ww

w.k

it.ed

u

∑∑

==

−−

+=

−p

i

qj

jt

jt

it

it

ZZ

YY

11 θ

φ

AICC

GIC

CIC

AR

MA

mo

de

lling

of G

NS

S re

sid

ua

ls u

sin

gd

iffere

nt m

od

el id

en

tifica

tion

crite

ria

Ge

od

etic

Institu

te

X. L

uo

, M. M

aye

r, B. H

eck

KIT

−T

he

Co

op

era

tion

of F

ors

ch

un

gs

ze

ntru

mK

arls

ruh

e G

mb

H a

nd

Un

ive

rsitä

tK

arls

ruh

e (T

H)

Ge

od

etic

W

ee

k 2

00

9S

ep

tem

be

r 2

2-2

4,

Ka

rlsru

he

S6

: T

he

ore

tica

l G

eo

de

sy

Ge

od

etic

Ins

titute

: X. L

uo

, M. M

aye

r, B. H

ec

k –

luo

@g

ik.u

ni-k

arls

ruh

e.d

eA

RM

A m

od

ellin

g o

f GN

SS

res

idu

als

us

ing

diffe

ren

t mo

de

l ide

ntific

atio

n c

riteria

2

AR

MA

(p,q

) mo

del

Intro

du

ctio

n

AR

MA

Ap

plic

atio

ns

ge

ne

ratin

g p

rog

no

stic

mo

de

ls (e

.g. in

eco

no

mic

scie

nce

s)

an

aly

sin

g p

hysic

ally

co

rrela

ted

pro

cesses (e

.g. in

ge

oscie

nce

s)

mo

de

lling

tem

po

ral c

orre

latio

ns o

f GN

SS

ob

se

rva

tion

s (m

otiv

atio

n)

AR

MA

: Au

toR

eg

res

siv

eM

ovin

g A

vera

ge

)0(

)(

21

11

1,σ

WN

ZZ

ZZ

YY

Yt

qt

qt

tp

tp

tt

~ ,

−−

−−

++

+=

−−

−θ

θφ

φL

L

:)

,(

qp

o

rde

r pa

ram

ete

rs:)

,,

,,

,(

11

Tq

pθ

θφ

φK

K=β

mo

de

l co

effic

ien

ts

:2

σw

hite

no

ise

(WN

) va

rian

ce

Mo

del id

en

tificatio

n c

riteria

su

bje

ctiv

e m

eth

od

s (s

tatis

tica

l tests

, gra

ph

ics, e

tc.)

ob

jectiv

e m

eth

od

s (s

pe

cifie

d d

ecis

ion

crite

ria)

Ge

od

etic

Ins

titute

: X. L

uo

, M. M

aye

r, B. H

ec

k –

luo

@g

ik.u

ni-k

arls

ruh

e.d

eA

RM

A m

od

ellin

g o

f GN

SS

res

idu

als

us

ing

diffe

ren

t mo

de

l ide

ntific

atio

n c

riteria

3

Crite

rion

AIC

C

Assu

mp

tion

s a

nd

a p

riori c

on

ditio

ns

:)

,,

(1

Tn

nX

XK

=X

ob

se

rvatio

ns fro

m a

Ga

ussia

nA

RM

A(p

,q) p

roce

ss

:)

,(

2σ

βψ

=tru

e m

od

el p

ara

me

ters

of A

RM

A(p

,q) p

roce

ss

:)

ˆ, ˆ

(ˆ

2σ

βψ

=m

axim

um

like

liho

od

estim

ato

ro

f ψ

ba

se

d o

n

nX

:)

,,

(1

Tn

nY

YK

=Y

ind

ep

end

en

t rea

lisa

tion

of A

RM

A(p

,q) p

roce

ss w

ith ψ

L: G

au

ssia

n lik

elih

oo

d fu

nctio

n o

f an

AR

MA

(p,q

) pro

ce

ss, e

.g.

−

−=

∑=

−

−−

−nj

j

jj

nn

Xr

XX

rr

L

11

2

2

2/1

10

2/2

2)

ˆ(

ˆ2

1ex

p)

()

ˆ2(

)ˆ

, ˆ(

σσπ

σ

Lβ

Ku

llback-L

eib

ler

ind

ex (K

LI)

()

)(

)ˆ

()

ˆ(

ln2

)|

ˆ(

ψψ

ψψ

ψψ

YY

YL

LL

E

to

rela

tive

of

K

LI

−

=∆

(1)

(2)

()∑

=−

−−

==

nj

jj

jX

rX

Xn

Sn

1

12

12

)ˆ

(1

ˆˆ

β

σ

Ma

xim

um

like

liho

od

estim

ato

rs:

on

e-s

tep

pre

dic

tor fo

r: re

-sc

ale

d m

ea

n s

qu

are

d e

rrors

of

:ˆ

jX

1−j

r;j

Xj

X̂

(3)

