Federated Markov Imputation in Clinical ICU Data

• Federated Markov Imputation is a privacy-preserving temporal imputation method that uses global Markov-chain transition matrices to harmonize irregular ICU time series across multiple sites.
• It leverages secure aggregation of discretized transition counts to mitigate the limitations of local imputation, ensuring better consistency and predictive performance in federated learning.
• Evaluated on the MIMIC-IV dataset for sepsis onset prediction, FMI outperforms local mean and local Markov imputation baselines, particularly under irregular sampling conditions.

Federated Markov Imputation (FMI) is a privacy-preserving temporal imputation method designed for federated clinical settings, especially multi-ICU environments where patient time series are incomplete and sampled at different temporal resolutions. It addresses a practical obstacle in federated learning on electronic health records: hospitals cannot share raw data, yet downstream predictive models such as sepsis onset predictors still require a consistent time-series format. FMI therefore lets multiple Intensive Care Units collaboratively build global transition models for temporal imputation without exposing patient-level trajectories, and it is evaluated on a real-world sepsis onset prediction task using the MIMIC-IV dataset, where it outperforms local imputation baselines, especially in scenarios with irregular sampling intervals across ICUs (Düsing et al., 25 Sep 2025).

1. Problem setting in federated multi-ICU time series

In a typical ICU, measurements such as lab values, vitals, and interventions are collected irregularly because clinicians order tests based on need rather than on a fixed schedule. In federated multi-centric settings, this irregularity is compounded by heterogeneity across sites: one ICU may record hourly, another every 2 hours, another every 3 hours. Since federated training requires a shared input format, the data often must be aligned to a common temporal grid. Sites with coarser sampling must then fill in many more missing values to match the global resolution (Düsing et al., 25 Sep 2025).

The paper frames this as a setting in which simple local imputation methods that rely only on a site’s own data can degrade badly when their temporal resolution is low, and models trained on imputed sequences may lose clinically relevant temporal dynamics. FMI is motivated by this exact issue. It is not presented as a generic missing-data routine detached from downstream learning; rather, it is a preprocessing component intended to make federated learning on electronic health records more practical when hospitals need a consistent time-series representation but cannot exchange raw patient trajectories (Düsing et al., 25 Sep 2025).

A central misconception addressed by the formulation is that temporal imputation in federated learning is merely a local data-cleaning step. The method instead treats temporal imputation as a federation-wide estimation problem, because the temporal resolution mismatch itself is distributed across institutions. This suggests that the quality of imputation depends not only on within-site signal but also on whether cross-site temporal dynamics can be aggregated without violating privacy.

2. Markov-chain formulation and federated aggregation

At its core, FMI uses first-order Markov chains over discretized feature values. Each ICU discretizes each feature into $n$ bins and constructs a local transition matrix $T_c \in \mathbb{R}^{n \times n}$, where each entry describes the empirical probability of moving from one bin to another between adjacent time steps. If $b_t$ denotes the bin at time $t$, then the local transition estimate at site $c$ is based on observed adjacent pairs $(b_{t-1}, b_t)$ in the non-missing data (Düsing et al., 25 Sep 2025).

A compact expression of the model is

$P(b_t=j \mid b_{t-1}=i) = T_{\text{fed}}(i,j),$

with federated transition entries estimated from aggregated counts:

$$T_{\text{fed}}(i,j)=\frac{N_{\text{fed}}(i,j)}{\sum_{k=1}^n N_{\text{fed}}(i,k)}.$$

Locally, each ICU computes

$T_c(i,j)=\frac{N_c(i,j)}{\sum_{k=1}^n N_c(i,k)},$

and secure aggregation produces

$N_{\text{fed}}(i,j)=\sum_c N_c(i,j)$

without exposing any individual $T_c \in \mathbb{R}^{n \times n}$0 (Düsing et al., 25 Sep 2025).

The federated step uses Secure Aggregation so that each site contributes masked transition counts rather than raw data. The server only receives the sum of these masked counts, enabling the construction of a global transition matrix $T_c \in \mathbb{R}^{n \times n}$1 without revealing any hospital-specific statistics. The method therefore does not federate model parameters in the usual deep-learning sense; it federates empirical transition statistics, which are then used for imputation. The privacy mechanism is aligned with that structure: only aggregated counts are shared, and secure aggregation prevents the server from inspecting site-level transitions (Düsing et al., 25 Sep 2025).

This design is methodologically significant because the shared object is not a predictor but a population-level temporal prior. A plausible implication is that FMI occupies an intermediate layer between data harmonization and model training: it is neither purely local preprocessing nor a standard federated optimization algorithm over prediction parameters.

3. Imputation rule and two-stage pipeline

The imputation rule is a Markov-based maximum-probability reconstruction. If both neighboring bins around a missing value are known, FMI chooses the missing bin that maximizes the product of the left and right transition probabilities:

$T_c \in \mathbb{R}^{n \times n}$2

If only one neighbor is known, it uses a one-sided transition probability. For runs of consecutive missing values, it infers the most probable path recursively. After the most likely bin is found, the imputed value is taken as the midpoint of that bin. The paper characterizes the method as simple, interpretable, and lightweight compared with neural imputation methods (Düsing et al., 25 Sep 2025).

The pipeline is explicitly two-stage. First, each site discretizes its data, computes local transition counts, and participates in the secure aggregation protocol to obtain the global transition matrix. Second, using that matrix, each ICU imputes missing values in its local sequences before training the downstream federated predictor. FMI is thus not the prediction model itself; it is the preprocessing and imputation component that feeds better time series into the federated classifier (Düsing et al., 25 Sep 2025).

