Review the basics of ODEs/PDEs, the Kaluza-Klein Theory, and the DSPA materials.

Outline: The SOCR Data Science Fundamentals project explores new theoretical representations and analytical strategies to understand large and complex data. It utilizes information measures, entropy, KL divergence, PDEs, and Dirac’s bra-ket operators. This fundamentals of data science research project employs time-complexity and inferential uncertainty for representation, modeling, analysis and interpretation of large, heterogeneous, multi-source, multi-scale, incomplete, incongruent, and longitudinal data.

See The Enigmatic Kime: Time Complexity in Data Science Video, a recording at the University of Michigan Institute for Data Science (MIDAS) Seminar Series, a PDF Slidedeck is available here.

Mission and Objectives

Internet of Things (IoT)

Defining Characteristics of Big Datasets

High-dimensional Data

Scientific Inference and Forecasting

Data science

Artificial Intelligence

Examples of Driving Motivational Challenges

Neuroimaging-genetics

Census-like Population Studies

4D Nucleome

Climate Change

Problems of Time

Definition of Kime and Kime-phases Circular distribution plots

The Non-Euclidean Kime Manifold

Economic Forecasting via Spacekime Analytics

Wavefunctions

Dirac bra-ket notation

Operators

Commutator

Non-Trivial commutator (position/momentum)

Trivial commutator (energy/momentum)

Introduction

Wavefunctions and the Fourier Transformation

Fourier Amplitudes and Phases

Phase Equivalence

Amplitude Equivalence

Effects of the Fourier Transform on Phases and Magnitudes

Minkowski spacetime

Events

Kaluza-Klein Theory

Coordinate Transformations, Covariance, Contravariance, and Invariance

Kime, Kevents and the Spacekime Metric

Some Problems of Time

Kime-solutions to Time-problems

The Kime-Phase Problem

Common use of Time

Rate of change

Velocity

Newton’s equations of motion

Position (x) and Momentum (p)

Wavefunctions

Schrödinger equation

Wave equation

Lorentz transformation

Euler–Lagrange equation

Wheeler-DeWitt equation

Analogous Kime Extensions

Rate of change

Kime motion equations

Kime-dynamics (kynamics)

Lorentz transformation in spacekime

Properties of the general spacekime transformations

Backwards motion in the two-dimensional kime

Rotations in kime and space hyperplanes

Velocity-addition law

Generalization of the principle of invariance of the speed of light

Heisenberg's Uncertainty Principle

5D spacekime manifold Waves and the Doppler effect

Kime calculus of differentiation and integration

The Copenhagen vs. Spacekime Interpretations

Space-Kime Formalism

Antiparticle in spacekime

The causal structure of spacekime

Distributional Derivatives

Reproducing Kernel Hilbert Spaces (RKHS) and Temporal Distribution Dynamics (TDD)

Radon-Nikodym Derivatives, Kimemeasures, and Kime Operator

Kime Applications in Data Science

Kime Philosophy

General Formulation of fMRI inference

Likelihood based inference

Magnitude-only fMRI intensity inference

Complex-valued fMRI time-series inference

Complex-valued Kime-indexed fMRI kintensity inference

Interactive kime-surface parametric plot

Kime-series/kime-surfaces reconstruction, visualization, and predictive analytics

Laplace-transformation duality between longitudinal data (time-series) and their spacekime counterparts (kimesurface models)

Synergies between the Laplace Transform and Meijer-G Functions

Observables (Datasets)

Inference Function

Inner Product

Eigenspectra (Eigenvalues and Eigenfunctions)

Uncertainty in 5D Spacekime

Fundamental Law of Data Science Inference

Superposition Principle

Terminology

Spacetime IID vs. Spacekime Sampling

Bayesian Formulation of Spacekime Analytics

Uncertainty in Data Science

Quantum Mechanics Formulation

Statistics Formulation

Decision Science Formulation

Information Theoretic Formulation

Data Science Formulation

TCIU/Spacekime Analytics Tutorial: Basic TCIU Protocol for Predictive Spacekime Analytics using Longitudinal Data

Kime-Phases Circular distribution plots

Image Processing

fMRI Data

Exogenous Feature Time-series analysis

Structured Big Data Analytics Case-Study (UKBB)

Spacekime Analytics of Financial Market and Economics Data and 3D Scene of the longitudinal EU Econometrics Indicators

UKBB Case-study

fMRI Data

Air-Quality Data

The authors are profoundly indebted to all of their mentors, advisors, and collaborators for inspiring the study, guiding the courses of their careers, nurturing their curiosity, and providing constructive and critical feedback. Among these scholars are Gencho Skordev (Sofia University), colleagues at Burgas Technical University), Kenneth Kuttler (Michigan Tech University, De Witt L. Sumners and Fred Huffer (Florida State University), Jan de Leeuw, Nicolas Christou, and Michael Mega (UCLA), Arthur Toga (USC), Brian Athey, Kathleen Potempa, Janet Larson, Patricia Hurn, Gilbert Omenn, and Eric Michielssen (University of Michigan).

Many other colleagues, students, researchers, and fellows have shared their expertise, creativity, valuable time, and critical assessment for generating, validating, and enhancing these open-science resources. Among these are Yufei Yang, Yuming Sun, Lingcong Xu, Simeone Marino, Yi Zhao, Nina Zhou, Alexandr Kalinin, Syed Husain, and many others. In addition, colleagues from the Statistics Online Computational Resource (SOCR) and the Michigan Institute for Data Science (MIDAS) provided encouragement and valuable suggestions.

Special thanks to Yueyang Shen, Yuxin Wang, Zijing Li, Yongkai Qiu, Daxuan Deng, Yufei Yang, Zhe Yin, Jinwen Cao, Rongqian Zhang, Yunjie Guo, Yupeng Zhang, and Yuyao Liu for their substantial efforts in developing, packaging, documenting, and validating the TCIU R source code, and proofreading the material.

These research and development efforts were partially supported by the US National Science Foundation (grants 1916425, 1734853, 1636840, 1416953, 0716055 and 1023115), US National Institutes of Health (grants UL1TR002240, R01CA233487, R01MH121079, R01MH126137, T32GM141746), the Burgas University “Prof. Dr. A. Zlatarov”, and the University of Michigan.