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.

Terminology and notation

1. Feature selection in Katrutsa A.M., Strijov V.V. 2017. Comprehensive study of feature selection methods to solve multicollinearity problem according to evaluation criteria // Expert Systems with Applications DOI
2. Tensor decomposition in in Motrenko A.P., Strijov V.V. 2018. Multi-way feature selection for ECoG-based brain-computer interface // Expert Systems with Applications DOI
3. Signal decoding in R.V.IsachenkoV.V.Strijov. 2022. Quadratic programming feature selection for multicorrelated signal decoding with partial least squares // Expert Systems with Applications DOI
4. Forecasting schedule and horizon in Uvarov N.D. et al. 2018. Selecting the Superpositioning of Models // Computational Mathematics and Cybernetics DOI

Topics

\\Part I

• Energy forecasting example
• Regression
• Linear model
• Model selection call
• Forecasting protocol
• Error functions
• Singular spectrum analysis
• SSA forecasting
• Forecasting protocols and verification (before AR)
• Autoregression
• Singular values decomposition (PCA, AE, Kar-Lo)
• QPFS model selection
• Auto, cross-correlation, cointegration
• Diagrams for ML and PLS
• Projection to latent space and relation to PCA, canonical-correlation analysis
• PLS-QPFS model selection
• Higher-order SSA
• Tensor decomposition
• Tensor model selection
• HOPLS
• Granger causality test
• Convergent cross mapping
• HOCCM to invent
• Taken’s theorem
• ResNet, Neural ODE
• Adjoint and back-propagation
• Flows and forecasting
• Space state models
• S4, Hippo, SaShiMi models
• (if room) Neural PDE, Lagrangian, Hamiltonian nns.

\\ To include

• RNN, LSTM, attention, transformer models
• Directional regression
• Harmonic functions
• Phase extraction
• Non-parametric regression and customer demands forecasting
• Graph earth prediction

\\ Part II

• Convolutional models
• Graph convolutions and spectrum
• Fourier transform and phase retrieval problem
• Radon transform and tomography reconstruction
• 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
• Statistics on Riemannian spaces
• Statistics on stratified spaces

General references

• Kolmogorov, A.N and Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis, Dover Publications, 1999.
• David Bachman: A Geometric Approach to Differential Forms, Birkhauser Boston, 2006.
• At the Interface of Algebra and Statistics by Tai-Danae Bradley, 2020

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