Nonlinear reconciliation: Error reduction theorems

TL;DR

This work addresses the problem of RMSE reduction when reconciling forecasts under nonlinear, manifold-defined constraints $M=\{z: f(z)=0\}$. It develops a geometric framework with deterministic RMSE-reduction theorems for constant-sign curvature hypersurfaces and extensions to non-constant curvature and higher codimension via a probabilistic approach, including a Monte Carlo estimator and calibration bounds. A Newton–Raphson augmented Lagrangian solver is discussed, and a practical, open-source JAX-based package $JNLR$ is released to implement the theorems and reconciliation procedures. Empirical validation across hypersurfaces and higher-codimension manifolds shows strong calibration and zero false positives for the deterministic guarantees, while the probabilistic framework provides actionable guidance for when reconciliation is likely to improve forecast accuracy. Overall, the paper bridges geometric theory and forecasting practice, enabling reliable RMSE improvements in nonlinear constraint settings through principled, data-driven decision rules.

Abstract

Forecast reconciliation, an ex-post technique applied to forecasts that must satisfy constraints, has been a prominent topic in the forecasting literature over the past two decades. Recently, several efforts have sought to extend reconciliation methods to the probabilistic settings. Nevertheless, formal theorems demonstrating error reduction in nonlinear constraints, analogous to those presented in Panagiotelis et al.(2021), are still lacking. This paper addresses that gap by establishing such theorems for various classes of nonlinear hypersurfaces and vector-valued functions. Specifically, we derive an exact analog of Theorem 3.1 from Panagiotelis et al.(2021) for hypersurfaces with constant-sign curvature. Additionally, we provide an error reduction theorem for the broader case of hypersurfaces with non-constant-sign curvature and for general manifolds with codimension > 1. To support reproducibility and practical adoption, we release a JAX-based Python package, JNLR, implementing the presented theorems and reconciliation procedures.

Figures (13)

• Figure 1: Conceptual plot for Theorem \ref{['theorem_1']} ($\hat{z}$ with negative $f(\hat{z})$ in the bottom right) and Theorem \ref{['theorem_3']} ($\hat{z}$ with positive $f(\hat{z})$, shown along atoms of its predictive distribution and their projection onto $M$).
• Figure 2: Concept for Theorem \ref{['theorem_2b']} with $f=(f_1,f_2)=[x-y,\;x^2-z]$. The zero level sets $f_1=0$ and $f_2=0$ are shown in blue and violet. The intersection $C=\{f_1\le0\}\cap\{f_2\le0\}$ is in green. If the projection direction is a positive combination of the normals $\nabla f_1(\tilde{z}),\nabla f_2(\tilde{z})$, it defines a supporting hyperplane for $C$ (gray).
• Figure 3: Left: probability of observing a positive condition from Theorem \ref{['theorem_1']} and \ref{['theorem_2b']} under isotropic Gaussian perturbation of points on the manifold. Right: empirical probability of reducing RMSE by reconciling. The x axis shows the noise level $\sigma_I$.
• Figure 4: Conceptual diagram for the probabilistic RMSE reduction. The unknown true distribution $\nu_t$ and the predictive distribution $\tilde{\nu}_t$ are mapped by $\phi_t$ to the scalar probability density functions of ${\color{nu}\mu_t} := \phi_{t,\#}{\color{nu}\nu_t}$ and ${\color{nu_tilde}\tilde{\mu}_t} := \phi_{t,\#}{\color{nu_tilde}\tilde{\nu}_t}$. Applying $h$ yields the true and predicted probabilities of RMSE reduction $e_t = \mathbb{E}_{f_{\mu_t}}[h(\Phi_t)]$ and $\tilde{e}_t = \mathbb{E}_{f_{\tilde{\mu}_t}}[h(\tilde{\Phi}_t)]$, respectively. In the depicted realization, $60\%$ of $f_{\mu_t}$ is below the dashed line defined by $\phi_t(z_t) = 0$, so the true RMSE reduction probability is $e_t=0.6$. However, the predictive distribution entirely lies below the dashed line, resulting in an estimated probability $\tilde{e}_t=1$. The observed $z_t$, lies above the dashed line, so $\phi_t(z_t) < 0$ and the projection to $\tilde{z}_t$ would increase the RMSE at time $t$ (and thus $y_t=0$).
• Figure 5: Examples of test cases in terms of tuples of predictions (green dots), ground truth (red triangles), and samples from predictive and reconciled distributions (gray and blue dots). Left: paraboloid case, Right: vector-valued function. In this case, just one tuple is plotted, but two points are visible since $f:\mathbb{R}^4\rightarrow \mathbb{R}^2$.

Theorems & Definitions (3)

• Proof 1
• Proof 2
• Proof 3