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Qronoseirèc - Eisagwg
Montèlo grammik c pollapl c palindrìmhshc:
x1, x2, . . . , xk anex�rthtec metablhtèc,
y : exarthmènh metablht
y = β0 + β1x1 + β2x2 + · · ·+ βkxk + ε,
E[ε] = 0, σ2e = Var[ε] genik�
y = f (x1, x2, . . . , xk) + ε, y = f (x1, x2, . . . , xk , ε)
Montèlo Qronoseir¸n: Qronoseir� {xt}nt=1 = {x1, . . . , xn}y −→ xtx1 −→ xt−1, x2 −→ xt−2, . . . , xk −→ xt−k
xt = f (xt−1, xt−2, . . . , xt−k) + ε, xt = f (xt−1, xt−2, . . . , xt−k , ε)
Dhm trhc Kougioumtz c Qronoseirèc
Qarakthristik� qronoseir¸n
Qronoseir� {xt}nt=1 = {x1, . . . , xn}qrìnoc deigmatolhyÐac stajerìc
1. Stasimìthta, phgèc mh-stasimìthtac:
t�seic, periodikìthta (epoqikìthta)
2. AitiokratÐa kai stoqastikìthta:DiereÔnhsh kai taÔtish tou aitiokratikoÔ mèrouc tou
sust matoc pou par�gei th qronoseir�?
3. Grammikìthta kai mh-grammikìthta- Grammik� aitiokratik� dunamik� sust mata èqoun aplèc
lÔseic (stajerì shmeÐo, periodik� shmeÐa troqièc)
- Entopismìc mh-grammikìthtac eÐnai dÔskoloc se mia
parathroÔmenh stoqastik sÔsthma.
'Ara ta pijan� sust mata pou endiaferìmaste na
diereun soume:
grammik stoqastik diadikasÐamh-grammikì dunamikì (pijan¸c qaotikì) sÔsthma
Dhm trhc Kougioumtz c Qronoseirèc
Stasimìthta, t�sh kai periodikìthta
0 100 200 300 400 500100
200
300
400
500
600
700
800
month t
x(t)
Iron production in Asutralia
0 2000 4000 6000 8000 10000200
250
300
350
400
450
500
550
day t
x(t)
Ozon
Stasimìthta ?
austhr stasimìthta
spp f (xt , xt+1, . . . , xt+τ ) stajer gia k�je t kai τ
asjen c stasimìthta
E[xt ], Var[xt ] kai Cov[xt , xt+τ ] stajer� gia k�je t kai τ
Dhm trhc Kougioumtz c Qronoseirèc
Apaloif t�shc kai periodikìthtac
Diaqwrismìc thc qronoseir�c
xt = µt + st + yt ,
I µt : sunist¸sa t�shc
I st : sunist¸sa periodikìthtac ( epoqikìthtac) gia perÐodo
d (st−d = st)
I yt : qronoseir� twn upoloÐpwn
Dhm trhc Kougioumtz c Qronoseirèc
Par�deigma: GICP
01 02 03 04 05 06100
105
110
115
120
125
years
GIC
PGICP, period Jan 2001 − Aug 2005
01 02 03 04 05 06−3
−2
−1
0
1
2
3
years
detr
ende
d G
ICP
GICP, linear trend is subtracted
01 02 03 04 05 06−3
−2
−1
0
1
2
3
years
year
cyc
le o
f GIC
P
GICP, year cycle
01 02 03 04 05 06−3
−2
−1
0
1
2
3
years
detr
ende
d an
d de
seas
oned
GIC
P
GICP, trend and period comp. subtracted
Dhm trhc Kougioumtz c Qronoseirèc
Par�deigma: GICP (sunèqeia)
{xt}56t=1
EktÐmhsh t�shc: µt = 103.9 + 0.31t
01 02 03 04 05 06100
105
110
115
120
125
years
GIC
P
GICP, period Jan 2001 − Aug 2005
x ′t = xt − µt
01 02 03 04 05 06−3
−2
−1
0
1
2
3
yearsde
tren
ded
GIC
P
GICP, linear trend is subtracted
Dhm trhc Kougioumtz c Qronoseirèc
Par�deigma: GICP (sunèqeia)
{st}56t=1 gia d = 12
01 02 03 04 05 06−3
−2
−1
0
1
2
3
years
