Prediction Noise in Forecasting Models

• Prediction noise is the random deviation in forecasts, arising from data uncertainty, environmental randomness, and intrinsic model limitations.
• It is mathematically modeled using additive Gaussian processes and filtering techniques such as Wiener–Kolmogorov optimal filtering for robust calibration.
• Recent advances include noise injection and self-supervised methods, which both mitigate noise effects and exploit them for improved prediction reliability.

Prediction noise refers to the stochastic deviations or uncertainties that arise in the process of forecasting system behavior, model outputs, or physical phenomena, as a result of the interplay between data uncertainty, environmental randomness, input perturbations, and model limitations. In contemporary research literature, prediction noise appears in diverse contexts such as time series forecasting, signal processing, model calibration, physical noise emissions, and uncertainty quantification in machine learning. Prediction noise may manifest as label noise in supervised tasks, input or measurement noise in dynamical systems, feature noise in regression/classification, or as a fundamental limitation in spectroscopic and physical measurement setups.

1. Mathematical Foundations and Taxonomy of Prediction Noise

Prediction noise is rigorously defined via the modeling of stochastic perturbations in the system inputs, outputs, or labels. In time series and trajectory prediction, noise is often modeled as additive, temporally uncorrelated random variables: $$x_{\text{noisy}}(t) = x^{\text{Noiseless}}(t) + \eta(t),\quad \eta(t)\sim\mathcal{N}(0,\sigma_N^2)$$ as in chaotic system forecasting (López-Caraballo et al., 2015). In trajectory modeling, spatial noise is injected as i.i.d. Gaussian perturbations: $$\tilde{\mathbf{x}}_t = \mathbf{x}_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, \sigma^2 I)$$ with a tunable amplitude or "noise factor" (Chib et al., 2023). In diffusion forecasting and uncertainty quantification, the forward process is formalized as

$$x_{n}^{\mathrm{ta}} = \sqrt{\bar{\alpha}_n} x_0^{\mathrm{ta}} + \sqrt{1-\bar{\alpha}_n} \epsilon,\quad \epsilon \sim \mathcal{N}(0, I)$$

with subsequent reverse denoising steps (Sheng et al., 23 Jan 2025).

A taxonomy distinguishes between:

• Label noise: Random mislabeling or perturbation in supervised targets (classification, regression).
• Feature/input noise: Corruption of the observed or measured predictors.
• Process/physical noise: Intrinsic randomness in physical phenomena affecting observed outcomes.
• Prediction set/distributional noise: Statistical spread in construction of predictive intervals or sets.

2. Theoretical Models: Filtering, Robustness, and Regularization

Optimal information extraction in the presence of prediction noise is fundamentally framed as a noise filtering and prediction problem. The Wiener–Kolmogorov theory defines the optimal linear filter for estimating a signal corrupted by additive noise by minimizing mean-square error (MMSE), resulting in an analytically specified filter kernel: $$H_{\rm WK}(\omega) = \frac{S_{ss}(\omega)}{S_{ss}(\omega) + S_{nn}(\omega)}$$ with MMSE

$$\mathrm{MMSE} = \int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{S_{ss}(\omega)S_{nn}(\omega)}{S_{ss}(\omega)+S_{nn}(\omega)}$$

and associated information bounds (Hathcock et al., 2016).

Regularization and robustification techniques are crucial for stabilizing machine learning models in the presence of prediction noise. Input noise injection during training,

$$u_{\text{in}}(t) = u(t) + \eta(t),\quad \eta(t)\sim\mathcal{N}(0, \sigma^2 I)$$

damps sensitivity to perturbations and guides the model to function robustly in a neighborhood of the training trajectory. The Linearized Multi-Noise Training (LMNT) approach deterministically penalizes the parameter sensitivity to such perturbations via a structured Jacobian-based penalty (Wikner et al., 2022).

3. Noise-Robust Forecasting and Self-Supervised Methods

Modern deep models often explicitly account for prediction noise through self-supervised auxiliary tasks. In trajectory forecasting, the Self-Supervised Waypoint Noise Prediction (SSWNP) paradigm introduces parallel clean and noise-augmented input views, compels prediction heads to agree via a spatial-consistency penalty, and trains a separate noise-prediction network to estimate the amount of corruption in the input (Chib et al., 2023). The total loss is

$$\mathcal{L} = \mathcal{L}_{\text{traj}}^{\text{clean}} + \mathcal{L}_{\text{traj}}^{\text{noisy}} + \lambda_c\,\mathcal{L}_{\text{consistency}} + \lambda_n\,\mathcal{L}_{\text{noise}}$$

incorporating trajectory accuracy and explicit noise estimation.

