Bred vectors characterize the nonlinear instability of dynamical systems and so far have been computed only for systems with known evolution equations. In this article, bred vectors are computed from a single time series data using time-delay embedding, with a new technique, nearest-neighbor breeding. Since the dynamical properties of the standard and nearest-neighbor breeding are shown to be similar, this provides a new and novel way to model and predict sudden transitions in systems represented by time series data alone.

Prediction of sudden regime changes in the evolution of
dynamical systems is a challenging problem. For systems with known dynamical
models, such as the Earth's atmosphere, simulated trajectories under
judiciously chosen, finite-size perturbations can provide useful information
regarding regime changes by detecting fast-growing instabilities along the
model representation of the system evolution, called the “control”.
Breeding is a technique to generate an ensemble of such perturbations,
developed for operational ensemble forecasting of the numerical weather
prediction

Models of most natural systems like the Earth's atmosphere are described by
a very large number of dynamical variables and thus are high dimensional.
However, the variables or degrees of freedom are nonlinearly coupled, and
consequently in dissipative systems the dimensionality of the phase space is
significantly reduced. This is the basis for the time-delay embedding method
in the reconstruction of phase space

This paper presents a novel extension of the original breeding technique to
the phase space reconstructed from time series data using the time-delay
embedding method. Because dynamic instabilities are intrinsically low
dimensional

To extend the breeding technique to a system represented by a time series

Having defined the reconstructed phase space by the time-delayed embedding,
the new approach to breeding is in essence a matter of selecting the
perturbed trajectories that capture the unstable directions along the
control. Over a breeding cycle with window size

The first column of figures depicts the growth rates of bred vectors
in the Lorenz system using three different methods:

To test whether the nearest-neighbor breeding shares with the standard
breeding the ability to predict regime changes in the reconstructed phase
space, we use the 3-D Lorenz (1963) system

The regime transitions are analyzed by performing and comparing the following
three breeding experiments. Experiment (a) is the standard breeding in the
3-D model phase space using Eq. (

In all experiments, the growth rate

The left column of Fig.

As pointed out by

Table

We note that, in addition to large bred vector growth rate, two other methods
have been also proposed to predict regime changes in the Lorenz three
variable system. In his original paper

Contingency tables based on the rule that regime change will occur
in the orbit following the appearance of high-growth-rate bred vectors using
three different methods. In (b) and (c) using the nearest-neighbor breeding,
high-growth-rate points in orbits with absolute values of extrema above 1 are
excluded. OBS and FCST stand for observed and forecast, respectively;
(a)–(c) are the same as in Fig.

Measures of forecast accuracy in terms of the hit rate (HR), threat
score (TS), and false alarm rate (FAR); (a)–(c) are the same as in
Fig.

For the binary forecasts,

The ability to predict regime change in a dynamical system using
the time series data of just one of its many variables, demonstrated in this
paper, has important implications. For most systems in nature and in
laboratory, the time series observations of only a limited number of physical
variables, often a single variable, are available. In many cases even the
actual number of variables is not known. This paper presents and demonstrates
that the nearest-neighbor breeding enables the prediction of regime change in
systems for which regime change follows the appearance of instabilities, thus
extending the predictive capability beyond the cases whose time evolution
equations are known. Further, when regime change is associated with large
changes in the dynamical states, this technique can lead to the prediction of
large or extreme events in the cases where nonlinear dynamical predictions
are made using time series data, e.g., in the Earth's magnetosphere and space
weather

This work is supported by JPL grant 1485739, CICS grants NOAA-CISC-EKEK_12 and CICS-EKEK_14, NSF/IIP grant 1338634, and ONR grant N00014-10-1-0557. This paper is in part Contribution No. 3465 of the Virginia Institute of Marine Science, College of William & Mary. Edited by: S. Vannitsem Reviewed by: four anonymous referees