Disco e y o mo i s o o ecas ou lie occu ence in ime se ies
F. Ma ínez–Ál a ez , A. T oncoso, J.C. Riquelme , J.S. Aguila –Ruiz
Keywo ds:
Time se ies o ecas ing
Pa e n ecogni ion
Mo i s
Ou lie s
abs ac
The o ecas ing p ocess o eal-wo ld ime se ies has o deal wi h especially unexpec ed alues, com-
monly known as ou lie s. Ou lie s in ime se ies can lead o un eliable modeling and poo o ecas s.
The e o e, he iden ifica ion o u u e ou lie occu ence is an essen ial ask in ime se ies analysis o
educe he a e age o ecas ing e o . The main goal o his wo k is o p edic he occu ence o ou lie s
in ime se ies, based on he disco e y o mo i s. In his sense, mo i s will be hose pa e n sequences p e-
ceding ce ain da a ma ked as anomalous by he p oposed me aheu is ic in a aining se . Once he mo i s
a e disco e ed, i da a o be p edic ed a e p eceded by any o hem, such da a a e iden ified as ou lie s,
and ea ed sepa a ely om he es o egula da a. The o ecas ing o ou lie occu ence has been added
as an addi ional s ep in an exis ing ime se ies o ecas ing algo i hm (PSF), which was based on pa e n
sequence simila i ies. Robus s a is ical me hods ha e been used o e alua e he accu acy o he p oposed
app oach ega ding he o ecas ing o bo h occu ence o ou lie s and hei co esponding alues. Finally,
he me hodology has been es ed on six elec ici y- ela ed ime se ies, in which mos o he ou lie s we e
p ope ly ound and o ecas ed.
1. In oduc ion
This wo k p oposes a new s a egy o p edic he occu ence o
ou lying da a in ime se ies, as well as p o iding accu a e o ecas s
o hem. I is wo h highligh ing ha he goal o his me hodology
is o o ecas hei appea ance, ins ead o de ec ing hem in an
al eady known se o alues, which is a common goal in obus s a-
is ics (Ma onna e al., 2007). The majo i y o obus s a is ical
echniques pe o m a pos e io i de ec ion, ha is, hey de e mine
whe he a da um is an ou lie o no , bu once i has al eady
occu ed. Howe e , a compa ison wi h hese echniques will be
o he u mos impo ance in o de o e alua e he accu acy o he
p oposed me aheu is ic.
A gene al-pu pose o ecas ing algo i hm, called PSF, was p e-
sen ed in Ma ínez-Ál a ez e al. (in p ess). I s main ea u e lied
in pe o ming a disc e iza ion o he ime se ies by means o ce -
ain clus e ing echnique. Then, i only used he gene a ed labels
o make p edic ions. Based on ha p e ious disc e iza ion, his
wo k a emp s o disco e pa e n sequences (hence o h called
mo i s) in he his o ical da a o o ecas he occu ence o ou lie s
and hei associa ed alues. This no el me hodology is inse ed in
he gene al scheme o PSF.
The p edic ion o ou lie s plays an impo an ole as w ong mod-
els and poo o ecas s a e ob ained when igno ing ou lie s. This
wo k p esen s a me aheu is ic o disco e mo i s in ime se ies
and, hen,i da a obep edic eda ep ecededbyanyo hesedisco -
e ed mo i s, conside hese da a as ou lie . The mo i s a e de e -
mined du ing he aining phase as hose pa e n sequences ha
p ecede da a wi h ema kable o ecas ing e o . Thus, he exis ing
PSF algo i hm is modified by adding a new mo i ex ac ion s ep.
The enhanced e sion is capable o p edic ing he appea ance o
such ou lie s wi h g ea eliabili y when he mo i s ex ac ion s ep
is added. In ac , he app oach has been success ully es ed on six
eal-wo ld elec ici y- ela ed ime se ies, in pa icula , on ene gy
p ices and demand o h ee di e en ma ke s, eaching sensi i i y
alues g ea e han 82%, and specifici y alues g ea e han 95%.
Fu he mo e, esul s abou he e ec o ou lie s on he a e age
o ecas ing e o s a e epo ed o all he six ime se ies, exhibi ing
ema kable o ecas ing e o educ ion.
Despi e he as a ie y o wo ks ela ed o ou lie s de ec ion
and mo i s disco e y in ime se ies, he e is no app oach in ime
se ies in o de o o ecas he occu ence o ou lie s, o he au ho s’
knowledge.
The emaining o he pape is o ganized as ollows. A e iew o
he mos ecen ly published wo ks ega ding ene gy ime se ies
o ecas ing, mo i s disco e y and ou lie s de ec ion can be ound
in Sec ion 2. Sec ion 3p o ides o mal desc ip ion o sensi i e
e ms, and p esen s a b ie explana ion o he o iginal algo i hm.
As o Sec ion 4, i in oduces he p oposed me hodology, showing
how o inse he ou lie occu ence o ecas ing in he o iginal algo-
i hm’s gene al scheme. The esul s ob ained o he six elec ici y
p ices and demand ime se ies a e epo ed and discussedin Sec ion
5. Finally, Sec ion 6summa izes he main conclusions achie ed.
2. Rela ed wo k
This sec ion p o ides use ul and ecen e e ences abou he
h ee main opics in ol ed in his pape : Ene gy ime se ies o e-
cas ing, mo i s disco e y in empo al da a and obus s a is ical
me hods o de ec ou lie s. Fo he sake o cla i y, hese opics ha e
been sepa a ed in h ee di e en sec ions.
2.1. Ene gy ime se ies o ecas ing
The in e es o analyzing elec ici y p ice ime se ies esides in
he p og essi e de egula ion o elec ic powe ma ke s. Fu he -
mo e, elec ici y p ice ime se ies possess ce ain ea u es ha u n
he p edic ion in o a di ficul ask: non-cons an mean/ a iance
and equen ly ou lie occu ences. Fo his eason, elec ici y-p o-
duce companies wan op imized bidding s a egies as well as
needing assessmen abou he isk o us ing o ecas s (Plazas
e al., 2005).
On he o he hand, he p ocess o o ecas ing he quan i y o
elec ici y equi ed o a specific geog aphical a ea du ing a ime
pe iod is called load o ecas ing o demand o ecas ing. This p o-
cess is key since cu en echnology allows o s o e a limi ed
amoun o elec ici y in ba e ies. The e o e, he demand o ecas -
