Projected Normal Distribution

• Projected Normal Distribution is a model for circular data that projects a multivariate normal onto a unit circle or sphere, capturing both symmetric and asymmetric features.
• It employs nonlinear change-of-variable techniques and latent variable methods to estimate parameters, accommodating dependencies in circular-linear datasets.
• Its applications span meteorology, animal movement, and geology, providing interpretable insights into mean direction and data spread while addressing non-identifiability issues.

The projected normal distribution is a probabilistic model for circular (directional or angular) data, constructed by projecting a multivariate normal distribution onto the unit sphere or circle. It is widely employed in directional statistics, including modeling of wind directions, biological orientation data, geological features, and, more generally, for poly-cylindrical (mixed circular-linear) data where standard linear models fail to capture the geometry of the sample space.

1. Mathematical Formulation and Construction

The projected normal distribution arises by projecting a multivariate normal random vector onto the unit sphere or unit circle. Consider the 2-dimensional case, which is prototypical for angular variables in $\mathbb{R}^2$ or for modeling a direction on the circle:

Let $$\mathbf{Z} \sim N_2(\boldsymbol{\mu}, \boldsymbol{\Sigma})$$, where $\mathbf{Z} = (Z_1, Z_2)^\top$. Define the projected angle as: $$X = \arctan^*\left(\frac{Z_2}{Z_1}\right) \in [0, 2\pi)$$ Here, $\arctan^*(\cdot)$ denotes the quadrant-corrected arctangent. The distribution of $X$ is termed the two-dimensional projected normal distribution, often denoted $PN_2(\mu_1, \mu_2, \sigma_1^2, \rho)$, where $\boldsymbol{\mu} = (\mu_1, \mu_2)$ and

$$\boldsymbol{\Sigma} = \begin{pmatrix} \sigma_1^2 & \sigma_1 \rho \ \sigma_1 \rho & 1 \end{pmatrix}$$

The density of $X$ is induced from the joint density $f_{\mathbf{Z}}(\mathbf{z})$ via nonlinear change of variables from $(Z_1, Z_2)$ to $(R, X)$, $R=\|\mathbf{Z}\|$. The marginal density of $X$ is then obtained by integrating over the hidden $R$: $$f(x) = \int_{0}^{\infty} r\, \phi_2 (r \mathbf{w}_x | \boldsymbol{\mu}, \boldsymbol{\Sigma}) dr$$ where $\mathbf{w}_x = (\cos x, \sin x)^\top$.

For the multivariate case, the projected normal for angles $\{\Theta_i\}_{i=1}^p$ is defined analogously by projecting a $2p$-variate normal vector onto $p$ circles: $$\Theta_i = \arctan^*(W_{i2}/W_{i1}),\quad \mathbf{W}_i \in \mathbb{R}^2;\quad \mathbf{W} \sim N_{2p}(\boldsymbol{\mu}_w, \boldsymbol{\Sigma}_w)$$

This framework generalizes naturally to the $n$-sphere for data in $\mathbb{R}^{n+1}$, where the projection to the sphere is $x \mapsto x/\|x\|$.

2. Distributional Properties and Statistical Features

The projected normal distribution possesses several desirable properties for modeling circular data:

• Flexibility: Capable of representing symmetric, asymmetric, unimodal, or bimodal circular densities.
• Parameter Interpretability: For the 2D case, the parameters $\mu_1$ and $\mu_2$ relate to the mean direction (modulo scale), while $\sigma_1^2$ and $\rho$ control spread and shape.
• Dependence structure: In the multivariate setting, the covariance matrix $\boldsymbol{\Sigma}_w$ fully captures the dependencies among angular variables. Arbitrary dependence structures can be accommodated.
• Rotation/reflection invariance: The marginal distribution on the angle is invariant to rotations or reflections of the coordinate axes.
• Closure under marginalization: Any subset of the angles retains the projected normal form.

Non-identifiability arises because scaling a normal random vector does not change the projected angle. Therefore, constraints (e.g., unit variance for a specific component) are imposed to ensure identifiability in estimation procedures.

3. Projected Normal Distribution for Poly-Cylindrical and Mixed Data

The projected normal serves as the foundational building block for models involving circular-linear and poly-cylindrical data. Within this context:

• Circular-linear General Projected Normal (CL-GPN): Augments the projected normal with conditional (often Gaussian) models for associated linear variables, permitting regimes with jointly distributed circular and linear measurements. For the $k$-th regime in a Hidden Markov Model (HMM), the joint emission distribution is

$$f(x_t, y_t | \xi_{tk}=1) = \int_{0}^{\infty} N\left(y_t; \gamma_{k0} + \gamma_{k1} r_t \cos x_t + \gamma_{k2} r_t \sin x_t, \sigma_{ky}^2 \right) \phi_2(r_t \mathbf{w}_t | \boldsymbol{\mu}_k, \boldsymbol{\Sigma}_k) r_t dr_t$$

where $x_t$ is circular, $y_t$ linear, $r_t$ is the latent radius, and $w_t = (\cos x_t, \sin x_t)^\top$.

