102 2022, XXV, 2
Finance
10.15240/ ul/001/2022-2-007
NONLINEAR ANALYSIS AND PREDICTION
OFBITCOINRETURN’SVOLATILITY
Tao Yin1, Yiming Wang2
1 Eas China Uni e si y o Poli ical Science and Law, Business School, China, ORCID: 0000-0001-6971-7374,
17011[email p o ec ed];
2 Peking Uni e si y, School o Economics, China, [email p o ec ed].
Abs ac : This pape mainly s udies he ma ke nonlinea i y and he p edic ion model based on
he in insic gene a ion mechanism (chaos) o Bi coin’s daily e u n’s ola ili y om June 27, 2013
o No embe 7, 2019 wi h an econophysics pe spec i e, so as o a oid he o ecas ing model
misspeci ica ion. Fi s ly, his pape s udies he mul i ac al and chao ic nonlinea cha ac e is ics
o Bi coin ola ili y by using mul i ac al de ended luc ua ion analysis (MFDFA) and la ges
Lyapuno exponen (LLE) me hods. Then, om he pe spec i e o nonlinea i y, he measu ed
alues o mul i ac al and chaos show ha he ola ili y o Bi coin has sho - e m p edic abili y.
The s udy o chaos and mul i ac al dynamics in nonlinea sys ems is e y impo an in e ms o
hei p edic abili y. The chaos signals may ha e sho - e m p edic abili y, while mul i ac als and
sel -simila i y can inc ease he likelihood o accu a ely p edic ing u u e sequences o hese
signals. Finally, we cons uc ed a numbe o chao ic a i icial neu al ne wo k models o o ecas
he Bi coin e u n’s ola ili y a oiding he model misspeci ica ion. The esul s show ha chao ic
a i icial neu al ne wo k models ha e good p edic ion e ec by compa ing hese models wi h he
exis ing A i icial Neu al Ne wo k (ANN) models. This is because he chao ic a i icial neu al ne wo k
models can ex ac hidden pa e ns and accu a ely model ime se ies om po en ial signals, while
he benchma k ANN models a e based on Gaussian ke nel local app oxima ion o non-s a iona y
signals, so hey canno app oach he global model wi h chao ic cha ac e is ics. A he same ime,
he mul i ac al pa ame e s a e u he mined o ob ain mo e ma ke in o ma ion o guide inancial
p ac ice. These abo e indings ma e o in es o s (especially o in es o s in quan i a i e ading)
as well as e ec i e supe ision o inancial ins i u ions by go e nmen .
Keywo ds: Nonlinea , mul i ac al, chaos, Bi coin, p edic ion.
JEL Classi ica ion: A10, E44, F37.
APA S yle Ci a ion: Yin, T., & Wang, Y.-M. (2022). Nonlinea Analysis and P edic ion o
Bi coin Re u n’s Vola ili y. E&M Economics and Managemen , 25(2), 102–117. h ps://doi.
o g/10.15240/ ul/001/2022-2-007
In oduc ion
Since i was p oposed by Sa oshi Nakamo o
(2008) a he end o 2008, Bi coin, as an
al e na i e o con en ional cu encies, has
quickly gained wide a en ion om he media,
in es o s and schola s.
This a en ion is a ibu ed o i s anspa ency,
simplici y, inc easing popula i y, decen alized
pee - o-pee sys em and sel - egula ion. The e
is a g owing in e es in s udying he gene al
dynamics o Bi coin ma ke . Fo ins ance,
di e si ica ion was measu ed (B iè e e al., 2015;
Bou i e al., 2017; U quha & Zhang, 2019;
Chaim & Lau ini, 2018; Lahmi i e al., 2018),
s a is ical p ope ies and ma ke e iciency
we e examined (Ba i ie a e al., 2017; Ca bone
e al., 2004; Ma inez e al., 2018; McCa hy,
2009; Symi si & Chal a zis, 2018), liquidi y and
mic os uc u e we e explo ed (Kou mos, 2018;
Dyh be g e al., 2018; Donie & Bona , 2015),
specula i e bubble and isk we e in es iga ed
(Os e iede & Lo enz, 2017; Bouoiyou e al.,
2015; Klein e al., 2018), egula ion was s udied
(Dwye , 2015; Tasca & Liu, 2018; Ka siampa,
EM_2_2022.indd 102 1.6.2022 17:52:28
103
2, XXV, 2022
Finance
2017) whils op imal ading was sc u inized
(Ajaz & Kuma , 2018; Li & Tou in, 2016; Yi e
al., 2018).
