Ci a ion: Smejkalo á, V.; Šomplák, R.;
Rosecký, M.; Š amko á, K. Machine
Lea ning Me hod o Changepoin
De ec ion in Sho Time Se ies Da a.
Mach. Lea n. Knowl. Ex . 2023,5,
1407–1432. h ps://doi.o g/
10.3390/make5040071
Academic Edi o : Pie angela
Sama a i
Recei ed: 30 Augus 2023
Re ised: 2 Oc obe 2023
Accep ed: 3 Oc obe 2023
Published: 5 Oc obe 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
machine lea ning &
knowledge ex ac ion
A icle
Machine Lea ning Me hod o Changepoin De ec ion in Sho
Time Se ies Da a
Ve onika Smejkalo á1, Rado an Šomplák1,* , Ma in Rosecký2and K is ína Š amko á3
1Facul y o Mechanical Enginee ing, Ins i u e o P ocess Enginee ing, B no Uni e si y o Technology,
Technická2896/2, 616 69 B no, Czech Republic; e onika.smejkalo a1@ u b .cz
2Czech Ma h, a.s., Šuma ská416/15, 602 00 B no, Czech Republic; [email p o ec ed]
3Facul y o Mechanical Enginee ing, Ins i u e o Ma hema ics, B no Uni e si y o Technology,
Technická2896/2, 616 69 B no, Czech Republic; k is ina.s amko a@ u b .cz
*Co espondence: ado an.somplak@ u b .cz
Abs ac :
Analysis o da a is c ucial in was e managemen o imp o e e ec i e planning om bo h
sho - and long- e m pe spec i es. Real-wo ld da a o en p esen s anomalies, bu in he was e
managemen sec o , anomaly de ec ion is seldom pe o med. The main goal and con ibu ion o his
pape is a p oposal o a complex machine lea ning amewo k o changepoin de ec ion in a la ge
numbe o sho ime se ies om was e managemen . In such a case, i is no possible o use only
an expe -based app oach due o he ime-consuming na u e o his p ocess and subjec i i y. The
p oposed amewo k consis s o wo s eps: (1) ou lie de ec ion ia ou lie es o end-adjus ed da a,
and (2) changepoin s a e iden i ied ia compa ison o linea model pa ame e s. In o de o use he
p oposed me hod, i is necessa y o ha e a su icien numbe o expe s’ assessmen s o he p esence
o anomalies in ime se ies. The p oposed amewo k is demons a ed on was e managemen da a
om he Czech Republic. I is obse ed ha ce ain was e ca ego ies in speci ic egions equen ly
exhibi changepoin s. On he mic o- egional le el, app oxima ely 31.1% o ime se ies con ain a leas
one ou lie and 16.4% exhibi changepoin s. Ce ain g oups o was e a e mo e p one o he occu ence
o anomalies. The esul s indica e ha e en in he case o agg ega ed da a, anomalies a e no a e,
and hei p esence should always be checked.
Keywo ds:
machine lea ning o ime se ies; was e gene a ion; sho ime se ies; anomaly de ec ion;
ou lie ; changepoin
1. In oduc ion
Theo e ical p ocedu es o making o ecas s a e al eady known and desc ibed in
de ail. An example is he ex ensi e heo e ical and p ac ical o e iew gi en by Pe opoulos
e al. [
1
]. Howe e , each o ecas mus be app oached indi idually wi h ega d o he
cha ac e o he da a (scope, p obabili y dis ibu ion, e c.). A c ucial pa o he o ecas
is da a p e-p ocessing [
2
]. I is app op ia e o conside a possible ans o ma ion o he
da a wi h espec o he e oskedas ici y [
3
]. Ano he impo an poin o he o ecas is he
iden i ica ion o anomalies, i.e., gene al s angeness in he da a se ies.
An example o anomalies in da a is shown in Figu e 1; i is chosen o esemble was e
gene a ion da ase s. Ob iously, he da a in Figu e 1show signs o anomalies. When an
ou lie is iden i ied, his poin is emo ed o u he wo k wi h da a. In he case o s ep
changepoin and end changepoin in he da a, espec i ely (see Sec ion 1.2), only a pa o
he ime se ies a e his anomaly is conside ed u he . When compa ing he common linea
end, signi ican di e ences in u u e o ecas s can be seen in Figu e 1(blue: 2010–2018,
ed: 2014–2018, and g een: 2015–2018) acco ding o he iden i ied anomalies. Iden i ying
ou lie s in he i s s ep is c ucial bu no always s aigh o wa d. Subjec i ely, i can be
de e mined ha he e is an ou lie alue in 2014 and ha he p oduc ion end has changed
since 2015 (g een). I can be assumed ha he end change is a pa o cyclicali y, and a
Mach. Lea n. Knowl. Ex . 2023,5, 1407–1432. h ps://doi.o g/10.3390/make5040071 h ps://www.mdpi.com/jou nal/make
Mach. Lea n. Knowl. Ex . 2023,51408
decline may be ollowed by an inc ease in he u u e. In he case o cyclicali y, he da a
would oscilla e a ound he end shown by he blue line. Howe e , cyclicali y canno be
clea ly demons a ed in such a sho ime se ies. I he alue in 2014 was no e alua ed as
an ou lie , he changepoin be ween 2013 and 2014 can be iden i ied ( ed). The au ho s a e
con inced ha he mos p obable upcoming end is he g een line (da a 2015–2018). This
hypo hesis o he au ho s was subsequen ly es ed in Sec ion 3a e he in oduc ion o
new me hods o iden i ica ion o anomalies in da a. Howe e , his opinion may no p o e
o be co ec in he u u e, and i is necessa y o e ise i wi h new da a. This is a majo
con ibu ion o his con ibu ion.
Mach. Lea n. Knowl. Ex . 2023, 5, FOR PEER REVIEW 2
has changed since 2015 (g een). I can be assumed ha he end change is a pa o
cyclicali y, and a decline may be ollowed by an inc ease in he u u e. In he case o
cyclicali y, he da a would oscilla e a ound he end shown by he blue line. Howe e ,
cyclicali y canno be clea ly demons a ed in such a sho ime se ies. I he alue in 2014
was no e alua ed as an ou lie , he changepoin be ween 2013 and 2014 can be iden i ied
( ed). The au ho s a e con inced ha he mos p obable upcoming end is he g een line
(da a 2015–2018). This hypo hesis o he au ho s was subsequen ly es ed in Sec ion 3 a e
he in oduc ion o new me hods o iden i ica ion o anomalies in da a. Howe e , his
opinion may no p o e o be co ec in he u u e, and i is necessa y o e ise i wi h new
da a. This is a majo con ibu ion o his con ibu ion.
Figu e 1. Example o p oblema ic changepoin de ec ion due o ou lie p esence.
The example in Figu e 1 shows a si ua ion based on ac ual was e gene a ion da a. As
men ioned abo e, he indi idual colo s o he linea ends co espond o diffe en
app oaches o anomalies in he da a. I is he e o e e iden ha he app oach o anomalies
can signi ican ly in luence he o ecas o he gene a ion o was e p oduc ion. This ype o
o ecas ing is essen ial o planning in was e managemen [4]. To be able o handle was e
adequa ely and ul ill he legisla ion se by he EU, i is necessa y o build he necessa y
in as uc u e [5]. In he case o a low-quali y o ecas , he e is a isk ha he in as uc u e
will be insufficien o excessi e capaci y. The di ec consequence is hen inancial losses
and non- ul illmen o he legisla i e goal. The pu pose o he analysis o anomalies in he
his o ical was e managemen da a is he p e-p ocessing o he da a so ha quali y
o ecas s can be made.
Diffe en kinds o was e a e analyzed qui e o en o suppo decision-making in
was e managemen (WM) and subsequen use o was e. In some cases, WM da a a e
a ailable on a daily o weekly basis, mainly o la ge ci ies like Zag eb [6], Helsinki [7],
o Teh an and Mashad [8]. Howe e , annual da a a e s ill mo e common, especially o
highe e i o ial le els. In such a case, he numbe o da a poin s is low, e.g., o he
coun ies o he Eu opean Union, da a was a ailable o 23 yea s [9]. Howe e , longe ime
se ies a e o en no adequa e as da a collec ion me hodologies change. Small da ase s a e
usually difficul o deal wi h, as common s a is ical ools aim o a sample size o 30 o
mo e. In all cases, accu a e o ecas s and unde s anding o was e gene a ion a e c ucial
o u he s eps and hei eal-wo ld implemen a ion. S udy [6] ocused on sho - e m
o ecas s, while he main goal o Kannanga a e al. [10] o Niska and Se kkola [7] was o
unde s and he socio-economic impac s on was e gene a ion. Howe e , e en in s udies
Figu e 1. Example o p oblema ic changepoin de ec ion due o ou lie p esence.
The example in Figu e 1shows a si ua ion based on ac ual was e gene a ion da a.
As men ioned abo e, he indi idual colo s o he linea ends co espond o di e en
app oaches o anomalies in he da a. I is he e o e e iden ha he app oach o anomalies
can signi ican ly in luence he o ecas o he gene a ion o was e p oduc ion. This ype o
o ecas ing is essen ial o planning in was e managemen [
4
]. To be able o handle was e
adequa ely and ul ill he legisla ion se by he EU, i is necessa y o build he necessa y
in as uc u e [
5
]. In he case o a low-quali y o ecas , he e is a isk ha he in as uc u e
will be insu icien o excessi e capaci y. The di ec consequence is hen inancial losses
and non- ul illmen o he legisla i e goal. The pu pose o he analysis o anomalies in he
his o ical was e managemen da a is he p e-p ocessing o he da a so ha quali y o ecas s
can be made.
