# Probabilistic Sensitivity Amplification Control for Lower Extremity Exoskeleton
## 2 Distributed Hidden-State Identification
### Distributed Identification Process
The gPoE distributed scheme provides more reasonable identification of the hidden states than any other models in Figure 4.  
Moreover, Figure 5 shows two aspects of the distributed identification framework with the different number of GP experts: (1) a comparison of approximation quality; (2) computation expense. Figure 4: Comparison of four data fusion methods for distributed hidden-state identification. In each subfigure, the confidence interval of the comparison algorithm is filled between two green lines. Furthermore, the GP-RTSS (full GP) to be approximated is shown in the blue shaded area, representing 95% of the confidence interval, and the ground truth is given in the red dashed line. (**a**) PoE; (**b**) gPoE ; (**c**) BCM; (**d**) rBCM. According to the information in Figure 5, for a certain set of training data, with the increase of the GP experts, the computation time significantly decreases. However, the computation expense reaches a minimum when there are five GP experts. Besides, the interesting properties are that there is no significant difference (root-mean-square-error (RSME): 0.01 and negative log-likelihood (NLL): 0.2) between the identification framework with one GP expert (full GP) and more than one expert. Furthermore, note that a better approximation quality can be expected with ten GP experts with the minima of RSME or with eight GP experts according to the NLL criterion. Thus, although we do not argue that our proposed hidden-state identification scheme outperforms the full GP identification algorithm concerning precision, the computation expense has been dramatically reduced. Moreover, the overall identification procedure is concluded in Algorithm 2. ```
input : Trained data \(\mathcal{D}=\mathbf{X}^{(m)},\mathbf{y}^{(m)}\), where \(m\in 1,...,M\), Hyper-parameters \(\mathbf{\theta}\) and observations; output : The mean \(\mathbf{\mu}_{*}\) and variance \(\mathbf{\sigma}_{*}\) of the identified system states; for\(m\gets 1\) to \(M\)do for\(k\gets 1\) to \(T\)do // Forward filtering; \((\mathbf{\mu}_{\mathbf{k}|\mathbf{k}}^{x})^{m}=(\mathbf{\mu}_{\mathbf{k}|\mathbf{k}-\mathbf{1}}^{x})^{m}+( \mathbf{C}_{\mathbf{k}|\mathbf{k}-\mathbf{1}}^{x})^{m}((\mathbf{C}_{\mathbf{k}|\mathbf{k}-\mathbf{1}}^{x})^{m} )^{-1}(z_{k}-(\mathbf{\mu}_{\mathbf{k}|\mathbf{k}-\mathbf{1}}^{x})^{m})\); \((\mathbf{C}_{\mathbf{k}|\mathbf{k}}^{x})^{m}=(\mathbf{C}_{\mathbf{k}|\mathbf{k}-\mathbf{1}}^{x})^{m}-(\bm {C}_{\mathbf{k}|\mathbf{k}-\mathbf{1}}^{x})^{m}((\mathbf{C}_{\mathbf{k}|\mathbf{k}-\mathbf{1}}^{x})^{m})^{- 1}((\mathbf{C}_{\mathbf{k}|\mathbf{k}-\mathbf{1}}^{x})^{m})^{T}\);  end for for\(k\gets T\)to\(1\)do // Backward smoothing; \((\mathbf{\mu}_{\mathbf{k}-\mathbf{1}|\mathbf{T}}^{x})^{m}=(\mathbf{\mu}_{\mathbf{k}-\mathbf{1}|\mathbf{k}}^{x})^{ m}+(\mathbf{I}_{\mathbf{k}-\mathbf{1}})^{m}((\mathbf{\mu}_{\mathbf{k}|\mathbf{T}})^{m}-(\mathbf{\mu}_{\mathbf{ k}|\mathbf{k}}^{x})^{m})\); \((\mathbf{C}_{\mathbf{k}-\mathbf{1}|\mathbf{T}}^{x})^{m}=(\mathbf{C}_{\mathbf{k}-\mathbf{1}|\mathbf{k}}^{x})^{m} +(\mathbf{I}_{\mathbf{k}-\mathbf{1}})^{m}((\mathbf{C}_{\mathbf{k}|\mathbf{T}}^{x})^{m}-(\mathbf{C}_{\mathbf{k} |\mathbf{k}}^{x})^{m})((\mathbf{J}_{\mathbf{k}-\mathbf{1}})^{m})^{T}\);  end for // Fusion with Generalized PoE ; for\(k\gets 1\)to\(T\)do \(\mathbf{\mu}_{s}=\mathbf{\sigma}_{*}^{-2}\sum_{m}\beta_{m}\mathbf{\sigma}((\mathbf{C}_{\mathbf{k}| \mathbf{T}}^{x})^{m})^{-2}(\mathbf{\mu}_{\mathbf{k}|\mathbf{T}}^{x})^{m})\); \(\mathbf{\sigma}_{*}^{-2}=\sum_{m}\beta_{m}(\mathbf{C}_{\mathbf{k}|\mathbf{T}}^{x})^{m}\);  end for
```  
**Algorithm 2**Inference with Monte Carlo Markov chain. Figure 3:0 Computation time, RSME and NLL for the different number of GP experts. The computation time is measured in MATLAB 2015b (MathWorks Company, USA) on a laptop ( 2.6 GHz cores and 8 GB RAM) with a Ubuntu 16.04 system (Canonical Ltd., UK). Each case of different numbers of GP experts is tested 20 times for the purpose of reducing the stochastic data influence.