  Real-world systems are often complex, dynamic, and nonlinear. Understanding
the dynamics of a system from its observed time series is key to the prediction
and control of the system's behavior. While most existing techniques tacitly
assume some form of stationarity or continuity, abrupt changes, which are often
due to external disturbances or sudden changes in the intrinsic dynamics, are
common in time series. Structural breaks, which are time points at which the
statistical patterns of a time series change, pose considerable challenges to
data analysis. Without identification of such break points, the same dynamic
rule would be applied to the whole period of observation, whereas false
identification of structural breaks may lead to overfitting. In this paper, we
cast the problem of decomposing a time series into its trend and seasonal
components as an optimization problem. This problem is ill-posed due to the
arbitrariness in the number of parameters. To overcome this difficulty, we
propose the addition of a penalty function (i.e., a regularization term) that
accounts for the number of parameters. Our approach simultaneously identifies
seasonality and trend without the need of iterations, and allows the reliable
detection of structural breaks. The method is applied to recorded data on fish
populations and sea surface temperature, where it detects structural breaks
that would have been neglected otherwise. This suggests that our method can
lead to a general approach for the monitoring, prediction, and prevention of
structural changes in real systems.
