time series analysis notes heheheheh.pdf

yo

1. NIT 3 : Rondom
component
TAKig 'expectution o:6:S
secuar Trend CTA)
seasonoL Noriotion cSt)
cyic voriation
Rond om variation R4
FYOM eq©
ECCAKEAN*: ELttk -KGEt+k-| * ¢tk-2 random IerrorY
component
varjote disference memod
+product erms]
Let us sUPPOse that ne series can be represennted as sumof Functjonal
.
pOart and rondom componett as Qven beiow For produo erms,.
NCettk) +<KeNCE¢tK-+ +NCEetk-2 t .
where e's are iid dlm ECEi
=
o
va -
Hence an estimoate of Y is Biven b9
=
NAY CORS+) caks) -NCe)= N CSay)
se inou hat for a polynomial se of mt deg ree
in t
Hs to be noted thort we calculate CARN) and not
tne, obserned voiance
of
a+ since ElakJJ=
0. Hence , once the value of k is knoun tne vorionce v
were DSt St*1-St
Can be
estimat ed.
Hence as we proceed uih ditferetrag difterencin9 in eqr 0, he roandom
In order to
obtain the
valve »f R and. thot 6f the estimate of errOr
variance, eiennent becomes more and more Predominant untiL Finaily the systematic
we coy OUt tve test of nyothesis
<ompone is compieteiy eminoted. Fom e4nO,e get
set H Yet
ne PproPriate tert Statistic " u. given b Pros. o. Ade rson os
we hove AEt= <E-1) Et C:E= +aJ
LER e,E+k<Ek-2-.
constt b Prof
6 Titner
Ln hiS boOR Yvtote ats
menod,

2. N111112
M

3. a536 = 355.6667
STATIONARY TIMNE SERIEES
24 24
e) a*9t
= 7424
= 7S7.5652
ODeterministic process
23
LF the Nayes Sbserved of process are
exacely aetemined
oy some
co) =2C SE*= 45232 20556
matnenmatieal FUnctiorn 6Ct) ,then the.PrOce ss is caied determinisic
22 22-
stocastic process
/NOn-eberministic Pracess Rardom ?rocess
2 Fthe ooserved vaue of a proceess are vandom in noure
henthe eroçess
21
's
sojdto.be Stocmostic. A time.ser.ies
ProesS,SGahed stochasic eroce ss.
8536 -78333 s'ince oefore"tne aetUal
realizaion
of.an observotion ,ue. Ao,
not
know
24x2 wth ceTtainty Os to udhat valve the
prOCe ss
s
oin9 to tAke at:any
= 424 t3H3 12 2609 pantieuAY eme t.
nence,tne Y in hea a sime series
S not-
determinis
23 bUt Stochaa sti e,
-
> 2056 o28
6G3
20 .
tionary Process
coA)= 3436S : 12.7653
A ime semes Pocess is said +o be stationary. i nere
bs . nO. systematic,
21x70
anange
iw ts
mean arianse gnd iE.Pereaic yoriatons have
oeen. removed .
on
stationGry time
sertes process can duDYs
be
transfor med
iwto
R N-V2 a ,2s 77.33 33 -i26.2601 4065 2 623 > 1.6
stationery proces s
77 8333
2 -V3 ,25 = 126.609 -1O2-8 x 9 2.1367>6
strit stotLOnary Processs
26.2609
strict stationarity S on
As
spitic ovtcoe of a specific
Ra 3-V4x 3, 25
02-3 2.76 53 x 12-9445 263 <i6
doto
generatn9 Process Foral:tie.t.ad remainS in euituGriUm about o. 02.8
omyon
mean Nave say
V4 are not si9nificantiy different and eiher of tne two can be t9ken as
he
2
Compones which cayse ack of stationarity are trend and
seasonali stmates of eror variance.
n itting statonary, me series. models, ise: First,
eiminate trend ond
the seasonal component from the riginol time series.
vCt)
cov C A ttk) FcK
not o T o

