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Mixed moving average field guided learning for spatio-temporal data

Imma Valentina Curato, Orkun Furat, Lorenzo Proietti, Bennet Stroeh

TL;DR

This work addresses forecasting spatio-temporal data when the predictive distribution is unknown by grounding learning in influenced mixed moving average fields (MMAFs). It introduces a theory-guided ML framework, MMAF-guided learning, that uses a spatio-temporal embedding and a generalized Bayesian (randomized) estimator over Lipschitz predictors to produce ensemble, one-time ahead forecasts with a causal interpretation. The authors establish fixed-time and any-time PAC Bayesian bounds for data generated by $\theta$-lex weakly dependent MMAFs, and they derive practical embedding strategies and estimator designs (including a randomized Gibbs estimator) to optimize generalization performance. Validation on simulated spatio-temporal Ornstein-Uhlenbeck processes demonstrates that the resulting ensemble forecasts yield narrow interquartile ranges that contain the true test values and offer a transparent, causally interpretable forecasting framework for raster data cubes. The approach provides a principled path to non-vacuous generalization guarantees in dependent spatio-temporal settings and offers guidelines for selecting embeddings and hyperparameters to balance dependence structure and learning efficiency. Overall, MMAF-guided learning contributes a theoretically grounded, causality-aware methodology for interpretable spatio-temporal forecasting with explicit uncertainty quantification.

Abstract

Influenced mixed moving average fields are a versatile modeling class for spatio-temporal data. However, their predictive distribution is not generally known. Under this modeling assumption, we define a novel spatio-temporal embedding and a theory-guided machine learning approach that employs a generalized Bayesian algorithm to make ensemble forecasts. We use Lipschitz predictors and determine fixed-time and any-time PAC Bayesian bounds in the batch learning setting. Performing causal forecast is a highlight of our methodology as its potential application to data with spatial and temporal short and long-range dependence. We then test the performance of our learning methodology by using linear predictors and data sets simulated from a spatio-temporal Ornstein-Uhlenbeck process.

Mixed moving average field guided learning for spatio-temporal data

TL;DR

This work addresses forecasting spatio-temporal data when the predictive distribution is unknown by grounding learning in influenced mixed moving average fields (MMAFs). It introduces a theory-guided ML framework, MMAF-guided learning, that uses a spatio-temporal embedding and a generalized Bayesian (randomized) estimator over Lipschitz predictors to produce ensemble, one-time ahead forecasts with a causal interpretation. The authors establish fixed-time and any-time PAC Bayesian bounds for data generated by -lex weakly dependent MMAFs, and they derive practical embedding strategies and estimator designs (including a randomized Gibbs estimator) to optimize generalization performance. Validation on simulated spatio-temporal Ornstein-Uhlenbeck processes demonstrates that the resulting ensemble forecasts yield narrow interquartile ranges that contain the true test values and offer a transparent, causally interpretable forecasting framework for raster data cubes. The approach provides a principled path to non-vacuous generalization guarantees in dependent spatio-temporal settings and offers guidelines for selecting embeddings and hyperparameters to balance dependence structure and learning efficiency. Overall, MMAF-guided learning contributes a theoretically grounded, causality-aware methodology for interpretable spatio-temporal forecasting with explicit uncertainty quantification.

Abstract

Influenced mixed moving average fields are a versatile modeling class for spatio-temporal data. However, their predictive distribution is not generally known. Under this modeling assumption, we define a novel spatio-temporal embedding and a theory-guided machine learning approach that employs a generalized Bayesian algorithm to make ensemble forecasts. We use Lipschitz predictors and determine fixed-time and any-time PAC Bayesian bounds in the batch learning setting. Performing causal forecast is a highlight of our methodology as its potential application to data with spatial and temporal short and long-range dependence. We then test the performance of our learning methodology by using linear predictors and data sets simulated from a spatio-temporal Ornstein-Uhlenbeck process.
Paper Structure (24 sections, 16 theorems, 104 equations, 7 figures, 10 tables)

