An algorithm for spectral factorization using random search techniques

Pierre Bernard and Claude Bonnemoy

Maitre de Conferences, Universit~ de Clermont II D~partement de Mathematiques Appliqu~es, Les Cbzeaux, B.P. 45 - 63170 Aubiere, France

An approximation method for the energy spectrum of a stationary stochastic dynamical system is presented, which allows approximate functional rational factorization.

This paper is in three parts. The first deals with a theoretical problem of approximation in Hardy Spaces, whose main result is the following:

Let S ("1, S be positive functions belonging to L1 (~,), such that log S ("~ and log S belong to L 1 (N .... dco/1 +o.)2).

Let h ~"~, h be the outer functions of the Ha rdy Space//2(1-I + ) such that S~"~- - Ih~")l 2 and S = IhJ ~ on iN.

If S ~"~ ~ z S in L~(~o), and log S (~ ~z log S in L~(gi,,,dco/1 + co2), then: h I~) n'~ h in H 2 ( f I +). The second part describes an effective algorithm, using random search methods, and gives an

almost sure convergence result for it. The third part treats numerically two examples, permitting comparison of this algorithm with

others (whenever there are...): the first example is a problem of approximation for a nonrational process (turbulence) that was considered in Ref. 22; the second example is a problem of model reduction (automatic) considered in Ref. 4.

I N T R O D U C T I O N

Many physical systems driven by random forces can be considered as driven by a zero mean stationary real Gaussian stochastic process: ¢ = ( ~ t , t ~ ) , physically realizable, which is equivalent to the condition:

f ~' Ilog S¢(o))[ de) < o9 ~, 1 +0'.) 2

(l)

where S¢ is the spectral density of 4, and is also equivalent to the fact that ¢ can be obtained by linearly-filtering a white noise by a causal filter in Lz(E+), i.e.,

~,= f_ ~ k ( t - u ) d W , (2)

Wiener integral, (Wt, t eN ) being a Brownian motion extended to ~. Hence, the Laplace Transform H(p) of k belongs to the Hardy Space He(H+) on the half-plane rI + = {z~C/Ne(z)> 0}.

These two presentations are related by the factorization of S¢:

S¢=lH(iog)t 2

In view of the quantitative study of the dynamics of such systems by simulation, or of the theoretical study using stochastic Ito or Stratonovitch differential equations, whose theory are very much developed nowadays, a technique of Markovian realization, described in Ref. 5, will be used.

Paper accepted October 1988. Discussion ends December 1989.

~') Computational Mechanics Publications 1989

66 Probabilistic Engineering Mechanics, 1989, Vol. 4, No. 2

This can be applied exactly when S¢ is a rational function, but if not, an approximation must be used, named in Ref. 5 approximated Markovianization.

In practical applications, this will be the case for turbulence, whose spectrum follows in the inertial domain Kolmogorov's 2/3 law. This problem is present in civil engineering, for dynamics of a structure loaded by turbulent wind (horizontal turbulence here, see Ref. l 1); or, in aeronautics, when considering airplane-guidance with vertical turbulence 18

The approximation of such a process with nonrational spectrum by a process with rational spectrum is a standard problem as long as the factorization (3) of the spectral density is known. However, there are numerical difficulties (no convexity . . . . ). The technique proposed here can be used in this case, and this has been done in Ref. 11. But the main practical case is when only S¢ is known, and not its factorization. The algorithm described in this paper provides an approximation of the process by a rational process as well as an approximated factorization of S¢.

I O N E RESULT O N A P P R O X I M A T I O N IN H A R D Y SPACES

Using the Moebius Transform z= w - 1 / w + 1, mapping conformly the half plane FI + onto the unit disc ~3 of the complex plane, and the boundary i~ ofH + onto the circle T, linear filtering problems for second order continuous time processes are reduced to discrete time problems 7. Therefore, the proofs will be given in discrete time, and the results translated to continuous time.

An algorithm for spectral factorization: P. Bernard and C. Bonnemoy

H2([D) is the second order Hardy Space of the unit disc ~3, i.e., the space of all complex holomorphic functions on I) whose Taylor coefficients about 0 constitute a 12(N) - sequence. To H2(D) is obviously associated the space H2(T) of functions on the unit circle T belonging to L2(T) and whose Fourier Series only contain positive rank terms.

