# Multi-Agent-Supported Tracking Based on Hidden Markov Model with Weighted Entropy
## 3 Multi-Agent-Supported Tracking Based on Hidden Markov Model with Weighted Entropy
### Kalman Filter Reconstruction Through HMM
Leveraging Equation (8), the unconditional mean of \(\mathbf{x}_{t}\) is derived through a recursive formulation as follows:  
\[\prod_{k=2}^{t}\mathbf{F}_{k}\mathbf{x}_{1}+\sum_{k=2}^{t}\prod_{l=k+1}^{t} \mathbf{F}_{l}\mathbf{B}_{k} \tag{10}\]  
And the unconditional co-variance is computed as follows:  
\[\mathbf{P}_{t}=\mathbb{E}[\mathbf{x}_{t}\mathbf{x}_{t}^{\mathsf{T}}] \tag{11}\]  
With the constrains on independence, exclusion of control inputs, and Equation (8), we have  
\[\mathbf{P}_{t}=\mathbf{F}_{t}\mathbf{P}_{t-1}\mathbf{F}_{t}^{\mathsf{T}}+ \mathbf{Q}_{t} \tag{12}\]  
It establishes an equation for the unconditional covariance and is formally recognized as the Lyapunov equation. The next task is to estimate the state \(\mathbf{x}_{t}\) utilizing a subset of output sequences \(\mathbf{z}_{0},\ldots,\mathbf{z}_{t}\). In standard Kalman filter notation, the governing equations for this estimation process are formulated as follows:  
\[\mathbf{\hat{x}}_{t|t}\triangleq\mathbb{E}[\mathbf{x}_{t}\mid\mathbf{z}_{0}, \ldots,\mathbf{z}_{t}] \tag{13}\]  
\[\mathbf{P}_{t|t} \triangleq \mathbb{E}[(\mathbf{x}_{t}-\mathbf{\hat{x}}_{t|t})(\mathbf{x}_{ t}-\mathbf{\hat{x}}_{t|t})^{\mathsf{T}}\mid\mathbf{z}_{0},\ldots,\mathbf{z}_{t}] \tag{14}\]  
The time propagation process is structured into two sequential stages to preserve the classical formulations of Kalman filters:  
* Time update: \[\mathbb{P}[\mathbf{x}_{t}\mid\mathbf{z}_{0},\ldots,\mathbf{z}_{t}]\to \mathbb{P}[\mathbf{x}_{t+1}\mid\mathbf{z}_{0},\ldots,\mathbf{z}_{t}]\] (15)
* Measurement update: \[\mathbf{v}_{t}\sim\mathcal{N}(0,\mathbf{R}_{t})\]0 (16)  
Conditioned on the output sequence \(\mathbf{z}_{0},\ldots,\mathbf{z}_{t}\), the variables \(\mathbf{x}_{t+1}\) and \(\mathbf{z}_{t+1}\) follow a joint Gaussian distribution with the mean and covariance matrix respectively denoted by  
\[\left[ \begin{array}{c}\mathbf{\hat{x}}_{t+1|t}\\ \mathbf{H}_{t+1}\mathbf{\hat{x}}_{t+1|t}\end{array} \right] \tag{17}\]  
\[\left[ \begin{array}{cc}\mathbf{P}_{t+1|t}&\mathbf{P}_{t+1|t}\mathbf{H}_{t+1 }^{\mathsf{T}}\\ \mathbf{H}_{t+1}\mathbf{P}_{t+1|t}&\mathbf{H}_{t+1}\mathbf{P}_{t+1|t}\mathbf{ H}_{t+1}^{\mathsf{T}}+\mathbf{R}_{t+1}\end{array} \right] \tag{18}\]  
Further, with \(\mathbf{z}_{0},\ldots,\mathbf{z}_{t}+1\), the Kalman filter can be formally represented through the following recursive equations:  
\[\mathbf{\hat{x}}_{t|t-1}=\mathbf{F}_{t}\mathbf{\hat{x}}_{t-1|t-1} \tag{19}\]  
\[\mathbf{P}_{t|t-1}=\mathbf{F}_{t}\mathbf{P}_{t-1|t-1}\mathbf{F}_{t}^{\mathsf{ T}}+\mathbf{Q}_{t} \tag{20}\]  
\[\mathbf{\hat{x}}_{t+1|t+1}=\mathbf{\hat{x}}_{t+1|t}+\mathbf{P}_{t+1|t}\mathbf{ H}_{t+1}^{\mathsf{T}}\Big{(}\mathbf{H}_{t+1}\mathbf{P}_{t+1|t}\mathbf{H}_{t+1}^{ \mathsf{T}}+\mathbf{R}_{t+1}\Big{)}^{-1}(\mathbf{z}_{t+1}-\mathbf{H}_{t+1} \mathbf{\hat{x}}_{t+1|t}) \tag{21}\]  
\[\mathbf{P}_{t+1|t+1}=\mathbf{P}_{t+1|t}-\mathbf{P}_{t+1|t}\mathbf{H}_{t+1}^{ \mathsf{T}}\Big{(}\mathbf{H}_{t+1}\mathbf{P}_{t+1|t}\mathbf{H}_{t+1}^{\mathsf{ T}}+\mathbf{R}_{t+1}\Big{)}^{-1}\mathbf{H}_{t+1}\mathbf{P}_{t+1|t} \tag{22}\]  
To infer the final posterior distribution \(\mathbb{P}(\mathbf{x}_{t}\mid\mathbf{z}_{1},\ldots,\mathbf{z}_{T})\), the following formulation is employed:  
\[\mathbf{\hat{x}}_{t|T}=\mathbf{P}_{t|T}(\mathbf{P}_{t|t}^{-1}\mathbf{\hat{x}}_ {t|t}+\mathbf{P}_{t|t+1}^{-1}\mathbf{\hat{x}}_{t|t+1}) \tag{23}\]  
\[\mathbf{P}_{t|T}=\Big{(}\mathbf{P}_{t|t}^{-1}+\mathbf{P}_{t|t+1}^{-1}-\Sigma_ {t}^{-1}\Big{)}^{-1} \tag{24}\]  
Thus, we formalize the Kalman filter as a special case of the HMM framework.