Structure Identification of Recursive TSK Particle Filtering via Type-2 Intuitionistic Fuzzy Decision

Abstract

To reduce the dimension and realize the adaptive adjustment of the fuzzy rules in the Takagi–Sugeno–Kang (TSK) model, a recursive TSK particle filtering algorithm based on type-2 intuitionistic fuzzy decision is proposed. In the proposed algorithm, we mainly emphasize on the structure identification of the TSK fuzzy model. Firstly, the similarity of features is calculated by the ridge reduction distribution function. Secondly, the type-2 fuzzy membership degree is applied to redefined a new intuitionistic fuzzy set, which is used to calculate the decision score (type-2 intuitionistic fuzzy membership degree) of each attribute feature. And we select the appropriate features (premise variables) by the threshold decision principle. Thirdly, the importance density function of particle filtering is constructed with the estimation results of the proposed TSK fuzzy model, which can effectively reduce the particle degradation phenomenon. Finally, the results of experiments on datasets show that the proposed algorithm is efficient and is excellent at handling the nonlinear non-Gaussian problem in the maneuvering target tracking system.

Appendix

According to the Eqs. (9) and (10), the objective functions $J_{1} ,J_{2}$ of the following Type-2 intuitionistic fuzzy algorithm are obtained :

$$J_{1} = \sum\limits_{i = 1}^{Q + 1} {\sum\limits_{j = 1}^{C} { - \left( {\mu_{ij}^{1} \left( {M_{i} } \right)} \right)^{{m_{1} }} \cdot f_{{_{ij} }}^{*} \left( {M_{i} } \right)} } + \lambda \sum\limits_{j = 1}^{C} {\mu_{ij}^{1} \left( {M_{i} } \right)}$$

$$J_{2} = \sum\limits_{i = 1}^{Q + 1} {\sum\limits_{j = 1}^{C} { - \left( {\mu_{ij}^{2} \left( {M_{i} } \right)} \right)^{{m_{2} }} \cdot f_{{_{ij} }}^{*} \left( {M_{i} } \right)} } + \lambda \sum\limits_{j = 1}^{C} {\mu_{ij}^{2} \left( {M_{i} } \right)}$$

Derivation of membership degree $\mu_{ij}^{1}$ by objective function $J_{1}$.

$$\frac{\partial J}{{\partial \mu_{ij}^{1} }} = - m_{1} \left( {\mu_{ij}^{1} \left( {M_{i} } \right)} \right)^{{m_{1} - 1}} f_{{_{ij} }}^{*} \left( {M_{i} } \right) + \lambda = 0$$

$$\mu_{ij}^{1} \left( {M_{i} } \right) = \left( {\frac{\lambda }{{m_{1} f_{{_{ij} }}^{*} \left( {M_{i} } \right)}}} \right)^{{\frac{1}{{m_{1} - 1}}}}$$

According to the constraint conditions $\sum\nolimits_{j = 1}^{C} {\mu_{ij}^{1} \left( {M_{i} } \right)} = 1$, the results are obtained.

$$\sum\limits_{r = 1}^{C} {\left( {\frac{\lambda }{{m_{1} f_{{_{ir} }}^{*} \left( {M_{i} } \right)}}} \right)^{{\frac{1}{{m_{1} - 1}}}} } = 1$$

$$\left( {\frac{\lambda }{{m_{1} }}} \right)^{{\frac{1}{{m_{1} - 1}}}} = \frac{1}{{\sum\limits_{r = 1}^{C} {\left( {\frac{1}{{f_{{_{ir} }}^{*} \left( {M_{i} } \right)}}} \right)^{{\frac{1}{{m_{1} - 1}}}} } }}$$

Therefore,

$$\mu_{ij}^{1} \left( {M_{i} } \right) = \frac{1}{{\sum\limits_{r = 1}^{C} {\left( {\frac{1}{{f_{{_{ir} }}^{*} \left( {M_{i} } \right)}}} \right)^{{\frac{1}{{m_{1} - 1}}}} } }}\left( {\frac{1}{{f_{{_{ij} }}^{*} \left( {M_{i} } \right)}}} \right)^{{\frac{1}{{m_{1} - 1}}}} = \frac{1}{{\sum\limits_{r = 1}^{C} {\left[ {f_{{_{ij} }}^{*} \left( {M_{i} } \right)/f_{{_{ir} }}^{*} \left( {M_{i} } \right)} \right]^{{1/\left( {m_{1} - 1} \right)}} } }}$$