The following Hilbert spaces are naturally isomorphic:

12(~) H 2 ( D ) H 2 ( T )

(a, ,n~N) h(z)= ~ a,z" f(0)-= ~ a,e ~° n=0 n=O

being harmonic functions in ~3. v(resp, v t"~) being the harmonic conjugate function of u(resp, u ~"~) chosen to be 0 at 0 (Ref. 2).

1 n II ht", - h [I ~ = ~ f'_ n lexp{u(e'°) + iv(e'°)}

- exp{ut"~(e i°) + ivt")(ei°)}] 2 dO

Ilh'"'- h]12 <~ 2(1, +J , )

with:

Following Beurling, let us recall that the function f e H2(T) with unit modulus are named inner functions; and those functions f e H 2 ( T ) such that log]fit)] is integrable and

log[f(0)[ = log[f (t)] dt (3) H

are outer functions. Every function f 6 H 2 ( T ) admits an unique splitting

into a product

1 (.1 .J_ [exp{ u(e;°) + I " = ~ n iv(ei°)}

_ exp{u(elO) + ivt,~(eiO)} [2 dO

1 n J. = ~ f_ ~ [exp{ u(ei°) + ivt"'(ei°)}

- exp{ut")(e i°) + ivt")(ei°)}12 dO

1 ('n J" =21-I J -n texp{u(e'°)} -exp{ut"'(e'°)}t2 dO

f = M I ' Qs

where M : is inner and Q: outer 2'3'8. The links between second order stationary processes

and their spectral densities, as well as the characterization of processes with a Markovian representation as rational spectrum processes are supposed to be known t.

The Szeg6-Kolmogorov-Krein theorem 2'3's implies that a necessary and sufficient condition for S~L~(T) to be the spectral density of a physically realizable process is that log S is integrable. Then, there exists h E H2(T) such that S= ]hi 2. But h is not unique, and the factorization result given before shows that it will be necessary to consider outer functions, defining in signal theory minimum phase filters.

The problem here consists in proving the regularity of the mapping from S to h, admitting the explicit form:

1 ~ d°+z ] h(z)=exp ~ J - n e'°~z l°gS(e'°)dO (4)

Theorem l Let S t"), S be positive functions in LI(T), such that

log S t") and log S belong to L*(T). Let h t") and h be their respective associated outer functions. Suppose S t") converges to S and logS t") to log S in U(T) as n---~ oo.

Then h t") converges to h in H2(D).

Proof

[[ht.,_h L2 = 1 F i n ]ht"'(e'°)-h(e'°)[ 2 dO 2FIJ_ n

1 H J . = ~ - fl_ n ] S " ~ - ~ 1 2 d0

J. IIs- St"'IIL ,T, (because: for a , b ~ + )

1 ('11 d-|n exp{2u(ei°)}lexp i{v(ei°) - vt")(ei°)} - 1] 2 dO I,

~ 2 H

The operator mapping its conjugate function to an harmonic function of h~(~) is of (1 -1) strong-weak type 2"9, and: there exists a constant K such that

L @ t")- v] > s ) ~ K/~II ut"'- u [l~

where L is the Lebesgue measure on T. It follows that v t")---, v in L-measure, so that e itv - v'"') - 1 converges to 0 in L-measure as n ~ oo. The sequence of integrands in I, is bounded in modulus by 2 exp{2u(ei°)} which equals 2S(co) and is integrable. As this sequence converges to 0 in probability, it is uniformly integrable, and I, n ~ 0.

In view of applications, we give a weaker result, avoiding verification of the convergence of logarithms.

Corollary 1 Let S ~") be a sequence of positive functions in U (T) and

Se D(T), m> 0 a constant and no an integer such that:

O<m<~S(O) VO~ T

and

Define u, u t"), v, v t"~ as follows:

h = exp(u + iv) h t") = exp(# ") + ivt"));

u = logth ] u t"'= loglht"'l

n>~n o O<m<~S~"~(O) VO~ T

Let us assume that S~")---. S in D (T) - norm. If h t"~, h are the outer functions defined in theorem 1, then h t")~ h in H 2 ( ~ ) - norm.

