Spatial-Temporal Modelling of Road Traffic Data in Seoul City

2 002 , V ol. 13, N o.2 p p . 2 61～2 70

S p ati al - T e m poral M o de llin g o f R o ad T raffic D at a in S e ou l City

S an g y e ol L e e ¹⁾ , S o oh an A h n ^{2 )} , Ch an g y i P ark ^{3 )} an d Jon g w o o Je o n ^{4 )}

A b s tra c t

Recent ly , t h e dem an d of th e In t ellig ent T r an sport at ion S y st em (IT S ) h a s b een in cr ea sed t o a lar g e ex t en t , an d a r eal - tim e tr affic inform at ion serv ice b a s ed on th e in t ern et sy st em b ecam e v ery im p ort ant . W h en IT S com p anies carry ou t r eal - t im e t raffic serv ices , th ey fin d som e t r affic dat a m is sin g , an d u se t h e conv en t ion al m eth od of recon st ru ct in g m is sin g v alu es by calcu latin g av er ag e t im e tr en d. H ow ev er , t h e m eth od is foun d un sat isfa ct ory , so t h at w e dev elop a n ew m et h od b a sed t h e sp atial an d spatial- t em por al m odels . A cr os s - v alidat ion t echn iqu e sh ow s t h at th e spatial- t em por al m odel ou tp erform s th e ot h er s .

K e y w o rd s : In t ellig ent T r an sport at ion S y st em (IT S ), sp at ial m odel, spatial- t em por al m odel, spat ial n eighb orh ood stru ctu r e, t im e serie s m odel, cr os s - v alidat ion m et h od .

1 . In tro du c ti on

In r ecent y ear s , t h e dem an d of t h e IT S h a s b een r apidly in cr ea sed, an d t h e t a sk of collect in g r eal - tim e t r affic in form ation h a s b ecom e cru cial in run nin g t h e IT S bu sin es s . F or obt ain in g a ccur at e inform at ion , IT S com panies h av e m ade effort s for an aly zin g t h e t r affic dat a in lin e s w it h t h e int ern et bu sin es s . H ow ev er , su ch an effort oft en en d s u p w it h u n desir able re sult s . T h e ROT IS , an IT S com pan y locat ed in th e m et r op olit an ar ea s of S eoul cit y , h a s b een collect in g th e r eal- t im e tr affic dat a . T h e dat a con sist s of t h e in form ation obt ain ed t hr ou g h t h e b eacon - b a s ed in fr a stru ctu r e at ev ery 15 m inu t e s . It pr ov ides cu st om er s w ith u s eful 1. A s s ociat e Pr ofes s or , Departm ent of St at istics , S eoul Nat ional Univ er sity , Seoul, 151- 742.

E - m ail: sylee@st at s .snu .ac .kr

2. P ost doct or , Depar tm ent of St atist ics , Seoul N ational Univ er s ity , Seoul, 151- 742.

3. Ph .d student , Departm ent of St atistics , Seoul National Univ er sity , S eoul, 151- 742.

4. Pr ofes s or , Departm ent of St at istics , S eoul Nat ional Univ er sity , Seoul, 151- 742.

t r affic inform at ion v ia calcu lat in g th e t r av elin g v elocity b et w een t w o prea s sign ed location s , m ea su r ed fr om t h e r oa d - side b eacon s an d pr ob e car s equipped w ith in - v ehicle m odu le. H ow ev er , s om e portion s of dat a ar e foun d m is sin g sin ce t h e t r affic in form at ion s ar e collect ed on ly from pr ob e car s .

