and (±1 , ±2 ) = (0.9, ’0.8) (R).

coef¬cients generate more long runs than those

with small lag-1 correlation coef¬cients, and vice

versa (see Figure 10.8).

0 whenever g (ω) = 0. By using the identity

sin(4π ω) = 2 sin(2π ω) cos(2π ω), we ¬nd that

11.2.6 The Spectrum of an AR(2) process.

g (ω) = ’2π ±1 (1 ’ ±2 ) sin(2π ω)

The power spectrum of an AR(2) process with

parameters (±1 , ±2 ) (11.20) is given by ’ 4π±2 sin(4π ω)

= (’2π ) sin(2πω)

σZ

2

(ω) = — ±1 (1 ’ ±2 ) + 4±2 cos(2π ω) .

1 + ±1 + ±2 ’ 2g(ω)

2 2

Since sin(2π ω) = 0 for all ω ∈ (0, 1/2), (ω) =

where

0 when

g(ω) = ±1 (1 ’ ±2 ) cos(2π ω) + ±2 cos(4πω).

cos(2πω) = ’±1 (1 ’ ±2 )/(4±2 ). (11.24)

Depending upon the parameters, this spectrum can This last equation has a solution ω ∈ (0, 1/2)

have a minimum or a maximum in the interior of when |±1 (1 ’ ±2 )| < 4|±2 |. This solution

the interval [0, 1/2]. Figure 11.3b displays spectra represents a spectral maximum when ±2 < 0 and

of both types. a spectral minimum when ±2 > 0.

When its derivative is zero, (ω) has a Equation (11.24) has solutions ω ∈ (0, 1/2) for

maximum or minimum, and we note that (ω) = both spectra shown in Figure 11.3b. When ± =

(0.3, 0.3), a minimum is located at ω ≈ 0.28.

6 There are exceptions to this statement. For example, annual

When ± = (0.9, ’0.8) a maximum occurs at

layer thickness in ice cores can be modelled as ˜blue noise™ (see,

ω ≈ 0.17.

for example, Fisher et al. [118]).

11.2: The Spectrum 225

δ-functions, functions that are in¬nitely large at

11.2.7 The Spectrum of a Linearly Filtered Pro-

cess. We described the auto-covariance function the frequencies of the oscillations they represent

of a linearly ¬ltered process Yt = k ak Xt+k in and zero everywhere else. By suitably generalizing

the de¬nition of integration, the δ-function can

[11.1.13]. The spectrum of such a process is (see

(C.15, 11.15)): be given an intensity such that the integral over

the δ-function is equal to the variance of the

= |F {a}|2 xx . oscillation.

yy

11.2.9 Interpretation: Periodic Weakly Sta-

11.2.8 Interpretation: General. Literal inter-

tionary Processes.8 Suppose a periodic stochastic

pretation of equations (11.16, 11.17) leads to the

process Xt can be represented as

incorrect notion that all weakly stationary stochas-

tic processes can be represented as a combination n

Xt = Z j e2πiω j t + Nt ,

of a ¬nite number of oscillating signals with (11.25)

random amplitude and phase. j=’n

However, a special class of weakly stationary

where ω j = 1/T j , j = ’n, . . . , n, T j ∈ Z

processes that behave in just this way can

are ¬xed frequencies, Z j , j = ’n, . . . , n, are

be constructed. An example of this type of

complex random variables, and Nt represents a

process is sea level measured at a given tide

noise term that is independent of the Z j . For

gauge. These measurements contain a tide signal

simplicity we assume that Nt is white in time, but

made up of a (practically) ¬nite number of

this assumption is easily generalized.

astronomically forced modes that is overlaid by

What conditions must be placed on the

irregular variations excited by weather.

frequencies ω j and random variables Z j to ensure

Such a process has in¬nitely long memory and

that Xt is real valued and weakly stationary?

is not ergodic. Its auto-covariance function does

To ensure that Xt is real for all t = 0, ±1, . . .,

not go to zero for increasing lag but instead

the frequencies ω j must be symmetric about zero

becomes periodic at long lags. Therefore, the

(i.e., ω’ j = ω j ), and for every j = 1, . . . , n,

Fourier transform of its auto-covariance function

random variables Z’ j and Z j must be complex

does not exist and the process has no auto-

conjugates.

spectrum in the sense of de¬nition [11.2.1]. A

Two conditions must be satis¬ed to ensure weak

different type of characteristic spectrum must be

stationarity. First, the mean of the process,

de¬ned for these processes, namely a discrete line

spectrum.7 We will discuss this type of process in n

µ X = E(Xt ) = E Z j e2πiω j t + E(Nt )

the next subsection.

j=’n

Ergodic weakly stationary processes have ¬nite

memory and summable auto-covariance functions should be independent of time. This means that

random variables Z j , j = 1, . . . , n, must have

with de¬ned Fourier transforms. Most time

series encountered in climate research are, to a mean zero.

reasonable approximation, of this type, at least Second, the auto-covariance function

E(Xt+„ Xt ) must be a function of „ alone.

after deterministic cycles such as the annual cycle

or the diurnal cycle have been subtracted. We The auto-covariance function is given by

will discuss the interpretation of spectra of such

γ („ ) = E(Xt+„ Xt )

processes in [11.2.10].

n

The two concepts of the power and line

E |Z j |2 e2πiω j „

= δ0,„ σ N +

2

spectra can be formally uni¬ed by de¬ning a

j=’n

generalized Fourier transform. The discrete part

n

of the spectrum is then represented by Dirac

E Z j Z— e2πiω j „

+ k

j=’n k= j

7 Note that the expression spectrum is used for a large variety

— e2πi(ω j ’ωk )t

of mathematical objects. Examples include the eigenvalue

spectrum of an EOF analysis, the power spectrum, and the

for all t = 0, ±1, . . ., where δ0,„ = 1 if „ = 0, and

line spectrum discussed here. Climatologists also use spatial

spectra that describe the distribution of energy to different

zero otherwise. Since the left hand side is constant

spatial scales. A common characteristic of these spectra is that

for all t, it follows that the random variables Z j ,

they are expressed as functions of a discrete or continuous set

j = 0, 1, . . . , n, must be uncorrelated.

of indices that are ordered on the basis of time scale (in case of

the power spectrum), relevance (eigenvalue spectrum), or other

8 Following Koopmans [229].

meaningful criteria.

11: Parameters of Univariate and Bivariate Time Series

226

Consequently, periodic weakly stationary pro- scales. Peaks in the spectra of ergodic, weakly

cesses (11.25) have periodic auto-covariance func- stationary processes do not re¬‚ect the presence of

an oscillatory component in the system.

tions of the form

However, peaks in the spectra of AR processes

n

2 e2πiω j „

γ („ ) = E |Z j | do indicate the presence of damped eigen-

oscillations in the system with eigenfrequencies