methods similar to multivariate regression analysis information that the experiment yielded was not

(see Section 8.4) can be used to relate model considered suf¬cient to ensure accuracy. Twenty-

response to the settings (see Gough and Welch ¬ve additional simulations were thus performed

[145, p. 782]). using parameter settings selected to be distant from

The initial experiment performed by Gough and the original 26 settings and also distant from one

Welch consisted of 26 simulations with parameter another; 15 of these converged to a steady state.

settings selected as follows. A range of values The ¬nal collection of 36 simulations success-

was identi¬ed for each parameter, which was fully captured most of the dependence between

divided into 25 equal length intervals. The 26 the model™s steady state circulation and the seven

values that delineate the boundaries of the intervals adjustable parameters. The resulting systematic

were recorded. The ¬rst combination of parameter description of the dependence between model

settings was obtained by randomly selecting one outputs and parameter settings makes it easier to

value from each of the seven sets of 26 values. tune the model to reproduce an observed circula-

The second combination of parameter settings is tion feature. Gough and Welch were also able to

obtained by randomly selecting a value from each study the interaction between pairs of parameters.

of the 25 remaining values, and so on. The result For example, they found that diapycnal eddy

is a random Latin hypercube design with seven diffusivity modi¬es the effect that the maximum

treatments and 26 levels (values) of each treatment, allowable isopycnal slope has on the number of

combined at random in such a way that every ocean points at which convection occurs. They

level of every treatment occurs once in the 26 thus demonstrated that this is a highly effective

combinations of parameter settings. The objective means of systematically exploring an unknown

is to obtain uniform (but necessarily sparse) parameter space.

8 Similar studies have been performed with an ice model

[79] and a simpli¬ed atmospheric model [58].

Part IV

Time Series

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195

Overview

In this part we deal with time series analysis, that is, the statistical description of stochastic processes

and the use of sample time series for the identi¬cation of properties and the estimation of parameters.

The motivation for our non-conventional development of the subject is explained in Section 10.1.

We introduce the concept of a stochastic process and its realizations, called time series, in Chapter 10.

Special emphasis is placed upon auto-regressive processes since they may be interpreted as discretized

linear differential equations with random forcing. At this stage we do not concern ourselves with the

tools needed to characterize such processes, namely the covariance function and the spectrum. Instead

we use a non-conventional non-parametric characterization, based on the frequency distribution of run

length, that is, the duration of excursions above or below the mean. It allows us to intuitively examine

characteristic properties of stochastic processes, such as memory or quasi-oscillatory behaviour, without

using more complex mathematical tools such as the Fourier transform. Also, we differentiate between

the variability caused by the internal dynamics of the process and that caused by the driving noise.

The conventional parametric characterization of a stochastic process, in terms of the auto- or

cross-covariance function and the spectrum, is introduced in Chapter 11. While the concept of the

covariance function poses no special problems, that of the spectrum is more dif¬cult. The spectrum

is often taken literally as the decomposition of a stochastic process into oscillations at a set of ¬xed

frequencies. This interpretation is only appropriate in certain limited circumstances when there are

good physical reasons to believe that the time series contains only a ¬nite number of regular oscillatory

signals. In general, though, the process will also contain noise, in which case the spectrum can not

be interpreted as glibly. For example, the white noise process does not contain regular or oscillatory

features; thus the interpretation of its spectrum as the decomposition of the white noise into equally

important oscillatory components is misleading.

This part of the book is completed with Chapter 12, in which we describe techniques for inferring

information about the true covariance function and spectrum.

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10 Time Series and Stochastic Processes

10.1 General Discussion

10.1.1 The Role of Noise. This part of the

book deals with stochastic processes and their

realizations, time series. We begin with a general

discussion of some of the basic ideas and

pitfalls. The language and terminology we use is

necessarily vague; more precise de¬nitions will

follow later in this chapter and in Chapters 11 and

12.

A time series Xt often consists of two compo-

nents, a dynamically determined component Dt

and a stochastic component Nt , such that

Xt = Dt + Nt .

Sometimes the time evolution of Dt is independent

of the presence of the stochastic component Nt ; in

such cases the evolution of Dt is deterministic.1

Examples are externally forced oscillations such

as the tides or the annual cycle. At other times

the dynamically determined part depends on

the random component. Such processes become

Figure 10.1: A realization of a process Xt = Dt +

deterministic when the stochastic component is

Nt in which the dynamical component Dt is not

absent. When the stochastic component (or noise)

affected by the stochastic component Nt .

is present, typical features, such as damped

Top: A dynamical component Dt made up of two

oscillations, are masked and therefore not clearly

oscillations.

detectable. One goal of time series analysis

Middle: A ˜white noise™ component nt .

is to detect and describe the characteristics of

Bottom: The sum of both components.

the dynamical component when the stochastic

component is present.

Figures 10.1 and 10.2 illustrate these concepts.

Figure 10.1 displays a purely deterministic negative. The distribution of the length of these

oscillation Dt , a realization of a white noise excursions is a characteristic of such processes.

process nt , and the sum Dt +nt . The addition of the When the dynamical component generates cyclical

noise introduces some uncertainty, but it does not features in the absence of noise, pieces of such

modify the period or phase of the oscillations. In cyclical features will also be present when the

contrast, Figure 10.2 illustrates a damped system noise is turned on. However, the ˜period™ will

in which Dt = ±xt’1 . Without noise (Nt = 0), ¬‚uctuate, often around the period of Dt when noise

any nonzero value decays to zero in a characteristic is absent, and the phase will vary unpredictably.

time. The addition of noise transforms this decay We refer to this as quasi-oscillatory behaviour.

into a stationary sequence of episodes (i.e., runs) The two types of stochastic processes differ

during which Dt is continuously positive or with respect to their predictability. Here, we

say a system is predictable at lead time „ if

1 We depart slightly from our standard notation by using D ,

t

the conditional distribution of Xt+„ given Dt

the dynamical component, to represent both the deterministic

is different from the unconditional distribution

and stochastic forms.

197

10: Time Series and Stochastic Processes

198

times t1 , . . . , t N . This is generally not practical.

Instead, the most important aspects of this

probabilistic structure are described with either

the auto-covariance function or, equivalently, the

spectrum. Both descriptions require that we make

a stationarity assumption of some sort about the

stochastic process, that is, we need to assume that

the statistical properties of the process are not time

dependent.

The spectrum is the Fourier transform (see Ap-

pendix C) of the auto-covariance function. While

both functions contain the same information, the

spectrum is often more useful than the auto-

covariance function for inferring the nature of the

dynamical part of the process. In particular, the

presence of multiple quasi-oscillatory components

in a process causes peaks in the spectrum. The

frequency at which a peak occurs often corre-

sponds to that of a periodicity in the deterministic

component of the process, and the width of the

peak is representative of the damping rate.

The truth of this is dif¬cult to deduce when

Figure 10.2: A realization of a process Xt = the spectrum is de¬ned as the Fourier transform

Dt + Nt for which the dynamical component Dt = of the auto-covariance function. Therefore con-

0.7Xt’1 is affected by the stochastic component ventional approaches for introducing the spec-