Modelling multiple time series with missing observations

M odelling M ultiple Tim e S e rie s w ith M issing O b serv atio n s By

Cheung King Chau S tu d e n t# 9151426

S u p e rv iso rs

Dr. Don P o s k itt and Dr. Ray C ham bers

SUBMITTED AS A PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF STATISTICS

a t th e

This dissertation is the original work of the author. All sources, authors or

other material to which reference was made have been duly acknowledged.

A cknow ledgem ent

Having sp e n t over one y e a r stu d y in g S t a ti s ti c s in th e A u s tra lia n N atio n al U n iv e rsity , I have had big im provem ent in th e o r e tic a l th in k in g , m a th e m a tic a l sk ills and com puting sk ills. Being e x tre m e ly in te r e s te d in tim e s e rie s a n a ly s is, th e w orking of th is th e s is allow s me to go one s te p f u r t h e r to th is in te r e s tin g and im p o rta n t to pic.

I m u st a tt r ib u t e th e su c ce ss o f th is th e s is to my s u p e rv is o rs , Dr. Don P o s k itt and Dr. Ray C ham bers. They have been a c o n s is te n t so u rc e o f en c o u ra g e m e n t, c ritic is m and guidance, w ith o u t w hose p a tie n c e and su p p o rt th is th e s is w ould not have been po ssib le.

I w ish to e x p re s s my g r a titu d e to Dr. W. K. Li, my f i r s t s u p e rv is o r o f my u n d e rg ra d u a te s tu d ie s a t th e U n iv e rsity of Hong Kong. His advice h as c u ltiv a te d my i n te r e s t in th e fie ld o f tim e s e r ie s a n a ly sis.

T his th e s is is d e d ic a te d to my p a r e n ts to whom I life lo n g in d eb te d n ess.

A b s tra c t

T his th e s is in tro d u c e s an a p p ro a c h to th e s ta t e sp ace m odelling o f tim e s e rie s t h a t may p o ssess m issin g o b se rv a tio n s. The p ro ce d u re s t a r t s by e s tim a tin g th e a u to c o v a ria n c e sequence u sin g an idea p ro p o sed by Parzen(1963) and S to ffe r(1 9 8 6 ). S uccessiv e Hankel m a tric e s a r e o b tain e d via A u to reg re ssiv e a p p ro x im a tio n s. The ra n k of th e Hankel m a tr ix is d e te rm in e d by a s in g u la r value decom po sitio n in c o n ju n c tio n w ith an a p p ro p r ia te m odel se le c tio n c rite r io n . An in te rn a lly b a lan ced s t a t e sp ace r e a lis a tio n of th e s e le c te d Hankel m a tr ix p ro v id es in itia l e s tim a te f o r m axim um likelihood e s tim a tio n . F inally, th e o r e tic a l e v a lu a tio n of th e F is h e r in fo rm a tio n m a tr ix w ith m issin g o b s e rv a tio n s is c o n sid e red .

C o n ten ts

C h a p te r 1. An O verview and a Sum m ary of th e th e s is .

C h a p te r 2. A Sum m ary of P a s t and P re s e n t developm ent of S ta te Space M odelling.

C h a p te r 3. A S urvey o f S t a ti s ti c a l T echniques to deal w ith

M issing O b se rv atio n s w ith a sp e cia l em phasis to Time S e rie s D ata.

C h a p te r 4. D e sc rip tio n of D a ta , Ig n o ra b ility A nalysis and

P re lim in a ry A nalysis of D e te rm in is tic p a r t o f th e D ata. C h a p te r 5. C o n stru c tio n of B alanced S ta te Space R e a lis a tio n s and

Model R eduction.

C h a p te r 6. Maximum L ikelihood E s tim a tio n and I ts A sym ptotic D is trib u tio n .

C h a p te r 7. D iffu se K alm an F ilte rin g .

C h a p te r 8. C onclusions and C o n trib u tio n s of th e th e s is . A ppendix A. G rap h s o f th e th e s is .

1. An O v e r v i e w and a S umm a r y of the thesis

1.1_____S u m m a r y of the T h e s i s

T h e aim of this thesis is to deve l op a me t h o d of fit t ing state space mo de l s to mul ti var iat e t i me series data cont ai ni ng missing observations. T h e model is i llust rated by using it to model the dynami c b eha vi our of the whi te blood cell count s of a L e u k a em i a pat i ent . T h e s e data consist of m e a s u r e m e n t s of whi te blood cell counts as well as ot her auxiliary i nformat ion, all of whi ch are categorical in nature. T h e s e data are d e f in e d in T a b l e S.l b el o w and figure 1.1 pres ent s the graphs for the blood cell co un t data y t . (see also C h a p . 4 on P.67) T h e proporti on of missing obs er vat ions of all t hr e e mai n variabl es are about 33.5%. However, t hey are not necessari l y mi ssi ng at t he s ame t i me, i.e. partial missing is possible.

Since the t hr e e whi te blood cell count s are cor rel at ed, a m u lt iv a r ia t e model for y t = ( N C , L C , O W B C ) ’ whe re O W B C = T W B C - N C - L C is d e v e l o p e d in w h a t follows. A major probl em with the model li ng process is the fact t h a t t h e r e are mi ssing data in t hes e cell counts.

1.2____ T r e a t m e n t of Mi ssing Data

T h r o u g h o u t it is as s umed t hat the missing data g en er at i ng process is i g n o r a b l e . By ignorability, we mean t hat the conditional e x p e c t a t i o n of the mi ssi ng dat a flag d e f i n e d as

(F.l) R =

t

f 1 if y is missing \ 0 if y is observed

gi ven t he auxi l i ary i nformat ion Z t is i n d e p e n d e n t of y t IZt (see Ru b i n and Li tt l e( 1987)). T h e as su m p ti on of i gnorability en ab l e s us to par ti t i on t he j oi nt l og-l ikel ihood of (yt , R t )IZt into the log-likelihood of y t IZt plus t hat of R t IZt p rovi ded t h a t t h ey have

T a b l e S. 1

M ain V ariab les

T o t a l W h i t e Blood Cell C o u n t ( TW B C ) N e u t r o p hy l l C o u n t (NC)

L y m p h o c y t e C o u n t (LC)

A uxiliary In f o r m a tio n

I nf ec t ion Obs er ved (0: no infection, 1: i nf ec t ed) Stage o f T r e a t m e n t (Stage from 1 to 4)

Cycl e wi thi n Stage (at most 8 cycles per stage) W e e k wi thi n Cycle

d is tin c t p a ra m e te rs . We, th u s, a re able to ‘ig n o re ’ th e in fo rm a tio n c o n ta in ed in th e lo g-lik eliho od due to R |Z . H ow ever,

t* 1 2 3 4 t

a d raw b a ck o f th is a ssu m p tio n is t h a t a ll f u tu r e a n a ly sis m u st be c o n d itio n a l on th e a u x ilia ry in fo rm a tio n Z^. (see section 3.1 on P.41)

One sim ple w ay to j u s tif y th e a ssu m p tio n of ig n o ra b ility is to check w h e th e r th e d is trib u tio n of th e c o n d itio n a l v a ria b le R IZ is c o n s ta n t o r n e a rly c o n s ta n t. T his can be b ased on a

t 1 t

p re lim in a ry lo g is tic a n a ly s is of th e m issing d a ta f la g in te r m s of th e a u x ilia ry v a ria b le s . A lo g istic a n a ly sis o f th e m issing d a ta f la g f o r th e w h ite blood cell c o u n ts d a ta in te rm s of th e a u x ilia ry v a ria b le s in d ic a te s a re d u c tio n o f 207.-30% in re s id u a l deviance. A lthough th is re d u c tio n is no t o u tsta n d in g , it is s u ff ic ie n t to in d ic a te t h a t th e a ssu m p tio n t h a t th e m issin g d a ta g e n e ra tin g p ro c e s s is ig n o ra b le is a re a s o n a b le one.

D efin itio n o f th e Model (see P.45-46; section 4.1 on P.68) In th e fo llo w in g developm ent, th e d a ta w ill be assu m ed to fo llo w th e s ta t e sp ace model

(F.2)

y = ß Z + Cx + e

Jt t t t

x = Ax + Be

t+i t t

w h ere

1. Z^ is th e v e c to r of a u x ilia ry v a ria b le s a t tim e t, 2. x^ is th e s t a t e v e c to r o f unknown dim ension, 3. e^ ~ N(0,£2) is i.i.d ., and

4. xq ~ N(/i,S) ind ep en d en t o f e^.

1.3 Rem oval of th e D e te rm in istic Com ponent

B efore th e s ta t e sp ace model (F.2) can be f it te d , th e d a ta m u st be ‘detrended*. In p a r tic u la r , a ro b u st re g re s s io n p ro c e d u re is used to sw eep out th e e f f e c t of th e a u x ilia ry v a ria b le s fro m th e s q u a re - ro o t tra n s fo r m e d tim e s e rie s

y = (VNC, VUC, v'ÖWBC)’.

