MPRA Munich Personal RePEc Archive Modeling hourly Elecriciy Spo Marke Prices as non saionary funcional imes series Dominik Liebl niversiy of Cologne Sepember 2010 Online a hp://mpra.ub.uni-muenchen.de/25017/ MPRA Paper No. 25017, posed 15. Sepember 2010 10:58 TC
Modeling hourly Elecriciy Spo Marke Prices as non saionary funcional imes series Dominik Liebl niversiy of Cologne Absrac The insananeous naure of elecriciy disinguishes is spo prices from spo prices for equiies and oher commodiies. p o now elecriciy canno be sored economically and herefore demand for elecriciy has an unempered eec on elecriciy prices. In paricular, hourly elecriciy spo prices show a vas range of dynamics which can change rapidly. In his paper we inroduce a robus version of funcional principal componen analysis for sparse daa. The funcional perspecive inerpres spo prices as funcions of demand for elecriciy and allows o esimae a single price curve for each day. Variaions in marke fundamenals such as commodiy prices are absorbed by he rs principal componens. Keywords: Funcional principal componen analysis, non saionary funcional ime series daa, sparse daa, elecriciy spo marke prices, European Elecriciy Exchange (EEX). 1. Inroducion Spo prices for elecriciy are peculiar. p o now elecriciy canno be sored economically herefore he amoun of elecriciy ha can be used for arbirage over ime can be negleced and demand for elecriciy has an unempered eec on elecriciy prices. The pricing in he power marke is based on marginal generaion coss of he las power plan ha is required o cover he demand. The supply curve is based on he increasing generaion coss of he insalled power Earlier versions of his paper have been presened a he DAGSa 2010 in Dormund and he Saisische Woche 2010 in Nürnberg. I wan o hank especially Prof. Alois Kneip for fruiful discussions, Prof. Pascal Sarda, who encouraged me consrucively on my way o wrie his paper during my say a he working group STAPH in Toulouse, and Prof. Wolfgang Härdle for consrucive discussion a he DATGSa 2010. Corresponding Auhor: Dominik Liebl (liebl@wiso.uni-koeln), Seminar für Wirschafs und Sozialsaisik, Lehrsuhl Prof. Mosler, niversiä zu Köln, Alberus MagnusPlaz, 50923 Köln Germany
plans, wih a seeply increasing, exponenial shape. sually base load plans such as nuclear and lignie plans cover he minimal load, i.e. demand for elecriciy, hroughou he year. Load following is mosly done by medium and peak load plans such as hard-coal and gas-red power plans. Pricing above marginal coss, so called peak load pricing, is ypical in markes wih non sorable producs. The reason behind his are opporuniy coss and incremenal coss from consrains, such as hose arising from emission and capaciy limis, which hen become marginal coss relevan (Cramon, 2004). This deviaion from variable cos pricing is also observable on he low demand side. Plan operaors ry o avoid shuing o power plans in order o avoid ramp up coss and herefore bid occasionally below variable coss, alhough laer becomes only visible if i is allowed o sell negaive prices are allowed o rade. In micro economic heory i is common o se prices equal o he marginal cos of producion and o deermine he equilibrium prices by he ineracion of demand and supply curves. Paricularly, elecriciy spo prices are usually modeled by he Courno model or he Supply Funcion Equilibrium model (Klemperer and Meyer, 1989). A recen example is he paper of Willems e al. (2009) an oher well known example is he paper of Green and Newbery (1992). Funcional daa analysis (fda) is able o share his perspecive, since is aomic saisical unis are funcions raher han poins and/or vecors. The books of Ramsay and Silverman (2005) and Ferray and Vieu (2006) give a broad overview o funcional daa analysis. As explained above, we have o disinguish beween pricing based on marginal generaion coss and pricing based on opporuniy coss. The funcional approach can model he marginal cos sysem well since marginal generaion coss are srongly conneced o he demand for elecriciy. The opporuniy coss sysem has o be modeled separaely and boh models may be conneced by a kind of regime swiching mechanism. Markov regime-swiching models, such as in Moun e al. (2006) and Kosaer and Mosler (2006), became one of he mos applied approaches in modeling and forecasing elecriciy prices. These ry o divide he series ino regimes wih own mean and covariance srucures. Bu he supply curve induces a coninuum of mean and covariance regimes and i is herefore dicul o assign prices o cerain regimes, even for he ofen used less volaile daily or half-daily averages of spo prices. One of he few high frequency analysis is done by Karakasani and Bunn (2008); Karakasani and Bunn model and forecas elecriciy spo prices from he K-Power Exchange. They divide heir daa ino sub samples for he 48 half hourly rading periods and addiionally separae weekdays from weekends. Wihin such sub samples, daily demand values for elecriciy have go a clear smooh sinoidal paern over he year wih higher/lower demand values during he winer/summer monhs. The demand paern is ranslaed ino a price paern by he ime invarian shape of he supply curve. This ranslaion causes addiional disorions and ha may be he reason why he auhors presen heir resuls for he rading periods 25 (12:30pm) and 35 (17:30pm) wih low varying demand paerns. Furhermore, elecriciy from volaile renewable 2
