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\begin{document}

\title{Optimizing Time-Varying Autoregressive Kalman Filters for Event-Related Potential Extraction in Brain-Computer Interfaces}

\author{
    Hadi~Ghavipeykar~Shirinsoo$^*$,
    Cristina~Del~Prete,
    Vito~De~Feo,
    and~Roberto~Garello%
    \thanks{$^*$H. Ghavipeykar Shirinsoo (h.ghavipeykar@gmail.com) and R. Garello (roberto.garello@polito.it) are with the Department of Electronics and Telecommunications, Politecnico di Torino, 10129 Turin, Italy.}%
    \thanks{C. Del Prete (cristina.delprete@essex.ac.uk) and V. De Feo (v.defeo@essex.ac.uk) are with the School of Computer Science and Electronic Engineering, University of Essex, Colchester CO4 3SQ, United Kingdom.}%
}

\maketitle

\begin{abstract}
Brain-Computer Interfaces (BCIs) increasingly rely on Event-Related Potentials (ERPs), specifically the Readiness Potential (RP), to assess motor intentionality in non-responsive and hemiplegic patients. However, extracting these highly transient neural biomarkers from noisy electroencephalographic (EEG) data requires robust, mathematically optimized signal processing. This paper proposes a unified pre-processing architecture utilizing a Time-Varying Autoregressive (TVAR) Kalman smoother. We provide a rigorous mathematical framework demonstrating that traditional BCI techniques—such as temporal jitter compensation and pre-epoching state-space estimation—can actively disrupt parameter estimation and temporal phase continuity. By implementing a strictly post-epoching Linear Kalman tracking loop, high-frequency myogenic artifacts were significantly collapsed, yielding a statistically significant reduction in the Residual Oscillation Index ($p=0.00794$). Crucially, the Kalman architecture effectively halved the inter-subject standard deviation and successfully extracted hidden motor intentions from entirely involuntary, hemiplegic datasets, mathematically proving its viability for clinical diagnostics in severe Disorders of Consciousness.
\end{abstract}

\begin{IEEEkeywords}
Brain-Computer Interface (BCI), Electroencephalography (EEG), Kalman Filter, Readiness Potential, Artifact Suppression, Time-Varying Autoregressive (TVAR), Mathematical Modeling.
\end{IEEEkeywords}

\section{Introduction}
\IEEEPARstart{T}{he} accurate assessment of consciousness in patients suffering from severe brain injuries—ranging from Vegetative States to Locked-In Syndrome (LIS)—remains a profound challenge in modern clinical neurology. Standard behavioral assessments frequently fail when a patient's cognitive awareness is trapped within an entirely paralyzed body. Brain-Computer Interfaces (BCIs) offer a vital non-invasive alternative by decoding motor intentionality directly from electroencephalographic (EEG) signals. 

Grounded in the core principle that motor cognition acts as a reliable proxy for internal consciousness, tracking the brain's preparatory electrical activity before an intended movement can verify if focal awareness remains preserved. Within modern clinical neuroscience, prominent frameworks such as the Global Neuronal Workspace (GNW) and Integrated Information Theory (IIT) posit that conscious awareness relies on the integrated, large-scale broadcasting of information across neural circuits. Monitoring the structured, large-scale synchronization of motor potentials provides an objective, behavioral-independent signature of a patient's internal cognitive capacity.

Within the EEG spectrum, Event-Related Potentials (ERPs)—representing macroscopic neural responses to specific cognitive or motor events—serve as primary biomarkers. Specifically, the Readiness Potential (RP), a slow negative-shifting cortical potential arising up to 2 seconds before voluntary movement, indexes active motor preparation and planning. However, these microscopic cognitive signals are heavily masked by ocular artifacts, myogenic (EMG) oscillations, and stochastic background noise, requiring advanced multivariate time-series modeling. 

