A deep learning integrated Cairns-Blake-Dowd (CBD) sytematic mortality risk model

Odhiambo, Joab; Weke, Pa ick; Nga e, Philip A icle A deep lea ning in eg a ed Cai ns-Blake-Dowd (CBD) sy ema ic mo ali y isk model Jou nal o Risk and Financial Managemen P o ided in Coope a ion wi h: MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel Sugges ed Ci a ion: Odhiambo, Joab; Weke, Pa ick; Nga e, Philip (2021) : A deep lea ning in eg a ed Cai ns-Blake-Dowd (CBD) sy ema ic mo ali y isk model, Jou nal o Risk and Financial Managemen , ISSN 1911-8074, MDPI, Basel, Vol. 14, Iss. 6, pp. 1-12, h ps://doi.o g/10.3390/j m14060259 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/239675 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ Jou nal o Risk and Financial Managemen A icle A Deep Lea ning In eg a ed Cai ns-Blake-Dowd (CBD) Sy ema ic Mo ali y Risk Model Joab Odhiambo * , Pa ick Weke and Philip Nga e   Ci a ion: Odhiambo, Joab, Pa ick Weke, and Philip Nga e. 2021. A Deep Lea ning In eg a ed Cai ns-Blake-Dowd (CBD) Sy ema ic Mo ali y Risk Model. Jou nal o Risk and Financial Managemen 14: 259. h ps://doi.o g/10.3390/j m14060259 Academic Edi o s: Michael McAlee and Shigeyuki Hamo i Recei ed: 8 May 2020 Accep ed: 11 June 2020 Published: 8 June 2021 Publishe ’s No e: MDPI s ays neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a il- ia ions. Copy igh : © 2021 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/). School o Ma hema ics, Uni e si y o Nai obi, Nai obi Ci y 30197-00100, Kenya; [email p o ec ed] (P.W.); [email p o ec ed] (P.N.) *Co espondence: [email p o ec ed] Abs ac : Many ac ua ial science esea che s on s ochas ic modeling and o ecas ing o sys ema ic mo ali y isk use Cai ns-Blake-Dowd (CBD) Model (2006) due o i s abili y o conside he coho e ec s. A h ee- ac o s ochas ic mo ali y model has h ee pa ame e s ha desc ibe he mo ali y ends o e ime when dealing wi h u u e beha io s. This s udy aims o p edic he ends o he model, k(2) by applying he Recu en Neu al Ne wo ks wi hin a Sho -Te m Long Memo y (an a i icial LSTM a chi ec u e) compa ed o adi ional s a is ical ARIMA (p,d,q) models. The no el deep lea ning (machine lea ning) echnique helps in eg a e he CBD model o enhance i s accu acy and p edic i e capaci y o u u e sys ema ic mo ali y isk in coun ies wi h limi ed da a a ailabili y, such as Kenya. The esul s show ha Long Sho -Te m Memo y ne wo k a chi ec u e had highe le els o p ecision when p edic ing he u u e sys ema ic mo ali y isks han adi ional me hods. Ul ima ely, he esul s can be implemen ed by Kenyan insu ance i ms when modeling and o ecas ing sys ema ic mo ali y isk help ul in he p icing o Annui ies and Assu ances. Keywo ds: sys ema ic mo ali y isk; deep lea ning; long sho - e m memo y; CBD; ecu en neu al ne wo ks 1. In oduc ion Since he s a o he 21s cen u y, mo ali y a es ha e been dec easing s eadily due o se e al ac o s such as imp o ed medical in en ions, obo ic su ge y, be e heal hca e sys ems, and be e die s, among many o he ac o s, see (Boo and Choi 2020;Chen 2020; Kilic 2020;Pou homayoun and Shakibi 2020). These ac o s ha e p omp ed ac ua ies, demog aphe s, and s a is icians o hink o no el ideas o do mo ali y modeling and o ecas ing o an inc eased le el o p ecision in he models. While his is a good idea o he gene al global popula ion, many go e nmen s, li e assu ance i ms, and li e pension companies ha e subs an ial inancial losses since hey canno make p ecise es ima ions when o e ing inancial se ices. Co ec mo ali y isk es ima ion is i al in hei inancial su i al, especially