Volatility Change Point Detection Using Stochastic Differential Equations and Time Series Control Charts

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1 Volaly Chage Po Deeco Usg Sochasc Dffereal Equaos ad Tme Seres Corol Chars M Kovar Absrac The arcle focuses o volaly chage po deeco usg SPC (Sascal Process Corol) specfcally hrough me seres corol chars ad sochasc dffereal equaos (SDEs) I he paper rece advaces mechage po for process volaly compoe sasfyg a sochasc dffereal equao (SDE) based o dscree observaos ad also usg me seres corol chars wll be revewed Theorecal par wll dscuss he mehodology of me seres corol chars ad SDEs drve by a Browa moo Research par wll demosrae he mehodologes usg a case sudy focusg o aalyss of Slova currecy durg he perod of 4 from he perspecve of s usefuless for geerag profs for compay maageme hrough me seres corol chars ad SDEs The am of he paper s o demosrae use of chage po deeco me seres of he Slova crow I also ams o hghlgh versaly of corol chars o oly maufacurg bu also maagg facal sably of cash flows Keywords Chage po Sascal process corol Sochasc dffereal equaos Tme seres corol chars I I INTRODUCTION face he volaly of he mare or of he asse prces plays a crucal role may aspecs For example opo prcg alhough he very basc Blac ad Scholes (973) [] ad Mero (974) [] model assumes a cosa volaly s well ow ha hs assumpo s urealsc whe oe wors wh real facal daa Ths fac causes well-ow effecs le mpled volaly ad volaly smles Chage po aalyss was ally roduced he framewor of depede ad decally dsrbued daa (see eg Hley 97 [3]; Icla ad Tao 994 [4]; Ba [5] [8]; Csörgö ad Horváh 997 [34]) ad qucly appled o he aalyss of me seres (see eg Km Cho ad Lee [6]; Lee Ha ad Na 3 [7]; Che Cho ad Zhou 5 [8]) Kuoyas (994 4) [39] [4]; Lee Nshyama ad Yoshda (6) [9] suded srucural chage po problems for he drf erm for couous observaos from ergodc dffuso processes Due o he fac ha volaly ca be esmaed whou error couous M Kovar s wh he Deparme of Sascs ad Quaave Mehods Tomas Baa Uversy Faculy of Maageme ad Ecoomcs Zl Czech republc (phoe: ; e-mal: ovarfame@sezamcz) me he chage po aalyss hs seup s o very eresg I hs paper here wll be revew rece advaces for he problem of mechage po for he volaly compoe of a process sasfyg a sochasc dffereal equao (SDE) based o dscree observaos (Chaper V) ad also usg me seres corol chars (Chaper IV) II LITERATURE REVIEW I fac chage po problems have orgally arse he coex of qualy corol bu he problem of abrup chages geeral arses may coexs le epdemology rhyhm aalyss elecrocardograms sesmc sgal processg sudy of archeologcal ses ad facal mares I parcular he aalyss of facal me seres he owledge of he chage he volaly srucure of he process uder cosderao s of a cera eres Varous auhors have suded chage po deeco problems usg paramerc ad o-paramerc procedures que exesvely I some cases he sudy was carred ou for ow uderlyg dsrbuos amely he bomal Posso Gaussa ad ormal dsrbuos amogs ohers Ths chaper dscusses some of he wor ha has bee doe o chage po deeco CUSUM s oe of he wdely used chage po deeco algorhms Bassevlle & Nforov (993) [] descrbed four dffere dervaos for he CUSUM algorhm The frs s more uo-based ad uses deas coeced o a smple egrao of sgals wh a adapve hreshold The secod dervao s based o a repeaed use of a sequeal probably rao es The hrd dervao comes from he use of he off-le po of vew for mulple hypoheses esg The fourh dervao s based upo he cocep of ope eded ess The prcple of CUSUM sems from sochasc hypohess esg mehod Nazaro Ramrez ad Tep (997) [] developed a sequeal es procedure for rase deecos a sochasc process ha ca be expressed as a auoregressve movg average (ARMA) model Prelmary aalyss shows ha f a ARMA(pq) me seres exhbs a rase behavor he s resduals behave as a ARMA(QQ) process where: Q p + q They showed ha resduals from he model before he parameer chage behave approxmaely as a sequece of Issue Volume 7 3

2 depede radom varables - afer a parameer chage he resduals become correlaed Based o hs fac hey derved a ew sequeal es o deerme whe a rase behavor occurs a gve ARMA me seres Blaze HogJoog Bors ad Alexader () [] developed effce adapve sequeal ad bach-sequeal mehods for a early deeco of aacs from he class of deal-of-servce aacs" Boh he sequeal ad bachsequeal algorhms used hresholdg of es sascs o acheve a fxed rae of false alarms The algorhms are developed o he bass of he chage po deeco heory: o deec a chage sascal models as soo as possble corollg he rae of false alarms There are hree aracve feaures of he approach Frs boh mehods are self-learg whch eables hem o adap o varous ewor loads ad usage paers Secod hey allow for deecg aacs wh a small average delay for a gve false alarm rae Thrd hey are compuaoally smple ad hece ca be mplemeed ole For more abou false alarm raes eg for he Shewhar corol chars see [9] Hua ad Md () [33] exame a process-moorg ool ha o oly provdes speedy deeco regardless of he magude of he process shf bu also provdes useful chage po sascs Lud Xaola Lu Reeves Gallagher ad Feg (7) [3] looed a he chage po deeco perodc ad auo correlaed me seres usg classc chage po ess based o sums of squared errors Ths mehod was successfully appled he aalyses of wo clmae chages Mosva ad Zhgljavsy (3) [4] developed a algorhm of chage po deeco me seres based o sequeal applcao of he sgular-specrum aalyss (SSA) The ma dea of SSA s performg sgular value decomposo (SVD) of he rajecory marx obaed from he orgal me seres wh a subseque recosruco of he seres Mboup Jo ad Fless (8) [5] preseed a chage po deeco mehod based o a drec ole esmao of he sgal s sgulary pos Usg a pecewse local polyomal represeao of he sgal he problem s cas o a delay esmao A chage po sa s characerzed as a soluo of a polyomal equao he coeffces of whch are composed by shor me wdow eraed egrals of he osy sgal The chage po deecor showed good robusess o varous ypes of oses Aure ad Aldrch () [6] used radom fores models o deec chage pos dyamc sysems We Hapg Yue ad Wag () [7] used Lyapuov expoe ad he chage po deeco heory o judge wheher aomales have happeed Aldrch ad Jemwa (7) [8] used phase mehods o deec chage complex process sysems Vce (998) [9] preseed a ew echque for he defcao of homogeees Caada emperaure seres The echque s based o he applcao of four lear regresso models order o deerme wheher he esed seres s homogeeous Vce s procedure s a ype of forward regresso algorhm ha he sgfcace of he o-chage po parameers he regresso model s assessed before (ad afer) a possble chage po s roduced I he ed he mos parsmoous model s used o descrbe he daa The chose model s he used o geerae resduals I uses he auocorrelao he resduals o deerme wheher here are homogeees he esed seres A frs cosders he ere perod of me ad he sysemacally dvdes he seres o homogeeous segmes Each segme s defed by some chage pos ad each chage po correspods o eher a abrup chage mea level or a chage he behavor of he red III PROBLEM FORMULATION The am of he paper s o demosrae use of chage po deeco me seres of he Slova crow hrough he me seres corol chars ad SDEs Slova crow was aalyzed due o accessbly of prmary sascal daa (currecy raes eres raes o whole me scale from Overgh o year) ad also due o avalably of fas commucao chaels wh he sascal deparme of he Naoal ba of Slovaa case here was a eed for ay clarfcao Box-Jes mehodology was ulzed as he ma aalycal ad mahemacal-sascal mehod o aalyze me seres The oupu wll be used o cosruc SPC corol chars for deeco of chage varace srucure Furhermore hs approach wll be cofroed wh chage po deeco by modelg he SDEs covered Chaper VI The followg heorecal chapers (Chaper IV ad Chaper V) provde a descrpo of mahemacal-sascal ools used o verfy he formulaed research am IV TIME SERIES CONTROL CHARTS A ARIMA corol chars Classc Shewhar SPC cocep assumes he measured daa are o auocorrelaed Eve very low degree of auocorrelao causes falure of he classc Shewhar corol chars a form of a large umber of pos ousde he regulaory lms corol dagram [3] The pheomeo s o uque o couous processes where he era processes deerme he auocorrelao daa me (chemcal ad clmae processes ec) Auocorrelao of daa becomes creasgly freque dscree processes parcular maufacurg wh shor produco cycles hgh speed produco wh hgh degree of produco auomao ad also es ad corol operaos I hese codos s possble o oba daa abou each produc wh he cosequece ha he me erval bewee measuremes (recordg) of wo cosecuve values of he moored varables s very shor Oe of he ways o acle auocorrelaed daa s he cocep of sochasc modelg of me seres usg auoregressve egraed movg average models he ARIMA model Lear sochasc auoregressve models (models AR) movg Issue Volume 7 3

3 average (model MA) mxed models (he ARMA models) ad ARIMA models based o Box-Jes mehodology are see as a me seres realzao of sochasc process Box-Jes mehodology represes a moder aalyss of saoary ad osaoary me seres based o probably heory Lear models AR ARMA ad MA are modelg ool for he saoary processes These models have a characersc shape of he auocorrelao fuco (Auocorrelao Fuco ACF) ad paral auocorrelao fuco (Paral Auocorrelao Fuco PACF) esseal ools for provdg formao abou he sochasc process ACF ad PACF esmaes are used o defy he me seres model Very ofe here are o-saoary processes pracce Nosaoary ca be prese due o he mea value chagg over me or process varace chagg over me If afer he rasformao of osaoary process varace of "radom wal" (so-called egraed process) usg dffereal d-h order s he fal process model o descrbe he saoary ARMA (p q) he orgal egraed process s called a auoregressve egraed movg average process of order p d q e ARIMA (p d q) [44] ARIMA corol char (Auoregresve Iegraed Movg Average) s based o he prcple of fdg a suable me