Semiparametric prediction models for variables related with energy production

González-Man eiga e al. Jou nal o Ma hema ics in Indus y (2018) 8:7
h ps://doi.o g/10.1186/s13362-018-0049-0
RESEARCH Open Access
Semipa ame ic p edic ion models o
a iables ela ed wi h ene gy p oduc ion
Wenceslao González-Man eiga1,2†, Manuel Feb e o-Bande1,2*†and Ma ía Piñei o-Lamas3†
*Co espondence:
manuel. eb e [email protected]
1MODESTYA g oup, Technological
Ins i u e o Indus ial Ma hema ics
(ITMATI), San iago de Compos ela,
Spain
2Dep . o S a is ics, Ma hema ical
Analysis and Op imiza ion, Fac. o
Ma hema ics, Uni e sidade de
San iago de Compos ela, San iago
de Compos ela, Spain
Full lis o au ho in o ma ion is
a ailable a he end o he a icle
†Equal con ibu o s
Abs ac
In his pape a e iew o semipa ame ic models de eloped h oughou he yea s
hanks o an ex ensi e collabo a ion be ween he Depa men o S a is ics and
Ope a ions Resea ch o he Uni e si y o San iago de Compos ela and a powe s a ion
loca ed in As Pon es (A Co uña, Spain) p ope y o Endesa Gene a ion, SA, is shown. In
pa icula hese models we e used o p edic he le els o sulphu dioxide in he
en i onmen o his powe s a ion wi h hal an hou in ad ance. In his pape also a
new mul idimensional semipa ame ic model is conside ed. This model is a
gene aliza ion o he p e ious models and akes in o accoun he co ela ion
s uc u e o e o s. I s beha iou is illus a ed in a simula ion s udy and wi h he
p edic ion o he le els o wo impo an pollu ion indica o s in he en i onmen o
he powe s a ion: sulphu dioxide and ni ogen oxides.
Keywo ds: Semipa ame ic p edic ion models; Pollu ion indica o s; Coin eg a ion
1 In oduc ion: an en i onmen al p oblem
The coal-fi ed powe s a ion in As Pon es is one o he p oduc ion cen e s owned by En-
desa Gene a ion SA in he Ibe ian Peninsula. I is loca ed in he own o As Pon es de
Ga cía Rod íguez, no heas o A Co uña p o ince.
This powe s a ion was designed and buil o make use o ligni e om he mine loca ed
in i s icini y. This solid uel was cha ac e ized by i s high mois u e and sulphu con en s
andi slowcalo ific alue.Th oughou heyea s heplan hasunde gonese e al ans o -
ma ion p ocesses in hei acili ies wi h he aim o educing emissions o sulphu dioxide
(SO2). The powe s a ion comple ed i s las adap a ion in 2008 o consume, as p ima y
uel, impo ed subbi uminous coal, cha ac e ized by i s low sulphu and ash con en s.
Theloca iono hepowe plan close ona u alsi eso highecological alue,suchas he
Na u alPa k AsF agasdoEumeandexis inglegisla ion,mean ha i hasexis edsince he
beginning a g ea conce n o i s impac on he en i onmen . The e o e he s a ion has a
Supplemen a y Con ol Sys em o Ai Quali y ha allows i o make changes in ope a ing
condi ions in o de o educe emissions when he wea he condi ions a e ad e se o he
sp ead o he emi ed smoke plume,specifically con aining SO2,and he ea esignifican
episodeso impai edai quali y.Spanishlaw,by ulesand egula ions,se smaximumcon-
cen a ions ha canbeachie ed o hesegasesinagi enpe iodo ime.Inpa icula , o
his plan he only limi ha migh be exceeded a any ime, is one ha is es ablished on
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ided you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons license, and
indica e i changes we e made.
González-Man eiga e al. Jou nal o Ma hema ics in Indus y (2018) 8:7 Page 2 o 16
he hou ly mean (con inuously compu ed) om he concen a ion o SO2in he soil, he
alue o 350 μg/m3.
The p oblem is o be able o p edic , using he in o ma ion ecei ed con inuously a
sampling s a ions and he pas in o ma ion, he u u e alues o SO2le els. S a is ical
o ecas models a e he key o ge hese p edic ions and sugges a cou se o ac ion o he
plan ope a o s.
