CIplus
Band 7/2016
Da a P ep ocessing: A New Algo i hm
o Uni a ia e Impu a ion Designed
Specifically o Indus ial Needs
Sowmya Chand aseka an, Ma in Zae e e , S e en Mo i z,
Jö g S o k, Ma ina F iese, And eas Fischbach,
Thomas Ba z-Beiels ein
Da a P ep ocessing: A New Algo i hm o
Uni a ia e Impu a ion Designed Speci ically o
Indus ial Needs
Sowmya Chand aseka an, Ma in Zae e e , S e en Mo i z,
Jö g S o k, Ma ina F iese, And eas Fischbach,
Thomas Ba z-Beiels ein
SPOTSe en Lab, TH Köln
S einmülle allee 1, 51643 Gumme sbach
E-Mail: {sowmya.chand aseka an, ma in.zae e e , s e en.mo i z, joe g.s o k,
ma ina. iese, and eas. ischbach, homas.ba z-beiels ein}@ h-koeln.de
Abs ac
Da a p e-p ocessing is a key esea ch opic in da a mining because i plays a
c ucial ole in imp o ing he accu acy o any da a mining algo i hm. In mos
eal wo ld cases, a signi ican amoun o he eco ded da a is ound missing
due o mos di e se e o s. This loss o da a is nea ly always una oidable.
Reco e y o missing da a plays a i al ole in a oiding inaccu a e da a
mining decisions. Mos mul i a ia e impu a ion me hods a e no compa ible
o uni a ia e da ase s and he adi ional uni a ia e impu a ion echniques
become highly biased as he missing da a gap inc eases. Wi h he cu en
echnological ad ancemen s abundan da a is being cap u ed e e y second.
Hence, we in end o de elop a new algo i hm ha enables maximum
u iliza ion o he a ailable big da ase s o impu a ion. In his pape , we
p esen a Seasonal and T end decomposi ion using Loess (STL) based
Seasonal Mo ing Window Algo i hm, which is capable o handling pa e ns
wi h end as well as cyclic cha ac e is ics. We show ha he algo i hm is
highly sui able o p e-p ocessing o la ge da ase s.
1 In oduc ion
Da a p e-p ocessing in ol es emo al o noise and ou lie s om a da ase ,
handling o missing alues, da a edundancy and da a inconsis ency. One o
P oc. 26. Wo kshop Compu a ional In elligence, Do mund, 24.-25.11.2016 1
he mos challenging ask among hem is o impu e he missing alues wi h
en ies ha easonably comple e he da ase s. The loss o da a may be due
o senso e o s, ansmission e o s, e o s o he ope a o and o he e o s.
Reco e y o hese missing alues hea ily a ec s he pe o mance o he
da a mining models. The accu acy o o ecas ing, classi ica ion, es ima ion,
and pa e n de ec ion o any da a mining algo i hm depends signi ican ly
on he accu acy o da a used in modeling. Thus, inaccu a e aining and
es ing da a may in oduce bias in o he models and p o ide misleading
conclusions [
14
]. In eali y, da ase s a e commonly uni a ia e. Mo eo e ,
mul i a ia e da ase s may lack co ela ion. The need o uni a ia e ime se-
ies analysis is p e alen in many ields, o example, online da a moni o ing
and pa e n de ec ion in in ensi e ca e uni s [
1
], o ecas ing in hyd ology
and en i onmen al managemen ields [
2
], unc ional magne ic esonance
imaging s a is ical analysis [3], o ecas ing in a day a i als a a call cen-
e [
4
], o ecas ing elec ici y spo -p ices [
5
], o ecas ing mac oeconomic
ime se ies [6].
Wi h uni a ia e ime se ies da a, he complexi y o eplacing hese missing
alue inc eases as no co ela ed a iables a e a ailable. Almos all o he well
known s anda d echniques ail o handle uni a ia e ime se ies da a as hei
scheme is based on he in e -a ibu e co ela ions in es ima ing he alues
o he missing da a. Also, he exis ing mul i a ia e algo i hms ei he canno
be applied o pe o m poo ly. Fu he mo e, some adi ional impu a ion
echniques pe o m well wi h end da ase s, while some echniques pe o m
only well i he da ase is seasonal. The e exis s no single uni a ia e
impu a ion echnique ha is sui able o all ypes o da a pa e ns [
7
]. This
is owing o he eason ha mos o he exis ing algo i hms a e designed
ex ensi ely o handle ei he seasonali y o end and no bo h.