seres model ad he use of corol char for he model s resduals (devaos from he values acually measured values from calculaed values wh he model use) The geeral shape of he model ARIMA (p d q) s such Φ B d x = Θ B ε () where p ( ) ( ) p q ( ) ( p B φb φb φ p B ) Φ = s auoregressve polyomal of p-h order q Θ B = θ B θ B θ B s movg averages q ( ) ( q ) polyomal of q-h order operaor deoed a bacward dfferece (roduced whe he model exhbs osaoary of he process) d dfferece order me B x = B bac shf operaor ( ) x φ φ φ parameers of auoregressve model p θ θ θ parameers of movg averages model q ε s a whe ose upredcable ormally dsrbued flucuaos he daa wh mea equal o zero ad cosa varace ad ucorrelaed values If x s a esmae of emprcal value of ˆ x calculaed wh help of rgh chose ARIMA model resduals of he model e = x xˆ wll be ucorrelaed ormally dsrbued radom varables Mos commoly used applcaos are ARIMA models Le us cosder he model x = ξ + φx + ε () where ξ a ϕ ( φ ) < < are uow cosas ad ε s ormally dsrbued ad ucorrelaed varable wh he mea equal o zero ad he cosa sadard devao σ Ths model s called auoregressve model of he frs order ad s deoed as AR() The values of he referece mar of qualy whch are muually shfed of me perods (x ad x ) have he correlao coeffce ϕ Ths meas ha auocorrelao fuco ACF should fall expoeally If we expad he prevous equao he form x = ξ + φ x + φ x + ε (3) we ge equao of secod order auoregressve model AR() Geerally varable x s drecly depede o he values precedg x x ec he auoregressve model AR (p) If we model he depedece of daa usg he radom compoe ε he we ge movg average model MA (q) Movg average model frs order has a equao: x = µ + ε θε (4) There s some correlao oly bewee wo values x ad x I ca be descrbed as follows: ρ = θ / ( + θ ) Ths correspods o he shape of he auocorrelao fuco ACF [44] For modelg of praccal problems s ofe suable o model compoud coag boh auoregressve ad he movg averages compoe The model s geerally ow as ARMA (p q) [45] ARMA of he frs order e ARMA ( ) s descrbed by he equao: x = ξ + φx + ε θε (5) I s ofe suable for chemcal ad oher couous processes where may qualy characerscs ca be easly modeled by AR () Measureme errors are descrbed by model s radom compoe we assume o be radom ad ucorrelaed The ARMA model assumes process saoary e ha he characer qualy referece values are aroud a sable mea Bu ofe pracce here are processes (eg he chemcal dusry) where he values of moored varable are "rug away" I s he covee o model processes usg approprae model wh he operaor of bacward dfferece such as he ARIMA model ( ) whose equao s x = x + ε θε (6) x µ ε ARIMA are dffere from Shewhar model ( = + for = ) However f we pu φ = he equao x = ξ + φ x + ε or θ = he equao x = µ + ε θε we ge he Shewhar model process Aoher mpora sep he use of ARIMA models s he choce of he approprae SPC corol char Whe resduals esg are deemed o auocorrelaed ad comg from a ormal dsrbuo s possble o use hem o verfy wheher or o he process s sascally sable Because he umber of observaos equals oe (orgal emprcal values x were deeced by each u) corol chars for dvdual values ad movg rage aes prory Locao of Issue Volume 7 3 3

4 he mea value CL ad upper ad lower corol lms (UCL LCL) for he ARIMA char for dvdual values ca be deermed from he formula CL = e (7) ( ) 3 UCL = e + R (8) 8 l 3 LCL = e R (9) 8 l where e s average value of resduals R s average movg rage l Values CL UCL ad LCL ca be calculaed as follows CL = R l () UCL = 3 67 R () l LCL = () To crease he sesvy of corol chars ARIMA s recommeded o use wo-sded CUSUM corol char wh he decso erval ±H or sadard EWMA char boh appled o he resduals of he model If we pursue more qualy characerscs smulaeously o a sgle produc o mulple we ca apply Hoellg char or CUSUM or EWMA chars for more varables [44] B CUSUM corol char CUSUM (Cumulave Sum) s a sequeal aalyss echque ha s used he deeco of abrup chages [3] Whe applyg CUSUM mehod a dagram s cosruced O x axs of hs dagram a selecve order s recorded O axs y es crero values Y are recorded The value of es crero afer -h selecve y ca be defed by formula ( µ ) ( µ ) C = x = C + x C = (3) j j= where s he selecve order = ) x j s he selecve average from values of regulaed value j-h seleco (j = ) The sudy of CUSUM wll o aswer he queso wheher he chage of dagram progresso sgals a sgfca devao (dcag a defable fluece o process) or wheher he fluece s radom Therefore decso crera eed o be added There are wo basc ypes of crera hrough whch a decso ca be made wheher he process s sascally vable or o These are: I decso mas II decso erval These procedures are descrbed deal publcao [7] CUSUM char for dvdual values ad for samples meas from ormally dsrbued daa Values of x are depede wh he same ormal dsrbuo N(µ σ ) wh he ow populao mea ad wh he ow populao sadard devao σ We assume logcal subgroups wh he same volume Cumulave