energy sources like wind and solar is ofen provided wih a purchase guaranee. If his is he case, hen no he smooh demand paerns are relevan for he price paerns bu he rougher adjused demand paerns wih need ou hourly elecriciy infeeds from renewable energy sources. Regarding he role of he supply curve as a diuser of he (adjused) demand paern, we propose o focus on he esimaion of is shape raher han on he esimaion of he price paern direcly. We propose o use (funcional) principal componen analysis o a low dimensional facor model o he daily supply curves. From he mehodological perspecive, he Dynamic Semiparameric Facor Model (DSFM) of Park e al. (2009) and he follow up applicaion o elecriciy spo prices Härdle and Trück (2010) are very close o our approach. Park e al. use an ieraing opimizaion algorihm o an orhogonal facor model o he daa and argue ha (funcional) principal componen analysis may no be able o handle sparse and non saionary (funcional) ime series daa. We exend he procedure of Yao e al. (2005), ha is able o handle sparse daa, o he conex of non saionary ime series daa. Härdle and Trück esimae a facor model o he daily N dimensional elecriciy spo price vecors Y = (Y 1,..., Y N ) and repor ha a hree dimensional facor model explains abou 80% of he variaion in hourly spo prices a he European Elecriciy Exchange (EEX). As he above cied papers, his applicaion of he DSFM model focuses on modeling and forecasing he price paern direcly and has o use a regime swiching mechanism in order o cope wih he vas price-diusions from he ineracions of demand paerns wih he supply curve. The esimaion of he supply curves has a limiaion ha is imposed by he aucion design. Even hough for each day here are N supply curves, one for each rading period h = 1,..., N, we can only esimae he mean supply curve of day. The general aucion design is similar for he greaes elecriciy markes places like in he Neherlands, Germany, Ausria, Scandinavian counries, France, and California. I is a wo-sided single-price aucion, which means ha here are bids from he purchase and sell side, ha are mached a a singe marke clearing price. The price for each rading period, h = 1,..., 24, of a day is seled by a separae aucion, and all period specic aucions of a day are conduced simulaneously he day before. The raders regiser he amouns of elecriciy hey are willing o sell/purchase for individually selecable price inervals in a rading ool, where each rading period is represened by a new inpu line. Noe ha hey base heir bid-decisions on he same informaion se for all period specic aucions. The price selemen mechanism deermines for each rading period, h, he spo price, Y h, by he poin of inersecion of he (over all bids) aggregaed supply and demand curves. More deails abou aucion designs a power exchanges can be found among ohers in he book of Rafal Weron (2006). The horizonally shifs of he N demand curves reveal he shape a daily mean supply curve by he price vecor, Y = (Y 1,..., Y N ), and he according (residual) demand values. Figure 1 shows he raw daa of 3
hree consecuive days of wo dieren (arbirary) weeks obviously, he supply curves possess depend on former supply curves and form a (funcional) ime series daa se. ER/MWh 0 20 40 60 80 100 120 140 40000 50000 60000 70000 Adjused Demand (MW) Figure 1: Three consecuive days from wo dieren arbirary weeks. Since we esimae he supply curves from he hourly spo prices, Y h, a he European Elecriciy Exchange (EEX), we will generally refer o he curves as price curves. The marke of he EEX has go a high share of producers of elecriciy from volaile renewable energy sources (mainly wind), who feed heir elecriciy direcly ino he grid and receive a cerain guaraneed price. Adjused demand, u, shall reec he price relevan residual amoun of elecriciy ha is demanded from he convenional marke paricipans. We assume ha he hourly spo prices for elecriciy a he EEX come from an underlying smooh process, such ha Y i = X (u i ) + ε i, (1) where X (.) is a smooh monoone funcion of adjused demand u, where is a closed and bounded subspace of R, we will se, wihou loss of generaliy, = [0, 1]. The index i = 1,..., 24 in u i refers o he i-h order saisic of he hourly adjused demand values dened as u h = d h p h, where d h is he gross demand for elecriciy and p h is he corresponding infeed of elecriciy from wind energy a day = {0, ±, 1, ±2,... } in hour h = 1,..., N, wih N = 24. The noise erm, ε i, is assumed o be independenly disribued wihin and beween each day, wih E(ε i ) = 0 and a heeroscedasic V ar(ε i ) = σ 2 ε(u i ). The wihin independence is realisic since he hourly prices of he day are deermined conemporaneously a 12 o'clock a day 1. The beween independence 4
