Eu hum, Maximilian; Sche e , Ma hias; Ungolo, F ancesco
A icle — Published Ve sion
A neu al ne wo k app oach o he mo ali y analysis o
mul iple popula ions: a case s udy on da a o he I alian
popula ion
Eu opean Ac ua ial Jou nal
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Sugges ed Ci a ion: Eu hum, Maximilian; Sche e , Ma hias; Ungolo, F ancesco (2024) : A neu al
ne wo k app oach o he mo ali y analysis o mul iple popula ions: a case s udy on da a o he
I alian popula ion, Eu opean Ac ua ial Jou nal, ISSN 2190-9741, Sp inge , Be lin, Heidelbe g, Vol.
14, Iss. 2, pp. 495-524,
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1 3
CASE STUDY
A neu al ne wo k app oach o hemo ali y analysis
o mul iple popula ions: acase s udy onda a o heI alian
popula ion
MaximilianEu hum1,2· Ma hiasSche e 1 · F ancescoUngolo1,3,4
Recei ed: 27 Feb ua y 2023 / Re ised: 14 July 2023 / Accep ed: 19 Decembe 2023 /
Published online: 6 Ma ch 2024
© The Au ho (s) 2024
Abs ac
A Neu al Ne wo k (NN) app oach o he modelling o mo ali y a es in a mul i-
popula ion amewo k is compa ed o h ee classical mo ali y models. The NN
se up con ains wo ins ances o Recu en NNs, including Long Sho -Te m Mem-
o y (LSTM) and Ga ed Recu en Uni s (GRU) ne wo ks. The s ochas ic app oaches
comp ise he Li and Lee model, he Common Age E ec model o Kleinow, and
he model o Pla . All models a e applied and compa ed in a la ge case s udy on
decades o da a o he I alian popula ion as di ided in coun ies. In his case s udy, a
new index o mul iple dep i a ion is in oduced and used o classi y all I alian coun-
ies based on socio-economic indica o s, sou ced om he local o ice o na ional
s a is ics (ISTAT). The a o emen ioned models a e hen used o model and p edic
mo ali y a es o g oups o di e en socio-economic cha ac e is ics, sex, and age.
Keywo ds Case S udy on Mo ali y· Longe i y Risk· Neu al Ne wo k· Mul i-
popula ion· Dep i a ion Index· Socio-economic cha ac e is ics· I alian da a
* Ma hias Sche e
[email p o ec ed]
1 Chai o Ma hema ical Finance, Technical Uni e si y o Munich, Ga chingbeiMünchen,
Ge many
2 Munich Reinsu ance Company (Munich RE), Munich, Ge many
3 School o Risk andAc ua ial S udies, Uni e si y o New Sou h Wales, Kensing on, NSW, 2052,
Aus alia
4 ARC Cen e o Excellence inPopula ion Ageing Resea ch, Uni e si y o New Sou h Wales,
Kensing on, NSW2052, Aus alia
496
M.Eu hum e al.
1 3
1 In oduc ion
1.1 Mo ali y modelling: mo i a ion, backg ound, andli e a u e
Since he seminal wo k o Lee and Ca e [17], se e al s ochas ic models o he
es ima ion and p ojec ion o mo ali y a es we e de eloped du ing he las dec-
ades, see, e.g., he con ibu ions o B ouhns e al. [2], Renshaw and Habe man
[31], Cai ns e al. [3], and Pla [28]. While hese pionee ing app oaches analyzed
single popula ions, models o mul iple popula ions gained conside able impo -
ance in subsequen yea s; a e i was ound ha he mo ali y p o iles o mul iple
popula ions ended o con e ge (c . Wilson [40]). Indeed, he mul i-popula ion
pa adigm o en has ad an ages o e modelling mo ali y a es o each popula ion
sepa a ely. Mos no ably, mul i-popula ion mo ali y models can cap u e common
ea u es o he mo ali y p o ile o simila popula ions such as neighbou ing coun-
ies, popula ions showing simila socio-economic-, en i onmen al-, o biological
cha ac e is ics, while simul aneously e lec ing popula ion-speci ic ea u es. This
mo i a es hei use o p oducing cohe en p ojec ions o mo ali y a es.
Recen ly, Neu al Ne wo k (NN)-based app oaches o mo ali y modelling
we e p oposed as an appealing al e na i e o classical s ochas ic models. Among
he i s examples is he wo k o Richman and Wü h ich [33], who analyses he
Swiss popula ion, and Hainau [11] who uses F ench, UK, and US mo ali y a es
o compa e a NN app oach o he Lee–Ca e model w/wo coho e ec s. Ano he
con ibu ion is Nig i e al. [23], who use a deep lea ning algo i hm based on a
wo-s ep Recu en Neu al Ne wo k (RNN) o enhance he o ecas s ob ainable
unde he Lee–Ca e model. Indeed, he e ha e been many ecen de elopmen s
in he use o NNs in he con ex o mul i-popula ion mo ali y modelling, such as
Pe la e al. [27], which conside s he use o one-dimensional (pe iod e ec only)
RNN wi h Long Sho -Te m Memo y (LSTM) and o Con olu ional Neu al Ne -
wo ks (CNN) o p o ide di ec o ecas s o he mo ali y a es compa ed o he
wo-s ep app oach o Nig i e al. [23]. Lindholm and Palmbo g [19] conside sim-
ila models wi h a ocus on he op imal use o da a o p ojec ion. Schnü ch and
Ko n [35] ex end he RNN and CNN by p oposing a wo-dimensional app oach
in ol ing age and pe iod. In a sligh ly di e en ashion, Scognamiglio [36] p o-
poses a NN a chi ec u e o he join calib a ion o indi idual Lee–Ca e models
based on i s classical log-no mal ep esen a ion as well as he Poisson Lee–Ca e
e sion o B ouhns e al. [2]. An app oach ha allows o cohe en p edic ions
wi hin sub-g oups o simila popula ion is p o ided in Pe la and Scognamiglio
[26]. Finally, Wang e al. [38] de elop a amewo k which ‘augmen s’ he mo al-
i y da ase o cons uc an image o neighbo hood mo ali y da a a ound he cen-
al dea h a e and use wo CNN app oaches o p ojec ing mo ali y a es.
497
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A neu al ne wo k app oach o hemo ali y analysis o mul iple…
1.2 Applica ions o (mul i‑popula ion) mo ali y models
A key applica ion o mul i-popula ion mo ali y modelling app oaches is he
analysis o he mo ali y le els based on socio-economic cha ac e is ics. Among
o he s, unde s anding mo ali y is ele an o policymake s in o de o p opose
and plan sus ainable s a e pension e o ms and budge s, o o add ess dispa i ies
be ween socio-economic g oups. Unde s anding mo ali y om a s a is ical poin -
o - iew is also ele an o p i a e sec o playe s such as insu ance companies
and pension unds, o e ing mo ali y-linked p oduc s like annui ies and pensions
and designing e ec i e solu ions o longe i y isk managemen and ans e .