The use o f th e s q u a re ro o t tr a n s f o r m a tio n is to s ta b lis e th e re s id u a l v a ria n c e . S e lec tio n of th e a u x ilia ry v a ria b le s to include a s r e g r e s s o r s is b ased on th e fo llo w in g s t a t i s t i c w hich is a n alo g o u s to th e F - s t a t i s t i c in le a s t sq u a re s a n a ly sis (see Huber(1981), page 196-197):

(F.3)

1 p - q

2

1 n -p

K.2^ i//( r ^/<r) 2 cr

( - j r E ^ ' ( r / o

- ) ) 2

H ere

1. y and y a r e v e c to rs of f i t t e d v alues using th e m odels w hich

p q

include p a u x ilia ry v a ria b le s (p-m odel) and q a u x ilia ry v a ria b le s (q-m odel) re sp e c tiv e ly .

2. r is th e re s id u a l of th e p-m odel. 1

3. n is th e num ber of o b se rv a tio n s.

4. t//(x) = m in{c,m ax(-c,x)> f o r some c o n s ta n t c > 0.

5. K = 1 + — —, m = th e re la tiv e fre q u e n c y of r s a tis fy in g I r I < c.

J l1

6. <r is th e m edian a b s o lu te d e v ia tio n of I r I . 1 i1

A ccording to Huber(1981), th e d is tr ib u tio n o f th e s t a t i s t i c (F.3) 2

is w ell a p p ro x im a te d by t h a t o f a x v a ria b le . p -q

p a ra m e te r e s tim a te s w ere th e n used to provide an in itia l e s tim a te o f ß. (see section 4.2 on P.70-74)

1.4 S u ccessive AR a p p ro x im a tio n s

A fte r d e te rm in in g an in itia l e s tim a te of ß along th e lin es d e sc rib e d above, th e m ethod p ro ceed s by d e te rm in in g su ccessiv e AR a p p ro x im a tio n s to th e d e tre n d e d tim e s e rie s = y^-ßZ^. The p th o rd e r AR a p p ro x im a tio n is d efin ed by f it ti n g th e fo llo w in g m odel to th e d a ta :

(F.4) w h ere

A y + A y + • • r t - i 2 t- 2

i.i.d .. It can be

+ A y + e p t - p t

show n t h a t th e e ’s t N (0,S ) is

t p

a p p e a rin g in (F.4) a re th e sam e a s t h a t a p p e a rin g in th e (F.2)(see section 2.6 on P.25) (see H annan and D eistler(1988)). T h ere a r e tw o s ta n d a r d w ays of

e s tim a tin g th e c o e ff ic ie n ts A ,...,A m aking up th is m odel. T hese

1 P

a r e th e L ev in so n -W h ittle (LW) a lg o rith m and B urg’s a lg o rith m . A lthough B urg ’s a lg o rith m is ty p ic a lly p r e f e r r e d , th e p re se n c e of m issin g d a ta im plies t h a t only viable o p tion f o r m odelling th e s e d a ta is th e LW a lg o rith m , w hich only r e q u ir e s th e a u to c o v a ria n c e sequence a s in p u t. The LW a lg o rith m lead s to e s tim a te s of A ,...,A and S w hich we w r ite a s A

l p p l

• ,A

s a tis f y th e Yule W alker eq u a tio n s

£

k = 0

A(p)R

0 if j > 0 ^ S‘ if j

p

w h e re R is th e e s tim a te d la g -k a u to c o v a ria n c e of y a lg o rith m can be fou n d, f o r exam ple,

Hannan(1986). (see section 5.1 on P.76)

in Jones(1978) T his

and

The e s tim a te d la g -k a u to c o v a ria n c e R is given by k

f o r i ,j = 1,2... h, T - k

£ a t+k( i ) a t ( j ) < y t+k( i ) - y ( i ) > < y t ( j ) - y ( j ) >

(F.5) . y u )

t = l

T - k

I

a ( i ) a ( j ) t +k t

h=dim (yt ), a k(i)= ={ 0

1 i f y ( i ) is observed

and y(k)=—

£ a ^ (k ) y ^ (k ) o th e rw is e

£ at<k)

t = i

T h is e s tim a te is known to be c o n s is te n t when some of th e d a ta a re m issin g and th e tim e s e rie s {a } is a sy m p to tic a lly s ta tio n a r y , see

k

Parzen(1963) and S to ffe r(1 9 8 6 ). (see se c tio n 3.3 on P.52-54)

F o r th e blood cell count d a ta , th e above p ro ce d u re w as used to f i t an AR(p) a p p ro x im a tio n to th e d a ta f o r p = l,2 ,...,m a x w h e re m ax ^ loglogT w as d e te rm in e d a s th e e s tim a te d dim ension of th e s t a t e v e c to r x^ sto p p e d changing. F o r each value o f p, hp b a la n c e d s t a t e sp ace r e p r e s e n ta tio n s of AR(p) w e re th e n c o n s tru c te d along th e lin es d e sc rib e d below and, fin a lly , a ll hpxm ax m odels th u s f i t t e d w e re co m p ared using th e H annan-Q uinn(1980) c r ite r io n to d e te rm in e th e fin a l dim ension of th e s ta t e v e c to r x .

t

1.5 B alanced S ta te Space R e p re se n ta tio n o f th e AR Models

F o r each valu es o f p = l,2 ,..., th e AR(p) a p p ro x im a tio n can be w r itte n a s A(p)(z *)y

A<p,(z~l )

e p w h ere t

I - A(p>z -1 - A(p)z ' p.

By em ploying an a lg o rith m d e sc rib e d in Robinson(1967), p.160-162, a d jA (p)(z l ) and detA (p)(z *) can be d e te rm in e d w here

(F.6) [ A(p\ z _1) ]-1

a d j A(p)(z l )

d e t A(p)(z *)

A m ethod b a se d on p a r t i a l f r a c tio n , see T ra n te r(1 9 6 0 ), can th e n be used to in v e rt th e polynom ial detA (p)(z *) by assu m in g t h a t it does n o t p o sse ss any r e p e a te d ro o ts . Polynom ial m u ltip lic a tio n can

(p) th e n be u sed to c a lc u la te th e c o e ffic ie n ts n

p o w er s e rie s e x p an sio n

r £ 0 of th e

(F.7) [ a(p)(z- ‘ ) ] - ' =

r^O

7T(p)z r . (see se c tio n 5.2 on P.77-79)

The p th o rd e r Hankel m a tr ix H p c o rre sp o n d in g to th e p

(F.8)

r n ( p )

1 r r ( p ) 2 • • TT( P )_P 7T( P ) 7T( P ) • • TT( P )

( p) _ 2 3 p + 1

p

7T( P ) 7T( P ) • • TT( P )

P p + 1 2 p - l

In th is s e ttin g , H annan and D eistler(1988) proved t h a t th e ra n k of H (p) is ^ hp. F u rth e rm o re , O tter(1985) show ed t h a t a s t a t e sp ace r e a lis a tio n w ith s ta t e v e c to r dim ension equal to th e ra n k of p

p

e x is ts p ro v ided t h a t th is ra n k is f in ite . Thus, we r e q u ir e an e s tim a te o f th e ra n k v , l^v ^hp, o f th e Hankel m a tr ix H (p) w hich,

p p p

in tu r n , w ill allow us to e s tim a te th e dim ension o f th e s ta t e v e c to r x

A co nvenient w ay of e s tim a tin g th e ra n k of a m a tr ix is to c a r r y o u t a s in g u la r value d ecom position of th e m a trix . L et H (p) = U S V’ d eno te th e s in g u la r value decom po sition on H (p).

p p p p p

H ere U an d V a r e o rth o n o rm a l m a tric e s (i.e. U ’U = V’V = I) and

p p p p p p

S = d ia g (s , . . . , s ) is th e diag o n al m a tr ix of s in g u la r v alues of

P( ) 1 hp

H p . Due to th e noisy n a tu r e of th e d a ta , th e m agn itud e o f th e s e

p

s in g u la r v alu es ty p ic a lly d e c re a s e s to z e ro very sm oothly. We t h e r e f o re r e q u ir e a c r ite r io n f o r deciding how m any o f th e s e s in g u la r v a lu e s can be c o n sid e red e ffe c tiv e ly z e ro . Since th e u ltim a te o b je c tiv e is to provide a b e s t f i t to th e d a ta , th e c r ite r io n used f o r th is p u rp o se is th e w ell-kn ow n c o n s is te n t c r ite r io n su g g e ste d by H annan and Quinn(1980), see H annan(1986). E ffe c tiv e ly , each one o f th e p o ssib le hp b alan ced s t a t e sp a ce m odels c o rre sp o n d in g to th e AR(p) a p p ro x im a tio n w e re com puted and t h a t m odel (and hence value v ) chosen w hich m inim ised th is

p

c r ite r io n , (see P.81-85)

F o r v = 1 ,2 ,...,h p , each b a la n c e d s ta t e sp ace model w as d e fin e d via th e s te p s :

1. P ut S = d iag (s , . . . , s ).

l v 1 V

2. d e fin e th e c o rre sp o n d in g p a r titio n s U = (U U ) and

p i p 2p

V = (V V ).