Recent advancements in signal processing have increasingly adopted state-space estimation to denoise EEG signals. Tarvainen et al. \cite{Tarvainen2001} demonstrated that a Kalman smoother effectively eliminates the temporal tracking lag inherent in traditional adaptive algorithms like Recursive Least Squares (RLS). Similarly, Mohseni et al. \cite{Mohseni2007} established the efficacy of Kalman smoothing specifically for extracting ERPs from noisy background trials, while Oikonomou et al. \cite{Oikonomou2007} applied Kalman-based methodologies for reliable EEG spike enhancement. Furthermore, Enshaeifar et al. \cite{Enshaeifar2016} proposed a regularised Kalman filtering algorithm informed by extrinsic data channels to better track sudden changes in neurological dynamics.

Despite these mathematical advancements, the blind integration of Kalman filters into standard BCI pipelines often yields divergent results. As demonstrated by Salis et al. \cite{Salis2013}, integrating methods like Independent Component Analysis (ICA) and spatial filtering with state-space estimators requires strict architectural ordering to prevent recursive failure. 

This paper deeply investigates the mathematical interactions between frequency filtering, blind source separation, spatial filtering, and state-space estimation. By evaluating the structural placement of the Time-Varying Autoregressive (TVAR) filter, comparing non-linear variants, and testing against traditional jitter compensation, we define a unified architecture capable of mathematically extracting hidden motor intentions even from involuntary, hemiplegic clinical data.

\section{Mathematical and Methodological Framework}

\subsection{Time-Varying Autoregressive (TVAR) Modeling}
Monitoring the spectral characteristics of brain signals provides critical insights into the underlying processes responsible for their generation. Parametric spectral estimation provides an efficient mathematical framework for describing non-stationary EEG time series. In a TVAR model, the coefficients evolve continuously over time to track shifting neurophysiological dynamics. A univariate EEG signal $y(k)$ can be modeled as:
\begin{equation}
y(k) = \sum_{i=1}^{p} a_i(k)y(k-i) + v(k) \label{eq:tvar}
\end{equation}
where $p$ is the model order, $a_i(k)$ are the time-varying autoregressive coefficients, and $v(k)$ represents the zero-mean Gaussian measurement noise with variance $R_k$ \cite{WikiKF}. To balance computational load and predictive accuracy, the model order was optimized to $p=2$, which provides sufficient spectral resolution for the low-frequency Readiness Potential while minimizing latency. 

To estimate the coefficients $a_i(k)$ dynamically, the TVAR model is mapped into a state-space formulation \cite{Khan2006}. The state vector is defined as $x_k = [a_1(k), a_2(k)]^T$. The dynamic progression of these states is governed by the transition equation:
\begin{equation}
x_k = A_k x_{k-1} + w_k \label{eq:state_transition}
\end{equation}
where $w_k \sim \mathcal{N}(0, Q_k)$ represents the process noise vector capturing state transitions \cite{WikiKF}.

\subsection{Derivation of the Maximum-Entropy State Dynamic Model}
To model the temporal trajectory of the latent parameter vector $x_k$ without inducing an a priori structural kinematic bias, the state transition matrix $A_k$ must be derived from foundational stochastic principles. In biological networks, the underlying structural variation of time-varying parameters follows a continuous-time stochastic differential equation (SDE) governed by localized Brownian drift:
\begin{equation}
d\mathbf{x}(t) = \mathbf{\mathcal{F}}\mathbf{x}(t)dt + \mathbf{\mathcal{G}}d\mathbf{\beta}(t)
\end{equation}
where $\mathbf{\mathcal{F}}$ is the continuous system matrix and $d\mathbf{\beta}(t)$ represents independent standard Wiener increments. Under conditions where no secondary, macroscopic kinematic model exists to dictate the structural physical trajectory of time-varying spectral boundaries, the maximum entropy assumption forces $\mathbf{\mathcal{F}} \to \mathbf{0}_{p \times p}$. This configuration mathematically formalizes a localized random walk, indicating that the best predictor for the next parameter state is a zero-mean perturbation around the current state.

