Circularlinear

Circular-linear, often written circular–linear, refers to statistical methods for analyzing relationships between circular (angular) data and linear data. Circular data consist of angles measured modulo 360 degrees (or 2π radians) and require special treatment to respect wrap-around; linear data are real-valued observations.

There are two primary modeling directions in circular–linear analysis.

- Circular response with linear predictors: circular regression. The outcome is an angle θ that is modeled with a circular distribution (commonly von Mises) whose mean direction μ depends on the predictors, for example μ = β0 + β1X1 + ... (taken modulo 2π). Estimation is typically via maximum likelihood. A practical alternative is to regress on the sine and cosine of θ, fitting a linear model to [cos θ, sin θ] and converting back to an angle for interpretation.

- Linear response with circular predictors: linear regression with circular covariates. A standard approach is to represent a circular predictor θ by its sine and cosine components and fit Y = β0 + β1 cos θ + β2 sin θ + ε. The cyclic effect is characterized by the amplitude A = sqrt(β1^2 + β2^2) and the phase φ = atan2(β2, β1), which describe the strength and preferred direction of the circular influence.

Estimation and software commonly rely on specialized packages in statistics environments. In R, packages such as circular provide tools for circular statistics, including regression with circular data, while Python has libraries like pycircstat for related analyses.

Applications of circular–linear methods span ecology, chronobiology, neuroscience, and meteorology, where relationships involve angles or times of day in conjunction with linear measurements. Cautions include appropriate handling of wrap-around, choosing the correct model for the data, and ensuring adequate sample size for stable estimates. See also directional statistics, von Mises distribution, and circular regression.