In ecen yea s, new s a is ical models ha e been designed o ob ain he simul aneous
p edic ion o wo pollu ion indica o s in he en i onmen due o he changes in he en i-
onmen allegisla ion,in hepowe s a ioni sel ,and hecons uc iono anewna u algas
combined cycle s a ion in he icini y. The uels ha a e going o be used make ha he
mainin e es liesinp edic ing he alueso heni ogenoxides(NOx)whichisemi edby
bo h acili ies simul aneously wi h he alues o SO2which is only emi ed by he powe
s a ion.
All hese changes ha e c ea ed a new p oblem: p edic ing hou ly mean concen a ions
o sulphu dioxideandni ogenoxides,measu edin heen i onmen o he wo acili ies.
Faced wi h his new app oach, he s a is ical o ecas models a e again an effec i e ool.
Thus, a mul idimensional p edic ion gene al model is designed (see Sec . 3).
2 Me hods: one-dimensional p edic i e models
2.1 Models designed o sol e he en i onmen al p oblem
Resul ing om hecollabo a iono e hepas yea sbe ween heDepa men o S a is ics
and Ope a ions Resea ch a he Uni e si y o San iago de Compos ela and he En i on-
men Sec ion o he powe s a ion, he In eg a ed Sys em o S a is ical P edic ion o he
Immision (SIPEI, in Spanish) ha e been c ea ed employing s a is ical models o p o ide
p edic ions o he le els o SO2wi h a hal an hou ho izon.
Due o da a a ailabili y wi h minu al equency in eal- ime and cu en legisla ion, he
hou ly mean is conside ed om bo h o he alues o SO2and NOx, o p edic ions o
u u e alues o bo h pollu an s. Thus, wo ime se ies a e cons uc ed, X1, and X2, , o
which hesubsc ip ep esen saminu alins an ,andeach aluewillbeana e ageo he
ac ual alues o he las hou :
X1, =1
60
59

i=0 SO2( –i)andX2, =1
60
59

i=0 NOx( –i),
whe e SO2( )andNO
x( ) ep esen he concen a ion o SO2and NOx, espec i ely,a
ime ,measu edinμg/m3.
The se ies o hou ly SO2means has a cha ac e is ic beha iou , highly influenced by
wea he condi ions and local opog aphy. I akes alues close o ze o o long pe iods
o ime, and i can suddenly and sha ply inc ease (episodes) in bad me eo ological condi-
ions o he dispe sion o he smoke plume. Nowadays, he se ies o hou ly NOxmeans
has a simila beha iou o ha o SO2, bu on a smalle scale (see Fig. 1). The main objec-
i eo hede elopeds a is icalmodelsis op edic heepisodes,soou in e es iscen ed
on he alues ha occu less equen ly along he ime se ies.
Becauseo his,akindo memo ycalledHis o icalMa ixwasdesigned(P ada-Sánchez
and Feb e o-Bande [14]), whichwillbeessen ial o hebeha iou o all de eloped models
so a . This ma ix is composed o a la ge numbe o ec o s based on (X –l,...,X ,X +k):
González-Man eiga e al. Jou nal o Ma hema ics in Indus y (2018) 8:7 Page 3 o 16
Figu e 1 Episode depic ed in one o sampling s a ions. The one hou mean o SO2and NOxa e,
espec i ely, d awn in ed and o ange
ealda ao bihou lySO2o NOxmeans,chosensoas oco e he ull angeo a iablein
ques ion and make he ole o his o ical memo y. To ensu e ha co e he en i e ange o
he a iable, he ma ix is di ided in o blocks acco ding o he le el o he esponse a i-
able, X +k. To upda e he memo y, in e e y ins an , when a new obse a ion is ecei ed,
he his o ical ma ix is enewed in he ollowing way: he class o which he new obse -
a ion belongs is ound and hen he oldes da um in such class lea es he ma ix and he
new obse a ion en e s i . Wi h a sample buil his way, makes su e ha always ha e up-
da ed in o ma ion on he ull a ia ion ange o he in e es a iable, and o e he yea s
his concep has been adap ed o he diffe en s a is ical echniques used.
