*3.4.ComplexPolynomialChaos*

While polynomial chaos has been well-studied and applied to a various number of applications in R*<sup>n</sup>*, alterations must be made for the restricted space S*n* due to its circular nature. A linear approximation can be made with little error when a circular variable's uncertainty is small; however, as the uncertainty increases, the linearization can impose significant error. Figure 2 shows the effects of projecting two wrapped normal distributions with drastically different standard deviations onto a tangent plane. The two wrapped normal distributions are shown in Figure 2a,b, with USTDs of 0.25 and 0.4 rad, respectively. Clearly, even relatively small USTDs result in approximately uniform wrapped pdfs.

One of the most basic projections is an orthogonal projection from an *n*-dimensional space onto an *n* − 1 dimensional plane. In this case, the wrapped normal pdf is projected orthogonally onto the plane (1, *x*, *<sup>z</sup>*), which lies tangent to the unit circle at the point (1,0), coinciding with the mean direction of both pdfs. The plane, and the projection of the pdf onto this plane are shown in Figure 2c,d. Approximating the circular pdf as the projected planar pdf comes with an associated loss of information. At the tangent point, there is obviously no information loss; however, when the physical distance from the original point to the projected point is considered, the error associated with the projected point increases. As is the case with many projection methods concerning circular and spherical bodies, all none of the information from the far side of the body is available in the projection. The darkness of the shading in all of Figure 2 comes from the distance of the projection where white is no distance, and black is a distance value of least one (implying the location is on the hemisphere directly opposite the mean direction).

To better indicate the error induced by this type of projection, Figure 2e,f also include a measure that shows how far the pdf has been projected as a percentage of the overall probability at a given point. At the tangent point, there is no projection required, therefore the circular pdf has to be shifted 0% in the *x* direction. As the pdf curves away from the tangent plane, the pdf has to be projected farther. The difference between Figure 2e and Figure 2f is that the probability approaches zero nearing *y* = ±1 in Figure 2e; therefore, the effect of the error due to projection is minimal. In cases where the information is closely concentrated about one point, tangent plane projections can be good assumptions. Contrarily, in Figure 2f the pdf does not approach zero, and therefore the approximation begins to become invalid. Accordingly, the red error line approaches the actual pdf, indicating that the majority of the pdf has been significantly altered in the projection.

**Figure 2.** Error induced by approximating a circular distribution as linear, tangent at the mean. (**a**) Wrapped normal distribution with pdf *pwn*(*<sup>θ</sup>*; 0, 0.25<sup>2</sup>). (**b**) Wrapped normal distribution approaching wrapped uniform distribution with pdf *pwn*(*<sup>θ</sup>*; 0, 2.4<sup>2</sup>). (**c**) Projection of *pwn*(*<sup>θ</sup>*; 0, 0.25<sup>2</sup>) onto a plane tangent to the unit circle. (**d**) Projection of *pwn*(*<sup>θ</sup>*; 0, 2.4<sup>2</sup>) onto a plane tangent to the unit circle. (**e**) Error associated with projecting *pwn*(*<sup>θ</sup>*; 0, 0.25<sup>2</sup>) onto a plane tangent to the unit circle. (**f**) Error associated with projecting *pwn*(*<sup>θ</sup>*; 0, 2.4<sup>2</sup>) onto a plane tangent to the unit circle.

In addition to restricting the space to the unit circle, most calculations required when dealing with angles take place in the complex field. In truth, the bulk of expanding polynomial chaos to be suitable for angular random variables is generalizing it to complex vector spaces. Previous work by the authors [27] has shown that a stochastic angular random variable can be expressed using a polynomial

chaos expansion. Specifically, the chaos expansion is one that uses polynomials that are orthogonal with respect to probability measures on the complex unit circle as opposed to the real line.
