Standardized Kalman Filter (SKF)

• Standardized Kalman Filter (SKF) is a dynamic Bayesian method for EEG source localization that employs linear-Gaussian state-space modeling with a post-hoc depth compensation step.
• It integrates temporal dynamics and Gaussian priors to correct for the systematic underestimation of deep sources while detecting both cortical and subcortical activities.
• Enhancements like RTS smoothing and change-rate tracking improve source separation and tracking stability in the inherently ill-posed EEG inverse problem.

Standardized Kalman Filter (SKF) is a dynamic Bayesian method for EEG source localization in which temporally evolving source amplitudes are inferred with a linear-Gaussian state-space model and then corrected for depth bias by a post-hoc standardization step. In the formulation studied for simultaneous cortical and sub-cortical activity detection, the unknown cortical and subcortical current amplitudes form the state, EEG electrode potentials form the measurement, and prior information enters through Gaussian covariance models whose scale is controlled by prior-to-measurement and evolution-prior signal-to-noise ratios. Within this framework, SKF is designed to combine temporal modeling, Gaussian priors, and depth compensation so that deep activity becomes detectable alongside superficial cortical sources in an otherwise ill-posed inverse problem (Prasikala et al., 31 Jul 2025).

1. Problem setting and defining characteristics

EEG source localization seeks to recover neural current distributions from scalp potentials, but the inverse problem lacks a unique solution unless constrained by prior knowledge. In the Bayesian setting, such knowledge is encoded through prior models, and different source-localization algorithms differ substantially in how these priors are imposed. The Standardized Kalman Filter occupies the subset of approaches that are both dynamic and Gaussian: it models source trajectories over time and applies Kalman filtering to update posterior means and covariances sequentially (Prasikala et al., 31 Jul 2025).

In the EEG-specific SKF literature, the defining feature is not merely Kalman recursion but the standardization applied after the covariance-form update. This post-hoc rescaling is described as sLORETA-style and time-varying; it is intended to equalize sensitivity across cortical depths and thereby counter the systematic underestimation of deep generators caused by the lead field. The same conceptual structure is retained in later variants, including change-rate tracking models that augment the state with temporal derivatives while preserving the diagonal standardization of the current-density estimate (Lahtinen, 29 Mar 2025).

This places SKF in a distinct methodological niche. Relative to static minimum-norm methods, it introduces explicit temporal dynamics. Relative to generic Kalman filters, it adds a depth-compensating normalization step that is central rather than auxiliary. Relative to other Bayesian EEG inverse schemes, it remains within a linear-Gaussian framework and therefore admits explicit prediction, innovation, gain, correction, and optional smoothing recursions (Prasikala et al., 31 Jul 2025).

2. Linear-Gaussian state-space formulation and prior parametrization

In the SKF framework used for EEG source localization, the time-indexed state vector is $x_t \in \mathbb{R}^n$, representing cortical and subcortical current amplitudes, and the measurement is $y_t \in \mathbb{R}^m$, representing EEG electrode potentials. Under the linear-Gaussian assumption, the model is

$$x_t = A_{t-1}x_{t-1} + q_{t-1}, \qquad q_{t-1} \sim \mathcal{N}(0,Q_{t-1}),$$

$$y_t = Lx_t + r_t, \qquad r_t \sim \mathcal{N}(0,R_t),$$

with initial prior $x_0 \sim \mathcal{N}(m_0,P_0)$. In the reported SKF study, $A_{t-1}=I$ is used, so the dynamics reduce to a random-walk prior. The lead-field matrix $L \in \mathbb{R}^{m \times n}$ is fixed by the head model, $m_0$ is typically zero, and $P_0$ is diagonal (Prasikala et al., 31 Jul 2025).

The prior covariances are parameterized by two decibel-scaled quantities. PM-SNR fixes the total prior variance $\theta_0$ per dipole through

$y_t \in \mathbb{R}^m$0

where $y_t \in \mathbb{R}^m$1 is the measurement-noise variance, $y_t \in \mathbb{R}^m$2 the expected signal amplitude, and $y_t \in \mathbb{R}^m$3 the number of dipoles; in practice, $y_t \in \mathbb{R}^m$4. EP-SNR fixes the incremental variance $y_t \in \mathbb{R}^m$5 per time step via

$y_t \in \mathbb{R}^m$6

Using $y_t \in \mathbb{R}^m$7 time steps, each process-noise variance is $y_t \in \mathbb{R}^m$8, giving $y_t \in \mathbb{R}^m$9 (Prasikala et al., 31 Jul 2025).