The nonlinea i y o he Bi coin ma ke is
a e y impo an opic in he exis ing li e a u e.
As a as we know, hese schola s (U quha ,
2016; Nada ajah & Chu, 2017) mainly s udied
he ma ke nonlinea i y o he Bi coin p ice and
e u n. Bu , he ma ke nonlinea o Bi coin
ola ili y has a ely been s udied. Vola ili y,
which is also known as e u n’s ola ili y,
plays an impo an ole in isk modeling and
e alua ion as well as in he p icing o complex
inancial p oduc s. In ac , Bi coin’s e u n is
highly ola ile. I s e u n depends la gely on
he sho age o he Bi coin and people’s us in
hem (U quha , 2016), which a ec s i s alue,
causing e u n o luc ua e wildly. The e o e,
his pape a emp s o discuss he nonlinea i y
o he Bi coin ola ili y ma ke , in an a emp o
ill he gap in his ho spo .
A p esen , mul i ac al heo y and chaos
heo y a e mainly used o s udy he nonlinea
cha ac e is ics o inancial ma ke . Mul i ac als
and chaos heo y e eal he nonlinea i y o
inancial ma ke om di e en pe spec i es.
To be speci ic, mul i ac al heo y (especially
mul i ac al de ended luc ua ion analysis
me hod, MFDFA) e eals he spa ial o ganiza ion
p ocess o inancial ma ke and he long- e m
co ela ion and sel -simila i y o inancial ime
se ies. The MFDFA me hod is widely used in
he ield o economy and inance (Riz i e al.,
2014; Cao e al., 2013; Bouoiyou e al., 2018;
Uddin e al., 2018), which can no only ind he
mul i ac ali y and nonlinea i y o he ma ke , bu
also exca a e mo e ma ke in o ma ion o guide
inancial p ac ice. Meanwhile, chaos heo y
(Lahmi i, 2017; Ad angi & Cha a h, 2001;
Ozun e al., 2010) p o ides he ime e olu ion
p ocess o he inancial ma ke , e eals ha
he in e nal s uc u e o he ime se ies o he
inancial ma ke is in insically de e minis ic and
nonlinea , and shows ha he inancial ime
se ies is in insically gene a i e and can be
u he p edic ed in a sho ime.
The s udy o mul i ac als and chaos in
nonlinea sys ems is also o g ea signi icance
in hei p edic abili y. On he one hand, a chao ic
sys em (signal) may ha e limi ed sho -
e m p edic abili y, while on he o he hand,
mul i ac als and sel -simila i y can inc ease he
likelihood o accu a e p edic ion o u u e ime
se ies (signal).
Ano he issue in he pape is o o ecas he
Bi coin ola ili y. P edic ing ola ili y o inancial
ime se ies can help in es o s a oid isks, which
is a ho and challenging opic in he inancial
ield. The e a e many p edic ion models o
he Bi coin ma ke in he exis ing li e a u e.
In pa icula , A i icial Neu al Ne wo k (ANN)
models can deal wi h bo h linea and nonlinea
da a, so many esea che s apply ANN models
o p edic he Bi coin ma ke (Hung e al., 2020;
Tiwa i e al., 2019; Seo & Hwang, 2018). To
ou knowledge, ew esea che s ha e used
he in e nal gene a ion mechanism o ime
se ies and ANN echnologies o build p edic i e
models o Bi coin ola ili y.