Di e en kinds o was e a e analyzed qui e o en o suppo decision-making in was e
managemen (WM) and subsequen use o was e. In some cases, WM da a a e a ailable on
a daily o weekly basis, mainly o la ge ci ies like Zag eb [
6
], Helsinki [
7
], o Teh an and
Mashad [
8
]. Howe e , annual da a a e s ill mo e common, especially o highe e i o ial
le els. In such a case, he numbe o da a poin s is low, e.g., o he coun ies o he
Eu opean Union, da a was a ailable o 23 yea s [
9
]. Howe e , longe ime se ies a e o en
no adequa e as da a collec ion me hodologies change. Small da ase s a e usually di icul
o deal wi h, as common s a is ical ools aim o a sample size o 30 o mo e. In all cases,
accu a e o ecas s and unde s anding o was e gene a ion a e c ucial o u he s eps and
hei eal-wo ld implemen a ion. S udy [
6
] ocused on sho - e m o ecas s, while he
main goal o Kannanga a e al. [
10
] o Niska and Se kkola [
7
] was o unde s and he socio-
economic impac s on was e gene a ion. Howe e , e en in s udies like [
11
] o [
12
], when
was e gene a ion is no analyzed di ec ly, WM da a quali y is impo an o achie e eliable
esul s, e.g., when e-was e is o in e es [
13
], i is easy o imagine ha apid de elopmen in
a numbe o e-was e de ices can in oduce anomalies in o da a. The same is e en mo e
Mach. Lea n. Knowl. Ex . 2023,51409
impo an ou side o municipal solid was e (MSW), whe e was e gene a ion can beha e
much mo e i egula ly ac oss he yea s. An example o such a case is ag icul u e- ela ed
was e p ocessing and op imiza ion [
14
]. E en mo e s able kinds o was e, like soil was e [
15
],
can be a ec ed by mine closu es and openings. I is known ha anomalies (ou lie s o
changepoin s) a e p esen in WM da ase s, a leas in some cases. Causes o hese p oblems
a e such as w ong en y (ou lie ) o s uc u al change (changepoin ). S uc u al changes can
be caused by legisla i e o o he changes in he WM sys em [
16
]. Al hough changepoin
de ec ion in ime se ies is qui e a common opic in he scien i ic li e a u e, see, e.g., [
17
], no
pape dealing wi h his issue in he WM ield was ound.
No au oma ic anomaly de ec ion me hod can ou pe o m an expe , especially In he
case o sho ime se ies. An expe can e en ind he eason o anomaly due o domain
knowledge. Howe e , a la ge numbe o was e ypes ha need special ea men exis
(e.g., a ious kinds o haza dous was e). In case mul iple ypes o was e a e analyzed o
many e i o ies, he numbe o such sho ime se ies is g owing as . I is no possible
o examine, e.g., ens o housands o ime se ies by an expe . Au oma ic de ec ion o
anomalies is hus a needed bu complica ed ask. In o de o iden i y changepoin s (i.e.,
s ep changes and end changes) in he da a, i is ad isable o emo e he ou lie s i s .
The e a e a numbe o me hods o ou lie analysis. Howe e , da a on was e p oduc ion
ha e hei speci ic cha ac e . Based on a e iew o exis ing me hods, sui able me hods we e
selec ed and es ed on ep esen a i e samples. The aim is o selec an app op ia e p ocedu e
o iden i ying ou lie s, conside ing his pa icula da a ype, Sec ion 2. In he nex s ep,
and his is he main con ibu ion o his pape , he a ailable app oaches o iden i ying
changepoin s a e assessed (Sec ion 2.3). The exis ing app oaches we e compa ed wi h he
app oach p oposed by he au ho s o his pape , and hei pe o mance was e alua ed. The
new app oach was u he applied in he case s udy o all was es p oduced in he Czech
Republic o he pe iod 2010–2018 (Sec ion 3).
1.1. Ou lie s
E en be o e anomaly de ec ion, some o ms o da a p e-p ocessing should be con-
side ed [
1
]. These include he emo al o ex e nal e ec s (e.g., he COVID pandemic is
likely o in oduce a signi ican inc ease in he amoun o packaging was e), agg ega ion
o was e ypes/ e i o ies (e.g., when a ious was e ypes a e ea ed as he same o only
highe e i o ial uni is o in e es ), ede ini ion o a iable o in e es (e.g., amoun o
sepa a ed was e o a io o he sepa a ed o o al amoun o was e) o da a ans o ma ion
(e.g., loga i hmic). These can bo h educe he numbe o anomalies and help o answe he
igh ques ions. Howe e , hese adjus men s a e no i ial since, e.g., a sui able p oxy o
ex e nal e ec is needed, and i is no commonly a ailable (o i is a ailable only o highe
e i o ial uni s).
Ou lie de ec ion is a opic ha has a ac ed a lo o in e es in ecen yea s in gene al,
e.g., [
18
], as well as in a ime se ies con ex . Only gene al ime se ies wo ks will be
discussed he e since no pape ocused speci ically on sho - ime se ies was ound. B aei
and Wagne [
19
] ocus on uni a ia e ime se ies only and o e a classi ica ion o me hods
in o h ee g oups: (1) s a is ical me hods, (2) classical machine lea ning me hods, and
(3) deep lea ning me hods.
Choi e al. [
20
] aimed owa d mul i a ia e ime se ies, which a e mo e common
nowadays. I men ions he challenges o classical app oaches, namely he lack o labels and
complexi y o da a and explains he bene i s o deep lea ning app oaches. These include
he possibili y o es ablishing ela ionships be ween a iables and modeling o empo al
con ex . This pape men ions h ee possible ways o ob ain an anomaly sco e om deep
lea ning me hods: (1) econs uc ion e o , (2) p edic i e e o , and (3) dissimila i y.
Blázquez-Ga cía e al. [
21
] deals wi h bo h uni a ia e and mul i a ia e case. The
men ioned uni a ia e and mul i a ia e me hods a e basically in ag eemen wi h al eady
men ioned e iews [
19
,
20
]. Howe e , he usage o uni a ia e me hods o mul i a ia e cases
is also discussed. The main di ec ion in how o le e age well-de eloped uni a ia e me hods
Mach. Lea n. Knowl. Ex . 2023,51410
is o i s apply a dimensionali y educ ion echnique (e.g., PCA—p incipal componen
analysis) and hen one o he adi ional me hods. This me hod is able o compa e he
municipali ies in indi idual yea s and iden i y ou lie s. I is, he e o e, no a ques ion
o e alua ing ou lie s in a ime se ies bu a he o iden i ying locali ies wi h anomalous
beha io . This app oach is no sui able o he applica ion equi ed in his s udy, whe e
ou lie emo al is o p ecede changepoin de ec ion.
Mos o he cu en esea ch in ou lie de ec ion in empo al da a aims o sol e p ob-
lems in ol ing mul i a ia e da ase s o eno mous sizes. In such a case, da a a e collec ed
au oma ically, and some imes eal- ime p ocessing is needed. Chalapa hy and Chawla [
22
]
p o ide examples o hese applica ions, e.g., in usion de ec ion, aud de ec ion, malwa e
de ec ion, medical anomaly de ec ion o indus ial anomalies de ec ion. Deep lea ning
app oaches a e usually success ul in hese cases. Howe e , au oma ic (and high- equency)
da a collec ion is s ill no common in WM. Insu icien leng hs o indi idual ime se ies
s ongly es ic he possible ange o me hods in such a case.
A di e en iew is o e ed by an ex ensi e e iew [
23
], which ecognizes hese
g oups o app oaches: classi ica ion, nea es neighbo , clus e ing, s a is ical echniques,
in o ma ion– heo e ical and spec al echniques. S a is ical echniques use s anda d s a is i-
cal es s (e.g., G ubbs’s, Dixon’s o Rosne ’s es , z-sco e, boxplo ule) o s a iona y ime
se ies. Fo non-s a iona y da a, he end- emo ing model is i ed, and he esiduals o his
model a e subsequen ly analyzed. To de ec ou lie s in esiduals, some a bi a y h eshold
o model- ela ed measu es like he Akaike in o ma ion c i e ion a e commonly used. The
las g oup o s a is ical app oaches can be called dis ibu ion es ima ion- ela ed. These
me hods assume ei he ha a sample is a composi ion o mul iple known pa ame ic dis i-
bu ions (mul iple “no mal” dis ibu ions o “no mal” and “anomalous” mix u es) o y o
es ima e he dis ibu ion based on he da a using non-pa ame ic me hods (his og ams o
ke nel densi y es ima ion). In o ma ion– heo e ical app oaches (using, e.g., Kolmogo o
complexi y, en opy, ela i e en opy) use a sliding window. Spec al echniques use me h-
ods like p incipal componen analysis, and hey y o cap u e he bulk o a iabili y in he
da a by combining he da a a ibu es.
Ou lie de ec ion echniques used in WM include he boxplo ule [
10
], exclusion o
op and bo om deciles [
16
] and z-sco es [
24
]. Howe e , Kannanga a e al. [
10
] do no
ake ime de elopmen in o accoun and he usage o deciles [
16
] is likely o be oo ough.
Rybo áe al. [
24
] deal only wi h one yea o annual da a (so he ime componen is no ele-
an ). Fo he de ec ion o ou lie s, an app oach combining common me hods is p esen ed
in Sec ion 2.1.
1.2. Changepoin s
The e m changepoin is used in his pape o e e o wo ypes o anomalies: (1) s ep
changepoin s; and (2) end changepoin s.
S ep changepoin s a e sudden changes a a ce ain poin in he ime se ies. These
changes may include changes in he a e age o o he pa ame e s o he p ocess ha c ea e
he ime se ies. Thanks o he de ec ion o s ep changepoin s in he da a, i is possible o ind
ou when he s uc u e o he ime se ies changes and hus be e unde s and he analyzed
p oblem. Va ious me hods make i possible o ind ou whe he he e a e signi ican s ep
changepoin s in he ime se ies and also de e mine hei posi ion. Inco ec iden i ica ion o
s ep changepoin s in he da a can lead o e oneous conclusions and inaccu a e p edic ion
models. The example o he s ep changepoin is shown in Figu e 2a; he posi ion o he s ep
changepoin is be ween he yea s 2012 and 2013, whe eas he yea 2012 will be called he
beginning o he s ep changepoin and he yea 2013 as he end o he s ep changepoin .
T end changepoin s a e sudden changes in he beha io and p ope ies o he obse ed
se ies. Speci ically, a end changepoin indica es a si ua ion wi h a signi ican change in
he angen . An example o he end changepoin is shown in Figu e 2b, whe e he end
changepoin is obse ed in 2013.
Mach. Lea n. Knowl. Ex . 2023,51411
Mach. Lea n. Knowl. Ex . 2023, 5, FOR PEER REVIEW 5
Figu e 2(a); he posi ion o he s ep changepoin is be ween he yea s 2012 and 2013,
whe eas he yea 2012 will be called he beginning o he s ep changepoin and he yea
2013 as he end o he s ep changepoin .
Figu e 2. Examples o changepoin s: (a) s ep changepoin ; (b) end changepoin .
T end changepoin s a e sudden changes in he beha io and p ope ies o he
obse ed se ies. Speci ically, a end changepoin indica es a si ua ion wi h a signi ican
change in he angen . An example o he end changepoin is shown in Figu e 2 (b),
whe e he end changepoin is obse ed in 2013.
In he li e a u e, hese wo kinds o changepoin s a e some imes analyzed sepa a ely.
