co ela ion JhðÞ¼JðÞðand JlðÞ¼0wi h c ¼0:5) ( igh ). Empi ical
e sions o bo h cha ac e is ics a e highligh ed in ed, heo e ical alues in black. G ey shading shows he fi h-la ges and
smalles en elope alues based on 199 andom simula ions acco ding o he null hypo hesis o andom labeling o unc ions
o e fixed poin loca ions.
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ealisa ions o any o he ini ial poin configu a ions. This sugges s ha he ejec ion o he null
hypo hesis is unlikely in absence o in e - unc ion in e ac ions. No e ha hese significan cases,
whe e he empi ical unc ion lies ou side he en elopes, co espond o andom excu sions o he
empi ical unc ion ou side he en elopes a he han significan ends o his unc ion. In di ec
con as , o he scena ios wi h posi i e/nega i e in e ac ion e ec s be ween unc ions, he pe -
cen age o imes bo h empi ical unc ions lie ou side he en elopes is always la ge han 78%
(assuming he empi ical unc ion lying abo e o bellow he en elopes) showing ha ou new
summa y cha ac e is ics de ec he p esence o hese in e - unc ion dependencies. Fo ins ance,
o he scena ios wi h nega i e in e ac ion e ec s, and any ini ial poin configu a ion, empi ical
c oss- unc ion ma k a iog am and co ela ion unc ions lie 100% o he simula ions abo e he
en elopes, confi ming he alidi y o ou new app oach o de ec spa ial dependencies be ween
unc ions.
5 Applica ions
5.1 Applica ion o Swiss T ee Da a
We fi s conside ee measu emen s eco ded a an annual basis o e 14 yea s ha o igina es
om a long- e m i iga ion expe imen loca ed in P ynwald, he cen al pa o he P yn-Finges
na ional pa k in Swi ze land (Schaub e al., 2016). Ini ia ed in 2003, he expe imen aimed o
in es iga e he e ec o inc eased wa e a ailabili y on he indi idual ees and he ecosys em
in a na u ally d y Sco s pine (Pinus syl es is L.) o es . The s udy egion co e s an a ea o
1.2 ha and is loca ed in one o he d ies inne -Alpine alleys o he Eu opean Alps (see Bose
e al., 2022, o de ailed summa y). The da a a hand was p o ided as open da a unde an Open
Da abase Licence and has been made a ailable publicly a h ps://openda a.swiss. I co e s he
ee-specific spa ial coo dina es, he ini ial assignmen in o he ea men o con ol g oup and
di e en ee cha ac e is ics o 900 ees. F om his sou ce, we ini ially selec ed he annual o al
c own de olia ion (TCD) om he p o ided lis o ee cha ac e is ics and also he exac poin
loca ions o he indi idual ee s ands. The TCD pa ame e is a commonly used pa ame e in
o es moni o ing s udies o quan i y he loss o needles o lea es o a gi en ee ela i e o a
local e e ence ee. Wi hin he applica ion, we conside ed he e ie ed TCD in o ma ion as
unc ion- alued ee a ibu e and assigned i as a ma k o he ee loca ions in a subsequen ac-
ion. Res ic ing he da a o comple e cases, we excluded any ees wi h incomple e o missing
TCD in o ma ion om he da a yielding a final sample o 799 ees wi h annual TCD eco ds
o e all 14 yea s. In a nex s ep, we compu ed he local pai wise co ela ion unc ion o all ees
o he educed sample which desc ibes he con ibu ion o he indi idual poin o he empi ical
Table 1. Pe cen age o imes he simula ed empi ical c oss- unc ion ma k a iog am (Ma k Va .) and co ela ion (Ma k Co .)
lie ou side he en elopes, based on he simula ion o 100 da ase s o he homogeneous Poisson p ocess (Poisson) (as defined
in Figu e 1, he Thomas p ocess (as defined in Figu e 2) and he S auss p ocess (as defined in Figu e 3), assuming no
in e ac ion e ec s, and, posi i e and nega i e in e - unc ion in e ac ion e ec s (as defined in Figu e 1). A and B deno e ha he
empi ical unc ion lies abo e o below he simula ed en elopes, espec i ely.
