Time-Varying Offset Estimation (TVOE)

• Time-Varying Offset Estimation (TVOE) is the process of tracking an offset parameter that evolves over time across applications like clock synchronization and sensor calibration.
• Researchers employ dynamic models such as Gauss-Markov random walks and recursive filtering to account for oscillator drift, mobility, or transmission delays.
• Empirical results demonstrate that TVOE methods reduce estimation errors and improve performance in areas like visual-inertial odometry, radar synchronization, and multivariate time series analysis.

Searching arXiv for recent and relevant papers on Time-Varying Offset Estimation and related temporal calibration/synchronization problems. Time-Varying Offset Estimation (TVOE) denotes the estimation of an offset parameter that is allowed, or required, to evolve over time rather than remain constant over an observation window. Across the literature, the term appears in several closely related settings: clock synchronization in wireless sensor and mobile networks, online temporal calibration in visual-inertial and radar-inertial odometry, delay tracking between sensor signals, synchronization for multicarrier communication systems, and continuously varying lag estimation between coupled multivariate time series. In all of these formulations, the common technical premise is that a static-offset model is physically mismatched when oscillator imperfections, mobility, transmission delays, rolling-shutter effects, sensor latency, or changing system dynamics induce gradual or irregular temporal drift. Representative formulations include Gauss-Markov random walks for clock parameters (Ahmad et al., 2012), per-frame random-walk time offsets in visual-inertial optimization (Ling et al., 2018), frame-wise online offset evolution models in VIO (Xiong et al., 2024), and state-space tracking of timing and frequency residuals in bistatic ISAC (Wang et al., 12 Jul 2025).

1. Definitions and recurring formulations

In the most general relative-measurement setting, each node $u$ has an unknown scalar variable $x_u \in \mathbb{R}$, and neighboring nodes obtain noisy relative measurements

$\zeta_{u,v}(k) = x_u - x_v + \epsilon_{u,v}(k).$

When $x_u$ is interpreted as a clock offset, TVOE becomes a distributed estimation problem over time-varying graphs (Liao et al., 2013).

In clock synchronization, the local clock is often modeled as

$\tau_u(t) = \alpha_u t + \beta_u,$

where $\alpha_u$ is clock skew and $\beta_u$ is clock offset (Liao et al., 2013). A related two-way timing formulation rewrites the problem in terms of

$$\xi \triangleq d+\theta,\qquad \psi \triangleq d-\theta,$$

with clock offset recovered as

$\theta = \frac{\xi-\psi}{2},$

and then models $\xi_k,\psi_k$ as time-varying hidden states (Ahmad et al., 2012).

In camera–IMU temporal calibration, the actual image capture time is modeled as

$x_u \in \mathbb{R}$0

where $x_u \in \mathbb{R}$1 is an unknown camera–IMU time offset for frame $x_u \in \mathbb{R}$2 (Ling et al., 2018). In high-dynamic-motion VIO, the temporal relation is written as

$x_u \in \mathbb{R}$3

with the offset evolving according to

$x_u \in \mathbb{R}$4

to represent cumulative drift and white Gaussian noise (Xiong et al., 2024).

In radar–IMU fusion, the relative time offset is defined as

$x_u \in \mathbb{R}$5

and incorporated directly into the EKF state vector (Kim et al., 2 Feb 2025). In bistatic ISAC, timing and carrier-frequency residuals are grouped into a 2D hidden state,

$x_u \in \mathbb{R}$6

and tracked frame to frame (Wang et al., 12 Jul 2025).

A broader interpretation appears in multivariate time-series analysis, where TVOE denotes a continuously varying lead-lag offset $x_u \in \mathbb{R}$7 between series $x_u \in \mathbb{R}$8 and $x_u \in \mathbb{R}$9, constrained to a finite search window $\zeta_{u,v}(k) = x_u - x_v + \epsilon_{u,v}(k).$0 (James et al., 2021). This suggests that TVOE is best regarded as a modeling class rather than a single algorithmic family.