A sim ple w ay t o s olv e t h e m is sin g dat a problem is t o u se a t im e t ren d . F or illu str at ion , let u s a s sum e th at du rin g 7 :01- 7 :15 p .m ., th e t r affic dat a on A r oad is m is sin g . In t his ca se, a conv ent ion al m eth od t o est im at e t h e m is sin g dat a is t o u se a t im e tr en d , calculat ed a s th e av era g e of ob serv ed p a st dat a corr espon din g t o 7 :01- 7 :15 p .m . on A r oad . A n ot h er w ay is t o u se t h e tr affic dat a on A r oad obt ain ed durin g , s ay , 6 :46 - 7 :00 p .m . an d pr ediction alg orit hm of fitt ed t im e s eries m odel for dat a on A r oad . A ctu ally , th is is t h e w ay th at th e ROT IS h an dles t h e pr oblem . H ow ev er , in ut ilizin g t h e m et h od on e u su ally fin d s a lar g e g ap b et w een t h e r eal an d pr edict ed v alu e s . T h e n eg at iv e r esult is du e t o t h e fact t h at t h e t r affic flow does n ot alw ay s follow on e patt ern in t im e . In t h at ca s e, an ex t r a inform at ion sh ou ld b e t ak en for com p en s at ion fr om an ot h er sou r ce. T h erefor e, w e con sider u t ilizin g th e dat a on ot h er r oad s t o set u p m or e soph ist icat ed st at ist ical m odel, esp ecially t h e dat a on th e n eigh b orin g road s . F r om t his r ea s on in g , a spatial m odel is t ak en int o con sider at ion . S o is t h e sp at ial - t em p or al m odel sin ce t r affic dat a h a s a st och a st ic t r en d b oth in t im e an d space .

T h e sp at ial m odels ar e w idely u sed in m an y applicat ion field s . Cliff an d Or d [9]

su g g est ed th e spat ial aut or eg r es siv e, m ov in g av er ag e an d r egr es siv e m odel an d an aly zed th e dat a Ca s ett i an d S em ple [5]. A li [2] an aly zed t h e S T A R (sp at ial t em por al au t o- r egr es sion ) pr oces s w it h un it sp atial or der . P feifer an d D eut sch [12], [13] an d Deu t sch an d P feifer [14] ex t en ded th e S T A R m odel t o t h e S T A RM A (spat ial t em por al aut o- r eg r es siv e m ov in g av er ag e ) m odel. S ee also Nu i an d T ia o [11], Cr es sie an d M aju r e [7] an d H u an g an d Cr es sie [10].

T h is pap er is or g anized a s follow s . In S ection 2, w e int r odu ce a sp at ial m odel w it h cert ain sp atial n eigh b orh ood st ru ctu r e su it able for tr affic dat a . B a sed on t his , w e also int r odu ce a spatial- t em por al m odel, an d dev elop a predict ion pr ocedu r e. In S ection 3, w e com p ar e six predict ion pr ocedur es th rou g h r eal dat a an aly sis . T h ey are t h e pr ocedu r es u sin g th e av er ag e t im e tr en d, th e t im e series m odel, t h e spat ial m odel, an d t h e spat ial- t em por al m odel. T h e dat a u sed h er e is t h e t r affic v elocit y dat a at Gan gn am - Gu , a dow n t ow n ar ea of S eou l cit y w it h 64 link s . It is sh ow n t h at t h e m et h od b a s ed on spat ial - t em p or al m odel out perform s t h e oth er m et h od s .

2 . P re di cti on of m i s s in g d at a

Befor e w e pr oceed, w e int r odu ce s om e n ot ation s an d t erm in olog ie s orig in at ed

fr om t h e t r an sport at ion en g in eerin g (cf. Ban k s [3]). A cr os sr oad is called a n od e

an d is giv en a un iqu e id nu m b er . A r oa d b et w een t w o cont igu ou s n odes is called

a link . It is im p ort ant t o n ot ice t h at t h er e ar e t w o link s b et w een con tigu ou s n odes .

F or ex am ple, for n odes A an d B , t h ere is t h e lin k fr om A t o B (A - B lin k ) an d

t h e on e fr om B t o A (B - A link ). In th e m eant im e, w e defin e N i at link i a s th e fam ily of n eig hb orin g lin k s eit h er t hr ou gh w hich car s can r each t h e lin k i or w h ich on e can r each t hr ou g h t h e lin k i. Ev ery t im e th ey pa s s a lin k , pr ob e car s s en d t h e con tr ol cent er a v elocit y dat a . N ow let X _i ( t) , i=1,…, L , t =1,…, J, den ot e

t h e ob serv ed v elocit y (km/ h ) dat a at link i an d tim e t. In our an aly sis , w e con sider t h e m ean - a dju st ed r .v .' s

Y i ( t) = X i ( t) - i ( t) , i =1,… L ,, t =1,…, J, (1) w h ere _i ( t) is th e tim e tr en d at link i t im e t.