No in e ac ion Posi i e in e ac ion Nega i e in e ac ion
Ma k Va . Ma k Co . Ma k Va . Ma k Co . Ma k Va . Ma k Co .
ABABAB A BA BA B
Poisson 24 28 25 21 44 97 100 76 100 63 100 57
Thomas 28 25 30 24 49 78 100 66 100 60 100 58
S auss 31 29 25 28 32 100 100 59 100 50 100 44
167Summa y cha ac e is ics o mul i a ia e unc ion- alued spa ial poin p ocess a ibu es
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pai co ela ion unc ion, ha is, i s pai co ela ion unc ion based local indica o o spa ial as-
socia ion. The local in o ma ion was hen used as a second unc ion- alued ma k in ou appli-
ca ion such ha each ee was ma ked by wo dis inc unc ion- alued quan i ies. The esul ing
poin pa e n wi h bo h unc ion- alued ma ks and classic second-o de summa y cha ac e is ics
o he poin s a e shown in Figu e 4. While no conside ed he e, we no e ha he da a also allows
o c oss- unc ion c oss- ype e sions as ou lined on Sec ion 2.3.2 by aking addi ionally he
ee-specific assignmen in o ea men o con ol g oup in o accoun . Such ad anced ma k
Figu e 4. Obse ed unc ion- alued ma ks and classic second-o de poin p ocess summa y cha ac e is ics o he Swiss ee
pa e ns. Top panel: spa ial dis ibu ion o Sco s pines o he P ynwald da a wi h obse ed o al c own de olia ion (le ), a
magnifica ion o he s udy egion (cen e ), and pai co ela ion unc ion and heo e ical en elopes unde he independen
ma k hypo hesis (le ). Bo om panel: spa ial dis ibu ion o Sco s pines o he P ynwald da a wi h local pai co ela ion unc-
ions as unc ion- alued ma ks (le ), a magnifica ion o he s udy egion (cen e ), and Ripley’sK ðÞ unc ion minus π2and
heo e ical en elopes ( igh ) compu ed om he poin loca ions. Empi ical e sions o bo h cha ac e is ics a e highligh ed in
ed, heo e ical alues in black, and blue lines a e he unc ion- alued ma ks. G ey shading shows he fi h-la ges and
smalles en elope alues based on 199 andom simula ions acco ding o he null hypo hesis o comple e spa ial andomness
(Poisson poin andomisa ions).
168 ECKARDT ET AL.
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cha ac e is ics migh help o in es iga e he complex in e play o he TCD and local pai co e-
la ion unc ion cu es wi h he e ec o addi ional wa e supply.
As expec ed by he la ge numbe o ees, he sampled poin pa e n eflec s some clea s uc-
u e and a endency o clus e ing among he poin s. This imp ession is suppo ed by he pai co -
ela ion unc ion ( op- igh panel) and also Ripley’sK unc ion bo om- igh panel) which show
a clea posi i e shi o he empi ical cu es om he heo e ical lines unde he comple e spa ial
andomness hypo hesis which indica es a clea endency o clus e ing.
Nex , o e alua e he findings o he p oposed au o- and c oss- unc ion summa y cha ac e is-
ics wi h he classic summa y cha ac e is ics o scala - alued ma ks commonly used a p esen ,
we ans o med he unc ion- alued ma ks in o unc ion-wise a e ages and compu ed he ma k
a iog am and S oyan’s ma k co ela ion unc ion om he a e aged quan i ies (see Figu e 5).
Figu e 5. Classic ma k summa y cha ac e is ics o he P ynwald ee da a wi h a e aged unc ion- alued poin a ibu es
ea ed as scala - alued ma ks. Ma k a iog am and ma k co ela ion unc ions o he mean TCD ( op) and mean local pai
co ela ion unc ion (bo om). Empi ical e sions o bo h cha ac e is ics a e highligh ed in ed, heo e ical alues in black.
G ey shading shows he fi h-la ges and smalles en elope alues based on 199 andom simula ions acco ding o he null
hypo hesis o andom labeling o ma ks o e fixed poin loca ions.
