Kime-Phase Tomography and Manifold Representation of Longitudinal Processes over a Complex-Time Domain

Variability in repeated measurement longitudinal processes is subject to intrinsic (domain space) latent phase states stochasticity and extrinsic (range space) chronological time dispersion typically modeled as additive noise. This talk introduces a complex-time (kime) representation of time-varying processes as parametric manifolds. We'll discuss kime-phase tomography (KPT), a mathematical statistics framework that extends the representation of longitudinal processes from the classical time, \( t \in \mathbb{R}^+_0 \), to kime, \( \kappa = t e^{i\theta} \). The kime phase \( \theta \in [0, 2\pi) \) tracks the intrinsic kime-phase domain stochasticity, due to cross-sectional variation in the repeated sampling. The phase is complementary to the classical extrinsic additive noise in the signal range space. We will discuss a probabilistic formulation of the kime domain \( \mathcal{M} = [0, T] \times S^1 \), equipped with different metrics, and introduce a family of non-commutative operators that satisfy canonical commutation relations analogous to quantum mechanics. This operator-theoretic foundation enables tomographic reconstruction of latent phase distributions from mixed-moment statistics via spectral deconvolution on the circle. The results establish identifiability conditions under which the phase distribution is uniquely recoverable, propose efficient expectation-maximization algorithms with \( \mathcal{O}(N \log N) \) complexity, and derive nonparametric convergence rates and information-theoretic lower bounds for estimation accuracy.