1,124 research outputs found
Fast computation of Tukey trimmed regions and median in dimension
Given data in , a Tukey -trimmed region is the set of
all points that have at least Tukey depth w.r.t. the data. As they are
visual, affine equivariant and robust, Tukey regions are useful tools in
nonparametric multivariate analysis. While these regions are easily defined and
interpreted, their practical use in applications has been impeded so far by the
lack of efficient computational procedures in dimension . We construct
two novel algorithms to compute a Tukey -trimmed region, a na\"{i}ve
one and a more sophisticated one that is much faster than known algorithms.
Further, a strict bound on the number of facets of a Tukey region is derived.
In a large simulation study the novel fast algorithm is compared with the
na\"{i}ve one, which is slower and by construction exact, yielding in every
case the same correct results. Finally, the approach is extended to an
algorithm that calculates the innermost Tukey region and its barycenter, the
Tukey median
Multidimensional trimming based on projection depth
As estimators of location parameters, univariate trimmed means are well known
for their robustness and efficiency. They can serve as robust alternatives to
the sample mean while possessing high efficiencies at normal as well as
heavy-tailed models. This paper introduces multidimensional trimmed means based
on projection depth induced regions. Robustness of these depth trimmed means is
investigated in terms of the influence function and finite sample breakdown
point. The influence function captures the local robustness whereas the
breakdown point measures the global robustness of estimators. It is found that
the projection depth trimmed means are highly robust locally as well as
globally. Asymptotics of the depth trimmed means are investigated via those of
the directional radius of the depth induced regions. The strong consistency,
asymptotic representation and limiting distribution of the depth trimmed means
are obtained. Relative to the mean and other leading competitors, the depth
trimmed means are highly efficient at normal or symmetric models and
overwhelmingly more efficient when these models are contaminated. Simulation
studies confirm the validity of the asymptotic efficiency results at finite
samples.Comment: Published at http://dx.doi.org/10.1214/009053606000000713 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Regression Depth and Center Points
We show that, for any set of n points in d dimensions, there exists a
hyperplane with regression depth at least ceiling(n/(d+1)). as had been
conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n
hyperplanes in d dimensions there exists a point that cannot escape to infinity
without crossing at least ceiling(n/(d+1)) hyperplanes. We also apply our
approach to related questions on the existence of partitions of the data into
subsets such that a common plane has nonzero regression depth in each subset,
and to the computational complexity of regression depth problems.Comment: 14 pages, 3 figure
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Projective Bundle Adjustment from Arbitrary Initialization Using the Variable Projection Method
Bundle adjustment is used in structure-from-motion pipelines as final refinement stage requiring a sufficiently good initialization to reach a useful local mininum. Starting from an arbitrary initialization almost always gets trapped in a poor minimum. In this work we aim to obtain an initialization-free approach which returns global minima from a large proportion of purely random starting points. Our key inspiration lies in the success of the Variable Projection (VarPro) method for affine factorization problems, which have close to 100% chance of reaching a global minimum from random initialization. We find empirically that this desirable behaviour does not directly carry over to the projective case, and we consequently design and evaluate strategies to overcome this limitation. Also, by unifying the affine and the projective camera settings, we obtain numerically better conditioned reformulations of original bundle adjustment algorithms
Projective Bundle Adjustment from Arbitrary Initialization Using the Variable Projection Method
Bundle adjustment is used in structure-from-motion pipelines as final refinement stage requiring a sufficiently good initialization to reach a useful local mininum. Starting from an arbitrary initialization almost always gets trapped in a poor minimum. In this work we aim to obtain an initialization-free approach which returns global minima from a large proportion of purely random starting points. Our key inspiration lies in the success of the Variable Projection (VarPro) method for affine factorization problems, which have close to 100% chance of reaching a global minimum from random initialization. We find empirically that this desirable behaviour does not directly carry over to the projective case, and we consequently design and evaluate strategies to overcome this limitation. Also, by unifying the affine and the projective camera settings, we obtain numerically better conditioned reformulations of original bundle adjustment algorithms
A data driven equivariant approach to constrained Gaussian mixture modeling
Maximum likelihood estimation of Gaussian mixture models with different
class-specific covariance matrices is known to be problematic. This is due to
the unboundedness of the likelihood, together with the presence of spurious
maximizers. Existing methods to bypass this obstacle are based on the fact that
unboundedness is avoided if the eigenvalues of the covariance matrices are
bounded away from zero. This can be done imposing some constraints on the
covariance matrices, i.e. by incorporating a priori information on the
covariance structure of the mixture components. The present work introduces a
constrained equivariant approach, where the class conditional covariance
matrices are shrunk towards a pre-specified matrix Psi. Data-driven choices of
the matrix Psi, when a priori information is not available, and the optimal
amount of shrinkage are investigated. The effectiveness of the proposal is
evaluated on the basis of a simulation study and an empirical example
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