The majo mo i a ion o his wo k is he GECCO Indus ial Challenge 2015.
The ask was o eco e he missing da a in hea ing sys ems (
h p://www.
spo se en.de/gecco/gecco-challenge/gecco-challenge-2015/
). The
da a con ains 606,837 obse a ions o ou pa ame e s sampled e e y minu e
om eal indus ial hea ing sys ems. The mos challenging aspec o his
challenge is o impu e he missing da a o a la ge in e al wi h he all
da a missing. This is a commonly occu ing eal wo ld scena io, when
he e is some da a ansmission ailu e, ile o e w i ing o da a sa ing
issue equi ing uni a ia e impu a ion. Also, such scena ios lead o la ge
in e als o missing da a o which he s anda d impu a ion echniques
pe o m poo ly.
2P oc. 26. Wo kshop Compu a ional In elligence, Do mund, 24.-25.11.2016
The emainde o his pape is o ganized as ollows. Sec ion 2 ocuses on
he p oposed impu a ion echnique. Sec ion 3 illus a es he expe imen al
s udy and pe o mance compa isons among a ious impu a ion schemes.
Sec ion 4 p esen s ou concluding ema ks.
2 P oposed Algo i hm
The new algo i hm en i led Seasonal Mo ing Window Algo i hm (SMWA)
is p oposed mainly o la ge in e als o missing da a, especially seasonal
and cyclic da a. The key aspec o ou algo i hm is ha s ong seasonali y
exis s in almos all p ac ical applica ions. Mo eo e , his seasonal beha io
has o be conside ed as cyclic. The e is no gua an ee ha he beha io
o a sys em a a speci ic ime is iden ical on wo di e en days. Al hough
i is e y likely ha he sys em beha io will be simila , ega dless o
he exac ime. SMWA u ilizes Seasonal and T end decomposi ion using
Loess (STL) [
10
] o decompose he da a. Ou p oposed app oach di e s
om o he exis ing STL based echnique [
8
] in how we handle impu a ion
a e pe o ming STL decomposi ion: The decomposed end componen is
linea ly in e pola ed. The seasonal and emainde componen is i ed wi h
bes pa e n iden i ied om he pas a ailable da a. Then, he impu ed
decomposed da a is ecomposed o o m he comple e da ase .
SMWA ini ially iden i ies he missing in e al as missing da a. Then i
selec s a ini e se o da a be o e he missing in e al as head and in a
simila ashion chooses a ini e se o da a a e he missing in e al as
ail. The combina ion o head,missing da a, and ail o ms he window as
shown in Figu e 1. This window hen slides h ough he pas da a and he
bes ma ching window wi h he minimum oo mean squa e e o (RMSE)
is impu ed in he missing da a.
The da a p epa a ion o SMWA impu a ion is explained in Algo i hm 1.
A uni a ia e ime se ies
𝑡𝑠
is gi en as he inpu da ase . The inpu ime
se ies is i s alida ed o he p esence o missing alues. Then, he indexes
o he missing da a a e iden i ied.
To implemen he algo i hm, we i s pe o m seasonal decomposi ion wi h
s l
[
10
]. As we equi e a comple e da ase o
s l
, linea in e pola ion is
pe o med as desc ibed in [
11
]. A e he da a is decomposed in o seasonal,
end, and i egula componen s, he end componen is sepa a ed. Then,
P oc. 26. Wo kshop Compu a ional In elligence, Do mund, 24.-25.11.2016 3
ail
window
missing da a
head
SMWA Technique
Figu e 1: Pic o ial ep esen a ion o he SMWA Technique
SMWA is applied o he emaining componen (i.e., seasonal and i egula )
deno ed as SMWAInpu . In he sepa a ed end componen , missing alues
a e illed in using linea in e pola ion as in [
11
]. Since he end ep esen s
he long e m ise o d op in da a, simple in e pola ion is su icien o i
he missing da a in he end componen .
