Time Series: Analysis and Modelling - Environmental Modelling: Finding Simplicity in Complexity - page 26

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Time Series: Analysis
and Modelling

Bruce D. Malamud 1 and Donald L. Turcotte 2
1 Department of Geography, King's College London, UK
2 Geology Department, University of California, USA

frequency-size distribution of values (how many values
at a given size), and (ii) the correlation or persistence of
those values (how successive values cluster together, or
the 'memory' in the time series). In persistent time series,
large values are more likely to follow large values, and
small values are more likely to follow small ones. Ideally,
values in a time series are either continuous or at equally
spaced time intervals. However, actual measurements
may not be equally spaced and there may be missing
data. In this chapter, we will give a brief overview of two
major attributes of time series that are frequently used in
time-series modelling: the frequency-size distribution of
values in a time series and the persistence of those values.
This chapter will be organized as follows. In Section 3.2,
we give some examples of environmental time series. In
Section 3.3 we will discuss frequency-size distributions of
time series. In addition to characterizing and modelling
the frequency-size distribution of all values in a time
series, from the smallest to the largest, an important
consideration is the distribution of the extreme values
in a time series, which are also known as the 'tails' of
the frequency-size distribution. The tails of a distribution
may be characterized as thin, where the largest values are
unlikely to occur, or fat, where catastrophic events are
relatively common.
In Section 3.4 we will discuss the concepts of uncor-
related values in a time series (white noises), and their
running sums (Brownian motions). Then, in Section 3.5,
we introduce measures of persistence and models that

3.1 Introduction

Characterizing environmental systems and their change
through time is a central part of the environmental
sciences. One way to approach this characterization is
through the analysis and modelling of time series. A time
series is a set of numerical values of any variable that
changes with time. Time series can be either continu-
ous or discrete . An example of a continuous time series
is temperature at a specified location. An example of a
discrete time series is the sequence of maximum daily
temperatures at this same location. Time series often
contain periodicities and trends . A temperature time series
has daily and yearly periodicities. Global warming super-
imposed on a temperature time series would result in
a trend.
A deterministic time series is one that can be predicted
exactly. In contrast, a stochastic time series is one where
values are drawn randomly from a probability distribu-
tion. After periodicities and trends have been removed
from a time series (i.e. the deterministic part of the
time series), the stochastic component of the time series
remains. This component is often referred to as 'random
noise'. However, there may be a degree of order in the
complexity of a random noise, such that successive values
in the time series are correlated with each other.
The stochastic (noise) component of a time series can
be broadly broken up into two parts: (i) the statistical