a e he ha d imes o he global Co id-19 pandemic see (Pou homay- oun and Shakibi 2020) ha is likely o lead o massi e global economic ecession a ec ing many na ions, bo h i s -wo ld and hi d-wo ld coun ies. Today, in he ac ua ial li e a u e, we ha e many e ined echniques ha many ac ua ies, s a is icians, and demog aphe s use when o ecas ing u u e mo ali y and sys ema ic longe i y isks. We es ima e he comple e li e expec a ions o hose who wish o buy annui ies and li e assu ance p oduc s sold in he ma ke . F om (Lee and Ca e 1992), he e a e many s ochas ic mo ali y models cu en ly used when modeling and o ecas ing sys ema ic mo ali y isk. Howe e , hese models ha e di e en s eng hs and weaknesses depending on he da a a ailabili y and he numbe o a ailable pa ame e s ha need o be de e mined o es ima ed. (Cai ns e al. 2006) model imp o es some o he weaknesses o he (Lee and Ca e 1992) model by inco po a ing he coho e ec s and double pa ame e abili y. Modeling a sys ema ic mo ali y isk ha is cons an o e age while p e en ing o e lapping o he age lines du ing o ecas ing leading desi able esul s as a pa simonious J. Risk Financial Manag. 2021,14, 259. h ps://doi.o g/10.3390/j m14060259 h ps://www.mdpi.com/jou nal/j m J. Risk Financial Manag. 2021,14, 259 2 o 12 model was demons a ed by (Cai ns e al. 2011), which leads o high ce ain y le els and accep abili y in he esul s. Many esea che s, including (Hainau 2018) in his pape , p oposed a neu al ne wo k capable o p edic ing and simula ing u u e sys ema ic mo ali y isk. Du ing his esea ch, he au ho used a neu al analyze when de ec ing la en ime p ocesses while di ec ly p edic ing mo ali y. The app oach did allow o iden i ica ion and duplica ion o non- linea i y obse ed in he changes o logi o ces o mo ali y. In addi ion, (Dep ez e al. 2017) used some machine lea ning echniques o imp o e he es ima ion p ocess o he logi mo ali y isk. This wo k was ex ended by (Le an esi and Pizzo usso 2019) o he amewo k o mo ali y o ecas ing as in he [3] model. Fu he mo e, a ecen pape by (Richman and Wü h ich 2018) p oposed mul iple-dimensional popula ions o (Lee and Ca e 1992) model whe e i es ima ed he pa ame e s using a i icial neu al ne wo ks. Many o he ele an machine lea ning uses and applica ions in an ac ua ial ield a e discussed by (Cas ellani e al. 2018) and (Gab ielli e al. 2020), especially when looking a he u u e o sys ema ic mo ali y isk modeling me hodologies. In his esea ch s udy, we use a deep lea ning echnique o imp o e he p edic i e capabili y o he (Cai ns e al. 2006) model. To be mo e speci ic, ou app oach aims a In eg a ing he o iginal (Cai ns e al. 2006) o mula ion by he in oduc ion o an a i icial Recu en Neu al Ne wo ks wi h Long Sho -Te m Memo y o LSTM a chi ec u e when o ecas ing u u e e olu ion o he k(2) pa ame e hus o e coming he challenges showed by he adi ional ARIMA (p , d , q) ime se ies p ocess. The choice o he CBD model ins ead o o he s anda d mo ali y models is based on he ac ha CBD sol es he p oblem o coho e ec in mo ali y synonymous wi h o he mo ali y models. In addi ion, i inco po a es he e ec o coho s in models compa ed o o he s used in modeling o sys ema ic mo ali y isk. Using LSTM allows mo e cohe ency when de e mining mo ali y o ecas s wi h high dynamism o obse ed mo ali y, especially when dealing wi h nonlinea mo ali y ends. To be mo e p ecise, he LTSM ne wo k is s uc u ed o help elabo a e long da a sequences o o m a memo y capable o p ese ing he i al ela ionships be ween he a ailable da a and e e y de ia ion wi hin hese sequences. In a