Resea ch A icle
Neu al Models o Impu a ion o Missing Ozone Da a in
Ai -Quali y Da ase s
Ángel A oyo ,1Ál a o He e o,1Ve ónica T icio,2
Emilio Co chado,3and MichaBWo niak4
1Depa men o Ci il Enginee ing, Uni e si y o Bu gos, Bu gos, Spain
2Depa men o Physics, Uni e si y o Bu gos, Bu gos, Spain
3Depa amen o de In o m´
a ica y Au om´
a ica, Uni e si y o Salamanca, Salamanca, Spain
4Depa men o Sys ems and Compu e Ne wo ks, W ocław Uni e si y o Science and Technology, W ocław, Poland
Co espondence should be add essed o ´
Angel A oyo; aa oyo[email p o ec ed]
Recei ed 5 Decembe 2017; Accep ed 31 Janua y 2018; Published 8 Ma ch 2018
Academic Edi o : Eloy I igoyen
Copy igh ©2018´
Angel A oyo e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License,
which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
Ozone is one o he pollu an s wi h mos nega i e e ec s on human heal h and in gene al on he biosphe e. Many da a-acquisi ion
ne wo ks collec da a abou ozone alues in bo h u ban and backg ound a eas. Usually, hese da a a e incomple e o co up and he
impu a ion o he missing alues is a p io i y in o de o ob ain comple e da ase s, sol ing he unce ain y and agueness o exis ing
p oblems o manage complexi y. In he p esen pape , mul iple- eg ession echniques and A i icial Neu al Ne wo k models a e
applied o app oxima e he absen ozone alues om i e explana o y a iables con aining ai -quali y in o ma ion. To compa e he
di e en impu a ion me hods, eal-li e da a om six da a-acquisi ion s a ions om he egion o Cas illa y Le´
on (Spain) a e ga he ed
in di e en ways and hen analyzed. The esul s ob ained in he es ima ion o he missing alues by applying hese echniques and
models a e compa ed, analyzing he possible causes o he gi en esponse.
1. In oduc ion and Rela ed Wo k
The ozone (O3) is an odo less, colo less, and highly eac i e
gas composed o h ee oxygen a oms. I is o med bo h
in he Ea h’s uppe a mosphe e (s a osphe ic ozone) and
a g ound le el ( oposphe ic ozone). I can be “good” o
“bad” o people’s heal h and o he en i onmen , depending
on i s concen a ion le els and loca ion in he a mosphe e
[1].
S a osphe ic O3is o med na u ally h ough he in e -
ac ion o sola Ul aViole (UV) adia ion wi h molecula
oxygen (O2). G ound-le el o “bad” ozone is no emi ed
di ec ly in o he ai . In he 1950s, hyd oca bons and ni o-
gen oxides (NO𝑥) we e iden i ied as he wo key chemi-
cal p ecu so s o pho ochemical smog and i s concomi an
high concen a ions o O3and o he pho ochemical oxidan
[2]. The majo i y o g ound-le el O3is o med om he
pho ochemical oxida ion o Vola ile O ganic Compounds
(VOCs) in he p esence o NO and o he NO𝑥. Signi ican
sou ces o VOCs a e chemical plan s, gasoline pumps, oil-
based pain s, au obody shops, and p in shops. NO𝑥 esul
p ima ily om high empe a u e combus ion, and i s mos
signi ican sou ces a e powe plan s, indus ial u naces and
boile s, and mo o ehicles [3].
1.1. Impo ance o Ozone. The O3exposi ion can cause dam-
age in di e en ways. In he s a osphe e, educed O3le els
asa esul o O
3laye deple ion mean less p o ec ion om
he sun’s ays and mo e exposu e o Ul aViole B (sho wa e)
ays (UVB) adia ion a he Ea h’s su ace [4]. The e ec s
on human heal h o he O3laye deple ion ha e been much
analyzed, inc easing he amoun o UVB ha eaches he
Ea h’s su ace. UVB causes nonmelanoma skin cance and
plays a majo ole in malignan melanoma de elopmen .
In addi ion, UVB has been linked o he de elopmen o
ce ain ca a ac s, nega i e e ec s in pa ien s wi h as hma, and
Hindawi
Complexi y
Volume 2018, A icle ID 7238015, 14 pages
h ps://doi.o g/10.1155/2018/7238015
2Complexi y
o he ch onic espi a o y disease. Wi h espec o g ound-
le el O3, and i s e ec s on human heal h, b ea hing O3can
igge a a ie yo heal hp oblems.Peoplewi has hmaand
o he ch onic espi a o y disease a e a la ge and g owing
segmen o he popula ion and a e also known o be especially
suscep ible o he e ec s o O3exposu e. On days wi h high
le els o O3, people wi h as hma end o expe ience inc eased
espi a o y symp oms [3]. The laye O3deple ion has also
nega i e e ec s on he p ocess o he de elopmen o plan s,
e ec s on he ma ine ecosys ems like a di ec educ ion in
phy oplank on p oduc ion, nega i e e ec s on ma e ials like
biopolyme s, and so o h. T oposphe ic O3does no p o ide
hep o ec i e unc ion ha i ul illsin hes a osphe e,being
high eac i i y. I s s ong oxidizing capaci y, when i s le els
ise abo e he na u al backg ound, can cause ad e se e ec s
in ma e ials (de i ed om i s co osi e e ec s), on ege a ion
and ecosys ems.
The p esen wo k ocuses on oposphe ic O3,whichis
a isk o heai quali y[3].Gi en heinc easeinO
3le els
in he oposphe e, i is cu en ly conside ed one o he mos
impo an a mosphe ic pollu an s.
