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Qronoseirèc

Dhm trhc Kougioumtz c

10 Maòou 2011

Dhm trhc Kougioumtz c Qronoseirèc

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)


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


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


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


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


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



{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



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


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


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


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.


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


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


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