Linear Filer
Le x y : T denoe a bivariae ime erie wih zero mean.
Suppoe ha he ime erie {y : T} i conruced a follow: y a x The ime erie {y : T} i aid o be conruced from {x : T} by mean of a Linear Filer.
Schemaic Diagram of a Linear Filer X Inpu Proce Linear Filer Y Oupu Proce
h yy h a a ' ' ' d f e a e i h i d f A e h i The auocovariance funcion of he filered erie
Thu he pecral deniy of he ime erie {y : T} i: f yy i a e f A f
Commen A: A a e i i called he Tranfer funcion of he linear filer. A i called he Gain of he filer while arg A i called he Phae Shif of he filer.
Alo h xy y x E h h a d f A e h i
Thu cro pecrum of he bivariae ime erie T y x : i: i xy f A f e f a
Definiion: K ij f ii f ij f jj = Squared Coherency funcion Noe: K ij
Commen : K xy f f xy f yy = Squared Coherency funcion. f A f A f if {y : T} i conruced from {x : T} by mean of a linear filer
Linear Filer wih addiive noie a he oupu
denoe a bivariae ime erie wih zero mean. Le =..., -, -,,,,... T y x : Suppoe ha he ime erie {y : T} i conruced a follow: v x a y The noie {v : T} i independen of he erie {x : T} (may be whie)
Schemaic Diagram of a Linear Filer wih Noie X Inpu Proce Linear Filer Y Oupu Proce N Addiive Noie a Oupu
h yy y y E h h h a a vv ' ' ' d f e d f e a e vv h i i h i The auocovariance funcion of he filered erie wih added noie
coninuing h ih e A f f yy where A a e vv i d Thu he pecral deniy of he ime erie {y : T} i: f yy A f f vv
Alo h xy y x E h h a d f e a h i d f A e h i
Thu cro pecrum of he bivariae ime erie T y x : i: i xy f A f e f a
Thu K xy f f xy f yy = Squared Coherency funcion. f A f A f f A f vv f vv Noie o Signal Raio
ox-jenkin Parameric Modelling of a Linear Filer
Conider he Linear Filer of he ime erie {X : T}: Y a X a X where a a and A i a e a e i = he Tranfer funcion of he filer.
{a : T} i called he impule repone funcion of he filer ince if X =and X = for, hen : Y a X a for T X a Linear Filer
Alo Noe: X a X a Y Y Y X X a a X
Hence {Y } and {X } are relaed by he ame Linear Filer. Definiion The Linear Filer Y a X i aid o be able if : a X a a converge for all.
Dicree Dynamic Model:
Many phyical yem whoe oupu i repreened by Y() are modeled by he following differenial equaion: I d d... r d d d d r r Y I d d d d... d d Where X() i he forcing funcion. X
If X and Y are meaured a dicree ime hi equaion can be replaced by: I... r r Y I... X b where = I- denoe he differencing operaor.
Thi equaion can in urn be repreened wih he operaor. r r Y I... b X I... b X I... or X Y b b b b... where
Thi equaion can alo be wrien in he form a a Linear filer a Y X a X Sabiliy: I can eaily be hown ha hi filer i able if he roo of (x) = lie ouide he uni circle.
Deermining he Impule Repone funcion from he Parameer of he Filer:
Now a or a Hence...... a a I a I r r b b b b...
Equaing coefficien reul in he following concluion: a j = for j < b. a j - a j- - a j- -...- r a j-r = j or a j = j- + a j- +...+ r a j-r + j for b j b+. and a j - a j- - a j- -...- r a j-r = or a j = a j- + a j- +...+ r a j-r for j > b+.
Thu he coefficien of he ranfer funcion, a, a, a,..., aify he following properie ) b zeroe a, a, a,..., a b- ) No paern for he nex -r+ value a b, a b+, a b+,..., a b+-r 3) The remaining value a b+-r+, a b+-r+, a b+-r+3,... follow he paern of an r h order difference equaion a j = a j- + a j- +...+ r a j-r
Example r =, =, b=3, = a = a = a = a 3 = a + = a 4 = a 3 + = + a 5 = a 4 + = [ + ] + = w + + a j = a j- for j 6.
