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Alone We Can Do So Little; Together We Cannot Be Detected^{ †}

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## Abstract

**:**

## 1. Introduction

**cl**uster

**o**ver-time

**s**tability

**e**valuation measure called CLOSE [15] that is significantly less computationally expensive. Moreover, the results are based on a much more intuitive threshold that incorporates the cluster membership of time series. We compare the obtained results with those of Tatusch et al. [16].

#### 1.1. Notation and Definitions

**Definition**

**1**

**.**A time series $T{S}_{x}={x}_{1},\dots ,{x}_{n}$ is an ordered set of n real-valued data points of any dimension. The data points are ordered chronologically by time. The order is represented by the corresponding indices of the data points.

**Definition**

**2**

**.**A subsequence ${S}_{x,[k,m]}={x}_{k},\dots ,{x}_{m}$ of a time series $T{S}_{x}$ is an ordered subset of $m-k$ real-value data points of $T{S}_{x}$ with $k<m$ and $\forall {x}_{l}\in T{S}_{x}:k<l<m:{x}_{l}\in {S}_{x,[k,m]}$.

**Definition**

**3**

**.**A data set $D=\{T{S}_{1},\dots ,T{S}_{m}\}$ is a set of m time series of the same length n and equivalent points in time.

**Definition**

**4**

**.**A cluster ${C}_{i,t}$ at time t with $i\in 1,\dots ,p$ being an unique identifier, is a set of similar data points, identified by a clustering algorithm. All clusters have distinct labels regardless of time.

**Definition**

**5**

**.**A cluster ${C}_{i,t}$ at time t with $i\in 1,\dots ,p$ being an unique identifier, is a set of similar data points identified by a clustering algorithm. All clusters have distinct labels regardless of time.

## 2. Related Work

## 3. Method

## 4. Experiments

#### 4.1. Eikon Financial Data Set

#### 4.2. Airline On-Time Performance Data Set

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DOOTS | Detecton of outliers in time series (method) |

CLOSE | cluster over-time stability evaluation (measure) |

DTW | dynamic time warping |

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**Figure 2.**(

**a**) Result of Tatusch et al. Colors: cluster memberships, red dashed boxes: outlier by distance, black dashed boxes: intuitive outliers. (

**b**) Result of our method. Colors: cluster memberships, red dashed boxes: outliers, black dashed boxes: preoutliers.

**Figure 3.**Airline on-time performance data set. (

**a**) Outlier detection result of the method of Tatusch et al. (

**b**) Result of our method.

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## Share and Cite

**MDPI and ACS Style**

Korlakov, S.; Klassen, G.; Bravidor, M.; Conrad, S.
Alone We Can Do So Little; Together We Cannot Be Detected. *Eng. Proc.* **2022**, *18*, 3.
https://doi.org/10.3390/engproc2022018003

**AMA Style**

Korlakov S, Klassen G, Bravidor M, Conrad S.
Alone We Can Do So Little; Together We Cannot Be Detected. *Engineering Proceedings*. 2022; 18(1):3.
https://doi.org/10.3390/engproc2022018003

**Chicago/Turabian Style**

Korlakov, Sergej, Gerhard Klassen, Marcus Bravidor, and Stefan Conrad.
2022. "Alone We Can Do So Little; Together We Cannot Be Detected" *Engineering Proceedings* 18, no. 1: 3.
https://doi.org/10.3390/engproc2022018003