1.2. Ozone Le el Moni o ing. A ound he wo ld he e a e nu-
me ous da a-acquisi ion ne wo ks o he measu emen o O3
le els and o he pollu an s, which consis o many s a ions
in di e en loca ions whe e di e en senso s measu e co e-
sponding magni udes. These ne wo k s a ions acqui e da a a
pe iodic in e als o ime (pe iods be ween en and i een
minu es a e he mos equen ones) bu equen ly appea
missing o co up ed da a. In Eu ope, da a a e conside ed as
co up ed when no mee ing he Council Decision 97/101/EC
o Janua y 27, 1997 [5], which es ablish a ecip ocal exchange
o in o ma ion and da a om ne wo ks and indi idual
s a ions measu ing ambien ai pollu ion wi hin he Membe
S a es. Some o hese ne wo ks p o ide in o ma ion abou he
alidi y o he da a, indica ing h ough codes i he da a is
co ec , i has no been possible o acqui e, o i is co up ,
bu in o he occasions his ype o in o ma ion is no p o ided
while he da a a e s ill missing. Some easons o such ailu es
ha ebeenpinpoin ed[6],namely,adamagedcable, helosso
p ope elec ical g ounding, hal -mel ed os o snow on he
dome, communica ions ailu e, and so o h. Some o hese
causes a e empo a y and may disappea spon aneously, bu
o he ones equi e he in e en ion o a main enance ask
o ce, and he e o e e o s pe sis o di e en pe iods o
ime.Theabsenceo alidda amayalsobedue o easons
such as he ollowing: mishandling o samples, low signal-
o-noise a io, measu emen e o , non esponse, o dele ed
abe an alue [7]. This is a p oblem o he analysis o he
in o ma ion coming om he measu emen ne wo ks, and
he impu a ion o hese missing da a [8] is necessa y. Any o
he a iables acqui ed in ne wo k s a ions may su e om
he p oblem o he absence o da a. I many da a a iables a e
omi ed o co up ed in he same eco d, he whole sample
mus be wi hd awn, when some models a e applied [9],
o subsequen asks such as con ol, classi ica ion, o ecas .
Al e na i ely, i da a o he same pollu an a e missing in
se e al adjacen ows, emo ing ha a iable may also be an
al e na i e solu ion. In conclusion, ha ing a comple e se o
da a is necessa y o pe o m a eliable s udy and o apply some
models ha canno deal wi h missing da a.
1.3. Missing Values and Rela ed Wo k. The s anda d classi i-
ca ion o missing da a phenomenon [10] includes di e en
si ua ions:
(i) Missing Comple ely A Random (MCAR), when he
p obabili y o an ins ance (case) ha ing a missing
alue o a a iable does no depend on ei he he
known alues o he missing da a.
(ii) Missing A Random (MAR), when he p obabili y o
an ins ance ha ing a missing alue o a a iable may
depend on he known alues bu no on he alue o
he missing da a i sel .
(iii) No Missing A Random (NMAR), when he p ob-
abili y o an ins ance ha ing a missing alue o a
a iable could depend on he alue o ha a iable.
As p e ious au ho s ha e poin ed ou , he complexi y
a ies be ween hese pa e ns o missing da a [11]. Usually, in
hecaseo ai -quali yda a,missing aluesa eassocia edwi h
MAR o MCAR. The ci cums ances ha may in e e e wi h
he acquisi ion o he da a a e many and no easily p edic able
[12].
To sol e he missing da a p oblem, a wide a ie y o
di e en me hodsha ebeenappliedup onow[8,10,13].
These impu a ion me hods (IMs) a e usually classi ied as
ollows:
(i) Single impu a ion (SI): he me hod ills in one alue
o each missing one [12].
(ii) Mul iple impu a ion (MI): mul iple simula ed alues
a e gene a ed a he same ime [14].
The uni a ia e and mul i a ia e impu a ion me hods
di e in which he app oxima ion o he missing alues o he
a iable unde s udy a e calcula ed om he es o he alues
o he e y same a iable (uni a ia e) o using alues o he
es o he a iables (mul i a ia e) [12].
Wi h he aim o educing he complexi y o o he MI
applied me hods [11], he p esen pape ocuses on single and
mul i a ia e impu a ion o he O3magni ude in ai pollu ion
da ase s. To do so, mul iple- eg ession (linea and nonlinea )
echniques oge he wi h A i icial Neu al Ne wo ks (ANN)
a e applied o eal-li e da ase s ob ained om public ai -
quali y ne wo ks.
Up o now, di e en A i icial-In elligence (AI) ech-
niques ha e been applied o impu a ion o missing da a. In
[7] impu a ion me hods based on six di e en echniques
a e compa ed: K-Nea es Neighbo s (KNN), Fuzzy K-Means
(FKM), Singula Value Decomposi ion, Bayesian P incipal
Componen Analysis (bPCA) and Mul iple Impu a ions by
Chained Equa ions. These me hods a e applied o ou
da ase s spli in o wo g oups o a ious sizes: small da ase s
(I is and E. coli) and la ge da ase s (b eas cance s 1 and 2).
bPCA and FKM appea ed o be he mos obus impu a ion
me hods in he es ed condi ions.
In [15] he accu acy o di e en impu a ion me hods is
e alua ed: MissFo es (MF) and Mul iple Impu a ion based
Complexi y 3
on Expec a ion-Maximiza ion (MIEM), along wi h wo o he
impu a ion me hods: Sequen ial Ho -Deck and Mul iple
Impu a ion based on Logis ic Reg ession (MILR). The mod-
els a e applied o e ou een bina y da ase s, wi h a ange o
missing da a a es be ween 5% and 50%. The esul s om 10-
old C oss-Valida ion (CV) show ha he pe o mance o he
impu a ion me hods a ies subs an ially be ween di e en
classi ie s and a di e en a es o missing alues.
Al hough many impu a ion me hods ha e been p oposed
up o now, scan a en ion has been paid o alida e ANN o
such a ask, aking ad an age o hei eg ession capabili y
[16]. Among hese p e ious s udies, ANN ha e been applied
o he es ima ion o los alues in [17], whe e he main goal
is iden i ying Lea ning Disabili ies (LD) in child en a ea ly
s ages. In [18], au ho s p oposed a SI app oach elying on a
Mul ilaye Pe cep on (MLP) whose aining is conduc ed
wi h di e en lea ning ules, and a MI app oach based on
he combina ion o MLP and KNN. 24 eal and simula ed
da ase s om he UCI eposi o y, he P omise eposi o y, and
mlda a.o g we e exposed o a pe u ba ion expe imen wi h
andom gene a ion o mono one missing da a pa e n.
