Delayed-State Kalman Filter

• Delayed-State Kalman Filter is a generalized formulation that incorporates delayed measurements by explicitly modeling cross-correlations, avoiding state augmentation.
• It is mathematically equivalent to stochastic cloning while requiring fewer computations and reduced memory, making it ideal for high-dimensional state estimation.
• The approach clarifies misconceptions by demonstrating that proper Kalman filtering can accurately handle delay-induced measurement correlations through a refined update derivation.

Searching arXiv for the specified papers and topic to ground the response in recent literature. arXiv search: delayed-state Kalman filter stochastic cloning delayed-state filtering The delayed-state Kalman filter is a Kalman-filter formulation for measurements that depend on prior states rather than only on the current state. In the treatment revisited by "Remarks on stochastic cloning and delayed-state filtering" (Mina et al., 28 Aug 2025), it is presented as a properly derived filter that accounts explicitly for the correlations induced by delayed or relative-state measurements, yields exactly the same state and covariance update as stochastic cloning, and does so without augmenting the state vector. The same paper describes this formulation as a long-established but often overlooked alternative, also referred to as the generalized Kalman filter, and uses it to clarify a recurring misconception: the limitation lies not in Kalman filtering per se, but in naive formulations that neglect the relevant cross-correlations (Mina et al., 28 Aug 2025).

1. Problem setting and definition

Many estimation problems in robotics and navigation involve measurements that depend on previous states. A prominent example is odometry, which measures the relative change between states over time. In such settings, a measurement acquired at time $k$ may depend on both the current state and a prior state $j$, so the standard Markov-style update assumption of the conventional Kalman filter is no longer sufficient by itself (Mina et al., 28 Aug 2025).

The conventional linear-Gaussian prediction model is written as

$x_k^- = \Phi_{k,j} x_j^+ + B_k u_k,$

$P_k^- = \Phi_{k,j} P_j^+ \Phi_{k,j}^T + S_k,$

with process-noise covariance $S_k = G_k Q_k G_k^T$. The standard update assumes a measurement model of the form

$y_k = H_k x_k + v_k,$

together with the absence of cross-correlation between the current state error and the measurement noise. The delayed-state case violates that simplifying assumption when the measurement depends on a prior state estimate whose uncertainty is already correlated with subsequent process noise and prediction errors (Mina et al., 28 Aug 2025).

The delayed-state Kalman filter addresses this case directly. Rather than restoring Markov structure by augmenting the state vector, it keeps a single-state representation and incorporates the non-zero cross-terms that arise from delayed-state dependence. In the paper’s framing, this is not a heuristic correction but a proper Kalman-filter derivation for delayed or relative-state measurements (Mina et al., 28 Aug 2025).

2. Measurement model and generalized update

The delayed-state measurement model considered in the paper is

$y_k = H_{j,k} x_j + H_{k,k} x_k + v_k,$

and, for relative measurements such as odometry,

$y_k = (p_k - p_j) + v_k.$

To derive a recursive filter in the current state alone, the prior state is re-expressed through the transition model: $x_j = \Phi_{j,k} (x_k - B_k u_k).$ Substituting this into the innovation shows that the residual is correlated with the process noise at time $k$ and with prior prediction errors. The paper identifies that point as the key reason why the naive Kalman update is incorrect in this setting (Mina et al., 28 Aug 2025).

The resulting gain is

$j$0

where $j$1. The state update is

$j$2

and the covariance update is

$j$3

A central structural property is that the standard Kalman filter is recovered when $j$4. In that case, the delayed-state dependence disappears, the cross-covariance term vanishes, and the generalized formulation reduces to the usual update. This is the sense in which the delayed-state Kalman filter extends rather than replaces the conventional filter (Mina et al., 28 Aug 2025).

3. Relation to stochastic cloning

Stochastic cloning handles delayed-state measurements by augmenting the state vector with the relevant prior state,

$j$5

and then applying standard Kalman-filter operations to the augmented system. The associated measurement model is

$j$6

This construction restores a Markov structure, but it does so at the expense of larger state and covariance matrices (Mina et al., 28 Aug 2025).

