Principal components are a well established tool in dimension reduction. The extension to principal curves allows for general smooth curves which pass through the middle of a multidimensional data cloud. In this paper local principal curves are introduced, which are based on the localization of principal component analysis. The proposed algorithm is able to identify closed curves as well as multiple curves which may or may not be connected. For the evaluation of the performance of principal curves as tool for data reduction a measure of coverage is suggested. By use of simulated and real data sets the approach is compared to various alternative concepts of principal curves.

Blood lactate markers are used as summary measures of the underlying model of an athlete's blood lactate response to increasing work rate. Exercise physiologists use these endurance markers, typically corresponding to a work rate in the region of high curvature in the lactate curve, to predict and compare endurance ability. A short theoretical background of the commonly used markers is given and algorithms provided for their calculation. To date, no free software exists that allows the sports scientist to calculate these markers. In this paper, software is introduced for precisely this purpose that will calculate a variety of lactate markers for an individual athlete, an athlete at different instants (e.g. across a season), and simultaneously for a squad.

For speed-flow data, which are intensively discussed in transportation science, common nonparametric regression models of the type y D m.x/ C noise turn out to be inadequate since simple functional models cannot capture the essential relationship between the predictor and response. Instead a more general setting is required, allowing for multifunctions rather than functions. The tool proposed is conditional modes estimation which, in the form of local modes, yields several branches that correspond to the local modes. A simple algorithm for computing the branches is derived. This is based on a conditional mean shift algorithm and is shown to work well in the application that is considered.

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We propose weighted repeated median filters and smoothers for robust non-parametric regression in general and for robust signal extraction from time series in particular. The proposed methods allow to remove outlying sequences and to preserve discontinuities (shifts) in the underlying regression function (the signal) in the presence of local linear trends. Suitable weighting of the observations according to their distances in the design space reduces the bias arising from non-linearities. It also allows to improve the efficiency of (unweighted) repeated median filters using larger bandwidths, keeping their properties for distinguishing between outlier sequences and long-term shifts. Robust smoothers based on weighted L 1 -regression are included for the reason of comparison.