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Rethinking Irregular Time Series Forecasting from the Perspective of Basis Functions

Rongwen Li, Changjian Chen

arXiv:2608.17284v1cs.LGcs.AI

TL;DR

Irregular time-series forecasting must learn from sparse, non-uniform observations, while existing basis-function methods can suffer sampling-induced bias and limited adaptability. DNBNet addresses these issues with density correction, learnable neural basis functions, and multi-scale mass-aware representations, achieving superior performance across five real-world benchmarks, including the best result on four of five datasets and an approximately 3.18% average relative MSE reduction versus Warpformer.

  • Problem

    Sparse, irregular measurements make reliable representation learning and accurate forecasting difficult, while existing methods may incur sampling-induced bias and limited basis adaptability.

  • Method

    DNBNet uses density-corrected neural basis responses with multi-scale decomposition, mass-aware fusion, and a dual-branch decoder for irregular forecasting.

  • Results

    DNBNet achieves the best performance on four of five datasets, second-best on PhysioNet, and approximately 3.18% lower average relative MSE than Warpformer.

  • Takeaways & Limitations

    Across five real-world benchmarks, DNBNet consistently outperforms strong baselines in irregular time-series forecasting.

  • Takeaways & Limitations

    The asymptotic-bias analysis establishes vanishing bias only when timestamp density approaches uniformity, corresponding to regular sampling.

Abstract

from arXiv · show

Irregular time series forecasting is crucial in many domains, such as healthcare and meteorological observation. However, due to the inherent characteristics of irregular time series, including sparse observations and non-uniform sampling, accurately predicting future dynamics remains challenging. In light of these two characteristics, many existing methods aggregate irregular observations into fixed-dimensional estimated response coefficients through predefined basis functions and use these coefficients as sequence representations. Nevertheless, this modeling paradigm still suffers from two key limitations: (i) a potential non-vanishing asymptotic bias caused by ignoring the sampling density of timestamps; and (ii) the limited adaptability of predefined basis functions to diverse temporal patterns. In this study, we propose a Debiased Neural Basis-Function Network (DNBNet) to address these challenges. Its core is a debiased neural basis-function response mechanism, which corrects asymptotic bias through importance sampling while parameterizing basis functions with neural networks to adapt to diverse temporal patterns. In addition, considering the sparsity of irregular data, we design a novel multi-scale decomposition module based on average pooling, together with a mass-aware fusion mechanism, to obtain richer representations. Finally, a dual-branch decoder is employed for forecasting. Extensive experiments on multiple real-world datasets demonstrate the effectiveness of DNBNet and its strong generalizability across diverse irregular time series scenarios. Our code can be obtained at https://github.com/hnu-vis/DNBNet.

1. Introduction

The introduction frames irregular multivariate time series forecasting as difficult because sparse, asynchronous measurements obscure continuous-time dynamics. It presents DNBNet as a debiased, neural basis-function approach augmented with multi-scale representation learning and validated on multiple benchmarks.

  • Motivation: Irregular multivariate time series contain asynchronous, non-uniform, variable-specific observations with unequal sampling intervals and missing values.They arise in domains including medical monitoring, sensor networks, and meteorological observations.
  • Motivation: Sparse and irregular measurements make reliable representation learning and accurate prediction challenging despite underlying continuous-time dynamics.The data-generating processes are continuous in time but observed only through limited irregular measurements.
  • Existing methods: Many existing methods map irregular observations to fixed-dimensional representations that can be interpreted as responses to predefined temporal basis functions.The resulting coefficients encode how strongly observations align with each basis and summarize potential temporal patterns.
  • Limitations: Existing approaches face potential non-vanishing asymptotic bias under irregular sampling and limited adaptability of predefined bases to diverse temporal patterns.Discrete coefficient approximations ignore timestamp distributions, while fixed bases may not flexibly model complex dynamics.
  • Proposed method: DNBNet corrects sampling-induced bias with KDE-based importance adjustment, learns adaptive neural basis functions, and adds multi-scale decomposition with mass-aware fusion.The model further uses a dual-branch decoder, while experiments on multiple benchmarks report superior predictive performance against competitive methods.

2. Related Work

Related work on irregular multivariate time-series forecasting includes continuous-time ODE/CDE models and basis-function approaches that represent temporal signals through response coefficients. Existing methods use mechanisms such as latent ODE construction, direct solution modeling, FFT, attention-based basis functions, and Gaussian temporal kernels.

