Definition

A metafeature is an abstraction of some substructure that is observed within the data. In the temporal domain, the substructures we are interested in are sub-events, like the LoudRuns of the Tech Support domain. Formally, a metafeature has two parts:

• A tuple of parameters $p= ( p_1,...,p_k )$, which represents a particular instantiated feature. Let $P_1$ represent the set of possible values of the parameter $p_1$. Let $P$ be the space of possible parameter values, i.e. $P = P_1 \times P_2 \times ... \times P_k$. $P$ is termed the parameter space.
• An extraction function $f$ which takes an instance $s \in \ensuremath{\mathit{SS}}$ (reminder: $\mathit{SS}$ is the set of all possible streams) for the domain and returns a set of instantiated features from $P$, i.e. $f: \ensuremath{\mathit{SS}}\rightarrow \mathbb{P}(P)$.

For every instance in the training set $\mathit{class}_T$ that we apply a metafeature to, a set of points in the parameter space are returned. Each point returned represents an ``occurrence'' or ``instance'' of the metafeatures, which we will term an instantiated feature. In other words, an instantiated feature is a tuple $(v_1 \in P_1, v_2 \in P_2, ... v_k \in P_k)$, where $v_1,...,v_k$ are values for each of the parameters.

The parameter space represents all the possible instantiated features we could possibly generate or observe in the data. Hence all instantiated features lie in the parameter space.

A simple example is the LoudRun metafeature. As previously outlined, the LoudRun metafeature has two parameters for each instance: the starting time $t$ and the duration $d$. Hence, our parameter space is two dimensional. In fact, it is easy to show that our parameter space is $\mathbb{N}^2$, although sometimes it will be convenient to treat it as $\mathbb{R}^2$.

In this case, it is easy to define the LoudRun extraction function as:

$$f(v) = \{ (t,d)\ \vert \ v(t-1)=L\$$

$$\wedge v(t)=v(t+1)=...=v(t+d-1)=H$$

$$\wedge v(t+d)=L \}$$

In other words, the extraction function $f$, when applied to an training stream finds all the subsections of the channel when there is an extended period of high-volume conversation, flanked by low-volume conversation on either side. This matches our intuition of a loud run.