# Probabilistic Reduced-Dimensional Vector Autoregressive Modeling with Oblique Projections

arXiv: https://arxiv.org/abs/2401.07206 · HTML: https://mlsanity.com/paper/2401.07206/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 52 references that are in the map.
- First posted on arXiv 14 Jan 2024.

## Builds on (most cited first, 12 of 12)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [AR Identification of Latent-variable Graphical Models](https://mlsanity.com/paper/1405.0027/) (2014)
- [Prediction error identification of linear dynamic networks with rank-reduced noise](https://mlsanity.com/paper/1711.06369/) (2017)
- [Sparse plus Low rank Network Identification: A Nonparametric Approach](https://mlsanity.com/paper/1510.02961/) (2015)
- [Identification of Low Rank Vector Processes](https://mlsanity.com/paper/2111.10899/) (2021)
- [Forecasting Multiple Time Series with One-Sided Dynamic Principal Components](https://mlsanity.com/paper/1708.04705/) (2017)
- [Factor Models with Real Data: a Robust Estimation of the Number of Factors](https://mlsanity.com/paper/1709.01168/) (2017)
- [The Existence and Uniqueness of Solutions for Kernel-Based System Identification](https://mlsanity.com/paper/2204.08092/) (2022)
- [A Structural-Factor Approach to Modeling High-Dimensional Time Series and Space-Time Data](https://mlsanity.com/paper/1808.06518/) (2018)
- [Dynamic probabilistic predictable feature analysis for multivariate temporal process monitoring](https://mlsanity.com/paper/2109.14666/) (2021)
- [A Generalized Probabilistic Monitoring Model with Both Random and Sequential Data](https://mlsanity.com/paper/2206.13437/) (2022)
- [Probabilistic Reduced-Dimensional Vector Autoregressive Modeling for Dynamics Prediction and Reconstruction with Oblique Projections](https://mlsanity.com/paper/2309.01161/) (2023)

## Abstract (arXiv)

In this paper, we propose a probabilistic reduced-dimensional vector autoregressive (PredVAR) model to extract low-dimensional dynamics from high-dimensional noisy data. The model utilizes an oblique projection to partition the measurement space into a subspace that accommodates the reduced-dimensional dynamics and a complementary static subspace. An optimal oblique decomposition is derived for the best predictability regarding prediction error covariance. Building on this, we develop an iterative PredVAR algorithm using maximum likelihood and the expectation-maximization (EM) framework. This algorithm alternately updates the estimates of the latent dynamics and optimal oblique projection, yielding dynamic latent variables with rank-ordered predictability and an explicit latent VAR model that is consistent with the outer projection model. The superior performance and efficiency of the proposed approach are demonstrated using data sets from a synthesized Lorenz system and an industrial process from Eastman Chemical.