2 .1 S p ati al m o de l

2 .1 .1 F ir s t ord e r N e ig h b orh o o d s t ru c tu re

F or t=1,…,96, w e con sider th e spatial m odel : Y i ( t) =

j N

c ¹ i , j ( ¹ ( t) ) Y j ( t) + i ( t) , i = 1, … , L , (2)

cf. Cr es sie [6]) w h ere { _i ( t) , i = 1 , , L } is a fam ily of in depen dent r .v . ' s w it h m ean 0 an d v arian ce ² . N ot e t h at c ¹ i , j ( ¹ ( t) ) , w hich ar e t h e spat ial dep en den ce param et er , can b e defin ed a s t h e (i,j )- t h com pon en t of t h e sy m m et ric

L L m atrix C ¹ ( ¹ ( t) ) :

C ¹ ( ¹ ( t) ) = ¹ ₁ ( t) C ¹ ₁ + ₂ ¹ ( t) C ¹ ₂ + ₃ ¹ ( t) C ¹ ₃ , (3 ) w h ere ¹ ( t) = ( ¹ ₁ ( t) , ¹ ₂ ( t) , ₃ ¹ ( t) ) ' an d t h e (i,j )- th elem en t of C ¹ ₁ , C ₂ ¹ , C ¹ ₃ is a s follow s :

( C ¹ ₁ ) _{i , j} = { ^{0 1} if i an d j ar e lin k ed b y g oin g st r aig ht

o.w.

( C ¹ ₂ ) _{i , j} = { ^{0 1} if i an d j ar e link ed by t urnin g t o th e rig ht

o.w. (4)

( C ¹ 3 ) i , j = { 0 ¹

if i an d j ar e link ed by t urnin g t o th e left o.w.

T h en , w e can w rit e

[I - C ¹ ( ¹ ( t) ) ]Y (t ) = (t ) (5 )

w h ere Y (t ) = (Y

(t ),…,Y

(t ))' an d (t ) = (

(t ),…,

(t )) ' .

2 .1 .2 S e c o n d o rde r N e i g h b o rh o o d s t ru c t u re

H ere w e con sider t h e sp atial m odel for each t im e t : Y i ( t) =

j c ² i , j ( ² ( t) ) Y j ( t) + i ( t) , i = 1 , … , L , (6 )

w h ere { i ( t) ; i = 1 , … , L } is a fam ily of in dep en den t r .v .' s w it h m ean 0 an d

v arian ce ² , an d c ² _{i , j} ( ² ( t) ) is th e (i,j )- th com p on ent of t h e follow in g sy m m etric L L m atrix :

C ² ( ² ( t) ) C ² ( ² ( t) ) =

9 k = 1

2

k ( t) C ^{2 ,} _k (7)

w h ere ² ( t) = ( ² ₁ ( t) , … , ² ₉ ( t) ) ' an d t h e (i,j )- t h elem ent of t h e sy m m et ric m at rices C ² ₁ , , C ² ₉ ar e defin ed a s follow s :

( C ² ₁ ) _{i , j} = { ^{0 1} if i an d j ar e link ed by g oin g st raig ht

o.w.

( C ² ₂ ) _{i , j} = { ^{0 1} if i an d j ar e link ed by t urn in g t o t h e rig ht

o.w. (8)

( C ² ₃ ) _{i , j} = { ^{0 1} if i an d j ar e link ed by t urn in g t o t h e left

o.w.

( C ₄ ² ) _{i , j} = { ^{0 1} if i an d j ar e link ed by g oin g st raig ht t w ice

o.w.

( C ² ₅ ) _{i , j} = { ^{0 1}

if i an d j ar e link ed by tu rnin g t o t h e righ t an d on e - t im e g oin g st r ait

o.w.

( C ² ₆ ) _{i , j} = { ^{0 1}

if i an d j ar e link ed by t urn in g t o t h e left an d t h en g oin g str aigh t

o.w.

( C ² ₇ ) _{i , j} = { ^{0 1}

if i an d j ar e link ed by t u rn in g t o th e rig ht an d t h en tu rnin g t o t h e left

o.w.

