Observed data sets often do not satisfy the assumptions required by many standard analysis methods, such as stationarity or homogeneity. As most data sets contains both coherent and highly irregular features, the estimation of the structure of interest must account for the underlying variability of the data, as well as represent the underlying structure well.

Starting from the problem of designing the `best' analysis tools to extract local structure, I will provide a precise formulation of the properties of a well-behaved local analysis function. With such in mind, I will derive the optimal analysis tools for estimating the features of a given class of non-stationary phenomena. I will explicitly use these functions to formulate statistical estimation procedures and will motivate the methodological developments with real data sets.

The core of this family of methods is to appropriate transform the data to best represent its structure, and then to suitably average the transformed data, to reduce variability. Care must be taken in both these steps.

This is joint work with Georgios Metikas and Andrew Walden.