The main theoretical result of (Mina et al., 28 Aug 2025) is that a properly derived delayed-state Kalman filter is mathematically equivalent to stochastic cloning in its state and covariance updates. The paper states that it provides a formal, inductive proof that, under linearity and invertibility of the transition model, the two approaches yield identical state and covariance estimates. The distinction is therefore representational and computational rather than inferential.

Approach	Mechanism	Consequence
Standard KF (naive)	Assumes measurement depends only on current state	Does not handle delayed-state cross-correlation
Stochastic cloning	Augments the state with prior state(s)	Handles correlations via augmentation
Delayed-state KF	Single state with explicit cross-covariance terms	Same state and covariance update as stochastic cloning

This equivalence is used in (Mina et al., 28 Aug 2025) to rebut the claim that Kalman filters are inherently unable to process correlated delayed-state measurements without state augmentation. The paper’s clarification is sharper: the failure belongs to an incomplete formulation, not to Kalman filtering as a class.

4. Computational and memory characteristics

The same paper gives explicit arithmetic-complexity expressions for stochastic cloning and the delayed-state Kalman filter. As summarized in its Table 1, stochastic cloning has complexity

$j$7

whereas the delayed-state Kalman filter has complexity

$j$8

The paper concludes that the delayed-state Kalman filter consistently requires fewer floating point operations per iteration than stochastic cloning, and states that this is confirmed by its Figure 1 (Mina et al., 28 Aug 2025).

Its Table 2 likewise reports memory requirements. Stochastic cloning is given as

$j$9

while the delayed-state Kalman filter is given as

$x_k^- = \Phi_{k,j} x_j^+ + B_k u_k,$0

The paper states that the delayed-state Kalman filter often requires less memory, especially when the state dimension $x_k^- = \Phi_{k,j} x_j^+ + B_k u_k,$1 is large compared to the measurement dimension $x_k^- = \Phi_{k,j} x_j^+ + B_k u_k,$2, with Figure 2 cited for that comparison (Mina et al., 28 Aug 2025).

These results motivate the paper’s characterization of the delayed-state Kalman filter as computationally advantageous and memory-efficient for higher-dimensional states. A plausible implication is that the representation choice becomes increasingly important when augmentation would enlarge already substantial covariance structures.

5. Delayed-state filtering, delayed update, and terminological overlap

The term delayed appears in multiple filtering contexts, and the literature summarized here distinguishes two different mechanisms. In the delayed-state Kalman filter, the delay is in the measurement dependence: a measurement at time $x_k^- = \Phi_{k,j} x_j^+ + B_k u_k,$3 depends on both $x_k^- = \Phi_{k,j} x_j^+ + B_k u_k,$4 and $x_k^- = \Phi_{k,j} x_j^+ + B_k u_k,$5 (Mina et al., 28 Aug 2025). In contrast, the Multi-state Constraint Kalman Filter literature uses delayed update to describe an update schedule in which feature observations are accumulated and incorporated only when a feature leaves the field of view (Zhang et al., 2024).

In the longstanding, standard MSCKF, feature observations are accumulated as a tracked feature is observed over time, and filtering updates are performed only when a feature is lost from the field of view. The measurement model is linearized as

$x_k^- = \Phi_{k,j} x_j^+ + B_k u_k,$6

and then projected into the left nullspace of $x_k^- = \Phi_{k,j} x_j^+ + B_k u_k,$7,

$x_k^- = \Phi_{k,j} x_j^+ + B_k u_k,$8

The delayed-update correction is performed only once, after stacking constraints from lost features (Zhang et al., 2024).

The immediate update strategy proposed in (Zhang et al., 2024) performs filter updates as soon as new feature measurements and a new pose become available. The paper states that immediate update helps construct more observation constraints and employ more filtering updates than delayed update, improves the linearization point of the measurement model, and therefore enhances estimation accuracy. By time $x_k^- = \Phi_{k,j} x_j^+ + B_k u_k,$9, the immediate strategy accumulates roughly $P_k^- = \Phi_{k,j} P_j^+ \Phi_{k,j}^T + S_k,$0 measurement constraints, while the delayed update incorporates $P_k^- = \Phi_{k,j} P_j^+ \Phi_{k,j}^T + S_k,$1 constraints at once; the paper also states that immediate update results in a smaller final covariance because its updated covariance includes the cumulative Fisher information from all corrections (Zhang et al., 2024).