169Summa y cha ac e is ics o mul i a ia e unc ion- alued spa ial poin p ocess a ibu es
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The empi ical e sions o bo h ma k cha ac e is ics show a clea de ia ion om he heo e ical
en elopes o he a e age TCD ( op panel). While he ma k a iog am ( op-le panel) sugges s
ha he mean TCD alues exhibi less pai wise a ia ion as expec ed unde he independen
ma k hypo hesis, we ound a clea posi i e shi o he empi ical pai wise p oduc o TCD a -
e ages as conside ed by he ma k co ela ion unc ion ( op- igh panel) om he heo e ical en-
elopes. In compa ison wi h he TCD, bo h empi ical ma k cha ac e is ics show almos no de-
ia ions om he independen ma k hypo hesis in case o he a e aged local pai co ela ion
unc ion (bo om panels). Excep o only some nega i e shi o he ma k a iog am (le ) a
small dis ances, bo h es ima ed cha ac e is ics a e co e ed by he en elopes.
Figu e 6. Au o- and c oss- unc ion ma k summa y cha ac e is ics compu ed om he Swiss ee da a. Top: au o- unc ion
ma k a iog am (le ) and ma k co ela ion unc ion ( igh ) o he o al c own de olia ion cu es. Cen al panel:
au o- unc ion ma k a iog am (le ) and ma k co ela ion unc ion ( igh ) o he local pai co ela ion unc ions. Bo om:
c oss- unc ion (le ) and ma k co ela ion unc ion ( igh ) be ween he o al c own de olia ion and he local pai co ela ion
unc ions. Empi ical e sions o bo h cha ac e is ics a e highligh ed in ed, heo e ical alues in black. G ey shading shows
he fi h-la ges and smalles en elope alues based on 199 andom simula ions acco ding o he null hypo hesis o andom
labeling o unc ions o e fixed poin loca ions.
170 ECKARDT ET AL.
In e na ional S a is ical Re iew (2025),93, 1, 150–178
© 2024 The Au ho (s). In e na ional S a is ical Re iew published by John Wiley & Sons L d on behal o In e na ional S a is ical Ins i u e.
Di e en om he classic ma k cha ac e is ics, all au o- and c oss- unc ion ma k a iog ams
and co ela ion unc ions o Figu e 6, excep he au o- unc ion ma k co ela ion o he local pai
co ela ion (cen al- igh panel), show significan esul s. As al eady indica ed by he classic
cha ac e is ics, he op panel co esponding o he TCD cu es eflec s again a nega i e de ia-
ion o he empi ical au o- unc ion ma k a iog am ( op-le panel) con as ed wi h a clea pos-
i i e shi o he empi ical au o- unc ion ma k co ela ion unc ion ( op- igh panel) om he
heo e ical lines unde he independen ma k hypo hesis. This indica es ha he obse ed
TCD cu es show less spa ial a ia ion among pai s o neighbou ing poin s. A he same ime,
Figu e 7. Obse ed unc ion- alued ma ks and classic second-o de poin p ocess summa y cha ac e is ics compu ed om
84 municipali ies o he p o ince o Albace e. Top panel: spa ial dis ibu ion o Spanish municipali ies and he yea ly di e -
ences o he e e ence yea 2022 o he business ( op-le ) and popula ion ( op- igh ) eco ds as unc ion- alued ma ks. Bo -
om panel: pai co ela ion unc ion and heo e ical en elopes unde he independen ma k hypo hesis (le ), and Ripley’s
K ðÞ unc ion minus π2and heo e ical en elopes ( igh ) compu ed om he poin loca ions. Empi ical e sions o bo h cha -
ac e is ics a e highligh ed in ed, heo e ical alues in black, and blue lines a e he unc ion- alued ma ks. G ey shading
shows he fi h-la ges and smalles en elope alues based on 199 andom simula ions acco ding o he null hypo hesis o
comple e spa ial andomness (Poisson poin andomisa ions).