3. P u t

r * , p , i_{i 2} • _{p + 1} 1

7T( P ) TT( P ) • • 7T( P )

x ( p > = B

2

, K ( p ) = ( n ( p ) •

A 1 • n ( p ) ) _P and K ( p | 1 _P =

3 p + 2

TT ( P) L P J

* , p ’ •

L p + 1 Zp J

A b alan ced s t a t e sp ace r e p r e s e n ta tio n of th e AR(p) a p p ro x im a tio n to y w hich assu m es ra n k (H (p)) = v can th e n be c o n s tru c te d by

t p

s e ttin g

(F.9)

x = A x + B c t + l l v t l v t

C x + e l v t t w h ere

1. v = dim (x ) t

2. A = S~1/2U’ K (p )tV S ' 1/2 l v l v l p p lp l v

3. B = S~1/2U’ K (p) l v l v l p B

4. C = K (p)V S"1/2. l v C lp l v

It can be show n t h a t a ll m odels c o n s tru c te d in th is w ay a r e n u m erica lly s ta b le , c o n tro lla b le and o b se rv a b le so t h a t th e e f f e c t due to u n c e rta in ty o f th e in itia l s t a t e xq is m inim ised, see C rabb and Young(1989). F o r each value o f p, th e r e f o r e , th e r e a r e hp b a la n c e d s t a t e sp ace m odels re p r e s e n tin g th e AR(p) p ro c e ss w ith th e dim ension of th e s t a t e v e c to r x^ ra n g in g fro m 1 to hp.

The a p p ro p r ia te v alue of v is th e n chosen so t h a t th e P

H annan and Quinn(1979) (HQ) c r ite r io n

(F.10) HQ(v) = -21 o g-likelihood + Zvj’loglogT]

A pplying th is p ro ce d u re f o r each value of p, values v , p p = l,2 ,...,m a x , a r e o b tain ed . The o rd e r of th e s ta t e sp ace m odel (F.2) fin a lly chosen to re p r e s e n t th e d a ta is th e n such t h a t HQ(v ) is m inimum . T his b alan ced s ta t e sp ace model is also used to

p

pro vid e in itia l e s tim a te s of th e s tr u c tu r a l p a ra m e te rs A, B, and C. The in itia l e s tim a te of Q is provided by th e value o f Sf f o r

p th e chosen v alue o f v .

p

1.6 Maximum L ikelihood E stim atio n

Having d e te rm in e d th e dim ension o f th e s ta t e v e c to r of th e model (F .2 ), m axim um likelihood e s tim a tio n o f th e p a ra m e te rs

0 = (v e c (A )\v e c (B )\v e c (C )\v e c (fi V ,v e c (ß )’)’

o f th is m odel is c a r r ie d o u t using n u m erica l o p tim isa tio n . In tu rn , th is r e q u ir e s an in itia l e s tim a te of 0 a s w ell a s a lg o rith m s f o r g e n e ra tin g th e likelihood s u rf a c e and its g ra d ie n t.

The in itia l e s tim a te of 0 is com puted along th e lines d e sc rib e d in th e p rev io u s se c tio n s. Thus, th e in itia l e s tim a te of ß is o b tain e d f rom a ro b u s t re g re s s io n of y in te r m s of th e a u x ilia ry v a ria b le s Z^, w hile th e in itia l e s tim a te s o f th e o th e r com ponents o f 0 a r e o b tain e d a s a b y -p ro d u c t o f th e p ro c e d u re used to d e te rm in e th e dim ension o f th e s ta t e v e c to r x .

t

The c o m p u ta tio n of th e likelihood s u rf a c e f o r th e m odel (F.2) is b a sed on th e fo llo w in g m odified v e rsio n o f th e K alm an F ilte r w hich a llo w s f o r m issin g d a ta :

M odified K alm an F ilte r

Given an in itia l e s tim a te of 0, assum e t h a t p = 0 and Z = {I-A®A> vec(BfiB’). The s ta tio n a r ity and s ta b ility c o n d itio n s imply t h a t Z s a ti s f i e s Z = AEA’+BQB’ and th e e f f e c t o f s e ttin g p= 0 is m inim ised by assu m in g t h a t th e model (F.2) is o b se rv a b le and c o n tro lla b le .

C Observed part of y if y is partially or fully observed

y (t) = < _ 1 1 .

1 null if y is completely missing

2. e^t) is the subvector of c^ corresponding to the observed part

of y

t 4 . H t - i 5 . Q

i t 6 . C

i t 7 . A

n

= E{(x -x )(x -x )’>

t t t t

E{e (t)e’> and

l t l i t E<e (t)c (t)’>.l l

is the -21og-likelihood of the model (F.2) at the given

parameter values for the first n observations.

Starting Conditions: x^ = Aji, Hq = AEA’+BfiB’ and Aq = 0.

For n = 1,2...T, do the following recursion:

Case 1: When some, not necessarily all, of the components of y

n

are observed,

Q = Q + C H C

n l l n In n- 1 In

K = H C Q"1

n n-1 In n

P = (I - K C )H

n n In n-1

x = x + K (y (n) - C x )

n n n 1 In n

x = Ax + B£T Q X(y (n) - C x )

n+l n l n n 1 In n

-1.

H = AP A’+BQB’-AK Q B'-BQ’ K’A’-BQ’ Q Q B’

n n n In In n In n In

A = A +log IQ |+(y (n)-C x )’Q X(y (n)-C x )

n n-1 1 n 1 1 I n n n 1 I n n

Case 2: When y is completely missing,

n

n-1

Ax

H = AP A’ + B^B’

n n

end the for loop.

The final value

model at the given parameter values. Without missing data, the The final value of A, A^, is then the -21og-likelihood of the

matrices Q , n , y (n) and C appearing in the above recursion

l l n l n 1 In

are replaced by Q, y and C respectively. In this case, the above

B alakrishnan(1984). T h a t is, th is m odified K alm an F ilte r is c o n s is te n t w ith th e K alm an F ilte r given in B alakrishnan(1984) in th e sen se t h a t th e dim ensions of a ll q u a n titie s have been m odified to be c o n s is te n t w ith th e observ ed d a ta , (see P.30 and P.59 for comparison)

O bserve t h a t if we re g a r d th e s ta t e s x^, t = 0 , l ... T, as p o te n tia lly ‘o b s e rv a b le ’, th e n th e ‘c o m p le te - d a ta ’ likelihood may be e x p re s s e d as

T

(F. 11) -21ogL = Tlog I QI + £ (yt -ß Z t -C x t )’n -1(y -ß Z -C x ). t = i

A s im ila r id ea w as used by Shum way and S to ffe r(1 9 8 2 ). The sc o re fu n c tio n (o r th e g r a d ie n t of th e likelihood) is th e n com puted via m a tr ix d if f e r e n tia tio n of (F. 11) and th e fo rm u la

(F .i2) s c( e| o ) = E<Sc(e|o ) | o }

w h ere 0 and 0 a r e th e observed d a ta and com plete d a ta s e ts

s c

re s p e c tiv e ly and S c(* |C M and Sc( • | CM a r e th e s c o re fu n c tio n s d efin ed by th e s e ‘o b se rv e d ’ d a ta and ‘c o m p le te ’ d a ta s e ts , (see Louis(1982) and T anner(1990)) (see section 6.1 on P.88-91)

In o r d e r to use (F.12), we need to e v a lu a te th e e x p e c ta tio n o f th e ‘co m p le te ’ d a ta sc o re fu n c tio n given th e o b served d a ta . In tu r n , th is r e q u ir e s t h a t we e v a lu a te q u a n titie s such a s x (T) = E(x |0 ) and P (T) = E{(x - x (T ))(x - x (T ))’>.

n n ' S n n n n n

T his is c a r r ie d out u sin g th e K alm an Sm oother: (see P.30-33) Kalm an S m ooth er

In itia l C onditions: x (T) = x and P (T) = P .T i l l

F o r n = T - 1 ,...,0 , do th e fo llo w in g re c u rsio n : S = P A’H 1

n n n

X (T) = X + S (x ( T) - X )

n n n n+1 n+1

P (T) = P + S (P (T)-H )S ’

n n n n+1 n n

end th e f o r loop.