and a model for he heeroscedasiciy follows from he error-in-variables discussion in he nex paragraph. As i can be seen in gure 1 he model from equaion (1) is suppored by he raw daa. There are some remarkable srong deviaions, especially for high and low values of adjused demand. This comes from an inheren inaccuracy of adjused demand values, u i. The spo marke of elecriciy is acually an oneday-ahead fuure marke, and he marke paricipans (i.e. he raders) base heir decisions on heir own hourly forecas values of adjused demand. Insead of hese price relevan bu unobservable forecas values we have o form he adjused demand values, u i, from he acual realized values of gross demand, d i, and acual producion of elecriciy from renewable energy sources, p i. Formally, we have o deal wih an error-in-variables problem and formalize he noisy covariaes as u i = w i + ν i, where w i are he unobservable price relevan adjused demand values, and he noise erm ν i is assumed o have E(ν i ) = 0 and V ar(ν i ) = σν 2 for all = {0, ±1, ±2,... } and i = {1,..., 24}. This inaccuracy causes sronger disorions a low and high values of adjused demand where he price curves have go higher slopes han for moderae values of adjused demand. Acually, his is a degeneraed case of an error-in-variable problem, since he dependen variable, Y i, is observed nearly wihou noise. We assume ha he noise in he observaions of Y i is negligible and ha we can ranslae he error-in-variables problem ino an usual esimaion problem wih heeroscedasic error erms. A more sophisicaed esimaion procedure is beyond he scope of our paper, bu migh be a opic for fuure sudies. Our aim is o esimae he daily price funcions X from he discree daa vecor Y = (Y 1,..., Y N ). We use N o refer o he amoun of prices per day ha are used o esimae he funcion X since some prices are assigned o he opporuniy regime. We use a parsimonious ex-pos assignmen, price above a cerain hreshold are classied o he opporuniy regime, all ohers are classied o he marginal cos regime. A reasonable hreshold seems o be 145 ER, since for prices > 145 ER he raders lose heir coninuous marginal cos reference and begin o bid in clusers (150 ER, 200 ER, 250 ER,... ) of prices (see lef panel of gure 2). The righ panel of gure 2 shows all prices classied o he marginal cos regime. Generically, we assume he daily price curves o come from a sochasic process (X ) for = {0, 1, 2,... } wih realizaions in he space of square inegrable funcions H = L 2 () on a compac se R. As in mulivariae saisics, saionary funcional sochasic process are usually described by heir ime invarian mean funcion and covariance operaor. However he series of price curves has go a clear sochasic, non saionary rend. The curves inheri his propery from he non saionary prices for raw maerials (such as gas, coal, and Co2-cericaes) ha are needed o produce elecriciy. The (funcional) random walk model, X = δ + X 1 + e, wih = {0, 1, 2,... } (2) 5
0 50 100 150 200 250 300 0.0000 0.0005 0.0010 0.0015 Spo Prices 0.0 0.2 0.4 0.6 0.8 1.0 0 50 100 150 Adjused demand ER/MWh Figure 2: Lef Panel: Non parameric densiy esimaion of elecriciy spo prices (Noe: Only prices 325 ER are ploed, bu here are prices up o 2437 ER). Righ Panel: Pooled spo prices classied o he marginal regime, wih wo highlighed price vecors from wo arbirary days. wih a linear rend, where δ H, an iniial value from a random funcion, Z 0, ha is normally disribued wih mean, µ Z, and covariance operaor, Γ Z, and a whie noise process, (e ) H, wih mean zero and covariance operaor, Γ e, seems mos appropriae. Noe ha, as in he univariae case he EX = µ Z for all, bu he covariance operaor depends on, such ha he process dened in (2) is non saionary. This is a special case of he so called ARH(1) model, i.e. an auo regressive model in H. Noe ha any ARH(p), wih p > 1, model can be ransformed ino an ARH(1) model, such ha he above model is no necessarily a resricion wih respec o he lag order (see Bosq (2000)). Furhermore, as usual pracice in mulivariae ime series analysis we do no apply funcional