*Article* **Angular Correlation Using Rogers-Szeg ˝o-Chaos**

#### **Christine Schmid 1,\* and Kyle J. DeMars 2**


Received: 30 December 2019; Accepted: 21 January 2020; Published: 1 February 2020

**Abstract:** Polynomial chaos expresses a probability density function (pdf) as a linear combination of basis polynomials. If the density and basis polynomials are over the same field, any set of basis polynomials can describe the pdf; however, the most logical choice of polynomials is the family that is orthogonal with respect to the pdf. This problem is well-studied over the field of real numbers and has been shown to be valid for the complex unit circle in one dimension. The current framework for circular polynomial chaos is extended to multiple angular dimensions with the inclusion of correlation terms. Uncertainty propagation of heading angle and angular velocity is investigated using polynomial chaos and compared against Monte Carlo simulation.

**Keywords:** polynomial chaos; Szeg˝o polynomials; directional statistics; Rogers-Szeg˝o; state estimation