  • Continuous-time modeling: Recent IMTS forecasting methods are broadly divided into three categories, including approaches that directly model underlying continuous time with ODEs or CDEs.Latent ODE constructs continuous-time latent sequences for interpolation and extrapolation, while Neural Flow directly models the ODE solution to avoid inefficient numerical integration.
  • Basis-function methods: Basis functions decompose complex continuous signals into response coefficients over temporal patterns and have been widely used in regularly sampled time-series analysis.This provides a classical perspective for characterizing temporal signals through fixed response coefficients.
  • Basis-function methods: Some time-series methods use the Fast Fourier Transform to capture periodic sequence components.The passage cites Zhou et al. (2022) and Qiu et al. (2025) as examples.
  • Basis-function methods: IMTS methods also extract temporal information with basis functions, including mTAN’s attention-based functions with predefined reference points.mTAN aggregates irregular observations through a time-aware attention mechanism.
  • Basis-function methods: KAFNet uses Gaussian temporal kernel aggregation to construct temporal representations.The supplied passage introduces this method but ends before detailing its construction further.

3. Preliminaries and Motivation

This section formalizes irregular time series forecasting and basis-function response estimation, then shows that discrete coefficients can retain asymptotic bias under irregular sampling. The analysis motivates explicitly debiased response estimation using timestamp-density information.

  • Bias motivation: These findings motivate a debiased response mechanism that explicitly accounts for the sampling density of observed timestamps.The motivation follows from the persistent irregular-sampling term in the estimation error bound.
  • Problem formulation: Irregular forecasting predicts future values at query timestamps from historical observations and their timestamps.The model maps historical observations and query timestamps to corresponding predictions.
  • Basis-function response: Existing methods approximate continuous basis-function response coefficients by summing basis evaluations over observed data.The section focuses on weighted average responses, while projection-response analysis is similar.
  • Bias motivation: Under irregular sampling, the discrete response approximation can incur asymptotic bias despite increasing observation counts.The analysis assumes bounded signals and basis functions with nontrivial mass under uniform time and timestamp sampling densities.
  • Bias motivation: The non-vanishing bias disappears only when timestamp density approaches the uniform distribution, corresponding to regular sampling.The finite-sample estimation term decreases with observation count, whereas the sampling-density term is independent of it.

4. Debiased Neural Basis-Function Network

DNBNet combines time-aware multi-scale pooling, debiased neural basis responses, mass-aware fusion, and a dual-branch decoder to forecast irregular time series. The architecture addresses sparse information, sampling bias, and the limited adaptability of predefined basis functions.

  • Multi-scale decomposition: DNBNet generates subsequences at multiple temporal scales through time-aware average pooling to capture heterogeneous patterns and enrich representations despite sparse observations.Window sizes and strides are defined by absolute time spans, and representation learning is primarily channel-independent.
  • Debiased neural basis response: Importance sampling introduces density correction into basis-response estimation, asymptotically removing sampling bias by weighting observations with 1/p(t).The weighting suppresses overemphasis on densely sampled regions, while the Gaussian-kernel bandwidth is learned as h = softplus(ρ).
  • Debiased neural basis response: Learnable neural basis functions replace fixed predefined bases, improving adaptability to diverse temporal patterns and reducing dependence on dataset-specific expert selection.Softmax bounds and normalizes basis responses across basis functions to improve computational stability.
  • Mass-aware fusion: Mass-aware fusion adaptively aggregates scale-specific coefficients according to basis support, then concatenates and projects them into a latent representation.Logarithmic transformation prevents excessively large basis masses from dominating fusion weights.
  • Dual-branch forecasting decoder: A dual-branch decoder combines feature-based prediction with explicit basis reconstruction, using a learnable sigmoid gate to balance their contributions.The basis branch extrapolates future values by evaluating learned basis functions at query timestamps and combining them with fused coefficients.

5. Experiments

Experiments across five real-world irregular multivariate time-series datasets show that DNBNet generally outperforms competing methods. Ablations and sensitivity analyses further support its debiasing mechanism, learnable basis functions, and multi-scale design.