ing plays an impo an ole o elec ici y powe supplie s because
bo h excess and insu ficien ene gy p oduc ion may lead o in-
c eased cos s and a significan educ ion o p ofi s.
The pu sui o accu a e o ecas ing in elec ici y p ice ime se -
ies has mo i a ed esea ch wo ks by many au ho s (Agga wal e al.,
2009). Thus, he use o mixed models was p oposed in Ga cía-
Ma os e al. (2007) o o ecas p ices o di e en ho izons o
p edic ion. Also ema kable was he wo k in oduced in T oncoso
e al. (2007) ha , by means o weigh ed nea es neighbo s me hod-
ology, o ecas ed nex -day elec ici y p ices. Also, he use o an
a ificial neu al ne wo k o ulfil he same goal can be ound in Pino
e al. (2008). E en he use o classical au o eg essi e models has
been ecen ly used o o ecas p ices in se e al ma ke s (We on
and Misio ek, 2008).
On he con a y, he au ho s in T oncoso e al. (2004) p oposed
a weigh ed nea es neighbo s-based me hodology o o ecas elec-
ici y demand. The p oposed app oach was es ed o e he nex -
day Spanish load o ecas ing. Also, he au ho s in El-Telbany and
El-Ka mi (2008) o ecas ed he Jo danian elec ici y demand wi h
an a ificial neu al ne wo k, which was ained by means o pa i-
cle swa m op imiza ion echniques. This ma ke was also s udied
in Bad an e al. (2008) bu , his ime, he au ho s p e e ed o con-
cen a e on sho and medium- e m load o ecas ing by using
eg ession models. Finally, Wang and Wang (2008) p oposed a
new p edic ion app oach based on suppo ec o machines
(SVM) echniques wi h a p e ious selec ion o ea u es om da a
se s by using an e olu iona y me hod.
The disco e y o ou lie s in elec ici y p ice ime se ies has also
been widely discussed in li e a u e. Thus, he au ho s in Lu e al.
(2005) p oposed a model based on he analysis o se e al a iables
(among which he elec ici y demand is highligh ed) by means o
Bayesian classifica ion (BC) and simila i y sea ching echniques.
A hyb id me hodology ha combined SVMs and BC was de eloped
in Wu e al. (2006) o classi y bo h spikes and no mal elec ici y
p ices. Al e na i ely, a da a mining amewo k based on SVM and
p obabili y classifie s was desc ibed in Zhao e al. (2007) wi h
he aim o o ecas ing spikes in p ices accu a ely.
By con as , he p edic ion o peaks in elec ici y demand was
add essed in Saini (2008). This wo k o ecas ed demand peaks up
o se en days ahead using eed o wa d neu al ne wo k and adap-
i e backp opaga ion lea ning me hods. In he wo k in oduced in
Ismail e al. (2009), he au ho s de eloped a ule-based me hod
ha combined eg ession models and uzzy sys ems o analyze
daily elec ici y peak load demands in Malaysia. Also, he au ho s
in Hyndman and Fan (2010) desc ibed a semi-pa ame ic addi i e
model o disco e ela ionships be ween he demand and exogen
a iables. The app oach was applied o long- e m peaks o he
Sou h Aus alian ma ke .
2.2. Mo i s disco e y
The disco e y o mo i s in con inuous da a, also known as unc-
ional da a in many wo ks (Valde ama, 2008), was o iginally o -
malized in Lin e al. (2002), in which he au ho s in oduced
se e al algo i hms o mine mo i s in ime se ies, among which
he k-mo i algo i hm highligh s. Howe e , he main d awback o
his algo i hm is i s dependence on a p e-fixed pa e n leng h.
La e , he au ho s in Tang and Liao (2008) p oposed a modified
e sion ha imp o ed, p ecisely, his ea u e. Fu he mo e, hey
gene a ed o iginal pa e ns by conside ing he disco e ed mo i s.
The de ec ion o on-line mo i s in con inuous da a has also been
add essed. Pa icula ly, a new me hodology o de ec on-line
mo i s in ime se ies by combining p obabilis ic models and poly-
nomial leas -squa es app oxima ions was p oposed in Fuchs e al.
(2009). This opic was also s udied in Mueen and Keogh (2010) in
which he au ho s ound and main ained ime se ies mo i s om
obo ics, online comp ession and wildli e managemen domains.
Besides hese goals, di e en objec i es ha e been ulfilled by
disco e ing mo i s ecen ly. Hence, he wo k p esen ed in Tanaka
e al. (2005) p oposed an algo i hm de o ed o disco e mo i s
based on he minimum desc ip ion leng h p inciple. The app oach
also allowed o ob ain mo i s om mul i-dimensional ime se ies
da a by using p incipal componen analysis. Mueen e al. (2009)
p oposed in 2009 an exac algo i hm o find ime se ies mo i s
much as e han b u e- o ce sea ching s a egies do. Also, an ap-
p oach based on ee-cons uc ion sea ch o disco e mo i s in
mul i a ia e ime se ies was p oposed in Wang e al. (2010) and
applied o senso y da ase s.
Since S o mo (2000) fi s e iewed s a egies o find DNA mo i s
(meaning ul base sequence pa e ns ha iden i y binding si es
esponsible o ansc ip ion ac o s) in 2000, a la ge amoun o
algo i hms ha e been de eloped. Thus, an ensemble algo i hm
a emp ing o disco e egula o y mo i s in DNA sequences was
p oposed in Hu and Kiha a (2006). Ano he algo i hm was p o-
posed in Wijaya e al. (2007) which, gi en a se o sequences, exe-
cu es mdi e en mo i finde s, each o hem epo ing nmo i s.
Finally, Sha o and Mino u (2009) p esen ed CisFinde , a so wa e
ha gene a es a comp ehensi e lis o mo i s en iched in a se o
DNA sequences and desc ibes hem wi h posi ion equency
ma ices.
2.3. Robus s a is ical me hods o de ec ou lie s
The p oblem o a pos e io i ou lie s de ec ion in ime se ies has
been widely s udied in he li e a u e, and aced om many di e -
en poin s o iew. In ac , he exis ence o e en ew ou lie s
usually leads o inaccu a e models and no sa is ac o y o ecas s
(Galeano e al., 2006), since hey may deeply influence he
es ima es ha classical me hods p opose (Ca ne o e al., 2007).