This comparison suggests a useful terminological distinction. Delayed-state filtering concerns how delayed-state measurements are modeled within the Kalman update. Delayed update in MSCKF concerns when measurement corrections are applied. The former is a measurement-correlation problem; the latter is an update-frequency and linearization problem.

6. Relation to smoothing and retrospective estimation

A broader state-space perspective is provided by "Particle Filters and Data Assimilation" (Fearnhead et al., 2017). That paper defines filtering, prediction, and smoothing distributions for latent Markov state-space models: $P_k^- = \Phi_{k,j} P_j^+ \Phi_{k,j}^T + S_k,$2

$P_k^- = \Phi_{k,j} P_j^+ \Phi_{k,j}^T + S_k,$3

$P_k^- = \Phi_{k,j} P_j^+ \Phi_{k,j}^T + S_k,$4

It also describes smoothing as retrospective or delayed-state inference, including full smoothing, fixed-lag smoothing, and marginal smoothing (Fearnhead et al., 2017).

Within that literature, delayed-state estimation typically refers to inferring past states given present and future observations, often through particle smoothers, forward-backward procedures, two-filter smoothers, or ensemble Kalman smoother variants. Most particle and EnKF methods natively provide only filtering, and smoothing requires dedicated algorithms. The paper emphasizes path degeneracy for naive particle-history storage and gives forward-backward smoothing as a way to mitigate that issue, albeit at greater computational cost (Fearnhead et al., 2017).

This suggests a second distinction that is often useful in technical discussions. The delayed-state Kalman filter of (Mina et al., 28 Aug 2025) is not a smoothing algorithm in the sense of (Fearnhead et al., 2017). It is a recursive filter for current-state estimation in the presence of measurements that depend on prior states. Smoothing, by contrast, targets the posterior of past states given an observation history extending beyond those states. The two are related by their treatment of temporal dependence, but they solve different estimation problems.

7. Misconceptions, scope, and implications

The most explicit misconception addressed in (Mina et al., 28 Aug 2025) is the claim that Kalman filter variants cannot properly account for correlations in delayed or relative-state measurement scenarios without state augmentation. The paper rejects that claim directly, stating that such assertions are “simply not true,” and attributes the confusion to a standard, naive formulation that neglects non-zero cross-terms. In its account, the delayed-state Kalman filter, stemming from the original work of Brown and Hartman (1968) and revisited through Brown and Hwang (1997), is a correct Kalman-filter derivation for this case (Mina et al., 28 Aug 2025).

The paper’s scope conditions are also clear. The equivalence to stochastic cloning is stated under linearity and invertibility of the transition model. Within that regime, the delayed-state Kalman filter is presented as mathematically exact, not approximate, and as fully capable of capturing the correlations between prior states, process noise, and current innovations through explicit cross-covariance terms (Mina et al., 28 Aug 2025).

A further implication emerges when the delayed-state formulation is read alongside the MSCKF results of (Zhang et al., 2024). Even when delayed information can be handled correctly at the level of measurement correlation, estimation quality may still depend strongly on update timing and linearization strategy. The MSCKF paper reports that immediate update strategies improve pose accuracy by 29% and 18% on EuRoC for all-cam and 3-cam variants, and by 30% and 22% on the KAIST VIO Dataset, while increasing processing time per frame from 0.0076 s for delayed update to 0.0128 s for 3-cam and 0.0216 s for all-cam on EuRoC, and to 0.038 s per frame for all-cam versus 0.013 s for 3-cam on KAIST (Zhang et al., 2024). Those results do not alter the theory of delayed-state filtering, but they underscore that correlation modeling, linearization accuracy, and computational structure are distinct design dimensions in practical estimators.

Taken together, these works place the delayed-state Kalman filter in a precise position within estimation theory. It is a generalized Kalman-filter formulation for delayed or relative-state measurements, equivalent in state and covariance update to stochastic cloning, more efficient in arithmetic and often in memory, and conceptually distinct both from delayed-update scheduling in sliding-window visual-inertial filters and from smoothing as retrospective state inference (Mina et al., 28 Aug 2025).