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A structural equation formulation for general quasi-periodic Gaussian processes

Unnati Nigam, Radhendushka Srivastava, Faezeh Marzbanrad, Michael Burke

Abstract

This paper introduces a structural equation formulation that gives rise to a new family of quasi-periodic Gaussian processes, useful to process a broad class of natural and physiological signals. The proposed formulation simplifies generation and forecasting, and provides hyperparameter estimates, which we exploit in a convergent and consistent iterative estimation algorithm. A bootstrap approach for standard error estimation and confidence intervals is also provided. We demonstrate the computational and scaling benefits of the proposed approach on a broad class of problems, including water level tidal analysis, CO$_{2}$ emission data, and sunspot numbers data. By leveraging the structural equations, our method reduces the cost of likelihood evaluations and predictions from $\mathcal{O}(k^2 p^2)$ to $\mathcal{O}(p^2)$, significantly improving scalability.

A structural equation formulation for general quasi-periodic Gaussian processes

Abstract

This paper introduces a structural equation formulation that gives rise to a new family of quasi-periodic Gaussian processes, useful to process a broad class of natural and physiological signals. The proposed formulation simplifies generation and forecasting, and provides hyperparameter estimates, which we exploit in a convergent and consistent iterative estimation algorithm. A bootstrap approach for standard error estimation and confidence intervals is also provided. We demonstrate the computational and scaling benefits of the proposed approach on a broad class of problems, including water level tidal analysis, CO emission data, and sunspot numbers data. By leveraging the structural equations, our method reduces the cost of likelihood evaluations and predictions from to , significantly improving scalability.

Paper Structure

This paper contains 24 sections, 6 theorems, 91 equations, 7 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Quasi-Periodic Gaussian Processes
  3. Dynamical Equation Model for QPGP
  4. Estimation strategy
  5. Two stage fast estimation algorithm
  6. Prediction of QPGP
  7. Uncertainty quantification
  8. Simulation Study
  9. Faster likelihood and prediction evaluation
  10. Finite sample performance of proposed estimator
  11. Finite sample performance of bootstrap standard errors
  12. Case Studies
  13. Carbon Dioxide Emission Signal
  14. Sunspot Numbers Data
  15. Water Level Signal
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\{Y_t, \ t\in\mathbb{Z}^+\}$ be a zero mean Quasi-Periodic Gaussian Process with parameters $p, \kappa_p$ and $\omega$. Then, for $s\le t \in \mathbb{Z}^+$, we have where $l(t)\triangleq t-\lfloor t/p\rfloor p$, $l(s)\triangleq s-\lfloor s/p\rfloor p$ with $l(t),l(s)\in\{1,2,\ldots,p\}$ and $\lfloor \cdot \rfloor$ denotes the floor function. $\blacksquare$

Figures (7)

  • Figure 1: The box plots of bootstrap standard errors (computed from $M=1000$ bootstrap samples) of $\hat{\omega}$, $\hat{\theta}$ and $\hat{\sigma}^2$, based on $1000$ simulation runs of standard QPGP with period $p=10$, $\omega=0.5$ and Mackay's periodic kernel with $( \theta=1, \sigma^2=1)$, are shown in left, center and right panel, respectively. Each panel consists of three box plots corresponding to sample sizes $n=600$, $3000$, and $10000$. The empirical standard error of estimators across simulation runs is shown in a dashed horizontal red line.
  • Figure 2: The plot shows the estimates of general $\kappa_p(\cdot)$ against lag in black solid line corresponding to the CO$_2$ dataset, along with the 95% bootstrap confidence limits in grey-dashed lines.
  • Figure 3: The plot shows the detrended carbon dioxide emission levels vs. year in black solid line together with fitting standard QPGP with $p=12$ and general kernel in dashed red line. The grey-shaded region corresponds to the estimated $95\%$ prediction intervals using plug-in estimates.
  • Figure 4: The plot shows sunspot numbers vs. year in a black solid line together with fitting standard QPGP with period $p=11$ and Matérn kernel in a dashed red line. The grey-shaded region corresponds to the estimated $95\%$ prediction intervals using plug-in estimates.
  • Figure 5: The plot shows the water level in a black solid line against days, together with fitted standard QPGP (with $p=148$ and $\kappa_p$ periodic Matérn kernel) in a dashed red line. The grey-shaded region corresponds to the estimated $95\%$ prediction intervals using plug-in estimates.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1: QPGP
  • Theorem 1
  • Proposition 1
  • Definition 2: Standard QPGP
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2