Algorithm 1. Proposed fully adaptive PF algorithm

Initiation step: Sample N particles from initial distributions of states and parameters: $$x_0^i~p\left(x_0\right)i=1,2,\dots ,N$$ $${\theta }_0^i~p\left({\theta }_0\right)i=1,2,\dots ,N$$ Assign initial equal weights to all the particles $$w_0^i~\frac{1}{N}i=1,2,\dots ,N$$ Recursive steps: Prediction: Estimate $m_{k-1}^i$ for each parameter using the shrinkage rule in Equation (4) Draw new samples for parameter vector from: $${\theta }_k^i~\mathcal{N}\left(·|m_{k-1}^i,a^2{\gamma }_{k-1}\right)$$ Propagate each particle one step forward using state process model with new sampled parameter: $$x_k^i~p(·|x_{k-1}^i,{\theta }_k^i)$$ Update: Compute the predicted observations using the tentative measurement model: $${\tilde{y}}_k=\tilde{h}\left(x_k,{\phi }_k,{\tau }_k\right)$$ Update the parameters of $\tilde{h}$ to minimize the KLD: $${\phi }_k=\arg \min \left\{D_{KL}\left(P||Q\right)\right\}=\arg \min \left\{{\displaystyle \int }P\left(y_k|x_k^i,{\theta }_k^i\right)\log \left[\frac{P(y_k|x_k^i,{\theta }_k^i)}{Q\left({\tilde{y}}_k^i|x_k^i,{\theta }_k^i,{\phi }_k\right)}\right]\right\}$$ Calculate the weights for each particle as new measurement ($y_k$) becomes available: $$w_k^i=w_{k-1}^i·p\left(y_k|x_k^i,{\theta }_k^i,{\phi }_k\right)$$ Normalize the weights: $$w_k^i=\frac{w_k^i}{{{\displaystyle \sum }}_{i=1}^N{w}_k^i}$$ Estimate: Estimate the expected state: $${\overline{x}}_k={\displaystyle \sum }_{i=1}^N{w}_k^i·x_k^i$$ Resample: Resample (with replacement) new set of particles for states and process model parameters ${\left\{x_k^i,{\theta }_k^i\right\}}_{i=1}^N$ based on calculated weights ($w_k^i$).