Time Series: Analysis and Modelling - Environmental Modelling: Finding Simplicity in Complexity - page 33

The values in a log-normal distribution can be obtained
directly by taking the logarithm of the values in a nor-
mal (Gaussian) distribution. An important parameter in
the log-norm al distribution is the coefficient of varia-
tion, c v = σ x / x .If c v is small, the distribution is nearly
Gaussian, i.e. the frequency-size distribution of values
observed in Figure 3.1b ( c v = 0 . 53). If c v is large, the dis-
tribution of values is strongly skewed, i.e. the distribution
of values in Figures 3.1c ( c v = 1 . 54) and 3.1e ( c v = 2 . 14).
A log-normal white noise can be constructed just as we
previously constructed a Gaussian white noise, by at each
time step randomly choosing a value from a log-normal
cumulative distribution function, and projecting this to
the horizontal axis.

where one multiplies a given value of the time series x n
(minus the mean) with x n + τ , a va lu e

τ

steps later (the
lag), and again from Equation 3.1, x is the mean, σ

x the
variance, and N the number of values in the time series
(Box et al ., 1994).
For zero lag ( 0 in Equation 3.6), and using the

definition for variance (Equation 3.1), the autocorre-
lation function is C (0) = 1 . 0. As the lag,

τ =

τ , increases,

τ = 1, 2, 3, ... ,( N − 1), the autocorrelation function

C ( τ ) decreases as the correlation between x n + τ and x n

decreases. Positive values of C ( ) indicate persistence,

negative values indicate antipersistence, and zero values
indicate no correlation. Various statistical tests exist
(e.g. the Q K statistic, Box and Pierce, 1970) that take
into account the sample size of the time series, and
values of C (

τ

calculated, to determine the
significance of the time series not being correlated. A plot
of C ( τ ) versus τ is known as a correlogram . A rapid decay
of the correlogram indicates short-range correlations,
and a slow decay indicates long-range correlations. In
Section 3.5.2 we will introduce time-series models for
short-range correlations (persistence), and in Section
3.5.4, models for long-range correlations.

τ ) for those τ

3.5 Persistence

The values in the Gaussian white noise illustrated in
Figure 3.4a are uncorrelated. That is, at any given time
n , the value x n in the time series is not related to any
previous values. In contrast, the values in the Brownian
motion illustrated in Figure 3.4b are correlated. Any
given value x n in the time series is related to earlier
values through Equation 3.4. In this (strongly) correlated
time series, large values tend to follow large values,
and small values tend to follow small values - there
is strong persistence in the time series. The values in
a Brownian motion have a ' memory ' of the previous
values.
If a time series has persistence, the persistence can be
short-range (a finite series of values are correlated with
one another) or long-range (all values are correlated
with one another). Persistence can have a strength that
varies from weak to very strong. If a time series has
antipersistence, large values tend to follow small ones,
and small values large ones. In this section, we will
discuss the autocorrelation function, models for short-
range persistence, spectral analysis, and then models for
long-range persistence.

3.5.2 Models for short-rangepersistence

Several empirical models have been used to generate
time series with short-range correlations (persistence)
(Huggett, 1985; Box et al ., 1994). A number of fields,
for example hydrology, use time-series models based on
short-range persistence (e.g., Bras and Rodriguez-Iturbe,
1993). We will illustrate this approach to time-series
modelling using the autoregressive (AR) model .Inthis
time series model, the values in the time series x n are
generated using the relation

p

x n = x + ε n +

1 φ j ( x n − j − x )

(3.7)

j =

where n = 1, 2, 3, ... , N and p < N . Typically the ε n are
selected randomly from a Gaussian distribution with zero
mean and standard deviation σ ε , that is a white nois e as
illustrated in Figure 3.4a. The user defines a constant x to
be the mean they would like for the resultant time series,
x n . The values of

3.5.1 Autocorrelationfunction

One technique by which the persistence (or antipersis-
tence) of a time series can be quantified is with the
autocorrelation function . The autocorrelation function
C ( τ ), for a given lag τ , is defined as:

, p ) are prescribed
coefficients relating x n to the p previous values of t he
deviation of the time series from the mean, x n − j − x .
When n

φ j ( j = 1, 2, 3, ...

0.
As a specific example, we will consider the case p

− j ≤ 0, then x n − j =

1;
the term x n is related only to the previous value x n − 1 ,
which then amplifies through the time series, with

=

N − τ

1
σ

1
N − τ

C ( τ ) = ( x n − x )( x n + τ − x ) (3.6)

x

φ 1

n = 1