close to that of the peak.

n

= Var(Z0 )+4 E |Z j |2 cos(2πω j „ ). To understand this, we return to (10.14),

j=1

p

β j y ’„ ,

for |„ | ≥ 1. X t+„ = j

Hence, only very special weakly station- j=1

ary stochastic processes can be represented

which describes the evolution of an AR( p) process

as a ¬nite sum of discrete signals. In con-

Xt from a given state X t’1 , . . . , X t’ p when the

trast with the ARMA processes described in

noise is turned off. The constants β j depend upon

Sections 10.3 and 10.5, these processes have

the process parameters ±1 , . . . , ± p and the initial

periodic auto-covariance functions for which

lim„ ’∞ „=0 γ ( )2 = ∞. Therefore, the sum- state X t’1 , . . . , X t’ p . The constants y j are the

roots of the characteristic polynomial (10.11)

mation (11.18) does not converge and the spectrum

does not exist. In conceptual terms: the system has p

in¬nitely long memory.

1’ ±k B k . (11.26)

Even though the power spectrum does not exist, k=1

we can de¬ne a spectrum that distributes the

Since there are p such roots, there are p sets of

variance of Nt with time scale in the usual way

initial states I k = {X t’1,k , . . . , X t’ p,k }, k =

and adds a speci¬c amount of variance, E |Z j |2 ,

1, . . . , p, for which β j = δ jk .9 It can be shown

at the discrete frequencies ω j , j = ’k, . . . , k.

that the initial states are given by

The discrete part of this spectrum is called a line

spectrum. As noted above, this kind of spectrum

X t’„,k = yk ’1 , „ = 1, . . . , p.

„

can be given a density function interpretation by

resorting to Dirac δ-functions.

Hence each set of states I k represents a ¬nite

segment of a time series that is either a damped

11.2.10 Interpretation: Ergodic Weakly Sta-

oscillation (if there are a pair of complex conjugate

tionary Processes. Processes with limited mem-

roots yk ) or simply decays exponentially (if yk is a

ory, that is,

real root). (See Figure 10.12.)

„ Equation (10.14) shows that the evolution of the

γ ( )2 < ∞,

lim system in the absence of noise is determined by

„ ’∞

=0 the mixture of ˜initial states™ I k . In particular, if

the initial state is one of the decaying exponential,

can not be periodic in the sense of (11.25). A

or damped oscillatory, states I k , it will stay in that

speci¬c amount of variability can not be attributed

state and continue to display the same behaviour

to a speci¬c frequency, otherwise we would again

in the future. In that sense, the roots of the

have a process with a discrete periodic component

characteristic polynomial represent eigensolutions

and in¬nite memory. Instead, variance is attributed

of the system. However, since noise is continually

to time scale ranges or frequency intervals. Given

two frequencies 0 ¤ ω1 < ω2 ¤ 1/2, we interpret added to the system, we see variability on all time

scales. The eigenmodes of the system determine

ω2

the way in which the variability in the input noise

(ω) dω

evolves into the future. When a sequences of states

ω1

evolves that is close to one of the eigenmodes

as the variability generated by the process in the

of the system, that mode tends to persist more

time scale range (1/ω2 , 1/ω1 ).

strongly than other sequences of states. These

preferences are, in turn, re¬‚ected in the tendency

11.2.11 Interpretation: Spectra of AR Proces- for there to be more variance in some parts of the

ses. In general, a peak in a spectrum indicates spectrum than others.

only that more variability is concentrated at time

9 For details, refer to [10.3.5,6].

scales near that of the peak than at other time

11.2: The Spectrum 227

We saw in [11.1.9] that the auto-covariance

function of the AR( p) process may be written as

(11.10)

cos(„ φk + ψk )

ai

ρ(„ ) = + ak

yi„ „

rk

i k

where the ¬rst sum is taken over the real roots

and the second is taken over the complex roots.

The complex roots yk are expressed in polar

coordinates as yk = rk eiφk . Thus, the auto-

covariance function is a sum of auto-covariance

functions of AR(1) and AR(2) processes, which

correspond to the initial states I j discussed above.

Figure 11.4: Power spectra of the AR(1) processes

The spectrum of the AR( p) process is then the

shown in Figure 11.3 displayed in log-log format.

Fourier transform of the sum of auto-covariance

Note that the derivative is zero at the origin.

functions, or, because of the linearity of the Fourier

transform, the sum of autospectra of AR(1) and

AR(2) processes. Thus, any peak in the spectrum

of the AR( p) process must originate from a peak spectral, or variance, density is expressed in units

of m2 — month, and the frequency in month’1 .

in an AR(2) spectrum, and we have seen that such

A peak at ω = 0.2/month represents a period of

peaks just correspond to ¬rst-order approximations

1/ω = 5 months.

of the eigen-oscillations of the AR(2) process (cf.

[10.3.5,6]).

Things are relatively clear in this context

because we have complete knowledge about the

11.2.13 Plotting Formats. An important practi-

process to guide us in the interpretation of the

cal aspect of spectral analysis concerns the format

spectrum. However, interpretation is much more

in which spectra are displayed. So far, we have

dif¬cult when spectra are estimated from ¬nite

used the plain format with the frequency ω as the

time series. The estimates are uncertain because

abscissa and the spectral density as the ordinate.

they are affected by sampling variability. They are

The log-log presentation, in which the logarithm

also affected by the properties of the estimator

of the frequency is plotted against the logarithm

itself and the way in which those properties (such

of the spectral density, is another common display

as bias) are affected by the true spectrum. So we

format. Spectra displayed in this way look rather

must attempt to interpret a noisy version of the true

different. This can be seen from Figure 11.4, which

spectrum that is viewed through ˜rose coloured

shows the same AR(1) spectra as Figure 11.3 but

glasses.™ Moreover, in practice cases this usually

in log-log format. Note that both spectra become

must be done without complete knowledge of the

˜white™ for frequencies close to the origin.

nature of the process that generated the spectrum.

An advantage of this format for theoreticians is

11.2.12 Units. Suppose that the process Xt that certain power laws, such as (ω) ∼ ω’k ,

is expressed in, say, units of A, and the time appear as straight lines with a slope of ’k. A

increment „ in units of δ. disadvantage with this format is that the area under

the curve as it is perceived by the eye is no longer