This pape will mainly ocus on he ollowing
wo aspec s: (1) We a emp o assess he
p edic abili y o Bi coin ola ili y by examining
i s inhe en nonlinea cha ac e is ics, including
inhe en chaos and mul i ac als. The chao ic
and mul i ac al cha ac e is ics o Bi coin
ola ili y a e de ec ed by using he la ges
Lyapuno exponen (LLE) and MFDFA based
on he ex ac ed gene alized Hu s exponen
o ime se ies. Speci ically, he o me allows
o es he exis ence o nonlinea de e minis ic
mapping, while he la e e eals he exis ence
o long- e m co ela ion in he case o non-
s a iona i y. (2) Ou goal is o use a special
a i icial neu al ne wo k o opology he hidden
dynamical sys em and au oma ically ex ac
he unde lying dynamical model o e eal he
nonlinea cha ac e is ics o i s ime se ies. In
o he wo ds, a chao ic in elligen signal da a
mining and p edic ion sys em (i.e., chao ic
a i icial neu al ne wo k) is cons uc ed h ough
he neu al ne wo k opology. We expec he
p edic ion accu acy o chao ic a i icial neu al
ne wo k model o be highe han ha o he
exis ing neu al ne wo k benchma k model.
In a wo d, he esul s om he nonlinea
pe spec i e a e expec ed o show ha he
p edic abili y o Bi coin ola ili y in he sho e m
depends on he measu ed alues o mul i ac al
and chaos, and he esul s o in oducing
chao ic a i icial neu al ne wo k a e expec ed
o p o e he consis ency and accu acy o i s
p edic ion abili y.
This pape imp o es and complemen s
p e ious li e a u e on Bi coin in ou aspec s:
(1) Bi coin and o he c yp ocu encies ha e
ecei ed much a en ion in he economic
and inancial li e a u e. Se e al ela ed
p oblems a e deba ed. The nonlinea i y o
EM_2_2022.indd 103 1.6.2022 17:52:28
104 2022, XXV, 2
Finance
c yp ocu encies a e e y impo an issues
add essed in he exis ing li e a u e. The p esen
pape a emp s o discuss he nonlinea i y o
he Bi coin ola ili y ma ke , in an a emp o
ill he gap in his ho spo . (2) This a icle will
demons a e a new pe spec i e on p edic ion:
econophysics. While using MFDFA me hod o
s udy he mul i ac als o Bi coin ola ili y, mo e
pa ame e in o ma ion is also mined o guide
ma ke p ac ice. (3) We es he obus ness o
he la ges Lyapuno exponen by boo s ap
me hod. (4) This pape is he i s ime o apply
chaos heo y o he Bi coin ola ili y ma ke ,
exca a es he in e nal gene a ion mechanism
o he ma ke , builds p edic ion models based
on i s in e nal gene a ion mechanism, and
p o es ha i s p edic ion e ec is be e han
ha o a i icial neu al ne wo k (ANN) model. In
he u u e, ou wo k will u he compa e o he
p edic ion models (such as GARCH model) o
show he supe io i y o he model.
The pape is o ganized as ollows. In he
Sec ion 1, he mul i ac al de ended luc ua ion
analysis (MFDFA) and he la ges Lyapuno
exponen me hodologies a e p oposed; da a
and model se ings a e in oduced in he Sec ion
2; Sec ion 3 p esen ed and analyzed he
empi ical esul s; Sec ion 4 u he discusses
he p edic ion and compa ison; he conclusion
and economic implica ions a e ou lined in he
Sec ion 5.
1. Me hodology
1.1 De e miningChaosbyLa ges 
Lyapuno Exponen (LLE)
Chaos is de e mined by la ges Lyapuno
exponen (LLE) (Rosens ein e al., 1993; Wol
e al., 1985):
1. The ime se ies wi h leng h N is {xi : i =
1,2, …, N}. A new m-dimensional phase
space sequence Xi = {xi, xi+τ, … , xi+(m–1)τ}
can be ob ained h ough he phase space
econs uc ion me hod. The o al numbe
o obse ed sample da a N, he numbe o
phase poin s is M = N – (m – 1)τ he delay
ime is τ and he embedding dimension is m;
2. Cons uc ini ial ec o :
L( 0) = min
║
X1 – Xj
║
j
(1)
whe e: 0 – he ini ial ime; X1 – he ini ial
phase poin ; and Xj – he es o he phase
poin se ;
3. The linea exponen ial g ow h a e can be
ob ained:
(2)
whe e: λ1 – linea exponen ial g ow h a e;
L( 1) – he ime 1 ec o dis ance; k – he
ime s ep;
4. Successi ely inc ease he embedding
dimension m, epea (2) and (3) un il he
la ges Lyapuno exponen becomes
s able wi h he change, and he calcula ed
esul is he es ima ed alue o he LLE. I
should be no ed ha i LLE > 0, i indica es
a ime se ies wi h chao ic dynamics. On
he con a y, i LLE < 0, i indica es ha he
ime se ies does no ha e chao ic dynamic
cha ac e is ics.
