one-step back forecast made only with Xt would

spectrum is a continuous function of frequency. In

still depend upon Xt+1 . Since Xt’1 and Xt+1 are

contrast, the periodogram is always discrete.

dependent, the errors would also be dependent and

It is important to note that our purpose in

factorization (11.14) would not be possible.

In general, ±„,„ is the correlation between the this chapter is to describe the spectrum as a

characteristic of a stochastic process (hence the

error of a one-step ahead forecast of Xt made

use of the word ˜parameter™ in the title). Spectral

3 If X is an AR(1) process with parameter ± then both

estimation is dealt with in Chapter 12.

t 1

Xt ’ ±1 Xt’1 and Xt ’ ±1 Xt+1 are white noise processes. To

con¬rm that Nt = Xt ’ ±1 Xt+1 is a white noise process, show 4 We have assumed, for mathematical convenience, that T is

that E(Nt Nt+„ ) = 0 for all „ = 0. odd. The expansion is slightly more complex when T is even.

11.2: The Spectrum 223

11.2.1 De¬nition of the Spectrum. Let Xt be an 5 The spectrum describes the distribution of

ergodic weakly stationary stochastic process with variance across time scales. In particular,

auto-covariance function γ („ ), „ = 0, ±1, . . .. 1

Then the spectrum (or power spectrum) of Xt is 2

Var(Xt ) = γ (0) = 2 (ω) dω. (11.19)

the Fourier transform5 F of the auto-covariance 0

function γ . That is

6 The spectrum is a linear function of the

auto-covariance function. That is, if γ is

(ω) = F {γ }(ω) (11.18)

decomposed into two functions, γ („ ) =

∞

γ („ )e’2πi„ ω

= ±1 γ1 („ ) + ±2 γ2 („ ), then

„ =’∞

(ω) = ±1 1 (ω) + ±2 2 (ω)

for all ω ∈ [’1/2, 1/2].

= F {γi }.

where

Note that since γ is an even function of „ , i

∞ 11.2.3 Theorem: The Spectra of AR( p) and

(ω) = γ (0) + 2 γ („ ) cos(2π „ ω). MA(q) Processes.

„ =1

1 The spectrum of an AR( p) process with

Note also that the spectrum and the auto- process parameters {±1 , . . . , ± p } and noise

covariance function are parameters of the stochas- variance Var(Zt ) = σ Z is

2

tic process Xt . When the process parameters are

σZ

2

known (not estimated from data), the spectrum is

(ω) = . (11.20)

well-de¬ned and not contaminated by any uncer- p

±k e’2πikω |2

|1 ’ k=1

tainty.

As is our practice with the auto-covariance and 2 The spectrum of an MA(q) process with

process parameters {β1 , . . . , βq } and noise

auto-correlation functions, we will use the notation

variance Var(Zt ) = σ Z is

2

x x to identify as the spectrum of Xt when

required by the context.

q

βl e’2πilω |2 .

(ω) = σ Z |1 +

2

(11.21)

11.2.2 Properties. l=1

1 The spectrum of a real-valued process is Proofs can be found in standard textbooks such as

[195] or [60].

symmetric. That is

(’ω) = (ω). 11.2.4 The Spectrum of a White Noise Process.

The spectrum of a white noise process Zt is

easily computed from (11.21). Since γ (0) = σ Z2

2 The spectrum is continuous and differentiable

and γ („ ) = 0 for nonzero „ , the spectrum is

everywhere in the interval [’1/2, 1/2].

independent of ω. That is

Consequently

Z (ω) = σ Z for all ω ∈ [’1/2, 1/2].

2

(11.22)

d

(ω)|ω=0 = 0.

3 The spectrum is drawn as a horizontal line,

dω

indicating that no time scale of variation is

4 The auto-covariance function can be recon- preferred, hence the allusion to white light. This

structed from the spectrum by using the agrees with the analysis of the run length L

inverse Fourier transform (C.6) to obtain discussed in [10.3.3].

1

11.2.5 The Spectrum of an AR(1) Process.

2

(ω)e2iπ ω„ dω.

γ („ ) =

The power spectrum of an AR(1) process with

’1

2

lag-1 correlation coef¬cient ±1 is

5 Note the speci¬c mathematical character of the discrete

σZ2

(ω) =

Fourier transform. It operates on the set of in¬nite, summable,

|1 ’ ±1 e’2πiω |2

real-valued series and generates complex-valued functions that

are de¬ned on the real interval [’1/2, 1/2]. See Appendix

σZ

2

= .

C. For more reading about the Fourier transform see standard (11.23)

1 + ±1 ’ 2±1 cos(2πω)

2

textbooks, such as [195].

11: Parameters of Univariate and Bivariate Time Series

224

This spectrum has no extremes in the interior of

the interval [0, 1/2] because, everywhere inside

the interval, the derivative

d

(ω) = ’2±1 (ω)2 sin(2π ω) = 0.

dω

The sign of the derivative is determined by ±1 .

Thus the spectrum has a minimum at one end of

the interval [0, 1/2] and a maximum at the other

end.

When ±1 > 0, the ˜spectral peak™ is located

at frequency ω = 0. Such processes are often

referred to as red noise processes.

AR(1) processes with ±1 < 0, which are

sometimes called blue noise processes, are of little

practical importance in climate research because

they tend to change sign every time step. In most

climate research contexts, the observed process

evolves continuously. Thus ±1 will be positive

given a suf¬ciently small time step.6

Figure 11.3 shows the spectra of the two ˜red™

AR(1) processes that were discussed in [10.3.3]

and [11.1.7]. The ±1 = 0.9 spectrum (right hand

axis in Figure 11.3a) is more energetic on long Figure 11.3: Power spectra of various AR

time scales (ω’1 greater than approximately seven processes. The left hand axis applies to spectra

time steps) than the ±1 = 0.3 spectrum (left hand labelled ˜L™ and the right hand axis applies to

axis in Figure 11.3a). At short time scales the those labelled ˜R.™

±1 = 0.3 process is somewhat more energetic. a) AR(1) processes with ±1 = 0.3 (L) and ±1 = 0.9

This interpretation is consistent with the ¬nding (R),

b) AR(2) processes with (±1 , ±2 ) = (0.3, 0.3) (L)