2.1.1 The fi s semipa ame ic model
In he ea ly yea s o de elopmen , he da a ansmission equency o SIPEI was pen-
aminu al, and also, he legisla ion in o ce a ha ime es ablished he limi alues o
he wo hou mean o he SO2. Fo his eason, he p edic ion models o SO2le els ini-
ially wo ked wi h se ies o bihou ly means. The objec i e was o ob ain he p edic ion,
wi h a hal an hou ho izon, o his ime se ies. The e o e, each ime i ecei es a new
obse a ion, X ,i has op edic he aluea six imesahead,X +6.
Asemipa ame ic app oachwas conside ed (Ga cía-Ju ado e al. [8]) which gene alizes
he adi ional Box–Jenkins models as ollows:
X +κ=ϕκ(X ,X –l)+Z +κ,κ,l∈Z+,
whe e Z has an ARIMA s uc u e o mean ze o independen o X (Box e al. [1]).
González-Man eiga e al. Jou nal o Ma hema ics in Indus y (2018) 8:7 Page 4 o 16
In pa icula a each ime , he eg ession unc ion ϕ6(X ,X –1)=E(X +6/X ,X –1)ises-
ima edwi h hewell-known Nada aya–Wa son ke nel ype es ima o (seeNada aya[13]
and Wa son [19]) using he in o ma ion p o ided by he his o ical ma ix. The second
s epis o calcula e he esidual imese ies ˆ
Z –64,...,ˆ
Z ela i e o he las sixhou s, whe e
ˆ
Zi=Xi–ˆ
E(Xi/Xi–6,Xi–7) o eachiandfi sanapp op ia eARIMAmodel o i .Finallywe
ge he Box–Jenkins p edic ion o ˆ
Z +6. The final poin p edic ion p oposed is gi en by:
ˆ
E(X +6/X ,X –1)+ˆ
Z +6.
2.1.2 Pa ially linea model
Thein o ma ionused by hep e ious semipa ame ic models oob ain hep edic ionsis
hepas o he imese ies;howe e i migh beuse ul oin oduceaddi ionalin o ma ion
in o de o imp o e hese p edic ions. Specifically, me eo ological and emission a iables
ha e been used wi h, he so-called pa ially linea models (P ada-Sánchez e al. [15]) o
es ima e bihou ly mean alues o SO2wi h one hou in ad ance.
Da ain he o mo (V ,Z ,Y )isconside ed,whe eV isa ec o o exogenous a iables,
Z =(X ,X –l)andY =X +12 being X he se ies o bihou ly SO2means; and i is assumed
ha hisse iescon o m o he ollowingpa iallylinea model:Y =V
β+ϕ(Z )+ ,whe e
 isane o e mo meanequals oze o.
This model can easily es ima ed ollowing Speckman [18] and allow us o ex end he
ho izon o one hou main aining he same le el o accu acy as he semipa ame ic model
o hal an hou ho izon. In any case, he inco po a ion o ex e nal in o ma ion sligh ly
imp o es hep edic ion because he measu e poin o he me eo ological a iables is lo-
ca ed a 80 m o e g ound le el which is ela i ely a away (and so, unco ela ed) espec
o he ypical heigh o he emi ed smoke plume (abo e 800 m o e g ound le el). Emis-
sionin o ma ionisalsoo li lein e es because hesesignalsa ealmos cons an specially
when he acili y is wo king no desc ibing a all he easons ha make he smoke plume
alls o heg ound.By hese easons,me eo ologicalo emissionin o ma ionwasno con-
side ed in he ollowing models.
2.1.3 Neu al ne wo ks
The change in he in e es se ies es ablished by he Eu opean Council Di ec i e
1999/30/CE, om bihou ly means o hou ly means, causes he ime se ies o be less
smoo h. A he beginning, he p e ious semipa ame ic model was adap ed o wo k on
henewse ieso hou lymeans.The esul sshowedaconside ableinc easein e mso he
a iabili y o he gi en p edic ions, ega ding he esul s usually ob ained o he se ies o
wohou means.
In an a emp o imp o e he esponse gi en by he SIPEI, and in pa icula , i s poin
p edic ions wi h hal an hou ho izon, new p edic o s based on neu al ne wo ks models
we e de eloped (Fe nández de Cas o e al. [6]).
