On-line EM Algorithm for the Normalized Gaussian Network

A normalized gaussian network (NGnet) (Moody & Darken, 1989) is a network of local linear regression units. The model softly partitions the input space by normalized gaussian functions, and each local unit linearly approximates the output within the partition. In this article, we propose a new on-line EMalgorithm for the NGnet, which is derived from the batch EMalgorithm (Xu, Jordan, &Hinton 1995), by introducing a discount factor. We show that the on-line EM algorithm is equivalent to the batch EM algorithm if a specific scheduling of the discount factor is employed. In addition, we show that the on-line EM algorithm can be considered as a stochastic approximation method to find the maximum likelihood estimator. A new regularization method is proposed in order to deal with a singular input distribution. In order to manage dynamic environments, where the input-output distribution of data changes over time, unit manipulation mechanisms such as unit production, unit deletion, and unit division are also introduced based on probabilistic interpretation. Experimental results show that our approach is suitable for function approximation problems in dynamic environments. We also apply our on-line EM algorithm to robot dynamics problems and compare our algorithm with the mixtures-of-experts family.