( C ² 8 ) i , j = { 0 ¹

if i an d j ar e link ed by t urn in g t o t h e rig ht t w ice o.w.

( C ² 9 ) i , j = { 0 ¹

if i an d j ar e link ed by t urn in g t o t h e left t w ice o.w.

T h en , w e obt ain

[I - C ² ( ² ( t) ) ]Y (t ) = (t ) (9 )

w h ere Y (t ) = (Y

(t ), ,Y

(t )) ' an d (t ) = (

(t ), ,

(t ))' 2 .1 .3 P ara m e t e r e s tim at i on an d p re di c t i on

T h e est im at or s ^v ( t) of , ^v ( t) v =1, 2, ar e obt ain ed a s t h e on es th at m inim ize t h e err or sum of squ ar es , th at is ,

v ( t) = ar gm in Y ' (t )[I - C ( ^v ( t) ) ] ' [I - C ( ^v ( t) ) ]Y (t )

(10) If w e set N ₁ = 3 an d N ₂ = 9 , it follow s t h at

v ( t) = ( A ^v ) ^{- 1} b ^v , v =1,2, (11)

w h ere A ^v is an N v N v m at rix an d it s (l,m )- t h com pon en t is Y ' ( t)( C ^v _l ) ' ( C ^v _m ) Y ( t) an d

b ^v = ( b ^v ₁ , … , b ^v _N

) ' , b ^v _k = Y ' ( t)( C ^v _k ) ' Y ( t) , k = 1 , … , N _v , v = 1 , 2 .

N ow let u s con sider t h e ca s e t h at dat a m is sin g occu r s . Let M den ot e t h e set of link s w h er e t h e dat a is m is sin g , an d let Y ^{( M )} ( t) den ot e t h e v ect or of ob serv ed dat a on ly . T h en , t ak en in t o con sideration Y ^{( M )} ( t) , redefin e t h e m at rix C ^v , v = 1 , 2 , in E qu ation s (3 ) an d (7 ), an d t h e m atrices C _k ^v , k = 1 , … , N _{v ,} v = 1 , 2 , in E qu ation s (4 ) an d (8). T h at is , if i M , t h en w e r em ov e t h e i- t h colu m n an d r ow fr om th e m atrices C ^{v '} s an d C _k ^{v '} s. D en ot e th em

b y C ^{v , M} an d C _k ^{v , M} , k = 1 , … , N _{v ,} v = 1 , 2 , an d fin d t h e e st im at or s ^{v ( M )} by

m inim izin g t h e err or sum of squ are s :

v ( M )

=ar gm in ( Y ^{( M )} ( t) ) ' ( I - C ^{v , M} ) ' ( I - C ^{v , M} ) Y ^{( M )} ( t) (12) F in ally , w e predict t h e m is sin g dat a X i ( t) of t h e lin k i M a s follow s :

X i v , S

( t) = i ( t) +

j M ,j N

c ^v i , j ( ^{v ( M )} ( t) ) Y j ( t) , v = 1 , 2 . (13 )

2 .2 S p at ia l - t e m p o ral m o d e l

In t his su b section , w e con sider t h e sp at ial - t em p or al m odel:

Z ^v i ( t) = Y i ( t) -

j N

c ^v i , j ( ^v ( t) ) Y i ( t) , Z ^v _i ( t) -

p l = 1

v

i , l Z ^v _i ( t - l) = _i ( t) +

q m = 1

v i , m

v

i ( t - m ) , v = 1, 2 (14 ) w h ere {Z ^v _i ( t) } are in depen dent of {Z _j ^v ( t) } for j i, an d { ^v i ( t) , t = 1 , 2 , … , J }, a fam ily of i.i.d. r .v .' s w it h t h e m ean 0 an d v arian ce ( ^v i ) ² , are in dep en den t of { j ^v ( t) , t = 1, 2 , … , J } for j i. W e call Z i ( t) th e spat ial r esidu al of link i at t im e t.

N ot e t h at t h e fir st equ at ion in (14) is t h e spat ial m odel con sider ed in S ection 2.1 an d t h e s econ d equ at ion form s a st at ion ary A RM A ( p , q) t im e series m odel.

