Efficient Tensor Implementation of Vector Autoregression (VAR) Model in JAX

This project provides an efficient implementation of the Vector Autoregression (VAR) model using tensor decomposition techniques. The VAR model is widely used in time series analysis to capture linear interdependencies among multiple time series.

The VAR model of order $P \geq 1$ is defined as:

$$ y_t = A_1 y_{t-1} + \ldots + A_P y_{t-P} + \varepsilon_t $$

where:

• $y_t \in \mathbb{R}^N$ are observed time series,
• $\varepsilon_t$ are $N$-dimensional innovations (centered, finite variance, i.i.d.),
• $A_t$ are $N \times N$ transition matrices to be estimated.

However, in a high-dimensional setting, whether it’s in $N$ or in $P$, the number of interest variables grows like $N^2P$, thus making the regression procedure both computationally and memory-expensive. Wang et al. (2020) introduce a rearrangement of the VAR model which leverages a tensor decomposition technique to cast the regression problem into a lower dimensional space, critically depending on the structure of the $A_i$, i.e., their sparsity and their spanned vector spaces.

Traditional estimation of VAR models can be computationally expensive in terms of both time and memory. This package addresses these challenges by leveraging tensor decomposition methods, as described in Wang et al. (2020).

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Why is this useful?

Vector Autoregression (VAR) models are essential tools for analyzing multivariate time series data. They are widely used in various industries, including:

Finance:

• Portfolio Optimization: Model the relationships between asset returns to optimize investment strategies.
• Risk Management: Forecast financial risks by analyzing interdependencies between economic indicators.
• Macroeconomic Forecasting: Predict economic variables such as GDP, inflation, and interest rates.

Pharmaceuticals:

• Drug Interaction Analysis: Model the relationships between different biological markers over time.
• Clinical Trial Forecasting: Predict patient outcomes based on longitudinal data.

Technology:

• Sensor Data Analysis: Analyze interdependencies in data from IoT devices or sensors.
• User Behavior Modeling: Predict user engagement or churn based on historical data.

Supply Chain and Logistics:

• Demand Forecasting: Model the relationships between supply chain variables to optimize inventory levels.
• Delivery Route Optimization: Predict delivery times and optimize logistics networks.

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What’s Included?

This package provides the following features:

• Alternating Least Squares (ALS) Algorithm: For VAR estimation via tensor decomposition.
• SHORR Algorithm: Lasso-penalized regression for VAR estimation via tensor decomposition.
• Higher-Order Singular Value Decomposition (HOSVD): For efficient tensor decomposition.
• Sparse and Orthogonal Regression Subroutines: For improved model interpretability.
• Sampling Procedures: Tools to generate synthetic VAR data for testing and validation.
• Tensor Algebra Utilities: Helper functions for tensor operations.

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How to Use This Package

Installation:

This package requires jax and numpy. You can install it using:

pip install -e .

References

• Di Wang, Yao Zheng, Heng Lian, and Guodong Li,
  "High-dimensional vector autoregressive time series modeling via tensor decomposition",
  https://arxiv.org/abs/1909.06624

• Rongjie Lai and Stanley Osher,
  "A splitting method for orthogonality-constrained problems",
  Journal of Scientific Computing, 2014.
  https://doi.org/10.1007/s10915-013-9740-x

For a complete list of references, see ref.bib.