  • Main results: DNBNet achieves the best MSE on four of five datasets and the second-best result on PhysioNet, reducing average relative MSE by approximately 3.18% versus Warpformer.MSE results are reported across healthcare, biomechanics, and climate benchmarks.
  • Ablation study: Removing the density factor p(t) causes consistent performance degradation, confirming the effectiveness of DNBNet’s debiased response mechanism.The ablation study evaluates DNBNet on three irregular time-series datasets.
  • Ablation study: Adding the density debiasing factor 1/p(t) to KAFNet and APN produces consistent, stable improvements with almost no additional complexity.The comparison uses the original biased weighted-average aggregation as the reference.
  • Sensitivity analysis: Using more basis functions does not necessarily improve performance; relatively small values such as K = 16 or K = 32 are often sufficient for strong results.The sensitivity study varies the number of basis functions K and scales S on three datasets.
  • Basis-function visualization: On PhysioNet, learned basis functions resemble localized RBFs while adapting their centers, bandwidths, and peak locations.This visualization highlights adaptive local response ranges that predefined RBF bases cannot readily provide.

6. Conclusion

The paper identifies non-vanishing asymptotic estimation bias and limited adaptability as limitations of existing basis-based irregular time series forecasting methods. It proposes DNBNet, which combines density correction, learnable neural basis functions, multi-scale response extraction, mass-aware fusion, and a dual-branch decoder.

  • Ignoring timestamp density can cause existing basis-based irregular time series forecasting methods to exhibit non-vanishing asymptotic estimation bias.
  • Predefined basis functions may limit the adaptability of existing methods to diverse irregular temporal patterns.
  • DNBNet integrates density correction, learnable neural basis functions, multi-scale response extraction, and mass-aware fusion for robust and adaptive representation learning.
  • Its dual-branch decoder combines implicit latent prediction with explicit basis-function reconstruction.

A. Proof of Theorem 1

The proof establishes that the uncorrected discrete basis-function estimator differs from the ideal continuous-time coefficient because of finite-sample error and non-uniform timestamp sampling. It bounds both terms and combines them to complete the theorem.

  • A. Proof of Theorem 1: The proof defines the ideal continuous-time response coefficient and the corresponding uncorrected discrete estimator for each basis function.These definitions establish the quantities whose discrepancy is analyzed.
  • A. Proof of Theorem 1: Because timestamps follow density p(t), the discrete estimator approximates a sampling-density-weighted coefficient rather than the ideal coefficient.This mismatch creates a sampling bias under non-uniform timestamp density.
  • A. Proof of Theorem 1: The coefficient error is decomposed into finite-sample approximation error and sampling bias, which are bounded separately.The finite-sample term is controlled using Hoeffding’s inequality and a union bound over all basis functions and auxiliary averages.
  • A. Proof of Theorem 1: Combining the finite-sample approximation bound with the sampling-bias bound yields the theorem’s final result and completes the proof.The argument first compares the relevant ratios, then bounds the sampling-bias term before combining both contributions.

B. Debiasing Property of Density-Corrected Estimation · C. Datasets and Baseline Model Details

The density-corrected estimator uses importance weighting to compensate for non-uniform timestamp sampling, directly approximating the desired coefficient and removing the systematic sampling bias. Under a lower-bounded sampling density and bounded signals and basis functions, a finite-sample concentration bound follows.

  • B. Debiasing Property of Density-Corrected Estimation: Under non-uniform timestamp density p(t), the uncorrected estimator converges to a sampling-density-weighted coefficient, causing non-vanishing bias.
  • B. Debiasing Property of Density-Corrected Estimation: Importance weighting by the inverse timestamp density compensates for irregular sampling in the density-corrected estimator.
  • B. Debiasing Property of Density-Corrected Estimation: The corrected estimator’s numerator and denominator directly estimate continuous-time integrals over the uniform time domain.
  • B. Debiasing Property of Density-Corrected Estimation: The density-corrected estimator approximates the desired coefficient c_k rather than the sampling-density-weighted coefficient c_p.
  • B. Debiasing Property of Density-Corrected Estimation: Assuming p(t) ≥ ρ > 0, |x(t)| ≤ 1, and 0 ≤ ϕ_k(t) ≤ m ensures bounded importance weights and supports a finite-sample bound.
  • B. Debiasing Property of Density-Corrected Estimation: The corrected estimator’s bound excludes the non-vanishing term involving ∥p(t) − 1∥L1, thereby removing systematic bias from irregular timestamp sampling.