2. Dynamic models of offset evolution

A central distinction in TVOE research is whether the offset itself is modeled as dynamic, or whether time variation is induced by other latent parameters such as skew or drift. In Bayesian clock-offset estimation for two-way message exchange, the hidden parameters follow a Gauss-Markov random walk,

$\zeta_{u,v}(k) = x_u - x_v + \epsilon_{u,v}(k).$1

with uniform priors on the initial states (Ahmad et al., 2012). The same random-walk structure is specialized to exponentially distributed delays in a factor-graph treatment of time-varying clock offset (Ahmad et al., 2012).

Optimization-based visual-inertial odometry models the per-frame camera–IMU offset as a slowly varying latent variable with continuous-time prior

$\zeta_{u,v}(k) = x_u - x_v + \epsilon_{u,v}(k).$2

which discretizes to

$\zeta_{u,v}(k) = x_u - x_v + \epsilon_{u,v}(k).$3

and contributes the regularization term

$\zeta_{u,v}(k) = x_u - x_v + \epsilon_{u,v}(k).$4

to the objective (Ling et al., 2018). The corresponding interpretation is that the offset varies slowly enough to be tracked, but not assumed constant.

TON-VIO uses a related but more explicitly learned formulation. Rather than relying solely on a hand-crafted stochastic prior, it treats the offset sequence across optimization windows as learnable temporal structure via

$\zeta_{u,v}(k) = x_u - x_v + \epsilon_{u,v}(k).$5

with the previous-window estimate normalized to $\zeta_{u,v}(k) = x_u - x_v + \epsilon_{u,v}(k).$6 and processed by an LSTM (Xiong et al., 2024). A plausible implication is that TVOE methods increasingly separate into model-based stochastic filters and data-driven sequence priors.

In radar-inertial odometry, the nominal offset dynamics are

$\zeta_{u,v}(k) = x_u - x_v + \epsilon_{u,v}(k).$7

while the error-state model includes random-walk noise $\zeta_{u,v}(k) = x_u - x_v + \epsilon_{u,v}(k).$8, so the practical effect is a constant nominal state with random-walk uncertainty (Kim et al., 2 Feb 2025). Bistatic ISAC adopts the state evolution

$\zeta_{u,v}(k) = x_u - x_v + \epsilon_{u,v}(k).$9

for joint TO/CFO tracking (Wang et al., 12 Jul 2025).

By contrast, the UWB synchronization formulation

$x_u$0

keeps $x_u$1 constant while allowing the effective timing mismatch to evolve linearly because of drift $x_u$2 (Mallat et al., 2014). The paper explicitly frames this as a time-varying synchronization error caused by drift rather than an arbitrarily time-varying offset function.

3. Observation models and sources of temporal information

TVOE methods differ most sharply in how offset information enters the likelihood or residual model. In two-way timing exchange, measurements are generated from forward and reverse message timestamps: $x_u$3 or equivalently

$x_u$4

with Gaussian, exponential, or log-normal likelihoods in the factor-graph formulation (Ahmad et al., 2012). In the exponential case, the support constraints

$x_u$5

play a crucial role in deriving closed-form max-product recursions (Ahmad et al., 2012, Ahmad et al., 2012).

In distributed mobile-network synchronization, each edge provides a noisy relative measurement of the offset or log-skew. For offset estimation,

$x_u$6

so the problem reduces to scalar relative-measurement estimation (Liao et al., 2013). The same paper notes an important distinction: offset noise can be biased because

$x_u$7

In VIO, the observation side is usually visual. TON-VIO adopts the feature-shift model

$x_u$8

so the offset is inferred through the image-plane displacement required for temporal alignment (Xiong et al., 2024). The paper’s critique is that conventional estimates of

$x_u$9

fail when tracking is unstable, motivating learned feature-velocity models (Xiong et al., 2024).

Optimization-based monocular VIO propagates pose to the actual capture time $\tau_u(t) = \alpha_u t + \beta_u,$0, for example through

$\tau_u(t) = \alpha_u t + \beta_u,$1

so the visual reprojection residual becomes directly sensitive to $\tau_u(t) = \alpha_u t + \beta_u,$2 (Ling et al., 2018).

