Gradient boosting for Dirichlet regression models

Balze , Michael; Be ghe , Elisabe h; Hu e , Swen; Hepp, Tobias
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G adien boos ing o Di ichle eg ession models
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ORIGINAL PAPER
G adien boos ing o Di ichle eg ession models
MichaelBalze 1 · Elisabe hBe ghe 2· SwenHu e 3,4· TobiasHepp2,5
Recei ed: 19 Ap il 2024 / Accep ed: 19 Ma ch 2025
© The Au ho (s) 2025
Abs ac
In a ious eal-wo ld applica ions, esea che s o en wo k wi h composi ional da a
which appea s as p opo ions, amoun s o a es. As a amewo k o dealing wi h he
unique na u e o composi ional da a, Di ichle eg ession models ha e been in o-
duced. In his a icle, we p opose a no el model-based g adien boos ing app oach
o Di ichle eg ession models embedded in he amewo k o gene alized addi i e
models o loca ion, scale and shape. This app oach allows o da a-d i en a iable
selec ion in low- as well as high-dimensional da a se ings. Mo eo e , he imple-
men a ion enables he di ec calcula ion o ma ginal e ec s o di e en p edic o
a iables. Thus, i p o ides an al e na i e es ima ion p ocedu e besides he well-
es ablished app oach based on he maximum likelihood p inciple. A e conduc ing
de ailed simula ion s udies o e alua e he pe o mance o he es ima ion p ocedu e
ega ding p edic ion accu acy and a iable selec ion in low- and high-dimensional
se ings, we p esen a eal-wo ld applica ion conce ning he changes in elec ion
esul s in he G ea Recession u ilizing a la ge-scale Eu opean da ase . Using ou
p oposed app oach, we in es iga e he e ec o p o es s on o ing p opo ions o dis-
inc pa y amilies while iden i ying impo an socioeconomic a iables and hei
e ec on hose o ing p opo ions ia a iable selec ion.
Keywo ds Di ichle eg ession models· G adien boos ing· S a is ical lea ning·
G ea Recession· Elec ions
1 In oduc ion
In many eal-wo ld applica ions, applied esea che s ha e o deal wi h composi ional
da a. Fo ins ance, in ecology, economics and medicine composi ional da a occu s
as p opo ions, amoun s o a es. The da a ma ix o en has a leas wo columns and
he en ies a e bounded in he in e al (0,1) which a e also sum-cons ained such
ha each ow sums up o one ep esen ing one obse a ion. A common app oach o
deal wi h composi ional da a is o ans o m he columns such ha log- a io anal-
ysis is possible (Ai chison 2003; Maie 2014). La ely, he amewo k o Di ichle
Ex ended au ho in o ma ion a ailable on he las page o he a icle
M.Balze e al.
eg ession models has been es ablished as an al e na i e o modeling composi ional
da a on he o iginal scale. Speci ically, Di ichle eg ession models can be desc ibed
in he amewo k o mul i a ia e gene alized addi i e models wi h he Di ichle dis-
ibu ion p o iding he unknown pa ame e s o be es ima ed h ough he addi i e
p edic o s (McCullagh and Nelde 1989; Maie 2014).
Conside ing ha he numbe o addi i e p edic o s in Di ichle eg ession models
di ec ly depends on he ca dinali y o he se o p opo ions o be analyzed, s a e-
gies ega ding model choice and a iable selec ion play a e y impo an ole in he
applica ion o hese models. To his end, di e en c i e ia ha e been de eloped in
o de o ob ain a quan i iable alue o deciding be ween mul iple candida e mod-
els o a ying complexi y. Pa icula ly, i model es ima ion is based on maximum
likelihood (ML), a common and popula op ion is o calcula e he Akaike in o ma-
ion c i e ion ia he likelihood p inciple. Ins ead o a c i e ia-based app oach, how-
e e , a iable selec ion and model choice can also be achie ed using egula izing
cons ain s. Such echniques include idge eg ession and he leas absolu e sh ink-
age and selec ion ope a o (lasso) as well as g adien boos ing algo i hms (Fah mei
e al. 2013; Hepp e al. 2016). O iginally es ablished in he domain o machine lea n-
ing, he concep behind boos ing algo i hms has been quickly adap ed o he ame-
wo k o ypical s a is ical eg ession models unde he names o componen -wise,
model-based o s a is ical boos ing (May e al. 2014). The implemen a ion o he
s a is ical boos ing algo i hm is a he s aigh o wa d and s a s wi h he speci ica-
ion o a sui able loss unc ion o he es ima ion p oblem. Then, pa ame e es ima-
ion educes o i e a i ely i ing (o en eg ession- ype) base-lea ne s o he nega i e
g adien o he loss and selec ing only he bes pe o ming one in each i e a ion. Fol-
lowing his i e a i e p ocedu e yields in e p e able esul s and, in combina ion wi h
ea ly s opping, he possibili y o a iable selec ion. The g adien boos ing algo-
i hm is also pa icula ly use ul in a high-dimensional se ing, whe e he numbe o
p edic o s exceeds he numbe o obse a ions (Bühlmann and Ho ho n 2007). E en
hough g adien boos ing algo i hms ha e been p oposed o a a ie y o eg ession
models and applica ions (see, o example, Robinzono e al. (2012); Zume a-Olas-
koaga e al. (2023)), o he bes o ou knowledge, he e does no exis a g adien
boos ing algo i hm o he amewo k o Di ichle eg ession models. The e o e, we
aim o ex end he algo i hms in he gamboos LSS (Ho ne e al. 2023) package o
boos ing gene alized addi i e models o loca ion, scale and shape (GAMLSS) o
Di ichle eg ession models in he p og amming language R (R Co e Team 2023).
To his end, we conduc simula ion s udies in low- and high-dimensional lin-
ea se ings, whe e he ocus is on he e alua ion o he pe o mance o es ima ion,
a iable selec ion and p edic ion. Addi ionally, he pe o mance is e alua ed wi h
espec o nonlinea base-lea ne s. Finally, we use he implemen ed algo i hm o
examine changes in o ing p opo ions in he Eu opean Union. We ake up he s udy
by B eme e al. (2020) who ound ha he le el o economic p o es s igge ed by
he economic c isis had a signi ican impac on elec ion esul s in he G ea Reces-
sion whe e he au ho s mainly ocus on he e ec o p o es s on he changes in elec-
o al loss o incumben pa ies. Based on he Di ichle eg ession amewo k, we
aim o ex end he scope o he analysis o examine he e ec o p o es s on elec ion
esul s o all majo Eu opean pa y amilies, enabling us a mo e di e en ia ed iew
G adien boos ing o Di ichle eg ession models
on how he G ea Recession a ec ed he s uc u e o pa y sys ems ac oss he Eu o-
pean Union.
Ou s uc u e o his a icle is as ollows: We i s desc ibe he heo e ical back-
g ound ega ding he Di ichle dis ibu ion, eg ession and g adien boos ing. Fu -
he mo e, we documen he implemen a ion o g adien boos ing in R in Sec .2. We
hen pe o m simula ion s udies in Sec .3. A e an in-dep h desc ip ion o he con-
ex , da ase and a iables, we pe o m es ima ion by g adien boos ing o Di ichle
eg ession models wi h he eal-wo ld da ase whe e we commen on he impac o
p o es s on elec ion esul s in he G ea Recession in Sec .4. Finally, we p o ide a
discussion in Sec .5.
2 Me hodology
2.1 Boos ing Di ichle eg ession models
In gene al, boos ing can be seen as he s eepes descen in a unc ion space (F ied-
man 2001), ha is, he algo i hm aims o i e a i ely educe he empi ical isk using
so-called base-lea ne s. While he o iginal algo i hms u ilized he same p edic ion
unc ion as base-lea ne s on eweigh ed obse a ions, componen -wise boos ing
algo i hms mainly ely on using eg ession unc ions o he sepa a e inpu a iables
in he da a, he eby esul ing in an addi i e model o mula (see Ho ne e al. (2014)
o a comp ehensi e o e iew o a ailable base-lea ne s). Mo eo e , he loss unc-
ion can be exp essed in e ms o a nega i e log-likelihood (NLL) unc ion, which
allows o desc ibe he ou come in a e y simila way as gene alized linea models
(Has ie and Tibshi ani 1990; Has ie e al. 2009; May e al. 2014). The model is hen
es ima ed by s a ing he algo i hm wi h an emp y model and i ing all p o ided
base-lea ne s o he nega i e g adien o he speci ied loss unc ion. In p inciple, he
model is only upda ed by a small p opo ion o he bes pe o ming base-lea ne .
