Smoothing. Backward smoother: At any give T, replace the observation yt by a combination of observations at & before T

Transcription

1 Smoohing Consan process Separae signal & noise Smooh he daa: Backward smooher: A an give, replace he observaion b a combinaion of observaions a & before Simple smooher : replace he curren observaion wih he bes esimae for How o obain i? Use he leas squares crierion SS E ˆ

2 No Consan process Can we use? ˆ his smooher accumulaes more and more daa poins and gains some sor of ineria So i canno reac o he change Obviousl, if he process change, earlier daa do no carr he informaion abou he i Use a smooher ha disregards he old values and reacs faser o he change Simple moving average

3 N N N N N M Simple moving average Choice of span N: -If N small :reacs faser o he change - large N:consan process N M Var ) ( he moving averages will be auocorrelaed since successive moving averages conain he same N- observaions N k N k N k k,, 3 Bigger variance for small N

4 Firs order exponenial smoohing Idea: give geomericall decreasing weighs o he pas observaions Obain a smooher ha reacs faser o he change However, Adjus he smooher, b mulipling ) ( ~ Simple or firs order exponenial smooher 4

5 ~ ~ Firs-order exponenial smoohing: linear combinaion of he curren observaion & he smoohed observaion a he previous ime uni ) ~ ( ~ ~ ~ Discoun facor : weigh on he las observaion : weigh on he smoohed value of he previous observaions ) ( 5

6 he iniial value ~ ~ ~ ~ ~ ) ~ ( ~ ~ ~ ~ As ges large, he conribuion of o becomes negligible ~ ~

7 ~ he iniial value commonl used esimaes of ~ ~ If he changes in he process are expeced o occur earl & fas ~ If he process a he beginning is consan 7

8 he value of As ges closer o, & more emphasis is given in he las observaions : he smoohed values follow he original values more closel : ~ ~ : ~ he smoohed values equal o a consan he leas smoohed version of he original ime series..4 Values recommended Choice of he smoohing consan: Subjecive: depending of willingness o have fas adapivi or more rigidi. Choice advocaed b Brown (invenor of he mehod): λ =.7 3 Objecive: consan chosen o minimize he sum of squared forecas errors 8

9 Use exponenial smoohers for model esimaion General class of models f ; vecor of unknown parameers uncorrelaed errors e.g. consan process Esimae parameers SS E If no all observaions have equal influence on he sum : inroduce weighs ha geomericall decrease in ime SS * E, * dss d E ake he derivaive ˆ he soluion ˆ For large ˆ ( ) ~ 9

10 Second order exponenial smoohing uncorrelaed wih mean & consan variance Use single exponenial smoohing: under/over esimae he acual values Linear rend model ) ( ~ E E E Given E ) ( ~ E ~ E E Simple exponenial smooher:biased esimae for he linear rend model Wh? bias & ) (

11 Soluions: a) Use a large value : Smoohed values ver close o he observed: fails o saisfacoril smooh he daa b) Use a mehod based on adapive updaing of he discoun facor c) Use second order exponenial smoohing

12 Second order exponenial smoohing Appl simple exponenial smoohing on ~ ~ ( ) ~ () ( ) ~ () ˆ ) () ~( ~ Unbiased esimae of main issues: choice of iniial values for he smoohers& he discoun facors Iniial values ~ ~ () ˆ ˆ,, ˆ, ˆ, Esimae parameers hrough leas squares

13 Hol s Mehod Divide ime series ino Level and rend L =λ +( α)(l + ) =β(l +L - )+( β) - F + =L + = acual value in ime λ = consan-process smoohing consan β = rend-smoohing consan L = smoohed consan-process value for period = smoohed rend value for period F = forecas value for period + = curren ime period 3

14 Use exponenial smoohers for forecasing A ime, we wish o forecas observaion a ime +, or a ime + ˆ sep ahead forecas Consan Process Can be esimaed b he firs-order exponenial smooher Forecas : ˆ ( ) ~ A ime + ˆ ~ ˆ ( ) ˆ ˆ ˆ () ˆ ( ) ˆ () e () e ( ) ˆ ( ) One-sep-ahead forecas error 4

15 ˆ ˆ ( ) () ˆ () e () ˆ Our forecas for he nex observaion is our previous forecas for he curren observaion plus a fracion of he forecas error we made in forecasing he curren observaion 5

16 Choice of Sum of he squared one-sep-ahead forecas errors SS E e Calculae for various values of Pick he one ha gives he smalles sum of squared forecas errors 6

17 Predicion Inervals Consan process ~ Z a ˆ e ~ firs-order exponenial smooher Z a ˆ e (-a/) percenile of he sandard normal disribuion Esimae of he sandard deviaion of he forecas errors 7

18 Linear rend Process forecas ahead sep ˆ,,,,,, ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ () () () () () () ~ ~ ~ ~ ~ ~ ˆ In erms of he exponenial smoohers 8

19 Parameer Esimaes b,+ = l ( + l) + + ( - l) çb, + b, b,+ = ( ) + - l l - l b,+ - b, æ è ( ) ( - l) b, ö ø (-α/) predicion inervals for an lead ime τ æ ç + l è - l ö æ () -ç+ l ø è - l ö c () ± Z a/ s e ø c l c i =+ é -4l + 5l ( - l) 3 ë ( ) + il 4-3l ( ) + i l ù û 9