Probabilistic Engineering Mechanics, 1989, Vol. 4, No. 2 67

An algorithm for spectral factorization: P. Bernard and C. Bonnemoy

Proof From Lebesgue's Theorem on positive reals (where the

function log is differentiable), we get

[log S ~")- log S I ~< 1 ISt.,_ SI

and so, if Is "'-sl 0 in D(T), so does

[log S ("1- log S[

The assumptions of theorem 1 are then satisfied. In the following, it will be proved that S E U (T)can be

approximated by a sequence of rational functions Stn)~ D(T) satisfying the assumptions of theorem 1.

A rational function belongs to H2([~) if and only if its poles are in D~ = {z~C/tz t > 1}, and is outer if and only if its roots are out of D.

Lemma 1 The set E of rational functions admitting only simple

poles all in D~ is dense in Hz([D). Note that E = ~ , ~ ~ E, where E, is the set of rational

functions with degree ( n - 1, n) whose poles are simple.

Proof H2(~) is a Hilbert space. To prove that a set E is dense,

it is sufficient to prove that if f~HZ([D) is orthogonal to any element of a family of generators of E, then f = O.

A family of generators of the set E of rational functions on C admitting only simple poles in D~ is the family of functions

1 / / ~--,> - -

1 - Z u

for Z such that IzI < 1. (This follows from the classical decomposition of rational functions).

So, with the scalar product on H2(9), the orthogonality condition is

is the limit in U(T) of a sequence S("~= Ihtn)[ 2, h ~"~ being, for every n s N, a rational function with all poles simple and belonging to D,, and all roots out of D. Moreover, if S is symmetric, then h~")= h ¢").

P r o o . f

(a) S can be written S=[hl 2, with h~HZ(D) outer and unique (up to a multiplicative constant of modulus 1 ). By Lemma l, there exists a sequence h t"~ of rational functions with all poles simple and belonging to D~, and all roots out of D, converging to h in H2(D) norm.

Each h I") has a decomposition Mh~., "Qh'"' in an outer and an inner component. It is well known 3 that Mh(n), can be obtained from h t"l by multiplication of h t"l by factors: 1@~ e - Z ~ 1 - ( t Z for all ~'s zero of h (n>. The new sequence of rational functionals obtained by this procedure satisfies the criteria, and has the same modulus on T as h I"~.

(b) Let H~ be the subspace of H = H 2 ([D) consisting of all 'real' elements of H, i.e., elements h such that h = h. Then it follows from a result of Ref. 7 that the real vector space Er= {x+x; x~E} is dense in H~. Note that E, is then the space generated by the rational functions

1 1 eN and zeC'xR

1 -a t ' 1 - 2(Re z)t + IzlZt TM

For continuous time processes, we obtain the following:

Theorem 2 Let S t"l, S be positive functions in L~([~,,,), such that

logS tn~ and log S belong to D(N,~, d~o/l +e)2). Let h (n) and h be the outer functions associated to Stn)

and S. Assume that S I"~ n ~ S in L ~ ([~,,,, d~o) and

1 fn f(e i°) 2II n l -Ze- i °dO=O (5)

LogS~"~n~clogS in LI (~,,~, ~d~j~ -)

Now, as fEH2([D), it is the Cauchy integral of its boundary value: if IzI < 1

1 f_ f(e i°) f ( Z ) = 2H io dO n 1 - Z e -

Then htn~n~ h in H2(lq+).

Corollary 2 Let S (n~, S be as in theorem 2; m > 0 a constant and n o an

integer such that:

It follows from (5) that f = 0. That is the result. The problem of finding the best approximant of

feHE(D) in E, has been studied by J. Della-Dora 14, using interpolation in self-reproducing spaces.

Another method will be presented in the second part of this paper.

Complex conjugation (an) ~ (an) induces on H2(~) and HZ(T) a conjugation denoted h ~ h, where if

and

0<m ~ (1 + (02)S(e9) Vcoe~

Vn>~n 0 0 < m ~< (1 + o)2)Stn)(o)) Vo)E~

If S("I---~S in L1(N~,,), then h("l---~h in H2(•+).

h(Z)= ~ a.Z", h(Z)= ~ anZn n - O n - O

Lemma 2 Let S be a positive function, without root on T,

integrable and whose logarithm is integrable on T. Then S

II R A N D O M S E A R C H O P T I M I Z A T I O N

In order to find the constructive counterpart of theorem 2, let us solve the following preliminary optimization problem.

