Partially-Observed Time Series (POTS)

• POTS is a framework for time series with incomplete observations, characterized by missing values and irregular sampling that necessitate specialized inference methods.
• It encompasses various formalizations, including masked multivariate data, latent dynamical systems, continuous-time models, and graph-structured observations.
• Advanced techniques in POTS involve reconstruction and imputation methods, neural continuous-time modeling, and control strategies to effectively analyze and forecast latent processes.

Partially-observed time series (POTS) denotes time-indexed data in which the underlying process is not fully available to the analyst. In the multivariate setting emphasized by PyPOTS, a POTS sample is an incomplete time series with missing values, A.K.A. irregularlysampled time series, commonly represented by a data tensor together with a binary mask that marks observed and missing entries (Du, 2023). In dynamical-systems formulations, partial observation may mean that only a subset of state coordinates is measured, as in $x(jh)=(z(jh),z^\dagger(jh))$ with $z$ observed and $z^\dagger$ completely missing, or that observations arrive at irregular times with a mask $\mathbf M_{t_i}$ over node-feature measurements (Okuno et al., 2023). In functional data analysis, the same idea appears as $Y_i(t)=O_i(t)X_i(t)$, where $O_i(t)$ is an observation indicator over the functional domain (Bastian et al., 30 Jun 2026). Across these settings, the central technical problem is to infer, forecast, classify, control, or otherwise analyze a latent temporal process from incomplete observations.

1. Formalizations of partial observability

A recurring formalization treats POTS as masked multivariate data. For each sample $i$, one has $X^{(i)}\in\mathbb R^{T_i\times D}$ together with $M^{(i)}\in\{0,1\}^{T_i\times D}$, where $m^{(i)}_{j,d}=1$ if feature $z$0 is observed at time $z$1 and $z$2 otherwise (Du, 2023). The irregular-sampling component is captured by variable gaps $z$3, while missingness is encoded at the feature level.

A second formalization begins from a latent dynamical system. In the autoregressive with slack time series model, the full state $z$4 evolves under a time-invariant system with first-order approximation $z$5, but only $z$6 coordinates are observed. Writing $z$7 with $z$8, the available series is $z$9, and the missing coordinates $z^\dagger$0 must be imputed jointly with the dynamics (Okuno et al., 2023).

A third formalization is continuous-time and graph-structured. In TG-NODE, one observes partial components of $z^\dagger$1 at irregular times $z^\dagger$2, with a binary mask $z^\dagger$3 specifying which node-feature entries are present (Zou et al., 2024). This makes partial observability simultaneously temporal, spatial, and feature-specific.

A fourth formalization is coarse observation of a latent Markov process. In continuous-time inference for partially observed Markov processes, the hidden state $z^\dagger$4 lives on a finite state space $z^\dagger$5, while the observation is a coarse indicator such as $z^\dagger$6. The posterior occupancy probability is then expressed as

$z^\dagger$7

with forward and backward messages defined over intervals of constant observation (Chen et al., 2023).

In functional settings, the observed object is itself a partially observed random function. The model $z^\dagger$8, $z^\dagger$9, with $\mathbf M_{t_i}$0, places POTS in the Banach space $\mathbf M_{t_i}$1, where pointwise evaluation and simultaneous confidence bands are well-defined (Bastian et al., 30 Jun 2026). This suggests that POTS is not a single data format but a family of observation models unified by latent temporal dependence and incomplete measurement.

2. Reconstruction, imputation, and latent-state recovery

One line of work reconstructs missing state components explicitly. The ARS model defines a completed state

$\mathbf M_{t_i}$2

and jointly estimates a linear operator $\mathbf M_{t_i}$3 and the slack series $\mathbf M_{t_i}$4 by minimizing

$\mathbf M_{t_i}$5

For fixed slack variables, $\mathbf M_{t_i}$6 is partially out by $\mathbf M_{t_i}$7, yielding a marginal objective $\mathbf M_{t_i}$8. Proposition 1 states that, in the $\mathbf M_{t_i}$9-dimensional time-invariant and linear case with one observed and one missing coordinate, any global minimizer reproduces the exact future observed sequence for sufficiently large $Y_i(t)=O_i(t)X_i(t)$0 (Okuno et al., 2023). The same source also notes limitations: the formulation is overparameterized $Y_i(t)=O_i(t)X_i(t)$1 free parameters), uses no explicit regularizer, and optimization can be slow or unstable if slack is initialized poorly.