F inally, a q u a si-N e w to n o p tim isa tio n a lg o rith m , see Davidon, F le tc h e r and Pow ell(1963), is used to lo c a te th e MLE. The a d v a n ta g e o f th e a lg o rith m over th e m ore conven tion al N ew ton-R ap h so n a lg o rith m is t h a t it does n o t re q u ire in v ersio n of th e H essian m a tr ix a t each s te p , see Rao(1977). (see P.87-88)

An im m ediate b y -p ro d u c t of th e K alm an F ilte r and Kalman S m o o ther is t h a t one s te p ah ead p re d ic tio n s and in te rp o la te d v alu es f o r m issin g d a ta a r e e a sily com puted via th e fo rm u la e : y = ßZ + Cx and y (T) = ßZ + Cx (T) + c (T) re sp e c tiv e ly . Here,

t t t ^ t t t

yt =E{y^ I yi (n),l^n<t>, y^(T)=E{y^ | O J and e^(T)=E{et | CM. G rap h s of th e s e p re d ic te d s e rie s f o r th e blood cell count d a ta a r e show n in f ig u r e 1.2.

1.7 E stim a tin g th e a c c u ra c y of th e MLE

It is w ell known t h a t th e a sy m p to tic v a ria n c e o f th e MLE is p ro p o rtio n a l to th e in v erse of th e F ish e r In fo rm a tio n M atrix

(F.13) 3 = E { S c(0 |O s )S c (e |O s )’>.

One d isa d v a n ta g e of th is a p p ro a c h is t h a t th is e x p e c ta tio n m u st be com puted e le m e n t-w ise , w hich is n u m erica lly in e ffic ie n t. T h e re is scope, th e r e f o r e , f o r f u r t h e r r e s e a rc h on developing a m ore e f f ic ie n t a lg o rith m f o r th is problem . F o r m ore d e ta il on co m p u ta tio n o f 3, see C h a p te r 6 and Appendix B o f th e th e s is .

1.8 D iscussion

The e s s e n tia l d if f e re n c e s b etw een ‘com plete d a t a ’ c a se and ‘in co m p lete d a t a ’ c a se a r e t h a t

1. L arg e s c a le m a tr ix c o m p u ta tio n is a f a c t of life w hen m issin g d a ta a r e p re s e n t w hile th is can be la rg e ly avoided w ith com plete d a ta . As a consequence, n u m erica l e r r o r s can a c cu m u la te in th e in co m p lete d a ta case.

2. The p re se n c e o f m issin g d a ta in tro d u c e s tim e -v a ry in g ele m en ts in to th e a n a ly s is, in c re a s in g th e d im e n sio n a lity of th e prob lem and f u r t h e r in c re a s in g p ro b lem s a s s o c ia te d w ith n u m erica l s ta b ility w hen im p lem en ting K alm an F ilte r.

in c re a s e s th e p re d ic tio n e r r o r of th e f i t t e d model. 1.9 D iffu se Kalm an F ilte rin g

The re g re s s io n p a ra m e te r ß in th e s ta t e sp ace m odel (F.2) is fix e d . In some a p p lic a tio n s, it may be m ore re a so n a b le to co n sid e r th is p a ra m e te r to be a r e a lis a tio n of a random v e c to r. T h is lead s to th e m odified model

y.

t+i

ß Z + Gx + e

t t t

F x + He

t t

w h ere

1. e ~ N(0, n) is i.i.d . t

2. xq = 0 and ß = b+Br w ith z ~ N(0,C).

3. z and e ’s a r e u n c o rre la te d . 4. C is n o n sin g u la r u n less C = 0.

A sp e c ia l c a se of m odel (F.14) h as been c a lled ‘d if f u s e ’ by Jong(1991). T his is when C = 0 w hich, in a B ayesian sen se, is s im ila r to th e c a se of a n o n in fo rm a tiv e p r io r and r e f l e c t s ou r ig n o ran ce ab o u t th e p a ra m e te r ß. T his se c tio n is con cern ed w ith m axim um likelihood e s tim a tio n of model (F.14) in such a ‘d if f u s e ’ s itu a tio n . In o r d e r to c a r r y o u t th e m axim um likelihood e s tim a tio n in such a s itu a tio n , we re q u ire an a lg o rith m f o r g e n e ra tin g th e likelihood and an in itia l e s tim a te o f th e model p a ra m e te r

0 = (v ec(F )’,vec(G )’,vec(H )’,v ec(b )’,vec(B )’,v e c (n )’ )’.

In itia l e s tim a te s f o r F, G, H, Q and b a r e a v a ila b le fro m m axim um likelihood e s tim a tio n of th e m odel (F.2). T h a t is, th e in itia l e s tim a te s of F, G, H and fi a r e provided by th e f i t t e d b a lan ced s t a t e sp ace m odel (F.2) w hile t h a t of b is provid ed by th e r o b u s t r e g re s s io n of y on th e a u x ilia ry v a ria b le s Z^. In o rd e r to allo w th e e f f e c t o f th e rand om v a ria b le z , an in itia l

e s tim a te o f B equ al to a m a tr ix of l ’s is a r b i t r a r i l y chosen.

a p p ro p ria te ly red u c in g th e dim ensions of th e v a rio u s m a tric e s involved to ta k e a cco u n t of th e m issing d a ta , in a way t h a t is c o n s is te n t w ith th e p ro o f of th e d iffu s e Kalm an F ilte r, (see P. 109-110)

Again, f o r e a se of e x p la n a tio n , we ad o p t th e fo llo w in g n o ta tio n :

1. y^(t) is th e observ ed p a r t of y .

2. B^ and a r e th e ro w s of B and b re s p e c tiv e ly w hich c o rre sp o n d to th e ob serv ed p a r t o f y .

3. G is th e ro w s o f G c o rre sp o n d in g to th e observed p a r t of y . 4. e^(t) is a su b v e c to r o f co rre sp o n d in g to th e o b served p a r t

o f yt .

5. Q = E{e ( t ) c ’> and Q = E{e (t)c ( t) ’>.

it i t lit l i

The m od ified d iffu s e K alm an F ilte r w hich allo w s f o r m issin g d a ta is th e n d e fin e d by th e fo llo w in g re c u rsio n :

M odified D iffu se Kalm an F ilte r

In itia l C onditions: A =0, Q =0, t$=0 and vecP ={I-F®F> SrectHQH’).

l i l

F or t = 1,2... T, do th e fo llo w in g re c u rsio n :

Case 1: When som e o f th e com ponents of y a r e observed , 1. e = (B Z ,y (t )—b Z ) - G A

t t t i t t i t t

2. D = G P G’ + Q

t it t it lit

3. K = (FP G’ + HQ’ )D_1

t t it it t

4. A = FA + K e

t+i t t t

5. P = (F-K G )P F ’+HfiH’-K fi H’

Case 2:

t+i

6. *9 =

7. Q. When y

1. A

t it t

u + log I I = Q + e ’D_1e

t t t t

is co m p letely m issing, = FA

t it

t+i t

2. P = FP F ’ + HfiH’

t+i t

end th e f o r loop.

(F.15)

w it h q a s c a la r.

s ’ -,

q

G iven th e above a lg o r ith m f o r g e n e ra tin g th e lik e lih o o d

and an i n i t i a l e s tim a te o f 0, th e q u a s i-N e w to n a lg o r ith m is invoked a g a in to lo c a te th e MLE o f th e m odel (F.14). H ow ever, in

th is case, th e g ra d ie n t o f th e lo g - lik e lih o o d is e s tim a te d by th e

f i n i t e d iffe r e n c e m ethod, i.e . by n u m e ric a l a p p ro x im a tio n .

By d e fin in g

(F. 16)

r s s ’ -|

th e p re d ic te d s e rie s based on th e m odel (F.14) f i t t e d by ML can be

e v a lu a te d by means o f th e fo r m u la

(F.17) y = (b +B y )Z + Cx

t t t t t

~ _

w h e re y = S s and x = A { - y 1)’ . G ra p h s o f these p re d ic te d

t t t t t t

s e rie s f o r th e b lo o d c e ll c o u n t d a ta a re show n in fig u r e 1.3. (see P. 113)

D iscu ssio n

A fe a tu r e o f t h is ‘ d if f u s e ’ a n a ly s is is t h a t th e f i t t e d s e rie s is e x tre m e ly s im ila r to t h a t o b ta in e d w hen th e o r d in a r y

K a lm a n F ilt e r is used. T h is im p lie s th a t, p r a c t ic a lly speaking ,

th e choice b e tw een th e o r d in a r y K a lm a n F ilt e r and th e d iffu s e

K a lm a n F ilt e r is r e a lly a m a tte r o f p re fe re n c e . H ow ever, f o r th e

s p e c ific d a ta s e t c o n s id e re d in th is th e s is , th e o r d in a r y K a lm a n

F ilt e r is c le a r ly p r e fe r r a b le since

1. These d a ta c o rre s p o n d to th e w h ite b lo o d c e ll c o u n ts o f ju s t

one p a tie n t. I t is d i f f i c u l t to j u s t i f y a d iffu s e d is t r ib u t io n

f o r ß in th is case.

