The matrix profile of the δ18O time series is computed using the stomp (Scalable Time series Ordered-search Matrix Profile) algorithm with a sub-sequence length (window size m) of 2500 years (125 data points). The exclusion zone w/m, defined as a proportion of the sub-sequence length, is set to 1, which is equal to the size of the sub-sequence length, in order to exclude trivial matches (similarity of a sub-sequence to another sub-sequence with data values in common). For each ith sub-sequence consisting of the time series values from i to i+m-1 with i=1,2,…n-m+1, its z-normalised Euclidean distance to every other jth sub-sequence is computed. The matrix profile is defined as the smallest value from that set of distances, while the profile index records the location of the most similar (non-trivial) sub-sequence, or in other words the value of j. Table  provides an illustrative example of the matrix profile and the corresponding profile index computed for the δ18O time series. The first column contains the data point corresponding to the start of the sub-sequence, i.e. the 3044th value of the time series (and the corresponding time) and so forth. The middle column contains the matrix profile, i.e. the minimum value of all the distances between sub-sequence 3044 and all other sub-sequences in the time series. The third column gives the profile index, which for the first row indicates that the most similar sub-sequence to sub-sequence 3044 is the sub-sequence starting at the 3250th value of the time series – all other sub-sequences in the time series are separated from sub-sequence 3044 by a value higher than 2.37.

Figure  depicts the complete matrix profile for the δ18O time series, which represents the distance of each sub-sequence to its most similar (non-trivial) sub-sequence (same as the middle column in Table 1). The global minimum value of the matrix profile is 2.34, occurring at 71.14 ka (indicated by the vertical dashed line in Fig.  and the bold numbers in Table 1). Of all the sub-sequences in the δ18O time series, the one starting at 71.14 ka exhibits the shortest distance to its most similar sub-sequence, i.e. the one starting at 75.26 ka (Table ). It should be noted that this sub-sequence does not necessarily correspond to the next minimum value of the matrix profile. Rather, it is the sub-sequence that is most similar to the one extracted from the global minimum and not necessarily the sub-sequence with the second-lowest matrix profile distance.

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. . This motif pair corresponds to the canonical DO events DO-19 and DO-20, which are most similar to each other. While the sub-sequence starting at 75.26 ka is most similar to the top sub-sequence corresponding to the matrix profile minimum, at 71.14 ka, other sub-sequences, while not the most similar ones, are still not far off. All the sub-sequences for which the distance to the top sub-sequence at 71.14 ka is within R times the distance from the top motif pair can be considered neighbouring motifs. Table  presents the motifs obtained for the δ18O time series using values of R equal to 2 and 3. Setting a value of R=2 enables the extraction of three neighbour motifs to the top motif, while setting R=3 allows for the additional extraction of five further neighbours. It should be noted that the distance represented in Table  is the distance of the motif sub-sequence to the top motif and not its matrix profile distance. An alternative approach to the use of a fixed value of the parameter R is to set a maximum distance between a neighbour motif and the main motif based on the variability of the matrix profile. In this study, neighbouring motifs are considered to be separated by less than twice the global standard deviation of the matrix profile (equal to 2×2.38=4.76), which in this case corresponds to the neighbours that would be extracted by considering R=2 (Fig. ). This aspect of selecting neighbour sequences to the main motif is further discussed in Sect. . The last column of Table  indicates the canonical DO event that coincides with each motif. The seventh motif (12.98–10.48 ka) does not correspond to a DO event, but it does include the Younger Dryas cooling event (GS-1) and the transition to the Holocene.

Figure  depicts the normalised sub-sequences corresponding to the motifs extracted from the δ18O time series. The top motif represents the patterns corresponding to the canonical DO-19 and DO-20. These aforementioned sub-sequences and their neighbour motifs represent abrupt transitions to warmer conditions, preceded by an approximately stable stadial level and followed by a slow decrease towards stadial conditions.

A comparable analysis is conducted for the Ca2+ record employing the same methodology as that employed for the δ18O time series. Figure  depicts the matrix profile for the Ca2+ record. While the overall pattern of the Ca2+ series is similar to that of the matrix profile of the δ18O record in Fig. , the peaks are sharper due to the lower noise level of the Ca2+ series. Moreover, the minimum value of the matrix profile for the Ca2+ series, indicated by the vertical dashed line in Fig. , occurs at a different time (at 37.14 ka) than for δ18O. Consequently, the sub-sequence with the shortest distance to its most similar sub-sequence in the Ca2+ record starts at 37.14 ka, corresponding to DO-8. From the profile index, the most similar sub-sequence, which constitutes the top motif pair for the Ca2+ time series, starts at 45.78 ka, corresponding to DO-12. The aforementioned top motif and its neighbour motifs are summarised in Table . The criterion of selecting as neighbour motifs the sub-sequences separated from the main motif by less than twice the matrix profile's standard deviation (2×2.50=5) yields all neighbour motifs for a radius of R=2 and the first neighbour motif for a value of R=3. This is illustrated in Fig. .