In he nex i e a ion, he g adien is ee alua ed using he upda ed model and again
i ed by all base-lea ne s. This p ocess is hen epea ed un il he desi ed numbe o
boos ing i e a ions is eached. Using he
L2
-loss hen esul s in he boos ing algo-
i hm o con e ge o he same solu ion as o dina y leas squa es (Bühlmann and
Ho ho n 2007). Al hough he e seems o be no p oo o mo e gene al loss unc ions,
empi ical esul s using he NLL as a loss unc ion indica e a e y simila beha io
wi h espec o he maximum likelihood solu ion o many gene alized linea models.
Appendix 2 p o ides a demons a ion o his p ope y o ou p oposed algo i hm.
A e y con enien ad an age o using boos ing algo i hms o es ima e eg ession
models is ha by i ing sepa a e base-lea ne s o all inpu a iables in he da a, he
algo i hm can in ac be used in high-dimensional se ings whe e he numbe o a i-
ables exceeds he numbe o obse a ions. In his con ex , Bühlmann (2006) p o ed
consis ency o boos ing wi h squa ed e o loss o high-dimensional linea models.
Mo eo e , a ho ough s a is ical pe spec i e on he boos ing amewo k is gi en in
Bühlmann and Ho ho n (2007). Ano he impo an connec ion can be d awn be ween
boos ing and he lasso (Tibshi ani 1996), as he s epwise model-building p ocess pe -
o med by boos ing in ac esul s in he es ima ion o a egula ized eg ession model.
M.Balze e al.
Fu he mo e, Has ie e al. (2007) demons a ed ha boos ing app oxima es he solu-
ion pa h o a s ic ly mono one
L1
- egula ized eg ession model o gene al con ex loss
unc ions. This means ha he boos ing solu ions a e op imal wi h espec o he so-
called
L1
-a c-leng h, which is he eason o a a he “g eedy” selec ion beha io due o
he inabili y o deselec a a iable once i has been added o he ac i e se o co a ia es,
which esul s in a highe numbe o alse posi i e selec ions compa ed o he
L1
-no m
op imal solu ions o he lasso (Hepp e al. 2016).
Whe e he lasso con ols he amoun o egula iza ion wi h an explici penal y e m,
egula iza ion ia boos ing algo i hms wo ks by s opping he algo i hm delibe a ely
du ing model i ing be o e con e gence. The e o e, he numbe o i e a ions
ms op
ac s as he main uning pa ame e ha p e en s o e i ing and imp o es p edic ion
accu acy o he o e all model by con olling he bias- a iance ade-o (May e al.
2012a). I
ms op
is chosen su icien ly la ge, he algo i hm e en ually eaches a solu ion
whe e addi ional upda es ha e no meaning ul impac on he loss ( ha is, o dina y leas
squa es o
L2
-loss). By s opping he upda es ea ly, spa se models and addi ional a i-
able selec ion can be achie ed.
I he dis ibu ion has mul iple pa ame e s, a mul iple dimension e sion o he g a-
dien boos ing algo i hm has o be u ilized since he aim is o en o boos all pa ame e s
o he dis ibu ion. In his case, he de aul algo i hm is he so-called cyclical compo-
nen -wise g adien boos ing algo i hm, in which he g id sea ch o iden i ying he op i-
mal s opping c i e ia scales exponen ially wi h he absolu e numbe o hese c i e ia
(Schmid e al. 2010; May e al. 2012a). To ci cum en he p oblems associa ed wi h
inc eased complexi y and compu a ional demand, he noncyclical componen -wise
boos ing algo i hm is an app op ia e al e na i e. I has been no ed ha his me hod is
mo e lexible in a iable selec ion and ime e icien in compa ison wi h he cyclical
algo i hm (Thomas e al. 2018).
Fo he implemen a ion o model-based g adien boos ing in he amewo k o Di -
ichle eg ession models, de i a ions o he NLL and he nega i e g adien ec o a e
necessa y. The e o e, cha ac e is ics o he Di ichle dis ibu ion a e desc ibed and nec-
essa y exp essions o he ing edien s in he boos ing algo i hms a e p o ided. Le
y
be
a andom ec o ep esen ing a se o p opo ions. Then,
yi
is he p opo ion o i em o
ca ego y
i∈{1, …,K}
wi h K being he maximum numbe o i ems o ca ego ies in
he da ase . In gene al, a andom ec o is called Di ichle dis ibu ed
whe e
𝜶=(𝛼1,…,𝛼K)
is he pa ame e ec o wi h
𝛼i
>
0
i and only i ollowing
h ee assump ions hold
Se s o p opo ions can addi ionally be combined, leading o a ma ix
D={y1,…,yN}
whe e N is he numbe o obse a ions in he espec i e da ase .
y=(y1,…,yK)∼Di (𝜶)
y=(y
1
,…,y
K
)
yi∈[0, 1]
K
∑
i=1
yi=1.

G adien boos ing o Di ichle eg ession models
The sum o all
𝛼i
is called he p ecision pa ame e and is gi en by
𝛼0
=
∑K
i=1
𝛼
i
wi h
which he expec a ions o he p opo ions can be quan i ied ia
In p inciple, he p ecision pa ame e go e ns he posi ion o he p obabili y densi y
in he
(K−1)
-simplex. Mo e densi y is concen a ed nea he expec a ion ec o , i
he p ecision is high. In gene al, he p obabili y densi y unc ion o he common
pa ame iza ion as in oduced in his pape is w i en as
Applying he loga i hm o he p obabili y densi y unc ion, u ilizing loga i hm ules
and nega ing yields he NLL as
The nega i e g adien ec o can be ob ained as he pa ial de i a i es o he NLL
wi h espec o each pa ame e
𝛼i
which is gi en by
whe e
𝜓(
⋅
)
is he digamma unc ion (Na ayanan 1991; Minka 2000; Gueo guie a
e al. 2008).
Based on he in oduced exp essions, Di ichle eg ession models can be
implemen ed which ely on he no ion ha he andom ec o
y=(y1,…,yK)
has
a condi ional Di ichle dis ibu ion wi h densi y
p(y|x)=p(y|𝜶(x))
and dis ibu-
ional pa ame e s
𝛼i(x)
which a e dependen on a iables. These dis ibu ional
pa ame e s a e connec ed o he addi i e p edic o s
𝜂
𝛼
i
ia a log link unc ion
g(𝛼
i
(x)) = log(𝛼
i
(x)) = 𝜂
𝛼
i
. The co esponding esponse unc ion is hus gi en by
g−1
(𝜂𝛼
i
)=exp(𝜂𝛼
i
)=𝛼i(x
)
which ensu es es ic ions on he pa ame e space o
he a iable dependen dis ibu ional pa ame e s. The addi i e p edic o s can be
exp essed as
in which
𝛽i0
a e he in e cep s,
ij,j={1, …,Qi}
a e unc ional e ec s o independ-
en a iables
x
and
Qi
he maximum numbe o hese unc ional e ec s o e e y
dis ibu ional pa ame e (Fah mei e al. 2013; Maie 2014).
(1)
𝔼
(yi)=
𝛼
i
𝛼0
.
(2)
p(y
�
𝜶)=
Γ
�∑
K
i=1𝛼i
�
∏
K
i=1
Γ(𝛼
i
)
K
�
i=1
y𝛼i−1
i
.
(3)
l
(y
|
𝜶)=−log p(y
|
𝜶)=−log Γ
(K
∑
i=1
𝛼i
)
+
K
∑
i=1
log Γ(𝛼i)−
K
∑
i=1
(𝛼i−1)log yi
.
(4)
−
𝜕
𝜕𝛼i
l(y
|
𝜶)=𝜓
(K
∑
i=1
𝛼i
)
−𝜓(𝛼i)+log yi∀i∈{1, 2, …,K}
,
𝜂
𝛼i=𝛽i0+
Q
i
∑
j=1
ij(x), o i=1, …,K
,
M.Balze e al.