For S given as in the theorem, we have to minimize the

68 Probabilistic Enyineerin9 Mechanics, 1989, Vol. 4, No. 2

function

g(h) = IS(w)-]h(w)}l z dw oo

for h~ E,. Hence, if the set of coefficients X of h describes some

constraint subset C of EN, we solve, for g:EN___, ~, the problem

(a) Find m = inf{g(X):X ~ C} (b) Find any x* ~ C:g(x*) <~ 2 = m + epsilon

The principle of the implemented algorithm is as follows: Choose first, in any manner, a point x°~C, or even in ~N \C (however this might be more expensive in computing time). A random sequence of points V ~ is generated (not necessarily independently) following a Gaussian law on EN, centred at x °, and restrained to C. The covariance matrices used for each generation may be chosen deterministically or at random as a measurable function of the previous observations. For each generation, V ~ C is tested, and if the answer is affirmative, g(W) is evaluated.

The first descent time is noted dl:

dl=inf{i>>.O:V~eC and g(W)<g(x°)}

Then define the random sequences X k ~ ~N and Uk ~ R:

X ~ . = x ° and Uk=g(X k) for0~<k<d~ (failure time)~

X dl = V al and Ual=g(X al) (success time)

During the search, a sequence F k of covariance matrices is used with a strategy consisting in decreasing Fk (in the sense of the definite positive order) to obtain a descent, or, if the search is too long in increasing F k to escape from eventual local minima. Let Z k = (X k, Fk) be the state of the system, which is defined in such a way that it carries all the useful information available at time k.

Then the same procedure is followed starting from the observations x el and g(x da) of X d~ and Unl until the second time d2 of success and so on.

It will be shown that under weak hypothesis, the sequence Uk, is almost-everywhere convergent to the minimum m of g on C.

For some optimization problems, as the one solved in this article, it is possible to introduce in the above pure random search some particular deterministic method for some part of the unknowns, for which the necessary assumptions are satisfied (gradient methods, etc.). This deterministic method will be used in alternance with a random search on the other unknowns. Such a mixed algorithm will be noted [AM], the pure random search algorithm will be noted [AI].

We now define a general algorithm [AI] and give the main results ~2, (for the proofs, see the appendix).

The first convergence results are due to J. Matyas ~, who obtained a probabilistic convergence of X k to X*, under hypothesis of unicity of X*. A.L. Guimier 16 obtained an a.e. convergence for independent V%. The state variable Z k was introduced by L .P . Devroye ~5. N. Baba 1°, F. J. Solis and R. J. B. Wets 2~ obtained an a.e.-convergence result for the constrained problem.

An algorithm for spectral factorization: P. Bernard and C. Bonnemoy

Lemma 4 Let (U k, k ~> 0) be a decreasing sequence of real random

variables on the probability space (fL G, P). For every 2 real, we define:

O~k(2)= P(Uk <2/Uk_ 1 >~ )~) (6)

Suppose that, for some )~ real

~ k ( 2 ) = + ~ (7) k = l

Then, almost everywhere, there exists a finite integer k o > 0 such that for all k >>- k o, U k < 2.

Moreover, if (Uk) has a finite lower bound m and if (7) holds for every 2 > m, then U k ~ m a.e.

Algorithm [AI] First let Qk be the sub a-algebra of G generated by V i,

0~< i ~< k, and Zk = (X k, Z~) a given measurable application from (fL Qk) to an euclidean measurable product space (R u x E', fln@fl')' Z~ contains the information at time k not contained in X k and necessary to a given method.

(a) Initialization k=0 , Zo,~.(x°,Z'o)eC xE' , Uoa~. g(x°), V°a~.x °

(b) k/> 0: V k + 1 is generated following a given law restrained to C, conditioned by Z k = z k, then

Xk+l= Xk+ if g(V k+l)>~g(Xk) (failure) V k a else (success i.e. descent)

(d)

Uk+l=g(Xk+l), Zk+l=c~(Zk, Vk+l), where the application ~b has to be defined for each particular method (see next examples). In case of success a stopping rule is applied ~ STOP or (d) k = k + 1, goto (b)

Examples of state variables As previously said, suppose that V k+l is generated

following a Gaussian law N(x k, Fk) and put Z k = (X k, Z'k = Fk) Then the covariance matrices may be chosen in any of the following manners:

(1) Fk is a deterministic function of k (increasing, decreasing or constant).