sum CUSUM C s defed for dvdual values ( = ) as: O a base of orgal scale: C = ( x j µ ) (4) j= O a base of ormal dsrbuo where he mea µ = ad he sadard devao σ = : ( x j µ ) U j = S = U (5) σ j = The CUSUM C s almos he same as CUSUM S measured he us of sadard devao σ Equao for C ca be rewre recurrely []: C = C = C + (x µ); ad wh he same prcple for S : S = S = S + U Suppose ha he orgal dsrbuo of observed varable N(µ σ ) chages o N(µ + δ σ ) dsrbuo for eger (a arbrary mome) I meas ha he populao mea µ wll face a cera shf of δ I also meas ha he shf sars a po (m C m ) ad grows learly wh he slope δ Bu he populao meas shf ca be more complcaed The CUSUM corol char ca reflec all hese chages [] CUSUM for sample meas We have cosdered maly he dvdual values ul ow Now we wll cosder subgroups wh m observaos ad calculae he sample meas from subgroups We have o wor wh he sample mea sadard devao σ σ x = m A shf of mea wll o be measured he us of σ bu he us of σ hs case We wll subsue he x dvdual values of x wh he sample meas x ad he j process sadard devao σ wh he sample mea sadard devao σ abovemeoed formulas [] x New Process Mea Esmae If here s a shf we ca esmae a ew process mea from he ex formula: + CI + µ + K+ pro C + I > H ˆ N µ = (6) CI µ K+ pro C I < H N where N + ad N s a umber of seleced pos from a mome [3] whe C = resp whe C = + Comparso of CUSUM ad Shewhar s Corol Chars Ths example shows praccally sesvy of he CUSUM corol char comparso wh he Shewhar s corol char for he sample meas The CUSUM corol char deecs process mea devao owards he lower values (aroud he subgroup see Fg ) whle he Shewhar s corol char does o deec hs devao [] I does o deec a shf owards he upper values (aroud he subgroup 56) I oly Issue Volume 7 3 4

5 deecs a bg shf aroud he subgroup 7 (see boh Fg ad Fg ) [] Fg Shewhar s Corol Char Source: QC Exper 5Cz Fg Corol Char CUSUM Source: QC Exper 5Cz C EWMA corol chars Smlarly o CUSUM dagram also EWMA dagram s feasble for suaos whch he process s mpaced by sudde small bu prevalg chages ad he values of observed characersc are o depede Ule sadard dagrams he regulao bouds deped o he selecve mome The auhor of he EWMA dagram s Robers (959) [4] Ths dagram wors wh es crero Y The value afer -h seleco s defed as follows: j ( λ) λ ( λ) ( j ) y = Y + f x (7) j= for j = ad for < λ < f(x j ) s he value of selecve characersc s he seleco order Y s he requred level of dsrbuo parameer of regulaed value Whe dealg wh Shewhar regulao dagrams for dvdual values we observed ha he dagram for dvdual values s very sesve o daa o-ormaly he sese ha real uder he ARL (ARL ) corol would be sgfcaly smaller ha he expeced value based o he assumpo of ormal dsrbuo Borror Mogomery ad Ruger (999) [5] compared he behavor of ARL Shewhar dagrams for dvdual values ad EWMA dagrams for he case of asymmerc dsrbuo Feaures of dvduals corol chars for Burr dsrbued ad Webull dsrbued daa see [3] I he followg ex a specfc gamma dsrbuo represeg he case of asymmerc dsrbuo ad dsrbuo represeg symmercal dsrbuo N( ) wll be ulzed ARL values of Shewhar regulao dagrams for dvdual values ad EWMA dagrams for hese dsrbuos are deoed he followg ables Two aspecs hese ables are bewlderg Frsly eve slgh o-ormaly dsrbuo leads o sgfca decrease of ARL value Shewhar dagram for dvdual values Subsequely he umber of false alarms creases Secodly EWMA wh λ = 5 or λ = ad appropraely seleced regulao boud ca wor very well o boh symmerc ad asymmerc dsrbuo Wh parameers λ = 5 ad K = 49 he ARL value for EWMA s approxmaely 8% of he lm emphaszed by heory of ormal dsrbuo of value ARL = 37 wh he excepo of exreme cases [6] Tab ARL values for he EWMA char ad a char of dvdual values for dffere gamma dsrbuo Source: [6] EWMA Shewhar λ 5 K Sadard Gamma (4 ) Gamma (3 ) Gamma ( ) Gamma ( ) Gamma (5 ) Tab ARL values for he EWMA char ad a char of dvdual values for dffere dsrbuo Source: [6] EWMA Shewhar λ 5 K Sadard Based o hs formao he properly desged EWMA s recommeded as a corol char for dvdual values a wde rage of applcaos parcularly process moorg I s almos a compleely oparamerc (depede of dsrbuo) procedure I addo EWMA chars are defely beer ha Shewhar chars for dvdual values as well as for he feaures of he mea shf deeco [6] [7] Issue Volume 7 3 5