20 Esimaion of he variance of he forecas errors, σ e Assumpions: model correc & consan in ime Appl he model o he hisoric daa & obain he forecas errors s e = = å = å = e () ( ( ) ) Updae he variance of he forecas errors when have more daa s e,+ ( ) = + s e, + e + () s e,+ = å e ( ) - + = ( )

21 Define mean absolue deviaion Δ D = E( e - E( e) ) Assuming ha he model is correc D = d e ( ) + ( -d)d - s e, =.5D

22 Model assessmen Check if he forecas errors are uncorrelaed Sample auocorrelaion funcion of he forecas errors - åéë e ( ) - eù û éë e -k =k r k = - éë ù û å = e ( ) - e ( ) - e ù û e n e -If he one-sep-ahead forecas errors are uncorrelaed, he sample auocorrelaions for an lag k should be around wih a sandard error /sq() -A sample auocorrelaion for an lag k ha lies ouside he Limis will require furher invesigaion of he model /

23 Exponenial Smoohing for Seasonal daa Seasonal ime Series Addiive Seasonal Model = L + S +e L = b + b S = S +s = S +s = L : linear rend componen S : seasonal adjusmen s: he period of he lengh of ccles ε : uncorrelaed wih mean zero & consan variance σ ε s Consrain: ås = = 3

24 Forecasing - Sar from curren observaion - Updae he esimae L L = l ( - S -s ) + ( - l )( L - + b,- ), < l < - Updae he esimae of β b, = l ( L - L - ) + ( - l )b,-, < l < S - Updae he esimae of S = l 3 - L ( ) + ( - l 3 )S -s, < l 3 < - τ- sep-ahead foreceas + () = L + b, + S ( - s) 4

25 Esimae he iniial values of he smooher Use leas-squares s- å ( ) = b + b + g i I,i - I,s { i=, =i,i+s,i+s I,i =, oherwise +e b, = L = b b, = b S j-s = g j, j s - S = - s- g j å j= 5

26 Exponenial Smoohing for Seasonal daa Seasonal ime Series Muliplicaive Seasonal Model = L S +e L = b + b S = S +s = S +s = L : linear rend componen S : seasonal adjusmen s: he period of he lengh of ccles ε : uncorrelaed wih mean zero & consan variance σ ε s Consrain: ås = = 6

27 Forecasing - Sar from curren observaion - Updae he esimae L L = l ( / S -s ) + ( - l )( L - + b,- ), < l < - Updae he esimae of β b, = l ( L - L - ) + ( - l )b,-, < l < S - Updae he esimae of S = l 3 / L ( ) + ( - l 3 )S -s, < l 3 < - τ- sep-ahead foreceas + () = L + b, + S ( - s) 7

28 Esimae he iniial values of he smooher From he hisorical daa, obain he iniial values b, = L = - n (n -)s, i = s is å = ( i-)+ b, = - s b, S j-s = s S * j S * = n s å i= n å = S i *, j s (-)s+ j - ((s +) / - j)b 8

29 Exponenial smoohing & ARIMA models = l +(- l) - e = - - e - = e - (- l)e - = = e -qe - (- B) = (-qb)e 9

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35 . Make a ime series plo of he daa. Use a simple exponenial smoohing wih lambda= o smooh he firs 4 ime period of his daa. How well does his smoohing procedure work? 3. Make one-sep-ahead forecass of he las observaions. Deermine he forecas errors 35

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37 = l +(- l) - = l + (- l), = = l + (- l) Series Series

38 Forecas : ˆ ( ) ~ One-sep-ahead error : e ( ) ˆ ( ) ˆ ~ obs acual forecas Forecas error

39 . Make a ime series plo of he daa. Use simple exponenial smoohing wih lambda=. o smooh he firs 3 ime periods of his daa. How well does his smoohing procedure work? 3. Make one-sep-ahead forecass of he las observaions. Deermine he forecas errors 39

40 Series 4

41 Series Series Series Series he smoohing procedures works well; i capures he rend of he daa and reduces he variance 4

42 obs acual Forecas Error

43 Shewhar chars 43

44 Shewhar chars A plo of he forecas errors vs ime: Conains a cener line ha represens he average of he forecas errors A se of conrol limis ha are designed o provide an indicaion ha he forecasing model Performance has changed Cener line: eiher zero or he average forecas error Conrol limis : placed a he hree sandard deviaions of he forecas errors n å MR = e i= ( ) - e - ( ) s e ( ) =.8865 MR 44

45 he forecas error falls near he 3σ limi. he forecasing procedure works well ; however he ime series is ver nois as he forecas errors s e ( ) = = Sum Aver LCL= UCL=.4 45

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47 Plo he firs differences of his daa?. Has differencing removed he rend? 3. Use exponenial smoohing on he firs differences. Insead of forecasing he original daa, forecas he firs differences for he remaining daa using exponenial smoohing and use hese forecass of he firs differences o obain forecass from he original daa 47

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50 Find he auocorrelaion funcion. Does i show an rend? Do ou obain an informaion abou he smoohing parameer ou could use? 5

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