Tranfer funcion {a }
Idenificaion of he ox-jenkin Tranfer Model wih r=
Recall he oluion o he econd order difference equaion a j = a j- + a j- follow he following paern: ) Mixure of exponenial if he roo of - x - x = are real. ) Damped Coine wave if he roo o - x - x = are complex. Thee are he paern of he Impule Repone funcion one look for when idenifying b,r and.
Eimaion of he Impule Repone funcion, a j (wihou pre-whiening).
Suppoe ha {Y : T} and {X : T}are weakly aionary ime erie aifying he following equaion: N X a Y Alo aume ha {N : T} i a weakly aionary "noie" ime erie, uncorrelaed wih {X : T}. Then h h XY N X a X E X Y E h
Suppoe ha for > M, a =. Then a, a,...,a M can be found olving he following equaion: h N X E X X E a h a
xy xy xy a a a M a a a M M a M a M a If he Cro auocovariance funcion, XY (h), and he Auocovariance funcion, XX (h), are unknown hey can be replaced by heir ample eimae C XY (h) and C XX (h), yeilding eimae of he implue repone funcion aˆ, aˆ,, ˆ M M M a M
In marix noaion hi e of linear equaion can be wrien: M xy xy xy a a a M M M M M
If he Cro auocovariance funcion, XY (h), and he Auocovariance funcion, XX (h), are unknown hey can be replaced by heir ample eimae C XY (h) and C XX (h), yeilding eimae of he implue repone funcion C C C xy xy xy aˆ C aˆ C aˆ C M aˆ C aˆ C aˆ C M M aˆ C M aˆ C M aˆ C M M ˆ aˆ,, a M, ˆ a M
Eimaion of he Impule Repone funcion, a j (wih pre-whiening).
Suppoe ha {Y : T} and {X : T}are weakly aionary ime erie aifying he following equaion: Y a Alo aume ha {N : T} i a weakly aionary "noie" ime erie, uncorrelaed wih {X : T}. X N
In addiion aume ha {X : T}, he weakly aionary ime erie ha been idenified a an ARMA(p,q) erie, eimaed and found o aify he following equaion: ()X = ()u where {u : T} i a whie noie ime erie. Then [()] - ()X = u ranform he Time erie {X : T} ino he whie noie ime erie{u : T}.
Thi proce i called Pre-whiening he Inpu erie. Applying hi ranformaion o he Oupu erie {Y : T} yeild: Y ) ( )] ( [ N X a ) ( )] ( [ ) ( )] ( [
or y a u n where y ) [ ( )] ( Y and n ) [ ( )] ( N
In hi cae he equaion for he impule repone funcion - a, a,...,a M - become (auming ha for > M, a = ): uy uy uy a a uu M a M uu uu or a k uy uu k and aˆ k C C uy uu k
Summary Idenificaion and Eimaion of ox-jenkin ranfer model
To idenify he erie we need o deermine b, r and. The fir ep i o compue. he ACF and he cro CF C (h) and C xy (h). Eimae he impule repone funcion uing C C C xy xy xy aˆ C aˆ C aˆ C M aˆ C aˆ C aˆ C M M aˆ C M aˆ C M aˆ C M M M
3. Deermine he value of b, r and from he paern of he impule repone funcion The Impule repone funcion {a } b - r + Paern of an r h order difference equaion
3. Deermine preliminary eimae of he ox- Jenkin ranfer funcion parameer uing: i. for j > b+.. ii. a j = a j- + a j- +...+ r a j-r for b j b+ a j = j- + a j- +...+ r a j-r + j 4. Deermine preliminary eimae of he ARMA parameer of he inpu ime erie {x }
5. Deermine preliminary eimae of he ARIMA parameer of he noie ime erie { } b y x b y x y y y r r xbxb xb y y y x x x r r b b b
Maximum Likelihood eimaion of he parameer of he ox-jenkin Tranfer funcion model