year
cyc
le o
f GIC
P
GICP, year cycle
yt = x ′t − st = xt − µt − st
01 02 03 04 05 06−3
−2
−1
0
1
2
3
years
detr
ende
d an
d de
seas
oned
GIC
P
GICP, trend and period comp. subtracted
Dhm trhc Kougioumtz c Qronoseirèc
Par�deigma: GICP (sunèqeia)
Prìbleyh gia xn+1 ? n + 1 = 57, Septèmbrioc 2005
Proèktash t�shc gia n + 1:µn+1 = 103.9 + 0.31(n+ 1) ⇒ µ57 = 103.9 + 0.31 · 57 = 121.7
EktÐmhsh periodik c sun�rthshc: sn+1 = s57 = s9 = 0.16
Prìbleyh: x̂57 = x56(1) = µ57 + s57 = 121.7 + 0.16 = 121.86
01 02 03 04 05 06100
105
110
115
120
125
years
GIC
P
GICP, period Jan 2001 − Aug 2005
Dhm trhc Kougioumtz c Qronoseirèc
Apaloif t�shc me pr¸tec diaforèc
stoqastik t�sh: de mporeÐ na perigrafeÐ wc k�poia gnwst
sun�rthsh tou qrìnou
pr¸tec diaforèc: yt = ∇xt = xt − xt−1
0 100 200 300 400 500−150
−100
−50
0
50
100
150
200
250
month t
y(t)
Iron production, first differences
0 2000 4000 6000 8000 10000−150
−100
−50
0
50
100
150
day t
y(t)
Ozon, first differences
Dhm trhc Kougioumtz c Qronoseirèc
Susqètish se qronoseir�, Leukìc jìruboc
xt , xt+1, . . . , xt+τ anex�rthtec tuqaÐec metablhtèc meÐdia katanom (iid):
I fxt (x) = fxt+1(x) = . . . = fxt+τ (x) = f (x)
I fxt ,xt+j (x , y) = fxt (x)fxt+j (y) gia j = 1, 2, . . .
Mia iid qronoseir� eÐnai entel¸c tuqaÐa (den perièqei
autosusqetÐseic grammikèc mh-grammikèc)
Mia iid qronoseir� lègetai kai leukìc jìrubocWN(0, σ2ε )
An ta iid stoiqeÐa thc qronoseir�c akoloujoÔn Gkaousian
katanom , tìte h qronoseir� lègetai Gkaousianìc leukìcjìruboc
Dhm trhc Kougioumtz c Qronoseirèc
TuqaÐoc perÐpatoc
tuqaÐoc perÐpatoc:
xt = xt−1 + εt , εt ∼WN(0, σ2ε )
ArqÐzontac apì k�poia tim µ (gia t = 0):
xt = µ+t∑
j=1
εj ,
O tuqaÐoc perÐpatoc eÐnai mh-st�simh qronoseir� me
stoqastikèc t�seic.
PaÐrnontac tic pr¸tec diaforèc prokÔptei h st�simh
qronoseir� tou leukoÔ jorÔbou.
Dhm trhc Kougioumtz c Qronoseirèc
Grammik susqètish
DÔo t.m. X ,Y :
I sundiaspor�
σXY ≡ Cov[X ,Y ]
I suntelest c susqètishc
ρXY ≡ Corr[X ,Y ]
Qronoseir� {xt}:
I autodiaspor� Cov[xt , xt−τ ] giak�poia ustèrhsh τ
I (grammik ) autosusqètish gia
k�poia ustèrhsh τ :ρτ = Corr[xt , xt−τ ]
EktÐmhsh thc autosusqètishc
ρ̂τ = rτ = Corr(xt , xt−τ ) =
∑nt=τ+1(xt − x̄)(xt−τ − x̄)∑n
t=τ+1(xt − x̄)2
Dhm trhc Kougioumtz c Qronoseirèc
ParadeÐgmata autosusqètishc
xt ∼WN(0, σ2x) =⇒ rτ ∼ N(0, 1/n)
�statistik� mhdenik � autosusqètish an rτ ∈ [−2/√n, 2/√n]
0 5 10 15 200
0.2
0.4
0.6
0.8
1
τ
r(τ)
Iron production, autocorrelation
0 5 10 15 20−0.5
0
0.5
τ
r(τ)
First differences of iron production, autocorrelation
0 5 10 15 20 25 30
−0.4
−0.2
0
0.2
0.4
0.6
τ
r(τ)
Detrended GICP, autocorrelation
2 4 6 8 10−0.5
0
0.5
τ
r(τ)
Detrended and deseasoned GICP, autocorrelation
Dhm trhc Kougioumtz c Qronoseirèc