CL-GPN: Circular-Linear Projected Normal

• Circular-Linear General Projected Normal (CL-GPN) is a statistical distribution for jointly modeling directional and linear data via multivariate normal projections.
• It enables flexible modeling of dependencies, skewness, and multimodality in circular–linear datasets applied to time series, clustering, and regression.
• Bayesian estimation leverages latent augmentation and conjugate updating to efficiently overcome identifiability issues and scale constraints.

The Circular-Linear General Projected Normal (CL-GPN) distribution is a statistical model designed for the joint analysis of circular (directional) and linear data. It extends the projected normal family, allowing for the flexible modeling of dependencies, skewness, and multimodality in datasets that simultaneously display circular and linear characteristics. The CL-GPN is central to modern approaches for Bayesian inference in circular-linear time series, clustering, and regression, with rigorous treatments of identifiability and conjugate Bayesian updating schemas.

1. Mathematical Definition and Foundations

The CL-GPN is defined via the projection of a multivariate normal variable onto a product of circles and Euclidean spaces. Let $z = (x,y) \in \mathbb{R}^2 \times \mathbb{R}^q$ where $x$ is intended to be projected onto the unit circle (yielding angle $\theta$) and $y \in \mathbb{R}^q$ is linear. The latent Gaussian construction is:

$z \sim N_{2+q} (\mu, \Sigma)$

where $\mu = (\mu_x, \mu_y)$, $$\Sigma = \begin{pmatrix} \Sigma_{xx} & \Sigma_{xy} \ \Sigma_{yx} & \Sigma_{yy} \end{pmatrix}$$. Define the polar coordinate transformation:

• $r = \|x\|$
• $u = x / r = (\cos\theta, \sin\theta)^\top$, with $\theta \in [0,2\pi)$

The joint density for $x$0, incorporating the Jacobian of transformation, is:

$x$1

Integration out of $x$2 yields the marginal for $x$3. For $x$4, explicit closed-form expressions are available (Zou et al., 2022).

2. Properties, Marginalization, and Dependence Structures

The CL-GPN family includes several key properties:

• Closure under Marginalization: Any subvector of circular or linear variables (i.e., any marginal of $x$5) maintains the CL-GPN structure with appropriately reduced parameters.
• Explicit Modeling of Dependence: $x$6 (or $x$7) parameterizes the linear association between the circular and linear components. The resulting joint law can capture conditional correlations, circumventing independence assumptions frequent in earlier cylindrical models (Mastrantonio et al., 2014).
• Determinants of Modality and Skewness: Bimodality and skewness in the circular marginals arise from nonzero $x$8 (location for projected normal part) or strong off-diagonal dependencies in $x$9.
• Dependence Measures:
  • Circular-circular correlation: Jammalamadaka–Sarma, calculated as

  $\theta$0 - Circular-linear association: Mardia's correlation,

  $\theta$1 - Linear-linear: classical Pearson correlation (Mastrantonio, 2017).

3. Identifiability and Parameter Constraints

A fundamental identifiability issue is inherited from the projected normal: for any $\theta$2, scaling $\theta$3 by $\theta$4 leaves the induced $\theta$5 unchanged. Thus, $\theta$6 are only identified up to scale. Standard solutions are:

• Explicit parameter constraint: Fixing an element of $\theta$7, typically $\theta$8.
• Post-processing for Bayes inference: MCMC draws for unconstrained $\theta$9 are normalized a posteriori via diagonal matrix rescaling with $y \in \mathbb{R}^q$0, yielding identified $y \in \mathbb{R}^q$1 and $y \in \mathbb{R}^q$2 for inference and reporting (Mastrantonio, 2017).

To maintain conjugacy in Bayesian setups, the conditional inverse-Wishart (CIW) prior is employed for $y \in \mathbb{R}^q$3, with block constraints and conditional sampling updating formulas (Zou et al., 2022).

