滋賀大学学術情報リポジトリ

245

ON THE APPLICATIONS OF

KM,O-LANGEVIN EQUATIONS

YUJI NAKANO

gl. INTRODUCTION

Ever since Box and Jenkins put together their ground-breaking book, time series analysis has been increasing in a accelated growth path. Fur-ther, time series analysis made rapid progress by the idea of AIC. Now we can apply their theories to many fields containing technology, economics,

etc. But, up-date time series analysis based upon AR models or ARMA

models has encountered a good deal of criticism as "measurement without theory", from macro econometric model analysis based upon traditional simultaneous equations models, in the econometric' study of random phe-nomena, because of their lack of global weakly stationary property. Their main criticisms are the followings.

(1.1) Almost all methods of time series analysis assume the global weak stationality of the model. But, it is doubtful that actual data is a realization of a global weak stationary process. (1.2) Almost all researchers in time series analysis have used AR

models or ARMA models. But they are global weak stationary

processes.

(1.3) The distribution of the model is assumed.

(1.4) The diagonistic tests in the fitting of the model are not

powerful.

246 *a*" ffoA?SXNms-gE'ef.i}t:•taIStee (ng2ss•2sg-51)

series of papers. The theory of KM20-Langevin equation is for the discrete time and local weakly stationary processes. We believe that this theory will answer not only those criticisms but also another problems containing temporal aggregation, missing observation, etc. In [4], we discussed the test of local weak stationality, predictions, non-linear equations, etc. The purpose of this paper is to introduce the theory of KM20-Langevin equation developed in [3] and show some applications. Section two introduces the theory of KM20-Langevin equation. Section three reviews the detailed properties of AR processes. The representation of a local weak stationary process by a normalized white noise is discussed in section four.

g2. KM,O-LANGEVINEQUATION

By Prof. Okabe [3], the theory of KM20-Langevin equation was

found and discussed. In this section, we introduce this theory. The termi-ogies, defintions and theorems of this section follow [3] and [4]. Let X :=

(X(n) ; lnl $ N) be a d-dimensional local weak stationary process on a probability space (O, 6, P) with mean vector zero and covariance func-•

tion R' '

'

(2.1) R(n) - E(X(n)'X(O)) (lnlS N).

For any n G N, 1 $ n $ N, we define a block Toeplitz matrix S. e M

(nd; R) by

R(O) R(1) --••••---• R(n-1)

(2.2) Sn= ' ' '

`R(n-2) --••-••••-• R(O) R(1)

ONTHEAPPLICATIONSOFKM20-LANGEVINEQUATIONS 247

(2.3) 'R(n) = R(-n) (lnlS IV)

and either of the following (2.4) and (2.5) hold : (2.4) S. E GL(nd; R) for any n 6 {1, ••••••, IV}.

(2.5) There exists 7Voe {1, ••'•••, IV-1} such that S, e GL(nd; R) for any n e {1, ••••••, IVo} and S. e GL(nd ; R) for any n e

{No+1, ''"'', N}.

Let M, Mo'(n) and Mo-(n) (O S n S- IV) be the closed linear

sub-spaces of L2(l2, J(9, P) defined by

(2.6) M = the linear hull of {X,(m) ; 1 $i$ d, Im1$ N)

(2.7) Mo'(n) = the linear hull of {X,(m) ; 1 =< iS d, O5 m =<

N)

(2.8) Mo-(n) = the linear hull of{X,(-m) ; 1 S- i <= d, O$m$

N),

where X(m) = '(XJ(m), ••••••, Xd(m)) (lml $ Ar). Then we introduce

two d-dimensional time series v+ = (v+(n) ; O S n $ N) and vr =

(v-(n) ; O$ n f-{ N) by

(2.9) v+(n) = X(n) - PM"(.)X(n)

(2.10) v-(n) =X(-n) -Pna,-(.)X(-n),

where Mo'(-1) = Mo-(-1) = {O} and PM,+c.-" (resp. PM,-(.-i)) stands for

the orthogonal projection on the space Me'(-1) (resp. MoL(n-1)). In

particular, it holds that

(2.11) v. (O) = v- (O) = X(O),

(2.12) v+ (resp. vrr) is an orthogonal time series with mean vector

zero,

(2.13) Mo'(n) = the linear hull of { v+,(m) ; 1 S- 7' <... d, O $ m $

N),

(2.14) Mo-(n) = the linear hull of{vm,(m) ; 1 i <= d, O$m

248 *** ftkeieSXtXgEE-Ef,f>..:E-tslstn (ng258•259e)

where v+(m) = '(v+i(m), ••'''', v+d(m)) and vun (m) = `(v-i(m), ''''•', V-d(M)).

