Sim u lation E xp erim en ts

In document Some aspects of time series frequency estimation (Page 75-78)

4.7.1 Im p lem en ta tio n o f th e proced ure.

Some co m p u ta tio n s have been done using sim u la ted d a ta to in v estig ate the frequency e stim a tio n p ro ced u re described in §4.1. T h e au to reg ressiv e p a ra m e te r e stim a te s w ere fo u n d using B u rg ’s alg o rith m , an d th e o rd e r was e stim a te d using th e A I C criterion.

T h e selected set of auto reg ressiv e p a ra m e te rs was used to co m p u te R i k ( j )

according to (4.1.11), th e n hk(0) a n d its first a n d second derivatives calculated using (4.1.12) a n d its derivatives, so th a t N ew to n ’s m e th o d was a p p lied using trig o n o m etric polynom ials as th e derivatives.

T h e in itial value for th e N ew to n ’s m e th o d c o m p u ta tio n was selected by eval­ u a tin g hk(0) a t a g rid of 200 equally-spaced p o in ts betw een 0 a n d 7r, a n d choos­ ing as th e s ta rtin g frequency th a t asso ciated w ith th e sm allest value of hk(0) on th is grid. In p ra c tice th is seem s to lead to th e ab so lu te m in im u m of hk{6) quite reliably. N ew to n ’s a lg o rith m is ite ra te d u n til successive values of th e e stim ate differ by less th a n some p rescrib ed to leran ce, a n d for these sim ulations I — uii+11 < 0.00005 was selected.

Sim ulations w ere perform ed using relativ ely sh o rt tim e series, N = 32,6 4 , 256; an d longer series, N = 1024 a n d 1800. For each value of iV, 100 rep licatio n s of th e e stim a tio n p ro ced u re were perfo rm ed on in d e p en d e n t sim u lated series. T h e first 100 p o in ts of each sim u lated series were d iscard ed to elim in ate th e s ta r tu p effects, a n d th e n e x t N p o in ts used for th a t replication.

A m ax im u m o rd er h as to be p rescrib ed for th e o rd e r selection p ro ced u re; b o th th e o re tic al a n d com m onsense considerations req u ire th a t th is b e no t to o large in rela tio n to th e len g th N of th e tim e series. For th e th re e sm aller values of JV, a m ax im u m o rd er of 14 was fitte d , a n d for th e larg er values, a m a x im u m of 38. T h e

A I C p ro ced u re generally chose orders k less th a n th ese m ax im a, except in th e

case N = 256 w hen 14 was th e m o d al o rd er selected.

4.7.2 In pu t d ata m od el

T h e following c rite ria governed selection of an in p u t d a ta m odel to in v estig ate th e p ro ced u re of th is c h ap te r.

(1) It should n o t be a fin ite-o rd er autoregression;

(2) T h e frequency a t w hich th e sp ec tra l d en sity of th e m odel has its m ax im u m should b e know n exactly, so th a t th e p erfo rm an ce of th e e stim a to r can be ev alu ated . It is no t sufficient for o u r p u rp o ses to know only th e p h ase angle of a ro o t of th e tra n sfe r fu n ctio n , close to th e u n it circle, n o r is it ad eq u a te to use th e cyclical m odel u sed in econom ics,

y ( n) = <f>(n) cos won + 7/’(n ) sin won + e(n)

<Kn ) = P<£(n - 1) + f ( n )

ip(n) = px^{n — 1) + 77(71)

w here e(n), f ( n ) a n d 77(71) are u n c o rre la ted w h ite noises, since as p m oves away from 1 (we w ould use \p\ < 1), th e sp ec tra l m ax im u m is again n o t ex actly at

(3) Of course, th e m odel should satisfy th e conditions of T h eo rem (4.4.1), so we can n o t satisfy req u irem en t (2) in th e way th a t is usually done in the engineering lite ra tu re by using y ( n ) = /> cos u;0n + e(n); how ever, ARM A processes certain ly satisfy th ese conditions. W e shall be in v estig atin g the case y (n ) = pc o sio0n + e(n) in th e n ext c h ap ter.

A n a p p ro p ria te A RM A m odel was designed as follows:

1. an MA tra n sfe r fu n ctio n was selected to have a single obvious p eak at a frequency n o t to o close to 0, 7r/2 or 7r.

2. A n A R tra n sfe r fu n ctio n w ith a sh arp p eak was c o n stru c te d according to

a(z) 2 COSO) * z2

1 + e Z + (1 + e)2 (4.7.1)

(see for exam ple C la erb o u t (1976)). \a(eld)\2 h as a m in im u m value a t 9 = uq

w hich is re la te d to u>* by

cos uj o = (1 + e2/( 2 -(- 2e)} cosu;*

so th a t for th e ujq deriv ed from th e MA tra n sfe r fu n ctio n an a p p ro p ria te e

a n d uj* can be chosen so th a t th is A R tra n sfe r fu n ctio n gives rise to a sp ectral den sity w ith a sh arp peak, a n d th e A R p a ra m e te rs co m p u ted from 4.6.1. T h e m odel chosen by th is p ro ced u re was an A R M A (2,6) w ith sp e c tra l peak a t ujq = 1.24232; th a t is, we have

2 6

J 2 a ( j ) y ( n - j ) = ^ ß ( j ) e ( n - j ) (4.7 2)

j = o j = 0

w ith { < * (0 ),a (l),a (2 )} = { 1 .0 ,-0 .6 0 1 6 ,+ 0 .8 7 3 4 } , an d {/?(0) , . . . , 0 (6 )} = {1.0, 3.000, 0.3000, —6.6000, —4.2789,3.6631, 3.0525}. T h e sp ec tra l d en sity of th is m odel is show n in F igure 4.1.

F IG U R E 4.2

R a d i a n F r e q u e n c y

In document Some aspects of time series frequency estimation (Page 75-78)