A neu al ne wo k model has been designed o p o ide p edic ions o one hou mean
alues o SO2wi h hal an hou in ad ance. I consis s o an inpu laye , one hidden laye
and an ou pu laye . The numbe o nodes in he ou pu laye is de e mined by he size o
he esponse o be ob ained om he ne wo k; in his case in e es ed in a p edic ion o
X +6.Asinpu o hene wo ki hasbeen aken hebidimensional ec o (X –3,X )and he
nodesin hehiddenlaye ha ebeen akenas heac i a ion unc iono alogis ic unc ion,
and in he ou pu laye , he iden i y unc ion.
González-Man eiga e al. Jou nal o Ma hema ics in Indus y (2018) 8:7 Page 5 o 16
Figu e 2 Episode o SO2depic ed in one o sampling s a ions ( ed) join ly wi h he p edic ion p o ided by
he neu al ne wo k (blue) (Fe nández de Cas o e al. [6])
The p edic o gi en by he neu al ne wo k has he ollowing exp ession:
ˆ
X +6 =o1=L

j=1
ωo
1j h
jθh
j+ωh
j1X –3 +ωh
j2X 
wi h h
j(z)= 1
1+e–z.
Theweigh s {ωh
j1,ωh
j2,ωo
1j;j=1,...,L}and he ends{θh
j;j=1,...,L}a ede e mineddu -
ing he aining p ocess, as well as he final Lnumbe o hidden laye nodes, ha is cho-
sen like he alue which neu al ne wo k p o ides be e esul s, a e ha ing ained ne -
wo kswi hiden icala chi ec u eanddiffe en alueso L.Todesign he ainingse o he
neu al ne wo k i ha e been conside ed his o ical ma ices, o me ly in oduced, sui ably
adap ed.
Figu e 2shows he o ecas s gi en hal an hou be o e by he neu al ne wo k wi h 50
nodes in i s hidden laye o an episode depic ed in one o he measu ing s a ions. The
good beha iou o he o ecas (do ed line) can easily be seen. The p ocedu es based on
neu al ne wo ks accu a ely p edic he eal one hou mean SO2ai quali y alues (solid
line). These models we e op imized la e wi h boos ing lea ning echniques (Fe nández
de Cas o and González-Man eiga [4]).
2.1.4 Func ional da a model
Theonehou mean alueso SO2canbe ea edasobse a ionso as ochas icp ocessin
con inuous ime. Thein e es is,asi wasdiscussedabo e, op edic ahal -hou ho izon,
so ha each o he cu es is an in e pola ed da a on hal an hou . In his case cu es we e
ob ainedbyconside ingsixpen aminu alconsecu i eobse a ions,wi hsamplingpoin s
o each unc ional da a. The e o e, we use andom a iables wi h alues in Hilbe space
H=L2([0,6]) wi h he o m X (u)=x(6 +u).

González-Man eiga e al. Jou nal o Ma hema ics in Indus y (2018) 8:7 Page 6 o 16
The ollowing s a is ical model is conside ed X =ρ(X –1)+ ,whe e is a Hilbe ian
s ong whi e noise and ρ:H→His he ope a o o es ima e. Fo he es ima ion o ρ,
a unc ional ke nel es ima o has been used in he au o eg essi e Hilbe ian o o de -one
amewo k.Fu he mo e,i hasbeencon enien lyadap ed heconcep o his o icalma ix
o he case whe e he da a a e cu es (Fe nández de Cas o e al. [5]).
2.1.5 O he app oaches designed o p edic p obabili ies
Themodelsdesc ibed,so a ,p o idepoin p edic ionso SO2,bu o he echniquesha e
also been de eloped in o de o p edic p obabili ies. The aim o hese al e na i e models
is o es ima e he p obabili y ha he se ies o bihou ly SO2measu es exceeds a ce ain
le el wi h an hou an icipa ion, namely in ou case, we p edic P(Z )=P(X +12 > |Z )
being Z =(X ,X –X –3). To do i addi i e models wi h an unknown link unc ion (Roca-
Pa diñas e al. [17]) ha e been used.
I has also been conside ed mo e complex gene alized addi i e models (GAM) wi h
second-o de in e ac ion e ms (Roca-Pa diñas e al. [16]). They ha e shown ha he
GAMwi hin e ac ionsde ec s heonse o episodesea lie hani doesGAMoni sown.
