Stem-and-leaf plots of frequency estimates for N = 1900 Order fitted by AIC

In document Some aspects of time series frequency estimation (Page 85-90)

= 1.6

0.5

9

111112222333

555556666666666677777788888888999999999999999999999

00000011112222222222333444

5556666678

3

0.01

9

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)

uj

0

(p2/

2) cos(fc — l)o;o

cr2

-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 following

T 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

_ ____________ cr2[{l + (k + 1M 2 - a2Ffc+i]_____________

{1 + (k -+• l ) a } { l ■!" {k — l)a} 4- a2Djt+1(2 cos kuj

q

— D k+i)

(5.2.10)

While we actually need these complete representations for later analysis, the

implications of this result are clearer if the leading terms only are considered, for

large fc, when it can be seen that

<T2R(k)~l = J*+1 + 0 ( k ~ 1)

—2k

_

(Xk(j) = , 2

~

1cosjojQ + 0 ( k ~ 2)

_ f

(5.2.11)

= —T- cosju)0 + 0 { k 2)

a 2k = <72{ l + 0 ( k - 1 )} .

The proofs of theorem 5.2.1, and of other theorems and lemmas in this chapter

In document Some aspects of time series frequency estimation (Page 85-90)