Fo his eason, he e is a la ge amily o obus s a is ical
me hods (Rousseeuw and Hube , 2011) ha deal wi h ou lie s
and, pa icula ly, p opose app oaches o de ec hei exis ence in
he da ase s subjec ed o analysis. Gelpe e al. p oposed an
adap ed e sion o he classical exponen ial and Hol -Win e s
smoo hing me hodologies, p o iding hem wi h obus ness
(Gelpe e al., 2010). Ano he e sion o a obus mul i a ia e
exponen ial smoo hing applied o ime se ies can be ound in
C oux e al. (2010). Following wi h classical me hods, a wo k ha
enhanced ARMA by adding obus ness can be ound in Mule e al.
(2009), in which he au ho s succeeded in limi ing he e ec o
ou lying da a o he ime s amp in which hey happen.
Suppo ec o machines (SVM) ha e also been adap ed o deal
wi h ou lie s. Ac ually an app oach o model ime se ies using o-
bus SVM was p oposed in Camps-Valls e al. (2004). In pa icula ,
he au ho s claimed ha hei p oposal p o ides s able models
and allows he analysis o models’ memo y dep h. Recen ly, a wo-
s eps me hodology was p oposed in Chuang and Lee (2011) ha
combined heuseo obus SVM o emo eanomalousobse a ions,
and non- obus SVM o ob ain es ima es om ha educed da ase .
Many o hese p oposals ha e been implemen ed and eely dis-
ibu ed in so wa e packages. Bu om all o hem he e a e wo
ha highligh . LIBRA (Ve bo en and Hube , 2005) is a Ma lab li-
b a y o obus analysis, ha con ains (among o he s) obus
co a iance es ima ion, eg ession, p incipal componen analysis,
p incipal componen eg ession o pa ial leas squa es, as well
as me hodologies o de ec ou lying obse a ions in da ase s. The
TOMCAT oolbox Daszykowski e al. (2007), also de eloped in he
Ma lab en i onmen , includes almos he same me hods ha
LIBRA does, bu also includes a g aphical in e ace.
3. Fundamen als
This Sec ion fi s defines some e ms in o de o p e en possi-
ble misin e p e a ions in sensi i e e ms. Since he p oposed
me hodology is based on an exis ing algo i hm, his Sec ion also
p o ides a b ie summa y o he ma hema ical undamen als
unde lying he PSF algo i hm. No e ha a mo e de ailed explana-
ion can be ound in Ma ínez-Ál a ez e al. (in p ess).
3.1. Defini ions
This wo k uses ce ain concep s –such as ou lie o mo i – ha
can be in e p e ed in many di e en senses, depending on he
applica ion o e en he au ho . Fo his eason, his Sec ion p o-
ides a o mal defini ion o hese sensi i e e ms.
Defini ion 1 (Hou ly ime se ies). An hou ly ime se ies Tis a se o
eal- alued da a in successi e o de , occu ing e e y hou . In his
wo k, T=[
1
,...,
p
], whe e pis he leng h o he ime se ies and
usually a mul iple o 24.
Defini ion 2 (Daily ime se ies). F om an hou ly ime se ies, a daily
ime se ies Dis o med by uples in R
24
,D=[d
1
,...,d
p/24
], whe e
d
i
=[
24(i1)+1
,...,
24i
].
Defini ion 3 (Label). In his wo k, he e m label is used o iden i y
a se o possible ca ego ical alues. Thus, L¼ l
1
;...;l
K
g, whe e Kis
a p e-fixed numbe .
Defini ion 4 (Sequence). A sequence Sis a se o labels occu ing in
successi e o de . In his wo k, S=[s
1
,...,s
q
], whe e qis he leng h
o he sequence and s
i
2L.
Defini ion 5 (Ou lie ). Gi en a es se , an ou lie is an obse a ion
which appea s o be inconsis en wi h he es o he da a, ela i e
o an assumed model (E e i , 2006). Ou lie s a e usually
ep esen ed by a bina y andom a iable
i
, o i=1,...,p ha
models hei occu ence (
i
= 1 i i occu s, and
i
= 0 o he wise),
and by ano he eal andom a iable z
i
, o i=1,...,p ha models
hei magni ude (Ma onna e al., 2007). Al hough di e en ou lie
ypes can be ound in he li e a u e, only he addi i e ou lie model
is conside ed in his wo k, due o he na u e o he s udied da a:
i
¼x
i
þ
i
z
i
;ð1Þ
whe e
i
he obse ed alue, and x
i
he i  h cleaned da a modeled
by any app oach.
Defini ion 6 (Mo i ). A mo i M
W
is a sequence o Wconsecu i e
labels conside ed o occu jus be o e an ou lie , whe e Wis he
p e-fixed leng h o he sequence. In addi ion, M
W
is a subsequence
ound in Sand, consequen ly: M
W
¼½s
0
1
;...;s
0
W
, whe e s
0
i
2L.
3.2. Time se ies o ecas ing: he PSF algo i hm
The PSF algo i hm is a gene al-pu pose ime se ies o ecas ing
algo i hm whose main ea u e is ha i only makes use o ce ain
labels –ob ained by means o a clus e ing p ocess– o o ecas a bi-
a y ho izons o p edic ion. Howe e , he ou pu is no composed
by labels bu by eal alues.
PSF can deal wi h an a bi a y numbe o samples pe day. How-
e e , specifically in his pape , he ime se ies conside ed consis s
o wen y- ou samples pe day. Tha is, gi en he hou ly alues up
o day i o a ime se ies, he PSF algo i hm p o ides he 24 hou ly
alues co esponding o day i+ 1. Fo mally, le d
i
2R
24
be a ec o
ha comp ises he 24 hou ly alues o a ce ain day, i.
Fi s , PSF applies clus e ing echniques o such da a in o de o
assign a label o each day. Fo mally, i uses a unc ion F
K
ha as-
signs a label l
i
2L o he alues d
i
2Do each day by means o a
clus e ing p ocess, F
K
:D!L, ha is, e e y 24 h a e iden ified by
a label. Once Kis fixed, his p ocess ans o ms he daily ime se ies
Din o a sequence o labels S, hus disc e izing he o iginal da a. Le
l
i
be he label assigned o he day iob ained by means o he appli-
ca ion o a clus e ing echnique. Le S
i
W
be he labels’ subsequence
o Wconsecu i e days, om day ibackwa d:
S
i
W
¼½l
iðW1Þ
;l
iðW2Þ
;...;l
i1
;l
i
ð2Þ
whe e he leng h o he window, W, is a pa ame e o be
de e mined.
Le W
⁄
be he leng h o he window de e mined by PSF. Fo a day i
and leng h o window W
⁄
, he PSF algo i hm sea ches o he subse-
quences o labels which a e exac ly equals o S
i
W

