Statistics for experiments

MAGI

recall

fast special

In is of

case

a :

,

. Yq

31 yz Bq

Y Bo y1 32.

1 = = =

=

= ...,

,

,

o

↑ Yq

q 0

= =... =

+ 1 + 9

=> B

1

E(y w

+ = +

04/1

VI ) 00

+ =

= -2

+- e

et if

Bijen

Ven =

=

i

e e lu

, if q

o

= Vere e

pi e 2

- if 2

,

1

Pe 9

=

= = , ....

Yt-el

PIX MA19)

2 p

if This

Per represented

bow ACF is

is of

+

o 1

+, = = e Malg

l lu

lave

Marg1 stating

In all -AC F

Per fo of

,

1 2 grem Per

1

+

we 9

a g

=

:

= =

, ..., ,

, MArq

Recall the

ARPI

four lave to valif No

that solespomdence

shown also gour

ome

one-

We

MA/1) B

MA(1) twee e

Pr between

not evespendence

) C

= is one-tene am

= =

to

In to

it westwist

the souvespendence

to domain B

the

obtain

auter is of

necessauly

ome

one- ,

stationality

11 simila

condition

11

in fo is

-

, . MAq

It What

MANI but

is about general

you

easy a

,

Invertibility

There beginitions inveufibibility

of

mode

a re .

SPEX AR

be

could represented

Definition invertible it

is as

as

a :

: assumption

usual

↓ component

vanbom

Y 2yt

nY

+ +

+ + uf

w +

= + 2

1 . . .

-

↑

comstamt

and

02) .

Where WN/o convergent

square

j mean

, , i

fandonly

invertible

MAGI

the is avvateustis

its equation

s

drem : Bl Bq1"

Bl lave

1 module

+...

+ >

+

+ 0 .

1

=

Summary at end

always

is

vandom component

definitions : ↑

HYNAR(P) - 02

ARPI /0

x2Yt

+

X i

xPT i b

Where NWN

+ +

2 +

· uf U

w

: = p ,

.

... .

.

- - JurWNo

: Se

MargY

Magl Where

-1 Batt Bqq

· M+ ... ,

= ↳ component

vandom

just explanation at

it's the it end

thue

be

should

were fou ,

it A/L)

AR(P) depend the

stationality Moots of

> 0

· : = on =

MA/q stationality

always

=

AR/P) always invertible

invertibility =

· : MA(q1 depends the BIL)

roots

it

= of 0

on =

ARMA(p qu

, and

AR/p)

Meuge Marqu

MYNARMAp g) Y constant

&pY-p

+Y only

2Y.2 and

only

+ one one

=

, ... component

Bq4

324 vandom

Bu + +

+ + uf

+ + +

1 . . . g

-

- -

two

The clawasteristics equations

xpli B/1)

All)

x22 factous

lave

and

21

1 m e

0 common

=

-

- -

- ... Bq(9

B

B

,

1 + +... +

+ 0 decibe

= general

mandataly we can

in

not

is ,

w to mot

w du

use

↑ situation

depends the

on

g)

ARMAP

If keep

stationally constant the

the

will

is model

in

we w

, ,

9)

ARMAP not

If stationary

is usually smitted Whiel

,

there

wowever special in

is some cases

w a re

,

, ;

, , the

present

though not

G constant and

the

ARMAP is

stationally

is sucle

model

in in

even cases

,

,

,

bift mame

It's

called

it fling

the

is another

With

same

. =

"de-meaned"

If then

to

the the

related Y

variable

so-called is

specification

model is w

-w

+

, ,

the the

model when

not stationary

also

included is

in possess :

,

* aft =

apft +

Bq4 +

By + +

+

+ + w

+ u >

= +

+

p +

1 1 g

. . . . . .

-

- -

g

The ARMAP be me-whilter follows

o p cam as :

,

All(y B((uf

+

w

+ = /MA/I

91

ARMA/p component

the to

stationally

is stationally

ARI)

related stati

is always

is

>

, .

AR/)

denemds by

only

mavy , MANDI

ARMA/P .

If 9) be converted With

stationary if in

is Nj convergent

square

mean

can

, ,

Is !

ARMAP 9) invertible

, AR/)

91

ARMA/P invertible converted

them With

be

it in ij

is square

mean

can

, , b

convergent 91 expementially

ARMA/P decays

it

ACF of > 0.

as

:

,

.

Not stationary so

What Recall the If

?

stationally !

importance stationality Sp not

not

SP

if is is

of

a a

f

"twamsgrum to stationally

ader it

make

stationally in

we cam

, : 02) We

10

Fou NEW Y Yt i demonstrated

d wave

where ufNWN %

example +ut

= 1 ,

.

. .

.

-

not to

flat stationary

IP

this How "transfrum" make it

is auber

it in

we can

.