• Joint Projected Normal and Skew-Normal (JPSN) (1711.10463): Extends the projected normal for use in higher-dimensional mixed circular-linear-vector settings by coupling with a multivariate skew-normal for the linear part. The full model integrates latent variables for efficient posterior sampling and supports arbitrary dependence among all variable types.

4. Estimation and Computational Considerations

Inference for the projected normal and its joint extensions typically relies on Bayesian methods using data augmentation with latent radii and (for skew-normal extensions) skewness variables:

• Gibbs Sampling: Latent variables (radii and skewness) are introduced to retain conditionally normal likelihoods, facilitating joint updates of mean vectors and covariance matrices via conjugate priors.
• Non-identifiability resolution: Sampling is performed in an unconstrained space, with post-processing of posterior draws via linear transformations to enforce constraints (e.g., variance normalization).
• Numerical Integration: For likelihoods involving marginalized latent radii, either analytical approximations or numerical integration techniques are used when closed-form expressions are unavailable (as is generally the case for $p>1$).
• Computational Effort: For instance, fitting HMMs with circular-linear GPN emissions in long time series (e.g., $T=2000$) may require several hours but yields stable convergence when implemented with efficient MCMC (occasionally using adaptive Metropolis steps for select parameters).

5. Parameter Estimation from Projected Data

For the case of isotropic normal distributions projected onto spheres (2410.22384), intrinsic statistics can be exploited:

• Mean estimation: The projected intrinsic mean (Fréchet mean) on the sphere commutes with the projection for isotropic covariance, i.e., the mean direction estimated on the sphere coincides with the projection of the ambient-space mean.
• Variance estimation: There exists a one-to-one correspondence between the normalized variance $\lambda = \sigma^2/\|\mu\|^2$ in the ambient normal and the intrinsic covariance of the projected data; estimation is effected by inverting an explicit bijection $f$ linking the two. Only the mean direction and the variance-to-mean ratio are identifiable from sphere-projected observations; separate recovery of $\|\mu\|$ and $\sigma^2$ is not possible.

Estimation proceeds via computation of the empirical Fréchet mean and empirical intrinsic covariance, followed by numerical inversion to solve for the normalized variance. Consistency of estimators and $\sqrt{n}$ convergence rates are established under regularity conditions.

6. Applications and Extensions

The projected normal distribution is extensively employed in:

• Hidden Markov Models with circular-linear observations: As demonstrated in wind regime classification, where wind direction (circular) and speed (linear) are modeled jointly, allowing dependence between variables.
• Animal movement analysis: As in joint modeling of zebras’ step-lengths (linear) and turning angles (circular) via the JPSN distribution, with improved recovery of joint dependence structure over cylindrical (independent) models.
• Directional statistics: Applications span meteorology (wind and wave modeling), geology, structural biology, robotics, and wireless communication (e.g., spherical/antenna orientation statistics).
• Spherical data parameter recovery: The projected normal framework allows consistent recovery of underlying normal distribution parameters when only directions are observed, a scenario common in various applied domains.

7. Key Formulas and Table

Aspect	Formula / Description
Projected Normal (2D)	$X = \arctan^*\left(\frac{Z_2}{Z_1}\right)$; $$\mathbf{Z} \sim N_2(\boldsymbol{\mu}, \boldsymbol{\Sigma})$$
Marginal Joint Density	$$f(\boldsymbol{\theta}, \mathbf{r}) = \prod_{i=1}^p r_i\, \phi_{2p}(\mathbf{w}| \boldsymbol{\mu}_w, \boldsymbol{\Sigma}_w)$$
Intrinsic Covariance Link	$v = f(\lambda)$, $f$ bijective; estimate $\lambda = \sigma^2 / \|\mu\|^2$ from projected covariance
JPSN Joint Density (aug)	$$f(\boldsymbol{\theta}, \mathbf{r}, \mathbf{y}, \mathbf{d}) = 2^q\,\phi_{2p+q}(...)\prod_i r_i\, \phi_q(\mathbf{d}|0,I)$$

These methodological developments enable flexible, robust, and interpretable modeling and inference for a wide range of problems in which standard Euclidean or cylindrical distributions are inadequate due to the rich geometry of the sample spaces involved.