in he da ase , p o-
iding he equal subsequences se , ES, defined by he equa ion,
ESði;W

Þ¼ days j2Dsuch ha S
j
W

¼S
i
W

no
ð3Þ
I is wo h ema king ha i no subsequence equal o S
i
W

was ound
in he da ase , ha is, ES(i,W
⁄
)=;, he leng h o he window would
dec ease by one uni , W
0
=W
⁄
1, and he PSF would sea ch o
subsequences equal o S
i
W
0
. This p ocess may be epea ed un il
any subsequence is ound, ha is, ES(i,W)–;.
The e o e, he W
⁄
consecu i e labels ha p ecede he day o be
p edic ed a e ex ac ed and sea ched o in he his o ical da a.
Once all occu ences o S
i
W

a e ound, he 24 hou ly alues o he
day i+ 1 a e p edic ed by a e aging he eal alues ound immedi-
a ely a e each S
i
W

ma ch. Ma hema ically,
^
d
iþ1
ðW

Þ¼ 1
#ESði;W

ÞX
j2ESði;W

Þ
d
jþ1
ð4Þ
Finally, he daily e o o any day iis defined by:
e
day
ði;W

Þ¼j
^
d
i
ðW

Þd
i
jð5Þ
4. Ou lie o ecas ing in ime se ies
This sec ion explains he me hodology p oposed o imp o e he
o ecas ing p ocess p o ided by he PSF algo i hm. The disco e y o
mo i s is included in he a o emen ioned algo i hm as a c ucial
s ep o o ecas ing he occu ence o ou lie s and, hen, p o iding
accu a e es ima es o such anomalous obse a ions.
The alue o wo pa ame e s had o be de e mined in he PSF
p ocess: The numbe o clus e s Kand he leng h o he window
W. Wi h ega d o K, he new app oach ac s exac ly he same as
wha was p oposed in he o iginal PSF, ha is, i applies h ee
well-known alidi y indices –Silhoue e, Dunn and Da ies–Boul-
din– and de e mines he op imal numbe o clus e s by means o
a majo i y o e sys em.
On he o he hand, he n old c oss- alida ion is used o ob-
ain he op imal alue o W. Twel e olds ha e been c ea ed in his
wo k (n= 12) o all he da ase s, whe e each old ep esen s a
mon h. The e o e he aining se consis s o one yea . The
12  old c oss– alida ion is hen e alua ed. The o ecas ing e o s
a e calcula ed in e e y old by a ying he leng h o W. Fo each
window size W, he mon hly e o s a e deno ed by e
mon h
(W) and
a e calcula ed as ollows:
e
mon h
ðWÞ¼ 1
#mon h X
i2mon h
e
day
ði;WÞð6Þ
o W=1,...,W
max
and W
max
= 10, since no longe sequences we e
ound in daily ime se ies. Then, he a e age e o s a e calcula ed
o each window size as ollows,

eðWÞ¼1
nX
mon h
e
mon h
ðWÞð7Þ
whe e n= 12 and mon h ={Jan,...,Dec}.
The W
⁄
selec ed is he one ha minimizes he a e age e o co -
esponding o he 12 olds (mon hs) e alua ed.
W

¼a gmin 
eðWÞg wi h W¼1;...;W
max
ð8Þ
I is now –jus a e he aining s ep and be o e he p edic ion p o-
cess– ha he disco e y o mo i s plays a c ucial ole, as i a emp s
a o ecas ing he occu ence o anomalous days in ime se ies.
Specifically, he mo i s o be ound a e hose which gene a e a
p edic ion e o g ea e han he a e age e o in he c oss– alida-
ion p ocess. The e o e, a se o days ibelonging o he aining se
(TS) ha sa isfies e
day
ði;W

Þ>
eðW

Þis cons uc ed. This se , CS o
candida es se , ga he s all he candida e days o be p eceded by a
sequence ha will e en ually be a mo i . Fo mally,
CS ¼ i2TS such ha e
day
ði;W

Þ>
eðW

Þg ð9Þ
Ne e heless, no all he sequences ha p ecede hese candida es
ha e he same p obabili y o be e en ually conside ed as ou lie
p ecu so s, since he associa ed e o s ange om alues close o
he mean e o ( hese candida e sequences should be e en ually
disca ded by he app oach) o significan ly high alues. Fo his ea-
son, each candida e is co-labeled by using clus e ing echniques,
mo e specifically, he K-means algo i hm. The decision on how
many clus e s ha e o be c ea ed is always an open ques ion and
many indices could be used. Howe e , i is wo hless o ha e a la ge
numbe o clus e s and he e o e, only h ee concep ual classes will
be c ea ed: A class ha ga he s days wi h low e o s (C
l
) o nea es
e o s o he 
eðW

Þ, a class con aining medium e o s (C
m
) and, fi-
nally, a class de o ed o iden i y high e o s (C
h
) o a hes e o s
o he 
eðW

Þ. Thus, o all days i2C
l
,j2C
m
and k2C
h
he ollowing
inequali ies a e ulfilled:
e
day
ðk;W

Þ>e
day
ðj;W

Þ>e
day
ði;W

Þ>
eðW

Þð10Þ
Fig. 1 illus a es an imagina y e o dis ibu ion in he TS, de e min-
ing he candida es ha will e en ually o m CS as he union o can-
dida es in C
l
,C
m
and C
h
. In o he wo ds, CS =C
l
SC
m
SC
h
.
A p io i easoning e eals ha hose candida es belonging o C
h
mus be mo e p obable o be p eceded by mo i s p eceding ou lie s
han he candida es in C
l
o C
m
. Resul s co esponding o each clus-
e o da a will be sepa a ely analyzed in Sec ion 5.
The nex s ep consis s o compu ing he sequences o labels
occu ing be o e he candida es in o de o de e mine which se-
quences will be conside ed ou lie p ecu so s, since no all hese
sequences will be mo i s. Hence, he app oach has o decide
whe he he sequences p eceding he candida es a e mo i s
esponsible o ou lie s o no . In pa icula , he sequences ha
only appea be o e he candida es will be conside ed mo i s p e-
ceding an ou lie occu ence. Tha is, i a sequence o labels p eced-
ing a candida e is ound p io o any o he day ha does no belong
o he CS, he sequence is disca ded and no conside ed o be a mo-
i . Thus, a se o mo i s MS is defined by:
MS ¼S
i
W