2.2 Al e na i e one-dimensional models: addi i e models
In he s a is ical li e a u e he e is a wide ange o one-dimensional models which can
be used o p edic he le els o SO2. We will ocus on he echniques we will use in he
nex sec ion o cons uc ou mul idimensional model: addi i e models o con inuous
esponse.
The e ha e been a numbe o p oposals o fi ing he addi i e models. F iedman and
S ue zle [7] in oduced a backfi ing algo i hm and Buja e al. [2] s udied i s p ope ies.
Mammen e al. [12] p oposed he so called smoo h backfi ing by employing p ojec ion
a gumen s. Le {(Y ,Z )}T
=1 be a andom sample o a s ic ly s a iona y ime se ies, wi h
Y one-dimensional and Z q-dimensional ollowing he model:
Y =m(Z )+ , ∈Z,(1)
whe e { }is a whi e noise p ocess and E[ |Z ]=0.
Typically,i isassumed ha he unc ionmisaddi i ewi hcomponen unc ionsmj, o
j=0,...,q, hus
Y =m0+m1(Z1, )+···+mq(Zq, )+ .(2)
A gene alized ke nel nonpa ame ic es ima ion can be gi en using smoo h backfi ing
o he unc ions m1,...,mq(see again he abo e men ioned pape s).
Inall hemodelsdesc ibedabo ei isusuallynecessa y heselec iono a egula iza ion
pa ame e (bandwid h wi h ke nel smoo hing, numbe o neu ons in he hidden laye
o neu alne wo ks,...).Thecalib a iono hispa ame e wasde elopedusingc oss-
alida ion echniques wi h he in o ma ion o he upda ed His o ical Ma ix.
3 Me hods: mul idimensional semipa ame ic p edic ion
Thenewgoalis oinco po a e hep edic iono NOxwi hhal anhou inad ance,aswell
as o con inuege ing hep edic ions o SO2,ashasal eadybeencommen ed.Theideais
González-Man eiga e al. Jou nal o Ma hema ics in Indus y (2018) 8:7 Page 7 o 16
o gene alize he one-dimensional semipa ame ic app oach p oposed by Ga cía-Ju ado
e al. [8] aking in o accoun he s uc u e o co ela ion be ween he ec o ial se ies ha
is in ended o p edic .
3.1 The model
Be (Y,Z)=(Yl,Zl), l=0,±1,±2,... a ec o ials ic lys a iona y ime se ies, whe e Yl
is a -dimensional esponse se ies and Zlis a q-dimensional co a iables se ies and, le
{(Y ,Z )}T
=1 be a andom sample o (Y,Z). The ollowing model is conside ed
Y =ϕ(Z )+E ,(3)
whe e Y =(Y1, ,...,Y , ) ,Z =(Z1, ,...,Zq, ) and E =(E1, ,...,E , ) . Le us conside wo
possible s uc u es o he mul idimensional esiduals se ies:
P1. Each Ek, is a s a iona y AR(pk) p ocess o he o m
Ek, =
pk

i=1
φi
kEk, –i+ξk, o all ∈Z,k=1,...,
independen o Z ,whe eξk, isawhi enoisep ocesswi h a ianceσ2
k, o k=1,..., .
P2. E has a VAR(p) s uc u e o he o m
E =p

i=1
iE –i+ξ o all ∈Z,
independen o Z ,whe e heia e fixed ( × )coefficien sma icesandξ is a
-dimensional whi e noise p ocess, i.e. E(ξ )=0,E(ξ ξ
)=ξand E(ξ ξ
s)=0 o
=s.
Ou main objec i e is o p edic Y using a sample o size T,κins an s ahead. The p e-
dic ion o Y +κis hen defined by
˙
Y +κ=ˆϕκ(Z )+ ˙
E +κ,(4)
whe e ˆϕκ(Z ) is a nonpa ame ic es ima e o ϕκ(Z )=E[Y +κ/Z ]and ˙
E +κ he p edic ion
gi en, κins an s ahead, o he esidual se ies cons uc ed as ˆ
E +κ=Y +κ–ˆϕκ(Z ).