Radar-inertial odometry uses a single-scan radar ego-velocity model,

$\tau_u(t) = \alpha_u t + \beta_u,$3

and applies the measurement at the aligned time

$\tau_u(t) = \alpha_u t + \beta_u,$4

so that temporal calibration affects both propagation and update scheduling (Kim et al., 2 Feb 2025).

In bistatic ISAC, the observation channel is the line-of-sight path. The measured LoS delay and Doppler are compared against geometric predictions,

$\tau_u(t) = \alpha_u t + \beta_u,$5

to form residual observations

$\tau_u(t) = \alpha_u t + \beta_u,$6

(Wang et al., 12 Jul 2025).

A different observation mechanism appears in adaptive delay estimation between signals. There the delay is encoded as the phase of an all-pass response

$\tau_u(t) = \alpha_u t + \beta_u,$7

and extracted from FIR coefficients by

$\tau_u(t) = \alpha_u t + \beta_u,$8

The offset is thus not located through cross-correlation peak search but through adaptive filter identification (Jelfs et al., 2021).

4. Inference architectures

The literature contains four dominant inference architectures: exact or closed-form Bayesian inference on tree-structured graphs, distributed consensus-like algorithms, recursive filtering, and optimization-based batch or sliding-window estimation.

Factor graphs and message passing

For two-way timing exchange, the posterior factorizes into two chain-structured factor graphs, one for $\tau_u(t) = \alpha_u t + \beta_u,$9 and one for $\alpha_u$0, with transition factors $\alpha_u$1, $\alpha_u$2 and measurement factors $\alpha_u$3, $\alpha_u$4 (Ahmad et al., 2012). Because the graph is a tree, max-product message passing yields the exact MAP estimate. The resulting recursion leads to closed forms such as

$\alpha_u$5

and the final estimator

$\alpha_u$6

(Ahmad et al., 2012). In the exponential-delay specialization, the estimator simplifies to nested minima of affine-transformed timestamps (Ahmad et al., 2012).

Distributed averaging and MJLS analysis

In mobile networks, non-reference nodes update according to

$\alpha_u$7

while reference nodes remain pinned to the true value (Liao et al., 2013). The corresponding error dynamics form a leader-follower consensus-like system under Markovian switching topology, and mean-square behavior is analyzed with Markov jump linear system tools (Liao et al., 2013).

Kalman and extended Kalman filtering

Recursive filtering is prevalent when the offset is treated as a latent state with process noise. In bistatic ISAC, the paper calls the method an EKF, but the written equations are the standard linear Kalman recursion: $\alpha_u$8 followed by the update

$\alpha_u$9

for $\beta_u$0 (Wang et al., 12 Jul 2025).

In radar-inertial odometry, the ES-EKF augments the state with $\beta_u$1 and derives a dedicated measurement Jacobian term

$\beta_u$2

by chain rule through orientation and velocity dependence on the aligned time $\beta_u$3 (Kim et al., 2 Feb 2025). The paper emphasizes that temporal calibration is not just an additive state but changes when the radar update is applied in the IMU time stream.

Sliding-window nonlinear optimization

Optimization-based VIO estimates time offset jointly with pose, velocity, biases, extrinsics, and feature positions. The state is extended from

$\beta_u$4

to

$\beta_u$5

and optimized in a MAP objective combining priors, inertial residuals, visual reprojection residuals, and offset smoothness priors (Ling et al., 2018). TON-VIO similarly augments the estimation objective with a learned offset prior term

$\beta_u$6

and modifies visual residual construction through FVON (Xiong et al., 2024).

5. Domain-specific realizations

The diversity of TVOE applications is not accidental; each domain instantiates the same abstract problem with different latent variables, excitation sources, and performance criteria.

Domain	Offset quantity	Core mechanism
Wireless clock synchronization	Clock offset / skew	Two-way timing exchange or relative measurements (Ahmad et al., 2012, Liao et al., 2013)
Visual-inertial odometry	Camera–IMU time offset	Visual feature shifting and pose propagation to actual capture time (Ling et al., 2018, Xiong et al., 2024)
Radar-inertial odometry	Radar–IMU time offset	Ego-velocity update aligned to IMU time stream (Kim et al., 2 Feb 2025)
Bistatic ISAC	TO/CFO residuals	LoS delay/Doppler residual tracking (Wang et al., 12 Jul 2025)
Signal processing	Time-varying delay	Adaptive all-pass filtering (Jelfs et al., 2021)
Multivariate epidemiological series	Continuously varying lag	Profile, affinity, inner-product, and loss-based lag tracking (James et al., 2021)

In wireless sensor networks, factor-graph TVOE is motivated by oscillator imperfections that render the offset time-varying (Ahmad et al., 2012, Ahmad et al., 2012). In mobile networks, the communication graph itself varies according to a finite-state homogeneous Markov chain, and the union-graph condition determines convergence (Liao et al., 2013).