simila sense, wi hin he con ex o adi ional ime se ies, he LSTM gi es oom o p edic ing u u e mo ali y o e ime by conside ing he subs an ial in luence o he his o ical sys ema ic mo ali y isk ends be o e adequa ely ep oducing i in o he o ecas ed end. In addi ion, he powe o LSTM is by p ese ing he in o ma ion o e a gi en pe iod, he e o e blocking he olde signals om slowly disappea ing du ing p ocessing. While he esea ch ocuses on o ecas ing sys ema ic mo ali y isk ends, pa ame e es ima ion me hodology emains simila as o (Lee and Ca e 1992). The pape does in oduce a new me hod o mo ali y i ing su ace as by (Hainau 2018) ha applies he use o neu al ne wo ks o deep lea ning echnique o i ing mo ali y a es as opposed o he con en ional SVD me hod (Singula Value Decomposi ion). This s udy in oduces a no el me hodology s uc u e based on he LSTM ne wo k when modeling u u e common ends o sys ema ic mo ali y isk. 2. Cai ns-Blake-Dowd (CBD) Model De ini ion 1. Le he (Cai ns e al. 2006) be; logi µ(x, )=α(1) xk(1) +α(2) xk(2) +α(3) xw(3) −x(1) The coho e ec in luence, w(3) −x , o any age-speci ic coho has been assumed o educe o ze o wi h ime. α(3) x dec eases wi h x as opposed o being a cons an i.e., α(3) x = c whe e cis a cons an i sel . The e o e, his will gi e us he model as logi µ(x, )=α(1) xk(1) +α(2) xk(2) +α(3) xw(3) −x(2) J. Risk Financial Manag. 2021,14, 259 3 o 12 whe e α(1) x= 1, α(2) x= (x−¯ x) , α(3) x= (xc−x ). Wi h he eplacemen o he alues, we ha e: logi µ(x, )=k(1) +k(2) (x−¯ x) + w(3) −x(xc−x)(3) Du ing he analysis, we ha e o use he cons ain ∞ ∑ i=1 w(3) −x= 0 o p e en in oducing he iden i iabili y p oblem du ing he p ocess o es ima ion as well as p ojec ion. The model has no p oblems o iden i ica ion. In he o iginal (Cai ns e al. 2006) model, esea che s o en used SVD o singula alue decomposi ion when es ima ing pa ame e s as pe he 2-s age p ocedu e. This is done by applying i o he ma ix o logi µ(x, ) as a way o inding alues o k(1) o hus ob aining alues k(2) and (x−¯ x) espec i ely. Secondly, o ensu e ha obse ed dea hs coincide wi h he es ima ed dea hs, k(2) is e i ed. Lemma 1. As pe he adi ional (Cai ns e al. 2006) o mula ion, k(2) is o en modeled using an Au o- eg essi e In eg a ed Mo ing A e age (0, 1, 0)as; k(2) =k(2) −1+δ+w(3) −x(4) whe eas δis de ined d i pa ame e and w(3) −xa e he andomness e m and w(3) −x∼N(0,σ2 k). 3. The Neu al Ne wo k Model 3.1. A i icial Neu al Ne wo k De ini ion De ini ion 2. An ANN (A i icial Neu al Ne wo k) is a se ies o algo i hms ha endea o s o iden i y unde lying ela ionships in a gi en da a se ia a p ocess capable o mimicking how a human b ain wo ks. A i icial neu al ne wo k a chi ec u e includes neu ons, he synap ic connec ions, which link he neu ons, and lea ning algo i hms. ANN is o med h ough 3 ca ego ies o laye s, known as hidden, inpu , and an ou pu laye espec i ely whe e each one o he laye s is made up o se e al neu ons (Hassoun e al. 1995). E e y uni in an a i icial ne wo k ob ains “p opo ional” in o ma ion ia synap ic links om many o he well connec ed ones a he same ime e u ning an ou pu h ough using an a i icial ac i a ion unc ion ha ans o ms hese p opo ional o als o he inpu signals. 