Conside ing speci ic popula ions ha ha e al eady been in es iga ed in he li -
e a u e, le us men ion Wen e al. [39], who compa e se e al s ochas ic mo ali y
models o i mo ali y a es o small geog aphic a eas in he UK (Lowe Laye
Supe Ou pu A eas), g ouped in deciles o hei Index o Mul iple Dep i a ion.
Cai ns e al. [6] de elop a mul i-popula ion mo ali y modelling app oach o he
analysis o he Danish popula ion on he basis o he deciles o a newly c ea ed
a luence index, which accoun s o indi idual in o ma ion on income and weal h.
1.3 Con ibu ions: me hodology andin es iga ed popula ion
This pape in es iga es he use o NNs o join ly model he mo ali y a es o
mul iple popula ions and compa es he empi ical esul s o classical s ochas ic
mo ali y models. The model o he dynamics o mo ali y a es d aws on he
wo k o Richman and Wü h ich [32], whe e hey p opose he use o Recu en
Neu al Ne wo ks (RNN) such as Long Sho -Te m Memo y (LSTM) and Ga ed
Recu en Uni (GRU). Al hough compu a ionally expensi e, we ocus on RNNs
o exploi hei sui abili y o deal wi h he ime-se ies s uc u e o mo ali y a es.
Indeed, due o hei ecu en connec ions, RNNs allow o main ain memo y o
pas obse a ions, since hese le he in o ma ion pass om pas ime s eps o he
cu en one. In his way, i is possible o model he in o ma ion h oughou he
en i e ime se ies. Fu he mo e, RNNs can handle ime se ies o a iable leng h,
due o hei sequen ial p ocessing o he da a, c . Hsu [12]. The NN app oaches
a e hen compa ed o well-es ablished s ochas ic mo ali y models o mul iple
popula ions such as he Lee–Ca e ex ension o Li and Lee [18] and he Common
Age E ec model o Kleinow [16], as well as o he single-popula ion app oach
ep esen ed by he Pla [28] model.
In ou case s udy, we in es iga e I alian mo ali y da a ha is g ouped acco d-
ing o socio-economic cha ac e is ics. Mo e p ecisely, we i s p opose a new dep-
i a ion index on he basis o i e a iables. This index allows sepa a ing he 106
I alian coun ies in o nine g oups o di e en socio-economic le el, which a e hen
used as popula ions o he empi ical analysis. To he bes o ou knowledge, his
s udy is he i s o explici ly add ess he use o NNs in he con ex o he analysis
o mul iple popula ions on he basis o socio-economic cha ac e is ics.
498
M.Eu hum e al.
1 3
The s uc u e o he emaining pape is as ollows: Sec . 2 in oduces and
explains he I alian mo ali y da a used o ou analysis and he c ea ion o he
new Index o Mul iple Dep i a ion. Sec ion3 desc ibes how NNs a e used in ou
s udy. Sec ion4 b ie ly in oduces he models used o compa ison. Then, Sec .5
shows he esul s o he empi ical analysis and, inally, Sec .6 concludes.
2 Da a
The collec ed mo ali y da a comp ises he ‘numbe o dea hs’ and ‘exposu e-a - isk
yea s’ o males and emales aged 50 o 95 li ing in I aly. I spans o e he calenda
yea s 1982–2018 and is g anula o he le el o he 106 I alian coun ies (called p o -
inces). Ou main sou ce o da a is he local o ice o na ional s a is ics (ISTAT).1 The
da a we e u he p ocessed o accoun o spli s in some coun ies o e he las hi y
yea s (see Eu hum [10] o de ails). Fu he elabo a ions we e needed o calcula e
he cen al exposu e o isk om he popula ion da a gi en o Janua y 1s o each
yea .2 We assume ha he ne mig a ion e ec does no bias ou indings abou mo -
ali y a es.3 Fu he mo e, we collec ed a se o wel e indica o s o socio-economic
cha ac e is ics o each coun y, which we used o c ea e an index o mul iple dep i-
a ion. Fo u he de ails ega ding peculia i ies and adjus men o da a, we e e o
he supplemen a y ma e ial4.
2.1 Index o mul iple dep i a ion
Fo each coun y, we o iginally collec ed a se o wel e indica o s ep esen ing di -
e en aspec s o he quali y o li e. Ou o hese indica o s, a subse o i e was ul i-
ma ely selec ed, e lec ing a p ope mix o di e en ypes o socio-economic ac o s.
These ha e been agg ega ed o a so-called ‘Index o Mul iple Dep i a ion’ (IMD).
In his way, he 106 p o inces5 om wen y egions could be classi ied o di e en
g oups based on hei socio-economic indica o s. The chosen a iables a e:
1. Rela i e po e y (in pe cen ): The pe cen age o households wi h a consump ion
expendi u e lowe han he a e age pe -capi a, as es ima ed by ISTAT
6;
1 The popula ion da a was collec ed om he websi e www. is a . i . The numbe o dea hs om 1982 o
2002 we e p o ided by he o ice o s a is ics.
2 To app oxima e he cen al exposu e o yea i, popula ions om Janua y 1s o yea i and yea
i+1
we e a e aged. Fu he , some da a cleaning was pe o med, o de ails we e e o Eu hum [10].
3 This assump ion appea s easonable on he g ounds o he ela i ely sho leng h o he ime se ies o
a ailable da a, especially i we conside ha censuses a e conduc ed e e y en yea s only, and also in
conside a ion o he age- ange we analyse, since mig a ion (in gene al and so) be ween I alian p o inces
ypically a ec s younge people. A de ailed accoun o his obse a ion is discussed in Cai ns e al. [5]
based on popula ion da a om England and Wales. Mo eo e , we a e no awa e o he exis ence o da a
ha allows o ack mig a ion be ween I alian p o inces.
4 See he co esponding Gi hub eposi o y o his pape .
5 ‘Sa degna’ has been kep o ou p o inces ins ead o i e o keep hings simple o his o ic da a.