In [19] six di e en ypes o ANN a e p oposed as
IM: MLP and i s a ia ions ( he Time-Lagged Feed o wa d
Ne wo k (TLFN)), he Gene alized Radial-Basis-Func ion
(GRBF) ne wo k, he Recu en Neu al Ne wo k (RNN), and
i s a ia ions ( he Time Delay Recu en Neu al Ne wo k
(TDRNN)). Addi ionally, he Coun e p opaga ion Fuzzy-
Neu al Ne wo k (CFNN) along wi h di e en op imiza ion
me hods is applied o in illing missing daily o al p ecipi a-
ion and ex eme empe a u e se ies om 15 wea he s a ions.
The s anda d MLP and TLFN appea o p o ide he mos
accu a e econs uc ion o missing p ecipi a ion and daily
ex eme empe a u es eco ds wi h esul s o he Rco ela-
ion coe icien be ween he obse ed and he econs uc ed
daily se ies close o 1.
In [20] a no el nonpa ame ic algo i hm named Gen-
e alized eg ession neu al ne wo k Ensemble o Mul iple
Impu a ion (GEMI) is p oposed. Addi ionally, a SI e sion o
his app oach (GESI) is p oposed. The algo i hms we e es ed
on 98 syn he ic and eal-wo ld da ase s. All simula ion esul s
show head an ageso GEMIascompa edwi hcon en ional
algo i hms. GEMI has hea y memo y s o age equi emen s
bu ou pe o med o he SI algo i hms.
In [21] i een eal and simula ed da ase s a e exposed o a
pe u ba ion expe imen , based on he andom gene a ion o
missing alues. Se e al a chi ec u es and lea ning algo i hms
o he MLP a e es ed and compa ed wi h h ee classic
impu a ion p ocedu es: mean/mode impu a ion, eg ession,
and ho -deck [22].
In [23] a me hodology based on Gaussian Mix u e Model
(GMM) and Ex eme Lea ning Machine (ELM) is de eloped
and es edonsomeda ase s om heUCIMachineLea ning
Reposi o y and he LIACC eg ession eposi o y. GMM is
used o model he da a dis ibu ion which is adap ed o
handle missing alues, while ELM enables de ising a Mul iple
Impu a ion s a egy o inal es ima ion. The combina ion
o GMMandELMisshown obesupe io inalmos all
es ed cases o e he me hod based on condi ional mean
impu a ion.
In[24]aSIapp oach elyingonaMLPandaMIapp oach
based on he combina ion o MLP and K-NN is p oposed.
The models a e applied o 18 eal and simula ed da ase s
like domains such as biology, medicine, chemis y, elec on-
ics, social su eys, census, and business. Fo da ase s wi h
only quan i a i e a iables MIMLP model p o ided he bes
esul s, wi h IMLP being he bes me hod o da ase s wi h
ca ego ical a iables.
In [25] a wo-s age hyb id model o illing he missing
aluesusing uzzyc-meansclus e ingandMLPisp oposed.
I is applied o a Wine da ase wi h a 1% o 5% o gene a ed
missing alues and he accu acy o he model is checked using
heMeanAbsolu ePe cen ageE o (MAPE).TheMAPE
ob ained o s age 2 (MLP eg ession o he ob ained da ase
as a esul o applying uzzy c-means in s age 1) is 4.95% o
1% missing- alue eco ds and 8.36% o 5% missing- alue
eco ds.
In he case o ai -quali y da a, ew impu a ion me hods
ha e been p oposed up o now. In [13], an impo an se o
SI: Lis wise, Uncondi ional mean, Modi ied Median, P in-
cipal Componen -based, Expec a ion-Maximiza ion (EM)
(Regula ized-EM), and MI me hods a e applied o h ee
da ase s wi h he mos impo an pollu an a iables (NO,
NO2,NO
𝑥,CO,O
3, PM10, and PM2.5) and a pe cen age o
missing da a among he 3.85% and he 23.52% depending
on he yea . Missing da a o he eigh a iables a e impu ed
in o de o assess he e ec i eness o he me hods applied.
In gene al, MI ends o yield mo e sca e ed alues han i s
coun e pa s, mainly when he a iables ha e many oids and
hey co ela e poo ly o he o he a iables like CO wi h 43.5%
o missing da a in 2006 and hey co ela e poo ly o he o he
a iables.
In [11] some me hods o he impu a ion o missing ai -
quali y da a a e compa ed: in he con ex o SI (linea , spline,
and nea es neighbo in e pola ions), MI ( eg ession-based
impu a ion, mul i a ia e nea es neighbo , Sel -O ganizing
Maps (SOM), and Mul ilaye Backp opaga ion (MLBP) ne s)
and hyb id me hods o he a o emen ioned. The da ase uses
he mos common pollu an s: NO𝑥,NO
2,O
3, PM10, SO2,
and CO concen a ions, all on a ime-scale o one pe hou
(hou ly a e aged), oge he wi h ou me eo ological pa am-
e e s. The pe o mance o he p oposed uni a ia e missing
da a in e pola ion was limi ed, and in gene al hey we e able
o illonly e ysho gapso con iguousmissingda a.The
gene al pe o mance o he applied impu a ion me hods was
ai good when conside ing he pollu an s (NO𝑥,NO
2,O
3,
PM10, SO2, and CO) which a e he mos impo an ones in
e ms o ai -quali y modelling, bu no so good ega ding
me eo ological a iables. The esul s sugges ed ha SOM
andMLBPa e heme hodso choice o ai -quali yda a
impu a ion and e en be e esul s can be achie ed by using
he MI.
1.4. Main Con ibu ions. The main con ibu ions o his wo k
a e as ollows:
(i) Deep s udy o he eal-li e human heal h p o ec ion
ask in Spanish egion o Cas illa y Le´
on.