PVe denote V+(n) (resp. V-(n)) the covariance matrix of v+(n)

(resp. v-(n)) (O f+{! n <-- N) :

(2.i5) V+(n) = E(v+(n)'v.(n))

(2.16) V-(n) =E(v-(n)'v-(n)).

We treat the case where condition (2.4) holds. We have

Theorem 2.1. There exists a unique system {y+(n, k),7-(n, k),6+(m), 6- (m) ; 1 S k $ n $ N, 1 S m $ N} of members in M(d ; R) such t)hat

for any n e {1, ••••••, N},

}t-1

(2.17) X(n) == -27+(n, k)X(k) - 6+(m)X(0) + v+(n)

}e =1

n;1

(2.18) X(-n) = -27ua(n, k)X(-k) - 6-(m)X(O) + v-(n).

lt=1

We call equation (2.17) (resp. (2,18)) a forward <resp. backward)

KM20-Langevin equation for X. Furthermore, we designate the system { V +(l), V-(l), 7+(n, k), y-(n, k), 6+ (m), 67(m) ; 1$ l KL N, 1 <= k< n

$ .N, 1 S- m <-- Ai } a KM20-Langevin data associated with the covariance function R of X. We obtain the following fundamental recursive relations

among the KM20-Langevin data associated with R.

Theorem 2.2. For any n, ke IV, 1 <= k < n <-- N,

(2.19) 7+(n,k) == 7+(n-1,k-1) + 6+(n)7-(n, k)

(2.20) y-(n, k) = yT(n-1, k-1) + 6-(n)7+(n, k)

(2.21) V+(n) == (I - 6+ (n) 6- (n)) V+ (n-1)

(2.22) V-(n) = (I - 6u(n)6+(n))Vr(n-1)

(2.23) 6-(n) V+(n-1) == V-(n-1)t6+(n)

(2.24) 6-(n) V+(n) = V=(n)'6.(n)

ONTHEAPPLICATIONSOFKML)O-LANGEVINEQUATIONS 249

A bijective correspondence among 6+(•), 6-(•) and R(•) is given by

Theorem 2.3. For any ne N, 1 $ n $ N,

n-2

(2.28) 6+(n) = -(R(n) + 27.(n, k)R(k+1))V-(n-1)-i

Je--O

n-2

le==o

Remark 2.1

(2.30) 6.,.(1) = -R(l)R(O)-i

(2.31) 6-(1) =-tR(1)R(O)"

(2.32) V.(O) = V-(O) =R(O)

Remark 2.2 in this case where d = 1, we can see that

(2•33) [ 21 (( 1 ), .-) 6: (;-)( ., .)

kV.(•) = V-(•).

g3. AUTOREGRESSIONPROCESSES

Time series analysis, especially AIC, frequently use AR (Auto regres-sion) processes in fitting of the model. In this section, we review the properties of AR processes.

Definition 3.1. The process {Z(n) ; n e Z} is said to be a white noise with mean O and covariance matrix 2 if

'

(3.1) E(Z(n) ) = O for n e Z,

and

(3.2) E(z(n)'z(nz)) = [3 l.i ". =t M..

We shall write {Z(n) ; n e Z} -- WN (O, 2).

Definition 3.2. The process {Z(n) ; n e Z} is said to be an AR(M) process if {X(n) ; n E Z} is stationary and if for every n,

2sO a'eStt tfok;"nvatXLtE:'e,"...E'fasck (ng258•259'il')

(3.3) X(n) + diiX(n-1) + •••'•• + ipMX(n-M) = Z(n),

where dii, ••••••, diM are real dxd matrices

and

{Z(n) ; ne Z} -v WN (O, 2).

In this section, we assume d = 1. Let {X(n) ; n e Z} be an one-dimensional AR(M) process. The auxiliary equation of (3.3) is defined by

(3.4) x" + q5iu"-i + •''''' + q5M = O.

