7. Early Classification of Time Series#
Early classification of time series is the task of performing a classification as early as possible for an incoming time series, and decision about when to trigger the decision is part of the prediction process.
7.1. Early Classification Cost Function#
Dachraoui et al. [1] introduces a composite loss function for early classification of time series that balances earliness and accuracy.
The cost function is of the following form:
where \(\mathcal{L}_c(\cdot,\cdot,\cdot)\) is a classification loss and \(t\) is the time at which a decision is triggered by the system (\(\mathbf{x}_{\rightarrow t}\) is time series \(\mathbf{x}\) observed up to time \(t\)). In this setting, \(\alpha\) drives the tradeoff between accuracy and earliness and is supposed to be a hyper-parameter of the method.
The authors rely on (i) a clustering of the training time series and (ii) individual classifiers \(m_t(\cdot)\) trained at all possible timestamps, so as to be able to predict, at time \(t\), an expected cost for all future times \(t + \tau\) with \(\tau \geq 0\):
where:
\(P(C_k | \mathbf{x}_{\rightarrow t})\) is a soft-assignment weight of \(\mathbf{x}_{\rightarrow t}\) to cluster \(C_k\);
\(P(y=i | C_k)\) is obtained from a contingency table that stores the number of training time series of each class in each cluster;
\(P_{t+\tau}(\hat{y} = j | y=i, C_k)\) is obtained through training time confusion matrices built on time series from cluster \(C_k\) using classifier \(m_{t+\tau}(\cdot)\).
At test time, if a series is observed up to time \(t\) and if, for all positive \(\tau\) we have \(f_\tau(\mathbf{x}_{\rightarrow t}, y) \geq f_0(\mathbf{x}_{\rightarrow t}, y)\), then a decision is made using classifier \(m_t(\cdot)\).
Early classification. At test time, prediction is made at a timestamp such that the expected earliness-accuracy is optimized, which can hence vary between time series.#
To use this early classifier in tslearn, one can rely on the
tslearn.early_classification.NonMyopicEarlyClassifier class:
from tslearn.early_classification import NonMyopicEarlyClassifier
early_clf = NonMyopicEarlyClassifier(n_clusters=3,
cost_time_parameter=1e-3,
lamb=1e2,
random_state=0)
early_clf.fit(X_train, y_train)
preds, times = early_clf.predict_class_and_earliness(X_test)
where cost_time_parameter is the \(\alpha\) parameter presented above
and lamb is a trade-off parameter for the soft-assignment of partial series
to clusters \(P(C_k | \mathbf{x}_{\rightarrow t})\) (when lamb tends to
infinity, the assignment tends to hard-assignment, and when lamb is set to
0, equal probabilities are obtained for all clusters).
