Overview

The SOCR group at the University of Michigan is developing a novel theoretical foundation to extend the notion of time to the complex plane. This approach lifts the concept of time from a positive real number representing event ordering to a 2D complex-time (kime) comprising a pair of coordinates - time (t) and phase (φ). This approach enables powerful data-driven analytical strategies for large longitudinal data. The 5D spacekime analytics utilize information measures, entropy, KL divergence, PDEs, Dirac’s bra-ket operators, and the Fourier transform. This data science fundamentals research project explores time-complexity and inferential uncertainty in modeling, analyzing, and interpreting large, heterogeneous, multi-source, multi-scale, incomplete, incongruent, and longitudinal data.

In 4D spacetime, classical time-series are represented as real-valued functions defined over the positive real time domain. In the 5D spacekime manifold, time-series (curves) extend to kime-series, which are represented geometrically as surfaces. This animation illustrates the surface corresponding to one such kime-series at a fixed spatial location. At any given kime, i.e., for a pair of arguments kime-magnitude (t) and the kime-phase (φ), the height of the kime-surface represents the intensity of the kime-series and is coded in rainbow color. Notice the 2D parametric kime grid superimposed on the surface of the kime-series. There are kime-phase aggregating operators that can be used to transform standard time-series curves to spacekime kime-surfaces. These kimesurfaces that can be modeled, interpreted, analyzed, and predicted using advanced spacekime analytics.
The Spacekime Community Site has a blog tracking some of the spacekime activities and includes generative AI textual and imaging interpretations of complex time and spacekime.

Kime-series (kimesurfaces)

In many disciplines like music, economics, biomedicine, and health sciences, the flow of data and information is time-dependent. This includes the longitudinal progression of events, frequencies, motion, states, etc. In the 5D spacekime manifold, these time-courses, temporal events, or time-series, are actually complex-time surfaces (kimesurfaces) whose realizations in the lower-dimensional 4D spacetime are observed simply as curves over time. The gif animations above illustrate the relation between the commonly observed longitudinal time-series and their reconstructed kimesurface counterparts. Note that observable time-series are just intersections of the more general kimesurfaces with 2D leaf foliation manifolds, e.g., planes. Specifically, the third animation shows an idealized (over-simplified and smooth) version of a complex-valued and kime-indexed fMRI at one spatial location. Note that the domain is indexed by the kime magnitude (time) and phase (direction), and the amplitude and the phase of the fMRI intensity are encoded respectively by the kimesurface height (z-axis) and its color (RGB).

For a more concrete example, imagine a symphony performance in a concert hall. At roughly the same time and the same spatial location (ignore periphery seats and bleacher rows for now), all attendees (observers) receive the same longitudinal input (e.g., temporal audio-visual-tactile stimuli). However, at the end of the performance, all attendees leave the concert hall with relatively different experiences, distinct impressions, and unique opinions about the adventure. At each instance of time, all observers share the same spacetime location, but their differential experiences correspond to random sampling through the kime-phase space (Φ), which explains their uniquely individual encounters. After the performance concludes, communication and stratification of the individual experiences lead to review(s) reflecting the “overall consensus” capturing the joint experience, performance quality, impact, and value of the encounter. In other words, kime-averaging of all impressions gives us a yardstick measure of overall performance quality that can be used to compare, contrast, or calibrate other similar past or future performances.