3.2. Deep Lea ning Modeling De ini ion 3. Le Q deno e a single neu on called pe cep on de ined by; Q=Θ(ZTy+c)(5) whe e yeR is he inpu and ZeR is he connec ed synap ic weigh , ψeN a e numbe s o he inpu signals and Θ is he ac i a ion unc ion. We ep esen his e m c as he bias ha is associa ed wi h he model known as ac i a ion e ge o h eshold. The use mus no e ha he unc ion, Θ , should ha e a di e en ial because he lea ning equa ions ha e g adien s (Minsky and Pape 2017). We in oduce Mul ilaye Pe cep on ( MLP ) used in nonlinea sepa able p oblems such as Exclusi e o (XOR) since ANN wi h a single laye is always inapp op ia e, hus sol ing he s a ed p oblem. In addi ion, mos neu ons in MLP a e p edisposed on a wide a ie y o laye s, wi h e e y uni ully connec ed o hose o he p eceding laye , as illus a ed by (Good ellow e al. 2016). The synapses connec uni s by de ining di e en ypes o a ailable ne wo ks in he sys em. In an ANN classical pa e n like eed- o wa d ANN, he in o ma ion mo es in a unila e al di ec ion om an inpu o an ou pu laye a he same ime he Recu en Neu al Ne wo ks (commonly known as RNNs) p ocesses he in o ma ion cyclically using he ex a synapses o ensu e ha he ep ocessed ou is as a esul o he en i e elabo a ion p ocess. J. Risk Financial Manag. 2021,14, 259 4 o 12 Figu e 1below shows he s anda d ep esen a ion o eed- o wa d ANN. A neu on is ep esen ed in e e y node, connec ed om one o he o he using a cs ep esen ing all synapses. Addi ionally, he g aph ep esen s he gene al inpu , la en , as well as ou pu a iables. The Schema ical iew o an a i icial neu al ne wo k (ANN) below has ci cles ep e- sen ing neu ons wi h lines ep esen ing synapses. The Synapses ake he indi idual inpu s be o e mul iplying hem by a “weigh ” commonly known as inpu “s eng h” o de e mine he gene al ou pu . In addi ion, Neu ons a e added o hese ou pu s om all a ailable synapses be o e applying he ac i a ion unc ion. Figu e 1. A No mal ep esen a ion o eed- o wa d ANN. De ini ion 4. F om he ou pu , le QeRkh deno e a gene ic hidden laye ha ing kh neu ons de ined as; Q1=Θ(ZTy+c)(6) whe e ZeRψ∗kh is de ined as a weigh ma ix and ceRkh is called he biases ec o . Acco ding o MLP scheme, he hidden laye ou pu becomes he inpu ins umen o he ollowing laye . Lemma 2. Conside ing a gi en p oblem o eg ession, whe e eN de ined as he numbe hidden laye s, hen he ou pu o ˆ yeRcan be calcula ed by: Q1=Θ1(ZT 1y+c1) Q2=Θ2(ZT 2Q1+c2) Q3=Θ3(ZT 3Q2+c3) ......... ˆ y=Θ (ZT Q −1+c ) whe e Z1 , Z2 , Z3 , ... Z deno e weigh ma ix ec o s, c1 , c2 , c3 , ... c deno e bias ec o s, and Φ1 , Φ2 , Φ3, ...Φ deno e ac i a ion unc ions ha needs no be di e en om one ano he . I is i al o no e ha all measu emen s o he weigh ma ices and bias ec o s do ely on he uni numbe wi hin he hidden laye s; hence, by enhancing hese hidden laye s in numbe s, he abs ac ion le els o he inpu da a also inc ease signi ican ly. 3.3. Backwa d P opaga ion o E o s De ini ion 5. Backp opaga ion is an algo i hm used o supe ised lea ning o a i icial neu al ne wo ks h ough g adien descen . P o ided an a i icial neu al ne wo k ( ANN ) and an e o unc ion, his me hod is capable o calcula ing he e o unc ion g adien wi h espec o he espec i e weigh s o neu al ne wo ks. J. Risk Financial Manag. 2021,14, 259 5 o 12 ANN aining in ol es he use o a gi en uncons ained op imiza ion p oblem wi h he aim o minimizing a unc ion wi hin he high dimensional space. We s a by de ining a loss unc ion as: B= ∑ i=1 ( i−ˆ )2 2(7) This loss unc ion measu es he de ia ions o p edic ed alues ˆ om he obse ed ones i.e., i ob ains he absolu e e o e ms be ween hese p edic ed alues o ˆ as well as obse ed alues o . The quan i y B also elies on he weigh s o he ma ices namely Z1 , Z2 , Z3 , ... Z , which ul ima ely in luences he alues o p edic ed ˆ . Consequen ly, he aim o he me hod is o ind he exac synap ic weigh alues, which minimizes he alue o quan i y B. While machine lea ning has many algo i hms