1.2 MFDFA Fo malism
The mul i ac al de ended luc ua ion analysis
(MFDFA), p oposed by Kan elha d e al.
(2002), is a use ul ool o de ec ing mul i ac al
beha io s. We can conduc he MFDFA analysis
wi h he ollowing s eps:
1. Suppose is inancial ime se ies o
leng h N; whe e: uj – he j h alue in he ime
se ies; j – he o de ing in he ime se ies.
2. Calcula e and di ide he p o ile
whe e uj.
3. De e mine he a iance:
(3)
whe e is he i ing polynomial in
segmen .
4. Calcula e he q h o de luc ua ion unc ion
;
whe e s – he numbe in each segmen ;
and he log-log plo s wq(s) e sus s o
di e en q can be desc ibed by wq(s)~
sH(q). When he se ies is mul i ac al,
a signi ican dependence o H(q) on q
should be obse ed. H(2) is he classic
Hu s exponen . I H(2) > 0.5, i indica es
ha he end change is pe sis en (long-
ange memo y); i i is an i-pe sis en ,
EM_2_2022.indd 104 1.6.2022 17:52:29
105
2, XXV, 2022
Finance
H(2) < 0.5; and H(2) = 0.5 o he andom
walk p ocess.
In addi ion, he singula i y s eng h α and
he singula i y spec um (α) can be calcula ed
ia Legend e ans o m (α) = qα – τ(q) =
= 1 + q[α – H(q)]. (α) desc ibes he ac al
dimension o he ensemble o med by all he
poin s ha sha e he same singula i y exponen
α. F ac al dimension (α) ~ α is shaped like
a single-peaked bell. The di e ence be ween
αmax and αmin , ∆α = αmax – αmin, is called he
mul i ac al spec um wid h, ha ep esen s
he in e al be ween he maximum p obabili y
and he minimum p obabili y and measu es
he deg ee o he mul i ac ali y p ope y. ∆ =
(αmin) – (αmax) is g ea e han 0 means ha
he chances o he sample being a he op
a e g ea e han he chances o being a he
bo om, and ice e sa. I is wo h men ioning
ha in es o s can look o in es men
oppo uni ies acco ding o he size o ∆ . This
pape chooses o apply MFDFA me hod, which
can no only de e mine he ma ke nonlinea i y
and mul i ac ali y, bu also ob ain o he by-
p oduc s, such as he disco e y o in es men
chances.
2. Da a Desc ip ion
and Model Se ings
2.1 Da a
In his pape , we use daily p ice o Bi coin
om May 8, 2013 o No embe 7, 2019. The
da a sou ce is h ps://coinma ke cap.com/. Fo
con enience, we deno e he ime sequence o
each da a se as and he co esponding p ice
sequence as p( ), whe e = 1, 2, …, 2385.
2.2 De ini iono Vola ili y
We es ima e he annualized ola ili y σ using 60
da apoin s sliding window. This olling sample
app oach wo ks as ollows: we compu e he
annualized ola ili y o he i s 60 e u ns, hen
we disca d he i s e u n and add he ollowing
e u n o he ime se ies, and con inue his way
un il he end o da a. Thus, each σ es ima e is
calcula ed om da a samples o he same size. We
ob ained an a e age o 2385 annualized ola ili y
σ om May 8, 2013 o No embe 7, 2019.
Fo each ime se ies, we de ine he daily
e u n ( ) as ollows: ( ) = lnp( ) – lnp( –1).
