A Statistical Approach for Modeling Irregular Multivariate Time Series with Missing Observations

Dingyi Nie; Yixing Wu; C. -C. Jay Kuo

cs.LGFeb 23, 2026

A Statistical Approach for Modeling Irregular Multivariate Time Series with Missing Observations

Authors: Dingyi Nie, Yixing Wu, C. -C. Jay Kuo

Organizations: University of Southern California

Abstract

Irregular multivariate time series with missing values present significant challenges for predictive modeling in domains such as healthcare. While deep learning approaches often focus on temporal interpolation or complex architectures to handle irregularities, we propose a simpler yet effective alternative: extracting time-agnostic summary statistics to eliminate the temporal axis. Our method computes four key features per variable-mean and standard deviation of observed values, as well as the mean and variability of changes between consecutive observations to create a fixed-dimensional representation. These features are then utilized with standard classifiers, such as logistic regression and XGBoost. Evaluated on four biomedical datasets (PhysioNet Challenge 2012, 2019, PAMAP2, and MIMIC-III), our approach achieves state-of-the-art performance, surpassing recent transformer and graph-based models by 0.5-1.7% in AUROC/AUPRC and 1.1-1.7% in accuracy/F1-score, while reducing computational complexity. Ablation studies demonstrate that feature extraction-not classifier choice-drives performance gains, and our summary statistics outperform raw/imputed input in most benchmarks. In particular, we identify scenarios where missing patterns themselves encode predictive signals, as in sepsis prediction (PhysioNet, 2019), where missing indicators alone can achieve 94.2% AUROC with XGBoost, only 1.6% lower than using original raw data as input. Our results challenge the necessity of complex temporal modeling when task objectives permit time-agnostic representations, providing an efficient and interpretable solution for irregular time series classification.

Figures & tables

Methods	P19		P12		PAM
	AUROC	AUPRC	AUROC	AUPRC	Accuracy	Precision	Recall	F1 score
Transformer	80.7 $\pm$ 3.8	42.7 $\pm$ 7.7	83.3 $\pm$ 0.7	47.9 $\pm$ 3.6	83.5 $\pm$ 1.5	84.8 $\pm$ 1.5	86.0 $\pm$ 1.2	85.0 $\pm$ 1.3
Trans-mean	83.7 $\pm$ 1.8	45.8 $\pm$ 3.2	82.6 $\pm$ 2.0	46.3 $\pm$ 4.0	83.7 $\pm$ 2.3	84.9 $\pm$ 2.6	86.4 $\pm$ 2.1	85.4 $\pm$ 2.4
GRU-D	83.9 $\pm$ 1.7	46.9 $\pm$ 2.1	81.9 $\pm$ 2.1	46.1 $\pm$ 4.7	83.3 $\pm$ 1.6	84.6 $\pm$ 1.6	85.2 $\pm$ 1.3	84.8 $\pm$ 1.2
SeFT	81.2 $\pm$ 2.3	41.9 $\pm$ 3.1	73.9 $\pm$ 2.5	31.1 $\pm$ 1.4	67.1 $\pm$ 2.2	70.0 $\pm$ 2.4	68.2 $\pm$ 1.5	68.5 $\pm$ 1.8
mTAND	84.4 $\pm$ 1.3	50.6 $\pm$ 2.4	84.2 $\pm$ 0.8	48.2 $\pm$ 3.4	74.6 $\pm$ 4.3	74.3 $\pm$ 4.0	79.5 $\pm$ 2.8	76.8 $\pm$ 3.4

Table 1: Performance comparison of our method with various state-of-the-art deep learning methods across three datasets (P19, P12, and PAM).

Methods	MIMIC-III
	AUROC	AUPRC
GRU-D	85.7 $\pm$ 0.2	52.0 $\pm$ 0.8
GRU-SIMPLE	82.8 $\pm$ 0.0	43.6 $\pm$ 0.4
IP-NETS	83.2 $\pm$ 0.5	48.3 $\pm$ 0.4
PHASED-LSTM	80.3 $\pm$ 0.4	37.1 $\pm$ 0.5
TRANSFORMER	82.1 $\pm$ 0.3	42.6 $\pm$ 1.0

Table 2: Performance comparison of our method with various deep learning methods on MIMIC-III Benchmarks: mortality prediction subset.