This paper contains 24 sections, 16 theorems, 104 equations, 7 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Setting
  3. Contributions and Outline
  4. Mixed moving average fields
  5. Notations
  6. Definition and properties of MMAF
  7. Weak dependence coefficients in MMAF-guided learning
  8. Mixed moving average field guided learning
  9. Pre-processing $N$ frames
  10. Spatio-temporal embedding
  11. Study of the dependence structure of $\boldsymbol{L}$ and $\boldsymbol{L}^{\epsilon}$
  12. PAC Bayesian bounds for MMAF generated data
  13. Ensemble Forecasts using MMAF-guided learning
  14. Practical implementation and casual forecast
  15. Linear predictors: an example with simulated data
  16. ...and 9 more sections

Key Result

Proposition 2.17

Let $\Lambda$ be an $\mathbb{R}$-valued Lévy basis with characteristic quadruplet $(\gamma, \sigma^2,\nu,\pi)$ and $f:H\times\mathbb{R}\rightarrow\mathbb{R}$ a $\mathcal{B}(H\times\mathbb{R})$-measurable function not depending on the spatial dimension, i.e.,

Figures (7)

  • Figure 1: Raster data cube's spatio-temporal index set with origin in $(t_0,x_0)$.
  • Figure 2: (a) Observed raster data cube. (b) Spatio-temporal embedding: the red pixel identifies the spatial point $x^*$, and for $i \in \mathbb{Z}$ the set $\mathcal{I}(t_0+ia,x^*)$ defined in (\ref{['index']}) is represented by the pixels in the green boxes. The parameters in use in this sampling are $c=\sqrt{2}$, $p_t=2$, $a_t=3$, $h_t=h_s=1$. (c) A realization $S$ from the cone-shaped sampling process.
  • Figure 3: The x- and y-axes represent the time and spatial dimension, respectively. We picture the last $3$ frames of a data set with spatial dimension $d=1$ where the blue stars represent the pixels used in the definition of the training data set, and the violet stars represent the space-time points where it is possible to provide forecasts with MMAF-guided learning for $p_t=c=h_t=1$. Note that the forecast in the time-spatial position $(4,3)$ lies in the intersection (red area) of the future lightcones $A_2(5)^+$, $A_2(4)^+$ and $A_2(3)^+$ as defined in (\ref{['future_lightcone']}) and represented with green cones.
  • Figure 4: Min-max and inter-quartile range of a $50$-member ensemble forecast for the training data sets $S^{x^*}_m$ described in Tables \ref{['akmi']} and \ref{['akmii']}. Dark grey and orange color represent the ranges related to the use of Table \ref{['akmi']}, whereas the light grey and red color represent the ranges related to the use of Table \ref{['akmii']}. The test set is depicted with a black thick line, while the forecasts obtained using the baseline estimator (\ref{['ols']}) are represented with a violet thick line. The training data set $S^{x^*}_m$ described in Table \ref{['akmii']} are used for computing the baseline estimates with respect to the GAU1A4, NIG1A4, GAU10, and NIG10 data sets in (a), (b), (c) and (d), respectively. The $x$-axis represents the pixels in $\mathbb{L}^{\prime}$, whereas the forecast values are along the $y$-axis.
  • Figure 5: Inter-quartile range of a 50-member ensemble forecast using $S^{1,x^*}_m$ (in violet) and $S^{2,x^*}_m$ (in green) as defined in Table \ref{['akmCompi']} (a) and Table \ref{['akmCompii']} (b). The test set and the baseline forecasts are depicted with a black and a violet thick line, respectively. The training data sets $S^{2,x^*}_m$ in Table \ref{['akmCompi']} and \ref{['akmCompii']} are respectively used as baseline estimates in (a) and (b). The $x$-axis represents the pixels in $\mathbb{L}^{\prime}$, whereas the forecast values are along the $y$-axis.
  • ...and 2 more figures

Theorems & Definitions (63)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Example 2.2: Gaussian Lévy basis
  • Example 2.3: Normal Inverse Gaussian Lévy basis
  • Definition 2.4
  • Definition 2.5: Spatio-temporal stationarity
  • Definition 2.6: MMAF
  • Remark 2.7
  • Example 2.8: STOU process
  • ...and 53 more