4. DiFferet
senernestmat ac.oynt. For oscilletorg moverent in a
s+ationary Stnct Time series
PrOcess.
There ore prinmorily
Schemes not account 4or oscilatorS MeVemert
nstoti onan Process.
sAean eenstan at itsGeVOTTanee. of*AEElen oftke T
perins and Sti+ 92tk,
**
Stn tk
are
identca liy d istr'itovte d
.
ci)
E ec*
of,.
movng verages o
the random
conmPonent
ciN Sun of aUmoer o eyclical comOne nts
Derict etene¥ imfties seeond rdrSTRONeH
cii Auto regre ssve equatons a series
Neak StafonaS
A procesS s sajd to be weak stotionary
/second order strtî enaryr
covanance
stoti on ars
f HS mean. isonstant and tts covorionces
ae a fun cti on
Fa series is
such.that its yaNÚes
at time tt depends
on me Previous kt
vaves acc. to the relation
of lag9 Period R 1 e,
ECS)= consit
and coYCY Nt tk)= fC)
where f is a matnemotical fn and
A striot stationarity implie S
SeCond order sationo
\»yt ts iverse
is not+trve Rt is a r.v.tnen tne seri es
iS called auto regre ssive AfreeRssiS-tenee SaNoAA¥S
mean totiorary:
A
PYoce ss
iS mean statonary
iF
ECY)
N t This. series Under cénain coneitions ic of oscinatory tyee.
TOESS ItovTomce strtoraT
c eKect o monn9 averages n me random comeonent.
Vari ance stationary
Process is variance statîo nary
f
movin9 aver ORE of re9 andom series enerotes an osciorto ry
EC-P32= 4 series nim varsing periods and ompiitud es
Cov ariance stationarN:
A pocess is coyariance
.
stat lonary
i
cii sum of a no of cye\lcal comeonewts
ostatory serjes con olso be Sum of a no of harmoni c terms N}h varWS
periods ona omnpitudes. TMUs i t is the oscWatory series im periodicrtiesS
F above 3 stationariies are sartissfed, t impies
weak stortion orrty
ond ampitudes A,A2,.
hen st can boe written as
A* AcoT 20, si
i,N2, S)
PCA)* impiies veak H ioov.
AUTO
cAVARIANCE AND TS FVCTION
Tne conariance bet ween y4 ard tt here k:.is th A9 Peribd:is
esined as

5. -
R :
Nhen R=0
, To
=Nar
(dN +)
n-
The coNoriance fn obtained For ierent valves F K, KD,\,2, .
caled tne auto covoriance Function CACYF) CorYelogram
The mortrîx of autocoatance funetion is commonly 'sed too fOY
meckin adomness in a Aota set.
The orapn ostained y P tiAS
he diFferentt vodes.
of, r 8nst he,
°n-
valves of
K-0,\,2,.
. iS calec a
corelogram.
arance coN aTIance This is a device or an mportant tooL for. Udgng the scheme resgonsibl
s
motrlx for tne osci\1a tory moyement în a
stattlonary time seies.
-
Gorrelogrom
F
MOvg average
n the randonm component
PUtocorrelation
and ts unction et there se o obs. n a ime
series, we
can Form n-r poirs r a
\a9
The usvel
correlationcoekf. ol
St and qa
were k is te aPernod peniod
of R S4Sttr), C9+ Nztt
is defineà as Consider a sinMple
moNmg average *extem 'm' on the
random Conmpe net
Co C94,S+fr)
RCEt). e khou mat E(R)= o .NCR)= *2 » ECR*)
CoyCRt,R') = o-EReRe')
Th coYrelation fUnction that we oktained *or
di.
NaIves o k is ca\leed
wedefme
7.(R¢4 R+ + Rt +m-1)
the autosorreation ntien
ca.c.f.)
an +tk Kt+* *RttKti* **tK+m-1)
Nhen k=o, To= To=
when k: -S, vs= CoNg8tJt-s)
s
since
e \a9
Period is
CovC Rttk) E *¢tK)
Autocorrelation
Nar)Naretk VEC)EA+K*)
E
E E ti/ E )
The
natri ef avtocorrelátion is
ECtK)
m m
cON. eMS t
caacelle

6. Tne cause
of tne
,ose\at ars rovement .is
thecfect of simple movIng aV8
on the roindom eOmponent.
.
The osiilatory MOvemerts of an dbseryed time ser\es
mybe attrilovted to one
OFthe tollowing shemes .
) moving verage ot ran dom component
ciime series represented by
a inear Combinaton of harmonic element s
an avto regressive series
tighteaN
aerag
coYrelgram anaais enablies s o decide any Par+icular cace as to the
E t+k)= Cm-k)
m
m-Ko
cause of oscilation
oherusise
For amoving werage of
extent
m
uim weignts, 2 .
Cm-k)7I m
oF random cemponents EL , izt,2,
. .
The generated
series is given by
nerpretation ditKiitk*) 2titkA2 *** * am-k Tim + +m Ei+k+m
shere i'sare:iid NO),thuS
m
'
ECS-0 ECsik)
For a stationary time series, the co Fre I am , S a. staight\Ang .originart ng
rom Co,i) an ending at CM, o)
correloqram
JvcyvSiAm