Inn

ovatio

ns

alg

orith

m

Ge

od

etic

Ins

titute

: X. L

uo

, M. M

aye

r, B. H

ec

k –

luo

@g

ik.u

ni-k

arls

ruh

e.d

eA

RM

A m

od

ellin

g o

f GN

SS

res

idu

als

us

ing

diffe

ren

t mo

de

l ide

ntific

atio

n c

riteria

4

Crite

rion

AIC

C

Estim

atio

n o

f KL

I

()n

SL

LY

XY

−+

−=

−2

ˆˆ

)ˆ

(ln

2)

ˆ(

ln2

σβ

ψψ

()

()

()(

)n

SE

LE

LE

YX

Y−

+−

=−

==

2ˆ

ˆ)

ˆ(

ln2

)ˆ

(ln

2)

|ˆ

(σ

∆β

ψψ

ψψ

ψψ

ψ

:K

LI

AIC

C: A

kaik

eIn

form

atio

n C

riterio

n (A

IC) w

ith s

mall s

am

ple

Co

rrectio

n

()(

))

2(

)1

(2

ˆˆ

2−

−−

++

≈q

pn

nq

pS

EY

σβ

ψ

AIC

C c

riterio

n (H

urv

ich

and

Tsa

i 19

89

):

()

2 )1

(2

)(

,ln

2:

)(

−−

−+

++

−=

qp

n

nq

pn

SL

XX

ββ

βA

ICC

(5)

(6)

Usin

g la

rge

-sa

mp

le a

pp

roxim

atio

ns:

(4)

Crite

rion

AIC

(Akaik

e1973)

AIC

C a

nd

AIC

asym

pto

tica

lly e

qu

iva

lent a

s

AIC

C p

artic

ula

rly a

dvis

ab

le fo

r sm

all s

am

ple

siz

es

()

)1

(2

)(

,ln

2:

)(

qp

nS

LX

X+

++

−=

ββ

βA

IC

∞→

n∞→

n

(7)

Ge

od

etic

Ins

titute

: X. L

uo

, M. M

aye

r, B. H

ec

k –

luo

@g

ik.u

ni-k

arls

ruh

e.d

eA

RM

A m

od

ellin

g o

f GN

SS

res

idu

als

us

ing

diffe

ren

t mo

de

l ide

ntific

atio

n c

riteria

5

Crite

ria C

IC a

nd

GIC

CIC

for A

R(p

) ord

er s

ele

ctio

n (K

lees

an

d B

roers

en

2002)

−+

−−

+−

−+

++

=∏

∑=

=

p

k

p

kk

nk

n

kn

pR

ES

p

11

1 13

,1)

1/(

11

)1

/(1

1m

ax)

(ln

)(

C

IC

with

the

resid

ua

l (estim

ate

d W

N) v

aria

nce

22

1

ˆˆ

()

(1),

: Yu

le-W

alk

er a

nd

Bu

rg e

stim

ate

s

p

kk

k

RE

Sp

σϕ

ϕ=

=−

∏

CIC

: Co

mb

ined

Info

rmatio

n C

riterio

n; G

IC: G

en

era

lised

Info

rmatio

n C

riterio

n

(8)

(9)

GIC

for M

A(q

) resp

. AR

MA

(p,q

) ord

er s

ele

ctio

n

qp

mq

pq

mq

+=

=:

),

(,

:)

( A

RM

AM

A

{}

(,

)ln

()

3,

()

:G

IC

Du

rbin

's m

eth

od

sm

mR

ES

mR

ES

mn

α=

+(1

0)

Mo

del id

en

tificatio

n b

ased

on

pre

dic

tion

erro

r (PE

)

Nm

Nm

mR

ES

mP

Ek

n

kn

pR

ES

pP

EA

Rpk

AR

AR

/1

/1

)(

)(

,)1

/(1

1

)1

/(1

1)

ˆ(

)ˆ

(

ˆ

1− +

⋅=

−+

−−

++

=∏

=

qp

mq

pq

mq

MA

ˆˆ

:)

,(

,ˆ

:)

(+

==

AR

MA

MA

(11

)

Ge

od

etic

Ins

titute

: X. L

uo

, M. M

aye

r, B. H

ec

k –

luo

@g

ik.u

ni-k

arls

ruh

e.d

eA

RM

A m

od

ellin

g o

f GN

SS

res

idu

als

us

ing

diffe

ren

t mo

de

l ide

ntific

atio

n c

riteria

6

Ca

se s

tud

y: d

ata

ba

se

HE

DA

54.1

km

TA

AF

53.7

km

RA

TA

203.7

km

SIB

I

42.5

km

AF

LO

32.4

km

Mu

ltipath

(MP

): stro

ng

MP

: weak

Fig

. 1: S

AP

OS

®n

etw

ork

in th

e a

rea o

f the s

tate

o

f Bad

en

-Wü

rttem

berg

(So

uth

west G

erm

an

y)