T h e m odel in dicat es t h at t h e sp at ial r esidu als obt ain ed aft er rem ov in g t h e effect s

of n eigh b orin g link s st ill h av e a correlat ion , b ut on ly in tim e n ot in sp ace. F or

e st im at in g t h e par am et er s , w e u se a sort of t h e plu g - in m et h od. T h at is , w e

a dju st th e m odel in (14 ) a s follow s :

Z _i ^{v *} ( t) = Y _i ( t) -

j N

c ^v _{i , j} ( ^v ( t) ) Y _i ( t) ,

Z _i ^{v *} ( t) -

p l = 1

v

i , l Z _i ^{v *} ( t - l) = _i ( t) +

q m = 1

v i , m

v

i ( t - m ) , v = 1 , 2 (15 )

w h ere ^v ( t) is t h e on e defin ed in E qu ation (11). In h an dlin g m is sin g v alu es , w e r eplace ^v ( t) b y ^{v ( M )} ( t) defin ed in E qu at ion (12).

F or each link , w e det erm in e th e or der s p an d q u sin g spat ial r esidu als {Z i ^{v *} ( t) , t = 1 , 2 , ¨ , J } defin ed in E qu ation (15 ) b a s ed on A k aik e ' s In form at ion Crit erion (A IC ). T h at is , w e con sider A RM A ( p , q) m odels , p , q = 1 , … , 6 , at ea ch

lin k an d ch oose t h e A RM A ( p ^{v ,} _i q ^v _i ) m odel w hich h a s th e sm allest A IC v alu e am on g t h ose m odels . On ce w e ch oos e an opt im al m odel, w e can e st im at e t h e p ar am et er s ' s , ' s an d ' s b a s ed on t h e lea st squ are s m eth od .

N ow , in pr edict ion of m is sin g v alu es w e u s e t h e K alm an filt er alg orith m . F or t his , w e n eed a form of a st at e spa ce m odel for t h e secon d equ at ion of M odel (15 ) (cf. H arv ey [1]). T h en , sim ilarly t o (13 ), w e can pr edict m is sin g dat a . T h e det ails are om it t ed for br ev it y .

3 . D at a A n aly s i s

F r om t h e y ear 1998, t h e ROT IS h a s b een collect in g t h e real- t im e t r affic inform at ion s in S eoul m et r opolit an ar ea s u sin g ab out 15,000 link s t hr ou g h t h e b eacon - b a s ed infr a st ru ct u re . A t pre sen t , ab out 5 m illion n um b er of dat a are g at h er ed ev ery day . In t his

section , w e com pare our pr oposed pr edict ion procedur e w ith th ose b a sed on th e t im e t r en d m eth od an d tim e s eries an aly sis . F or t h e com p aris on , w e con sider only dat a ob serv ed in th e 64 link s of t h e Gan g n am - Gu ar ea in S eou l fr om F ebru ary 7 t o F eb ru ary 28 in 2001 y ear ex cept S at u rday s an d S un day s , 16 day s in t ot al. W e fir st obt ain t h e t im e t ren d i ( t) in E qu at ion (1) an d th e t im e s eries p ar am et er s in S ection 2.2 for each link u sin g dat a .

' s an d ' s an d ^{2 '} s

A . A v e ra g e t im e t re n d m o de l

T his m odel pr edict s X _i ( t) on ly u sin g _i ( t) of dat a obt ain ed at link i an d t im e t.

T . T im e s e rie s m o d e l

F or each link i , w e ch oos e an opt im al A RM A ( p , q) m odel ou t of t h e A RM A m odels w it h p , q = 0 , … , 6 , for { Y _i ( t) } , i = 1 , … , L , w h er e { Y i ( t) } is defin ed in E qu at ion (1). If Y i ( t|t - 1) is th e on e - st ep pr edict or , t h e pr edict ed v alu e X _i ^T ( t) of X _i ( t) is g iv en a s follow s (cf. H arv ey [1]):

X _i ^T ( t) = _i ( t) + Y _i ( t| t - 1) . (16 )

S 1. S p a ti al m o d e l w it h t h e f ir s t ord e r n e i g h b o rh o o d s t ru c t u re

W e pr edict th e m is sin g dat a u sin g spatial m odel w it h t h e fir st or der n eighb orh ood st ru ctu r e in S ect ion 2.1. T h e pr edict or is X i

1 , S

( t) a s in S ection 2.1.