Methods	P19		P12		MIMIC-III
	AUROC	AUPRC	AUROC	AUPRC	AUROC	AUPRC
Mean-Imp + LR	74.4 $\pm$ 1.5	43.4 $\pm$ 2.7	64.4 $\pm$ 1.3	26.9 $\pm$ 2.0	70.4 $\pm$ 1.4	28.5 $\pm$ 1.5
For-Imp + LR	83.0 $\pm$ 0.7	54.7 $\pm$ 2.3	70.7 $\pm$ 2.0	33.0 $\pm$ 3.1	76.1 $\pm$ 0.7	35.1 $\pm$ 0.9
Lin-Imp + LR	83.4 $\pm$ 0.9	55.7 $\pm$ 1.6	73.2 $\pm$ 1.8	34.2 $\pm$ 2.8	76.4 $\pm$ 0.5	36.2 $\pm$ 0.9
Proposed + LR	79.5 $\pm$ 1.0	28.0 $\pm$ 2.2	84.3 $\pm$ 1.7	50.0 $\pm$ 4.0	83.4 $\pm$ 0.6	45.8 $\pm$ 1.6
Raw + XGB	95.6 $\pm$ 0.3	80.1 $\pm$ 0.7	81.1 $\pm$ 1.2	44.0 $\pm$ 2.2	83.3 $\pm$ 0.9	48.2 $\pm$ 2.2

Table 3: Performance comparison: our proposed statistical features vs. raw input and imputed features; on P19, P12, and MIMIC-III datasets.

Methods	PAM
	Accuracy	Precision	Recall	F1
Mean-Imp + LR	54.7 $\pm$ 2.1	58.4 $\pm$ 2.7	53.3 $\pm$ 2.2	52.0 $\pm$ 2.0
For-Imp + LR	57.2 $\pm$ 1.7	62.0 $\pm$ 1.4	54.7 $\pm$ 1.2	53.6 $\pm$ 1.3
Lin-Imp + LR	60.1 $\pm$ 2.2	63.0 $\pm$ 1.9	58.4 $\pm$ 1.6	57.8 $\pm$ 1.7
Proposed + LR	91.0 $\pm$ 0.9	92.2 $\pm$ 0.6	92.6 $\pm$ 0.8	92.3 $\pm$ 0.7
Raw + XGB	85.1 $\pm$ 0.7	88.9 $\pm$ 0.5	84.1 $\pm$ 1.2	85.7 $\pm$ 0.9

Table 4: Performance comparison: our proposed statistical features vs. raw input and imputed features; on PAMAP2.

Experiments	P19 AUROC	P12 AUROC	MIMIC-III AUROC	PAM F1
$\boldsymbol{M}$ + LR	85.7 $\pm$ 0.6	56.4 $\pm$ 2.1	60.3 $\pm$ 1.0	14.6 $\pm$ 1.0
$\boldsymbol{F}$ + LR	79.5 $\pm$ 1.0	84.3 $\pm$ 1.7	83.4 $\pm$ 0.6	92.3 $\pm$ 0.7
$\boldsymbol{M}$ + XGB	94.2 $\pm$ 0.3	70.9 $\pm$ 1.0	70.2 $\pm$ 1.3	8.5 $\pm$ 0.4
$\boldsymbol{X}$ + XGB	95.6 $\pm$ 0.3	81.1 $\pm$ 1.2	83.3 $\pm$ 0.9	85.7 $\pm$ 0.9
$\boldsymbol{F}$ + XGB	90.0 $\pm$ 0.6	85.7 $\pm$ 0.9	85.9 $\pm$ 0.8	97.6 $\pm$ 0.4

Table 5: Extensive experiments on discriminative power of mask array

\boldsymbol{M}

across four datasets.

Figure 1: Feature importance analysis on the P19 dataset using XGBoost. The importance metric is Total Gain. For each of the four feature types— $\mu^{(0)}$ (Mean), $\sigma^{(0)}$ (STD), $\mu^{(1)}$ (Mean Change), and $\sigma^{(1)}$ (Change Variability)—we summed the gains of all corresponding features, averaged over 5-fold cross-validation.

Irregular Time-Series Modeling

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Irregular time series, characterized by non-uniform sampling intervals, missing observations, and variable lengths, are ubiquitous in healthcare, mobility, and environmental monitoring, yet effective and interpretable classifiers for this setting are limited. Existing approaches often rely on imputation, which can obscure the temporal structure of the data, or require complex neural architectures that are opaque and difficult to explain. In this work, we extend the Bag-Of-Receptive-Fields (BORF), a fast, deterministic, and interpretable transform for time series, to the irregular setting. Our key contribution is a time-weighted normalization scheme in which each observation is weighted proportionally to its associated time delta, making pattern extraction sensitive to the actual temporal distribution of samples rather than only their index position. This requires deriving an efficient sliding-window recurrence for the time-weighted standard deviation, preserving the linear time complexity of BORF. We benchmark the resulting method against state-of-the-art irregular time series classifiers on datasets from the PYRREGULAR repository, demonstrating competitive classification performance with the added benefit of human-interpretable explanations.

Francesco Spinnato

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⋆Division of Information Science and Engineering, KTH Royal Institute of Technology, Stockholm, Sweden

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Abstract

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