We hen calcula e he 60-day s anda d
de ia ion as ollows:
,
whe e . A las , he 60-day ac ual
Fig. 1: The his o ical ola ili y o Bi coin’s 60-day annualized yield
Sou ce: own
No e: The ho izon al axis ep esen s ime and he e ical axis is he ola ili y o Bi coin.
EM_2_2022.indd 105 1.6.2022 17:52:30
106 2022, XXV, 2
Finance
annualized ola ili y can be ob ained as ollows:
.
Fig. 1 exhibi s he his o ical ola ili y o Bi coin’s
60-day annualized yield ime se ies anging om
June 27, 2013 o No embe 7, 2019 including
2325 da a. The ho izon al axis ep esen s he
ime axis, and he e ical axis is he ola ili y pe
60-day le els. These es ima es a e ob ained as
desc ibed in he abo e wi h a window wid h o 60-
day. I shows ha he e a e conside able a iabili y
and i egula i y in his o ical ola ili y in Fig. 1.
Wha ollows, in o de o examine he
nonlinea i y, mul i ac ali y and p edic abili y o
he Bi coin ma ke , we will espec i ely discuss
he mul i ac al p ope ies and chaos p ope ies
o he his o ical ola ili y ime se ies.
2.3 MFDFA Model Se ings
The i s s ep o se he MFDFA model, which
means ha i is necessa y o se he inpu
pa ame e s m, q, and scale o MFDFA analysis.
No mally, he alue o m should be be ween 1
and 3 when he smalles segmen sizes con ain
10–20 samples. A e compa ison o he
mul i ac al spec um wi h di e en m alues,
we choose m = 1 in he MFDFA model, in
o de o p e en o e i ing o polynomial end.
As no ed by Lashe mes e al. (2004), q-o de s
be ween −5 and 5 a e su icien in mos cases.
Acco ding o Zhou (2009) and Ihlen (2012), we
se 8 as he minimum segmen size, 23 as he
maximum segmen size in MFDFA model.
3. Empi ical Resul s
3.1 TheMul i ac alo  heBi coin
Vola ili y
The Hu s exponen and mul i ac al spec um
o he Bi coin ola ili y a e shown in Fig. 2.
The line in Fig. 2A e e s o he q-o de Hu s
exponen H(q) o Bi coin ola ili y. Conside ing
he p eceding model, q-o de based on
gene alized Hu s exponen H(q) is an indica o
o mul i ac al p ope ies. The igu e shows ha
he alue o H(q) is appa en ly dependen on
q alues. The dec easing H(q) indica es ha
he ola ili y se ies o he Bi coin has signi ican
mul i ac al p ope ies. When q = –5, H(q) is
1.8, highe han 0.5, and dec eases smoo hly
wi h a ising q alue be ween −5 and 5. This
indica es signi ican pe sis en p ope ies
o small luc ua ions. When he alue o
becomes posi i e, H(q) s ays sligh ly abo e
0.5. In pa icula , H(2) > 0.5, which implies
pe sis ence and long- ange memo y s uc u e.
Fig. 2: The Hu s exponen and mul i ac al spec um o he Bi coin ola ili y
Sou ce: own
No e: H(q) – he gene alized Hu s exponen ; q – he o de o luc ua ion unc ion; α – he singula i y s eng h; (α) – he
singula i y spec um.
EM_2_2022.indd 106 1.6.2022 17:52:31

107
2, XXV, 2022
Finance
By looking in o he mul i ac al spec um,
he line in Fig. 2B gi es mo e in o ma ion
abou mul i ac al ea u es. The shape o
he mul i ac al spec um has a long-le ail,
meaning he Bi coin ola ili y has a mul i ac al
s uc u e sensi i e o he local luc ua ions wi h
la ge magni udes, bu insensi i e o he local
luc ua ions wi h small magni udes. And h ough
calcula ion, Δ = –0.7 is less han 0, indica ing
he chances o he Bi coin ola ili y being a
he bo om a e g ea e han he chances o
being a he op. By calcula ing he wid h (e.g.,
he alue Δα = αmax – αmin), we lea n ha he
mul i ac ali y deg ee is 1.78. The Δα alue is
a om 0, indica ing he Bi coin ola ili y ma ke
wi h highe mul i ac ali y. These cha ac e is ics
ully show ha he ma ke is nonlinea and he
main conclusions a e lis ed in Tab. 1.
