# Probabilistic Sensitivity Amplification Control for Lower Extremity Exoskeleton
## 2 Distributed Hidden-State Identification
### Model Learning  
Although the Gaussian process has a substantial impact in many fields, the crucial limit is that the computation expense scales in \(\mathcal{O}(N^{3})\) for training, while regarding prediction, the computed expense is \(\mathcal{O}(N^{2})\) if caching the kernel matrix. Thus, we employ the distributed Gaussian process [20] to address this issue, since this method can handle arbitrarily large datasets with sufficient hierarchical product-of-expert (PoE) models. The Gaussian process, fully specified by a mean function \(\mu_{f}\) and a covariance function \(k_{f}\), is a probabilistic distribution over functions \(f\). The latent function \(y_{i}=f(\mathbf{x_{i}})+\omega,\omega\sim\mathcal{N}(\mathbf{0},\mathbf{\sigma_{\omega}^{ 2}})\) can be learned from a training dataset \(\{\mathbf{X},\mathbf{z}\}\), where \(\mathbf{X}=[\mathbf{x_{1},...,x_{n}}]\) and \(\mathbf{z}=[z_{1},...,z_{n}]\). In addition, the posterior predictive distribution of the function \(f_{*}=f(x_{*})\), by a given set of hyper-parameters \(\mathbf{\theta}\) is defined with mean and covariance, respectively  
\[E[f_{*}] =m(\mathbf{x}_{*})=\mathbf{k}_{*}^{T}(\mathbf{K}+\sigma^{2}\mathbf{I})^{-1}\mathbf{z} =\mathbf{k}_{*}\mathbf{\beta}, \tag{1}\] \[\mathbf{var}[f_{*}] =\sigma^{2}(x_{*})=k_{**}-\mathbf{k}_{*}^{T}(\mathbf{K}+\sigma^{2}\mathbf{I} )^{-1}\mathbf{k}_{*}, \tag{2}\]  
With \(\mathbf{x}_{*}\) defining input, \(\mathbf{k}_{*}=k(\mathbf{X},\mathbf{x}_{*})\) and \(k_{**}=k(\mathbf{x}_{*},\mathbf{x}_{*})\). Note that the hyper-parameters \(\mathbf{\theta}\) of the covariance function consist of the length-scale \(I\), the signal variance \(\mathbf{\sigma_{f}^{2}}\) and the noise variance \(\mathbf{\sigma_{\omega}^{2}}\). Besides, the hyper-parameters are of significance to the covariance function defining the non-parametric model behavior [21]. According to the independent distributed assumption, the marginal likelihood \(p(\mathbf{y}|\mathbf{X},\mathbf{\theta})\) can be decomposed into \(M\) individual products:  
\[\log p(\mathbf{y}|\mathbf{X},\mathbf{\theta})\approx\sum_{m=1}^{M}\log p(\mathbf{y}^{(m)}|\bm {X}^{(m)},\theta), \tag{3}\]  
With \(M\) the total number of experts. In the model learning process, the hyper-parameters \(\mathbf{\theta}\) are shared for all \(M\) GPs. Moreover, a usual method for optimizing the hyper-parameters is to maximize the evidence maximization. Nevertheless, we apply a more efficient method in this paper, i.e., expectation maximization with Monte Carlo sampling. Therefore, the general class of latent state-space models with the Gaussian process is given below:  
\[\mathbf{x}_{k} =h(\mathbf{x_{k-1},a_{k-1}})+\mathbf{\omega}_{k}, \mathbf{\omega}_{k} \sim\mathcal{N}(\mathbf{0},\mathbf{Q}), \tag{4}\] \[\mathbf{z}_{k} =g(\mathbf{x}_{k})+\mathbf{v}_{k}, \mathbf{v}_{k} \sim\mathcal{N}(\mathbf{0},\mathbf{R}), \tag{5}\]  
With \(k=1,...,T\). The \(\mathbf{x}\in R^{D}\) is a latent state that evolves over time, while \(\mathbf{z}\in R^{D}\) can be read from actual measurement data. Both additive system noise, as well as measurement noise are assumed i.i.d. If not stated otherwise, both the measurement function and dynamic function are implicit, but instead can be described with the probability distributions over these Gaussian functions. Here, the functions are a Gaussian process non-parametric model, defined as \(h\sim\mathcal{GP}_{h}\) and \(g\sim\mathcal{GP}_{g}\), separately. Limited by the space of this paper, we briefly explain the learning process of the dynamic function \(h\sim\mathcal{GP}_{h}\). In the expectation step, we compute the posterior distribution of the system states \(q(\mathbf{X})=p(\mathbf{X}^{(m)}|\mathbf{y}^{(m)},\mathbf{\theta})\), where \(\mathbf{X}^{(m)}=(\mathbf{x}_{k-1},\mathbf{a}_{k-1})\) and \(\mathbf{y}^{(m)}=\mathbf{x}_{k}\). Nevertheless, the posterior distribution is not a Gaussian process and hence is not easy to obtain analytically.