?

Stationary Y-Y Whe

NWN is stationally

= .

-

Af It

A weak

stationary stationary

is

is weak

ut ut

s >

= -

X X

At + -

= 1

- Dyf

Tyc

Note flue IS this Sys

in case : ....,

,

Not qu

stationality ARMA/p

in ,

g the

Recall stationally ALLI greater

fluat ARMAP noots lave

is all mobile

of to

a

,

these

In noots

tham lave

general

1 complex

. may :

, the sizele)

gueateu flan

1 unit

motule autside

1 ;

. bata

flue circlel the

to will

2 unit real

module con find

in

1

equal we

: ,

. to

Noots equal only

one : the

circle)

1/withim

less won't

. flon

3 data

module unit

the real

in we

: ,

this

don't

type

this gind

find roots Usually

of we .

case

. ALI

the polynonial

becompose

Thus gollows

as

cam

, :

we 218/2)

11

A(l) = -

where : d

ANI it's

lad roots module and degwee

of

1

of P

.

· ,

AllI stationality), deguee

1 .

it's

greater tham

Was noots module and p-d

of of

· I stationality

have

we it'

don't stationary

if

know

Ii

T

BILI

The 91 ALLIX not stationary

SP is

ARMAp It

: = ;

, 11-116/n arma

But "transformed" 9)

the recess :

,

Arma-Äl//1-219Y Biclut is stationary

+= 210y 1y

G V

11

if

> 0 +

+

- > = =

= -

b Y

2(X

(1

if Xt

1

- +

= =

- - 1

-

11247

(1 11 V +

(2)

G 24 +

if 22

2 +

+

-

> + +

=

= 2

- = - 1

- -

qu

(p d

Sp Arima ,

, integrated

I .

stands gou

#Narimalp alllin-218y

g)

d Bill

+

w uf

+=

,

,

GunwN/o se

Where ·

: , XP(P

x12-x222-

Alll 1

· -

= - . . .

(

B(L) Bq29

B B

1 +

· + +...

+

= BIL1

Alll Moots

lave

and 0

0

· common

ne

=

=

All thom

greater

las With

Noots module .

all 1

· o depends

stationary o

o

Remarks

TNARIMAP 9) stationary d

6 is > a

=

,

,

MINARIMA/d g) the

is Marg) component

invertible is invertible

p ;

, ,

MONARIMAP 9) q to

equivalent ARMAP

to

is weets

6 With I equal .

1

,

, ,

another component of sp

↓

↑

Seasonal is

data

lot

SD Seasonal

seasonal

of

a T component

The time indicated by

contain component .

5

often seasonal

series also a ,

Four example : monthly monthly time

series repaut

the

seres

S elease for

· 12 >

: =

time

daily +

series S

· : =

Time lauws 5 24

series

· =

:

1) Pure AR(1) with 5

Seasonal

examples = 12

=

Y 21Y

+ + uf

+

+

w 12

= -

21(12) /

(1 uf

- =

2) Pure with

RW 7

S

Seasonal =

X

↑ uf

+

+

= -

27)x

(1 uf

=

-

Pure seasonal ARIMA

Giu QSally11-191X B(%) uf

Pure ARIMAP

Seasonal D w +

=

, , It's

where => whose

process

a

:

Gui ser meat

tends to

WN/o

~ mogwession be

· , auber P

the of

AR similar

steristico process

of

not. itself

Cava in way

a

↑ - p

Ally x120 to

ascouting

x22 centain

· 1 - a

= -

- ... duter Q pecificity

MA of

elvavadeustive the constant

of

pol Mosess

.

↑ Bals

B22

lo

%

Bl B

· +

1 + +...

+

= ,

Was tham

queater

A/L1 rests module

all of 1

· o

=

%= % lave roots

B11

All and

· 0 no commen

0 =

Remarks

MYNPue QIs

ARIMALP D

stationary

is

Seasonal D =

- >

,

, /V .

Pure

If the ARIMA/P due to

stationality 15

Seasonal Q1S is In

D21 fast

1-26

D

of

no = :

, ,

2 >L to

with

1 5 complex equal

modules

voots all .

1 1

=

0

= =

- = , 11-115 10

Please between

to

attention 11-10

tee difference and As :

=

pay =

, (1 (0) 13y 11 y

27y y

15 +

+

- +

+ s

=

=

- -

= - -

1)/

2) ° / /1

1

11 <A

- =

= -

Both but

not stationally

generates processes :

,

At to

was equal

all

s Noots 1

· ,

different

las to

neets

As .

which 1

equal

is

s only

· one

among

,

~ Pure the

invertible

Q

ARIMA/P that is

Seasonal component invertible

is

D is

MA ;

, ,

/

B/

% tan

lave modules .

1

g