A related 2025 variant augments the state with a change-rate variable $$x_t = A_{t-1}x_{t-1} + q_{t-1}, \qquad q_{t-1} \sim \mathcal{N}(0,Q_{t-1}),$$0 and writes $$x_t = A_{t-1}x_{t-1} + q_{t-1}, \qquad q_{t-1} \sim \mathcal{N}(0,Q_{t-1}),$$1. In that model, the observation $$x_t = A_{t-1}x_{t-1} + q_{t-1}, \qquad q_{t-1} \sim \mathcal{N}(0,Q_{t-1}),$$2 is supplemented by a pseudo-observation derived from a second-order backward-difference formula,

$$x_t = A_{t-1}x_{t-1} + q_{t-1}, \qquad q_{t-1} \sim \mathcal{N}(0,Q_{t-1}),$$3

and the state transition becomes

$$x_t = A_{t-1}x_{t-1} + q_{t-1}, \qquad q_{t-1} \sim \mathcal{N}(0,Q_{t-1}),$$4

For this BDF2-CR-SKF, the process-noise blocks are matched as $$x_t = A_{t-1}x_{t-1} + q_{t-1}, \qquad q_{t-1} \sim \mathcal{N}(0,Q_{t-1}),$$5 and $$x_t = A_{t-1}x_{t-1} + q_{t-1}, \qquad q_{t-1} \sim \mathcal{N}(0,Q_{t-1}),$$6, while the velocity-noise covariance is chosen as $$x_t = A_{t-1}x_{t-1} + q_{t-1}, \qquad q_{t-1} \sim \mathcal{N}(0,Q_{t-1}),$$7 (Lahtinen, 29 Mar 2025).

These parametrizations make explicit how prior strength and temporal smoothness enter SKF. A plausible implication is that SKF performance is governed as much by covariance design as by the filtering recursion itself, because PM-SNR, EP-SNR, or $$x_t = A_{t-1}x_{t-1} + q_{t-1}, \qquad q_{t-1} \sim \mathcal{N}(0,Q_{t-1}),$$8 alter both posterior uncertainty and the effective weighting between measurement and temporal continuity.

3. Kalman recursion and the standardization operator

Given the linear-Gaussian model, SKF uses the standard Kalman forward recursions. The prediction step is

$$x_t = A_{t-1}x_{t-1} + q_{t-1}, \qquad q_{t-1} \sim \mathcal{N}(0,Q_{t-1}),$$9

followed by innovation and gain computation,

$$y_t = Lx_t + r_t, \qquad r_t \sim \mathcal{N}(0,R_t),$$0

and then the correction step,

$$y_t = Lx_t + r_t, \qquad r_t \sim \mathcal{N}(0,R_t),$$1

The prior parameters enter directly through $$y_t = Lx_t + r_t, \qquad r_t \sim \mathcal{N}(0,R_t),$$2 and $$y_t = Lx_t + r_t, \qquad r_t \sim \mathcal{N}(0,R_t),$$3, and thereby determine the predicted covariance $$y_t = Lx_t + r_t, \qquad r_t \sim \mathcal{N}(0,R_t),$$4 and Kalman gain $$y_t = Lx_t + r_t, \qquad r_t \sim \mathcal{N}(0,R_t),$$5 (Prasikala et al., 31 Jul 2025).

The defining SKF operation follows the update. Let $$y_t = Lx_t + r_t, \qquad r_t \sim \mathcal{N}(0,R_t),$$6 be the unweighted posterior mean. The diagonal reweighting matrix is constructed as

$$y_t = Lx_t + r_t, \qquad r_t \sim \mathcal{N}(0,R_t),$$7

and the standardized estimate is

$$y_t = Lx_t + r_t, \qquad r_t \sim \mathcal{N}(0,R_t),$$8

In the related change-rate formulation, the same idea appears in the equivalent expression

$$y_t = Lx_t + r_t, \qquad r_t \sim \mathcal{N}(0,R_t),$$9

again emphasizing a diagonal whitening that equalizes variance across depths (Lahtinen, 29 Mar 2025).