Acco ding o T uong e al. [25], me hods o changepoin de ec ion can be di ided in o
wo main b anches: 1. online; and 2. offline. Online me hods a e based on eal- ime
de ec ion o changes. Offline me hods de ec changes in he en i e da ase a once, so hey
e ospec i ely iden i y he loca ion o ab up changes. Fo he pu poses o his pape , a
mo e de ailed desc ip ion o offline me hods wi h machine lea ning p inciples will be
sufficien .
1.3. Machine Lea ning in Changepoin De ec ion
Acco ding o he ex ensi e o e iew o me hods by Aminikhanghahi and Cook [17],
machine lea ning algo i hms can be di ided in o 1. supe ised; and 2. unsupe ised.
Supe ised me hods a e machine lea ning asks whe e he algo i hm analyses he
aining da ase , which consis s o pai s o inpu and ou pu a iables. The goal o he
algo i hm is o lea n he mapping unc ion om he inpu o he ou pu . When he
supe ised me hods a e applied o he p oblem o changepoin de ec ion, algo i hms a e
ained as classi ie s (bina y o mul i-s a e). A e de e mining he numbe o s a es and
s a e bounda ies, he me hods wo k on he p inciple o a sliding window passing h ough
he ime se ies and looking o he occu ence o changepoin s [17]. A summa y o possible
mul i-class classi ie s: decision ees, nea es neighbo , suppo ec o machine, Naï e
Bayes, Bayesian ne , hidden Ma ko Model, condi ional andom ield and Gaussian
Mix u e Model can be ound in he s udy [17]. Also, an o e iew o bina y class classi ie s
is offe ed [17]: suppo ec o machine, Naï e Bayes and logis ic eg ession. A mo e
de ailed e iew o online changepoin de ec ion me hods is offe ed in he s udy [17,26].
Supe ised machine lea ning app oaches a e usually ai ly accu a e models wi h simple
c ea ion. The main disad an age is he dependence on he quali y o he aining da a.
Figu e 2. Examples o changepoin s: (a) s ep changepoin ; (b) end changepoin .
In he li e a u e, hese wo kinds o changepoin s a e some imes analyzed sepa a ely.
Acco ding o T uong e al. [
25
], me hods o changepoin de ec ion can be di ided in o
wo main b anches: (1) online; and (2) o line. Online me hods a e based on eal- ime
de ec ion o changes. O line me hods de ec changes in he en i e da ase a once, so
hey e ospec i ely iden i y he loca ion o ab up changes. Fo he pu poses o his
pape , a mo e de ailed desc ip ion o o line me hods wi h machine lea ning p inciples will
be su icien .
1.3. Machine Lea ning in Changepoin De ec ion
Acco ding o he ex ensi e o e iew o me hods by Aminikhanghahi and Cook [
17
],
machine lea ning algo i hms can be di ided in o (1) supe ised; and (2) unsupe ised.
Supe ised me hods a e machine lea ning asks whe e he algo i hm analyses he
aining da ase , which consis s o pai s o inpu and ou pu a iables. The goal o he
algo i hm is o lea n he mapping unc ion om he inpu o he ou pu . When he su-
pe ised me hods a e applied o he p oblem o changepoin de ec ion, algo i hms a e
ained as classi ie s (bina y o mul i-s a e). A e de e mining he numbe o s a es and
s a e bounda ies, he me hods wo k on he p inciple o a sliding window passing h ough
he ime se ies and looking o he occu ence o changepoin s [
17
]. A summa y o possible
mul i-class classi ie s: decision ees, nea es neighbo , suppo ec o machine, Naï e
Bayes, Bayesian ne , hidden Ma ko Model, condi ional andom ield and Gaussian Mix-
u e Model can be ound in he s udy [
17
]. Also, an o e iew o bina y class classi ie s is
o e ed [
17
]: suppo ec o machine, Naï e Bayes and logis ic eg ession. A mo e de ailed
e iew o online changepoin de ec ion me hods is o e ed in he s udy [
17
,
26
]. Supe ised
machine lea ning app oaches a e usually ai ly accu a e models wi h simple c ea ion. The
main disad an age is he dependence on he quali y o he aining da a.
Unsupe ised me hods a e algo i hms ha disco e hidden pa e ns based on s a is i-
cal ea u es, no da a labeling. The ypical app oach is o conside p obabili y dis ibu ions
om which da a in he pas and p esen a e gene a ed. These wo dis ibu ions a e s a-
is ically es ed o de e mine whe he hey a e equal o signi ican ly di e en . This kind
o app oach is based on he likelihood a io. This is he a io be ween wo p obabili y
densi ies calcula ed in wo consecu i e in e als [
27
]. The second line o app oach is
subspace iden i ica ion; his is based on he analysis o subspaces in which ime se ies
sequences a e cons ained [
28
]. The las app oach is called p obabilis ic me hods, which
is di ided in o wo lines: (1) Bayesian; and (2) Gaussian. The assump ion o Bayesian
me hods is ha a ime se ies may be di ided in o non-o e lapping s a e pa i ions, and he

Mach. Lea n. Knowl. Ex . 2023,51412
da a wi hin each s a e a e om some p obabili y dis ibu ion. Compa ed o he likelihood
a io me hods, Bayesian me hods conside no only pai s o consecu i e in e als bu
all p e ious in e als [
29
]. In he Gaussian me hods (also called he Gaussian p ocess),
ime se ies obse a ions a e de ined as a noisy e sion o Gaussian dis ibu ion unc ion
alues. Gaussian p ocess unc ion is used o make a no mal dis ibu ion p edic ion a ime
using obse a ions a ailable h ough ime (
−
1). Then, he p- alue is calcula ed o he
ac ual obse a ion unde he e e ence dis ibu ion. The
α
- h eshold is used o e alua e
he p- alue, and an algo i hm de e mines whe he he ac ual obse a ion does no ollow
he p edic i e dis ibu ion, which indica es a changepoin [
30
]. A mo e de ailed e iew o
o line changepoin de ec ion me hods is o e ed in s udies [17,25].
I should be no ed ha changepoin de ec ion is usually done on much longe ime
se ies (hund eds o housands o da a poin s). The use o common me hods o annual
da a (e.g., in WM) is he e o e e y limi ed.
1.4. Summa y o Li e a u e Re iew and No el y
No pape dealing wi h changepoin de ec ion in WM was ound. Thus, his pape
aims o p opose a complex amewo k o changepoin de ec ion in WM ime se ies.
Gene ally, he e is li le o no a en ion o ou lie de ec ion in sho ime se ies (say,
leng h < 30). Ou lie s a e made using a combina ion o known me hods. P e ious s udies
ha e no add essed his app oach, and his is he i s poin o no el y in his s udy. Di e en
app oaches o ou lie s we e es ed, and he solu ion ha achie ed he bes esul s was
ecommended (Sec ion 2.1).
A sui able me hod was no ound o he de ec ion o changepoin s on WM da a,
which is also p o en by es ing in Sec ion 2.5. Wi h ega d o he inapp op ia eness o
using exis ing app oaches o he WM da a, a comple ely new app oach o he analysis
o changepoin s is p esen ed, aking in o accoun he speci ic na u e o da a in he WM
a ea (Sec ion 2.3). The main goal is o de elop an app oach ha allows (1) au oma ed
upda ing o esul s o changepoin de ec ion and (2) lea ning o he model wi h he help
o new expe e alua ions. The p oposed amewo k o changepoin de ec ion can be
help ul o e e ybody dealing wi h WM da a. Was e gene a ion da a a e essen ial o WM
in as uc u e, capaci y alloca ion and shi o he ci cula economy.
2. Ma e ial and Me hods
This sec ion in oduces me hods used o bo h ou lie and changepoin de ec ion.
The me hods we e es ed on was e p oduc ion da a om he Czech Republic, da ase
includes annual da a o he pe iod 2010–2018. Was e p oduc ion in he Czech Republic
is egis e ed unde ca alog numbe s, which a e classi ied in o 20 g oups acco ding o he
o igin o he was e. Was e g oup 3 (was e om wood ea men ) and was e g oup 20
(municipal solid was e) we e chosen o es ing. Was e g oup 20 consis s o 15,320 ime
se ies and ep esen s was e wi h a ela i ely s able end. Con e sely, g oup 3 consis s o
737 ime se ies and ep esen s highly a iable was e p oduc ion (see Appendix A). Since
he p oposed me hods inco po a e expe judgmen , smalle subse s we e used o es ing
da a. As al eady men ioned in Sec ion 1, li le o no wo k was done in anomaly de ec ion in
WM. Thus, he changepoin de ec ion amewo k is a e y impo an ool o da a analysis
in WM.
Da a ans o ma ions can e en imp o e he p ope ies o he ime se ies unde exami-
na ion. On he o he hand, commonly used ans o ma ions like loga i hms can b ing in
some p oblems i ze o was e gene a ion is possible o a pa icula was e ype. In such a
case, ze oes need o be eplaced, bu he selec ion o he igh alue o eplacemen can be
a di icul ask wi h a big impac on subsequen analysis. Thus, none o hese adjus men s
was used he e. This p oblem emains an op ion o u u e esea ch.
The p oposed me hodology o anomaly de ec ion consis s o wo subsequen s eps:
1. Ou lie de ec ion (Sec ions 2.1 and 2.2);
2. Changepoin de ec ion (Sec ions 2.3–2.5).
Mach. Lea n. Knowl. Ex . 2023,51413
Ou lie de ec ion is a necessa y p e equisi e o subsequen changepoin de ec ion. A
combina ion o common app oaches o da a p ocessing is used o ou lie de ec ion. A new
me hod, including a machine lea ning app oach, was de eloped o changepoin de ec ion;
he e o e, a signi ican ly la ge space is de o ed o his pa .
Unless o he wise s a ed, all o he compu a ions, da a manipula ion and isualiza ion
we e conduc ed ia R so wa e (R Co e Team, 2021, [31]) wi h de aul pa ame e se ing.
2.1. Ou lie De ec ion—Me hods Desc ip ion
This sec ion p o ides a sho desc ip ion o me hods selec ed o es ing based on
e iew. Some ypes o models o de ec ing ou lie s a e sui able o a speci ic ype o da a.