6 Accessible a h ps:// www. is a . i / en/ analy sis- and- p odu c s/ da ab ases/ s a b ase
499
1 3
A neu al ne wo k app oach o hemo ali y analysis o mul iple…
2. P ima y ca e and esiden ial and semi- esiden ial acili ies (measu ed in beds pe
10,000 inhabi an s). This in o ma ion is indica i e o he expendi u e in heal h
ca e by he egion whe e he coun y is loca ed, which in i s u n is a ec ed by i s
weal h7;
3. Social se ices and bene i s o municipali ies, measu ed as cos s in Eu os pe
capi a (2017 da a). Se ices include, e.g., day nu se y, socio-educa ional se ices
o ea ly childhood, and so on. I is assumed ha highe cos s co espond o mo e
in es men s in social se ices by municipali ies and hence mo e bene i s om a
sociological poin o iew o he popula ion;
4. Unemploymen a e (in pe cen o he popula ion aged o e 15);
5. Numbe o elonies commi ed by pe sons con ic ed by inal judgemen (pe 1,000
inhabi an s). F om a s a is ical poin o iew, his a iable is no signi ican ly
co ela ed wi h he o he indica o s used o building he index. I is assumed
ha a high ela i e numbe o elonies commi ed (by pe sons con ic ed by inal
judgemen ) sugges s wo se li ing condi ions om a sociological poin o iew,
bu also comes along wi h a a he de icien economic si ua ion.
A de ailed desc ip ion o hese a iables can be ound on he ‘S a base po al’ o
Is a , which p o ides access o a la ge amoun o da a on he I alian popula ion,8
which is also he sou ce o he unde lying da a (downloaded on
1s
o Ma ch 2021
o yea 2018).
The a iables ha e been chosen based on a co ela ion analysis be ween hem.
The selec ion was hen alida ed by a anking measu e, Kendall’s au, o check
whe he he ank o p o inces changes signi ican ly when omi ing a a iable om
he selec ion. We ound ha Kendall’s au does no signi ican ly change when omi -
ing o adding a u he a iable o he i e we selec ed.
Ou index was cons uc ed using z-sco es o scale he i e di e en a iables o
a compa able le el and uni ; he same app oach is used in Osse a o io della salu e
[24]. This allowed o agg ega e he single z-sco es pe p o ince o a o al alue and
inally pe mi ed a anking o he p o inces. Fo each p o ince, he z-sco e was cal-
cula ed as
whe e i indica es he espec i e socio-economic classi ie , x
j
i
i s alue, j he coun y,
mi
he mean o his classi ie o e all coun ies, and
si
he (unbiased) sample de ia-
ion o e all coun ies.
The in e p e a ion is in ui i e: he highe he alue
zj
o a coun y, he wo se is i s
socio-economic si ua ion, o in o he wo ds, he dep i a ion in ha a ea is highe .
Implici ly, we assume ha he s anda dized alue o each a iable has he same
impac on he z-sco e (which could be easily gene alized by in oducing weigh s).
z
j=
5
∑
i=1
zj
i
=
5
∑
i=1
x
j
i
−mi
s
i
,
7 Since ou index is c ea ed wi h he in e p e a ion o ‘smalle alues co esponding wi h be e li ing
condi ions’, we changed he sign o his co a ia e and also he nex one.
8 Accessible a h ps:// www. is a . i / i / da i- anali si-e- p odo i/ banche- da i/ s a b ase
500
M.Eu hum e al.
1 3
Then, he aim is o ank he coun ies on he basis o he alues o hei z-sco e.
Hence, we c ea ed nine g oups o coun ies, each wi h homogeneous popula ions,
anging be ween 6 and 7 million people. This spli does no accoun o sex and
age o he I alian popula ion. This implici ly assumes ha he coun ies ha e simila
dis ibu ed popula ion by age and sex. I , by using his c i e ion, we ound ha wo
coun ies had he same z-sco e alue,9 we alloca e hese in he same g oup. In his
way, we aim a c ea ing g oups o compa able size, whe e he coun ies he ein ha e
simila socio-economic cha ac e is ics. The use o socio-economic indica o s o he
calenda yea 2018 only, implies ha he anking and he co esponding g oups do
no change o e ime. This assump ion may no e lec he socio-economic de elop-
men s in I aly ac oss he las 36 yea s. Ne e heless, we ind his assump ion ea-
sonable in conside a ion o he esul s ob ained in he p elimina y analysis o he
mo ali y and socio-economic da a. Indeed, mo e dep i ed socio-economic coun ies
a e loca ed in he sou h o I aly, compa ed o he his o ically weal hie a eas in he
no h, as also demons a ed by he ime se ies o he unemploymen a e, and his
e idence is u he con i med when analysing he e olu ion o he aw mo ali y
a es.
We pu posely op ed o a simple and in e p e able app oach o c ea e he index,
ha migh be e ined in u he s udies. Ou main ocus is o elabo a e on mo ali y
models as desc ibed la e , and his simple and in ui i e app oach p o ides e y plau-
sible esul s. In any case, we ema k how he c ea ion o he g oups was ca ied ou
on he basis o hei o de , a he han hei index alue.
To ob ain a geog aphical imp ession o he index-based subdi ision, he ollow-
ing map is epo ed, see Fig.1. The colo -scheme is as ollows: B igh e colo s indi-
ca e a lowe index alue, e lec ing be e socio-economic condi ion based on he
IMD de ined abo e. The o iginal map was aken om he De Agos ini websi e10 and
colo ed wi h s anda d g aphic ools.
To ob ain a i s imp ession o mo ali y a es, he c ude pe iod dea h a es
m
(x, ,i)=
d(x, ,i)
E
c
(x, ,i)
be ween 1982 and 2018 a e plo ed in Fig. 2 o he male and
emale popula ion aged 68 and 83 yea s old, espec i ely. We obse e:
• In gene al, mo ali y a es dec ease o e ime and inc ease wi h age; which is
consis en wi h he li e a u e and he biological ageing p ocess;
• Mo ali y a es o emales a e lowe han o males, as widely obse ed in many
o he na ional popula ion ables;
• The mos dep i ed subpopula ions (g1 in da k blue) appea o ha e highe mo -
ali y a es o e he pe iod analysed. The di e ence and o de ing in subg oups is
mo e p onounced o he emale subpopula ion, while o males, he di e ence
in mo ali y a es is less e iden o weal hie subg oups. This could be due o
he chosen index o due o he unde lying popula ion. This e ec may also be he
consequence o a signi ican no h–sou h di ision in e ms o socio-economic
well-being, while people s ill li ing longe in sou he n pa s o he coun y, as
9 As pe wo decimal digi s.
10 h ps:// blog. geog a ia. deasc uola. i / a ic oli/ che- ine- a an no- le- p o i nce- i ali ane
501
1 3
A neu al ne wo k app oach o hemo ali y analysis o mul iple…
i is he case o egions like Sa dinia and Calab ia, which a e amous o hei
excep ional longe i y (see Poulain e al. [29]);
• In he ea lie yea s o he s udy, dep i a ion ends and di e ences a e ha de
o de ec . This may be caused by he ac ha he socio-economic analysis
Fig. 1 Subdi ision o he I alian p o inces based on he new index. We obse e he endency o no he n
p o inces ha ing smalle index alues, i.e. be e li ing condi ions
502
M.Eu hum e al.
