Page 113 - MATLAB Recipes for Earth Sciences

106 5 Time-Series Analysis

plot(f,phase)
xlabel('Frequency')
ylabel('Phase angle')
title('Phase spectrum')

The phase shift at a frequency of f=0.01 (period 100 kyr)

interp1(f,phase,0.01)
which produces the output of

ans = 0.2796

The phase spectrum is normalized to a full period τ=2›. Hence, a phase
shift of 0.2796 equals (0.2796*100 kyr)/(2*›) = 4.45 kyr. This corresponds
roughly to the phase shift of 5 kyr introduced to the second data series with
respect to the fi rst series.
As a more comfortable tool for spectral analysis, the Signal Processing
Toolbox also contains a GUI function named sptool, which stands for
Signal Processing Tool.

5.6 Nonlinear Time-Series Analysis (by N. Marwan)

The methods described in the previous sections detect linear relationships
in the data. However, natural processes on the Earth often show a more
complex and chaotic behavior. Methods based on linear techniques may
therefore yield unsatisfying results. In the last decades, new techniques of
nonlinear data analysis derived from chaos theory have become increasingly
popular. As an example, methods have been employed to describe nonlinear

behavior by defining, e.g., scaling laws and fractal dimensions of natural
processes (Turcotte 1997, Kantz and Schreiber 1997). However, most meth-
ods of nonlinear data analysis need either long or stationary data series.
These requirements are often not satisfied in the earth sciences. While most

nonlinear techniques work well on synthetic data, these methods fail to de-
scribe nonlinear behavior in real data.
In the last decade, recurrence plots as a new method of nonlinear data
analysis have become very popular in science and engineering (Eckmann
1987). Recurrence is a fundamental property of dissipative dynamical sy-
stems. Although small disturbations of such a system cause exponentially
divergence of its state, after some time the system will come back to a state
that is arbitrary close to a former state and pass through a similar evolution.
Recurrence plots allow to visualize such a recurrent behavior of dynamical