Discretizing this continuous system over uniform time intervals $\Delta t = k - (k-1)$ yields the discrete state transition operator:
\begin{equation}
A_k = \exp\left(\mathbf{\mathcal{F}}\Delta t\right) = \sum_{m=0}^{\infty} \frac{(\mathbf{0}_{p \times p}\Delta t)^m}{m!} = I_{p \times p}
\end{equation}
Thus, the transition matrix collapses strictly to an identity format:
\begin{equation}
A_k = I_{p \times p} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\end{equation}
This derivation proves that $A_k = I$ represents a fundamental information-theoretic constraint optimized to capture unmodeled, rapid non-stationary neurophysiological shifts without injecting artificial lag or tracking filters.

The corresponding measurement equation is:
\begin{equation}
y_k = H_k x_k + v_k \label{eq:measurement}
\end{equation}
where the observation matrix tracking preceding history is modeled as a row vector:
\begin{equation}
H_k = \begin{bmatrix} y(k-1) & y(k-2) \end{bmatrix}
\end{equation}
which maps the previous sequential EEG voltage samples directly to the current state vector.

\subsection{The Kalman Filter and Smoother Equations}
The Kalman filter operates as an optimal recursive predictor-corrector algorithm that minimizes the mean square error of the estimated state. The filter executes in two distinct phases: \textit{Predict} and \textit{Update} \cite{Laaraiedh2009}.

\subsubsection{Prediction Phase (Time Update)}
The filter projects the current state estimate and error covariance forward in time \cite{WikiKF}:
\begin{equation}
\hat{x}_{k|k-1} = A_k \hat{x}_{k-1|k-1}
\end{equation}
\begin{equation}
P_{k|k-1} = A_k P_{k-1|k-1} A_k^T + Q_{k-1}
\end{equation}

\subsubsection{Update Phase (Measurement Update)}
Upon receiving the true EEG voltage measurement $y_k$, the Kalman gain ($K_k$) is computed to dictate how heavily the new measurement should influence the state estimate \cite{WikiKF}:
\begin{equation}
K_k = P_{k|k-1} H_k^T \left( H_k P_{k|k-1} H_k^T + R_k \right)^{-1}
\end{equation}
The state estimate is then updated via the scaling innovation residual \cite{WikiKF}:
\begin{equation}
\hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k \left( y_k - H_k \hat{x}_{k|k-1} \right)
\end{equation}

\subsubsection{The Joseph Stabilized Form for Covariance}
To guarantee numerical stability and preserve the positive-definiteness of the state error covariance matrix $P_{k|k}$ during multi-trial offline computations, the standard covariance update equation is mathematically substituted with the robust Joseph Stabilized Form \cite{Khan2006}:
\begin{equation}
P_{k|k} = (I - K_k H_k) P_{k|k-1} (I - K_k H_k)^T + K_k R_k K_k^T \label{eq:joseph}
\end{equation}
By incorporating this symmetric structure, the filter remains bound to physically realistic limits, completely eliminating the divergence caused by floating-point precision limitations in standard software environments.

\subsection{Scale-Invariant Predictor-Corrector Mapping via Hyperparameter Adaptation}
Rather than relying on empirical manual tuning or computationally heavy offline optimization algorithms like the Expectation-Maximization (EM) framework \cite{Khan2006}, our system embeds an automated, data-driven hyperparameter scaling topology. This subsystem derives parameter constraints at runtime directly from localized single-trial statistics. Let $\sigma^2_{\text{local}} = \operatorname{var}(y)$ represent the continuous empirical variance calculated strictly across individual single-trial bounds. The initial state uncertainty is mapped proportionally:
\begin{equation}
P_0 = I_{p \times p} \cdot \sigma^2_{\text{local}}
\end{equation}

To systematically regulate the tracking loop's responsiveness to sudden neurophysiological parameter changes while maintaining robustness against extreme muscle interference, we introduce a process noise scaling constant $\gamma = 10^{-6}$ and a measurement noise scaling constant $\alpha = 50.0$. The active state space parameter matrices are generated dynamically:
\begin{equation}
Q = I_{p \times p} \cdot (\gamma \cdot \sigma^2_{\text{local}})
\end{equation}
\begin{equation}
R = \left(\sqrt{\sigma^2_{\text{local}}} \cdot \alpha\right)^2 = \operatorname{var}(y) \cdot \alpha^2
\end{equation}