such ha i2CS and ESði;WÞ#CS
no
ð11Þ
No e ha MS can be also exp essed as ollows: MS =M
l
SM
m
SM
h
,
whe e M
l
,M
m
and M
h
a e he disco e ed mo i s associa ed o he se-
quences ound in he classes C
l
,C
m
, and C
h
espec i ely.
Fig. 4 depic s he p ocess o disco e ing he sequences o labels
ha ep esen he mo i s p eceding ou lie s. This figu e shows an
illus a i e example in which six clus e s we e c ea ed, K=6
(labels a e digi s 1 o 6). The e o e, he ime se ies appea s disc e -
ized, making only use o six di e en labels o be assigned one o
each day. The labels in bold e e o hose days ini ially included
in CS, ha is, hose ha ob ained o ecas ing e o g ea e han
he a e age. By con as , he labels ollowed by a bulle a e hose
days ha do no belong o CS bu a e p eceded by a sequence equal
o ano he ha p ecedes a candida e. Then, he h ee labels p eced-
ing each day in CS (W
⁄
= 3 in his example) a e ex ac ed. Finally,
he cases in which he sequences be o e he candida es also
appea ed be o e any day no belonging o CS we e disca ded.
O he wise, hese sequences a e conside ed mo i s o pa e n se-
quences p eceding an especially unexpec ed alue. Fo his pa ic-
ula example, no e ha only he sequence {1,5,4} would ha e been
conside ed as mo i .
The goals a e now: (i) o o ecas he occu ence o an ou lie
and (ii) p o ide accu a e es ima ion o i . Thus, once MS is
50 100 150 200 250 300 350
0
5
10
15
Days
E o (%)
E o dis ibu ion in TS
Cl
Ch
Cm
MEAN
Fig. 1. Illus a i e dis ibu ion o candida es in C
l
,C
m
and C
h
.
cons uc ed, he gene al scheme o o ecas ing is as ollows. O igi-
nal PSF jus ex ac ed S
W

and sea ched o i in he his o ical da a.
Bu now, be o e i s sea ch, i has o be de e mined i his pa e n
sequence ma ches any o he mo i s o ming MS. Gi en his si ua-
ion, wo cases may a ise:
(1) S
W

does no ma ch any o he mo i s in MS. The app oach
de e mines ha he day o be o ecas ed is no an ou lie
and i would con inue wi h no mal PSF p ocedu e. Tha is,
^
d
iþ1
ðW

Þis calcula ed as PSF does om Eq. (4).
(2) S
W

ma ches any o he mo i s in MS. The app oach de e -
mines ha he day o be o ecas ed is an ou lie . In his case
he challenge is o p o ide accu a e es ima ions o hese
obse a ions, i.e. o p o ide accu a e alues o z
i
. To ulfill
his goal, a simple s a egy is p oposed: To a e age he alue
o he ou lie s ound by he p oposed app oach in he his o -
ical da a. Fo mally, he se o ou lying days om he aining
se is defined by:
OS ¼ i2TS such ha S
i
W

2MSgð12Þ
Then, he o ecas o he ou lie is based on he a pos e io i
de ec ed ou lie s in he aining se :
^
d
iþ1
ðW

Þ¼ 1
#OS X
i2OS
d
iþ1
ð13Þ
whe e #OS is he numbe o ou lie s de ec ed by he app oach
in he his o ical da a ( he numbe o elemen s in OS), and ^
d
i
he alues o he ime se ies o hese ou lying days ha o m
OS.
Fig. 2 illus a es he en i e p ocess o p edic ion when he dis-
co e y o mo i s is included in he PSF algo i hm. No e ha his
s ep has o be pe o med immedia ely a e he clus e ing (c ea ion
o he sequence o labels) and be o e he o ecas ing. In addi ion,
he s eps co esponding o disco e y o mo i s and p edic ion a e
u he de ailed in Fig. 3.
Finally, he pseudocode o he p oposed me hodology is p e-
sen ed in Fig. 5, and ha o he disco e y o mo i s p ocess in Fig. 6.
5. Resul s
This sec ion p esen s he esul s ob ained by he applica ion o
he p oposed me hodology o six ime se ies. The mo i s disco e y
p ocess o six eal-wo ld ime se ies is desc ibed in Sec ion 5.1.
Then, a s a is ical analysis has been ca ied ou o de e mine he
alidi y o he assump ions made when o ecas ing he occu ence
o ou lie s in he ime se ies. This analysis can be ound in Sec ion
5.2. Finally, o compa e he esul s ob ained, Sec ion 5.3 epo s
a e age o ecas ing e o s o he new me hodology and o he
echniques.
5.1. Mo i s disco e y in eal-wo ld ime se ies
The disco e y o mo i s on eal-wo ld ime se ies is now de-
sc ibed. In pa icula , six public elec ici y- ela ed ( h ee o p ices
and h ee o demand) ime se ies ha e been conside ed o show
ha he p oposed me hodology p ope ly wo ks on di e en da a-
se s. Thus, he new app oach has been applied o he Spanish
(OMEL), New Yo ke (NYISO) and Aus alian (ANEM) ma ke s,
whose da a a e a ailable on-line in Spanish Elec ici y P ice Ma ke
Ope a o (h p://www.omel.es), The New Yo k Independen Sys-
em Ope a o (h p://www.nyiso.com) and Aus alia’s Na ional
Elec ici y Ma ke (h p://www.nemmco.com.au), espec i ely.
The o ecas ing p ocess is applied o he yea 2006 o he h ee
ma ke s, wi h a his o ical da a o one yea and wi h a ho izon o
p edic ion o one mon h. As wel e mon hs a e going o be e alu-
a ed o each ma ke , he me hodology is going o be es ed on 72
da ase s. Gi en his si ua ion, e e y ime a mon h is o ecas ed he
aining se changes. Fo ins ance, when Janua y 2006 is o ecas ed,
he aining se comp ises he whole yea o 2005. Howe e , when
Feb ua y 2006 is o ecas ed he his o ical da a anges om Feb u-
a y 2005 o Janua y 2006, and so on.
These changes in he aining se in ol e changes in he config-
u a ion o PSF. Fi s o all, bo h Kand Wha e o be de e mined
acco ding o he me hodology p esen ed in Sec ion 4.Table 1 sum-
ma izes he alues o hese pa ame e s o he six ma ke s in he
yea 2006.
The esul s a e he mo i s ex ac ion s ep o he h ee ma ke s
a e summa ized inTables2–4.Tha is, a summa yo all encoun e ed
classes, sequences and mo i s can be ound in heses Tables o he
Spanish,NewYo ke andAus alianma ke s, espec i ely.Howe e ,
onlyelec ici y p ices esul so Janua y 2006 o heAus alianma -
ke a e now desc ibed as he explana ion o he emaining ele en
mon hs o each yea and ma ke is simila . The e o e, all he com-
men s abou he esul s p o ided below e e o p ices shown in
Table 4. Fi s , he pa ame e s o be se in he PSF a e equal o:
(K,W) (3,6), acco ding o Table 1. The CS can be now cons uc ed.
Fo hispu pose, he 
eðWÞ(seeEq.(7))has o beconside ed since he
candida es a e hose days belonging o he aining se (Janua y o
Decembe 2005) ha ob ained an e o g ea e han 
eðWÞ.The alue
o he mean e o , calcula ed acco ding o he me hodology in
Sec ion 4is 
eð6Þ¼5:81%. The e o e, CS would be o med by
all days in 2005 wi h o ecas ing e o g ea e han 5.81%.
Now he h ee classes a e cons uc ed by applying K-means,
wi h K= 3 as men ioned in Sec ion 4. The h ee clus e s a e defined
as: C
l
is he class ha con ains he candida es days wi h e o om
5.81% o 7.13%, C
m
he one ha ga he s he candida es wi h e o s
anging om 7.13% o 9.47% and C
h
he class ha con ains he can-
dida es wi h e o s g ea e han 9.47%. The e o dis ibu ion,
acco ding o hese h ee clus e s, is shown in Fig. 7.
F om he365 dayso 2005 ha comp ise he ainingse ,137(see
Table 4, ow 1: #C
l
+#C
m
+#C
h
= 101 + 32 + 4 = 137) had an e o
g ea e han 5.81% so he cons uc ed CS con ains 137 candida es
days. Once he candida es a e selec ed, he numbe o di e en se-
quences ha gene a ed hem a e conside ed. F om he candida es
in C
l
, 5 di e en sequences we e ound (S
l
= 5); om he candida es
in C
m
,3(S
m
= 3) and om he candida es in C
h
,2(S
h
= 2). This ac in-
ol es ha om all he K
W
= 729 possible sequences, only 10 caused
e o s g ea e han he a e age.
No e ha he e we e si ua ions in which a pa icula sequence
appea ed be o e di e en candida es ha belong o di e en
classes. Fo hese cases, only he sequence ha appea ed in he
class wi h highe associa ed e o (C
l
was de o ed o include days
wi h lowe e o s, C
m
o medium e o s and C
h
o highe ones) was
coun ed.
Finally, he numbe o mo i s ha iden i y ou lie s a e de e -
mined. F om he sequences C
l
, only one appea ed exclusi ely as
Da a Clus e ing MOTIFS
DISCOVERY P edic ion
Labeled da a
Mo i s
Inse
p edic ed sample
Mo e
days?
Yes
End
No
Fig. 2. Illus a ion o he p oposed me hodology.

