The detection of breakpoints is based on the estimation of a piecewise constant function with the Adaptive Weights Smoothing (AWS) procedure (Polzehl and Spokoiny, 2002): AWS is an iterative, data-adaptive smoothing technique that was designed for smoothing in regression problems involving discontinuous regression function. The regression function is approximated by a simple local constant gaussian model and estimated as a weighted Maximum Likelihood Estimate (MLE), the choice of the weights being completely data-adaptive. The weighted MLE is of the form:

The AWS procedure allows the computation of the weights through an iterative procedure: at each iteration , the increase in defines a new larger neighborhood around each , which is used to calculate the new MLE of . For each location , the estimation is improved by computing the new weights taking into account:

- the distance between and in terms of geographical proximity
- the distance between and in terms of statistical comparison

The new weight is calculated as a function of where kernels and are non-increasing functions and must fulfill .

**Summary of the AWS procedure:****Initialization:**- set all
to the mean of the
and all weights
to 1
**Iteration:**- for each location :
- Start by looking at its closest neighbors
- Recompute the new weights (in a neighborhood of radius )
- Increase the size of neighborhood at each step

**Stop:**- when is greater than a fixed threshold

Philippe Hupé 2004-11-19