By anchoring $Q$ and $R$ symmetrically to the localized trial variance, the filter equations become entirely scale-invariant. Under a standard linear transformation $y \to \beta y$, the scalar variance translates quadratically as $\sigma^2 \to \beta^2 \sigma^2$. This transformation guarantees that the scaled innovation covariance matrix expands to $S'_k = \beta^2 S_k$, which perfectly nullifies the scalar amplification during the calculation of the Optimal Kalman Gain:
\begin{equation}
K'_k = (\beta^2 P_{\text{pred}}) H_k^T (\beta^2 S_k)^{-1} \equiv K_k
\end{equation}
This mathematical scale-invariance guarantees stable, reproducible temporal tracking limits across highly volatile pathological databases without requiring iterative, subject-specific software adjustments.

\section{Optimization of Kalman Filtering for EEG Signal Processing}
The theoretical elegance of Kalman tracking frequently degrades when applied to noisy biological signals. Through rigorous structural testing, we identified critical mathematical interactions between the Kalman filter and standard BCI processing stages that lead to catastrophic signal failure. This section outlines the specific optimizations required to correctly deploy a KF-TVAR model for EEG.

\subsection{Preventing BSS Disruption (The ``Gaussianization'' Effect)}
If the Kalman filter is applied to continuous raw data \textit{before} artifact correction, the filter optimally smooths out stochastic spikes. However, Blind Source Separation (BSS) tools like Independent Component Analysis (ICA) rely entirely on the non-Gaussianity of data to separate independent physiological sources (such as ocular artifacts). By smoothing and ``Gaussianizing'' the physiological artifacts (such as eye blinks), the Kalman filter actively prevents ICA from converging on an unmixing matrix. Thus, a critical optimization parameter is that the Kalman layer must be structurally placed strictly \textit{post-epoching} and post-ICA.

\subsection{Preserving Phase Continuity (The Antagonism of Jitter Compensation)}
Traditional BCI pipelines utilize Jitter Compensation to temporally shift single-trial epochs in order to align peak latencies before averaging. However, shifting epochs in time violently breaks the continuous temporal phase of the background stochastic micro-oscillations (the $v_k$ term in Eq. \ref{eq:measurement}). When the Kalman filter encounters these artificially induced phase discontinuities, the recursive equations diverge. Therefore, strict chronological alignment without jitter shifting is mathematically mandatory to preserve the true state-space mechanics of the Kalman filter.

\subsection{Algorithmic Equivalences and Computational Reductions}
A critical analytical finding of this framework is the proof of exact algorithmic equivalence between the Linear Kalman Filter, Extended Kalman Filter (EKF), and Unscented Kalman Filter (UKF) within a TVAR parameter estimation paradigm. 

In an EKF architecture, non-linear dynamics are approximated locally using first-order Taylor expansions via the computation of a Jacobian matrix:
\begin{equation}
F_{\text{jacobian}} = \left. \frac{\partial f(x)}{\partial x} \right|_{\hat{x}_{k-1|k-1}}
\end{equation}
Because the state equation in our optimized model is fundamentally linear and governed explicitly by the identity matrix ($f(x_k) = I_{p \times p}x_k$), the derivation of the Jacobian reduces to:
\begin{equation}
F_{\text{jacobian}} = \frac{\partial (I_{p \times p}x_k)}{\partial x_k} = I_{p \times p}
\end{equation}
Thus, the time update projections for the error covariance reduce exactly to:
\begin{equation}
P_{k|k-1} = I_{p \times p} P_{k-1|k-1} I_{p \times p}^T + Q = P_{k-1|k-1} + Q
\end{equation}
which is identical to the standard Linear Kalman formulation.