moving average erms in order o avoid idenicaion problems wih he AR erms. The mean funcion of he funcional random walk is independen of and we can invesigae, wihou loss of generaliy, he properies of he demeaned process (X ) = (X µ Z ). This yields he same funcional random walk process as in equaion (2) bu wih an iniial funcional random variable Z 0 ha has he zero funcion as i's mean; we wrie X = δ + X 1 + e. As already noed in Park e al. (2009), we need saionary of he ime series in order o use he well developed funcional principal componen analysis. Tradiional ransformaion procedures such as diereniaion of he ime series, (X ), canno be used, because he prices are observed a non equidisan adjused demand values, u i. We propose a new ransformaion procedure ha decomposes he original series, (X ), ino is saionary funcional componen, ( X ), and is non saionary univariae random walk componen, (Θ ). The decomposiion is moivaed by he uni sphere projecion of funcional daa (see Locanore e al. 6
(1999) and Gervini (2008)). In order o use a noaion, ha corresponds o he marix noaion of mulivariae saisics, we inroduce he ensor produc noaion dened as X X (u, v) = X (u) X (v), for (u, v) R 2 and = {0, 1, 2,... }. Then, he covariance funcion [ of a saionary funcional series, ( X ), can be wrien as ρ(u, v) = E ( X µ ) ( X ] µ ) (u, v). The covariance operaor is dened as Γ X f(u) = ρ(u, v)f(v)dv, for any funcion f H. Is specral decomposiion by is eigenvalues, λ1 > λ 2 >..., and corresponding eigenfuncions, φ 1, φ 2,..., wih he usual resricions φ2 k = 1 and φ kφ m = 0 for m < k, allows us o wrie he funcionals X by he well known Karhunen-Loève decomposiion as X (u) = µ Z (u) + β k φ k (u). Where µ Z = E( X ) and β k = ( X µ Z )φ k are he principal componen scores wih E(β k ) = 0 and E(βk 2 ) = λ k. No leas because of he bes basis propery (in he mean square error sense) of he Karhunen-Loève decomposiion, ofen a relaively small number of eigenfuncions provides a good o he sample curves. One problemaic fac regarding our daa is ha he adjused demand values u j are no uniformly disribued over he whole domain, bu may be clusered a sub-inervals wihin. This makes i dicul o approximae he inegrals in β k = ( X µ Z )φ k by radiional mehods like he rapezoidal rule where β k N j=1 Y jφ k (u j )(u j u,j 1 ), wih u 0 = 0. This is he same problem as in he so called sparse daa problem in funcional daa analysis where i is dicul o approximae inegrals from insucienly many daa poins (see e.g. Yao e al. (2005)). Yao e al. sugges o esimae he principal componen scoresβ k by heir condiional expecaion given he sparse daa {u 1,..., u N }. This procedure works very well for our non uniform disribued u j. In he nex secion we inroduce a new decomposiion of a non saionary funcional ime series ino is saionary spherical componen and ino is non saionary scaling componen. In he secion 3 we discuss how o esimae he mean funcion, µ Z, he covariance funcion, ρ(u, v), which is he kernel of he covariance operaor, Γ X f(u) = ρ(u, v)f(v)dv, and he sandard deviaion, σ. In subsecion 3.1 we esimae he eigenfuncions of he covariance operaor and discus heir inerpreaions. The subsecion 3.2 adaps he condiional esimaion of principal componen scores from Yao e al. (2005) in order o esimae he pc-scores based on non uniformly (on ) disribued daa, u i. Finally, we demonsrae he goodness of of our esimaion procedure in subsecion 3.3. 7
2. Principal componen analysis for non saionary daa From a pracical poin of view i would be a grea advanage o projec he innie dimensional ARH(1) process ino a nie dimensional funcional space, P, spanned by K basis funcions, φ 1,..., φ K, such ha he mean squared error of he projecion, T N { } 2 X (u j ) X,K (u j ) wih X,K = =1 j=1 K β k φ k and β k = X φ k, (3) is minimized, where (X ) = (X µ Z ). If he series (X ) corresponds (a leas wih high accuracy) o a K dimensional funcional ime series, (X,K ), we could ransform he innie dimensional process, (X ), ino K univariae ime series, (β 1 ), (β 2 )..., (β K ), ha are orhogonal o each oher. The well known Karhunen-Loève heorem suggess o use he K eigenfuncions ha correspond o he K highes eigenvalues of he covariance operaor of he process (X ) as basis funcions. Esimaion of he eigenfuncions works perfecly for iid or saionary daa, bu in he case of non saionary processes we face he problem ha each elemen of (X ) has go a dieren covariance operaor. Neverheless, we can show ha each covariance operaor, Γ X of X for all = {1, 2,... }, is an elemen of he same space (see heorem 2.1). Theorem 2.1. Wihou loss of generaliy, given a de-meand version (X ) = (X µ Z ) of he random walk process in equaion (2). a) The covariance operaors, Γ X = EX X for = {1, 2,... }, are elemens of he same space. As a consequence, he eigenfuncions of he covariance operaors are he same for all = {1, 2,... }. b) The covariance operaors, Γ X = EX X for = {1, 2,... }, are asympoically idenical, apar from scale dierences. This characerisics moivaed us o ransformaion he process by he unisphere projecion, ha is usually used in mulivariae robus saisics of iid samples (see e.g. Huber and Ronchei (2009), Locanore e al. (1999) and Gervini (2008)). We propose o decompose he series, (X ), ino a funcional componen, ( X ) H, and an univariae componen, (Θ ) R, such ha (X ) = ( X Θ ). We call he s componen spherical componen and he laer scaling componen. Deniion The spherical componen of a funcional random walk as in equaion (2) is given by πx = πδ + πx 1 + πe. Where π = (.)/. 2, is he uni-sphere projecion operaor, wih. 2 = (.)2, is he L 2 norm in H. Deniion The scaling componen of a funcional random walk as in equaion (2) is given by X 2 = δ 2 + X 1 2 + e 2. Wih. 2 = (.)2, he L 2 norm in H. 8
I can be shown ha he spherical componen, compacly wrien as X = δ + X 1 + ẽ, is saionary and ha he covariance operaors of each elemen in ( X ) has go he same eigenfuncions as is non saionary counerpar in (X ) (see heorem 2.2). Noe ha he scaling componen, compacly wrien as Θ = α + Θ 1 + ɛ, is a sandard univariae random walk wih drif α R and whie noise process (ɛ ) N(0, σ ɛ ). Theorem 2.2. Wihou loss of generaliy, given a de-meand version (X ) = (X µ Z ) of he random walk process in equaion (2). a) Is spherical componen, ( X ) = (πx ), is a saionary process. b) The covariance operaors, Γ X = E X X for = {1, 2,... }, are elemens of he same space as he non spherical counerpars, Γ X = EX X. As a consequence, he eigenfuncions of he covariance operaors, Γ X, are he same as of he covariance operaors, Γ X, for all = {1, 2,... }. As a consequence of heorem 2.2, asympoically, he covariance operaors, Γ X, of he non saionary original process, (X ), are he same as he covariance operaors, Γ X, of he spherical process, ( X ) excep for scale dierences. Therefore, we can esimae he original covariance operaors from he saionary spherical series, ( X ). And rescale he esimaed covariance operaor by he scaling componen, (Θ ), ha has absorbed he scale dierences. The K eigenfuncions ha belong o he rs K eigenvalues, λ 1,..., λ K, of he spherical covariance operaor will fulll he opimaliy crierion in (3). 3. Esimaion of he mean, covariance funcion, and sandard deviaion We esimae he mean funcion by local linear polynomial smoohing as proposed by Yao e al. (2005). The measuremen errors are balanced when all prices are pooled and herefore he esimaion of he mean funcion ˆµ(u) = Sm[u, (u i, Y i ), T, N, h µ ] says saisfacory, where S[v, (u i, Y i ), T, N, h µ ] denoes he resul of he local polynomial smoohing procedure of he pooled daa (u i, Y (u i )), for i = 1,..., N, and = 1,..., T, evaluaed a v R wih smoohing parameer h µ. All smoohing parameers are deermined such ha hey are he minimizing he generalized cross validaion crierion (Silverman (1984)). Explici formulas of he esimaors of he mean funcion and he covariance funcion are given in he Appendix A. In gure 3 he esimaed mean funcion, ˆµ, is ploed along wih all pooled daa poins, (u i, Y i ) for = 1,..., T and i = 1,..., N, ha are classied o he marginal cos regime (Y i 145 ER). Furhermore, all prices wih 9
adjused demand values smaller han 34, 000 (MW) are omied because hey are no dense enough o guaranee subsequen sable local polynomial smoohing. ER/MWh 0.0 0.2 0.4 0.6 0.8 1.0 0 50 100 150 Figure 3: Esimaed mean funcion, ˆµ, and all pooled prices minus ouliers. The esimaion of he covariance funcion uses he above explained spherical componen, ( X ), of he original series, (X ). The spherical esimaor of he covariance funcion is given in equaion 4, where Sm [ u, v, (, G ), T, h ρ, linear ] denoes he resul of he local linear polynomial surface smoohing procedure of he pooled daa (, G ), = 1,..., T, evaluaed a (u, v) R 2 wih smoohing parameer h ρ and. E denoes he sandard euclidean norm, ρ n (u, v) = Sm [ u, v, (, G ), T, h ρ, linear ], (4) wih: G = [ (Y (u i ) ˆµ(u i )) (Y (u j ) ˆµ(u j )) Y (u i ) ˆµ(u i ) E Y (u j ) ˆµ(u j ) E ] i,j=1,...,n for all i j. (5) We use he subscrip n o denoe he esimaor of ρ in order o avoid messy superscrips. One should exercise cauion in esimaion of he covariance funcion ρ. As equaion 1 indicaes, we have o ake he noise erm ino accoun oherwise he esimaor of he diagonal, ρ(v = u, u) = ρ(u), would be biased. A sraigh forward soluion, originally proposed by Saniswalis and Lee (1998), is 10
