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docs/thesis/md/related_work.md

@@ -91,7 +91,11 @@ Hand washing compliance can be measured using different tools. Jain et al. @jain
 A study by Li et al. @li_wristwash_2018 is able to recognize 13 steps of a hand washing procedure with an accuracy of $85\,\%$. They employ a sliding window feature based hidden markov model approach. Wang et al. explore using sensor armbands to assess the users compliance with given hand washing hygiene guidelines @wang_accurate_2020. They run a classifier using XGBoost and are mostly able to separate the different steps of the scripted hand washing routine.
 Added to that, Cao et al. @cao_awash_2021 developed a system that similarly detects different steps of a scripted hand washing routine and prompts the user, if they confuse the order of the steps or forget one of the steps. The technology is aimed at elderly patients with dementia. Their system is able to detect which step of hand washing is currently conducted based on wrist motion data using an LSTM based neural network. However, none of the three systems mentioned in this paragraph are meant to separate hand washing from other activities. 
 
-Mondol et al. employ a simple feed forward neural network consisting of a few linear layers to detect hand washing @sayeed_mondol_hawad_2020. Their method seeks to specifically eliminate false positives by trying to detect out of distribution (OOD) samples, i.e. samples that are very different from the ones seen by the model during training. They apply a conditional Gaussian distribution of the network's features of the last layer before the output layer (penultimate layer). They use the said features of all positive class samples to calculate the mean $\boldsymbol{\mu}$ and covariance matrix $\mathbf{S}$ of the feature distribution. Based on these measures, one can compute each sample's distance to the distribution using the Mahalanobis distance (as seen in equation \ref{eqn:mahala}). If during test time, the model predicts a sample to belong to the positive class, the distance is calculated. If the distance is bigger than a threshold ($d_{th}$), the sample is classified as a negative. The threshold $d_{th}$ can be derived by selecting it fittingly in order to include almost all positive samples seen during training. On their own data set (HAWAD data set) they reach F1-Scores of over 90% for hand washing detection.
+Mondol et al. employ a simple feed forward neural network consisting of a few linear layers to detect hand washing @sayeed_mondol_hawad_2020. Their method seeks to specifically eliminate false positives by trying to detect out of distribution (OOD) samples, i.e. samples that are very different from the ones seen by the model during training. They apply a conditional Gaussian distribution of the network's features of the last layer before the output layer (penultimate layer).
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+![Steps of HAWAD for parameter estimation and inference, taken from @sayeed_mondol_hawad_2020](img/HAWAD_filter.png){width=98% #fig:HAWAD}
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+They use the said features of all positive class samples to calculate the mean $\boldsymbol{\mu}$ and covariance matrix $\mathbf{S}$ of the feature distribution. Based on these measures, one can compute each sample's distance to the distribution using the Maharanis distance (as seen in equation \ref{eqn:mahala}). If during test time, the model predicts a sample to belong to the positive class, the distance is calculated. If the distance is bigger than a threshold ($d_{th}$), the sample is classified as a negative. The threshold $d_{th}$ can be derived by selecting it fittingly in order to include almost all positive samples seen during training. The parameter estimation and hand washing steps performed in the HAWAD paper can be seen in @fig:HAWAD. On their own data set (HAWAD data set) they reach F1-Scores of over 90% for hand washing detection.
 
 \begin{figure}
 \begin{align}
@@ -101,4 +105,10 @@ D_M(\mathbf{x}) = \sqrt{(\mathbf{x}- \boldsymbol{\mu})^T\mathbf{S}^{-1}(\mathbf{
 \caption*{Equation \ref*{eqn:mahala}: Mahalanobis distance}
 \end{figure}
 
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 TODO: graphic representations where missing
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