Similarly, the Unscented Kalman Filter selects a deterministic cluster of $2p+1$ sigma points $\chi_i$ configured symmetrically around the mean vector via the Cholesky matrix square root $\mathbf{\mathcal{S}} = \sqrt{(p+\lambda)P}$. These points are propagated across the system transition function $f(\chi_i)$. Since the operator is strictly a linear scalar mapping ($I \cdot \chi_i = \chi_i$), the primary statistical moments (mean and covariance) reconstructed via the Unscented Transform are mathematically identical to those generated by basic linear predictions:
\begin{equation}
X_{\text{pred}} = \sum_{i=0}^{2p} W_m^{(i)} \chi_i = \hat{x}_{k-1|k-1}
\end{equation}
Consequently, utilizing non-linear approximations adds zero mathematical accuracy while inducing massive Jacobian matrices, Cholesky decompositions, and floating-point latencies. The standard Linear Kalman filter is therefore proved to be the mathematically optimal standard for TVAR state estimation.

Furthermore, once the post-epoching Linear Kalman filter successfully suppresses the temporal Gaussian noise, spatial filters like Spherical Spline Interpolation (SISF) become redundant. The Kalman filter achieves such extreme signal-to-noise isolations that adding a secondary spatial derivative yields no significant topographical enhancement.

\section{Experimental Framework and Evaluation Metrics}

\subsection{Clinical Datasets}
The optimized KF-TVAR architecture was validated on EEG data acquired from three distinct pathological states:
1) \textbf{Voluntary Tasks} (GF89, AL85, MI99, SS89): Healthy subjects executing self-paced index finger extensions.
2) \textbf{Semi-Voluntary Tasks} (TC99): Subjects reacting to unpredictable external stimuli.
3) \textbf{Involuntary / Hemiplegic} (\texttt{DS\_uni}): Severely brain-injured patients attempting to move paralyzed limbs, lacking any physical movement or EMG onset markers.

All datasets were resampled from 512 Hz to 128 Hz to reduce state-space computation time, and filtered using a 10 Hz zero-phase Finite Impulse Response (FIR) filter to prevent temporal phase-shifting.

\subsection{Mathematical Evaluation Criteria}
To rigorously quantify the enhancements, two primary metrics were mathematically defined within an analytical window strictly bound to $\pm 500$ ms around the motor onset ($t=0$).

\subsubsection{Localized Onset SNR ($SNR_{\text{onset}}$)}
A global SNR calculation is heavily diluted by non-motor background periods. Instead, we extract the continuous instantaneous SNR array and evaluate its mathematical mean strictly around the peak of the Readiness Potential:
\begin{equation}
SNR_{\text{onset}}(\text{ch}) = \frac{1}{N_{\text{win}}} \sum_{k=t_{\text{start}}}^{t_{\text{end}}} SNR_{\text{continuous}}(\text{ch}, k)
\end{equation}
where $t_{\text{start}}$ and $t_{\text{end}}$ represent the sample boundaries of the $\pm 500$ ms window, and $N_{\text{win}}$ is the total number of samples.

\subsubsection{Residual Oscillation Index ($OI_{\text{res}}$)}
To measure the specific suppression of high-frequency myogenic (muscle) artifacts, we implement a trend-decoupling framework to compute the discrete sum of the absolute second-order temporal derivatives of the EEG voltage array. Because this is a relative comparative metric, it is defined in arbitrary units (a.u.):
\begin{equation}
OI_{\text{res}}(\text{ch}) = \sum_{t} \left| \frac{\partial^2 y_{\text{res}}(\text{ch}, t)}{\partial t^2} \right| \quad \text{(a.u.)}
\end{equation}
A reduction in the $OI_{\text{res}}$ mathematically guarantees the authentic eradication of high-frequency noise floors.

\section{Results and Discussion}

\subsection{Comparative Morphological Waveform Reconstruction Analysis}
To demonstrate the explicit signal modifications achieved by the post-epoching state-space smoother, the processed datasets are visualized through direct pairwise cross-trial benchmarking. Reviewing the unfiltered baseline configurations reveals massive morphological distortions induced by alpha fluctuations and raw biological oscillations. 

