Table 3 SI_CPSF algorithm

Initialization:

Set \(\alpha\),\({{\varvec{\uptau}}}\)

Calculate \({{\mathbb{C}}}_{{p_{{\mathbf{u}}} \left( {\mathbf{x}} \right)}}^{\alpha }\)\({{\mathbb{C}}}_{{p\left( {{\mathbf{x}}_{0} } \right)}}^{\alpha }\)

Implement partitioning sampling technique

\(\left\{ {{\hat{\mathbf{X}}}_{0} ,{{\varvec{\upomega}}}_{0} } \right\}_{{N_{0} }} \leftarrow PST\left( {p\left( {{\mathbf{x}}_{0} } \right),{{\mathbb{C}}}_{{p\left( {{\mathbf{x}}_{0} } \right)}}^{\alpha } ,{{\varvec{\uptau}}}} \right)\)

//Overall time steps:

For \(k \leftarrow 1\) to \(K\) do

1): Get the expression of the posterior PDF at time step k (see Eq. 15);

2): Get the estimation of the prior CPS (see Eq. 18);

3): Get the estimation of the posterior CPS (see Eq. 19);

4): Implement partitioning sampling technique:

\(\left\{ {{\tilde{\mathbf{X}}}_{k} ,{\tilde{\mathbf{\omega }}}_{k} } \right\}_{{\tilde{N}_{k} }} \leftarrow PST\left( {p\left( {{\mathbf{x}}_{k} \left| {{\mathbf{y}}_{1:k} } \right.} \right),{{\tilde{\mathbb{C}}}}_{{p\left( {{\mathbf{x}}_{k} \left| {{\mathbf{y}}_{1:k} } \right.} \right)}}^{\alpha } ,{{\varvec{\uptau}}}} \right)\)

5): Space contraction: Obtain \(\left\{ {{\hat{\mathbf{X}}}_{k} ,{{\varvec{\upomega}}}_{k} } \right\}_{{N_{k} }}\) according to (20)

6): State estimation (see Eq. 17)

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