Algorithm 4: A diffusion distributed Kalman filter separating diffusion update.

For the node $i\in V$ and node $j\in N_i$,   Initialize with:     ${\widehat{x}}_{i,0\mid 0}^{\mathrm{loc}}=\mathbb{E}{x}_0$,     $P_{i,0\mid 0}^{\mathrm{loc}}=\mathbb{E}\left[(x_0-\mathbb{E}{x}_0){(x_0-\mathbb{E}{x}_0)}^{\mathrm{T}}\right]$,     ${\widehat{x}}_{i,0\mid 0}=\mathbb{E}{x}_0$,     $P_{i,0\mid 0}=\mathbb{E}\left[(x_0-\mathbb{E}{x}_0){(x_0-\mathbb{E}{x}_0)}^{\mathrm{T}}\right];$   Local update:     ${\widehat{x}}_{i,t\mid t-1}^{\mathrm{loc}}=A{\widehat{x}}_{i,t-1\mid t-1}^{\mathrm{loc}}$,     $P_{i,t\mid t-1}^{\mathrm{loc}}=A{P}_{i,t-1\mid t-1}^{\mathrm{loc}}{A}^{\mathrm{T}}+\mathsf{\Gamma }Q{\mathsf{\Gamma }}^{\mathrm{T}}$,     ${\left(P_{i,t\mid t}^{\mathrm{loc}}\right)}^{-1}={\left(P_{i,t\mid t-1}^{\mathrm{loc}}\right)}^{-1}+H_i^{\mathrm{T}}{R}_i^{-1}{H}_i$,     ${\widehat{x}}_{i,t\mid t}^{\mathrm{loc}}={\widehat{x}}_{i,t\mid t-1}^{\mathrm{loc}}+P_{i,t\mid t}^{\mathrm{loc}}{H}_i^{\mathrm{T}}{R}_i^{-1}(y_{i,t}-H_i{\widehat{x}}_{i,t\mid t-1}^{\mathrm{loc}})$;   Diffusion incremental Update:     ${\widehat{x}}_{i,t\mid t-1}=A{\widehat{x}}_{i,t-1\mid t-1}$,     $P_{i,t\mid t-1}=A{P}_{i,t-1\mid t-1}{A}^{\mathrm{T}}+\mathsf{\Gamma }Q{\mathsf{\Gamma }}^{\mathrm{T}}$,     ${\left(P_{i,t\mid t}^{\mathrm{diffusion}}\right)}^{-1}=P_{i,t\mid t-1}^{-1}+H_i^{\mathrm{T}}{R}_i^{-1}{H}_i$,     ${\widehat{x}}_{i,t\mid t}^{\mathrm{diffusion}}={\widehat{x}}_{i,t\mid t-1}+P_{i,t\mid t}^{\mathrm{diffusion}}{H}_i^{\mathrm{T}}{R}_i^{-1}(y_{i,t}-H_i{\widehat{x}}_{i,t\mid t-1})$;   Communication and fusion update:     Send ${\widehat{x}}_{j,t\mid t}^{\mathrm{diffusion}}$ and $P_{j,t\mid t}^{\mathrm{diffusion}}$ to adjacent node i,     $P_{i,t\mid t}^{-1}={\sum }_{j\in N_i}{w}_j{\left(P_{i,t\mid t}^{\mathrm{diffusion}}\right)}^{-1}+w_i{\left(P_{i,t\mid t}^{\mathrm{loc}}\right)}^{-1}$,     ${\widehat{x}}_{i,t\mid t}=P_{i,t\mid t}\left[{\sum }_{j\in N_i}{w}_j{\left(P_{i,t\mid t}^{\mathrm{diffusion}}\right)}^{-1}{\widehat{x}}_{j,t\mid t}^{\mathrm{diffusion}}+w_i{\left(P_{i,t\mid t}^{\mathrm{loc}}\right)}^{-1}{\widehat{x}}_{i,t\mid t}^{\mathrm{loc}}\right],$   where $w_j$ and $w_i$ is calculated by Equation (5).