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SOCR TCIU: Chapter 3: The Kime-Phase Problem
Kime-phase Estimation and Kime-Operators in Spacekime Representation
SOCR Team
04/23/2025
1 The effects of Kime-Magnitudes and Kime-Phases
Jointly, the amplitude spectrum (magnitudes) and the phase spectrum (phases) uniquely describe the spacetime representation of a signal. However, the importance of each of these two spectra is not equivalent. In general, the effect of the phase spectrum is more important compared to the corresponding effects of the amplitude spectrum. In other words, the magnitudes are less susceptible to noise or the accuracy of their estimations. The effects of magnitude perturbations are less critical relative to proportional changes in the phase spectrum. For instance, particularly in terms of spacetime locations where the signal is zero, the signal can be reconstructed (by the IFT) relatively accurately using incorrect magnitudes solely by using the correct phases REF. For a real valued signal \(f\), suppose the amplitude of its Fourier transform, \(FT(f)=\hat{f}\), is \(A(\omega) > 0, \forall \omega\), then: \[f(x)=IFT(\hat{f})=Re\left (\frac{1}{2\pi}\int_{R} \underbrace{A(\omega)e^{i\phi(\omega)}}_{\hat{f}(\omega)}\ e^{i\omega x}d\omega \right)= Re\left (\frac{1}{2\pi}\int_{R}A(\omega)e^{i(\phi(\omega)+\omega x)}d\omega\right) = \frac{1}{2\pi}\int_{R} {A(\omega) \cos(\phi(\omega)+\omega x)}d\omega.\]
Thus, the zeros of \(f(x)\) occur for \(\omega x+ \phi(\omega)=\pm k\frac{\pi}{2}\), \(k= 1,2,3,.\).
A solely amplitude driven reconstruction \(\left ( f_A(x)=IFT(\hat{f})=\frac{1}{2\pi}\int_{R}\underbrace{A(\omega)}_{no\ phase}\ e^{i\omega x}d\omega \right)\) would yield worse results than a solely-phase based reconstruction \(\left ( f_{\phi}(x)=IFT(\hat{f})=\frac{1}{2\pi} \int_{R}\underbrace{e^{i\phi(\omega)}}_{no\ amplitude}\ e^{i\omega x}d\omega\right )\). The latter would have a different total energy from the original signal, however, it would include some signal recognizable features as the zeroth-level curves of the original \(f\) and the phase-only reconstruction \(f_{\phi}\) signals will be preserved. This suggests that the Fourier phase of a signal is more informative than the Fourier amplitude, i.e., the magnitudes are robust to errors or perturbations.
In X-ray crystallography, crystal structures are bombarded by particles/waves, which are diffracted by the crystal to yield the observed diffraction spots or patterns. Each diffraction spot corresponds to a point in the reciprocal lattice and represents a particle wave with some specific amplitude and a relative phase. Probabilistically, as the particles (e.g., gamma-rays or photons) are reflected from the crystal, their scatter directions are proportional to the square of the wave amplitude, i.e., the square of the wave Fourier magnitude. X-rays capture these amplitudes as counts of particle directions, but miss all information about the relative phases of different diffraction patterns.
Spacekime analytics are analogous to X-ray crystallography, DNA helix modeling, and other applications, where only the Fourier magnitudes (time), i.e., power spectrum, is only observed, but not the phases (kime-directions), which need to be estimated to correctly reconstruct the intrinsic 3D object structure REF, in our case, the correct spacekime analytical inference. Clearly, signal reconstruction based solely on either the amplitudes or the phases is an ill-posed problem, i.e., there will be many alternative solutions. In practice, such signal or inference reconstructions are always application-specific, rely on some a priori knowledge on the process (or objective function), or depend an information-theoretic criteria to derive conditional solutions. Frequently, such solutions are obtained via least squares, maximum entropy criteria, maximum a posterior distributions, Bayesian estimations, or simply by approximating the unknown amplitudes or phases using prior observations, similar processes, or theoretical models.
2 Solving the Missing Kime-Phase Problem
There are many alternative solutions to the problem of estimating the unobserved kime-phases. All solutions depend on the quality of the data (e.g., noise), the signal energy (e.g., strength of association between covariates and outcomes), and the general experimental design. There can be rather large errors in the phase reconstructions, which will in turn affect the final spacekime analytic results. Most phase-problem solutions are based on the idea that having some prior knowledge about the characteristics of the experimental design (case-study phenomenon) and the desired inference (spacekime analytics). For instance, if we artificially load the energy of the case-study (e.g., by lowering the noise, increasing the SNR, or increasing the strength of the relation between explanatory and outcome variables), the phases computed from the this stronger-signal dataset will be more accurate representations than the original phase estimates. Examples of phase-problem solutions include energy modification and fitting and refinement methods.
