Course outline. Financial Time Series Analysis. Overview. Electricity demand forecasts. Forecasting models. Demand seasonality
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1 Financial Time Serie Analyi Parick McSharry Triniy Term 4 Mahemaical Iniue Univeriy of Oxford Coure ouline. Daa analyi, probabiliy, correlaion, viualiaion echnique. Time erie analyi, random walk, auoregreion, moving average 3. Technical analyi, rend following, mean reverion 4. Nonlinear ime erie analyi 5. Nonlinear modelling, regime wiching, neural nework 6. Parameer eimaion, model elecion, foreca evaluaion 7. Volailiy forecaing, GARCH, leverage effec 8. Rik analyi, value a rik, quanile regreion 9. Energy conumpion, demand forecaing. Enemble predicion, wind power generaion. Weaher derivaive, index-baed inurance. Quaniaive rading raegie, algorihmic rading Lecure 9 Copyrigh 4 Parick McSharry Overview Elecriciy demand Seaonaliy Weaher dependence Long-erm mulivariae foreca Deniy foreca Univariae hor-erm foreca Elecriciy demand foreca Demand i imporan for eimaing capaciy required o avoid black-ou Conrol of generaion and diribuion Abiliy o make informed deciion Reduce rik and minimie co Implemenaion depend on he foreca horizon: Shor-erm: enuring yem abiliy Medium-erm: mainenance cheduling Long-erm: capial planning Required for udying elecriciy price Deign of efficien elecriciy marke Forecaing model Demand eaonaliy Require differen model for differen foreca lead ime Are weaher foreca available and reliable? Univariae for hor-erm (< day Mulivariae wih weaher variable included for medium-erm Mulivariae wih imulaed weaher for longerm
2 Influence on demand Deerminiic eaonaliy: daily and weekly eaonaliy (low demand a weekend Annual eaonaliy (low demand during he ummer Calendar effec: Summer holiday, Chrima holiday Bank holiday Bridge day, elecion, rike, Weaher effec: Temperaure Wind peed Luminoiy Nonlinear dependence on emperaure Demand end o increae wih decreaing emperaure due o he ue of heaing equipmen Demand alo end o increae wih increaing emperaure due o he ue of cooling equipmen uch a air condiioner For he Neherland he opimal emperaure giving rie o he lowe demand wa 5 o C Cooling power Cooling power i defined o decribe he influence of draugh on demand uing a nonlinear funcion of emperaure and wind peed Cooling power, C, on day : W C = / (8.3 T where W i he wind peed and T i he emperaure on day. o if T < 8.3 C o if T 8.3 C The demand model Demand, D, on day : where τ i a ime of year variable giving he deerminiic eaonaliy, I i he luminoiy and he δ are he dummy variable. Model deail Demand growh: quadraic dependence on ime Fourh order polynomial ued o decribe he deerminiic annual eaonaliy Separae decripion (level and eaonaliy for Friday, Saurday and Sunday Nonlinear weaher dependence on emperaure, cooling power and luminoiy Srucural uncerainy Model inadequacy Model mi-pecificaion A priori rucural bia of he uer: I like NN I already have code for worked la ime are univeral approximaor
3 Foreca For each year beween 995 and The four previou year were ued o fi he model parameer Ou-of-ample predicion were generaed for ha year Thee year ahead foreca ar from he 3 of December of he previou year Evaluaion aiic Normalied Roo Mean Square Error (NRMSE ( Dˆ D NRMSE = σ where σ i he andard deviaion of he demand Mean Abolue Percenage Error (MAPE MAPE = Dˆ D D Fuure weaher Weaher foreca are only of value ou o wo week ahead Require weaher value for one year ahead Need o preerve linear auocorrelaion and cro-correlaion beween he weaher variable Alo require imilar hiogram of value Weaher imulaion Diribuion i mainained by huffling he original ime erie Auocorrelaion i preerved by fixing he ampliude of he power pecrum Calculae he Fa Fourier Tranform (FFT Fix he ampliude and cramble he phae (conaining he nonlinear informaion Tranform back o he ime domain Each replicae i known a a urrogae Temperaure urrogae Year ahead demand foreca 3
