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Robust measures of dispersion for circular data with an anomaly detection rule

Houyem Demni, Mia Hubert, Giovanni C. Porzio, Peter J. Rousseeuw

TL;DR

A robust circular anomaly detection procedure is developed, and employed to visualize outliers through a circular violin plot, and robust estimators of parameters of circular distributions can be derived.

Abstract

Circular variables that represent directions or periodic observations arise in many fields, such as biology and environmental sciences. An important issue when dealing with circular data is how to estimate their dispersion robustly, avoiding undue effects of anomalies. This work extends three robust dispersion measures from the line to the circle. Their robustness is studied via their influence functions and relative bias curves. From these dispersion measures, robust estimators of parameters of circular distributions can be derived. This yields robust estimators for the concentration parameter of the von Mises distribution and the dispersion parameter of the wrapped normal distribution. Their breakdown values and statistical efficiencies are obtained, and they are compared in a simulation study. Building on the best performing estimator, a robust circular anomaly detection procedure is developed, and employed to visualize outliers through a circular violin plot. Three real datasets are analyzed.

Robust measures of dispersion for circular data with an anomaly detection rule

TL;DR

A robust circular anomaly detection procedure is developed, and employed to visualize outliers through a circular violin plot, and robust estimators of parameters of circular distributions can be derived.

Abstract

Circular variables that represent directions or periodic observations arise in many fields, such as biology and environmental sciences. An important issue when dealing with circular data is how to estimate their dispersion robustly, avoiding undue effects of anomalies. This work extends three robust dispersion measures from the line to the circle. Their robustness is studied via their influence functions and relative bias curves. From these dispersion measures, robust estimators of parameters of circular distributions can be derived. This yields robust estimators for the concentration parameter of the von Mises distribution and the dispersion parameter of the wrapped normal distribution. Their breakdown values and statistical efficiencies are obtained, and they are compared in a simulation study. Building on the best performing estimator, a robust circular anomaly detection procedure is developed, and employed to visualize outliers through a circular violin plot. Three real datasets are analyzed.
Paper Structure (24 sections, 24 theorems, 120 equations, 19 figures)

This paper contains 24 sections, 24 theorems, 120 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Background material
  3. Location and dispersion of circular random variables
  4. The von Mises and wrapped normal distributions
  5. Three measures of circular dispersion
  6. The Circular Median Absolute Deviation (CMAD)
  7. The Circular Least Median Spread (CLMS)
  8. The Circular Least Trimmed Standard deviation (CLTS)
  9. Values, infimum, and supremum at von Mises and wrapped normal
  10. Influence functions
  11. Bias curves
  12. Robust estimation of parameters of circular distributions
  13. Robust consistent estimators of kappa and sigma
  14. Breakdown values of CMAD, CLMS, and CLTS
  15. Efficiencies of psi hat CMAD, psi hat CLMS, psi hat CLTS
  16. ...and 9 more sections

Key Result

Lemma 1

Let $\Theta \in {\mathcal{M}}_1$ be a circular random variable, and $\theta \in [-\pi, \pi)$.

Figures (19)

  • Figure 1: Circular plot of the directions traveled by Sardinian sea stars. Two unusual observations are colored red.
  • Figure 2: Influence function of $\textnormal{CMAD}$ and $\textnormal{CLMS}$ at the von Mises distribution with $\mu=0$ and $\kappa = 1/2$.
  • Figure 3: Influence function of $\textnormal{CLTS}$ at the von Mises distribution with $\mu=0$ and $\kappa = 1/2$.
  • Figure 4: Relative bias curves of $\textnormal{CMAD}$, $\textnormal{CLMS}$, $\textnormal{CLTS}$ and $\textnormal{CSD}$ at von Mises distributions with $\mu=0$. The rows are for concentration $\kappa = 2,5$. The columns have contamination rate $\varepsilon= 5\%, 10\%, 20\%$.
  • Figure 5: Values of the $\textnormal{CMAD}$, $\textnormal{CLMS}$, $\textnormal{CLTS}$ as a function of the population $\textnormal{CSD}$ at the von Mises distribution (left) with $0 < \kappa \leqslant 10$, and at the wrapped normal distribution with $0 < \sigma \leqslant 3$. Dashed lines correspond to the infimum and to the supremum of these circular measures.
  • ...and 14 more figures

Theorems & Definitions (33)

  • Definition 1: Circular Median Set
  • Definition 2: Circular Variance
  • Definition 3: CMAD
  • Lemma 1: Values of $\textnormal{CMAD}$
  • Theorem 1
  • Theorem 2: Value of $\textnormal{CMAD}$ under reflectional symmetry
  • Definition 4: CLMS
  • Lemma 2: Conditions for $\textnormal{CLMS}=0$
  • Lemma 3: Maximum of $\textnormal{CLMS}$
  • Theorem 3
  • ...and 23 more