4. Bayesian Estimation and Computational Techniques

Fully Bayesian inference for the CL-GPN leverages augmentation and conjugate updates:

• Latent Augmentation: Introduction of radii $y \in \mathbb{R}^q$4 (for each sample) renders the likelihood in terms of conditionally Gaussian variables, significantly simplifying the structure of the complete-data posterior.
• Gibbs Sampling: All conditionals—means, covariance, and augmented variables—can be updated analytically (Gibbs sampler) or in a Metropolis-within-Gibbs framework depending on mixture complexity and priors.
• CIW priors: For block-constrained $y \in \mathbb{R}^q$5 matrices needed by identifiability, closed-form CIW full conditionals underpin efficient posterior sampling (Zou et al., 2022, Mastrantonio, 2017).
• Label switching: In mixture or HMM cases, label swapping can be addressed by post-hoc relabeling.
• Convergence diagnostics: Posterior draws are validated using trace-plots and Gelman–Rubin statistics, with mixing improvement via adaptive proposals and marginalization techniques for mixture parameters (Mastrantonio et al., 2014).

5. Extensions: Mixtures, Nonparametrics, and Hidden Markov Models

The CL-GPN is extensible in several directions:

• Finite and Infinite Mixture Models: The CL-GPN kernel serves as the emission distribution for mixtures of arbitrary size. Dirichlet process mixtures (DPSPN) offer flexible nonparametric Bayesian clustering of circular–linear data without pre-specifying the number of components (Zou et al., 2022).
• Poly-cylindrical (multivariate) generalizations: The joint projected normal–skew normal (JPSN) extends CL-GPN for higher-dimensional circular and linear combinations, maintaining closure properties and exploiting Gaussian augmentation for computation (Mastrantonio, 2017).
• Bayesian Hidden Markov Models: In time-series contexts, the CL-GPN can define the regime-specific emission law in a Markov switching structure, with state transitions and emission parameters estimated via Markov chain Monte Carlo. This relaxes independence assumptions often made between circular and linear elements and handles temporal clustering (Mastrantonio et al., 2014).

6. Applications and Simulation Studies

Empirical and simulation analyses validate the flexibility of the CL-GPN:

• Real Data Modeling: In ecological movement (e.g., GPS tracking of animals), turning angles (directional) and log step-lengths (linear) for multiple subjects are jointly modeled. The CL-GPN accurately recovers cross-domain dependencies, as demonstrated in the modeling of four zebras’ trajectories, capturing complex marginal and joint shapes, including bimodality and skewness in angular components (Mastrantonio, 2017).
• Wind Series Modeling: Time-series of wind direction and speed exemplify the advantage of the CL-GPN over independence assumptions, as meaningful regime-dependent dependencies between direction and speed are captured in hidden Markov formulations (Mastrantonio et al., 2014).
• Comparison with Alternatives: Predictive performance (using CRPS and other criteria) surpasses that of cylindrical models lacking circular–linear dependence, supporting the suitability of the approach for heterogeneous data sources (Mastrantonio, 2017).
• Clustering and Density Estimation: Dirichlet process CL-GPN mixtures outperform other models in flexibility and clustering accuracy for directional-linear data with complex dependencies (Zou et al., 2022).

7. Significance, Limitations, and Future Directions

The CL-GPN model represents a mathematically rigorous solution for joint modeling of circular and linear phenomena within a Gaussian-augmented framework. Its closure under marginalization, explicit dependence measurement, and adaptability to Bayesian nonparametric and hidden Markov settings make it a reference model for cylindrical data analysis.

Among the key limitations is the non-identifiability in the projected normal component, though Bayesian post-processing and parameter constraints offer fully effective remedies. Computational scaling for complex mixtures or state-switching models is addressed by analytic conjugacy and latent augmentation, but further research may focus on computational acceleration and generalization to higher compositional dimensions or other non-Euclidean supports (Mastrantonio, 2017, Zou et al., 2022).