Transformation-cost time-series method for analyzing irregularly sampled data

Irregular sampling of data sets is one of the challenges often encountered in time-series analysis, since traditional methods cannot be applied and the frequently used interpolation approach can corrupt the data and bias the subsequence analysis. Here we present the TrAnsformation-Cost Time-Series (TACTS) method, which allows us to analyze irregularly sampled data sets without degenerating the quality of the data set. Instead of using interpolation we consider time-series segments and determine how close they are to each other by determining the cost needed to transform one segment into the following one. Using a limited set of operations—with associated costs—to transform the time series segments, we determine a new time series, that is our transformation-cost time series. This cost time series is regularly sampled and can be analyzed using standard methods. While our main interest is the analysis of paleoclimate data, we develop our method using numerical examples like the logistic map and the Rössler oscillator. The numerical data allows us to test the stability of our method against noise and for different irregular samplings. In addition we provide guidance on how to choose the associated costs based on the time series at hand. The usefulness of the TACTS method is demonstrated using speleothem data from the Secret Cave in Borneo that is a good proxy for paleoclimatic variability in the monsoon activity around the maritime continent.

Figure 1

Illustration of the transformation of $S_a$ to $S_b$. In total $S_a$ undergoes seven steps numbered $S_1,S_2,...,S_6$. Note that $S_7=S_b$. The path shown is a minimal-cost path and all the steps are elementary steps, like moving a spike or deleting and creating.

Figure 2

RQA analyses for logistic map: (a) Lyapunov exponent of the logistic map; (b) and (c) determinism calculated from TACTS for (b) various levels of deleting and (c) two measurement noise levels ($\sigma =0.05$ and $0.1$) and for two different rates of irregularity ($\gamma =5\%$ and $10\%$). For details, see text as well as the legends of the figures.

Figure 3

Determinism against noise level $\sigma$: $\gamma =10\%$ and $r=3.5$ (periodic, black line) and $r=4$ (chaotic, red line).

Figure 4

The time series of Rössler: Panel (a) shows the $y$ component with maxima highlighted as bullets; panel (b) shows the result of the maximum map $\tilde{Y}$ acting on $y$; panel (c) shows the time series for a skewness of 0.3; and panel (d) shows the time series for a skewness of 2.0. In panels (b)–(d) we show the interpolated time series that we draw from as a black dashed line. Panel (e) offers a comparison between the different time series and highlights their irregularity.

Figure 5

RQA analyses for Rössler map distance time series: In panel (a) the Lyapunov exponent $\lambda$ is given as a reference over the whole $b$ control-parameter range. In panel (b) the determinism DET determined from TACTS is shown for the maximum time series $\widehat{Y}$ (black line) as a reference. The other shown data results from the $\gamma$-distributed time series with skewness 0.3 (red dashed line), skewness 0.5 (green dash-dot line), skewness 1.0 (blue dotted line), and skewness 2.0 (pink line).

Figure 6

Upper graph: ${\delta }^{18}\text{O}$ record of the Secret Cave, Borneo; lower graph DET determined from the corresponding transformation-cost time series. Horizontal lines H1–H6 indicate the six Heinrich events, while the most recent line determines the Younger Dryas, a cold period in the northern hemisphere most likely caused by a collapse of the North American ice sheet. The light red band of the DET indicates the 90% confidence interval.