3.2 Es ima ions
We suppose ha he model (3) is e ified. The fi s s ep is o make a nonpa ame ic es i-
ma ion o ϕindependen ly o eacho he componen s o Y :ϕ(Z )=(ϕ1(Z ),...,ϕ (Z )).
Fu he mo e, we assume ha he unc ions ϕka e addi i e wi h componen unc ions ϕj
k,
o k=1,..., and j=0,...,q, hus
ϕk(Z )=ϕ0
k+ϕ1
k(Z1, )+···+ϕq
k(Zq, ), k=1,..., .(5)
The e o e, addi i e models wi h qco a ia es a e es ima ed using he smoo h backfi -
ing echnique. We ha e o ake in o accoun ha he p ocess E is no obse able since
he unc ion ϕis no known. Thus, we ha e o eplace E by he esiduals
ˆ
E =Y –ˆϕ(Z )
González-Man eiga e al. Jou nal o Ma hema ics in Indus y (2018) 8:7 Page 8 o 16
and use hese app oxima ions o E in he maximum likelihood es ima ions la e defined.
To es ima e he pa ame ic pa o he model, we mus conside he wo possible e o
s uc u es p oposed abo e:
P1. The pa ame e s φk=(φ1
k,...,φpk
k)o he e o p ocess {Ek, }a e es ima ed by s an-
da d maximum likelihood me hods. In pa icula , we use a condi ional maximum
likelihood es ima o o e e y componen o he o m
ˆ
φk=a gmax
φk∈ˆ
l(φk),
whe e is a compac pa ame e space and ˆ
lis he condi ional log-likelihood gi en
by
ˆ
lφk,σ2
k=–T
2log(2π)+1
2logσ–2
k–1
2
T

=pk+1ˆ
Ek, –ˆ
Ek, (φk)/σk2
wi h ˆ
Ek, (φk)=pk
i=1 φi
kˆ
Ek, –i.
P2. The coefficien s ma ices (1,...,p)o he -dimensional e o p ocess {E }a e
also es ima ed by gene alized maximum likelihood me hods (Hamil on [10]). Fi s ,
we need o es ablish he ollowing no a ion:  =[12...p]deno e he ( × p)
coefficien sma ix,le X bea( p×1) ec o con ainingplagso eacho heelemen s
o E :X
=[E
–1 E
–2 ...E
–p].
The heo e icalcondi ionallog-likelihood unc ion obeop imizedhas he ollow-
ing exp ession:
l(,ξ)=– T
2log(2π)+
2log–1
ξ–1
2
T

=1 E – X  –1
ξE – X .
Thus he condi ional log-likelihood is:
ˆ
l(ˆ
,ˆ
ξ)=– T
2log(2π)+
2logˆ
–1
ξ–1
2
T

=1 ˆ
E –ˆ
 ˆ
X  ˆ
–1
ξˆ
E –ˆ
 ˆ
X .
3.3 O he conside a ions: he phenomenon o coin eg a ion
Some imes he ec o ialp ocessescanbecoin eg a ed,soonehas o akein oaccoun he
s uc u e o co ela ion be ween he se ies. The no ion o coin eg a ion has been one o
he mos impo an concep s in ime se ies since G ange [9] and Engle and G ange [3]
ha o mally de eloped i . The issue has b oad applica ions in he analysis o economic
da a as well as se e al publica ions in he economic li e a u e.
Le Y =(Y1, ,...,Y , ) be a ec o o ime se ies in eg a ed o o de 1 (I(1)). Y is said
o be coin eg a ed i a linea combina ion o hem exis s ha i is s a iona y (I(0)), i.e., i
he e exis s a ec o β=(β1,...,β ) such as
β Y =β1Y1, +···+β Y , ∼I(0).
The ec o βis called he coin eg a ion ec o . This ec o is no unique since o any
scala c he linea combina ion cβ Y =β∗ Y ∼I(0). The e o e, no maliza ion is o en
assumed o iden i y an unique β. A ypical no maliza ion is β=(1,–β2,...,–β ) .
González-Man eiga e al. Jou nal o Ma hema ics in Indus y (2018) 8:7 Page 9 o 16
Johansen [11] add esses he issue o he coin eg a ion wi hin an e o co ec ion model
in he amewo k o ec o au o eg essi e models (VAR). Conside hen a gene al model
VAR(p) o he ec o o se ies Y
Y =0D +1Y –1 +···+pY –p+ξ , =1,...,T,
whe e D con ains de e minis ic e ms (cons an , end, ...).