S 2. S p a ti al m o d e l w it h t h e s e c o n d o rde r n e i g h b o rh o o d s tru c t u re

W e pr edict th e m is sin g dat a u sin g spatial m odel w it h th e secon d or der n eighb orh ood st ru ct u re in S ection 2.1. T h e pr edict or is X _i ^{2 , S} ( t) a s in S ection 2.1.

S T 1. S p at ia l - t e m p o ral m o d e l w it h t h e f ir s t ord e r n e ig h b o rh o o d s t ru c tu re W e pr edict th e m is sin g dat a u sin g spat ial- t im e m odel w it h th e fir st or der

n eighb orh ood st ru ct ur e in S ect ion 2.2. T h e pr edict or is X _i ^{1 , S T} ( t) a s in S ect ion 2.2.

S T 2. S p at ia l - t e m p o ral m o d e l w it h t h e s e c o n d o rde r n e i g h b o rh o o d s tru c t u re W e pr edict th e m is sin g dat a u sin g sp atial - t im e m odel w ith t h e secon d or der n eigh b orh ood st ru ct u re in S ection 2.2. W e den ot e t h e pr edict or by

X _i ^{2 , S T} ( t) .

A T S 1 S T 1 S 2 S T 2

CRV 7.24 7.13 6.84 6.78 6.24 6.19

r at io of CRV (% ) 100 98.6 94.5 93.7 86.3 85.5

T able 1.: Com parison of cr os s - V alidation st atist ic s of 6 m odels

F or t h e com p aris on , w e u s e th e cros s - v alidation t ech niqu e for th e ab ov e 6

m odels . If w e let X i , d ( t) , i = 1 , … , 64 , t = 1 , … , 96 , d = 1 , … , 16 , b e th e

ob serv ation at t t im e an d i lin k on d - th day , th e cr os s - v alidat ion is

im plem ent ed by rem ov in g t h e dat a X _{i , d} ( t) an d pr edict in g it fr om th e r em ainin g

dat a . H er e w e u se t h e cr os s - v alidat ion st at ist ics (CRV ) in H u an g an d Cr es sie [10]

a s follow s :

CR V L , i = 1

16 d = 1 |M _{i , d} |

16 d = 1

96 t = 1 , t M

[ X i , d ( t) - X d

( - i , - t)

] ²

1 2

, (17 )

CR V = 1

16 d = 1

64 i = 1 |M _{i , d} |

16 d = 1

64 i = 1

96 t = 1 , t M

[ X i , d ( t) - X d

( - i , - t)

] ²

1 2

(18 )

w h ere X _d ( - i , - t)

den ot e s t h e pr edict or of X _{i , d} ( t) by r em ov in g X _{i , d} ( t) an d

M _{i , d} , |M _{i , d} | ar e th e set of tim es w h en ob serv ation s are n ot m is sin g at lin k i on

d - th day an d t h e n um b er of elem en t s of M _{i , d} , r espectiv ely .

In T able 1, w e sum m arized t h e CRV v alu es of all t h ose m et h ods an d t h e r at ios of t h e CRV of ea ch m et h od t o t h at of M et h od A . T able 1 sh ow s t h at t h e spat ial- t em por al m odel u sin g t h e secon d or der n eighb orh ood st ru ct ur e, is b ett er t h an t h e oth er s . In av er ag e, th er e w a s ab ou t 14% im pr ov em ent in CRV w h en u sin g t h e spat ial- t em poral appr oach w it h s econ d order n eighb orh ood stru ctu r e.

T his st ron g ly r ecom m en d s t o u se a sp at ial - t em p or al m odel for h an dlin g m is sin g v alu es .