The SMWA impu a ion is explained in Algo i hm 2. I uses SMWAInpu
ob ained om Algo i hm 1. Based on he leng h o da a and he missing
alues a alue o
𝑙
(
𝑙
=head= ail), which ep esen s he common size o he
da ase o bo h head and ail, has o be p o ided. The minimum leng h
o he numbe o missing da a samples
𝑔
, o which he impu a ion has o
be done, has o be p o ided. Fo smalle missing in e als, i.e., less han
𝑔
missing samples, linea in e pola ion is pe o med. The use is also ee
o choose he maximum leng h o pas window
𝑤
o be conside ed o he
e alua ion. The de aul sugges ions based on p elimina y expe imen s a e
pas window size
𝑤
o
𝑛
3
and
𝑙
o
𝑤
12
o smalle da ase s, whe e
𝑛
ep esen s
he leng h o he da ase .
Le
eq
(
𝑡𝑠
)deno e he equency o he da ase . Fo la ge da ase s, e.g,
𝑛 >
100
,
000, de aul sugges ions a e he pas window
𝑤
o size
𝑛
30
and
𝑙
o
𝑛
( eq(𝑡𝑠))
o minu ely da a, o hou ly da a
𝑙
o
𝑛
(60× eq(𝑡𝑠))
, and o o he
4P oc. 26. Wo kshop Compu a ional In elligence, Do mund, 24.-25.11.2016
Algo i hm 1.: Da a p epa a ion o he Seasonal Mo ing Window Algo i hm
(SMWA)
Inpu : uni a ia e Time Se ies 𝑡𝑠
1: Valida e he inpu da a
2: De e mine he indexes o missing samples: index
3: Pe o m linea in e pola ion on 𝑡𝑠
4: Decompose in e pola ed 𝑡𝑠 in o 𝑆𝑒𝑎𝑠,𝑇 𝑟𝑒𝑛𝑑,𝐼𝑟𝑟𝑒𝑔 componen s wi h 𝑠𝑡𝑙
5: Sepa a e he 𝑇 𝑟𝑒𝑛𝑑, ill in NA in index, pe o m linea in e pola ion
6: Add 𝑆𝑒𝑎𝑠,𝐼𝑟𝑟𝑒𝑔 as SMWAInpu
Ou pu : SMWAInpu ,𝑇 𝑟𝑒𝑛𝑑,
Algo i hm 2.: Seasonal Mo ing Window Algo i hm (SMWA)
Inpu :
uni a ia e Time Se ies 𝑡𝑠
𝑙 ◁ size o da ase o head and ail
𝑔 ◁ minimum leng h o missing gap o be impu ed
𝑤 ◁ maximum leng h o pas window
𝑜𝑝𝑡𝑖𝑜𝑛 ◁ accep s s ing head, ail,bo h
SMWAInpu and T end om Algo i hm 1
1: Calcula e 𝑛as leng h o 𝑡𝑠
2: o 𝑖in 1:𝑛do
3: Iden i y he 𝑖𝑛𝑑𝑖𝑐𝑒𝑠 wi h missing in e als ≥𝑔in 𝑡𝑠
4: o 𝑒𝑎𝑐ℎ𝑖𝑛𝑑𝑒𝑥 in 𝑖𝑛𝑑𝑖𝑐𝑒𝑠 do
5: Iden i y he ac ual 𝑚𝑖𝑠𝑠𝑖𝑛𝑔 𝑑𝑎𝑡𝑎 in 𝑡𝑠
6:
Fo mula e window as
ℎ𝑒𝑎𝑑
+
𝑚𝑖𝑠𝑠𝑖𝑛𝑔 𝑑𝑎𝑡𝑎
+
𝑡𝑎𝑖𝑙 ◁
as pe speci ica ions
in 𝑜𝑝𝑡𝑖𝑜𝑛
7: o 𝑗in 1:𝑤do
8: T y
{
◁
y-ca ch o ensu e i pas window o speci ied leng h ’
𝑤
’ exis
be o e missing da a
9:
Slide he window in he pas by
𝑗
in
𝑡𝑠
and calcula e he RMSE o
head and ail}
10: Ca ch{
11: No i y e o }
12:
Find he bes i ing pas window wi h leas RMSE o head and ail
e u n The bes i ing window
13:
SMWAImpu ed
←
Impu e he alues om he bes i ing window in he
missing da a in 𝑡𝑠
Ou pu : SMWAResul ←T end +SMWAImpu ed ◁Final SMWA impu ed
ime se ies
P oc. 26. Wo kshop Compu a ional In elligence, Do mund, 24.-25.11.2016 5
equencies
𝑙
o
𝑤
10
. Conside ing he compu a ional ime, we ecommend
his algo i hm o ai ly la ge
𝑔
. This algo i hm can be compu ed wi h
ei he head only o ail only, o bo h. The pe o mance o he me hod
depends on he choice o uning pa ame e s
𝑙
,
𝑔
and
𝑤
. The alue o he
inpu pa ame e s depends on he leng h o he da ase and he pe cen age
o he missing alues and hence i migh a y o each da ase . The window
is o med wi h head,missing da a, and ail o each o he missing in e als
g ea e han
𝑔
. Then we slide his window h ough pas da a, bu no ea lie
han
𝑤
s eps be o e he window. The bes ma ching pas window wi h he
smalles oo mean squa ed e o (RMSE) is iden i ied and he alues o
his bes ma ch a e impu ed in o he gap. Finally, he end componen is
added back in o he impu ed da ase .