(ii) Mul isenso o O3da a analysis.
4Complexi y
(iii) Expe imen al e alua ion o he p oposed app oach
based on mul iple- eg ession echniques oge he
wi h ANN models.
To he bes o au ho s knowledge, his is he i s app oach
o impu a ion me hods o O3based on bo h MLP and Radial-
Basis-Func ion Ne wo ks.
The es o his pape is o ganized as ollows. Sec ion 2
p esen s he echniques and models applied. Sec ion 3 de ails
he eal-li e case s udy ha is add essed in p esen wo k,
while Sec ion 4 desc ibes he expe imen s and esul s. Finally,
Sec ion 5 se s ou he main conclusions and u u e wo k.
2. Reg ession Techniques and ANN Models
In o de o ill missing o co up ed alues o O3in high
dimensional da ase s wi h ai -quali y in o ma ion, wo e-
g ession echniques and wo ANN models ha e been applied
in p esen s udy. This se o echniques applied as impu a ion
me hods is desc ibed in his sec ion.
2.1. Reg ession Techniques. Linea eg ession a emp s o
model he ela ionship be ween wo a iables by i ing a
linea equa ion o obse ed da a. One a iable is conside ed
o be an explana o y a iable, and he o he is conside ed o
be a dependen a iable [26].
The gene al pu pose o mul iple eg essions [27] is o
lea n mo e abou he ela ionship be ween se e al indepen-
den o p edic o a iables and a dependen o c i e ion
a iable.
2.1.1. Mul iple Linea Reg ession. Mul iple linea eg ession
(MLR) a emp s o model he ela ionship be ween wo o
mo e explana o y a iables and a esponse a iable by i ing
a linea equa ion o obse ed da a [28]. E e y alue o he
independen a iable (𝑥) is associa ed wi h a alue o he
dependen a iable (𝑦). The popula ion eg ession line o 𝑝
explana o y a iables
𝑥1,𝑥2,...,𝑥𝑝(1)
isde ined obe
𝑢𝑦=𝛽0+𝛽1𝑥1+𝛽2𝑥2+⋅⋅⋅+𝛽𝑝𝑥𝑝.(2)
This line desc ibes how he mean esponse 𝑢𝑦changes
wi h he explana o y a iables. The obse ed alues o y
a y abou hei means 𝑢𝑦anda eassumed oha e hesame
s anda d de ia ion 𝜎.The i ed alues𝑏0,𝑏1,...,𝑏𝑝es ima e
he pa ame e s 𝛽0,𝛽1,...,𝛽𝑝o he popula ion eg ession
line.
Since he obse ed alues o y a y abou hei means
u𝑦, he mul iple- eg ession models include a e m o his
a ia ion. The model is exp essed as DATA = FIT + RESID-
UAL, whe e he “FIT” e m ep esen s he exp ession 𝛽0+
𝛽1𝑥1+𝛽2𝑥2+⋅⋅⋅+𝛽𝑝𝑥𝑝. The “RESIDUAL” e m ep esen s
he de ia ions o he obse ed alues 𝑦 om hei means 𝑢𝑦,
which a e no mally dis ibu ed wi h mean 0 and a iance 𝜎.
The no a ion o he model de ia ions is 𝜀.
Fo mally, he model o mul iple linea eg ession, gi en
nobse a ions, is [28]
𝑌𝑖=𝛽0+𝛽1𝑥𝑖1 +𝛽2𝑥𝑖2 +⋅⋅⋅+𝛽𝑝𝑥𝑖𝑝 +𝜀𝑖
o 𝑖=1,2,...,𝑛. (3)
2.1.2. Mul iple Nonlinea Reg ession. AMul ipleNonlinea
Reg ession (MN-LR) is a o m o eg ession analysis in which
obse a ional da a a e modelled by a unc ion which is a
nonlinea combina ion o he model pa ame e s and depends
on one o mo e independen a iables [29]. The da a a e i ed
by a me hod o successi e app oxima ions.
The pa ame e s can ake he o m o an exponen ial,
igonome ic, powe , o any o he nonlinea unc ion. To
de e mine he nonlinea pa ame e es ima es, an i e a i e
algo i hm is ypically used.
𝑦=𝑓(𝑋,𝐵)+𝜀, (4)
whe e 𝐵 ep esen s nonlinea pa ame e es ima es o be
compu ed, 𝑋is he dependen o c i e ion a iables, and
𝜀 ep esen s he e o e ms.
2.2. A i icial Neu al Ne wo ks. A i icial Neu al Ne wo ks
(ANN), also known as A i icial Neu al Sys ems (ANS),
connec ionis sys ems, adap i e ne wo ks, and dis ibu ed
and pa allel p ocessing a e simpli ied models o na u al
neu al sys ems. The ollowing de ini ion, gi en by Hech -
Nielsenin1989[30], o malizes heconcep o ANN:
An ANN is a pa allel p ocessing compu e sys em
dis ibu ed, consis ing o a se o elemen a y p o-
cessing uni s equipped wi h a small local memo y
and in e connec ed in a ne wo k h ough connec-
ions wi h associa ed weigh s. Each p ocessing uni
has one o mo e inpu connec ions and a single
ou pu connec ion ha links o many colla e al
connec ions as desi ed. All p ocessing associa ed
wi h an elemen a y uni is a local, i.e. depends
only on he alues ha ake inpu signals om he
uni and he in e nal s a e o he same.
2.2.1. Mul ilaye Pe cep on (MLP). TheMLPconsis so a
sys em o simple in e connec ed neu ons o nodes. The nodes
a e connec ed by weigh s and ou pu signals which a e a
unc ion o he sum o he inpu s o he node modi ied
by a simple nonlinea ans e , o ac i a ion, unc ion. The
a chi ec u e consis s o se e al laye s o neu ons; he inpu
laye se es opass heinpu ec o o hene wo k.The e ms
“inpu ec o s” and “ou pu ec o s” e e o he inpu s and
ou pu so heMLPandcanbe ep esen edassingle ec o s
[31]. A MLP may ha e one o mo e hidden laye s and inally
anou pu laye .MLPa e ullyconnec ed,wi heachnode
connec ed o e e y node in he nex and p e ious laye .