We can men ion, o example, he widely used seman ic models ha canno be di ec ly used
o he issue o ime se ies [
32
]. The i s g oup o conside ed me hods o sho ime se ies
consis s o i ing a simple model and subsequen analysis o esiduals. The Hol me hod
was selec ed as sui able o end emo al. Since i is gene ally sligh ly mo e lexible han,
e.g., linea i , bu no as lexible as polynomial o nonlinea i s. Such a balance is needed
o a oid signi ican unde o o e i ing. No e ha in his s ep, he main goal is no o i
he da a pe ec ly bu a he o es ima e and emo e gene al ends om he da a o use
o s a iona y da a me hods. Th ee me hods we e selec ed o he analysis o esiduals,
namely he Dixon es , G ubbs es and z-sco e. The elec ed es s (G ubbs and Dixon)
should be able o deal e en wi h small da ase s. Z-sco e is a common me hod o compa e
esiduals wi h a no mal dis ibu ion, which uses quan iles o no mal dis ibu ion o assess
possible ou lie s. “Fa away” usually means wo o h ee s anda d de ia ions om he
mean, which co esponds o he p obabili y o obse ing a leas such a dis an poin wi h
a p obabili y o 4.6 and 0.27%, espec i ely (based on he assump ion ha da a come om
no mal dis ibu ion). Howe e , his me hod is mo e o an unw i en ule han an exac es .
In con as , bo h he G ubbs and Dixon es s a e exac es s used o ou lie de ec ion. These
es s should be sui able e en o small da ase s, which a e e y bene icial in ou p oblem
se up, see [33,34].
O he common me hods selec ed o es ing (LOF, GLOSH and kNN dis ance) a e
mo e o less connec ed o he no ion o densi y. The kNN dis ance echnique o ou lie
de ec ion consis s o i s c ea ing a kNN g aph and hen. The anomaly sco e is de ined
as he dis ance o he k- h nea es neighbo . The main idea o LOF is also based on he
kNN echnique (B eunig e al., 2000) [
35
]. I uses eachabili y dis ance, which is he ac ual
dis ance o wo poin s (e.g., A and B). Howe e , i he poin s a e close enough (i.e., B lies
wi hin a adius de ined by dis ance om A o i s k- h nea es neighbo ), he dis ance o
k- h nea es neighbo (so-called k- h dis ance) is used ins ead o he ac ual dis ance. LOF
measu es local eachabili y densi y (i.e., he in e se o he a e age eachabili y dis ance o
he objec om i s neighbo s). Then, he LOF is he a e age local eachabili y densi y o he
neighbo s di ided by he objec ’s own local eachabili y densi y.
Finally, GLOSH is he me hod ha uni ies bo h he global and local la o s o he
ou lie de ec ion p oblem in o a single de ini ion o an ou lie de ec ion measu e [
35
].
S a ing om he usual assump ion ha he e a e one o mo e da a-gene a ing p ocesses
deemed nonsuspicious and no icing ha clus e s a e na u al candida es o model such
gene a o (s). The scope o he e e ence se can be adjus ed o each objec based on
he closes clus e (in a densi y-based pe spec i e) wi hin he densi y-based hie a chy.
The e o e, hie a chical DBSCAN (densi y-based spa ial clus e ing o applica ions wi h
noise) o HDBSCAN (hie a chical DBSCAN) clus e ing is done i s . Then,
e(xi)
is he
lowes adius a which
xi
s ill belongs o i s clus e (and below which
xi
is labeled as
noise).
emax(xi)
is he lowes adius a which his clus e o any o i s subclus e s s ill exis
(and below which all i s objec s a e labeled as noise). The GLOSH sco e is hen de ined
as ollows:
GLOSH(xi)=1−max(xi)
e(xi)(1)
Mach. Lea n. Knowl. Ex . 2023,51414
Bo h ime and was e p oduc ion we e escaled o [0,1] in e als o e e y ime se ies
o allow a gene al se ing o h esholds o kNN dis ance, LOF and GLOSH. As in he case
o he z-sco e, hese echniques a e no s a is ical es s, so a sui able h eshold needs o
be ound o each speci ic p oblem. The gene al ecommenda ion o he kNN dis ance
h eshold is qui e di icul , so i is ecommended o explo e he his og am o alues and
i e a i ely adjus he h eshold. Fo LOF, poin s wi h a sco e “la ge ” han 1 can be
conside ed ou lie s. Howe e , in some cases, 1.2 is la ge enough; in o he cases, 3 is no .
Sco es a e om he ange [0,1] in he case o GLOSH, while alues close o 1 a e suspicious.
Th eshold iden i ica ion
The Dixon and G ubbs es s a e used in he usual way wi h a se signi icance le el
(a common alue o 0.05 is used he e). Fo o he me hods, a sui able h eshold has o be
de e mined. I is necessa y o ind a sui able h eshold o o he conside ed app oaches
(z-sco e, kNNd, LOF, and GLOSH). The e a e gene al ecommenda ions o se ing alues,
bu no speci ic alues a e a ailable. Fo he z-sco e, he limi alue o 3 is mos o en used,
some imes 2, bu hese alues we e no sui able o he es ed da a. So, he limi alues we e
adjus ed o he z-sco e as well. Fi s , ou lie s a e de ec ed using he selec ed algo i hm. The
ollowing p ocedu e was applied o e i y he algo i hm (A) and ind a sui able pa ame e
se ing (B):
A. Ve i ica ion o algo i hmic solu ion based on isual assessmen :
1.
Rep esen a i e examples we e selec ed o indi idual ypes o was e o expe
assessmen . Fo each ype o was e, 10 suspicious se ies we e selec ed (i
a ailable), which, acco ding o he au ho s, ep esen ed he la ges possible
spec um o anomalous cases. In o al, 230 ime se ies we e assessed. This
p ocedu e should con ibu e o a mo e gene al pa ame e se ing;
2.
Fi e expe s independen ly e alua ed he ou lying se e i y on a scale o 1
(ce ainly no )–4 (ce ainly yes). The p oposed p ocedu e can be speci ied in
he u u e by g ea e in ol emen o expe knowledge and mo e sensi i e
ea men o indi idual was e ac ions.
B. Se ing he pa ame e s o algo i hms conside ing expe judgmen :
1.
The expe e alua ion om poin 2 was used o se he limi alue o each
app oach, om which he obse a ion is al eady conside ed ou lying.
The h eshold was se as he sample o he z-sco e, as shown in Figu e 3. Fi s , he
median o he expe e alua ion was calcula ed o each e alua ed se ies. Due o he odd
numbe o expe s, ca ego ies 1–4 we e main ained. Subsequen ly, he median a ing was
compa ed wi h he c i e ion alue o he suspec ed poin . A g aphic ep esen a ion is
p o ided in Figu e 3. The esul ing limi alue o 2.17 was de e mined as he median o he
z-sco e o he median expe e alua ion (e_med = 3). I is also easy o see om Figu e 3
ha he di e ences in medians be ween o igina o s a e ypically no e y la ge; he e o e,
i is possible o use a uni e sal pa ame e o bo h o igina o s.
2.2. Ou lie De ec ion—Pe o mance E alua ion
As al eady men ioned, a he simple me hods we e selec ed o ou lie de ec ion wi h
ega d o he leng h o he in es iga ed ime se ies. These include he p ocedu e consis ing
o da a i ing by he Hol me hod ( end) and he subsequen analysis o esiduals (Dixon
es , G ubbs es , z-sco e). These s eps a e epea ed i e a i ely i an ou lie was ound in he
p e ious i e a ion. O he common echniques like LOF, GLOSH and kNN dis ance we e
also conside ed. A o al o six a ian s o me hods will be es ed. Subsequen ly, a isual
assessmen o he success o he app oach o ou lie sea ch was pe o med.
Mach. Lea n. Knowl. Ex . 2023,51415
Mach. Lea n. Knowl. Ex . 2023, 5, FOR PEER REVIEW 9
3 ha he diffe ences in medians be ween o igina o s a e ypically no e y la ge;
he e o e, i is possible o use a uni e sal pa ame e o bo h o igina o s.
Figu e 3. Boxplo o z-sco es o he Hol me hod by he median o expe e alua ion and
o igina o .
Rema k: Do s indica e ou lie s.
2.2. Ou lie De ec ion—Pe o mance E alua ion
As al eady men ioned, a he simple me hods we e selec ed o ou lie de ec ion wi h
ega d o he leng h o he in es iga ed ime se ies. These include he p ocedu e consis ing
o da a i ing by he Hol me hod ( end) and he subsequen analysis o esiduals (Dixon
es , G ubbs es , z-sco e). These s eps a e epea ed i e a i ely i an ou lie was ound in
he p e ious i e a ion. O he common echniques like LOF, GLOSH and kNN dis ance
we e also conside ed. A o al o six a ian s o me hods will be es ed. Subsequen ly, a
isual assessmen o he success o he app oach o ou lie sea ch was pe o med.
Fo he e alua ion o he applied me hods, 100 ime se ies o g oup 3 and 200 ime
se ies o g oup 20 we e selec ed a andom. The p esence o ou lie s was e alua ed by
isual assessmen and hen compa ed wi h indi idual me hods. The esul s a e p esen ed
poin wise since poin wise ou lie de ec ion is o in e es he e (see Table 1). O he wise,
cases like he w ong loca ion (co ec ly iden i ying ha ime se ies con ains ou lie bu a
he w ong place) o pa ial success (ei he no all o he p esen ou lie s a e iden i ied, o
a highe numbe o ou lie s is p edic ed) would occu . In o al, 1100 poin s o g oup 3 and
2200 poin s o g oup 20 we e assessed (11 poin s in each ime se ies). The poin s we e
di ided in o ou g oups:
• T ue posi i es (TP)— he ou lie was co ec ly iden i ied;
• False posi i es (FP)— he ou lie has been iden i ied, bu based on isual assessmen ,
i does no occu ,
• False nega i es (FN)— he ou lie has no been iden i ied bu based on isual
assessmen , i occu s;
• T ue nega i es (TN)— he ou lie has no been iden i ied, no does i occu in he da a,
The alues in Table 1 indica e he numbe o poin s assigned o he TP, FP, FN o TN
g oups. The si ua ion is summa ized by mul iple pe o mance measu es sui able o
imbalanced classi ica ion asks [36].
Table 1. Accu acy o me hods o ou lie de ec ion sampled om he se o ime se ies.
TP FP FN TN P ecision Recall F1 GM Jacca d
Figu e 3.
Boxplo o z-sco es o he Hol me hod by he median o expe e alua ion and o igina o .
Rema k: Do s indica e ou lie s.