3 Expe imen al S udy
In all ou da ase s, as he p obabili y o missing da a does no depend
upon he obse ed o he unobse ed da a hese a e classi ied as missing
comple ely a andom (MCAR) [
9
]. As his algo i hm is p oposed mainly
o la ge in e als o missing da a, we examine his ea u e by emo ing
ela i ely la ge da a in e als. As he pe o mance o he algo i hms may
depend on he posi ion o missing alues, he missing in e als we e chosen
andomly each ime based on 30 di e en andom seeds. The pe o mance
o he algo i hm is e alua ed wi h a ious es scena ios. Fo each es
scena io he ollowing s eps a e pe o med:
1. Load a comple e ime se ies sComple e
2.
Randomly emo e alues in sComple e as pe each scena io equi e-
men s and ob ain swi hNAs
3. Apply an impu a ion algo i hm o swi hNAs o ge sImpu ed
4.
Compa e sComple e and sImpu ed by using a sui able accu acy o
e o measu e
3.1 Compa ison F amewo k
To analyze he e iciency o he SMWA algo i hm, i s pe o mance is
compa ed wi h se e al s a e o he a impu a ion echniques which we e
6P oc. 26. Wo kshop Compu a ional In elligence, Do mund, 24.-25.11.2016
implemen ed in he s a is ical p og amming language R. The RMSE was
chosen as a pe o mance measu e, i.e.,
𝑅𝑀𝑆𝐸(𝑧, 𝑧imp) = √︂∑︀𝑛
𝑡=1(𝑧−𝑧imp)2
𝑛(1)
whe e
𝑧imp
is he impu ed alue and
𝑧
deno es he ac ual alue o he ime
se ies. Mainly, me hods om impu eTS [
12
], zoo [
11
], and o ecas [
8
]
packages in R a e used o expe imen s as he desc ibed below:
∙Spline in e pola ion
: This me hod om he impu eTS package
uses
na.in e pola ion
o implemen he spline in e pola ion o
missing alues.
∙Seasonal decomposi ion
: This me hod is also om he impu eTS
package. I ini ially sepa a es he seasonal componen om he
ime se ies, hen pe o ms impu a ion on he end and i egula
componen s and inally adds he seasonal componen again. The
me hod used is
na.seadec
. The algo i hm in e nally uses mean
impu a ion o nonseasonal se ies.
∙Seasonal spli
: This is he hi d me hod om he impu eTS package.
I spli s he imes se ies in o seasons and hen pe o ms impu a ion
sepa a ely o each o he season. The algo i hm used is
na.seaspli
.
The algo i hm in e nally uses mean impu a ion o non seasonal se ies.
∙LOCF
: LOCF s ands o las obse a ion ca ied o wa d. I is a
me hod om he zoo package. I eplaces each missing alue wi h
he mos ecen non missing alue p io o i . The algo i hm used is
na.loc .
∙Mean Impu a ion
: This me hod is om he zoo package. I ills he
missing alues wi h mean alue o a ime se ies using
na.agg ega e
.
∙Linea in e pola ion
: I is a me hod om he zoo package. I
eplaces he missing alues wi h in e pola ed alues using
na.app ox
.
∙S uc u al ime se ies model
: I is a me hod om he zoo package.