Ind

ivid

ual a

bso

lute

calib

ratio

nA

nt. c

orre

ctio

n

Based

on

sig

nal q

uality

measu

res (S

NR

*) O

bs. w

eig

htin

g

1-H

z G

PS

ph

ase d

ou

ble

diffe

ren

ces

Ob

serv

atio

ns

Nie

lldry

(a p

riori), M

F**: N

iellw

et(1

996)

Atm

os. m

od

ellin

g

Calc

ula

ting

ep

och

-wis

e m

ean

valu

es o

fh

igh

ly c

orre

late

d S

DD

R tim

e s

erie

sT

ren

d m

od

ellin

g

Stu

den

tised

ph

ase d

ou

ble

diffe

ren

ce

resid

uals

(SD

DR

) in s

idere

al

time

Tim

e s

erie

s d

ata

(No

. 210, n

=3600)

Stro

ng

(HE

DA

), weak (o

ther b

aselin

es)

Mu

ltipath

imp

act

Tab

. 1: G

PS

pro

cessin

g s

trate

gie

s a

nd

data

ch

ara

cte

ristic

s

AR

MA

sel

(Bro

ers

en

2000)

Min

imum

of

CIC

, GIC

and

pre

dic

tion e

rrors

Burg

’s a

lgorith

m

Durb

in’s

meth

ods

(least-s

quare

s)

CIC

, GIC

ITS

M2000

(Bro

ckw

ell

and D

avis

2002)

Min

imum

of

AIC

C v

alu

es

Hannen-R

issannen

Innovatio

ns a

lgorith

m

(maxim

um

likelih

ood)

AIC

C

So

ftware

packag

e

AR

MA

iden

tificatio

n

Para

mete

r

estim

atio

nC

riterio

n

Tab

. 2: A

RM

A m

od

ellin

g u

sin

g d

iffere

nt id

en

tificatio

n c

riteria

*SN

R: S

igna

l-to-N

ois

e R

atio

, **MF

: mappin

g fu

nctio

n

Ge

od

etic

Ins

titute

: X. L

uo

, M. M

aye

r, B. H

ec

k –

luo

@g

ik.u

ni-k

arls

ruh

e.d

eA

RM

A m

od

ellin

g o

f GN

SS

res

idu

als

us

ing

diffe

ren

t mo

de

l ide

ntific

atio

n c

riteria

7

searc

hare

a:

searc

hare

a:

p=

q+

1p

=q

AIC

CC

IC, G

IC

}1

0,

,1{

}1

0,

,1{

K K

∈ ∈q p

1

}1

0,

,1{

−= ∈

pq p

K

Ord

er s

ele

ctio

n

Fig

. 2: C

om

paris

on

of o

rder s

ele

ctio

n u

sin

g d

iffere

nt m

od

el id

en

tificatio

n c

riteria

10

max

max

==

qp

MP

stro

ng

Ge

od

etic

Ins

titute

: X. L

uo

, M. M

aye

r, B. H

ec

k –

luo

@g

ik.u

ni-k

arls

ruh

e.d

eA

RM

A m

od

ellin

g o

f GN

SS

res

idu

als

us

ing

diffe

ren

t mo

de

l ide

ntific

atio

n c

riteria

8

∑≈

→=

03

.

0|

|5.

02

vv

Au

toc

orre

latio

n fu

nc

tion

(AC

F)

SD

DR

: TA

AF

18261

68

mu

ltipath

: weak

SD

DR

: HE

DA

1826168

mu

ltipath

: stro

ng

Fig

. 3: C

om

paris

on

of a

uto

co

rrela

tion

fun

ctio

n u

sin

g d

iffere

nt m

od

el id

en

tificatio

n c

riteria

Ge

od

etic

Ins

titute

: X. L

uo

, M. M

aye

r, B. H

ec

k –

luo

@g

ik.u

ni-k

arls

ruh

e.d

eA

RM

A m

od

ellin

g o

f GN

SS

res

idu

als

us

ing

diffe

ren

t mo

de

l ide

ntific

atio

n c

riteria

9

∑≈

→=

dB

/Hz

6.0

||

10

02

vv

Po

wer s

pe

ctra

l de

nsity

(PS

D)