1 3
was based on indica o s o he yea 2018, while di e en p o inces e ol ed
di e en ly o e decades;
• The spike in he mo ali y a es o males and emales aged 83 in 2003 is
likely o be due o he massi e hea wa e o he summe in 2003 (Johnson
e al. [15]). Fu he discussion is included in Sec .5.
Th oughou he analysis, we use he mo ali y a es o he i s 33 calenda yea s
1982–2014 when aining he models ( aining se ), while hose o 2015–2018
a e used o p edic ions and o ecas s ( es se ). We op ed o his spli in abou
90%–10% due o he sho leng h o he ime-se ies o a ailable da a. We belie e
his spli is a easonable comp omise be ween he need o su icien da a o i
he models (especially on a es ic ed age- ange o a 50–95 yea s old popula-
ion), whils using he la es yea s o assess hei p edic i e abili y.
−4.8
−4.4
−4.0
1990 2000 2010
Yea
Dea h Ra e (Log)
G oup
g1
g2
g3
g4
g5
g6
g7
g8
g9
Age 68, Female
−4.0
−3.5
−3.0
1990 2000 2010
Yea
Dea h Ra e (Log)
G oup
g1
g2
g3
g4
g5
g6
g7
g8
g9
Age 68, Male
−2.8
−2.4
−2.0
1990 2000 2010
Yea
Dea h Ra e (Log)
G oup
g1
g2
g3
g4
g5
g6
g7
g8
g9
Age 83, Female
−2.50
−2.25
−2.00
−1.75
1990 2000 2010
Yea
Dea h Ra e (Log)
G oup
g1
g2
g3
g4
g5
g6
g7
g8
g9
Age 83, Male
Fig. 2 C ude dea h a es in log-scale o ages 68 esp. 83, emale and male popula ion. G oup 1 (g1) is
he wo s g oup socio-economically speaking
509
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A neu al ne wo k app oach o hemo ali y analysis o mul iple…
4.1 Li andLee (LL) model
This model ex ends he single-popula ion Lee and Ca e [17] model o he analysis
o mul iple popula ions. The Li and Lee [18] model in oduces a se o pa ame e s
which a e common o he se o analysed g oups, as well as g oup-speci ic pa am-
e e s cap u ing he unexplained a iance.
Model 4.1 (Li and Lee model) Fo age (in yea s) x, ime pe iod , and g oup i, he Li
& Lee model desc ibes he loga i hm o he ‘cen al mo ali y a e’ m(x, ,i) as
He e, m(x, , i) can be app oxima ed as he a io be ween he dea h coun s,
deno ed as D(x, ,i), and he cen al exposu e a isk
Ec(x, ,i)
.
𝛼(x,i)
,
𝛽(x,i)
, and
𝜅( ,i)
a e g oup-speci ic pa ame e s.
𝛼(x,i)
indica es he a e -
age o e ime o he log mo ali y a e. The common unc ion K( ) explains he e o-
lu ion o he mo ali y o e ime o all g oups, and B(x) is a global age modula ing
pa ame e , indica ing how a es change by age o changes in he ime ac o K( ).
𝛽(x,i)
and
𝜅( ,i)
ha e he same ole as B(x) and K( ), bu ac on a g oup-speci ic
le el.
4.2 Common age e ec (CAE) model
The CAE model o Kleinow [16] assumes ha age e ec s a e common o all popu-
la ions, ollowing he assump ion ha age e ec s may be e y simila in coun ies
sha ing a simila socio-economic s uc u e.
Model 4.2 (Kleinow model) Fo age x, ime pe iod , and g oup i, he Kleinow
model o o de p assumes he ollowing model o he loga i hm o he cen al dea h
a e
The o de p ollows om he allowance o u he age– ime in e ac ion pa am-
e e s. In ou analysis we se
p=2
.
4.3 Pla model
The hi d model we use o compa ison was ini ially p oposed by Pla [28]. How-
e e , in his wo k we use a simpli ied e sion, which includes wo pe iod-speci ic
ac o s, wi hou accoun ing o he coho e ec .
Model 4.3 (Pla model) Fo age x, ime pe iod , and g oup i, he Pla model (wi h-
ou coho e ec s) models he loga i hm o he cen al dea h a es as
(4.1)
log (
m(x, ,i)
)
=𝛼(x,i)+B(x)K( )+𝛽(x,i)𝜅( ,i),
min
≤ ≤
min
+T−
1.
(4.2)
log (
m(x, ,i)
)
=𝛼(x,i)+𝛽
(1)
(x)𝜅
(1)
( ,i)+…+𝛽
(p)
(x)𝜅
(p)
( ,i)
.
510
M.Eu hum e al.
1 3
whe e
x
deno es he a e age age o he obse ed age ange. The i s s ochas ic
componen
𝜅(1)
ep esen s changes in he le el o mo ali y o all ages, while
𝜅(2)
allows o changes in mo ali y o a y be ween ages.
4.3.1 Fo ecas ing
Fo he models o Li & Lee, Kleinow, and Pla , we pe o med o ecas s ia a clas-
sical ime se ies app oach. In ac , ime dependen
𝜅
-p ocesses a e modelled as a
s ochas ic ime se ies o p edic mo ali y a es h ough ARIMA (Au o Reg essi e
In eg a ed Mo ing A e age) models. These can be eadily implemen ed in R, e.g.
by using he unc ion au o.a ima om he package o ecas (Hyndman and
Khandaka [14]).
5 Empi ical esul s
All models a e i ed based on da a spanning om 1982 o 2014. Howe e , i ed
alues a e jus compa ed o he yea s 1992 o 2014, since he NN-based app oach
does no deli e alues o he i s en yea s, see Richman and Wü h ich [33]. In
wha ollows, hen
i∈{1, …,9},x∈{50, …, 95}
, and
∈{1992, …, 2014}
. Fu -
he mo e, we g aphically inspec he models by using he s anda dized esiduals, as
de ined in Wen e al. [39], see also Table4.
5.1 In‑sample i
We i s compa e he h ee compe ing s ochas ic mo ali y models, ha we b ie ly
in oduced in Sec .4, in e ms o hei explana ion a io. Then we compa e hese o
he a es ob ained by using he NN models o his pape .