7. point Cm,o) onwards. mUs, he correlogram
II oscilate between the points co,) and Cm,0)
and
then coincide WHh the x-axis Wheh
Rm
$OME SPEcIAL PRo CESSES
k-0, ro=|
k=m,m 0 Moyin9 aNes9e CMA) f":
2) Autoregressi ve ( AR)
3 AutoregresSiNe Moving ayerage CARMt)
Tmese all 3 Proce sses elp
s to; deter mine hpW time series
is
evolved.
myb)
-
OR
non pAst atalobs. or past radom shocks. evoived in tame series
'
in
partilor, if a= l/m ie1,2).
mthen r Cm-k) (\/ ma) Suppos e yoy are át Some P oint t, *hen R means yoU nave t-t Past
bseyoation s
Si, Ya,...
.
-. 4-
ime series obs Upto time t-i
AAH
E . .. &t-random shooES UPto time t-
= I ,uNhich'
Äs
, the. eåM
of. si raî gt
tine in
kt random sMock at time
intercep form NOU there are 3 possibilities For tne evalWation of
s (
ci Procest has
memory f random
shocks components of where. t was (random
shocks cofres poniwa
o past vaiues of v) bt no
memory oF where tt wasS
co
ci) process has memory oF Where t CPAst. vayes of ) Was t no main oryo
Andom sho cks corres ponding to past vowes of 3
(no
ThUs,
tor a series qnerot ed oy an. m
Point sim : simple moying:
Ny.
of random omeonents,te correlqran
consis ot Jtod ghe liA
ch Process nas memory ot where it past vales ot ) Nas as we as mernory f
otnn
te points CMo) and co, qether wim the
axi fro Me
an dom shocks corres pondno9 to past valves oF Y.

8. M M

9. o
O
SMM

10. AUTORES RESSIVE MODEL iven
St-, Je
become \hdependent of
t-2, 3
and so on enen nis propere
of StoCnastic erocess is caled Markovjan properby.in nich present, and
Atpregressive (AR) process FUtUre becomes independet OF the post
Conseructed y regres sing current vajve of v ahjable on Þast yave
.
current val ve Mean= H
sbppose SE it depends on or is directly offected by past valvesre, St-1Je-2
-
Take S-0, then P=o
Ses reqress ion oF the vatiabe against tseF, Tmus, it is caued as avtoregres Sion
Meon of the process is o.
means gainst s
ASSume hat the proces Nanance sttionary
Prediots
future behayioUr based
on past kehaviour UseA For
Ore castn 9 hen
NSE=VC-1)::
there is corel oation bhw ourrent an the preceeding
alues
C9t)=NC9E-)=
ARC) Process Then E[Pt-+E ¢EC )+ECE)+2,E4-+)
t is deined asS
9 +<+Et -0
Where : intercePt erm
{&;tET}=°urely random rOcess, N(9,
e,4,
Porameters of AR(E) ProcesS
AC VF and ACF
-ECYL
.
ELN -+2t-2t
+
OpSt-p+ {+ Et
-9
CI-1-2 -P) =
-92*-
FUrthe
Mean 0F Ehe ARCP) Proces
E Sl =0
*R-\2,
ARCI) proei S, p=
Sa-U-t-Kl

11. ACF Of ARCE) proeSs
P k 2
To
=4 since 19,1<1, Pndeclines geometicaly
If 90, then 9 is tve +K, then Pk s +ve +k
ve
.
ARC) processeS
0, =0, p 2
arian ce of the procesj
Etb-Yt]+0,Elt-2 3t +E( &t t)
- + + *-O
ELLC 4e143ei+E)
.
O+0+
EL+t-iJ
= EL&t-+AJt-2*t)3t-
ESt Jt-21
= ElSt-2 9iSe-+ ¢-1+£t)J
APter solving tnese3: eyabions, vne can.
ostain To,,3,
-