6 V CHANGE POINT DETECTION USING STOCHASTIC DIFFERENTIAL EQUATIONS Chage po esmao cosss he defcao of he sa whch a chage occurs he parameer of some model There are several approaches o he soluo of hs problem ad here we cosder a leas squares soluo (see eg [4] [5] [8]) bu oher approaches such as maxmum lelhood chage po esmao are also possble (see eg [8] [34]) We assume we have a dffuso process soluo o dx = b X d + θσ X dw (8) ( ) ( ) where b( ) ad σ( ) are ow fucos ad θ Θ R s he parameer of eres As [36] gve dscree observaos from (8) o [ T = ] we wa o defy rerospecvely f ad whe a chage value of he parameer θ occurred ad esmae cossely he parameer before ad afer he chage po The asympocs s as ad = T fxed For smplcy we assume ha he chage occurs a sa whch s oe of he egers Ths s a problem of volaly chage po esmao ha frequely occurs face applcaos We assume ha θ = θ before he me chage ad θ = θ afer he me chage wh θ < θ (bu hs does o maer he fal resuls) I order o oba a smple leas squares esmaor we use Euler approxmao So from ow o we assume all he hypoheses ecessary o have he Euler approxmao place Namely we ca wre he Euler scheme as X = X + b X + θσ X W W ( ) ( )( ) + + ad roduce he sadardzed resduals ( X + X ) b( X ) ( W + W ) Z = = θ σ X ( ) The Z 's are d (depede ad decally dsrbued) Gaussa radom varables The chage po esmaor s obaed as ˆ = arg m m ( Z θ ) + ( Z θ ) (9) θ θ = = + wh = We deoe by [x] he eger par of he τ = τ real x ad somemes we wre = [ ] ad [ ] ( ) τ τ o dcae he chage po he couous mescale Defe he paral sums S = Z S = Z S = Z = ad deoe by of θ ad = = + θ ad θ he al leas squares esmaors θ for ay gve value of (9) For couous-me observaos hs problem was suded [9] A bayesa approach for dscree-me observaos ca be foud [35] For ergodc dffuso processes ad = T uder addoal mld regulary codos he resuls meoed here are sll vald θ S = Z = = ad θ S = = Z = + These esmaors wll be refed oce a cosse esmaor of s obaed Deoe by U he quay ( θ ) ( θ ) = + = = + U Z Z The ˆ s defed as ˆ = arg m U To sudy he asympoc properes of rewre as ( ) U = Z Z V = where Z ad = = Z ( θ θ ) ( ) S D V = = wh S D = S ( ) U s beer o Ths represeaos of U s obaed by leghy bu sraghforward algebra ad s raher useful because mmzao of U s equvale o he maxmzao of V ad hece of D So s easer o cosder he followg esmaor of ( ( )) ˆ = arg max D = arg max V () As a sde remar ca be oed ha for fxed (ad uder suable hypoheses) D s also a approxmae lelhood rao sasc for esg he ull hypohess of o chage volaly (see eg [4]) Oce ˆ has bee obaed he followg esmaors of he parameers θ ad θ ca be used: S ˆ ˆ θ = () ˆ S ˆ ˆ θ = () ˆ Nex resuls provde cossecy of ˆ ˆ θ ad ˆ θ as well as her asympoc dsrbuos Issue Volume 7 3 6

7 H θ θ Fac ([34]) Uder : = = we have ha where ( ) D d W ( τ ) { W τ τ } (3) s he Gaussa sochasc process Browa brdge whch s useful whe we eed o model varable whch sars a some po ad ha mus reur o a specfc po a a specfc me he fuure The asympoc resul above s useful o es f a chage po does exs I parcular s possble o oba he asympoc crcal values for he dsrbuo of he sasc by meas of he same argumes used [34] ˆ Fac ([34]) The esmaor ˆ τ = sasfes / ˆ τ τ = θ θ O log (4) ( ) p ( ) Moreover for ay β ( / ) p β ( ˆ τ τ ) Fally ˆ τ τ = O (5) p ( θ θ ) I s also eresg o ow he asympoc dsrbuo of ˆ τ for small dscrepaces bewee θ ad θ The case ϑ = θ θ equal o a cosa s less eresg because whe ϑ s large he esmae of s que precse Assumpo ϑ such a way ha ϑ log Assumpo ad Fac mply he cossecy of ˆ τ Fac 3 ([34]) Uder Assumpo for as we have ha ϑ ( ˆ τ τ ) d v arg max W ( v) (6) v ɶ θ ( ) where W ( u ) s a wo-sded Browa moo W ( u) u < W ( u) = W ( u) u (7) wh W ad W wo depede Browa moos ad ɶ θ a cosse esmaor for θ or θ Fally we have he asympoc dsrbuos for he esmaors ˆ θ ad ˆ θ defed () a () We deoe by θ he commo lmg value of θ ad θ Fac 4 ([34]) Uder Assumpo we have ha ˆ θ θ d N ( Σ) (8) ˆ θ θ where 4 τ θ Σ = ( τ ) 4 θ (9) Esmao of he chage po wh uow drf σ x are uow s ecessary o Whe boh b( x ) ad ( ) assume ha a leas σ ( x) s cosa ad hece we cosder he sochasc dffereal equao dx = b X d + θ dw (3) ( ) The b( x ) ca be esmaed oparamercally wh bˆ ( x) = = x X ad Z are esmaed as ˆ X + X Z = bˆ ( X ) ( ) K X + X h x X K = h (3) Remar I Sao s approach [34] he wo esmaors are ohg bu Nadaraya-Waso erel regresso esmaors of he followg codoal expecaos b( x) = lm Ε{ X x X = x} σ ( x) = lm Ε{ ( X x) X = x} σ x are see as I hs approach b( x ) ad ( ) saaeous codoal meas ad varaces of he process whe X = x The wo quaes ca be rewre for fxed as o b( x) = Ε{ X + X X = x} + { } ( ) ( ) σ + ( ) o x = Ε X X X = x + ( ) So f we have esmaed resdues Z hs case we ca use he followg coras o defy he chage po: ˆ ˆ ˆ S ˆ S ɶ = arg m Z Z = + = + (3) where Sˆ = Zˆ a ˆ ˆ S = Z = = + We oba he ew sasc ( ) ˆ ˆ ˆ ˆ ˆ S S SD = = V where ˆ ˆ S D = Sˆ ( ) Issue Volume 7 3 7