The cons uc ion o he model-based g adien boos ing algo i hms o Di ichle
eg ession models is based on he ex ension o he algo i hms o GAMLSS p o-
posed in May e  al. (2012a) and Thomas e  al. (2018). In p inciple, only mino
adjus men s o he o me algo i hms a e necessa y o ensu e p ope unc ionali y.
Fi s , he algo i hms o GAMLSS a e designed o a maximum numbe o ou dis-
ibu ional pa ame e s. Howe e , in he con ex o Di ichle eg ession models, he
numbe o dis ibu ional pa ame e s depends on he ca dinali y o he se o p opo -
ions gi en by he esponse ma ix. Thus, he algo i hms a e ex ended o accommo-
da e mo e han ou dis ibu ional pa ame e s. Second, in he ini ializa ion o he
algo i hms, o se alues o he addi i e p edic o s 
𝜼[0]
=(𝜂
[0]
𝛼1
,…,𝜂
[0]
𝛼K
)
ha e o be
speci ied. The algo i hms a e qui e gene al in his con ex and e en allow o use -
speci ic inpu s. Howe e , in Di ichle eg ession models he o se alues a e se as
he minimum o he a ailable p opo ions in he esponse ma ix o each dis ibu-
ional pa ame e (Ronning 1989). Aside om adjus men s o he numbe o dis i-
bu ional pa ame e s and o se alues, he algo i hmic ou line emains unchanged.
The adjus ed algo i hms can be ound in he Appendix 1. Fo mo e gene al de ails
on he speci ic componen s in he algo i hms, we e e o May e al. (2012a) and
Thomas e al. (2018).
2.2 S abili y selec ion o boos ed Di ichle eg ession models
In gene al, model choice and a iable selec ion in model-based g adien boos ing
is pe o med ia ea ly s opping o he algo i hms by means o he s opping c i e-
ion
ms op
. Howe e , he choice o he s opping c i e ion ia k- old c oss- alida ion,
subsampling o boo s apping ends o include oo many (non-in o ma i e) a iables
esul ing in a non-pa simonious model (May e al. 2012b; Thomas e al. 2018).
To mi iga e he consequences o he inclusion o oo many non-in o ma i e
a iables in he inal model, Meinshausen and Bühlmann (2010) p opose a gene ic
algo i hm called s abili y selec ion which can be applied o any possible a iable
selec ion echnique, including model-based g adien boos ing. The goal o s abil-
i y selec ion is o gain e o con ol o he non-in o ma i e a iables. In p inciple,
deno e by V he numbe o non-in o ma i e ( alse posi i e) a iables, hen he e o
con ol is ob ained by unning he s abili y selec ion algo i hm on mul iple subsam-
ples o he da a whe e he so-called pe - amily e o a e (PFER)
𝔼(V)
is con olled.
In all subsamples, he in o ma i e base-lea ne s should almos always be selec ed by
he algo i hm (Meinshausen and Bühlmann 2010; Shah and Samwo h 2013; Ho ne
e al. 2015).
The s abili y selec ion algo i hm o boos ed Di ichle eg ession models is
iden ical o he algo i hm de eloped o componen -wise g adien boos ing o he
GAMLSS. The gene al algo i hmic ou line as p oposed by Thomas e al. (2018) ol-
lows h ee s eps:
1. D aw andom subse s B o he da a wi h size
⌊n∕2⌋
whe e n is he o iginal numbe
o obse a ions in he da a and i a boos ed model o each subse un il he numbe
G adien boos ing o Di ichle eg ession models
o chosen base-lea ne s eaches he maximum q o he i e a ions he s opping
c i e ion
ms op
.
2. Calcula e he selec ion equency
𝜋 j
o each base-lea ne de ined as
whe e

Sb
is he se o base-lea ne s chosen in i e a ion b.
3. Include a iables in a s able se (and hus in he model) i he calcula ed selec ion
equency
𝜋 j
is g ea e o equal o a use -speci ied h eshold
𝜋 h
An uppe -bound o he PFER is hen ob ained ia
whe e P is he numbe o a iables. A conside able challenge lies in he speci ica-
ion o
q,𝜋 h
and PFER since he choice is no i ial o he p ac i ione al hough
only wo ou o h ee pa ame e s ha e o be speci ied. Typically, q has o be cho-
sen a leas as la ge as he numbe o in o ma i e a iables in he da ase whe eas
he h eshold
𝜋 h
should be be ween (0.6,0.9) (Meinshausen and Bühlmann 2010).
Then, he PFER can be compu ed based on Eq.5. Un o una ely, he in o ma i e
a iables in a p ac ical se ing a e gene ally unknown. To ci cum en he p oblem, q
should be chosen in ui i ely by he p ac i ione . A e wa d, he idea is o look o a
b eak-poin in he dec easing o de o alues o he selec ion equencies
𝜋 j
. Finally,
a h eshold is chosen whe e all a iables a e included i he selec ion equencies
a e la ge han he b eak-poin (Thomas e al. 2018). Al e na i ely, i is possible o
se an uppe -bound o he PFER which hen is he expec ed alue o alse posi i e
base-lea ne s (Ho ne e al. 2015).
3 Simula ion s udy
To e alua e he pe o mance o ou implemen ed model-based g adien boos ing
app oach o Di ichle eg ession models, simula ion s udies o he i a ia e and
sep i a ia e Di ichle amilies wi h ing edien s as s a ed in Sec .2 a e conduc ed.
Speci ically, he ocus is on he e alua ion o he pe o mance ega ding es ima ion,
a iable selec ion and p edic ion. Al hough he implemen ed Di ichle amily can
handle any numbe o dis ibu ional pa ame e s, he choice o he sep i a ia e Di -
ichle amily in he simula ion s udy is mo i a ed based on he applica ion in he
case s udy in Sec .4. Since he unc ional o m o he a iables in he applica ion is
assumed o be s ic ly linea , nonlinea base-lea ne s a e e alua ed using he i a i-
a e Di ichle amily, which eases he compu a ional bu den in es ima ion. The lea n-
ing a e is se o alue o
𝜈=sl =0.1
because ha is he usual p ac ice (see, o
𝜋
j=1
B
B
∑
b=1
𝕀{j∈
Sb
}

Ss able
={j∶𝜋
j
≥𝜋
h
}
.
(5)
𝔼
(V)≤q
2
(2
𝜋
h −1)P
M.Balze e al.
example, Schmid and Ho ho n (2008); May e al. (2012b); Ho ne e al. (2014).
Rega ding he choice o he upda ing s a egy, he adjus ed noncyclical algo i hm
is u ilized due o he ad an ages s a ed in Sec .2. Ne e heless, we poin ou ha
he cyclical algo i hm can also be easily u ilized ins ead. The sea ch o he op imal
main uning pa ame e ec o
ms op
is conduc ed by minimizing he empi ical isk
ia 25- old subsampling in he linea se ing and ia an addi ional alida ion se wi h
N al =1000
obse a ions in he nonlinea se ing. In e e y simula ion se ing, a o al
o 100 epe i ions a e pe o med.
Rega ding he e alua ion o he a iable selec ion, he ocus is on he c i e ia
based on he con usion ma ix (S ehman 1997). Pa icula ly, he choice alls on he
ue posi i e a e (TPR), which is he p opo ion o co ec ly selec ed a iables ou
o all ue in o ma i e a iables, ue nega i e a e (TNR), which is he p opo ion
o co ec ly nonselec ed a iables ou o all ue non-in o ma i e a iables, he alse
disco e y a e (FDR), which is he p opo ion o non-in o ma i e a iables in he se
o all selec ed a iables, he posi i e p edic i e alue (PPV), which is he p opo ion
o co ec ly selec ed a iables ou o all selec ed a iables and he nega i e p edic-
i e alue (NPV), which is he co ec ly nonselec ed a iables ou o all nonselec ed
a iables.
Fu he mo e, o he e alua ion o he p edic i e pe o mance, an addi ional es
da ase wi h he same numbe o obse a ions as he ain da ase om he same Di -
ichle dis ibu ion is d awn. Rega ding he c i e ia o he p edic i e pe o mance,
he NLL which is calcula ed on he es da ase is u ilized. Fu he mo e, he ma -
ginal oo mean squa ed e o o p edic ion (RMSEP) de ined as
is addi ionally p o ided. Fo bo h NLL and RMSEP, lowe alues a e always
p e e ed.
