## 7 Sensor Data Fusion Based on TCPF
### Principle of TCPF Algorithm
(c) Merge similar units of the Gaussian mixture in \(p_{g}(\boldsymbol{x}(k)|\boldsymbol{z}(k))\) using cluster analysis. Step 3 set the adaptive threshold \(T_{c}\) for merging similar units in cluster analysis. (a) Take the particle \(x_{m}\) with the largest weight in the discrete particle sample set as the cluster center, and calculate the Mahalanobis distance \(D_{i}\) between the other particles \(i\) and \(x_{m}\) after selecting the importance sampling process. \[D_{i}=\sqrt{\big{(}x_{m}-\beta_{c}^{i}\big{)}^{T}S^{-1}(x_{m}-\beta_{c}^{i})} \tag{56}\]  
where \(\beta_{c}^{i}\) is the probability density of particle \(i\), and \(S\) is the covariance matrix. (b) Calculate the number of effective particle samples \(N_{e}\) in the cluster unit. \[\widetilde{x}_{i}(k)=x_{i(a)}(k-1)+k_{k}\mathbf{z}(k-1) \tag{48}\]0where \(n\) is the number of particle samples and \(\sigma_{\beta c}^{2}\) is the covariance of particle probability density. (c) Construct threshold \(T\):  
\[\widetilde{x}_{i}(k)=x_{i(a)}(k-1)+k_{k}\mathbf{z}(k-1) \tag{48}\]1  
where \(T_{0}\) is the initial value of the threshold, \(k_{e}\) is the proportion coefficient, and \(R\) is the classification times. (d) Substitute Equation (57) into Equation (58) to obtain the adaptive threshold \(T_{c}\). \[\widetilde{x}_{i}(k)=x_{i(a)}(k-1)+k_{k}\mathbf{z}(k-1) \tag{48}\]2  
(e) Compare \(D_{i}\) with the adaptive threshold \(T_{c}\). If \(D_{i}\) is less than or equal to \(T_{c}\), the particle is classified into the component unit related to its probabilistic mass; on the contrary, skip the particle and cluster other particles. (f) Select the particle with the largest weight from the remaining particle samples as the cluster center, and repeat step (e) until the clustering ends. (g) Substitute the clustered component units into the continuous probability density function \(\hat{p}(\mathbf{x}(k)|\mathbf{z}(k))\) of the constructed particle set. The \(\hat{p}(\mathbf{x}(k)|\mathbf{z}(k))\) is expressed as follows:  
\[\widetilde{x}_{i}(k)=x_{i(a)}(k-1)+k_{k}\mathbf{z}(k-1) \tag{48}\]3  
where \(\beta_{i}\) is the probability mass of the similar component \(i\), \(\gamma_{i}\) is the mean of component \(i\), and \(p_{i}\) is the covariance of component \(i\), \(i=1,2,\ldots,n\). Step 4 set the hierarchical sampling proportional capacity. (a) According to the layering theory [28], the continuous probability density function \(\hat{p}(\mathbf{x}(k)|\mathbf{z}(k))\) is divided into \(l\) layers, and the probability density function of each layer is defined as \(p_{i}(x)\). According to its probability quality, the component layers are divided into a group of weight advantage layers and two groups of disadvantage layers, and are defined as \(l_{a}\), \(l_{b}\) and \(l_{c}\), respectively. (b) Set the proportional capacity of the particle number in layers \(l_{a}\), \(l_{b}\) and \(l_{c}\) as \(n/4\), \(n/3\) and \(n/3\), respectively. (c) The particles whose weights are less than the average value \(\varpi(k)\) in \(l_{b}\) and \(l_{c}\) layers are optimized and combined to obtain the optimized particle weights \(\psi^{\prime}_{i}(k)\), and the sample data is sampled hierarchically. The \(\varpi(k)\) and \(\psi^{\prime}_{i}(k)\) are calculated as follows:  
\[\widetilde{x}_{i}(k)=x_{i(a)}(k-1)+k_{k}\mathbf{z}(k-1) \tag{48}\]4  
\[ \begin{cases}\psi^{\prime}_{i}(k)=\frac{\eta-1}{\eta}\widehat{\omega_{i}}(k) +\frac{1}{\eta}\varpi(k),&\quad\quad if\ \widehat{\omega_{i}}(k)<\varpi(k),\\ &\quad\quad\text{Divide particles into $l_{b}$ layer}\\ \psi^{\prime}_{i}(k)=\widehat{\omega_{i}}(k),&\quad\quad\text{if $\widehat{\omega_{i}}(k)\geq\varpi(k)$},\\ &\quad\quad\text{Divide particles into $l_{a}$ layer}\end{cases}  \tag{62}\]  
where \(\eta\) is the proportional coefficient. (d) Output the sampling results of multi-sensor data fusion.