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A Unified and Computationally Efficient Non-Gaussian Statistical Modeling Framework

David Bolin, Xiaotian Jin, Alexandre B. Simas, Jonas Wallin

Abstract

Datasets that exhibit non-Gaussian characteristics are common in many fields, while the current modeling framework and available software for non-Gaussian models is limited. We introduce Linear Latent Non-Gaussian Models (LLnGMs), a unified and computationally efficient statistical modeling framework that extends a class of latent Gaussian models to allow for latent non-Gaussian processes. The framework unifies several popular models, from simple temporal models to complex spatial-temporal and multivariate models, facilitating natural non-Gaussian extensions. Computationally efficient Bayesian inference, with theoretical guarantees, is developed based on stochastic gradient descent estimation. The R package \texttt{ngme2}, which implements the framework, is presented and demonstrated through a wide range of applications including novel non-Gaussian spatial and spatio-temporal models.

A Unified and Computationally Efficient Non-Gaussian Statistical Modeling Framework

Abstract

Datasets that exhibit non-Gaussian characteristics are common in many fields, while the current modeling framework and available software for non-Gaussian models is limited. We introduce Linear Latent Non-Gaussian Models (LLnGMs), a unified and computationally efficient statistical modeling framework that extends a class of latent Gaussian models to allow for latent non-Gaussian processes. The framework unifies several popular models, from simple temporal models to complex spatial-temporal and multivariate models, facilitating natural non-Gaussian extensions. Computationally efficient Bayesian inference, with theoretical guarantees, is developed based on stochastic gradient descent estimation. The R package \texttt{ngme2}, which implements the framework, is presented and demonstrated through a wide range of applications including novel non-Gaussian spatial and spatio-temporal models.
Paper Structure (31 sections, 2 theorems, 67 equations, 7 figures, 13 tables, 1 algorithm)

This paper contains 31 sections, 2 theorems, 67 equations, 7 figures, 13 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The LLnGM Framework
  3. The model formulation
  4. Flexible but unified process models
  5. Inference and prediction
  6. Parameter Estimation
  7. Convergence Diagnostics
  8. Full Bayesian Inference
  9. Model Prediction
  10. The ngme2 Package: Implementation of the Framework
  11. Applications
  12. The Grasshopper population data
  13. Longitudinal data analysis
  14. Climate Reanalysis Data
  15. Space-time bivariate model for the Wind Data
  16. ...and 16 more sections

Key Result

Proposition 1

Under the LLnGM model eq:framework, the conditional distribution of $\mathbf{W}|\mathbf{V}, \mathbf{Y}$ is given by $\mathbf{W}|\mathbf{V}, \mathbf{Y} \sim \mathcal{N}(\boldsymbol{\mu}_W, \boldsymbol{\Sigma}_W)$, where $\boldsymbol{\Sigma}_W^{-1} = \mathbf{K}^{\top} \mathbf{D}_{\mathbf{V}^W}^{-1} \m and $\mathbf{D}_{\mathbf{V}^W} = \text{diag}(\boldsymbol{\sigma}^W \odot \boldsymbol{\sigma}^W \odo

Figures (7)

  • Figure 1: Grasshopper Population Analysis: (a) Time Series Data and (b) Rolling Window 95% Prediction Intervals from Gaussian and NIG AR(1) Models
  • Figure 2: Longitudinal eGFR Trajectories for 7 Randomly Selected Patients Showing Individual Patterns of Kidney Function Change over Follow-Up Time
  • Figure 3: Transformed Precipitation Data for the United States with the Triangulation Mesh Used for Spatial Modeling.
  • Figure 4: Wind Data over the Balkan Peninsula Region. Arrows Indicate Wind Direction, While Colors Represent Wind Speed Magnitude.
  • Figure 5: Traceplots of the MAP Estimation for the Parameters in the NIG-AR(1) Model using ngme2. The red lines are the average of 4 parallel runs. The blue lines indicate the true values.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Proposition 1: Conditional distribution of $\mathbf{W}|\mathbf{V}, \mathbf{Y}$
  • proof
  • Proposition 2: Conditional distributions of mixing variables
  • proof