325 research outputs found
Robust online scale estimation in time series: A regression-free approach.
This paper presents variance extraction procedures for univariate time series. The volatility of a times series is monitored allowing for non-linearities, jumps and outliers in the level. The volatility is measured using the height of triangles formed by consecutive observations of the time series. This idea was proposed by Rousseeuw and Hubert (1996, Regression-free and robust estimation of scale for bivariate data, Computational Statistics and Data Analysis, 21, 67{85) in the bivariate setting. This paper extends their procedure to apply for online scale estimation in time series analysis. The statistical properties of the new methods are derived and nite sample properties are given. A nancial and a medical application illustrate the use of the procedures.Breakdown point; Inuence function; Online monitoring; Outliers; Robust scale estimation;
breakdown and groups
The concept of breakdown point was introduced by Hodges (1967) and Hampel (1968, 1971) and still plays an important though at times a controversial role in robust statistics. It has proved most successful in the context of location, scale and regression problems. In this paper we argue that this success is intimately connected to the fact that the translation and affine groups act on the sample space and give rise to a definition of equivariance for statistical functionals. For such functionals a nontrivial upper bound for the breakdown point can be shown. In the absence of such a group structure a breakdown point of one is attainable and this is perhaps the decisive reason why the concept of breakdown point in other situations has not proved as successful. Even if a natural group is present it is often not sufficiently large to allow a nontrivial upper bound for the breakdown point. One exception to this is the problem of the autocorrelation structure of time series where we derive a nontrivial upper breakdown point using the group of realizable linear filters. The paper is formulated in an abstract manner to emphasize the role of the group and the resulting equivariance structure
Nonparametric regression as an example of model choice
Nonparametric regression can be considered as a problem of model choice. In this paper
we present the results of a simulation study in which several nonparametric regression
techniques including wavelets and kernel methods are compared with respect to their
behaviour on different test beds. We also include the taut-string method whose aim is
not to minimize the distance of an estimator to some “true” generating function f but to
provide a simple adequate approximation to the data. Test beds are situations where a
“true” generating f exists and in this situation it is possible to compare the estimates of
f with f itself. The measures of performance we use are the L^2 and the L^infinity norms and
the ability to identify peaks
Constructing a regular histogram - a comparison of methods
Even for a well-trained statistician the construction of a histogram for a given
real-valued data set is a difficult problem. It is even more difficult to construct
a fully automatic procedure which specifies the number and widths of the bins in
a satisfactory manner for a wide range of data sets. In this paper we compare
several histogram construction methods by means of a simulation study. The study
includes plug-in methods, cross-validation, penalized maximum likelihood and the
taut string procedure. Their performance on different test beds is measured by the
Hellinger distance and the ability to identify the modes of the underlying density
Residual-based localization and quantification of peaks in x-ray diffractograms
We consider data consisting of photon counts of diffracted x-ray radiation as
a function of the angle of diffraction. The problem is to determine the
positions, powers and shapes of the relevant peaks. An additional difficulty is
that the power of the peaks is to be measured from a baseline which itself must
be identified. Most methods of de-noising data of this kind do not explicitly
take into account the modality of the final estimate. The residual-based
procedure we propose uses the so-called taut string method, which minimizes the
number of peaks subject to a tube constraint on the integrated data. The
baseline is identified by combining the result of the taut string with an
estimate of the first derivative of the baseline obtained using a weighted
smoothing spline. Finally, each individual peak is expressed as the finite sum
of kernels chosen from a parametric family.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS181 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Methods and Algorithms for Robust Filtering
We discuss filtering procedures for robust extraction of a signal from noisy time series. Moving averages and running medians are standard methods for this, but they have shortcomings when large spikes (outliers) respectively trends occur. Modified trimmed means and linear median hybrid filters combine advantages of both approaches, but they do not completely overcome the difficulties. Improvements can be achieved by using robust regression methods, which work even in real time because of increased computational power and faster algorithms. Extending recent work we present filters for robust online signal extraction and discuss their merits for preserving trends, abrupt shifts and extremes and for the removal of spikes
Regression-Based, Regression-Free and Model-Free Approaches for Robust Online Scale Estimation
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