Businesses, banks and ins i u ions can hold
Bi coin, because Bi coin p o ides use s wi h
lowe legi ima e ansac ion cos s (Kim, 2017).
F om he pe spec i e o he policy make s who
egula e he Bi coin ma ke , he e a e po en ial
easons o he exis ence o long- ange memo y
beha io : he lack o clea egula o y laws and
egula o y au ho i ies. The e o e, compa ing
wi h he adi ional inancial and commodi y
ma ke s, policy make s should s eng hen
ma ke supe ision, o mula e ele an laws
and egula ions and es ablish e o m measu es
o educe he long- ange memo y le el.
The long- ange memo y p ope y means
ha he Bi coin ola ili y ma ke can be p edic ed
in sho e m o cap u e specula i e p o i s. The
highe mul i ac ali y shows ha he ola ili y
ma ke will change g ea ly, and he ma ke is
e y complex. These esul s ha e implica ions
o economic en i y. In es o s can p edic he
u u e ola ili y o analyze p ice luc ua ions and
ca y ou isk con ol.
The e a e also impo an p ac ical
implica ions o Bi coin luc ua ions being mo e
likely o be a he bo om han a he op: Risk
a e se in es o s will con inue o hold posi ions
o inc ease posi ions app op ia ely o maximize
p o i s based on he highe p obabili y ha he
ola ili y o Bi coin is low. Howe e , he isk
appe i e in es o s end o ha e a s ong isk
ole ance, in he hope o highe expec ed e u n
on in es men , will educe he app op ia e
posi ions. These, which a e also impo an by-
p oduc s o he MFDFA app oach, can help
in es o s (especially o in es o s in quan i a i e
ading) cap u e he a bi age oppo uni y
and manage isk. All hese cha ac e is ics will
p o ide good judgmen o in es men decision
make s, isk con ol manage s and go e nmen
egula o s.
3.2 TheChaoso  heBi coinVola ili y
As he Bi coin ma ke is p edic able, in o de
o make a mo e accu a e p edic ion, we
wan o know whe he he in e nal gene a ion
mechanism o Bi coin ola ili y ime se ies is
chao ic. In o de o iden i y he chaos o he
Bi coin ola ili y ime se ies, he i s s ep is o
econs uc he phase space, which equi es
he de e mina ion o wo pa ame e s: he
embedding dimension m and he delay ime
τ. Fi s ly, we ob ain he delay ime τ = 6 wi h
he mu ual in o ma ion unc ion me hod (F ase
& Swinney, 1986), and can de e mine he
embedding dimension m = 3 by Cao me hod
(Gao & Zheng, 1993). Then, he la ges
Lyapuno exponen LLE = 0.0091 is g ea e
han 0 bu e y close o ze o acco ding o
he Wol algo i hm. A he las , we apply he
Model-based boo s ap me hod (Da ison &
Hinkley, 1997; Yin & Wang, 2019) o e i y he
obus ness o LLE. The idea o his me hod is
shown in he Appendix A1. When he con idence
alue is 95%, he con idence in e al [0.1279;
0.1733] can be ob ained. Since he quan iles
o he empi ical dis ibu ion a e all la ge han
H(2) > 0.5 Δ = –0.7 < 0Δα = 1.78 > 0
Long- ange memo y The chances o he Bi coin ola ili y
being a he bo om a e g ea e han
he chances o being a he op
The Bi coin ola ili y ma ke wi h
highe mul i ac ali y
Sou ce: own
No e: H(2) – he classic Hu s exponen ; Δ = (αmin) – (αmax); Δα = αmax – αmin.