o leave ou he diagonal elemens, as done in equaion 5. The variance of he curves X (u) is usually reeced as a prominen feaure along he diagonal of he covariance funcion ρ and may be under esimaed by he above explained esimaion procedure. Following Yao e al. (2003) we roae he coordinaes, (u i, u j ), of each elemen of G, clockwise by 45, ( u i u j ) = ( 2 2 2 2 2 2 2 2 ) ( ) ui, u j and esimae he surface again wih a local quadraic polynomial smooher, [ ] ρ n (u, v ) = Sm u, v, (, Ġ), T, h ρ, quadraic (6) [ (Y (u i wih: Ġ = ) ˆµ(u i )) ( Y (u j ) ˆµ(u j )) ] Y (u i ) ˆµ(u i ) E Y (u j ) ˆµ(u j ). (7) E i,j=1,...,n for all i j The quadraic orhogonal o he diagonal of he covariance funcion approximaes he variance of he funcions beer. The diagonal of he esimaed covariance funcion, ρ n (u, v = u) = ρ n (u) is se equal o ρ n (0, u/ 2) for all u wih ρ n (u) < ρ n (0, u/ 2); we denoe his adjused esimaion of he covariance funcion classically by ˆρ. In he lef panel of gure 4 he esimaed covariance funcion, ˆρ, is shown; he sharpened diagonal is hardly visible bu exisen. The righ panel of gure 4 gives a comparison of he hree esimaed diagonals of he covariance funcion, ρ n (u), ρ n (u), and D n (u), where he laer uses only he noisy diagonal elemens (see equaion (8)). [ D n (u) = Sm u, (, G ] D ), T, h D, linear (8) [ ] (Y (u i ) ˆµ(u i )) 2 wih: GD = Y (u i ) ˆµ(u i ) 2 E i=1,...,n. The dashed line refers o he esimaor ρ n (u), from equaion (4), he doed line refers o he esimaor ρ n (u), from equaion (6) wih roaed coordinaes and quadraic smoohing, and he solid line is refers o he esimaor, Dn (u), from smoohing only he diagonal componens. I is clearly visible ha he inclusion of he diagonal elemens leads o a srong disorion of he covariance diagonal, because of he noise erm. The dierence beween he diagonal esimaion wih diagonal componens, D, and wihou diagonal componens bu roaed coordinaes, ρ, can be used o esimae he variance of he noise erm, σ. 3.1. Specral decomposiion of he covariance operaor Given an esimaion of he spherical covariance funcion, ˆρ, we can derive he se of eigenvalues, {ˆλ},...K, and he se of eigenfuncions, { ˆφ k },...K, by he 11
0.05 0.04 0.03 0.02 0.01 1.0 0.8 0.6 l 0.4 0.2 0.0 0.0 0.2 0.4 r 0.6 0.8 1.0 0.02 0.04 0.06 0.08 0.10 Diagonal Terms only Diagonal Terms excluded wih Roaion Diagonal Terms excluded wihou Roaion 0.0 0.2 0.4 0.6 0.8 1.0 Figure 4: Lef Panel: Esimae covariance funcion, ˆρ. Righ Panel: The resuls of he hree versions of diagonal esimaion of he covariance funcion. specral decomposiion of he covariance operaor, ˆΓf(u) = ˆρ(u, v)f(v)dv, wih f(u) L 2 (), from he soluions of he eigenequaions ˆρ(u, v) ˆφ k (u)du = ˆλ k ˆφk (v), wih he usual resricions ˆφ 2 k = 1 and ˆφ k ˆφl = 0 for all k < l. The sandard procedure is o discreize he covariance funcion, ˆρ, a an equidisan grid (u d 1,..., u d n) (u d 1,..., u d n) and hen o use rouines from he mulivariae specral decomposiion of marices (see e.g. Ramsay and Silverman (2005) for a deailed explanaion). From heorems 2.1 and 2.2 above, we know ha he eigenfuncions of he spherical covariance operaors, Γ X, are he same as he eigenfuncions of all original covariance operaors, Γ X, such ha hese fulll he bes basis propery of equaion (3). Noe ha, hey are no unique in his characerisic. There may be an oher se of eigenfuncions ha is as well ecien in he mean squared error sense bu is beer o inerpre. Again, mehods from he mulivariae saisics can be used here o produce new eigenfuncions. Given a discreized se of esimaed eigenfuncions, [φ d 1,..., φ d K ] wih ˆφ d k = ( ˆφ k (u d 1),..., ˆφ k (u d n)) R n, every K K roaion marix R, wih R R = RR = I, leads o a new orhonormal se of basis vecors [ψ d 1,..., ψ d K ] = R[φd 1,..., φ d K ]. Ofen, eigenfuncions are only inerpreable afer a suiable roaion scheme. The well known VARIMAX roaion ries o maximize he variance of each discreized eigenfuncion ˆφ d = ( ˆφ(u d 1),..., ˆφ(u d n)) by eiher scaling he values ˆφ(u d i ) agains zero or agains very high absolue values. Figure 5 shows he four roaed eigenfuncions ha belong o he four highes eigenvalues. The 12
