On some properties of weighted averaging with variable weights

Density-based means have been recently proposed as a method for dealing with outliers in the stream processing of data. Derived from a weighted arithmetic mean with variable weights that depend on the location of all data samples, these functions are not monotonic and hence cannot be classified as aggregation functions. In this article we establish the weak monotonicity of this class of averaging functions and use this to establish robust generalisations of these means. Specifically, we find that as proposed, the density based means are only robust to isolated outliers. However, by using penalty based formalisms of averaging functions and applying more sophisticated and robust density estimators, we are able to define a broader family of density based means that are more effective at filtering both isolated and clustered outliers.