The ML es ima ion o he Di ichle eg ession model in he low-dimensional lin-
ea se ing is pe o med ia he Di ichle Reg package based on heo e ical esul
and de i a ions discussed in Maie (2014). The lexible boos ing app oaches o
GAMLSS can be ound in he co esponding gamboos LSS package which is
ex ended o Di ichle eg ession models. An implemen a ion o any Di ichle dis-
ibu ion ia he co esponding Di ichle amily and he co esponding R code
o ep oducibili y o all simula ion s udies can be ound in he Gi Hub eposi o y
h ps:// gi hub. com/ micba lz/ Di ic hle R egBoo s .
3.1 S udy design
Fo he linea se ing, he numbe o obse a ions in he simula ed da ase s is ixed a
N=150
. In con as , he numbe o independen a iables is a ied be ween
P=10
and
P=300
which yields wo simula ion se ings. Pa icula ly, a low- (
N>P
) and
high-dimensional (
N<P
) linea se ing is conside ed in which he ue p edic o s in
he da a gene a ing p ocess a e chosen as
RMSEP
(Yk)=
√
√
√
√
1
N es
N es
∑
i
=1
(yki −yki)
2
G adien boos ing o Di ichle eg ession models
he ue unc ional o m. Fu he mo e, he p e iously no eached ex ema a e
now app oxima ed much be e . This sugges s ha applica ion se ings in ol ing
highly nonlinea unc ional e ec s o a iables ( o example, squa e- oo ans o -
ma ions) can be e ec i ely es ima ed using sui able base-lea ne s wi hin g adien
boos ing.
Fig. 4 Pa hs o he ue nonlinea e ec s ( ed) and es ima ed pa hs by he addi i e boos ing model o he
nonlinea se ing wi h N = 200, P = 10 and 100 epe i ions (colou igu e online)
Fig. 5 Pa hs o he ue nonlinea e ec s ( ed) and es ima ed pa hs by he addi i e boos ing model o he
nonlinea se ing wi h N = 1000, P = 10 and 100 epe i ions (colou igu e online)

M.Balze e al.
4 Case s udy: Changes inelec ion esul s in heG ea Recession
in heEu opean union
The G ea Recession las ing om he yea s 2007 o 2009 was accompanied wi h
se e e and de as a ing economic down u n in coun ies in he Eu opean Union
whe e mac oeconomic indica o s changed signi ican ly. Pa icula ly, he all in
agg ega e demand caused a ecession which in u n led o a dec eased ou pu and
hus a s agna ing economic g ow h in se e al coun ies a ound he wo ld. Fu he -
mo e, he ecession led o se e e inancial and so e eign deb c ises (Ball 2014).
Looking a he labo ma ke , he ecession caused a sha p inc ease in unemploymen
( an Ou s 2015). Sociologis s in e ed ha nega i e economic e ec s o in olun-
a ily unemploymen would spill o e o well-being and heal h o he indi iduals
(Bu ga d and Kalouso a 2015). Mac oeconomic heo y sugges s ha policymake s
can eac o hese exogenous shocks in o de o cushion hem. Al hough he go e n-
men s and cen al banks bailed ou indus ies, p o ided elie p og ams, in oduced
ax cu s and o he iscal s imuli, hey had o pay a p ice in doing so. Pa icula ly,
hese policies a e all connec ed o a sha p inc ease in public deb due o an inc ease
in go e nmen spending (Danzige 2013). As a esponse o hese policies and he
wo sening economic pe o mance, people ook i massi ely o he s ee s, oicing
hei economic and poli ical dissa is ac ion. (B eme e al. 2020; K iesi e al. 2020;
Po os and Ca alho 2022). In ou applica ion, we build on he s udy o B eme
e al. (2020) who aimed o show i changes in he economic condi ions in he G ea
Recession had an e ec on he p o es landscape. Fu he mo e, he au ho s elabo-
a ed i he in ensi y o hese economic e ec s we e as s ong as in he case o he
elec o al poli ics. Mo eo e , he aim was o es ablish o wha ex en p o es poli ics
had d i en he massi e elec o al punishmen a ha ime.
To his end, a la ge-scale eal-wo ld Eu opean da ase is u ilized, combining elec-
o al esul s wi h he p o es e en da a om he Eu opean Obse a o y o Poli ical
Con lic and Democ acy (PolDem) (K iesi and e al.. 2020). Al hough B eme e al.
(2020) al eady es ablished ha economic p o es s ha e an e ec on elec ions using
he da ase , he au ho s ocus mainly on elec o al loss o incumben s, ha is, he di -
e ence in he o ing p opo ion o he p ime minis e ’s pa y a ime and
+1
as
hei esponse a iable. In pa icula , hey showed ha poo economic pe o mance
led o elec o al losses and an inc ease in p o es s using o dina y leas squa es es i-
ma ion. In his case, incumben s we e s onge punished when he numbe o p o-
es s was la ge (B eme e al. 2020). To ge a mo e di e en ia ed pic u e on how
he G ea Recession has a ec ed he s uc u e o Eu ope’s pa y sys ems, i is also
possible o es ablish he e ec o independen a iables on o ing p opo ions o all
pa y amilies simul aneously. We can iew he o ing p opo ions as he dependen
a iables and enable he possibili y o wo k in he amewo k o Di ichle eg ession
models. This app oach is much mo e lexible since indi idual changes in pa y ami-
lies can be obse ed and i allows o an in e p e a ion o ma ginal e ec s o p o es s
on indi idual pa ies pe o mance (Maie 2014).
In o de o main ain compa abili y be ween he esul s o B eme e al. (2020)
and ou analysis, we u ilize he exac same da ase . The o iginal da ase consis s
G adien boos ing o Di ichle eg ession models
o 900 obse a ions, whe e each obse a ion co esponds o he o ing p opo -
ion o an indi idual pa y in an elec ion. Mo eo e , 57 a iables a e addi ion-
ally included which a e in o ma ion on o ing p opo ions, numbe o p o es s,
numbe o pa icipan s in he p o es e en s, pa ies, deb change, g oss domes ic
p oduc (GDP) change, unemploymen change as well as dummy a iables ha
indica e i he coun y is in Eas e n Eu ope, was bailed ou o was in he G ea
Recession. Al hough a di e se ange o a iables can be u ilized, he lis ed a i-
ables a e o main in e es in ou analysis. In Tables4and 5, he desc ip ion o
dependen and independen a iables can be obse ed.
Du ing he examina ion pe iod las ing om he yea s 2000 un il he end o 2015,
we ha e access o 117 elec ions. In o de o main ain a dis inc empo al ela ionship
o he G ea Recession, he elec ions a e classi ied as ollows: Elec ions be o e Oc o-
be 2008 a e classi ied as p ec isis elec ions, he i s elec ion a e Oc obe 2008—
when he US-Ame ican bank Lehman B o he s collapsed—as i s -c isis elec ion
and all elec ions a e wa d as la e -c isis elec ions. The da ase includes all pa ies in
30 coun ies1 ha ha e ob ained a leas h ee pe cen in o ing p opo ions and ha e
a leas one sea in he espec i e coun ies pa liamen .
Since he o iginal da ase is on he le el o indi idual pa ies pe elec ion, agg e-
ga ion and combina ion o pa ies in o pa y amilies a e necessa y o ob ain he
desi ed da a. Pa icula ly, he pa ies a e classi ied in o dis inc pa y amilies on
which he analysis is conduc ed. The classi ica ion o indi idual pa ies in o pa y
amilies leads o a disc epancy in he obse a ions o he o iginal da ase and ou
inal da ase since in some coun ies mo e han one indi idual pa y can be assigned
o he exac same pa y amily o which he p opo ions a e summed up. Rega d-
ing he classi ica ion o he pa ies in di e en coun ies, we ollow he p oposed
Pa lGo classi ica ion by Dö ing and Manow (2019), allowing us o di e en ia e he
majo Eu opean pa ies amilies om bo h igh and le . This limi s he esul s o
his pa icula de ini ion o he pa y amilies. Following a di e en de ini ion and
classi ica ion o he pa y amilies leads o a change in agg ega ion o he pa ies
in di e en coun ies and hus o po en ially a ying esul s in he analysis. We do
no omi any elec ion and hus conside a holis ic app oach. Fu he p ep ocessing
is also necessa y o ob ain a da ase which is sui able o he analysis. Pa icula ly,
o nonexis en pa y amilies in elec ions, he o ing p opo ions a e assigned o
ze o. Fu he mo e, o ing p opo ions a e o iginally in pe cen which a e di ided by
100 o ob ain a p opo ional scale. Finally, ow sums do no pe ec ly sum up o one
such ha we ollow he no maliza ion and ans o ma ion sugges ed by Smi hson
and Ve kuilen (2006) and gene alized by Maie (2014) which is de ined as
y
∗
i=
yi(N−1)+
1
K
N
.