4 . Co n c lu din g R e m ark s

In th is paper , w e pr opos ed an alg orit hm u sin g t h e sp at ial - t em p or al m odel t o r econ st ru ct th e m is sin g v alu es . A sign ificant im pr ov em ent in CRV w a s seen in u sin g t h e spat ial m odel com p ar ed w ith a t im e series m odel. A s w e can see in S ection 3, t h e spatial- t em por al m odel out perform s t h e pu r e spat ial m odel, bu t t h e differ en ce of im pr ov em en t in CRV b et w een t w o m odels is n ot r em ark able. T his in dicat es t h at sp atial effect s dom in at e t em por al effect s in our t r affic dat a an aly sis , an d t h e con cept of sp atial m odelin g sh ould b e t ak en int o con sideration . T o our k n ow ledg e , ou r w ork is t h e fir st at t em pt t o u se spat ial- t em por al m odel in IT S r elat ed fields in K or ea . W e b eliev e t h at th e r esu lt obt ain ed h ere giv es a fu n ct ion al t ool t o solv e t h e m is sin g v alu e pr oblem .

A c k n o w le dg e m e n t s . W e ar e g rat efu l t o t h e ROT IS In c. for pr ov idin g dat a an d r elev ant t r affic inform at ion s . W e also w ish t o th ank th e edit or an d t h e t w o r efer ees . T h is w ork w a s support ed by K or ea Resear ch F oun dation Gr ant (KRF - 99 - 042- D0021 D 1200).

R e f e re n c e s

1. A .C. H arv ey (1993 ). T im e S eries m od els . T h e M IT pr es s , Cam bridg e, M es sach u s ett s .

2. A li, M . M . (1979 ). A n aly sis of st at ion ary sp atial - t em por al pr oces s es : E stim at ion an d pr edict ion . B iom e tr ika , 6 6 , 513 - 518.

3. Ban k s , J . H . (1997 ) I n trod uction t o T ransp orta tion E ng in e ering . N ew - Y ork , M cGr aw - H ill.

4. B esag , J . (1974 ). S p atial in t er act ion s an d th e st at ist ical an aly sis of lat t ice dat a . J ournal of the R oy al S ta t is tical S ocie ty B , 3 6 , 192- 225.

5. Ca sett ie , E . an d S em ple , R . K . (1969). Con cern in g th e t est in g of spatial diffu sion

h y pot h e ses . Ge og rap hical A naly s is , 1 , 254 - 259.

6. Cr es sie , N . (1993 ). S ta tis t ics f or Sp a t ial D a ta . R ev is ed edit ion , N ew Y ork , W iley an d S on s .

7. Cre s sie, N . an d M aju r e, J . J . (1997 ). S pat io - T em por al S t at istical M odelin g of Liv est ock W a st e in S tr eam s . J ournal of A g ricalt ural, B iolog ical, an d E nv ir onm en tal S ta t is t ics , 2 , 24- 47.

8. Cressie, N., Kaiser, M. S., Daniels, M. J.,Aldworth, J., Lee, J., Lahiri, S. N., and Cox, L.

H . (1998 ). S patial an aly sis of part iculant m at t er in an u rb an en v ironm en t . Un pu blish ed m an u s cript .

9. Cliff, A . D . an d Ord , J . K . (1975). S pace - t im e m odelin g w it h an applicat ion t o r eg ion al for eca stin g .

T ransactions an d P ap e rs , I ns t itut ion of B rit is h Ge og rap hers , 6 6 , 119 - 128.

10. Hu an g , H . C. an d Cr es sie, N . (1996 ). S pat io - t em p or al pr edict ion of sn ow w at er equ iv alen t u sin g th e K alm an filt er . Com p uta t ional S ta t is tics and D a ta A naly s is , 2 2 , 159 - 175.

11. Niu , X . an d T iao, G. C. (1995). M odelin g sat ellit e ozon e dat a . J ournal of the A m er ican S ta t is t ical A ss ocia t ion , 9 0 , 969 - 983.

12. P feifer , P . E . an d Deu t sch , S . J . (1980). A t hr ee - st ag e it er at iv e pr ocedu r e for space - t im e m odelin g . T e chn om e tr ics , 2 2 , 35 - 47.

13. P feifer , P . E . an d Deu t s ch , S . J . (1980). Ident ificat ion an d int erpr et at ion of fir st or der sp ace - t im e A RM A m odels . T echn om e tr ics , 2 2 , 397 - 408.

14. D eut sch , S . J . an d P feifer , P . E . (1981). S pace - tim e A RM A m odelin g w ith cont em por an eou sly corr elat ed in n ov at ion s . T e chn om e tr ics , 23 , 401- 409.

[ 2002년 10월 접수, 2002년 10월 채택 ]