To pe o m a comp ehensi e compa ison, he MLP is
ained wi h he ollowing algo i hms:
(1) Le enbe g-Ma qua d backp opaga ion (LM)
Complexi y 5
(2) G adien Descen wi h momen um and adap i e
lea ning a e backp opaga ion (GDX) [32]
(3) Ba ch T aining wi h weigh and bias lea ning ules
(TB)
(4) Scaled Conjuga e G adien backp opaga ion (SCG)
(5) Bayesian Regula iza ion backp opaga ion (BR).
2.2.2. Radial-Basis-Func ion Ne wo ks (RBFN). In a RBFN
[33] each uni in he hidden laye o his ne wo k has i s
own cen oid, and, o each inpu ec o 𝑥=(𝑥𝑙,𝑥2,...,𝑥𝑛),
i compu es he dis ance be ween 𝑥and i s cen oid. I s
ou pu o heuni iscalcula edasanonlinea unc iono his
dis ance.
Assuming ha he e a e inpu nodes and mou pu
nodes, he o e all esponse unc ion wi hou conside ing
nonlinea i y in an ou pu node has he ollowing o m [34]:
𝑀
∑
𝑖=1𝑊𝑖∗𝐾(𝑥−𝑧𝑖
𝜎𝑖)=𝑀
∑
𝑖=1𝑊𝑖∗𝑔(󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑥−𝑧𝑖󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨
𝜎𝑖), (5)
whe e 𝑀∈Nis he numbe o uni s in he hidden laye ,
𝑊𝑖∈R𝑚is he ec o o weigh s linking he 𝑖 h hidden-laye
uni o he ou pu nodes, xis an inpu ec o , Kis a adially
symme ic ke nel unc ion o a uni in he hidden laye , z𝑖
and 𝜎𝑖a e he cen oid and smoo hing ac o o he 𝑖 h ke nel
node, espec i ely, and 𝑔:[0,∞)→Ris a unc ion called he
ac i a ion unc ion, which cha ac e izes he ke nel shape.
3. Case S udy
In p esen s udy, da a om ai -quali y s a ions in Cas illa y
Le´
on (CyL) a e analyzed. CyL is a Spanish egion loca ed a
he no h-cen e o he Ibe ian Peninsula. I is composed o
nine p o inces and i is he mos ex ensi e egion o Spain
wi ha o alsu aceo 94,226squa ekilome e sand hesix h
wi h mo e popula ion: 2,435,797 habi an s. G oss Domes ic
P oduc (GDP) in CyL ep esen s he 5.3% o coun y’s
GDP [35]. Clima e in CyL app oaches wha is known as
he con inen al ocean, cha ac e ized by cold win e s and ho
summe s wi h sho sp ing and au umn pe iods.
CyL egion p o ides a wide ne wo k o s a ions [36]
o he acquisi ion o ai -quali y da a. These da a a e public
a ailable acco ding o he Open Da a Ini ia i e om he
Spanish Go e nmen [37].
S a ions om his ne wo k ha e some in e es ing cha ac-
e is ics:
(1) S a ions a e classi ied in ypes: u ban, backg ound,
ando ien ed o he ege a ionp o ec ion[36].
(2) These s a ions collec he undamen al ai -quali y
pollu an s, and among hem is he O3,whichis he
objec i e pollu an o his s udy. Daily a e ages da a
[38] o each pollu an a e p o ided in each loca ion.
(3) This da a p esen s emp y o co up ed da a in all o i s
a iablesinsome owsandina easonablepe cen age
o be es ima ed.
Figu e 1: Loca ion o he six selec ed s a ions in CyL, by Google
Maps.
In hep esen s udy,pollu an da a eco dedinsix
di e en s a ions om he CyL ne wo k a e analyzed. Daily
da a a e ages om yea s 2000 o 2008 ha e been selec ed.
Fo some pe iods o ime wi hin he selec ed ime window,
da a a e no a ailable o all he a iables and, hus, he
whole example is ejec ed o he s udy. Th ee o he s a ions
a e loca ed in he cen e o he ci ies and labeled as u ban
s a ions; hese s a ions a e o ien ed o he p o ec ion o
he human heal h. The o he h ee s a ions a e backg ound
s a ions and a e also o ien ed o he p o ec ion o he
human heal h. These s a ions measu e a g ea e numbe
o pollu an s han he o he ype o s a ions and a e he
mos impo an ones in e ms o ai quali y, and many o
hema eno collec eda hes a ions o he ege a ion
p o ec ion. This ac is impo an o he de e mina ion o he
O3missing alues,as hisgasisespeciallyha m ul o human
heal h.
The h ee u ban s a ions conside ed in p esen s udy a e
as ollows:
(1) ´
A ila. “Bus S a ion” s a ion. Geog aphical coo di-
na es: 40.65914, −4.68237; 1150 me e s abo e sea le el
(masl).
(2) A anda de Due o. “Ja dines de Don Diego” s a ion.
Geog aphical coo dina es: 41.67111, −3.68388; 801
masl.
(3) Le´
on.“A da.SanIgnaciodeLoyola”s a ion.Geo-
g aphical coo dina es: 42.60388, −5.58722; 838 masl.
The h ee backg ound s a ions a e as ollows:
(1) Bu gos. “Fuen es Blancas” s a ion. Geog aphical co-
o dina es: 42.33611, −3.63611; 929 masl.
(2) Sego ia. “Acueduc o” s a ion. Geog aphical coo di-
na es: 40.95555, −4.11055; 951 masl.
(3) Medina del Campo (Valladolid). “Bus S a ion” s a ion.
Geog aphical coo dina es: 41.31638, −4.90916; 721
masl.
Figu e1shows heloca iono hesixselec eds a ions ha
ha e been s udied in he p esen pape .
