Local probabilistic neural networks in hydrology

One of the many types of neural networks that found application in hydrology is the probabilistic neural networks. Probabilistic neural networks are based upon the Parzen approximation of probability densities by (Gaussian) kernels. The advantages of probabilistic neural networks are that they learn extremely quickly, give probabilistic interpretation and by this not only produce estimation of the mean but also give insight into the other statistics of the errors.

When (in higher dimensions) the observations tend to cluster around lower dimensional subspaces, the classical approach fails by not being able to take this into account. The solution proposed here is to use a local version, based upon Gaussian kernels with locally estimated covariances. This concept resembles the “local and global embedding dimension” used in (classical) deterministic time series analysis.

As an example, results on predicting discharges in a small catchment will be presented. Inputs are lagged discharges. If the time discretisation scale is rather small, and one uses many lags, the input space becomes high dimensional but the observations by the mutual dependence between the components of the input fill only a lower dimensional subspace of this. It will be shown that this new technique offers better results in these cases.