Tab. 1: Mul i ac al es esul s o he ime se ies o Bi coin ola ili y
EM_2_2022.indd 107 1.6.2022 17:52:31
108 2022, XXV, 2
Finance
he la ges Lyapuno exponen s LLE = 0.0091,
his means ha he alue o LLE is signi ican ly
g ea e han ze o. I can be de e mined ha he
ime se ies o Bi coin ola ili y is chao ic using he
chaos heo y, as shown in he ollowing Tab. 2.
This shows ha he ola ili y ime se ies o
Bi coin has an in insic de e minis ic gene a ion
mechanism, namely chaos. The e o e, in he
ollowing sec ion, we will ocus on whe he he
p edic ion model based on in insic gene a ion
mechanism (chaos) can signi ican ly imp o e
he exis ing p edic ion model wi h chaos.
4. P edic ion and Compa ison
Accu a e p edic ion o Bi coin ola ili y means
high e u ns o in es o s, isk managemen and
con ol and e ec i e egula ion o he inancial
ma ke by go e nmen depa men s, so we
hen build mul iple p edic ion models based on
he endogenous s uc u e (chaos) o he ime
se ies o Bi coin ola ili y and compa e wi h he
wo exis ing models wi hou chaos. I is hoped
ha he p edic ion accu acy o he chao ic
p edic ion models will be highe , a oiding he
o ecas ing model misspeci ica ion. Speci ically,
we in oduce h ee kinds o p edic ion models,
namely, chaos + ANN (a i icial neu al
ne wo k)- ype, chaos- ype and ANN- ype. The
chaos + ANN- ype con ains a hyb id based
RBF neu al ne wo k model wi h chaos (RBF-
CHAOS model) and a hyb id based BP neu al
ne wo k wi h chaos (BP-CHAOS model); he
chaos- ype is he weigh ed i s -o de Local
Region (LR-CHAOS model), which is a model
based on chaos wi hou ANN; he ANN- ype
includes wo exis ing p edic ion models only
wi h ANN, namely, RBF model and BP model.
Fo he cons uc ion o hese p edic ion models,
please e e o Appendix A2–A4 o li e a u e.
In he simula ion expe imen , he o iginal
samples and he p edic ed samples a e x(n)
and xp(n) espec i ely, and he absolu e e o
e(n) = xp(n) – x(n), he mean absolu e e o
(MAE) and he pe cen age e o (Pe ) a e
used as he e alua ion c i e ia o he p edic ion
accu acy. The smalle MAE and Pe alues
a e, he mo e p edic i e e ec o he model is,
whe e he MAE and he Pe a e espec i ely
de ined as:
,
whe e Np ep esen s he numbe o p edic ed
samples.
In his sec ion, he i s 2,315 da a o
Bi coin ola ili y ime se ies a e used as aining
samples o p edic he Bi coin ola ili y in he
nex 10 days. Fig. 4–8 show he p edic ion
images o he i e models espec i ely. The
ho izon al axis ep esen s he p edic ed days,
and he e ical axis is he ola ili y pe 60-day
le els. The ed line ep esen s he p edic ed
alue and he blue line is he measu ed alue.
The close hese wo lines a e, he close he
p edic ed alue and he measu ed alue a e.
The absolu e alue o e(n) o hese i e igu es
does no exceed 1.5 × 10–3. I ollows om
Fig. 3–7 ha hese i e models can p edic
p ices o he Bi coin ola ili y, and being mo e
accu a e in he sho e m and la ge e o s in
he long e m. I should be emphasized ha
LR-CHAOS model can simula e he measu ed
alue well.
The MAE alue and Pe alue a e lis ing
in Tab. 3, and we so hese 5 models based
on MAE and Pe alue. The smalle he MAE
alue and Pe alue a e, he highe he o de o
he co esponding model is, as shown in Tab. 3.
We can see om Tab. 3 ha :
(1) I can be concluded ha he o de ing o all
p edic ion models is LR-CHAOS > RBF-CHAOS
> BP-CHAOS > RBF > BP, indica ing ha he
op imal p edic ion model is LR-CHAOS among
hese models. I ollows om he abo e ha
he model which eally imp o es he p edic ion
accu acy is based on he in e nal gene a ion
mechanism o ime se ies, and his ma e s o
he ele an in es men ins i u ions (in es o s).