applied in i s applica ion, backp opa- ga ion is among he mos commonly used eed- o wa d aining ANNs. The algo i hm wo ks by compa ing he p edic ed alues e sus he expec ed ones acco ding o modi ying he synap ic weigh s h ough back-p opaga ing he loss unc ion’s g adien . F om Figu e 1, he p ocedu e con inuously al e na es o wa d wi h backwa d p op- aga ion in he ollowing s eps, namely in he o wa d s ep, he p edic ed alues o ˆ a e calcula ed by ixing he espec i e synap ic weigh s, and in his backwa d s ep, he adjus weigh s hus educing he e o B o he ne wo k. I is impo an o no e ha ANN can i e a i ely pe o m bo h o wa d and backwa d p opaga ion by modi ying he weigh s o ind he combina ion, which minimizes he o e all loss unc ion. De ini ion 6. Analy ically, backp opaga ion algo i hm upda es all weigh s o Z in he las laye by he ule o del a as ollows; ∆Z =−i∂B ∂Z0 (8) whe e i is called he lea ning a e. As o o he p eceding laye s, we di e en ia e using p oduc o chain ule o di e en ia ion. The o he weigh s ma ix Z −1a e de e mined as: ∆Z −1=−i∂B ∂Q −1 ∗∂Q −1 ∂Z −1 (9) and he p ocess con inuous on o many o he laye s in he sys em. We look in o he same idea in a igu a i e way, jus like a g adien o slope descen simila o a “climbing down a s eep hill” so long as i eaches a local minimum o global limi . Howe e , a e e y upda e, he sea ch does mo e in he g adien ’s opposi e di ec ion while he slope o he g adien and lea ning a e is de e mined by he Mo emen ampli ude (Baydin e al. 2017). Mo eo e , he choice o a e i is a i al elemen , as a small alue can lead o se e al i e a ions simul aneously; la ge alues migh pe mi con e gence, especially o a global minimum. We choose om a wide ange o a chi ec u e, including he hidden laye s numbe s, uni s o e e y laye , and he hype -pa ame e alues like lea ning a e, epochs, and ac i a ion unc ion, which emain ano he heu is ic p oblem o ANN use s. I is impo an o no e ha he choice will always depend on he da a ype a ailable, which migh be a di icul s ep o jus easy. An ini ial ound o he hype -pa ame e s uning, especially be o e he es ing, migh be highly needed. Addi ional ex ensi e desc ip ions o ANNs and back-p opaga ion algo i hms a e explained well (Alpaydin 2016) and (James e al. 2013). 3.4. Recu en Neu al Ne wo k Using a Long Sho -Te m Memo y A chi ec u e P oposi ion 1. We inco po a e he concep o Deep Lea ning echniques in s ochas ic mo ali y modelling o inc ease hei p edic abili y and o ecas ing accu acy. J. Risk Financial Manag. 2021,14, 259 6 o 12 P oo . The eed o wa d ANNs, which always ep esen a powe ul ool o analysis, can be insu icien when e ec i ely managing ime sequences o he a ailable da a. Howe e , he ecu en connec ions be ween nodes ha ha e ea u ed he RNNs allow o an ac i e analysis o he gi en sequen ial da a. Ne e heless, h ough applying he gi en RNN s uc u e, we o en ace he massi e p oblem o g adien s disappea ing and weigh s change, be o e becoming iny as o show no e ec . Consequen ly, he ne wo k will g adually lose i s capabili y o lea ning om he pas o become ope a ionally insu icien o he mo e p olonged da a sequences analysis and hus helping in making excellen p edic ions. I is why we say ha RNNs possess a sho memo y only. As a way o o e coming he s a ed p oblem, (Hoch ei e and Schmidhube 1997) had come up wi h he Long Sho -Te m Memo y, commonly abb e ia ed as LSTM. The LSTM is a e sion o RNN whose a chi ec u e can allow conside a e ela ionships be ween he sequence o da a, e en i i happens in he long un, hus e adica ing he anishing g adien p oblem in he p ocess. Simila