A Comprehensive Framework for Uncertainty Quantification of Voxel-wise Supervised Models in IVIM MRI

Accurate estimation of intravoxel incoherent motion (IVIM) parameters from diffusion-weighted MRI remains challenging due to the ill-posed nature of the inverse problem and high sensitivity to noise, particularly in the perfusion compartment. In this work, we propose a probabilistic deep learning framework based on Deep Ensembles (DE) of Mixture Density Networks (MDNs), enabling estimation of total predictive uncertainty and decomposition into aleatoric (AU) and epistemic (EU) components. The method was benchmarked against non probabilistic neural networks, a Bayesian fitting approach and a probabilistic network with single Gaussian parametrization. Supervised training was performed on synthetic data, and evaluation was conducted on both simulated and two in vivo datasets. The reliability of the quantified uncertainties was assessed using calibration curves, output distribution sharpness, and the Continuous Ranked Probability Score (CRPS). MDNs produced more calibrated and sharper predictive distributions for the D and f parameters, although slight overconfidence was observed in D*. The Robust Coefficient of Variation (RCV) indicated smoother in vivo estimates for D* with MDNs compared to Gaussian model. Despite the training data covering the expected physiological range, elevated EU in vivo suggests a mismatch with real acquisition conditions, highlighting the importance of incorporating EU, which was allowed by DE. Overall, we present a comprehensive framework for IVIM fitting with uncertainty quantification, which enables the identification and interpretation of unreliable estimates. The proposed approach can also be adopted for fitting other physical models through appropriate architectural and simulation adjustments.

NBMLSS: probabilistic forecasting of electricity prices via Neural Basis Models for Location Scale and Shape

Forecasters using flexible neural networks (NN) in multi-horizon distributional regression setups often struggle to gain detailed insights into the underlying mechanisms that lead to the predicted feature-conditioned distribution parameters. In this work, we deploy a Neural Basis Model for Location, Scale and Shape, that blends the principled interpretability of GAMLSS with a computationally scalable shared basis decomposition, combined by linear projections supporting dedicated stepwise and parameter-wise feature shape functions aggregations. Experiments have been conducted on multiple market regions, achieving probabilistic forecasting performance comparable to that of distributional neural networks, while providing more insights into the model behavior through the learned nonlinear feature level maps to the distribution parameters across the prediction steps.

On-line conformalized neural networks ensembles for probabilistic forecasting of day-ahead electricity prices

Probabilistic electricity price forecasting (PEPF) is subject of increasing interest, following the demand for proper quantification of prediction uncertainty, to support the operation in complex power markets with increasing share of renewable generation. Distributional neural networks ensembles have been recently shown to outperform state of the art PEPF benchmarks. Still, they require critical reliability enhancements, as fail to pass the coverage tests at various steps on the prediction horizon. In this work, we propose a novel approach to PEPF, extending the state of the art neural networks ensembles based methods through conformal inference based techniques, deployed within an on-line recalibration procedure. Experiments have been conducted on multiple market regions, achieving day-ahead forecasts with improved hourly coverage and stable probabilistic scores.

Probabilistic electric load forecasting through Bayesian Mixture Density Networks

Probabilistic load forecasting (PLF) is a key component in the extended tool-chain required for efficient management of smart energy grids. Neural networks are widely considered to achieve improved prediction performances, supporting highly flexible mappings of complex relationships between the target and the conditioning variables set. However, obtaining comprehensive predictive uncertainties from such black-box models is still a challenging and unsolved problem. In this work, we propose a novel PLF approach, framed on Bayesian Mixture Density Networks. Both aleatoric and epistemic uncertainty sources are encompassed within the model predictions, inferring general conditional densities, depending on the input features, within an end-to-end training framework. To achieve reliable and computationally scalable estimators of the posterior distributions, both Mean Field variational inference and deep ensembles are integrated. Experiments have been performed on household short-term load forecasting tasks, showing the capability of the proposed method to achieve robust performances in different operating conditions.

Estimation of Switched Markov Polynomial NARX models

This work targets the identification of a class of models for hybrid dynamical systems characterized by nonlinear autoregressive exogenous (NARX) components, with finite-dimensional polynomial expansions, and by a Markovian switching mechanism. The estimation of the model parameters is performed under a probabilistic framework via Expectation Maximization, including submodel coefficients, hidden state values and transition probabilities. Discrete mode classification and NARX regression tasks are disentangled within the iterations. Soft-labels are assigned to latent states on the trajectories by averaging over the state posteriors and updated using the parametrization obtained from the previous maximization phase. Then, NARXs parameters are repeatedly fitted by solving weighted regression subproblems through a cyclical coordinate descent approach with coordinate-wise minimization. Moreover, we investigate a two stage selection scheme, based on a l1-norm bridge estimation followed by hard-thresholding, to achieve parsimonious models through selection of the polynomial expansion. The proposed approach is demonstrated on a SMNARX problem composed by three nonlinear sub-models with specific regressors.

Identification of Probability weighted ARX models with arbitrary domains

Hybrid system identification is a key tool to achieve reliable models of Cyber-Physical Systems from data. PieceWise Affine models guarantees universal approximation, local linearity and equivalence to other classes of hybrid system. Still, PWA identification is a challenging problem, requiring the concurrent solution of regression and classification tasks. In this work, we focus on the identification of PieceWise Auto Regressive with eXogenous input models with arbitrary regions (NPWARX), thus not restricted to polyhedral domains, and characterized by discontinuous maps. To this end, we propose a method based on a probabilistic mixture model, where the discrete state is represented through a multinomial distribution conditioned by the input regressors. The architecture is conceived following the Mixture of Expert concept, developed within the machine learning field. To achieve nonlinear partitioning, we parametrize the discriminant function using a neural network. Then, the parameters of both the ARX submodels and the classifier are concurrently estimated by maximizing the likelihood of the overall model using Expectation Maximization. The proposed method is demonstrated on a nonlinear piece-wise problem with discontinuous maps.