Palaeoclimate time series, reflecting the state of Earth's climate in the distant past, occasionally display very large and rapid shifts showing abrupt climate variability. The identification and characterisation of these abrupt transitions in palaeoclimate records is of particular interest as this allows for understanding of millennial climate variability and the identification of potential tipping points in the context of current climate change. Methods that are able to characterise these events in an objective and automatic way, in a single time series, or across two proxy records are therefore of particular interest. In our study the matrix profile approach is used to describe Dansgaard–Oeschger (DO) events, abrupt warmings detected in the Greenland ice core, and Northern Hemisphere marine and continental records. The results indicate that canonical events DO-19 and DO-20, occurring at around 72 and 76 ka, are the most similar events over the past 110 000 years. These transitions are characterised by matching transitions corresponding to events DO-1, DO-8, and DO-12. They are abrupt, resulting in a rapid shift to warmer conditions, followed by a gradual return to cold conditions. The joint analysis of the

Palaeoclimate time series reflect Earth's climate in the distant past based mainly on proxies derived from records such as sediments, ice cores, or speleothems. One of the most extensively studied palaeoclimate time series is that of oxygen isotope ratios (

The ice cores retrieved from the Greenland ice sheet revealed the occurrence of rapid warming events, which occurred over a few decades. Such abrupt transitions are designated Dansgaard–Oeschger (DO) events, during which climate conditions alternated between fully glacial (so-called stadial) and relatively mild (interstadial) conditions

DO events are particularly observable in the Greenland ice core records, but similar transitions were identified in diverse palaeoclimate records (e.g.

The original identification of DO events was conducted by visual inspection of the NGRIP

Given that DO events can be considered a recurring pattern in a palaeoclimate time series, algorithmic methods for the extraction of similar patterns from a time series can be applied to characterise these particular patterns. In this study the matrix profile approach

The matrix profile approach is described in detail in

For a real-value time series

The brute-force obvious algorithm to compute the matrix profile requires the computation of a large number of Euclidean distances equal to

The matrix profile enables the identification of motifs in the time series, corresponding to sub-sequences that are highly similar to each other. The (tying) lowest points in the matrix profile correspond to the locations of the optimal time series motif pair, i.e. the pair of sub-sequences that are most similar

For a single time series

The Python code stumpy

The analysis presented here applies to the 20-year-resolution time series of

Time series of

The results of the matrix profile analysis of the palaeoclimate time series are presented initially in terms of the characterisation of DO events from a single time series, the

The matrix profile of the

Snippet of the matrix profile and profile index of the

Figure

Matrix profile of the NGRIP

The top motif pair, representing the two most similar sub-sequences in the time series starting at 71.14 and 75.26 ka, respectively, is displayed in Fig.

Figure

Top motifs for the NGRIP

Normalised motifs for the NGRIP

Motifs extracted from the matrix profile of the

A comparable analysis is conducted for the Ca

Matrix profile of the NGRIP Ca

Top motifs for the NGRIP Ca

Motifs extracted from the matrix profile of the Ca

Figure

Normalised motifs for the NGRIP Ca

Figure

Join matrix profile for the

Snippet of the join matrix profile and the profile index of the

The minimum value of the join matrix profile indicates the location of the top motif, i.e. the sub-sequence in the

Normalised top motif across the NGRIP

While the matrix profile is an algorithmic approach that enables the extraction of recurring patterns in a time series, its results are dependent on parameters that are prescribed empirically, often through a trial-and-error process, and they are dependent on the specific application and purpose of the analysis. These constraints are discussed in Sect.

The matrix profile is dependent on a single parameter, the sub-sequence length

In our study, a window size of 2500 years was selected as adequate for the extraction of motifs with durations typical of DO events. The matrix profile obtained with this window size is compared in Fig.

Matrix profile of the NGRIP

Motifs extracted from the matrix profile of the NGRIP

The top motif is derived directly from the lowest values of the matrix profile and therefore depends only on the specified sub-sequence length. However, the extraction of neighbouring sequences to the top motif and of other motifs depends not only on the window size, but also on the tolerance (distance-wise) with which a sub-sequence is considered to match a pattern given by the value of parameter

The identification of higher-order motifs, other than the top (global) motif corresponding to the matrix profile minimum, is more challenging due to the dependence on the constraints that must be set in terms of the maximum distance for which sub-sequences are taken as matching. As illustrated in the preceding section, varying the radius parameter yields both matching and dissimilar patterns. Therefore, we have adopted a more flexible criterion than the radius parameter

The comparison of sub-sequence similarity is performed here based on the Euclidean distance metric. An alternative would be to consider instead dynamic time warping (DTW) as a more robust distance measure for time series

Here we only discuss the influence on matrix profile results of methodological options in terms of parameters and metric selection. A further extension would be to assess how differences in the time series data would impact the matrix profile results, e.g. by applying surrogate time series methods such as the approach of

Normalised motifs for the NGRIP

Various time series of Ca

Join matrix profile for the

The most analogous pattern (top motif) across two time series does not need to occur at the same time in the two series. This fact is illustrated by considering the same

Top motif across

For the original version and the two versions of Ca

Snippet of the matrix profile and profile index for the shifted Ca

Snippet of the matrix profile and profile index for the trimmed Ca

In this study, the algorithmic matrix profile approach was employed to identify recurring patterns in the well-studied NGRIP palaeoclimate record. The matrix profile is dependent on a single parameter, the sub-sequence length. This is generally set by considering the typical duration of the patterns of interest, as there are no stringent criteria for its specification. In this analysis, a window size of 2500 years was considered. Shorter patterns exist in the time series and are of interest, but short-length sub-sequences can be harder to identify in terms of recurring patterns, and thus this study focused on patterns spanning around 2500 years. Consistent patterns were obtained for window sizes of 3000 and 3500 years, indicating that the results are robust to window sizes within this range.

The objective of the matrix profile approach was not to identify the abrupt DO transitions or to determine their precise timing. Rather, the objective was to characterise the abrupt transitions in a purely data-driven manner, based on the shape of the corresponding DO patterns. For the

The matrix profile method has also been employed to identify the most analogous pattern across the two different

All the data and software code used in this work are publicly available at

SB: conceptualisation; formal analysis; writing – original draft preparation. MES: writing – review and editing. DDR: writing – review and editing.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This work is TiPES contribution no. 281.

This research has been supported by the European Union's Horizon 2020 research and innovation programme (grant no. 820970) and the Fundação para a Ciência e a Tecnologia (grant no. LA/P/0063/2020).

This paper was edited by Kira Rehfeld and reviewed by two anonymous referees.