ly, RNNs need bo h long- and sho -memo y, hus managing o gene a e an ex ao dina y pe o mance in he analysis o ime se ies. Howe e , se e al imp o emen s in he o iginal wo k, LSTM, ha e been imp o ed h ough a se ies o s udies such as (Bahdanau e al. 2014) and (Cho e al. 2014). Ul ima ely, one can de ine an excellen ly composed basic s uc u e as anilla LSTM. De ini ion 7. F om Figu e 2, Le =g , i = and o deno e he ou pu ha would be impo an in RNN analysis. Le ou pu o he auxilia y-ou pu ga e be de ined as; g = (Z y +U Q −1+c )(10) = (Ziy +UiQ −1+ci)(11) o = (Zoy +UoQ −1+co)(12) j = (Zjy +UjQ −1+cj)(13) The o ge ga e ou pu g as de ined by Equa ion (10), illus a es ac s om he p e- ceding cell s a e as well as he one o igina ing om he p esen inpu a e mixed wi hin a nonlinea way h ough a sigmoid ac i a ion unc ion. A e wa ds, g is mixed h ough a poin -wise p oduc especially wi hin i s p e ious memo y s a e c( − 1 ) . I s inpu ga e , as de ined in Equa ion (11), uses an ac i e sigmoid ac i a ion, which pe mi ing o decisions when in o ma ion is ecei ed be o e i is upda ed. The ou pu ga e o , as de ined in Equa ion (12), plays he ole o p e en ing non-signi ican memo y con en ansmission ha is s o ed in o ma ion wi hin he o he blocks. I s ole as a sigmoid unc ion is o pass app op ia e memo y in o ma ion. As a way o egula ing p ocessed da a low, he inpu ga e i does combines wi h ha de i ed om all linked auxilia y NN j as de ined in Equa ion (13). De ini ion 8. Le deno e he en i e inpu block p ocessing p ocedu e ha pa icipa es in cons uc- ion o he p esen memo y cell s a e as: c( ) = c( −1)~g + ~j To ge he cu en ou pu , which is a combina ion in be ween he de ined unc ion in abo e equa ion; Q=Φ(c( )) ∗o (14) F om Equa ion (14), his LSTM a chi ec u e o e s an ou s anding ool when dealing wi h o ecas ing ime se ies, pa icula ly in cases o longe ime lag connec ions, ca ching andomness, and managemen o he noise. Ne e heless, any use o LSTM, jus ANNs in gene al, mus ha e he ace o he classical p oblems ha conce n he hype pa ame- e s choices. J. Risk Financial Manag. 2021,14, 259 7 o 12 Figu e 2. A LSTM Block S uc u e wi h I s In e nal In o ma ion Fo wa d Flow Design. 4. Ma hema ical Applica ion and Resul s In his a ea, we in oduce he LSTM and RNN a chi ec u es wi hin he s anda d scheme o he CBD model. Mo e dis inc ly, he s udy’s objec i e is o exploi he ad an ages and unc ionali ies o he LSTM a chi ec u e o imp o e he CBD model p edic i e capaci y. Fo his aim, we design se e al expe imen s o es LSTM skills in o ecas ing u u e sys ema ic mo ali y isk o e ime be o e compa ing i s pe o mance wi h he esul s de i ed om he model o ARIMA. Thus, he analysis o he s udy will conce n on he ime index k(2) end p edic ion, bea ing in mind he ARIMA (p , d , q) model as he o ecas ed benchma k, whe eas o he pa- ame e s k(1) and (x−¯ x) a e de e mined as pe he es ima ion me hod by (Cai ns e al. 2006). Dis inc ly, he CBD model ha applies a simple andom walk p ocess wi h d i is i al o calib a e he bes ARIMA (p,d,q), as illus a ed by (Hyndman and Khandaka 2007). This p ocedu e checks he ime se ies s a iona i y in he ini ial ound using a sui able uni a y oo es be o e choosing he di e encing o de d. The 2nd s age de e mines he au o- eg essi e bes alues and mo ing a e age o de , like p and q, espec i ely, using exac in o ma ion c i e ia o AIC o BIC. In mos cases, he implemen ed algo i hm u ilizing he unc ion, which is p esen in he py hon package o o ecas ing (Hyndman and Khandaka 2007); and (Baue e al. 2020). P oposi ion 2. The pe o mance