Figure  displays the normalised motifs for the Ca2+ time series. These motifs correspond to the same canonical DOs as the motifs of the δ18O time series, with the addition of DO-21. However, the ordering of the motifs differs. In the case of the Ca2+ record, considered on the reverse logarithmic scale, the recurring patterns, constituted by the top motif and its neighbour motifs, represent abrupt decreases in the atmospheric dust concentration. These decreases are preceded by slightly decreasing stadial normalised values and are followed by an approximately stable interstadial level.

Figure  depicts the join matrix profile of the δ18O and Ca2+ time series, which is defined as the distance between each sub-sequence in the δ18O record and its most similar sub-sequence in the Ca2+ record. In this join case, the necessity for an exclusion zone to avoid trivial matches is negated by the fact that the sub-sequences originate from different time series. The join matrix profile (Fig. ) differs from the previously defined individual profiles but exhibits similarities to the matrix profile for the single δ18O time series (Fig. ). This is not unexpected given that the temporal variability of the Ca2+ series closely follows that of the δ18O series. For illustrative purposes, a snippet of the join matrix profile and the profile index is presented in Table .

The minimum value of the join matrix profile indicates the location of the top motif, i.e. the sub-sequence in the δ18O time series which is most similar (distance-wise) to a sub-sequence in the Ca2+ time series (whatever it is). The global minimum of the join matrix profile occurs at 71.28 ka, which is in close proximity to the previously defined minimum for δ18O at 71.14 ka. Table  indicates that the sub-sequence in the δ18O time series starting at 71.28 ka exhibits the greatest similarity to the sub-sequence in the Ca2+ record also starting at 71.28 ka. The most analogous sub-sequences across the δ18O and Ca2+ records are presented in Fig. , still with Ca2+ displayed on the reverse logarithmic scale. In this particular case, the location (time) of the top motif is the same for the two records, and thus normalised motifs are represented on the temporal scale rather than plotted as a function of the index value from m=1 to m=125, as in Figs.  and . This motif is identical to the previously identified motif in the individual analysis of the δ18O time series (Sect. ). This transition, occurring approximately 70 000 years ago (the canonical DO-19), exhibits the most analogous patterns of warming and cooling across δ18O and less dusty or dustier Ca2+ records. Additionally, it bears the closest resemblance in shape to the transition in the δ18O record immediately after, approximately 73 000 years ago.

The matrix profile is dependent on a single parameter, the sub-sequence length m. Consequently, it is a method that can be readily applied to any time series, as it only requires the appropriate specification of a window size (see Sect. ). However, the matrix profile results are highly contingent upon the value selected for m. There are no universal formulas or rigorous criteria for selecting the window size, as this depends on the objective of the analysis and the type of motif being investigated. The sub-sequence length is typically determined by considering the length of the patterns of interest in the time series being analysed. In the absence of prior knowledge regarding the length of the motif of interest in a time series, an extension of the matrix profile, which computes nearest-neighbour information for all sub-sequences of all lengths, can be considered .

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.  with the matrix profile for window sizes of 3000 and 3500 years. The matrix profile values increase with increasing values of m, yet the results remain remarkably consistent across this range of window sizes, exhibiting a similar overall pattern and the lowest distance at around 70 ka. The highest values of the matrix profile (at around 90, 66, and 26 ka) reflect the flatter parts of the δ18O time series, which lack discernible features and thus comprise more disparate sub-sequences. The motifs obtained for the different window sizes are presented in Fig. , which provides further evidence of the robustness of the recurring patterns extracted in the considered window range.

