Variational Quantization for Forecasting with State Space Models

Core Concepts

A new forecasting model that combines discrete state space hidden Markov models with neural network architectures and variational training procedures to enable accurate predictions on large datasets with diverse time series.

Abstract

The paper introduces a new forecasting model that combines discrete state space hidden Markov models with recent neural network architectures and training procedures inspired by vector quantized variational autoencoders.
The key aspects of the proposed approach are:

Modeling time series as being governed by a hidden Markov process, where the hidden states determine the emission distribution of the observed time series.

Learning a collection of emission laws that can be activated depending on the dynamics of the hidden process. This allows the model to capture a rich variety of time series behaviors.

Introducing a variational discrete posterior distribution of the latent states and using a two-stage training procedure to alternatively learn the parameters of the latent states and the emission distributions.

The proposed training approach based on the Evidence Lower Bound (ELBO) avoids the computational challenges of traditional Expectation-Maximization algorithms used for hidden Markov models.

The authors evaluate the performance of the proposed method on several datasets, including a large fashion time series dataset and a collection of reference forecasting benchmarks. The results show that the model outperforms other state-of-the-art solutions, especially on non-stationary time series and when leveraging available external signals.

Stats

The fashion dataset contains 10,000 weekly time series representing the evolution of the visibility of garments on social media, along with external signals representing the visibility of the same garments on a sample of influencer users.
The reference datasets include 8 widely used time series forecasting benchmarks covering a variety of domains such as weather, traffic, electricity load, and influenza-like illness.

Quotes

"Forecasting tasks using large datasets gathering thousands of heterogeneous time series is a crucial statistical problem in numerous sectors."
"The main challenge is to model a rich variety of time series, leverage any available external signals and provide sharp predictions with statistical guarantees."
"By learning a collection of emission laws and temporarily activating them depending on the hidden process dynamics, the proposed method allows to explore large datasets and leverage available external signals."

Key Insights Distilled From

Variational quantization for state space models

by Etienne Davi... at arxiv.org 04-18-2024

https://arxiv.org/pdf/2404.11117.pdf

Deeper Inquiries

How could the proposed model be extended to handle more complex dependencies between the hidden states and the observed time series, such as non-linear relationships or long-term dependencies

To handle more complex dependencies between the hidden states and the observed time series, the proposed model could be extended in several ways:

Non-linear Relationships: Introducing non-linear activation functions in the neural networks used for modeling the emission laws can capture more complex relationships between the hidden states and the observed time series. Functions like ReLU, sigmoid, or tanh can help model non-linear patterns effectively.

Deep Neural Networks: Increasing the depth of the neural networks can allow the model to learn hierarchical representations of the data, enabling it to capture intricate dependencies. Deep architectures like deep residual networks or transformers can enhance the model's capacity to capture complex relationships.

Attention Mechanisms: Incorporating attention mechanisms in the model can help focus on relevant parts of the input sequence, allowing the model to learn long-range dependencies more effectively. Attention mechanisms have been successful in capturing complex patterns in sequential data.

Gated Recurrent Units: Using gated recurrent units (GRUs) or long short-term memory (LSTM) units can help the model capture long-term dependencies in the data. These units are designed to retain information over long sequences, making them suitable for modeling complex relationships.

By incorporating these extensions, the model can better handle non-linear relationships and long-term dependencies between the hidden states and the observed time series, enhancing its forecasting capabilities.

What are the potential limitations of the variational training approach, and how could it be further improved to ensure robust and stable convergence across different datasets and forecasting tasks

The variational training approach, while effective, has some potential limitations that could be addressed for improved robustness and stability across different datasets and forecasting tasks:

Initialization Sensitivity: Variational training can be sensitive to initialization, leading to suboptimal solutions or convergence issues. Using advanced initialization techniques like Xavier or He initialization can help mitigate this sensitivity and improve convergence.

Hyperparameter Tuning: The performance of variational methods can be influenced by hyperparameters such as learning rates, batch sizes, and regularization parameters. Conducting thorough hyperparameter tuning using techniques like grid search or Bayesian optimization can optimize the model's performance.

Regularization Techniques: Incorporating regularization techniques like dropout or weight decay can prevent overfitting and improve the generalization of the model across different datasets. Regularization helps in reducing model complexity and enhancing stability.

Ensemble Methods: Utilizing ensemble methods by training multiple instances of the model with different initializations can improve robustness and stability. Ensemble averaging can help reduce variance and enhance the model's predictive performance.

By addressing these limitations and implementing strategies to improve initialization, hyperparameter tuning, regularization, and ensemble methods, the variational training approach can be further enhanced for more robust and stable convergence in various forecasting tasks.

Given the model's ability to capture non-stationary behaviors, how could it be applied to other domains beyond time series forecasting, such as anomaly detection or change point analysis

The model's capability to capture non-stationary behaviors can be applied to domains beyond time series forecasting, such as anomaly detection or change point analysis, by leveraging its ability to detect shifts in underlying patterns. Here's how it could be applied:

Anomaly Detection: The model can be used to identify anomalies in sequential data by detecting deviations from normal patterns. By training the model on normal data sequences, it can learn to recognize unusual behaviors or outliers that deviate significantly from the learned patterns, indicating potential anomalies.

Change Point Analysis: The model can be employed to detect change points in time series data where the underlying distribution shifts. By monitoring the hidden states and their transitions, the model can identify abrupt changes in the data, signaling a change point. This can be valuable in various applications, including signal processing and quality control.

Event Detection: The model can be adapted to detect specific events or occurrences in sequential data by associating certain hidden states with particular events. By training the model on labeled event data, it can learn to recognize and predict the onset of specific events based on the observed time series data.

By applying the model's ability to capture non-stationary behaviors in these domains, it can enhance anomaly detection, change point analysis, and event detection tasks, providing valuable insights and predictive capabilities in diverse applications.