We assume that all absolute values of the roots of (3.4) are less than one.

Then

(3.5) R(n) + diiR(n-1) + ''•••• + dwR(n-M) =Ofor n) 1.

of M terms whe're

1. For every real and distinct root P, a term of the form

(3.6) ap"

is included.

2. For every real root q of order l, a term of the form (3.7) (bi + b2" + •••••• + binH)q"

is included.

3. For each pair of unrepeated complex conjugate roots m and m, a term of the form

(3.8) am" + iiiln

is included. Here ev is a complex number.

4. For a pair of complex conjugate roots 7 and 7 repeated s times, a term of the form

(3.9) (Bi + P2n + •••••• + 6,nS-i) 7" + (Bi + B2n + •••••• + P,ns-i)7n

is included. Here Pi, ••••••, P, are complex numbers.

ONTHEAPPLICATIONSOFKM20-LANGEVINEQUATIONS 251

Conversely, let R be given by a. sum of the form from (3.6) to (3.9). Namely, for n e N',

tl t2

(3.10) R(n) == 2aiP," + 2(b,i + •••••• + b,t,n'"i)q," +

1=:1 1=1

t3 t,

2(aim," + iiinm,") + 2 (P,i + •••••• + P.,nSirri) 7," + 2 (Pn +

1=1

1=1 1=1

and

(3.11) R(-n) =R(n).

We assume

(3.12) lp,I< 1(i = 1, ••••••, ti),lq,l< 1(i -- 1, ••--••, t2),

lm,1< 1(i = 1, -••-, t3) and17,1< 1(i -- 1, ••'''', t4)•

ti t4

Let M == ti + 21, + 2t3 + 22s,.

t=1 t=1

Then, there exist dii, ••••••, diM such that

tl tl t3 t"

(3.13) R'(x-P,) ff (x-q,) ZZ (x-m,) (x--m-,) iO' (x-7,) (x-7,) iiE

t==I t=1 i=1 z=1

xM + .45ixM-i + ''''" + diM•

It is easy to get

(3.15) R(n) + ipiR(n-1) +••••••+ diMR(n-M) =Oforn)M.

The conditions that R becomes an auto covariance of an AR process are -decided by the following Yule-Walker equations,

M

(3.16) 2 dijR (i-j) = O for 1 S i S M.

j=o

Here, dio == 1. Further, if

(3•17) f(1) : = i., .=3nl.e-t"AR(n) l O for all 1 6 [-z, z], from general theory we get

Theorem 3.1. R ( • ) is an autocovan'ance function of an AR (M) Process. The proof of Theorem 3.1 is also easily verified from (2.19) and (2.20) of

2s2 "7f*" ifosci'nNikXLtEE'eR..E'ftscft ceg2s8•259-Sl-)

KM20-Langevin equation.

Example 3.1. For n e Z, let

(3.18) R(n) = cip]i"i + c2pL,i"i + c3p3ini, where (i) c,(i -- 1, 2, 3) are non-zero real numbers,

(ii) P,(i = 1, 2, 3) are non-zero real numbers which absolute values are

less than one.

We set

(3.l9) dii = -(Pi+P2+P3), di2 = (PiP2+P2P,3+P3Pi),

di3 - -pip2p3• ci - 7,i(pP,3ipP,2))((iipP,LpP;)((iipP,g2)) ,

op,2 (P,-P,) (1-P,P,) (1-p,2)

and c3 = c.

C2 == p,2(p,-p,) (1-P,P,) (1-P,2)

If c3(P3-Pi) (P3-P2) > O, then R(t) is an autocovariance function of an AR(3) process.

Example 3.2. We applied a tent function in [4]. We define for each P

e (O, 1) a mapping Åët,p from [O, 1] onto [O, 1] by

(3.2o) Åë,., (.) - [ ei-ixp) -, (i-.) li : [ [gl e;.

It is known that dit,p is a mixing transformation with Lebesgue measure as its unique invariant probability measure, which is called a tent function.

We define an one-dimensional strongly stationary time series T p =

(T,(n) ; O $ n < oo) on aProbability space ([O, 1], 6[O, 1], dx) by

n-tlmes

-and (3.21) Yt,,(n) (x) = dit,po••••••odit,p

(3.22) Tp(n) == Yt.p(n) -E(Yt,p(n)).