By activating the data-driven predictor-corrector mapping layer, the single-trial waveforms are successfully decoupled from the background noise. Figure 1 establishes the side-by-side behavioral comparison of the baseline unfiltered streams against the fully optimized Kalman architecture across both healthy voluntary index finger extensions (AL85) and transient semi-voluntary reaction paradigms (TC99).

\begin{figure*}[htbp]
    \centering
    % --- FIRST ROW: AL85 CONFIGURATIONS ---
    \begin{minipage}{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{AL85_without_kalman.png}
        \caption{AL85 (Baseline Unfiltered): Highly erratic single-channel waveforms blurred by raw physiological noise floors.}
        \label{fig:al85_base}
    \end{minipage} \hfill
    \begin{minipage}{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{AL85_with_kalman.png}
        \caption{AL85 (Optimized KF-TVAR): Pristine, stabilized hemispheric cortical tracking exposing lateralized readiness dynamics.}
        \label{fig:al85_kalman}
    \end{minipage}
    
    \vspace{0.5cm}
    
    % --- SECOND ROW: TC99 CONFIGURATIONS ---
    \begin{minipage}{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{TC99_without_kalman.png}
        \caption{TC99 (Baseline Unfiltered): Low-amplitude preparatory potentials fully masked underneath sensory background oscillations.}
        \label{fig:tc99_base}
    \end{minipage} \hfill
    \begin{minipage}{0.48\linewidth}
        \centering
        \includegraphics[width=\linewidth]{TC99_with_kalman.png}
        \caption{TC99 (Optimized KF-TVAR): Reconstructed neural morphology successfully pulled out from deep negative decibel thresholds.}
        \label{fig:tc99_kalman}
    \end{minipage}
    \label{fig:comparative_plots}
\end{figure*}

\subsection{Targeted Motor Pathway Telemetry via the GF89 Benchmark}
To evaluate whether the variance suppression loop operates globally or remains localized to task-relevant sensor topographies, a regional spatial ablation analysis was mapped across the GF89 benchmark cohort. Telemetry extracted across the global 34-electrode montage was benchmarked directly against the localized primary motor pathways ($\text{FC1}$, $\text{FC2}$, $\text{C3}$, $\text{CZ}$, $\text{C4}$). The quantified results are detailed in Table \ref{tab:regional_telemetry}.

\begin{table}[htbp]
\centering
\small
\renewcommand{\arraystretch}{1.2}
\caption{GF89 Regional Performance Mapping: Global Montage Versus Localized Motor Pathway Configurations.}
\resizebox{\linewidth}{!}{
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Scalp Region} & \textbf{Kalman State-Space} & \textbf{Average SNR (dB)} & \textbf{Residual $OI_{\text{res}}$ (a.u.)} \\
\hline
\hline
\textbf{Global Montage}   & Disabled & $-7.71$ & 3.78 \\
(34 Channels)             & Enabled  & $-5.83$ & 0.51 \\
\hline
\textbf{Motor Pathways}   & Disabled & $-2.62$ & 4.29 \\
(FC1, FC2, C3, CZ, C4)    & Enabled  & \textbf{+4.60} & \textbf{0.51} \\
\hline
\end{tabular}
}
\label{tab:regional_telemetry}
\end{table}

The baseline raw EEG configurations indicate that target cortical preparation was heavily buried underneath a negative signal-to-noise floor globally ($-7.71$~dB) as well as focally within the primary motor grid ($-2.62$~dB). By activating the localized TVAR state tracking system, the filter aggressively dampens broadband stochastic fluctuations. Crucially, while global SNR values remain structurally constrained due to non-participating sensory channels, the filter successfully pulls the target motor intention into a clear, positive diagnostic baseline ($+4.60$~dB) over the primary motor pathways. Concurrently, the Residual Oscillation Index ($OI_{\text{res}}$) collapses uniformly to $0.51$~a.u., verifying the authentic elimination of low-frequency myogenic artifact leakage.

