Base de Datos

In d e scrib in g a tim e series u sin g sp ectral an aly sis th e fu n d am e n ta l com p o n en ts are taken to be sin u so id al w aves of the form Rcos{o)t+(f>), w hich for a given an g u lar freq u en cy co, 0 < CD < ;r, is specified by its

a m p litu d e R > 0 and phase 0, 0 < 0 < 2k. T hus in a tim e series of n

observations it is not possible to d istin g u ish m ore th an n j l in d e p e n d e n t sinusoidal com ponents. The frequency range 0 < ct) < ;r is lim ited to a sh o rte st w av elen g th of 2 sam p lin g u n its because any w ave of higher frequency is indistinguishable u p o n sam pling (or is ab ased w ith) a w ave w ithin this range. Spectral analysis follows the idea th at for a tim e series m ade up of a finite num ber of sine w aves the am plitude of any com ponent at frequency O) is given to order 1 / n by:

7 1 N

"Zxtexplicot) t=i

For a series this is d e fin e d as: f ( c o ) = — ixiexp(icot) t=i

2 the iTcn

scaling factor

being chosen so that 2 j f{cD)dcD = cr^, ie. the spectrum indicates how o j , the 0

sam ple

variance of the series is distributed over com ponents in the frequency range

0 < CO < K.

It m ay be dem o n strated that f*{co) is equivalently defined in term s of the sam ple autocorrelation function (acf) rj^ of the series as:

. ^ 1 r n - l ]

f { c o) = — ^ C Q + 2 ^ C ) ^ c o s k c o >

Chapter 5: Data Acquisition and Analysis.

If the series Xt contains a determ inistic sin u so id al com ponent of am plitude R f this will be rev ealed in the sam ple sp e ctru m as a sh a rp p eak of a p p ro x im ate w id th njn an d height (nl2K)R^. This is called th e discrete p a rt of the spectrum , the variance R?- associated w ith this com ponent being in effect concentrated at a single frequency.

O n the other h an d if the series is purely stochastic (stationary w ith acf r%), th en w ith increasing sam ple size the expected value of f*{d) converges to the theoretical spectrum - the continuous part:

f{(o) = — \ y Q + 2 ^ y ) ^ c o s k o ) \ , w h e r e yj^ a r e th e t h e o r e t i c a l

'2.71 [ k=l )

autocovariances.

The sam ple spectrum does not, how ever, converge to this value b u t at each freq u e n cy p o in t flu ctu ate s a b o u t th e th e o re tic a l sp e c tru m w ith an exponential distribution, being in d ep en d en t a t frequencies separated by an interval of I n l n or m ore. V arious techniques are therefore em ployed to sm ooth the sam ple spectrum an d red u ce its variability. M uch of the strength of spectral analysis derives from the fact th at the erro r lim its are m ultiplicative so that features m ay still show u p as significant in a p art of the spectrum w hich has a generally low level, w hereas they are com pletely m asked by other com ponents in the original series. The spectrum can help to distinguish determ inistic cyclical com ponents from the stochastic quasi­ cycle com ponents which produce a broader peak in the spectrum .

A large discrete com ponent in a spectrum can d istort the continuous part of a large frequency range surro u n d in g the corresponding peak. This m ay be alleviated at the cost of slightly broadening the peak by tapering a portion of the d ata at each end of the series w ith w eights w hich decay sm oothly to zero. It is usual to correct for the m ean of the series and for any linear trend by sim ple regression, since these w ould sim ilarly distort the spectrum .

The pro g ram "spec.f", w ritten by D. L uthra, w as u se d to perfo rm the spectral calculations. The supplied tim e series m ay be m ean corrected and

tapered, the tapering factors being those of the split cosine bell: ( f 1 - c o s V V 1-cos T J j l ^ t ^ T , n + l - T < t < n JJ 1, otherwise w h e re T = 2 s a m p le s p e c tru m f*(co) = - ^

an d p is th e ta p e rin g p ro p o rtio n . The u n sm o o th ed 2 exp(i(ot) t=i 27tk , û)jç — 77~/ —0,1,..., £ 2 is th e n c a lc u la te d for

, w here [] denotes the integer part.

The sm oothing is defined by a trapezium w indow w hose shape is supplied W (a) = l, \cc\^p

YJ{a) = -—1^, p < |a |< l by the function:

The w idth of the w indow is fixed as 27t/M an d a set of averaging w eights constructed:

n

w here g is the norm alising constant an d the sm oothed spectrum obtained i s / ( u / ) = I Wlt/"(u< + (üjt).

Chapter 5: Data Acquisition and Analysis.