Table1 shows he explana ion a ios o he models o Li & Lee, Kleinow, and
Pla , de ined o g oup i and model M as
whe e
𝛼
c(x,i) ∶=
1
T
∑
log
d(x, ,i)
Ec(x, ,i
)
is he a e age log c ude dea h a e o e ime. The
explana ion a io is use ul o analysing how much in o ma ion abou he c ude
dea h a es
d(x, ,i)
E
c
(x, ,i)
is explained by he espec i e model.
We obse e how he Li & Lee model pe o ms bes in e ms o explana ion
a ios o eigh ou o nine emales subg oups, and he same is no ed o he
(4.3)
log (
m(x, ,i)
)
=𝛼(x,i)+𝜅
(1)
( ,i)+𝜅
(2)
( ,i)(x−x)
,
(5.1)
R
M
i=1−
∑
x,
�
log d(x, ,i)
Ec(x, ,i)−log (m(x, ,i))
�2
∑
x, �
log d(x, ,i)
Ec(x, ,i)−𝛼c(x,i)
�
2
,
511
1 3
A neu al ne wo k app oach o hemo ali y analysis o mul iple…
Kleinow model when looking a he male popula ion. The Pla model, analysed a
he le el o a single popula ion, s ill yield compa able explana ion a ios, despi e
being lowe compa ed o he Li & Lee and he CAE model o Kleinow. Fu -
he mo e, he Pla model has a lowe numbe o pa ame e s, which may in pa
explain hese esul s. Ou esul s a e u he con i med by he analysis o hei
Akaike In o ma ion C i e ion (AIC) (Akaike [1]), shown in Table2, which indi-
ca es he ela i e quali y o a s a is ical model wi h a penal y e m o he numbe
o pa ame e s. The AIC is calcula ed as
whe e
𝓁(
𝜃)
indica es he log-likelihood alue a (op imal) pa ame e
𝜃
and p deno es
he numbe o pa ame e s o he espec i e model (see Table9).
Tables3 shows he mean squa ed e o o model M and subpopula ion i, cal-
cula ed as ollows:
whe e
m(x, ,i)M
deno es he i ed mo ali y a e de i ed om model M. Table 4
shows he MSE o he RNN models analysed in his pape .
(5.2)
AIC
=−
2
⋅𝓁(
𝜃)+
2
⋅
p,
(5.3)
MSE
M
i=
1
n⋅T
∑
x,
(
m(x, ,i)− m(x, ,i)M
)
2
,
Table 1 Explana ion a ios o di e en models
The highes a ios o a speci ic sex a e emphasized in bold
Dep i a ion Female Male Combined
G oup
RLL
i
RCAE
i
RPla
i
R
LL
i
RCAE
i
RPla
i
RLL
i
RCAE
i
RPla
i
10.862 0.853 0.848 0.782 0.823 0.811 0.881 0.867 0.874
20.868 0.856 0.838 0.822 0.810 0.804 0.877 0.854 0.870
30.862 0.857 0.822 0.842 0.846 0.826 0.893 0.879 0.875
40.859 0.848 0.802 0.855 0.856 0.836 0.892 0.881 0.872
50.831 0.831 0.774 0.868 0.872 0.842 0.886 0.879 0.852
60.876 0.866 0.826 0.896 0.903 0.882 0.917 0.910 0.901
7 0.876 0.882 0.842 0.888 0.912 0.894 0.929 0.924 0.917
80.866 0.857 0.826 0.880 0.890 0.864 0.910 0.902 0.895
90.866 0.861 0.831 0.885 0.898 0.866 0.914 0.906 0.909
A e age 0.863 0.857 0.823 0.857 0.868 0.847 0.900 0.889 0.885
Table 2 AIC alues o di e en
models
Lowes alues o speci ic sex in bold
Model Female Male Bo h sexes
Li and Lee -66,185,533 -67,260,680 -134,742,870
Kleinow -66,188,446 -67,256,443 -134,750,703
Pla -66,194,573 -67,261,519 -134,749,793
512
M.Eu hum e al.
1 3
Table 3 Mean squa ed e o s o di e en models and g oups
Lowes alues o speci ic sex in bold. Scale:
10
−
4
Dep i a ion Female Male Combined
G oup LL CAE Pla LL CAE Pla LL CAE Pla
11.3327 1.4275 1.3743 3.6259 3.0236 2.7788 1.4856 1.6027 1.4515
20.9692 1.2805 1.2509 2.0578 1.8407 1.8716 1.1946 1.2721 1.1912
3 0.7436 0.7396 0.8416 2.4992 1.9319 2.0930 0.9395 0.9666 0.9221
40.6853 0.8357 0.8987 2.8050 2.1437 2.3239 0.8530 0.9327 0.9533
50.8433 0.9518 1.0570 2.4233 2.1329 1.8905 0.8928 1.0593 1.0390
60.5896 0.6464 0.7379 2.6853 2.3437 2.1192 0.7630 0.8736 0.7749
70.6317 0.6708 0.7788 3.7611 3.6867 2.8752 0.7681 0.8645 0.7284
80.7108 0.7783 0.7892 3.4809 2.4448 2.6673 0.8677 1.0031 0.8662
90.6903 0.7570 0.8233 4.4645 2.8037 3.3212 0.8259 0.9085 0.7876
Sum 7.1964 8.0876 8.5518 27.8031 22.3517 21.9406 8.5902 9.4831 8.7140
Table 4 Mean squa ed e o s o di e en Recu en ne wo k models and g oups
Lowes alues o speci ic sex and ne wo k ype in bold. Scale:
10
−
4
Dep i a ion Female Male Combined
G oup
ELSTM3
i
ELSTM2
i
ELSTM1
i
ELSTM3
i
ELSTM2
i
ELSTM1
i
ELSTM3
i
ELSTM2
i
ELSTM1
i
1 0.6477 0.6242 0.6495 1.1999 1.1560 1.2223 0.5309 0.5071 0.5362
2 0.5710 0.5521 0.5907 0.6545 0.6475 0.6853 0.4347 0.4295 0.4485
3 0.4006 0.3876 0.4139 0.8020 0.7556 0.8101 0.3449 0.3294 0.3451
4 0.3114 0.2992 0.3196 0.6739 0.6364 0.6783 0.2379 0.2245 0.2413
5 0.4202 0.4039 0.4270 0.8119 0.7816 0.8481 0.3581 0.3511 0.3709
6 0.3363 0.3257 0.3411 0.7937 0.7621 0.8044 0.3046 0.2900 0.3096
7 0.3213 0.3123 0.3215 0.9426 0.8860 0.9429 0.2753 0.2599 0.2771
8 0.2922 0.2781 0.3000 0.8838 0.8420 0.8981 0.2760 0.2591 0.2801
9 0.2979 0.2827 0.3007 0.7111 0.6564 0.7067 0.2772 0.2580 0.2766
Sum 3.5986 3.4658 3.6639 7.4734 7.1236 7.5963 3.0395 2.9086 3.0853
EGRU3
i
EGRU2
i
EGRU1
i
EGRU3
i
EGRU2
i
EGRU1
i
EGRU3
i
EGRU2
i
EGRU1
i