2. The d e c o m p o s itio n o f ß in to th e p a ra m e te rs b and B im p lie d by

(F.14) reduces th e degrees o f fre e d o m f o r f i t t i n g th e m odel so

m uch t h a t i t te n d s to r e s u lt in n u m e ric a l in s t a b ilit y .

On th e o th e r hand, i f o u r d a ta c o rre s p o n d to th e w h ite

b lood c e ll c o u n ts o f a g ro u p o f p a tie n ts , th e n th e d iffu s e K a lm a n

1. A lth o u g h d if f e r e n t p a tie n ts m ay have d if f e r e n t values o f ß, i t

m ay be re a so n a b le to assume t h a t th e re is a ‘ u n iv e r s a l’ ß f o r a ll p a tie n ts w h ic h we m ay re g a rd as ‘ d if f u s e ’ in o rd e r to

r e f le c t o u r ig n o ra n c e a b o u t it s value.

2. M ore p a tie n ts lead to m ore d a ta . In th is case th e loss o f

degrees o f fre e d o m th ro u g h th e in c lu s io n o f a d d itio n a l

p a ra m e te rs re p re s e n ts a re a so n a b le p ric e to pay f o r th e

a d d itio n a l e x p la n a to ry p o w e r p ro v id e d by these p a ra m e te rs .

1.10 C o nclusion

T h is th e s is d e m o n s tra te s t h a t s ta te space m o d e llin g is a

c o n ve n ie n t w a y o f m o d e llin g tim e s e rie s w it h ig n o ra b le m is s in g

d a ta . H ow ever, th e presence o f m is s in g o b s e rv a tio n s f u r t h e r

in cre a se s th e n o n - lin e a r it y o f th e lik e lih o o d fu n c tio n . T h e re fo re ,

as Jones(1986) in d ic a te d , i t can ta k e a lo t o f e f f o r t to com pute

th e MLE o f th e m odel (F .2 ) o r m odel (F.14). By a p p r o p r ia te ly

re d u c in g th e dim e n sio n s o f th e m a tric e s in v o lv e d , th e m o d ifie d K a lm a n F ilt e r and th e m o d ifie d K a lm a n S m o o th e r p ro ve d to be a v e ry

e f f ic ie n t w a y to g e n e ra te th e lik e lih o o d s u rfa c e o f m odel (F .2 )

and m odel (F.14) re s p e c tiv e ly . F in a lly , c o n s id e ra tio n has been g ive n to e s tim a tin g th e a c c u ra c y o f th e r e s u ltin g MLE. A lth o u g h th e

T

im

e

P

lo

ts

f

o

r

Le

v

el

W

h

it

e

B

lo

o

d

Cell

C

o

u

n

ts

Ne

u

tro

p

h

yl

l

C

o

u

n

t

F

ig

u

re

1

2. S ta te Space M odelling

Since Box and Jenkins(1969) ad v o cated th e use of ARIMA m odelling, we have w itn e sse d an in c re a sin g a c c e p ta n c e o f a n o th e r fam ily of m odels deriv ed fro m e n g in eerin g lit e r a tu r e c a lled ‘S ta te Space M odelling’. S ta te Space M odelling is e s se n tia lly a M arkovian r e p r e s e n ta tio n of o u tp u t s e rie s given p e rh a p s an in p u t/o u tp u t map (Kalm an(1969)). Kalman(1968) proved th e e x is te n c e o f s ta t e sp ace r e a lis a tio n f o r given t r a n s f e r fu n c tio n o r im pluse resp o n se m apping. T his, in e f f e c t, j u s t i f i e s th e use of s ta t e sp a ce m odelling in m odelling m ost tim e s e rie s d a ta . M oreover, S ta te Space M odelling (SSM) also o f f e r s se v e ra l f le x ib ilitie s w hich a re n o t s h a re d by ARIMA m odelling. They can be d e scrib ed a s follow s: 1. SSM is v e c to r in n a tu r e a s co m pared w ith u n iv a ria te e x te n sio n

o f V ector ARMA m odelling, see Aoki(1991). 2. SSM allo w tim e v a ry in g p a ra m e te rs .

3. SSM allo w s ir r e g u la r ly o b serv ed d a ta (Bucy and Joseph(1968)). To su m m a risin g th is in te r e s tin g and im p o rta n t s u b je c t, we begin by d e sc rib in g t r a n s f e r fu n c tio n model and s ta t e sp ace m odel. The equiv alence b etw een th e tw o fa m ilie s o f m odels a r e show n by a m inim al r e a lis a tio n th e o re m given in Kalman(1968). Then, d if f e r e n t fo rm s of s t a t e sp ace m odels a r e in tro d u c e d b u t th e m o st im p o rta n t of a ll is th e p re d ic tio n e r r o r fo rm w hich w ill be c o n c e n tra te d in f u tu r e a n a ly sis. C onversion b etw een ARMA m odels and S ta te sp ace m odels a r e d iscu ssed . We, th e n , come to th e Kalm an F ilte r and th e Kalm an S m o other. F in ally , m axim um likelihood e s tim a tio n a s w ell as th e c a lc u la tio n o f th e F is h e r in fo rm a tio n m a tr ix a r e discu ssed .

2.1 D e fin itio n s o f T r a n s f e r F u n ctio n Model D efin itio n 2.1.1

A dynam ical in p u t- o u tp u t sy ste m o r T r a n s f e r fu n c tio n m odel (SI/O ) is d e fin e d by th e e x is te n c e of an in p u t-o u tp u t m ap F w hich m aps th e in p u t in to th e o u tp u t.

R em arks:

1. A SI/O is c a lled lin e a r if F is lin e a r.

2. If th e u n d erly in g tim e index is s u b s e t of 1 it is c a lle d d is c re te .

3. A SI/O is c a lle d tim e - in v a r ia n t if FS =S F w h e re S u(x)= u(t+ x)

t t t

It is c le a r fro m d e fin itio n (2.1.1) t h a t th e sy stem fu n c tio n F p ro v id es th e key b etw een in p u ts and o u tp u ts. In g e n e ra l, f o r a lin e a r, d is c r e te , tim e - in v a r ia n t ZI/O , F can be d e sc rib e d in th e fo llo w in g as

00

(2.1.2) y (t) = jT L( j ) u ( t - j )

j= -o o

w h ere y (t) a r e o u tp u ts , u (t) a re in p u ts and L (j) a r e c o e ffic ie n ts of th e t r a n s f e r fu n c tio n £(z)=£ L (j)z , zeC. F or sim p lic ity , we can w r ite F = { L (j)|j€ Z } w hich is also c a lled im pulse re sp o n se fu n ctio n . In ad d itio n , a sy stem is c a lled c a u sa l if L (j)= 0 f o r j<0. Those r e a d e r s who a r e f a m ilia r w ith tim e s e rie s a n a ly sis w ould u n d e rs ta n d t h a t a ll s ta tio n a r y d is c re te sy ste m s a d m it c a u sa l t r a n s f e r fu n c tio n (see Hannan and D eistler(1988)). We sh a ll be m ainly co n cern ed w ith c a u sa l t r a n s f e r fu n c tio n only.

D efin itio n 2.1.3

A dynam ical sy ste m w ith s t a t e sp ace is d e fin e d by th e e x is te n c e of

2

th e s ta t e evo lu tio n fu n c tio n <p:T w h e re x is th e s e t o f all

+

p o ssib le s ta t e s , T 2 = {(t , t )eT 2 | t > t >, U is th e s e t of a ll

K + 1 o 1 l o

in p u ts and T is th e tim e index, w hich s a ti s f i e s th e fo llo w in g p ro p e rtie s :

(i) *l(to’t o’x o>u)=xo- V *

(ii) y ( t , t ,y ( t , t ,x ,u ) ,u H p ( t , t ,x ,u)

(iii) if f o r u ,u e l/, u (t)= u (t) f o r t <= t < t th en

1 2 1 2 0 1

^V W 'V ^'V W V

(iv) r is th e re a d in g fu n c tio n w ith r:*xUxT-»Y.

In w o rd s, y ( t , t ,x q,u) is th e s ta t e o b ta in e d a t tim e t^ by s ta r ti n g fro m x q and ap ply in g in p u t u.

Rem ark:

1. Zm is c a lled lin e a r if y ( t ,t q,... h^xl/-»* is lin e a r and r(.,.,t):^ x U ^ Y is lin e a r f o r a ll t.

2. Zm is c a lle d s ta tio n a r y if ^p(t + t,t + t,x ,u )= y (t .t ^ x ^ S ^ u ) and r ( x ,u ,t ) is in d ep en d en t o f t.

