Complex-time Representation of Longitudinal Processes and Topological Kime-Surface Analysis

Complex-time (kime) extends the traditional representation of temporal processes into the complex plane and captures the dynamics of both classical longitudinal time and repeated-sampling process variability. Novel approaches for analyzing longitudinal data can be developed that build on the 2D parametric manifold representations of time-varying processes repeatedly observed under controlled conditions. Longitudinal processes that are typically modeled using time series are transformed into multidimensional surfaces called kime-surfaces, which jointly encode the internal dynamics of the processes as well as sampling variability. There are alternative strategies to transform classical time-courses to kime-surfaces. The spacekime framework facilitates the application of advanced topological methods, such as persistent homology, to these kime-surfaces. Topological kime-surface analysis involves studying the topological features of kime-surfaces, such as connected components, loops, and voids, which remain invariant under continuous deformations. These topological invariants can be used to classify different types of time-varying processes, detect anomalies, and uncover hidden patterns that are not apparent in traditional time-series analysis.

New AI models can be developed to predict, classify, tesselate, and forecast the behavior of high-dimensional longitudinal data, such as functional magnetic resonance imaging (fMRI), by leveraging complex-time representation of time-varying processes and topological analysis. Kime-surfaces represent mathematically-rich and computationally-tractable data objects that can be interrogated via statistical-learning and artificial intelligence techniques. Spacekime analytics has broad applicability, ranging from personalized medicine to environmental monitoring, and statistical obfuscation of sensitive information.

complex-time, kime

spacekime analytics

AI

statistical learning

topological analysis