4 Probabiliic forecaing Accoun for uncerainy: Model uncerainy Weaher uncerainy Run Mone Carlo imulaion Differen weaher cenario Differen error realiaion Calculae he foreca PDF Foreca reul Foreca Training Teing Teing Weaher Acual Acual Simulaed Year NRMSE MAPE NRMSE MAPE NRMSE MAPE Average Peak demand PDF Foreca of peak iming Generally occur on a weekday: Monday hrough Thurday Uually around he middle of December during he pre-chrima ruh PDF foreca of peak magniude Non-normal aymmeric foreca PDF Shor-erm (< day foreca Compare and evaluae four univariae mehod muliple hypohee approach Ue inra-day daa (3 week for he ae of Rio in Brazil (hourly and for England and Wale (half-hourly. Rio: 3,36 obervaion for eimaion and,68 for evaluaion. England and Wale: 6,7 obervaion for eimaion and 3,36 for evaluaion. 4
5 England and Wale erie: wihin-day cycle (48 period and a wihin-week cycle (336 period. Demand (MW Half-hour Monday 7 March o Sunday Ocober Demand (MW Rio erie: wihin-day cycle (4 period, wihin-week cycle ( Hour Sunday 5 May 996 o Saurday 3 November 996 Demand (MW Two week of daa... Week Week Half-hour Rio Demand (MW Week Week Half-hour England & Wale Mehod ( Box and Jenkin SARIMA-good rack record; ( an exponenial moohing alernaive - robu, aracive for online demand forecaing (Taylor, 3; (3 an arificial neural nework implemenaion ha performed well on imilar daa (Darbellay and Slama, ; (4 a principal componen analyi approach - combine decompoiion and regreion, bu reduce # of poible regreion: focu on PC. Double Seaonal ARIMA (ARIMA ( p, d, q ( P, D, Q ( P,, D Q Sandard muliplicaive ARIMA: φ p d D ( L ΦP( L X = θq( L ΘQ( L exended o model wo eaonaliie in he demand ( = 4, 48 and = 68, 336 : φ p ( L Φ ( L Ω ( L P P The Double SARIMA model d = θ D q D X ε ( L ΘQ ( L ΨQ ( L ε Selecion Crieria: lowe SBC, reidual Rio: ARIMA(3,,3 (3,,3 4 (3,,3 68. England and Wale: ARIMA(,, (,, 48 (,, 336 5
6 Double Seaonal Exponenial Smoohing Similarly, exend Hol-Winer: Level: Trend: ˆ S Seaonaliy : y ( k = ( S + k T D + k W + k = T α ( X ( D S W + ( α( S + T = γ ( S S + ( γ T D = δ δ ( X ( S W + ( D ( Seaonaliy : W = ω X ( S D + ( ω W Double Seaonal Exponenial Smoohing Accuracy improved by a adjuing order auocorrelaion: AR( model, e = λe - + ξ, i fied o he -ep-ahead in-ample error (reidual, e. k-ep-ahead foreca from foreca origin τ are hen modified by adding he erm λ k e τ. S = α ( X ( D S W ( ( + α S + T T = γ ( S S + ( γ T D = δ ( X ( S W + ( δ D W = ω X ( S D + ( ω W Double Seaonal: Parameer Eimaed ( week Level α ( Trend γ Wihin-day eaonaliy δ Wihin-week eaonaliy ω Rio England and Wale AR λ Arificial Neural Nework Single hidden layer feed-forward nework: oupu i demand and inpu (x i are lag demand: f ( x, v, w g m g = v j k j= i= w ji xi g ( and g ( are acivaion funcion (igmoidal and linear w ji and v j are he weigh (parameer. Weigh are eimaed by: min v, w n ( y + + f ( x, v, w λ wji λ vi = j= i= j= n where n i he number of in-ample obervaion, and λ and λ are regulariaion parameer ha penalie complexiy Saionariy => eaonal and fir difference m k m Following he hold-ou mehod: England &Wale: λ =λ =.5 and m=4, and he following lag:, 48, 49, 96, 44, 9, 4, 88, 336, 384, 43, 48, 64 and 67. Rio: λ =λ =.5 and m=, and he following lag in our model:,, 4, 5, 48, 7, 96,, 44, 68, 9, 6, 4, 64, 3 and
7 PCA Baed Mehod PCA reduce he daa and hu capure he eence of he day-o-day imilariy Each PC i modelled for he weekly effec: regreion wih dummy variable for he weekday effec, linear and quadraic erm on # of week o eimae he componen and he # of componen need o be decided erially correlaed error => model e( MAPE: England & Wale -week poample.5%.%.5%.%.5%.% MAPE Lead ime (half-hour Arifical Neural Nework Double SARIMA PCA wih Error Model Double Seaonal Exponenial Smoohing wih AR( Adjumen MAPE: Rio, -week po-ample 5.% 4.% 3.%.%.%.% MAPE Lead ime (hour Arifical Neural Nework Double SARIMA PCA wih Error Model Double Seaonal Exponenial Smoohing (AR( Error Reference McSharry, PE, Bouwman, S and Bloemhof, G. (5 Probabiliic foreca of he iming and magniude of peak elecriciy demand. IEEE Tranacion on Power Syem, (: 66-7 Taylor, JT, de Meneze, LM and McSharry, PE (5 A comparion of mehod for forecaing elecriciy demand up o a day ahead. Inernaional Journal of Forecaing, (: -6 Taylor, JT and McSharry, PE (7 Shor-Term Load Forecaing Mehod: An Evaluaion Baed on European Daa. IEEE Tranacion on Power Syem, (4: 3-9 7
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