The paper contrasts FMI with two baselines. Local mean imputation fills missing values using a site’s feature-wise mean and ignores temporal structure entirely. Local Markov Imputation (LMI) uses each ICU’s own transition matrix only. LMI can work in the regular setting, but it becomes infeasible in the irregular setting for sites that do not have hourly data, because they lack the fine-grained temporal resolution needed to estimate the same transitions as hourly sites. FMI overcomes this by learning a global transition model from all sites (Düsing et al., 25 Sep 2025).

4. Experimental setting and reported performance

The empirical evaluation uses MIMIC-IV and focuses on 28,610 patients relevant to sepsis prediction, assigned to seven ICU types: MICU, MICU/SICU, SICU, TSICU, CCU, CVICU, and NSICU. For each patient, 25 clinically relevant features are extracted and aggregated into six 1-hour windows after admission. Labels follow the Sepsis-3 definition: a patient is positive if sepsis develops within the first 30 hours of ICU admission. Two temporal settings are considered. In the Regular setting, all ICUs retain the hourly data. In the Irregular setting, heterogeneity is simulated by randomly assigning two ICUs to 2-hour intervals and two ICUs to 3-hour intervals, removing intermediate values accordingly. Performance is evaluated primarily with AUC (Düsing et al., 25 Sep 2025).

The downstream model is a federated LSTM network: an input layer, followed by three LSTM layers with 16 units each, each with batch normalization and dropout of 0.2, then a linear layer down to 8 units and a final binary output layer. Training proceeds for 50 rounds with FedAvg. The baselines are local mean imputation and LMI (Düsing et al., 25 Sep 2025).

Setting	FMI mean AUC	Baseline means
Regular	0.8878	LMI: 0.8710; local mean imputation: 0.8634
Irregular	0.8629	local mean imputation: 0.7961

In the Regular setting, all methods exceed AUC 0.8, which the paper treats as a threshold for clinical applicability. FMI is better than LMI on 5 of the 7 ICUs, though two ICUs slightly favor LMI. In the Irregular setting, performance drops overall, as expected, and local mean imputation degrades notably, with mean AUC falling below 0.8. The gains are especially pronounced for ICU groups with 3-hour intervals, such as MICU/SICU and NSICU, where FMI appears to mitigate the loss from coarse sampling. For some 1-hour sites, performance is similar to or slightly better than in the regular setting, showing that the irregularity primarily harms coarse-resolution sites (Düsing et al., 25 Sep 2025).

The paper’s explanation is that a global transition matrix provides more robust temporal statistics than a local one, especially when local data are sparse or coarse. Local Markov estimates can be unstable when a site has fewer observed transitions, and mean imputation ignores sequential structure altogether. By pooling transition counts across sites, FMI gets a richer estimate of likely state changes while preserving privacy.

5. Relation to adjacent frameworks and acronym ambiguity

The acronym “FMI” is not unique across arXiv literature. In "Automatic integration of SystemC in the FMI standard for Software-defined Vehicle design," FMI means Functional Mock-up Interface, not Federated Markov Imputation. That paper uses FMI as a standardized, open, tool-agnostic interface for exchanging and co-simulating dynamic models across simulation tools, with the corresponding packaged artifact called the Functional Mock-up Unit (FMU) (Pollo et al., 27 Aug 2025). In the context of federated clinical time-series imputation, by contrast, FMI denotes a Markov-chain–based imputation method and has no relation to the co-simulation standard.

FMI is also adjacent to, but distinct from, federated EM-based imputation. "Federated Expectation Maximization with heterogeneity mitigation and variance reduction" introduces FedEM and its imputation specialization FedMissEM, where missing values are handled through conditional expectations under the current model and the federated system exchanges compressed sufficient-statistic information rather than raw data (Dieuleveut et al., 2021). That framework is iterative rather than Markovian. It does not introduce a Markov chain or Markov transition model for the missing-data process. If FMI is intended specifically as Federated Markov Imputation, FedMissEM is conceptually related through federated imputation, but not via a Markov model or a Markov transition kernel (Dieuleveut et al., 2021).

These distinctions matter because they separate three different senses of shared structure: a co-simulation interface standard, an EM-based latent-variable imputation framework, and a federated temporal imputer based on first-order transition statistics.

6. Limitations, privacy assumptions, and proposed extensions

The paper is explicit that the evaluation is preliminary. The method is only tested on a single task and dataset, with comparisons limited to two baselines. FMI is also based on a first-order Markov assumption and discretization into bins, which may lose fine-grained dynamics and may not capture long-range temporal dependencies. It is not yet evaluated against more sophisticated imputers such as neural sequence models, and the authors explicitly note the need for more detailed analyses and broader comparisons (Düsing et al., 25 Sep 2025).

On the privacy side, secure aggregation protects individual transition counts, but FMI still depends on sharing aggregated statistics, so it inherits the usual federated-learning assumptions about honest-but-curious servers and secure protocol execution. The method also does not fully address potential leakage from highly specific aggregate patterns in small cohorts, although secure aggregation reduces direct exposure substantially (Düsing et al., 25 Sep 2025).

The proposed extensions follow directly from these limitations. The paper suggests testing more baselines, validating the method in real clinical deployments, and likely extending the approach to richer temporal models or higher-order dynamics. It also names adaptive binning, feature-specific Markov models, and integrating FMI with more advanced federated sequence models as natural directions. Another promising direction would be to generalize beyond simple imputation toward jointly learning the imputer and predictor in a federated pipeline (Düsing et al., 25 Sep 2025).

Even in its current form, FMI is significant because it tackles a practical but under-addressed problem: how to harmonize irregular ICU time series across hospitals without sharing sensitive patient data, and how to do so in a way that improves downstream prediction performance rather than just reconstructing missing values in isolation.