dieren ypes of power plans become apparen surprisingly well. The greaes par (70.88%) of he oal variaions of he spo prices, which belong o he marginal cos regime, is beween 50, 200 MW and 62, 000 MW. This region is generally supplied by coal power plans, which face he mos price volaile resource commodiies (hard coal, brow coal, and CO2 cericaes). The second greaes par (22.16%), beween 62, 000 MW and 73, 720 MW, generally can be assigned o gas and oil power plans ha are ofen used in hours wih peak demands for elecriciy. The hird par (5.49%) of he oal variance, beween 34, 520 MW and 50, 200 MW, can be assigned o he base power plans, mosly nuclear power plans. VARIMX Eigenfun. 1 70.88 % VARIMX Eigenfun. 2 22.16 % ER/MWh 0 50 100 150 ER/MWh 0 50 100 150 0.0 0.2 0.4 0.6 0.8 1.0 Adjused Demand (MW) 0.0 0.2 0.4 0.6 0.8 1.0 Adjused Demand (MW) VARIMX Eigenfun. 3 5.49 % VARIMX Eigenfun. 4 0.3 % ER/MWh 0 50 100 150 ER/MWh 0 50 100 150 0.0 0.2 0.4 0.6 0.8 1.0 Adjused Demand (MW) 0.0 0.2 0.4 0.6 0.8 1.0 Adjused Demand (MW) Figure 5: VARIMAX roaed eigenfuncions. 3.2. Condiional esimaion of he principal componen scores From he heorem 2.2 we know ha he spherical covariance operaor, Γ X, is asympoically he same as he covariance operaor, Γ X, excep for scale dierences. This allows us o model, rs, he spherical sample curves, ( X ), as a K dimensional process (see equaion (3)), X,K = K ξ k φ k, and hen o rescale hem o heir original size, (X ), by heir scaling componen, (θ ). For simpliciy, we do no disinguish noaionally beween sample and generic versions of (X ). The usual esimaion of he pc-scores approximaes he inegral, ξ k = φ k X. Given he non uniformly disribued daa, 13
u 1,..., u N, over he domain,, we canno adequaely approximae he pcscores by numerical inegraion procedures. The PACE approach of Yao e al. (2005) uses he condiional expecaion, E(β k X ), given a join normal disribuion of he random vecor (β k, X ). This procedure can be applied o our problem, when we use he assumpion ha he spherical scores, ξk, and he discree spherical curve values, Ỹ = Y µ Y µ E wih Y = (Y,1,..., Y N ) and µ = (µ(u 1 ),..., µ(u N )), come from a join Gaussian disribuion of ( β k, X ). We esimae he condiional principal componen scores, ξ k c, given he non uniformly disribued discree observed curve daa, Ỹ, by ξ c k = E[ β k Ỹ ] = λ k φ k Σ 1 Ỹ (Ỹ ), (9) where Σ Y = [ ρ(u i, u j )] i,j=1,...,n + σ 2 I N is a N N symmeric marix and φ k = (φ(u 1 ),..., φ(u N )) a N dimensional vecor. 3.3. Fied Curves Dieren from radiional mehods our esimaion procedure does no focus on esimaion of he hourly spo prices direcly, bu on he esimaion of daily mean price curves (or supply curves, respecively). The lef panel of gure 6 shows he esimaed price curve of Thursday he 9 h February in 2006. The circle poins are assigned o he marginal cos regime and conribue o he esimaion procedure. The wo prices corresponding o he wo riangle poins are assigned o he opporuniy regime and do no conribue o he esimaion procedure. The righ panel of gure 6 shows he whole week from Monday he 6 h o Sunday he 12 h February in 2006. Here, he prices are ploed in he radiional in correspondence o heir rading periods. We wan o emphasize ha our separaion of he daa ino a marginal cos regime and an opporuniy cos regime is more fundamenal han sandard regime swich models ha usually swich beween wo or more (ofen comparable) ime series models. (See Jong (2006) for an overview of classical regime swich models in he conex of elecriciy spo marke daa.) Here, we base he regime swich on a change in he bidding behavior of he raders. Ex pos his is easily done wih a hard hreshold price; see discussion o gure 2. Ex ane his is no a rivial hing o do, since every rader may be forced o swich ino he opporuniy cos regime on basis of privae informaion such as delivery obligaions and unexpeced changes in power plan capaciies. On basis of public available daa i will be hardly possible o predic individual regime swiches, bu i migh be possible o predic siuaions in which (nearly) all marke paricipans will have o swich ino he opporuniy regime. This will be par of fuure research. 14
ER/MWh 50 100 150 200 250 300 0.0 0.2 0.4 0.6 0.8 1.0 MW Figure 6: Esimaed price curve, X (u), of Th. 9/Feb./06; wih superimposed 24 prices, Y = {Y 1,..., Y 24 }, of ha day. The wo prices corresponding o he wo riangle poins are assigned o he opporuniy regime. 4. Conclusion In his paper we suppor a new angle of vision in modeling hourly elecriciy spo marke daa. We argue ha he inra-day seasonaliy canno be esimaed by radiional ime series models, ha are based on he assumpion of a hourly updaing informaion se (as already done by Huisman e al. (2007)). This assumpion is ofen no valid because mos elecriciy exchanges use a singe price aucion where he hourly price vecors, {Y h } h=1,...,24 for day, are deermined simulaneously he day before a 1. We use a funcional ime series model and esimae daily mean supply funcions by funcional principal componen analysis. Here, he inra-daily raw daa is no he radiional consecuive price vecor, {Y h } h=1,...,24, bu he re-ordered price vecor, {Y i } i=1,...,24, corresponding o he covariae vecor of adjused demand values, u 1 < < u i < < u 24. This inroduces wo problems, rs, he daa looses is equidisan design, acually he adjused demand values, u i, even are no uniformly disribued wihin he domain R. Second, (funcional) principal componen analysis needs iid daa or a leas saionary daa, bu our daa se is non saionary. The rs problem is solved by an adapion of he principal componen analysis for spaces daa (see Saniswalis and Lee (1998) and Yao e al. (2005)). The second problem is solved by he inroducion of a new decomposiion of he funcional imes series ino a saionary spherical componen and a non saionary scaling componen. The laer is one of our main conribuions ha migh be very useful for many oher funcional imes series esimaion problems. Furhermore, 15