(2) Let I N be the unit N × N matrix and give/~0 >>/~ > 0, choose

Fk =#f in

with

/~k = O#max{#, min{/~ k if success

_c IIv +'- 1121} else

which is a possible choice if one is near the solution m. Etc . . . .

Theorem 3 If:

(hl) Denoting by L the Lebesgue measure on R N, let us suppose: g is measurable, C is a Borelian set of R"

Probabilistic Engineering Mechanics, 1989, Vol. 4, No. 2 69

An algorithm .]br spectral factorization: P. Bernard and C. Bonnemoy

with: L(C) > 0, m > - z is a real number,

LIT~(g))>O where T~,(g)= {xe C:g(x)<2}

(h2) For 2 > m let:

Ak().): {X k+l ¢ T),(g)}

Bk(2) = { X k e ~N \ T~(g)}

(h3)For every 2>m, there exists an integer k o and a divergent series whose general term ~k(2)~> 0 is such that:

PO'[AR()o)]>~ek(2 ) on B,(2) for k ~ k o

Then the sequence U k converges a.e. to m.

The next result gives a very rude, sufficient condition for convergence of the particular algorithm used in part III (but other kind of methods are possible 12,13),

Corollary 3 If the regular conditional law of V k+l knowing that

Zk=Z k has a continuous density Sk(V/Zk) of the form (ak(V--~), whose support is the whole space R" and if9 is continuous and bounded from below, then the hypothesis (h3) is satisfied.

Application For the examples in part III we use Sk(1)/z k) where (Dk is

the density of a non degenerate Gaussian law N(0, Fk) and where the covariance matrices are diagonal with deterministic evolution.

Mixed methods: algorithm [AM] If we note S~(w)=lh(w)] 2 belonging to the rational

fractions set E defined in part I, Lemma 1, we have to minimize for X e C:

NI(x)= fR IS <w)- S(w)l dw

where S is given. Let us explain how convergence can, in the present

case, be improved by introducing an auxiliary quadratic optimization problem, with the same constraints, concerning:

N2(X) = fR IS <w)- S<w)12 dw

The evaluation of the criterion will be done on a bounded interval [ - b , b ] where b is chosen > 0 large enough to make negligible

fb + ~ S(w) dw

as for usual computing procedures for undefined integrals.

If S(w)~>s>O on [ - b , b ] , then the hypothesis of corollary 2 is satisfied.

The frequency interval [ - b , b] is discretized and the

criterion is computed by a Simpson rule. So we have to minimize, for a = 1

£ : E A lSx(w )- Stw )l Ok-K

For ~=2, N2 is of the type:

'(Ak(S(wk) ~=~ xj(wkY-1 ) 2 giX)=k=o - o+j/wO where {Wk, S(Wk)} O<<.k<~K are the observations.

Note that the corresponding optimization problem is quadratic with respect to the part XD = (Xl . . . . . Xv) of the unknowns.

Set XL= (Xo+ 1 . . . . . XO+L), X = (Xo, XL) and N = D + L. Note also that the constraints on X are C = Co x CL

where Co is a cube in R N (for example xj > 0 1 ~<j ~< D) and CL is of any sort.

Algorithm [AM] (a) First start as for the [AI] algorithm with a pure

random search on the N unknowns of the main criterion e = 1 until a cost-test becomes true for a descent time d (cf. example in part III). At this time X d = (Xo, Xc).

(b) Then, fixing XLeCL, find with a deterministic method (for instance Gauss-Seidel with constraints) XoeCv such that

~?2(xo, XL)< N2(Xo, xL)

(c) If N1 (XD, XD < NI (Xo, XD. Then fixing Xo, find by [AI] method XLeCL such that

N1 (Xo, XD < -NI (Xo, XL) and put X e' = (Xo, XL)

Else apply [AI] pure random search on the N unknowns for c~ = 1, starting from X a to obtain a new descent point X d', after d ' - d - 1 generations.

(d) And so on, until stopping rule becomes true.

NB: Another way is to minimize only the criterion for c~ = 2. The one for c( = I is bounded by a decreasing sequence and the solutions obtained are quite close in those examples.

Stopping rule It is a weakness of those methods. Statistical rules may

be used I 3, but for the examples in part III the following is done: as m>0, the algorithm is performed to pass through a positive level 2>0 , unless the number of generations between two successes is too large (here 5000). The convergence to a local minimum is possible (one intent to escape is by increasing Fk).

