= 1.6
0.5
9
111112222333
555556666666666677777788888888999999999999999999999
00000011112222222222333444
5556666678
3
0.019
0001222334444
555556666666677777888888889999
0000001111111111122222234444444
55566666667778899
0011122
7
= 3.0
0.5
9
44
556666667777788888888888888999999999999999
000000000000000000011111111111111111111111122222222223
0.01
3
55567888999999
0000011222222222333333344444
55555555555666666666667777777888888889999
000001122334
5556
2.9998
O n th e o th e r h an d , th e following p ro ced u re is a d v o cated in th e engineering lite ra tu re as being c o m p u ta tio n ally less burd en so m e (Jack so n et al. (1978));
A lg o r it h m 5 .2 : • e stim a te autoregressive p a ra m e te rs ak
• choose o rd e r k by AIC o r o th e r p ro ced u re
• find th e zeros of th e tra n sfe r fu n ctio n czjfc(z) using a su itab le alg o rith m
• lo cate th e zero w ith m odulus closest to 1 an d e stim ate a?o by th e arg u m en t of th a t zero.
T his a lg o rith m clearly dep en d s on th e a lg o rith m th a t lo cates th e zeros of
cLk(z). Such alg o rith m s are know n to be ill-conditioned w hen th e polynom ial ro o ts are re p e a te d or close to g e th e r, a n d we a p p e a r to have no in fo rm atio n ab o u t w h e th e r th is is so for ajt(z), o r n o t.
W e focus p a rtic u la rly on A lg o rith m 5.1 since p a rt of th e a ttra c tio n of th e auto reg ressiv e e stim atio n p ro ced u re is th a t it is valid w h e th er th e sp e c tra l density is ab so lu tely continuous o r n o t, a n d one w ould wish in principle to use an algorithm w ith th is p ro p erty . A lg o rith m 5.2 will n o t lo cate th e m a x im u m of th e sp ectral d e n sity in th e case w here th e zero w ith m o d u lu s closest to 1 is very far from the u n it circle w hich m ay be th e case for an ab so lu tely continuous sp e c tru m even w ith q u ite a sh arp peak. In th is la tte r case, th e frequency e x tra c te d by A lg o rith m 5.2 w ould have no im m ed iate re la tio n to th e s p e c tra l d en sity m axim a.
5 .1 A p p e n d ix — S im u la tio n E x p e r im e n t s
T h e sim u latio n ex p erim en ts described in th is section a n d su b seq u en tly were p e rfo rm ed as follows.
T h e in p u t d a ta were g e n erate d using in d e p en d e n t G au ssian rai^>m variables e(n) w ith m eans zero a n d u n it variances, g e n erate d by th e double-precision NAG
lib ra ry su b ro u tin e G 05D D F, ad d ed to a sinusoidal signal w ith a m p litu d e p = 20, a n d vary in g frequencies loq and phases d>, according to
y ( n) = p cos(uj0n + (f>) + e(n) .
T h e signal-to-noise ra tio SN R is defined in engineering p ap ers on a logarithm ic scale, so in th is case
SN R = 1 0 lo g 10 p2/ 2ct2 = 23.01 .
No e x p e rim e n ta tio n was done on th e effect of varying SN R on th e p ro p e rtie s of th e p ro c e d u re, since th is h as been in v estig ated in o th e r w ork (see for exam ple M arple (1979)).
T h e value used is n o t considered to rep resen t a high signal-to-noise ra tio , so th e e x p erim en tal evidence p re sen te d here is relevant to th e case w here th e signal is noisy.
T h e frequency estim atio n p ro ced u re was im p lem en ted ex actly as described in sectio n 4.6.1 for ab so lu tely con tin u o u s sp ec tru m in p u t. T h e o rd er was selected u sing AIC alth o u g h as ex p lain ed in section 5.6 below th is is n o t very satisfactory. H ow ever, th e a ttra c tio n of th e A R e stim atio n p ro ced u re is th a t it rem ain s valid w h e th e r th e sp ec tru m is ab so lu tely continuous or n o t, so th a t one sh ould use a p ro c e d u re w hich has th e m ost general applicability. In real d a ta it seem s likely th a t p u re line sp e c tra will be relatively rare, so th e A IC criterio n was used.
A n ex p erim en t was co n d u cted w here, for given IV, a m odel of fixed o rd er was fitted ; th e frequency e stim ate s th u s p ro d u c e d were ludicrously in a c c u ra te , an d this suggests th a t som e qualification m ight be m ad e to th e resu lts of K ay a n d M arple (1981) w ho used th e sam e o rd e r for each of th e ir A R sp ec tra l estim tes. O n th e o th e r h a n d , th e y chose as th e order, w hich is, as will be show n subsequently, a p p ro x im a te ly th e sam e as AIC in th e signal-plus-noise case.
