on the time scale of the quasi-periodicity. A realization of a white noise process is shown in

Two important goals of time series analysis are Figure 10.1.

to identify characteristic time scales in stochastic The characteristic time „ M = 1, since for any

processes, and to determine whether two time nonzero „ (10.2)

series share common information.

P (Xt+„ > 0|Xt > 0) =

In the following we consider exclusively time

series samples in discrete time. Also, for the sake ∞∞

0 0 f ts (x, y) d x dy

of brevity, the time step between two consecutive =

∞

0 f t (x) d x

data is arbitrarily set to 1.

∞ ∞

0 f N (x) d x — 0 f N (x) d x

= 0.5.

10.2.2 Stochastic Processes. We have, so far, ∞

0 f N (y) dy

used the expression ˜time series™ rather informally.

Time series may be seen as randomly selected The probability of observing a run (i.e., a sequence

¬nite sections of in¬nitely long sequences of of consecutive xs s of the same sign) of length L

random numbers. In that sense, a time series is a beginning at an arbitrary time t is obtained from an

random sample of a stochastic process, an ordered independence argument. Runs are observed when

’Xt’1 , Xt , . . . , Xt+L’1 and ’Xt+L all have the

set of random variables Xt indexed with an integer

t (which usually represents time). same sign. Therefore, since two signs are possible,

In general, the state Xt of the process at any

P (L = L) = 2 — 2’(L+2) = 2’(L+1) . (10.3)

speci¬c time t depends on the state of the process

at all other ˜times™ s. In particular, for any pair of

Note that

˜times™ (t, s), there is a bivariate density function

f ts such that

P (L = 0) = 1 ’ P (L > 0) = 1/2.

P (Xt ∈ [a, b] and Xs ∈ [c, d]) (10.2) That is, there is probability 1/2 that a run does

bd

not begin at time t. The probability of observing

= f ts (x, y) d x d y.

a run of length L = L, given that a run begins

ac

at time t, is 2’L . Thus the probability that a run

The marginal density functions derived from f ts beginning at a given time will become exactly

(see [2.8.3]) are, of course, the density functions L = 3 time units in length is 2’3 = 0.125. The

of Xt and Xs , given by probability that the run will last at least three time

steps is ∞ 2’L = 0.25. The corresponding

∞ L=3

f t (x) = f ts (x, y) dy probabilities for L = 10 are only 0.01 and 0.02.

’∞

∞

f s (y) = f ts (x, y) d x. 10.2.4 De¬nition: Stationary Processes. A

’∞

stochastic process {Xt : t ∈ Z} is said to be

stationary if all stochastic properties are indepen-

4 Note that the direction of the inequalities in (10.1) does not

affect the de¬nition of „ M . dent of index t.

10.2: Basic De¬nitions and Examples 201

It follows that if {Xt } is stationary, then:

1 Xt has the same distribution function F for

all t, and 1958-77

2 for all t and s, the parameters of the joint

distribution function of Xt and Xs depend

only on |t ’ s|.

10.2.5 Weakly Stationary Processes. For most

purposes, the assumption of strict stationarity

can usually be replaced with the less stringent

assumption that the process is weakly stationary,

in which case

• the mean of the process, E(Xt ), is indepen-

dent of time, that is, the mean is constant, and

• the second moments E(Xs Xt ) are a function

only of the time difference |t ’ s|.

Year

A consequence of the last condition is that the

variance of the process, Var(Xt ), does not change

with time. Figure 10.4: 1958“77 time series of monthly mean

The two conditions required for weak station- atmospheric CO2 concentration measured at the

arity are less restrictive than the conditions enu- Mauna Loa Observatory in Hawaii.

merated in [10.2.4], and are often suf¬cient for

the methods used in climate research. Even so, the

weaker assumptions are often dif¬cult to verify. We refer to processes with such properties as

Provided there are not contradictory dynamical weakly cyclo-stationary processes. It follows from

arguments, it is generally assumed that the process the second condition that the variance is also

is weakly stationary. a function of the time within the externally

determined cycle. Cyclo-stationary behaviour can

10.2.6 Weakly Cyclo-stationary Processes. be seen in Figures 1.7, 1.8, and 10.4. Huang and

The assumption that a process is stationary, or North [189] and Huang, Cho, and North [188]

weakly stationary, is clearly too restrictive to rep- describe cyclo-stationary processes and cyclo-

resent many climatological processes accurately. stationary spectral analysis in detail.

Often we know that stochastic properties are linked The conditions for weak cyclo-stationarity

to an externally enforced deterministic cycle, such parallel those for ordinary weak stationarity,

as the annual cycle, the diurnal cycle, or the Mi- except that the parameters of interest are indexed

lankovitch cycles. When we deal with variations by the phase of the external cycle. Statistical

on time scales of months and years, the annual cy- inference problems that can be solved for weakly

cle is important. For time scales of hours and days stationary processes can generally also be solved

the diurnal cycle is important. For variations on for weakly cyclo-stationary processes. However,

time scales of thousands to hundreds of thousands the utility of these models is strongly constrained

of years, the Milankovitch cycle will affect the data by the very large demands they place on the

signi¬cantly. We therefore consider processes with data sets used for parameter estimation. Cyclo-

the following properties. stationary models generally have many more

parameters than their stationary counterparts and

1 The mean is a function of the time within the

all of these parameters must be estimated from the

external cycle, that is, E(Xt ) = µt|m , where

available data.

t|m = t mod L and L is the length of the

external cycle measured in units of observing

intervals. 10.2.7 Examples. Suppose Xt is a stationary

2 E (Xt ’ µt|m )(Xs ’ µs|m ) , the central sec- process. If a linear trend is added, the resulting

ond moment, is a function only of the time process Yt = Xt + ±t is no longer stationary:

difference |t ’ s| and the phase t|m of the its distribution function, FYt (y) = f X (y ’ ±t),

external cycle. depends on t.

10: Time Series and Stochastic Processes

202

is a non-stationary process. The ¬rst moment of Xt

is independent of time, but the variance increases

with time. In fact,

t

t

E(Xt ) = E = E Z j = 0,

j=1 Z j

j=1

and

t

Var(Xt ) = E ( j=1 Z j )

2

t t

= E Z j Zk = E Z2

j

j,k=1 j=1

= tσZ .

2

This stochastic process, a random walk, is

stationary with respect to the mean, but non-

stationary with respect to the variance.

This process describes the path of a particle

Figure 10.5: Scatter diagram of the bivariate that experiences random displacements. If a large

MJO index, in units of standard deviations. A number of such particles are considered, the centre

sub-segment of the full time series is shown in of gravity will not move, that is, E(Xt ) = 0, but the

Figure 10.3 [388]. scatter increases continuously. Thus the random

walk is sometimes a useful stochastic model for

describing the transport of atmospheric or oceanic