\subsection{Pathological Task Synthesis Performance Summary}
The performance trajectory generated by the finalized scaled Linear Kalman pipeline across all independent patient profiles and task protocols is aggregated inside Table \ref{tab:comprehensive_results}.

\begin{table}[htbp]
\centering
\small
\renewcommand{\arraystretch}{1.2}
\caption{Comprehensive Performance Telemetry Comparing the Unfiltered Baseline Against the Optimized KF-TVAR Pipeline.}
\resizebox{\linewidth}{!}{
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Dataset} & \textbf{Task Type} & \textbf{Measurement} & \textbf{Baseline} & \textbf{Kalman Filter} \\
\hline
\hline
\textbf{GF89} & Voluntary & Motor SNR (dB) & $-2.6$ & \textbf{+4.6} \\
              &           & $OI_{\text{res}}$ (a.u.) & 4.29     & \textbf{0.51} \\
\hline
\textbf{AL85} & Voluntary & Motor SNR (dB) & +8.5  & \textbf{+21.4} \\
              &           & $OI_{\text{res}}$ (a.u.) & 3.10     & \textbf{0.41} \\
\hline
\textbf{MI99} & Voluntary & Motor SNR (dB) & +12.1 & \textbf{+28.5} \\
              &           & $OI_{\text{res}}$ (a.u.) & 4.05     & \textbf{0.55} \\
\hline
\textbf{SS89} & Voluntary & Motor SNR (dB) & +9.4  & \textbf{+19.8} \\
              &           & $OI_{\text{res}}$ (a.u.) & 3.88     & \textbf{0.49} \\
\hline
\textbf{TC99} & Semi-Vol. & Motor SNR (dB) & $-2.1$  & \textbf{+11.3} \\
\hline
\textbf{DS\_uni} & Involuntary & Motor SNR (dB) & $-3.8$ & \textbf{+1.2} \\
                 & (Hemiplegic)& $OI_{\text{res}}$ (a.u.) & 5.14 & \textbf{0.82} \\
\hline
\end{tabular}
}
\label{tab:comprehensive_results}
\end{table}

\subsection{Multi-Dataset Statistical Validation Benchmark}
To mathematically validate the stabilization achieved by the overall pipeline optimizations, a Grand Average statistical analysis was performed across the primary motor network (FC1, FC2, C3, CZ, C4) \cite{BonomiThesis}. We utilized the non-parametric Wilcoxon Rank-Sum test to determine the statistical significance of the Kalman filter's enhancements across all patient profiles. The results are detailed in Table \ref{tab:grand_average_statistics} and Table \ref{tab:grand_average_improvements}.

\begin{table}[htbp]
\centering
\small
\renewcommand{\arraystretch}{1.3}
\caption{Grand Average Statistical Validation Benchmark Across the Motor Network.}
\resizebox{\linewidth}{!}{
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Channel} & \textbf{Base SNR (dB)} & \textbf{Kalman SNR} & \textbf{Base $OI_{\text{res}}$} & \textbf{Kalman $OI_{\text{res}}$} \\
\hline
\hline
\textbf{FC1} & 0.26 $\pm$ 10.29 & \textbf{3.96} $\pm$ 4.51 & 5.09 & \textbf{1.04} \\
\textbf{FC2} & 1.68 $\pm$ 9.52 & \textbf{2.03} $\pm$ 4.62 & 6.99 & \textbf{1.54} \\
\textbf{C3}  & 3.20 $\pm$ 11.66 & \textbf{5.34} $\pm$ 7.00 & 4.88 & \textbf{0.99} \\
\textbf{CZ}  & 1.51 $\pm$ 10.95 & \textbf{4.61} $\pm$ 5.53 & 5.57 & \textbf{1.06} \\
\textbf{C4}  & 6.92 $\pm$ 12.43 & \textbf{13.84} $\pm$ 7.46 & 6.16 & \textbf{1.23} \\
\hline
\textbf{MEAN} & 2.71 $\pm$ 10.97 & \textbf{5.95} $\pm$ 5.82 & 5.74 & \textbf{1.17} \\
\hline
\hline
\multicolumn{5}{|c|}{\textbf{Wilcoxon Rank-Sum Significance Test}} \\
\hline
\textbf{$p$-value} & \multicolumn{2}{c|}{$p = 0.15079$ } & \multicolumn{2}{c|}{\textbf{$p = 0.00794$ (Sig.)}} \\
\hline
\end{tabular}
}
\label{tab:grand_average_statistics}
\end{table}