8 ad he chage po s defed as he soluo o ˆ = arg max Dˆ Cossecy ad dsrbuoal resuls meoed [34] [37] ad [38] Ths paper wll be deal wh a chage po problem for he volaly of a process soluo o a sochasc dffereal equao whe observaos are colleced a dscree mes The sa of he chage volaly regme s defed rerospecvely by maxmum lelhood mehod o he approxmaed lelhood For couous me observaos of dffuso processes Lee Nshyama ad Yoshda (6) [9] cosdered he chage po esmao problem for he drf I he prese wor here wll be oly assume regulary codos o he drf process VI PROBLEM SOLUTION Mos daa aalyss procedures ad he resulg coclusos are depede o fulfllg basc assumpos o whch hese procedures were based If o me all oher sadard procedures such as calculag a average cofdece ervals perceles mos ess classcal Shewhar corol chars desg of models for he descrpo of me seres ec are quesoable ad mpugable They usually provde correc resuls ad coclusos A ypcal breach of codos for applcao of corol by Shewhar chars or dffere echologes bu also for he cosruco of models descrbg me seres s show [4] The assumpos mus be verfed by sascal ess whch volve he basc assumpos for sascal process corol: Fg 3 Tme seres of O/N rae of he Slova crow for he perod 8 Source: Ow Processg No-saoary of he me seres s also cofrmed by he shape of he ACF ad PACF ACF values fall very slowly ad he frs value as well as he PACF s close o oe The perodogram has a sgfca pea he zero (o-seasoal) frequecy Seasoaly dcaes eher he ACF ad PACF or he perodogram Fg 4 ACF of he orgal me seres Source: Ow Processg ormal daa dsrbuo symmery cosa mea of he process cosa varace (sadard devao) of daa depedece o-correlao of daa absece of oulers ad exreme values If hese assumpos are frged coveoal regresso mehods for he me seres aalyss provde based ad ofe correc resuls Le us ow aalyze he me seres of O/N rae of he Slova crow for he perod 8 (used [43]) For hs purpose he ARIMA model wll be used whch s appled f a resulg process s showg such auocorrelao ad paral auocorrelao afer he rasformao of he egraed process usg dffereao of d-h order ha s expressed he form of saoary ad verble ARMA model (pq) The course of me seres s llusraed he followg fgure The graph shows ha he me seres s o-saoary bu s o clear wheher coas a seasoal compoe Fg 5 PACF of he orgal me seres Source: Ow Processg Fg 6 Perodogram of he orgal me seres Source: Ow Processg Issue Volume 7 3 8

9 The me seres wll be aga saoarzed by he I oseasoal dfferece Le us sp he elemeary seps of he aalyss ad go drecly o he aalyss of wo deal models o descrbe hs me seres The frs model s afer he power learzao ARIMA () c ad he secod model s afer he logarhmc learzao of he orgal me seres SARIMA()() c Esmaed whe ose varace = 4347 Esmaed whe ose sadard devao = 96 Box-Perce Tes Tes based o frs 4 auocorrelaos Large sample es sasc = 33 P-value = 4336 AIC = -549 Ierpolao crera shows SARIMA ()() c as much more suable ha ARIMA () c for he descrpo of hs me seres The fgures below show graphs wh he me seres forecas auocorrelao ad paral auocorrelao fucos of usysemac compoe for esmaed model ad resdual perodogram Fg 7 Resdual perodogram for ARIMA model () c Source: Ow Processg Resdual perodogram for ARIMA model () c shows ha resduals are saoary Now he focus s o he exeded SARIMA model () () c followed by ables of esmaes ad erpolao crera ad he very esmaes of he model s parameers Tab 3 Esmaes of he erpolao crera of SARIMA model ()() c Source: Ow Processg Esmao Valdao Sasc Perod Perod RMSE MAE MAPE ME MPE Fg 8 Graph of he me seres wh forecas ad cofdece erval for forecas Source: Ow Processg Accordg o Aae formao crera he Box-Perce es of auocorrelao of usysemac compoe ad erpolao crera SARIMA model ()() c appears o be beer for he descrpo of hs me seres ha ARIMA model () c The followg fgure llusraes he resdual ACF ad PACF of he esmaed model P-value of he Box-Perce es ad boh of hese graphs dcae o-auocorrelao of usysemac compoe hus he esmaed model appears o be correc Tab 4 Esmaes of parameers of SARIMA model ()() c Source: Ow Processg Parameer Esmae Sad Error T P-value AR() MA() MA() SAR() SMA() Mea Cosa - Fg 9 Resdual ACF of SARIMA model ()() c Source: Ow Processg Issue Volume 7 3 9

10 Fg Resdual PACF of SARIMA model ()() c Source: Ow Processg As evde from he prevous mages he EWMA corol char ad CUSUM were able o deec chages he mea almos mmedaely (EWMA: 3 November ; CUSUM: 4 Sepember ) These are corol chars wh memory herefore formao o varably he me seres of SARIMA model ()() c was used for parameer esmao o cosruc hese dagrams very effecvely The ARIMA corol char deeced a shf of mea ad herefore hgh heeroscedascy 5 Augus whch correspods o flucuaos of Slova crow Fg Resdual perodogram of SARIMA model ()() c Source: Ow Processg Now ha we have formao o he process due o a sascal model ca be used o cosruc corol chars for deecg chages he mea Ths deeco wll be demosraed o he me seres of Overgh values of he Slova crow durg Jauary 6 Jue 4 Fg 4 ARIMA corol char for volaly chage po deeco Source: Ow Processg The we loo a he volaly chage po deeco usg he SDEs leas squares approach Fg EWMA corol char for mea shf deeco wh λ = 6 Source: Ow Processg Fg 5 Overgh values chage po aalyss of Overgh values of he Slova crow he perod Jauary 6 Jue 4 Source: Ow Processg Fg 3 CUSUM corol char for mea shf deeco 6σ Source: Ow Processg All he aalyss usg SDEs have bee doe usg he R sascal evrome (R Developme Core Team 9) ad he pacage sde (see [37]) ad Yuma (see [4]) Followg s he oupu of he R programmg laguage evrome usg he sde pacage Issue Volume 7 3 3