The 2025 prior-parameter study generalizes the standardization by replacing the fixed exponent with a tunable $x_0 \sim \mathcal{N}(m_0,P_0)$0:

$x_0 \sim \mathcal{N}(m_0,P_0)$1

$x_0 \sim \mathcal{N}(m_0,P_0)$2

Equivalently, if

$x_0 \sim \mathcal{N}(m_0,P_0)$3

then

$x_0 \sim \mathcal{N}(m_0,P_0)$4

which has the form $x_0 \sim \mathcal{N}(m_0,P_0)$5 with $x_0 \sim \mathcal{N}(m_0,P_0)$6 (Prasikala et al., 31 Jul 2025).

Empirically, the original exponent $x_0 \sim \mathcal{N}(m_0,P_0)$7 is not treated as universally optimal. The reported comparison states that $x_0 \sim \mathcal{N}(m_0,P_0)$8 under-compensates deep sources, $x_0 \sim \mathcal{N}(m_0,P_0)$9 over-amplifies them, and $A_{t-1}=I$0 yields nearly equal sensitivity across cortical depth (Prasikala et al., 31 Jul 2025). In this sense, SKF standardization is not a fixed normalization rule but a tunable bias-correction mechanism.

4. Smoothing, change-rate tracking, and stability-oriented variants

SKF can be used in pure filtering mode, but the EEG formulation also incorporates fixed-interval smoothing through the Rauch-Tung-Striebel (RTS) backward pass. Starting from $A_{t-1}=I$1 and $A_{t-1}=I$2, one computes for $A_{t-1}=I$3

$A_{t-1}=I$4

$A_{t-1}=I$5

$A_{t-1}=I$6

The same post-hoc standardization is then applied to the smoothed means $A_{t-1}=I$7 (Prasikala et al., 31 Jul 2025).

In the reported synthetic SEP experiments, RTS smoothing was not merely a denoising refinement. The paper states that smoothing plus $A_{t-1}=I$8 most effectively suppresses the “echo” of the deep-source peak at the time of the superficial peak, thereby enhancing temporal separability of overlapping sources. The averaged cortical maps at $A_{t-1}=I$9 and $L \in \mathbb{R}^{m \times n}$0 are described as confirming that smoothing plus $L \in \mathbb{R}^{m \times n}$1 suppresses residual activity at the wrong time point and yields the most spatially precise, depth-unbiased reconstructions (Prasikala et al., 31 Jul 2025).

A separate stability-oriented extension replaces the plain random-walk model by change-rate tracking. In BDF2-CR-SKF, the additional velocity measurement is described as acting as a pseudo-integral feedback, damping unbounded growth and reducing sensitivity to the choice of $L \in \mathbb{R}^{m \times n}$2. The informal stability condition is that the eigenvalues of the closed-loop update map satisfy $L \in \mathbb{R}^{m \times n}$3, and the paper reports that under extreme $L \in \mathbb{R}^{m \times n}$4 values of $L \in \mathbb{R}^{m \times n}$5 or $L \in \mathbb{R}^{m \times n}$6 the RW-SKF tracks collapse or oscillate, whereas the BDF2-CR-SKF remains qualitatively correct (Lahtinen, 29 Mar 2025).

These two extensions address different limitations. RTS smoothing exploits future data to improve fixed-interval estimates and source separability; BDF2-CR-SKF alters the forward model itself to improve tracking stability and reduce tuning sensitivity. This suggests that the contemporary SKF literature is moving in two complementary directions: post-estimation refinement and model-based stabilization.

5. Prior-parameter effects and empirical performance

The most detailed parameter study on SKF-based EEG source localization used synthetic data similar to the P20 / N20 component of somatosensory evoked potentials and evaluated simultaneous recovery of a deep thalamic source at $L \in \mathbb{R}^{m \times n}$7 and a superficial cortical source at $L \in \mathbb{R}^{m \times n}$8 under measurement-noise levels of $L \in \mathbb{R}^{m \times n}$9, $m_0$0, and $m_0$1. Reconstruction quality was assessed by how well the two sources were recovered without spurious echoes (Prasikala et al., 31 Jul 2025).

