Forecasting with Noise in Mind

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Author: Denis Avetisyan

A new loss function tackles the inherent uncertainty in time series data to deliver more accurate predictions.

This paper introduces RI-Loss, a learnable residual-informed loss function based on the Hilbert-Schmidt Independence Criterion, to improve time series forecasting by explicitly modeling noise structure and capturing temporal dependencies.

Despite advances in time series forecasting, standard approaches often fail to adequately address inherent noise and capture complex temporal dependencies. This paper introduces RI-Loss: A Learnable Residual-Informed Loss for Time Series Forecasting, a novel loss function grounded in the Hilbert-Schmidt Independence Criterion that explicitly models residual noise structure to improve predictive accuracy. By enforcing dependence between residuals and random time series, RI-Loss enables more robust, noise-aware representations and achieves optimal convergence rates through rigorous statistical analysis. Could this approach unlock more reliable forecasting across diverse real-world applications and pave the way for truly intelligent time series analysis?

The Inevitable Decay of Prediction: Confronting Noise in Time Series

The ability to accurately predict future values in a time series – a sequence of data points indexed in time order – underpins countless critical applications, ranging from financial market analysis and resource management to weather forecasting and healthcare monitoring. However, standard time series forecasting methods frequently encounter difficulties when confronted with real-world data, which is rarely pristine. Complex systems often generate data characterized by non-linearity, seasonality, and, crucially, noise. This noise, stemming from unpredictable external factors or measurement errors, can obscure the underlying patterns and significantly degrade the performance of traditional models like ARIMA or exponential smoothing. Consequently, while these methods provide a valuable starting point, their limitations necessitate the development of more robust and adaptable techniques capable of discerning genuine predictive signals from the pervasive influence of random variation.

Traditional time series forecasting often relies on loss functions such as Mean Squared Error ($MSE$), which, while computationally convenient, possess limitations when confronted with real-world data. $MSE$ treats all errors equally, meaning that a single outlier or anomalous data point can disproportionately influence the model’s learning process and lead to inaccurate predictions. This sensitivity stems from the function’s quadratic nature, heavily penalizing larger deviations. Furthermore, $MSE$ struggles to discern nuanced temporal relationships, particularly those involving non-linear patterns or long-range dependencies, as it primarily focuses on minimizing the overall magnitude of error without explicitly considering the sequential nature of the data. Consequently, models optimized with $MSE$ may fail to capture the subtle, yet critical, dependencies necessary for precise forecasting in complex systems.

Successfully forecasting time series data hinges on a fundamental, yet remarkably difficult, task: discerning meaningful patterns – the “signal” – from random fluctuations – the “noise”. Traditional methods often treat all data points equally, making them vulnerable to the distorting influence of outliers or transient events. Advanced techniques, therefore, focus on explicitly modeling the underlying data distribution, attempting to statistically characterize what constitutes typical behavior versus anomalous variation. This requires going beyond simple averages and employing complex probabilistic models, such as Gaussian processes or state-space models, capable of capturing the data’s inherent structure and dependencies. By accurately representing this distribution, these models can effectively filter out noise, identify genuine trends, and ultimately produce more reliable and robust forecasts, even in the face of substantial data complexity and uncertainty. The challenge isn’t merely prediction, but rather a nuanced understanding of the process generating the observed time series.

RI-Loss: Sculpting Signal from the Static

RI-Loss utilizes the Hilbert-Schmidt Independence Criterion (HSIC) as the foundation for a novel loss function aimed at explicitly quantifying the relationship between predictive model residuals – representing the signal – and the inherent noise within the data. HSIC is a kernel-based method used to measure statistical dependence; in this context, it assesses the dependence between the residuals and the noise. The loss function is formulated to minimize this dependence, effectively encouraging the forecasting model to learn representations that are independent of the noise component. Mathematically, HSIC calculates the squared distance between kernel mean embeddings of the residuals and noise, with the goal of driving this distance toward zero, indicating statistical independence. The kernel function, typically a Gaussian kernel, transforms the data into a higher-dimensional space where dependencies are more readily detectable.

RI-Loss aims to improve forecasting accuracy by minimizing the correlation between model residuals – the difference between predicted and actual values – and the inherent noise within the data. This is accomplished by formulating a loss function that penalizes statistical dependencies between these two components. Specifically, the loss function is designed to maximize the independence, as measured by the Hilbert-Schmidt Independence Criterion (HSIC), between the residual space and the noise distribution. By encouraging statistical independence, the model is incentivized to learn representations that capture the underlying signal, effectively reducing the impact of noise on the final prediction and improving generalization performance. This approach differs from traditional loss functions which primarily focus on minimizing prediction error without explicitly addressing noise characteristics.

Kernel methods are central to the implementation of RI-Loss, enabling the modeling of non-linear dependencies between data points without explicitly defining these relationships. These methods operate by mapping input data into a higher-dimensional feature space via a kernel function, $k(x_i, x_j)$, which calculates the similarity between data points $x_i$ and $x_j$. Common kernel choices include the Gaussian (Radial Basis Function) kernel and polynomial kernels. This transformation allows for the representation of complex relationships as inner products in the feature space, facilitating the calculation of the Hilbert-Schmidt Independence Criterion (HSIC) which quantifies the statistical independence between residuals and noise, without requiring explicit feature engineering or assumptions about the underlying data distribution.