an ou lie p ecu so , M
l
= 1. Wi h e e ence o he sequences in C
m
,
one ou o h ee, M
m
= 1. Las , bo h sequences in C
h
we e exclusi e,
M
h
=2.
Fig. 8 is p o ided o de e mine he use ulness o di iding he CS
in o h ee g oups. These his og ams show he mo i s dis ibu ion
along wi h C
l
,C
m
and C
h
and, o be p ecise, he ela ion be ween di -
e en sequences and mo i s o each class and ma ke , exp essed
as %. Thus, each ba is calcula ed by di iding he numbe o di e -
en sequences ha p ecede he candida es and he numbe o mo-
i s ha a e finally selec ed. As i is possible o obse e, he
p obabili y ha a sequence becomes a mo i is di ec ly ela ed
wi h he e o associa ed o he candida e o which i p ecedes.
Fo ins ance, he pe cen age o mo i s o he ANEM’s elec ici y
p ice ime se ies a e 20.26% o he sequences in C
l
, 29.82% o
he sequences in C
m
and 52.94% o he sequences in C
h
.
The mo i s ound in Janua y 2006, ep esen ed as a nume ical
sequence o labels, a e shown in Tables 5 and 6 co esponding o
p ices and demand, espec i ely. Fo ins ance, no e ha he fi s
mo i ound in ANEM’s p ices is M
1
l
¼ 1;3;2;2;3;1g. Each label
is ep esen ed by one o he Kclus e s gene a ed du ing he ain-
ing o he PSF (whe e K= 3 in his case) and iden ifies 24 h. As he
leng h o he window was se o W= 6, hese six labels in ac ep-
esen 144 h.
Figs. 9 and 10 illus a e he mos ep esen a i e mo i s ound in
all he ma ke s when o ecas ing Janua y 2006. Ac ually, hese mo-
i s ep esen he ime se ies alues ha will p ecede an ou lie . As
each mo i was ep esen ed by Wconsecu i e labels (see Defini ion
6), hese figu es depic he alues associa ed o e e y label, which
ha e been ob ained by means o clus e ing echniques.
Cons uc
CS
Clus e ing
CS
Find
mo i s
Candida es
DISCOVERY OF MOTIFS
{Cl,C
m,C
h}
Sea ch o S
in his o ical da a
PREDICTION
Mo i s se , MS
Window's leng h, W*
any mo i ?
NO
YES
Fo ecas s
S ma ches
w* w*
Sea ch o mo i s Fo ecas da a
om Eq. (13)
in his o ical da a
Fo ecas da a
om Eq. (4)
Fo ecas s
(Ou lie occu ence)
Mo i s se , MS
Window's leng h,
W
*
T aining se , TS
Numbe o clus e s, K Se W W*
Fig. 3. De ail o disco e y o mo i s and p edic ion s eps.
Fig. 4. Illus a i e example o mo i s disco e y.
Fig. 5. A gene al scheme o he p oposed me hodology.
Rega ding he elec ici y p ices, no e ha o he Spanish ma -
ke , six mo i s we e ound; six o New Yo k, and ou o he Aus-
alian ma ke . As o he elec ici y demand, he numbe o mo i s
ound we e se en, fi e and ou o he Spanish, New Yo k and
Aus alian ma ke s, espec i ely. In ac ual ac , he cu es in hese
figu es ep esen he a e age e olu ion o he fi e (OMEL), ou
(NYISO) and six (ANEM) days p io o an ou lie o ecas in p ices
ime se ies and he a e age e olu ion o he h ee (OMEL and NYI-
SO) and ou (ANEM) days p io o an ou lie o ecas in demand
ime se ies.
Fig. 6. Pseudocode o he mo i s disco e y.
Table 1
Se ing he PSF. OMEL e e s o he Spanish ma ke , NYISO o he New Yo ke ma ke and ANEM co esponds o he Aus alian ma ke .
Mon h P ices Demand
OMEL NYISO ANEM OMEL NYISO ANEM
KWKWKWKWKWKW
Janua y 4 5 5 4 3 6 7 3 4 3 6 4
Feb ua y 4 5 5 3 3 6 7 4 3 5 6 5
Ma ch 4 5 5 4 3 6 7 3 4 4 4 3
Ap il 4 5 5 4 4 6 6 4 4 5 4 4
May 45 63 46 54 44 64
June 6 4 5 3 3 6 6 3 4 5 5 5
July 5 5 6 4 3 5 6 4 3 5 6 4
Augus 6 4 6 3 4 6 5 4 4 3 5 4
Sep embe 6 4 5 3 3 6 7 3 5 4 6 5
Oc obe 6 4 5 4 3 6 5 4 5 4 5 4
No embe 6 4 5 4 3 6 5 3 4 4 6 4
Decembe 5 5 5 3 3 5 6 3 5 3 4 6
Table 2
Mo i s dis ibu ion o Spanish ma ke s.
Mon h P ices Demand
#C
l
(S
l
)[M
l
]#C
m
(S
m
)[M
m
]#C
h
(S
h
)[M
h
]#C
l
(S
l
)[M
l
]#C
m
(S
m
)[M
m
]#C
h
(S
h
)[M
h
]
Janua y 98(7)[1] 25(4)[3] 8(2)[2] 103(6)[2] 35(6)[2] 6(3)[3]