1 Aus ia, Belgium, Bulga ia, Cyp us, Czech Republic, Denma k, Es onia, Finland, F ance, Ge many,
G eece, Hunga y, Iceland, I eland, I aly, La ia, Li huania, Luxembou g, Mal a, No way, Ne he lands,
Poland, Po ugal, Romania, Slo akia, Slo enia, Spain, Sweden, Swi ze land and he UK.
M.Balze e al.
Table 4 Da a ype and desc ip ion o dependen and independen a iables
Va iable Type Desc ip ion
Vo e Composi ional Vo ing p opo ion o pa y amilies
P o es s Nume ic Numbe o p o es e en s in he legisla i e pe iod be o e he elec ion scaled by he du a ion o he
legisla i e pe iod
Unemploymen Nume ic Change in he qua e ly unemploymen a e (in pe cen ) be ween elec ions
Deb Nume ic Change in Go e nmen deb be ween elec ions
GDP Nume ic Change in annual GDP (in pe cen ) be ween elec ions
C isis Bina y Dummy a iable ha measu es whe he he elec ion ook place du ing he G ea Recession
Bailou Bina y Dummy a iable ha measu es whe he a coun y was bailed ou by he In e na ional Mone a y Fund o
he Eu opean Union
Eas Bina y Dummy a iable ha measu es whe he a coun y is in Eas e n Eu ope
G adien boos ing o Di ichle eg ession models
In gene al, he esul s a e p esen ed wi h a ocus on he es ima ion ia model-based
g adien boos ing. Since B eme e al. (2020) wo ked in a s ic ly linea se ing in he
o dina y leas squa e es ima ion, simple linea models a e u ilized as base-lea ne s
o compa abili y. Fu he mo e, all p esen ed a iables in Table4 a e included in he
analysis and addi ionally an in e cep o e e y dis ibu ional pa ame e . Rega ding he
lea ning a e, he na a i e in he simula ion s udy in Sec .3.1 is ollowed and a s ep-
leng h o 0.1 is chosen. As o he op imiza ion, he sea ch o he
ms op
is conduc ed
by minimizing he empi ical isk ia 25- old subsampling. A e ob aining he op imal
s opping c i e ion, he ollowing esul s which can be seen in Table6 a e ex ac ed.
I can be obse ed ha e e y
𝛼i
depends on di e en a iables, e en hough
he model equa ions o en sha e he same a iables. Pa icula ly, impo an o
he analysis is he ac ha he P o es s a iable is almos always included excep
in he case o
𝛼1
, ha is, he social democ a s. In con as o he esul s p esen ed
in B eme e al. (2020), he dis ibu ional pa ame e s in he Di ichle eg ession
amewo k
𝛼i
depend on socioeconomic a iables, namely
GDP,Unemploymen
and Deb . Thus, he esul s p esen ed in his case s udy unde line he di e ence
in he es ima ion amewo k wi h addi ional a iable selec ion echnique. Un o -
una ely, he model equa ions imply p ac ical es ic ions wi h ega d o he in e -
p e a ion o he e ec s o ce e is pa ibus changes in a iables. Pa icula ly, since
Table 5 Desc ip i e s a is ics
o o ing p opo ions o pa y
amilies be o e no maliza ion
and ans o ma ion
Pa y amily Ze o coun Median In e qua ile ange
𝛼1
Social Democ a ic 0 0.268 0.156
𝛼2
Conse a i e 1 0.343 0.2419
𝛼3
Libe als 29 0.098 0.221
𝛼4
G eens 66 0 0.073
𝛼5
Radical le 45 0.0399 0.0965
𝛼6
Radical igh 49 0.0437 0.139
𝛼7
O he s 5 0.0663 0.07
Table 6 Es ima ion esul s ia model-based g adien boos ing o Di ichle eg ession models in he case
s udy
𝛼1
𝛼2
𝛼3
𝛼4
𝛼5
𝛼6
𝛼7
In e cep 0.84 0.78 − 0.22 − 0.65 − 0.59 − 0.67 − 0.24
P o es s 0.011 − 0.010 − 0.032 0.029 0.0073 0.026
Bailou − 0.24
C isis − 0.049 0.024 0.1
Unemploymen − 0.0069 0.0029
Deb − 0.008 0.0039 − 0.0023
GDP 0.0062
Eas 0.2 − 0.439 − 0.484 0.084
M.Balze e al.
he o ing p opo ions a e no di ec ly modeled, he e ec o changes in he a i-
ables can only be in e p e ed on he dis ibu ional pa ame e s
𝛼i
(Maie 2014).
Gi en he esul s o he boos ed Di ichle eg ession model, he in e -
p e a ion o he coe icien s is now s aigh o wa d. In gene al, low coe -
icien alues o mos o he a iables o all dis ibu ional pa ame e s a e
obse ed. Rega ding he absolu e magni ude, highe coe icien s can be
obse ed o he a iables Bailou and Eas , whe eas socioeconomic a i-
ables ha e a he low posi i e o nega i e coe icien alues. Fo ins ance, i
P o es s inc eases by one scaled e en ce e is pa ibus hen he
𝛼2
inc eases by
100
⋅(exp(𝛽
(𝛼
2
)
P o es s
−1)=100 ⋅(exp(0.011 −1)≈
1.106
pe cen on a e age. Simi-
la in e p e a ions can be made o e e y dis ibu ional pa ame e . Thus, wi h an
inc easing numbe o economic p o es s he dis ibu ional pa ame e
𝛼2
inc eases
in absolu e alue. In con as ,
𝛼3
will dec ease due o he nega i e sign o he
coe icien when he numbe o economic p o es s inc eases. Al hough implica-
ions can be ob ained h ough his in e p e a ion, ha is, he expec a ion o he
p opo ions o he g eens will dec ease wi h an inc easing numbe o p o es s, i
would be con enien o ob ain a me hod o ge a gene al o e iew o changes in
p opo ions o all pa y amilies di ec ly.
An al e na i e app oach elies on gene a ing a es da ase o eligible alues o
P o es s o any o he a iable o in e es . A e wa d, ma ginal p edic ions o dis-
ibu ional pa ame e s
𝛼i
a e compu ed. The gene al idea o he compu a ion o
he ma ginal e ec s is p o ided by he gamboos LSS package whe e he e ec s
o a co esponding base-lea ne o a a iable o in e es a e e u ned while he
o he con inuous a iables a e ixed a hei mean and ca ego ical a iables a
hei mode. U ilizing he exp ession gi en by Eq.1, he ma ginal e ec s o he
a iables on he expec a ions o he p opo ions o e e y obse a ion in he es
da ase can be quan i ied.
In Fig.6, he p og ess o he expec a ions o he o ing p opo ions o inc eas-
ing numbe o economic p o es s can be obse ed. Pa icula ly, he expec a ions o
he g eens and he libe als a e declining almos linea ly and s ongly i he numbe
o economic p o es s inc eases. Thus, he model sugges s ha bo h pa y amilies
ha e no p o i ed om ci izens’ oicing hei economic g ie ances in he s ee s. In
con as , he adical le in pa icula bu also o he pa ies a e gaining in expec ed
p opo ions as he numbe o p o es s inc eases. The inding unde lines he eme -
gence o an i-aus e i y pa ies om he le du ing he G ea Recession (DellaPo a
e al. 2017). Howe e , he ex en o economic p o es s has ha dly any e ec on he
elec o al pe o mance o he conse a i e and he adical igh . Addi ionally, a nega-
i e e ec is obse able in he expec ed p opo ions o he social democ a s. This is
su p ising since he coe icien o he dis ibu ional pa ame e is nonexis en such
ha he ma ginal p edic ion is always cons an . The beha io can be bes explained
by looking a Eq.1. Thus, i
𝛼1
emains cons an and
𝛼0
inc eases wi h he numbe o
economic p o es s, he expec ed p opo ions o social democ a s dec eases.