Noise level	Best PM-SNR / EP-SNR	Reported effect
$m_0$2	$m_0$3 / $m_0$4	Increasing either prior drive degraded spatial accuracy; EP-SNR had stronger negative impact
$m_0$5	$m_0$6 / $m_0$7	Similar optimum with reduced sensitivity to parameter changes
$m_0$8	$m_0$9 / $P_0$0, or vice versa	Best results before performance fell off again

Across all noise levels, the un-smoothed SKF retained a residual deep-source echo at the superficial peak. Incorporating RTS smoothing and increasing the standardization exponent from $P_0$1 to $P_0$2 effectively removed that echo, with the clearest benefit at $P_0$3, where smoothed SKF with $P_0$4 gave the cleanest separation of deep and superficial time courses (Prasikala et al., 31 Jul 2025).

The change-rate study used a different synthetic protocol: two-source Gaussians in thalamus and somatosensory cortex, separated by $P_0$5, a high-resolution forward model with mesh approximately $P_0$6 and $P_0$7 sources, Gaussian scalp noise at $P_0$8, $P_0$9, and $\theta_0$0, and $\theta_0$1 realizations. The metrics were localization correctness, $\theta_0$2–$\theta_0$3 quantile band width, peak-height difference, and track “enmeshment” (Lahtinen, 29 Mar 2025).

Within that setup, both SKF variants separated deep and surface sources better than sLORETA. At high SNR ($\theta_0$4), BDF2-CR-SKF quantile bands were narrower than RW-SKF but slightly wider than sLORETA. Under mis-tuned $\theta_0$5, BDF2-CR-SKF remained accurate while RW-SKF failed. In spatial focality comparisons, BDF2-CR-SKF was more focal than sLORETA and comparable to RW-SKF. In real somatosensory-evoked-potential data at $\theta_0$6, $\theta_0$7, $\theta_0$8, $\theta_0$9, and $y_t \in \mathbb{R}^m$00, all three methods located broadly correct cortical foci at $y_t \in \mathbb{R}^m$01 and $y_t \in \mathbb{R}^m$02, but only the two SKF variants yielded focal thalamic spots at $y_t \in \mathbb{R}^m$03 and $y_t \in \mathbb{R}^m$04 (Lahtinen, 29 Mar 2025).

Taken together, these results identify three empirically important control knobs in EEG SKF: the prior scale, the temporal-evolution scale, and the standardization exponent. They also indicate that model augmentation with change-rate information can broaden the regime in which acceptable reconstructions are obtained.

6. Numerical considerations, methodological boundaries, and nomenclature

The EEG SKF papers summarized here present covariance-form Kalman recursions. Separate Kalman-filter research emphasizes that conventional covariance-form implementations can be numerically unstable because direct Riccati propagation and explicit inversion can destroy symmetry or positive definiteness under round-off, whereas square-root filters propagate Cholesky factors and avoid these failure modes (Kulikova et al., 2013). This suggests that a square-root implementation could be relevant for large-scale or ill-conditioned SKF inverse problems, although such an implementation is not described in the EEG SKF sources provided here.

A second methodological boundary concerns the acronym itself. In the Kalman-filter literature, “SKF” also denotes the Schmidt–Kalman filter, in which nuisance or “considered” states are propagated but not corrected by the measurement, with gains forced so that the considered-state block is not updated (Ramos, 2022). A related map-based localization literature uses the same abbreviation for Schmidt–Kalman and its Cholesky-Schmidt-Kalman variants, where the map covariance is handled through sparse Cholesky factors rather than dense blocks (Dutoit et al., 2016). More recently, “SKF” has also been used for the SINDy Kalman Filter, where sparse regression coefficients are treated as latent states in an online learning problem (Pillonetto et al., 14 Nov 2025).

The Standardized Kalman Filter is therefore distinct both conceptually and operationally. Its defining move is not Schmidt-style omission of state updates, nor sparse-regression parameter tracking, but post-hoc variance standardization of Kalman source estimates to remove amplitude bias across depth, optionally combined with RTS smoothing or change-rate pseudo-observations in EEG source localization (Prasikala et al., 31 Jul 2025). Recognizing this distinction is important because the shared acronym masks substantially different state partitions, covariance structures, and objectives.

From the standpoint of EEG inverse methodology, the current evidence base supports a specific interpretation: SKF is a linear-Gaussian dynamic source-localization framework whose practical performance depends critically on prior-covariance tuning, whose depth compensation depends critically on the standardization operator and its exponent, and whose stability and separability can be improved either by smoothing or by augmenting the observation model with change-rate information (Lahtinen, 29 Mar 2025).