171Summa y cha ac e is ics o mul i a ia e unc ion- alued spa ial poin p ocess a ibu es
In e na ional S a is ical Re iew (2025),93, 1, 150–178
© 2024 The Au ho (s). In e na ional S a is ical Re iew published by John Wiley & Sons L d on behal o In e na ional S a is ical Ins i u e.
he p oduc o he TCD cu es clea ly exceeds he expec ed case, ha is, he non-spa ial unc-
ional mean squa ed. Fo he cen al panels showing he au o- unc ion cha ac e is ics compu ed
om he local pai co ela ion unc ions, he au o- unc ion ma k a iog am (le ) again sugges s
smalle a ia ion be ween he unc ion- alued ma ks compa ed wi h he independen ma k se -
ing o some small dis ances. Finally, looking a he c oss- unc ion cha ac e is ics o he TCD
and local pai co ela ion cu es, bo h esul s show a clea a ia ion om he independen ma k
en elopes. This would imply ha he pai wise spa ial a ia ion o bo h unc ions is smalle han
unde he limi ing case whe e he c oss- unc ion a iog am is equal o he co a iance, whe eas
he pai wise p oduc o he wo ma ks exceeds he limi ing case in which he pai wise p oduc o
he wo ma ks app oaches he p oduc o he unc ional means μhand μl.
5.2 Applica ion o Spanish Labou Da a
As second example o a spa ial poin pa e n wi h bi a ia e unc ion- alued ma ks, we con-
side da a on he o al numbe o companies as o 1 Janua y and he numbe o esiden s
Figu e 8. Classic ma k summa y cha ac e is ics o he Spanish municipali y da a compu ed om he a e aged business and
popula ion in o ma ion wi h a e aged unc ion- alued poin a ibu es ea ed as scala - alued ma ks. Ma k a iog am and
condi ional mean p oduc o ma ks o he mean business ( op) and mean popula ion unc ion (bo om). Empi ical e sions o
bo h cha ac e is ics a e highligh ed in ed, heo e ical alues in black. G ey shading shows he fi h-la ges and smalles en-
elope alues based on 199 andom simula ions acco ding o he null hypo hesis o andom labeling o ma ks o e fixed poin
loca ions.
172 ECKARDT ET AL.
In e na ional S a is ical Re iew (2025),93, 1, 150–178
© 2024 The Au ho (s). In e na ional S a is ical Re iew published by John Wiley & Sons L d on behal o In e na ional S a is ical Ins i u e.
eco ded annually a municipali y le el o he pe iod om 2012 o 2022. The da a o igina ed
om he o ficial da a epo s eleased by he Na ional S a is ics Ins i u e o Spain (INE) and
was made publicly a ailable a www.ine.es. The business in o ma ion was de i ed om he o -
ficial business egis e o INE and co esponds o he o al numbe o local companies o e di -
e en economic sec o s. The local assignmen o he companies o exac ly one municipali y was
pe o med by INE in a p e-p ocessing s ep using he egis e ed business add ess in o ma ion o
a oid po en ial inconsis encies in case o egionally wide sp eading business loca ions, o ex-
ample, ac o ies o business acili ies o one company in se e al dis inc municipali ies. F om
he p o ided da a, we ini ially selec ed a sample o 87 municipali ies ha all in o he bounda ies
o Albace e, a Spanish p o ince on La Mancha ( he Spanish Pla eau). The a ea o La Mancha is
Figu e 9. Au o- and c oss- unc ion ma k summa y cha ac e is ics compu ed om he Spanish municipali y da a. Top panel:
au o- unc ion ma k a iog am (le ) and ma k co ela ion unc ion ( igh ) o he business cu es. Cen al panel: au o- unc ion
ma k a iog am (le ) and ma k co ela ion unc ion ( igh ) o he popula ion cu es. Bo om panel: c oss- unc ion (le ) and
ma k co ela ion unc ion ( igh ) be ween he business and popula ion cu es. Empi ical e sions o bo h cha ac e is ics a e
highligh ed in ed, heo e ical alues in black. G ey shading shows he fi h-la ges and smalles en elope alues based on 199
andom simula ions acco ding o he null hypo hesis o andom labeling o unc ions o e fixed poin loca ions.
173Summa y cha ac e is ics o mul i a ia e unc ion- alued spa ial poin p ocess a ibu es
In e na ional S a is ical Re iew (2025),93, 1, 150–178
© 2024 The Au ho (s). In e na ional S a is ical Re iew published by John Wiley & Sons L d on behal o In e na ional S a is ical Ins i u e.