To p o ide an in-dep h analysis o he ma ginal e ec o socioeconomic a i-
ables, g aphical esul s o Deb and Unemploymen a e addi ionally p esen ed.
Based on Fig.7, we concu ha libe als lose in expec ed o ing p opo ions i

G adien boos ing o Di ichle eg ession models
he pe cen change in deb is posi i e and high, whe eas he adical le s a e win-
ning in expec ed p opo ions wi h an inc ease in deb change. Rega ding o he
pa ies, such a conclusi e esul canno be gi en. Pa icula ly, he expec ed p o-
po ions inc ease sligh ly o social democ a s, conse a i es, g eens and o he s
wi h inc easing pe cen age change in deb . Fo he adical igh , he expec ed p o-
po ion sligh ly dec eases. The e o e, he in e p e a ion o ma ginal e ec s has
o be pe o med ca e ully i only a ew dis ibu ional pa ame e s
𝛼i
depend on a
a iable o in e es . Since o ou pa ame e s, Deb is no included in he model
Fig. 6 Ma ginal e ec o he numbe o economic p o es s on he expec a ions o he o ing p opo ions
ob ained ia he boos ed Di ichle eg ession model
Fig. 7 Ma ginal e ec o deb change in pe cen on he expec a ions o he o ing p opo ions ob ained
ia boos ed Di ichle eg ession model
M.Balze e al.
equa ions, he ma ginal p edic ions emain cons an ega dless o he magni ude
in deb change. In u n, his beha io ampli ies he consequences o Eq.1 since
he absolu e inc ease in
𝛼0
emains o e all low.
A simila beha io can be obse ed in he ma ginal e ec s o unemploymen change
whe e e en less dis ibu ional pa ame e s depend on he unemploymen change a iable.
In Fig.8, he expec ed p opo ions o almos all pa y amilies emain cons an . Small
ma ginal e ec s can be obse ed in he expec ed o ing p opo ions o he adical igh s
and o he s. These e ec s can be mainly aced back o he co esponding dis ibu ional
pa ame e s since hese depend on he unemploymen change a iable. Pa icula ly, adi-
cal igh s lose whe eas o he pa ies gain in expec ed p opo ions as he pe cen change
in unemploymen inc eases. Al hough he model indica es a shi o o e s owa d o he
pa ies, he magni ude o change in expec ed p opo ions emains gene ally low.
Summing up he ob ained esul s, ollowing conclusion can be d awn om he
boos ed Di ichle eg ession model: Fi s , dis inc changes in o ing beha io o ci i-
zens can be obse ed compa ed o p e ecession elec ions. Pa icula ly, he model sug-
ges s ha an inc ease in he numbe o economic p o es s is accompanied by change
o expec ed o ing p opo ions o pa y amilies. Speci ically, people a e u ning hei
backs on g een and libe al pa ies when he numbe o economic p o es s is high,
whe eas adical le and o he pa ies a e p o i ing elec o ally unde hese condi ions.
Mo eo e , a shi o he igh in o ing beha io o di e en alues o economic p o-
es s canno be obse ed, implying ha adical igh s and conse a i es ail o p o ide
a ac i e al e na i es o si ua ions whe e he numbe o economic p o es s is high.
Second, he ma ginal e ec s o deb and unemploymen change a e conside ably low
o e all expec ed o ing p opo ions. Ne e heless, libe als end o lose in expec ed
p opo ions indica ing a shi o o e s o adical le pa ies i pe cen change in deb
inc eases. Once again adical le s seem o p o ide desi ed al e na i es o an inc eas-
ing deb bu den in he Eu opean Union. Since s ong posi i e inc eases in absolu e deb
we e obse ed in he G ea Recession, his esul implies ha people a e becoming
Fig. 8 Ma ginal e ec o unemploymen change in pe cen on he expec a ions o he o ing p opo ions
ob ained ia he boos ed Di ichle eg ession model
G adien boos ing o Di ichle eg ession models
inc easingly le -leaning when aced wi h such eali ies. Likewise, high numbe s o
unemploymen we e obse ed in he G ea Recession. Al hough he e ec o unem-
ploymen is o e all low, o e s om adical igh pa ies end o de lec o o he pa -
ies in such si ua ions. Mo e insigh in o he ma ginal e ec s o o he socioeconomic
a iables eco e ed om he boos ed Di ichle eg ession model can be obse ed in
Appendix 4.1. In he simula ion s udy, we obse e ha he g adien boos ing algo-
i hm exhibi s high FDR alues when elying solely on c oss- alida ion o de e mine
he op imal s opping c i e ion. To examine how combining g adien boos ing wi h s a-
bili y selec ion a ec s pe o mance in he case s udy, we p esen addi ional insigh s in
Appendix 4.2. The inal model sugges ed by he chosen pa ame e combina ion u he
emphasizes he impo ance o he numbe o p o es s, while o he a iables wi h less
impac such as unemploymen do no each he equi ed s abili y h eshold. Ne e he-
less, i should be no ed ha , as wi h s a is ical hypo hesis es s, he noninclusion o a
a iable in he model should no be aken as e idence ha he e is no ma ginal e ec
o his a iable on he change in he expec ed o ing p opo ions a all.
5 Discussion
Ou key indings and he main con ibu ions o his pape a e: (a) G adien boos -
ing can be ex ended o Di ichle eg ession by an adjus men o he algo i hms by
May e al. (2012a) and Thomas e al. (2018). (b) The implemen a ion is pe o med
by ex ending he package o boos ing GAMLSS called gamboos LSS (Ho ne e al.
2023). The ex ension is implemen ed by w i ing a new Di ichle amily which can
be speci ied in he es ima ion p ocedu e. (c) In he simula ion s udies, he es ima-
ion p ocedu e is accompanied wi h high TPR alues in he low-dimensional linea
se ing. The ue coe icien s a e es ima ed wi h a high deg ee o accu acy and he
p edic i e pe o mance is gene ally be e in he g adien boos ing app oach han in
he ML Di ichle eg ession. Fo he high-dimensional linea se ing, he g adien
boos ing es ima ion p ocedu e is also easible. Fu he mo e, nonlinea complex base-
lea ne s can also be u ilized in applica ion se ings since hey es ima e he ue o m
o nonlinea e ec s qui e well. (d) In summa y, p ac i ione s and applied s a is icians
ha e an al e na i e app oach o lexible es ima ion o Di ichle eg ession models.
This allows o da a-d i en a iable selec ion which leads o spa se and much mo e
in e p e able models. Al hough ou analysis ocuses on he i a ia e and sep i a ia e
Di ichle dis ibu ion, any Di ichle dis ibu ion is implemen ed, he eby allowing o
a po en ial applica ion in eal-wo ld se ings wi h an a bi a ily numbe o ca ego ies
and hus dis ibu ional pa ame e s. (e) Addi ionally, he new implemen ed me hod is
applied in a eal-wo ld applica ion se ing. Ou p oposed es ima ion p ocedu e allows
o a mo e di e en ia ed iew on he elec o al consequences o he G ea Recession.
Howe e , we wan o indica e ha u he esea ch is necessa y in his applica ion
se ing and we only iewed a limi ing case. Pa icula ly, di e en economic ques ions
can be in es iga ed using he da ase and g adien boos ing o Di ichle eg ession.
Na u ally, he e a e po en ial imp o emen s and ex ensions beyond he scope
o his pape . Al hough simula ion s udies ha e been conduc ed in low- and high-
dimensional se ings as well as o he es ima ion nonlinea e ec s, he scope o
M.Balze e al.
ou se ings is o cou se by no means exhaus i e. Fu he in e es ing scena ios may
include co ela ed noise a iables o di e en ypes o base-lea ne s, o example,
spa ial o andom e ec s. Mo eo e , he implemen a ion o he g adien boos ing
algo i hm is based on he so-called common pa ame iza ion o he Di ichle dis i-
bu ion and, as a consequence, he e ec o a iables on he expec a ions o depend-
en p opo ions canno be in e p e ed di ec ly (Maie 2014). This becomes pa icu-
la ly isible in he applica ion. Al hough we could conclude some esul s due o
ma ginal e ec s, a di ec quan i iable alue o any change in p o es s on he expec-
a ion o o ing p opo ions o di e en pa y amilies is s ill missing.
