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Model-based clustering using a new mixture of circular regressions

Sphiwe B. Skhosana, Najmeh Nakhaei Rad

TL;DR

This work introduces a finite mixture of circular regressions to address multimodal circular responses, expanding regression modelling beyond unimodal circular-linear frameworks. By grounding the model in von-Mises distributions and deriving EM-based maximum likelihood estimation, the approach enables both parameter estimation and data-driven clustering with component-specific circular regressions. Through simulations, the authors demonstrate robust performance in parameter recovery and clustering, with accuracy improving for well-separated components and some degradation under overlap. The wind-direction application showcases practical utility, delivering interpretable clustering and improved fit over single-component models, with uncertainty quantified via bootstrap. The framework offers a versatile tool for directional data analysis with covariate effects, applicable across biology, geology, meteorology, and wind-energy studies.

Abstract

Regression models, where the response variable is circular, are common in areas such as biology, geology and meteorology. A typical model assumes that the conditional distribution of the response follows a von-Mises distribution. However, this assumption is inadequate when the response variable is multimodal. For this reason, in this paper, a finite mixture of regressions model is proposed for the case of a circular response variable and a set of circular and/or linear covariates. Mixture models are very useful when the underlying population is multimodal. Despite the prevalence of multimodality in regression modelling of circular data, the use of mixtures of regressions has received no attention in the literature. This paper aims to close this knowledge gap. To estimate the proposed model, we develop a maximum likelihood estimation procedure via the Expectation-Maximization algorithm. An extensive simulation study is used to demonstrate the practical use and performance of the proposed model and estimation procedure. In addition, the model is shown to be useful as a model-based clustering tool. Lastly, the model is applied to a real dataset from a wind farm in South Africa.

Model-based clustering using a new mixture of circular regressions

TL;DR

This work introduces a finite mixture of circular regressions to address multimodal circular responses, expanding regression modelling beyond unimodal circular-linear frameworks. By grounding the model in von-Mises distributions and deriving EM-based maximum likelihood estimation, the approach enables both parameter estimation and data-driven clustering with component-specific circular regressions. Through simulations, the authors demonstrate robust performance in parameter recovery and clustering, with accuracy improving for well-separated components and some degradation under overlap. The wind-direction application showcases practical utility, delivering interpretable clustering and improved fit over single-component models, with uncertainty quantified via bootstrap. The framework offers a versatile tool for directional data analysis with covariate effects, applicable across biology, geology, meteorology, and wind-energy studies.

Abstract

Regression models, where the response variable is circular, are common in areas such as biology, geology and meteorology. A typical model assumes that the conditional distribution of the response follows a von-Mises distribution. However, this assumption is inadequate when the response variable is multimodal. For this reason, in this paper, a finite mixture of regressions model is proposed for the case of a circular response variable and a set of circular and/or linear covariates. Mixture models are very useful when the underlying population is multimodal. Despite the prevalence of multimodality in regression modelling of circular data, the use of mixtures of regressions has received no attention in the literature. This paper aims to close this knowledge gap. To estimate the proposed model, we develop a maximum likelihood estimation procedure via the Expectation-Maximization algorithm. An extensive simulation study is used to demonstrate the practical use and performance of the proposed model and estimation procedure. In addition, the model is shown to be useful as a model-based clustering tool. Lastly, the model is applied to a real dataset from a wind farm in South Africa.
Paper Structure (26 sections, 36 equations, 4 figures, 5 tables, 2 algorithms)

This paper contains 26 sections, 36 equations, 4 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. von-Mises distribution
  4. Circular regression model
  5. Circular-Linear regression model
  6. Circular-Circular regression model
  7. Circular-Circular-Linear regression model
  8. Maximum likelihood estimation
  9. Mixtures of circular regressions
  10. MLE via the Expectation-Maximization algorithm
  11. Modelling and computational details
  12. Model-based clustering rule
  13. Choosing the number of components
  14. Computing the mean direction
  15. Initialization strategy
  16. ...and 11 more sections

Figures (4)

  • Figure 1: (a) A circular plot of the 744 hourly average wind directions, measured in radians. (b) A scatter plot of the average wind directions and average wind speeds, in metres/second. (c) A scatter plot of the average wind directions and average air temperature, in degrees Celsius. (d) A scatter plot of the average wind directions and hour of the day, in radians. The data were recorded at a wind farm in the Eastern Cape province of South Africa.
  • Figure 2: Circular histograms of a typical sample of size $n=500$ generated using \ref{['sim1_data']} for the (a) non-overlapping and (b) overlapping case.
  • Figure 3: Circular histograms of a typical sample of size $n=500$ generated using \ref{['sim2_data']} for the (a) non-overlapping and (b) overlapping case.
  • Figure 4: (a) Circular plot of wind direction and (b)-(d) scatter plots of wind direction on wind speed, air temperature and hour of the day, respectively, with each point color coded based on the component it belongs to using the model-based clustering rule. The red points are associated with component 1 (east wind direction) and the blue points are associated with component 2 (west wind direction). The solid lines give the fitted regression functions for each component.