Fo he e alua ion o he applied me hods, 100 ime se ies o g oup 3 and 200 ime
se ies o g oup 20 we e selec ed a andom. The p esence o ou lie s was e alua ed by
isual assessmen and hen compa ed wi h indi idual me hods. The esul s a e p esen ed
poin wise since poin wise ou lie de ec ion is o in e es he e (see Table 1). O he wise, cases
like he w ong loca ion (co ec ly iden i ying ha ime se ies con ains ou lie bu a he
w ong place) o pa ial success (ei he no all o he p esen ou lie s a e iden i ied, o a
highe numbe o ou lie s is p edic ed) would occu . In o al, 1100 poin s o g oup 3 and
2200 poin s o g oup 20 we e assessed (11 poin s in each ime se ies). The poin s we e
di ided in o ou g oups:
•T ue posi i es (TP)— he ou lie was co ec ly iden i ied;
•
False posi i es (FP)— he ou lie has been iden i ied, bu based on isual assessmen , i
does no occu ,
•
False nega i es (FN)— he ou lie has no been iden i ied bu based on isual assess-
men , i occu s;
•
T ue nega i es (TN)— he ou lie has no been iden i ied, no does i occu in he da a,
Table 1. Accu acy o me hods o ou lie de ec ion sampled om he se o ime se ies.
TP FP FN TN P ecision Recall F1 GM Jacca d
G oup 3
Hol + G ubbs es 45 18 15 1022 0.71 0.75 0.73 0.84 0.58
Hol + Dixon es 44 8 16 1032 0.85 0.73 0.79 0.91 0.65
Hol + z-sco e 47 19 13 1021 0.71 0.78 0.75 0.84 0.60
LOF 39 15 21 1025 0.72 0.65 0.68 0.84 0.52
GLOSH 41 6 19 1034 0.87 0.68 0.77 0.93 0.62
kNNd 42 7 18 1033 0.86 0.70 0.77 0.92 0.63
G oup 20
Hol + G ubbs es 104 39 19 2038 0.73 0.85 0.78 0.85 0.64
Hol + Dixon es 90 30 33 2047 0.75 0.73 0.74 0.86 0.59
Hol + z-sco e 106 49 17 2028 0.68 0.86 0.76 0.82 0.62
LOF 91 36 32 2041 0.72 0.74 0.73 0.84 0.57
GLOSH 82 20 41 2057 0.80 0.67 0.73 0.89 0.57
kNNd 97 46 26 2031 0.68 0.79 0.73 0.82 0.57
Rema k, sou ce [
36
]: P ecision (PPV)—p opo ion o posi i e samples ha we e co ec ly classi ied o he o al
numbe o posi i e p edic ed samples:
PPV =TP
FP +TP
. Recall (TPR)— he a io o he co ec ly classi ied
nega i e samples o he o al numbe o nega i e samples:
TPR =TN
FP +TN
. F1—ha monic mean o p ecision
and ecall:
F
1
=2PPV ×TPR
PPV +TPR
. GM—measu e o balanced and imbalanced da a:
GM =√TPR ×TNR
, whe e
TNR =TN
FP +TN
. Jacca d—measu e igno es he co ec classi ica ion o nega i e samples:
Jacca d =TP
TP +FP +Fn
.
Mach. Lea n. Knowl. Ex . 2023,51422
Mach. Lea n. Knowl. Ex . 2023, 5, FOR PEER REVIEW 16
no es ed in he selec ed ime se ies ( he ime se ies is ei he oo noisy o he end is oo
clea ). As he da a se expands, he c i ical alues will be upda ed.
B. C i ical alues o angles
The e was a compu ed angle a each poin o he examined ime se ies. This means
ha each pai o consecu i e poin s was i ed by a s aigh line and he angle be ween
each pai o consecu i e lines was compu ed. The limi a ion o he size o hese angles was
based on he assump ion o he shape o he changepoin s. The changepoin s should be L-
shaped o Z-shaped (wi h igh o ob use angles) and no V-shaped o A-shaped (meaning
acu e angles). This was aken in o accoun by he expe s du ing he isual assessmen .
C i ical alues o angles we e se be ween 75°and 140°. The poin in he ime se ies wi h
an angle in his ange is lagged as suspicious in e ms o changepoin occu ence.
C. C i ical alues o slopes o lines
Fo each i ed line om he p ocedu e om pa ag aph B, he slope o he line was
s o ed, and he ules desc ibed in he Ma e ials and Me hods sec ion we e ou lined. The
se ing o he limi a ion o he slopes is demons a ed in Figu e 7. When his ime se ies
was analyzed, i wen h ough he equi emen o he coefficien o de e mina ion.
Simul aneously, ou poin s (yea s 2012–2016) wen h ough he equi emen o angle
size. The poin s in he ime se ies om he yea s 2013 and 2014 we e elimina ed based on
he equi emen o he same signs o slopes o all ou lines. A hese poin s, he signs o
he slopes we e no he same— wo we e posi i e, and wo we e nega i e. Howe e , a
poin om 2016 passed he equi emen o he same signs o slopes. The equi emen o
a leas h ee angles be ween he lines and he x-axis la ge han 45° was no me . As can
be seen om Figu e 7, he angles be ween he lines and he x-axis a his poin a e e y
low. The only poin ha passed bo h equi emen s is he yea 2012. This poin looks
suspicious in e ms o changepoin occu ence e en a e a isual e iew.
Figu e 7. Demons a ion o slope limi a ion.
D. C i ical alues o SMAPE
A e he de ec ion o suspicious poin s, i was necessa y o ind a sui able accu acy
measu emen o he inal alida ion o he de ec ed changepoin . The assump ion o he
app oach is ha he changepoin should spli he ime se ies in o wo pa s, which
sepa a ely ha e an almos pe ec i . Fo his pu pose, he me ic SMAPE p o ed o be
app op ia e. The o mula o he SMAPE me ic is he nume a o o he complex ac ion
om Equa ion (2). The o mula p o ides a esul be ween 0% and 200%. The denomina o
in Equa ion (2) was added o include in o ma ion on bo h pa s in he calcula ion and
speci ically o include he amoun o change be ween hese pa s ( a io o means).
Acco ding o he es ing da a, he c i ical alue o KRITsmape was se a 1.5. Scena ios
Figu e 7. Demons a ion o slope limi a ion.
D.
C i ical alues o SMAPE
A e he de ec ion o suspicious poin s, i was necessa y o ind a sui able accu acy
measu emen o he inal alida ion o he de ec ed changepoin . The assump ion o
he app oach is ha he changepoin should spli he ime se ies in o wo pa s, which
sepa a ely ha e an almos pe ec i . Fo his pu pose, he me ic SMAPE p o ed o be
app op ia e. The o mula o he SMAPE me ic is he nume a o o he complex ac ion
om Equa ion (2). The o mula p o ides a esul be ween 0% and 200%. The denomina o
in Equa ion (2) was added o include in o ma ion on bo h pa s in he calcula ion and
speci ically o include he amoun o change be ween hese pa s ( a io o means). Acco ding
o he es ing da a, he c i ical alue o KRITsmape was se a 1.5. Scena ios wi h a
KRITsmape alue below his c i ical alue a e conside ed admissible, and he scena io wi h
he lowes a e age o KRITsmape in bo h pa s is selec ed.
The model u he lea ns by adding addi ional expe opinions on he occu ence o
changepoin s. Each expe es ima e adds in o ma ion, and hen i is possible o adjus he
c i ical alues. Wi h he annual expansion o he da a se , he e should be a e-e alua ion
by expe s. Once he da a se is la ge enough, i will be possible o ain a eg ession
model wi h da a di ided in o aining, es ing and alida ion. A he momen , he e is
no enough da a a ailable, so he s ep-by-s ep p ocedu e has been designed (see Figu e 4).
A change in he end is iden i ied in such a ime se ies ha passes all c i e ia. The main
goals o he app oach a e au oma ed calcula ion upda es and model lea ning using new
expe e alua ions.
2.5. Changepoin De ec ion—Pe o mance E alua ion
The p esen ed me hod o changepoin de ec ion was es ed and compa ed o exis -
ing me hods (see Sec ion 1.3). The p ocedu e o es ing was analogous o he ou lie ’s
assessmen . So, a o al o 300 ime se ies we e e alua ed (100 o g oup 3 and 200 o
g oup 20) and his is he es da a. In he case o changepoin de ec ion, only 6 poin s we e
assessed o each ime se ies because he changepoin is no assumed in 2 i s poin s and
3 las poin s o he ime se ies. The occu ence o changepoin s was i s s a ed by isual
assessmen s and hen compa ed o compu ed esul s. As o ou lie s, he assessed ime
se ies we e di ided in o 4 g oups (TP, FP, FN, TN); he esul s a e summa ized in Table 3.

Mach. Lea n. Knowl. Ex . 2023,51423
Table 3. Accu acy o me hods o changepoin de ec ion.
TP FP FN TN P ecision Recall F1 GM Jacca d Suspicious
Time Se ies
G oup 3
New me hod 8 6 12 574 0.57 0.40 0.47 0.75 0.31 14%
bps 1 75 19 505 0.01 0.05 0.02 0.11 0.01 76%
bcp 1 36 19 544 0.03 0.05 0.04 0.16 0.02 37%
G oup 20
New me hod 3 8 14 1175 0.27 0.18 0.21 0.52 0.12 6%
bps 1 156 16 1027 0.01 0.06 0.01 0.08 0.01 79%
bcp 3 75 14 1108 0.04 0.18 0.06 0.20 0.03 39%
Rema k, sou ce [
36
]: P ecision (PPV)—p opo ion o posi i e samples ha we e co ec ly classi ied o he o al
numbe o posi i e p edic ed samples:
PPV =TP
FP +TP
. Recall (TPR)— he a io o he co ec ly classi ied neg-
a i e samples o he o al numbe o nega i e samples:
TPR =TN
FP +TN
. F1—ha monic mean o p ecision and
ecall:
F
1
=2PPV ×TPR
PPV +TPR
. GM—measu e o balanced and imbalanced da a:
GM =√TPR ×TNR
, whe e
TNR =TN
FP +TN
. Jacca d—measu e igno es he co ec classi ica ion o nega i e samples:
Jacca d =TP
TP +FP +Fn .
F om he es ing o he ou in es iga ed g oups (TP, FP, FN, TN), he new me hod
can be conside ed qui e success ul. As can be seen om he esul s, he new me hod has
a highe success in e ms o TP. The main ad an age o he new me hod is a signi ican
educ ion in FP compa ed o exis ing me hods. This e ec a ises because he new me hod
a oids iden i ying ime se ies wi h “A” o “V” shaped de elopmen as changepoin s. In
mos cases, his is caused by oscilla ion a ound he end, no a eal changepoin . In any
case, i is ecommended o app oach he esul s o his analysis ca e ully. Ideally, isually
e i y he ue p esence o he changepoin . This is possible i he numbe o ime se ies
suspec ed o he p esence o changepoin (TP and FP) is signi ican ly educed by pe o ming
he de ec ion. I should be no ed ha he s a ed alues assess he indi idual poin s o he
ime se ies. In ac , he e a e 100 ime se ies o g oup 3 and 200 ime se ies o g oup 20.
