Mathematical forecasting

Current version, October 2022, is here

Main topics

1. Autoregression and singular structure analysis
2. Tensor decomposition and spatial-time models
3. Signal decoding and multi-modeling
4. Continuous-time forecasting and Neural ODEs
5. Convergent cross mapping and dynamic systems
6. Space alignment
7. Riemann, Metrics learning
8. Diffusion-graph PDEs

Lab works

Lab work contains a report in the pynb or TeX format and a talk with a discussion

1. Title and motivated abstract
2. Problem statement
3. Model, problem solution
4. Code, analysis, and illustrative plots
5. References

Note: the model (and sometimes data) are personal contribution. The rest – infrastructure, error functions, and plots, are welcome to be created collectively and shared.

Topics of the lab works are

• Autoregressive forecasting (Singular Structure Analysis)
• Spatial-time forecasting (Tensor Decomposition)
• Signal decoding (Projection to Latent Space)
• Continuous-time forecasting (Neural Differential Equations)

Discussion and collaboration

• Telegram
• GitHub

Exam and grading

Four lab works within deadlines and the exam on topics with problems and discussion. Each lab gives 2pt, and the exam gives 2pt, so 2*4+2=10.

Motivation

Mathematical forecasting methods play a crucial role in scientific research and industry. The distinction between forecasting and machine learning methods lies in the algebraic structures. We build forecasting models not only in vector spaces but in vector fields. These fields include time and space and have continuous nature. We propose a holistic approach to teaching this course: we must consider mathematical methods that combine continuous-time high-dimensional vector and tensor fields. We discuss linear, differential, and non-linear models. We introduce model ensembles to reveal both the source and the target space dependencies.

Abstract

This course delivers methods of model selection in machine learning and forecasting. The models are linear, tensor, deep neural networks, and neural differential equations. The modeling data are videos, audios, encephalograms, fMRIs, and other measurements in natural science. The practical examples are brain-computer interfaces, weather forecasting and various spatial-time series forecasting. The lab works are organized as paper-with-code reports.

The course joins two parts of the problem statements in Machine Learning. The first part comes from the structure of the measured data. The data come from Physics, Chemistry, and Biology and have intrinsic algebraic structures. These structures are parts of the theory that stands behind the measurement. The second part comes from errors in the measurement. The stochastic nature of errors requires statistical methods of analysis. So this course joins algebra and statistics. It is devoted to the problem of predictive model selection.

Schedule

Date		N	Subject	Link
September	3	1	Probabilistic models	Slides
	10	2	Models: regression, encoders, and neural networks
	17	3	Processes: bayesian regression, generative and discriminative models
	24	4	Functional data analysis: decomposition of processes
	31	5	Spatiotemporal models
October	8	6	Convolutional models
	15	7	Talks for the fist part of lab-projects	The talk template
	22	8	Graph convolutions and spectrum
	29	9	Fourier transform and phase retrieval problem
	4	10	Radon transform and tomography reconstruction
November	12	11	Tensor decomposition and decoding problem
	19	12	Statistics on riemannian spaces
	26	13	Statistics on stratified spaces
December	3	14	Talks for the second part of lab-projects	The talk template
	4	15	Exam: problems and discussion	List of problems

Topics

• Forward and inverse problems, kernel regularisation
• Karhunen–Loeve theorem, FPCA
• Parametric and non-parametric models
• Reproductive kernel Hilbert space
• Integral operators and Mercer theorem Convolution theorem
• Graph convolution
• Manifolds and local models

L3 courses towards machine learning

• Functional analysis
• Differential geometry

References

1. Functional data analysis by James Ramsay, Bernard Silverman, 2020
2. Riemannian geometric statistics in medical image analysis. Edited by Xavier Pennec, Stefan Sommer, and Tom Fletcher, 2020
3. Manifolds, tensors and forms by Paul Renteln, 2014
4. Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators by Tailen Hsiing, Randall Eubank, 2013
5. At the Interface of Algebra and Statistics by Tai-Danae Bradley, 2020