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\section{Algorithms}
\label{sec:algorithms}
In Section~\ref{sec:uniqueness}, we showed that a nearest-neighbor classifier can accurately predict if a skeleton belongs to the same person if the error of the skeleton measurements is small. In this section, we suggest a probabilistic model for skeleton recognition. In this model, a skeleton is classified based on the distance from average skeleton profiles of people in the training set.
\subsection{Mixture of Gaussians}
\label{sec:mixture of Gaussians}
A mixture of Gaussians \cite{bishop06pattern} is a generative probabilistic model, which is typically applied to modeling problems where class densities are unimodal and the feature space is low-dimensional. The joint probability distribution of the model is given by:
\begin{align}
P(\bx, y) = \cN(\bx | \bar{\bx}_y, \Sigma) P(y),
\label{eq:mixture of Gaussians}
\end{align}
where $P(y)$ is the probability of class $y$ and $\cN(\bx | \bar{\bx}_y, \Sigma)$ is a multivariate normal distribution, which models the density of $\bx$ given $y$. The mean of the distribution is $\bar{\bx}_y$ and the variance of $\bx$ is captured by the covariance matrix $\Sigma$. The decision boundary between any two classes is known to be is linear when all conditionals $\cN(\bx | \bar{\bx}_y, \Sigma)$ have the same covariance matrix \cite{bishop06pattern}. In this setting, the mixture of Gaussians model can be viewed as a probabilistic variant of the nearest-neighbor (NN) classifier in Section~\ref{sec:uniqueness}.
The mixture of Gaussians model has many advantages. First, the model can be easily learned using maximum-likelihood (ML) estimation \cite{bishop06pattern}. In particular, $P(y)$ is the frequency of $y$ in the training set, $\bar{\bx}_y$ is the expectation of $\bx$ given $y$, and the covariance matrix is computed as $\Sigma = \sum_y P(y) \Sigma_y$, where $\Sigma_y$ represents the covariance of $\bx$ given $y$. Second, the inference in the model can be performed in a closed form. In particular, the model predicts $\hat{y} = \arg\max_y P(y | \bx)$, where:
\begin{align}
P(y | \bx) =
\frac{P(\bx | y) P(y)}{\sum_y P(\bx | y) P(y)} =
\frac{\cN(\bx | \bar{\bx}_y, \Sigma) P(y)}{\sum_y \cN(\bx | \bar{\bx}_y, \Sigma) P(y)}.
\label{eq:inference}
\end{align}
In practice, the prediction $\hat{y}$ is accepted when the classifier is confident. In other words, $P(\hat{y} | \bx) \! > \! \delta$, where $\delta \in (0, 1)$ is a threshold that controls the precision and recall of the classifier. In general, the higher the threshold $\delta$, the lower the recall and the higher the precision.
In this work, we use the mixture of Gaussians model for skeleton recognition. Skeleton measurements are represented by a vector $\bx$ and each person is assigned to one class $y$. In particlar, our dataset $\cD = \set{(\bx_1, y_1), \dots, (\bx_n, y_n)}$ consists of $n$ pairs $(\bx_i, y_i)$, where $y_i$ is the label of the skeleton $\bx_i$. To verify that our method is suitable for skeleton recognition, we plot for each skeleton feature $x_k$ (Section~\ref{sec:experiment}) the histogram of differences between all measurements and the expectation given the class $(\bx_i)_k - \E{}{x_k | y_i}$ (Figure~\ref{fig:error marginals}). All histograms look approximately normal. This indicates that all class conditionals $P(\bx | y)$ are multivariate normal and our generative model, although very simple, may be nearly optimal \cite{bishop06pattern}.
\begin{figure}[t]
\centering
\includegraphics[height=4.4in, angle=90, bb=4.5in 1.5in 6.5in 7in]{graphics/ErrorMarginals}
\caption{Histograms of differences between 9 skeleton measurements $x_k$ (Section~\ref{sec:experiment}) and their expectation given the class $y$.}
\label{fig:error marginals}
\end{figure}
\subsection{Sequential hypothesis testing}
\label{sec:SHT}
The mixture of Gaussians model can be extended to temporal inference through sequential hypothesis testing. Sequential hypothesis testing \cite{wald47sequential} is an established statistical framework, where a subject is sequentially tested for belonging to one of several classes. The probability that the sequence of data $\bx^{(1)}, \dots, \bx^{(t)}$ belongs to the class $y$ at time $t$ is given by:
\begin{align}
P(y | \bx^{(1)}, \dots, \bx^{(t)}) =
\frac{\prod_{i = 1}^t \cN(\bx^{(i)} | \bar{\bx}_y, \Sigma) P(y)}
{\sum_y \prod_{i = 1}^t \cN(\bx^{(i)} | \bar{\bx}_y, \Sigma) P(y)}.
\label{eq:SHT}
\end{align}
In practice, the prediction $\hat{y} = \arg\max_y P(y | \bx^{(1)}, \dots, \bx^{(t)})$ is accepted when the classifier is confident. In other words, $P(\hat{y} | \bx^{(1)}, \dots, \bx^{(t)}) > \delta$, where the threshold $\delta \in (0, 1)$ controls the precision and recall of the predictor. In general, the higher the threshold $\delta$, the higher the precision and the lower the recall.
Sequential hypothesis testing is a common technique for smoothing temporal predictions. In particular, note that the prediction at time $t$ depends on all data up to time $t$. This reduces the variance of predictions, especially when input data are noisy, such as in the domain of skeleton recognition.
In skeleton recognition, the sequence $\bx^{(1)}, \dots, \bx^{(t)}$ are skeleton measurements of a person walking towards the camera, for instance. If the camera detects more people, we use tracking to distinguish individual skeleton sequences.
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