Algorithm 1: The minimax PF algorithm for the standard PF in wireless sensor networks

□ Initialization

for $\kappa =1,\cdots ,N$, where N is the number of particles.

1. Random generation of initial particles:

${\psi }_0^{\kappa }\sim p\left({\psi }_0\right)$, and assign initial weights: ${\omega }_0^{\kappa }=\frac{1}{N}$.

end

□ Sequential update

for $t=1,\cdots ,T$, where T is the total time steps.

for $\kappa =1,\cdots ,N$,

2. Propagation of particles via a proposal density: $${\mathit{\psi }}_t^{\kappa }\sim q\left({\mathit{\psi }}_t^{}|{\mathit{\psi }}_{t-1}^{\kappa },{\mathit{m}}_t\right).$$ (32)

end

for $\kappa =1,\cdots ,N$,

3. Computing the weights of particles: $${\omega }_{i,t}^{\kappa }=p\left(m_{i,t}\right|{\mathit{\psi }}_t^{\kappa })\mathrm{for}i=1,\dots ,M$$ (33)

assuming the proposal density,

$q\left({\mathit{\psi }}_t^{}|{\mathit{\psi }}_{t-1}^{\kappa },{\mathit{m}}_t\right)=p\left({\mathit{\psi }}_k^{}|{\mathit{\psi }}_{t-1}^{\kappa }\right)$.

end

for $\kappa =1,\cdots ,N$,

4. Normalization. $${\overline{\omega }}_{i,t}^{\kappa }=\frac{{\omega }_{i,t}^{\kappa }}{{\sum }_{\kappa =1}^N{\omega }_{i,t}^{\kappa }}\mathrm{for}i=1,\dots ,M.$$ (34)

5. Selecting the minimum weight among M weights. $${\omega }_t^{\mathcal{M},\kappa }={\overline{\omega }}_{\tau ,t}^{\kappa },\mathrm{where}\tau =\underset{i\in \left\{1,\dots ,M\right\}}{argmin}{\overline{\omega }}_{i,t}^{\kappa }.$$

end

6. Normalization of the weights: $${\overline{\omega }}_t^{\mathcal{M},\kappa }=\frac{{\omega }_t^{\mathcal{M},\kappa }}{{\sum }_{\kappa =1}^N{\omega }_t^{\mathcal{M},\kappa }}\mathrm{for}\kappa =1,\cdots ,N.$$ (35)

7. Computing the estimate at the time step t: $${\widehat{\mathit{\psi }}}_t^{\mathcal{M}}=\sum _{\kappa =1}^N{\overline{\omega }}_t^{\mathcal{M},\kappa }{\mathit{\psi }}_t^{\kappa }.$$ (36)

8. Resampling N particles.

end