# Anomaly Detection in Paleoclimate Records Using Permutation Entropy

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

**Figure 1.**Water-isotope records for ${\delta}^{18}$O (top panel) and $\delta $D (bottom panel) taken from the West Antarctic Ice Sheet (WAIS) Divide ice core (WDC).

#### 2.2. Complexity Estimation

## 3. Results

**Figure 2.**Permutation entropies of the ${\delta}^{18}$O and $\delta $D data from Figure 1, calculated in rolling 2400-point windows (i.e., $W=2400$) with a word length $m=4$ and a delay $\tau =1$.

**Top panel**: weighted (WPE) and unweighted (PE) permutation entropy of ${\delta}^{18}$O in cyan and blue, respectively.

**Bottom panel**: WPE and PE of $\delta $D in orange and red, respectively.

**Figure 3.**Top panels: the remeasured $\delta $D (red) and ${\delta}^{18}$O (blue) data points in the range of ≈4.5–6.5 kybp, with the original records shown in gray. Bottom panels: a closeup of a small segment of those traces, marked with a gray box in the top panels. Note that the vertical scales here are different than in Figure 1.

**Figure 4.**A simple anomaly detection algorithm. Here, ${\sigma}^{2}$ is estimated on a rolling 2400-point overlapping window for $\delta $D (

**top**) and ${\delta}^{18}$O (

**bottom**). Neither the feature in the gray-shaded box in Figure 2 nor the other anomalies described later in this paper are brought out by this calculation.

**Figure 5.**PE and WPE of ${\delta}^{18}$O (

**top panel**) and $\delta $D (

**bottom panel**) using remeasured data from 1037–1368 m (≈4.5–6.5 kybp).

**Figure 6.**Top row: $\delta $D data from 47.5 to 47.8 kybp before and after removal of and interpolation over (black dashed line) a range of faulty values (shaded in gray). Bottom row: WPE calculated from the corresponding traces. The width of the square wave in the lower left plot is the size of the WPE calculation window (2400 points at 1/20th year per point) plus the width of the anomaly. The horizontal shift between the earliest faulty value and the rise in WPE is due to the windowed nature of the WPE calculation.

**Figure 7.**Isotope data near 38.7 kybp that produced a spike in PE but not in WPE. According to the lab records, the graphical user interface froze during the analysis of this segment of the ice, compromising the results.

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFA | Continuous Flow Analysis |

CRDS-CFA | Cavity Ring-Down Spectroscopy Continuous Flow Analysis |

NSF-ICF | National Science Foundation Ice Core Facility |

PE | Permutation Entropy |

WPE | Weighted Permutation Entropy |

WAIS | West Antarctic Ice Sheet |

WDC | WAIS Divide Ice Core |

## References and Notes

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**MDPI and ACS Style**

Garland, J.; Jones, T.R.; Neuder, M.; Morris, V.; White, J.W.C.; Bradley, E.
Anomaly Detection in Paleoclimate Records Using Permutation Entropy. *Entropy* **2018**, *20*, 931.
https://doi.org/10.3390/e20120931

**AMA Style**

Garland J, Jones TR, Neuder M, Morris V, White JWC, Bradley E.
Anomaly Detection in Paleoclimate Records Using Permutation Entropy. *Entropy*. 2018; 20(12):931.
https://doi.org/10.3390/e20120931

**Chicago/Turabian Style**

Garland, Joshua, Tyler R. Jones, Michael Neuder, Valerie Morris, James W. C. White, and Elizabeth Bradley.
2018. "Anomaly Detection in Paleoclimate Records Using Permutation Entropy" *Entropy* 20, no. 12: 931.
https://doi.org/10.3390/e20120931