Suppose Y is I(1) and possibly coin eg a ed. Then, he VAR ep esen a ion is no he
mos sui able ep esen a ion o analysis because he coin eg a ing ela ionships a e no
explici ly appa en . The coin eg a ing ela ionships become appa en i he VAR model is
ans o med o a ec o e o co ec ion model o o de p(VECM(p))
Y =0D +Y –1 +1Y –1 +···+p–1Y –p+1 +ξ ,
whe e =1+···+p–I ,k=–p
j=k+1 j,k=1,...,p–1andY =Y –Y –1.The
ma ix is called he long- un impac ma ix and ka e he sho - un impac ma ices.
Mo eo e , he anko hesingula ma ixp o idesin o ma ionon henumbe o coin-
eg a ion ela ions ha exis ,i.e., he anko coin eg a ion.Johansenp oposesasequen ial
p ocedu e o likelihood a io es s o es ima e his ange.
3.4 P edic ion scheme
We p esen now he p edic ion scheme s ep by s ep:
1. E e y ins an ,ϕκ(Z )is es ima ed wi h he smoo h backfi ing echnique
independen ly o each o componen s using he da a (Yl,Zl–κ),l=κ+1,...,T.
2. The esiduals se ies ˆ
E +κis compu ed by
ˆ
E +κ=Y +κ–ˆϕκ(Z ), =1,...,T–κ.
3. The ollowing s ep is o make an app op ia e adjus men on he model e o
s uc u e (VECM) and o ob ain he p edic ion κins an s ahead: ˙
ET+κ.
4. The p oposed final p edic ion is gi en by (4).
This scheme is a na u al gene aliza ion o he one-dimensional p edic ion models de-
sc ibed in Sec . 2.1.1. In he nex wo sec ions simula ion examples and eal da a analysis
a e conside ed.
4 Resul s and discussion
4.1 A simula ion s udy
To analyze he beha io o he p oposed p edic ion p ocedu e, a simula ion s udy has
been pe o med gene a ing samples om a ificial se ies and making a p edic ion s udy
o klags using, in all cases, Z =Y –1.
The ollowing models a e conside ed:
Se ies1. Two independen AR(3) wi h cons an end:
Y =ϕ+E1,
E2, ,
González-Man eiga e al. Jou nal o Ma hema ics in Indus y (2018) 8:7 Page 16 o 16
12. Mammen E, Lin on O, Nielsen J. The exis ence and asymp o ic p ope ies o a backfi ing p ojec ion algo i hm unde
weak condi ions. Ann S a . 1999;27(5):1443–90.
13. Nada aya EA. On es ima ing eg ession. Theo y P obab Appl. 1964;9(1):141–2.
14. P ada-Sánchez J, Feb e o-Bande M. Pa ame ic, non-pa ame ic and mixed app oaches o p edic ion o spa sely
dis ibu ed pollu ion inciden s: a case s udy. J Chemom. 1997;11(1):13–32.
15. P ada-Sánchez J, Feb e o-Bande M, Co os-Yáñez T, González-Man eiga W, Be múdez-Cela J, Lucas-Domínguez T.
P edic ion o SO2 pollu ion inciden s nea a powe s a ion using pa ially linea models and an his o ical ma ix o
p edic o - esponse ec o s. En i onme ics. 2000;11(2):209–25.
16. Roca-Pa diñas J, Cada so-Suá ez C, González-Man eiga W. Tes ing o in e ac ions in gene alized addi i e models:
applica ion o SO2 pollu ion da a. S a Compu . 2005;15(4):289–99.
17. Roca-Pa diñas J, González-Man eiga W, Feb e o-Bande M, P ada-Sánchez J, Cada so-Suá ez C. P edic ing bina y ime
se ies o SO2 using gene alized addi i e models wi h unknown link unc ion. En i onme ics. 2004;15(7):729–42.
18. Speckman P. Ke nel smoo hing in pa ial linea models. J R S a Soc, Se B, S a Me hodol. 1988;50:413–36.
19. Wa son GS. Smoo h eg ession analysis. Sankhya, Se A. 1964;26:359–72.