F u rth e r resu lts from th e sim ulations are given la te r in th is c h a p te r in the sections w hich th ey illu stra te .
5 .2 A n a ly t ic P r e lim in a r ie s
5 .2 .1 S o lu t io n o f t h e Y u le -W a lk e r e q u a t io n s
From th e d escrip tio n of F ( 0), th e sp ec tra l d istrib u tio n of { y (n )}, we see th a t its ra n g e co n tain s an infinite n u m b e r of p o in ts, a n d G re n an d e r an d Szegö (1958) show th a t in th is case, if one form s th e p ro je c tio n of y (n ) on {y(n — 1 ) , . . . , y(n — k)}
in L 2[dF(9)\ w ith p a ra m e te rs ce* = { a j t ( l ) , . . . , otk(k)}T , in th e sam e w ay as in
th e case w hen F(9) is ab so lu tely continuous, an d define
- 2
/*(«) =
i + E
M
> =1
/ 2x
; (5.2.1)th e n as k —► oo, f k(9) converges alm ost surely to th e alm ost-everyw here existing d eriv ativ e of F(9), th a t is, in th is case g(9).
F o r th e p u rp o se of e stim a tin g ujq a resu lt is needed th a t describes th e b e h a v io u r of fk{9) on a set of m easu re zero. From now on only th e special case
i = 1, g(9) = cr2/2tz will be investigated.
In th is case th e au to co v arian ce sequence for {y(n)} is given by
r(0 ) = E { y ( n ) 2} = a 2 + p2/ 2 (5.2.2)
r ( j ) = E { y ( n ) y ( n + j ) } = (p2/ 2) cos ju>0
w hich is th e sequence of F o u rier coefficients of th e d is trib u tio n th a t is a tra in of d e lta fu n ctio n s a t (2mr ±u>o) a d d ed to an u n d erly in g c o n sta n t, so th a t, as is well know n, in th e in terv al 0 E ( —7r, 7r), { y ( ^ ) } has th e generalized sp e c tra l density
f ( 8 ) = (<72/27r) + (p2/2){S(8 + wo) + 8(9 - w0)} . (5.2.3)
T h is fu n ctio n is in teg rab le since it is a.s. c o n sta n t, a n d it is b o u n d e d away fro m zero, so th a t f log f ( 9 ) d 9 > —oo, b u t f(Q) itself is u n b o u n d ed . In o rd er to
show th a t f k(9) —►f(&) as k —> oo, first th e p a ra m ete rs a k of th e p ro je c tio n are derived.
Define th e k x k au to co v arian ce m a trix R ( k — 1) (indexed by th e g re a te st lag of au to co v arian ce in th e m a trix , ra th e r th a n by th e m a trix dim ension), by
R ( k - 1)
cr2 + p2/2
(p2/2) cosu>o
. . . (p2/2 )cos(k — l)u>o
(p2/ 2 ) c o s u Q
a 2 + p2/2
. . . (p 2/2 ) cos(fc —2)
uj0
(p2/
2) cos(fc — l)o;ocr2
-f p2/ 2(5.2.4) a n d let = { r ( l ) , r ( 2 ) , . . . , r(fc)}.
Define £* = [1, e ~tuJ° ,. . . , e *(*—1)«*r°], ^ = (^*)T , w here * den o tes tra n sp o se of a m a trix a n d com plex co n ju g atio n of its elem ents. T h e n re p re se n t
R ( k - 1) = a 2 [l k + (p(5.2.5)<t2 ) [ £ £ * 4 + 777?*]] .
T h e p ro je c tio n coefficients ajt are found by solving th e Y ule-W alker eq u atio n s
R ( k - l ) a k = —r* , (5.2.6)
an d th e p re d ic tio n e rro r variance cr2. by solving th e a u g m en ted Y ule-W alker eq u a tions
R ( k )
1
Oik
(5.2.7) In tro d u ce th e n o ta tio n D k = sin ku>o/ sin üüq
,
Fk=
D \ (rela te d to th e D irichlet a n d Fejer kernels respectively*)using these, we have th e followingT h e o r e m 5 .2 .1 . Let {y(n)} be generated by (5.1.1) w ith conditions (5.1.3) hold
ing, for loq € (0,7r). Then, i f a = p2/4cr2
(i) a 2R { k ) ~ l
= Jfc+i -
^ 2 a [ l+
(k + l)a ][co s(r-
c)a;0]r,c,=i,...,Jt+i- 2 a 2D k+l[cos(k - ( r - 1) - (c - l))a^o]r>c,= i.... Jfc-hi
j j
(Ü) ock(j)
—2a{l + (k + l)a} cosjcoo + 2a2D k+i cos(k — j)u>o
{1 + (fc 4- l) a } { l + (k — l)a} + a2D fc+1(2cos ku0 - D k+
1)
(5.2.9)
^ 2