The las column o Table 3, “Suspicious ime se ies”, summa izes he pe cen age o ime
se ies ha a e suspicious. In his espec , he e is a signi ican bene i in using he new
me hod, which can se e as an indica o o he de ec ion o ime se ies ecommended o
isual assessmen and a inal decision on he p esence o changepoin . Based on he inal
con ol o he esul s, he ime se ies classi ied as FN by a new me hod a e, in mos cases,
ambiguous, wi h he assessed alues close o he c i ical alues (see Sec ion 2.4). The e o e,
hese a e no signi ican changepoin s, and hei neglec does no ha e a nega i e impac
on u he wo k wi h da a.
Highe accu acy o he new me hod can be achie ed by se ing pa ame e s indi idually
o a speci ic da ase . The di e en was e ac ions can a y signi ican ly in hei cha ac e ,
and i can be bene icial o adap he me hod. The new me hod allows mo e deg ees o
eedom o indi idual se ings compa ed o exis ing me hods. Fo his eason, e en
mo e success ul esul s a e expec ed o he a iable da a. This is e iden by he TP
alue compa ed o he o he me hods in g oup 3, which ep esen s he a iable da a in
his es ing.
3. Resul s and Discussion
The case s udy is ealized o he Czech Republic. The da ase o he case s udy
consis s o annual da a om he pe iod 2010–2018. The da ase con ains a b oade ange
(mo e han 750) o was e ypes, which can be g ouped in o subg oups and g oups (20; see
Appendix A o de ails). The mic o- egional le el o da a has been used o analysis. The
Czech Republic consis s o 206 mic o- egions. The o al has been p ocessed abou 9600 ime
se ies because some ypes o was e a e p oduced only in some mic o- egions. The ollowing
esul s a e p esen ed o agg ega ed was e ypes (in o 20 was e g oups). Fu he mo e, no e
ha hese esul s a e displayed only o he egional le el ( e i o ies ‘CZ0XY’) and na ional
le el ( e i o y ‘c ’) da a o be e cla i y (see Appendix B o de ails).
Mach. Lea n. Knowl. Ex . 2023,51424
3.1. Ou lie De ec ion
Figu es 8–10 p o ide a g aphical ep esen a ion o he esul s using he p oposed
solu ion (Hol + G ubbs es ).
Mach. Lea n. Knowl. Ex . 2023, 5, FOR PEER REVIEW 18
ecommended o isual assessmen and a inal decision on he p esence o changepoin .
Based on he inal con ol o he esul s, he ime se ies classi ied as FN by a new me hod
a e, in mos cases, ambiguous, wi h he assessed alues close o he c i ical alues (see
Sec ion 2.4). The e o e, hese a e no signi ican changepoin s, and hei neglec does no
ha e a nega i e impac on u he wo k wi h da a.
Highe accu acy o he new me hod can be achie ed by se ing pa ame e s
indi idually o a speci ic da ase . The diffe en was e ac ions can a y signi ican ly in
hei cha ac e , and i can be bene icial o adap he me hod. The new me hod allows mo e
deg ees o eedom o indi idual se ings compa ed o exis ing me hods. Fo his eason,
e en mo e success ul esul s a e expec ed o he a iable da a. This is e iden by he TP
alue compa ed o he o he me hods in g oup 3, which ep esen s he a iable da a in
his es ing.
3. Resul s and Discussion
The case s udy is ealized o he Czech Republic. The da ase o he case s udy
consis s o annual da a om he pe iod 2010–2018. The da ase con ains a b oade ange
(mo e han 750) o was e ypes, which can be g ouped in o subg oups and g oups (20; see
Appendix A o de ails). The mic o- egional le el o da a has been used o analysis. The
Czech Republic consis s o 206 mic o- egions. The o al has been p ocessed abou 9600
ime se ies because some ypes o was e a e p oduced only in some mic o- egions. The
ollowing esul s a e p esen ed o agg ega ed was e ypes (in o 20 was e g oups).
Fu he mo e, no e ha hese esul s a e displayed only o he egional le el ( e i o ies
‘CZ0XY’) and na ional le el ( e i o y ‘c ’) da a o be e cla i y (see Appendix B o
de ails).
3.1. Ou lie De ec ion
Figu es 8–10 p o ide a g aphical ep esen a ion o he esul s using he p oposed
solu ion (Hol + G ubbs es ).
Figu e 8. Pe cen age o ime se ies con aining ou lie by was e g oup (see he Appendix o de ails)
and yea .
Figu e 8.
Pe cen age o ime se ies con aining ou lie by was e g oup (see Appendix Aand B o
de ails) and yea .
Mach. Lea n. Knowl. Ex . 2023, 5, FOR PEER REVIEW 19
Figu e 9. Pe cen age o ime se ies con aining ou lie s by egion and yea .
Figu e 10. Pe cen age o ime se ies con aining ou lie by egion and was e g oup (see Appendix o
de ails).
Figu e 8 shows he pe cen age o ime se ies in a gi en was e g oup (g oup
desc ip ion is included in he appendix) con aining ou lie s in a gi en yea . This kind o
g aph allows us o quickly iden i y p oblema ic was e (g oup) bo h o e all (high
pe cen age o ou lie s ac oss he yea s) and indi idual yea s. Indi idual g oups o was e
a e lis ed e ically (G oup 1–20); he yea s a e shown ho izon ally (2010–2018). The ed
colo in he g aph indica es ha a la ge pa o he ime se ies o he gi en g oup and yea
shows an ou lie . G oups 1 and 5 gene ally con ain a highe pe cen age o ou lie s, and
also, he single wo s case is o g oup 5 in 2014. I should be men ioned ha he da a in
g oups 1 and 5 a e signi ican ly a iable, and i is app op ia e o deal wi h he eason o
he ou lie in hese g oups. On he o he hand, he p oduc ion o packaging was e (g oup
15) and municipal solid was e (g oup 20) a e s able. I can he e o e be obse ed ha some
Figu e 9. Pe cen age o ime se ies con aining ou lie s by egion and yea .
Mach. Lea n. Knowl. Ex . 2023,51425
Mach. Lea n. Knowl. Ex . 2023, 5, FOR PEER REVIEW 19
Figu e 9. Pe cen age o ime se ies con aining ou lie s by egion and yea .
Figu e 10. Pe cen age o ime se ies con aining ou lie by egion and was e g oup (see Appendix o
de ails).
Figu e 8 shows he pe cen age o ime se ies in a gi en was e g oup (g oup
desc ip ion is included in he appendix) con aining ou lie s in a gi en yea . This kind o
g aph allows us o quickly iden i y p oblema ic was e (g oup) bo h o e all (high
pe cen age o ou lie s ac oss he yea s) and indi idual yea s. Indi idual g oups o was e
a e lis ed e ically (G oup 1–20); he yea s a e shown ho izon ally (2010–2018). The ed
colo in he g aph indica es ha a la ge pa o he ime se ies o he gi en g oup and yea
shows an ou lie . G oups 1 and 5 gene ally con ain a highe pe cen age o ou lie s, and
also, he single wo s case is o g oup 5 in 2014. I should be men ioned ha he da a in
g oups 1 and 5 a e signi ican ly a iable, and i is app op ia e o deal wi h he eason o
he ou lie in hese g oups. On he o he hand, he p oduc ion o packaging was e (g oup
15) and municipal solid was e (g oup 20) a e s able. I can he e o e be obse ed ha some
Figu e 10.
Pe cen age o ime se ies con aining ou lie by egion and was e g oup (see
Appendix Aand B o de ails).
Figu e 8shows he pe cen age o ime se ies in a gi en was e g oup (g oup desc ip ion
is included in Appendix A) con aining ou lie s in a gi en yea . This kind o g aph allows
us o quickly iden i y p oblema ic was e (g oup) bo h o e all (high pe cen age o ou lie s
ac oss he yea s) and indi idual yea s. Indi idual g oups o was e a e lis ed e ically
(G oup 1–20); he yea s a e shown ho izon ally (2010–2018). The ed colo in he g aph
indica es ha a la ge pa o he ime se ies o he gi en g oup and yea shows an ou lie .
G oups 1 and 5 gene ally con ain a highe pe cen age o ou lie s, and also, he single wo s
case is o g oup 5 in 2014. I should be men ioned ha he da a in g oups 1 and 5 a e
signi ican ly a iable, and i is app op ia e o deal wi h he eason o he ou lie in hese
g oups. On he o he hand, he p oduc ion o packaging was e (g oup 15) and municipal
solid was e (g oup 20) a e s able. I can he e o e be obse ed ha some g oups o was e
ha e a highe occu ence o ou lie s in he long e m. In he las yea (2018), he ou lie was
iden i ied wi h less equency. The eason is he less eliable iden i ica ion o ou lie s a he
las poin o he ime se ies. The esul s show ha 83% o he ime se ies om g oup 15 and
76% ime se ies om g oup 20 a e wi hou ou lie s.
Figu e 9demons a es he impac o ime and e i o y on he p esence o ou lie s. The
cha sys em is he same as Figu e 8, wi h he di e ence ha mic o- egions a e agg ega ed
in o egions on he e ical axis. The designa ion o he egions is gi en in Appendix B. F om
his kind o g aph, a e i o y con aining an unusual numbe o ou lie s (also wi h espec
o yea ) can be iden i ied. This allows o quick iden i ica ion o po en ially p oblema ic
combina ions o e i o y and yea . A non-sys ema ic dis ibu ion o colo s in he cha is
ine, e.g., o he e i o y o CZ051 in 2015, mo e ou lie s han usual we e iden i ied. This
may ha e been caused by he cu en condi ions in he gi en locali y, a egis a ion e o ,
e c., which did no a ec o he egions. In he case o such a p oblema ic pai ( e i o y
and yea ), i should be ques ioned whe he he e is a sys ema ic e o in he eco ds o
his yea was ac ually speci ic in a gi en e i o y (in his case, ‘ou lie s’ could be only
‘ex emes’ and should no be emo ed o co ec ed). In his case, CZ010 (see Appendix B)
shows a high numbe o ime se ies con aining ou lie s in 2010. Due o he high incidence
o ou lie s in 2010, i may be use ul o omi his yea ’s da a o u he analysis i he e is no
p oblem wi h he amoun o da a o a pa icula analysis. Howe e , he e seems o be no
big di e ence be ween egions, wi h egional a e ages (o e he whole pe iod) anging
om 4.5% o 6.5%.