10.4052 0.4219 0.5124 0.5439 0.5636 0.8325 0.3051 0.3396 0.3979
20.3478 0.3599 0.4389 0.3925 0.4212 0.5785 0.2744 0.2982 0.3658
30.2756 0.2951 0.3586 0.4108 0.4216 0.5702 0.2111 0.2458 0.2905
40.2633 0.2686 0.2928 0.3824 0.3978 0.5464 0.1955 0.2099 0.2358
50.3323 0.3449 0.3901 0.4831 0.5321 0.6863 0.2736 0.3038 0.3401
60.2736 0.2774 0.3059 0.4036 0.4290 0.5686 0.2219 0.2499 0.2778
7 0.2577 0.2562 0.2752 0.4324 0.4461 0.6428 0.2101 0.2247 0.2450
80.2291 0.2327 0.2633 0.4676 0.4806 0.6703 0.1826 0.2038 0.2341
90.2387 0.2398 0.2612 0.4237 0.4296 0.5115 0.1967 0.2160 0.2408
Sum 2.6232 2.6966 3.0984 3.9399 4.1217 5.6069 2.0710 2.2918 2.6278
513
1 3
A neu al ne wo k app oach o hemo ali y analysis o mul iple…
F om he esul s in Table3, we obse e ha i is no possible o d aw a decisi e
conclusion abou he bes pe o ming model, since o di e en g oups and sex he e
is a mixed e idence abou he bes pe o ming model. When he da a o bo h males
and emales a e combined, hen we obse e ha he Pla model has a be e pe o -
mance in he wo ex emes o he dep i a ion g oup, o in o he wo ds, shows a be -
e pe o mance o he leas and mos dep i ed coun y g oups.
The wo RNN models seem o sensibly imp o e he in-sample i compa ed o he
h ee compe ing s ochas ic models. In mo e de ail, he wo-laye LSTM ou pe o ms
he o he wo LSTM models o e all socio-economic g oups o males, emales, and
combined. On he o he hand, he GRU models wi h h ee laye s u n ou o pe o m
e en be e . Fo emale and bo h sexes combined mo ali y a es, he mos dep i ed
g oups show highe in-sample e o s compa ed o less dep i ed ones. A simila e i-
dence can be obse ed o he LL, CAE, and Pla models. This may be indica i e o
he issues o mo ali y models in gene al when i ing mo e dep i ed subpopula ions
ia a mul i-popula ion app oach. This can also be he e ec o he c ea ed index in
Sec .2 and o he unde lying da a. Ne e heless, we no e ha he di e ence be ween
emales and males is smalle o he RNN app oach compa ed o he compe ing
models.
In conclusion, he mul i-popula ion RNN models ou pe o m he well es ablished
s ochas ic mo ali y models when analysing coun y-based socio-economic sub-
g oups o he I alian popula ion, based on he mean squa ed e o . The mean squa ed
e o s o males a e highe compa ed o emales and bo h sexes combined.
Fo a deepe inspec ion o hese esul s we es ic he MSE analysis o a sho e
age ange o males and emales, in a way such ha hey bo h ha e he same li e
expec ancy. These anges we e chosen based on emaining li e expec ancy o I al-
ian males a age 50 in 2017 (32.08 yea s) and I alian emales a age 54 in yea 2017
(app ox. 32 yea s). The e o e, based on he I alian li e ables om he Human Mo -
ali yDa abase [13], we es ic he analysis o he male popula ion aged 50 o 82
and o he emales aged 54 o 86.
Table 5 Mean squa ed e o s o
speci ically ailo ed age anges,
emales (54–86), and males
(50–82), SMM models
Scale:
10
−
4
Dep i a ion Female Male
G oup LL CAE Pla LL CAE Pla
1 0.0996 0.0982 0.1025 0.1113 0.1117 0.1147
20.0855 0.0904 0.0925 0.1004 0.0969 0.1026
30.0795 0.0851 0.0828 0.1006 0.0985 0.1035
40.0859 0.0875 0.0886 0.0899 0.0869 0.0922
5 0.0915 0.0884 0.0885 0.1052 0.1036 0.1059
60.0770 0.0792 0.0800 0.1178 0.1169 0.1189
70.0790 0.0806 0.0825 0.1081 0.1067 0.1067
80.0804 0.0867 0.0853 0.1098 0.1059 0.1090
90.0782 0.0793 0.0802 0.1072 0.1097 0.1115
Sum 0.7566 0.7755 0.7830 0.9503 0.9367 0.9650
514
M.Eu hum e al.
1 3
Again, we ecalcula ed he dep i a ion g oup-speci ic MSE o males and emales
o he es ic ed age- anges, s ill based on he model i ed o he o iginal da ase
(males and emales aged 50 o 95). The esul s a e shown in Table5 (LL, CAE, and
Pla models) and Table6 ( o RNN).
5.2 Mean squa ed e o s o speci ic ages
Fo he selec ed age anges, we obse e ha o he h ee compe ing s ochas ic mo -
ali y models, he mean squa ed e o is less han 0.1 o he mean squa ed e o o
all ages, e en i hese educed age anges co e 72% o he modelled yea s.
Again, we obse e ha o he emale popula ion he MSE is la ge o he mos
dep i ed socio-economic g oups o bo h he compe ing models as well as o he
RNN-based ones. A simila e idence is ob ained o he males a a lesse ex en .
Fo he models i ed using emale subpopula ion da a, we s ill ace mo e di icul ies
when conside ing he mos dep i ed g oups compa ed o less dep i ed ones, espe-
cially o SMM models. I is also e iden , ha all models wi hin one model class
(SMM o RNN) ha e simila mean squa ed e o s. This means, he models pe o m
e y well o hese ages h ough all selec ed models. Le us also men ion he ac
ha mean squa ed e o s in he RNN case a e by a smalle han in he SMM case.
