The AWS procedure (Fig 1)

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 $ \hat{\theta}(X_i)=\hat{\theta}_i$ is of the form:

$\displaystyle \hat{\theta}_i$ $\displaystyle =$ $\displaystyle \underset{\theta \in \Theta}{\textrm{min}} \frac{1}{2\sigma^2}\sum_{j=1}^N w_{ij}(Y_j-\theta)^2$  

The AWS procedure allows the computation of the weights $ w_{ij}$ through an iterative procedure: at each iteration $ k$, the increase in $ h^{(k-1)}$ defines a new larger neighborhood around each $ X_i$, which is used to calculate the new MLE of $ \theta_i$. For each location $ X_i$, the estimation $ \theta^{(k-1)}_i$ is improved by computing the new weights taking into account:

The new weight $ w_{ij}^{(k)}$ is calculated as a function of $ K_l\big(\boldsymbol{l}_{ij}^{(k)}\big)K_s\big(\boldsymbol{s}_{ij}^{(k)}\big)$ where kernels $ K_s$ and $ K_l$ are non-increasing functions and must fulfill $ K_s(0)=K_l(0)=1$.

Summary of the AWS procedure:
set all $ \theta_i^{(0)}$ to the mean of the $ \textrm{log}_2\textrm{-ratio's }$$ Y_i$ and all weights $ w_{ij}^{(0)}$ to 1

for each location $ X_i$:

when $ h^{(k)}$ is greater than a fixed threshold

Figure 1: Aws results

Philippe Hupé 2004-11-19