The pollu an s ga he ed in he abo e-men ioned s a ions
andanalyzedin hep esen s udya eas ollows:

6Complexi y
Table 1: Co ela ion ma ix o he six a iables in he da ase .
O3CO NO NO2PM10 SO2
O31.000 −0.123 −0.161 −0.202 0.072 −0.013
CO −0.123 1.000 0.360 0.412 0.358 0.299
NO −0.161 0.360 1.000 0.540 0.233 0.330
NO2−0.202 0.412 0.540 1.000 0.330 0.257
PM10 0.072 0.358 0.233 0.330 1.000 0.251
SO2−0.013 0.299 0.330 0.257 0.251 1.000
Table 2: Pe cen age o missing and co up ed da a o each one o he analyzed a iables.
O3NO NO2CO PM10 SO2
Missing 8.104% 8.020% 8.034% 8.554% 9.131% 8.196%
Co up ed 1.857% 2.047% 1.815% 2.926% 2.413% 1.801%
To al 9.961% 10.067% 9.849% 11.480% 11.544% 9.997%
(1) Ozone (O3), 𝜇g/m3, seconda y pollu an . See Sec-
ion 1.
(2) Ca bon monoxide (CO), mg/m3,p ima ypollu an .
I is an odo less, colo less gas o med by he incom-
ple e combus ion o uels. When people a e exposed
o CO gas, he CO molecules will displace he oxygen
in hei bodies and lead o poisoning [39].
(3) Ni ic oxide (NO), 𝜇g/m3,p ima ypollu an .NOis
a colo less gas which eac s wi h ozone unde going
apid oxida ion o NO2, p edominan in he a mo-
sphe e [39].
(4) Ni ogen dioxide (NO2), 𝜇g/m3,p ima ypollu an .
F om he s andpoin o heal h p o ec ion, ni ogen
dioxide has se exposu e limi s o long and sho
du a ion [39].
(5) Pa icula e ma e (PM10), 𝜇g/m3,p ima ypollu an .
These pa icles emain s able in he ai o long pe i-
ods o ime wi hou alling o he g ound and can be
mo ed signi ican dis ances by he wind. I is de ined
by he ISO as ollows: “pa icles which pass h ough
a size-selec i e inle wi h a 50% e iciency cu -o a
10 𝜇m ae odynamic diame e . PM10 co esponds o
he ‘ ho acic con en ion’ as de ined in ISO 7708:1995,
Clause 6” [40].
(6) Sulphu dioxide (SO2), 𝜇g/m3,p ima ypollu an .I
is a gas. I smells like bu n ma ches. I s smell is
also su oca ing. SO2is p oduced by olcanoes and in
a ious indus ial p ocesses. In he ood indus y, i
is also used o p o ec wine om oxygen and bac e ia
[39].
P ima y pollu an s a e injec ed in o he a mosphe e di-
ec ly. Seconda y pollu an s a e o med in he a mosphe e
h ough chemical and pho ochemical eac ions om he
p ima y pollu an s [36].
All da a om hese six a iables we e no malized o he
s udy. On he o he hand, all o hem a e highly deco ela ed.
Table 1 shows he co ela ion ma ix o he six pollu an s o
he case s udy.
I is wo h men ioning ha O3is he mos independen
pollu an , as i s co ela ion coe icien s wi h he es o he
a iables a e close o ze o.
The e a e a o al o 13,526 samples, as one sample pe
day (daily a e age) was collec ed o he wel e mon hs o
e e y yea , be ween yea s 2000 and 2008, in he six s a ions
analyzed in his s udy. Missing o co up ed da a appea in all
he a iables in some ows, which a e omi ed o he s udy.
Table 2 shows he pe cen age o missing o co up ed da a
p esen edineach a iablein hewholeda ase .
All hesampleswi ha leas onemissingo co up ed
alue we e emo ed om he da ase .
4. Expe imen s, Resul s, and Discussion
The main a ge o his pape is o ill missing O3 alues in ai
pollu ion da ase s. To do so, se e al impu a ion me hods a e
comp ehensi ely compa ed as desc ibed below.
4.1. Expe imen al Se ings. The impu a ion me hods desc ib-
ed in Sec ion 2 a e applied o di e en da ase s, all o hem
wi h he six a iables desc ibed in Sec ion 3:
(1) The Whole Da ase (WD), comp ising he 13,526 sam-
ples: esul s o his da ase s a e shown in Sec ion 4.2.
(2) The Season Da ase (SD): samples in WD a e spli
in ou subse s acco ding o he ou seasons o he
yea : sp ing (3,453 samples), summe (3,349 samples),
au umn (3,295 samples), and win e (3,429 samples).
Resul s o his da ase a e shown in Sec ion 4.3.
(3) The Type s a ion Da ase (TD): samples in WD a e
spli in o wo subse s acco ding o he ype o he
s a ion whe e he da a come om; “u ban” (6,763
samples) o “backg ound” (6,763 samples). Resul s o
his da ase s a e shown in Sec ion 4.4.
Fo he h ee da ase s, bo h s a is ical and neu al impu a-
ion me hods we e applied and he pe o mance is calcula ed
h ough n- old C oss-Valida ion (CV). The main idea behind
CV is o spli da a, no mally many imes, o es ima ing he
Complexi y 7
Table 3: Linea eg ession and nonlinea eg ession esul s o he WD.
Me hod MSE Time (s)
Mean STD Mean STD
MLR 5.490𝐸−06 2.311𝐸−08 0.089 0.216
MN-LR 5.415E −06 2.437𝐸−08 2.143 0.254
Table 4: Radial-basis unc ion ne wo k esul s o he WD.