If T={t eZ I k = 0 ,l,2 ,...} we c a n d e fin e th e o n e -s te p evolu tion

k 1

fu n c tio n a s <p(t ,t ,x ( t ),u (t ))=f (x (t ),u (t )). Then th e

r k+i k k k k k k

(2.1.4)

x ( t )=f (x (t ),u (t ))

k+1 k k k

x ( t ) = X

0 0

y ( t )= r(x (t ),u (t ) ,t )

k k k k

M oreover, if 2m is lin e a r th e n f and r ( . , . , t ) can be re p re s e n te d by m a tric e s and we have th e fo llo w in g model d e sc rip tio n :

x ( t )=A (k)x(t )+B(k)u(t )

k+l k k

(2.1.5) x ( t )=x

0 0

y ( t )=C (k)x(t )+D(k)u(t )

J k k k

We c a ll (2.1.5) s ta t e sp ace model w ith tim e vary in g p a ra m e te rs . H ow ever, a s f a r as o u r p u rp o se is concerned, we a r e only in te r e s te d in c o n s ta n t p a ra m e te rs , i.e. A(k)=A, B(k)=B, C(k)=C and D(k)=D.

2 .2 C oncepts o f C o n tro lla b ility and O b serv ab ility D efin itio n 2.2.1

An event (x ,x ) is c o n tro lla b le i f f th e r e e x is ts a t€ T and an ueU such t h a t <p(t,T,x,u)=0. In w o rd s, an event is c o n tro lla b le i f f it can be t r a n s f e r r e d to 0 in f in ite tim e by an a p p ro p r ia te choice of in p u t fu n c tio n u.

D e fin itio n 2 .2 .2

An event (t,x) is re a c h a b le i f f th e r e is a s e T and an ueU such t h a t x=<p(r,s,0,u).

R e su lts

The fo llo w in g r e s u lts assu m e a r e a l, d is c r e te -tim e , n -d im e n sio n a l, lin e a r, c o n s ta n t dynam ical sy ste m Z=(A,B,-):

(2 .2 .3 ) A re a c h a b le s ta t e is alw a y s c o n tro lla b le and th e con v erse is alw ay s t r u e w henever d et(A )*0.

(2 .2 .4) A s t a t e x of 2 is re a c h a b le i f f x € span(B ,A B ,..,A n *B)

r i ” 1

and hence 2 is co m pletely re a c h a b le i f f rank(B ,A B ,..,A B) is n. (2.2.5) The s e t o f a ll re a c h a b le s ta t e s fo rm a v e c to r su b sp ace.

As e a sily in f e r r e d fro m th e above r e s u lts , r e a c h a b ility is eq u iv a len t to c o n tro lla b ility in ou r c o n te x t a lth o u g h th e y r e p r e s e n t d i f f e r e n t c o n c ep tu a l m eanings. In f a c t c o n tr o lla b ility r e q u ir e s t h a t th e map fro m in p u t fu n c tio n s to s ta t e s be s u rje c tiv e and we can w r ite

t

(2 .2 .6 ) x (t) = V Aj 1B u (t-j)+ A t x(0).

Thus, if th e in p u ts u is u n d er "control" th e n we can move x (t) in to any s ta t e given any in itia l s ta te , (see Bucy and Joseph(1968) and Hannan and D eistler(1988)) T his also im plies t h a t th e e f f e c t due to u n c e rta in ty in th e in itia l s ta t e x(0) w ould be m inim al if o u r sy ste m is c o n tro lla b le .

D e fin itio n 2 .2 .7

An event (t,x) in a r e a l, f in ite dim ensional, lin e a r dynam ical sy ste m Z=(A ,-,C) is u n o b serv ab le i f f CA x= 0 f o r a ll k >= x.

D efin itio n 2 .2 .8

W ith re s p e c t to th e sam e sy stem , an event (x ,x ) is u n c o n s tru c tib le i f f CAkx=0 f o r a ll k <= x.

R e su lts

The fo llo w in g r e s u lts a r e r e f e r r e d to a r e a l, d is c r e te -tim e , n -d im e n sio n a l, lin e a r, c o n s ta n t dynam ical sy ste m E=(A,-,C):

(2.2 .9 ) The s e t of a ll u n o b serv ab le s ta t e s fo rm s a v e c to r su b sp ace. Hence, we say t h a t Z is co m p letely o b se rv a b le i f f th e su b sp ace o f u n o b serv ab le s t a t e s c o n ta in s only th e z e ro elem ent.

(2.2.10) An u n o b serv ab le s t a t e is alw ay s u n c o n s tru c tib le and vice v e rs a w henever det(A )*0. Thus, C om plete o b s e rv a b ility is alw a y s e q u iv a len t to co m plete c o n s tr u c ta b ility when det(A )*0.

(2.2.11) The sy ste m £ is com pletely o b se rv a b le i f f ran k (C , ,A’C’,..,A ,n 1C’) is n.

O b se rv ab ility e n s u re s t h a t th e m ap fro m p r e s e n t s ta t e s to p re s e n t and p a s t o u tp u t fu n c tio n s be in je c tiv e . Thus, o b s e rv a b ility a lso m eans t h a t any given p a ir o f in p u t-o u tp u t sequ ences (u^.y^), k = 0 ,l,... uniquely d e te rm in e an in itia l s t a t e x(0 ) by solving th e sy stem of e q u a tio n s CAt x (0 )= y (t), t = 0 ,l ,2 ,. ., n - l . T o g e th e r o b se rv a b ility and c o n tr o lla b ility m akes s u re t h a t o u r m o d e llin g /filte rin g pro b lem is w ell posed.

T heorem (2.2.12) (C anonical D ecom position)(K alm an(1968))

Every r e a l (con tin u ou s o r d is c r e te tim e), f in ite d im ension al, c o n s ta n t, lin e a r dynam ical sy ste m m ay be can o n ic a lly decom posed in to fo u r p a r t s , o f w hich only one p a r t, t h a t w hich is co m p letely c o n tro lla b le and co m p letely o b se rv a b le, is involved in th e in p u t/o u tp u t b eh av io u r o f th e sy stem .

r e s t r i c t i o n to co n sid e r only th e p a r t w hich c o rre sp o n d s to c o m p le te ly o b serv ab le and com pletely c o n tro lla b le . T hus, we can a ssu m e com plete o b s e rv a b ility and com plete c o n tro lla b ility when p e rfo rm in g ou r d a ta a n a ly sis. T his is im p o rta n t as th ey jo in tly g u a r a n te e n ot only m inimum d im en sio n ality b u t also th e e x is te n c e o f s t a t e sp a ce r e a lis a tio n .

2 .3 R e a lis a tio n T heory S ta te m e n t of th e problem :

Given th e im pluse resp o n se m a tr ix o r t r a n s f e r fu n c tio n o f SI/O , c a n w e fin d a s ta t e sp ace model w ith th e sam e b eh av io u r a s th e given sy ste m ?

T h is pro b lem w as solved by Kalman(1968) via th e fo llo w in g m inimum r e a lis a tio n th eo re m f o r r e a l, c o n tin u o u s-tim e , f in ite dim en sio nal, lin e a r d ynam ical system :

T h e o rem 2 .3 .1 (Kalm an(1968))

Given th e im pulse re sp o n se m a tr ix W o f a r e a l, c o n tin u o u s-tim e , f in i te d im ension al, lin e a r dynam ical sy stem , th e r e e x is ts a r e a l, c o n tin u o u s tim e, f in ite d im en sio n al, lin e a r dynam ical sy ste m w hich

(a) r e a lis e s W: i.e. th e im pluse resp o n se m a tr ix o f £ w is eq u al to W;

(b) h a s m inim al dim ension in th e c la s s of lin e a r sy ste m s s a tis f y in g (a);

(c) is co m p letely c o n tro lla b le and com pletely ob serv ab le; C o ro lla ry 2 .3 .2

If W com es fro m a c o n s ta n t sy ste m , th e r e is a c o n s ta n t w hich s a t i s f i e s (a), (b) and (c).

C o ro lla ry 2 .3 .3

All c la im s of C o ro lla ry 2 .3 .2 co n tin u e s to hold if W is re p la c e d by t r a n s f e r fu n c tio n m a tr ix of a c o n s ta n t, f in ite dim en sional sy ste m .

