8.4 Applications 285 product were given by Eq. (8.15), with the potentials given by Eq. (8.3) using the Gaussian smoothing function. However, the metric suffers from the same limitation as van Rossum's distance because it implicitly assumes Poisson pro- cesses. With our methodology the metric can be easily extended. Moreover, notice that Schreiber and colleagues' "correlation measure" is only a pre- metric since it does not verify the triangle inequality. In d CS this is ensured by the arc cosine function. 8.4 APPLICATIONS To exemplify the importance of the developments shown here, in the follow- ing we derive two fundamental machine learning algorithms for spike trains: principal component analysis (PCA) and Fisher linear discriminant (FLD). These two algorithms serve to demonstrate that the framework presented here allows the development of both supervised and unsupervised learning meth-