Fu he mo e, i is in e es ing o obse e an alignmen o emale and male a e i -
ing (as i was aimed h ough he analysis o di e en age anges): he SMM mean
squa ed e o o he male popula ion we e on a e age app oxima ely 3.02 imes
compa ed o emales. When es ic ing he age ange obse a ion, his ac o educes
o abou 1.23, which means ha o ou da a bo h male and emale a es a e almos
i ed equally well on a e age o an age ange wi h simila emaining li e expec a-
ion. Fo he h ee selec ed RNN models om abo e, he ac o s a e 1.52 and 1.01
espec i ely, ha is he alignmen o emale and male a es i ing abili y is almos
pe ec in he RNN case.
Table 6 Mean squa ed e o s o
speci ically ailo ed age anges,
emales (54–86), and males
(50–82), selec ed RNN models
Scale:
10
−
4
Dep i a ion Female Male
G oup GRU3 GRU2 LSTM1 GRU3 GRU2 LSTM1
1 0.0322 0.0318 0.0327 0.0287 0.0289 0.0293
2 0.0186 0.0185 0.0195 0.0193 0.0199 0.0189
3 0.0209 0.0211 0.0211 0.0173 0.0178 0.0178
4 0.0163 0.0164 0.0167 0.0159 0.0165 0.0154
5 0.0169 0.0168 0.0171 0.0171 0.0173 0.0180
6 0.0153 0.0151 0.0153 0.0184 0.0187 0.0181
7 0.0188 0.0185 0.0195 0.0208 0.0207 0.0199
8 0.0122 0.0122 0.0124 0.0157 0.0164 0.0162
9 0.0174 0.0173 0.0173 0.0167 0.0173 0.0172
Sum 0.1689 0.1677 0.1715 0.1700 0.1735 0.1707
515
1 3
A neu al ne wo k app oach o hemo ali y analysis o mul iple…
Concluding, i seems ha he oldes people (o age 87–95) d i e mos o he mean
squa ed e o . In addi ion, one should be awa e o looking a di e en age anges o
emales and males based on he e idence ha li e expec ancy signi ican ly di e s
o hese wo g oups. The poin in ime o olde ages, whe e mo ali y is ha de o
i due o mo e ola ili y in dea h and exposu e da a, s a s ea lie o males which
could explain a highe mean squa ed e o when looking a all ages.
Ano he possible eason o la ge e o s in he male popula ion h ough all mod-
els may come om he di e ence in popula ion sizes o he unde lying dep i a-
ion g oups. These a e highe in he emale case, since in he obse ed age anges
emales ou li e males. A la ge sample size could gi e he s a is ical model mo e
s abili y when es ima ing pa ame e s. Howe e , he di e ence is no e y la ge15.
5.2.1 S anda dized esiduals
We pe o m a g aphical analysis o hese esul s by in es iga ing he s anda dized
esiduals. When a model i s he da a easonably well, s anda dized esiduals should
no exhibi any pa e n based on yea s o ages. To quo e Cai ns e al. [4], “i he
model i s he da a well, hen he s anda dized esiduals should be independen o
50
60
70
80
90
1980 1990 2000 2010
Yea
Age
−5
0
5
10
Res
Female − LiLee − G oup 7
50
60
70
80
90
1980 1990 2000 2010
Yea
Age
−5
0
5
10
Res
Male − Kleinow − G oup 3
Fig. 5 Selec ed esidual hea map plo s pe g oup
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−2.5
0.0
2.5
5.0
50 60 70 80 90
Age
Residuals
1990
2000
2010
Yea
Female − LiLee − G oup 8
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−0.3
−0.2
−0.1
0.0
0.1
0.2
50 60 70 80 90
Age
Residuals
1995
2000
2005
2010
Yea
Male − LSTM3 − G oup 6
Fig. 6 Selec ed esidual plo s unc ion o age, pe g oup, bo h sexes
15 O he popula ion in I aly aged 50–95 on Janua y
1s
2021, 46.31% we e males (Is a S a base).
516
M.Eu hum e al.
1 3
each o he , meaning ha he hea plo should exhibi a high deg ee o andomness,
wi h no disce nible pa e ns”.
Figu e5 plo s he hea maps o he s anda dized esiduals o he i ed a es unde
he Li and Lee model o g oup 7 and o he g oup 3 o he CAE model. The plo s
o he o he g oups unde he h ee SMMs a e simila . These a e calcula ed o
model M as
whe e
m(x, ,i)M
deno es he i ed mo ali y a e unde model M.
The wo plo s show a diagonal line, which is indica i e o a coho e ec o hose
indi iduals bo n a ound 1918, which co esponds o he end o he i s wo ld wa
and he Spanish lu pandemic. We discuss hese poin s in mo e de ail in Appendix
A.2. Fo all o he ages and yea s, he s anda dized esiduals appea o lack any spe-
ci ic pa e n (Figs.6, 7).
Fo RNN models, he s anda dized esiduals shown in Fig.8 ( o he o he g oups
we ha e simila e idences) seem o be sp ead homogeneously ac oss he ages and
yea s. The e a e no isible coho e ec s in esiduals as in ea lie models and esidu-
als seem o be much smalle han in he p e ious models. These obse a ions sug-
ges ha RNN models used in his wo k a e able o cap u e coho e ec s in he da a
and yield a be e in-sample i o obse ed yea s. In any case, we obse e ha o
ages 70+/80+ in 2003, esiduals a e la ge and posi i e, meaning ha he obse ed
mo ali y is highe han expec ed. Con e sely, o 2004 mo ali y a es seem o be
o e -es ima ed by he models. A possible explana ion o his unde -es ima ion in
2003 could be he massi e Eu opean hea wa e in summe 2003, see Johnson e al.
[15], he ho es summe in Eu ope o cen u ies. ISTAT epo ed o e 18,000 dea hs
in ha summe compa ed o he yea be o e (+ 11.6%). As epo ed in Ma one [21],
91.8% o hese we e aged 75 and olde . This could explain in pa he la ge numbe
o nega i e esiduals a olde ages in 2004. These ex ao dina y ci cums ances may
explain why he models could no ully de ec such a sudden inc ease in he numbe
(5.4)
Z
(x, ,i)M=d(x, ,i)−E
c
(x, ,i)⋅m(x, ,i)
M
√
Ec(x, ,i)⋅m(x, ,i)M
,
−4
−2
0
2
4
6
1990 2000 2010
Yea
Residuals
50
60
70
80
90
Age
Male − LiLee − G oup 9
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
1995 2000 2005 2010 2015
Yea
Residuals
50
60
70
80
90
Age
Female − GRU2 − G oup 5
Fig. 7 Selec ed esidual plo s unc ion o yea , pe g oup, bo h sexes
517
1 3
A neu al ne wo k app oach o hemo ali y analysis o mul iple…
o dea hs. A simila esul migh be obse ed in 2020 o la e ollowing he COVID-
19 pandemic.
