Foreca Repone Variable When he value in a repone column are ordered equeniall over ime, i i ofen of inere o foreca heir behavior beond he end of he daa. Thi procedure fi a parameric ARIMA ime erie model o a ample of daa and foreca i fuure behavior. The daa for hi anali coni of n equeniall ordered obervaion. Le x = obervaion a ime, =,,, n n = number of obervaion Acce Highligh: one Repone column. A Time index column ma alo be eleced o poiion he poin along he X axi. Selec: Foreca from he main menu. Oupu Page : Foreca from he fied model. Oupu Page : Auocorrelaion funcion (ACF) for he reidual from he fied model. Oupu Page 3: Plo of he reidual. Opion An ARIMA ime erie model i ued o foreca he daa. The general form of hi model i mo eail expreed in erm of he backward operaor B, which operae on he ime index of a daa value uch ha B j = -j. Uing hi operaor, he model ake he form p P d D ( B B... B )( B B... B )( B) ( B ) Z q Q ( B B... B )( B B B ) a =... () where Z μ () = and a i a random error or hock o he em a ime, uuall aumed o be random obervaion from a normal diribuion wih mean 0 and andard deviaion σ a. For a aionar erie, μ repreen he proce mean. Oherwie, i i relaed o he lope of he foreca funcion. μ i omeime aumed o equal 0. 006 b SaPoin, Inc. Foreca Repone Variable -
The above model i ofen referred o a an ARIMA(p,d,q)x(P,D,Q) model. I coni of everal erm:. A noneaonal auoregreive erm of order p.. Noneaonal differencing of order d. 3. A noneaonal moving average erm of order q. 4. A eaonal auoregreive erm of order P 5. Seaonal differencing of order D. 6. A eaonal moving average erm of order Q. While he general model look formidable, he mo commonl ued model are relaivel imple pecial cae. Thee include: AR() auoregreive of order The obervaion a ime i expreed a a mean plu a muliple of he deviaion from he mean a he previou ime period plu a random hock: ( ) a = μ + φ μ + (3) AR() auoregreive of order The obervaion a ime i expreed a a mean plu muliple of he deviaion from he mean a he previou ime period plu a random hock: ( μ ) + φ ( μ ) a = μ + φ + (4) MA() moving average of order The obervaion a ime i expreed a a mean plu a random hock a he curren ime period plu a muliple of he random hock a he previou ime period: = + a θa μ (5) MA() moving average of order The obervaion a ime i expreed a a mean plu a random hock a he curren ime period plu muliple of he random hock a he previou ime period: = + a θa θ a μ (6) ARMA(,) mixed model wih fir order erm The obervaion a ime i expreed a a mean plu a muliple of he deviaion from he mean a he previou ime period plu a random hock a he curren ime period plu a muliple of he random hock a he previou ime period: ( μ ) + a θa = + φ μ (7) 006 b SaPoin, Inc. Foreca Repone Variable -
ARIMA(0,,) moving average of order applied o he fir difference The difference beween he curren period and he previou period i expreed a a random hock a he curren ime period plu a muliple of he random hock a he previou ime period: = a θ a (8) I can be hown ha hi model i equivalen o a Simple Exponenial Smoohing model. ARIMA(0,,) wih conan moving average of order applied o he fir difference wih a conan Thi model add a conan o he model defined above: = μ + a θ a (9) The addiion of a conan caue he foreca o follow a linear rend. ARIMA(0,,) moving average of order applied o he econd difference The difference of he fir difference i expreed a a random hock a he curren ime period plu muliple of he random hock a he previou ime period: ( ) ( ) = a a θ a θ (0) Thi model i equivalen o Hol Linear Exponenial Smoohing model. ARIMA(0,,)x(0,,) eaonal and noneaonal MA erm of order The obervaion a ime i expreed a a combinaion of he obervaion one eaon ago plu he difference beween he obervaion la period and i counerpar one eaon ago plu muliple of he hock o hi he em hi period, la period, and wo period one eaon ago: = + + a a Θa + θθa θ () Man economic ime erie wih a eaonal componen can be well repreened b hi model. The pe of ARIMA model o be fi i pecified b:. Acceing he Properie dialog box for he Repone column conaining he ime erie daa.. On he Pred. ab, elec he deired model. 006 b SaPoin, Inc. Foreca Repone Variable - 3
Specif he order for each erm in he model, and he lengh of eaonali if he daa i eaonal. Alo pecif in he Percenile field he pe of foreca limi deired (percenage and pe). I i omeime helpful o ranform he daa uing a logarihm or ome oher power ranformaion. To pecif a ranformaion:. Acce he Properie dialog box column conaining he ime erie daa b doubleclicking on he column header.. On he Di. ab, elec he aumed diribuion. The defaul elecion aume ha he daa wih no ranformaion follow a normal diribuion. If ou elec Lognormal, he logarihm of he daa will be aumed o follow a normal diribuion. If ou elec Power normal, he daa will be aumed o follow a normal diribuion afer raiing hem o he indicaed power. When forecaing he daa, he obervaion are ranformed o he meric in which he are normall diribued, foreca and foreca limi are calculaed in he ranformed meric, and he invere ranformaion i made o he foreca o pu hem back in heir original uni. 006 b SaPoin, Inc. Foreca Repone Variable - 4
