Detection of local mixing in time-series data using permutation entropy

Mixing of neighboring data points in a sequence is a common, but understudied, effect in physical experiments. This can occur in the measurement apparatus (if material from multiple time points is pulled into a measurement chamber simultaneously, for instance) or the system itself, e.g., via diffusion of isotopes in an ice sheet. We propose a model-free technique to detect this kind of local mixing in time-series data using an information-theoretic technique called permutation entropy. By varying the temporal resolution of the calculation and analyzing the patterns in the results, we can determine whether the data are mixed locally, and on what scale. This can be used by practitioners to choose appropriate lower bounds on scales at which to measure or report data. After validating this technique on several synthetic examples, we demonstrate its effectiveness on data from a chemistry experiment, methane records from Mauna Loa, and an Antarctic ice core.

Figure 1

Permutation entropy example. Top: a short segment of a chaotic signal from the Lorenz system. Bottom: permutation entropy of the Lorenz time series in the top panel.

Figure 2

Lorenz permutation entropy. (a) Permutation entropy (PE) of the full version of the Lorenz signal from the top panel of Fig. 1, calculated with a range of values of $\tau$, the spacing between data points used to construct the permutations. (The trace in the bottom panel of Fig. 1 is a short segment of the $\tau =1$ trace in this image.) (b) If local mixing effects are artificially added to the same signal, the ordering of the PE traces is reversed. (c) A simple bin-averaging operation removes the local mixing effects, restoring the normal order of the traces.

Figure 3

Mixing and permutation entropy. If the values of neighboring points are not independent and the $\tau$ value is small, then the points used to construct each permutation are effectively drawn from a single local distribution, schematized here with different shades. In this case, even though the underlying data may be deterministic, the relationships of these draws will be random and the permutation entropy will be artificially high. Higher $\tau$ values widen the span of the calculation beyond the spread of the local distributions, thereby mitigating this effect.

Figure 4

Data. The three experimental time-series data sets used as examples in this paper: chemical sensor data from a gas-mixing experiment, water-isotope data from an Antarctic ice core (WDC), and atmospheric methane records from Mauna Loa (MLO). Insets are provided for WDC and MLO to see finer details of each time series, e.g., seasonal temperature variation in the WDC signal.

Figure 5

Bin size and mixing scales. Plots of $\overline{\mathcal{R}}$ versus bin size for the (a) Lorenz and (b) Mackey-Glass examples.

Figure 6

Reversal results in the Mackey-Glass system. PE traces from the Mackey-Glass system with $1\le \tau \le 6$ for (a) the original time series (b) with added local-mixing effects (c) after a bin-averaging operation was applied to remove those effects.

Figure 7

Reversal results in experimental data sets. From top to bottom: PE traces for the gas mixture, WAIS Divide, and Mauna Loa data. In all cases, the images in the left and right columns show the calculations for the raw and bin-averaged data, respectively. The traces in the latter are temporally offset by the width of the binning window times the width of the PE calculation window. (This offset is essentially invisible in the Mauna Loa data because of the number of data points in the time series.) (a) Gas mixture: raw. (b) Gas mixture: binned with $j=15$. (c) WDC: raw. (d) WDC: binned with $j=15$. (e) Mauna Loa: raw. (f) Mauna Loa: binned with $j=4$.

Figure 8

$\overline{\mathcal{R}}$ as a function of bin size. (a) Gas mixture, (b) WAIS Divide, and (c) Mauna Loa methane.

Figure 9

Windowed $\overline{\mathcal{R}}$ analysis for the Mauna Loa data. Each point corresponds to the value of $\overline{\mathcal{R}}$ over the previous 5000 points.