The SOM algorithm is very astonishing. On the one hand, it is very simple to write down and to simulate, its practical properties are clear and easy to observe. But, on the other hand, its theoretical properties still remain without proof in the general case, despite the great efforts of several authors. In this paper, we pass in review the last results and provide some conjectures for the future work.

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Many authors use feedforward neural networks for modeling and forecasting time series. Most of these applications are mainly experimental, and it is often difficult to extract a general methodology from the published studies. In particular, the choice of architecture is a tricky problem. We try to combine the statistical techniques of linear and nonlinear time series with the connectionist approach. The asymptotical properties of the estimators lead us to propose a systematic methodology to determine which weights are nonsignificant and to eliminate them to simplify the architecture. This method (SSM or statistical stepwise method) is compared to other pruning techniques and is applied to some artificial series, to the famous Sunspots benchmark, and to daily electrical consumption data.

Neural Gas (NG) constitutes a very robust clustering algorithm given euclidian data which does not suffer from the problem of local minima like simple vector quantization, or topological restrictions like the self-organizing map. Based on the cost function of NG, we introduce a batch variant of NG which shows much faster convergence and which can be interpreted as an optimization of the cost function by the Newton method. This formulation has the additional benefit that, based on the notion of the generalized median in analogy to Median SOM, a variant for non-vectorial proximity data can be introduced. We prove convergence of batch and median versions of NG, SOM, and k-means in a unified formulation, and we investigate the behavior of the algorithms in several experiments.