Mach. Lea n. Knowl. Ex . 2023,51426
The las o he p esen ed g aphs (Figu e 10) shows he impac o was e g oup and
e i o y on ou lie s ega dless o he yea . I can be seen ha , e.g., a high numbe o ime
se ies con aining ou lie s is p esen in g oups 1 and 5 ac oss he e i o ies. G oup 14 (was e
o ganic sol en s, e ige an s and p opulsion media (excluding was es lis ed in g oups 7
and 8) seem o be p oblema ic o e i o ies CZ071 and CZ072. This is was e ha can be
closely linked o indus y in a gi en a ea. On he o he hand, he ime se ies om g oup 15
does no con ain a high numbe o ‘p oblema ic’ ime se ies. I should also be no ed ha
pai s p esen ed in Figu e 10 con ain up o 80% o ime se ies wi h ou lie s. Such a high
numbe is suspicious and should be examined closely.
In summa y, he p esence o an ou lie was iden i ied in app oxima ely 16% o ime
se ies. In he was e p oduc ion o municipali ies, conside able s abili y can be obse ed
in he pe cen age o iden i ied se ies o indi idual was e ac ions. Fo ac ions whe e
a leas 100 se ies we e a ailable, all anged be ween 10% and 20%. The p oduc ion o
companies gene ally con ained a smalle numbe o se ies and hus also a g ea e a iabili y
o esul s. O he main, and he e o e mo e nume ous ac ions, pape , plas ic (bo h 28%),
and me als (36%, bu only 55 se ies a e a ailable) ha e he mos ou lie s.
3.2. Changepoin De ec ion
Simila plo s, as in he case o ou lie de ec ion, can be c ea ed. Yea s 2010 and
2016–2018 a e excluded due o p oblema ic iden i ica ion a he beginning and end o he
ime se ies. Figu e 11 shows he occu ence o changepoin s o was e g oups ( e ical axis)
and yea (ho izon al axis). Each was e g oup includes mul iple ime se ies; he e o e, he
occu ence o changepoin s is exp essed as a pe cen age o all ime se ies. Figu e 11 shows
a high numbe o changepoin s p esen in 2012 o mul iple was e g oups. The eason,
among o he hings, may be changes in he me hod o eco ding da a, as he da a used
a e eco ded annually by all was e p oduce s. Figu e 12 ( o e i o ies) ag ees wi h he
p e ious esul and shows ha 2012 is p oblema ic o mos o he egions; howe e , CZ010
con ains a pa icula ly high numbe o changepoin s. Howe e , i is no clea why 2012
con ains such a high numbe o ime se ies. Such cases need o be in es iga ed closely.
Mach. Lea n. Knowl. Ex . 2023, 5, FOR PEER REVIEW 21
case o ou lie s). As Figu e 13 shows, up o 40% ime se ies o some was e g oups has
iden i ied changepoin s. These esul s show ha changepoin de ec ion is essen ial o
u he wo k wi h da a, and p e-p ocessing is c ucial in ime se ies. The quali y s udies
should include his pa o da a analysis; howe e , usually he da a p e-p ocessing is no
pe o med a all o a mos ou lie de ec ion [9]. Ne e heless, he changepoin de ec ion
can ha e a undamen al impac on he quali y o he ou pu om he analyses, as shown
by he esul s o he case s udy on WM in he Czech Republic.
Figu e 11. Pe cen age o ime se ies con aining changepoin by was e g oup (see Appendix o
de ails) and yea .
Figu e 12. Pe cen age o ime se ies con aining changepoin by egion and yea .
Figu e 11.
Pe cen age o ime se ies con aining changepoin by was e g oup (see Appendix Aand B
o de ails) and yea .
Mach. Lea n. Knowl. Ex . 2023,51427
Mach. Lea n. Knowl. Ex . 2023, 5, FOR PEER REVIEW 21
case o ou lie s). As Figu e 13 shows, up o 40% ime se ies o some was e g oups has
iden i ied changepoin s. These esul s show ha changepoin de ec ion is essen ial o
u he wo k wi h da a, and p e-p ocessing is c ucial in ime se ies. The quali y s udies
should include his pa o da a analysis; howe e , usually he da a p e-p ocessing is no
pe o med a all o a mos ou lie de ec ion [9]. Ne e heless, he changepoin de ec ion
can ha e a undamen al impac on he quali y o he ou pu om he analyses, as shown
by he esul s o he case s udy on WM in he Czech Republic.
Figu e 11. Pe cen age o ime se ies con aining changepoin by was e g oup (see Appendix o
de ails) and yea .
Figu e 12. Pe cen age o ime se ies con aining changepoin by egion and yea .
Figu e 12. Pe cen age o ime se ies con aining changepoin by egion and yea .
Finally, Figu e 13 demons a es ha in mos cases, only one o h ee was e g oups
pe e i o y con ain mos o he changepoin s. Once again, he numbe o changepoin s
p esen in his g aph is much highe when compa ed o Figu es 11 and 12 (same as in he
case o ou lie s). As Figu e 13 shows, up o 40% ime se ies o some was e g oups has
iden i ied changepoin s. These esul s show ha changepoin de ec ion is essen ial o
u he wo k wi h da a, and p e-p ocessing is c ucial in ime se ies. The quali y s udies
should include his pa o da a analysis; howe e , usually he da a p e-p ocessing is no
pe o med a all o a mos ou lie de ec ion [
9
]. Ne e heless, he changepoin de ec ion
can ha e a undamen al impac on he quali y o he ou pu om he analyses, as shown by
he esul s o he case s udy on WM in he Czech Republic.
Mach. Lea n. Knowl. Ex . 2023, 5, FOR PEER REVIEW 22
Figu e 13. Pe cen age o ime se ies con aining changepoin by egion and was e g oup (see
Appendix o de ails).
Summa y
The p esen ed me hod o changepoin de ec ion achie ed he bes esul s compa ed
o he p e ious app oaches (bsp and bcp) (see Table 3 in Sec ion 2.5). Diffe en indica o s
we e e alua ed based on he TP, FP, FN and TN alues o wo diffe en was e g oups:
G oup 3 ( ep esen s highly a iable da a) and G oup 20 ( ep esen s ela i ely s able da a).
I is pa icula ly impo an o educe he numbe o FP using he new me hod. This is a
case whe e a changepoin is de ec ed bu does no appea in he da a. This p oblem was
signi ican ly elimina ed wi h he new me hod. This is a majo bene i because, in he case
o high alues o FP, some in o ma ion is neglec ed o u he wo k wi h he da a. By
using he p esen ed me hod, be e esul s can be achie ed du ing da a p e-p ocessing
and he eby imp o e he quali y o planning in he a ea o WM.
The i s s ep o de ec ing changepoin s is o de ec ou lie s. F om an o e all
pe spec i e, i seems ha he p oblem wi h ou lie s is a he diminishing in ime o mos
g oups (especially in he las 2 yea s). The pe cen age o ou lie s in he i s se en yea s
(2010–2016) is be ween 5 and 6%, while in he las 2 yea s, his numbe d ops unde 4%.
This beha io is expec ed since da a collec ion quali y and con ol mechanisms a e
imp o ing. Was e ime se ies om mining (g oup 1) and pe oleum indus y (g oup 5)
con ain he bigges pe cen age o ou lie s (abou 10%). On he o he hand, 4 o 5% o MSW
ime se ies (g oup 15 o 20 espec i ely) con ain ou lie s. As expec ed, hese numbe s a e
lowe o changepoin s, e.g., he a e age pe cen ages o ime se ies con aining
changepoin ange om 2 o 6.5% pe was e g oup (g oup 9 being he wo s ).
In o al, 31.1% o ime se ies con ained (a leas one) ou lie and 16.4% changepoin .
Anomalies we e de ec ed, especially o was e ac ions wi h an indis inc end (e.g.,
g oup 3). G oup 3 includes was e om wood p ocessing. In his ime se ies, a changepoin
can be o en seen in he pe iod o he ba k bee le calami y in he Czech Republic. This
change can be esponded o hanks o anomaly de ec ion. O he wise, he o ecas would
be o med wi hou eac ing o his pe iod. On he con a y, g oup 20 (was e om ci izens
and simila ) has a ela i ely cons an end. Howe e , changepoin s can o en appea as a
esul o legisla i e changes, echnological p og ess, in as uc u e changes, e c. Al hough
anomalies a e mo e common a lowe le els o e i o y, hey should be de ec ed o all
da a. Thus, mo e accu a e o ecas s can be achie ed.
Figu e 13.
Pe cen age o ime se ies con aining changepoin by egion and was e g oup (see
Appendix Aand B o de ails).

Mach. Lea n. Knowl. Ex . 2023,51428
Summa y
The p esen ed me hod o changepoin de ec ion achie ed he bes esul s compa ed
o he p e ious app oaches (bsp and bcp) (see Table 3in Sec ion 2.5). Di e en indica o s
we e e alua ed based on he TP, FP, FN and TN alues o wo di e en was e g oups:
G oup 3 ( ep esen s highly a iable da a) and G oup 20 ( ep esen s ela i ely s able da a).
I is pa icula ly impo an o educe he numbe o FP using he new me hod. This is a
case whe e a changepoin is de ec ed bu does no appea in he da a. This p oblem was
signi ican ly elimina ed wi h he new me hod. This is a majo bene i because, in he case
o high alues o FP, some in o ma ion is neglec ed o u he wo k wi h he da a. By
using he p esen ed me hod, be e esul s can be achie ed du ing da a p e-p ocessing and
he eby imp o e he quali y o planning in he a ea o WM.
The i s s ep o de ec ing changepoin s is o de ec ou lie s. F om an o e all pe spec-
i e, i seems ha he p oblem wi h ou lie s is a he diminishing in ime o mos g oups
(especially in he las 2 yea s). The pe cen age o ou lie s in he i s se en yea s (2010–2016)
is be ween 5 and 6%, while in he las 2 yea s, his numbe d ops unde 4%. This beha io
is expec ed since da a collec ion quali y and con ol mechanisms a e imp o ing. Was e
ime se ies om mining (g oup 1) and pe oleum indus y (g oup 5) con ain he bigges
pe cen age o ou lie s (abou 10%). On he o he hand, 4 o 5% o MSW ime se ies (g oup 15
o 20 espec i ely) con ain ou lie s. As expec ed, hese numbe s a e lowe o changepoin s,
e.g., he a e age pe cen ages o ime se ies con aining changepoin ange om 2 o 6.5%
pe was e g oup (g oup 9 being he wo s ).