Beyond Error Minimization: Foundations in Generalization and Robustness

The theoretical underpinnings of RI-Loss generalization performance are formalized through the use of Rademacher Complexity and U-Statistics. Rademacher Complexity, denoted as $R_S(H)$, quantifies the capacity of a hypothesis set $H$ to fit random noise on a dataset $S$, effectively measuring model complexity. U-Statistics are used to analyze the expected risk of RI-Loss, providing a means to decompose the empirical risk and establish bounds on the difference between training and test performance. Specifically, these statistical tools allow for the derivation of generalization bounds that relate the empirical risk, the Rademacher Complexity of the hypothesis space, and the size of the training dataset, demonstrating how RI-Loss contributes to controlling overfitting and improving robustness.

Theoretical bounds derived from Rademacher Complexity and U-Statistics establish that RI-Loss minimizes the expected difference between a model’s performance on the training data and its performance on unseen test data. Specifically, these bounds quantify the generalization error, demonstrating that RI-Loss effectively constrains this error even with complex models or challenging data distributions. This control over the generalization gap is achieved by regulating the model’s sensitivity to individual training examples, thereby promoting robustness to variations present in the test set. The resulting bounds provide a quantifiable measure of the model’s ability to perform consistently across different data samples, indicating improved performance on previously unseen data.

The RI-Loss framework facilitates analysis of the relationship between model complexity, data distribution, and generalization performance through the application of U-Statistics and Rademacher Complexity. Specifically, it provides a means to quantify how the capacity of a model – its complexity – interacts with the characteristics of the training data distribution to determine the expected difference between performance on the training set and an unseen test set. By bounding this generalization error, the framework allows for a systematic evaluation of how changes in model architecture or training data impact the model’s ability to perform reliably on new, previously unobserved data. This principled approach moves beyond empirical observation, enabling theoretically grounded predictions about model behavior and robustness.

Empirical Validation: A Consistent Signal Across Diverse Systems

Rigorous experimentation has confirmed that the proposed RI-Loss function consistently enhances the accuracy of time series forecasting across diverse model architectures. Studies utilizing both the complex, attention-based Transformer networks and the more straightforward Multilayer Perceptron (MLP) models reveal a significant improvement in predictive performance when RI-Loss is integrated into the training process. This suggests that the benefits of RI-Loss are not tied to a specific model type, but rather stem from its ability to refine the loss landscape and guide optimization towards more accurate forecasts. The consistent gains observed across these differing architectures underscore the broad applicability and potential of RI-Loss as a general-purpose enhancement for time series prediction tasks, offering a pathway to improved results regardless of the underlying model complexity.

Evaluations across diverse time series datasets reveal a consistent performance advantage when incorporating RI-Loss into forecasting models. Empirical results demonstrate that RI-Loss substantially reduces the Mean Squared Error (MSE) compared to standard loss functions; specifically, the Informer model achieves an average MSE reduction of 9.4%, while the DLinear model benefits from a 5.2% decrease. This improvement isn’t limited to a single architecture or dataset, suggesting that RI-Loss offers a generalizable method for enhancing the accuracy of time series forecasting, potentially by better capturing the underlying relationships within the data and mitigating the impact of noisy observations.

Rigorous testing across 160 diverse time series forecasting scenarios reveals the consistent advantage of RI-Loss over traditional Mean Squared Error (MSE). In a significant majority of cases – 133 out of 160 – RI-Loss demonstrably improved forecasting performance. This widespread success indicates a substantial degree of robustness and generalizability, extending beyond specific datasets or model configurations. Quantitative analysis further highlights these gains, with the Informer model achieving an average reduction of 6.9% in Mean Absolute Error (MAE) when utilizing RI-Loss, and the DLinear model showing a 4.4% improvement in MAE. These results collectively suggest that RI-Loss is not merely a specialized optimization technique, but a broadly applicable method for enhancing the accuracy of time series forecasting models.

The pursuit of accurate time series forecasting, as demonstrated in this work with RI-Loss, inherently acknowledges the limitations of any predictive model. It’s a recognition that simplification – distilling complex temporal dynamics into manageable parameters – carries a future cost. As David Hilbert observed, “We must be able to answer the question: What are the ultimate limits of our ability to compute?” This resonates deeply with the presented research; RI-Loss attempts to address the ‘noise’ inherent in time series data, acknowledging that perfect prediction is unattainable, and striving instead for graceful degradation through a more robust modeling of residual structures. The efficacy of RI-Loss, by explicitly incorporating noise characteristics, suggests a pragmatic acceptance of computational limits, rather than a naive pursuit of absolute accuracy.

What Lies Ahead?

The introduction of RI-Loss represents a versioning of the forecasting problem—a refinement, not a resolution. The explicit modeling of noise, while beneficial, merely pushes the inevitable decay of predictive power further down the timeline. All models are, ultimately, temporary bulwarks against the relentless arrow of time. The current formulation, predicated on the Hilbert-Schmidt Independence Criterion, invites scrutiny regarding its scalability with high-dimensional time series. The computational cost of kernel methods remains a practical limitation, a friction that will necessitate exploration of approximations or alternative independence measures.

Future work will likely concern itself with the interplay between RI-Loss and the inherent non-stationarity of real-world time series. Adaptively weighting the residual-informed component, perhaps through meta-learning, could allow the model to gracefully age as the underlying data distribution shifts. Further investigation into the Rademacher complexity of the loss function is warranted—understanding its generalization bounds will be crucial for deploying these models beyond controlled environments.

Ultimately, the pursuit of perfect forecasting is a Sisyphean task. The value lies not in achieving stasis, but in building systems that acknowledge and adapt to the inherent impermanence of the data they attempt to predict. Each refinement is merely a temporary stay against the entropy, a carefully constructed illusion of order in a fundamentally chaotic universe.

Original article: https://arxiv.org/pdf/2511.10130.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-11-16 14:42