C.1. Datasets · C.2. Baseline Model Details · D. Additional Experiments Results

The experiments use five public datasets spanning clinical, biomechanical, behavioral, and meteorological irregular time series. DNBNet is compared with 12 representative baselines covering forecasting, classification, and interpolation approaches.

  • C.1. Datasets: Five public datasets are used, with their sources and characteristics summarized for the experiments.The datasets cover multiple real-world irregular time-series scenarios.
  • C.1. Datasets: MIMIC contains 21,250 ICU samples with 96 variables from patients’ first 48 hours, while PhysioNet contains 11,981 samples with 36 variables from the same period.MIMIC includes vital signs, medications, and laboratory measurements; PhysioNet includes measurements such as serum glucose and heart rate.
  • C.1. Datasets: Human Activity contains 1,359 samples and 12 variables representing 3D positions recorded at non-uniform time intervals.The data come from five subjects performing various activities.
  • C.1. Datasets: StudentLife captures irregular human behavior sensing, whereas USHCN provides 1,114 meteorological samples with 5 variables and common missing observations.StudentLife irregularity reflects scheduling, device usage, and asynchronous sensing; USHCN includes temperature and precipitation variables.
  • C.2. Baseline Model Details: The study compares DNBNet with 12 representative baseline models spanning irregular time-series forecasting, classification, and interpolation.The listed baselines include PrimeNet, SeFT, mTAN, CRU, GNeuralFlow, Raindrop, tPatchGNN, GraFITi, Warpformer, Hi-Patch, KAFNet, and APN.
  • C.2. Baseline Model Details: The baselines employ diverse representations, including pretraining, set-function aggregation, multi-time attention, continuous-time recurrence, graph modeling, patching, temporal warping, and kernel aggregation.These approaches address irregularity through time alignment, sparse observation modeling, multi-scale dependencies, or latent representation learning.
  • C.2. Baseline Model Details: APN is a basis-function-based forecasting method that uses adaptive temporal aggregation to encode irregular observations for IMTS forecasting.It is described as a strong and efficient baseline.

D.1. Complete MAE Results · D.2. Varying Lookback Lengths and Forecast Horizons

DNBNet remains effective across evaluation metrics, application domains, lookback windows, and forecast horizons. It achieves leading or near-leading MAE results and maintains strong performance under varied temporal settings.

  • D.1. Complete MAE Results: DNBNet achieves the best MAE on USHCN, Human Activity, and MIMIC, while ranking second-best on Student Life and PhysioNet.These results are generally consistent with the MSE findings.
  • D.1. Complete MAE Results: The MAE results show that DNBNet remains effective under different evaluation metrics and across diverse application domains.
  • D.2. Varying Lookback Lengths and Forecast Horizons: DNBNet consistently delivers stable and competitive performance across different lookback windows.
  • D.2. Varying Lookback Lengths and Forecast Horizons: The model does not rely on a specific input length and extracts useful temporal patterns from both shorter and longer historical observations.
  • D.2. Varying Lookback Lengths and Forecast Horizons: Forecast-horizon results are reported in Table 8 using lookback lengths from Table 5 and horizons defined by each sequence’s remaining length.
  • D.2. Varying Lookback Lengths and Forecast Horizons: The evaluated horizons include 12 hours for MIMIC-III and PhysioNet, 300 milliseconds for Human Activity, the next 3 observations for USHCN, and 10 days for Student Life.
  • D.2. Varying Lookback Lengths and Forecast Horizons: DNBNet maintains strong performance under these forecast-horizon settings, further demonstrating the effectiveness of its debiased neural basis-function response.

D.3. Complete Ablation Results

Complete ablation experiments across five datasets show that the full DNBNet consistently outperforms its variants. Removing density correction, learnable bases, average pooling, or the basis-function decoder degrades performance, supporting each component’s contribution.

  • Complete Ablation Results: The full DNBNet consistently outperforms its variants across all five datasets.These complete ablation results verify the contribution of each component.
  • Complete Ablation Results: Removing density correction p(t), learnable basis functions, average pooling, or the basis-function decoder leads to performance degradation.The alternatives to learnable bases include predefined RBF and Fourier bases.
  • Complete Ablation Results: Table 8 evaluates five irregular multivariate time-series datasets using MSE and MAE.Prediction horizons are 12 hours for MIMIC-III and PhysioNet, 300 milliseconds for Human Activity, 3 observations for USHCN, and 10 days for Student Life.
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