# Anomaly Detection in Autonomous Deep-Space Navigation via Filter Bank Gating Networks
## 2 Related Work
Typically, the outputs from each estimator are scaled by their weight, and then they are all linearly combined to form a final state estimate (Figure 2). Alternatively, the weights can be used to select the estimator that is indicated to be the most likely model (i.e., highest weight), and then its solution will be used as the optimal MME solution. Care must be used in both situations since the size of the estimate vector may vary depending on the estimator's setup. However, if the user is only interested in certain parameters for prediction purposes (e.g., target state, dynamics parameters, etc. ), then these may be extracted to form the final estimate vector. Figure 2: Multiple Model Estimation (MME) filter bank with gating network. As defined, MME is a method that is very similar to the idea of MoE; however, MME is more focused on estimation problems (whereas MoE is more generally applicable) and MME gating networks can (and usually are) functions of the filters' outputs. This difference is significant because it means an MME does not need to be trained in the same way that an MoE does--though it is still possible to design a gating network that would be trained. A standard MME gating network relies more on statistical theory and the modeling put into the estimators as opposed to the MoE, which can be trained to act as desired. This may tend to mean that an MME is less flexible than an MoE (depending on how many experts are used), but it may be naturally more robust. To achieve robustness with an MoE, the operator needs to understand all of the possible failure scenarios, in order to make the proper adjustments and train the MoE accordingly. As mentioned, the weights from the standard gating network in the MME algorithm tend to measure the likelihood that their corresponding filter is the correct model. The original metric used by [PERSON] [35] is based on [PERSON] rule. Given a set of \(N\) estimators in an MME (\(\{\alpha_{1},\alpha_{2},\ldots,\alpha_{N}\}\)), [PERSON]'s weight quantifies the likelihood that any given model is the correct one. We compute this as shown in Equation (1). \[p_{k,i}=f(\alpha=\alpha_{i}|\mathcal{Y}_{1}^{k})=\frac{f(\mathcal{Y}=\mathcal{ Y}_{1}^{k}|\alpha_{i})f(\alpha=\alpha_{i})}{\sum_{j=1}^{N}f(\mathcal{Y}= \mathcal{Y}_{1}^{k}|\alpha_{j})f(\alpha=\alpha_{j})} \tag{1}\]  
\[\mathcal{Y}_{i}^{j}=\{\vec{y}_{i},\vec{y}_{i+1},\ldots,\vec{y}_{j-1},\vec{y}_ {j}\} \tag{2}\]  
In this equation, \(\mathcal{Y}_{i}^{j}\) is the set of all measurements between indices \(i\) and \(j\) (Equation (2)), \(f(\mathcal{Y}|\alpha_{i})\) is the conditional probability density function expressing the likelihood of receiving the measurement set \(\mathcal{Y}\) given estimator \(\alpha_{i}\), and \(f(\alpha)\) is the prior distribution describing the likelihood of a given estimator being the true model. It can be shown that \(\sum_{i=1}^{N}p_{k,i}=+1\), and it is required that \(\sum_{i=1}^{N}f(\alpha=\alpha_{i})=+1\). Generally, we assume that the measurements contain Gaussian error that is zero mean with known covariance (\(R_{i}\) being the covariance for measurement \(\vec{y}_{i}\)). We also assume that the MME estimators have a uniform prior distribution (\(f(\alpha=\alpha_{i})=1/N,i=1,2,\ldots,N\)). If we further assume that measurements are statistically independent in time (or at least over smaller mini-batches), then we can rewrite this weighting scheme in a more sequential notation as shown in Equation (3). \[p_{k,i}=f(\alpha=\alpha_{i}|\mathcal{Y})=\frac{f(\vec{y}=\vec{y}_{k}|\alpha_{ i})f(\alpha=\alpha_{i}|\mathcal{Y}_{1}^{k-1})}{\sum_{j=1}^{N}f(\vec{y}=\vec{y}_{k}| \alpha_{j})f(\alpha=\alpha_{j}|\mathcal{Y}_{1}^{k-1})}\] (  
function that computes the expected measurement for the \(k\)th measurement given the state of the \(i\)th estimator; and \(\hat{H}_{i,k}\) is the state vector-derivative of the \(k\)th expected measurement function for the \(i\)th estimator.