In o al, 31.1% o ime se ies con ained (a leas one) ou lie and 16.4% changepoin .
Anomalies we e de ec ed, especially o was e ac ions wi h an indis inc end (e.g.,
g oup 3
). G oup 3 includes was e om wood p ocessing. In his ime se ies, a changepoin
can be o en seen in he pe iod o he ba k bee le calami y in he Czech Republic. This
change can be esponded o hanks o anomaly de ec ion. O he wise, he o ecas would
be o med wi hou eac ing o his pe iod. On he con a y, g oup 20 (was e om ci izens
and simila ) has a ela i ely cons an end. Howe e , changepoin s can o en appea as a
esul o legisla i e changes, echnological p og ess, in as uc u e changes, e c. Al hough
anomalies a e mo e common a lowe le els o e i o y, hey should be de ec ed o all
da a. Thus, mo e accu a e o ecas s can be achie ed.
The limi a ion o he me hod is p ima ily ha i is necessa y o ha e a su icien
numbe o expe e alua ions. Thanks o his, he sys em has he oppo uni y o lea n and
achie e quali y esul s. Fu he mo e, he e alua ion mus be epea ed wi h each ex ension
o he ime se ies. Tha is, in he case o annual de ail, he upda e should ake place once
a yea . Conside ing he need o epea calcula ions, he p ocess is comple ely au oma ed.
The sys em is designed o place minimum equi emen s on he a ailable da a. I is hus
adap ed o a e y sho ime se ies ha may show an uns able end.
4. Conclusions and Fu u e Wo k
The p esen ed pape deal wi h he issue o anomaly de ec ion in a sho ime se ies
ocused on he WM ield. Al hough i is a well-known ac ha WM da a con ain anomalies,
li le o no a en ion was gi en o hei de ec ion p e iously. Only naï e me hods o ou lie
de ec ion in WM we e ound in he li e a u e, while no pape dealing wi h changepoin
de ec ion was ound. The Hol me hod combined wi h he G ubbs es is ecommended o
ou lie de ec ion based on he esul s p o ided in Sec ion 2.2.
The changepoin de ec ion me hods used in he li e a u e a e o en no applicable
o such a sho ime se ies. The e o e, he possibili y o c ea ing i s own me hod based
on basic s a is ical p ocedu es such as linea eg ession, coe icien o de e mina ion and
SMAPE, and se ing simple ules was explo ed. The c ucial aspec o he me hod is o
de ine app op ia e ules and se c i ical alues co ec ly o make de ec ion as accu a e as
possible and con enien o he leng h o ime se ies in he ield o WM. The c i ical alues
a e se using p inciples o supe ised machine lea ning. As he ime se ies ex ends, he
c i ical alues will be upda ed. The lea ning o he model is possible by addi ional isual
Mach. Lea n. Knowl. Ex . 2023,51429
assessmen . Acco ding o he cu en da a, he me hod p o ed o be success ul and eliable.
The p obabili y o ype I e o is a maximum o 2% in all es ed was e g oups, and he
p obabili y o ype II e o is less han 8%. In summa y, o abou 90% o he ime se ies,
he changepoin s we e co ec ly iden i ied.
Howe e , i should be no ed ha no objec i e in o ma ion exis s abou he p esence o
anomalies in he da ase unde examina ion. The e is no absolu ely co ec me hod wi h
which o compa e esul s. Thus, expe judgmen is he only way o assess he esul s,
and i needs o inco po a e knowledge o legisla i e changes and o he ex e nal ac o s
in luencing was e gene a ion. In e e y case, anomalies iden i ied by p oposed algo i hms
should be judged by an expe . The new changepoin de ec ion app oach was compa ed
wi h exis ing app oaches ha a e usable o sho ime se ies (Sec ion 2.5). The new
app oach is conside ed o be he mos success ul o his ype o da a om he a ailable
me hods, as i comes closes o expe e alua ion. The main bene i o he p oposed
app oach is he possibili y o au oma ing he p ocess o deciding on he p esence o an
anomaly and he possibili y o lea ning. Thanks o machine lea ning, he sys em will lea n
based on he opinions o se e al expe s. When a comp ehensi e da a se is achie ed, he
app oach will be less dependen on he subjec i e opinion o he indi idual.
In he ield o was e managemen , his con ibu ion ep esen s a undamen al con i-
bu ion leading o he imp o emen o p oduc ion o ecas s and/o was e managemen .
Wi hou high-quali y o ecas s, i is impossible o c ea e adequa e plans o he was e
economy and o ul ill legisla i e goals. The bene i o he con ibu ion can, he e o e also
be pe cei ed om he poin o iew o suppo ing he sus ainabili y o he use o na u al
esou ces. F om he pe spec i e o da a p ocessing heo y, he p oposed me hod ep esen s
a unique app oach o ime se ies p ocessing. The me hod, he e o e, has he po en ial o
be used in o he a eas o da a p ocessing ha show a simila cha ac e , especially sho
ime se ies.
The need o a su icien amoun o expe e alua ion is one o he main limi a ions o
he app oach. Wi h a highe numbe o a ings, a highe quali y o esul s can be achie ed.
This is he main poin o u he esea ch. The me hod should mo e om s ep-by-s ep
c i e ia o iden i ying change o a eg ession model. Howe e , a la ge numbe o expe
e alua ions a e needed. In ansi ion sec ions, i may be use ul o use a combina ion o
hese app oaches. Ano he possible di ec ion o u u e esea ch is he usage o some
ans o ma ion o he o iginal da a. E.g., loga i hmic ans o ma ion educes he numbe o
ou lie s as well as imp o es o he p ope ies o da a. Loga i hmic ans o ma ion is widely
used o p ocessing da a om a ious ields [
39
]. Howe e , in such a case, a ze o- alue
eplacemen s a egy needs o be de eloped and examined ho oughly. A la ge numbe o
suspicious cases in es iga ed will be bene icial, as well as a wide ange o expe s. Ano he
way o imp o e he whole amewo k is by an enhancemen o he algo i hmic de ec ion
by expe judgmen o adjus he pa ame e s o he p oposed me hod.
Au ho Con ibu ions:
Concep ualiza ion, R.Š.; me hodology, M.R. and K.Š.; in es iga ion, M.R.,
and V.S.; da a cu a ion, K.Š.; w i ing—o iginal d a p epa a ion, M.R. and V.S.; w i ing— e iew
and edi ing, V.S. and R.Š.; isualiza ion M.R. and K.Š.; supe ision, R.Š.; p ojec adminis a ion, R.Š.;
unding acquisi ion, R.Š. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
The a icle was w i en as pa o he p ojec TIRSMZP719 (P ognosis o was e p oduc ion
and de e mina ion o he composi ion o municipal was e). The au ho s g a e ully acknowledge
he suppo p o ided by TACR (Technology Agency o he Czech Republic) and he Minis y o he
En i onmen o he Czech Republic. This wo k was also suppo ed by g an No. SS02030008 “Cen e
o En i onmen al Resea ch: Was e managemen , ci cula economy and en i onmen al secu i y”.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen :
The ou pu o he p ojec is a me hodology. The da a on which he
de elopmen was ca ied ou is no public.
Mach. Lea n. Knowl. Ex . 2023,51430
Acknowledgmen s:
We acknowledge he inancial suppo ecei ed om he Technology Agency o
he Czech Republic.
Con lic s o In e es :
The au ho s decla e no con lic o in e es . The unde s had no ole in he design
o he s udy; in he collec ion, analyses, o in e p e a ion o da a; in he w i ing o he manusc ip , o
in he decision o publish he esul s.
Appendix A
Table A1. Was e g oup numbe s.
Was e G oup Numbe Desc ip ion
1
Was es om geological explo a ion, mining, ea men and u he p ocessing o mine als and s one
2
Was es om ag icul u e, ho icul u e, ishe ies, o es y, hun ing and ood p oduc ion and p ocessing
3
Was es om wood p ocessing and manu ac u e o boa ds, u ni u e, cellulose, pape and pape boa d
4 Was es om he lea he , u and ex ile indus ies
5 Was es om oil e ining, na u al gas e ining and py oly ic coal p ocessing
6 Was es om ino ganic chemical p ocesses
7 Was es om o ganic chemical p ocesses
8Was es om he manu ac u e, p ocessing, dis ibu ion and use o pain s (pain s, a nishes and
enamels), adhesi es, sealan s and p in ing inks
9 Was es om he pho og aphic indus y
10 Was es om he mal p ocesses
11 Was es om chemical su aces, su aces o me al and o he ma e ials and hyd ome allu gy o
non- e ous me als
12 Was es om shaping and physical and mechanical su ace ea men o me als and plas ics
13 Oil was es and was es o liquid uels (excluding edible oils and was es o g oups 5, 12 and 19)
14
Was e o ganic sol en s, e ige an s and p opulsion media (excluding was es lis ed in g oups 7 and 8)
15 Was e packaging; abso ben s, cleaning clo hs, il e ma e ials and p o ec i e clo hing, no elsewhe e
speci ied o included
16 Was es no o he wise speci ied in he ca alogue
17 Cons uc ion and demoli ion was es (including exca a ed soil om con amina ed si es)
18 Was es om heal h ca e and e e ina y ca e and/o esea ch ela ed o hem (excluding ki chen
was e and was e om ca e ing acili ies, which a e non-cen al heal h)
19
Was es om was e ea men plan s (use and disposal), was ewa e ea men plan s o he ea men
o hese wa e s and he p oduc ion o wa e o human consump ion and indus ial wa e
20 Municipal was es (household was es and simila ade, indus ial and o icial was es), including
sepa a ely collec ed componen s
Appendix B
Te i o y Code Te i o y Le el Te i o y Name
CZ Coun y Czech Republic
CZ010 Region P ague he Capi al ci y
CZ020 Region Cen al Bohemian Region
CZ031 Region Sou h Bohemian Region
CZ032 Region Pilsen Region
CZ041 Region Ka lo y Va y Region
CZ042 Region Ús ínad Labem Region
CZ051 Region Libe ec Region
CZ052 Region H adec K álo éRegion
CZ053 Region Pa dubice Region
CZ063 Region Vysoˇcina Region
CZ064 Region Sou h Mo a ian Region
CZ071 Region Olomouc Region
CZ072 Region Zlín Region
CZ080 Region Mo a ian-Silesian Region
Mach. Lea n. Knowl. Ex . 2023,51431
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