Mathematical forecasting

One-year plan, eight sections

1. Autoregression and singular structure analysis
2. Tensor decomposition
3. Signal decoding
4. Continuous-time forecasting
5. Convergent cross mapping
6. Alignment
7. Metrics learning
8. Diffusion-graph PDEs

This 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 structure. This stricture is part of the theory that stands behind the measurement. The second part comes from errors of the measurement. The stochastic nature errors request the statistical methods of analysis. So this course joins algebra and statistics. It is devoted to the problem of predictive model selection.

The course holds two semesters: Fall 2020 and Spring 2021. It contains lectures and practical works. Out of schedule cuts off half the score. The scoring, max:

1. Questionnaires during lectures (3)
2. Two application projects (2+2)
3. The final exam: problems with discussion (3)

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