Then RT,(n) =- E(T,(n) T,(O))

ONTHEAPPLICATIONSOFKM20-LANGEVINEQUATIONS 253

(-1) "12-i I2p -11 for P e (O, 1/2)

= 12-i6o,n for P=1/2

12-i12p-11 for Pe(1/2,1).

This implies that (Tp(n) ,' O $ n < oo) has a white noise property or a simple Markovian property, according to the case P = 1/2 or P = 1/2. If

P = 1/2, we define RT,(-n) = RT,(n) for n e N. From the theory of

KM20-Langevin equation, there exists an orthogonal time series v == (v(n) ; O $ n < oo) such that

(3•23) Tp(") = [q"T(O,)(..1) + .(.) i:i Z =t= Oo,

here

(3•24) E(v2(n))= [llli(i-q,) lgi Z=t Oo,

and

(3.2s) q-[i,S2e,1ii igi ;[[2)g,/3.

As previously stated, autocovariance functions of AR processes have special forms. Therefore, the class of AR processes is not so wide in weakly

stationary processes. We think that it is not so efficient to use AR processes

in the fitting of models for random phenomena [4]. The theory of

KM20-Langevin equation, we believe, must be applicable in data analysis.

g4. REPRESENTATION

We assume d l 1. From the theory of KM20-Langevin equation, a

locai weakly stationary process {X(n) ; n e Z} is represented as a linear sum of an normalized white noise whose mean is zero and covariance is the unit matrix I.

2s4 g'R tftAS-dikNLtEEE,A.uS'fiptM (ng2ss•2sgg)

(4.1) x(o) = lt7;(6>-g.(o) = rm4-(o).

(4.2) X(1) == -7.(l,o),A71J-((5>-4.(o) + rmg.(1),

and

(4.3) X(-1) = -7m(1,o)A7I-(6Ye.(o) + vEl7:-CDUg-(1).

For n) 2,

n-2

(4.4) X(n) = Åí{(-1)7+(n, l) + 2 (-1)M'i7+(n, lei)

l=O ISmSn-1

n>k,> >k.>t

Å~7+(ki, k2)'"'"''7+(fe,., l)}A7-IJ-(Z7g+(l) - 7+(n, n-1)

Å~wt4+(n-1) + s/7;(ii)'e+(n),

and

n-2

(4.5) X(-n) = 2{(-1)7=(n, l) + 2 (-1)'"']7-(n, fei)

l=O ISmSn-1

n>le,> >k.>l

Å~oi-(kb k2)t'••••••oi-(le., l) }vf=l7t:-(ZIFg-(l) - 7tr(n, n-1)

Å~A7r(iFJs-(n-1) + A7r(M4-(n),

where {e+(n) ; nE N'} and {4- (n) ; ne N'} -- WA[ (O, I).

Proof. For neN", let 4+(n) = A7I-(Mrmiv+(n) and 4-(n) ==

vr"VT(M- i v- ( n) .

Then, immediately we get Theorem 4.1. (Q. E. D.) Conversely, it is easy to prove

Theorem 4.2. Let covan'ance matrices {R(n) ; n e Z} and a normal-ized white noise {4+(n), e-(n) ; n e N*} be given. By tafeing a linear sum of the white noise, we can construct a local zveakly stationa7y Process {X(n) ; n e Z} whose covan'ance fzanction is R.

Proof. From {R(n); nEZ}, we can get 7+(•, *), 7-(', *), V+(')

and V-(•). Therefore we can define {X(n) ; n e Z} as in

Theorem 4.1.

ONTHEAPPLICATIONSOFKM20-LANGEVINEQUATIONS 255

References

[1] P.J. Brockwell and R.A. Davis: Time series: Theory and Methods. Verlag (1987).

[2] W. A. Fuller: Introduction to statistical time series, Wiley (1976).

[3] Y. Okabe: On stochastic difference equations for the multi-dimensional weakly

statlonary tlme serles.

Prospect of Algebraic Analysis. Academic Press (1988).

[4] Y. Okabe and Y. Nakano: On the the theory of KM20-Langevin equations with

applications to 'data analysis (I).

To appear in Hokkaido Mathematical Journal,