Su. 120 100 80 60 40 Th. Fr. Sa. ER/MWh 120 100 80 60 40 Orig. Prices Fied Prices Mo. Tu. We. 120 100 80 60 40 0 5 10 15 20 25 0 5 10 15 20 25 Hours Figure 7: Original and ed prices of he week: Mo. 6/Feb./06 Su. 12/Feb./06. Here, he prices are re-ordered ino he radiional hourly perspecive. our approach handles he informaion se correcly as a daily updaing informaion se. The resul ha we need only hree componens in order o explain 98.53% of he oal variaion suppors he imporance o accoun for he correc consideraion of he daa generaing process. Oher sudies ha work wih similar approaches bu use he radiional hourly price vecors, {Y h } h=1,...,24, need higher numbers of principal componens for comparable fracions of explained variance (see e.g. Wolak (1997) and Härdle and Trück (2010)). 16
Appendix A. Explici Formulas Mean funcion µ(u): ˆµ(u) = Sm[u, (u i, Y i ), T, N, h µ ] ˆµ(u) = ˆβ 0 (u) Wih ˆβ T N 0 (u) from: min K 1 ( u u i )[Y i β 0 β 1 (u u i )] 2 β 0,β 1 h µ i Wih ˆβ 0 (u, v) from: [ ρ n (u, v) = Sm u, v, (, G ] ), T, h ρ, linear [ (Y (u i ) ˆµ(u i )) (Y (u j ) ˆµ(u j )) wih: G = Y (u i ) ˆµ(u i ) E Y (u j ) ˆµ(u j ) E ρ(u, v) = ˆβ 0 (u, v) T min β 0,β 11,β 12 K 2 ( u u i h G 1 i j N, v u j h G ) ] i,j=1,...,n for all i j [G (u i, u j ) β 0 β 11 (u u i ) β 12 (v u j )] 2 { (1 w Where K 1 (w) = ) 3 w < 1 0 oherwise Where: w = u ui h µ K 2 (w, x) = K 1 (w)k 1 (x) Where: w = u ui h G and x = v ui h G (Or any oher valid univar. kernelfuncion.) The bandwidhs are deermined by Generalized Cross Validaion (CGV). These rouines are already implemened in he R package locfi (Loader, 2010). Appendix B. Proofs Proof of heorem 2.1. Par a): From he deniion of he covariance operaor and he random walk. Γ X (u, v) = E(X X (u, v)) = E(X (u) X (v)) 1 1 Γ X (u, v) = E(( e i (u) + Z0 (u))( e i (v) + Z0 (v))). i=0 Wih Γ e (u, v) = K λe k φe k φe k (u, v) we can wrie e (u) = βe k φe k (u) for all = {1, 2,... }, where βk e N(0, λe k ). And similar for Z 0, wih Γ Z (u, v) = i=0 17
K λz k φz k φz k (u, v), we can wrie Z 0(u) = βz k φz k (u), where βz k N(0, λ Z k ). This yields, Γ X (u, v) = Γ e (u, v) + Γ Z (u, v), (B.1) which corresponds o he usual univariae and mulivariae random walk characeric of an wih O() increasing covariance. Given he specral decomposiions of he covariance operaors, we have, Γ X (u, v) = λ e kφ e k φ e k(u, v) + λ Z k φ Z k φ Z k (u, v). Noe ha Γ X (u, v) is an elemen of an addiion of wo vecor spaces, P P + Q Q = {p p = α p,k φ e k φ e k, (α p,k ) 2 <, α p,k R k} + {q q = α q,k φ e k φ e k, (α q,k ) 2 <, α q,k R k}. Wihou loss of generaliy we can invesigae he degeneraed case where λ Z k = 0 for all k. Then we have Γ X (u, v) = λ e kφ e k φ e k(u, v) P P. Wih he propery ha vecor spaces are closed wih respec o scalar muliplicaion we can direcly show ha P P = P +i P +i, P P = span{ φ e 1 φ e 1, φ e 2 φ e 2,... } = {p p = α k φ e k φ e k and α k R k} = {p p = γ k ( + i) φ e k φ e k and γ k R k} where γ = (α k )/( + i) = span{( + i) φ e 1 φ e 1, ( + i) φ e 2 φ e 2,... } = P +i P +i = P P, for arbirary, i = {1, 2,... }. Therefore each covariance operaor, Γ X (u, v), is an elemen of he same space K K P P = {p p = α p,k φ e k φ e k, (α p,k ) 2 <, α p,k R k}. This shows par a) of heorem 2.1. 18
Par b): From equaion (B.1) we have, ( ) lim, s ΓX (/s) = cons. = Γ X +s lim (, s (/s) = cons. ( Γ e + (1/) Γ Z ) ( + s) ( Γ e + (1/( + s)) Γ Z ) ) = 1 (1 + cons.). This shows par b) of heorem 2.1. Proof of heorem 2.2. Par a): Wihou loss of generaliy, we invesigae he de-meand process, ( X ), given by π(x µ Z ) = δ + π(x 1 µ Z ) + πe = 1 πe + π(z 0 µ Z ) i=0 X = δ + X 1 + ẽ wih X = (X µ Z ) and (.) = π(.) = (.) (.) 2 1 = δ + ẽ + Z 0 wih Z0 = (Z 0 µ Z ) and Z 0 = πz0 i=0 We proof ha he spherical componen, ( X ), is a (weak) saionary process. I.e. (i) has go consan mean funcion for all = {0, 1, 2,... }, (ii) nie covariance operaor, and (iii) auocovariance operaors ha are independen of (see any inroducory ime series book, such as Shumway and Soer (2006)). Given he funcional random walk process dened by equaion (2). Wihou loss of generaliy, we se δ equal o he null funcion such ha, Condiion (i): X = X 1 + ẽ 19