A lthough th e r e e x is ts s t a t e sp a c e r e a lis a tio n o f given t r a n s f e r fu n c tio n , i t is by no m eans unique. C onsider th e fo llo w in g s ta t e sp a ce sy ste m (2.3 .4 ):

-1 - 1 -1

I ts t r a n s f e r fu n c tio n can be show n to be z C(I-Az ) B+I w h ere

-1

z is th e lag o p e ra to r. F o r any n o n -sin g u la r m a tr ix T w ith

- 1 -1

c o m p atib le dim ension, th e sy stem (TAT ,TB,CT ) can be seen to have th e sam e t r a n s f e r fu n c tio n . T h e re fo re , th e e x is te n c e of s ta t e sp ace r e a lis a tio n is n o t unique b u t it is unique if we r e s t r i c t to equivalence c la s s e s d e fin e d by th e fo llo w ing eq uivalence re la tio n : (A ,B ,C ) (A ,B ,C ) i f f th ey have th e sam e t r a n s f e r fu n ctio n .

1 1 1 ~ 2 2 2

We c o n sid e r a c a u sa l t r a n s f e r fu n c tio n r e p r e s e n ta tio n of in p u t-o u tp u t model:

00

(2.3.5) y (t) = V U i ) u ( t - i ) i= 0

w h ere y (t)e[Rp is th e o u tp u t v e c to r and u(t)e[Rq is in p u t v e c to r. Then we im m ed iately have th e fo llo w ing re la tio n :

r y (k + l) ' [G G G • • • ]

1 2 3 ' u(k) ' rG0 0 0 • • • ' ’ u (k + l) 1 y(k+2) G G ...

2 3 u (k -l) Gl G0 0 • • • u(k+2)

G ...

3 + G2 Gl G0

* •

4 l j

o r Y+ = Hu + J u + w h e re H and 7 a re c a lled Hankel m a tr ix and T o e p litz m a tr ix re sp e c tiv e ly . It is c le a r fro m th e above re la tio n t h a t H r e p r e s e n ts th e re la tio n s h ip b etw een f u tu r e o u tp u ts and p a s t in p u ts. If rank (H )= n < co th is re la tio n s h ip can be d e sc rib e d by an n -d im e n sio n a l m em ory v e c to r c a lled s ta t e v e c to r.

R esu lt (2 .3 .6 ) (H annan and D eistler(1988))

1. The ra n k o f H is f in ite i f f th e t r a n s f e r fu n c tio n is r a tio n a l. 2. If th e t r a n s f e r fu n c tio n is r a tio n a l and can be w r itte n a s a b

w h ere a and b a r e polynom ial m a tric e s th e n th e d e g re e n o f th e polynom ial d e t(a ) is equ al to th e num ber of lin e a rly

indep end en t ro w s o f H. If H h a s ra n k n, th e n Hn also h a s ra n k n w h ere

n

G G G • • • G

1 2 3 n

G G ... G

2 3 n + 1

G G n n+1 \

G 2 n - l

/

T heorem (2.3.7)

Given th e im pulse resp o n se m a tr ix o r t r a n s f e r fu n c tio n {G } o f a k

r e a l, d is c r e te -tim e , f in ite dim ensional, c o n s ta n t, lin e a r dynam ical sy stem , we have

(1) th e r e is (A,B,C) of sy ste m (2.3 .4 ) such t h a t dim(x)<oo and th e t r a n s f e r fu n c tio n of (2 .3 .4 ) is equal to {G^} i f f ran k(H ) is fin ite .

(2) th e r e is a r e a lis a tio n w ith dim (x)= rank(K ).

(3) Suppose rank (H ) is f in ite , d im (x)= rank(K ) i f f th e r e a lis a tio n is o b se rv a b le and c o n tro lla b le i f f th e r e a lis a tio n is m inim al. An im m ediate consequence o f th e o re m (2.3.7) and r e s u lt (2 .3 .6 ) is t h a t any c a u sa l r a tio n a l t r a n s f e r fu n c tio n can be e x p re s s e d as

- 1 -1

z C(I-Az )B+I w h ere (A,B,C) is o b serv ab le and c o n tro lla b le . T his r e s u lt is im p o rta n t b ecau se th e r e is alw ay s a m inim al s ta t e sp ace r e a lis a tio n f o r given c a u sa l r a tio n a l fu n c tio n w hich a p p ro x im a te s th e t r a n s f e r fu n c tio n o f o u r d a ta s e rie s . It also j u s t i f i e s th e u se o f s ta t e sp a ce m odelling in a p p ro x im a tin g th e dynam ics o f ou r d a ta s e rie s . M oreover, th e Hankel m a tr ix H is e x tre m e ly im p o rta n t in n o t only fo rm a lis in g th e re la tio n s h ip b etw een th e f u tu r e o u tp u ts and p a s t in p u ts b u t also d e te rm in in g th e dim ension of s ta t e sp a ce re a lis a tio n .

2 .4 S ta b ility

We s h a ll n o t d isc u ss s ta b ility in g r e a t d e p th such a s d if f e r e n t c o n c ep ts o f s ta b ility e x c e p t to rec o g n ise t h a t s ta b ility is a p r e - r e q u is ite co n d itio n to have a c o n v erg en t c a u sa l r a tio n a l t r a n s f e r fu n c tio n . T his can e a sily be seen fro m e x p re s s io n o f th e

-1 ”1 \ “1

t r a n s f e r fu n c tio n o f sy ste m (2 .3 .4 ) d e fin e d a s z C(I-Az ) B+I and (I-A z *) l - Y ° Ajz J. The in v erse e x is ts i f f d e t(I-A z X)*0 f o r

Lt j =o

| z | ^ l w hich is c le a rly e q u iv a len t to det(A -A I)*0 f o r |A|2=1 and th is is e q u iv a le n t to m ax( | A^ | )^1 w h ere A^ is th e ith eigen value o f A. T h e re fo re we have th e fo llo w in g re s u lt:

R esu lt 2.4.1

The tim e - in v a r ia n t lin e a r d is c r e te sy ste m (2 .3 .4 ) is s ta b le in th e sen se t h a t it h a s a co n v e rg e n t t r a n s f e r fu n c tio n o u tsid e th e u n it disk i f f I A I ^ 1 f o r a ll i.

1 i1

R em ark: f o r co n tin u o u s sy ste m th e co n d itio n w ould be Re(Aj) ^ 0 f o r a ll i in s te a d o f I A I ^ 1 f o r a ll i.

By e x p a n d in g th e t r a n s f e r fu n c tio n d e fin e d above, we have th e

s te a d y s ta te s o lu tio n o f system (2 .3 .4 ) w r it t e n as fo llo w s :

00

(2 .4 .2 ) x ( t ) = £ AJ_1B e ( t - j) J = i

00

(2 .4 .3 ) y ( t ) = £ CAJ_1B e ( t - j) + e ( t) J = i

M o re o ve r, s t a b ilit y is c r u c ia l in p ro v in g th e e x is te n c e o f

s o lu tio n o f th e L ia p u n o v e q u a tio n P -F P F ’ =Q w h ic h p ro v id e s th e

s t a r t in g c o n d itio n f o r o u r f i l t e r i n g a lg o r ith m developed by K a lm a n and Bucy(1961). By a ssum ing P is s ta b le , th e s o lu tio n o f L ia p u n o v

e q u a tio n can be w r it t e n as fo llo w s (2 .4 .4 ):

vec(P) = { I - (F®F)> V e c tQ ), o r

00

P = V FJQ FJ . j= 0

2.5 P re d ic tio n E r r o r F o rm and S ta te Space M odels

In th e fo llo w in g we s h a ll c o n s id e r in p u t s e rie s as unobse rve d e r r o r and re g a rd th e m as w h ite noise. We r e f e r t h a t e ( t) is w h ite

noise i f f E ( e (t))= 0 and E ( e ( t) e ( s ) ’ )=ö Q. G iven th e b in a r y n a tu re ts

o f o u r observed in p u t s e rie s , w e w i l l c o n s id e r in c lu d in g th e m in to

o u r s ta te space m odel in la t e r s e c tio n and w o u ld lik e to ig n o re

th e m a t th e m om ent. T h ro u g h o u t th e lit e r a t u r e , s ta te space r e p re s e n ta tio n can come in d if f e r e n t fo rm s . F o r e xa m p le S hum w ay and S to ffe r(1 9 8 2 ) c o n s id e re d a p p lic a tio n o f EM a lg o r ith m to

id e n t if y th e fo llo w in g s ta te space m odel (2.5.1):

x ( k + l) = A x (k ) + As(k)

y (k ) = C x (k ) + u>(k)

w h e re 1. y(k)elR p is th e observed o u tp u t s e rie s

2. x(k)e[Rq is th e unobse rve d s ta te v e c to r w it h

x ( 0 ) is a N (/i,E ) r .v .

3. *$(k),uji(s) a re indep e n d e n t w h ite noise

4. x ( 0 ) and u (k ) o r u>(s) a re indepe ndent.