I ills missing alues using seasonal Kalman il e using
na.S uc TS
.
∙STL based in e pola ion
: I is a me hod om he o ecas pack-
age, which uses linea in e pola ion o non-seasonal se ies and a
pe iodic STL decomposi ion wi h seasonal se ies. The me hod used
is na.in e p.
P oc. 26. Wo kshop Compu a ional In elligence, Do mund, 24.-25.11.2016 7
0.0
2.5
5.0
7.5
agg ega e
app ox
in e p
kalman
loc
seadec
seaspli
SMWA
spline
S uc TS
Impu a ion Algo i hm
RMSE (Roo Mean Squa e E o )
Impu a ion esul s o he Bee sales da ase
Figu e 5: Impu a ion esul o he Bee sales da ase wi h 10% o missing da a
a andom loca ions based on 30 di e en seeds
Table 3: Compa ison o RMSE (mean) & Compu a ion ime (mean)- Bee sales
da ase o a ious impu a ion echniques o 10% o missing da a a
andom loca ions based on 30 di e en seeds. Smalle alues a e be e .
Bes alues a e shown in bold ace.
Impu a ion Me hod RMSE Compu a ion ime (s)
Mean Impu a ion 1.88 0.001
Seasonal spli 0.69 0.01
Seasonal decomposi ion 0.72 0.03
Spline in e pola ion 3.85 0.009
LOCF 2.61 0.008
STL based in e pola ion 0.55 0.05
Linea in e pola ion 2.33 0.012
SMWA impu a ion 0.93 0.42
S uc u al ime se ies model 2.37 1.17
Kalman smoo hing 0.55 4.61
14 P oc. 26. Wo kshop Compu a ional In elligence, Do mund, 24.-25.11.2016
50
100
150
200
250
agg ega e
app ox
in e p
kalman
loc
seadec
seaspli
SMWA
spline
S uc TS
Impu a ion Algo i hm
RMSE (Roo Mean Squa e E o )
Impu a ion esul s o he SP da ase
Figu e 6: Impu a ion esul o he SP da ase wi h 10% o missing da a a
andom loca ions based on 30 di e en seeds
Table 4: Compa ison o RMSE (mean) & Compu a ion ime (mean) - SP
da ase o a ious impu a ion echniques o 10% o missing da a a
andom loca ions based on 30 di e en seeds. Smalle alues a e be e .
Bes alues a e shown in bold ace.
Impu a ion Me hod RMSE Compu a ion ime (s)
Mean Impu a ion 124.16 0.001
Seasonal spli 124.36 0.001
Seasonal decomposi ion 124.17 0.02
Spline in e pola ion 106.45 0.008
LOCF 129.55 0.009
STL based in e pola ion 59.76 0.03
Linea in e pola ion 59.89 0.01
SMWA impu a ion 60.40 0.34
S uc u al ime se ies model 60.42 0.22
Kalman smoo hing 49.78 1.01
P oc. 26. Wo kshop Compu a ional In elligence, Do mund, 24.-25.11.2016 15
10
20
30
40
agg ega e
app ox
in e p
kalman
loc
seadec
seaspli
SMWA
spline
Impu a ion Algo i hm
RMSE (Roo Mean Squa e E o )
Impu a ion esul s o he Re u n Tempe a u e da ase
Figu e 7:
Impu a ion esul o Re u n Tempe a u e da ase wi h missing gap o
size 100 a 10 andom loca ions based on 30 di e en seeds
3.4 Case III: La ge eal-wo ld da a se wi h inc eased da a gaps
In his es case, he comple e Re u n Tempe a u e da ase om he GECCO
Indus ial challenge is aken and 100 con inuous da a a e emo ed andomly
a 10 di e en loca ions o 30 uns. This pa icula scena io occu s in
indus ies when da a ansmission o a senso ails. The pa ame e se ings
o SMWA a e
𝑙
= 100,
𝑔
= 100, and
𝑤
= 20
,
000. Then SMWA is used o
impu e hese la ge missing gaps along wi h o he algo i hms. The me ics
in Table 5 clea ly show he impo ance and pe o mance o SMWA o
e y la ge indus ial da ase s. The SMWA bags he i s posi ion wi h a
e y low mean RMSE o 6.8 as in Figu e 7. The Kalman Smoo hing which
se ed bes in smalle da a se s canno pe o m well wi h la ge in e als
o missing da a al hough i ook e y high compu a ional ime. Also, STL
based in e pola ion, which showed be e pe o mance in smalle da ase s
anks in second posi ion ollowing SMWA. I is o be no ed ha s uc u al
ime se ies model is no e alua ed wi h such la ge da ase s o mul iple uns
as i akes e y la ge p ocessing ime. In gene al, he pe o mance o linea
in e pola ion, STL based in e pola ion and SMWA a e highly con incing
16 P oc. 26. Wo kshop Compu a ional In elligence, Do mund, 24.-25.11.2016
Table 5: Compa ison o RMSE (mean) & Compu a ion ime (mean) - Re u n
Tempe a u e da ase o a ious impu a ion echniques wi h missing
gap o size 100 a 10 andom loca ions based on 30 di e en seeds.