In VIO, the offset is particularly consequential under high dynamic motion. TON-VIO argues that a few milliseconds of temporal misalignment can produce inconsistent visual-inertial constraints and large drift, especially when feature tracking is unstable (Xiong et al., 2024). The 2018 optimization-based VIO paper further uses the offset variable to absorb part of the rolling-shutter effect via an effective capture-time approximation (Ling et al., 2018).

In communications, synchronization problems often couple timing offset with CFO. AFDM derives joint ML estimators of symbol time offset and CFO from chirp-periodic prefix redundancy, using a correlation metric

$\beta_u$7

and then either joint or stepwise ML recovery (Tang et al., 2023). OTFS similarly decomposes timing offset as

$\beta_u$8

and performs a two-stage TO estimate followed by GCE-BEM-based ML CFO refinement over a linear time-varying channel (Bayat et al., 2023). These are synchronization problems rather than generic offset-tracking frameworks, but they belong to the same technical family because the unknown temporal misalignment is estimated under explicit channel time variation.

The multivariate time-series formulation in epidemiology is conceptually distinct but methodologically informative. It defines maps such as

$\beta_u$9

and

$$\xi \triangleq d+\theta,\qquad \psi \triangleq d-\theta,$$0

thereby treating TVOE as lag tracking between evolving multivariate structures rather than calibration of sensing hardware (James et al., 2021).

6. Convergence, optimality, and reported empirical behavior

The strongest formal convergence result in the supplied literature is the mobile-network theorem: if $$\xi \triangleq d+\theta,\qquad \psi \triangleq d-\theta,$$1 is governed by an ergodic homogeneous Markov chain with $$\xi \triangleq d+\theta,\qquad \psi \triangleq d-\theta,$$2, then the estimation error is mean-square convergent if and only if the union graph

$$\xi \triangleq d+\theta,\qquad \psi \triangleq d-\theta,$$3

is connected (Liao et al., 2013). Mean-square convergence is defined there as convergence of both

$$\xi \triangleq d+\theta,\qquad \psi \triangleq d-\theta,$$4

with asymptotic expressions derived using MJLS theory (Liao et al., 2013).

For two-way timing-exchange TVOE, the factor-graph estimator is reported to be near-optimal. In the Gaussian likelihood case, the FGE matches the BCRB; in the log-normal case it matches the Bayesian lower bounds; in the exponential case it is close to the BCHRB (Ahmad et al., 2012). The same work states that as $$\xi \triangleq d+\theta,\qquad \psi \triangleq d-\theta,$$5, the Bayesian TVOE estimator reduces to the corresponding classical ML estimator (Ahmad et al., 2012). This establishes a precise connection between dynamic Bayesian estimation and static synchronization estimators.

In adaptive delay estimation, the normalized all-pass algorithm satisfies the convergence condition

$$\xi \triangleq d+\theta,\qquad \psi \triangleq d-\theta,$$6

and the minimum achievable mean square error is

$$\xi \triangleq d+\theta,\qquad \psi \triangleq d-\theta,$$7

under the stated LMS-type assumptions (Jelfs et al., 2021). Synthetic experiments show that NAAP tracks both small and large delay changes robustly across all SNRs, while ETDE fails on the largest step in the large-step scenario and Sun can become numerically unstable because of learning-rate sensitivity (Jelfs et al., 2021).