A distinct strategy reconstructs a state-like representation from delays of the observed signal. In the delay embedded echo-state network framework, a scalar observation $Y_i(t)=O_i(t)X_i(t)$2 is transformed into a delay vector

$Y_i(t)=O_i(t)X_i(t)$3

Theoretical support comes from Takens’ embedding theorem: if $Y_i(t)=O_i(t)X_i(t)$4, the map $Y_i(t)=O_i(t)X_i(t)$5 is an embedding of the attractor. Strong observability further implies the existence of a continuous predictor $Y_i(t)=O_i(t)X_i(t)$6 such that $Y_i(t)=O_i(t)X_i(t)$7. An ESN with leaky-integrator update

$Y_i(t)=O_i(t)X_i(t)$8

and linear readout $Y_i(t)=O_i(t)X_i(t)$9 is then trained by ridge regression to approximate this predictor (Goswami, 2022).

When partial observability concerns regime labels rather than missing features, the PHMC-LAR model occupies an intermediate position between fully hidden and fully observed regime-switching models. The latent state $O_i(t)$0 follows a partially hidden Markov chain constrained by label sets $O_i(t)$1, while the observation satisfies a regime-specific linear autoregression

$O_i(t)$2

Learning proceeds by an EM algorithm whose E-step uses a backward-forward-backward recursion adapted to $O_i(t)$3, and whose M-step updates transition probabilities and autoregressive parameters in closed form. The paper reports that partially observed states decrease EM convergence times and that the model is robust to labelling errors in both inference and prediction tasks (Dama et al., 2021).

For continuous-time finite-state systems, latent-state recovery can be exact at the level of conditional probabilities. The continuously-sampled sum-product formulation solves forward and backward linear ODEs on each interval of constant observation, stitches them by rate-matrix boundary updates, and yields posterior marginals $O_i(t)$4 by normalization of $O_i(t)$5. Because each interval admits a matrix-exponential solution, the method gives explicitly solvable continuous-time inference for coarse measurements of a hidden Markov process (Chen et al., 2023).

3. Continuous-time latent dynamics and neural architectures

Neural continuous-time models extend POTS from missing-value completion to latent-flow modeling. Continuous Latent Process Flows assumes a latent process $O_i(t)$6 governed by an SDE,

$O_i(t)$7

and decodes a continuous observable process through a time-indexed normalizing flow $O_i(t)$8, where $O_i(t)$9 is an Ornstein–Uhlenbeck base process. Since the latent SDE has no closed-form transition density, inference uses a piecewise variational posterior process over each interval $i$0, together with a variational lower bound derived using trajectory re-weighting. The paper reports that CLPF consistently achieved the lowest test negative log-likelihood on four synthetic benchmarks and matched or beat state-of-the-art continuous-time models on three irregularized real-world datasets (Deng et al., 2021).

For networked dynamical systems, TG-NODE combines a graph Neural ODE with a graph-convolutional GRU. The latent state $i$1 evolves between observations according to

$i$2

where the nodewise dynamics obey

$i$3

At observation times, missing entries are first imputed by

$i$4

and the hidden state is updated by a graph-GRU. Two additional mechanisms are explicit: a reliability matrix $i$5, which down-weights less-trusted imputations, and a time-aware forget gate

$i$6

which discounts stale latent states when $i$7 is large. Training alternates between an imputation loss on observed entries and a weighted forecasting loss on generated dense paths (Zou et al., 2024).

SeqLink addresses the failure mode in which a single latent ODE trajectory drifts during long gaps. It retains the ODE-RNN backbone but augments each series with cross-sample linked trajectories. If $i$8 denotes a candidate latent vector from another sample $i$9, link weights $X^{(i)}\in\mathbb R^{T_i\times D}$0 are computed by an attention module, and an aggregated correction

$X^{(i)}\in\mathbb R^{T_i\times D}$1

is injected into the update of sample $X^{(i)}\in\mathbb R^{T_i\times D}$2. Training is staged: first an ODE-RNN autoencoder with cutout corruption learns latent trajectories, then the full SeqLink model is fine-tuned for imputation, forecasting, or classification (Abushaqra et al., 2022).

