Graphics Programs Reference
In-Depth Information
samples equally spaced along the sediment core are therefore not equally
spaced on the time axis. In this case, the quantity
where
T
is the full length of the time series and
N
is the number of data points,
represents only an average sampling interval. In general, a time series
y
(
t
k
)
of a process can be represented as a linear sum of a long-term component or
trend
y
tr
(
t
k
), a periodic component
y
p
(
t
k
) and a random noise
y
n
(
t
k
).
The long-term component is a linear or higher-degree trend that can be ex-
tracted by fi tting a polynomial of a certain degree and subtracting the values
of this polynomial from the data (see Chapter 4). Noise removal will be
described in Chapter 6. The periodic - or cyclic in a mathematically less
rigorous sense - component can be approximated by a linear combination
of cosine (or sine) waves that have different amplitudes
A
i
, frequencies
f
i
and
phase angles
ψ
i
.
The phase angle
helps to detect temporal shifts between signals of the
same frequency. Two signals
y
1
and
y
2
of the same period are out of phase if
the difference between
ψ
ψ
2
is not zero (Fig. 5.2).
The frequency
f
of a periodic signal is the inverse of the period
ψ
1
and
. The
Nyquist frequency f
Nyq
is half the sampling frequency
f
s
and provides a maxi-
mum frequency the data can produce. As an example, audio compact disks
(CDs) are sampled at frequencies of 44,100 Hz (Hertz, which is 1/second).
The corresponding Nyquist frequency is 22,050 Hz, which is the highest
frequency a CD player can theoretically produce. The limited performance
of anti-alias fi lters used by CD players again reduce the frequency band and
cause a cutoff frequency at around 20,050 Hz, which is the true upper fre-
quency limit of a CD player.
We generate synthetic signals to illustrate the use of time-series analysis
tools. While using synthetic data we know in advance which features the
time series contains, such as periodic or stochastic components, and we can
introduce artifacts such as a linear trend or gaps. This knowledge is particu-
larly important if you are new to time series analysis. The user encounters
plenty of possible effects of parameter settings, potential artifacts and errors
τ