SD

DR

: TA

AF

18261

68

mu

ltipath

: weak

SD

DR

: HE

DA

1826168

mu

ltipath

: stro

ng

Fig

. 4: C

om

paris

on

of p

ow

er s

pectra

l den

sity

usin

g d

iffere

nt m

od

el id

en

tificatio

n c

riteria

Ge

od

etic

Ins

titute

: X. L

uo

, M. M

aye

r, B. H

ec

k –

luo

@g

ik.u

ni-k

arls

ruh

e.d

eA

RM

A m

od

ellin

g o

f GN

SS

res

idu

als

us

ing

diffe

ren

t mo

de

l ide

ntific

atio

n c

riteria

10

Wh

ite n

ois

eσ̂

Sate

llite p

air: P

RN

0917

Site

: Ta

ub

erb

isch

ofs

heim

AR

MA

sim

ula

tion

SD

DR

data

: TA

AF

18

26168 (M

P: w

eak)

Fig

. 5: C

om

paris

on

of A

RM

A s

imu

latio

n b

ase

d o

n d

iffere

nt m

od

el id

en

tificatio

n c

riteria

Sim

ula

tion

(AIC

C): A

RM

A(3

, 3)

Sim

ula

tion

(CIC

, GIC

): AR

MA

(2, 1

)

SD

DR

data

: HE

DA

18

26168 (M

P: s

tron

g)

Sim

ula

tion

(AIC

C): A

RM

A(5

, 10)

Sim

ula

tion

(CIC

, GIC

): AR

MA

(3, 2

)

Ge

od

etic

Ins

titute

: X. L

uo

, M. M

aye

r, B. H

ec

k –

luo

@g

ik.u

ni-k

arls

ruh

e.d

eA

RM

A m

od

ellin

g o

f GN

SS

res

idu

als

us

ing

diffe

ren

t mo

de

l ide

ntific

atio

n c

riteria

11

Facto

rs im

pactin

g th

e id

en

tificatio

n p

erfo

rman

ce

GN

SS

ob

se

rva

tion

al d

ata

(e.g

. da

ta q

ua

lity, s

am

ple

siz

e)

ap

plie

d a

lgo

rithm

s fo

r pa

ram

ete

r estim

atio

n

the

hig

he

st c

an

did

ate

ord

er fo

r se

lectio

n

Co

mp

aris

on

of th

e u

sed

iden

tificatio

n c

riteria

Crite

rion

AIC

C

hig

he

r sele

cte

d o

rde

rs w

ith s

trong

va

riability

be

tter p

erfo

rman

ce

in th

e c

ase o

f low

-qua

lity d

ata

larg

e o

rde

r searc

h a

rea

→tim

e-c

on

su

min

g c

om

pu

tatio

n

Crite

ria C

IC, G

IC

low

er s

ele

cte

d o

rde

rs w

ith c

om

pa

rable

mo

dellin

g re

su

lts

rea

so

na

ble

redu

ctio

n o

f ord

er s

ea

rch

are

a →

less c

om

pu

tatio

nal c

ost

No

t the v

ery

best, b

ut th

e m

ost re

liab

le a

nd

effic

ien

t crite

rion

!

Co

nclu

sio

ns

iden

tical w

ithin

th

is c

ase s

tud

y

Ge

od

etic

Ins

titute

: X. L

uo

, M. M

aye

r, B. H

ec

k –

luo

@g

ik.u

ni-k

arls

ruh

e.d

eA

RM

A m

od

ellin

g o

f GN

SS

res

idu

als

us

ing

diffe

ren

t mo

de

l ide

ntific

atio

n c

riteria

12

Ge

od

etic

Ins

titute

−U

niv

ers

ität

Ka

rlsru

he

(TH

) −E

ng

lers

traß

e7

, 76

13

1 K

arls

ruh

e, G

erm

an

yT

el.: +

49

(0)7

21

60

8 3

668

, Fa

x: +

49

(0)7

21

60

8 6

808

, ho

me

pa

ge

: ww

w.g

ik.u

ni-k

arls

ruh

e.d

e

GP

S s

yste

m m

od

ern

isatio

ns +

ad

van

ce

d m

ath

em

atic

al m

od

ellin

g =

ac

cu

rate

an

d re

liab

le p

ositio

nin

g re

su

lts

Th

an

k y

ou

very

mu

ch

for y

ou

r atte

ntio

n!

Qu

estio

ns

& c

om

men

ts

Th

e p

roje

ct “Im

pro

vin

g th

e s

toch

astic

mo

de

l of G

PS

ob

se

rva

tion

s b

y

mo

de

lling

p

hysic

al

co

rrela

tion

s”

(HE

14

33

/16

-1/2

) is

su

ppo

rted

by

the

De

uts

ch

e F

ors

ch

ung

sge

me

insch

aft

(DF

G).