Smalle alues a e be e . Bes alues a e shown in bold ace.
Impu a ion Me hod RMSE Compu a ion ime (s)
Mean Impu a ion 10.25 0.02
Seasonal spli 9.02 1.45
Seasonal decomposi ion 9.97 3.25
Spline in e pola ion 13.47 1.49
LOCF 9.73 0.45
STL based in e pola ion 7.53 3.63
Linea in e pola ion 7.35 0.46
SMWA impu a ion 6.868.28
Kalman smoo hing 13.87 4110.01
S uc u al ime se ies model NA NA
in e ms o RMSE wi h such la ge in e als o missing da a. Though
linea in e pola ion gi es simila RMSE alues compa ed o SMWA, hey
eplace long gaps wi h a s aigh line, while SMWA ies o ep oduce he
simila complex pa e n as in unde lying eal da a. This pa e n eco e y
combined wi h be e accu acy and ela i ely less compu a ional ime
makes he SMWA echnique sui able o eal ime indus ial da a.
4 Conclusion
The SMWA is specially p oposed o la ge in e als o missing da a, which is
a equen ly occu ing scena io in eal indus ial applica ions. The p oposed
SMWA combines good impu a ion accu acy wi h quick compu a ional ime.
I ocuses on ex ac ing he bes possible pa e ns om he a ailable pas
da a and u ilizing i illing in he missing in e al. The addi ional posi i e
e ec o using SMWA echnique is o he abili y o he algo i hm o
p ese e he unde lying eal pa e ns be e han o he echniques.
I is also shown ha he SMWA algo i hm is well sui ed o wo k wi h
a ious kinds o uni a ia e da ase s. Al hough algo i hms like Kalman
smoo hing a e highly obus o smalle da ase s, hey ail o pe o m
P oc. 26. Wo kshop Compu a ional In elligence, Do mund, 24.-25.11.2016 17
well wi h la ge in e als o missing da a. Also, due o e y expensi e
compu a ional ime, Kalman smoo hing canno be used in p ac ice o
la ge indus ial da ase s.
Fu he mo e, he SMWA echnique can be u ilized o i he missing da a
e en om he a ailable u u e ime se ies. This can be easily done by
unning he algo i hm wi h a e e sed ime se ies. Fu he imp o emen
could be gained by app oxima ing he esul s o bo h, pas and u u e da a.
This algo i hm can be easily used on op o any o he impu a ion algo i hm,
especially o la ge missing in e als.
5 Acknowlegmen
Pa s o his wo k ha e been de eloped in he p ojec ’IMP o T: In elligen e
Mess e ah en zu P ozessop imie ung on T inkwasse be ei s ellung und
- e eilung’ ( e e ence numbe : 03ET1387A). Kindly suppo ed by he
Fede al Minis y o Economic A ai s and Ene gy o he Fede al Republic
o Ge many.
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Kon ak /Imp essum
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He ausgebe / Edi o ship
P o . D . Thomas Ba z-Beiels ein,
P o . D . Wol gang Konen,
P o . D . Bo is Naujoks,
P o . D . Ho s S enzel
Ins i u e o Compu e Science,
Facul y o Compu e Science and Enginee ing Science,
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Sch i lei ung und Ansp echpa ne / Con ac edi o ’s office
P o . D . Thomas Ba z-Beiels ein,
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Facul y o Compu e Science and Enginee ing Science,
TH Köln,
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ISSN (online) 2194-2870
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