We u he compa e he esiduals as unc ion o he age (Fig.6), and as a unc ion
o he calenda yea (Fig.7), whe e g oups and sex ha e been chosen andomly and
50
60
70
80
90
1995 2000 2005 2010 2015
Yea
Age
−0.2
0.0
0.2
Res
Female − LSTM1 − G oup 1
50
60
70
80
90
1995 2000 2005 2010 2015
Yea
Age
−0.2
−0.1
0.0
0.1
0.2
Res
Female − GRU3 − G oup 8
50
60
70
80
90
1995 2000 2005 2010 2015
Yea
Age
−0.2
−0.1
0.0
0.1
0.2
Res
Male − GRU1 − G oup 6
50
60
70
80
90
1995 2000 2005 2010 2015
Yea
Age
−0.2
−0.1
0.0
0.1
0.2
Res
Male − LSTM2 − G oup 3
Fig. 8 Randomly selec ed esidual hea map plo s pe g oup
Table 7 Ou -o -sample mean squa ed e o s o di e en models and g oups
Lowes alues o speci ic sex in bold. Scale:
10
−
4
Dep i a ion Female Male Bo h Sexes
G oup
ELL
i
EKL
i
EPla
i
ELL
i
EKL
i
EPla
i
ELL
i
EKL
i
EPla
i
1 1.0038 0.9955 2.1453 7.7344 4.2412 4.8872 1.1059 1.0278 2.5211
20.6560 0.7621 1.4115 2.3270 0.7531 1.4893 0.7326 5.1101 1.1578
30.7433 1.1019 1.9106 3.2611 2.1388 1.7976 1.0087 9.9613 1.4781
40.5366 0.6045 16.3794 9.7053 1.8354 16.1711 8.8701 0.8926 13.5179
5 1.0720 0.9237 2.3226 2.8685 2.9456 3.2342 0.8967 1.1394 2.0517
6 1.4249 1.3634 2.7108 3.5877 4.0032 2.2568 1.3117 1.5394 2.0675
70.9823 1.9311 2.0970 2.8236 6.2054 2.3788 1.0659 0.9724 1.7426
8 1.2298 1.1849 2.3117 3.9409 3.5925 1.6364 1.0616 1.4791 1.7348
91.3697 2.2630 2.1508 5.4839 2.3254 1.3812 1.2547 1.1114 1.5911
Sum 9.0183 11.1301 33.4397 41.7324 28.0406 35.2326 17.3079 23.2334 27.8627
518
M.Eu hum e al.
1 3
whe e we exclude he da a o he coho s bo n in 1916, 1917, and 1920, based on
obse a ions in Appendix A.2.
Pa e ns a e no obse able i no o yellow poin s which indica e age 90+. In
gene al, esiduals o he Li and Lee model ( he same holds o he o he models o
ype SMM) sp ead on much highe le els han hose o RNN models, by a ac o
la ge han en. Howe e , bo h model ypes sugges e enly dis ibu ed esiduals wi h
excep ion o he oldes yea s which seem ha de o model. P esumably, his la e
obse a ion is d i en by he smalle exposu e o olde ages.
5.2.2 Ou ‑o ‑sample measu es
We analyse he ou -o -sample pe o mance o he implemen ed models by using a
o ecas pe iod o ou yea s, wi h
n=46
and
T=4
. All igu es ha e been ob ained
on he basis o he p ocedu e desc ibed in Sec . 3 o RNN and Sec . 4 o he
SMMs. Resul s a e shown in Tables7 and8.
Table 8 Ou -o -sample mean squa ed e o s o di e en RNNs and g oups
Lowes alues o speci ic sex and ne wo k ype in bold. Scale:
10
−
4
Dep i a ion Female Male Bo h sexes
G oup
ELSTM3
i
ELSTM2
i
ELSTM1
i
ELSTM3
i
ELSTM2
i
ELSTM1
i
ELSTM3
i
E
LSTM2
i
ELSTM1
i
1 1.0916 0.9873 1.0225 2.0643 2.1807 2.0877 1.3548 1.3108 1.2765
2 0.6805 0.6004 0.6265 1.0510 1.1554 1.1369 0.7402 0.7240 0.7030
3 0.7193 0.6268 0.6695 0.9951 1.0019 0.9764 0.6845 0.6732 0.6642
4 0.7020 0.6073 0.6596 1.1020 1.1364 1.0489 0.7590 0.7397 0.6967
5 0.7966 0.6839 0.7631 1.1623 1.2082 1.0256 0.8945 0.8979 0.8581
6 1.2054 1.0858 1.1674 1.3938 1.5168 1.4414 1.1915 1.1963 1.1657
7 0.8209 0.7249 0.8058 1.1760 1.2602 1.1435 0.7999 0.8020 0.7914
8 1.0526 0.9342 0.9979 1.2587 1.2837 1.1041 1.0766 1.0483 0.9947
9 1.0315 0.8979 0.9617 1.2335 1.3099 1.2287 1.0079 0.9829 0.9457
Sum 8.1005 7.1485 7.6741 11.4367 12.0532 11.1932 8.5090 8.3750 8.0959
EGRU3
i
EGRU2
i
EGRU1
i
EGRU3
i
EGRU2
i
EGRU1
i
EGRU3
i
E
GRU2
i
EGRU1
i
1 0.9718 0.8977 1.1457 2.1442 2.2496 2.2774 1.1988 1.1511 1.2861
2 0.5922 0.5266 0.7010 1.1787 1.1653 1.1986 0.6606 0.6301 0.7223
3 0.6240 0.5631 0.7322 1.0778 1.0036 1.0708 0.6095 0.5845 0.6802
4 0.5709 0.5095 0.6293 1.0881 1.1330 1.1358 0.6224 0.5863 0.6684
5 0.6083 0.5181 0.7028 1.3581 1.4528 1.2105 0.6910 0.6191 0.7283
6 1.0879 1.0308 1.1805 1.5759 1.6477 1.6592 1.1063 1.0329 1.1135
7 0.6631 0.5915 0.6908 1.1824 1.2642 1.2480 0.6614 0.6132 0.6786
8 0.8777 0.7958 0.9119 1.3260 1.4557 1.3781 0.9272 0.8538 0.9174
9 0.8758 0.8304 0.8947 1.3269 1.4161 1.3662 0.9129 0.8486 0.8722
Sum 6.8717 6.2636 7.5890 12.2580 12.7880 12.5448 7.3902 6.9196 7.6669
