Given distinct climatic periods in the various facets of the Earth's climate system, many attempts have been made to determine the exact timing of 'change points' or regime boundaries. However, identification of change points is not always a simple task. A time series containing N data points has approximately N k distinct placements of k change points, rendering brute force enumeration futile as the length of the time series increases. Moreover, how certain are we that any one placement of change points is superior to the rest? This paper introduces a Bayesian Change Point algorithm which provides uncertainty estimates both in the number and location of change points through an efficient probabilistic solution to the multiple change point problem. To illustrate its versatility, the Bayesian Change Point algorithm is used to analyse both the NOAA/NCDC annual global surface temperature anomalies time series and the much longer δ 18 BackgroundThe time series that record various aspects of the Earth's climate system are widely recognized as being nonstationary (Hays et al., 1976; Imbrie et al., 1992;Karl et al., 2000;Tomé and Miranda, 2004;Raymo et al., 2006;Beaulieu et al., 2010; among others). Several methods have been implemented to solve the 'change point' problem for shorter climatic time series. For example, Karl et al. (2000) fixes the number of discontinuities and then uses both Haar (square) wavelets and a brute force minimization of the residual squared error for the placement of piecewise continuous line segments. Similar to this second approach, Tomé and Miranda (2004) automate the creation of a matrix of over-determined linear equations and consecutively solve this system for every possible combination of change points that satisfies their constraints. To deal with the exponentially increasing number of change point solutions associated with longer time series, dynamic programming change point algorithms have been developed that reduce the computational burden to a more manageable size (Ruggieri et al., 2009 (2004) first identify change points by visual inspection and then refine their location so as to: (1) minimize the number of change points; (2) be consistent with previous research; and (3) have support from an iterative non-parametric statistical method (Lanzante, 1996). This iterative approach adds one change point at a time, testing each for statistical significance. In an attempt to minimize the a priori assumptions on the number and location of change points, Menne (2006) proposes a semi-hierarchic splitting algorithm to place the change points. Here, the placement of a change point splits the time series, but each splitting step is followed by a merge step to determine whether change points chosen earlier are still significant.Each of these methods returns a single, 'optimal' solution. But if there are ∼N k possible placements of k change points in a time series of length N, how confident are we that this one solution is vastly superior to any other, especially one that may only ...
1] Although different paleoenvironmental time series resolve past climatic change at different time scales, nearly all share one characteristic: they are nonstationary over the length of the record sampled. We describe a recursive dynamic programming change point algorithm that is well suited to identify shifts in the Earth system's variability, as it represents a nonstationary time series as a series of regimes, each of which is homogeneous. The algorithm fits the data by minimizing squared errors not only over the parameters of the models for each subsequence but also over an arbitrary number of boundary points without restrictions on the lengths of regimes. The versatility of the algorithm is illustrated by an application to 5 Ma of Plio-Pleisotcene d 18 O variations. We seek to identify either the single dominant ''Milankovitch'' frequency or linear combinations of frequencies and consistently identify changes $780 ka and $2.7 Ma, among others, in each analysis done. Our applications also provide support to the recent hypothesis that obliquity-based Milankovitch terms can account for the circa 100 ka cycle that empirically dominates the most recent 1 million years.Citation: Ruggieri, E., T. Herbert, K. T. Lawrence, and C. E. Lawrence (2009), Change point method for detecting regime shifts in paleoclimatic time series: Application to d 18 O time series of the Plio-Pleistocene, Paleoceanography, 24, PA1204,
Change point models seek to fit a piecewise regression model with unknown breakpoints to a data set whose parameters are suspected to change through time. However, the exponential number of possible solutions to a multiple change point problem requires an efficient algorithm if long time series are to be analyzed. A sequential Bayesian change point algorithm is introduced that provides uncertainty bounds on both the number and location of change points. The algorithm is able to quickly update itself in linear time as each new data point is recorded and uses the exact posterior distribution to infer whether or not a change point has been observed. Simulation studies illustrate how the algorithm performs under various parameter settings, including detection speeds and error rates, and allow for comparison with several existing multiple change point algorithms. The algorithm is then used to analyze two real data sets, including global surface temperature anomalies over the last 130 years.
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