NONLINEAR ANALYSIS AND PREDICTION OF BITCOIN RETURN’S VOLATILITY

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