Sample Daa The file price.gm conain a ime erie of n = 6 obervaion, paced 5 minue apar. The fir everal obervaion are hown below: Row Time Price 0:00 444.5 0:5 445 3 0:30 446. 4 0:45 447. 5 :00 446.7 6 :5 446. 7 :30 445.6 8 :45 446.4 9 :00 446.3 0 ;5 446.5 006 b SaPoin, Inc. Foreca Repone Variable - 5
Foreca STATGRAPHICS Mobile Rev. 4/7/006 The iniial page dipla he foreca generaed from he fied model. I plo:. Poin mbol for he obervaion ued o fi he model.. A line hrough he obervaion howing he one-ahead foreca. The one-ahead foreca are he foreca for each ime period made uing all of he informaion available a he previou ime period. If F (k) = foreca for ime +k made a ime hen he one-ahead foreca error i F (). 3. Foreca F n (k) for everal period beond he la daa value. 4. Predicion limi for he foreca. Thee limi are creaed a he Percenage level pecified on he Pred. ab of he Column Properie dialog box. 006 b SaPoin, Inc. Foreca Repone Variable - 6
Below he graph are he forecaed value for he nex 4 period beond he end of he daa, and ummar aiic howing how well he model wa able o predic he oberved value. The aiic are from he one-ahead foreca error e in each period, which equal e = - F - () () Depending on he pe of model eleced, ome number of iniial value m mu be deermined from he daa before foreca can be generaed. The one-ahead foreca error are hu defined for = m+, m+,, n. The error aiic diplaed are: Roo mean quared error RMSE = Mean error n e = m+ n m (3) ME = n e = m+ n m (4) Mean abolue percenage error MAPE = 00 n = m+ e / n m % (5) Mean percenage error MPE = 00 n = m+ ( e / ) n m % (6) The MAPE and MPE are onl diplaed if all daa value are poiive. If he example above, he MAPE indicae ha he error in forecaing he nex daa value average abou %. 006 b SaPoin, Inc. Foreca Repone Variable - 7
ACF (Auocorrelaion Funcion) The procedure alo plo he eimaed auocorrelaion funcion of he one-ahead foreca error. The auocorrelaion a lag k, defined b r k n k = m+ = n ( e e )( e e ) ( e e ) = m+ + k, k =,,, w (7) eimae he correlaion beween foreca error k uni apar. If he model fi he daa well, hen all of he bar hould fall wihin he probabili limi, which are drawn a horizonal line around 0. In addiion, a P-value i diplaed for he Box-Pierce aiic Q = n w r i i= (8) where w i he number of auocorrelaion ha have been eimaed. A mall P-value (below 0.05) would indicae ignifican auocorrelaion remaining in he error, which migh neceiae uing a differen ARIMA model. 006 b SaPoin, Inc. Foreca Repone Variable - 8
Reidual Thi plo how he one-ahead foreca error. Horizonal line are drawn a -igma and 3-igma o help in idenifing oulier. 006 b SaPoin, Inc. Foreca Repone Variable - 9
Miing Value ARIMA model require ha daa be available a equall paced inerval of ime. Conequenl, i i aumed ha he ime or diance beween each row i he ame (i.e., 5 minue, one da, one buine da, one inch, ec.) Conequenl, pecial echnique are needed o handle miing daa. The rule for handling miing daa are a follow: Miing value a he op of he column are ignored. Miing value a he end of he column are reaed a poin for which foreca are deired. Miing value in he middle of he column are replaced wih inerpolaed value according o he following rule, a long a here are no oo man miing value cloe ogeher:. If, he obervaion a ime, i miing, find he wo obervaion in he ame eaon ha precede ime ( - and - ) and he wo obervaion in he ame eaon ha come afer ime ( + and + ).. If none of he four obervaion are miing, hen he replacemen value for i: 3 + + + 3+ = (9) 8 3. If + i miing bu he oher hree are no, hen he replacemen value for i: + 3 + + = (0) 3 4. If + i miing bu he oher hree are no, hen he replacemen value for i: 3 + 8 + + = () 6 5. If - i miing bu he oher hree are no, hen he replacemen value for i: + 8+ 3+ = () 6 6. If - i miing bu he oher hree are no, hen he replacemen value for i: 3 + + + = (3) 3 7. If + and + are miing bu he oher wo are no, hen he replacemen value for i: 006 b SaPoin, Inc. Foreca Repone Variable - 0
STATGRAPHICS Mobile Rev. 4/7/006 = + (4) 8. If - and + are miing bu he oher wo are no, hen he replacemen value for i: + + = (5) 3 9. If - and + are miing bu he oher wo are no, hen he replacemen value for i: + + = (6) 0. If - and + are miing bu he oher wo are no, hen he replacemen value for i: + = + (7). If - and + are miing bu he oher wo are no, hen he replacemen value for i: + = + (8) 3. If - and - are miing bu he oher wo are no, hen he replacemen value for i: + + = (9) If more han of he four obervaion are miing, an error meage will be diplaed and he anali will no be performed. The inerpolaed value are deigned o perfecl reproduce a quadraic rend (if onl one obervaion i miing) or a linear rend (i wo obervaion are miing), provided no noie i preen. 006 b SaPoin, Inc. Foreca Repone Variable -