Partial observability also arises in spatio-temporal PDE learning. The latent neural PDE model assumes a continuous latent field $X^{(i)}\in\mathbb R^{T_i\times D}$3, observed only through noisy partial measurements $X^{(i)}\in\mathbb R^{T_i\times D}$4. Spatial derivatives are learned implicitly by a neural dynamics function over local neighborhoods $X^{(i)}\in\mathbb R^{T_i\times D}$5, producing an ODE system over node values. The framework combines collocation and the method of lines, amortized variational inference, and multiple shooting, and is described as grid-independent because neighborhoods are defined by absolute coordinates rather than the data mesh (Iakovlev et al., 2023). A plausible implication is that POTS methods increasingly blur the classical line between missing-data modeling and latent scientific-model discovery.

4. Simultaneous inference, function spaces, and irreversibility

POTS is not limited to prediction or representation learning. In dependent functional data analysis, partially observed functional time series are modeled on the space of bounded functions equipped with the supremum norm. With $X^{(i)}\in\mathbb R^{T_i\times D}$6, $X^{(i)}\in\mathbb R^{T_i\times D}$7, and physical-dependence conditions on both $X^{(i)}\in\mathbb R^{T_i\times D}$8 and $X^{(i)}\in\mathbb R^{T_i\times D}$9, simultaneous inference is based on the pointwise estimator

$M^{(i)}\in\{0,1\}^{T_i\times D}$0

Theorem 2.1 establishes asymptotic simultaneous coverage of confidence bands over the entire domain $M^{(i)}\in\{0,1\}^{T_i\times D}$1, with uniform band-width of order $M^{(i)}\in\{0,1\}^{T_i\times D}$2 and no extra log-factors. The same framework extends multiscale statistics to test stationarity, piecewise constancy, and threshold exceedance in non-stationary partially observed functional time series (Bastian et al., 30 Jun 2026). This is significant because the paper explicitly contrasts $M^{(i)}\in\{0,1\}^{T_i\times D}$3 with $M^{(i)}\in\{0,1\}^{T_i\times D}$4: pointwise values and simultaneous confidence bands are natural in the former and unavailable in the latter.

In stochastic thermodynamics, partial observation changes what can be inferred about nonequilibrium dissipation. For a continuous-time Markov chain observed only after coarse-graining, the entropy production rate is estimated through lower bounds based on the Kullback–Leibler divergence between forward and backward path ensembles. In second-order semi-Markov form, the bound decomposes into an affinity term and a waiting-time-distribution term,

$M^{(i)}\in\{0,1\}^{T_i\times D}$5

The paper presents a hierarchy of lower bounds as more information becomes observable and introduces a “Transformed semi-CG” representation in which runs of hidden states are relabeled by their length $M^{(i)}\in\{0,1\}^{T_i\times D}$6, allowing tighter use of waiting-time asymmetries (Kapustin et al., 2022). The reported conclusion is that the Transformed semi-CG KLD bound is the tightest among the discussed bounds and is especially informative near stalling conditions where current-based asymmetries vanish.

These two lines of work show that POTS methodology can target uncertainty quantification and physical irreversibility, not only imputation or forecasting. This suggests a broader interpretation of POTS as a statistical regime in which the observed process may fail to inherit the full probabilistic structure of the latent process.

5. Control, decision-making, and parameter learning under partial observability

Partial observability also enters control through belief-state optimization. In the multi-time-scale nonlinear stochastic system inspired by leukemia treatment, the latent state evolves on a time index set $M^{(i)}\in\{0,1\}^{T_i\times D}$7, observations arrive only on $M^{(i)}\in\{0,1\}^{T_i\times D}$8, and controls may be chosen only on $M^{(i)}\in\{0,1\}^{T_i\times D}$9. The model is

$m^{(i)}_{j,d}=1$0

To value measurements for parameter identification, the framework introduces a Fisher information matrix built from output sensitivities and augments the state to $m^{(i)}_{j,d}=1$1. Under regularity assumptions, the resulting model is shown to be a regular nonstationary POMDP with an equivalent belief-space MDP and an existence theorem for an optimal Markov policy in belief space (Chapman et al., 2022).