#o neu ons MSE Time (s)
Mean STD Mean STD
10 5.104E −06 2.723𝐸−08 0.050 1.091
30 5.108𝐸−06 1.273𝐸−08 0.050 1.091
50 5.105𝐸−06 2.513𝐸−08 0.047 0.098
isk, e o , o pe o mance o each algo i hm. Pa o da a ( he
aining samples) is used o aining each algo i hm, and he
emaining pa ( he alida ion samples) is used o alida ing
he algo i hm(s). Then, CV selec s he algo i hm wi h he
smalles es ima ed isk [41]. CV p e en s om o e i ing
because he aining sample is independen o he alida ion
sample. The numbe o he 𝑘pa ame e s (da a pa i ions) was
10 o all he expe imen s in he p esen s udy. I means ha
90% o he da a a e used o aining and 10% o alida ion.
In he case o neu al models, he aining p ocess is epea ed
en imes (one o each old). In he case o MLP, aining is
also epea ed o each aining algo i hm (see Sec ion 2.2).
Fo all he expe imen s he Mean and he S anda d De ia ion
(STD) o he Mean Squa e E o (MSE) o he en olds
a ep esen edinTables3–11.TheMeanand heSTDo he
execu ion ime (in seconds) a e also p esen ed in Tables 3–11
o he 10 olds.
Fo MLP and RBFN di e en ne wo k opologies ha e
been applied: combina ions o 10, 20, and 30 neu ons in he
hidden laye . Addi ionally, in he case o MLP, he model is
ained 10 imes wi h he same combina ion o pa ame e s
o educe he e ec o andomness and ge mo e s a is ically
signi ican esul s.
4.2. Resul s om he Whole Da ase . In his sec ion, esul s in
e ms o MSE and execu ion ime when applying MLR, MN-
LR, RBFN, and MLP o he WD a e p esen ed.
In Tables 3 and 4, i can be obse ed ha he MSE Mean
alues o he de e mina ion o he O3a e e y simila o
he h ee applied me hods (MLR, MN-LR, and RBFN). In he
case o RBFN, sligh ly lowe alues o MSE a e ob ained, wi h
he lowes one being ob ained wi h 10 neu ons in he hidden
laye . Rega ding execu ion imes, he MN-LR me hod u ns
ou o be he slowes and RBFN he quicke . The high alues
o STD o he un imein hecaseo RBFNa edue o he ac
ha i g ea ly a ies om one old o he o he s.
As i can be seen in Table 5, he LM, SCG, and BR aining
algo i hms p esen he lowes alues o MSE Mean in all cases
(10, 30, and 50 neu ons) and e y close o hose shown in
Tables 3 and 4. The lowes alue o MSE was ob ained wi h
heLMlea ningalgo i hmand50neu ons.Thelea ning
algo i hm ha a ained he wo s esul s (in e ms o MSE)
is GDX. Wi h espec o execu ion ime, he SCG algo i hm
a ained he bes esul s, while LM and BR a e he second
bes ones, while TB was he slowes o he i e algo i hms.
Ob iously, he aining algo i hms ake mo e ime when 50
neu ons a e de ined in he hidden laye , he TB algo i hm
being he one wi h g ea es e ec .
4.3. Resul s om he Season Da ase . In Tables 6–8 esul s o
applying MLR, MN-LR, RBFN, and MLP o subse s wi h da a
om he ou seasons o he yea (sp ing, summe , au umn,
and win e ) a e p esen ed.
In Tables 6 and 7 he 3 me hods p esen simila alues
in MSE Mean, and he lowes MSE Mean is achie ed by he
RBFN wi h 50 neu ons in he hidden laye o he summe
season. The MSE Mean alues a e highe han ha obse ed
o heWD.Theseasono heyea wi h helowes alues
o MSE Mean is he summe . One eason may be ha he e
a e ew a ia ions in pollu ion condi ions du ing summe
ime. This is due o he small a ia ion in wea he condi ions
du ing summe as well as low indus ial ac i i y and a ic in
u ban a eas due o aca ion ime. Fu he mo e, co ela ion
coe icien s in mo e han 20 pollu an s analyzed in [42] a e
highe o measu emen s in he summe compa ed wi h
co ela ions o measu emen s o e all days combined. The
season o he yea wi h he wo s esul s in he calcula ion
o he MSE has been he au umn in he case o he wo
eg ession echniques and RBFN, al hough he di e ences
be ween he h ee seasons (sp ing, summe , and au umn)
is no signi ican . In e ms o execu ion ime, i is p obed
once again ha MN-LR is he slowes me hod, while RBFN
is he quickes one, e u ning e y simila esul s o he ou
seasons o he yea .
In Table 8, simila ly o Table 5, he aining algo i hms
ha achie e he bes esul s in e ms o MSE Mean a e LM,
SCG, and BR. LM achie ed he bes alue o MSE Mean in
10 o he 12 cases shown in Table 8, being exceeded by BR
by a minimum alue o he win e and sp ing seasons wi h
a con igu a ion o 10 neu ons. GDX eco ds he wo s MSE
aluesin he12casesshowninTable8.Again, hebes MSE
Mean is ob ained o he summe season, educing he MSE
Mean in compa ison wi h hose egis e ed by RBFN. The
season o he yea wi h he wo s esul s in he calcula ion o
8Complexi y
Table 5: Mul ilaye pe cep on esul s o he WD.
# o neu ons T aining algo i hm MSE Time (s)
Mean STD Mean STD
10
LM 4.731𝐸−06 5.143𝐸−08 0.070 0.287
GDX 1.129𝐸−04 6.825𝐸−05 0.317 0.287
TB 5.889𝐸−05 4.973𝐸−05 0.642 0.022
SCG 5.216𝐸−06 1.514𝐸−07 0.060 0.001
BR 4.775𝐸−06 1.092𝐸−07 0.074 0.003
30
LM 4.599𝐸−06 1.015𝐸−07 0.102 0.442
GDX 4.045𝐸−04 3.523𝐸−04 0.481 0.442
TB 4.223𝐸−05 2.087𝐸−05 1.420 0.025
SCG 5.162𝐸−06 1.19𝐸−07 0.063 0.001
BR 4.727𝐸−06 5.667𝐸−08 0.102 0.005
50
LM 4.512E −06 8.952𝐸−08 0.160 1.080
GDX 1.541𝐸−04 3.722𝐸−04 0.648 1.080
TB 4.812𝐸−05 3.032𝐸−05 2.156 0.051
SCG 5.014𝐸−06 8.322𝐸−08 0.068 0.001
BR 4.731𝐸−06 1.099𝐸−07 0.161 0.010
Table 6: Linea eg ession and nonlinea eg ession esul s o he Season Da ase .