H ow ever, Aoki(1991) o fte n r e f e r r e d to th e fo llo w in g s ta te space

m odel (2 .5 .2 ):

x ( k + l) = A x (k ) + B e(k)

w h e re 1. y(k) and x(k) a r e s im ila rly defined

2. e(k) is independent w hite noise w ith E(e(t)e(s))=<5 n ts 3. x(0) and e(k) a r e u n c o rr e la te d .

For m ore v a r i e ty of s t a t e space r e p r e s e n t a ti o n , we r e f e r to Otter(1985). In f a c t , s t a t e space model can, in g e n e ra l, be w r i t t e n in th e follow ing f o rm (2.5.3):

x(t+ l) = Ax(t) + £(t) y (t) = Cx(t) + T) (t )

w h e re 1. y (t) and x (t) a r e s im ila rly defin ed 2. £ (t) and 7)(t) a r e e r r o r s s a ti s f y i n g

E ( £ ( t ) \ T ) ( t n = 0

£(s)

7)(s) ( ^ ( t ) >,7)(t)>)=5

' Q S 1 S ’ R

We im m ediately see t h a t s y ste m (2.5.3) is m ore g e n e ra l th a n s y ste m (2.5.1) c o n s id e re d in Shumway and Stoffer(1 98 2). On t h e o t h e r hand, sy s te m (2.5.2) is c a lled P re d ic tio n E r r o r Form and can be shown t h a t every s t a t e sp a ce s y ste m (2.5.3) can be t r a n s f o r m e d into th is f o rm . The follo w in g d e ta ils a r e e x t r a c t i o n f ro m Hannan and Deistler(1988):

Suppose th e s t a b i l i t y cond itio n of (2.4.1) is s a t i s f i e d and we have th e fo llo w in g ste a d y s t a t e so lution analogous to (2.4.2):

(2.5.4) x ( t) = £ Aj_1£ ( t - j ) J = i

00

and hence y ( t) = £ CAj ^ ( t - j ) + 7)(t) J = i

Define H (t) = span{y( j) | j^t> and le t x ( t | s ) be th e o rth o g o n a l p r o je c ti o n of x ( t) on H (s). Then f ro m th e s p e c if ic a tio n of s y ste m

y

(2.5.3), we have x (t+ l | t)= A x (t | t)= A x(t | t-l)+A {x(t | t ) - x ( t | t-l)>. Moreover, we de fin e e (t) by y (t)= C x (t | t - l ) + e ( t ) . Since e (t) is o rth o g o n a l to H ( t —1) and H (t) is also span ned by e ( t ) and y ( j) ,

y y

j< t, we see t h a t x ( t | t ) - x ( t 11-1) e s p a n (e (t)). Thus t h e r e e x i s t s B such t h a t sy s te m (2.5.3) is t r a n s f o r m e d into

x ( t + l | t ) = A x ( t | t - 1 ) + Be(t) y ( t) = C x ( t | t - 1 ) + e (t) a f o rm of s y s te m (2.5.2).

a p p e a r s in th e t r a n s f e r fu n ctio n r e p r e s e n t a t i o n (2.3.5) of y(t). Because of all of th e s e a d v a n ta g e s we a r e going to c o n c e n t r a t e mainly on P re d ic tio n E r r o r Form.

2.6 S t a t e Space Model and Causal ARMA model

In th is sectio n, we sh a ll d iscuss th e re la tio n s h ip o r connection b e tw e en S t a t e Space Models and m ore t r a d i t i o n a l (to S t a t i s t i c i a n s ) c a u sa l ARMA models. Consider th e follow ing Causal ARMA(p.q) model (p^q) (2.6.1):

A y ( t - j ) + e (t) +

j

t

j = i

w h e re y (t) is th e o u tp u t p ro c e s s and e (t) is s t a n d a r d w h ite noise. We define

x (t) = A x ( t —1) + K e ( t - l )

p p i p

x (t) = A x (t —1) + x ( t —1) + K e ( t - l )

p - i p - i l p p - i

x (t)

2

X (t)

1

X ( t ) 1

w h ere

A x (t-1) + x (t-1) + K e ( t - l )

2 1 3 2

A x (t-1) + x (t-1) + K e ( t - l )

1 1 2 1

y (t) - e (t)

f B + A i=l , 2 , . . . , q

i 1 M

A i = q + l , . .. , p

Then t h e ARMA model (2.6.1) can be r e p r e s e n t e d by th e follow ing s t a t e sp a ce model (2.6.2) in p r e d ic tio n e r r o r form :

x(t+ l) = Ax(t) + Ke(t) x(0) = 0

y (t) = Cx(t) + e (t)

w h ere x ( t) = (x ( t ) \ . . . , x ( t ) ’)’, C = ( I,0 ,...,0 ) , K = ( K \ . .. ,K ’ )’,

l p l p

A I • • • 0 l

A 0

p

Conversely, if given a c o n tr o lla b le and o bserv able, i.e. minimal, s t a t e space model w ith dim(x)=n:

x(t+ l) = Ax(t) + Be(t) y ( t) = Cx(t) + e (t)

\

y t ' c ' ' I

r \

e

t * f l

= CA

x (t) +

CB I e

t+i

CAn CAn_1B • • • • CB I c

{ t + n

Since, by o b s e rv a b ility , (C, ,(C A )\...,(C A n V ) ’ is a lre a d y of fu ll ra n k n, th e r e e x is ts a , . . . , a such t h a t

1 n

CAn = a C + a CA + • • • + a CAn_1

n n- 1 1

Then

y lt+ n l- a ^ y lt+ n -l)- _{_a y*}

n t e t +n +Q e 1 t +n- 1+ +Q en t f o r som e Q ,Q ... Q .

1 2 n

T hus, we see t h a t d is c r e te - tim e s ta t e sp ace m odels is in f a c t e q u iv a len t to C ausal ARMA m odels. How ever, w ith r e g a r d to allow in g irr e g u la rly o bserved d a ta , s ta t e sp ace m odels o f f e r a very e f f ic ie n t r e p r e s e n ta tio n in te r m s of bo th e v a lu a tin g th e likelihood fu n c tio n and o f a c tin g a s o n -lin e p ro ce d u re a s d a ta com es in.

2.7 K alm an F ilte r and K alm an S m oother

T his se c tio n is co n cern ed w ith u n d e rsta n d in g and d e riv a tio n of th e c e le b ra te d K alm an F ilte r and its S m oother developed by Kalm an and Bucy(1961). As H annan and D eistler(1988) p o in ted o u t, Kalm an F ilte rin g in tro d u c e d a new e r a and found a lo t o f r e a l tim e a p p lic a tio n s e sp ec ially when th e f i l t e r is n ot r e s t r i c te d to s ta tio n a r y s e rie s . M oreover, Kalman(1968) d e sc rib e d t h a t W iener-K olm ogorov F ilte r + th e o ry of fin ite -d im e n s io n a l lin e a r dynam ical sy ste m s = K alm an F ilte r. T hus, we see t h a t K alm an F ilte r is a very in te r e s tin g and p o w e rfu l to o ls in reso lv in g pro b lem s of s t a t i s t i c a l p re d ic tio n and f ilte r in g .

How ever, th e r e a re few a u th o rs who a c tu a lly a d d re sse d th e problem of e v a lu a tin g th e a sy m p to tic v a ria n c e o f th e p a ra m e te rs e s tim a te . One o f th em is M ehra(1976) who co n sid e red expanding th e dim ension of s t a t e v e c to r to e v a lu a te th e F ish e r in fo rm a tio n m a tr ix th e in v erse of w hich d e te rm in e s th e a sy m p to tic v a ria n c e o f MLE. With r e g a r d to v a ria tio n f o r m issing o b se rv a tio n s o r e v a lu a tio n of F ish e r in fo rm a tio n m a tr ix w ith m issin g o b se rv a tio n s, we sh a ll d isc u ss th em in la t e r c h a p te r.

As s u ffic e d to o u r p u rp o se, we sh a ll c o n c e n tra te on stu d y in g a s ta b le and s ta tio n a r y s ta t e sp ace sy stem (2.7.1):

x(k+l) = Ax(k) + Be(k) y(k) = Cx(k) + e(k) w h ere

1. y(k)€lRn is th e o u tp u t v e c to r

2. xfklelR™ is th e u nobserved s ta t e v e c to r 3. x(0) is d is tr ib u te d a s N(/i,Z)

4. e(k) is in d ep en d en t w h ite noise no rm ally d is tr ib u te d w ith m ean 0 and v a ria n c e Q.

S ta te m e n t of Problem :

Given s ta t e sp a ce sy stem (2.7.1) and f o r t ^ t fin d an o p tim al e s tim a to r x ( t ) of th e u n o bserv ed s ta t e v e c to r x (t ) in th e sen se

i i

t h a t i ts m ean s q u a re e r r o r E{(x(t ) - x ( t ))(x (t ) - x ( t ))’> is

l l i i

minimum.