Delay ime Embedding
dimension
La ges Lyapuno
exponen (LLE)
Con idence in e al
o 95% o LLE Chaos
τ = 6 m = 3 LLE = 0.0091 > 0[0.1279; 0.1733] Yes
Sou ce: own
Tab. 2: Chaos es esul s o he ime se ies o Bi coin ola ili y
EM_2_2022.indd 108 1.6.2022 17:52:31
109
2, XXV, 2022
Finance
Fig. 3: P edic ed by RBF-CHAOS model
Sou ce: own
No e: x(n) – he o iginal p ice; xp(n) – he p edic ed p ice; e(n) = xp(n) – x(n).
Fig. 4: P edic ed by BP-CHAOS model
Sou ce: own
No e: x(n) – he o iginal p ice; xp(n) – he p edic ed p ice; e(n) = xp(n) – x(n).
EM_2_2022.indd 109 1.6.2022 17:52:32
110 2022, XXV, 2
Finance
Fig. 5: P edic ed by LR-CHAOS model
Sou ce: own
No e: x(n) – he o iginal p ice; xp(n) – he p edic ed p ice; e(n) = xp(n) – x(n).
Fig. 6: P edic ed by RBF model
Sou ce: own
No e: x(n) – he o iginal p ice; xp(n) – he p edic ed p ice; e(n) = xp(n) – x(n).
EM_2_2022.indd 110 1.6.2022 17:52:34
117
2, XXV, 2022
Finance
laye . The speci ic s uc u e o RBF model has been men ioned in many li e a u es, so i will no be
epea ed he e.
A hyb id based RBF neu al ne wo k model wi h Chaos (RBF-CHAOS model) can be cons uc ed
by ollowing s eps:
S ep 1: Take he embedded dimension as he inpu numbe o RBF ne wo k, and le he ou pu
numbe as 1;
S ep 2: Take he adial basis unc ion o m as:
whe e c is called he wid h alue, he inpu ec o o he ne wo k is is called
a adial basis unc ion; ep esen s no m; signi ies he cen e o he adial basis unc ion.
A3.BPModelandBP-CHAOSModel
Back P opaga ion neu al ne wo k (BP model), including inpu laye , hidden laye and ou pu laye ,
is using minimum a iance lea ning me hod (Rumelha e al., 1986; Yang, 1996). A he same ime,
i is a kind o supe ised lea ning neu al ne wo k, which con ains h ee o mo e laye s o neu al
ne wo ks.
The hyb id based BP neu al ne wo k model wi h Chaos (BP-CHAOS model) can be cons uc ed
by ollowing s eps:
S ep 1: Take he embedded dimension as he inpu numbe o BP ne wo k, and le he ou pu
numbe as 1;
S ep 2: Take inpu and ou pu o laye nodes, espec i ely as ,
, j = 1, 2, …, n; whe e is he ou pu o hidden
laye , he connec ion weigh om he inpu laye o he hidden laye is wij, he h eshold alue o
hidden laye node is ep esen ed by hj.
The connec ion weigh om he hidden laye o he ou pu laye is uj, and b is he h eshold o
he ou pu laye .
No e ha he wo models (RBF-CHAOS and BP-CHAOS) a e based on ANN and chaos, while
he o he wo models (RBF and BP) a e only based on ANN.
A4.LR-CHAOSModel
The LR-CHAOS model (Zhang, 2010) can be cons uc ed by ollowing s eps:
S ep 1: Se he neighbo ing poin Mki o he cen e poin Mk , i = 1,2, …, q, and le di = Mki – Mk ,
whe e ep esen s no m.
S ep 2: Se dm = min {di}, and de ine he weigh πi o he poin Mki as:
1
exp( ( ))
exp( ( ))
im
iq
im
k
dd
dd
π
=
−−
=−−
∑
.
S ep 3: Make he linea i as Mki + 1 = ae + bMki o es ima e he coe icien s a and b, whe e
e = (1, …, 1)T.
No e ha he LR-CHAOS model is only based on chaos.
EM_2_2022.indd 117 1.6.2022 17:52:37