Subse Me hod MSE Time (s)
Mean STD Mean STD
Sp ing MLR 1.895𝐸−05 1.406𝐸−07 0.085 0.208
MN-LR 1.895𝐸−05 1.242𝐸−07 0.169 0.298
Summe MLR 2.101𝐸−05 1.447𝐸−07 0.085 0.215
MN-LR 1.343E −05 1.365𝐸−07 0.665 0.321
Au umn MLR 2.106𝐸−05 2.079𝐸−07 0.085 0.208
MN-LR 2.101𝐸−05 1.447𝐸−07 0.677 0.259
Win e MLR 1.895𝐸−05 1.406𝐸−07 0.088 0.214
MN-LR 1.895𝐸−05 1.242𝐸−07 0.168 0.274
Table 7: Radial-basis unc ion ne wo k esul s o he Season Da ase .
Subse # o neu ons MSE Time (s)
Mean STD Mean STD
Sp ing
10 1.845𝐸−05 1.687𝐸−07 0.046 0.096
30 1.847𝐸−05 1.06𝐸−07 0.050 0.098
50 1.846𝐸−05 1.297𝐸−07 0.047 0.096
Summe
10 1.308𝐸−05 1.549𝐸−07 0.045 0.098
30 1.310𝐸−05 1.389𝐸−07 0.045 0.096
50 1.306E −05 1.344𝐸−07 0.047 0.096
Au umn
10 1.986𝐸−05 1.176𝐸−07 0.046 0.096
30 1.987𝐸−05 2.323𝐸−07 0.046 0.097
50 1.987𝐸−05 2.199𝐸−07 0.045 0.095
Win e
10 1.845𝐸−05 1.687𝐸−07 0.046 0.100
30 1.847𝐸−05 1.060𝐸−07 0.045 0.093
50 1.846𝐸−05 1.297𝐸−07 0.046 0.096
Complexi y 9
Table 8: Mul ilaye pe cep on esul s o he Season Da ase .
Subse # o neu ons T aining algo i hm MSE Time (s)
Mean STD Mean STD
Sp ing
10
LM 1.696𝐸−05 2.550𝐸−07 0.063 0.086
GDX 3.298𝐸−04 8.39𝐸−04 0.183 0.086
TB 5.870𝐸−05 2.590𝐸−05 0.383 0.049
SCG 1.959𝐸−05 3.439𝐸−07 0.056 0.001
BR 1.695𝐸−05 1.988𝐸−07 0.076 0.015
30
LM 1.531𝐸−05 4.107𝐸−07 0.068 0.217
GDX 6.652𝐸−04 1.051𝐸−03 0.210 0.217
TB 1.069𝐸−04 3.909𝐸−05 0.473 0.019
SCG 1.870𝐸−05 2.857𝐸−07 0.055 0.005
BR 1.562𝐸−05 3.915𝐸−07 0.071 0.225
50
LM 1.473𝐸−05 4.138𝐸−07 0.080 0.181
GDX 6.286𝐸−04 1.200𝐸−03 0.253 0.181
TB 1.564𝐸−04 1.402𝐸−04 0.770 0.075
SCG 1.809𝐸−05 3.768𝐸−07 0.056 0.001
BR 1.580𝐸−05 3.697𝐸−07 0.089 0.490
Summe
10
LM 1.009𝐸−05 1.203𝐸−07 0.059 0.0663
GDX 4.718𝐸−04 4.791𝐸−04 0.166 0.0663
TB 7.65𝐸−05 6.231𝐸−05 0.355 0.0396
SCG 1.217𝐸−05 2.749𝐸−07 0.053 0.0007
BR 1.010𝐸−05 1.436𝐸−07 0.064 0.0059
30
LM 9.171𝐸−06 2.713𝐸−07 0.065 0.440
GDX 8.070𝐸−04 1.446𝐸−03 0.202 0.440
TB 9.117𝐸−05 5.051𝐸−05 0.500 0.047
SCG 1.118𝐸−05 4.118𝐸−07 0.056 0.001
BR 9.822𝐸−06 3.107𝐸−07 0.073 0.004
50
LM 8.673E −06 3.284𝐸−07 0.083 0.225
GDX 4.572𝐸−04 9.864𝐸−04 0.253 0.225
TB 1.269𝐸−04 6.38𝐸−05 0.743 0.019
SCG 1.089𝐸−05 1.561𝐸−07 0.057 0.001
BR 9.851𝐸−06 2.165𝐸−07 0.085 0.003
Au umn
10
LM 1.622𝐸−05 2.146𝐸−07 0.061 0.101
GDX 1.598𝐸−04 3.096𝐸−04 0.168 0.101
TB 7.248𝐸−05 3.589𝐸−05 0.351 0.058
SCG 1.904𝐸−05 2.617𝐸−07 0.055 0.001
BR 1.628𝐸−05 7.680𝐸−07 0.071 0.009
30
LM 1.495𝐸−05 3.564𝐸−07 0.067 0.196
GDX 1.045𝐸−03 1.520𝐸−03 0.204 0.196
TB 1.09𝐸−04 6.737𝐸−05 0.506 0.048
SCG 1.808𝐸−05 2.756𝐸−07 0.054 0.001
BR 1.522𝐸−05 3.456𝐸−07 0.069 0.001
50
LM 1.401𝐸−05 1.926𝐸−06 0.079 0.307
GDX 5.676𝐸−04 1.700𝐸−03 0.240 0.307
TB 1.005𝐸−04 6.447𝐸−05 0.734 0.029
SCG 1.758𝐸−05 5.509𝐸−07 0.054 0.001
BR 1.559𝐸−05 6.591𝐸−07 0.083 0.103