A P P E N D I X

Proof of lemma 4 Note Yt()O=l{c,k<;, where 14 is the characteristic

function of A. The sequence (YkO'), k>~O) is then a nonhomogeneous Markov chain with two states 0 and l, whose state transition matrice from k - 1 to k is:

70 Probabilistic Engineering Mechanics, 1989, Vol. 4, No. 2

Define

k

:~(,~)= FI p,(,E) i = i

An algorithm for spectral factorization: P. Bernard and C. Bonnemoy

Q(") will be searched, in fact, in the form:

[q/2] Y~ (z - z y ) ( z - ~y)r(">(z)

1=1

Then, PR(2)= ~(1-~i(~)) 1 - I=IH (~ --~i(/~)) 1

If ~ L ~ ~,(2) = + ~ then ~I~:~ (1 - ~-()o))--~ 0. The chain is then ergodic and the state 1 is absorbing,

which implies the first assertion. For the second part, as shown in Ref. 16, note that the

sequence (Uk), a.e. decreasing and lower bounded, is a.e. convergent to U with Ua~.m.

If f~0 = { U = m}, then D\f2 o = U ;L ~ (u > m + l/p) and applying the first part of the lemma with 2 = m + l/p:

so that (Uk)--,m almost everywhere.

Proof of theorem 3 By construction, the sequence (Uk) is decreasing. We

only have to show that, for every 2 > m, there exists an integer k0 such that ~K(2)/> ~tk(2) for k t> ko, which implies (7) for every 2 > m, and then the result by lemma 4. Now

O~k + l ('~):" P(Uk + l < ~/Uk ) A )

= p(Xk+I~ T~(g), xk¢ T~(g))/p(XR¢ T; (g))

= P(Ak(2)~BR(2))/P(Bk(2))

_ 1 ~ PO~(Ak(2)) dP P(Bk(;t)) .Ja~(~)

Proof of corollary 3 Let To(g)= {x~R": g(x)<g(x°)} which is bounded by

hypothesis. Note that for the [AI] descent method, X k ~ To(g ) for every k > 0.

As we may suppose that P(Bk(2))> 0 one can write

= ~ ( ; , ) > 0

III E X A M P L E S

With the previous notations, the approximating sequence will be of the form:

where

P(")(iog) 2 Sx,.,(o)=

p<q

with

P q e(")(z) = ~ al"'z z, Q(")(z)= ~ b}")z i,

/=0 j=O

al "), b}"'e N; b(q ")= 1

r(,)(z ) = ~ 1 for even q z--Z(q ") for odd q, Z~q")~

So (in those examples), solutions with at most one real pole only are considered.

The unknowns will be:

z(")-'(")+ifll ") (ill ")=0 for odd q)

X(o " '= (a~0 ") . . . . . @")), X(s")= {(el"',fll"')1% l<~ [q/2]}u{%}

X(n)_ iX(n) x (nhE R N -~ o , L J N = ( p + l )+q

and the constraints:

C = {xe NU/real part of the roots of/~") and Q(")<0}

Example 122 8 2

1 ] +3 (1.339m) s(~o) =

2 (1 + (1.339¢o)2) l 1/6

The data are: K=40 , b=10, c~= 1 and c~=2. We both use [AI] and [AM] methods in the two following cases:

(a) p = 2, q = 3, (N = 6). For which the approximation Sx of S is of the form:

where

S ~ , ~ ( ~ ) = y3o9 4 + y 2 c o 2 +y~

((p2 -- 0)2)2 ..~ S2(D2)(O)2 _it_ X2)

Yl = x~, x 2 x~ - 2xlx 3 Y3= 3, Y2= p 2 = 2 2 x4+xs=~lz+ /~ , s=2x4=2~ l,

with the above notations.

X6 ~ ~2

0+4

) . 3

1.2

: I I I 1, 2 . 3 . 4 .

÷ + $ x x S a n ( 2 , a ) e o s o= [z,z) Example 1.

Fig. 1.