Looking a he a e age selec ion a es, ou algo i hm e eals a ela i ely high FDR,
indica ing ha many non-in o ma i e a iables wi h mino impo ance a e selec ed.
The FDR is e en highe when he op imal s opping c i e ion is ound ia en old c oss-
alida ion ins ead o 25- old subsampling as can be seen in appendix 3.1.1, 3.1.2 and
3.1.3. The e o e, we conduc ed addi ional simula ions o assess he applicabili y o s a-
bili y selec ion which allows con ol o e he amoun o alse posi i e selec ions (Mein-
shausen and Bühlmann 2010; Ho ne e al. 2015). Using pe mu a ion echniques also
allows he compu a ion o p alues o indi idual base-lea ne s wi hou he e o con-
ol p o ided by s abili y selec ion Hepp e al. (2019). Al e na i ely, implemen ing a
ecen ly p oposed s a egy o deselec base-lea ne s wi h low impac in o de o ob ain
spa se models can be conside ed (S öme e al. 2022). Fu he mo e, he po en ially
inc easing numbe o dis ibu ional pa ame e s due o a ying Di ichle dis ibu ions
leads o an inc eased demand in compu a ional esou ces o unning he algo i hms.
The e o e, al e na i es o ea ly s opping could be conside ed. P obing is al eady such
an app oach o ea ly s opping o uni a ia e loca ion models, whe e he algo i hm is
s opped when he i s andomly shu led e sion o he obse ed a iables is added o
he da ase , he eby also esul ing in ela i ely spa se solu ions (Thomas e al. 2017).
An impo an aspec no ed in boos ing algo i hms o dis ibu ional eg ession
models is ha using he same ixed s ep-leng h o e e y p edic o unc ion may
esul in imbalances ha p e en ce ain pa ame e s om being upda ed. Sugges ed
s a egies in ol e scaling he ou come a iables o he nega i e g adien ec o
(Ho ne e al. 2016; S öme e al. 2023 o using adap i e s ep-leng hs Zhang e al.
(2022). Howe e , since he scale o he g adien s o all
𝛼i
in he Di ichle eg ession
amewo k emains iden ical ega dless o i, hese issues should be o mino conce n
o he implemen a ion discussed in his a icle.
Thus, he nex goal would be o implemen a model-based g adien boos ing algo-
i hm o he al e na i e pa ame iza ion o he Di ichle dis ibu ion. This app oach
enables he di ec modeling o expec ed o ing p opo ions using a logi link unc-
ion (Fe a i and C iba i-Ne o 2004; Maie 2014). Rega ding he case s udy, u he
esea ch is needed o examine he d i e s o o ing p opo ions du ing c ises in coun-
ies ou side he Eu opean Union. Addi ionally, modeling andom e ec s emains a
key a ea o in e es . An upda ed da ase co e ing mo e ecen ecessions could p o-
ide new insigh s in o p o es e ec s, allowing o meaning ul compa isons wi h p e-
ious indings. Mo eo e , econside ing de ini ions o pa y amily classi ica ions
could e ine he analysis. Beyond he sociology con ex , he newly implemen ed g adi-
en boos ing me hod o Di ichle eg ession models has b oad applicabili y in ields
such as economics, medicine and ecology. We encou age p ac i ione s and applied
G adien boos ing o Di ichle eg ession models
Table 12 Resul s o he a e age
selec ion a es in he high-
dimensional linea se ing (N =
150, P = 300) o 50 epe i ions
and i a ia e Di ichle amily
Repo ed a e he ue posi i e (TPR) alse disco e y (FDR), ue
nega i e a e (TNR), posi i e p edic i e (PPV) and nega i e p edic-
i e alue (NPV)
TPR (%) TNR (%) FDR (%) PPV (%) NPV (%)
𝛼1
100 94.53 83.41 16.59 100
𝛼2
100 94.49 84.16 15.84 100
𝛼3
99.33 96.07 77.47 22.53 99.99
Table 13 Resul s o he
p edic i e pe o mance on
independen es da ase in
he high-dimensional linea
se ing (N = 150, P = 300) o
50 epe i ions and i a ia e
Di ichle amily
Repo ed a e he oo mean squa ed e o o p edic ion (RMSEP)
and nega i e log-likelihood (NLL)
Boos ing Di ichle
eg es-
sion
RMSEP(
Y1
) 0.015135604 –
RMSEP(
Y2
) 0.010652941 –
RMSEP(
Y3
) 0.009215892 –
NLL − 408.3796 –
Fig. 11 Resul s o he es ima ed linea e ec s o he i s en a iables in he high-dimensional linea se -
ing (N = 150, P = 300) wi h i a ia e Di ichle amily o 50 epe i ions. Ho izon al ed lines ep esen
he ue alues (colou igu e online)

M.Balze e al.
(A) Balanced case: E e y
𝛼i
depends on wo in o ma i e a iables bu all o hem
a e sha ed. In o al, we ha e
7⋅2=14
in o ma i e a iables. This leads o
(B) Unbalanced case: E e y
𝛼i
depends on wo in o ma i e a iables bu none o
hem a e sha ed. In o al, we ha e
7⋅2=14
in o ma i e a iables. This co -
esponds o:
log
(𝛼1)=𝛽
(1)
1X1−𝛽
(1)
2X
2
log
(𝛼2)=𝛽(2)
1X1+𝛽(2)
2X
2
log
(𝛼3)=𝛽(3)
1X1−𝛽(3)
2X
2
log
(𝛼4)=𝛽(4)
1X1−𝛽(4)
2X
2
log
(𝛼5)=𝛽(5)
1X1+𝛽(5)
2X
2
log
(𝛼6)=𝛽(6)
1X1−𝛽(6)
2X
2
log
(𝛼
7
)=𝛽(7)
1
X
1
+𝛽(7)
2
X
2
Fig. 12 Resul s o he in es iga ion o he egula iza ion beha io wi h changing signs o he ue coe -
icien s in he i a ia e Di ichle amily o
𝐦s op
=
𝟏𝟓𝟎
. Red line ep esen s he cen al coe icien s o
in e es (colou igu e online)
G adien boos ing o Di ichle eg ession models
Table 14 Resul s o he a e age
selec ion a es in he low-
dimensional linea se ing (N =
150, P = 49) o 50 epe i ions
and sep i a ia e Di ichle amily
Repo ed a e he ue posi i e (TPR) alse disco e y (FDR), ue
nega i e a e (TNR), posi i e p edic i e (PPV) and nega i e p edic-
i e alue (NPV)
TPR (%) TNR (%) FDR (%) PPV (%) NPV (%)
𝛼1
100 81.61 72.56 27.44 100
𝛼2
100 95.26 37.14 62.86 100
𝛼3
94.00 94.39 42.03 57.97 99.60
𝛼4
100 95.57 34.30 65.70 100
𝛼5
95.33 93.39 47.00 53.00 99.68
𝛼6
92.67 96.87 28.68 71.32 99.52
𝛼7
100 81.00 72.81 27.19 100
Table 15 Resul s o he
p edic i e pe o mance on
independen es da ase in
he low-dimensional linea
se ing (N = 150, P = 49) o
50 epe i ions and sep i a ai e
Di ichle amily
Repo ed a e he oo mean squa ed e o o p edic ion (RMSEP)
and nega i e log-likelihood (NLL)
Boos ing Di ichle eg ession
RMSEP(
Y1
) 0.06856661 0.09017981
RMSEP(
Y2
) 0.04404223 0.07814164
RMSEP(
Y3
) 0.03376675 0.04628070
RMSEP(
Y4
) 0.03275246 0.05141709
RMSEP(
Y5
) 0.04976701 0.07501433
RMSEP(
Y6
) 0.02130379 0.04965494
RMSEP(
Y7
) 0.07476324 0.09988373
NLL − 3385.321 83745487845
Fig. 13 Resul s o he es ima ed linea e ec s o he (N = 150, P = 49) linea se ing o 50 epe i ions
and sep i a ia e Di ichle amily. Ho izon al ed lines ep esen he ue alues (colou igu e online)
M.Balze e al.