The VIO literature emphasizes empirical rather than closed-form convergence. TON-VIO reports about 19% TPE reduction for the VINS-Fusion-based variant, about 46% TPE reduction for the OpenVINS-based variant on EuRoC with injected time-varying offset noise, and an average APE reduction of about 32% on the SCube dataset (Xiong et al., 2024). The 2018 varying-offset VIO paper reports that its uncertainty-aware initialization improves success from 14/20 to 18/20, and that its varying-offset model outperforms the constant-offset version on Samsung Galaxy S8 data (Ling et al., 2018).

Radar-inertial odometry reports self-collected-sequence estimates converging to approximately $$\xi \triangleq d+\theta,\qquad \psi \triangleq d-\theta,$$8 s and $$\xi \triangleq d+\theta,\qquad \psi \triangleq d-\theta,$$9 s on example sequences, with average self-collected offset $\theta = \frac{\xi-\psi}{2},$0 s, and reductions of 56% in APE translation, 75% in APE rotation, 50% in RPE translation, and 58% in RPE rotation relative to EKF-RIO (Kim et al., 2 Feb 2025).

Bistatic ISAC reports that TVOE keeps TO RMSE below 25 ns at low SINR, while a static method exceeds 40 ns, and that at high SINR the proposed method achieves nearly 60% lower TO RMSE than the static LoS baseline (Wang et al., 12 Jul 2025). This is specifically a TO-estimation result; the downstream significance is correction of NLoS delay-Doppler parameters for sensing (Wang et al., 12 Jul 2025).

These results suggest that TVOE is most beneficial when the underlying misalignment is neither negligible nor strictly static, and when the sensing or communication task is strongly nonlinear in timing error.

7. Assumptions, limitations, and recurring misconceptions

A recurring misconception is that TVOE always means the offset parameter itself is a free arbitrary function of time. Several papers make a narrower assumption. In the UWB affine-clock model, $\theta = \frac{\xi-\psi}{2},$1 is constant while time-varying mismatch is induced by drift $\theta = \frac{\xi-\psi}{2},$2 (Mallat et al., 2014). In radar-inertial odometry, the nominal $\theta = \frac{\xi-\psi}{2},$3 is constant in the dynamics and only the uncertainty model permits random-walk adaptation (Kim et al., 2 Feb 2025). In two-way timing-exchange factor graphs, the time variation is specifically Gauss-Markov, not unconstrained (Ahmad et al., 2012, Ahmad et al., 2012).

Another misconception is that improved offset estimation can be isolated from the observation model. Several papers argue the opposite. TON-VIO explicitly states that both the observation side and the prediction side must be modeled: FVON improves feature-velocity estimates for unstable tracking, while TPN learns the temporal evolution of the offset (Xiong et al., 2024). Similarly, optimization-based VIO requires repeated IMU integration over variable-length intervals because the offset alters the effective measurement time, making computational structure part of the estimator design (Ling et al., 2018).

Identifiability is often motion- or topology-dependent. The radar-inertial work states that if the platform is stationary, the offset cannot be determined, and slow motion makes estimation harder (Kim et al., 2 Feb 2025). The mobile-network result requires connectivity of the union graph to propagate reference information (Liao et al., 2013). In bistatic ISAC, the method assumes a strong and reliable LoS component, known or measurable UAV trajectory, and smooth clock drift (Wang et al., 12 Jul 2025). In temperature-varying synchronization, the neural estimator generalizes well only when test conditions fall within the thermal range seen during training (Mongelli et al., 2022).

The non-Gaussian case also raises a methodological limitation for classical filters. Under temperature variations, the induced skew perturbation $\theta = \frac{\xi-\psi}{2},$4 is described as “not Gaussian; it is actually a multi-modal distribution, with significant asymmetry among the peaks,” so regular Kalman filtering becomes suboptimal (Mongelli et al., 2022). The paper therefore formulates a Bayesian functional optimization problem and approximates the optimal estimator with splines and neural networks (Mongelli et al., 2022). This suggests a broader point: TVOE is often not only a state-estimation problem, but also a model-mismatch problem.

Across the cited literature, the unifying pattern is that temporal misalignment becomes a state-like quantity once physical conditions invalidate constant-offset assumptions. TVOE methods then differ mainly in three choices: the stochastic model assigned to offset evolution, the sensing modality that provides temporal information, and the inference architecture used to couple dynamic prediction with measurement correction.