11 $ [] 655 $au [] 655 ( Sepember ) $hea [] $hea [] 4977 Loog a he prevous fgure aoher chage po may be prese So we reaalyze he frs par of he seres o spo he secod chage po $ [] 76 $au [] 76 (3 February ) $hea [] $hea [] Boh chage pos (3 February ad Sepember ) correspod o flucuaos of Slova crow VII DISCUSSION Mos radoal corol charg procedures are grouded o he assumpo ha he process observaos beg moored are depede ad decally dsrbued Wh he adve of hgh-speed daa colleco schemes he assumpo of depedece s usually volaed Tha s auocorrelao amog measuremes becomes a here characersc of a sable process Ths auocorrelao causes sgfca deerorao corol charg performace To address hs problem several approaches for hadlg auocorrelaed processes have bee proposed The mos popular procedure ulzes eher a Shewhar CUSUM or EWMA char of he resduals of he appropraely fed ARMA model However procedures of hs ype possess poor sesvy especally whe dealg wh posvely auocorrelaed processes As a alerave we have explored he applcao of he sascs used a me seres procedure for deecg oulers ad level shfs process moorg The sudy focused o he deeco of level shfs of auocorrelaed processes wh parcular emphass o he mpora AR() model The resuls preseed showed ha me seres chars are foud o be sesve deecg small shfs ad we ulze he fac ha hese corol chars ca be used cera suaos where he daa are auocorrelaed As for he wo sub-seres defed by he chage po esmae ad esmaors of hese parameers: boh are cosse ad asympocally ormal a he usual rae wh he umber of observaos The leas squares esmaor ˆ τ seems o have a good performace erms of bas ad varably for models wh cosa or bouded drf whle behaves badly he presece of ubouded drf as me T grows VIII CONCLUSION I facal mares s crucal o have a accurae descrpo of he volaly of he mare ad/or he dffere facal producs All prcg formulas mae use of he hsorcal values of he volaly as a fudameal grede I s well ow however ha volaly s o cosa over me eve shor me ad hus he moorg of he volaly s oe of he prmary ass emprcal face The sascal way o moor srucural chages s called chage po aalyss or chage po esmao The paper deal wh he corol chars ad SDEs applcaos facal daa Ths d of daa s very sesve o mea shfg ad srog auocorrelao appears very ofe Therefore we pu a focus o dyamc regulao chars CUSUM EWMA ad ARIMA models We hghlghed he versaly of corol chars o oly maufacurg bu also maagg he facal sably of facal flows A refed defcao of he ype of erveo affecg he process wll allow users o effecvely rac he source of a ou-of-corol suao whch s a mpora sep elmag he specal causes of varao I s also mpora o oe ha he proposed procedure ca also be appled whe dealg wh a more geeral auoregressve egraed movg average model Auocorrelaed process observaos maly arse uder auomaed daa colleco schemes Such colleco schemes are ypcally corolled by sofware upgradeable o hadle SPC fucos Uder such a egraed scheme he usefuless of he proposed procedure wll be opmzed Based o formao from chaper 4 we would recommed a properly desged me seres corol chars as corol chars for dvdual measuremes a wde rage of applcaos They are almos perfecly oparamerc (dsrbuo-free) procedures Sochasc dffereal equaos are amog he mos used sochasc models o descrbe couous me facal me seres Alhough daa are colleced dscree me he uderlyg srucure of he couous model allows for very dealed aalyss of hese daa I he aalyss of he Slova crow he approach revealed (ule me seres corol chars) aoher chage po he srucure of varably hus appearg o be preferable for he chage po deeco I also shows ha he use of SDEs provdes robusess he esmao resuls REFERENCES [] F Blac ad M S Scholes The Prcg of Opos ad Corporae Lables (Boo syle) Joural of Polcal Ecoomy vol pp [] R C Mero Theory of Raoal Opo Prcg (Boo syle) Bell Joural of Ecoomcs ad Maageme Scece vol pp 4 83 [3] D V Hley Iferece abou he chage-po from cumulave sum ess Bomera vol pp 59 3 Issue Volume 7 3 3

12 [4] C Icla ad G C Tao Use of Cumulave Sums of Squares for Rerospecve Deeco of Chage of Varace Joural of he Amerca Sascal Assocao vol pp 93 3 [5] J Ba Leas Squares Esmao of a Shf Lear Processes Joural of Tmes Seres Aalyss vol pp [6] S Km S Cho ad S Lee O he Cusum Tes for Parameer Chages GARCH() Models Commucaos Sascs - Theory ad Mehods vol 9 pp [7] S Lee J Ha O Na ad S Na The Cusum Tes for Parameer Chage Tme Seres Models Scadava Joural of Sascs vol 3 3 pp [8] G Che Y K Cho ad Y Zhou Noparamerc Esmao of Srucural Chage Pos Volaly Models for Tme Seres Joural of Ecoomercs vol 6 5 pp [9] S Lee Y Nshyama N Yoshda Tes for Parameer Chage Dffuso Processes by Cusum Sascs Based o Oe-Sep Esmaors Aals of he Isue of Sascal Mahemacs vol 58 6 pp [] M Bassevlle ad I Nforov Deeco of abrup chages: heory ad applcao Eglewood Clffs Ed New