The ox- Jenkin model i wrien y a x a x b x r where I... r and I... The parameer of he model are:,,..., r and,,,..., In addiion. he ARMA parameer of he inpu erie {x }. The ARIMA parameer of he noie erie { }
The model for he noie { }erie can be wrien u p where I... q and I... q p
Given aring value for {y }, {x }, and x, y and u u and he parameer of he ranfer funcion model and he noie model δωβ, and α We can calculae ucceively: u u δωβ,, α, x, y, u The maximum likelihood eimae are he value ˆ ˆ δωβ, ˆ and αˆ ha minimize: S δωβ,, α u
Fiing a ranfer funcion model Example: Monhly Sale (Y) and Monhly Adveriing expendiure
The Daa Mon Adver Sale Mon Adver Sale Mon Adver Sale Mon Adver Sale 78.54 35 5.8 763 47.35 43 5 8.3 4 34.8 34 5 87.95 84 8.6 99 5 84.6 348 3 6.38 57 53 8.8 679 3.5 6 53.9 658 4 5.3 58 54 93.78 756 4 3.8 997 54 65 64 5.8 66 55 7. 766 5 96.46 45 55 89.86 5 6.54 44 56 6. 73 6 47.96 98 56 9.53 466 7 53. 94 57 8.74 498 7 4.87 89 57 89 78 8 54.58 5 58 9.54 597 8 9.97 58 8.6 597 9 63.6 8 59 34.59 9 9.95 94 59 89.9 553 46.6 7 6 7.67 6 75.54 354 6 58. 576 57.49 544 6 9.6 94.58 9 6 7.36 63 5.65 598 6 5.73 74 9.4 5 6 7.83 556 3 55.8 93 63 7.76 9 3 59.5 9 63 73.4 554 4 34.5 758 64.35 7 4 67.83 647 64 7.84 75 5 45.9 855 65 87 36 5 93. 67 65 8.8 4 6 93.54 67 66 9.78 978 6 78.4 63 66 6.55 63 7 89.59 755 67 83.73 7 7.9 95 67 89.5 344 8 35.8 63 68 64. 977 8 9.3 3 68 5.83 7 9 65.66 559 69 85.7 67 9 6.48 37 69.57 437 64 5 7.3 63 6.77 437 7 34.75 9 9. 849 7 5.95 66 69.3 446 7 56.88 34 56.3 559 7 7.55 33 5.6 48 7 36.69 54 3 33.3 787 73 7.99 449 3 48.4 34 73 65.53 77 4 34.64 998 74 4.8 7 4 65.86 69 74 45.3 64 5. 459 75 3.9 6 5 64.9 48 75 47.97 84 6.6 659 76 76.5 896 6 4.46 35 76 53.66 735 7 3.36 454 77 79.89 3 7 48.73 976 77.47 8 4.49 538 78 44.64 863 8 45.87 49 78 77.5 937
Uing SAS Available in he Ar compuer lab
The Sar up window for SAS
To impor daa - Chooe File -> Impor daa
The following window appear
rowe for he file o be impored
Idenify he file in SAS
The nex creen (no imporan) click Finih
The finihing creen
You can now run analyi by yping code ino he Edi window or elecing he analyi form he menu To fi a ranfer funcion model we need o idenify he model Deermine he order of differencing o achieve Saionariy Deermine he value of b, r and.
To deermine he degree of differencing we look a ACF and PACF for variou order of differencing
To produce he ACF, PACF ype he following command ino he Edior window- Pre Run buon
To idenify he ranfer funcion model we need o eimae he impule repone funcion uing: C C C xy xy xy aˆ C aˆ C aˆ C M aˆ C aˆ C aˆ C M M aˆ C M aˆ C M aˆ C For hi we need he ACF of he inpu erie and he cro ACF of he inpu wih he oupu M M M
To produce he Cro correlaion funcion ype he following command ino he Edior window
he impule repone funcion uing can be deermined uing ome oher package (i.e. Excel) 8 r, = b = 4 6 4 3 4 5 6 7 8 9 345678934 -
To Eimae he ranfer funcion model ype he following command ino he Edior window
To eimae he following model b y x where I... r and I... Ue inpu=( b $ ( -lag ) / ( -lag) x) In SAS and... r
The Oupu
The Oupu