o ARIMA (p,d,q) is compa ed wi h ha o LSTM. The LSTM looks like a smoo h, na u al compe i o o ARIMA (p,d,q) because i can cap u e a long- e m sequence o pa e n wi hin sequen ial da a. We s a building an LSTM model, which enume a es he s a ed unc ion linking k(2) o he ime lags, as: k(2) = (k(2) −1,k(2) −2,k(2) −3,k(2) −4, ....k(2) −j) + w(3) −x(15) whe e jeN is de ined as he numbe o ime lags being conside ed and w(3) −x is he homoschedas ic e o o andomness e m. P oo . The LSTM ne wo k, jus like many o he s anda d machine lea ning me hods, needs he da ase di iding in o es ing and aining se s. The aining se o en ep esen s supe ised lea ning, whe eas es ing is o he alida ion o he model. Table 1shows a supe ised lea ning da ase , which is help ul o p edic ion. Upon comple ion o aining, he ne wo k will ha e lea ned he inpu -ou pu unc ional ela ionship, hus p edic ing u u e alues o k(2) by using only he inpu . To be mo e p ac ical, aking he inpu as (m"J) ma ix wi h ime lags o k(2) as well as he ou pu as he (m" 1 ) ec o o bes cu en alues, wi h meNis he numbe as in Table 1. J. Risk Financial Manag. 2021,14, 259 8 o 12 Table 1. Supe ised Lea ning Da aSe . Ou pu Inpu k(2) k(2) −1k(2) −2.... k(2) −j k(2) +1k(2) k(2) −1.... k(2) −j+1 k(2) +2k(2) +1k(2) .... k(2) +j−2 k(2) +3k(2) +2k(2) +1.... k(2) +j−3 .... .... .... .... .... k(2) +mk(2) +m−1k(2) +m−2.... k(2) +m−j The p edic ed k(2) alues, a ime m+ 1, m+ 2, m+ 3, ..., m+J , a e done ecu si ely. Gene ally, he p edic ed alues o k(2) in a gene ic ime m+ is de e mined using he alues o k(2) wi h = (m+λ− 1, m+λ− 2, m+λ− 3, ..., m+λ−J ) as inpu . The alues o k(2) a e de e mined by he p edic ed as opposed o obse ed alues. We s a by es ima ing he CBD model pa ame e s k(1) , (x−¯ x) and k(2) using he SVD me hod. The ex ac ed ime se ies o k(2) is deno ed as he i s base o ou analysis. The da a is hen spli in o aining se and es ing se as pe 80% aining and 20% es ing ule. Consequen ly, we de e mine he las yea T o obse a ion. We ha e done he analysis o he U.K. and Kenya di e en ia ing h ough gende wi h one- ime lag (j=1)in Table 2. Table 2. Tes ing se yea s as pe Na ions. Na ions Numbe o Yea s Yea s o Tes ing Se U.K. 1930–2018 1998–2018 Kenya 2010–2020 2010–2020 When selec ing he op imum hype pa ame e s combina ion o he neu al ne wo k, i is essen ial o ca y a p elimina y ine- uning ound o all hese coun ies while dis in- guishing hem by gende (see Table 3). In his s ep, we can ge combina ions, which will be used du ing LSTM calib a ion du ing he o ecas ing p ocedu e. On he uning esul s, we ha e disco e ed ha his a chi ec u e ha ing one hidden laye does pe ume be e han o he s on ou da a and he numbe o neu ons depending on he coun y. Using a Rec i ied Linea Uni (ReLU) as an ac i a ion unc ion ou pe o med many o he unc ions when es ing many o he coun ies. Mo eo e , he e is no clea e idence on he in luence o he pe o mance o hype -pa ame e s. Table 3. ARIMA by Na ion and Gende . Na ion ARIMA Model (p,d,q) U.K. Males ARIMA (1,1,0) Females ARIMA (1,1,0) Kenya Males ARIMA (0,1,3) Females ARIMA (0,1,3) A e he calib a ion s ep, he pape ’s analysis will include nume ical and g aphical p ocessing and p esen a ion o he goodness o i . To be speci ic, he s udy will ollow he app oach o ou o sample, which deno es he es ing s ep wi hin he ield o machine lea ning. The es ima ion o pa ame e k(2) pa ame e is de e mined using SVD, as o male and emale espec i ely. Figu e 3dashed e ical line shows a sepa a ion o he o ecas ed pe iod compa ed o one used in aining he LSTM ne wo k. As o ARIMA models, i