#
Nonparametric Regression Estimation for Circular Data^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Kernel-Type Estimators

## 4. Simulation Study

- R1.
- $\mathsf{\Theta}=(\mathrm{atan}2(6{X}_{1}^{5}-2{X}_{1}^{3}-1,-2{X}_{2}^{5}-3{X}_{2}-1)+\epsilon )\left(\mathrm{mod}2\pi \right)$
- R2.
- $\mathsf{\Theta}=(\mathrm{acos}({X}_{1}^{5}-1)+{\displaystyle \frac{3}{2}}\mathrm{asin}({X}_{2}^{3}-{X}_{2}+1)+\epsilon )\left(\mathrm{mod}2\pi \right)$

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

- Fisher, N.I. Statistical Analysis of Circular Data; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Mardia, K.V.; Jupp, P.E. Directional Statistics; John Wiley & Sons: New York, NY, USA, 2009. [Google Scholar]
- Di Marzio, M.; Panzera, A.; Taylor, C.C. Non-parametric Regression for Circular Responses. Scand. J. Stat.
**2013**, 40, 142–149. [Google Scholar] [CrossRef]

**Table 1.**Means of errors $\frac{1}{n}{\sum}_{i=1}^{n}\{1-cos[m\left({\mathbf{X}}_{i}\right)-\widehat{m}\left({\mathbf{X}}_{i}\right)]\}$ over 500 simulations, for models R1 and R2, using Nadaraya-Watson and local linear fits.

Model R1 | Model R2 | ||||||
---|---|---|---|---|---|---|---|

$\mathit{\kappa}$ | $\mathit{\kappa}$ | ||||||

Estimator | n | 5 | 10 | 15 | 5 | 10 | 15 |

Nadaraya−Watson | 64 | 0.0226 | 0.0121 | 0.0089 | 0.0367 | 0.0213 | 0.0165 |

100 | 0.0171 | 0.0108 | 0.0080 | 0.0388 | 0.0024 | 0.0152 | |

225 | 0.0058 | 0.0049 | 0.0038 | 0.0185 | 0.0125 | 0.0108 | |

400 | 0.0056 | 0.0035 | 0.0026 | 0.0129 | 0.0080 | 0.0062 | |

$\mathrm{Local}\phantom{\rule{4.pt}{0ex}}\mathrm{linear}$ | 64 | 0.0234 | 0.0125 | 0.0089 | 0.0283 | 0.0144 | 0.0107 |

100 | 0.0165 | 0.0086 | 0.0061 | 0.0209 | 0.0013 | 0.0083 | |

225 | 0.0050 | 0.0039 | 0.0029 | 0.0103 | 0.0061 | 0.0047 | |

400 | 0.0050 | 0.0026 | 0.0018 | 0.0074 | 0.0043 | 0.0033 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Meilán-Vila, A.; Francisco-Fernández, M.; Crujeiras, R.M.; Panzera, A.
Nonparametric Regression Estimation for Circular Data. *Proceedings* **2019**, *21*, 27.
https://doi.org/10.3390/proceedings2019021027

**AMA Style**

Meilán-Vila A, Francisco-Fernández M, Crujeiras RM, Panzera A.
Nonparametric Regression Estimation for Circular Data. *Proceedings*. 2019; 21(1):27.
https://doi.org/10.3390/proceedings2019021027

**Chicago/Turabian Style**

Meilán-Vila, Andrea, Mario Francisco-Fernández, Rosa M. Crujeiras, and Agnese Panzera.
2019. "Nonparametric Regression Estimation for Circular Data" *Proceedings* 21, no. 1: 27.
https://doi.org/10.3390/proceedings2019021027