Phasesmoments

Phasesmoments are a set of statistics used to describe the distribution of the phase angle of a complex-valued quantity. Let Z be a complex random variable with polar form Z = R e^{i θ}, where θ is the phase in [0, 2π). The k-th phasesmoment is defined as m_k = E[e^{i k θ}] for k ∈ Z. The collection of moments {m_k} is called the phasesmoments of θ. The zeroth moment m_0 = 1 when θ is defined and the distribution is normalized; higher moments capture concentration and symmetry of the phase distribution.

These moments capture the concentration and symmetry of the phase distribution. The magnitude |m_k| indicates how strongly the phase tends to a preferred direction; the angle arg(m_k) reflects the dominant phase orientation. For a phase distribution that is invariant under rotation (uniform on [0, 2π)), m_k = 0 for all k ≠ 0, and m_0 = 1.

From samples θ_1, ..., θ_N, estimate m_k as m_k_hat = (1/N) ∑ e^{i k θ_n}. Larger k require more data to achieve stable estimates. Confidence intervals can be derived from circular statistics theory.

The phasesmoments are the Fourier coefficients of the phase distribution f_θ(θ). If f_θ is expanded as f_θ(θ) = (1/2π) ∑ m_k e^{-i k θ}, the phasesmoments encode the distribution completely (in the sense of sufficient density information) when all moments are known.

Phasesmoments appear in signal processing to analyze phase-based modulation and in neuroscience to study phase locking. They are also used in optics and communications to characterize phase noise and synchronization.

Estimating higher-order moments can be sensitive to outliers and requires large samples. For highly multimodal phase distributions, many moments may be needed to reconstruct the density. Careful handling of circular data is essential to avoid bias. See also: circular statistics, phase distribution, mean resultant length, Fourier analysis.