# Mathematical forecasting

## Contents

## Motivation

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.

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.

## Lectures

### Main topics

- Autoregression and singular structure analysis
- Tensor decomposition and spatial-time models
- Signal decoding and multi-modeling
- Space alignment
- Convergent cross-mapping and dynamic systems
- Continuous-time forecasting and Neural ODEs

### Fall semester

**Introduction**- Semester overview, motivation, homework labs, exams
- Time and space in forecasting application problems
- Linear, neural, and memory forecasting models

**Phase space approximate**- Singular spectrum analysis and forecasting
- k-linear forms, Principal component analysis
- Singular values decomposition

**Basic models**- Cross-correlation
- Stochastic processes, autoregression, GARCH
- Non-parametric regression and kernels
- Error functions, residue convolution model, and analysis

**Fourier transform**- Discrete transforms, wavelet transform
- Gabor transform and spectrogram
- 2d transform, Gerchberg–Saxton algorithm

**Higher-order linear models**- Tensors and Penrose notation
- Tucker decomposition and alternated least squares
- Higher-order singular values decomposition

**Neural models**- Convolutions for time and space
- Recursive, Hopefield, and Memory models
- Sequential models with attention

**Canonical correlation analysis**- Projection to latent space
- PLS as SVD, model optimization, and selection
- Higher-order PLS

**Time and space alignment**- Dynamic time warping
- Dynamic barycenter averaging
- Self-modeling regression

**Causality detection**- Granger test
- Convergent cross-mapping
- Dynamic system and Taken's theorem

**Differential models**- Residual neural networks
- Neuro-ODE and its solution
- Splines, Controlled neuro-ODE

**State-space representation**- Linear differential models
- Partial differential models
- Memory models

**Forecasting and control**- Control models
- Controllability and feedback
- Proportional integral derivative controller

## Lab works

**Current labworks, October 2022, is here**
Lab work contains a report in the pynb or TeX format and a talk with a discussion

- Title and motivated abstract
- Problem statement
- Model, problem solution
- Code, analysis, and illustrative plots
- References

Note: **the model** is the **personal** contribution. The infrastructure: data acquisition, data uploads, error functions, and plots are welcome to be created **collectively** and shared.

### Topics of the lab works (Fall)

- Autoregressive forecasting – Singular structure Analysis
- Spatial-time forecasting – Tensor decomposition
- Signal decoding – Projection to latent space
- Continuous-time forecasting – Neural differential equations

### Example of the lab report

- Put here

### Format of lab works

- Create a .pynb or .py file Surname2022Lab in the folder
- The report also could be in the .tex file.
- Find the format of your report above.
- The computational experiment contains common part and individual part.
- Common part:
- use four short sample set [airplane], [electricity], [accelerometer hand motion], [video hand motion],
- prepare the design matrix and target a scalar/vector for each time sample (in the form time, vecx, vecy),
- set the forecast horizon, plot the forecast and estimate the error.

- Individual part:
- select a lab work and specify your model (you can adopt any code available for),
- tune parameters, make your forecast according the horizon,
- write the report.

- Error analysis is a part of the report:
- plot of the forecast,
- MAPE error (and your optimization error, if available) and its standard deviation,
**prove your model has the optimal structure**, try various structure parameters.

Details:

- time refers to each sample (in unix or any useful format),
- the horizon is an expected fundamental period,
**note that**the historical time ends before the forecasting period, it means we could use either historical data or the forecasted data (the historical data are not updated after history ends),- the forecasting protocol is in parer, text, slides by Nikita Uvarov.

Examples:

- Old format of the report
- Code and project
- Previous project from Sourceforge.net

## Discussion and collaboration

## 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.

## Terminology and notation

*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*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*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*Forecasting schedule and horizon*in Uvarov N.D. et al. 2018. Selecting the Superpositioning of Models // Computational Mathematics and Cybernetics DOI

## Topics

### Fall

- 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

### Spring

- Space state models
- S4, Hippo, SaShiMi models
- RNN, LSTM, attention, transformer models
- Neural PDE, Lagrangian, Hamiltonian nns.
- Directional regression
- Harmonic functions
- Phase extraction
- Non-parametric regression and customer demands forecasting
- Graph earth prediction
- 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

Appendix to Spring

- Probabilistic diffusion and Graphs
- Graph convolution, graph representation
- Neural diffusion and PDEs, GRAND
- Tensors and Ricci flow, and PDE
- Remmannian, Ricci tensors
- Differential forms
- Metrics learning and SDP
- Taken's ODE

## Catch-up 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
- Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators by Tailen Hsiing, Randall Eubank, 2013