Time Series: Analysis and Modelling - Environmental Modelling: Finding Simplicity in Complexity - page 41

are found in environmental applications, particularly
where complex behaviour is involved. In Figure 3.1, we
gave five examples that illustrate various properties of
environmental time series. The variability of the time
series of atmospheric CO 2 given in Figure 3.1a illustrates
the roles of periodicity and a trend. The periodicity
is the annual variability and the trend is attributed to
anthropogenic CO 2 emissions. The values in the time
series of global seismicity given in Figure 3.1b are close
to a Gaussian distribution of values with a very weak
strength of persistence (i.e. close to a white noise). The
time series of river-discharge given in Figure 3.1c includes
a strong periodic annual component and a highly skewed
distribut io n (a large value of the coefficient of variation,
c v = σ x /

noises. Examples are given in Figure 3.8. A property
of fractional noise models is that the power-spectral
density (Equation 3.12) has a power-law dependence on
frequency (Equation 3.13). Fractional Gaussian noises are
an example of a (weakly) stationary time series, where the
mean and standard deviation of the values in the time
series are independent of the segment length considered.
It is also important to consider nonstationary time
series. The classic example is Brownian motion, which
is obtained by taking the running sum of a Gaussian
white noise. Examples of Brownian motion are given in
Figure 3.4b,c. Fractional Brownian motions can be gen-
erated by taking the running sum of fractional Gaussian
noises. Examples are given in Figure 3.8. Fractional Brow-
nian motions exhibit drift and their origin is not defined.
Many models that generate time series are statistical
in nature. A distribution of values is determined from
a specified statistical distribution. Correlations, long-
and/or short-range, are specified using models such as
the autoregressive (AR) model or the Fourier filtering
model that generates fractional noises and motions.
Much of classical physics was developed on the basis
of linear differential equations. These equations cannot
generate a stochastic (noisy) behaviour. One approach to
the generation of complex time series is to use stochastic
differential equations. In Equation 3.17, we introduce this
with the Langevin equation. It is accepted that determin-
istic Navier-Stokes equations can generate turbulence
which is characterized by complex time series. However,
relevant solutions have not been found. Lorenz (1963)
introduced a set of deterministic equations that generated
complex time series, and he associated this behaviour with
the concept of deterministic chaos. Subsequently other
examples have been presented.
The behaviour of time series has many important
consequences in the environmental sciences, including
risk analysis, prediction and forecasting, and the better
understanding of underlying processes. The frequency-
size distribution and persistence properties of time series,
focussed on in this chapter, have many applications to
the modelling of the climate and weather, hydrology,
ecology, geomorphology, natural hazards, and manage-
ment and policy, amongst the many disciplines in the
environmental sciences.

= .

54). The time series of tree rings given in
Figure 3.1d has a near Gaussian distribution of values and
is well approximated by a long-range persistent fractional
Gaussian noise with β = 0 . 8. The time series of daily
precipitation given in Figure 3.1e has a highly skewed
distribution of values, and the time series is sparse, with
nonzero values unequally spaced, due to there beingmany
days with zero rainfall.
We have emphasized in this chapter the role of the
frequency-size distribution of values in a time series
and the role of persistence. Any statistical distribution
can be taken as representative of the values (e.g. see
Stedinger et al ., 1993). Examples include the Gaussian
(normal) and the log-normal. The tails of the specified
distribution controls the extreme event statistics, both
very large and very small. Extreme events are rarewith thin
tailed (exponential) distributions and occur with greater
frequency with fat-tailed (power-law) distributions.
Persistence (or anti-persistence) is ameasure of correla-
tions between the values in a time series. Many models for
time series are primarily concerned with the generation
of persistence. One class of models has short-range per-
sistence. Each value in the time series is related to a finite
number of adjacent values. As an example of this type of
model we have considered the autoregressive (AR) model
in some detail. Time series generated using this model are
illustrated in Figure 3.5. We use the autocorrelation func-
tion (Equation 3.6) to quantify short-range persistence,
with an example given in Figure 3.6.
Many time series exhibit long-range persistence, where
each value in the series is related to all previous and
subsequent values. Spectral analysis is one method often
used to identify and quantify the strength of long-range
persistence, and is also used as amethod for the generation
of time series models with long-range persistence. One
such method uses spectral filtering to generate fractional

x 1

References

Adas, A. (1997) Traffic models in broadband networks. Commu-
nications Magazine, IEEE 35 , 82-89. DOI: 10.1109/35.601746.