Time Series: Analysis and Modelling - Environmental Modelling: Finding Simplicity in Complexity - page 34

controlling the amount of short-range persistence. For
this example, Equation 3.7 becomes:

In the time domain we see how a function or variable
evolves over time. In the frequency domain , we see how
much of the variation in the amplitude of the signal is
contained within given frequency bands. When going
from the temporal to the frequency domain we lose some
information (so in terms of information, the 'frequency'
domain is a subset of the 'time' domain), but it enables us
to examine our data in a very different way. Time series
often contain certain frequencies that are more dominant
than others.
Many methods in the broad environmental sciences
use the frequency domain when examining time series.
Take an example such as temperature, which tends to
have higher values during the daytime and lower values
during the evenings. Although exactly the 'same' low
and high temperatures are not repeated day after day,
there is a cycle (period) of approximately 24 hours from
'low' to 'high' to 'low', and 'high' to 'low' to 'high'.
Pulling out the dominant cycles (if they exist) at different
scales, has a range of environmental applications ranging
from characterization to 'prediction', and underlies many
geostatistical and time-series techniques for examining
data and models. Probably the best known technique
for examining the frequency domain is spectral analysis,
especially using the Fourier transform.
The Fourier transform X m of an equally spaced time
series x n , n

x n = x + ε n + φ 1 ( x n − 1 − x ) (3.8)

The theoretical mean for this (short-range) correlated
time series is again x and its theoretical variance is
given by

2
ε

σ

x

σ = (3.9)

1

− φ

1

The autocorrelation function for this AR[1] process, at
lag

, is theoretically given by (Box et al ., 1994; Swan and
Sandilands, 1995):

τ

= φ 1 C ( τ ) (3.10)

Examples of this AR[1] time series are given in Figure 3.5,
with 0, we again

have a Gaussian white noise. With increasing values of
φ 1 , the persistence becomes stronger, as evidenced by
large values becoming more likely to follow large ones,
and small values followed by small ones. We also apply
the autocorrelation function C ( τ ) (Equation 3.6) to each
time series given in Figure 3.5, and give the resulting
correlograms in Figure 3.6. These results are in excellent
agreement with Equation 3.10.
Other examples of empirical models for short-range
persistence in time series include the moving-average
(MA) model, and the combination of the AR and MA
models to create the ARMA model. Reviews of many of
these models are given in Box et al . (1994) and Chat-
field (1996). Just as a stationary Gaussian white noise
can be summed to give a Brownian motion (Figure 3.4),
a stationary ARMA model can be summed to give the
nonstationary autoregressive integrated moving-average
(ARIMA) model. There are many applications of short-
range persistence models in the social and physical
sciences, ranging from river flows (e.g. Salas, 1993),
and ecology (e.g. Ives et al ., 2010) to telecommunication
networks (e.g. Adas, 1997).

φ 1 = 0 . 0, 0.2, 0.4, and 0.8. With φ 1 =

, N , results in an equivalent rep-
resentation of that time series in the frequency domain.
It is defined as:

= 1, 2, 3, ...

N

x n exp (2 π inm / N ) , m = 1, 2, 3, ... , N (3.11)

X m = δ

n = 1

where are the intervals (including units of time) between

successive x n ,and i is the square root of − 1. The resul-
tant Fourier coefficients X m corresponds to the frequencies
f m = m / ( N δ ). The time series is prescribed in the fre-
quency domain by the N values of X m , and the total
length of time series we are considering is T = N δ .The
X m values are symmetric around X N / 2 (i.e., X m =

δ

X N − m , ,
for m = 1, 2, 3, ... , N / 2), so in practice we consider only
the unique Fourier coefficients X m , , m

= ... /

2.
The quantities X m are complex numbers, X m = (a +
b i ), with the modulus

1, 2, 3, , N

3.5.3 Spectral analysis

b 2 ) 0 . 5 , which is a
measure of the amplitude of the signal at frequency f m .
Small values of m indicate low frequencies (long periods)
and large values of m indicate high frequencies (short
periods). It is standard practice to use

(a 2

|

X m |=

+

It is common in the environmental sciences and other
disciplines to examine a time series, not in the time
domain , but in the frequency (spectral) domain , with
common transformations including the Fourier, Hilbert ,
or wavelet transforms . As an illustration, we will consider
in some detail, the Fourier transform and spectral analysis ,
and then relate these to long-range persistence .

2 as a measure
of the amplitude of a time series. This measure is referred
to as the energy distribution of the time series as a

|

X m |