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 R. Once more, there are no specific criteria for specifying R, other than it should be a small integer value. The larger the value of R, the greater the distance between sub-sequences considered to match the motif and therefore the lower the similarity between potential matches. The use of smaller values of R reduces the potential for considering as matching sub-sequences that are dissimilar. This is demonstrated in Fig. , which depicts the normalised motifs for the NGRIP δ18O time series using R=2 and the supplementary motifs obtained using R=3. To facilitate visualisation, the first two neighbour motifs for R=3 are plotted separately from the subsequent neighbour motifs (see Table ). The discrepancy between the top motif pattern and the neighbour motifs assigned increases with the higher-order motifs indicating a value of R=3 is not optimal in this case. Notwithstanding, the neighbour motifs 6 and 7, derived from the use of R=3, remain in close proximity to the primary motif. An alternative approach to enhance the flexibility of the extraction of neighbour motifs is to consider a maximum distance calculated not from a radius R but rather from a metric computed from the overall matrix profile variability. In this study we adopt this approach, whereby neighbour motifs are selected as the patterns that are separated from the top motif by less than twice the standard deviation of the matrix profile. For the δ18O time series, the motifs obtained using this criterion are identical to those obtained by setting R=2. However, this approach yields the same neighbour motifs for the Ca2+ time series as for R=2, in addition to the first neighbour motif corresponding to R=3.

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 R in order to ensure appropriately constrained results and interpretable motifs. While the matrix profile calculation is only dependent on the sub-sequence length and the top motif is objectively obtained from the global minimum value of the matrix profile, subsequent motifs or neighbours depend on the adopted distance constraints. To circumvent this limitation, this study focuses on the extraction of the top motif from the palaeoclimate records. In general, the extraction of motifs has to take into account the purpose of the analysis and is very much problem-dependent in terms of what the patterns of interest are in a time series (which may differ, even for the same time series, depending on the goal of the analysis). In the case of very large datasets for which information on the patterns of interest is not available, the statistical significance of motifs can be assessed to evaluate which additional motifs are significant, other than the top motif .

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 . However, empirical comparisons showed that the Euclidean distance is competitive with or superior to more complex measures . Therefore, the Euclidean distance between z-normalised sub-sequences is employed as the similarity measure in this study.

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 . Extending the methodology to account for uncertainty in dating and proxy values would be a further methodological extension particularly relevant for palaeoclimate time series.

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 δ18O and Ca2+ proxy records but artificially changing the Ca2+ time series in two distinct ways. The first case introduces an artificial shift in time of 500 data points (10 ka) by adding to the beginning of the record 500 points with the same value as the mean of the Ca2+ time series. The second case removes the first 500 data points of the Ca2+ series, thus obtaining a shorter record of a different length than the δ18O series. The time series are displayed in Fig. , and the join matrix profile between the δ18O series and these versions of the Ca2+ series is presented in Fig. . The outcomes are largely analogous, with only minor discrepancies in the join matrix profile for the different versions of the Ca2+ time series. In the case of the shifted version of the Ca2+ series (Fig. b), the difference between this version and the original is observed in the highest values of the join matrix profile, which are flatter. The matrix profile peaks correspond to parts of the series with no discernible temporal structure, typically a featureless noise level. In such cases the most similar sub-sequences are the stable level values (equal to the mean of the time series) that were introduced at the beginning of the record. In the case of the trimmed version of the Ca2+ series (Fig. c), it should be noted that the join matrix profile has the same length, despite the reduced length of the Ca2+ series. This is because the join matrix profile contains the distance of every sub-sequence in the δ18O series to the trimmed Ca2+ series (of a smaller size). Consequently, the number of sub-sequences (in the δ18O series) remains constant, yet the distances are calculated between each of these sub-sequences and a smaller number of sub-sequences in the Ca2+ series. The principal distinction between the join matrix profile values in this particular case is observed at the beginning of the record, as the initial sub-sequences of the δ18O series are most similar to the initial sub-sequences in the Ca2+ series, which no longer exist, having been matched to other sub-sequences in the Ca2+ time series.

For the original version and the two versions of Ca2+ under consideration, the minimum value of the join matrix profile occurs at the same time corresponding to the top motif previously identified in the δ18O record at around 71 ka. Tables  and present excerpts of the join matrix profile and profile index for the shifted and trimmed Ca2+ time series, respectively. In the former case, the minimum value of the matrix profile occurs at the same index value of the δ18O time series as previously defined. However, the most similar sub-sequence in the Ca2+ record is no longer coincidental, although it is correctly identified at 10 ka later. A comparable outcome is observed in the latter case. The profile index indicates that the most similar sub-sequence occurs at the index value of the time series corresponding to the correct time. Figure  displays the top motif across the δ18O time series and the different versions of the Ca2+ series. It can be seen that the most similar sub-sequence across the two records corresponds to the canonical DO-19. Furthermore, the same results are obtained with shifted and trimmed versions of the Ca2+ time series, demonstrating the robustness of the approach.