Feb ua y 87(6)[0] 31(7)[2] 5(2)[2] 121(8)[1] 22(5)[3] 3(2)[1]
Ma ch 73(5)[2] 16(3)[1] 8(1)[1] 99(7)[1] 30(7)[4] 6(1)[0]
Ap il 103(9)[1] 30(6)[1] 6(3)[2] 83(4)[0] 17(5)[2] 5(3)[3]
May 65(4)[0] 51(6)[0] 10(4)[2] 101(6)[2] 26(8)[3] 5(3)[3]
June 97(6)[0] 38(5)[2] 4(0)[0] 124(7)[3] 27(5)[1] 7(1)[1]
July 180(8)[2] 27(3)[1] 12(5)[3] 97(8)[2] 13(3)[0] 8(4)[3]
Augus 101(8)[3] 26(5)[0] 9(4)[2] 113(11)[4] 29(6)[3] 11(3)[2]
Sep embe 110(8)[1] 25(5)[3] 5(0)[0] 105(9)[2] 19(5)[4] 3(2)[2]
Oc obe 108(7)[1] 23(4)[2] 6(1)[0] 98(8)[3] 32(3)[2] 4(1)[1]
No embe 120(9)[1] 40(6)[0] 6(3)[1] 99(9)[2] 20(5)[3] 9(3)[3]
Decembe 169(10)[2] 38(9)[3] 10(3)[1] 114(8)[2] 24(3)[2] 12(2)[0]
Fig. 11 shows ANEM’s elec ici y p ices o July 2006. This
mon h is illus a ed since i p esen s la ge ou lie s. As o his
mon h W= 5 (see Table 1), he numbe o labels conside ed o ou -
lie occu ence o ecas ing is fi e. Also, he numbe o mo i s ound
in TS we e fi e (see Table 4): M
1
l
¼ 1;2;3;3;1g;M
2
l
¼ 2;2;
3;1;3g;M
3
l
¼ 3;1;1;1;1g;M
4
h
¼ 2;1;3;3;1gand M
5
h
¼ 2;1;1;
1;3gThus, g ey ba s ep esen ou lie s a pos e io i de ec ed by
means o he obus s a is ical me hod p esen ed in Gelpe e al.
(2010), ha is, days 3, 13, 18 and 22 July we e he ou lie s iden i-
fied. I can be app ecia ed ha sequences M
3
l
;M
4
h
and M
1
l
a e h ee
o he mo i s ound in TS ha e en ually p eceded days 13, 18 and
22 July, espec i ely. Fu he mo e, 3 d July was p eceded by he
mo i M
5
h
. Only he las wo labels o his mo i (1,3) a e depic ed
in Fig. 11 because he fi s h ee labels (2,1,1) co espond o days
in June. Finally, no e ha one o he mo i s ound in TS, M
2
l
, did no
occu when o ecas ing July 2006.
5.2. E alua ion o ou lie occu ence o ecas ing
Once all he MS ha e been cons uc ed, he p oposed me hod
p edic a p io i i he day o be o ecas ed will be an ou lie . This
Sec ion is de o ed o s a is ically quan i y he ou lie occu ences
Table 3
Mo i s dis ibu ion o New Yo k ma ke s.
Mon h P ices Demand
#C
l
(S
l
)[M
l
]#C
m
(S
m
)[M
m
]#C
h
(S
h
)[M
h
]#C
l
(S
l
)[M
l
]#C
m
(S
m
)[M
m
]#C
h
(S
h
)[M
h
]
Janua y 101(8)[3] 34(3)[1] 12(2)[2] 96(9)[1] 28(3)[1] 14(4)[3]
Feb ua y 92(11)[2] 36(4)[1] 14(3)[2] 88(9)[3] 14(2)[1] 7(2)[2]
Ma ch 89(7)[2] 45(5)[1] 11(2)[1] 101(9)[2] 33(2)[0] 9(3)[2]
Ap il 110(13)[3] 21(4)[1] 9(5)[4] 93(8)[2] 29(3)[2] 11(4)[2]
May 121(7)[2] 31(2)[1] 6(3)[1] 114(9)[3] 30(4)[2] 13(6)[3]
June 142(5)[1] 32(0)[0] 7(0)[0] 103(7)[0] 26(3)[1] 10(4)[2]
July 92(10)[3] 41(5)[1] 18(4)[4] 86(5)[1] 29(4)[2] 8(3)[2]
Augus 84(7)[2] 39(6)[2] 9(6)[5] 76(7)[4] 14(2)[2] 4(1)[1]
Sep embe 107(7)[0] 40(4)[0] 10(1)[1] 84(8)[1] 21(4)[1] 8(3)[3]
Oc obe 141(9)[2] 28(3)[0] 4(0)[0] 95(6)[0] 32(4)[2] 9(3)[1]
No embe 99(12)[3] 32(8)[2] 15(4)[3] 115(8)[3] 38(6)[2] 15(6)[2]
Decembe 87(8)[1] 44(3)[1] 8(0)[0] 109(8)[3] 29(4)[2] 12(4)[3]
Table 4
Mo i s dis ibu ion o Aus alian ma ke s.
Mon h P ices Demand
#C
l
(S
l
)[M
l
]#C
m
(S
m
)[M
m
]#C
h
(S
h
)[M
h
]#C
l
(S
l
)[M
l
]#C
m
(S
m
)[M
m
]#C
h
(S
h
)[M
h
]
Janua y 101(5)[1] 32(3)[1] 4(2)[2] 131(7)[0] 46(7)[2] 5(3)[2]
Feb ua y 165(5)[0] 25(6)[1] 9(1)[1] 115(6)[1] 56(7)[3] 6(3)[3]
Ma ch 133(8)[2] 13(2)[0] 3(0)[0] 92(6)[0] 72(9)[1] 7(3)[3]
Ap il 190(13)[3] 8(2)[0] 11(4)[2] 102(7)[2] 64(5)[0] 5(4)[3]
May 187(17)[5] 13(4)[2] 6(3)[1] 123(7)[2] 41(3)[1] 11(4)[0]
June 169(12)[5] 22(5)[1] 3(3)[1] 183(16)[3] 33(3)[0] 4(4)[2]
July 172(22)[3] 9(0)[0] 2(2)[2] 117(9)[1] 40(5)[0] 8(5)[3]
Augus 142(20)[2] 34(3)[2] 6(4)[2] 139(11)[2] 41(4)[3] 7(3)[1]
Sep embe 102(18)[4] 20(8)[3] 12(2)[1] 110(8)[2] 27(3)[2] 9(3)[3]
Oc obe 81(13)[2] 53(13)[4] 9(2)[0] 142(12)[3] 42(8)[3] 5(1)[1]