All a iables a e independen ly d awn om he uni o m dis ibu ion U(0,1). The
c i e ia o he pe o mance e alua ion emain iden ical o he main ex . Fo he
h eshold
𝜋 h
, a ying alues be ween 0.55 and 0.99 in 0.01 s eps a e conside ed.
Fo he maximum numbe o included base-lea ne s, h ee di e en alues, namely
q∈{15, 25, 50}
a e chosen. Finally, he numbe o noise a iables is a ied be ween
P=50, 100
and 150.
S a ing wi h he balanced case (Fig. 14), esul s a e epo ed o he di e en
combina ions o P and q. Gene ally, he esul s sugges ha he numbe o TP is
high i he h eshold is low and g adually dec eases as he h eshold inc eases.
Un o una ely, he numbe o in o ma i e a iables o 14 is no a ained in any se -
ing. The closes case achie ing a numbe o TP o 13 is
q=50
and
P=50
which
is a he unlikely in p ac ical applica ions. Howe e , he numbe o FP is low o ou -
igh ze o o mos o he conside ed simula ions. Fo ins ance, only o an inc eas-
ing numbe o noise a iables P in combina ion wi h a huge numbe o base-lea ne s
q as well a low alues o
𝜋 h
, he numbe o FP achie es a maximum alue o wo.
Thus, he se ing a aining he highes numbe o FP is
P=150
and
q=50
which
is a high-dimensional case in combina ion wi h a low h eshold. A e wa d, wi h an
inc easing alue o
𝜋 h
he numbe o FP eaches ze o s a ing om a h eshold o
𝜋 h =0.7
.
The esul s o he unbalanced case (Fig.15) a e gene ally in line wi h he esul s
o he balanced case, al hough small changes can be no ed. Pa icula ly, he numbe
o TP is highe ac oss e e y simula ion se ing compa ed o he balanced case. Fu -
he mo e, e en wi h an inc easing h eshold
𝜋 h
, he numbe o TP dec eases slowly,
some imes e en emaining cons an (see, o example,
q=50
and
P=15
). Addi-
ionally, he closes case achie ing a numbe o TP o 13 is again
q=50
and
P=50
.
Rega ding he FP, he esul s sugges o e all low o ou igh ze o inclusion o addi-
ional non-in o ma i e a iable. Pa icula ly, e en o la ge alue o noise a iables
and base-lea ne s in combina ion wi h low alues o he h eshold, he FP emains
ze o. Only o he edge case o
q=50
and
P=50
, he maximum numbe o FP o
wo is a ained.
log
(𝛼1)=𝛽
(1)
1X1−𝛽
(1)
2X2
log
(𝛼2)=𝛽(2)
1X3+𝛽(2)
2X4
log
(𝛼3)=𝛽(3)
1X5−𝛽(3)
2X6
log
(𝛼4)=𝛽(4)
1X7−𝛽(4)
2X8
log
(𝛼5)=𝛽(5)
1X9+𝛽(5)
2X10
log
(𝛼6)=𝛽(6)
1X11 −𝛽(6)
2X
12
log
(𝛼
7
)=𝛽(7)
1
X
13
+𝛽(7)
2
X
14
G adien boos ing o Di ichle eg ession models
Appendix4: Fu he esul s in hecase s udy
Appendix4.1: Ma ginal e ec s o socioeconomic a iables
See Figs.16, 17, 18 and 19.
Appendix4.2: Coe icien s andselec ion equencies ins abili y selec ion
In he main ex , we p esen he esul s o he applica ion se ing whe e he op i-
mal s opping is ound ia 25- old subsampling. Howe e , as e idenced in Sec .3.2.1
model-based g adien boos ing wi h he s opping c i e ion ound by means o c oss-
alida ion yields a high FDR. To imp o e he FDR, we p opose o u ilize s abil-
i y selec ion which shows conside able imp o emen when applied o he a i icial
da a in he simula ion s udy. Consequen ly, he esul s in he applica ion migh also
include many non-in o ma i e a iables. The e o e, we in es iga e he esul s o he
case s udy unde conside a ion o s abili y selec ion. To his end, we se he maxi-
mum numbe o base-lea ne s
q=40
and he
PFER =5
. Howe e , since he speci ic
choice o he h ee pa ame e s q,
𝜋 h
and PFER is no i ial and depends on he
p ac i ione , explo ing di e en combina ions is always an op ion. The maximum
selec ion equencies o he i s 15 a iables o he speci ied pa ame e s can be
seen in Fig.20.
In gene al, he esul s indica e ha he numbe o chosen a iables has dec eased
subs an ially in compa ison wi h he model ob ained by c oss- alida ion. Pa icu-
la ly, he P o es s a iable is s ill p esen in he inal model o he conse a i es,
g eens, adical le s and o he s, whe eas mos o he socioeconomic a iables ha e
been d opped. Ne e heless, he GDP a iable o conse a i es s ill emains in he
model. Fu he mo e, he bina y dummy a iable Eas is included o g eens and ad-
ical Le s. The upda ed esul s o he coe icien s can be now seen in Table16.
Fig. 14 S abili y selec ion in he balanced Case (Scena io A) wi h sep i a ia e Di ichle amily. Black
solid lines ep esen he p og ess o he numbe o ue posi i e a iables (TP). Blue dashed lines ep e-
sen he p og ess o he numbe o alse posi i e a iables (FP) (colou igu e online)
M.Balze e al.
In p inciple, hese esul s a e ai ly unsu p ising because he p esen ed ma ginal
e ec s al eady hin owa d a iables which ha e been emo ed by s abili y selec ion.
Pa icula ly, he le o e coe icien s o P o es s a e exhibi ing la ge changes in he
o ing p opo ions as e idenced by he ma ginal e ec s in Fig.6. Addi ionally, only
he conse a i es exhibi subs an ial inc eases in o ing p opo ions as he pe cen
change o GDP inc eases in Fig.16. Finally, he la ges changes in ma ginal e ec s
o he Eas a iable a e seen in g eens and adical le s as indica ed by Fig.19.
Fig. 15 S abili y selec ion in he unbalanced case (Scena io B) wi h sep i a ia e Di ichle amily. Black
solid lines ep esen he p og ess o he numbe o ue posi i e a iables (TP). Blue dashed lines ep e-
sen he p og ess o he numbe o alse posi i e a iables (FP) (colou igu e online)
Fig. 16 Ma ginal e ec o GDP change in pe cen on he expec a ions o he o ing p opo ions ob ained
ia he boos ed Di ichle eg ession model

G adien boos ing o Di ichle eg ession models
Fig. 17 Ma ginal e ec o c isis elec ions on he expec a ions o he o ing p opo ions ob ained ia he
boos ed Di ichle eg ession model
Fig. 18 Ma ginal e ec o bailou s on he expec a ions o he o ing p opo ions ob ained ia he boos ed
Di ichle eg ession model
M.Balze e al.
Fig. 19 Ma ginal e ec i coun ies a e in Eas e n Eu ope on he expec a ions o he o ing p opo ions
ob ained ia he boos ed Di ichle eg ession model
Fig. 20 Maximum selec ion equencies o he i s 15 a iables a e s abili y selec ion unde pa ame e
choices o
q
=
40
and
PFER =5
G adien boos ing o Di ichle eg ession models
Acknowledgemen s We g ea ly hank Benjamin Ho ne o his suppo in in eg a ing ou algo i hms
in o he gamboos LSS package. The wo k on his a icle was suppo ed by he Volkswagen Founda ion
(F eigeis Fellowship) and Deu sche Fo schungsgemeinscha (DFG, Ge man Resea ch Founda ion)
wi hin p ojec s 51701299 and 492988838.
Funding Open Access unding enabled and o ganized by P ojek DEAL.
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s abili y selec ion o Di ichle
eg ession models in he case
s udy
𝛼1
𝛼2
𝛼3
𝛼4
𝛼5
𝛼6
𝛼7
In e cep 0.84 0.78 − 0.65 − 0.59 − 0.67
P o es s 0.011 − 0.032 0.029 0.026
GDP 0.0062
Eas − 0.439 − 0.484
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