## 2 Materials and Methods
### Formulation of IBIF Model Based on a Kalman Filter  
Even though the IBIF algorithm performs well in laboratory settings where the calibration procedure is done with a Rothenberg mask, there are uncertainties related to the application of the IBIF filter in ambulatory settings. First, the position and arrangement of the sensor during in field monitoring might not match laboratory specifications, so the subject-specific parameters could change slightly. One approach for tracking relevant latent signals (i.e., GVV) of a given process (i.e., IBIF) based on related noisy/perturbed observations (i.e., neck-skin acceleration) is the use of a Bayesian approach, which allows to simultaneously estimate both the unknown signal and its uncertainty [42]. Under the assumption of linearity and Gaussian distributions for the unknown states, a Kalman Filter is the optimal Bayesian estimator. In this work, we propose an alternative formulation of IBIF combining the state-space framework with the MA canonical form [43] obtained from the \(h(n)\) impulse response:  
\[\mathbf{x}(n+1) =\mathbf{A}\mathbf{x}(n)+\mathbf{w}(n), \tag{10}\] \[y(n) =\mathbf{C}\mathbf{x}(n)+v(n). \tag{11}\]  
What follows is the instantiation of a Kalman filter from the model (10) and (11), to our particular problem, where \(\mathbf{x}(n)\) is the state vector containing the GVV signal: \(\mathbf{x}(n)=[x(n-N+1)\quad x(n-N+2)\quad x(n-N+3)\quad\cdots\quad x(n)]^{T}\) where \(N\) is the length of the skin-impulse response. Following [43], the transition matrix \(\mathbf{A}\) is given by:  
\[\mathbf{A}= \begin{bmatrix}0&1&0&\ldots&0&0\\ 0&0&1&\ldots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&0&\ldots&0&1\\ 0&0&0&\ldots&0&0\end{bmatrix} \in\mathbb{R}^{N\times N},\]and \(\mathbf{w}(n)\) is a Gaussian process noise with zero mean and covariance matrix:  
\[\mathbf{R}_{\mathbf{w}}= \begin{bmatrix}0&0&\ldots&0&0\\ 0&0&\ldots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&\ldots&0&0\\ 0&0&\ldots&0&\sigma_{w}^{2}\end{bmatrix} \in\mathbb{R}^{N\times N}.\]  
The initial condition is specified with the mean \(\mathbf{m}_{0}=\mathbf{E}(\mathbf{x}_{0})\) and covariance \(\mathbf{P}_{0}=\mathbf{E}((\mathbf{x}_{0}-\mathbf{m}_{0})(\mathbf{x}_{0}- \mathbf{m}_{0})^{T})\) of the initial state \(\mathbf{x}_{0}\). The observation Equation (11) relates the accelerometer signal \(y(n)\) as the convolution between the unobserved state \(\mathbf{x}\) and the neck-skin impulse response \(h(n)\) with coefficients:  
\[\mathbf{C}= \begin{bmatrix}h(0)&h(1)&h(2)&\cdots&h(N-1)\end{bmatrix} ^{T}\in \mathbb{R}^{1\times N}.\]  
According to (11), Gaussian measurement noise \(v(n)\) with mean zero and variance \(\sigma_{v}^{2}\) is assumed as the additive perturbation to the observed signal. Implementation of the standard MA Kalman filter for a discrete-time set \(n=1,\ldots,T\) is described in Algorithm 1:  
```
1:procedureKalman(\(\mathbf{A},\mathbf{C},\mathbf{R}_{\mathbf{w}},\mathbf{R}_{v},\mu_{0},\Sigma_{0},y(n)\))
2: Initialization: Set \(\mathbf{x}(0|0)=\mu_{0}\) and \(\mathbf{P}(0|0)=\mathbf{\Sigma}_{0}\)
3: Filtering: For \(n=1,2\ldots T\)
4: Prediction equations:
5:\(\mathbf{x}(n|n-1)=\mathbf{Ax}(n-1|n-1)\)
6:\(\mathbf{P}(n|n-1)=\mathbf{AP}(n-1|n-1)\mathbf{A}^{T}+\mathbf{R}_{\mathbf{w}}\)
7: Update equations:
8:\(\mathbf{K}(n)=\mathbf{P}(n|n-1)\mathbf{C}^{T}(\mathbf{CP}(n|n-1)\mathbf{C}^ {T}+\mathbf{R}_{v})^{-1}\)
9:\(\mathbf{x}(n|n)=\mathbf{x}(n|n-1)+\mathbf{K}(n)(y(n)-\mathbf{C}^{T}\mathbf{x }(n|n-1))\)
10:\(\mathbf{P}(n|n)=\mathbf{P}(n|n-1)-\mathbf{K}(n)\mathbf{CP}(n|n-1)\)
```  
**Algorithm 1** Kalman Filter Algorithm. The state matrix \(\mathbf{A}\) is circular, and the state vector is defined by including the glottal flow for different delays. Therefore, when the filter is applied, states with different delays \(n-N+1,n-N+2,\ldots,n-1\) are estimated conditioned on the observations up to the current time index \(n\), i.e., future information is used in the inference process.