S .

f

Probabilistic Engineering Mechanics, 1989, Vol. 4, No. 2 71

An algorithm for spectral )dctorization: P. Bernard and C. Bonnemoy

Table 1.

x ° AI2. 3 AM2.3 AI1,2 AM1.2

~1 0.386 0.0555 0.0550 0.0643 0.0659 N 2 0.071 3.43 E - 4 3.39 E - 4 1.07 E - 3 5.7 E - 4 ST 0 21 15 32 19 KT 0 41068 17264 53921 17704 t min 0 17 7 19 4

Table 2. x ° AI AM Routh a-J3 Pade

~l 0.465 0.0197 0.0193 0.389 0.430 NE 0.080 1.29 E - 4 4.73 E - 5 1.66 E - 3 0.021 Number of descents ST 0 24 25 Numer of generations KT 0 17255 30039 Time t 0 8 mn 16 mn

(b) log s I

-3. -2. p = l , q = 2 , ( N = 4 )

s ~ , 2 ( ~ ) = Yl +Y2 0)2

(p2 _ ~02 )2 + $2(D2

where

ya=X~, y2=x~, p2-x2±x2 - - - 3 T 4~ S = 2 X 3

The results appear in Table 1. ST is the total number of descents, K T is the total number of generations, comput ing time is obtained for a Mini-6-96. The results for S2x '3 are, of course, better than those for S~ '2 and the cost of the [AM] method is less than for the [AI] method.

Examples of coefficients (constraints are always satisfied):

Solution AIz, 3 X* =(0.296, 1.378, 1.130, - 1.143, 0.044, - 0 . 324 )

Solution AM1, 2

X* = (0.738, 1.122, - 1.008, 0.0004)

Example 2*

S((o) = I G(ico)l 2

z 5 + 9z 4 + 22.25z 3 + 30.25z 2 + 23.25z + 9

G(z)- z6 + 10.5zS + 45z 4 + 100z 3 + 119z 2 + 69.5z + 15

whose poles are: ( -2_+i ) , - 0 . 5 , - 1 , - 2 , 3. A stable reduction of S(o)) may be obtained by:

y3 co4 + Y2 c°2 + Yl Sx(C°) - - ((p2 __ (.02)2 _~_ S2(D2)(~D 2 + X 2)

with the same coefficients as for example l(a) and:

p = 2, q = 3, (N = 6), K = 40, b = 20, c~ = 1 and 2

Several deterministic methods have been used in Ref. 4: moments method, Pad~'s approximation, c t - f l Routh development, direct Routh method, cont inuous fractions.

I-1" ~-X. X. IZ.

+ + S X X $ am

0 o Pad* o o Routh a-b Example 2.

log f

Fig. 2.

The best one of this g roup is the c~-/~ Routh, other ones being not even stable (unrespected C constraints).

[AI] and [AM] algorithms have been performed (deterministic method for xl, x2, x3 in [ A M ] ) with an initial diagonal matrix Fo with coefficients 0.1 6is.

x ° = (1, v/3, 1, - 1, 1, - 1 ) , N1 (x°) = 0.465, /~2(X0) = 0 .080

the level being fixed to 2 = 2 E - 2 for/~1. The results appear in Table 2. The mixed method

[AM] starts for the tenth descent s, the test INI(X°) - min0<.k.~ s NI(Xk)] /KT<5E-5 being satisfied; the random search being then performed for: x4, xs, xo only.

The C P U times are obtained for a M I N I - 6 96. The mixed method [AM] gives a better approximat ion but is more expensive that the [A11 method (random search for

X I , . . . , X 6 ) .

Solution AM

X* = (1.422, 1.244, 1.060, - 1.565, 1.854, - 0.402)~ C

72 Probabilistic Engineering Mechanics, 1989, Vol. 4, No. 2

An aloorithm for spectral factorization: P. Bernard and C. Bonnemoy

C O N C L U S I O N

An algori thm for spectral factorization of s tat ionary processes is obtained. This a lgori thm has been implemented on SM 90 in the Applied Mathematics Depar tmen t of the Universi ty of Clermont II, on the M I N I 6-96 of the Scientific Compute r Centre of this Universi ty, and on the C N U S C - C o m p u t e r in Montpell ier . In example 2, it gave better results than all a lgori thms proposed in Ref. 4. Moreover, this algori thm can easily be generalized to mult ivar ia te processes. We are working on this extension.

The use of a r a n d o m search technique gives us general advantages of this kind of method: great doma in of applications, easy p rogrammat ion , as well as their weakness: elevated cost, no good stop rule.

The authors thank Paul Kree for his care in reading this work.

R E F E R E N C E S

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Probabilistic Engineering Mechanics, 1989, Vol. 4, No. 2 73