Yor: Prece Hall 993 ch [] D Nazaro B Ramrez ad S Tep Trase deeco wh a applcao o a chemcal process Compuers d Eg 997 pp [] R B Blaze K HogJoog R Bors ad T Alexader A ovel approach o deeco of deal of servce aacs va adapve sequeal ad bach-sequeal chage-po deeco mehods Proceedgs of he IEEE pp -6 [3] R Lud L W Xaola Q Lu J Reeves C Gallagher ad Y Feg Chage-po deeco perodc ad auocorrelaed me seres Joural of Clmae () 7 pp [4] V Mosva ad A Zhgljavsy A algorhm based o sgular specrum aalyss for chagepo deeco Commucao Sascs Smulao ad Compuao vol 3 o 3 pp [5] M Mboup C Jo ad Ml Fless A ole chage-po deeco mehod Proceedgs of he Mederraea Coferece o Corol ad Auomao Cogress Cere Ajacco Frace: IEEE 8 pp 9-95 [6] L Aure ad C Aldrch Usupervsed process faul deeco wh radom foress Idusral ad Egeerg Chemsry Research vol 49(9) pp [7] X We H Hapg Y Yue ad Q Wag Aomaly deeco of ewor raffc based o he larges Lyapuov expoe IEEE pp [8] C Aldrch ad G T Jemwa Deecg chage complex process sysems wh phase space mehods The 7 Sprg Naoal Meeg 7 AIChE 7 pp - [9] L A Vce A echque for he defcao of homogeees Caada emperaure seres Joural of Clmae Chage 998 pp 94-4 [] M J Chadra Sascal Qualy Corol Ued Saes of Amerca: CRC Press LLC ch 8 [] T J Harrs ad W H Ross Sascal process corol procedures for correlaed observaos Caada Joural of Chemcal Egeerg vol pp [] C W Lu ad M R Reyolds EWMA corol chars for moorg he mea of auocorrelaed processes Joural of Qualy Techology vol pp [3] D S Chambers ad D J Wheeler Udersadg Sascal Process Corol d ed USA: SPC Press Ic 99 ch 3 [4] S W Robers Corol char ess Based o Geomerc Movg Averages Techomercs vol 959 pp 39-5 [5] C M Borror D C Mogomery ad G C Ruger Robusess of he EWMA Corol Char o No-ormaly Joural of Qualy Techology vol 3(3) 999 pp [6] D Mogomery Iroduco o Sascal Qualy Corol 6h edo New Yor: Joh Wley & Sos Ic 9 ch 9 [7] C W Lu ad M R Reyolds Corol chars for moorg he mea ad varace of auocorrelaed processes Joural of Qualy Techology vol 3 o pp [8] J Ba Esmao of a chage po mulple regresso models Rev Eco Sa vol pp [9] A S Jamal ad L JL False Alarm Raes for he Shewhar Corol Char wh Ierpreao Rules WSEAS Trasacos o Iformao Scece ad Applcaos vol 3 6 pp 7 [3] Ch Tg ad Ch Su Usg Coceps of Corol Chars o Esablsh he Idex of Evaluao for Tes Qualy WSEAS Trasacos o Commucaos vol 8 9 pp 6 [3] B Ka ad B Yazc The Idvduals Corol Chars for Burr Dsrbued ad Webull Dsrbued Daa WSEAS Trasacos o Mahemacs vol 5 6 pp 5 [3] Z M Nopah M N Bahar S Abdullah M I Kharr ad C K E Nzwa Abrup Chages Deeco Fague Daa Usg he Cumulave Sum Mehod WSEAS Trasacos o Mahemacs vol 7 8 pp [33] N K Hua ad H Md Robus Idvduals Corol Char for Chage Po Model WSEAS Trasacos o Mahemacs vol 9 pp 7 [34] M Csörgö ad L Horvah Lm Theorems Chage-po Aalyss New Yor: Wley 997 ch [35] J Lechy ad G Robers Marov cha Moe Carlo mehods for swchg dffuso models Bomera vol 88() pp [36] A De Gregoro ad S M Iacus (7) Leas squares volaly chage po esmao for parally observed dffuso processes Worg Paper Avalable: hp://arxvorg/abs/79967v [37] S M Iacus Smulao ad Iferece for Sochasc Dffereal Equaos: wh R Examples Sprger Seres Sascs New Yor: Sprger 8 ch 4 [38] S M Iacus ad N Yoshda (9) Esmao for he Chage Po of he Volaly a Sochasc Dffereal Equao Avalable: hp://arxvorg/abs/9638 [39] Y Kuoyas Idefcao of Dyamcal Sysems wh Small Nose Dordrech: Kluwer 994 ch 5 [4] Y Kuoyas Sascal Iferece for Ergodc Dffuso Processes Lodo: Sprger-Verlag 4 ch 4 [4] M Melou ad J Mly Kompedum sasceho zpracova da Praha: Academa Naladaelsv Aademe ved Cese republy 6 ch [4] YUIMA Projec Team () Yuma: The YUIMA Projec Pacage Avalable: hp://r-forger-projecorg/projecs/yuma [43] M Kral ad M Kovar Carry Trade s ed Zla: Georg ch 6 [44] D Nosevcova Vybrae meody sasce regulace procesu pro auoorelovaa daa Auoma vol o 8 pp 4-43 [45] R Huse Eoomerca aalyza Praha: Vysoa sola eoomca v Praze Naladaelsv Oecoomca 7 ch 6 M Kovar graduaed a he Faculy of Maageme ad Ecoomcs Tomas Baa Uversy Zl where he s lecurg a he Deparme of Sascs ad Quaave Mehods sce 9 He also graduaed a Faculy of Appled Iformacs Tomas Baa Uversy Zl he feld of Iformao Techology Auhor ad co-auhor of 7 boos ad 5 lecure oes hs research s focused o mahemacal ad sascal mehods qualy maageme ad compuaoally-esve sascal daa aalyses wh resuls publshed umerous peer-revewed jourals ad preseed a cofereces he Czech Republc as well as eraoally Mar Kováří s also a cosula of sascal daa aalyss applcao of sascal mehods qualy maageme ad quesoare-based surveys daa processg A parcpa several successful academc projecs specfcally Iovao of Follow-up Maser's degree programmes a he Faculy of Maageme ad Ecoomcs reg o CZ7//736 ad IGA's (Ieral Gra Agecy) Developme of mahemacal ad sascal mehods ulzao qualy maageme reg o IGA/73/FaME//D he also cooperaes wh orgazaos such as Barum Coeal s r o ad Tomas Baa Regoal Hospal a s Issue Volume 7 3 3