A complementary formulation develops continuous-time POMDPs for finite latent state and action spaces. There, the belief $m^{(i)}_{j,d}=1$2 follows a controlled jump-diffusion on the simplex,

$m^{(i)}_{j,d}=1$3

and the optimal discounted value function satisfies an HJB-type partial integro-differential equation in belief space. Two deep-learning solution methods are given: an offline collocation approach that fits a value network by minimizing HJB residuals, and an online advantage-updating algorithm that combines replay buffers with continuous-time temporal-difference structure (Alt et al., 2020).

The educational tutoring environment provides a finite-horizon POMDP in which the hidden state contains concept masteries and motivation, observations are quiz or probe outcomes, and actions include lectures, tutoring sessions, and probing interventions. The model explicitly defines a value-of-information criterion

$m^{(i)}_{j,d}=1$4

with the policy principle that probing should occur only if $m^{(i)}_{j,d}=1$5 exceeds the probe cost. The paper compares DQN with heuristic belief-based baselines and reports that probing interventions can reduce the difficulty of student estimation, while increasing hidden information makes the problem harder (Jiang et al., 19 Nov 2025).

Parameter learning under partial observability remains difficult when the latent process is non-Markovian in calendar time. For partially observed time-changed SDEs,

$m^{(i)}_{j,d}=1$6

with discrete noisy observations $m^{(i)}_{j,d}=1$7, the likelihood is intractable and is attacked through Girsanov-based decomposition, Euler–Maruyama path augmentation, unbiased score-based stochastic approximation, and multilevel MCMC. The multilevel Bayesian estimator is proved to achieve mean square error $m^{(i)}_{j,d}=1$8 with cost $m^{(i)}_{j,d}=1$9 (Zhao et al., 11 May 2026). Taken together, these works show that POTS is as much a control and inverse-problem setting as it is a missing-data setting.

6. Tasks, benchmarks, and software ecosystems

A software-centric view of POTS is provided by PyPOTS. The 2023 toolbox defines four core tasks—imputation, classification, clustering, and forecasting—and organizes them through task-specific base classes, metrics, dataset utilities, and model serialization (Du, 2023). The 2026 tutorial expands this to an end-to-end pipeline with four interacting modules: missingness simulation, data preprocessing, model training and inference, and evaluation and visualization. It gives explicit missingness models for MCAR, MAR, and MNAR, describes preprocessing steps such as time-bucketing, normalization over observed entries, and mask-and-delta encoding, and extends the task set to include anomaly detection (Du et al., 27 Apr 2026).

Evaluation protocols in the literature vary by task and observation model. In ARS, forecasting is assessed at horizons $z$00 using

$z$01

and reported as the mean relative error $z$02; in all synthetic circular and Lorenz-type settings described, the relative error is below $z$03, indicating improvement over standard AR(1) (Okuno et al., 2023). In TG-NODE, the metric is test MSE $z$04 over interpolation and extrapolation on an irregular 8-node oscillator network with $z$05, $z$06, and $z$07 observed entries; the proposed method is reported to uniformly outperform RNN$z$08, GRU-Decay, and Neural ODE (Zou et al., 2024). SeqLink evaluates imputation and forecasting by MSE and classification by AUC on toy data, Electricity load, ETT, Weather, and PhysioNet, reporting gains that are largest when missingness or gap length is largest and sequences are shortest (Abushaqra et al., 2022). CLPF uses test negative log-likelihood and sequential prediction error, and reports lowest test negative log-likelihood on irregular synthetic benchmarks as well as strong results on Mujoco-Hopper, Beijing air-quality, and PTB-ECG (Deng et al., 2021).

The software literature also emphasizes reproducibility and engineering discipline. PyPOTS is described as open-source, publicly available on GitHub, and designed with unit testing, continuous integration, continuous delivery, code coverage, maintainability evaluation, interactive tutorials, and parallelization (Du, 2023). The 2026 tutorial further recommends logging data splits, setting random seeds, recording software versions, registering custom metrics, and using contribution-ready engineering practices when extending models (Du et al., 27 Apr 2026).

Across these benchmark and software perspectives, a common pattern emerges: POTS research treats missingness handling, representation learning, forecasting, and downstream analysis as a single end-to-end problem rather than as separable preprocessing and modeling stages. This suggests why recent work places equal weight on observation masks, continuous-time dynamics, uncertainty-aware losses, and standardized evaluation pipelines.