In the same way, the objective function $J_{2}$ can be derived from the membership degree $\mu_{ij}^{2}$.

$$\mu_{ij}^{2} \left( {M_{i} } \right) = \frac{1}{{\sum\limits_{r = 1}^{C} {\left( {\frac{1}{{f_{{_{ir} }}^{*} \left( {M_{i} } \right)}}} \right)^{{\frac{1}{{m_{2} - 1}}}} } }}\left( {\frac{1}{{f_{{_{ij} }}^{*} \left( {M_{i} } \right)}}} \right)^{{\frac{1}{{m_{2} - 1}}}} = \frac{1}{{\sum\limits_{r = 1}^{C} {\left[ {f_{{_{ij} }}^{*} \left( {M_{i} } \right)/f_{{_{ir} }}^{*} \left( {M_{i} } \right)} \right]^{{1/\left( {m_{2} - 1} \right)}} } }}$$

According to the idea of Type-2 fuzzy C regression clustering, if $\frac{1}{{\sum\nolimits_{r = 1}^{C} {\left[ {f_{{_{ij} }}^{*} \left( {M_{i} } \right)/f_{{_{ir} }}^{*} \left( {M_{i} } \right)} \right]} }} < \frac{1}{C}$, the objective function $J_{1}$ is used to derive the membership degree $\mu_{ij}^{1}$. On the contrary, the objective function $J_{2}$ is used to derive the membership degree $\mu_{ij}^{2}$. The upper membership function shown in Eq. (11) can be obtained.

$$\bar{\mu }_{ij} = \left\{ {\begin{array}{*{20}l} {\frac{1}{{\sum\nolimits_{r = 1}^{C} {\left[ {f_{{_{ij} }}^{*} \left( {M_{i} } \right)/f_{{_{ir} }}^{*} \left( {M_{i} } \right)} \right]^{{1/\left( {m_{1} - 1} \right)}} } }},} \hfill & {\frac{1}{{\sum\nolimits_{r = 1}^{C} {\left[ {f_{{_{ij} }}^{*} \left( {M_{i} } \right)/f_{{_{ir} }}^{*} \left( {M_{i} } \right)} \right]} }} < \frac{1}{C}} \hfill \\ {\frac{1}{{\sum\nolimits_{r = 1}^{C} {\left[ {f_{{_{ij} }}^{*} \left( {M_{i} } \right)/f_{{_{ir} }}^{*} \left( {M_{i} } \right)} \right]^{{1/\left( {m_{2} - 1} \right)}} } }},} \hfill & {otherwise} \hfill \\ \end{array} } \right.$$

If $\frac{1}{{\sum\nolimits_{r = 1}^{C} {\left[ {f_{{_{ij} }}^{*} \left( {M_{i} } \right)/f_{{_{ir} }}^{*} \left( {M_{i} } \right)} \right]} }} \ge \frac{1}{C}$, the objective function $J_{1}$ is used to derive the membership degree $\mu_{ij}^{1}$. On the contrary, the objective function $J_{2}$ is used to derive the membership degree $\mu_{ij}^{2}$. The lower membership function shown in Eq. (12) can be obtained.

$$\mu_{ij}^{'} = \left\{ {\begin{array}{*{20}l} {\frac{1}{{\sum\nolimits_{r = 1}^{C} {\left[ {f_{{_{ij} }}^{*} \left( {M_{i} } \right)/f_{{_{ir} }}^{*} \left( {M_{i} } \right)} \right]^{{1/\left( {m_{1} - 1} \right)}} } }},} \hfill & {\frac{1}{{\sum\nolimits_{r = 1}^{C} {\left[ {f_{{_{ij} }}^{*} \left( {M_{i} } \right)/f_{{_{ir} }}^{*} \left( {M_{i} } \right)} \right]} }} \ge \frac{1}{C}} \hfill \\ {\frac{1}{{\sum\nolimits_{r = 1}^{C} {\left[ {f_{{_{ij} }}^{*} \left( {M_{i} } \right)/f_{{_{ir} }}^{*} \left( {M_{i} } \right)} \right]^{{1/\left( {m_{2} - 1} \right)}} } }},} \hfill & {otherwise} \hfill \\ \end{array} } \right.$$