No embe 112(9)[1] 43(7)[1] 14(5)[4] 137(12)[2] 38(7)[3] 7(3)[2]
Decembe 121(11)[3] 39(4)[2] 13(6)[2] 134(10)[4] 29(5)[2] 8(4)[2]
0 50 100 150 200 250 300 350
0
2
4
6
8
10
12
14
16
18
20
Days
E o (%)
9.47%
7.13%
5.81%
Ch
Cm
Cl
Fig. 7. Fo ecas ing e o s in TS dis ibu ed in C
l
,C
m
and C
h
ob ained by applying he K-means o he candida e se CS wi h K=3.
F. Ma ínez–Ál a ez e al. /Pa e n Recogni ion Le e s 32 (2011) 1652–1665 1659
p ope ly o ecas ed by he p oposed me hodology. Thus, he qual-
i y pa ame e s used o e alua e he accu acy o he app oach a e
fi s in oduced in Sec ion 5.2.1 and, hen, he conduc ed s a is ical
analysis is epo ed in Sec ion 5.2.2.
5.2.1. Pa ame e s o quali y
The pa ame e s used o assess he accu acy o he app oach a e
now in oduced. A pos e io i analysis has been ca ied ou o
de e mine he exis ence o ou lie s in all examined se ies. In pa -
icula , he obus me hod p oposed in Gelpe e al. (2010) (he ea -
e called RHW o simplici y) o de ec ou lie s in ime se ies has
been conside ed. Hence, a o ecas o an ou lie occu ence is said
o be p ope ly made by he p oposed app oach, i RHW also poin s
he obse a ion as anomalous. Thus, a p io i o ecas ing ( he p o-
posed app oach in his wo k) is compa ed o a pos e io i de ec ion
( he me hod p oposed in Gelpe e al. (2010)).
No e ha he au ho s in Gelpe e al. (2010) de e mined ha
ou lie s a e hose da a ha do no ulfil any o he wo bounds hey
define:
UB
¼^
y
þ2^
ð14Þ
LB
¼^
y
2^
ð15Þ
whe e UB
and LB
a e he uppe and lowe bounds espec i ely, ^
y
is
he fi ed alue, and ^
is he s anda d de ia ion o he eg ession
esiduals hey ob ain.
Hence, in subsequen equa ions, ue posi i es o TP is he num-
be ou lie occu ences p ope ly o ecas ed, ha is, he numbe o
days p eceded by a mo i in MS ha a e ou lie s acco ding o RHW;
ue nega i es o TN is he numbe o days ha we e no p eceded
by a mo i in MS and we e no conside ed ou lie by RHW ei he ;
alse posi i es o FP is he numbe o days p eceded by a mo i in
MS ha we e no conside ed ou lie by RHW; and alse nega i es
o FN is he numbe days no conside ed ou lie s (no p eceded
by any mo i in MS) and e en ually conside ed ou lie s by RHW.
Acco ding o hese defini ions, he sensi i i y is he p obabili y
ha a mo i disco e ed p ecedes a eal ou lie s. I s o mula is de-
fined as ollows:
Sensi i
i y ¼TP
TP þFN ð16Þ
Ano he ele an pa ame e is he specifici y, o he a io o se-
quences p eceding he day o be o ecas ed p ope ly disca ded by
he app oach. The ma hema ical exp ession is:
Speci ici y ¼TN
TN þFP ð17Þ
The posi i e p edic i e alue (PPV) is he p obabili y ha a o e-
cas ed ou lie is indeed a eal one. I s o mula is:
PPV ¼TP
TP þFP ð18Þ
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
80.00%
90.00%
OMEL
ANEM
NYISO
CmCh
Cl
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
80.00%
90.00%
OMEL
ANEM
NYISO
ClCmCh
Fig. 8. Mo i s dis ibu ion in C
l
,C
m
and C
h
.
Table 5
Mo i s ound in TS o he elec ici y p ice when
o ecas ing Janua y 2006.
Ma ke Mo i
OMEL M
1
l
¼ 1;1;3;4;4g
M
2
m
¼ 3;4;4;1;3g
M
3
m
¼ 3;4;4;2;1g
M
4
m
¼ 2;3;4;4;1g
M
5
h
¼ 1;1;1;2;3g
M
6
h
¼ 3;2;3;3;4g
NYISO M
1
l
¼ 1;3;5;4g
M
2
l
¼ 2;2;3;1g
M
3
l
¼ 5;1;1;3g
M
4
m
¼ 3;4;2;5g
M
5
h
¼ 3;2;4;1g
M
6
h
¼ 3;1;5;1g
ANEM M
1
l
¼ 1;3;2;2;3;1g
M
2
m
¼ 2;1;3;3;3;1g
M
3
h
¼ 3;2;1;1;2;3g
M
4
h
¼ 3;3;3;1;1;2g
Table 6
Mo i s ound in TS o he elec ici y demand when
o ecas ing Janua y 2006.
Ma ke Mo i
OMEL M
1
l
¼ 6;4;4g
M
2
l
¼ 7;1;7g
M
3
m
¼ 5;5;4g
M
4
m
¼ 4;7;5g
M
5
h
¼ 4;4;7g
M
6
h
¼ 6;5;6g
M
7
h
¼ 7;7;2g
NYISO M
1
l
¼ 2;3;3g
M
2
m
¼ 4;3;4g
M
3
h
¼ 1;3;2g
M
4
h
¼ 3;4;1g
M
5
h
¼ 2;2;3g
ANEM M
1
m
¼ 6;5;3;4g
M
2
m
¼ 2;6;3;4g
M
3
h
¼ 2;3;1;4g
M
4
h
¼ 4;6;6;5g