Diffusion-based methods for momentary or uncertainty-rich prediction decompose the overall prediction noise into a prior, drawn from interpretable system dynamics (periodic, local), and a residual, assigned to the neural denoiser (Sheng et al., 23 Jan 2025). This separation allows hybrid analytic-neural compensation, enhancing robustness and yielding faster, more accurate convergence in regimes with abrupt input shifts.

4. Prediction Noise in Uncertainty Quantification and Conformal Methods

In probabilistic prediction frameworks, prediction noise directly impacts the calibration, coverage, and efficiency of confidence intervals or prediction sets. Conformal prediction (CP) frameworks construct uncertainty sets $C(x)$ such that

$\Pr_{(x,y)}[y \in C(x)] \geq 1 - \alpha$

Under label noise, standard CP with noisy calibration labels yields conservative sets, worsening efficiency. For uniform (random-flip) noise, a conditional expectation–based conformal score

$$\hat{S}(x, \tilde{y}, \epsilon) = (1-\epsilon) S(x, \tilde{y}) + \epsilon \frac{1}{k} \sum_{i=1}^k S(x,i)$$

recovers near-oracle prediction set size and maintains coverage (Penso et al., 2024). In online scenarios, robust threshold adaptation (NR-OCP) using an unbiased pinball loss counteracts the coverage gap induced by noise: $$\tilde{\ell}_{1-\alpha}(\tau, \tilde{S}_t, \{S_{t,y}\}) = \ell_{1-\alpha}(\tau, \tilde{S}_t) - \frac{\eta}{K(1-\eta)} \sum_{y=1}^K \ell_{1-\alpha}(\tau, S_{t,y})$$ restoring empirical and expected coverage to target levels (Xi et al., 30 Jan 2025).

In regression, noise-free conformal thresholds can be recovered by deconvolution of the noisy empirical coverage, using a known or estimated noise kernel (Cohen et al., 18 Sep 2025).

5. Noise Prediction in Physical Systems and Engineering

Prediction noise is central to the understanding, mitigation, and exploitation of physical noise in engineering design:

• Rolling noise: Prediction models for rolling noise in buildings analytically relate geometric, material, and dynamic wheel/floor properties to radiated sound power via convolutional and impedance-based models; controlling prediction noise is vital for system design (Edwards et al., 2021).
• Aerodynamic noise: Analytical models interpret trailing/leading-edge noise from serrated blades as arising due to phase-interference mechanisms and utilize noise-prediction equations that capture geometric and flow-induced noise features (Fourier-based mode-sum reduction from complex infinite-band Wiener–Hopf solutions) (Lyu et al., 2019, Lyu et al., 2015).
• Optical fiber modal noise: The photometric noise in fibre-spectroscopy is analytically determined by the visibility of the modal interference pattern, which can be described via explicit relationships between fiber properties, optical configuration, and S/N ratio (Lemke et al., 2011).

6. Systematic Mitigation and Exploitation of Prediction Noise

Recent frameworks emphasize not only the suppression or filtering of prediction noise, but also its constructive exploitation to achieve regularization, diversity, and calibrated uncertainty:

• In dynamic system prediction, noise injection and LMNT regularization stabilizes long-term climate prediction of chaotic PDEs, outperforming ridge or Jacobian-only schemes (Wikner et al., 2022).
• In deep communication systems, dedicated neural predictors estimate channel noise rates from delayed feedback, drastically improving throughput and delay metrics over heuristic models (Cohen et al., 2021).
• In quality-of-service and medical inference, architectures with parallel prior/posterior branches detect and correct feature or label noise, refining predictions via KL-alignment of corrupted inputs to clean label-driven embeddings (Wang et al., 2023, Liu et al., 2024).

7. Empirical Findings, Quantitative Gains, and Limitations

Empirical studies confirm that explicit modeling, estimation, and compensation of prediction noise yields substantial performance improvements:

• SSWNP improves average and final displacement errors by 20–40% relative to baselines, and maintains robust performance under input perturbations (Chib et al., 2023).
• In mobile-traffic prediction, decomposing noise into interpretable priors and neural residuals yields 30–50% lower MAE/RMSE and more calibrated uncertainties (Sheng et al., 23 Jan 2025).
• In conformal prediction, robust conformal methods shrink inflated prediction sets (size drops by 30–50%) while preserving target coverage (Penso et al., 2024, Xi et al., 30 Jan 2025, Cohen et al., 18 Sep 2025).
• Regularization by noise injection or LMNT enables indefinite stable forecasting in chaotic systems where unregularized models are unstable (Wikner et al., 2022).

Limitations include the need for accurate specification or estimation of noise kernels, potential bias under adversarial or un-modeled noise, and possible degradation in the presence of systematic label/feature artifacts.

---

Prediction noise, through a combination of analytic, statistical, and algorithmic techniques, is now a quantifiable, controllable, and often beneficial component of modern prediction theory and practice—a central consideration in the development of robust, efficient, and uncertainty-aware modeling systems across scientific domains.