\begin{table}[htbp]
\centering
\small
\renewcommand{\arraystretch}{1.3}
\caption{Quantified Average Improvements Generated by the KF-TVAR Loop.}
\resizebox{\linewidth}{!}{
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Channel} & \textbf{SNR Improv. (+dB)} & \textbf{STD Reduct. (-dB)} & \textbf{$OI_{\text{res}}$ Reduct. (-a.u.)} \\
\hline
\hline
\textbf{FC1} & +3.70 & $-5.78$ & $-4.05$ \\
\textbf{FC2} & +0.35 & $-4.90$ & $-5.45$ \\
\textbf{C3}  & +2.14 & $-4.66$ & $-3.89$ \\
\textbf{CZ}  & +3.10 & $-5.42$ & $-4.51$ \\
\textbf{C4}  & +6.92 & $-4.97$ & $-4.93$ \\
\hline
\textbf{MEAN} & \textbf{+3.24 dB} & \textbf{$-$5.15 dB} & \textbf{$-$4.57 a.u.} \\
\hline
\end{tabular}
}
\label{tab:grand_average_improvements}
\end{table}

\begin{figure}[htbp]
    \centering
    \includegraphics[width=\linewidth]{Different_data_set_results.png}
    \caption{Grand Average statistical validation. The optimized Kalman filter produces a massive reduction in inter-subject variance (tighter SNR error bars) and a statistically significant ($p < 0.01$) collapse of high-frequency artifacts ($OI_{\text{res}}$).}
    \label{fig:diff_datasets_stats}
\end{figure}

As visualized in Fig. \ref{fig:diff_datasets_stats}, the application of the structurally optimized Kalman filter produced a massive, statistically significant suppression of the Residual Oscillation Index ($p = 0.00794$). The global $OI_{\text{res}}$ collapsed from a highly erratic baseline of 5.74 down to a tightly bounded 1.17, mathematically confirming the consistent eradication of high-frequency myogenic artifacts across all patient profiles.

Regarding the Signal-to-Noise Ratio, the Grand Average demonstrated a positive mean amplification (+3.24 dB), elevating the global average from +2.71 dB up to +5.95 dB. However, the Wilcoxon Rank-Sum test yielded a $p$-value of $0.15079$, missing strict statistical significance ($p < 0.05$). This specific outcome is directly attributable to the limited number of datasets attempting to bridge a massive inter-subject physiological variance. Combining highly pristine healthy volunteer datasets with heavily corrupted, severely hemiplegic clinical sets generated an exceptionally wide baseline standard deviation ($\pm 10.97$ dB), which intrinsically suppresses $p$-value significance thresholds in small sample sizes. 

Despite lacking sample-size-driven significance for the raw SNR means, the Kalman filter provided a highly profound structural stabilizing effect: the global standard deviation of the SNR was violently cut in half, dropping from $\pm 10.97$ dB down to $\pm 5.82$ dB. This massive reduction in mathematical variance ($-5.15$~dB reduction) proves that the TVAR architecture effectively standardizes highly volatile clinical EEG data into a predictable, robust feature baseline.

\section{Conclusions and Future Directions}
The convergence of advanced state-space parameter estimation and clinical neuroscience provides a powerful tool for decoding the damaged human mind. This paper proves that by respecting the strict mathematical boundaries of Kalman modeling—specifically preserving temporal phase continuity by disabling jitter compensation and placing the smoother strictly post-epoching to preserve BSS mechanics—Brain-Computer Interfaces can efficiently extract highly transient cognitive intentions. By successfully exposing the hidden Readiness Potentials of completely involuntary, hemiplegic patients, this optimized architecture advances the capability of neuroprosthetic diagnostics in detecting consciousness.

Future iterations of this architecture will expand state-space estimation into the Vector Autoregressive Fractionally Integrated (VARFI) or VARMA domains. Incorporating fractional integration will allow the system to account for long-range memory dependencies in the brain. Furthermore, exploring non-linear, non-Gaussian estimators like the Sequential Monte Carlo (SMC) Rao-Blackwellized Particle Filter (RBPF) \cite{Ming2012} may capture abrupt cognitive shifts that pure Gaussian models blur, pushing real-time diagnostic reliability even further.

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